1.1. Basic Mechanisms of Heat Transfer The basic mechanisms of heat transfer are generally considered to be conduction, convection, boiling, condensation, and radiation. Of these, radiation is usually significant only at temperatures higher than those ordinarily encountered in tubular process heat transfer equipment; therefore, radiation will not be considered in any great detail in this Manual. All of the others play a vital role in equipment design and will frequently appear in the discussion. In this section, the emphasis will be upon a qualitative description of the processes and a few very basic equations.

1.1.1. Conduction Mechanism Conduction in a metallic solid is largely due to the random movement of electrons through the metal. The electrons in the hot part of the solid have a higher kinetic energy than those in the cold part and give up some of this kinetic energy to the cold atoms, thus resulting in a transfer of heat from the hot surface to the cold. Since the free electrons are also responsible for the conduction of an electrical current through a metal, there is a qualitative similarity between the ability of a metal to conduct heat and to conduct electricity. In addition, some heat is transferred by interatomic vibrations. Fourier Equation The details of conduction are quite complicated but for engineering purposes may be handled by a simple equation, usually called Fourier's equation. For the steady flow of heat across a plane wall (Fig. 1.1) with the surfaces at temperatures of T1, and T2 where T1 is greater than T2 the heat flow Q per unit area of surface A (the heat flux) is:

XTk

XXTT

kqAQ

ΔΔ

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

==21

21 (1.1)

The quantity k is called the thermal conductivity and is an experimentally measured value for any material. Eq. (1.1) can be written in a more general form if the temperature gradient term is written as a differential:

dxdTk

AQ

−= (1.2)

The negative sign in the equation is introduced to account for the fact that heat is conducted from a high temperature to a low temperature, so that ( inherently negative; therefore the double negative indicates a positive flow of heat in the direction of decreasing temperature.

)dxdT

Conduction Through a Tube Wall The main advantage of Eq. (1.2) is that it can be integrated for those cases in which the cross-sectional area for heat transfer changes along the conduction path. A section of tube wall is shown in Fig. 1.2. Q is the total heat conducted

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through the tube wall per unit time. At the radial position r in the tube wall (r i rr o≤≤ ), the area for heat transfer for a tube of length L is A = 2π rL. Putting these into Eq. (1.2) gives

dxdTk

rLQ

−=π2

(1.3)

which may be integrated to

( )io

oi

rrTTkL

Qln

)(2 −=

π (1.4)

If Ti < To, Q comes out negative; this just means that the heat flow is inward, reversed from the sense in which we took it. For thin-walled tubes, the ratio of the outer to the inner radius is close to unity, and we can use the simpler equation,

io

oio

rrTTkLr

Q−

−=

)(2π (1.5)

with very small error. Conduction Through a Bimetallic Wall Sometimes, for reasons of corrosion, strength and/or economy, a tube is actually constructed out of two tightly fitting concentric cylinders of different metals as shown in Fig. 1.3. Note that r' is the outside radius of the inner tube and the inside radius of the outer tube, and T' is the corresponding temperature. From Eq. 1.4 we may write directly for the inner tube:

( )i

ii

rrTTkL

Q'ln

)'(2 −=

π (1.6)

and for the outer tube:

( )'ln)'(2

rrTTkL

Qo

oo −=

π (1.7)

Since the same amount of heat must flow through both tubes, the Q's are equal. Then the equations can be combined to eliminate the unknown temperature T', and Q is given by:

o

o

i

i

oi

kLrr

kLrr

TTQ

ππ 2)'/ln(

2)/'ln(+

−= (1.8)

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Contact Resistance In the previous section, the assumption was made that the outer surface of the inner cylinder and the inner surface of the outer cylinder were at the same temperature, implying that there was no resistance to heat transfer between the two. This assumption is essentially correct if the two surfaces are metallurgically bonded to one another, as can be achieved when one material is fused to the other or when they are bonded by a detonation wave. The assumption can be seriously in error if the two surfaces are merely in close physical contact, even at the very high pressures that can be exerted by shrink-fitting. Practical metal surfaces generally have roughnesses ranging from 10 to 180 microinches, the degree and the form of the roughness depending upon the metal and the method of forming the surface. When two such surfaces are placed in contact, the "hills" are touching while the "valleys" are filled with the ambient atmosphere, usually air. Because of the low thermal conductivity of gases, practically all of the heat is conducted through the points in metal-to-metal contact. At low pressures, this will be only a small portion of the surface perhaps less than one percent and the resulting constriction of the heat flow lines can lead to an interface resistance several times greater than in the metal slabs themselves. At higher contact pressures between the surfaces, the hills are flattened to give a greater surface area in contact in order to sustain the load, and the interface resistance decreases. Various methods are used to ensure good thermal contact including co-extrusions and shrink-fitting, but repeated thermal cycling in the normal operation of process equipment, together with creep, can cause long-term serious loss of efficiency. Unfortunately it is almost impossible to predict contact resistance in process equipment applications. Here, as in all process equipment design, the engineer must assess the consequences of being wrong and the possible alternatives.

1.1.2. Single Phase Convection Fluid Motion Near a Surface Convective heat transfer is closely connected to the mechanism of fluid flow near a surface, so the first matter of importance is to describe this flow. Single phase flow must be characterized by both the geometry of the duct through which the flow occurs and by the flow regime of the fluid as it goes through the duct. There are two basically different types of duct geometry: constant cross-section, in which the area available for flow to the fluid has both the same shape and the same area at each point along the duct, and varying cross-section, in which the shape and/or the area of the duct vary with length, usually in a regular and repeated way. The most common constant cross-section duct geometry that one deals with in process heat transfer applications is the cylindrical tube. In a cylindrical geometry, it is assumed that all parameters of the flow are a function only of the radial distance from the axis of the cylinder (or equivalently from the wall) and of the distance from the entrance (entrance effects). For flow in ducts of varying cross section (flow across tube banks is the case of interest here), another phenomenon occurs that is of the utmost importance in calculating the pressure drop and which adversely affects the efficiency of conversion of pressure drop to heat transfer. Whereas in a duct of constant cross section only the pressure effect due to the friction of the fluid moving relative to the surface (skin friction) needs to be taken into account, in a bank of tubes form drag is a major contributor to the pressure drop. Form drag arises when the pressure on the front surface of a tube is greater than the pressure on the back surface. This pressure difference appears as part of the total pressure drop across the tube bank; unlike skin friction, form drag is not very efficient in promoting heat transfer. Form drag also gives rise to boundary layer separation and wake formation, which may cause destructive vibration of the tubes in a tube bank.

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The type of flow in a duct can also be characterized by the flow regime; that is, laminar flow, turbulent flow, or some transition state having characteristics of both of the limiting regimes. All of the exact definitions of laminar flow are very complex, and an illustration (like Fig. 1.4) is much more useful. If we have a round tube with a liquid flowing in it at a steady rate, and if we inject a dye trace with a needle parallel to the axis of the tube, one of two things can happen: 1) The dye trace may flow smoothly down the tube as a well-defined line, only very slowly becoming thicker, or 2) The dye trace may flow irregularly down the tube moving back and forth across the diameter of the tube and

eventually becoming completely dispersed. The first case is laminar flow and the second is turbulent flow. Laminar flow corresponds to the smooth movement of layers of fluid past one another without mixing; turbulent flow is characterized by a rapid exchange of packets or elements of fluid in a radial direction from one part of the flow field to another through turbulent eddies. There are differences in the velocity pattern also. In laminar flow, the velocity at a given point is steady, whereas in turbulent flow the velocity fluctuates rapidly about an average value. If one measures the local velocity at various positions across the tube, one finds that laminar flow gives a parabolic velocity

distribution whereas turbulent flow gives a blunter velocity profile, as shown in Fig. 1.4. In both flows, the fluid velocity is zero at the wall and a maximum at the centerline. The flow regime that exists in a given case is ordinarily characterized by the Reynolds number. The Reynolds number has different definitions for flow in different geometries, but it is defined as in Eq. 1.9 for flow inside tubes:

μμρ GdVd ii ==Re (1.9)

where di is the inside diameter of the tube, V is the average velocity in the tube, ρ the density of the fluid, and μ the viscosity of the fluid. Laminar flow is characterized by low Reynolds numbers, turbulent flow by high Reynolds Numbers. For flow inside tubes, Reynolds numbers below about 2,000 result in laminar flow being the stable flow regime. Reynolds numbers above about 2100 give turbulent flow for pressure drop calculations, while Reynolds numbers above 10,000 give turbulent flow for heat transfer. The range from 2100 to 10,000 is generally referred to as the transition flow regime for heat transfer. Other geometries have different Reynolds number ranges to characterize flow regimes. Heat Transfer to a Flowing Fluid (Convection) Convection heat transfer can be defined as transport of heat from one point to another in a flowing fluid as a result of macroscopic motions of the fluid, the heat being carried as internal energy. The convection process has received a great deal of both experimental and analytical attention and, although we are mainly concerned with using the results of these studies, a cursory look should be taken at the physical process of convection, both to define terms and to establish some intuitive sense of what really the correlations we use are trying to represent.

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In laminar flow past a cold wall the heat is transferred to the tube surface from the fluid next to the wall. Within the fluid, heat is transferred from "layer" to "layer" of the fluid by conduction. There are no fluid motions perpendicular to the direction of flow to transport the heat by any other mechanism. Since the different "layers" of fluid are moving at different velocities, however, the conduction process is much more complex to analyze than for the solid wall previously discussed. If we look at a fluid in turbulent flow past a cold surface and mark a few representative elements of fluid in order to trace their paths, we would obtain a picture something like Fig. 1.5. The corresponding time-averaged velocity and temperature profiles might look like Figs. 1.6 and 1.7. The flow near the wall has only a few small eddies, so that the predominant mechanism for heat transfer is conduction. At the wall, the fluid velocity is zero and the fluid temperature is the same as the wall. The velocity and temperature gradients near the wall are much steeper than those in the bulk flow where eddy transport becomes dominant. It is important to note that when we refer without further qualification to the velocity or temperature of a stream, we mean the volume-mean or mixing-cup values shown on the figures as V and T f . (We will drop the bars henceforth.) However, it is important to remember that some portions of the fluid are at possibly significantly higher or lower temperatures, where thermal degradation or phase change might occur. For the case shown, if the fluid had a freezing temperature between Ts and Tf, a layer of solid would form on the wall, resulting in a major change in the heat transfer and fluid flow mechanisms. Film Heat Transfer Coefficients For many convective heat transfer processes, it is found that the local heat flux is approximately proportional to the temperature difference between the wall and the bulk of the fluid, i.e.,

)( sf TTAQ

−∝ (1.10)

which causes us to define a constant of proportionality, called the "film coefficient of heat transfer," usually denoted by h:

)( sf TThAQ

−= (1.11)

The value of h depends upon the geometry of the system, the physical properties and flow velocity of the fluid.

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The concept of a heat transfer coefficient is useful to the designer only if there exists a quantitative relationship between these variables and the heat transfer coefficient. It is important also that this relationship be reasonably valid for the conditions existing in the particular application. These relationships, or correlations, may come from either theoretical or experimental studies, or from a combination of both. The correlation may be expressed as an equation, a graph, a table of values, or a computational procedure. These forms are more or less readily convertible from one into another according to the needs and convenience of the user. In using the correlation, the designer needs to know, at least roughly, how accurate the results are likely to be in his application. We sometimes employ a film coefficient in cases where the flux is not even approximately proportional to the temperature difference (e.g., nucleate boiling), contradictory to the implication of Eq. (1.10). This practice offers a useful basis of comparison of the relative resistances of the several heat transfer processes in a given problem, but has no fundamental significance.

1.1.3. Two Phase (Liquid-Gas/Vapor) Flow Regimes of Two-Phase Flow In the present context, two-phase flow will usually refer to the simultaneous flow of a liquid and a gas or vapor through a duct. Such a flow occurs when a vapor is being condensed or a liquid is being vaporized; less commonly, a two-phase flow may involve a gas-liquid mixture (such as air and water) flowing together and being heated or cooled without any appreciable change of phase. The actual two-phase flow configuration, or regime, existing in a conduit in a given case depends upon the relative and absolute quantities and the physical properties of the fluids flowing, the geometric configuration of the conduit, and the kind of heat transfer process involved, if any. We will first consider the flow regimes observed by Alves (1) in his study of air-water flows in horizontal tubes. These regimes are diagrammed in rather idealized form in Fig. 1.8. The chief difference between flow regimes studied in a non-heat transfer situation and those existing during condensation is that a liquid film exists on the entire surface of the conduit during condensation. However, we may presume that this thin film of draining condensation does not cause any vital difference in the interaction between the vapor and the main inventory of liquid. For boiling flows, there are at least two additional flow regimes required. In one, bubbles are formed on the walls which disrupt the flow pattern in the immediate vicinity of the wall. In order for this to occur, the wall must be above the boiling point (though this does not guarantee that bubbles will form); if the bulk fluid temperature is saturated, the bubbles will be carried off downstream very much like the bubble flow regime shown in the Fig. 1.8, but if the bulk flow is subcooled, the bubbles will quickly collapse. In the latter case, there will be only a relatively small effect upon pressure drop. The second new flow regime is mist flow in which the liquid film on the wall in mist-annular flow has been evaporated and all the remaining liquid is carried as droplets in the vapor stream. We may view the flow regime as a consequence of the interaction of two forces, gravity and vapor shear, acting in different directions. At low vapor flow rates, gravity dominates and one obtains stratified, slug-plug, or bubble flow depending upon the relative amount of liquid present. At high vapor velocities, vapor shear dominates, giving rise to wavy, annular, or annular-mist flows. It would be desirable to have some way to predict the flow regime a priori, and many attempts have been made to do this in a general and consistent way. No attempt has succeeded, but the work of Baker (2) is considered to be generally the best available in the open literature even though it is a dimensional

11

representation and defies explanation in fundamental terms. The Baker map is shown in modified form as Fig. 1.9 and is useful in giving a general appreciation of the general kind of flow regime existing under given conditions. For two-phase flow inside vertical tubes, the stratified and wavy flow regimes cannot exist, and the flow regimes generally recognized in this case are bubble, slug, annular, and annular with mist. The Fair map (3) (Fig. 1.10) is generally recognized as the best for this configuration and is reproduced here in slightly modified form. It was originally developed for the analysis of thermosiphon reboilers and basically refers to a liquid stream entering at the

bottom of the tube and boiling as it flows upward. For condensation inside a vertical tube, the vapor generally enters at the top and the two-phase mixture resulting from condensation flows downward. At high condensing rates, where vapor shear dominates, most of the flow is in the annular flow regime. Fig. 1.10 can be used to estimate when this assumption breaks down. Our knowledge of two-phase flow patterns across tube banks, with or without baffles, is much more limited and indeed hardly extends beyond what intuition tells us. Fortunately, at least some correlations of pressure drop and related hydrodynamic effects are largely independent of a knowledge of flow pattern, and it is possible to make some quantitative calculations. Heat Transfer to a Two-Phase Flow The analysis of heat transfer to or from a two-phase flow is quite complex, involving the properties, quantities, and fluid mechanics of both phases. The design correlations resulting from these analyses are also subject to greater error than those for single phase heat transfer.

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For a gas-liquid flow involving no change of phase as a result of heat transfer, the flow is usually turbulent. The heat is transferred by turbulent eddies within each phase and across the gas-liquid and fluid-solid interfaces by turbulent boundary layer phenomena similar to turbulent flow in a pipe. The rate of heat transfer (and the pressure drop) is relatively high because of the strong turbulence created by the gas-phase shear on the liquid. Where phase changes are also involved, as in vaporization or condensation, additional heat transfer mechanisms come into play. These are discussed in more detail in subsequent sections.

Phase Relationships in Two-Phase (Liquid Gas/Vapor) Flow Two phase flows must also be characterized in terms of the composition and the resulting thermodynamic relationships between the two phases. Four cases can be distinguished:

a. The liquid and the gas are different pure components. The classic example is an air-water mixture, which is not a common industrial problem, but is very important because a great deal of what is known about two-phase flow has been determined on this system. While in general this information can be carried over to condensation and some boiling work, there are important differences that must be recognized and allowed for. Thermodynamically, the pressure and temperature can be independently varied over wide ranges in this system.

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b . The liquid and gas (vapor) are the same pure component. This is a common case in condenser design, occurring for example in condensers on columns separating and/or purifying a product. The pressure-temperature relationship in this case is the vapor pressure curve for the component. c. The liquid and gas (vapor) are multicomponents. The thermodynamic relationships are more complex, the temperature, for example, being variable over a range of values at a given pressure, but with a changing ratio of total liquid to total vapor and with changing composition of each phase. Prediction of the amount and composition of each phase is relatively well understood and easily done in few cases, as for mixtures of light hydrocarbons; other cases require laboratory thermodynamic data.

d. This case is identical to N or (c), but with a non-condensable gas present. e.g. [air in steam, or a solvent in an inert stripping gas]. This case is thermodynamically similar to (c), but the non-condensable gas does not appear in the liquid phase.

Other Two-Phase Systems Other two-phase systems occur in industrial processing and are briefly described below. In general, less is known about these systems than for liquid-gas/vapor systems, and the correlations are correspondingly fewer and less accurate. 1. Solid-gas systems. The most important application of solid-gas mixtures is in fluidized beds, in which the gas is

introduced at the bottom of the bed and flows upwards. If the gas flow rate is great enough, the forces on the solid particles (usually finely divided, such as catalyst particles or sand) cause the bed to expand. Then the particles no longer rest directly on one another and circulate freely through the bed; bubbles of gas rise through the bed and break through the upper surface, giving an appearance similar to boiling of a liquid. Heat transfer surface in the form of tubes may be put into the bed, and heat may also be transferred through the walls that contain the bed. Heat transfer rates in a fluidized bed are generally much higher than for a comparable flow of gas only.

If the gas rate is increased further, the solid particles are carried along with the gas. This is often done deliberately to give a "transfer line fluidized catalytic reactor." Heat transfer mechanisms in this case are qualitatively similar to those in gas liquid flow in a pipe, except that now the "eddies" in the solid phase are composed of masses of discrete particles all moving together, rather than a macroscopically continuous phase.

2. Solid-liquid systems. Liquid fluidized beds are also known; the major difference is that solid and liquid densities

are fairly similar, so the solid is more easily fluidized and transported with the liquid. Heat transfer rates are only marginally greater than for the comparable liquid-only flow.

3. Liquid-liquid systems. Mixtures of two immiscible liquids are sometimes encountered in heat exchangers and

flow patterns similar to those for gas-liquid flows are observed. The heat transfer coefficients for these systems are generally intermediate between the values that would be observed for the single liquids flowing alone at, the same average velocity.

1.1.4. Condensation Modes of Condensation Condensation is the process by which a vapor is changed to a liquid by removing the latent heat of condensation from the vapor. There are four basic modes or mechanisms of condensation generally recognized: dropwise, filmwise, direct contact, and homogeneous. 1. Dropwise condensation. In dropwise condensation, the drops of liquid form from the vapor at particularly-

favored locations, called nucleation sites, on a solid surface. These sites may be pits or scratches or any surface irregularity and there may be many thousands of them per square inch. These drops grow by continued condensa-tion from the vapor and by agglomeration of adjacent drops when they come into contact. Dropwise condensation

14

occurs only on surfaces which are not strongly wetted by the liquid and so the drops do not spread out over the surface. They grow in place until they become so large that they run off the surface by gravity or are blown off by the flowing vapor.

While dropwise condensation is alluring because of the high coefficients reported, it is not considered at this time to be suitable for deliberate employment in process equipment. Generally, contaminants must be continuously injected into the vapor, or special surface materials (often of low thermal conductivity) employed. Even so, the process is unstable and unpredictable, and of questionable efficacy under conditions of high vapor velocity and industrial practice.

2. Filmwise condensation. In filmwise condensation, the drops initially formed quickly coalesce to produce a

continuous liquid film on the surface through which heat must be transferred to condense more liquid. The actual heat transfer mechanism that operates in filmwise condensation is closely related to the two-phase flow mechanisms described in the previous section. Filmwise condensation is the usual mode that occurs in practice and that is assumed to exist for condenser design calculations.

3. Direct contact condensation. In direct contact condensation, the liquid coolant is sprayed directly into the vapor,

which condenses directly onto the surface of the spray drops. Direct contact condensation is a very efficient process, but it results in mixing the condensate and coolant. Therefore it is useful only in those cases where the condensate is easily separated, or where there is no desire to reuse the condensate, or where the coolant and condensate are the same substance.

4. Homogeneous condensation. In homogenous condensation, the liquid phase forms directly from supersaturated

vapor, away from any macroscopic surface. Since this requires subcooling the vapor on the order of a hundred degrees below the saturation temperature and since homogeneous condensation is actually observed at much smaller subcoolings, it is generally assumed that in practice there are sufficient numbers of dirt or mist particles present in the vapor to serve as nucleation sites. Homogeneous condensation is primarily of concern in fog for-mation in equipment and is not a design mode.

Condensation Outside Tubes. A common condenser design has the vapor outside a bank of tubes, with cold coolant flowing inside the tubes. The

15

vapor condenses on the outside of the tubes in the filmwise mode. Then the film flows under the influence of gravity to the bottom of the tube and drips off on to the next tube lower in the bank. If the condensate is very viscous, this filmwise flow may be in the laminar flow regime at least for the first few tubes (Fig. 1. 11a). More commonly, the con-densate is relatively inviscid (like water) and the film quickly becomes turbulent (Fig. 1.11b). The two cases lead to quite different predictions of the rate of heat transfer of the latent heat of condensation of the vapor through the liquid film.

If the vapor flow rate is high, vapor shear acting in crossflow on the condensate film becomes significant in blowing off the liquid, carrying it downstream as a spray, and causing the film to become turbulent even earlier than it would have under the influence of gravity alone (Fig.1. 11c).

Condensation Inside Tubes. Another common condenser configuration, especially in air-cooled condensers, has the vapor condensing inside tubes. The tubes are usually horizontal or only slightly slanted downwards to facilitate drainage, but vertical and inclined tube arrangements are also used. If the vapor flow rate is very low, the condensate forms on the cold wall and drains under the influence of gravity into a pool in the bottom of the tube; this pool in turn drains by gravity out the exit end of the tube (Fig. 1.12a). This phenomenon occurs under conditions, which favor the stratified or wavy flow regimes (see Figs. 1.8 and 1.9). Condensation of Mixtures. There are several important differences between the condensation of an essentially pure component and the condensation of a mixture. A basic analysis of the problem for a binary mixture was given by Colburn and Drew (4) who presented the diagram in Fig. 1.13. In a mixture, the heavier, less volatile components condense first, and only as the temperature of the remaining vapor is lowered do the lighter components condense. Thus, there are always sensible heat transfer effects to be considered in both the liquid and vapor phases. Because of the

16

low heat transfer coefficient associated with cooling the vapor, this process becomes a major resistance to heat transfer in the process, indicated in the figure by the temperature difference (Tv-Ti). The heavier component is enriched in the vicinity of the interface compared to the bulk (as indicated by the con-centration difference (yi-yv)) and counter-diffuses back into the bulk vapor. Since the compositions of both liquid and vapor phases are continually changing, there is the problem of changing physical properties to consider in evaluating the equations. The problem becomes even more complex for multicomponent mixtures, and approximation methods must be used to design condensers under these conditions.

1.1.5. Vaporization Mechanisms of Pool Boiling. There are several mechanisms, or processes, through which a liquid at the saturation temperature may be converted to a vapor by the addition of heat. If the boiling or vaporization occurs on a hot surface in a container in which the liquid is confined, the process is called "pool boiling." There are several quite different mechanisms by which pool boiling occurs depending upon the temperature difference between the surface and the liquid, and to a lesser extent, upon the nature of the surface and the liquid. These mechanisms are best discussed in connection with a curve of heat flux to the liquid. The classic curve of heat flux vs. temperature difference between surface and liquid saturation temperature for saturated pool boiling is reproduced in Fig. 1.14. The coordinates are logarithmic and the values shown are typical of a light hydrocarbon. The various regimes indicated on Fig. 1. 14 are: (a) The natural convection regime characterized

by a T less than about 10° F. In this region, the liquid in contact with the hot surface is superheated and rises by natural convection to the surface between the vapor and liquid where the superheat is released by quiescent vaporization of liquid. There is no vapor bubble formation in the bulk of the liquid and the heat transfer coefficients are characteristic of those of natural convection processes.

Δ

(b) The nucleate boiling regime, in which vapor

bubbles are formed at preferred nucleation sites - typically small pits or scratches - on the hot surface. The liquid is superheated by direct contact with the solid surface. Once a vapor nucleus forms at a nucleation site, the bubbles grow very rapidly by desuperheating the surrounding liquid until buoyant forces pull them free from the surface and cause them to rise to the vapor-liquid interface. There are various correlations that have been proposed in the literature for this region. However, because nucleate boiling phenomena are so strongly affected by the exact nature of the surface and the fluid, it is best to use experimental information when designing in this region.

(c) The point indicated by (c) is variously termed the maximum, peak, critical, or burnout heat flux: the highest

attainable heat flux for any reasonable surface temperature. At this point, the release of vapor is so vigorous that the flow of liquid to the surface is just sufficient to supply the vapor. Any further increase in surface temperature results in some of the vapor generated being unable to escape and the heat flux falls off. The peak heat flux

17

phenomenon is essentially a vapor hydrodynamic limit and is nearly independent of the exact nature of the surface. Various fundamentally-derived correlations of the peak heat flux exist.

(d) Transition boiling is an intermediate regime characterized by the occasional generation of a vapor film at the

surface, which insulates the surface from the cooling liquid, leading to local hot spots and unstable operation. The film is unstable in the transition boiling regime, and after a short period of time the liquid will flood back to cool the surface and temporarily go back into the nucleate boiling regime. Heat transfer equipment is ordinarily never intended to operate in the transition regime.

(e) Film boiling is the boiling regime that is stable at large temperature differences between the surface and the

saturation temperature. In film boiling, a stable, almost quiescent, film of vapor exists between the surface and the liquid pool. Heat transfers by conduction across the vapor to the liquid pool resulting in creation of more vapors. The film eventually becomes unstable and releases large vapor bubbles at relatively infrequent intervals. The bubbles rise through the pool to the interface. Film boiling is characterized by large temperature differences, generally very low heat fluxes, and correspondingly very low heat transfer coefficients. The surface may become hot enough to thermally degrade the substance being boiled. Fouling problems are also strongly accentuated in the film boiling regime because any fouling deposit that forms on the surface cannot be re-dissolved or washed away by liquid. In general, it is considered undesirable to operate in the film boiling regime.

Vaporization During Flow. The above boiling processes take place in a container or pool of liquid and are therefore referred to as pool boiling phenomena. Certain classes of vaporization equipment, notably thermosiphon and pump-through reboilers, operate with a net liquid velocity past the heat transfer surface. Under these conditions, the boiling processes are modified by a shear stress operating on the layer of liquid immediately adjacent to the hot surface. In general, natural convection boiling phenomena will be overshadowed by forced convection, and the nucleation process will be suppressed to some degree, possibly completely. With complete suppression, the superheated liquid is transported from the tube wall by turbulent eddies to the vapor-liquid interface, where vaporization takes place to form the vapor. The heat transfer coefficient under these conditions is greater than that which would exist if nucleate boiling only were operative. Film boiling is also possible under forced convection vaporization if the wall temperature is high enough. Frequently in these cases, one encounters mist flow, in which the liquid inventory is carried along in the vapor as tiny droplets, which are heated and vaporized by contact with the superheated vapor. This process is characterized by very low heat transfer coefficients and is never deliberately designed for in vaporization equipment.

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Correlations exist which allow the designer to estimate the effect of flow phenomena on the boiling process and if these are found to be significant, to calculate the heat transfer coefficient under forced convection conditions. Vaporization of Mixtures. Mixtures present a serious problem in the design of reboilers. Figs. 1.15 and 1.16 illustrate the problem. When a wide boiling range mixture enters a reboiler, the first vapor to be generated is rich in the low boiling components, leaving behind a liquid which is enriched in the high boiling components, especially in the immediate vicinity of the heat transfer surface. The light components then must diffuse through this barrier to the surface to vaporize. A greater driving force is required than is indicated by the mixed mean composition of the remaining liquid (the problem is entirely analogous to that of a wide condensing range mixture discussed in the previous section.) If one measures the heat flux as a function of the boiling surface temperature and compares it to the saturation temperature for the bulk mixture, the apparent result is a decrease in heat transfer coefficient at compositions intermediate to the pure components, as illustrated in Fig 1.17. However, it would seem that the real effect is a distortion of the temperature driving force and that the heat transfer coefficient itself is in fact a monotonically changing function from the pure light component to the pure heavy component. This however does not alter the basic problem which is how does one calculate the “true boiling heat transfer coefficient” -Δ T product, which is the heat flux, and hence gives the amount of area required in the condenser.

1.1.6. Radiation Source of Radiant Energy. All matter constantly radiates energy in the form of electromagnetic waves. The amount of energy emitted depends strongly upon the absolute temperature of the matter and to a lesser extent upon the nature of the surface of the matter. The basic law of radiation was derived by Stefan and Boltzmann and may be written for our purposes as:

4absT

AQ σε= (1.12)

where σ is the Stefan-Boltzmann constant (equal to 0. 1714 x 10 Btu/hr ft 2 (°R) ) and T8− 4

abs is the absolute temperature in °R. ε is the emissivity and has a value between 0 and 1. For a perfect reflector ε = 0 and for a perfect emitter, a so-called "black body", ε = 1. Highly polished metals have an emissivity of about 0.02 to 0.05, oxidized aluminum about 0.15, steel from 0.6 to 0.9 whether clean or rusted, and most paints from 0.8 to 0.98, largely independent of color. Since all surfaces that radiate heat will also absorb heat, it follows that all surfaces that can "see" each other are exchanging heat with one another, the net rate depending upon the absolute temperatures, the emissivities, the areas and

19

the spatial geometric relationships of the surfaces. The complete formulation of the problem is quite complex and will not be developed further here. Among the many excellent texts on the subject is "Radiation Heat Transfer" by L. Siegel and J. R. Howell (5). Radiation in Process Heat Transfer Problems At usual atmospheric temperatures, radiant heat transfer is relatively unimportant compared to most other heat transfer mechanisms, though there are a few areas where it makes a significant contribution, e.g., the loss of heat from non-insulated steam lines. At higher temperatures, radiation becomes relatively more important, and at temperatures above perhaps 1000 - 1500 °F (depending upon the other processes), it is usually essential to take radiation into account. However, for normal heat exchanger applications such elevated temperatures seldom exist. Therefore, the principles and equations of radiation heat transfer will not be developed further.

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1.2. Basic Heat Exchanger Equations

1.2.1. The Overall Heat Transfer Coefficient Consider the situation in Fig. (1.18). Heat is being transferred from the fluid inside (at a local bulk or average temperature of Ti), through a dirt or fouling film, through the tube wall, through another fouling film to the outside fluid at a local bulk temperature of To. Ai and Ao are respectively inside and outside surface areas for heat transfer for a given length of tube. For a plain or bare cylindrical tube,

i

o

i

o

i

o

rr

LrLr

AA

==ππ

22

(1.13)

The heat transfer rate between the fluid inside the tube and the surface of the inside fouling film is given by an equation of the form Q/A = h(Tf - Ts) where the area is Ai and similarly for the outside convective process where the area is Ao . The values of hi and ho have to be calculated from appropriate correlations. On most real heat exchanger surfaces in actual service, a film or deposit of sediment, scale, organic growth, etc., will sooner or later develop. A few fluids such as air or liquefied natural gas are usually clean enough that the fouling is absent or small enough to be neglected. Heat transfer across these films is predominantly by conduc-tion, but the designer seldom knows enough about either the thickness or the thermal conductivity of the film to treat the heat transfer resistance as a conduction problem. Rather, the designer estimates from a table of standard values or from experience a fouling factor Rf. Rf is defined in terms of the heat flux Q/A and the temperature difference across the fouling ΔTf by the equation:

AQT

R ff /

Δ= (1.14)

From Eq.1.14, it is clear that Rf is equivalent to a reciprocal heat transfer coefficient for the fouling, hf:

fff T

AQR

hΔ

==1

(1.15)

and in many books, the fouling is accounted for by a "fouling heat transfer coefficient," which is still an estimated quantity. The effect of including this additional resistance is to provide an exchanger somewhat larger than required when it is clean, so that the exchanger will still provide the desired service after it has been on stream for some time and some fouling has accumulated.

21

The rate of heat flow per unit length of tube must be the same across the inside fluid film, the inside dirt film, the wall, the outside dirt film, and the outside fluid film. If we require that the temperature differences across each of these resistances to heat transfer add up to the overall temperature difference, (Ti - To), we obtain for the case shown in Fig.1.18 the equation

( )ooo

fo

w

io

i

fi

ii

oi

AhAR

Lkrr

AR

Ah

TTQ

12

/ln1++++

−=

π

(1.16)

In writing Eq. (1.16), the fouling is assumed to have negligible thickness, so that the values of ri, ro, Ai and Ao are those of the clean tube and are independent of the buildup of fouling. Not only is this convenient – we don't know enough about the fouling to do anything else. Now we define an overall heat transfer coefficient U* based on any convenient reference area A*:

( oi TTAUQ −≡ ∗∗ ) (1.17) Comparing the last two equations gives:

( )ooo

fo

w

io

i

fi

ii AhA

AAR

LkrrA

AAR

AhA

U*****

*

2/ln

1

++++

=

π

(1.18)

Frequently, but not always, A* is chosen to be equal to Ao, in which case U* = Uo, and Eq. (1.18) becomes:

( )o

fow

ioo

i

ofi

ii

oo

hR

LkrrA

AAR

AhA

U1

2/ln

1

++++=

π

(1.19)

If the reference area A* is chosen to be A i , the corresponding overall heat transfer coefficient U i is given by:

( )oo

i

o

ifo

w

ioifi

i

i

AhA

AAR

LkrrA

Rh

U++++

=

π2/ln11

(1.20)

The equation as written applies only at the particular point where (Ti - To) is the driving force. The question of applying the equation to an exchanger in which Ti and To vary from point to point is considered in the next section. The wall resistance is ordinarily relatively small, and to a sufficient degree of precision for bare tubes, we may usually write

22

( )( )

( )( ) wio

i

w

ioi

wio

o

w

ioo

krr

XrLk

rrnA

krr

XrLk

rrnA

+

Δ≅

+

Δ≅

212

/;

212

/ππll

(1.21)

Inspection of the magnitudes of the terms in the denominator of Eqs. 1. 19 or 1.20 for any particular design case quickly reveals which term or terms (and therefore which heat transfer resistance) predominates. This term (or terms) controls the size of the heat exchanger and is the one upon which the designer should concentrate his attention. Perhaps the overall heat transfer coefficient can be significantly improved by a change in the design or operating conditions of the heat exchanger. In any case, the designer must give particular attention to calculating or estimating the value of the largest resistance, because any error or uncertainty in the data, the correlation, or the calculation of this term has a disproportionately large effect upon the size of the exchanger and/or the confidence that can be placed in its ability to do the job.

1.2.2. The Design Integral In the previous section, we obtained an equation that related the rate of heat transfer to the local temperature difference (T-t) and the heat transfer area A, through the use of an overall heat transfer coefficient U. In most exchanger applications, however, one or both of the stream temperatures change from point to point through the flow paths of the respective streams. The change in temperature of each stream is calculated from the heat (enthalpy) balance on that stream and is a problem in thermodynamics. Our next concern is to develop a method applying the equations already obtained to the case in which the temperature difference between the two streams is not constant. We first write Eq. (1. 17) in differential form

( )tTUdQdA

−= *

* (1.22)

and then formally integrate this equation over the entire heat duty of the exchanger, Q t :

( )∫ −= tQ

o tTUdQA *

* (1.23)

This is the basic heat exchanger design equation, or the design integral. U* and A* may be on any convenient consistent basis, but generally we will use Uo and Ao. U* may be, and in practice sometimes is, a function of the amount of heat exchanged. If 1/U*(T-t) may be calculated as a function of Q, then the area required may be calculated either numerically or graphically, as shown in Fig. (1.19).

23

The above procedure involving the evaluation of Eq. (1.23) is, within the stated assumptions, exact, and may always be used. It is also very tedious and time consuming. We may ask whether there is not a shorter and still acceptably accurate procedure that we could use. As it happens, if we make certain assumptions, Eq. (1.23) can be analytically integrated to the form of Eq. (1.24)

( )MTDUQ

A t*

* = (1.24)

where U* is the value (assumed constant) of the overall heat transfer coefficient and MTD is the "Mean Temperature Difference," which is discussed in detail in the following section.

24

1.3. The Mean Temperature Difference

1.3.1. The Logarithmic Mean Temperature Difference 1. Basic Assumptions. In the previous section, we observed that the design equation could be solved much easier if we could define a "Mean Temperature Difference" (MTD) such that:

)(**

MTDUQA t= (1.25)

In order to do so, we need to make some assumptions concerning the heat transfer process. One set of assumptions that is reasonably valid for a wide range of cases and leads to a very useful result is the following:

1. All elements of a given stream have the same thermal history. 2. The heat exchanger is at a steady state. 3. Each stream has a constant specific heat. 4. The overall heat transfer coefficient is constant. 5. There are no heat losses from the exchanger. 6. There is no longitudinal heat transfer within a given stream. 7. The flow is either cocurrent or counter-current.

The first assumption is worthy of some note because it is often omitted or stated in a less definitive way. It simply means that all elements of a given stream that enter an exchanger follow paths through the exchanger that have the same heat transfer characteristics and have the same exposure to heat transfer surface. In fact, in most heat exchangers, there are some flow paths that have less flow resistance than others and also present less heat transfer surface to the fluid. Then the fluid preferentially follows these paths and undergoes less heat transfer. Usually the differences are small and do not cause serious error, but occasionally the imbalance is so great that the exchanger is very seriously deficient. Detailed analysis of the problem is too complex to treat there, but the designer learns to recognize potentially troublesome configurations and avoid them. The second, third, fourth, fifth, and sixth assumptions are all straight- forward and are commonly satisfied in practice. It should be noted that an isothermal phase transition (boiling or condensing a pure component at constant pressure) corresponds to an infinite specific heat, which in turn satisfies the third assumption very well. The seventh assumption requires some illustration in terms of a common and simple heat exchanger configuration, the double pipe exchanger. 2. The Double Pipe Heat Exchanger. A double pipe heat exchanger essentially consists of one pipe concentrically located inside a second, larger one, as shown in Fig. 1.20. One fluid flows in the annulus between the inner and outer pipes and the other in the inner pipe. In Fig. 1.20, the two fluids are shown as entering at the same end, flowing in the same direction, and leaving at the other end; this con-figuration is called cocurrent. In Fig. 1.21, possible temperature profiles are drawn for the temperatures of the

25

fluids in this exchanger. (We have shown the hot fluid in the annulus and the cold fluid in the inner pipe, but the reverse situation is equally possible.) Notice that the outlet temperatures can only approach equilibrium with one another, sharply limiting the possible temperature change. If we had plotted the local temperatures vs. quantity of heat transferred, we would get straight lines, a consequence of the assumption that the specific heats are constant. A countercurrent heat exchanger is diagrammed in Fig.1.22 and a possible set of temperature profiles as a function of length is shown in Fig. 1.23. Also observe that the maximum temperature change is limited by one of the outlet temperatures equilibrating with the inlet temperature of the other stream, giving a basically more efficient heat exchanger for otherwise identical inlet conditions compared to the cocurrent arrangement. For this reason, the designer will almost always choose a countercurrent flow arrangement where possible. If one stream is isothermal, the two cases are equivalent and the choice of cocurrent or countercurrent flow is immaterial, at least on grounds of temperature profiles. 3. The Logarithmic Mean Temperature Difference. The analytical evaluation of the design integral Eq. 1.23 can be carried out for both cocurrent and countercurrent flow if the basic assumptions are valid. The details of the derivation are not relevant here and can be found in a number of standard textbooks (e.g. Ref. 6). For the cocurrent exchanger, the result is:

( ) ( )( )( )oo

ii

ooii

tTtT

tTtTMTD

−−

−−−=

ln

( ) ( )( )( )

(1.26)

and for the countercurrent case,

io

oi

iooi

tTtT

tTtTMTD

−−

−−−=

ln

( ) ( )iooi tTtTMTD −=−=

(1.27)

For the special case that (Ti – to) = (To – ti), eqn. (1.27) reduces to:

(1.28) The definitions of MTD's given in Eqns. (1.26) and (1.27) are the logarithmic means of the terminal temperature differences in each case. Because of its widespread importance in heat exchanger design, Eq. (1.27) is commonly referred to as "the logarithmic mean temperature difference," abbreviated as LMTD.

26

1.3.2. Configuration Correction Factors on the LMTD 1. Multiple Tube Side Passes. One of the assumptions of the LMTD derivation was that the flow was either completely cocurrent or completely countercurrent. For a variety of reasons, mixed, reversed or crossflow exchanger configurations may be preferred. A common case is shown in Fig. 1.24 - a one-shell-pass, two-tube-pass design (a 1-2 exchanger, for short): Note that on the first tube side pass, the tube fluid is in countercurrent flow to the shell-side fluid, whereas on the second tube pass, the tube fluid is in cocurrent flow with the shell-side fluid. A possible set of temperature profiles for this exchanger is given in Fig. 1.25.

Note that it is possible for the outlet tube side temperature to be somewhat greater than the outlet shell-side temperature. The resulting temperature profiles then might look like Fig. 1.26. The maximum possible tube outlet temperature that can be achieved in this case, assuming constant overall heat transfer coefficient, is

ioo tTt −= 2max, (1.29) Since this requires infinite area and all of the other assumptions being rigorously true, one would ordinarily stay well below this limit or look for another configuration. An alternative arrangement of a 1-2 exchanger is shown in Fig. 1.27, and a possible set of temperature profiles is given in Fig. 1.28. In this case t* cannot exceed To. In spite of the very different appearance of these two cases, it turns out that they give identical values of the effective temperature difference for identical temperatures.

27

The problem of computing an effective mean temperature difference for this configuration can be carried out along lines very similar to those used to obtain the LMTD. The basic assumptions are the same (except for the pure cocurrent or countercurrent limitation), though in addition it is assumed that each pass has the same amount of heat transfer area. Rather than calculate the MTD directly however, it is preferable to compute a correction factor F on the LMTD calculated assuming pure countercurrent flow, i.e.

LMTDMTDF = (1.30)

where F = 1 indicates the flow situation is equivalent to countercurrent flow, and lower values very clearly and directly show what penalty (ultimately expressed in area required) is being paid for the 1-2 configuration. It is im-portant to remember that the LMTD used in Eq. (1.30) is to be calculated for the countercurrent flow case, Eq. (1.27). The correction factor F is shown in Fig. 1.29 for a 1-2 exchanger as a function of two parameters R and P defined as (in terms of the nomenclature given on the chart):

(1.31a)

differenceetemperaturMaximumfluidtubeofRange

tTtt

P

fluidtubeofRangefluidshellofRange

ttTT

R

=−−

=

=−−

=

11

12

12

21

(1.31b)

The chart given here is adapted from the Standards of the Tubular Exchanger Manufacturers Association (9) and is almost identical to the one in Kern (7). The corresponding chart in McAdams (8) uses entirely different symbols, but is in fact identical to the one given here. However, there are other different (but finally equivalent) formulations and each one should be used carefully with its own definitions. Examination of the chart reveals that for each value of R, the curve becomes suddenly and extremely steep at some value of P. This is due to the tube-side temperature approaching one of the thermodynamic limits discussed above. It is extremely dangerous to design an exchanger on or near this steep region, because even a small failure of one of the basic assumptions can easily render the exchanger thermodynamically incapable of rendering the specified performance no matter how much excess surface is provided; the first assumption is especially critical in this case. Therefore, there is a generally accepted rule-of-thumb that no exchanger will be designed to F < 0.75. Besides, lower values of F result in large additional surface requirements and there is almost always some way to do it better. The discussion to this point has centered on the 1-2 exchanger. Larger numbers of tube-side passes are possible and frequently used. Kern discusses the problem briefly and points out that correction factors for any even number of tube-side passes are within about 2 percent of those for two passes, so it is common practice to use Fig. 1.29 for all 1-n exchangers where n is any even number. Other configurations will be discussed later. Kern, McAdams, and Perry's Handbook (10) give fairly extensive collections of F charts.

28

29

2. Multiple Shell-Side Passes. In an attempt to offset the disadvantage of values of F less than 1.0 resulting from the multiple tube side passes, some manufacturers regularly design shell and tube exchangers with longitudinal shell-side baffles as shown in Fig. 1.30. If one traces through the flow paths, one sees that the two streams are always countercurrent to one another, therefore superficially giving F = 1.0. The principle could be extended to multiple shell side passes to match multiple tube side passes but this is seldom or never done in practice. Even the provision of a single shell-side longitudinal baffle poses a number of fabrication, operation and maintenance problems. Without discussing all of the possibilities, we may observe that there may be, unless very special precautions are taken, will be, thermal leakage from the hot shell-side pass through the baffle to the other (cold) pass, which violates the 6th assumption. Further, there may even be physical leakage of fluid from the first shell-side pass to the second because of the pressure difference, and this violates the 1st assumption. A recent analysis has been made of the problem (Rozenmann and Taborek, Ref. 11), which warns one when the penalty may become severe.

3. Multiple Shells in Series. If we need to use multiple tube side passes (as we often do), and if the single shell pass configuration results in too low a value of F (or in fact is thermodynamically inoperable), what can we do? The usual solution is to use multiple shells in series, as diagrammed in Fig. 1.31 for a very simple case. More than two tube passes per shell may be used. The use of up to six shells in series is quite common, especially in heat recovery trains, but sooner or later pressure drop limits on one stream or the other limit the number of shells. Qualitatively, we may observe that the overall flow arrangement of the two streams is countercurrent, even though the flow within each shell is still mixed. Since, however, the temperature change of each stream in one shell is only a fraction of the total change, the departure from true countercurrent flow is less. A little reflection will show that as the number of shells in series becomes infinite, the heat transfer process approaches true counter- current flow and F

1.0. It is possible to analyze the thermal performance of a series of shells each having one shell pass and an even number of tube passes, by using heat balances and Fig. 1.29 applied to each shell. Such calculations quickly become very tedious and it is much more convenient to use charts derived specifically for various numbers of shells in series. Such charts are included in Chapter 2 of this Manual.

30

4. The Mean Temperature Difference in Crossflow Exchangers. Many heat exchangers - especially air-cooled heat exchangers (Fig. 1.32) - are arranged so that one fluid flows crosswise to the other fluid. The mean temperature difference in crossflow exchangers is calculated in much the same way and using the same assumptions as for shell and tube exchangers. That is,

MTD = F (LMTD) (1.32) where F is taken from Figure 1.33 for the configuration shown in Fig. 1.32. Recall that the LMTD is calculated on the basis that the two streams are in countercurrent flow.

31

1.4. Construction of Shell and Tube Heat Exchangers

1.4.1. Why a Shell and Tube Heat Exchanger? Shell and tube heat exchangers in their various construction modifications are probably the most widespread and commonly used basic heat exchanger configuration in the process industries. The reasons for this general acceptance are several. The shell and tube heat exchanger provides a comparatively large ratio of heat transfer area to volume and weight. It provides this surface in a form which is relatively easy to construct in a wide range of sizes and which is mechanically rugged enough to withstand normal shop fabrication stresses, shipping and field erection stresses, and normal operating conditions. There are many modifications of the basic configuration, which can be used to solve special problems. The shell and tube exchanger can be reasonably easily cleaned, and those components most subject to failure - gaskets and tubes – can be easily replaced. Finally, good design methods exist, and the expertise and shop facilities for the successful design and construction of shell and tube exchangers are available throughout the world. 1.4.2. Basic Components of Shell and Tube Heat Exchangers. While there is an enormous variety of specific design features that can be used in shell and tube exchangers, the number of basic components is relatively small. These are shown and identified in Fig. 1.34. 1. Tubes. The tubes are the basic component of the shell and tube exchanger, providing the heat transfer surface between one fluid flowing inside the tube and the other fluid flowing across the outside of the tubes. The tubes may be seamless or welded and most commonly made of copper or steel alloys. Other alloys of nickel, titanium, or aluminum may also be required for specific applications. The tubes may be either bare or with extended or enhanced surfaces on the outside. A typical Trufin Tube with extended surface is shown in Fig. 1.35. Extended or enhanced surface tubes are used when one fluid has a substantially lower heat transfer coefficient than the other fluid. Doubly enhanced tubes, that is, with enhancement both inside and outside are available that can reduce the size and cost of the exchanger. Extended surfaces, (finned tubes) provide two to four times as much heat transfer area on the outside as the corresponding bare tube, and this area ratio helps to offset a lower outside heat transfer coefficient. More recent developments are: a corrugated tube which has both inside and outside heat transfer enhancement, a finned tube which has integral inside turbulators as well as extended outside surface, and tubing which has outside surfaces designed to promote nucleate boiling. These and other developments are treated in detail in the last chapter of this book.

32

2. Tube Sheets. The tubes are held in place by being inserted into holes in the tube sheet and there either expanded into grooves cut into the holes or welded to the tube sheet where the tube protrudes from the surface. The tube sheet is usually a single round plate of metal that has been suitably drilled and grooved to take the tubes (in the desired pattern), the gaskets, the spacer rods, and the bolt circle where it is fastened to the shell. However, where mixing between the two fluids (in the event of leaks where the tube is sealed into the tube sheet) must be avoided, a double tube sheet such as is shown in Fig. 1.36 may be provided. The space between the tube sheets is open to the at-mosphere so any leakage of either fluid should be quickly detected. Triple tube sheets (to allow each fluid to leak separately to the atmosphere without mixing) and even more exotic designs with inert gas shrouds and/or leakage recycling systems are used in cases of extreme hazard or high value of the fluid. The tube sheet, in addition to its mechanical requirements, must withstand corrosive attack by both fluids in the heat exchanger and must be electrochemically compatible with the tube and all tube-side material. Tube sheets are sometimes made from low carbon steel with a thin layer of corrosion-resisting alloy metallurgically bonded to one side. 3. Shell and Shell-Side Nozzles. The shell is simply the container for the shell-side fluid, and the nozzles are the inlet and exit ports. The shell normally has a circular cross section and is commonly made by rolling a metal plate of the appropriate dimensions into a cylinder and welding the longitudinal joint ("rolled shells"). Small diameter shells (up to around 24 inches in diameter) can be made by cutting pipe of the desired diameter to the correct length ("pipe shells"). The roundness of the shell is important in fixing the maximum diameter of the baffles that can be inserted and therefore the effect of shell-to-baffle leakage. Pipe shells are more nearly round than rolled shells unless particular care is taken in rolling, In order to minimize out-of-roundness, small shells are occasionally expanded over a mandrel; in extreme cases, the shell is cast and then bored out on a boring mill. In large exchangers, the shell is made out of low carbon steel wherever possible for reasons of economy, though other alloys can be and are used when corrosion or high temperature strength demands must be met. The inlet nozzle often has an impingement plate (Fig. 1.37) set just below to divert the incoming fluid jet from impacting directly at high velocity on the top row of tubes. Such impact can cause erosion, cavitations, and/or vibra-tion. In order to put the impingement plate in and still leave enough flow area between the shell and plate for the flow to discharge without excessive pressure loss, it may be necessary to omit some tubes from the full circle pattern. Other more complex arrangements to distribute the entering flow, such as a slotted distributor plate and an enlarged annular distributor section, are occasionally employed. 4. Tube-Side Channels and Nozzles. Tube-side channels and nozzles simply control the flow of the tube-side fluid into and out of the tubes of the exchanger. Since the tube-side fluid is generally the more corrosive, these channels

33

and nozzles will often be made out of alloy materials (compatible with the tubes and tube sheets, of course). They may be clad instead of solid alloy. 5. Channel Covers. The channel covers are round plates that bolt to the channel flanges and can be removed for tube inspection without disturbing the tube-side piping. In smaller heat exchangers, bonnets with flanged nozzles or threaded connections for the tube-side piping are often used instead of channels and channel covers. 6. Pass Divider. A pass divider is needed in one channel or bonnet for an exchanger having two tube-side passes, and they are needed in both channels or bonnets for an exchanger having more than two passes. If the channels or bonnets are cast, the dividers are integrally cast and then faced to give a smooth bearing surface on the gasket between the divider and the tube sheet. If the channels are rolled from plate or built up from pipe, the dividers are welded in place. The arrangement of the dividers in multiple-pass exchangers is somewhat arbitrary, the usual intent being to provide nearly the same number of tubes in each pass, to minimize the number of tubes lost from the tube count, to minimize the pressure difference across any one pass divider (to minimize leakage and therefore the violation of the MTD derivation), to provide adequate bearing surface for the gasket and to minimize fabrication complexity and cost. Some pass divider arrangements are shown in Fig. 1.38. 7. Baffles. Baffles serve two functions: Most importantly, they support the tubes in the proper position during assembly and operation and prevent vibration of the tubes caused by flow-induced eddies, and secondly, they guide the shell-side flow back and forth across the tube field, increasing the velocity and the heat transfer coefficient. The most common baffle shape is the single segmental, shown in Fig. 1.39. The segment sheared off must be less than half of the diameter in order to insure that adjacent baffles overlap at least one full tube row. For liquid flows on the shell side, a baffle cut of 20 to 25 percent of the diameter is common; for low pressure gas flows, 40 to 45 percent (i.e., close to the maximum allowable cut) is more common, in order to minimize pressure drop.

The baffle spacing should be correspondingly chosen to make the free flow areas through the "window" (the area between the baffle edge and shell) and across the tube bank roughly equal. For many high velocity gas flows, the single segmental baffle configuration results in an undesirably high shell-side pressure drop. One way to retain the structural advantages of the segmental baffle and reduce the pressure drop (and,

34

regrettably, to some extent, the heat transfer coefficient, too) is to use the double segmental baffle, shown in Fig. 1.40. Exact comparisons must be made on a case-to-case basis, but the rough effect is to halve the local velocity and therefore to reduce the pressure drop by a factor of about 4 from a comparable size single segmental unit.

For sufficiently large units, it is possible to go to triple segmental arrangements and ultimately to strip and rod baffles, the important point being always to insure that every tube is positively constrained at periodic distances to prevent sagging and vibration. Special provisions must be made for supporting finned tubes passing through a baffle: 1) provide unfinned lands (of normal bare tube diameter) at the baffles; 2) use baffles thick enough that several fins fall within the baffle thickness and provide a solid bearing surface or 3) provide a thin metal wrap outside the fins in the vicinity of the baffle.

1.4.3. Provisions for Thermal Stress 1. The Thermal Stress Problem. Since, by its very purpose, the shell of the heat exchanger will be at a significantly different temperature than tubes, the shell will expand or contract relative to the tubes, resulting in stresses existing in both components and being transmitted through the tube sheets. The consequences of the thermal stress will vary with circumstances, but shells have been buckled or tubes pulled out of the tube sheet or simply pulled apart. The fixed tube sheet exchanger shown in Fig. 1.34 is especially vulnerable to this kind of damage because there is no provision made for accommodating differential expansion. There is a rough rule of thumb that says a simple fixed tube sheet configuration can only be used for cases where the inlet temperatures of the two streams do not differ by more than 100 'F. Obviously, there must be many qualifications made to such a flat statement, recognizing the differences in materials and their properties, temperature level of operation, start-up and cycling operational procedures, etc. 2. Expansion Joint on the Shell. The most obvious solu-tion to the thermal expansion problem is to put an ex-pansion roll or joint in the shell, as shown in Fig. 1.41. This becomes less attractive for large diameter shells and/or increasing shell-side pressure. However, very large diameter, near-atmospheric pressure shells have been designed with a partial ball-joint in the shell designed to allow the shell to partially "rotate" to accommodate stresses. 3. Internal Bellows. In recent years, an internal bellows design (Fig. 1.42) has become popular for such applications as waste heat vertical thermosiphon reboilers, where only one pass is permitted on the tube side. These bellows have been designed to operate successfully with high pressure boiling water on the tube side and high temperature reactor effluent gas on the shell.

35

4. The U-Tube Exchanger. One design variation that allows independent expansion of tubes and shell is the U-tube configuration shown schematically in Fig. 1.43. While this design solves the thermal expansion problem about as well as it can be solved, it has some drawbacks (e.g., inability to replace individual tubes except in the outer row, inability to clean around the bend) that render it unacceptable for some services.

5. Floating Head Designs. Several different designs of "floating head" shell and tube exchangers are in common use. The goal in each case, of course, is to solve the thermal stress problem and each design does accomplish that goal. Inevitably, however, something must be given up, and each configuration has a somewhat different set of drawbacks to be considered when choosing one. The simplest floating head design is the "pull-through bundle" type, shown in Fig. 1.44. One of the tube sheets is made small enough that it and its gasketed bonnet may be pulled completely through the shell for shell-side in-spection and cleaning. The tube side may be cleaned and individual tubes may be replaced without removing the bundle from the shell. Unfortunately, many tubes must be omitted from the edge of the full bundle to allow for the bonnet flange and bolt circle.

This objection is met in the "split-ring floating head" type (Fig. 1.45) by bolting the floating head bonnet to a split backing ring rather than to the tube sheet. At some cost in added mechanical complexity, most of the tubes lost from the bundle in the pull-through design have been restored, and the other features retained. Two other types, the "outside-packed lantern ring," (Fig. 1.46), and the "outside-packed stuffing box", (Fig. 1.47) are more prone to leakage to the atmosphere than the foregoing types and give up the advantage of positive sealing

36

so important in high pressure or hazardous fluid service. They have the advantage of allowing single tube side pass design.

1.4.4. Mechanical Stresses 1. Sources of Mechanical Stresses. Every exchanger is subject to mechanical stresses from a variety of sources in addition to temperature gradients. There are mechanical stresses which result from the construction techniques used on the exchanger, e.g., tube and tube sheet stresses resulting from rolling in the tubes. During the manufacture, shipping and installation of the exchanger there are many, frequently unforeseen stresses imposed. There are stresses caused by the support structure reacting to the weight of the exchanger, and stresses from the connecting piping; these stresses are generally very different during normal plant operation than during construction or shutdown. Finally, there are the stresses arising within the exchanger as a result of the process stream conditions - especially pressure – during operation. 2. Provision for Mechanical Stress. To protect the ex changer from permanent deformation or weakening from these mechanical stresses, it is necessary to design the exchanger so that any stress that can be reasonably expected to occur will not strain or deform the metal beyond the point where it will spontaneously return to its original condition. And it is necessary to insure that stresses greater than the design values do not occur. The analysis of stresses and strains in a heat exchanger is an extremely broad and complicated subject, and will not be developed in any detail here. The more obvious problems can be at least anticipated in a qualitative way by the thermal designer who can then seek the advice of a specialist in the subject.

1.4.5. The Vibration Problem A very serious problem in the mechanical design of heat exchangers is flow - induced vibration of the tubes. There are several possible consequences of tube vibration, all of them bad. The tubes may vibrate against the baffles, which can eventually cut holes in the tubes. In extreme cases, the tubes can strike adjacent tubes, literally knocking holes in each other. Or the repeated stressing of the tube near a rigid support such as a tube sheet can result in fatigue cracking of a tube, loosening of the tube joint, and accelerated corrosion. Vibration is caused by repeated unbalanced forces being applied to the tube. There are a number of such forces, but the most common one in heat exchangers is the eddying motion of the fluid in the wake of a tube as the fluid flows across the tube. The unbalanced forces are relatively small, but they occur tens, hundreds, or thousands of times a second, and their magnitudes increase rapidly with increased fluid velocity. Even so, these forces are ordinarily damped out with no damage to the tube. However, any body can vibrate much more easily at certain frequencies (called "natural frequencies") than at others. If the unbalanced forces are applied at "driving frequencies" that are at

37

or near these natural frequencies, resonance occurs; and even small forces can result in very strong vibration of the tube. Although progress is being made, the prediction of whether or not a given heat exchanger configuration will adequately resist vibration is not yet a well-developed science. The two best ways to avoid vibration problems are to support the tubes as rigidly as possible (e.g., close baffle spacing) and to keep the velocities low. Both of these often conflict with the desire to keep the cost of the exchanger down. For the present, experience is the best guide in this area.

1.4.6. Erosion Another essentially mechanical problem in heat exchanger design is that of erosion: the rapid removal of metal due to the friction of the fluid flowing in or across the tube. Erosion often occurs with and accelerates the effect of corrosion by stripping off the protective film formed on certain metals. The erosion rate depends upon the metal (the harder the metal, the less the erosion if other factors are equal), the velocity and density of the fluid, and the geometry of the system. Thus, erosion is usually more severe at the en-trance of a tube or in the bend of a U-tube, due to the additional shear stress associated with developing the boundary layer or turning the fluid. Other, more elusive effects are associated with the chemistry of the fluid and the tube metal, especially where corrosion is involved. There are some commonly used upper velocity limits for flow inside tubes of a given metal. These limits are shown in Table 1. 1.

1.4.7. Cost of Shell and Tube Heat Exchangers Some appreciation of the cost basis for shell and tube exchangers is essential to the proper selection of a con-figuration and allocation of streams. It is impossible to furnish here a precise and reasonably current correlation for estimating the cost of a given configuration, but we may at least identify the relative contributions to the total cost and how these change with certain design specifications. The cost of a shell and tube exchanger f.o.b. the fabrication shop is composed of the costs of the individual components (shell, tubes, etc.), plus the assembly cost. The cost of each component is the sum of the material cost, plus gross fabrication (e.g., rolling the shell), plus machining. The final price to the customer will also include engineering and other overheads, the fabricator's profit and shipping. Some of these, e.g. the profit, are roughly proportional to the total cost, while the shipping is more nearly proportional to the total weight. In order to arrive at the most economical unit (on a first cost basis), it is necessary to consider the effect of the special requirements of the unit on each of the component costs. For example, suppose one fluid requires a special alloy to resist corrosion. If that fluid is put in the shell, both the shell and the tubes must be made of that alloy; conversely, if the corrosive fluid is put in the tubes, only the tubes and tube side fittings must be alloy, and that cost can often be further reduced if the tube sheets and channels are only faced with the alloy. The same total heat transfer area may be put in a shell that is of small diameter and relatively long, or in one that has a larger diameter and shorter length. The cost of a shell - often the largest single cost in the total exchanger cost - increases very rapidly with diameter but only linearly (at most) with length. Therefore, unless space or pressure drop limits dictate otherwise, the most economical exchanger is usually one of relatively large length to diameter ratio - up to perhaps 12 to I for a very rough rule of thumb.

38

For the exchanger as a whole, only a complete cost analysis of several different designs can establish which is in fact the least expensive in first cost. There are however, a number of rules, which will tend to select the more economical designs out of the multitude of possibilities. These will be discussed further.

1.4.8. Allocation of Streams in a Shell and Tube Exchanger In principle, either stream entering a shell and tube ex changer may be put on either side-tube-side or shell-side -of the surface. However, there are four considerations, which exert a strong influence upon which choice will result in the most economical exchanger: 1. High pressure: If one of the streams is at a high pressure, it is desirable to put that stream inside the tubes. In this case, only the tubes and the tube-side fittings need be designed to withstand the high pressure, whereas the shell may be made of lighter weight metal. Obviously, if both streams are at high pressure, a heavy shell will be required and other considerations will dictate which fluid goes in the tube. In any case, high shell side pressure puts a premium on the design of long, small diameter exchangers. 2. Corrosion: Corrosion generally dictates the choice of material of construction, rather than exchanger design. However, since most corrosion - resistant alloys are more expensive than the ordinary materials of construction, the corrosive fluid will ordinarily be placed in the tubes so that at least the shell need not be made of corrosion - resistant material. If the corrosion cannot be effectively prevented but only slowed by choice of material, a design must be chosen in which corrodible components can be easily replaced (unless it is more economical to scrap the whole unit and start over.) 3. Fouling: Fouling enters into the design of almost every process exchanger to a measurable extent, but certain streams foul so badly that the entire design is dominated by features which seek a) to minimize fouling (e.g. high velocity, avoidance of dead or eddy flow regions) b) to facilitate cleaning (fouling fluid on tube-side, wide pitch and rotated square layout if shell-side fluid is fouling) or c) to extend operational life by multiple units. 4. Low heat transfer coefficient: If one stream has an inherently low heat transfer coefficient (such as low pressure gases or viscous liquids), this stream is preferentially put on the shell-side so that extended surface may be used to reduce the total cost of the heat exchanger.

39

1.5. Application of Extended Surfaces to Heat Exchangers

1.5.1. The Concept of the Controlling Resistance Examination of Eqs. (1.18), (1.19), or (1.20) shows that, if any one of the terms in the denominator on the right hand side is substantially larger than the others, it essentially fixes the value of U. Since each of these terms is a resistance to heat transfer, the term that dominates is called the "controlling resistance," and the designer needs to devote particular attention to its value. As noted in the previous section, one heat transfer coefficient may be much lower than the other, and this coefficient is the one that leads to a high resistance for heat transfer. Thus, a low coefficient corresponds to a high, and frequently controlling, resistance. One way to minimize the adverse effects of a low heat transfer coefficient is to place that stream on the shell side of a heat exchanger and use extended surface to increase the ratio of the outside to inside area (Ao/Ai) to offset the low value of ho.

1.5.2. Types of Extended Surface "Extended surface" is a term that covers many possibilities but the type that we shall be most concerned with is the round tube with round fins essentially transverse to the tube axis. Typical integral finned tubes are shown in Fig. 1.48. The low-finned tube commonly used in shell and tube exchangers provides about 3 to 4 times as much outside area as inside; i.e., Ao/Ai is about 3 to 4. In the Wolverine S/T type, fin counts of 16 to 40 per inch are available. The outside diameter of the fins is just slightly less than that of the bare tube at the ends, so the tube can be inserted through the tube sheet holes. The wall under the fins is controlled to a specified thickness, and the wall at the plain ends is about two gauges heavier. For tube support at baffles, unfinned sections ("lands") are available. A medium-fin height tube with 11 fins per inch is also used in shell and tube exchangers, and has an Ao/Ai ratio typically about 5. The high-fin tube is used to advantage when gases are to be heated or cooled or when a process stream is to be air-cooled. High-fin tubes come in a wide variety of fin heights, thicknesses, and spacings, giving values of Ao/Ai up to 25. For corrosion protection, a mechanically bonded liner tube may be used inside the finned tube. The liner can be made of corrosion-resistant alloy, while the outer tube and fins are made of a high conductivity metal such as copper or aluminum to improve heat transfer.

Fig 1.48. Several types of Wolverine Trufin Tubing

All of the finned tubes produced by Wolverine have integral fins, i.e., the fins are formed from the base tube by an extrusion process so that the final tube and its fins are one piece of metal. For high-finned tubes, other fin types exist in which the fin is a separate piece of metal wrapped on the tube. The fin is usually held in place only by the tension of the deformed fin metal or occasionally by soldering, welding or brazing the fin to the tube. In these finned tubes, contact resistance between the fin and the tube is an important consideration. Even if the fin is originally in good thermal contact, repeated thermal cycling can result in creep and partial separation of the

40

fin. Also, the crevice between fin and tube provides a possible site for corrosion with consequent increase in contact resistance and possibly accelerated failure of the tube.

1.5.3. Fin Efficiencies and Related Concepts Consideration of the heat flow path from the tube-side fluid through the tube wall, and fins, and into the fin side fluid reveals that the distance which the heat must flow' is longer than in the corresponding plain tube case. It is true that the additional distance is through the usually highly conductive fin metal, but there is still an additional resistance to the flow of heat, which partially offsets the advantage of the extended surface. In order to take account of this resistance in our calculations, we define a "fin efficiency, " φ, in the following manner: φ is the ratio of the total heat transferred from the actual fin to the total heat that would be transferred if the fin were isothermal at its base temperature. This definition is illustrated by Fig. 1.49. In this figure, the fin temperature Tf is higher than the bulk outer fluid temperature, To, so heat flows from the fin to the outer fluid. All the heat thus transferred however must flow into the fin through the base or root of the fin. Therefore, the temperature at the base of the fin, Tr, is the highest temperature in the fin. The temperature decreases continuously from that point as one proceeds outward in the fin in order to cause the heat to flow from the base to the fin. The rate of decrease depends upon the shape and thermal conductivity of the fin and the film heat transfer coefficient from the fin to the outer fluid. It is clear that the maximum amount of heat would be transferred from the fin if the fin were everywhere at Tr, and that could only happen if the fin metal had infinite thermal conductivity. No metal has infinite thermal conductivity, of course, but this makes a convenient reference point against which real fins can be compared, as in the above definition of fin efficiency. A complex mathematical analysis is necessary to obtain a solution for the fin efficiency, and this will not be carried out here. A very complete discussion of the problem and it solutions is given in the book, "Extended Surface Heat Transfer", by Kern and Kraus (12). For the kinds of fins that are considered here, a good equation to use over most of the range of interest is:

r

o

ddm

31

12

+

=Φ (1.33)

where

kyRh

Hm

foo

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=1

2 (1.34)

41

The geometrical variables are defined in Fig. 1.50, and ho and Rfo are respectively the actual convective heat transfer coefficient and the actual fouling resistance on the fin side, based on the fin area. An inspection of Eqs. (1.33) and (1.34) indicates that the fin efficiency is higher for the better-conducting fin materials, and higher as the outside heat transfer coefficient is lower. Finned tubes are not generally used in applications where the fin efficiency is less than 0.65. The fin efficiency is applicable only to the fin area of the tube. It is assumed that heat is transferred from the root area of the tube (between the fins) with an efficiency of 1.00. Thus, we may write the equation for the total heat transfer from the tube as:

( ) ( )Φ−+−= orfinoorrooto TTAhTTAhQ (1.35) where Aroot refers to the heat transfer area of the root portions between the fins and Afin to the heat transfer area of all the fins on the tube. This equation may be written as:

( )( )finrootoro AATThQ Φ+−= (1.36) or

( ) eqoro ATThQ −= (1.37) where Aeq, the equivalent or effective heat transfer area, is

finrooteq AAA Φ+= (1..38) If the outside of the tube (including the fins) has a fouling film on it, Eqs. (1.35), (1.36) and (1.37) must be written:

( ) ( )Φ−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

++−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+= orfin

foo

orroot

foo

TTAR

h

TTAR

h

Q1

11

1 (1.39)

or

( )( )finrootor

foo

AATTR

h

Q Φ+−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+= 1

1 (1.40)

or

( ) eqor

foo

ATTR

h

Q −

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+= 1

1 (1.41)

42

and Eq. (1.38) is unchanged. Aeq may be divided by the total outside heat transfer area of the tube Ao to give a weighted fin efficiency.

finroot

finrootw AA

AAE

+Φ+

= (1.42)

and values of this ratio can be calculated for a given tube as a function of material and

⎟⎟⎠

⎞⎜⎜⎝

⎛+ fo

o

Rh1

once and for all. This has been

done in Figures 1.51 a, b, c for the Wolverine Trufin Products. FFTSQHrBtu

fh foo

°××+

../,/1

1

1.5.4. The Fin Resistance Method Another way of representing the heat transfer effectiveness of a finned tube is by the "fin resistance method," as first developed by Carrier and Anderson (13) and further modified by Young and Ward (14). This is the method that will be generally used in the remainder of this Manual. The line of argument that led to Eq. (1.16) was based upon a plain tube in which all of the outer heat transfer was taken to be equivalent in its ability to transfer heat. The validity of that equation may be extended to finned tubes if we rewrite the equation introducing an additional resistance due to heat conduction through the fin, Rfin:

Fig. 1.51a Weighted Fin Efficiency of 19 Fin Type S/T Trufin

ooo

fo

o

fin

mw

w

i

fi

ii

oi

AhAR

AR

Akx

AR

Ah

TTQ11

+++Δ

++

−= (1.43)

The third term in the denominator, referring to the wall resistance, is evaluated for that portion of the tube wall that lies between the inside diameter and the root diameter, i.e., exclusive of the fins. Thus

( )ir ddx −=Δ 2/1

( )LxdA im Δ+≈

(1.44) and

π (1.45)

We now define an overall heat transfer coefficient Uo based on the entire outer surface of the finned tube, Ao:

( )oioo TTAUQ −=F°FTSQHrBtu

fh foo

××+

../,/1

1

(1.46) Fig 1.51b. Weighted Fin Efficiencies of 26 and 28 Fin Type S/T Trufin

43

Eliminating and ( between Eqs. (1.43) and (1.46) and rearranging, we have: Q )oi TT −

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

Δ+++=

i

o

ii

ofi

mw

ofinfo

oo AA

hAAR

AkxARR

hU111

(1.47)

As written, each term on the right-hand-side is a resistance to heat transfer for the particular process involved; the temperature drop across each part of the process is proportional to the magnitude of the corresponding resistance. If we take the first three terms on the right side of Eq. (1.47), the corresponding temperature drop is that from the root to the external fluid,

, and the corresponding heat flow is ( or TT − )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−=

finfoo

oor

RRh

ATTQ1

(1.48)

FFTSQHrBtu

fh foo

°××+

../,/1

1 This may be equated to Eq. (1.40) which uses the fin efficiency Φ, but which must give the same heat flow. With some rearrangement

Fig.1.51c Weighted Fin Efficiencies of 32 & 40 Fin Type S/T Trufin

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Φ+

Φ−= fo

o

fin

rootfin R

hAA

R 11

finR

FFTSQHRBturh foo

°××+

../,/1

1

Fig. 1.52a Fin Resistance of 19 Fin Type S/T Trufin

(1.49)

Using Eqs. (1.33, 1.34) to evaluate ,

may be calculated as a function of

Φ

⎟⎟⎠

⎞⎜⎜⎝

⎛

oh1+ foR

for various fin materials and geometries: results are given in Figures 1.52a, b, c for the standard Wolverine Trufin Products. An important feature of Rfin is that it is nearly constant for a given fin geometry and material out to high values of the abscissa (relative to the usual range of finned tube application.)

44

Fig. 1.52c Fin Resistance of 32 & 40 Fin Type S/T TrufinFig.1.52b Fin Resistance of 26 & 27 Fin Type S/T Trufin

1.5.5. Some Applications of Finned Tubes. One major application of high-finned tubes is in air-cooled heat exchangers. Atmospheric air, like all low pressure gases, gives a very low heat transfer coefficient at normal velocities. By contrast, the tube-side fluid, usually a liquid to be sensibly cooled or a vapor to be condensed may have a coefficient up to 100 times higher, or even more. Therefore, high-finned tubes are used in these exchangers to reduce the overall size of exchanger required. Even so, some of these installations cover several acres. Construction details and design methods for this kind of heat exchanger are given in Chapter 4 of this manual. Low- and medium-finned tubes are used in a variety of sensible, condensing, and boiling services in shell and tube exchangers. A typical sensible heat transfer application would be cooling a compressed gas in a compressor inter-cooler, using cooling water in the tubes. Low-finned tubes are used for condensing organic vapors, which have condensing coefficients only a third or a quarter of that of the cooling water inside the tubes. In addition to providing additional heat transfer area, the fins provide drip points that facilitate the drainage of the condensate. On the other hand, finned tubes are not used for condensing steam or other high surface tension fluids on the fin side. The high surface tension causes the liquid to hang on the surface, largely insulating it by a static film of liquid. Finned tubes are also used in boiling services, especially when condensing steam is the heating medium inside the tubes. The condensing coefficient in this case may be 2000 or 3000 Btu/hr.ft2.F, so even the high heat transfer coefficients commonly associated with nucleate boiling may be relatively small by comparison, and the design can benefit by the use of low-finned tubes.

45

1.6. Fouling in Heat Exchangers

1.6.1. Typical Fouling Resistances. "Fouling" is a general term that includes any kind of deposit of extraneous material that appears upon the heat transfer surface during the lifetime of the heat exchanger. Whatever the cause or exact nature of the deposit, an additional resistance to heat transfer is introduced and the operational capability of the heat exchanger is correspondingly reduced. In many cases, the deposit is heavy enough to significantly interfere with fluid flow and increase the pressure drop required to maintain the flow rate through the exchanger. The designer must consider the effect of fouling upon heat exchanger performance during the desired operational lifetime and make provisions in his design for sufficient extra capacity to insure that the exchanger will meet process specifications up to shutdown for cleaning. The designer must also consider what mechanical arrangements are necessary to permit easy cleaning. Allowance for fouling is largely a matter of experience. There are tables of typical values for various services (e.g., Ref.10 and the TEMA Standards (9), and there is a proprietary correlation available for cooling water. An adaptation of the TEMA table is included here as Table 1.2. However, fouling behavior is strongly dependent upon many variables and these interactions are very complex, so each problem really needs to be examined for unexpected pitfalls and special considerations. The best general survey of fouling in heat exchangers is by Taborek, et al. (15, 16). Other important contributions are Refs. (17) to (21).

1.6.2. Types of Fouling There are several different basic mechanisms by which fouling deposits may be created and each of them in general depends upon several variables. In addition, two or more fouling mechanisms can occur in conjunction in a given service. In this section we will identify the major mechanisms of fouling and the more important variables upon which they depend. a. Sedimentation fouling. Many streams and particularly cooling water contain suspended solids which can settle out upon the heat transfer surface. Usually the deposits thus formed do not adhere strongly to the surface and are self- limiting; i.e., the thicker a deposit becomes, the more likely it is to wash off (in patches) and thus attain some asymptotic average value over a period of time. Sedimentation fouling is strongly affected by velocity and less so by wall temperature. However, a deposit can "bake on" to a hot wall and become very difficult to remove. b. Inverse solubility fouling. Certain salts commonly found in natural waters - notably calcium sulfate - are less soluble in warm water than in cold. If such a stream encounters a wall at a temperature above that corresponding to saturation for the dissolved salt, the salt will crystallize on the surface. Crystallization will begin at special active points - nucleation sites – such as scratches and pits, often after a considerable induction period, and then spread to cover the entire surface. The buildup will continue as long as the surface in contact with the fluid has a temperature above saturation. The scale is strong and adherent and usually requires vigorous mechanical or chemical treatment to remove it. c. Chemical reaction fouling. The above fouling mechanisms involve primarily physical changes. A common source of fouling on the process stream side are chemical reactions that result in producing a solid phase at or near the surface. For example, a hot heat transfer surface may cause thermal degradation of one of the components of a process stream, resulting in carbonaceous deposits (commonly called "coke") on the surface. Or a surface may cause

46

polymerization to occur, resulting in a tough layer of low-grade plastic or synthetic rubber. These deposits are often extremely tenacious and may require such extreme measures as burning off the deposit in order to return the exchanger to satisfactory operation. d. Corrosion product fouling. If a stream corrodes the metal of the heat transfer surface, the corrosion products may be essential to protect the remaining metal against further corrosion, in which case any attempt to clean the surface may only result in accelerated corrosion and failure of the exchanger. e. Biological fouling. Many cooling water sources and a few process streams contain organisms that will attach to solid surfaces and grow. These organisms range from algae and microbial slimes to barnacles and mussels. Even when only a very thin film is present, the heat transfer resistance can be very great. Where macroscopic forms like mussels are present, the problem is no longer one of heat transfer - there won't be any through the animal - but rather of plugging up the flow channels. If biological fouling is thought to be a problem, the usual solution is to kill the life forms by chlorination, or to discourage their settling on the heat transfer surface by using 90-10 copper-nickel (Alloy C70600) or other high copper alloy tubes. As an alternative to continuous chlorination, intermittent "shock" chlorination may be successful. f. Combined mechanisms. Most of the above fouling processes can occur in combination. A common example is the combination of (a) and (b) in cooling tower water. Most surface waters contain both sediment and calcium carbonates, and the concentrations of these components rise as the water is re-circulated through the cooling system. It is therefore common to find deposits composed of crystals of inverse solubility salts together with finely divided sediments. The behavior of these deposits is intermediate between the two limiting cases: the crystals tend to hold the sediment in place, but there are planes of weakness in the structure that fail from time to time and cause the deposit to break off in patches.

1.6.3. Effect of Fouling on Heat Transfer As noted above the effect of fouling is to form an essentially solid deposit upon the surface, through which heat must be transfer red by conduction. If we knew both the thickness and the thermal conductivity of the fouling, we could treat the heat transfer problem simply as another conduction resistance in series with the wall. In general, we know neither of these quantities and the only possible technique is to introduce the additional resistance as fouling factors in computing the overall heat transfer coefficient as previously discussed. Fouling effects inside the tube usually cause no particular problems if allowance has been made for the reduction in heat transfer and the small increase in flow resistance. However, fouling on the outside of finned tubes can be a more complicated matter, because in extreme situations there is a possibility that the finite thickness of the fouling layer can effectively close off the flow through the fins. On the other hand, finned surfaces are sometimes found to be more resistant to fouling than plain surfaces (Ref. 21); the reasons for this are not well-established, though it may be that the expansion and contraction of the surface during normal operational cycles tends to break off brittle fouling films. Caution is advised, however, in applying finned tubes to services known to be significantly fouling. High-finned tubes are commonly used only with air and other low pressure and relatively clean gases. Such fouling as does occur is mostly dust deposition, which can easily be removed by blowing.

1.6.4. Materials Selection for Fouling Services Potential fouling problems may influence materials selection in one of three ways. Most obvious perhaps is the minimization of corrosion-type fouling by choosing a material of construction, which does not readily corrode or produce voluminous deposits of corrosion products. If chemical removal of the fouling deposit is planned, the material selected must also be resistant to attack by the cleaning solutions.

47

Secondly, biological fouling can be largely eliminated by the selection of copper-bearing alloys, such as 90-10 copper-nickel (UNS 70600) or 70-30 copper-nickel (UNS 71500). Generally, alloys containing copper in quantities greater than 70% are effective in preventing or minimizing biological fouling. Thirdly, some types of fouling can be controlled or minimized by using high-flow velocities. If this technique is to be employed, the possibility of metal erosion should be considered as it is important to restrict the velocity and/or its duration to values consistent with satisfactory tube life. Some metals, such as titanium or stainless steels, can be quite resistant to erosion by the high velocity effluent. Recently, a new copper-nickel alloy containing nominally 83% copper, 17% nickel and 0.4% chromium has been developed by the International Nickel Company (Alloy C72200). This alloy is similar to the other copper-nickels relative to its corrosion resistance, but, however, has a much greater resistance to velocity attack, being capable of operating in sea water at velocities approaching 25 feet per second.

1.6.5. Removal of Fouling If fouling cannot be prevented from forming, it is necessary to make some provision for its periodic removal. Some deposits can be removed by purely chemical means, e.g., removal of carbonate deposits by chlorination. The application of chemical cleaning techniques is a specialized art and should be undertaken only under the guidance of a specialist. However, since chemical cleaning ordinarily does not require removal of the equipment nor disassembly of the piping (if properly designed), it is the most convenient of the cleaning techniques in those cases where it can be used. There are a number of techniques for mechanical removal of fouling. Scraping or rotary brushing are limited to those surfaces that can be reached by the scraping tool – a problem that is eased on the shell side by the use of large clearances between tubes and/or the use of rotated square tube layout. (It should be noted that scraping should not be used on finned tubes.) Use of very high velocity water jets is very common both inside and outside the tubes, though for the shell side the jets will not be very effective deep inside a large tube bank. For situations where there is a high premium for maintaining a high degree of cleanliness, e.g., large power plant condensers, it may be possible to install a system for the continuous on-stream cleaning of the interior surfaces of the tube. The Amertap® System utilizes slightly oversized, sponge rubber balls which are continuously recirculated through the tubes on a random basis. As the balls pass through the tubes, they remove the accumulation of scale or corrosion products. A mesh basket in the outlet piping collects the balls and a ball pump re-injects them into the entering water flow. The type of balls having an abrasive secured to the outer surface should be used only with great caution as continuous abrasive action may shorten tube life due to the removal of the protective corrosion film formed in copper and copper-base alloys. The M.A.N.® System provides for on-stream cleaning by passage of brushes through the tubes. On heat exchangers, which use this system, plastic baskets or cages are installed at both ends of every tube. A plastic brush designed to fit the interior of the tube is installed into one basket. Periodically the water flow through the tubes is reversed, forcing the brushes through the tubes where they are caught in the basket at the far end of the tube. The cooling water flow is again reversed to its normal flow direction. The brush returns to the cage at the discharge end of the tube. Because of the relative simplicity of the system, it may be used on all sizes of heat exchangers provided the cooling water flow can be reversed.

48

TABLE 1.1

EROSION LIMITS: MAXIMUM DESIGN FLUID VELOCITIES FOR FLOW INSIDE TUBES

Water

Low carbon steel 10 ft/sec Stainless steel 15 ft/sec Aluminum 6 ft/sec Copper 6 ft/sec 90-10 Cupronickel 10 ft/sec 70-30 Cupronickel 15 ft/sec Titanium > 50 ft/sec

Other Liquids

[Allowable velocity for given liquid] = [Allowable velocity for water]2

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

liquidgivenofDensityWaterofDensity

Gases and Dry Vapors

The maximum design velocity for steel tubing may be estimated from:

[ ]))((1800sec/,

weightMolecularpsiainpressureAbsoluteftV =

Allowable velocities for other metals may be taken to be in the same ratio as for water.

49

TABLE 1.2

FOULING RESISTANCES FOR TYPICAL PROCESS APPLICATIONS

Adapted from Standards of Tubular Exchanger Manufacturers Association The values given in this table are typical of the situations described, but allowances must be made for substantial variations from one case to the next. The resistances are referenced to the surface area in contact with the fluid and are to be used in connection with equations of the form of (1.19) and (1.20).

The units on the resistances are . BtuFft /2° I. Water

Temperature of Heating Medium Up to 240°F 240°F - 400°F*

Temperature of Water 125°F or less Over 125°F

Water Velocity ft/sec

Water Velocity ft/sec Types of Water 3 ft

and less Over 3 ft

3 ft and less

Over 3 ft

Sea Water Brackish Water

0.005 0.002

0.001 0.001

0.003 0.003

0.002 0.002

Cooling Tower and Artificial Spray Pond:

Treated Makeup 0.001 0.001 0.002 0.002 Untreated 0.003 0.003 0.005 0.004 City or Well Water (Such as Great Lakes) 0.001 0.001 0.002 0.002 Great Lakes 0.001 0.001 0.002 0.002 River Water: Minimum 0.002 0.001 0.003 0.002 Mississippi 0.003 0.002 0.004 0.003 Delaware, Schuylkill 0.003 0.002 0.004 0.003 East River and New York 0.003 0.002 0.004 0.003 Bay Chicago Sanitary Canal 0.008 0.006 0.01 0.008

Muddy or Silty 0.003 0.003 0.005 0.005 Hard (Over 15 grains/gal) 0.003 0.003 0.005 0.005 Engine Jacket 0.001 0.001 0.001 0.001 Distilled 0.0005 0.0005 0.0005 0.0005 Treated Boiler Feedwater 0.001 0.0005 0.001 0.001 Boiler Blowdown 0.002 0.002 0.002 0.002

50

TABLE 1.2 (continued)

*Ratings in columns 3 and 4 are based on a temperature of the heating medium of 240°F-400°F. If the heating medium temperature is over 400°F and the cooling medium is known to scale, these ratings should be modified accordingly. II. General Industrial Fluids

Gases and Vapors Manufactured Gas .01 Engine Exhaust Gas .01 Steam (non-oil bearing) .0005 Exhaust Steam (oil bearing) .001 Refrigerant Vapors (oil bearing) .002 Compressed Air .002 Industrial Organic Heat Transfer Media .001

Liquids Refrigerant Liquids .001 Hydraulic Fluid .001 Industrial Organic Heat Transfer Media .001 Molten Heat Transfer Salts .0005 Fuel Oil .005 Transformer Oil .001 Engine Lube Oil .001 Quench Oil .004

III. Chemical Process Streams

Gases and Vapors Acid Gas .001 Solvent Vapors .001 Stable Overhead Products .001

Liquids MEA & DEA Solutions .002 DEG & TEG Solutions .002 Stable Side Draw and Bottom Product .001 Caustic Solutions .002 Vegetable Oils .003

IV. Natural Gas/Gasoline Process Streams

Gases and Vapors Natural Gas .001 Overhead Products .001

Liquids Lean Oil .002 Rich Oil .001 Natural Gasoline & Liquified Petroleum Gases .001

51

TABLE 1.2 (continued)

V. Oil Refinery Streams

Crude & Vacuum Unit Gases and Vapors Atmospheric Tower Overhead Vapors .001 Light Naphthas .001 Vacuum Overhead Vapors .002

Crude Oil

0-199 F 0-199 F

Velocity in ft./sec Velocity in ft./sec

Under 2 ft

2 – 4 ft

4 ft and over

Under2 ft

2 – 4 ft

4 ft and over

Dry 0.003 0.002 0.002 0.003 0.002 0.002

Salt* 0.003 0.002 0.002 0.005 0.004 0.004

300-399 F 400-499 F

Velocity in ft./sec Velocity in ft./sec

Under 2 ft

2 – 4 ft

4 ft and over

Under2 ft

2 – 4 ft

4 ft and over

Dry 0.004 0.003 0.002 0.005 0.004 0.003

Salt* 0.006 0.005 0.004 0.007 0.006 0.005

*Normally desalted below this temperature range. (Asterisk to apply to 200-299*F, 300-4990F, 500'F and over.)

Gasoline .001 Naphtha & Light Distillates .001 Kerosene .001 Light Gas Oil .002 Heavy Gas Oil .003 Heavy Fuel Oils .005 Asphalt & Residuum .010

52

TABLE 1.2 (continued)

Cracking & Coking Unit Streams Overhead Vapors .002 Light Cycle Oil .002 Heavy Cycle Oil .003 Light Coker Gas Oil .003 Heavy Coker Gas Oil .004 Bottoms Slurry Oil (4 ½ ft/sec minimum) .003 Light Liquid Products .002

Catalytic Reforming, Hydrocracking, & Hydrodesulfurization Streams Reformer Charge .002 Reformer Effluent .001 Hydrocracker Charge & Effluent** .002 Recycle Gas .001 Hydrodesulfurization Charge & Effluent** .002 Overhead Vapors .001 Liquid Product over 50* A.P.I. .001 Liquid Product 30'-50* A.P.I. .002

**Depending on charge characteristics and storage history, charge resistance may be many times this value.

Light Ends Processing Streams Overhead Vapors & Gases .001 Liquid Products .001 Absorption Oils .002 Alkylation Trace Acid Streams .003 Reboiler Streams .003

Lube Oil Processing Streams .003 Feed Stock .002 Solvent Feed Mix .002 Solvent .001 Extract* .003 Raffinate .001 Asphalt .005 Wax Slurries* 003 Refined Lube Oil .001

*Precautions must be taken to prevent wax deposition on cold tube walls.

53

Nomenclature Since the wide variety of geometries and correlation requires a bewildering variety of subscripts and multiple usages, a detailed listing and, where necessary, description of the specific symbols used in this section follows each major symbol definition.

A Surface area for heat transfer. Ao and Ai are the corresponding values

for the outside and inside surface, respectively, and Am denotes the logarithmic mean of Ao and Ai. A* is a generalized area, corresponding to U*. Afin is the total heat transfer area for the fins on a tube, Aroot is the area of the bare tube remaining between the fins, and Aeq is the com-bination of these two values defined by Eq. (1.38).

ft2

Cp, cp Specific heat of the flowing fluid. The capitalized symbol usually refers

to the hot fluid and the lower case to the cold fluid, but they may instead refer to fluid outside and inside the heat transfer surfaces, respectively.

Btu/lb°F

d Diameter of a tube. do and di are the outside and inside diameters,

respectively, and dm denotes the logarithmic mean. dr is the root diameter of a finned tube: the distance from the tube axis to the base of the fins. deq denotes an equivalent diameter defined for a non-circular conduit; there are numerous such definitions and they will be individually given and explained where they are used.

in. or ft.

Ew Weighted fin efficiency as defined by Eq. (1.42). dimensionless F Correction factor for the logarithmic mean temperature difference

(LMTD) to make it applicable to heat exchangers in which the flow is not entirely counter-current or co-current.

dimensionless

G Mass velocity (mass flow rate of fluid per unit cross-sectional area for

flow). Gv and are superficial mass velocities for vapor and liquid respectively; “superficial" means the values are calculated as if the given fluid were flowing alone, using the entire cross-sectional area.

lGlbm/ft2 hr

gc Gravitational conversion constant. 4.17x108 lbm ft/lbf hr2

H Fin height from root to tip. in. or ft. h Film heat transfer coefficient. ho and hi are the values for the outside

and the inside of the heat transfer surface, respectively. hf is an equivalent heat transfer coefficient for any fouling that may be present, equal to the reciprocal of the fouling resistance.

Btu/hr ft2°F

k Thermal conductivity of a material. kw refers to the wall material, while

ko and ki refer to the fluids on the outside and inside of the heat transfer surface, respectively.

Btu/hr ft2°F

L Length, usually of a tube. ft. MTD Mean temperature difference, defined by Eq. (1.24). °F

54

m Quantity characterizing fin geometry and properties, defined by Eq.

(1.34). dimensionless

P Parameter in MTD calculations, defined by Eq. (1.31b). dimensionless Q Heat flow. Btu/hr q Heat flux (Heat flow per unit area of heat transfer surface). qs is the

sensible heat flux and qλ is the latent heat flux respectively, in the Colburn-Drew analysis for condensing binary mixtures.

Btu/hr ft2

R Parameter in MTD calculations, defined by Eq. (1.31a). dimensionless Rf Resistance to heat transfer due to fouling. Rfo and Rfi are fouling

resistances on the outside and inside of a heat transfer surface, respectively.

hr ft2 oF/Btu

Rfin Resistance to heat transfer in a fin, given by Eq. (1.49). hr ft2°F/Btu Re Reynolds number for flow through or past a surface. There are many

different definitions depending upon the particular geometry involved. Eq. (1.9) is the definition for flow inside tubes.

dimensionless

r Radius of a tube. ro and ri are the outside and inside radii respectively;

rm is the logarithmic mean of ro and ri. r' is the outside radius of the inner tube and the inside radius of the outer tube in a bimetallic tube.

in. or ft.

T, t Temperatures. Both symbols (usually subscripted) are used more or

less interchangeably and for this reason every temperature must be carefully defined for each particular discussion. Usually, capital letters refer to the hot fluid and lower case to the cold fluid, but sometimes capitals refer to the outside fluid and lower case to the inside. Ti and ti usually refer to the inlet temperatures of the two streams and To and to to the outlet temperatures.

°F

Tabs Absolute temperature of a surface, used only in radiation calculations. °R U Overall heat transfer coefficient for heat transfer between two fluids

separated by a surface. U* is a generalized value defined by Eq. (1. 17). Uo and Ui are values referenced to the outside and inside surface areas, respectively.

Btu/hr ft2°F

V Mean velocity of a fluid flowing in a conduit. Vmax is the maximum

velocity of flow, which occurs at the centerline of a tube. For tube banks, Vmax is calculated as the mean velocity at the point where the tubes are closest together.

ft/sec

W, w Mass flow rates of the fluids in a heat exchanger. W and Cp are always

associated with one stream, and w and Cp with the other. lbm/hr

x Usually, a length variable, especially when it appears as Δxw, the wall

thickness of a tube. But see further. ft.

55

x For two-phase vapor-liquid flows, the quality of the flow: the weight

fraction of the flow that is vapor. dimensionless

Y Thickness of a fin. in. or ft. y Fraction of a component in the vapor phase. Yv is the composition in

the bulk vapor phase and Yi the composition at the interface. dimensionless

Z Fraction of a component in the liquid phase. Zl is the composition in the

bulk liquid phase and Zi the composition at the interface. dimensionless

GREEK ε Emissivity for radiation from a surface. dimensionless Λ Modified Baker parameter for two-phase flows in a horizontal tube.

( )vρρl=Λ Lbm/ft3

λ Latent heat of vaporization for a liquid. Btu/Ibm μ Viscosity of a fluid. μl, refers to liquid, μv to vapor. Ibm/ft hr ρ Density of a fluid. lρ refers to liquid density, vρ to vapor density. Ibm/ft3

σ Surface tension of a liquid. Also, in radiation, the Stefan-Boltzmann

constant. dyne/cm 0.173x10-8 Btu/hr ft2(oR)4

Φ Fin efficiency: the ratio of the total heat transferred from a real fin to that

transferred if the fin were isothermal at its base temperature. dimensionless

Ψ Modified Baker parameter for two-phase flows in a horizontal tube.

3/23/1 / ll σρμ=Ψ(ft5/3cm/ Ibm

1/3 hr1/3dyne)

56

Bibliography 1. Alves, G. E., Chem. Eng. Prog. 50, No. 9, 449 (1954). 2. Baker, 0., Oil and Gas J. 53, No. 12, 185 (1954). 3. Fair, J. R., Pet. Ref. 39, No. 2, 105 (1960). 4. Colburn, A. P., and Drew, T. B., Trans. AIChE 33 197 (1937). 5. Siegel, R., and Howell, J. R., Thermal Radiation Heat Transfer, 2nd Ed., Hemisphere Publ. Corp.,

Washington, D.C. (1981). 6. Parker, J. D., Boggs, J. H., and Blick, E. G., Introduction to Fluid Mechanics and Heat Transfer,

Addison-Wesley Publishing Co., Reading, Mass. (1969). 7. Kern, D. Q., Process Heat Transfer, McGraw-Hill Book Co., New York (1950). 8. McAdams, W. H., Heat Transmission, 3rd Edition, McGraw-Hill Book Co., New York (1954). 9. Standards, Tubular Exchanger Manufacturers Association, 6th Ed., New York (1978). 10. Perry, R.H., and Chilton, C.H., Eds, Chemical Engineers' Handbook, 5th Ed., McGraw-Hill Book Co.,

New York (1973). 11. Rozenman, T., and Taborek, J., AIChE Symp. Ser. No. 118, 68, 12 (1974). 12. Kern, D. Q., and Kraus, A. D., Extended Surface Heat Transfer, McGraw-Hill Book Company, New

York (1972). 13. Carrier, W. H., and Anderson, S. W., Heating, Piping, and Air Conditioning, 304 (May, 1944). 14. Young, E. H., and Ward, D. J., Ref. Eng. 29, No. 11 (October, 1957). 15. Taborek, J., Aoki, T., Ritter, R. B., Palen, J. W., and Knudsen, J. G., Chem. Eng. Prog. 68, No. 2, 59

(1972). 16. Taborek, J., Aoki, T., Ritter, R. B., Palen, J. W., and Knudsen, J. G., Chem. Eng. Prog. 68, No. 7, 69

(1972). 17. Kern, D.Q., and Seaton, R.E., Chem. Eng. Prog. 55, No. 6, 71 (1959). 18. Kern, D.Q., and Seaton, R.E., Chem. Eng. 66, 125 (Aug. 10, 1959). 19. Gilmour, C.H., Chem. Eng. Prog. 61, No. 7, 49 (1965). 20. Kern, D.Q., Proc Third Int. Heat Transfer Conf. (Chicago) Vol. 1, 170 (August, 1966). 21. Moore, I.A., "Fin Tubes Foil Fouling for Scaling Services," Chem. Proc., 8 (August 1974). 22. Schlunder, E.U., ed., Heat Exchanger Design Handbook, Hemisphere Publ. Corp., Washington, D.C.

(1983).

57

2.1. Heat Exchangers with Low- and Medium-Finned Trufin

2.1.1 Areas of Application In Chapter 1, we found that it is usually advantageous to use Trufin Tubes when one of the film heat transfer coefficients is significantly smaller than the other. The lower coefficient tends to dominate or control the magnitude of U, the overall heat transfer coefficient, resulting in a large required heat transfer area and a correspondingly large heat exchanger. We also showed that one can reduce the total length of tubing required and therefore the size of the heat exchanger if finned surface is used in contact with the fluid having the low film heat transfer coefficient. An approximately optimum design can be obtained under these conditions if the resistances of the two sensible heat transfer processes are approximately equal. This requirement may be stated as

ooii AhAh11

≈

or

o

i

i

o

hh

AA

≈

A large number of applications result in values of ranging from 2 to 10, and it is under these

conditions that types S/T and W/H Trufin are most applicable, for these tubes have values ranging from just under 3 up to over 6.

)/( oi hh)/( io AA

This section is devoted to applications where single-phase heat transfer is taking place on the finned surface of the tubes. Typical applications include (but are not limited to) the following: 1. Cooling of liquids and gaseous product with cooling water. It is frequently necessary to cool gas or

liquid products for storage, using cooling tower or naturally available water. Unless the product is very corrosive, the water will usually be in the tubes. The water coefficient will usually be about 1000

, whereas a typical shell-side coefficient will be from 50 for a moderate pressure gas to 300 to 350 for a non-aqueous, low-viscosity liquid. The use of high-finned tube (Type H/F) might be considered for the moderate pressure gas, but construction requirements will usually indicate a shell and tube exchanger with medium-finned type W/H or type S/T Trufin. For the liquids, one of the low-finned type S/T Trufin tubes is usually indicated.

FfthrBtu °2/

2. Cooling of compressed gases (either between the stages or when the compression is complete.)

These gas coefficients can vary from 25 to 100; the values are lower than in the previous case because pressure drops are often limited to low velocities through the cooler. Again, medium-finned type S/T or W/H Trufin is probably indicated because of its more favorable area ratios, but low-finned tubing is also often used.

3. Feed-effluent exchangers and similar arrangements for heat recovery. There is increasingly a need to

recover heat by using a hot effluent stream from a reactor or a distillation column to heat an incoming stream. One of these streams usually has intrinsically a higher heat transfer coefficient (for example, a hot liquid effluent stream) than the other, and exchanger design advantages often result if Trufin is used.

58

The above are only typical applications for S/T and W/H Trufin. As a general statement, Trufin should be used wherever the resulting exchanger is less expensive or more operationally convenient than the plain tube exchanger required for the same service. Often, only a comparison of the final designs of both the finned and the plain tube exchangers will reveal the advantages of the Trufin tube design.

2.1.2. Description of Low- and Medium-Finned Trufin 1. Type SIT Trufin® Low-Finned Tube. An

example of Type S/T Trufin tube is shown in Fig. 2. 1. The tube shown has 19 fins per in., but similar tubes are produced with 16, 26, 32, and 40 fins per in. The fin height for these tubes is approximately 1/16 inch, and these are the tubes commonly referred to as low finned tubes. The 40-fin product is also supplied with a .035 in. fin height. The 32-fin product has a fin height of .032 and is generally supplied in titanium.

2. Type SIT Trufin® Medium-Finn Tube. S.T Trufin

medium-finned tube is characterized by having 11 fins per in. and fins 1/8 in. high, resulting in outside to inside surface area ratios of about 5. A typical tube is shown in Fig. 2.2.

These tubes are supplied either with belled ends suitable for rolling into tube sheets or plain ends up to 3 in. in length.

3. Type S/T Turbo-Chil® Finned Tube. Type S/T

Turbo-Chil finned tube is illustrated in Fig. 2.3. This tube configuration combines the 19, 26, or 40 fins per in. external surface enhancement of conventional S/T Trufin with the enhancement of the inside heat transfer coefficient afforded by the spiral ridges. The turbulence level of the fluid in the tube is increased by the spiral ridges.

Because Turbo-Chil enhances both the inside and outside heat transfer, it is mainly useful in applications where the heat transfer coefficients on either side of a plain tube would be comparable in magnitude. Turbo-Chil then allows a sharply increased heat transfer rate per unit length of tube and can considerably reduce the volume of heat exchanger required for a particular service.

Special correlations are required for the in-tube heat transfer and pressure drop for Turbo-Chil.

59

2.2. Basic Equations for Heat Exchanger Design 2.2.1. The Basic Design Equation and Overall Heat Transfer Coefficient The basic heat exchanger equations applicable to shell and tube exchangers were developed in Chapter 1. Here, we will cite only those that are immediately useful for design in shell and tube heat exchangers with sensible heat transfer on the shell-side. Specifically, in this case, we will limit ourselves to the case when the overall heat transfer coefficient is constant and the other assumptions of the mean temperature difference concept apply. Then the basic design equation becomes:

(2.1) )(** LMTDFAUQT = where is the total heat load to be transferred, UTQ * is the overall heat transfer coefficient referred to the area A*, A* is any convenient heat transfer area, LMTD is the logarithmic mean temperature difference for the purely countercurrent flow configuration, and F is the configuration correction factor for multiple tube-side and/or shell-side passes. Charts of F for the common shell and tube exchanger configuration are discussed later. U* is most commonly referred to the total outside tube heat transfer area, including fins, in which case

it is written as and is related to the individual film coefficients, wall resistance, etc. by oA

oU

i

o

ii

ofi

mw

owfinfo

o

o

AA

hAAR

AkAxRR

h

U 111

++Δ

+++= (2.2)

where and are the outside and inside film heat transfer coefficients, respectively, and are

the outside and inside fouling resistances, oh ih foR fiR

wxΔ , and are the wall thickness (in the finned section) and

wall thermal conductivity, and is the resistance to heat transfer due to the presence of the fin. Since all of the low-and medium-finned tubes manufactured by Wolverine are integral (i.e., tube and fins are all one piece of metal), there is no need to include a contact resistance term.

wk

finR

Suitable correlations for and will be developed later in this section. The fouling resistances are ordinarily specified by the customer based upon experience with the streams in question, but typical values may be found in Chapter 1, Table 1.2.

oh ih

The mean wall heat transfer area is given with sufficient precision as mA

( rim ddLA +=2

)π (2.3)

60

If it is preferred to use an overall heat transfer coefficient based upon the inside heat transfer area the following relationship holds:

iA

(2.4) iioo AUAU =

It is of the greatest importance to always identify the reference area when quoting the value of a film or overall heat transfer coefficient. 2.2.2. Fin Efficiency and Fin Resistance The general concept of fin efficiency and fin resistance was developed in Chapter 1. Accordingly, we will only reiterate the major equations and concepts here. The value of for use in Eq (2.2) is given by finR

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Φ+

Φ−= fo

o

fin

rootfin R

hAAR 11

(2.5)

where Φ is the fin efficiency and is given by:

ro ddm /3

1

12

+=Φ (2.6)

where

YkRh

Hm

wfoo

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=1

2 (2.7)

Also,

sNLdYs

sLdA frrroot ππ =⎟⎠⎞

⎜⎝⎛

+= (2.8)

and

( ) frofin LNddA 22

42 −=π

(2.9)

Typical fin efficiencies for S/T Trufin are above 0.90 for virtually all applications, often approaching 1.00 for those applications in which low-finned Trufin is most valuable.

61

These are shown in Chapter I as a function of foo

Rh

+1

for the various metals out of which S/T Trufin is

manufactured. Use of these figures (1.51 and 1.52) shortcuts the need to carry out the calculation of Eqns. (2.5 to 2.9) for most design cases. 2.2.3. Mean Temperature Difference, F Factors We will use the Mean Temperature Difference (MTD) formulation for design of heat exchangers in this Manual. The MTD is related to the Logarithmic Mean Temperature Difference (LMTD) by the equation

MTD=F(LMTD) (2.10) where the LMTD is always defined as for the countercurrent flow arrangement shown in Fig. 2.4:

( ) ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−=

12

21

1221

tTtT

n

tTtTLMTD

l

( ) ( )

(2.11)

In the rare occasion that the heat exchanger is a purely cocurrent (parallel) flow arrangement, F = I and the LMTD is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−=

22

11

2211

tTtTn

tTtTLMTDl

(2.12)

where T and are the shellside and tubeside inlet temperatures, respectively, and T and are the corresponding outlet temperatures.

1 1t 2 2t

The value of F depends upon the exact arrangement of the streams within the exchangers, the number of ex changers in series, and two parameters defined in terms of the terminal temperatures of the two streams:

fluidtubeofRangefluidshellofRange

ttTT

R =−−

=12

21 (2.13)

differenceetemperaturMaximumfluidtubeofRange

tT

ttP =

−

−=

11

12 (2.14)

The mathematical relationships between F, R, and P have been reported in a number of places, e.g., Refs. (1, 2), but the graphical representations are of the greatest interest to us in this Manual. These are shown for the most important cases in Figs. 2.5 to 2.12, inclusive.

62

Once the terminal temperatures of both streams of a heat exchanger are specified or otherwise determined, R, P, and LMTD can be calculated, F found for the heat exchanger configuration, and finally the MTD can be calculated. Values of F below 0.8 or 0.75 at the lowest should not be used for three reasons: 1. The charts cannot be read accurately. 2. The low value of F means that substantial additional area must be supplied in the heat exchanger

to overcome the inefficient thermal profile. 3. Design in or near the steep portion of the curves indicates that the thermodynamically limiting

configuration is being approached, even if all the assumptions are perfectly satisfied. Violation of even one of the assumptions (e.g., excessive bypassing) by even a little bit may result in an ex-changer that is in fact thermodynamically incapable of meeting the specified temperatures.

If the value of F determined for the proposed configuration is too low, the use of additional shells in series will result in an improvement, as shown by the successive F charts for given values of R and P. Alternatively, it may be possible to redesign the exchangers to permit the use of fixed tube sheet units and purely countercurrent flow (for which F is unity.)

63

64

65

66

67

68

69

70

71

2.3. Heat Transfer and Pressure Drop During Flow Across Banks of Trufin Tubes

2.3.1. Heat Transfer In Trufin Tube Banks The design method for the shell side heat transfer coefficient and pressure drop of a baffled shell and tube heat exchanger using type S/T Trufin will be described in detail in the sample problem -following this Section. However, as an essential component of that calculation, we must have basic heat transfer and fluid flow data for flow across banks of finned tubes. Regrettably, there are very few data available in the open literature for banks of low and medium-finned tubes. * However, the data that are available indicate that, with a small modification, we can use the extensive data and correlations available on banks of plain tubes. The database that we shall use comes originally from Williams and Katz (3), but the definitive analysis for pre sent purposes was done by Briggs, et al. (4). The latter reference interprets the earlier data in the light of the Delaware method (5) for shell-side heat transfer and pressure drop. The design method proposed later in this Section is a modified version of the Delaware method, so the database and its incorporation into a design method are at least self-consistent. The heat transfer data for tube banks are correlated as plots of the Colburn j factor for heat transfer vs. the crossflow Reynolds number. The Colburn j factor for crossflow is defined as:

14.03/2

s

w

s

p

smp

os k

CGC

hj ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

μμμ

(2.15)

where is the mass velocity through the minimum free flow area between adjacent tubes: mG

m

sm S

WG = (2.16)

and the subscript s indicates the shell-side flow. In a circular tube bundle, will be defined as the minimum free flow area through one crossflow section (i.e., between adjacent baffles) at or near the centerline of the bundle.

mS

The method of calculating will be demonstrated by the examples given in this Section. mS The crossflow Reynolds number for a finned tube bank is defined as:

s

mrs

Gdμ

=Re (2.17)

72

where is the root diameter of the finned tube. When data for Trufin heat exchangers are compared (via the Delaware method) to those for an otherwise identical plain tube exchanger, Briggs, et al. (4) found results like those in Fig. 2.13, for water, oil, and glycerine.

rd

From Fig. 2.13, we see that there is essentially no difference between the curves for plain and Trufin tube banks at

values above about 500. Below 500, the Trufin tube bank performance falls off compared to the plain tube bank. Comparing all of their data for three different sets of tube banks, Briggs, et al. (4) prepared the graph in Fig. 2.14 which gives the ratio

sj

sRe

tubeplainjtubefinnedj

s

s

,,

s

sRe

as a function of below 1000. sRe It appears that at low Reynolds Numbers, the fins tend to trap the fluid between them, reducing the local velocity between the tubes and therefore the heat transfer coefficient. It should be noted that the Briggs, et al. (4) data were only for tubes having 19 fins/in. More recently, Rabas, et al. (11) have shown an effect in the heat transfer and pressure drop of low fin tube banks due to fin density. The designer should also note that the ratio appears to be approaching a minimum value of 0.5 at very low , Since the increase in area due to the fins is always greater than a factor of 2 (more usually 3 or 4), there is still a net increase in the value of

compared to plain tubes at a

comparable .

j

oo Ah

sRe *The equations used for banks of high-finned tubes cannot be applied to low-finned tubes, partly because the large amount of data on which they are based was only obtained for air over narrow ranges of flow

73

rates, temperatures, and pressures. Also, the geometrical proportions of the fins and tubes are quite different. Using the above data, we can now construct the basic design curves for the most common tube bank geometries. In Ref. (5), and more recently in a somewhat revised form (e.g., Perry's Chemical Engineer's Handbook (6)), curves of vs. have been presented for the most common tube layouts used in

shell-and-tube heat exchangers. Applying the ratio shown in Fig. 2.14, we generate the basic vs.

curves for Trufin Tube banks as shown in Fig. 2.15.

sj sRe

sj

sRe Note that the determining factor for which curve is used from Fig. 2.15 is the tube layout - equilateral triangular ( ), rotated square ( ), or inline square ( ). There are small differences among the various pitch ratios (the ratio of the distance between centers of adjacent tubes to the tube outside diameter), but these are secondary to other effects in the normal range (1.2-1.5) used in shell and tube exchangers.

°30 °45 °90

2.3.2. Pressure Drop During Flow Across Banks of Low-Finned Trufin Tubes The basis for the pressure drop correlation for banks of type S/T & W/H Trufin is essentially the same as for the heat transfer correlation: the Briggs, et al. paper (4) interpretation of the results of Williams and Katz (3) in the light of the Delaware method (5). The correlating quantity for pressure drop during crossflow is the friction factor defined by Eq. (2.18):

14.02

42

swc

mscss N

GgPf ⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ=

μμρ

(2.18)

where is the pressure drop for flow across a tube bundle, is defined by Eq. (2.16). and is

the number of major restrictions crossed by the fluid in one tube bundle. is equal to the number of rows of tubes in the bundle for inline square or equilateral triangular arrays, and equal to one less than the number of rows of tubes in a rotated square ( ) array. The viscosity gradient term is included to account for the non-isothermal effects,

sPΔ mG cN

cN

°45sρ is the density of the shell-side fluid, and is the gravitational

conversion constant. cg

The friction factor is correlated against the shell side or crossflow Reynolds number defined by Eq. (2.17). Typical results from that study are shown in Fig. 2.16. Unfortunately, the result is not as clear-cut and satisfying for pressure drop as for heat transfer. The basic friction factor curves are found to be about twice the value that would have been predicted from the Delaware work on plain tube banks. However, as can be seen from Fig. 2.16 the Trufin result is quite comparable to that found for the corresponding plain tube bank in the Briggs et al. study (4).

74

75

The explanation seems to lie in the fact that, in the reduction of the shell and tube

exchanger pressure drop data to ideal tube bank data, all of the errors implicit in the

correction factors are eventually concentrated in the final result, the

curves. In fact, pressure drop prediction is a more uncertain and inaccurate art than heat transfer prediction, and this result should not be surprising. However, it does indicate that

care should be taken in developing and using pressure drop correlations and in the interpretation of the result. The correlation that is proposed therefore, is to double the

values (at any given Reynolds number) reported from the original Delaware work.

These values are shown in Fig. 2.17, when these are used in the Delaware method, the predicted pressure drop should be either about right or on

the conservative side, possibly by as much as a factor of 2. In this figure, curves are shown for two different pitch ratios (based upon the outside tube diameter) but there is no dependence upon tube

layout. This is speculative, but appears to be consistent with the little information available.

sf

sf

2.3.3. Effect of Fouling on Trufin There are few data available on fouling on finned surfaces and even these data are contradictory. On the one hand there is a general belief that Trufin should not be used in severely fouling services because the fouling deposits may find a firmer foothold between the fins than on a plain tube and may close off the surface entirely. However, it is also true that a given weight of fouling material (in moderate quantity) must spread over a larger area and therefore have a proportionately smaller effect on Trufin than on a plain one. Also, it has been found that some brittle fouling deposits are cracked off from Trufin during thermal cycling as the tube expands and contracts. In sum, the final decision as to whether or not Trufin should be used in a given fouling service must be left to the designer's judgment. In general, the decision would be not to use Trufin in severely fouling service (but in that case the stream in question should probably go inside the tube anyway, to simplify cleaning.) For most services, however, Trufin will likely prove no worse than a plain tube in total deposit accumulated and has the advantage that the penalty against the area required is reduced by the increased ratio of outside (finned) area to inside area, compared to a plain tube.

76

2.4. Heat Transfer and Pressure Drop Inside Tubes 2.4.1 Heat Transfer And Pressure Drop In Single Phase Flow Inside Round Tubes 1. Flow Regimes for Heat Transfer. The fundamental mechanisms of heat transfer inside tubes were discussed in Chapter 1, and a few of the salient points will be repeated here in connection with presenting detailed correlations for calculating the heat transfer coefficient. The particular correlation to be used for calculating in tube heat transfer coefficients depends upon the flow regime existing inside the tube - laminar, turbulent, or transition. For the majority of applications for which Trufin is suitable, the flow will be turbulent; however, it is necessary to present a complete set of correlations to cover the entire range. The flow regime existing in a tube may be determined by calculating the Reynolds number, : iRe

i

iiii

Vdμρ

=Re (2.19)

where is the inside diameter of the finned portion of the tube, id iρ and iμ are respectively the density

and viscosity of the fluid flowing and is the average fluid velocity inside the finned portion. Any set of dimensions may be used in the equation as long as the final result is dimensionless.

iV

If for a given flow is less than about 2000, the flow is laminar, though some disturbances from the entrance of the tube or from an upstream pump may persist to significant distances down the tube.

iRe

If is greater than about 10,000, the flow is fully developed turbulent and good heat transfer correlations exist.

iRe

For between 2000 and 10,000, the heat transfer coefficient is between the values for laminar and turbulent and cannot be predicted with precision. This is the so called transition regime and it is generally recommended that the designer attempt to keep the flow conditions out of this range. This can usually be done by increasing the velocity sufficiently (e.g., going to multiple tube passes) to get into the fully developed turbulent flow regime.

iRe

2. Heat Transfer in Laminar Flow. A number of correlations exist for the laminar flow regime, but the one most widely recommended is the Hausen correlation:

[ ]

14.0

,3/2)/(PrRe04.01

)/(PrRe0668.065.3 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++=

iw

i

iii

iii

i

ii

LdLd

dk

hμμ

(2.20)

77

where ih is the mean coefficient for the entire length L of a single tube. is the Prandtl number of the fluid flowing defined as

iPr

i

ipii k

c μ=Pr (2.21)

Examination of Eq. (2.20) shows that the mean coefficient decreases with increasing length of tube, L. This is a consequence of the buildup of an adverse temperature gradient in laminar flow. The value of L to be used is the length of a single pass, or in a U-tube bundle, the length of the straight tube from the tube sheet to the tangent point of the bend. In other words, the adverse temperature gradient is assumed to be completely destroyed by the turnaround or by the strong secondary flow induced in the U-bend. The dependence of ih on L means that the usual design calculation is in principle reiterative; that is, L must

be estimated, ih calculated, and a revised value of L calculated from the area requirement based upon

that ih . This reiterative process ordinarily converges very rapidly. For hand calculations, sufficient accuracy is achieved by evaluating all physical properties (except wμ ) at the mean bulk temperature of

the stream t or T as defined in Fig. 2.18. The term included at the end of Eq. (2.20) is the so-called Sieder - Tate term, which corrects the coefficient for the effect of a viscosity difference between the bulk fluid and that at the wall. For example, if a liquid is being heated in the tube, the wall temperature and therefore the temperature of the liquid at the wall is higher than the bulk temperature. Thus the viscosity of the liquid at the wall is less than the bulk viscosity and the boundary layer of the liquid on the wall is thinner, resulting in a small increase in the film heat transfer coefficient over that calculated for the constant viscosity case. It is sufficiently accurate to calculate the mean wall temperature

14.0, )/( iwi μμ

wT , at which wμ is evaluated as shown in Fig. 2.18.

( tThUtT

i

iw −+= ) (2.22)

where is calculated based upon the inside surface area. iU One element of conservatism that has been introduced into the calculations of this section is the neglect of natural convection effects in calculating ih . These effects arise from the density differences caused by the temperature gradients and almost always act in heat exchangers to increase the heat transfer coefficients, the effect increasing with increasing temperature differences. However, the computation of

78

these effects adds complications to the calculation and it is usually much more convenient and somewhat conservative to neglect them. 3. Heat Transfer in Turbulent Flow. In the fully developed turbulent flow regime at above 10,000, the most widely applicable correlation for hi is the Sieder Tate equation:

iRe

14.0

,

3/18.0 PrRe023.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

wi

iii

i

ii d

khμμ

(2.23)

Actually, in various sources the prefactor constant is quoted variously at 0.019 to 0.027, and the 0.023 is a plausible average. This variation should be considered by the designer when deciding what confidence to place in an answer and the precision to which it should be calculated. For values less than 60 the mean heat transfer coefficient is somewhat higher than that given by Eq. (2.23) because of entrance effects; however, most exchangers greatly exceed that ratio and in any case it is usually conservative to ignore the improvement.

idL /

For water, which will very commonly be the tube-side fluid in Trufin applications, the Eagle-Ferguson chart (7) is regarded as both very convenient for hand calculations and more accurate than most others. This chart is given in Fig. 2.19. Recall that the values of the velocity and inside diameter referred to are those of the finned por-tion of the tube. 4. Heat Transfer in Transition Flow. For between 2000 and 10,000, the heat transfer coefficient is very unpredictable, being largely dependent upon the structuring of the flow caused by upstream flow conditions. There is some possibility of flow oscillations because of hydrodynamic instability between laminar and turbulent flow, and the designer is best advised to avoid this region if possible.

iRe

However, the coefficient will be bounded by Eqs. (2.20) and (2.23) and a plausible if not very exact procedure in this range is as follows:

A. Calculate ih by Eq. (2.20) as if the flow were laminar.

B. Calculate by Eq. (2.23) as if the flow were turbulent. ihC. Estimate Tih )(

[ ] ⎟⎠⎞

⎜⎝⎛ −

−+=8000

2000Re)( iiiiTi hhhh (2.23A)

5. Pressure Drop Inside Round Tubes. The pressure drop on the tube-side of the heat exchanger is composed of several different terms: the pressure losses in the inlet and outlet nozzles, the pressure losses in the headers or channels, the pressure losses associated with accelerating the fluid and establishing the flow profile in the tube and finally the frictional losses of the flow in the tube.

79

80

80

The latter term is usually the major contributor, and is the only one that can be calculated with a fair degree of accuracy. For the nozzle losses, it is usually sufficient to calculate the loss for each nozzle at about three times the velocity head in the nozzle, Eq. (2.24):

c

noznoz g

VP2

32ρ

=Δ (2.24)

where is calculated at the smallest cross-section area for flow (i.e., highest velocity) in the nozzle. nozV The combined header and tube entrance losses are estimated in a similar way, but using the velocity inside the tube, : iV

⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

c

ient g

VP

23

2ρ (2.25)

In a multipass exchanger, this loss is incurred for each pass, so Eq. (2.25) should be multiplied by the number of tube-side passes. The frictional losses inside the tube are calculated using a friction factor chart similar to Fig. 2.20. The ordinate of Fig. 2.20 is the Fanning friction factor, which is related to the frictional pressure loss by

14.022

i

w

ci

iiii gd

LVfP ⎟⎟

⎠

⎞⎜⎜⎝

⎛=Δ

μμρ

(2.26)

where L is the tube length; if there are n tube passes, the total in tube friction loss is n times the value given by (2.26). The differing behavior of laminar and turbulent flow is clearly indicated in Fig. (2.20) by the curves above and below

iRe = 2100. However, because of the disturbance of the flow caused by the entrance, the flow may

display higher pressure drops in the range above 600 than indicated by the laminar flow curve.

Therefore, the turbulent flow curves have been extrapolated to lower (dashed curve) until they intersect the laminar flow curve.

iRe

iRe

In the turbulent flow regime, the frictional pressure loss increases with the increasing roughness of the surface. For new tube, the friction factor is given by the "smooth" curve in Fig. 2.20.

81

82

82

2.4.2. Heat Transfer in Two-Phase Flow Inside Tubes 1. Heat Transfer During In-Tube Condensation. Frequently, type S/T Trufin will be used where the fluid in the tubes is condensing. Usually the condensing vapor will be steam, although other pure or nearly pure vapors will be involved. The condensation of a pure component under low pressure drop conditions (the usual case and the only one that can be practically handled without an elaborate computer program) is nearly isothermal, but there is a substantial variation in the local condensing heat transfer coefficient from the entrance (nearly all vapor and a high coefficient) to the exit (usually nearly all liquid and a low coefficient). For most heat exchanger design purposes at the hand calculation level, it is both necessary and sufficient to use an average heat transfer coefficient on the condensing side. One correlation that is simple to use, generally conservative, and sufficiently accurate for most purposes is due to Boyko and Kruzhilin (8):

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +=

2)/()/(

PrRe024.0 43.0,

8.0,

omimii

ii d

kh

ρρρρll

l

(2.27) where

iv

vim x⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+=

ρρρ

ρρ l1)/( (2.28)

and

( ) ov

vom x⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+=

ρρρ

ρρ l1/ (2.29)

In the above equations is the quality of the inlet vapor and is the quality of the outlet vapor. For the

usual case, = 1.00 and = 0; then the bracketed term in Eq. (2.27) reduces to ix ox

ix ox

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +

2/1 vρρl

The term in Eq. (2.27) requires some comment; it is the Reynolds number computed as if the entire condensing flow were liquid:

l,Rei

l

l

l

l μρ

πμ

ii

i

iii

Vd

d

wd=

⎟⎠⎞

⎜⎝⎛

=2

,

4

Re (2.30)

83

where is the total weight flow rate of fluid through one tube. is the Prandtl number for the liquid. iW l,Pri The frictional pressure drop during condensation can usually be estimated with sufficient accuracy to be about 1/4 to 1/2 of the pressure drop calculated as if the entire vapor flow traveled the total length of the tube without condensing. A more detailed analysis of flow and heat transfer during condensation is given in Chapter 3. 2. Heat Transfer During In-tube Boiling. Occasionally a boiling coolant is to be used in the tubes of a finned shell and tube exchanger. The general question of boiling heat transfer is an extremely complicated one. For non-critical situations the following remarks may suffice. The first matter is to ensure that the conditions for nucleating a vapor phase exist. Generally this condition will be satisfied if the wall temperature exceeds the local saturation temperature of the liquid by

to . The exact value depends upon the substance to be boiled, being greater for water than for non-aqueous liquids, and upon the pressure, becoming less as the pressure increases. Even if boiling doesn't occur immediately at the entrance, to the tube, the single phase heat transfer heats up the liquid until conditions for nucleation are satisfied. The Sieder - Tate equation (2.23) can be used to estimate the sensible heat transfer coefficient.

5 F°10

Once a vapor phase is formed, the Boyko-Kruzhilin equation (2.27) can be used to estimate the average coefficient for the boiling region up to an exit quality of about 0.5. Above this value, there is a significant probability that the boiling flow may go into the "dry-wall" regime, in which the liquid phase is mostly in the form of mist in the vapor flow and the wall is not completely wetted. This leads to very low heat transfer rates and possibly deposition of solid matter on the tube wall. Pressure drop in a boiling flow is very difficult to predict using methods suitable for hand calculation. The basic concepts and some applicable correlations are discussed in Chapter 5. 3. Heat Transfer During Gas-Liquid Flow. The in-tube flow may occasionally be a gas-liquid mixture in which little or no phase change (boiling or condensation) takes place. Any calculations on such a system must be very uncertain because in practical heat transfer equipment it is impossible to guarantee that the flow of each phase will be more-or-less uniformly distributed among each of the tubes. A rough estimate of the coefficient may be obtained from the constant-quality version of the Boyko-Kruzhilin equation:

xdk

hv

vi

ii ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+=

ρρρl

lll 1PrRe024.0 43.08.0

, (2.31)

84

2.5. Preliminary Design of Shell and Tube Heat Exchangers 2.5.1 Basic Principles of Design 1. Design Criteria for Process Heat Exchangers. The criteria that a process heat exchanger must satisfy are easily enough stated if we confine ourselves to rather broad statements: First, of course, the heat exchanger must meet the process requirements. That is, it must effect the desired change in the thermal condition of the process stream within the allowable pressure drops, and it must continue to do this until the next scheduled shutdown of the plant for maintenance. Second, the exchanger must withstand the service conditions of the plant environment. This includes the mechanical stresses of installation, startup, shutdown, normal operation, emergencies, and maintenance, and the thermal stresses induced by the temperature differences. It must also resist corrosion by the process and service streams (as well as by the environment); this is usually mainly a matter of choice of materials of construction, but mechanical design does have some effect. Desirably, the exchanger should also resist fouling, but there is not much the designer may do with confidence in this regard except keep the velocities as high as pressure drop and vibration limits permit. Third, the exchanger must be maintainable, which usually implies choosing a configuration that permits cleaning - tube-side and/or shell-side, as may be indicated – and replacement of tubes and any other components that may be especially vulnerable to corrosion, erosion, or vibration. This requirement may also place limitations on positioning the exchanger and in providing clear space around it. Fourth, the exchanger should cost as little as is consistent with the above requirements; in the present context, this refers to first cost or installed cost, since operating cost and the cost of lost production due to exchanger unavailability have already been considered by implication in the earlier and more important criteria. Finally, there may be limitations on exchanger diameter, length, weight and/or tube specifications due to site requirements, lifting and servicing capabilities, or inventory considerations. It is sometimes stated as a desirable feature that the exchanger design be specified with an eye to possible alternative uses in other applications. However, this has disturbing implications. Most heat exchangers are intended for projects having an expected life of five to twenty years - equal to or greater than the probable life of the exchanger. To suggest that a heat exchanger might become available sooner implies either that the exchanger or the process will prove unsatisfactory in its role. It is far better to labor under the positive compulsion that the only hope for success is by designing each item uniquely for the best performance in the task at hand. 2. Structure of the Heat Exchanger Design Problem. The basic logical structure of the process heat exchanger design procedure is shown in Fig. 2.21. The basic structure is the same whether we use hand design methods or computer design. First, the problem must be identified - as completely and unambiguously as possible. This includes data like flow rates, pressures, temperatures and compositions, and it also includes qualitative information such as the likelihood of fouling and the difficulty of cleaning, special materials requirements, and any unusual conditions to be encountered during operation.

85

It is at this point in the design process that the single most important design decision is made: the basic configuration of the heat exchanger, whether it is to be double pipe, shell-and-tube, plate, etc. The next step is to select a tentative set of major parameters for the exchanger: tube type, size, layout, shell diameter and length, baffle spacing, etc. A procedure for doing this will be given later in this chapter. Next, the thermal performance for the tentative configuration is rated using the procedure given later in this Chapter. That is, the overall heat transfer coefficient is calculated for the given flow rates in the design chosen; that value is combined with the required heat duty and the calculated value of the mean temperature difference to determine the heat exchanger area required. Finally, this value is compared to the area available in the chosen design. If the calculated area is reasonably close to the available area, the heat exchanger is acceptable from a thermal point of view and one may go on to the pressure drop calculation. However, if the areas do not corres-pond, it is necessary to adjust the tentative configuration parameters to increase or decrease the heat transfer area as required, and then the new configuration is re-rated. In the pressure drop calculation, the pressure drop of each stream must be less than - but not greatly less than – the allowable values. If the calculated pressure drop is much less than that allowed, it will probably prove possible to reduce the size of the exchanger. Once the thermal performance requirement and the pressure drop limitations are satisfied, one can go on to do a mechanical design and cost estimation. These stages are not included in this manual. 2.5.2. Preliminary Design Decisions 1. Allocation of Streams. Having selected a shell and tube exchanger, the designer's next decision is to decide which stream goes into the tubes and which into the shell. There are several different circumstances, which control this decision, among which the following are the most important: a. Possibility of using extended surface tubes. As we have repeatedly stressed, the use of Trufin tubes is generally advantageous when one fluid has a coefficient significantly lower than the other. In these cases, the fluid with the low coefficient will be allocated to the shell side. b. One fluid is highly corrosive. The solution to handling a corrosive fluid is to use an alloy which is resistant to the fluid. Since all corrosion resisting alloys are relatively expensive, it follows that the corrosive substance should be put in the tubes. Then only the tubes, the tube sheets (often simply faced

86

with the alloy), and the tube side channels and piping need to be made of the alloy. The shell, baffles, tie rods, etc., can be made of low carbon steel, at a considerable cost savings. c. One fluid is at high pressure. When one fluid is at a much higher pressure than the other, it should be placed in the tubes. Only the tube-side components have to be constructed to resist the high pressure, and particularly it is not necessary to use a heavy and expensive high pressure shell. d. One fluid is severely fouling. Since it is easier to clean the tube-side than the shell-side by mechanical methods (brushes, water jets, etc.), the more severely fouling fluid should be put on the tube-side. f. If the allowable pressure drop for one stream is very limited, that stream should normally go on the shell-side. Even though the "efficiency of conversion" of pressure drop into heat transfer coefficient is not as high on the shell-side (because of form drag), there is usually so much flexibility in the selection of design parameters on the shell side that is easier to accommodate the low pressure drop limitation. Clearly, there will frequently be conflicts between these "rules"; for example, one fluid may be at high pressure and the other one may be severely corrosive. The decision in this case may have to be delayed until complete designs have been prepared and priced for both cases. In general, the lower cost option would be selected, but safety considerations or differences in the reliability of operation might overrule a purely economic decision. 2. Selection of Shell Type. The most important factor in selecting a shell type is the thermal stress problem. As discussed in Chapter 1, thermal stresses arise because the tubes are at a different average temperature than the shell, and the differences in thermal expansion can result in a number of catastrophic events: the tubes may be pulled out of the tube sheets, or the tubes may be pulled apart or buckled, or the shell may buckle, or the tube sheets may be deformed enough to open up leaks through the gaskets. The determination of whether or not a serious thermal stress problem exists is a complex calculation and only a few rules of thumb can be given here: a. Fixed tube sheet exchangers having no specific arrangement to relieve or avoid thermal stresses can only be used when the difference in inlet temperatures of the two streams is less than about 100 ºF. b. Fixed tube sheet exchangers with rolled expansion joints in the shell can be used for inlet temperature differences of up to about 200 ºF for moderate pressure shells (say on the order of 150 psia). Expansion joints have been used too much higher inlet temperature differences for low pressure (i.e., thin-walled) shells. c. Where they can be used, U-tube bundles represent an essentially complete solution to the tube stress problem because each tube is free to expand or contract independently of the shell and, within wide limits, the other tubes. However, tube sheet thermal stress problems are not completely solved and must be further analyzed. The relative advantages and disadvantages of the various floating head shell designs were discussed in Chapter I and will not be further treated. However, most of the floating head designs have multiple tubes passes and require a configuration correction factor on the LMTD.

87

3. Selection of Shell Arrangement. It is often necessary to use more than one exchanger to accomplish a given service. The two basic arrangements are exchangers in parallel (Fig. 2.22) and exchangers in series (Fig. 2.23):

The parallel arrangement is mainly used when pressure drop limitations (coupled with length, diameter and baffle spacing limits) force a reduction in shell-side velocity and thus throughput per unit. For identical units, each exchanger may be separately analyzed using its proportional share of the flow rates. The purely series arrangement is mainly useful when

a. The single shell with multiple tube passes gives too low a value for F, as previously discussed in the Mean Temperature Difference Concept,

b. There are limitations on shell length and/or diameter, requiring the total area to be disposed in more

than one shell. The shells are usually identical for economy in manufacturing and ease and flexibility of installation, operation and maintenance. An infinite variety of series-parallel arrangements is possible. The most common is several exchanger trains in parallel (to split the flows down to rates that can be conveniently handled in the maximum acceptable exchanger size), with each train composed of several exchangers in series (to improve the Mean Temperature Difference). A single train can be analyzed using its proportion of the total flow and the total service determined by multiplying by the number of trains in parallel. An example of a more unusual series-parallel arrangement is shown in Fig. 2.24. Such an arrangement might be considered if the shell-side fluid had a severe pressure drop limitation (forcing parallel flow) and needed to be changed in temperature only over a narrow range, while the tube-side fluid had only a low

88

flow rate, thereby encouraging many tube-side passes to keep the velocity up. It should be noted that the two shells will have different outlet temperatures and hence there will be some loss of efficiency when the two effluent streams are mixed. In general, a trial and error solution of the heat balance and rate equations for each exchanger is required in mixed series-parallel arrangements, easily enough accomplished by a computer program, but rather tedious by hand for any significant number of shells. 2.5.3. Procedure for Approximate Size Estimation 1. Calculation of Q. For the usual design case, sufficient data are given or can be chosen to calculate Q and MTD. For sensible heat transfer on the shell-side, Q is calculated by (2.32) TCWQ pss Δ= or for the tube-side by (2.33) )( tcwQ pii Δ= 2. Calculation of MTD. The logarithmic mean temperature difference (LMTD) can be readily calculated from the terminal temperature differences by Eq. (2.11). Alternatively, for preliminary design, the LMTD can be estimated within a few percent; the LMTD is always smaller than the arithmetic mean, the difference being roughly proportional to the ratio of the larger terminal temperature difference to the smaller. The value of F can also be calculated from the terminal temperatures, using Figs. 2.5 to 2.12, depending upon the configuration. Or, for preliminary design of multiple tube pass designs, F may be estimated as 0.9, which is the average between the maximum possible value, 1.0, and the minimum recommended value, 0.8. This value may be shaded higher if the ratio of the terminal temperature differences is near unity and lower if the outlet stream temperatures are similar. In the latter case - and more especially if there is a temperature cross - the thermodynamic feasibility of the design should be checked before proceeding further. An absolute limit that may be quickly checked is for tube-side heating (2.34) 122 2 tTt −≤ for tube-side cooling (2.35) 122 2 tTt −≥ where is the outlet temperature on the tube-side, the inlet temperature on the tube-side, and the outlet temperature on the shell-side.

2t 1t 2T

When it is required to use multiple shells in series (as when two fluids are specified to exchange heat over a wide temperature range in a feed-effluent exchanger), there is a rapid graphical technique for estimating a sufficient number of shells in series. The procedure is shown in Fig. 2.25 and goes as follows:

a. The terminal temperatures of the two streams are plotted on the ordinates of ordinary arithmetic graph paper, the hot fluid inlet temperature and the cold fluid outlet temperature on the left hand ordinate and the hot fluid outlet and cold fluid inlet temperature on the right hand ordinate. The

89

distance between the ordinates is arbitrary, (corresponding to the total amount of heat exchanged between the two streams) and may be chosen to the convenience of the user.

b. If the specific heat of each stream is constant, straight lines ("operating lines") are drawn from the

inlet to the outlet temperature point for each stream.

If the specific heat of one or both streams varies, it is necessary to calculate the temperature of that stream as a function of the amount of heat added or removed by the other, resulting in one or both operating lines being curved. In this case, the procedure given here for finding a sufficient number of shells in series will still be valid, but the Mean Temperature Difference concept will not be. The exact design procedures to be followed in this case are beyond the scope of this Manual.

c. Starting with the cold fluid outlet

temperature (275°F in Fig. 2.25), a horizontal line is laid off until it intercepts the hot fluid line. From that point a vertical line is dropped to the cold fluid line (reaching it in our example at a cold fluid temperature of 228°F). This procedure defines a heat exchanger operation in which the hot fluid temperature is never less than any temperature reached by the cold; that is, there is no temperature cross and we know there is no thermodynamic difficulty that can arise if this operation is carried out in one shell.

d. The process is repeated until a vertical line

intercepts the cold fluid operating line at or below the cold fluid inlet temperature. (Alternatively, the process is continued until a horizontal line crosses the right hand ordinate.)

e. The number of horizontal lines (including

the one that intersects the right hand ordinate) is equal to the number of shells in series that is clearly sufficient to perform the duty. In the case of our example problem, this number is three.

We may do some calculations that give some quantitative feel for the present case:

The overall LMTD is:

Fn

LMTD °=⎟⎠⎞

⎜⎝⎛

−−

−−−= 4.56

80165275310

)80165()275310(

l

(2.11)

and

90

848.08031080275

=−−

=P (2.14)

744.080275

165310=

−−

=P (2.13)

From Fig. 2.7 for 3 shells in series F=0.81 This is a very acceptable value. For four shells in series, F = 0.91, and for two shells in series, F < 0.5 (in fact, this configuration is probably thermodynamically unworkable). Whether there would be any advantage in going to four shells rather than three would depend upon the particular circumstances of the case. 3. Estimation of . The step with the greatest uncertainty in preliminary calculations is estimating the overall heat transfer coefficient. Tables of U's typical of various services are widespread; the drawback to their use is that, in trying to include the entire range ever encountered in practice, these tables give a spread of values so great as to be almost meaningless for more or less optimal design. A table of for

various services is included here as Table 2. 1. The values given are for for Trufin tubes, with the controlling resistance on the finned side.

oU

oU

oU

A better procedure is to build up the value of from the individual h values, and wall, fin and fouling resistances using Eq. (2.2). It will be generally found that one or at most two terms will dominate the value of U, and attention can be focused upon these controlling values. It will generally be found that the range of reasonable values is far smaller than the range of possible values. It will also prove useful to make some estimate of the basic uncertainties in each value, i.e., the uncertainty in the value that would still exist even if the best available correlation and physical properties were used. This will often indicate rather clearly the futility of worrying too much about the precise value of a coefficient for preliminary design purposes.

oU

4. Estimation of and Key Exchanger Parameters. Once Q, MTD, and are known, the total outside

heat transfer area (including fin area) is readily found from oA oU

oA

)(MTDU

QAo

o = (2.36)

The next question is, "What set of heat exchanger dimensions will accommodate the heat transfer area?" Fig. 2.26 is an aid in answering that question. In this figure the ordinate is in , and the abscissa is the effective tube length (tube sheet to tube sheet for straight tube bundles, or length of a single straight section from tube sheet to tangent line for a U-tube bundle) in ft. The solid black lines in parameter are the commonly specified shell inside diameters in inches. Using standard tube count tables for 3/4 in.

oA 2ft

91

O.D., type S/T Trufin, 19 fins/in. tubes in a 15/16 in. triangular pitch for a fixed tube sheet heat exchanger with one tube pass, the total outside tube heat transfer area that can be fitted into a shell has been calculated. Therefore, once the required area is known, Fig. 2.26 shows immediately the combinations of tube length and shell diameter that will provide that area in a single shell for an exchanger of the given tube size and layout.

oA

The dashed lines shown in parameter and marked 3:1, 6: 1, etc, are ratios of tube length to shell inside diameter. These lines are included to give a rapid "feet" for the proper proportions of the exchanger. An exchanger with a length to diameter ratio too short - say, less than 3:1 - is likely to suffer from poor distribution of the streams and an excessive cost because of the large shell diameter. An exchanger with an excessively large length to diameter ratio is likely to be difficult to handle mechanically (the bundle must be very carefully supported when pulled, because of its springiness) and requires a wide clearway for pulling the bundle and retubing. An arbitrary ratio of 15:1 has been assigned to the probable upper limit. Most heat exchangers fall into the 6:1 to 8:1 range, though there has been a pronounced trend towards higher values as pressure drop estimation procedures have improved. 5. Generalization of Fig. 2.26. The usefulness of Fig. 2.26 can be greatly extended by defining an effective area A' by the equation (2.37) 4321' FFFFAA oo = where A’o is the area on the ordinate of Fig. 2.26

oA is the outside area of a finned tube heat exchanger as calculated from Eq. (2.36)

1F is the correction factor for the unit cell tube array (= 1.00 for 3/4 in. tubes on a 15/16 in. triangular pitch).

2F is the correction factor for the number of tube passes (= 1.00 for one tube pass).

3F is the correction factor for the shell construction/tube bundle layout type (= 1.00 for fixed tube sheet).

4F is the correction factor for the specific fin geometry and density (= 1.00 for S/T Trufin, 19 fins/in.) Tables for the various correction factors are given at the end of this chapter. 6. Application of the Preliminary Design Procedure. The use of the estimation method given in the preceding chapter is best illustrated by an example: Estimate the approximate size of an air compressor intercooler required to cool 13,000 SCFM (58,500 lb/hr) of air at 65 psig from to , using water at . The tubes are to be type S/T Trufin 3/4 in. O.D., 26 fins/in., of phosphorus deoxidized copper, and a U-tube bundle is desired. A tube layout on a 1 in. triangular pitch is specified.

F°350 F°125 F°80

92

93

First, calculate the heat load:

hrBtuFFlb

BtuhrlbQ /000,172,3)125350(241.0)/500,58( =°−⎟⎠⎞

⎜⎝⎛

°= (2.32)

Next, calculate the mean temperature difference; assuming an outlet water temperature of : F°110

Fn

LMTD °=⎟⎠⎞

⎜⎝⎛

−−

−−−= 5.116

80125110350

)80125()110350(

l

(2.11)

111.08035080110

=−−

=P (2.14)

5.780110125350

=−−

=R (2.13)

F = 0.91 (from Fig.2.5) MTD = 0.91 (116.5 °F) = 106 °F (2.10) From Table 2.1, we estimate to be about 25 Btu/hr °F. oU 2ft Then

2'Re 1197

)106)(25(000,172,3)( ftA dqo == (2.36)

Before entering Fig. 2.26, it is necessary to correct this area to the same basis as Fig. 2.26, by finding the proper factors to put into Eq. (2.37).

14.11 =F From Table 2.2, noting that a 3/4 in. O.D. by 1 in triangular pitch is specified

06.12 =F From Table 2.3, assuming that the shell diameter will be in the range of 13 1/4 in. to 17 1/4 in., and two tube passes will suffice. These assumptions have to be checked.

08.13 =F From Table 2.4, correcting for the U-tube design and that the shell diameter will be in the 13

1/4 to 21 1/4 inch range

79.04 =F From Table 2.5, for the tube specified. Then

94

(2.37) 22 1234)79.0)(08.1)(06.1)(14.1)(1197(' ftftA o == which is the value to be entered in Fig. 2.26 From Fig. 2.26, we find that this area can be accommodated in a 15 1/4 in. i.d. shell, by 12 ft long or a 17 1/4 in. i.d. shell, by 10 ft. long, or a 19 1/4 in. i.d. shell 8 ft. long. In order to proceed from this point a number of other calculations must be performed but these depend in part upon the Delaware Method of Shell-side Rating to be discussed next. We will return to this problem for a final design at the end of this chapter.

95

2.6. Delaware Method for Shell-Side Rating of Shell and Tube Heat Exchangers

2.6.1. Introduction The Delaware method for calculating the heat transfer coefficient and the pressure drop for a single-phase fluid flowing on the shell side of a shell and tube heat exchanger is based on the extensive experimental and analytical research program carried on in the Department of Chemical Engineering at the University of Delaware from 1946 to 1963. The procedure given here is modified from that originally presented in Ref. (5) and more recently in Ref. (6). As discussed previously, the method has been further modified to apply to Trufin tubes by following Ref. (4). The Delaware method assumes that the flow rate and the inlet and outlet temperatures (also pressures for a gas or vapor) of the shell-side fluid are specified and that the density, viscosity, thermal conductivity, and specific heat of the shell-side fluid are known or can be reasonably estimated as a function of temperature. The method also assumes that the following minimum set of shell-side geometry data are known or specified:

Tube root and outside diameters, and rd odFin spacing and thickness, s and Y Tube geometrical arrangement (unit cell) Shell inside diameter, iDShell outer tube limit (diameter), lotDEffective tube length (between tube sheets), L Baffle cut (distance from baffle tip to shell inside surface) cl

Baffle spacing (face-to-face). sl

Number of sealing strips/side, ssN From this geometrical information all remaining geometrical parameters needed in the shell-side calcula-tions can be calculated or estimated by methods given here. However, if additional specific information is available (e.g., tube-baffle clearance), the exact values of certain parameters may be used in the calculations probably with some improvement in accuracy. In order to complete the rating of a shell and tube exchanger, it is necessary to calculate the tube-side heat transfer and pressure drop characteristics from the methods given previously. Not all of the fluid flow rates and temperatures can be independently specified, but are connected through the heat balance on the exchanger. Similarly, the overall rate equation, Q = UA (MTD) must be satisfied, and it may well be that U calculated by this design method does not equal that required by the heat balance and rate equation. If this happens when one is designing an exchanger to perform a given service, it is necessary to change one or more of the geometrical parameters (the tube length is a particularly popular choice because changing it does not require complete recalculation of the coefficient) until the calculated and required performances are in substantial agreement. If an existing exchanger is

96

being rated, disagreement between calculated and required performance can only be resolved by changing flow rates and/or terminal specifications until agreement exists. Finally, it should be remembered that this method, though apparently generally the best in the open literature, is not extremely accurate, even for plain tubes. An exhaustive study by HTRI (Ref. 9) testing the various methods against 972 heat transfer data points and 1332 pressure drop data points covering a very wide range of fluids and geometrical parameters showed that this method predicted shell-side coefficients from about 50 per cent low to 100 percent high, while the pressure drop range was from about 50 per cent low to 200 per cent high. The mean error for heat transfer was about 15 per cent low (conservative) for all Reynolds numbers, while the mean error for pressure drop was from about 5 per cent low (unsafe) at Reynolds numbers above 1000 to about 100 percent high at Reynolds numbers below 10. It is necessary to be very careful about the units used in the following equations and graphs. The graphs and equations used for generating estimates of the geometrical parameters are in the units most commonly used in the U.S. for the quantity in question. But dimensional consistency is required in the equations used for calculating heat transfer coefficient and pressure drop. Therefore, it is advisable to check the units used in each equation before assigning units to the final answer. 2.6.2. Calculation of Shell-Side Geometrical Parameters 1. Total number of tubes in the exchanger, : If not known by direct count, find in the tube count table,

Table 2.6, as a function of the shell inside diameter , the tube pitch, p, and the layout. The tube count is for a fixed tube-sheet, fully-tubed bundle. In order to estimate the tube counts for a different bundle geometry, the factor F

tN

iD

3 from Table 2.4 may be used as a divider for the numbers in Table 2.6. Example: The tube count for 3/4 in. 0 D tubes on a 15/16 in. triangular pitch in a 39 in. ID shell, with a split backing ring floating head and 4 tube side passes can be estimated to be:

from table 2.6 tube count = 1338 from table 2.4 = 1.06 3F

1262)06.1(

133813383

==F

tubes

2. Tube-pitch parallel to flow, , and normal to flow

: pp

np These quantities are needed only for the purpose of estimating other parameters. If a detailed drawing of the exchanger is available, or if the exchanger itself can be conveniently examined, it is better to obtain these other parameters by direct count or calculation. The pitches are defined in Fig. 2.27 and some values tabulated in Table 2.7.

97

3. Number of tube rows crossed in one crossflow section, : Count from exchanger drawing or estimate from

cN

p

i

ci

c p

DD

N⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

l21

(2.38)

This is the number of rows of tubes between the tips of adjacent baffles, counting each row at the tip as one-half of a row. 4. Fraction of total tubes in crossflow, . can be calculated from cF cF

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −+= −−

lll

lll

ot

i

ot

ci

ot

cic D

cDD

DD

DF

2cos2

2cossin

221 11π

π (2.39)

where all the angles are read in radians. For convenience, has been plotted to an acceptable degree

of precision in Fig. 2.28 as a function of percent baffle cut and shell diameter . For pull-through floating head construction or other designs with large clearances between the shell inside diameter and the outer tube limit, ), the value of is a little higher than that shown.

cF

iD

loti DD −( cF 5. Number of effective crossflow rows in each window, . Estimate from: cwN

p

ccw p

N l8.0= (2.40)

This assumes that the tube field covers about 80 percent of the distance from the baffle cut to the shell, and that the flow on the average penetrates about half-way through this part of the tube field before turning to flow parallel to the tubes through the baffle cut and then turning again to begin the next crossflow section. 6. Number of baffles, Calculate from .bN

1−=s

bLNl

(2.41)

In design procedure, the length may not be (and need not be) specified precisely at this point. The heat transfer coefficient calculation does not (for most cases) require values of either L or , so it may be convenient to calculate the shell-side and overall coefficients, followed by the required length to satisfy the thermal specification. Then this length (rationalized to an integral number of baffle spaces) can be used to calculate and the pressure drops. Of course, it may then be necessary to choose a new shell

bN

bN

98

diameter and start over again if the allowable pressure drops have been exceeded, or if they are so much over the calculated value as to suggest that a smaller shell diameter might suffice.

99

7. Crossflow area at or near the centerline for one crossflow section, Estimate from eq. 2.42 or eq. 2.43.

.mS

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

++−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+−=

YssHdp

pdDDDS o

n

oototism 2)(lll (2.42)

for rotated and inline square layouts; or from

( )⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

++−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+−=

YssHdp

pdDDDS o

oototism 2lll (2.43)

for triangular layouts. 8. Fraction of crossflow area available for bypass flow, Estimate from .sbpF

( )m

sotisbp S

DDF ll−= (2.44)

This is the area between the outermost tubes and the shell and can constitute a major path for flow to largely escape contact with heat transfer surface. In constructions with a large clearance between the shell and the outer tube limit, it is practically essential to block this flow with sealing devices. 9. Tube-to-baffle leakage area for one baffle, Estimate from .tbS

2.),1(0152.0:.85 inFNSind cttbo +== (2.45)

2.)1(0184.0:.43 inFNSind cttbo +== (2.46)

(2.47) 2.)1(0245.0:.1 inFNSind cttbo +== These values are based on TEMA Class R construction which specifies 1/32 in. diametrical clearance between tube and baffle as the usual standard (Ref. 10): Values should be modified if extra tight or loose construction is specified, or if clogging by dirt is anticipated. 10. Shell-to-baffle leakage area for one baffle, If the diametrical clearance between the outside of

the baffle and the inside of the shell,

.sbS

sbδ , is known, can be calculated from sbS

21 .,2

1cos2

inD

DS

i

csbisb

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= − l

πδ

(2.48)

100

where the value of the term is in radians and is between 0 and 2. This area has been calculated and plotted in Fig. 2.29 as a function of percent baffle cut and inside diameter. Fig. 2.29 is based on TEMA Class R Standards (Ref. 10):

)/21(cos 1ic Dl−−

in.,Di Diametral shell-baffle clearance, in.

8 – 13 0.100 These values are for pipe 14 – 17 0.125 shells; if rolled shells are 18 – 23 0.150 used, add 0.125 in. 24 – 39 0.175 40 – 54 0.225 55 – 60 0.300

Since pipe shells are generally limited to diameters below 24 in., the larger sizes are shown using the rolled shell specification. Again, allowance should be made for especially tight or loose construction. No specification is given in the TEMA Standards for shells above 60 in. in diameter. 11. Area for parallel flow through window, This area is obtained as the difference between the gross

window area, and the window area occupied by tubes, :

.wS.wgS wtS

wtwgw SSS −= (2.49)

The value of can be calculated from wgS

2112

.,(21cossin)(21)(21cos4

inDDD

DSi

c

i

c

i

ciwg

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−−⎥

⎦

⎤⎢⎣

⎡−= −− lll

(2.50)

For convenience, however, the values of are plotted in Fig. 2.30 as a function of and . wgS )/( ic Dl iD The window area occupied by the tubes, , can be calculated from wtS

22 .,)1(8

indFNS oct

wt π−= (2.51)

For convenience, the value of Swt can be found from Fig. 2.31. To use this figure, enter the lower abscissa at the appropriate value of Nt and proceed vertically to the solid line corresponding to do, Then proceed horizontally to the dashed line corresponding to the estimated value of Fc and thence vertically to read the value of Swt on the upper abscissa. 12. Equivalent diameter of window, Dw. (Required only if laminar flow, defined as , exists.) 100Re ≤s

101

102

103

104

Calculate from

.,)1(

2

4in

DdFN

SD

ioct

ww

θπ+−

= (2.52)

where θ is the baffle cut angle and is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛−= −

i

c

Dl2

1cos2 1θ (2.53)

For convenience, θ is plotted in Fig. 2.32 as a function of ( ).%100⎟⎟⎠

⎞⎜⎜⎝

⎛

i

c

Dl

105

2.6.3. Shell-Side Heat Transfer Coefficient Calculation 1. The Delaware method first calculates the heat transfer coefficient for crossflow of the fluid (at the

design flow rate) in an ideal tube bank bounded by the shell on the sides and two adjacent baffles on the ends. This is done using the ideal tube bank correlation previously presented in this Chapter.

This coefficient must then be corrected for the effects of the baffle geometry, tube-to-baffle and shell-to-baffle leakage, bundle by-pass effects (including sealing strip effects) and, at low Reynolds numbers, the build-up of an adverse temperature gradient. The resulting coefficient is then the effective shell-side heat transfer coefficient and is used with the other heat transfer terms to calculate the overall heat transfer coefficient. The overall coefficient is then used with the heat duty and the Mean Temperature Difference to calculate the area required. If the effective tube length has not been previously specified, the required length may now be calculated to determine the feasibility of the design. If the length has been given (as in the case of rating an existing exchanger), the required area may be compared to the available area to determine the suitability of the given exchanger.

2. Calculate shell-side Reynolds number, The shell-side Reynolds number is defined as .Re s

ms

srs S

Wdμ

=Re (2.54)

where

Ws = weight flow rate of shell-side fluid, Ibm/hr

sμ = bulk viscosity of shell-side fluid, Ibm/ft hr dr = root dia. of Trufin tube, ft.

It is important to verify that the quantities used in Eq. (2.54) are in such units that the resulting value of Res is dimensionless. It is usually adequate to use the arithmetic mean bulk shell-side fluid temperature (i.e., halfway between the inlet and exit temperatures) to evaluate all bulk properties of the shell-side fluid. In the case of long temperature ranges or for a fluid whose viscosity is very sensitive to temperature change, special care must be taken (such as breaking the calculation into segments, each covering a more limited temperature range). Even then, the accuracy of the procedure is less than for more conventional cases.

3. Find js from the ideal tube bank curve for a given tube layout at the calculated value of Res, using Fig.

2.15. 4. Calculate the shell-side heat transfer coefficient for an ideal tube bank, ho,i

⎟⎟⎠

⎞⎜⎜⎝

⎛

°⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

FfthrBtu

Ck

SW

Cjhswspm

sspsio 2

14.03/2,, μ

μμ

(2.55)

5. Find the correction factor for baffle configuration effects, Jc. Jc is read from Fig. 2.33,as a function of

Fc.

106

6. Find the correction factor for baffle leakage effects, is found from Fig. 2.34 as a function of

the ratio of the total baffle leakage area,

.lJ lJ)( tbsb SS + , to the crossflow area, Sm, and of the ratio of the

shell-to-baffle leakage area Ssb to the tube-to-baffle leakage area, Stb. 7. Find the correction factor for bundle bypassing effects, Jb. Jb is found from Fig. 2.35 as a function of

Fsbp and of Nss/Nc (the ratio of the number of sealing strips per side to the number of rows crossed in one baffle crossflow section). The solid lines on Fig. 2.35 are for ; the dashed lines for Re

100Re ≥s

s < 100. Sealing strips are always mounted in pairs, arranged symmetrically with respect to both the bundle diameter parallel to the flow direction and the bundle diameter perpendicular to the flow direction. The best compromise between the improved heat transfer coefficient and the greater complexity of construction seems to be achieved when one pair of sealing strips is used for approximately every six tube rows.

8. Find the correction factor for adverse temperature gradient buildup at low Reynolds numbers, Jr. This

factor is equal to 1.00 if Res is equal to or greater than 100. For Res equal to or less than 20, the correction factor is fully effective and a function only of the total number of tube rows crossed. For Res between 20 and 100, a Iinear proportion rule is used.

Therefore:

a. If Res < 100, find J*r from Fig. 2.36, knowing Nb and (Nc+Ncw). b. If Res≤ 20, Jr = J*r. c. If 20 < Res < 100, find Jr from Fig. 2.37, knowing J*r and Res.

9. Calculate the shell-side heat transfer coefficient for the exchanger, ho Btu/hr ft2°F, from the equation: (2.56) rbcioo JJJJhh l,= 2.6.4. Shell-Side Pressure Drop Calculation 1. From the Delaware experimental work, we know the correlations for calculating the pressure drop for

a single ideal cross-flow section as well as for a single ideal window section. Further studies there indicated: a) that the pressure drop across the inlet and exit sections was affected (reduced) by bundle bypass but not by baffle leakage, b) that the pressure drop across internal crossflow sections was affected by both bundle bypass and baffle leakage, and c) that the pressure drop through a window was affected by baffle leakage but not by bypass.

The calculational structure is then to calculate the ideal crossflow and window pressure drops, correct each of those terms by the effective correction factors, then multiply the effective pressure drops by the number of sections of that kind found in the exchanger, and finally to sum the effects to give the total shell side pressure drop (exclusive of nozzles.)

107

108

109

110

111

If the resulting value is satisfactory, the exchanger is designed at least from a shell-side thermal-hydraulic point of view. If the required pressure drop is too large, it is necessary to redesign the exchanger, probably using a larger shell diameter. If the calculated pressure drop is much below the allowable, it is probably possible to reduce the shell diameter and redesign to a smaller and probably less expensive heat exchanger. 2. Find fs from the ideal tube bank friction factor curve for the given tube layout at the calculated value of

Res, using Fig. 2.17.

112

3. Calculate the pressure drop for an ideal crossflow section,

14.02

2

,24

s

w

mcs

cssib

SgNWf

P ⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

μμ

ρ (2.57)

The units on Eq. (2.57) must be checked to ensure that they are consistent. 4. Calculate the pressure drop for an ideal window section iwP ,Δ .

a. If Res ≥ 100:

swmc

cwsiw SSg

NWP

ρ2)6.02(2

,+

=Δ (2.58)

b. If Res < 100:

swmc

s

w

s

o

cw

swmc

ssiw SSg

wDdp

NSSg

wP

ρρμ 2

2, 26 +⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−=Δ

l (2.59)

5. Calculate correction factor for effect of baffle leakage on pressure drop, . Read from Fig. 2.38 as a

function of (SlR

sb + Stb)/Sm with parameter of Ssb/(Ssb + Stb). The curves are not to be extra polated beyond the points shown.

6. Find the correction factor bundle bypass, Rb. Read from Fig. 2.39 as a function of Fsbp and Nss/Nc.

The solid lines are for Res 100; the dashed lines are for Re≥ s < 100. 7. Calculate the pressure drop across the shell-side (excluding nozzles), ΔPs from:

[ ] )1(2))(1( ,,,c

cwbibiwbbibbs N

NRPRPNRPNP +Δ+Δ+Δ−=Δ l (2.60)

The application of the Delaware method to heat exchanger design is illustrated by the following example.

113

114

115

116

2.7. Examples of Design Problems for Low- and Medium-Finned Trufin in Shell and Tube Heat Exchangers

In this section two examples of heat exchanger design using Trufin are worked out to illustrate the use of the methods just described. The first problem is for a water cooled air compressor aftercooler, cooling air from 350°F to 125°F. This is the same problem that was previously used to illustrate the use of the preliminary design procedure, and the results of that problem will be used as the starting point for the solution. The second problem involves heat recovery, using a high temperature gas oil stream to preheat an incoming medium crude; this problem will be worked from scratch using the preliminary procedures already developed to get started. 2.7.1. Design Of A Compressor Aftercooler The problem is to design a heat exchanger to cool 13,000 SCFM (58,500 lb/hr) of air at 65 psig from 350 to 125°F, using cooling water available at 80°F. The unit was specified to be a U-tube configuration using type S/T Trufin 3/4 in. O.D., 26 fins per in. (catalog No. 65-265058), of phosphorus - deoxidized copper. The tubes are to be laid out on a I in. equilateral triangular pitch. The important tube dimensions are

do = 0.750 in. dr = 0.625 in. H = 0.0625 in. Y = 0.012 in. s = 0.026 in. Δxw = 0.058 in.

di = 0.509 in. Ao = 0.640 ft2/ft Ai = 0.133 ft2/ft Si = 0.206 in2. (inside cross-sectional flow

area) kw = 170 Btu/hr ft°F

The air properties are evaluated at 65 psia and at a mean air side temperature of 240°F, except for the wall viscosity which is evaluated at 130°F (a rough approximation ahead of time, but the solution is very insensitive to this value.) The values used are:

Density 0.300 lb/ft3

Specific heat 0.241 Btu/Ibm °F Viscosity (bulk) 5.40 X 10-2 lbm/ft hr Viscosity (wall) 4.68 x 10-2 Ibm/ft hr Thermal conductivity 0.0188 Btu/hr ft °F

The water outlet temperature will be assumed to be 110°F and the properties evaluated at a mean bulk temperature of 95°F, with an estimated mean wall temperature of 130°F. The values used are:

Density 62.0 lbm/ft3

Specific heat 1.00 Btu/Ibm °F Viscosity (bulk) 1.84 lbm/ft hr Viscosity (wall) 1.31 Ibm/ft hr Thermal conductivity 0.36 Btu/hr ft°F

117

The fouling factor of the water will be taken as 0.001 hr ft2 °F/Btu; no fouling factor will be assigned to the air. The preliminary design estimate suggested the following units:

Inside shell diameter, Effective tube length, in. ft.

15 1/4 12 17 1/4 10 19 1/4 8

For low pressure compressor intercoolers or aftercoolers, pressure drop of the gas is a major, consideration. This suggests looking at the larger diameter, shorter shells (contrary to the usual case for high pressure gases or liquids). Therefore, the following design is directed towards the 19 1/4 in. inside diameter shell. In the same philosophy, we select a baffle spacing near the maximum allowable under TEMA standards, which is the inside diameter of the shell; the value tried will be 18 in. Also, we select the maximum baffle cut, which provides only enough overlap to ensure that the central tube rows pass through all baffles. Since this is a U-tube bundle and some of the tubes near the centerline must be omitted because of bend radius restrictions, the maximum allowable cut will be about 8 in., which is the value chosen. We simply step through the various calculations in the order they come: Shell-side geometry data:

dr = 0.625 in. do = 0.750 in. s = 0.026 in. Y = 0.012 in. p = 1 in., equilateral triangular layout Di = 19 1/4 in.

lotD = 18 3/4 in. Effective tube length: This will be about 8 feet, but the exact determination will be left until after the heat transfer coefficients have been calculated. Then the required heat transfer area and the corresponding length will be calculated.

.4/318

.8

in

in

s

c

=

=

l

l

Nss = 0; This is indicated in this case for several reasons: The small clearance between Di and lotD the small crossflow distance due to the large baffle cut, and the fact that at least two of the center line rows are missing anyway due to the minimum bend radius. Calculation of shell-side geometrical parameters

118

1. Nt = 08.1

282 = 260; the fixed tube sheet, two-tube pass value of Nt is 282 by Table 2.6. This must be

corrected for the U-tube construction using F3 from Table 2.4. 2. pp = 0.866 in: pn = 0.500 in.

3. Nc = 24866.0

25.1982125.19

→=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

(2.38)

While Nc = 4 by the calculation, at least two tube rows will be lost by the minimum bend radius requirement for the U-tube construction. So Nc = 2 will be used for the remainder of the calculation. 4. 416.025.19/8/ ==ic Dl Fc = 0.25 from Fig.2.28. However, since at least two crossflow rows are lost near the centerline, Fc will be reduced to 0. 15 for the remainder of the calculations.

5. 84.7866.0

)8(8.0→==cwN (2.40)

6. Nb will be calculated after the tube length required for heat transfer area is known.

7. 2.119038.0026.0)0625.0(2)75.01(

175.075.1875.1825.1918 inSm =

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+−⎟

⎠⎞

⎜⎝⎛ −

+−= (2.43)

8. 076.0119

)18)(75.1825.19(=

−=sbpF (2.44)

9. Stb = 0.0148(260)(1.15) = 5.5 in2

10. Ssb = 2.5 in.2 from Fig. 2.29 11. From Fig. 2.30, Swg = 114.6 in.2

From Eq. (2.51), Swt = 48.8 in.2 Therefore, Sw = 65.8 in.2 from Eq. (2.49)

12. Not needed, since the shell-side flow is turbulent. Calculation of shell-side heat transfer characteristics:

1. 300,68)054.0)(119()144)(500,58(

12625.0Re =⎟

⎠⎞

⎜⎝⎛=s (2.54)

2. js = 0.0055 from Fig 2.15

119

3. FfthrBtuh io °=⎟⎠⎞

⎜⎝⎛×⎥

⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡= 2

14.03/2

, /1220468.00540.0

)054.0)(241.0(0188.0

119)144)(500,58()241.0(0055.0 (2.55)

4. Jc=0.65 (at Fc = 0.15 from Fig. 2.33)

5. 067.0119

50.250.5=

+=

+

m

sbtbS

SS

313.085.2==

+ sbtb

sbSS

S

34.2.86.0 FigfromJ =l

6. Jb = 0.91 from Fig. 2.35 7. Jr = 1.00 since Res > 100. 8. ho = 122 (0.65)(0.86)(0.91) (2.56) ho = 62.1 Btu/hr ft2°F Calculation of required heat transfer area: 1. Calculate heat load

QT = (58,500)(0.241)(350 – 125) = 3.17 x 106 Btu/hr (2.32) 2. Water flow rate, assuming an outlet temperature of 110°F

hrlbWi 700,105

)00.1)(80110(1017.3 6

=−×

= (2.33)

3. Water velocity, assuming two tube-side passes

sec/55.2)3600)(0.62)(206.0)(2/260(

)144(700,105 ftVi ==

This is possible value, but good design would generally call for a water velocity above 3 ft/sec in the tubes. If we go to four tube-side passes, Nt is about 240 tubes, and the tube-side velocity becomes

sec,/53.5)55.2()4/240()2/260( ft= which is better practice.

4. Using Fig. 2.19,

120

FfthrBtuhi °== /1300)015.1(1280 5. Using the equation developed in Chapter 1 for fin resistance, or from Fig. 1.52

Rfin = 8 x 10-5 hr ft2 °F/Btu 6. Using Eq. 2.2

FfthrBtu

Uo

°=

×+×+×+×=

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛+×+

=

−−−−

−

2

3452

5

/3.40

1051.81023.11081061.11

133.0640.0

13001001.0

148.0640.0

)170(12058.0)108(

1.621

1

7. Fn

LMTD °=⎟⎠⎞

⎜⎝⎛

−−

−−−= 5.116

80125110350

)80125()110350(

l

(2.11)

8. 111.08035080110

=−−

=P (2.14)

5.780350

125350=

−−

=R (2.13)

F = 0.9 From Fig. 2.5 9. MTD = 0.9(116.5) = 104.8°F (2.10)

10. 26

751)8.104(3.40

1017.3 ftAo =×

= (2.36)

which gives a required effective tube length of

.9.4)640.0(240

751 ft=

If we choose an effective tube length of 6 feet, then we require 3 baffles (= 4 baffle spaces, each 18 in. = 1 ½ feet baffle spacing.) This puts both nozzles on the same side of the shell, which we shall assume is satisfactory in this case. Calculation of shell-side pressure drops: 1. fs = 0.20 from Fig. 2.17.

121

2. 214.0

2

2

28

22

, /4.311040.51068.4

)119)(1017.4)(30.0(2)144)(2()500,58)(20.0(4 ftlbP fib =⎟

⎟⎠

⎞⎜⎜⎝

⎛

×

×

×=Δ

−

− (2.57)

3. [ ] 28

22

, /246)3.0)(8.65)(119)(1017.4(2

)144()8(6.02)500,58( ftlbP fiw =×

+=Δ (2.58)

4. .38.2.67.0 FigfromR =l 5. Rb = 0.73 from Fig. 2.39. This value is actually high since we have had to go to four tube passes. There will be an internal bypass channel, which in usual practice will be partially blocked by tie rods. The effect on heat transfer will be small, but it is possible that the effect on pressure drop might be to drop Rb to as low as 0.55. We will use the higher value here. 6. Then, using Eq.2.60 and Nb = 3:

[ ]

2./24.5

1441

281)73.0)(4.31(267.0)246(3)73.0)(4.31(2

inlbP

P

fs

s

=Δ

⎟⎠⎞

⎜⎝⎛

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +++=Δ

This is a feasible value though perhaps higher than we would like; but as discussed previously, this calculation is likely to be conservative. If we had to reduce this value, we could do any of the following:

a. Increase the shell diameter. But the chosen shell is already quite large in diameter compared to its length.

b. Use a TEMA J shell ("split flow"), which would reduce the pressure drop by about a factor of four

at the cost of substantially more heat transfer area.

c. Use double-segmental baffles, which would have roughly the same effect as using a J shell.

d. Use a "no-tubes-in-the-window-design", reducing the baffle cut somewhat and substantially increasing the number of crossflow tubes compared to the present design, and probably increasing the length substantially.

The Delaware method has not at this point been developed to apply to the geometry modifications sug-gested in b, c, and d above. Calculation of tube-side pressure drop:

1. 500,28)84.1(

)3600)(53.5)(0.62(12509.0Re =⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=i (2.19)

2. fi = 0.006 from Fig. 2.20.

122

3. ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+=

1275.18

212)6(4 πL

L = 28.9 ft.

where the last term is a conservative estimate of the added length of flow in the U-bends. 4. From Eq. (2.26)

( )22

14.02./19.3/459

84.131.1

2.3212509.0

)9.28()53.5)(0.62)(006.0(2 inlbftlbP ffi ==⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=Δ

5. The loss for two tube entrances is, from Eq. (2.25),

22

2./23.1177

)2.32(2)53.5)(0.62(32 inlb

ft

lbP f

fent ==

⎥⎥⎦

⎤

⎢⎢⎣

⎡=Δ

These values are well within standard practice. If the shell-side pressure drop of possibly as much as 5.3 lbf / in.2 is acceptable, the unit designed above will do the job. Summary of Major Design Parameters Shell dimensions: 19 1/4 in. ID x 6 ft. effective tube length (tube sheet face to tangent line.) Shell type: U-tube Baffles: Segmental, 41.6 percent cut; 3 baffles, spaced 18 in. apart. Tubes: Wolverine Type S/T Trufin, 65-265058-01 (3/4in. OD, 26 fins per in., 0.058 in. wall, phosphorous deoxidized copper). Four tube-side passes. Tube layout: 3/4 in. OD tubes on I in. triangular pitch. Sealing strips: None. 2.7.2. Design of A Gas Oil to Crude Heat Recovery Exchanger The problem is to design a split ring floating head exchanger to heat 49,800 bpd (597,000 lb/hr) of 34° API MidContinent Crude from 125°F to 180°F, using 13,200 bpd (152,000 lb/hr) of 28° API Gas Oil at 410°F, cooling it to 220°F. The unit will use type S/T Trufin I in. OD, 19 fins/in. (Catalog No. 60-197083), of low carbon steel. The tubes are to be laid out on a 1 1/4 in. rotated square. Pressure drop is limited to 15 psi on each side.

123

The important tube dimensions are:

do = 1.00 in. dr = 0.875 in. H = 0.0625 in. Y = 0.017 in. s = 0.036 in. Δxw = 0.083 in.

di = 0.709 in. Ao = 0.688 ft2/ft Ai = 0.186 ft2/ft Si = 0.395 in.2

Kw = 26 Btu/hr ft°F

The properties of the shell-side fluid (34° API crude) at a mean fluid temperature of 150°F are:

Density 51.2 Ibm/ft2

Specific Heat 0.51 Btu/Ibm°F Viscosity (bulk) 7.0 Ibm/ft hr Viscosity (wall, at 2000F) 4.4 Ibm/ft hr Thermal conductivity 0.071 Btu/hr ft°F

The properties of the tube-side fluid (28° API Gas Oil) at a mean fluid temperature of 315°F are:

Density 49.3 Ibm/ft, Specific heat 0.58 Btu/lbm °F Viscosity (bulk) 2.90 Ibm/ft hr Viscosity (wall, at 2000F) 7.50 Ibm/ft hr Thermal conductivity 0.061 Btu/hr ft°F

The fouling factor for both the crude and the gas oil will be taken as 0.002 hr ft2°F/Btu for each stream. The next step is to estimate the dimensions of the heat exchanger required, using the procedure for approximate size estimation given previously in this section.

Q = 597,000 (0.51)(180-125) = 1.67 x 107 Btu/hr for the crude (2.32)

Q = 152,000 (0.58)(410-220) = 1.68 x 107Btu/hr for the gas oil (2.33)

Fn

LMTD °=⎟⎠⎞

⎜⎝⎛

−−

−−−= 7.152

125220180410

)125220()180410(

l

(2.11)

667.0410125410220

=−−

=P (2.14)

289.0410220180125

=−−

=R (2.13)

From Fig. 2.5, F = 0.92

124

Uo may be estimated from Table 2.1, using the value for a medium organic fluid in the tubes and a heavy organic fluid on the shell. A median value of 30 Btu/hr-ft2°F will be sufficient for present purposes. Then the actual area required may be estimated from

27

3960)30)(92.0)(7.152(

1067.1 ftxAo == (2.36)

To obtain a value to enter Fig. 2.26, the following correction factors are needed: F1 = 1.54 for 1 in. tubes on a 1 1/4 in. rotated square pitch F2 = 1.03 for two tube passes and a shell inside diameter between 25 and 33 in. (to be checked later, if necessary.) F3 = 1.09 for a split ring floating head and a 23 1/4 - 35 in. shell inside diameter. F4 = 0.97 for I in. OD S/T Trufin, 19 fins/in. Ao = 3960(1.54)(1.03)(1.09)(0.97) = 6640 ft2 for entry into Fig. 2.26. (2.37) From Fig. 2.26, we see the following combinations answer to this requirement:

Shell inside Effective tube diameter, in. length, ft. L/D 37 10 3.2 35 11.5 3.9 33 13 4.7 31 14.5 5.6 29 17 7.0 27 19.5 8.7 25 23 11.0 23 1/4 27 13.9

Undoubtedly several of these could be chosen and designed to meet the thermal-hydraulic performance. Because of the high shell-side flow rate, let us choose the 31 in. ID shell for at least preliminary evaluation. Before proceeding through the complete Delaware method, we can check the tube-side velocity to ensure that it is within reasonable limits: For two passes:

38209.1

417417

3===

FNt

(Here we have taken the fixed tube sheet tube count of 417 from Table 2.6 for the given tube layout and divided it by F3 (= 1.09) for the split ring floating head configuration to obtain the estimate of 382 tubes in the bundle, or 191 per pass.) Then the tube-side velocity is

125

sec/63.1)3600)(3.49)(395.0)(191(

)144)(000,152( ftVi ==

This value is too low, and we can readily estimate that going to six tube-side passes would give a velocity of about 5 ft/sec, which would be a better design for fouling control and probably still be acceptable for tube-side pressure drop. So six tube-side passes will be used for the tube-side. Entering now upon the Delaware method, we list the following basic shell-side geometry values:

dr = 0.875 in. do = 1.00 in. Y = 0.017 in. p = 1 1/4 in., rotated square (45°) layout Di = 31 in.

lotD = 29 3/8 in. L (Effective tube length): As in the previous examples, this will be determined once the heat transfer coefficients have been calculated.

cl = 10.8 in. This is a "35 percent cut", based on the diameter. This must be adjusted up or down somewhat in the final design to correspond to the actual centerline of a row of tubes.

sl = 16 in. This is about half the shell diameter, a common first choice for liquid flow on the shell side. This value can be adjusted in either direction to give evenly spaced baffles in the shell, or to adjust the heat transfer coefficient or pressure drop up or down as needed.

Nss will be chosen later to give one pair of sealing strips for approximately every six rows of tubes in crossflow (Nc). Calculation of shell-side geometrical parameters:

1. Nt = 09.1

387 = 355, from Tables 2.6 and 2.4.

2. Pp = 0.884 in.; pn = 0.884 in., from Table 2.7.

3. [ ] 105.10884.0

)35.0(20131≈=

−=cN (2.38)

Use ;6

10≈ssN i.e., 2 pairs of sealing strips

4. Fc = 0.40 from Fig. 2.28.

5. 108.9884.0

)8.10(8.0≈==cwN (2.40)

6. Calculate Nb later

126

7. 24

183

83 .200

017.0036.0036.0)0625.0(211

884.075.029

293116 inSm =⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

++−

−+−=

(2.42)

8. 130.0200

16)2931( 83

=−

=sbpF (2.44)

9. Stb = 0.0245(355)(1.40) = 12.2 in.2 (2.47) 10. Ssb = 9.2 in.2 from Fig. 2.29 11. Swg = 235 in.2 from Fig. 2.30 Swt = 84 in.2 from Eq. 2.51 Sw = 235-84 = 151 in.2 12. Not needed for this case. Calculation of shell-side heat transfer coefficient:

1. 4480)200)(0.7(12

)144)(000,597(875.0Re ==s (2.54)

2. js = 1.1 x10-2 from Fig. 2.15.

3. FfthrBtuh io °=⎟⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡×= − 2

14.03/22

, /1894.40.7

)0.7(51.0071.0

200)144(000,597)51.0(101.1 (2.55)

4. Jc = 0.845 from Fig. 2.33.

5. 017.0200

2.122.9=

+=

+

m

tbsbS

SS

430.02.122.9

2.9=

+=

+ tbsb

sbSS

S

34.2.80.0 FigfromJ =l

6. 20.0102==

c

ss

NN

Jb = 0.95 from Fig. 2.35

7. ho = 189(0.845)(0.80)(0.95) (2.56)

= 121 Btu/hr-ft2°F

127

Calculation of tub-side and overall heat transfer coefficients: 1. Calculate tube-side velocity:

No. of tubes/pass = 6

355 = 59

sec

29.5)395.0)(59)(3.49(3600

)144(000,152 ftVi ==

2. Calculate tube-side Reynolds number:

100,19)90.2(12

)3600)(29.5)(3.49(709.0Re ==i (2.19)

3. Calculate the tube-side heat transfer coefficient:

FfthrBtuhi °=⎟⎠⎞⎜

⎝⎛

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡= 2

14.03/18.0 /167

50.790.2

061.0)90.2(58.0)100,19(

709.0)12(061.0023.0 (2.23)

4. Calculate the overall heat transfer coefficient, using Eq. (2.2). Rfin may be obtained from Chapter 1

as 4.9 x 10-4 hr ft2°F/Btu.

186.0688.0

1671

186.0688.0002.0

196.0688.0

)12(26083.0109.4002.0

1211

14 +++×++

=−oU

234433 1021.21040.71034.9109.4100.21026.8

1−−−−−− ×+×+×+×+×+×

=

= 24.3 Btu/hr ft2°F 5. Calculate required area and length of exchanger:

)7.152)(92.0(3.241068.1

)(

7×==

LMTDFUQA

oo (2.36)

Ao = 4920 ft2

.20)688.0(355

4920 ftL ==

For a 16 in baffle spacing, this corresponds to 15 baffle spaces or 14 baffles, putting the nozzles on opposite sides of the shell. If this were not satisfactory, a slightly shorter or longer baffle spacing could be investigated. Calculation of shell-side pressure drop.

128

1. From Fig. 2.17, at Res = 4480, fs = 0.38 2. Pressure drop across one ideal crossflow section:

2,

14.0

28

22

,

/6.61

0.74.4

)200)(1017.4)(2.51(2)144)(10()000,597)(38.0(4

ftlbP

P

fib

ib

=Δ

⎟⎠⎞

⎜⎝⎛

×=Δ

(2.57)

3. Pressure drop through one ideal window section:

[ ] ./8.45)2.51)(151)(200)(1017.4(2

)144()10(6.02)000,597( 28

22

, ftlbP fiw =×

+=Δ (2.58)

4. 38.2.58.0 FigfromR =l 5. Rb = 0.87 from Fig. 2.39 6. [ ] ( ) 22

1010 ./88.6/9901)87.0)(6.61(258.0)8.45(14)87.0)(6.61(13 inlbftlbP ffs ==+++=Δ

Even if nozzle losses are added this is well within allowable design limits. Calculation of tube-side pressure drop. 1. From Fig. 2.20, at Rei = 19,100.

fi = 0.007 2. From Eq. (2.26), with L = 6 x 20 = 120 ft.

14.02

4.40.7

)2.32(709.0)12)(120()29.5)(3.49)(007.0(2

⎟⎠⎞

⎜⎝⎛=Δ iP

22 ./02.9/1300 inlbftlbP ffi ==Δ Additionally, entrance/exit losses must be assessed at each nozzle and tube entrance (one per pass) by Eq. (2.24 and 2.25). These are

222

./57.3/514)2.32(2)29.5(3.49)3(8 inlbftlbP ffent ==

⎥⎥⎦

⎤

⎢⎢⎣

⎡=ΣΔ (2.24)

The total tube-side loss of 12.6 psi is within limits. Summary of Major Design Parameters:

129

Shell dimensions: 31 in. ID x 20 ft effective tube length Shell type: Split ring floating head Baffles: Segmental, 35 per cent cut; 14 baffles, spaced 16 in. apart Tubes: Wolverine Type S/T Trufin 60-197083-63 (1 in. OD, 19 fins per inch, 0.083 in. wall, carbon steel.) Six tube passes. Tube layout: 1 in. OD tubes on 1 ¼ in. rotated square pitch Sealing strips: Two pairs

TABLE 2.1 TYPICAL OVERALL DESIGN COEFFICIENTS FOR TRUFIN TUBED HEAT EXCHANGERS

TUBE-SIDE FLUID

SHELL-SIDE FLUID TOTAL FOULING

RESISTANCE IN hr ft2°F/Btu

Uo

Btu/hr ft2°F Water Gas, about 10 psig 0.002 15-20 Water Gas, about 100 psig 0.002 25-35 Water Gas, about 1000 psig 0.002 50-75 Water Light organic liquids 0.0025 70-120 Water Medium organic liquids 0.003 50-80 Water Heavy organic liquids 0.0035 30-65 Water Very heavy organic liquids (cooling) 0.005 5-30

Condensing Steam Gas, about 10 psig 0.0005 15-20 Condensing Steam Gas, about 100 psig 0.0005 25-40 Condensing Steam Gas, about 1000 psig 0.0005 60-85 Condensing Steam Light organic liquids 0.001 100-150 Condensing Steam Medium organic liquids 0.0015 75-130 Condensing Steam Heavy organic liquids 0.002 50-85 Condensing Steam Very heavy organic liquids 0.0035 10-40 Light organic liquids Light organic liquids 0.0017 60-90 Light organic liquids Medium organic liquids 0.0022 40-70 Light organic liquids Heavy organic liquids 0.0027 25-55 Light organic liquids Very heavy organic liquids 0.0042 5-25

Medium organic liquids Heavy organic liquids 0.0037 20-40 Medium organic liquids Very heavy organic liquids 0.0055 5-25

General Notes on Table 2. 1. 1. The total fouling resistance and the overall heat transfer coefficient are based on the total outside

tube area, including fins. 2. Allowable pressure drops on each side are assumed to be about 10 psi except for (a) low pressure

gas, when the pressure drop is assumed to be about 5 per cent of the absolute pressure, and (b) heavy organics where the allowable pressure drop is assumed to be about 20 to 30 psi.

130

3. Aqueous solutions give approximately the same coefficients as water. 4. Liquid ammonia gives about the same results as water. 5. "Light organic liquids" include liquids with viscosities less than 0.5 cp, such as hydrocarbons through

C8, gasoline, light alcohols and ketones, etc. 6. "Medium organic liquids" include liquids with viscosities between about 0.5 cp and 1.5 cp, such as

kerosene, straw oil, hot gas oil, absorber oil, and light crudes. 7. "Heavy organic liquids" include liquids with viscosities greater than 1.5 cp, but not over 50 cp, such as

cold gas oil, lube oils, fuel oils, and heavy and reduced crudes. 8. "Very heavy organic liquids" include tars, asphalts, polymer melts, greases, etc., having viscosities

greater than about 50 cp. Estimation of coefficients for these materials is very uncertain, and frequently their fouling characteristics are such as to render the use of finned tubes unwise. Values of Uo for heating these liquids are usually significantly higher than for cooling them.

TABLE 2.2 F1 FOR VARIOUS UNIT CELLS

Tube Outside Diameter, In. Tube Pitch, In. Layout F1

5/8 13/16 0.90 5/8 13/16 , 1.04 3/4 15/16 1.00 3/4 15/16 , 1.16 3/4 1 1.14 3/4 1 , 1.31 1 1 1/4 1.34 1 1 1/4 , 1.54

TABLE 2.3 F2, FOR VARIOUS NUMBERS OF TUBE-SIDE PASSES*

F2 NUMBER OF TUBE-SIDE PASSES INSIDE SHELL

DIAMETER, IN. 2 4 6 8 Up to 12** 1.20 1.40 1.80 -- 13 ¼ to 17 ¼** 1.06 1.18 1.25 1.50 19 ¼ to 23 ¼ 1.04 1.14 1.19 1.35 25 to 33 1.03 1.12 1.16 1.20 35 to 45 1.02 1.08 1.12 1.16 48 to 60 1.02 1.05 1.08 1.12 Above 60 1.01 1.03 1.04 1.06 * Since U-tube bundles must always have at least two passes, use of this table is essential for U-tube bundle

estimation.

131

** Use of this table for the small shell diameters gives very approximate answers.

TABLE 2.4 F3 FOR VARIOUS TUBE BUNDLE CONSTRUCTIONS

F3 INSIDE SHELL DIAMETER IN. TYPE OF TUBE BUNDLE

CONSTRUCTION UP TO 12

13 ¼ - 21 ¼

23 ¼ - 35

37 - 48

ABOVE 48

Split Backing Ring (TEMA S)

1.30

1.15

1.09

1.06

1.04

Outside Packed Floating Head (TEMA P)

1.30

1.15

1.09

1.06

1.04

U – Tube* (TEMS U) 1.12

1.08

1.03

1.01

1.01

Pull-Through Floating Head (TEMA T)

__ 1.40

1.25

1.18

1.15

* Since U-tube bundles must always have at least two tube-side passes, it is essential to use Table 2.3 also.

TABLE 2.5 F4 FOR VARIOUS TUBE AREA ENHANCEMENTS

TUBE DESCRIPTION F4 TUBE DESCRIPTION F4 Plain (unfinned) tube, any outside diameter*

2.56 S/T Trufin, 26 fins/in.: 5/8 in. O.D. 3/4 in. O.D. 1 in. O.D.

0.76 0.79 0.75

S/T Trufin, 11 fins/in.: 3/4 in. O.D. 7/8 in. O.D. 1 in. O.D.

0.87 0.83 0.81

S/T Trufin, 28 fins/in.: 3/4 in. O.D. 7/8 in. O.D. 1 in. O.D.

0.97 0.96 0.96

S/T Trufin, 16 fins/in.: 1/2 in. O.D. 5/8 in. O.D. 3/4 in. O.D. 7/8 in. O.D. 1 in. O.D.

1.29 1.23 1.20 1.17 1.16

S/T Trufin, 32 fins/in.: 5/8 in. O.D. 3/4 in. O.D. 7/8 in. O.D. 1 in. O.D.

1.01 1.00 0.99 0.99

S/T Trufin, 19 fins/in.: 3/8 in. O.D. 1/2 in. O.D. 5/8 in. O.D. 3/4 in. O.D. 7/8 in. O.D. 1 in. O.D.

1.12 1.05 1.02 1.00 0.99 0.97

S/T Trufin, 40 fins/in.: 3/4 in. O.D. 3/4 in. O.D.

0.78 0.54

*Outside diameter effects are taken into account for F1.

132

TABLE 2.6 TUBE COUNTS FOR FIXED TUBE SHEET EXCHANGERS, ADAPTED FROM REF. (9)

NUMBER OF TUBE PASSES SHELL ID

Di, IN.

TUBE OD,

do, IN.

TUBE PITCH, p. IN. AND LAYOUT

1 PASS

2 PASS

4 PASS

6 PASS

8 PASS

8

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

64 40 42 24 27

48 36 40 20 26

34 24 26 16 18

24 20 24 12 14

10

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

85 64 73 36 42

72 55 66 35 40

52 39 52 29 34

50 38 44 20 24

12

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

122 92

109 55 64

114 86

102 52 60

94 71 88 44 52

90 68 80 38 44

88 67 72 34 40

13 ¼

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

151 115 136 70 81

142 108 128 64 74

124 94

112 54 62

112 85

102 48 56

106 80 96 45 52

15 ¼

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

204 154 183 92

106

192 145 172 90

106

166 125 146 83 96

162 122 140 79 92

152 115 132 76 88

17 ¼

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

264 200 237 127 147

254 192 228 116 134

228 172 208 107 124

220 166 192 98

114

216 160 180 90

104 19 ¼

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

332 251 295 158 183

326 246 282 152 176

290 220 258 138 160

280 212 248 131 152

268 203 232 124 144

21 ¼

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

417 315 361 196 226

396 300 346 190 220

364 275 318 176 204

349 263 312 161 186

336 254 302 152 176

133

TABLE 2.6 (cont.)

NUMBER OF TUBE PASSES SHELL ID

Di, IN.

TUBE OD,

do, IN.

TUBE PITCH, p. IN. AND LAYOUT

1 PASS

2 PASS

4 PASS

6 PASS

8 PASS

23 ¼

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

495 375 438 232 268

478 362 416 226 262

430 325 382 204 236

420 318 372 197 228

408 309 360 190 220

25

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

579 439 507 273 316

554 419 486 261 302

512 387 448 237 274

488 369 440 226 272

472 357 430 214 248

27

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

676 512 592 324 375

648 490 574 311 360

602 456 536 290 336

584 442 516 280 324

560 424 496 266 308

29

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

785 594 692 372 430

762 577 668 360 416

704 533 632 337 390

688 521 604 329 380

664 503 580 318 368

31

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

909 688 796 428 495

878 665 774 417 482

814 616 732 391 452

792 600 708 387 448

776 587 680 374 432

33

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

1035 784 909 501 579

1002 759 886 479 554

944 715 836 450 520

920 696 812 463 504

896 678 780 422 488

35

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

1164 881 1023 558 645

1132 857 1002 538 622

1062 804 942 507 586

1036 784 920 498 576

1012 766 896 484 560

37

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

1304 987 1155 631 729

1270 962 1124 616 712

1200 909 1058 573 662

1168 884 1032 561 648

1136 860 1004 547 632

134

TABLE 2.6 (cont.)

NUMBER OF TUBE PASSES SHELL ID

Di, IN.

TUBE OD,

do, IN.

TUBE PITCH, p. IN. AND LAYOUT

1 PASS

2 PASS

4 PASS

6 PASS

8 PASS

39

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

1460 1106 1277 699 808

1422 1077 1254 685 792

1338 1013 1194 644 744

1320 1000 1164 633 732

1296 981 1120 623 720

42

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

1703 1290 1503 820 947

1664 1260 1466 795 918

1578 1195 1404 756 874

1552 1175 1372 751 868

1528 1157 1344 741 856

45

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

1960 1484 1726 948 1095

1918 1453 1690 924 1068

1830 1386 1622 885 1022

1800 1363 1588 866 1000

1776 1345 1552 845 976

48

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

2242 1698 1964 1074 1241

2196 1663 1936 1056 1220

2106 1595 1870 1018 1176

2060 1560 1828 994 1148

2032 1539 1792 963 1112

54

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

2861 2167 2519 1365 1577

2804 2124 2466 1361 1572

2682 2031 2380 1307 1510

2660 2015 2352 1281 1480

2632 1993 2320 1264 1460

60

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

3527 2671 3095 1700 1964

3476 2633 3058 1680 1940

3360 2545 2954 1629 1882

3300 2500 2928 1586 1832

3264 2472 2896 1548 1788

66

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

4292 3251 3769 2069 2390

4228 3203 3722 2045 2362

4088 3096 3618 1976 2282

4044 3063 3576 1957 2260

3967 3005 3508 1920 2217

72

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

5116 3875 4502 2477 2861

5044 3821 4448 2449 2828

4902 3713 4324 2378 2746

4868 3687 4280 2345 2708

4776 3618 4199 2300 2656

135

TABLE 2.6 (cont.)

NUMBER OF TUBE PASSES SHELL ID

Di, IN.

TUBE OD,

do, IN.

TUBE PITCH, p. IN. AND LAYOUT

1 PASS

2 PASS

4 PASS

6 PASS

8 PASS

78

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

6034 4571 5309 2916 3368

5964 4518 5252 2878 3324

5786 4383 5126 2802 3236

5740 4348 5068 2785 3216

5631 4265 4972 2732 3155

84

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

7005 5306 6162 3394 3920

6934 5253 6108 3361 3882

6766 5125 5964 3277 3784

6680 5060 5900 3235 3736

65553 4965 5788 3173 3665

90

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

8093 6131 7103 3896 4499

7998 6059 7040 3859 4456

7832 5933 6898 3784 4370

7708 5839 6800 3748 4328

7562 5729 6671 3677 4246

96

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

9203 6971 8093 4454 5144

9114 6904 8026 4420 5104

8896 6739 7848 4318 4986

8844 6700 7796 4274 4936

8677 6573 7648 4194 4842

108

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

11,696 8,860

10,260 5,669 6,546

11,618 8,801

10,206 5,623 6,494

11,336 8,587 9,992 5,507 6,360

11,268 8,536 9,940 5,455 6,300

11,055 8,375 9,752 5,353 6,181

120

¾ ¾ ¾ 1 1

15/16 1 1 1 ¼ 1 ¼

14,459 10,953 12,731 7,029 8,117

14,378 10,892 12,648 6,961 8,038

14,080 10,666 12,450 6,815 7,870

13,984 10,593 12,336 6,765 7,812

13,720 10,394 12,103 6,637 7,664

136

TABLE 2.7

TUBE PITCHES PARALLEL AND NORMAL TO FLOW

TUBE O.D. do, IN.

TUBE PITCH p, IN LAYOUT pp, IN pn, IN.

5/8 = 0.625

¾ = 0.750

¾ = 0.750

¾ = 0.750

¾ = 0.750

1

1

1

13/16 = 0.812

15/16 = 0.938

1.000

1.000

1.000

1 ¼ = 1.250

1 ¼ = 1.250

1 ¼ = 1.250

0.704

0.814

1.000

0.707

0.866

1.250

0.884

1.082

0.406

0.469

1.000

0.707

0.500

1.250

0.884

0.625

137

NOMENCLATURE A Heat transfer area. AI, inside tube heat transfer area; Ao, outside tube

heat transfer area, including fins; A'o, effective heat transfer area for use with Fig. 2.26; A*, arbitrary convenient reference area for heat transfer; Am, mean wall area for heat transfer, defined by eq. 2.3; Aroot, heat transfer area of the bare tube remaining between the fins; Afin is the total heat area of all of the fins on a tube.

in.2 or ft.2

cp Specific heat of fluid flowing. cp,i refers to the tube-side fluid; Cp,s refers to

the shell-side fluid. Btu/lbm°F

Di Shell inside diameter. in. or ft.

ltD0 Diameter of the outer tube limit. in. or ft.

Dw Equivalent diameter of the window. in. or ft. d Tube diameter. do, outside diameter of tube, or diameter over the fins; di,

inside tube diameter, dr, root diameter of finned tube. in. or ft.

F Configuration correction factor for the logarithmic mean temperature

difference. dimensionless

Fc Fraction of the total tubes that are in crossflow. dimensionless Fsbp Fraction of total crossflow area that is available for bypass flow around

tube bundle. dimensionless

f Friction factor. fi is the friction factor inside tubes; fs is the friction factor

for crossflow or shell-side flow. dimensionless

G Mass velocity. Gm is the mass velocity of the shell-side fluid through the

minimum free-flow area on the shell side, Sm. lbm/hr ft2

gc Gravitational conversion constant. 32.2 lbmft/lbf sec2

or 4.17 x 108 Ibm ft/lbf

H Fin height. This is the distance from the root to the outer tip of the fin. in. or ft. h Film heat transfer coefficient. hi is the coefficient based on inside area for

turbulent flow; ih is the average coefficient for laminar flow inside a tube of length L; (hi)T is the coefficient for flow in the transitional flow; ho is the coefficient based on outside heat transfer area; ho,i is the coefficient for ideal crossflow on the shell-side of an exchanger.

Btu/hr ft2°F

Jb Correction factor on the shell-side heat transfer coefficient to account for dimensionless

138

by-pass flow effects. Jc Correction factor on the shell-side heat transfer coefficient to account for

baffle configuration effects. dimensionless

lJ Correction factor on the shell-side heat transfer coefficient to account for baffle leakage effects.

dimensionless

Jr Correction factor on the shell-side heat transfer coefficient to account for

buildup of adverse temperature gradient. dimensionless

J*r Base correction factor on the shell-side heat transfer coefficient to

account for buildup of adverse temperature gradient. dimensionless

j Colburn j factor for heat transfer. js is the value for crossfiow or shell-side

flow, defined by Eq. (2.15). dimensionless

k Thermal conductivity. ki is the thermal conductivity of the fluid flowing

inside the tube; lk is the thermal conductivity of a liquid; ks is the thermal conductivity of shell side fluid; kw is the thermal conductivity of tube wall material.

Btu/hr ft2°F

L Tube length effective for heat transfer. in. or ft. LMTD Logarithmic mean temperature difference, defined for countercurrent flow

by Eq. (2.11). °F

cl Baffle cut, from baffle tip to inside of shell. in. or ft.

sl Baffle spacing, face to face. in. or ft.

m Parameter in fin efficiency and fin resistance calculations. Defined in Eq.

(2.7). dimensionless

Nb Number of baffles in exchanger. dimensionless Nc Number of major restrictions crossed in a tube bank on one cross-flow

section of a baffled shell and tube exchanger. dimensionless

Ncw Number of effective crossflow rows in each window or turnaround section

of an exchanger. dimensionless

Nf Number of fins per unit length of tube. (in.)-1 or (ft.)-1 Nss Number of pairs of sealing strips or equivalent obstructions to bypass

flow encountered by the stream in one crossflow section. dimensionless

Nt Total numbers of tubes in the exchanger. For a U-tube bundle, Nt is equal

to the number of holes in the tubesheet. dimensionless

139

n Number of tube-side passes in series. dimensionless P Parameter in MTD calculations defined by Eq. (2.14). dimensionless ΔP Pressure drop. ΔPb,i is the pressure drop across one ideal (no leakage or

bypass) cross-flow section in a baffled shell and tube exchanger. ΔPw,i is the pressure drop through one ideal (no leakage) window section of a baffled shell and tube exchanger. ΔPs is the pressure drop across an ideal tube bank. ΔPi is the frictional pressure loss inside a tube. ΔPent is the pressure loss due to acceleration and friction at the entrance of a tube, and ΔPnoz is the pressure loss during flow through one nozzle.

lbf/in2 or lbf/ft2

Pr Prandtl number defined by Eq. (2.21). Pri is the PrandtI number for the

fluid flowing inside a tube; Prs is the Prandtl number of the fluid on the shell-side; lPr refers to the liquid properties.

dimensionless

p Tube pitch: distance between centers of nearest tubes in tube layout; pn,

tube pitch normal to flow; distance between centers of tubes in the same tube row normal to the flow; pp tube pitch parallel to flow; distance bet-ween centers of adjacent tube rows in the direction of flow.

in. or ft.

Q Heat duty. QT is the total heat load to be transferred in an exchanger. Btu/hr R Parameter in MTD calculations defined by Eq. (2.13). dimensionless Rb Correction factor for effect of bundle bypass on pressure drop. dimensionless Rf Fouling resistance. Rfi, fouling resistance on inside tube surface; Rfo,

fouling resistance on outside tube surface. hr ft2°F/Btu

Rfin Fin resistance defined by Eq. (2.5). hr ft2°F/Btu

lR Correction factor for effect of baffle leakage on pressure drop. dimensionless

Re Reynolds number; Rei is the Reynolds number for flow inside round

tubes, Eq. (2.19); l,Re i refers to the liquid Reynolds number inside a tube; Res is the Reynolds number for shell-side flow, defined by Eq. (2.17).

dimensionless

S Cross-sectional area for flow. Sm is the minimum free flow area through

one crossflow section; for a circular tube field, Sm is evaluated at or near the centerline. Ssb is the shell to baffle leakage area for a single baffle; Stb is the tube to baffle leakage area for a single baffle. Sw is the area available for flow through a single window; Swg is the flow area through a single window with no tubes; Swt is the window area that is occupied by tubes.

ft2

s Fin spacing. in. or ft.

140

T,t Temperatures of the two streams in the beat exchanger. Usually, T refers

to the hot stream and t to the cold stream, though T may also be used to refer to the shell-side stream and t to the tube-side stream, irrespective of their relative values. Subscripts 1 and 2 usually refer to the inlet and outlet values, respectively. T and t are the mean temperatures at which

properties are evaluated, as defined by Fig. 2.18. wT is the mean wall temperature, evaluated by means of Eq. (2.22).

°F

U Overall heat transfer coefficient. U*, overall coefficient referenced to an

arbitrary heat transfer area A*. Ui, overall coefficient based upon inside heat transfer are; Uo, overall coefficient based upon total outside heat transfer area.

Btu/hr ft2°F

V Velocity of flow. Vi is the average velocity of a fluid flowing inside a tube;

l,iV is the velocity of a two-phase flow computed as if the entire flow were or liquid. Vmax is the velocity of a fluid flowing across a tube bank at the minimum crossflow are, Smin.

ft/sec or ft/hr

W Weight (or mass) flow rate. Ws denotes the shell-side flow rate; wi

denotes the tube-side flow rate. lb/hr

x Local quality of a two phase flow: ratio of mass of vapor to total mass

present; xi is inlet quality; xo is the outlet quality. dimensionless

Δxw Wall thickness of tube. Usually, in this Manual, the thickness of interest is

the finned section. in. or ft.

Y Mean thickness of a single fin. in. GREEK δsb Diametral clearance between baffle and shell. in. or ft. ε Roughness of tube/pipe inside surface. ft. θ Baffle cut angle. radians μ Fluid viscosity. μi refers to the tube-side fluid; μs refers to the shell-side

fluid; lμ refers to the liquid viscosity and μv to the vapor viscosity in a twophase flow.

Ibm/ft hr

ρ Fluid density. iρ refers to the tube-side fluid; sρ refers to the shell-side

fluid; lρ refers to the liquid density and vρ to the vapor density in a two phase flow.

lbm/ft3

Φ Fin efficiency. Dimensionless

141

BIBLIOGRAPHY

1. Kern, D.Q., Process Heat Transfer McGraw-Hill Book Co., New York (1948). 2. Mueller, A.C., "Heat Exchangers", Section 18 of Handbook of Heat Transfer, Edited by Rohsenow,

W.M., and Hartnett, J.P., McGraw-Hill Book Co., New York (1973). 3. Williams, R.B., and Katz, D.L., "Performance of Finned Tubes in Shell- and -Tube Heat Exchangers",

Trans. A.S.M.E. 74, 1307-1320 (1952).

4. Briggs, D.E., Katz, D.L., and Young, E.H., "How to Design Finned-tube Heat Exchangers", Chem. Eng. Prog. 59, No. 11, 49-59 (November, 1963).

5. Bell, K.J., "Final Report of the Cooperative Research Program on Shell and Tube Heat Exchangers", BulletinNo. 5, University of Delaware, Engineering Experiment Station (1963).

6. Perry, R.H., and Chilton, C.H., Eds., Chemical Engineers' Handbook, 5th Edition, 10-25-10-30, McGraw-Hill Book Co., New York (1973).

7. Eagle, A., and Ferguson, R.M., "On the Coefficient of Heat Transfer from the Internal Surface of Tube Walls", Proc. Royal Society A127, 540-566 (1930).

8. Boyko, L.D., and Kruzhilin, G.N., "Heat Transfer and Hydraulic Resistance During Condensation of Steam in a Horizontal Tube and in a Bundle of Tubes", Int. J. Heat and Mass Transfer, 10, 361-373 (1967).

9. Palen, J.W., and Taborek, J. "Solution of Shell Side Flow Pressure Drop and Heat Transfer by Stream Analysis Method", Chem. Eng. Prog. Symp. Series, No. 92, "Heat Transfer-Philadelphia", 65, 53-63 (1969).

10. Standards of Tubular Exchanger Manufacturers Association, 6th Ed., New York (1978). 11. Rabas,T.J., Eckels, R.W., and Sabatino, R.A., "The Effect of Fin Density on the Heat Transfer and

Pressure Drop Performance of Low Fin Tube Banks." ASME Paper 80-HT-97.

142

3.1. Trufin Tubes in Condensing Heat Transfer 3.1.1. Modes of Condensation Condensing is the heat transfer process by which a saturated vapor is changed into a liquid by means of removing the latent heat of condensation. Four basic mechanisms of condensation are generally recognized: dropwise, filmwise, direct contact, and homogeneous. In dropwise condensation, the drops of liquid form from the vapor at particular nucleation sites on a solid surface, and the drops remain separate during growth until carried away by gravity or vapor shear. In filmwise condensation, the drops initially formed quickly coalesce to produce a continuous liquid film on the surface through which heat must be transferred to condense more liquid. In direct contact condensation, the vapor condenses directly on the (liquid) coolant surface which is sprayed into the vapor space. In homogeneous condensation, the liquid phase forms directly from super saturated vapor, away from any macroscopic surface; it is however generally assumed that, in practice, there are sufficient numbers of dirt or mist particles present in the vapor to serve as nucleation sites. While dropwise condensation is alluring because of the high coefficients reported, it is not considered at this time to be suitable for deliberate employment in process equipment. Generally, contaminants must be continuously injected into the vapor, or special materials (often of low thermal conductivity) employed. Even so, the process is unstable and unpredictable, and of questionable efficacy under conditions of high vapor velocity and industrial practice. Direct contact condensation is a very efficient process, but it results in mixing the condensate and coolant. Therefore, it is useful only in those cases where the condensate is easily separated, or where there is no desire to reuse the condensate, or where the coolant and condensate are the same substance. Homogeneous condensation is primarily of concern in fog formation in equipment and is not a design mode. Therefore, all subsequent references to condensation will mean filmwise condensation, in which the heat transfer surface is covered with a thin film of condensate flowing under the influence of gravity, vapor shear, and/or surface tension forces. The necessary equations for calculating the heat transfer and pressure drop for condensing will be developed later in this Chapter. The case for in-tube condensing will be studied first then extended to cover condensing outside Trufin tubes. 3.1.2. Areas of Application In Chapter 1, it was pointed out that it is usually advantageous to use Trufin when one of the film heat transfer coefficients is significantly smaller than the other. The lower coefficient tends to control the magnitude of U, the overall heat transfer coefficient, and therefore the size of the heat exchanger. Hence, if Trufin is used, with the low coefficient fluid in contact with the higher heat transfer area of the fin, the total amount of tubing is reduced compared to the plain tube case; therefore the overall size of the heat exchanger is also reduced. The best design is generally obtained if the thermal resistances of the two fluid heat transfer processes are approximately equal. This condition is obtained when:

143

ooii AhAh11 ≈

o

i

i

o

hh

AA

≈

In a large number of condensing applications in the process and refrigeration industries, especially where water cooling is used, the value of hi/ho ranges from 2 to 5 or even 10. Since low- and medium-finned Trufin have Ao/Ai values from about 3 to 7, these tubes are often found to afford substantial savings in overall heat exchanger size and cost. In these applications the condensation takes place on the outside (fins) of the tubes. In other applications where air is used as the cooling medium, the air side heat transfer coefficients are much lower than the condensing coefficients. High-fin Trufin is used in these cases with the condensing taking place inside the tubes and the high outside area placed on the air side. 3.1.3. Types of Tubes Available 1. Type S/T Trufin Low-Finned Tube

Tubes of this type are made with 16 to 40 fins per inch and fin heights of approximately 1/16 inch. The diameter over the fins is equal to or less than the plain end diameter to allow the tube to be inserted through a tubesheet.

2. Medium-Finned Trufin

These tubes are characterized by having 11 fins per inch and fin heights of 1/8 inch. The tubes can be supplied with plain ends of a smaller diameter than the finned section (type W/H) or with belled ends suitable for rolling into tubesheets (type S/T).

3. Type S/T Turbo-Chil Finned Tubes

The outer surface of these tubes is similar to standard type S/T Trufin. In addition, the inner surface of the tube is provided with integral spiral ridges which enhance the internal heat transfer coefficient.

4. Koro-dense

This tube is a corrugated rather than a finned tube but is mentioned here because of its advantageous application in steam condensing. Two types are available: MHT, a medium corrugation severity affording maximum tube side performance if pressure drop permits, and LPD, a low corrugation severity for use when tube-side pressure drop is limiting.

Condensation generally takes place on the outside surface of the above tubes.

5. High-finned Trufin

Tubes of this type are made in both copper and aluminum. Fin counts range from 5 to 11 fins per inch with fin heights as high as 5/8 inch. The aluminum finned tube can be supplied with liners of various other metals.

3.2. Condensation of Vapor Inside High-Finned Trufin Tubes 3.2.1. Vapor-Liquid Two-Phase Flow 1. Phase Relationships of Two-Phase Condensing Flows. Condensation of a vapor is frequently carried out inside the tubes of an air-cooled heat exchanger employing high-finned Trufin tubes. Usually the tubes are horizontal or nearly so, but occasionally inclined or vertical orientations are used. The flow of a vapor-liquid two phase mixture is a good deal more complicated than single phase flow, and the correlations for heat transfer and pressure drop correspondingly less accurate. Since in general the two-phase heat transfer coefficients are quite high, this inaccuracy is not a serious matter for heat transfer in air-cooled condensers. However, the pressure drops are also quite high and care is required in interpreting the predicted values into design considerations. The first way to characterize a condensing two-phase flow is by its composition. Five cases are discernible:

a. The liquid and the gas are different pure components. The usual example is an air-water mixture, which is not a common industrial problem, but is very important because a great deal of what is known about two-phase flow has been determined on this system. While in general this information can be carried over to condensation and some boiling work, there are important dif-ferences that must be recognized and allowed for. Thermodynamically, the pressure and temperature can be independently varied over wide ranges in this system.

b. The liquid and gas (vapor) are the same pure component. This is a common case in condenser

design, occurring for example in condensers on columns separating and/or purifying a product. The pressure-temperature relationship in this case is the vapor pressure curve for the component.

c. The liquid and gas (vapor) are multi-component mixtures, each containing some of each of the

components. The thermodynamic relationships are more complex, the temperature, for example, being variable over a range of values at a given pressure, but with a changing ratio of total liquid to total vapor and with changing composition of each phase. Prediction of the amount and com-position of each phase is relatively well understood and easily done in a few cases, such as mixtures of light hydrocarbons; other cases require laboratory thermodynamic data.

d. Like case (b) or (c), but with a non-condensable gas present. The most common example cited

here is air in steam, but there are also applications as condensation of a solvent from an inert stripping gas.

e. Intermediate between (c) and (d), typified by condensation of light gasoline fractions from a wet

natural gas. Depending on the composition of the gas and the pressure and temperature of the condensation, some of the methane and ethane present will dissolve in the liquid, and a small amount of the heavier components will be present in the vapor.

2. Characterization of Two-Phase Vapor-Liquid Flows. This section will concentrate on case (b), condensation of a pure component, but many of the equations and procedures developed will be applicable to the other situations, as discussed later. There are a number of quantitative parameters of two-phase flows that characterize the flow and must be defined:

144

a. Weight flow rates: The weight flow rates of each phase (in typical units of lb/hr), and wlw v, are

generally either known or computable from process specifications. In a condensing flow, the total weight flow rate remains constant, the liquid rate increasing as the vapor rate decreases. The flow mechanisms of the condensation process may change as the ratio changes from entrance to exit. It is convenient to non-dimensionalize the ratio of the two phases by defining the quality x as:

lwww

xv

v

+= (3.1)

b. Volume flow rates: The volume flow rates are occasionally useful quantities and are defined as:

ϑϑ = ;l

l

ρw

ϑϑ = v

vwρ

(3.2)

c. Mass velocities: The mass velocity of the total two phase stream, G (lb/hr ft2) is constant in a

constant cross-section conduit. It is defined as:

Sww

G v+= l (3.3)

where S is the cross-sectional area of the conduit.

d. Superficial mass velocities: The superficial mss velocity of a given phase is defined as:

xv

v GSw

G == (3.4)

)1( xGSw

G −== ll (3.5)

The entire cross-sectional area of the conduit is used in calculating Gv and even though each phase alone occupies only a portion of the cross section. These quantities have no fundamental significance, but they are convenient and useful parameters for correlating two-phase flow data.

lG

e. Velocities: The true mean velocities of each phase are not simply or fundamentally related to the

other characterizing parameters of the system. The liquid, being generally preferentially located near the walls of the conduit and being more viscous and dense than the vapor, in general flows more slowly, giving rise to the phenomenon of "slip." if Vv is the actual mean velocity of the vapor in (ft/hr) at a given point in the conduit and , is the corresponding value for the liquid, VlV v is

usually greater than . It is customary to define a quantity, the slip ratio, as ; as noted above, the slip ratio is almost always greater than 1 and can reach 10 or 20 or even more. If V

lV lVVv /v =

, the flow is said to be homogeneous; this is not a very realistic case, but it is simple to use in that it permits densities, velocities, and weight and volume fractions to be directly related to each lV

145

other. In the lack of better information, homogeneous flow may be assumed taking precaution against non-conservative consequences of this assumption.

f. Phase volume fractions: The actual volume occupied by the vapor phase divided by the total con-duit volume is called the vapor phase volume fraction, Rv, or perhaps more commonly, the void fraction. The liquid volume fraction, , is equal to (1 – RlR v). For a steady flow, or one averaged over a sufficient period of time, these are also the fractions of the cross-sectional area that are occupied by the respective phases. By use of continuity, we may write

)1( xVV

x

x

Vw

Vw

Vw

SS

Rvv

vv

v

vv

v

vv

−+=

+==

llll

l

ρρ

ρρ

ρ (3.6)

In these terms the slip ratio is given by

⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −==

xx

RR

VV

sv

v

v

v1

1ρρl

l

(3.7)

g. Effective density: The effective density for a two-phase mixture, , is defined as the mass of a

unit Volume of the flowing mixture, effρ

lρρρ )1( vvveff RR −+= (3.8)

3. Correlations of the Vapor Volume Fraction. It is evident that data on the vapor volume fractions are needed in order to calculate some of the characteristic quantities of a two-phase flow - Many experimental studies have been made to determine values of R, and a number of correlations have been proposed (Ref. (1) surveys the work done up to about 1965). A generally usable correlation which seems to give reasonable results over the range of process applications was proposed by Martinelli and co-workers, mostly based on air-water systems at about atmospheric pressure. The correlation for volume fractions proposed by Martinelli and Nelson (2) seems to work as well as any other correlation proposed over the entire range of data available. The original correlation was intended for steam-water systems, but is easily generalized in terms of the reduced pressure. The modified correlation is given here as Fig. 3. 1. The abscissa of Fig. 3. 1 is ttx where

11.057.01

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

=v

vtt x

xxμμ

ρρ l

l

(3.9)

The ordinates are R, and Re; the parameter is the reduced pressure of the vapor

critR p

pP = (3.10)

The form of the parameter xtt is deduced by Martinelli from a particular model of two phase flow but it seems to be remarkably generally applicable. The limiting curve for PR = 0.005 is taken from the

146

experimental results at atmospheric pressure for air-water and the curve for PR = I is obtained from the requirement that the slip ratio go to unity as the two phases become more similar. The curves for other reduced pressures are interpolated by eye.

4. Regimes of Two-Phase Flow. To this point, we have said nothing of the actual physical appearance of a two phase flow. In some respects, such as void fraction calculations for process condensers, the physical description of the flow is relatively unimportant. In other regards, it is of some importance. The types and the detailed description of two-phase flow configurations (or regimes) depends upon the relative and absolute quantities and the physical properties of the fluids flowing, the geometric configuration of the conduit, and the kind of heat transfer process involved. The flow regimes observed by

147

Alves (3) in his study of air-water flows in horizontal tubes are diagrammed in rather idealized form in Fig. 3.2. The chief difference between flow regimes studied in a non-heat transfer situation and those existing during condensation is that a liquid film exists on the entire surface of the conduit during condensation; however, we may presume that this film of draining condensate does not cause any vital difference in the interaction between the vapor and the main inventory of liquid. We may view the flow regime as a consequence of the interaction of two forces-gravity and vapor shear acting in different directions. At low vapor flow rates, gravity dominates and one obtains stratified, slug-plug, or bubble flow depending upon the relative amount of liquid present. At high vapor velocities, vapor shear dominates, giving rise to wavy, annular, or annular-mist flows. It would be desirable to have some way to predict the flow regime a priori, and many attempts have been made to do this in a general and consistent way. No attempt has succeeded, but the work of Baker (4) is considered to be generally the best available even though it is a dimensional representation and defies explanation in fundamental terms. The Baker map is shown in modified form as Fig. 3.3 and is useful in giving a general appreciation of the general kind of flow regime existing under given conditions. During the course of condensation, the relative ratios of vapor and liquid change; these changes may be followed conveniently on the Baker map with the aid of Fig. 3.4, which is used in the following way: Calculate the values of the abscissa and ordinate on the Baker map corresponding to the half-condensed point (Gv =

l). Plot this point on Fig. 3.3. Place the 0.5 point (circled) of the curve of Fig. 3.4 on top of

the point plotted on Fig. 3.3 (it is helpful if a transparency of Fig. 3.4 is used) so that the coordinates are parallel on- the two figures. Then the curve on Fig. 3.4 traces the sequence of flow regimes as a function of vapor quality from pure vapor at the left infinity point to pure liquid at the bottom in finity point.

G

It may be assumed that condensing flow patterns in tubes inclined slightly downwards (in the direction of flow) are similar to those in horizontal tubes. Condensation is seriously reduced in tubes inclined slightly upwards (5) and care must be taken to insure that this does not occur, often by deliberately designing the condenser to a slight (2-3°) downward inclination. For two-phase flow inside vertical tubes, the stratified and wavy flow regimes cannot exist, and the flow regimes generally recognized in this case are bubble (low vapor rate), slug, annular, and annular with mist. There is no generally recognized flow regime map for condensing in vertical tubes at the low heat fluxes generally characteristic of air-cooled condensers. This is not a serious deficiency in predicting heat transfer coefficients and the same is also true of condensing in steeply inclined tubes.

148

5. Pressure Drop in Two-Phase In- Tube Flow. Pressure changes in two-phase flow arise from three sources.

a. Friction loss, which always causes a

pressure decrease in the direction of flow. The pressure loss due to friction in a two-phase flow is generally much higher than in a comparable single phase flow because of the roughness of the vapor-liquid interface. The pressure gradient due to friction depends upon local conditions, which change in a condensing flow. Therefore, the total pressure effect from friction depends upon the path of condensation.

b. Momentum effects, resulting from a change

in the velocity, and hence kinetic energy, of the stream. The pressure change from this cause is negative (pressure loss) if the flow accelerates as in boiling and positive (pressure gain) if the flow decelerates, as is the usual case in condensation. The total effect on pressure from this cause depends only upon terminal conditions, though in some cases it may be necessary to calculate the local pressure gradient contributed by velocity changes.

149

c. Hydrostatic effects, resulting from changes in elevation of the fluid. The pressure from this source

alone always decreases in an upward direction and is zero in a horizontal tube. The local gradient depends upon the local density; therefore, it is generally necessary to calculate this contribution along the path of condensation.

The algebraic sum of the three contributions is equal to the net total pressure effect. In process applications, the friction loss usually predominates in horizontal configurations and is usually comparable to the hydrostatic effect in vertical designs. However, in very high velocity condensing flows, the momentum effect may be greater than the other contributions, resulting in a pressure rise from inlet to outlet (e.g., Ref. (6)). Calculation of the pressure loss due to friction may be made using the Martinelli-Nelson correlation (2). First, we calculate the pressure gradient as if only the liquid were flowing in the conduit.

icf dgGf

ddp

l

ll

ll ρ

2

,

2−=⎟

⎠

⎞⎜⎝

⎛ (3.11)

where is read from the Fanning friction factor chart or Eq.(3.13) at a Reynolds number calculated from:

lf

l

ll μ

Gdi=Re (3.12)

if > 2100. If < 2100, use the vapor-phase based correlation given further on in this section. The smooth tube correlation may be used though it is more conservative to use the correlation for the relative roughness of the actual surface at high Reynolds numbers. Over the usual range of process applications,

may be computed from

lRe lRe

lf

4/1078.0

⎟⎠⎞⎜

⎝⎛

=

l

l

l

μGdi

f (3.13)

The pressure gradient due to friction for the two-phase flow is then calculated from

ll

ll ,

2

, ftt

TPFf ddp

ddp

⎟⎠⎞

⎜⎝⎛Φ=⎟

⎠⎞

⎜⎝⎛ (3.14)

where is read from Fig. 3.5 at the appropriate value of 2

yylΦ

11.057.01

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

=v

vtt x

xxμμ

ρρ l

l

(3.15)

If < 2100, and > 2100, the calculation proceeds similarly, but based on the vapor phase lRe vRe

150

ivc

vv

vf dgGf

ddp

ϑ

2

,

2−=⎟

⎠⎞

⎜⎝⎛l

(3.16)

where fv us taken from the Fanning chart at

v

viv

Gdμ

=Re (3.17)

or from

4/1078.0

⎟⎟⎠

⎞⎜⎜⎝

⎛=

v

viv

Gdf

μ

(3.18)

2ttlΦ is again read from Fig. 3.5 at the appropriate value of xtt and the two-phase friction pressure drop is

given by

vftttt

TPFf ddpx

ddp

,

75.12

,⎟⎠⎞

⎜⎝⎛Φ=⎟

⎠⎞

⎜⎝⎛

lll (3.19)

If both and are below 2100, an approximate value may be obtained by using the laminar flow friction factor equation for either phase:

lRe vRe

vorvorf

ll Re

16= (3.20)

However, in this case, the pressure drop will probably be so low as to be unimportant. The pressure effect due to momentum changes is given by

[ ]ll

l

dxvxvd

gG

ddp v

cTPFm

+−−=⎟

⎠⎞

⎜⎝⎛ )1(

, (3.21)

However, the total effect can be calculated from inlet and outlet conditions only:

[ ] [{ outletvinletvc

TPFm xvxvxvxvgGp +−−+−=Δ )1()1(, ll ] } (3.22)

If a dry saturated vapor enters and is completely condensed, Eq. (3.22) reduces to

[ ] ⎥⎦

⎤⎢⎣

⎡−=−=Δ

ll ρρ

112

,,,vc

outletinletvc

TPFm gGvv

gGp (3.23)

151

152

Recall that for condensation this is a pressure increase. The hydrostatic pressure effect is calculated from

ceff

TPFg g

g

ddp Θ

=⎟⎠

⎞⎜⎝

⎛ cos

,ρ

l (3.24)

where effϑ is given by Eq. (3.8) and θ is the angle between the vertical

and the axis of the tube ( θ = 0 for a vertical tube; θ = π/2 for a horizontal tube, see Fig. 3.6). The hydrostatic contribution to the pressure is always negative in the upward direction. The total local pressure gradient is the algebraic sum of the three effects:

TPFgTPFmTPFfTPFT ddp

ddp

ddp

ddp

,,,,⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

llll (3.25)

The total pressure difference from one end of the condenser to the other must be found by integration over the quality range

( ) ( )[ ] dxp ddx

TPFT

x

x ddp

TPFo

ill

/1,

∫=Δ (3.26)

Exact numerical evaluation of this integral is a rather tedious trial-and-error process, even on a computer, because the properties and the local quality depend upon the local pressure implicitly. For a condenser, however, one is usually concerned with complete condensation (x1 = 1; xo = 0) and with a very nearly linear condensation rate, (dx/ ) = 1/L. Then, a good approximation to Δpld TPF is obtained by a "pseudo-Simpson" rule:

Lddp

ddp

ddpp

xTPF

xTPF

xTPFTPF Δ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧⎟⎠

⎞⎜⎝

⎛+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛=Δ=== 5.09.01.0

4823

9625

lll (3.27)

6. Flooding Effects in Upward Flow. In some applications, variously termed reflux or knockback condensers, the vapor flows upward in the tube while the liquid flows downward on the walls. In this case, the vapor shear subtracts from the gravitational force causing the liquid to flow downwards, leading to a thickening of the film; if the film is in laminar flow, the heat transfer coefficient is also reduced, an effect analyzed by Nusselt (7). In general, the effect is rather to cause early turbulence, which does thicken the film, but also increases the coefficient. If the film would be turbulent even in the absence of vapor shear, the effect of vapor shear on heat transfer is not clear and probably not of critical importance. The main concern in knockback condensers is that flooding be avoided, i.e., that at no point in the condenser should the vapor shear be equal to the gravitational force of the liquid. The critical point is at the entrance to the lower end of the tube when both the liquid and vapor flow rates are maximum. Soliman, Schuster, and Berenson (8) have analyzed the problem in very fundamental terms, but the best design correlation for flooding is by Diehl and Koppany (9). In summary, their correlation for incipient flooding is

153

5.0

21'

⎟⎟⎠

⎞⎜⎜⎝

⎛=

vv FFV

ρσ 10

5.0

21 >⎟⎟⎠

⎞⎜⎜⎝

⎛

vFFif

ρσ (3.28)

15.15.0

21' 71.0

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛=

vv FFV

ρσ 10

5.0

21 <⎟⎟⎠

⎞⎜⎜⎝

⎛

vFFif

ρσ (3.29)

where V’v = superficial flooding velocity of vapor, ft/sec

( ) ( )[ ] 4.01 80//12 σidF = ( ) ( )[ ] 0.180//12 <σidif

0.1= ( ) ( )[ ] 0.180//12 ≤σidif (3.30)

25.02 )/( lGGF v= (3.31)

The equation is dimensional, so that it is essential to use the units specified (ρv in lb/ft3, di in ft, σ in dyne/cm). Since this is for incipient flooding, conservative design requires that the maximum design vapor velocity be somewhat below this; if the conditions are within the fairly wide range represented by the data in the Diehl-Koppany correlation, design to 70 percent of the predicted flooding velocity appears safe. Otherwise, the velocity should be limited to 50 percent of that calculated. In a few cases, condensers have been designed so that the vapor shear is great enough to carry the condensate up and out of the condenser. Total condensation is impossible in this case since there must be sufficient vapor flow at the top of the condenser to carry the liquid out. It has been observed that, as one increases the vapor flow above the Diehl-Koppany limit for flooding, the amount of liquid carryover increases slowly until virtually complete carryover occurs at about three times the flooding velocity. This value then becomes the design case at the top of the condenser for relatively foolproof operation. For intermediate values, the liquid that drains back must be carried up into the condenser again and blown through, a situation that offers opportunity for slugging, maldistribution, and generally unstable operation. 3.2.2. Condensation Heat Transfer Laminar Film Condensation of a Pure Component in a Vertical Tube. Laminar film condensation of a pure component from a saturated vapor was among the first heat transfer problems to be successfully analyzed from a fundamental point of view. The definitive work is by Nusselt (7) in two papers published two weeks apart in 1916. The analysis is readily available in a number of books, of which Jakob (10) and Kern (11) may be especially recommended. The original Nusselt analysis applies specifically to laminar flow of a condensing film on a vertical surface. However, it is possible to generalize the approach to apply to a number of other cases, including in-tube condensation in horizontal tubes. For this reason, it is worthwhile to examine the analysis in some detail here. There are a number of assumptions implicit in the basic Nusselt model. The key assumption is that the liquid film (Fig. 3.7) is in laminar flow and its hydrodynamics are controlled by the viscous terms in the Navier-Stokes equations. This allows the neglect of the inertial or kinetic energy terms and yields a simple equation relating film thickness and velocity profile to gravity and liquid viscosity. Relaxing this assumption yields complex equations that must be solved numerically; this has been done by several authors and the results indicate that the more

154

complex solution is negligibly different from the simple one for most process applications. Other assumptions of the Nusselt model include:

1. Saturated vapor. 2. The liquid and the vapor have the same

temperature (Tsat) at the interface. (No interfacial resistance.)

3. Heat is transferred by conduction only

through the liquid film.

4. The temperature profile is linear through the liquid film.

5. The liquid and the solid surface are at the

same temperature at their interface.

6. The solid surface is isothermal.

7. The liquid properties are not a function of temperature.

8. The vapor exerts neither shear nor normal

stresses on the liquid surface.

9. The liquid has zero velocity at the liquid-solid interface (no-slip condition).

10. The sensible heat of subcooling the liquid

is negligible compared to the latent heat load.

Before discussing the validity of these assumptions, let us look at the resulting equations. The local value of the film heat transfer coefficient at a distance x from the start of condensation is

4/13

)(4)(

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

=xTTgk

hwsat

vx

l

lll

μλρρρ

(3.32)

A far more useful quantity is the average coefficient fora surface of length L, which we identify for convenience as the condensing coefficient hc:

4/13

)()(

943.0⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

== ∫ wsat

vx

L

o

c TTLgk

dxhhl

lll

μλρρρ

(3.33)

155

Note that the heat transfer coefficient predicted by (3.33) decreases as L and (Tsat – Tw) increase. This is due to the increased resistance to conduction offered by a thickened film. The derivation of Eq. (3.33) was carried out in terms of a vertical plane surface. Since the condensate film is so thin compared to typical tube diameters, the result is applicable to condensation on the inside or outside of vertical tubes if the other assumptions of the derivation are satisfied. Strictly speaking, the dependence of hc on L and (Tsat – Tw) violates two of the assumptions underlying the validity of the logarithmic mean temperature difference usually employed in heat exchanger design. The effect can be shown to be not serious, and it is actually very slightly conservative to use the conventional F-LMTD formulation. The coefficient as given by Eq. (3.33) is useful if one knows the condenser tube length, but is awkward if one is trying to design a condenser for a given duty. The equation can be reworked to a more convenient form if one first defines a tube loading per linear foot of tube drainage perimeter Γ. That is, if w lb/hr are to be condensed on each tube,

tpw

=Γ (3.34)

where Pt= πd for a vertical tube. The total heat load per tube is

Q = λw = hcπdL(Tsat – Tw) (3.35) With some rearrangement, we obtain the desired equation:

( )3/13

924.0⎥⎥⎦

⎤

⎢⎢⎣

⎡

Γ−

=l

lll

μρρρ gk

h vc (3.36)

For later purposes, it is also desirable to define a condensate Reynolds number

lμΓ

=4Rec (3.37)

Substituting this into Eq. (3.36) gives

( ) 3/13/1

2

3Re47.1 −

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −= c

vc

gkh

l

lll

μ

ρρρ (3.38)

Before developing the treatment further, it is useful to re-examine the several assumptions of the Nusselt derivation. In general, the validity of the Nusselt equation has been established in experiments in which care has been taken to satisfy the assumptions. But what if those assumptions are not satisfied in an actual application? Does a departure from ideality completely invalidate the equation, or can the equation or its application be modified to still give useful design results? Only by considering each of the assumptions can we answer that.

156

The consequences of the violation of Nusselt's basic assumption, i.e., larninar flow of the condensate film, are very significant and will be examined at greater length later. Within this assumption, however, many workers have analytically and experimentally tested the effect of violation of the other assumptions listed above. We will now consider these briefly, with attention focused upon the design consequences of the result. The first assumption, that of saturated vapor, has been studied experimentally and theoretically; the weight of the evidence is that superheating effects in pure vapors are small and Nusselt's equation can be safely applied, if the sensible heat load required to desuperheat the vapor to saturation is added to the latent heat load in calculating condenser duty and if the mean temperature difference for the condenser is calculated using the vapor saturation temperature. This, of course, assumes that the con denser surface temperature is below saturation, so that condensation does occur. The method of calculating surface temperatures will be considered when desuperheating in condensers is discussed. The second assumption, that of no interfacial resistance, has been a subject of investigation in many areas of transport processes. The effect of an interfacial resistance becomes significant only at extremely high transfer rates for condensation processes; for example, interfacial resistance has been suspected as the phenomenon responsible for liquid metal condensation coefficients being "only" about 10,000 BTU/hr ft2°F, instead of the 100,000 or so predicted by Nusselt's equation. For air cooled condensers, it is completely immaterial whether the assumption is true or not. The third assumption, concerning the mechanism of heat transfer in the liquid film is exactly as valid as the assumption of laminar flow and breaks down only when the flow is no longer laminar. The fourth assumption, that of a linear temperature pro file in the film, can be relaxed by replacing λ by

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+

λλ

)(68.01 , wsatp TTC l (3.39)

This is a significant correction only when (Tsat – Tw) is relatively quite large, usually out of the range of process practice. Since the effect is to increase the calculated value of hc it is generally conservative to neglect this correction. The fifth assumption, equilibrium at the liquid-solid interface, has not been questioned on theoretical grounds, but in practical cases, allowance must be made for a dirt film resistance to heat transfer. This is handled in the conventional way and does not affect the Nusselt equations. The sixth assumption, an isothermal condensing surface, is generally not realized in practice. While the remaining features of the Nusselt analysis can be rigorously applied to a non-isothermal surface, the calculation requires a computer, and standard practice is to simply use the average computed surface temperature in a condenser. Even this in principle requires a reiterative calculation; assuming a surface temperature, calculating the coefficient, checking the surface temperature, etc. For the usual process case, the effect of an assumption of a mean wall temperature can be shown to be small and probably conservative. The seventh assumption, constant liquid properties, is never strictly valid, but is accepted because the alternatives are so formidable. The physical properties are generally taken at the arithmetic mean film temperature. Significant errors in the final result are unlikely to arise unless the temperature difference is

157

very great or the condensate has a very large temperature coefficient of viscosity. In case of doubt, use the viscosity at the surface temperature in the Nusselt equation. The assumption concerning vapor shear on the condensate on a vertical surface was examined by Nusselt himself (7) and is described in detail in Jakob (10). If the vapor and the condensate flow together vertically downwards, vapor shear somewhat enhances the condensing coefficient in laminar flow (the role of vapor shear in causing laminar flow to be destroyed is discussed later.) If the vapor and condensate flow in opposite directions, the condensate film is thickened and resistance increases; however, in this case a probable and significant consequence is that the film becomes rippled and/or turbulent, and entrainment, slugging, flooding, etc., occur. Assumption 9, no slip at the liquid-solid interface, is very strongly supported by many studies on laminar flow. The last assumption, that the heat load from subcooling the liquid is negligible compared to the latent heat, while generally true, can be easily and conservatively relaxed by adding this amount to the total heat to be transferred. The condensate film is subcooled (on the average) to (Tsc = 3/8 (Tsat – Tw)), so the added heat load is

wsatpsc TTwcQ −= (83

,l (3.40)

Possibly a more important and interesting point is that this degree of subcooling may be sufficient (if reheating is avoided) to satisfy the NPSH requirements of the con-densate pump and eliminate the need for a condensate subcooler. Turbulent Film Condensation in Vertical Tubes. Since Eqs. (3.33) and (3.38) are valid only for laminar flow of the condensate film in the vertical tube, we now consider the following questions:

1. When does the film cease to be laminar? 2. What correlation is valid when the film becomes

turbulent?

3. What is the effect of vapor shear on the conden-sation mechanisms and correlations?

With reference to the first question, there is no hard and fast answer to the criterion for transition from laminar to turbulent flow. Ripples can appear on the surface at quite low values of Rec (as diagrammed in Fig. 3.8) but these seem to have very little effect on the condensing coefficient. There is a definite break in the heat transfer behavior of films at Rec of 1600 to 2000 in the absence of vapor shear, and this number can be used here as the critical Reynolds number for the falling film.

158

The presence of vapor shear causes an early appearance of turbulence, Rec values as low as 250 to 300 being reported by Carpenter and Colburn (12). As it turns out, from a practical point of view, the question is not a serious one since the magnitude of the calculated coefficient itself indicates what flow regime exists. Condensation heat transfer coefficients under turbulent flow but low vapor shear conditions are correlated by a fundamental analysis due to Colburn (13) based on analogies between heat and momentum transfer in single phase turbulent flows. The Colburn approach required a numerical integration to get the mean coefficient so that a graphical representation of the final result is usually presented. For this purpose, it is convenient to note that the Nusselt solution can be plotted as

( ) cv

c vsgk

h Re3/1

3

2

⎥⎥⎦

⎤

⎢⎢⎣

⎡

− ρρρ

μ

lll

l

with a slope of -1/3 on log-log coordinates and an intercept of 1.47 at Rec = 1. The Colburn solution may also be plotted on these coordinates with

lPr as a parameter, giving the graph shown here as Fig. 3.9. The break at Rec = 2100 shows the point at which the film is presumed to become turbulent; Colburn used this value in his computation, and it is somewhat conservative. Notice also that the Colburn curves have a Prandtl number dependence. The correlation shown in Fig. 3.9 has been reasonably well verified experimentally, but it seems unwise to extrapolate the Prandtl number dependence to values higher than 5; available data do not go very much higher. (High Prandtl numbers usually arise from higher viscosity, which also causes lower Reynolds numbers, so the problem is in some sense self-limiting. However, the physical structure of a high-flow-rate, high viscosity condensate film may be a thick laminar layer close to the cold wall with a turbulent liquid film cascading down the outside. This could result in a higher heat transfer resistance than expected from a condensate layer whose properties are calculated at a mean temperature.)

Turning now to the case of a vapor-shear controlled condensing situation inside a vertical tube, there are several procedures to choose from in the literature. Carpenter and Colburn (12) did the first really comprehensive experimental study of this problem, correlating their average coefficients by

159

2/12/12/1

Pr065.0 vcc F

k

hl

ll

l =ρ

μ (3.41)

where

v

mvvc

fGF

ρ2

2,= (3.42)

2/12

,,,2,

, 3 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ ++=

ovovivivmv

GGGGG (3.43)

and f is the Fanning friction factor, given by

4/1,

078.0

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

v

mviGdf

μ

(3.44)

Gv,i and Gv,o are the vapor phase mass velocities at the inlet and outlet, respectively. If the vapor comes in dry and saturated and is completely condensed, Gv,m = 0.58 Gv,i. There are numerous other correlations for condensation in the presence of high vapor shear, and some of them are more accurate. However, the more accurate ones are also harder to use, and usually the additional accuracy is not required in air-cooled condensers, where the condensation is not the controlling resistance. The Carpenter-Colburn correlation is valid only under the condition that vapor shear controls the liquid film hydrodynamics and hence heat transfer. If vapor shear does not control, this correlation will give an unrealistically low coefficient. Therefore, in deciding which correlation to use,the following rule applies: Calculate condensing coefficients by the gravity-controlled correlation (Eq. 3.36) and by the vapor shear-controlled correlation (Eqs. 3.41 to 3.44) and take the higher value. Larninar Film Condensation Inside Horizontal Tubes. Essentially the same set of assumptions as previously used for vertical tubes may be applied to condensation in a horizontal tube. The only essential difference between the analyses for the two cases is that:

a. for the horizontal tube, the effective component of the gravitational acceleration, g sin θ, changes with position around the tube, and

b. the condensate will form a pool in the bottom

of the tube (Fig. 3. 10) and render that part of the tube surface ineffective.

160

The fraction of surface thus affected depends upon the properties of the condensate, the rate of condensation, the geometry of the tube, and the provision made for removing the condensate. If there were no condensate pool, Nusselt's analysis gives the heat transfer coefficient as Eq. (3.45)

4/13

)()(

725.0⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

=wsati

vc TTd

gkh

l

lll

μλρρρ

(3.45)

Kern (11) recast this equation in a form that introduces the condensate weight flow rate per tube Wt, eliminates the temperature difference, and introduces a penalty for the presence of the pool. His equation is:

3/13 )(761.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

l

lll

μρρρ

t

vc W

gLkh (3.46)

Again, more elegant equations are available, but the presumed additional precision is not essential. Eq. (3.46) is valid only at low vapor shear rates. At high condensing loads, with vapor shear dominating, the correlations should be independent of tube orientation, and this is in fact found to be the case. Therefore, the high-vapor-shear condensing coefficient may be calculated from the Carpenter-Colburn equations (Eqs. 3.41 to 3.44) previously given for this case. The selection of which correlation to apply in a given case is based on the following argument: The flow patterns, the hydrodynamics and consequently, the heat transfer processes are dominated by gravity at low tube loadings and by vapor shear at high loadings. Correlations are available for both limiting cases; each of these correlations predicts low coefficients (relative to the appropriate correlation) when it is applied to situations to which it is in fact not applicable. Therefore, in any case in which there is doubt as to the correct correlation, calculate the condensing coefficient by each correlation (Eq. (3.41 to 3.44) and Eq. (3.46)) and choose the higher value of hc. Filmwise Condensation Inside Inclined Tubes. In the foregoing discussion of condensation inside vertical and horizontal tubes, the argument was made that tube orientation was unimportant at high velocities where vapor shear controlled, and we would expect the same to be true of condensation in inclined tubes with vapor flow towards the lower end of the tube. In the gravity-controlled regime, however, inclination should make a difference. Very little information is available in the open literature on this problem and the best we can do here is suggest a procedure that will be responsive to the major effects and reduce to the correct limiting cases. If we start with a vertical tube θ = 0, as defined in Fig. 3.6), we would expect an increase in θ to reduce the effective component of the gravitational force acting on the draining film by the function g cos θ. Therefore, it is suggested that the condensing heat transfer coefficient for an inclined tube in the gravity-controlled regime be calculated from Fig. 3.9, using as the ordinate

161

3/1

3

2

cos)( ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

− θρρρ

μ

gkh

vc

lll

l

instead of that shown. But if that procedure is carried to nearly horizontal tubes, where θ → 90°, cos θ → 0 and hc → 0, which is physically unrealistic. It hardly seems likely that hc would fall below the value for a horizontal tube; indeed, even a few degrees of downward slope should aid greatly in the drainage and improve the coefficient above the horizontal case. Therefore, it seems conservative to place the lower bound on hc (of an inclined tube) as the value given by Eg. (3.46) for a horizontal tube at the same loading. Additionally, in each case, the value of hc predicted by Eqs. (3.41 to 3.44) should be calculated. If the value so calculated is greater than that obtained in the previous paragraph, the flow may be presumed to be predominantly in the vapor shear-controlled regime and the higher value of hc used for further calculations. 3.2.3. Mean Temperature Difference for In-Tube Condensation MTD for a Pure Saturated Vapor. For the usual conditions of condensing a pure saturated vapor in an air cooled exchanger, the correct value of the Mean Temperature Difference (MTD) is the logarithmic mean temperature difference, using the saturation temperature, Tsat, of the vapor at the nominal condensing pressure as the constant hot side temperature:

⎟⎠⎞

⎜⎝⎛

−==

−−

osat

isattTtT

io

n

ttLMTDMTD

l

(3.47)

Strictly speaking, this equation is only valid if the overall coefficient is constant and if the condensing fluid temperature is isothermal, i.e., if there is no desuperheating or subcooling and the vapor pressure drop is very small compared to the absolute pressure. In fact, these are usually not serious limitations. Since the condensation process is commonly only a relatively small part of the total resistance to heat transfer, the overall coeffi cient varies little about a mean value calculated on the basis of an average condensing coefficient, or for some correlations, a condensing coefficient calculated at the average quality of the condensing stream. The treatment of desuperheating and subcooling is given later. The other condition - negligible pressure drop in the condensing process may be relaxed approximately but satisfactorily by finding the saturation temperatures at the inlet and outlet pressures, Tsat,i and Tsat,o, calculating the LMTD as

⎟⎠⎞

⎜⎝⎛

−−−=

−−

osat

isattTtT

oisatiosat

n

tTtTLMTD

l

)()( ,, (3.48)

and finally calculating the mean temperature difference as

MTD = F(LMTD) (3.49)

162

where F is found from the curves in Chapter 4. This is a reasonably valid procedure only if the temperature approach is not too close and if the pressure drop in the condensing vapor is relatively small. Both of these conditions are usually met in air-cooled condensers. Most air-cooled condensers are designed so that the condensation takes place in a single pass from one header to the other. If this is not done, the liquid and vapor phases will tend to separate and mal-distribute in the turn around header, resulting in some tubes having a surplus of vapor (and, therefore, possibly not giving complete condensation) and other tubes having a surplus of liquid (and giving excessive subcooling.) However, it is usually necessary for the single condensing pass to include several vertical tube rows. Since the air becomes progressively hotter from row to row, the local temperature difference and the condensation rate correspondingly decrease. This means that if the vapor in the bottom row of tubes is just completely condensed (no subcooling), the vapor in the upper rows is in completely condensed, leading to a loss of vapor. In order to avoid this, Mueller (14) has derived a safety factor to be applied to assure that the vapor entering the top rows is completely condensed by the end of the tube (resulting in some subcooling of the condensate in the lower rows). The factor is

2)1()1(Knnn

Knnn

ncondesatioforrequiredlengthtubeTotal

exchangerinlengthtubeTotalE

−+

−+== (3.50)

where

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

airpairCWUAK exp1 (3.51)

and where U is the overall heat transfer coefficient, based on the total outside heat transfer area A (actually, any consistent combination of U and A may be used) and N is the number of rows of tubes in the exchanger. The heat transfer area (and therefore the tube length) required for complete condensation is calculated assuming a uniform condensing loading for each row of tubes, and is then multiplied by E from Eq. (3.50) to find the total area (and the additional tube length) required to assure complete condensation in the top row. Desuperheating of vapor. If the vapor entering a condenser is superheated, the sensible heat content of the vapor must be removed and transferred through the cooling surface before that vapor can be condensed. If the cold surface is above the saturation temperature of the vapor, the heat is removed by a convective sensible heat transfer mechanism, the coefficient for which can be calculated from correlations given in Chapter 2 employing vapor physical properties. However, if the cold surface is below the saturation temperature of the vapor at the existing pressure, vapor will condense directly upon the surface with essential thermodynamic equilibrium existing at the condensate-vapor interface and with the temperature gradient from the superheated state to saturation occurring in the vapor immediately adjacent to the interface. Available information indicates that the heat transfer coefficient for condensation directly from the superheated vapor is within a few percent of that for condensation from the saturated vapor, using in each case the saturation temperature of the vapor as the temperature driving force for heat transfer. Such

163

differences as have been observed are well within the present ability to predict condensing coefficients for a given situation. This fortuitous agreement is useful in the design of desuperheating condensers. The first matter is to establish a test to determine whether or not condensation will occur on a cold surface exposed to superheated vapor. If we assume that heat is transferred from the superheated vapor by sensible heat transfer, and if Tv is the local vapor temperature, Tsat the condensing or saturation temperature, and t the local coolant temperature, the wall temperature on the vapor side, T'w, is given by

s

vsvw h

tTUTT

)('

−−= (3.52)

where hs is the sensible heat transfer coefficient for the vapor stream and Us is the overall coefficient computed using hs, both referenced to the same surface. If T’w > Tsat, no condensation will occur, T’w is equal to the true wall surface temperature Tw and the heat transfer rate is given by (Q/A) = Us (Tv - t). If T'w < Tsat, condensation will occur and the heat transfer rate is given by (Q/A) = Uc (Tsat - t), where Uc is the overall coefficient computed assuming condensation does occur, and the true inside tube surface temperature is given by;

c

satcsatw h

tTUTT

)( −−= (3.53)

There is a further interesting consequence of the above discussion. Let U+ be the combined heat transfer coefficient for the wall and fin resistance, the coolant and any dirt films. U+ is essentially independent of the heat flux and whether or not condensation is occurring inside the tube. Then, the heat flux for the sensible heat transfer desuperheating case is (Q/A) = U+(T'w - t) and that for the condensing case is (Q/A) = U+(Tw - t). Since T'w ≥Tw, we conclude that condensation will occur directly from the superheated vapor, unless a higher heat flux is obtained by the sensible heat transfer mode. A corollary to this is that it is both simpler and more conservative (in the sense of calculating a larger condenser area) to assume that condensation will occur directly from the superheated vapor, using the saturation temperature and a condensing heat transfer coefficient in the rate equation, and of course, including the sensible heat in the heat load. However, in designing a desuperheating condenser where the desuperheating heat load is an appreciable fraction of the total heat load, the designer may wish to avail himself of the possible surface savings afforded by the higher heat flux of sensible desuperheating. This is a fairly complicated matter, since the heat balance and rate equations must be simultaneously balanced by increments in two directions - vertically through the tube bundle and longitudinally along the length of the tubes. This is strictly a computer solution; proprietary programs are available and they run quite rapidly in both the rating and design modes. It must be remembered that the computer solution is only as good as the data base and the skill of the engineer and systems analyst who put the program together, and even the best of the methods have a substantial, range of uncertainty in the final answer. Subcooling the Condensate. It is often desirable to further subcool the condensate beyond the few degrees achieved by the condensing process itself. This can be done by passing the condensate through one or more rows of tubes in the bottom of the air cooler. Again, the exact thermal analysis of this problem is quite complicated and can only be carried out by a computer program. An approximate calculation can be carried out by considering the condensing and subcooling heat transfer processes to be carried out in two separate heat exchangers in series. Thus one can calculate the mixed mean air temperature just after passing over the subcooling tubes, using the

164

methods outlined in Chapter 4 on air cooling. This mixed mean air temperature can then be used as the inlet air temperature to the condensing section, which is designed according to the methods in this chapter. There are several important cautions to be observed. First, the air off of the subcooling sections varies along the length of the tube, and this variation alters the condensing pattern, increasing the local temperature difference (compared t ' o the mixed mean assumption) and local condensation rate at one end and decreasing it at the other. The effect tends to cancel out, though whether the net effect is conservative or non-conservative depends upon the specifics of each problem. Certainly the absolute er-ror of the method increases as the process stream temperatures approach more closely the air temperatures. Another consideration is this: Almost certainly the design and the mechanical layout of the air-cooled exchanger will be dominated by the condensing process. Therefore, the subcoofing achieved will be at the mercy of the conditions set for the condensing process and not in dependently controllable. If close condensate subcooling control is required, it is much better to provide a separate heat exchanger, the operating conditions for which can be adjusted as required.

165

3.3. Condensation of Vapor Outside Low- and Medium-Finned Trufin Tubes

This section is concerned with the application of low- and medium-finned Trufin tubes to cases where condensation (including desuperheating and subcooling) is taking place on the finned surface of the tubes. Typical applications for outside condensing include: Condensing refrigerants in air conditioning, refrigeration and cryogenic processing systems with cooling water. Typical refrigerants have heat transfer coefficients from about 150 to 500 Btu/hrft2°F; the water coefficient will usually be from 1000 to 1500, so the advantages of using Trufin on the condensing side are clear. Condensing overhead streams for distillation columns in water-cooled condensers. An overhead stream from a distillation column may be composed of essentially one component or it may contain many components. In either case, the condensing coefficient is likely to be less than half of the water coefficient. If the condensing temperature range is very long (for a multi-component mixture, or for a vapor containing a non-condensable gas), a large portion of the total heat transfer may be sensible cooling of the remaining gas or vapor. This coefficient is almost always very low compared to the coolant coefficient, or even the condensing coefficient, making the use of Trufin even more attractive. Condensing vapor using a non-aqueous coolant. In this case, the coefficients of the two fluids may be nearly equal and there might seem to be little advantage in using standard Trufin. However, a doubly enhanced tube such as Turbo-Chil has internal spiral ridges as well as conventional external fins. These ridges enhance the tube side coefficient and the combined enhancement can lead to a very substantial reduction in heat exchanger size. One application for which Trufin is not recommended is the condensation of water vapor (steam). Because of its high surface tension, water tends to bridge the gaps between the fins and a thick layer is retained on the tube, severely reducing the effective heat transfer coefficient. For this application, Korodense can be advantageous. 3.3.1. Shell and Tube Heat Exchangers for Condensing Applications Shell and tube exchangers offer a mechanically feasible way of providing a large heat transfer area in a relatively compact volume. The configuration is strong enough to withstand a wide range of operating conditions commonly encountered in the process and power industries, and offers an enormous number and range of design options to meet most requirements. The basic construction of the shell and tube exchanger is described in Chapter 1, and we will not repeat that material here. However, there are several special configurations of shell and tube exchanger that are particularly interesting in condensing applications, and these are described in the following paragraphs (15). Fig. 3.11 shows a typical shell and tube condenser as employed in the process industries. Condensation occurs on the shell-side, the vapor entering through a nozzle at one end and the condensate being removed from the nozzle on the bottom side of the shell at the other. There is also a vent for non-condensable gases at the condensate exit end of the condenser. This configuration is known, by TEMA notation, as an E shell. The drawing shows a fixed tube sheet, two coolant pass arrangement. The baffling is shown in Section AA in the most usual configuration, that is, segmentally cut baffles with a ver-tical cut and the baffles notched on the bottom to allow drainage of the condensate from one compartment to the next and finally to the condensate exit. However, there has been a recent trend to

166

use horizontally-cut baffles so that the two-phase flow must go up and over; this arrangement is believed to minimize the possibility of stratification and liquid segregation on the shell-side, but at the cost of additional pressure drop. An impingement plate is shown at the vapor inlet to protect the tubes im-mediately adjacent to the vapor nozzle from erosion from liquid droplets carried along in the vapor. It is almost universal practice to use an impingement plate at the vapor inlet on a shell-side condenser.

It is often necessary in heat recovery service (e.g., feed effluent exchangers) to carry out condensation in several exchangers in series. Shell-side condensation in stacked horizontal E shells is the usually preferred design in this case (Fig. 3.12). This arrangement minimizes (but does not eliminate) the possibility of phase segregation and the resultant distortion of the multicomponent vapor-liquid equilibrium profile. It is still important to select a baffle spacing and cut to keep vapor velocities fairly high, and the baffle geometry will often be different for each shell. Fig. 3.13 shows a somewhat different configuration for shell-side condensation, referred to in the TEMA notation as a G shell. The flow is split into two streams which proceed more or less symmetrically from the center vapor

inlet to the ends of the tubes, being guided in this direction by the longitudinal baffle. The flow is then turned around in the end baffle sections and brought back through the lower portion of the tube field; the condensate exits from the central hot well of the condenser. Again, the standard practice is to make the baffle cuts vertical so that the flow is from side to side. Fig. 3.14 shows the TEMA J shell configured as a condenser. The baffle cuts are usually vertical. Dividing the flow and halving the flow distance (compared to the corresponding E shell) results

167

in a reduction of 60 to 80 percent in pressure drop, a consideration of especial importance in vacuum service. However, the temperature profiles require careful analysis because they are not identical for the two sections.

J shells can be stacked in series as shown in Fig. 3.15. As drawn, the coolant flow is in parallel to the two shells; this maximizes the effective temperature difference for condensing but can only be used when abundant coolant is available. Fig. 3.16 shows a further modification of shell-side condensing arrangements, commonly referred to as an H shell. This is effectively a double G shell arrangement, commonly, called a double split flow, and is used to reduce the pressure drop. These units are frequently used for vacuum condensing applications. The transverse baffles have vertical cuts and overlap only enough to insure tube support, i.e., the overlap will ordinarily be two rows of tubes at the vertical centerline of the shell. The configuration shown uses a U tube bundle in order to show possi-ble variant design configurations; however, fixed tube sheet/single tube-side pass and other design options are feasible with this particular shell-side geometry. If pressure drop is extremely limited, the usual design choice will be the X shell shown in Fig. 3.17. This arrangement provides that the vapor flow is essentially in cross-flow with the tube bank and is always arranged so that gravity will remove the condensate from the tube surfaces. There must be sufficient clearance between the top of the baffles and the shell to allow the vapor to disperse longitudinally across the entire length of the tubes. Sometimes this is accomplished by putting a large longitudinal vapor nozzle on top of the shell; this arrangement is descriptively referred to as a bathtub nozzle. If adequate space is provided between the tube field and the shell for longitudinal flow of the vapor, each transverse baffle can be a full baffle giving full tube support to the entire bundle. The baffles exist only for tube support, and their spacing is controlled by vibration requirements. The arrangement shown here has a four coolant pass arrangement. The vent for this design would ordinarily be located in the condensate hot well, which would be designed to maintain a vapor/liquid interface below the tube field. The non-condensables would tend to accumulate at the top of the hot well and could be vented from there. Vents at the far end of the

168

shell would also usually be required in order to permit removal of any non-condensable gases that might otherwise accumulate there and not be swept down to the hot well.

Another design to meet a special need is the shell-side reflux or knock-back condenser shown in Fig. 3.18. A typical application is to partially condense a wide condensing range vapor-gas mixture. The first liquid to condense is tarry but will dissolve in the lighter liquid condensed near the top of the bundle. By

169

keeping the gas/vapor velocity low (an essential in all reflux condensers), the lighter liquid drains downwards constantly washing off the tarry material. Fig. 3.19 shows a recent proprietary innovation in shell side condensation in the rod baffle design, licensed by Phillips Petroleum Company (16). In this case, each segmental baffle is replaced by a set of four individual baffles, each one composed of a ring that fits close to but inside the shell, with straight rods extending from one side of the ring baffle to the other. The straight rods are so arranged that they pass between the tube rows with minimum clearance. Every point at which a rod passes next to a tube provides a point of support for that tube in that direction. The rods are arranged so that there are two rows of tubes between each pair of rods in the baffle; the next baffle in line has offset rods passing between the two tube rows that were not individually supported by the previous baffle on that side. The individual baffles are spaced 6 to 8 in. apart, i.e., each tube being supported on all four sides every 24 to 32 in. along the length of the tube. This configuration was first introduced to prevent vibration in an extremely high velocity service but it is now recognized to be at least as valuable for condenser application because of the low pressure drop caused by this particular selection of rod geometry.

Some other shell-side condensing options have considerable application. Fig. 3.20 shows variable baffle spacing on the condensing side. The intention here is to maintain high vapor velocity and therefore shear enhancement of the condensing heat transfer coefficient. Clearly, a fair amount of knowledge on the effect of vapor shear and the ability to predict two-phase flow characteristics on the shell-side is required if this option is to be effectively exercised. This arrangement can be used only when there is a reasonable amount of pressure drop allowed on the condensing side to permit maintaining vapor velocities at a high enough value to get shear enhancement. Removal of non-condensable gases in this geometry is quite positive as long as a vent is located on the exit end of the shell, because this is where the natural flow of the vapors will carry the gas. A computer based rating method is essential to the effective use of this option, and detailed discussion of such applications is beyond the scope of this Manual.

170

Fig. 3.21 shows a vapor belt distributor, which is intended to uniformly distribute the vapor around the bundle and therefore reduce the excessively large pressure drops and vibration and erosion problems encountered at the vapor inlet end. The shell is extended underneath the vapor nozzle thereby acting as an impingement baffle. Usually, the extended shell within the vapor belt distributor has slots cut around the periphery in order to introduce the vapor into the tube field from all angles. In this case, the open end of the shell shown in the diagram is not needed. The vapor belt distributor is a relatively high-cost construction option but is particularly useful where pressure drop may be a critical consideration. All vapors introduced into a condenser will contain some amount of non-condensable gas, at least during certain portions of the operational period, whether this composition is shown in the process specifications or not. This fact gives rise to the basic rules for venting condensers: 1. All condensers must be vented. 2. Condensers must be designed to move the non-condensables to a particular point in the condenser by directing the vapor flow path positively through the use of baffles and other mechanical arrangements. 3. The vent must be located where the non-condensables are finally concentrated. 4. A vent condenser may be necessary, sometimes even using a refrigerated coolant, in order to recover as much of the remaining condensable vapor from the non-condensable gas as possible. It is fair to say that probably half of all operational condenser problems arise from a failure to recognize the basic principles of venting non-condensable gases. In condensation applications it is frequently necessary to subcool the condensate produced in the condenser. The required amount of subcooling may vary from a few degrees to prevent cavitation in a pump to many degrees required for cooling the condensate to a safe temperature for long-term atmospheric storage. The surest way to carry out the subcooling, especially if large amounts of subcooling are required, is to provide a separate subcooler as shown in Fig. 3.22. In this case, the subcooler is designed as a single-phase heat exchanger with far more precise design methods than are available for integral subcooling calculations. Additionally, the coolant flow rate may be separately controlled to the subcooler giving better control of effluent liquid temperatures. The obvious disadvantage is the generally higher first cost of the separate condenser and subcooler, compared to the cost of the con denser with integral subcooling. Integral subcooling can be accomplished in a horizontal shell-side condenser by allowing a stratified pool to accumulate in the shell and be sensibly cooled by the inlet coolant, as shown in Fig. 3.23. Some sort of level control (preferably adjustable) is required, and the design calculations are both fairly complex and not very accurate.

171

3.3.2. The Basic Design Equations The basic heat exchanger equations applicable to shell and tube exchangers were developed in Chapter 1, and the reader is referred to that material for the development. Here, we will cite only those that are immediately useful for design in shell and tube heat exchangers with condensing heat transfer on the shell side. Specifically, for the moment, we will limit ourselves to the case when the overall heat transfer coefficient is constant and the other assumptions of the mean temperature difference concept apply. (The important cases of condensation of multi-component vapor or a vapor with a non-condensable gas do not satisfy this requirement and are discussed later.) The basic design equation becomes:

QT = U*A*(LMTD) (3.54) where QT is the total heat load to be transferred, U* is the overall heat transfer coefficient referred to the area A*. A* is any convenient heat transfer area, and LMTD is the logarithmic mean temperature difference. U* is most commonly referred to Ao, the total outside tube heat transfer area, including fins, in which case it is written as Uo and is related to the individual film coefficients, wall resistance, etc. by

⎟⎠⎞⎜

⎝⎛+⎟

⎠⎞⎜

⎝⎛+⎟

⎠⎞⎜

⎝⎛+++

=Δ

i

o

ii

o

m

O

w

w

o AA

hAA

fiAA

kz

finfoh

oRRR

U11

1 (3.55)

where ho and hi are the outside and inside film heat transfer coefficients, respectively, Rfo and Rfi are the outside and inside fouling resistances, Δxw and kw are the wall thickness (in the finned section) and wall

172

thermal conductivity, and Rfin is the resistance to heat transfer due to the presence of the fin. Since all of the low and medium Trufin tubes manufactured by Wolverine are integral (i.e., tube and fins are all one piece of metal), there is no need to include a contact resistance term. Suitable correlations for ho will be developed in this section. Correlations for hi have been developed in Chapter 2. Equations and charts for fin resistance and Wall resistance calculations are found in Chapter 1. 3.3.3. Mean Temperature Difference The Mean Temperature Difference formulation is detailed in Chapter I but must be used very carefully in condensation applications. For the special but important case of condensation of a pure or nearly pure vapor with negligible pressure drop on the condensing side (i.e., isothermal condensation), the Logarithmic Mean Temperature Difference is not only valid but considerably simplified. If the condensing temperature is Tsat and the inlet and outlet coolant temperature are t1 and t2, then the LMTD (and the MTD) is given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

2

1

12

tTtT

n

ttLMTD

sat

satl

(3.56)

(Strictly speaking, since most condensing heat transfer correlations are functions of the local heat flux or the local temperature difference, the LMTD is not exactly valid. However, for practical cases, the effect is small and the use of the LMTD is generally conservative by engineering practice). If multi-component mixtures or vapors with non-condensable gases are being condensed, or if there are appreciable desuperheating or subcooling effects, the MTD/LMTD formulation is not valid. Special techniques need to be used for these cases; these are described later. 3.3.4. Condensation of a Superheated Vapor It is often necessary to condense a vapor whose inlet temperature is above the saturation temperature at the pressure existing in the condenser. Such a case can arise with exhaust vapor from a non-condensing turbine or a vapor throttled through a valve. There is some confusion in the literature about the proper way to design for condensing a superheated vapor, but the following argument [shortened from Ref. (17)] is generally accepted as correct. If the vapor entering a condenser is superheated, the sensible heat content of the vapor must be removed and transferred through the cooling surface before that vapor can be condensed. If the cold surface is above the saturation temperature of the vapor, the heat is removed by a convective sensible heat transfer mechanism, the coefficient for which can be calculated from a correlation applicable to the geometry involved and employing vapor physical properties. However, if the cold surface is below the saturation temperature of the vapor at the existing pressure, vapor will condense directly upon the surface with essential thermodynamic equilibrium existing at the condensate-vapor interface and with the temperature gradient from the superheated state to saturation occurring in the vapor immediately adjacent to the interface. The information available [Refs. (18,19)] indicates that the heat transfer coefficient for condensation directly from the superheated vapor is within a few percent of that for condensation from the saturated

173

vapor, using the saturation temperature of the vapor to calculate the temperature difference for heat transfer. To determine whether or not condensation will occur on a cold surface exposed to superheated vapor, first assume that heat is transferred from the superheated vapor by sensible heat transfer. If Tv is the local vapor temperature, Tsat the condensing or saturation temperature, and t the local coolant temperature, then the surface temperature on the vapor side, T's , is given by

s

vsVs h

tTUTT

)('

−−= (3.57)

where hs is the sensible heat transfer coefficient for the vapor stream and Us is the overall coefficient computed using hs. If T's >Tsat, no condensation will occur, T's is equal to the true surface temperature T, and the heat transfer rate is given by (Q/A) = Us, (Tv - t). If T’s < Tsat condensation will occur and the heat transfer rate is given by (Q/A) = Uc (Tsat - t), where Uc is the overall coefficient computed assuming condensation does occur, and the true surface temperature is given by

c

satcsats h

tTUTT

)( −−= (3.58)

In this case, Uc = Uo as given by Eq. (3.55), with ho in that equation being the same as h. in the above discussion. Correlations for the hc are given later. There is a further consequence. Define U' as the combined heat transfer coefficient for the wall resistance, the coolant, and any dirt film. U' is essentially independent of the heat flux and whether or not condensation is occurring on the outer surface. Then the heat flux for the sensible heat transfer desuperheating case is (Q/A)s = U' (T's - t) and that for the condensing case is (Q/A)c = U' (Ts - t). Since T's >Tsat ≥ Ts, (Q/A)s > (Q/A)c. Therefore, condensation will occur directly from the superheated vapor, unless a higher heat flux is obtained by the sensible heat transfer mode. A corollary to this is that it is both simpler and more conservative (in the sense of calculating a larger condenser area) to assume that condensation will occur directly from the superheated vapor, using the saturation temperature and a condensing heat transfer coefficient in the rate equation, and of course including the sensible heat in the heat load. 3.3.5. Condensation with Integral Subcooling On the Shellside Condensate subcooling is frequently required to provide the NPSH for a pump or to cool a product for a surge tank or for storage. Usually this subcooling is best done in a separate sensibly cooling heat exchanger, specifically designed for this purpose. It should also be remembered that filmwise condensation results in a liquid film that is subcooled on the order of 1/3 to 1/2 of (Tsat – Ts). If reheating can be avoided and the pump judiciously located, this subcooling may be sufficient for the purpose. Occasionally, however, it may be desirable, especially if a moderate degree of subcooling is required, to provide a region in a horizontal condenser shell where the condensate may exchange heat with the cold coolant entering. In this case, a rational approach to calculating the subcooling area required is necessary. Referring to Fig. 3.23, the surface of the liquid pool is shown as essentially horizontal, whereas in reality pressure differences may cause considerable hydrostatic gradient. In this case, the value of F calculated later must be understood to be the effective fraction of tubes flooded.

174

In the absence of better knowledge, the pool can be conservatively assumed to be isothermal at Tsc the desired subcooling temperature. The condenser therefore may be envisioned as composed of two regions, each isothermal on the shell side: 1) the condensing region at Tsat and 2) the subcooling region at Tsc. The heat load and temperature profile of the coolant as a function of length are then given by

Q(x) = wcp (t-ti) (3.59)

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

tTtT

n

ttaxU

sc

isc

isc

l

(3.60)

in the subcooling region, where t is the temperature of the coolant at any distance x down the tube. Solving for t gives

p

sceiscsc wc

axUtTTt −−−= )( (3.61)

At the end of the tube, x = L, and t is the outlet temperature for that pass, tbo in the case diagrammed in Fig. 3.23. An analogous equation holds for the condensing region, resulting in a tube outlet temperature of tao from the tubes in the first pass that are in condensing service. At the turnaround, the two streams from the first pass are presumed to mix completely to give tdo by heat balance, which becomes the inlet temperature for the next pass. In the use of Eq. (3.61) it is immaterial whether aL is taken to be the area per tube and w the coolant flow rate per tube or whether aL is the total area of all tubes in a pass and w the total coolant flow rate. In the following computational scheme, the latter viewpoint is adopted together with the fact that in a two-pass configuration (with equal tubes/pass) aL = AT/2, where AT is the total tube surface area in the bundle. The algorithm for this case proceeds by estimating the total area required (step 3) and determining whether the final outlet temperature of the coolant is equal to that required by the heat balance. The total area is then adjusted until the temperatures do match. Since step 3 provides only a very rough estimate, it is possible that the AT thus predicted will throw the first sequence of calculations into the wrong branch, especially if F (step 5) is near 1. This is readily detected by inspection during the calculations and a more realistic value of AT can be estimated at steps 9 or 15. The computational sequence is: 1. Calculate the heat loads.

Qc = Wλ (3.62)

Qsc = WCp(Tsat – Tsc) (3.63) 2. Calculate the coolant flow rate.

)( iop

sccttc

QQw

−+

= (3.64)

3. Estimate the total area.

175

)(

2

iscsc

sc

osat

isat

ioc

cT tTU

Q

tTtT

n

ttU

QA

−+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−

≈

l

(3.65)

4. Calculate the outlet temperature from the flooded tubes in the first pass.

p

Tsceiscscbo wc

AUtTTt

2)( −−−= (3.66)

5. Calculate the fraction of tubes in the first pass to be flooded, F.

)( ibop

scttwc

QF

−= (3.67)

If F > 1, the entire first pass is to be flooded; go to step 10 directly.

6. Calculate the outlet temperature from the unflooded tubes in the first pass.

p

Tcwc

AUeisatsatao tTTt 2)( −−−= (3.68)

7. Calculate the mixed mean outlet temperature from all the first pass tubes.

tdo = (1 – F)tao + Ftbo (3.69)

8. Calculate the outlet temperature from the second pass (and hence from the condenser.)

p

Tcwc

AUedosatsato tTTt 2

* )( −−−= (3.70)

9. Does t*o =to specified?

Yes: Total condenser – subcooler area required is AT and F of the first pass tubes are to be flooded. No: Assume new AT and go to step 4.

10. Calculate the exit temperature from the first-pass tubes, all of which are flooded.

p

Tscwc

AUeiscsceo tTTt 2)( −−−= (3.71)

11. Calculate the exit temperature from the second-pass tubes that are flooded.

p

Tscwc

AUeeoscscfo tTTt 2)( −−−= (3.72)

176

12. Calculate the exit temperature from the second-pass tubes in condensing service.

p

Tcwc

AUeeosatsatgo tTTt 2)( −−−= (3.73)

13. Calculate the fraction of second pass tubes that are in condensing service.

)('

eogop

cttwc

QF

−= (3.74)

14. Calculate the mixed-mean temperature from the second-pass tubes.

t* = F’tgo + (1 – F’)tfo (3.75) 15. Does t*o = to specified?

Yes: Total condenser-subcooler area required is AT and all but F' of the second pass tubes are to be flooded, i.e. (1-F'/2) of all the tubes in both passes are to be flooded.

No: Assume new AT and go to step 10.

If U tubes are used in the condenser-subcooler, no mixing of the tube side fluid occurs at the end of the first pass. Assume that the tubes are all the same length and have the same effective heat transfer area; this is not strictly true, but is a convenient assumption and one well within the normal bounds of error for both this case and for other U-tube applications. The first six steps in the computational scheme are identical with the previous analysis. Then the scheme proceeds as follows: 7. Calculate the outlet temperature from the second pass tubes that were flooded in the first pass.

p

Tcwc

AUebosatsatjo tTTt 2)( −−−= (3.76)

8. Calculate the outlet temperature from the second pass tubes that were in condensing service in the first pass.

p

Tcwc

AUeaosatsatko tTTt 2)( −−−= (3.77)

9. Calculate the mixed mean outlet temperature from all the second pass- tubes (and hence from the condenser).

t*o = Ftjo + (1 – F)tko (3.78) 9A. Does t*o = to specified?

Yes: Total condenser-subcooler area is AT and F of the first pass tubes are to be flooded.

177

No: Assume new AT and go to step 4.

Steps 10- 15 for this case (corresponding to all of the first pass tubes being flooded) are identical to those for the previous case. 3.3.6. Filmwise Condensation on Plain and Trufin Tubes The development of the equations for condensing outside of tubes is based on the same Nusselt model and assumptions that were discussed in detail in the analysis for in-tube condensing. These led to the development of Eq. (3.45) for condensing inside a horizontal tube. By replacing di in the equation with do the following equation is obtained:

4/13

)()(

725.0⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

=wsato

vc TTd

gkh

l

lll

μλρρρ

(3.79)

where hc is the average condensing heat transfer coefficient on the outside of the tube. Further, and lk

lμ are respectively the thermal conductivity and viscosity of the condensate, lρ and vρ , the densities of condensate and vapor respectively, λ the latent heat of condensation, g the gravitational acceleration, do the outside diameter of the tube, Tsat the saturation temperature of the vapor and Tw the wall temperature. The equation may also be written as:

3/13 )(951.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

WgLk

h vc

l

lll

μρρρ

(3.80)

where L is the length of the tube and W is the mass of vapor condensed on the tube per unit time. Other forms of the equation are possible. These equations are found to predict actual heat transfer coefficients on horizontal single plain round tubes quite closely, on the average about 15 percent lower than the experimental values. The difference is usually attributed to rippling of the film and early turbulence and drainage instabilities on the bottom side of the tube. The equation can also be used with some modification to predict condensing coefficients on Trufin tubes. Condensation on horizontal low-finned tubes was studied experimentally by Beatty and Katz (20) who found that the data could be correlated by a modified form of Eq. (3 79):

4/14/13 1)()(

689.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

=eqwsat

vc dTT

gkh

l

lll

μλρρρ

(3.81)

where

178

4/14/1

_

4/1113.11⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛Φ=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

req

root

eq

fin

eq dAA

LA

A

d (3.82)

where Φ is the fin efficiency, Afin is the total fin area, Aeq is the effective outside area of a finned tube, Aroot is the plain tube outside heat transfer area (based on root diameter), and dr is the root diameter. L is defined by:

o

fin

da

L = (3.83)

where afin is the area of one side of one fin,

)(4

22rofin dda −=

π (3.84)

The fin efficiency can be calculated from:

r

oddm

31

12

+

=Φ (3.85)

where

YkRHm

wfoho⎟⎠⎞⎜

⎝⎛ +

=1

2 (3.86)

also,

LNddA frofin )(2

22 −=π (3.87)

where Nf is the number of fins per unit length and L is the finned length of the tube,

⎟⎠⎞

⎜⎝⎛

++Φ=

YssLdAA rfineq π (3.88)

and

⎟⎠⎞

⎜⎝⎛

+=

YssLdA rroot π (3.89)

The above set of equations is quite accurate for calculating the heat transfer coefficient on Trufin. However, it is also rather awkward to use, and it has been found as a practical matter that a much simpler and nearly as good a result can be obtained by the use of Eq. (3.79) using dr as the diameter and applying the value of hc thus found to the entire outside finned tube surface Ao.

179

⎥⎦

⎤⎢⎣

⎡−+⎟

⎠⎞

⎜⎝⎛

+= froro Ndd

YssdLA )(

21 22π (3.90)

Alternatively, Eq. (3.80) can be used to calculate ho. Use of either equation is in principle a trial and error calculation, and the actual procedure is later illustrated by an example problem. In the application of all of these equations, it must be remembered that Trufin tubes should not be used for condensing steam. Water has a high surface tension and the condensate film will bridge the fins. The thick layer of liquid thus held in place on the tube acts as an insulator; the resulting heat transfer rate per unit length of tube is substantially less than for a plain tube. It is not clear at what value of the surface tension this effect becomes significant. Condensates with surface tensions as high as 25-30 dyne/cm have been condensed on finned tubes with no reported difficulty; this includes the usual design range for hydrocarbons, alcohols, refrigerants, etc., with surface tensions from 10 to 25 dyne/cm. However, there is a severe penalty for the condensation of water at atmospheric pressure, at which the surface tension is about 60 dyne/cm. No data or reported experience appears to exist in between. 3.3.7. Filmwise Condensation on Tube Banks 1. Effect of number of rows of tubes. In any practical condenser design the condensate formed on tubes near the top of the bundle will fall on tubes lower in the bank and presumably modifying the heat transfer coefficient. Nusselt (7) assumed undisturbed laminar flow on each successive tube and found theoretically that the average condensing heat transfer coefficient for a bank of horizontal plain tubes is given by:

hc,N = hc,1N-1/4 (3.91) where hc,1 is the heat transfer coefficient for one row calculated by the previous equations and N is the number of tubes in one vertical row. For a reasonable size process condenser N may be 20 or 30, and this equation predicts a very severe penalty compared to a single tube. Kern (21) recommended an exponent of (-1/6), again for plain tubes. However, Short and Brown (22) experimentally found no net penalty against the single tube coefficient in a single row 20 tubes high. The latter result is generally borne out in process experience, and current design practice is to assume that the average coefficient for the entire tube bank is the same as for a single tube. The explanation generally advanced is that rippling, splashing, and turbulence induced by liquid falling from one tube to the next overcomes the possible disadvantages of an increasing liquid loading on each tube. 2. Effect of vapor shear. It is well known that vapor flowing at a high velocity across the tubes can increase the heat transfer coefficient significantly above the theoretical gravity-driven flow model of Nusselt. However, very few data actually exist to estimate the significance of the effect and most of these data are proprietary. One

180

widely-cited graph, Fig. 3.24, indicates that the heat transfer coefficient becomes greater than the Nusselt value at vapor Reynolds numbers above about 30,000, increasing to about 10 times as much at Rev = 100,000. However, all data are for plain tube banks and often at vapor velocities so high that excessive pressure drops would be required to achieve significant improvement. Until more data are available, it appears prudent to calculate condensing coefficients as if the flows were purely gravity-driven. 3.3.8. Pressure Drop during Shell Side Condensation Very few data have been published for pressure drop during condensation on the shell-side of the shell and tube heat exchanger. Diehl and Unruh (23, 24) presented a correlation for two-phase adiabatic (i.e., no phase change) pressure drop across several tube banks in both vertical downwards and horizontal cross flow for an ideal tube bank. Making certain assumptions, Brooks (25) worked from the Diehl-Unruh correlations to obtain a correction fac-tor converting the calculated shell-side pressure drop for the all-vapor flow to the pressure drop for a condenser with a saturated vapor inlet and an exit vapor quality xo. This figure is given here as Fig. 3.25. In order to use Fig. 3.25, one must first calculate the shell side pressure drop for the all-vapor flow, using the method given in Chapter 2. Since that method is quite lengthy to present, it will not be repeated here. Then the pressure drop for the all-vapor flow is multiplied by the correction factor Φ2

gtt, from Fig. 3.25 for the appropriate exit quality from the condenser. Several assumptions have been made in obtaining Fig. 3.25, among the more important being: 1. The shell-side vapor flow is always turbulent. 2. All tube bank layouts have the same correction factor. (Diehl and Unruh found that the 45° banks give

somewhat higher pressure drops than the 60° and 90° banks in the liquid-rich end, where the pressure gradient is less anyway. The present curve is based on the mean of the results.)

3. The effect of two-phase flow on pressure drop in the windows is similar to that in crossflow. 4. The rate of condensation is constant through the length of the bundle. A partial condenser that has a

higher condensation rate near the entrance should give a lower pressure drop than estimated by Fig. 3.25 and vice versa.

It should be recognized that the calculation of shell-side pressure drop is at best a rough estimation and that too little testing has been done against actual condensers to allow for a more precise method.

181

3.4. Examples of Design Problems for Low- and Medium-Finned Trufin in Shell and Tube Condensers

3.4.1. Condenser Design for a Pure Component: Example Problem Statement of Problem Design a condenser to condense 2,630,000 lb/hr of propane at 148°F and 200 psia (saturation temperature is 105.2°F) using water at 86°F. Maximum allowable pressure drop is 10 psi for the propane and 25 psi for the water. Minimum water velocity is 8 ft/sec. Fouling resistances are 0.001 hrft2°F/Btu for the water and 0.0003 for the propane, each based upon the respective heat transfer areas. An E or J shell design is to be used, with maximum allowable baffle spacing and cut. Wolverine S/T Trufin tubes, 3/4 in. by 16 BWG, 19 fins/in., of 70/30 CuNi are to be used. Maximum allowable tube length is 80 feet. Fixed tube sheet construction is satisfactory. Some Comments Upon the Design This problem is similar to an actual design performed for the propane condensers for a large LNG plant using brackish water as a coolant. The high water velocity is intended to minimize fouling, and 70/30 CuNi is selected for its resistance to corrosion. The original design was for a rod baffle exchanger, but a conventionally baffled shell is chosen here solely to illustrate the shell side pressure drop calculation. Preliminary Estimate of Size 1. Heat duty The heat duty is composed of two parts: the sensible heat of desuperheating the vapor from 148°F to 105.2°F and the latent heat of condensation at 105.2°F. Desuperheat:

QDSH = WCp(TSH – TSAT) (3.92)

QDSH = (2,630,000 lb/hr)(0.39 Btu/lb°F)(148 – 105.2)°F = 4.39x107 Btu/hr

182

Laten heat:

Qc = (2,630,000 lb/hr)(138.1 Btu/lb) = 3.63x108 Btu/hr (3.62) Total duty = 4.07x108 Btu/hr It will be assumed that the total duty is transferred by condensation at a constant temperature of 105.2°F 2. Mean temperature difference A design outlet water temperature must be chosen. Try 95°F for a first set of calculations. Then

Fn

LMTD °=⎟⎠⎞

⎜⎝⎛

−−

−= 2.14

952.105862.105

8695

l

(3.56)

Since the condensing side is isothermal, F = 1.00. We may also calculate the water requirement at this time:

hrlbxFlbBtuF

hrBtuxw OH /1052.4

)/00.1)(0.9(/1007.4 7

8

2=

°°= (3.64)

3. Overall heat transfer coefficient Estimate the various coefficients and resistances, based on their respective heat transfer areas: Condensing: ho = 500 Btu/hr ft2°F, based on total outside heat transfer area. Fouling, outside: Rfo = 3.0 x 10-4 hr ft2°F/Btu, same basis Fin resistance: Rfin = 7.1 x 10-4 hr ft2°F/Btu, same basis 0.065 in.

Wall resistance: )/17)(/12(

.065.0

FfthrBtuftin

inRw °

=

= 0.00032 hr ft2°F/Btu, based on the mean wall diameter,

Water: hi = 1600 Btu/hr ft2°F, based on the inside tube area from Fig. 2.19 Fouling, inside: Rfi = 0.001 hr ft2°F/Btu, same basis Combining these, with the appropriate corrections to put the overall coefficient on the basis of the total outside heat transfer area gives:

183

FfthrBtu

x

Uo °=

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ ++

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

+++

=

−

2

4

/4.96

130.0503.0001.0

16001

12573.0

503.000032.0101.70003.0500

1

1

π

(3.55) based on total outside heat transfer area. 4. Calculation of area

252

81097.2

)2.14)(/4.96(

/1007.4ftx

FFfthrBtu

hrBtuxAo =

°°= (3.54)

5. Estimation of shell dimensions We may use the methods described in Chapter 2. Specifically, we may use Fig. 2.26, but first we must compute the effective area A’o with which to enter the chart:

A’o = AoF1F2F3F4 where Ao is the actual area required in the heat exchanger, 2.97x105 ft2 for the present case

F1 is a correction factor for the unit cell array. Since there is no apparent reason why a 3/4 in. OD tube on 15/16 in. triangular layout will not serve, F1 = 1.00.

F2 is a correction factor for the number of tube passes. If possible, we will use one pass, so F2 = 1.00

for present. F3 is a correction factor for the shell construction. Fixed tube sheet construction is satisfactory, so F3

= 1.00. F4 is the correction factor for the specific fin geometry and density. For 3/4 in. O.D. 19 fins/in.

construction, F4 = 1.00. so

Ao' = 2.97 X 105 ft2 (1.00)(1.00)(1.00)(1.00) = 2.97 x 105 ft2

Referring to Figure 2.26, we see that a combination of diameter and length (on the figure) that will meet this requirement is a 120 in. I.D. shell, with tubes about 41 feet long. Referring to Table 2.6, we see that the tube count for a one pass exchanger is about 14,500. This gives a total area of

Ao = (14,500 tubes)(0.503 ft2/ft)(41 ft) = 299,034 ft2, compared to the 297,000 ft2 estimated. This would allow a 40.33 ft long tube.

Check the water side velocity:

184

( )( )( )( )( ) sec/3.10

.195.0sec/3600/1.62500,14

/144/1052.423

227

2ft

inhrftlb

ftinhrlbxV OH ==

This velocity is above the stated minimum; check the pressure drop:

( )( )( )( )450,54

/79.1

sec/3600sec/3.10/1.62Re

312508.0

==hrftlb

hrftftlbfti (2.19)

fi = 0.0046 from Fig. 2.20

( )( )( ) ( )( )

psift

ftftftlbp

ftin

lbftlb

f

f

m

45.121442.32

33.40sec/4.10/1.620046.02

2

2

2sec12508.0

23

=

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ (2.26)

(neglecting the Sieder-Tate term) The entrance pressure loss is:

( )( )psi

ftftlbp

ftin

lbftlb

ent

f

m

1.21442.322

sec/3.10/1.623

2

2

2sec

23

=

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ (2.25)

so the calculated tube-side pressure drop is within the design limit. Proceed with this design. Shells of this diameter fall outside standard TEMA specifications on baffle spacing. It should be mechanically conservative to design to the maximum unsupported span of 52 in. (i.e., baffle spacing of 26 in. for a fully tubed bundle) recommended in Table R-4.52 of Ref. (1). If this gives excessive velocities or pressure drops, we may then switch the design to a J or an X shell, or consider a rod baffle design. Heat Transfer Calculation Having a general idea of the design, it is now necessary to check the specific values of the heat transfer coefficients. 1. The condensing coefficient First, calculate the coefficient by Eq. (3.80). W is the mass of vapor condensed per tube in unit time:

( ) tubehrlbtubes

hrlbW //3.181

500,14

/000,630,2==

Then

185

( )

( )FfthrBtu

ftxh

hrlb

hrftlb

lbhr

lbft

ftlb

ftlb

FfthrBtu

cf

m

°=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=°

2

3/18

3

/5113.181194.0

33.401071.44.273.29074.0951.0

233

We may check this against the Beatty-Katz prediction, Eq. 3.81 to 3.89). The various quantities are:

( ) 27.633.4012638.0 ftftftAroot =⎟

⎠⎞

⎜⎝⎛= π

( ) ( ) 2222

6.1533.401219144

638.0750.02

ftftft

finsftA fin =⎥⎦

⎤⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=π

( ) ( )11.1

170003.0

212056.0

12011.0

5111

2=

⎟⎠⎞

⎜⎝⎛⎥⎦

⎤⎢⎣

⎡+

⎟⎠⎞

⎜⎝⎛=

°°

ftftm

FfthrBtu

BtuFfthr

In principle, this quantity must be recalculated using the new value of hc until the solution converges.

692.01

1

.638.0

.750.03

)11.1( 2 =+

=Φ

inin

( ) 22 0.14.056.0.0265.0

33.4012638.0)6.15(692.0 ft

inin

ftftftAeq =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+= π

24222

105.8144

638.0750.04

ftxfta fin−=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −=π

( ) ftft

ftxL 0136.0

105.8

12750.0

24==

−

( ) 4/14/1

2

24/1

2

24/1

93.3638.0

12

0.14

7.60136.01

0.14

6.15692.03.11 −=⎟⎠

⎞⎜⎝

⎛+⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛ftft

ft

ft

ft

ftdeq

Estimate wall temperature to be 100°F

186

( )( )

)93.3(1002.105194.0

1.1381017.44.273.29074.0689.0 4/1

4/18

3

2332−°

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

°−⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

= ftF

xh

hrftlb

lbBtu

lbhr

lbft

ftlb

ftlb

FfthrBtu

cf

m

FfthrBtu °= 2/1000 This value should be converged as indicated above and will be somewhat smaller than indicated. Even so, it will be substantially higher than the 511 Btu/hr ft2°F obtained from the unmodified Nusselt equation. However, the effect on the overall coefficient will be quite small (about 5 per cent increase) because the condensing process is only a small part (about 15%) of the total resistance. In view of the other uncertainties in the calculations (and realizing that the water side fouling resistance is both the most uncertain term and the largest single resistance), this difference is probably not enough to worry about.

Now compute the overall heat transfer coefficient based upon the revised values:

Condensing: ho = 510 Btu/hr ft2°F, based on total outside heat transfer area,

Fouling, outside: Rfo = 0.0003 hr ft2 °F/Btu, same basis,

Fin resistance: Rfin = 7.1 x 10-4 hr ft2 °F/Btu, same basis,

Wall resistance: Rw = 0.00032 hr ft2 °F/Btu, based on mean wall diameter,

Water: hi = 1990 Btu/hr ft2 °F, based on inside area from Fig. 2.19

Fouling, inside: Rfi = 0.001 hr ft2 °F/Btu, same basis Then,

( ) ( )( )FfthrBtu

x

U °=

++⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+++

=−

2

130.0503.0

19901

12573.0

4510

1

0 /4.101

001.0503.000032.0101.70003.0

1

π

And the corresponding area is

22

8700,282

)2.14)(/4.101(/1007.4 ft

FFfthrBtuhrBtuxAo =

°°=

ftftfttubes

ftL 8.38

)/503.0)(500,14(

000,2822

2==

Say 39 ft. for further calculations.

187

Shell-Side Pressure Drop Calculation So far we have established that the heat transfer characteristics and the tube-side pressure drop and velocity are satisfactory. We still need to calculate the shell-side pressure drop. The basic procedure is to calculate the shell-side pressure drop as if the vapor phase flowed un condensed for the entire length (using the Delaware method as given in Chapter 2 and then to correct it by a two-phase multiplier as shown in Fig. 3.25. 1. Basic geometrical data

Tube outside diameter: do = 0.750 in.

Tube root diameter: dr = 0.638 in.

Fin spacing: s = 0.0265 in.

Fin thickness: Y = 0.011 in.

Tube layout: Equilateral triangular with 1.5/16 in. = 0.9375 in. pitch

Shell inside diameter: Di = 120 in.

Shell outer tube limit: Dotl = 119 in.

Effective tube length: L = 39 ft.

Baffle cut: = 58 in. cl

Baffle spacing: = 26 in. sl

Number of sealing strips per side: Nss = 0

2. Shell-side geometrical parameters

a. Total number of tubes in the exchanger: Nt = 14,5000 b. Tube pitch parallel to flow: pp = 0.814 in.

Tube pitch normal to flow: pn = 0.469 in.

c. Number of tube rows crossed in one crossflow section:

( ) ( )[ ]5

.814.0

21.120 .120.58

=−

=in

inN in

in

c (2.38)

d. Fraction of total tubes in crossflow:

48.0.120.58==

inin

Di

cl

188

From Fig. 2.28

Fc = 0.05

e. Number of effective crossflow rows in each window:

57.814.0.)58(8.0==

inin

Ncw (2.40)

f. Number of baffles:

( ) 18139

1226

=−=ft

Nb (2.41)

An even number also means that the nozzles are on opposite sides of the shell as required in this case.

g. Crossflow area at centerline:

( )

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−−

+−

×=

.011.0.0265.0.0265.0

.056.02.750.0.9375.0.9375.0

.750.0.119.119.120

.26

ininin

inininin

inininin

inSm

2.900 inSm =

h. Fraction of crossflow area available for bypass flow:

029.0).900(

.)26.)(119.120(2

=−

=in

inininFsbp (2.44)

i. Tube-to-baffle leakage area for one baffle:

Stb = 0.0184(14,500)(1 + 0.05) = 280 in.2 (2.46)

j. Shell-to-baffle leakage area for one baffle: Assume diametral shell-to-baffle clearance = 0.700 in.

21 .4.67.120.)58(2

1cos2

.)700.0.)(120(in

inininin

Ssb =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= −π

k. Area for flow through window:

189

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−= −−

.120.58

21cossin.120.58

21.120.58

21cos4

.)120( 112

inin

inin

ininin

Swg (2.50)

= 5,415 in.2

( ) ( ) 22 .043,3.750.005.018500,14 ininSwt =−= π (2.51)

Sw = 5,415 in.2 - 3,043 in.2 = 2,372 in.2 (2.49)

3. Shell-side pressure drop calculation

a. Calculate shell-side Reynolds number:

( )( )( )

6

2144900

120638.0

1007.1021.0

000,630,2Re ×=

⎟⎠⎞

⎜⎝⎛

=ft

ft

hrftlb

hrlb

s (2.54)

It is necessary to guess a friction factor for this case as being about 0.2. (See Fig. 2.17)

b. Pressure drop for an ideal crossflow section:

( )( ) ( )

( )psiaftlb

ftp f

lbhrftlb

ftlb

ib

f

m

2.3/4591017.485.12

5000,630,22.04 2

21449008

2

,

23

==

⎟⎟⎠

⎞⎜⎜⎝

⎛×⎟

⎠⎞

⎜⎝⎛

=Δ (2.57)

The viscosity gradient correction is ignored in condensing flows.

c. Pressure drop for an ideal window section:

( ) ( )( )

( )( )psiaftlb

ftft

hrlbp f

ftlb

lbhrftlb

iw

f

m

9.10/576,185.11017.42

576.02/000,630,2 2

2144372,22

1449008

2

,

22

==

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛×

+=Δ (2.58)

d. Correction factor for baffle leakage:

386.0.900

.280.4.672

22=

+=

+

in

ininS

SS

m

tbsb

19.0.280.4.67

.4.6722

2=

+=

+ inin

inSS

S

tbsb

sb

From Fig. 2.38

190

48.0=lR

e. Correction factor for bundle bypass:

Fsbp = 0.029 Nss = 0

From Fig. 2.39

Rb = 0.90

f. Total shell-side pressure drop for an all vapor flow:

( )( )( ) ( )[ ]( ) ( )( )( ) psipsipsipsips 215190.02.3248.09.101890.02.3118 5

57 =+++−=Δ (2.60)

for an all-vapor flow.

g. Correction for a condensing flow: From Fig. 3.25, we find that a totally-condensed flow has a pressure drop 0.29 times the pressure drop for the corresponding all vapor-flow, so the estimated pressure drop for the proposed con-denser design would be 62.2 psi.

This value is substantially greater than the allowable pressure drop, and some means must be found to reduce this value.

Alternative Designs for This Case Besides the excessive pressure drop calculated for the conventional E shell design in this case, there are several other concerns. Considering the high velocities, tube vibration is a definite possibility, especially near the inlet, and an annular distributor (vapor belt) should be specified for all E and J shells (including rod baffle designs). For an X shell (which may also be a rod baffle), a single over sized ("bathtub") nozzle should be used running nearly the full length of the top of the shell, or multiple conventional nozzles may be used along the top at some increase in the piping cost. To deal with shell-side pressure drop itself, a conventionally baffled J shell design (with otherwise the same geometry as the computed example) would probably suffice. A J shell will reduce the shell-side pressure drop by a factor of 6 to 8 compared to an identical E shell. The decrease in shell-side velocity will not affect the calculated (design) value of the condensing coefficient, because no credit is taken for vapor shear enhancement. Alternatively, an E shell using rod baffle construction would also reduce the pressure drop to within acceptable limits. This pressure drop could be estimated by calculating the longitudinal flow pressure drop through the tube array using an equivalent diameter based upon the outside diameter of the tubes; the same Δp correction factor for condensing flow across tubes, Fig. 3.25 could be used with reasonable accuracy. Finally, an X shell using same shell diameter and length could be used either with full circle tube supports or rod baffle supports, as long as the vapor was well distributed along the length of the bundle as noted above.

191

3.4.2. Condenser Design for a Multi-component Mixture: Example Problem Special Considerations of Multi-component Condensation The mechanisms in condensing a multi-component mixture were described qualitatively in Chapter 1. The important differences compared to pure component condensation may be summarized as follows: 1. The heavier components preferentially condense first so that the compositions of each phase are

changing from point to point. 2. As condensation proceeds, the condensing temperature decreases and the remaining vapor must be

sensibly cooled also. The corresponding sensible heat transfer duty must be transferred from the vapor by a sensible heat transfer coefficient which is generally quite low compared to the condensing coefficient.

The multi-component condensation process has been analyzed in fundamental terms by Krishna and Panchal (26), but the use of these methods in design has not yet been completed and in any case requires a computer. A more heuristic procedure was proposed by Silver (27) and put in suitable form for condensers with multiple coolant passes by Bell and Ghaly (28). Variations of this method are in use in various computer-based condenser rating procedures, but even the simplified method is extremely tedious for hand calculations. So in the following section a reduced form of the Silver method is illustrated; this method will be generally suitable for well-behaved cases - ones in which the condensing curve (of temperature vs. heat release) is nearly linear or slightly concave upwards, in which the properties do not change greatly from start to finish, and in which the coolant temperature does not approach the vapor temperature too closely. Under these conditions, the design integral from (27) can be replaced by the following approximation:

MTDQ

hZ

UA T

svoo ⎥

⎦

⎤⎢⎣

⎡+=

'1 (3.93)

The individual terms in Eq. 3.93 are defined as follows:

ofofin

w

o

w

w

i

ofi

ii

oo

hRR

AA

kx

AA

RAh

AU

11'

+++Δ

++= (3.94)

The condensing heat transfer coefficient, ho, is calculated from the methods of this section, such as Eq. (3.80).

T

svQQ

Z = (3.95)

where Qsv is the sensible heat duty removed by cooling the vapor and can be calculated approximately by the following equation:

( )[ ]( )outininout vvoutvvvpvsv TTWWWCQ −−+= 2

1 (3.96)

192

and QT is the total heat transferred in the condenser. Then, hsv is the sensible heat transfer coefficient calculated at the half-condensed point and as if only the vapor phase were flowing. Finally, the MTD is calculated from the LMTD and the configuration correction factor (for multiple coolant passes) using the vapor and coolant terminal temperatures. The application of these equations is illustrated in the following example problem. Statement of the Problem A saturated vapor mixture composed of 0.6 mol fraction nC5 and 0.4 nC4 at a pressure of 50 psia is to be totally condensed at the rate of 120,000 Ibm/hr on the shell-side of a horizontal E-shell and tube heat exchanger using Wolverine S/T low-finned Trufin with cooling water available at 85 F. Fixed tube sheet and low carbon steel construction will be acceptable. Pressure drop for the vapor side should not exceed 5 psi and for the water 10 psi. Fouling resistances are 0.001 hrft2°F/Btu for the water and 0.0005 for the hydrocarbon mixture. Wolverine S/T Trufin, 3/4 in. x 16 BWG, 19 fins/in. of low carbon steel are to be used. Physical properties: The physical properties for the pure components are taken from Perry (29) and the properties of the mixture are calculated using the mixing rules in the same source. These values will be assumed constant throughout the problem:

Liquid Vapor Density, Ibm/ft3 33.4 0.57 Viscosity, lbm/ft hr 0.375 0.017 Specific heat, Btu/Ibm°F 0.58 0.42 Thermal conductivity, Btu/hr ft°F 0.077 0.0098 The condensing curve can be constructed from vapor pressure and enthalpy data from the same source. This curve is shown in Fig. 3.26. Water properties are given at 100°F and assumed constant throughout. Density, lb./ft3 62.0 Viscosity, lbm/ft hr 1.63 Specific heat, Btu/IbmF 1.00 Thermal conductivity, Btu/hr ft°F 0.361

193

Tube Characteristics (nominal) Outside diameter Inside diameter Root diameter Fin height Fin thickness Outside heat transfer area Inside heat transfer area Outside/inside area ratio Inside flow area per tube Thermal conductivity Fin resistance

0.750 in. 0.495 in. 0.625 in. 0.052 in. 0.011 in. 0.503 ft2/ft 0.1303 ft2/ft 3.86 0.195 in.2

26 Btu/hr ft°F 3.1x10-4 hr ft2°F/Btu

194

Preliminary Estimate of Design 1. Heat duty

Q = 1.798 x 107 Btu/hr (from Fig. 3.26) 2. Mean temperature difference

Try a design outlet water temperature of 115°F.

( ) ( )( ) F

nLMTD °=

−−−=

−−

6.4385136115152

85136115152l

Assume multiple tube passes

448.08515285115

=−−

=P

533.085115

136152=

−−

=R

F = 0.955

MTD = 0.955 (43.6) = 41.6°F

3. Estimated overall heat transfer coefficient, based on outside surface (not to be confused with U'):

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛++

Δ+++

=

i

ofi

iw

o

w

wfinfo

o

o

AA

RhA

Akx

RRh

U11

1

( )

FfthrBtu °=

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ ++

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+×+×+

=−−

212506.0

44

/5.75

1303.0503.0001.0

1001503.0

2612065.0101.3105

2501

1

π

Note that in order to reflect the extra resistance of the vapor phase, the condensing coefficient was estimated at 250 Btu/hr ft2°F, rather than the 400 typical of pure component condensation under these conditions.

4. Calculation of area

( )2

75696

5.756.4110789.1 ftAo =

×= , estimated total outside finned surface.

5. Estimation of shell dimensions. Again, we may use Fig. 2.26 of chapter 2.

195

A'o = AoF1F2F3F4

Ao = 5696 ft2

F1 = 1.00 (3/4 in. tube on 15/16 in. triangular pitch)

F2 = 1.03 (Assume - for the moment - 2 tube passes in a 31 in. shell)

F3 = 1.00 (Fixed tube sheet)

F4 = 1.00 (S/T Trufin 19 fins/in.)

Ao’ = 5696 (1.00) (1.03)(1.00)(1.00)

Ao’ = 5867 ft2

The possibilities (from Fig. 2.26) are:

Shell ID, in. Effective Tube Length, ft.

35 10 33 11 31 13 29 15 27 17.5 25 20

23 1/4 23.5 Choose the 31 in. ID case for further analysis. Check water-side velocity: 2 passes:

hrlbFFlbBtu

hrBtuw

mOH /000,599

)30)(/00.1(/10798.1 7

2=

°°×

=

For two tube passes, Nt = 878, or 439/pass So:

( )( )( )( )( )

sec514sec/3600/0.62.195.0439

/.144/000,59932

22

2ft/ .

hrftlbin

ftinhrlbV OH ==

which is quite acceptable for the moment. Thermal Design The problem here is to calculate ho and hsv for the shell side and hi for the tube-side so that Eq. (3.93) and (3.94) can be evaluated.

196

1. Calculation of ho:

Using Eq. 3.80 and assuming L = 13 ft.

tubeperhrlbtubes

hrlbW /7.136878

/000,120==

( )

( )

3/1

28

33

3

7.136375.0

131017.48.324.33077.0

951.02

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞⎜

⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛×⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

=°

hrlb

hrftlb

FfthrBtu

o

fthr

ft

ftlb

ftlb

h

FftBtu °= 2/357 , based on outside finned tube surface area. 2. Calculation of hsv: The Delaware method will be used based upon the vapor flow rate at the half-condensed point, i.e., 60,000 lb/hr. For a preliminary calculation, a baffle spacing equal to 22.28 in. and a baffle cut of 35 percent of the diameter (10.85 in.) will be assumed. Then the shell-side parameters are: (From Chapter 2)

a. Nt = 878

b. pp = 0.814 in.

pn = 0.469 in.

c. 11.814.0

21.31 .31.85.10

=⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛−

=in

inN

inin

c

d. 35.0.31

.85.10==

inin

Di

cl

Fc = 40

e. ( )

11.814.0

.85.108.0==

inin

Ncw

f. bafflesft

Nftin

inb 6113

/.12.80.22

=−=

197

g. ( ) ( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=

.011.0.0416.0.0416.0

.052.02.750.0.9375.0.9375.0

.750.0.5.30.5.30.31.28.22

ininin

inininin

inininininSm

= 202 in.2

h. ( )

055.0.9.201

28.22.5.30.0.312

=−

=in

ininFsbp

i. Stb = 0.0184 (878) (1 + 0.40) = 22.6 in.2

j. Assume diametral clearance of 0.300 in.

( )( ) ( ) 21 .7.8

.31.85.102

1cos2

.300.0.31in

inininin

Ssb =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= −π

k. 2112

.4.235.31

.)85.10(21cossin.31

.)85.10(21.31

.)85.10(21cos4

.)31( inin

inin

inin

ininS wg =⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠

⎞⎜⎝

⎛ −×⎟⎠

⎞⎜⎝

⎛ −−⎟⎠

⎞⎜⎝

⎛ −= −−

( ) ( ) 22 .4.116.750.040.018

878 ininSwt =−= π

Sw = 235.4 in.2 - 116.4 in.2 = 119 in.2

The heat transfer calculation is as follows:

a. ( )( )

( )( ) 000,131.9.201/017.0

/.12/000,60.625.0Re

2==

inhrftlb

ftinhrlbins

b. js = 0.0039

c. ( ) ( )( )( )( )⎥⎦

⎤⎢⎣

⎡°

°

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ° hrftlbFlbBtu

FfthrBtu

in

ftinhrlbh Flb

Btuio /017.0/42.0

/0098.0

.9.201

/.144/000,6042.00039.0

2

22

,

= 86.6 Btu/hr ft2oF

d. Jc = 0.85

e. 155.0.9.201

.6.22.7.82

22=

+=

+

in

ininS

SS

m

tbsb

278.0.6.22.7.8

.7.822

2=

+=

+ ininin

SSS

tbsb

sb

198

775.0=lJ

f. 0=c

ssNN

; Fsbp = 0.055

Jb= 0.93

g. hsv = 86.6(0.85)(0.775)(0.93) Btu/hr ft2°F

= 53.1 Btu/hr ft2°F, based on outside tube surface area.

3. Calculation of hi: use Fig. 2.19.

hi = 1.04(1140 Btu/hr ft2°F) = 1190 Btu/hr ft2°F, based on inside tube surface area.

4. Calculation of U’o: Refer to Equation (3.94).

( )

FU o °=+×+×+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ +

=−−

2

44ftBtu/hr 8.86

3571105101.3

1303.0503.0

1226065.0

1303.0503.0001.0

11901

1'

5. Calculation of Z:

( ) hrBtuFFhrlb

FlbBtuQsv /200,4031361520000,120

21042.0 =°−°⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+

°=

0224.0/10798.1

/200,4037

=×

==hrBtu

hrBtuQQ

ZT

sv

6. Calculation of Ao: Refer to Eq. (3.93).

27

51626.4110798.1

1.530224.0

8.861 ftAo =⎟

⎟⎠

⎞⎜⎜⎝

⎛ ×⎟⎠⎞

⎜⎝⎛ +=

compared to the estimated 5696 ft2. The actual area provided by the design rated is:

Ao = (878 tubes)(13 ft)(0.503 ft2/ft) = 5741 ft2

which is 10 percent greater than the calculated required area. This is a reasonable safety factor in this kind of problem, and it is suggested that the designer not be tempted to pare the design down.

Shell-side Pressure Drop Calculations The method applied here is to calculate the pressure drop for vapor-only flowing through the shell-side using the Delaware method, followed by application of the Diehl Unruh two-phase correction factor for total condensation.

199

1. Delaware calculation of Δps, for vapor only.

a. 000,262).9.201)(/017.0(

)/.12)(/000,120.(625.0Re

2==

inhrftlb

ftinhrlbins

f = 0.25

b. ( )( ) ( ) psiftlb

ftin

in

hrlbp f

lbhrftlb

ftlb

ib

f

m

65.1/238

/144

9.2011017.457.02

11/000,12025.04 2

22

28

2

,

23

==

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛×⎟

⎠⎞

⎜⎝⎛

=Δ

c. ( ) ( )( ) psiftlb

ftin

in

ftin

in

hrlbp f

ftlb

lbhrftlb

iw

f

m

56.1/225

57.0/.144

119

/.144

9.2011017.42

116.02/000,120 2

22

2

22

28

2

,

32

==

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛×

+=Δ

d. = 0.54 lR

e. Rb = 0.80

f. Δps = [(6 – 1)(238 lbf/ft2)(0.80) + 6(225 lbf/ft2)] 0 .54 + 2(238 lbf/ft2)(0.80) + (1 + 1/11)

= 2005 lbf/ ft2 = 13.9 psi

This is the pressure drop for the case in which only vapor flows the entire length of the shell.

g. Correction for a condensing flow. From Fig. 3.25, we find that a totally condensing flow gives

about 0.3 times as much pressure drop as that for vapor flowing the entire distance, so we estimate the actual shell side (nozzle-to-nozzle) pressure drop as 0.3 (13.9 psi) = 4.2 psi which is slightly lower than the problem statement. However, it should be noted that these calculations are quite imprecise, and one could gain a small margin of safety (about 8 percent on the pressure drop) by specifying an exchanger with total tube length of 14 feet (with the same baffle spacing), giving an effective tube length of about 13.7 feet. Otherwise, the design seems to meet the requirements quite well.

200

NOMENCLATURE

A Heat transfer area. A*, reference area; Afin, fin heat transfer area; Aeq, effective area for a finned tube; Ai, Ao, inside and outside tube areas, respectively; Am, mean wall heat transfer area; Aroot, root heat transfer area for finned tube; AT, total heat transfer area in exchanger.

ft2

a Heat transfer area per foot of tube. ft2/ft afin Area of one side of one fin. ft2 Cp,cp Specific heat of vapor and coolant, respectively. Btu/lb°F

vpC , Mean specific heat of the vapor phase in a multi-component mixture. Btu/lb°F

d Diameter; deq, equivalent diameter of a finned tube in condensation; di,do,

inside and outside diameters of a tube; dr, root diameter of a finned tube. in. or ft.

E Mueller correction factor for total condensation, Eq. (3.50). dimensionless F Correction factor for the logarithmic mean temperature difference (LMTD) to

make it applicable to heat exchangers in which the flow is not entirely countercurrent or cocurrent.

dimensionless

F, F’ Fraction of tubes flooded in a condenser with integral subcooler. dimensionless F1, F2 Parameters in the Diehl-Koppany flooding velocity correlations, Eq. (3.30

and 3.3 1) dimensionless

F1, F2, F3, F4

Correction factors for approximate sizing procedures. dimensionless

Fc Chen two-phase convective heat transfer multiplier. dimensionless Fvc Parameter in Carpenter-Colburn equation for in-tube condensation, defined

by Eq. (3.83). Ibm/ft hr2

fi Friction factor for pressure drop inside tubes. dimensionless g Gravitational acceleration. ft/sec2

gc Gravitational conversion constant. 32.2 lbm ft/lbf sec2

G Mass velocity (mass flow rate of fluid per unit cross-sectional area for flow).

Gv and are superficial mass velocities for vapor and liquid respectively; “superficial" means the values are calculated as if the given fluid were flow-ing alone, using the entire cross-sectional area. G

lG

v,i, Gv,m, and Gv,o,. are respectively the inlet, mean, and outlet superficial vapor mass velocities for

Ibm/ft2 hr

201

the Carpenter-Colburn correlation for intube condensation. H Fin height. in. or ft. h Film heat transfer coefficient; hi, ho, inside and outside coefficients,

respectively; hc, average condensing coefficient; hc,N, average coefficient for a vertical bank of N tubes; svh , average sensible heat transfer coefficient for the vapor-gas mixture in multicomponent condensation.

Btu/hr ft2°F

k Thermal conductivity; , liquid thermal conductivity; klk v, vapor thermal

conductivity; kw, thermal conductivity of tube wall. Btu/hr ft2°F

K Parameter in Mueller total condensation analysis, Eq. (3.51) dimensionless L Tube length. ft. L Equivalent length of fin. ft. LMTD Logarithmic mean temperature difference. °F MTD True mean temperature difference, F (LMTD). °F m Parameter in fin efficiency equation. dimensionless N Number of tubes in a vertical row. dimensionless Nf Number of fins per unit length. fins/inch Nui Nusselt number for heat transfer inside a tube. dimensionless p Pressure lbf/in2absolute Pcrit Critical pressure of a fluid. lbf/in2absolute Pr Prandtl number for a fluid. dimensionless PR Reduced pressure, defined by Eq. (3.10). dimensionless Δp Pressure drop; Δpent, pressure drop for entrance to a tube; Δ, pressure drop

due to friction for flow inside a tube; Δp, pressure drop on the shell side, nozzle to nozzle.

lbf/ft2 or lbf/in2

ΔpTPF Total pressure effect in a two-phase flow. lbf/in2

Δpm,TPF Pressure effect due to momentum changes in two phase flow. lbf/in2

ll ,fddp

⎟⎠

⎞⎜⎝

⎛ Pressure gradient due to friction for liquid flowing in a conduit. (lbf/in2)/ft

202

TPFfddp

,⎟⎟⎠

⎞⎜⎜⎝

⎛l

Pressure gradient due to friction for a two-phase mixture flowing in a conduit.

(lbf/in2)/ft

vfddp

,⎟⎠

⎞⎜⎝

⎛l

Pressure gradient due to friction for a vapor flowing in a conduit. (lbf/in2)/ft

TPFgddp

,⎟⎠

⎞⎜⎝

⎛l

Pressure gradient due to hydrostatic effect in two-phase flow. (lbf/in2)/ft

TPFmddp

,⎟⎟⎠

⎞⎜⎜⎝

⎛l

Pressure gradient due to momentum changes in two-phase flow. (lbf/in2)/ft

TPFTddp

,⎟⎠

⎞⎜⎝

⎛l

Total pressure gradient in two-phase flow. (lbf/in2)/ft

Q Heat load; Qc, condensing heat load; Qsc subcooling heat load; Qsv heat heat

load for sensible cooling of a vapor; QT total heat load in a condenser. Btu/hr

Rfi, Rfo Fouling resistances, inside and outside surfaces, respectively. hr ft2°F/Btu Rfin Fin resistance to heat transfer. hr ft2°F/Btu

vR,Rl Liquid and vapor volume fractions, respectively: the actual volume occupied by a given phase, divided by the total volume of the system.

dimensionless

Rec Condensate Reynolds number defined by Eq. (3.37). dimensionless Rei Reynolds number for flow inside a tube. dimensionless s Spacing between fins. in. or ft. T Temperature on the condensing side; Ts, Ts’, surface temperatures, actual

and calculated, during desuperheating; Tsat, saturation temperature of vapor; Tsc, temperature of exiting subcooled condensate; Tw, surface temperature in condensing zone.

°F

t Temperature of coolant; t1 t2, inlet and outlet temperatures, respectively; t*,

to*, mixed mean turnaround and exit temperatures in condensers with subcooling zones; tao, tbo, tdo, teo, tfo, tgo, tjo, tko, local turnaround temperatures in condensers with subcooling zones.

°F

U Overall heat transfer coefficient; U*, overall coefficient based on a reference

area, A*; Ui, Uo, overall coefficients based upon the inside and outside tube areas, respectively; Uc, Us, and Usc, overall coefficients for the condensing zone, the sensible heat transfer zone, and the subcooling zone, respectively; U', and Uo’, partial overall coefficients, including condensate film wall, fouling, and coolant resistances.

Btu/hr ft2°F

Vi In-tube fluid velocity. ft/sec or ft/hr

203

Vv’ Superficial incipient flooding velocity of the vapor in a knockback condenser by the Diehl-Koppany correlation.

ft/sec

v,ϑϑl Volume flow rates of liquid and vapor phases, respectively, in a two-phase

flow. ft3/hr

W Vapor mass flow rate. , , inlet and outlet vapor mass flow rates,

respectively. invW

outvW lb/hr

Wt Condensate weight flow rate per tube defined in Eq. (3.46). lb/hr w Coolant mass flow rate. lb/hr x Distance from tube inlet. ft. x For two-phase vapor-liquid flows, the quality of the flow: the weight fraction

of the flow that is vapor. dimensionless

Xo Outlet vapor quali!y. dimensionless Δxw Wall thickness. in. or ft. Y Fin thickness. in. or ft. Z Fraction of total heat duty that is vapor/gas sensible cooling. dimensionless GREEK λ Latent heat of condensation. Btu/Ib μ Viscosity. , liquid phase viscosity; μlμ s, viscosity of fluid evaluated at

surface temperature.

Ibm/ft hr

ρ Density. lρ , liquid density; vρ , vapor density. Ibm/ft3

σ Surface tension of a liquid. dyne/cm Γ Condensate loading per foot of tube drainage perimeter, defined by Eq.

(3.34). Ibm/ft hr

δ Thickness of a condensate film. in. or ft. θ Angular orientation of a tube, Fig. 3.6. degrees Λ Modified Baker parameter for two-phase flows in a horizontal tube.

vρρl=Λ lb./ft3

Φ Fin efficiency. dimensionless

204

2gttΦ Multiplying factor for calculating total shell-side pressure drop in a condenser

from vapor phase pressure drop. dimensionless

2ttlΦ Martinelli-Nelson two-phase flow friction pressure drop multiplier. dimensionless

Xtt Martinelli-Nelson two-phase flow parameter defined by Eq. (3.15). dimensionless Ψ Modified Baker parameter for two-phase flows in a horizontal tube.

3/23/1 / ll σρμ=Ψ(ft5/3cm) / (Ibm

1/3 hr1/3dyne)

205

BIBLIOGRAPHY 1. Tong, L. S., "Boiling Heat Transfer and Two-Phase Flow", John Wiley and Sons, New York (1965). 2. Martinelli, R. C., and Nelson, D. B., Trans. ASME, 70, 695, (1948). 3. Alves, G. E., Chem. Eng. Prog. 50, No. 9, 449 (1954). 4. Baker, 0., Oil and Gas J. 53, No. 12, 185 (1954). 5. Nilsson, S. N., Paper 2.32, Proc X111. Int. Conf. Ref., Wash., D.C. (August, 1971). 6. Goodykoontz, J. H., and Dorsch, R. G., NASA TN D-3953 (1967). 7. Nusselt, W., Zeits VDI, 60, 541, 569 (1916). 8. Soliman, M., Schuster, J. R., and Berenson, P. J., J Heat Transfer 90, 267 (1968). 9. Diehl, J. E., and Koppany, C. R., Chem. Eng. Prog. Symp. Series No. 92, "Heat Transfer -

Philadelphia", 65, 77 (1969). 10. Jakob, M., Heat Transfer, J. W. Wiley and Sons, New York, Vol. 1 (1949). 11. Kern, D. Q., "Process Heat Transfer", McGraw-Hill Book Co., New York (1950). 12. Carpenter, E. F., and Colburn, A. P., General Discussion on Heat Transfer, London, 20, ASME, New

York (195 1). 13. Colburn, A. P., Trans. AIChE 30, 187 (1934). 14. Mueller, A. C., Private Communication (1971). 15. Standards of Tubular Exchangers Manufacturers Association, 6th Ed., White Plains, NY (1978). 16. Small, W. M., and Young, R. K., Heat Transfer Engineering, 1, No. 2, 21-27 (1979). 17. Bell, K. J., Chem. Eng. Prog., 68, No. 7, 81-82 (1972). 18. McAdams, W. H., Heat Transmission, 3rd Ed., McGraw-Hill , New York (1954). 19. Minkowycz, W. J., and Sparrow, E. M., Int. J. Heat Mass Transfer, 9, 1125 (1966). 20. Beatty, K. 0., and Katz, D. L., Chem. Eng. Prog., 44, No. 1, 55-70 (1948). 21. Kern, D. Q., AlChE J., 4 , No. 2, 157 (195 8). 22. Short, B. E., and Brown, H. E., General Discussion on Heat Transfer, London, 27, ASME (1951). 23. Diehl, J. E., Pet. Ref., 36, No. 10, 147 (1957).

206

24. Diehl, J. E., and Unruh, C. H., Pet. Ref., 37, No. 10, 124 (1958). 25. Brooks, B., "Two Phase Pressure Drop Estimation for Horizontal Crossflow Through Tubebanks",

Report, School of Chemical Engineering, Oklahoma State University (1982). 26. Krishna, R., Panchal, C. B., Webb, D. K., and Coward, L, Letters in Heat and Mass Transfer, 3 , 163

(1976). 27. Silver, L., Trans. 1. Chem. E., 25, 30 (1947). 28. Bell, K. J., and Ghaly, M. A., AlChE Symp. Ser., 69, No. 131, 72 (1973). 29. Perry, R. H., and Chilton, C. H., Eds., Chemical Engineers' Handbook, 5th Edition, McGraw - Hill

Book Co., New York (1972).

207

4.1. Heat Exchangers with High-Finned Trufin Tubes 4.1.1. Areas of Application It is frequently the case that one fluid in a heat exchange process has a much higher film heat transfer coefficient than the other, under the conditions of the given problem. Thus, water very commonly gives a value of 1000 to 1500 Btu/hr ft2°F, whereas air at atmospheric pressure usually gives a value of about 10. The consequence of this imbalance is that the size of the heat exchanger is almost completely controlled by the necessity of providing a large area in contact with the poor heat transfer medium. Often, the best way to provide this area without unduly increasing the overall size of the heat exchanger is to use banks of high-finned tubes such as shown in Fig. 4.1, with the poor heat transfer medium flowing across the finned surface and the other fluid inside the tube. High-finned Trufin is used in a wide variety of services, but the large majority of applications are for transferring heat to atmospheric air. Air has become increasingly important as the ultimate medium for rejecting waste heat, for a variety of reasons. In some areas, water is altogether lacking. Even when available, water may be too costly, or require too much treatment to minimize fouling or corrosion, or require too much reprocessing before it can be discharged to the environment. And in some situations, air is preferable to even readily available water as a cooling medium. The final result is a rapidly increasing need for heat exchangers specifically designed to handle, and transfer heat to, large quantities of air. In these exchangers, air is blown across banks of finned tubes, picking up heat from the stream on the tube-side, which is correspondingly cooled. The hot air is then usually dispersed into the atmosphere and the heat dissipated by mixing. Equipment designed for this purpose is commonly called an air-cooled heat exchanger or air cooler, and to some extent this term has become a generic term for most high-finned Trufin apparatus. However, it is important to remember that high-finned Trufin has a variety of other applications, as this section will indicate. Typically, in an air-cooled exchanger, the fluid on the tube-side may be a process liquid that needs to be sensibly cooled before going to storage or to the next step in the process, or the tube-side fluid may be a vapor that must be condensed. Further, the vapor may be originally superheated above its saturation temperature so that it needs to be de-superheated before condensation takes place. The vapor may be essentially a single component, or it may be a mixture of several components, not all of which are necessarily condensable. Subcooling of the condensate may also be performed in air-cooled equipment.

208

In the cases cited above, the cooling process was direct, in the sense that the heat was transferred directly from the process fluid, through the tube wall and fins, to the air. Indirect air cooling is also used. In this arrangement, the process streams are actually cooled in shell and tube heat exchangers with a closed water loop as the intermediate coolant. The water is then cooled in air-cooled heat exchangers. This arrangement allows the use of generally more compact water-cooled equipment in the immediate vicinity of the process units with ultimate heat rejection accomplished to the atmosphere with air-cooled equipment on the periphery of the plant. The quality of the intermediate coolant can be controlled to minimize corrosion and fouling problems. The equipment and piping arrangement is more extensive and complicated, and the temperature of the process fluids cannot be reduced as low as for the direct cycle. The indirect cycle can be very useful if air temperatures get so low as to cause the process fluids to freeze up or get exceedingly viscous upon direct cooling. The intermediate fluid in the indirect cycle can be a water-glycol mixture so chosen that it will not freeze under the most extreme conditions encountered. The temperature of the intermediate fluid can be controlled by bypassing a portion of it around the coolers, or by shutting down some or all of the fans. In addition to its use in rejecting heat from process plants, high-finned Trufin may also be used to reject heat from power plants (both direct and indirect cooling cycles have been proposed and constructed), and refrigeration and air-conditioning systems. In another application - space heating systems - it is the warm air off of the tubes that is the desired product; the heat source may be either condensing steam or a hot liquid inside the tubes. There are, finally, two applications areas in which the atmospheric air is the heat source rather than the heat sink. In the first of these, air is used to supply heat to a process fluid, either to sensibly warm it or to vaporize it. There is only a limited temperature range over which atmospheric air may be cooled, since if the fin surface temperature drops below the dew point of the air and below 32°F (0°C), ice will form on the fins and can rapidly lead to a restricted air flow and possible mechanical damage to the tube. In the other application, it is the cooled air (for air-conditioning or refrigeration) which is of interest, the cold sink being a boiling refrigerant or possibly a sensibly heating brine. In principle, the air-side calculations (i.e., on the tube bank finned surface) are the same for all of the above applications, though the specific ranges of parameters vary widely. However, the tube-side calculations are quite different, and are covered in other sections of this manual. 4.1.2. High-Finned Trufin All high-finned Trufin made by Wolverine has integral fins. That is, the fins are raised from the base tube metal in a fabricating operation so that the final tube and its fins are one piece of metal, except for the Trufin Type L/C which has an internal liner of a different metal. But here also, the outer tube and the fins are a single piece of metal. Integral firming ensures the maximum thermal efficiency of the tube since there is no possibility of the fins becoming partially or totally separated from the tube metal by environmental corrosion at the base of the fin or by repeated expansion and contraction in operation or by mechanical damage in handling. There are several different types of Wolverine high-finned tubes manufactured. The descriptions given below are intended only to indicate the major features of each type and certain general classes of applications that each lends itself to. Certain limitations of material availability and construction features are also indicated. For a detailed listing of sizes available, tolerances, material specifications, ordering information, etc., see Section 6.

209

Fig. 4.2 defines the major geometrical parameters common to Wolverine high-finned tubes. Type H/F Trufin (Fig. 4.3) is normally produced in alloy 122 (DHP copper) but is also available in some sizes of Alloy 706 (90/10 copper nickel). Standard size inside diameters range from 5/16 in. to 1 1/4 in. with corresponding fin diameters from 0.9 in. to 2.2 in. and fin counts of 7 and 9 fins per inch. Type H/R Trufin (Fig.4.4) is produced in 3003 aluminum. Standard root diameters range from 3/8 in. to I in. with corresponding fin diameters from 0.9 in. to 1.9 in. and standard fin counts of 5, 7, 9, and 11 fins per inch. Type I/L Trufin (Fig. 4.5) has internal longitudinal fins as well as helical fins on the outside. It is available only in 3003 aluminum and has the same fin configurations as Type H/R aluminum. Type L/C Trufin (Fig. 4.6) is a duplex finned tube in which the outer tube is integrally finned 3003 aluminum with the same outside fin configuration similar to Type H/R aluminum. The inner tube may be of any material including copper, admiralty, copper nickels, low carbon and stainless steels. The dimensions of the inner tube are standard dimensions for heat exchanger tubes. Type L/C Trufin is used when corrosion or pressure considerations require the use of a special material in contact with the process fluid. The aluminum assures good heat conductance through the fins into the air, as well as excellent resistance to atmospheric corrosion.

210

4.1.3. Description of Equipment 1. Basic Arrangements. Most large air-cooled heat ex-changers are essentially composed of a shallow (3 to 8 rows) tube bank across which a large quantity of air is blown or drawn at relatively low velocities by large fans. Two different configurations are shown in Figs. 4.7 and 4.8. Fig. 4.7 shows a horizontal tube, forced draft arrange-ment, in which the fan is mounted below the tube bank and blows air upwards past the tubes. This configuration is commonly used for both cooling liquids and con-densing vapors; especially in the latter case, the tubes may be slanted 2° or 3° downwards in the direction of flow to facilitate drainage of the condensate from the tube. The forced draft arrangement is mechanically attractive: the fan and drive may be supported directly on the ground with a fairly short shaft, easing stress and vibration problems and simplifying maintenance. However, the hot air leaves the top of the unit at a fairly low velocity and may tend to recirculate through this or nearby units and raise the inlet temperature, reducing the unit capacity. Fig. 4.8 shows the horizontal tube, induced draft arrange-ment. Induced draft produces generally more uniform airflow across the bundle and projects the hot air plume more positively into the atmosphere, reducing recircula-tion problems. However, in the arrangement shown, fan and driver are more difficult to secure and maintain, and the fan loses efficiency in handling the less-dense hot air. An alternative arrangement places the driver below the bundle, connected by a long shaft to the fan above the bundle; however, tubes must be left out of the bundle to allow the shaft through, with consequent bypassing problems and loss of surface, not to mention potential shaft vibration problems.

211

2. Tube Bundle Construction. The tubes are always arranged in a triangular layout or (less commonly) a rotated square layout, as shown in Fig. 4.9. Inline arrangements are never used because a major portion of the air can flow through the bundle in the clear channel between the tips of fins on adjacent tubes and mix only very poorly with the heated air flowing through the fin field. The effect is to reduce the apparent heat transfer coefficient to approximately half of that of a triangular array. (Refs. 1,2) Even in the staggered layouts the tubes are put as close together as possible without having the tubes vibrate against each other in normal operation. Typically, the tip to-tip clearance is 1/4 in.

Usually the tubes are inserted in box-type headers, Fig. 4.10. The tubes may be expanded into and/or welded to the tube sheet. In the simple box header, the flanged cover plate may be easily removed, exposing the ends of all of the tubes for inspection, leak testing, cleaning, replacement, rerolling or rewelding, or plugging. The integrally welded box header is designed for higher pressure operation; any operations on the tubes are carried out by removing the inspection plugs and working through the inspection hole.

A minimum of three rows of tubes is used in tube banks; the usual maximum is eight rows, though occasionally up to 12 are employed. The greater number of rows of tubes is used where a relatively small

212

airflow is required. Generally in this case the tube-side fluid is at high temperature and the air temperature can increase over a correspondingly greater range. In certain applications (e.g., condensation, especially multi-component mixtures) the tube bank is arranged for single pass; that is, the process fluid is introduced at one end and flows in parallel through all of the tubes in the bundle to the other end, where it exits from the exchanger. For other applications (e.g., cooling of a liquid over a wide temperature range), it is often better to let the tube side fluid flow back and forth through the tube rows making two, three, or occasionally more passes through the exchanger and across the air stream. In this case, the tube side fluid always starts at the top tube row and with successive passes, moves to lower tube rows in order to approximate as closely as possible countercurrent flow. The flow path is controlled by pass dividers or partition plates in the headers, as illustrated for a two pass unit in Fig. 4. 10b. Usually an integral number of tube rows (one or two) is taken for each pass, but by using vertical as well as horizontal dividers it is possible to split rows into frac-tional rows per pass, as shown in Fig. 4.11. However, in order to make room for the plates it may be necessary to omit tubes or distort the layout, which may result in air bypassing. Also, tube-side fluid distribution may be non-uniform. Since the assumptions may not be satisfied, such arrangements should not be used when close temperature approaches are called for. If the distance between headers is greater than about 4 to 6 feet, it is necessary to provide periodic tube support plates, through which the tubes pass with only enough clearance over the fin diameter to allow the tubes to be inserted easily during assembly. To provide a bearing surface for support, the space between fins may be filled (for the distance of a few fins) with a low melting alloy such as a solder cast in place or by a wraparound shroud, or the support plate may be made thick enough to hold several fins. The bundle is held together and stiffened by side members of appropriate size and shape (commonly U-channels) bolted or welded to the headers and the support plates and to the legs or tower support structure. The clearance between the outermost tubes and the side members should be as small as possible to minimize air bypassing. Auxiliary features such as plenums, shrouds, louver systems, and screens are fastened to the main structural members for support. 3. Fans and Drivers. Next to the tube bundle itself, the fan and its driver are the most important elements of the air-cooled exchanger. The fans are invariably axial flow propeller type, with four or six blades, up to 24 feet in diameter (larger ones are available). Especially in the larger sizes, adjustable pitch blades are used. Maximum static pressure is limited to one inch of water, with one half inch of water pressure drop a common design specification. The fan blades are often made of plastic if the air temperature does not exceed 175°, which includes many induced draft applications. Aluminum blades are usable up to around 300°F and steel at higher temperatures. Tip speed should not exceed about 12,000 ft/min in the larger sizes; apart from strength considerations, fan noise increase rapidly at higher speeds.

213

Both electric motor and steam turbines are used as drivers, either direct or indirect. In isolated locations, engine drivers can be used. For indirect drive, V-belts or reduction gears are used; temperature limitations must be observed in induced draft positioning. Hydraulic drives are also used. In estimating fan power requirements, a combined fan and driver efficiency of 60 percent is reasonable, with a range from 50 to 75 percent. In view of the many options open for fans and drives and the specialized nature of the field, detailed information and recommendations should be sought from the manufacturer for each application. 4. Other Components. Fan rings, shrouds and plenums are always used on air-cooled exchangers to guide the airflow and minimize peripheral escape, bypassing, and recirculation of air. There seems to be no general body of information available on design practices in these areas, and reasonable attention to the mechanics of airflow is probably sufficient to avoid catastrophic problems. A course screen underneath the fan is commonly used as a safety precaution in forced draft units where there is any possibility that people or animals may get close to the fan and also to protect the fan and tube bundle from ingesting flying debris. A variety of techniques and equipment is used to ensure that the process fluid does not freeze in the tubes under conditions of low air temperature. First, the fans can be shut off: as a rough rule of thumb, an air-cooled heat exchanger will transfer about one-third as much heat with the fans off. Second, louvers may be installed under the tube bank to further restrict airflow. Third, it is possible to arrange for some of the fans to be reversible and deliberately recirculate some of the warm air that has passed through the bundle back through the bundle into the fresh air supply. Fourth, some bundles have a separate row of tubes underneath the main bundle into which steam can be bled to warm the incoming air.

214

4.2. Heat Transfer with High-Finned Trufin Tubes 4.2.1. Fin Temperature Distribution and Fin Efficiency 1. Temperature Distribution in Fins. The temperature in a fin is not constant, due to the resistance to conductive heat transfer in the fin metal. A typical temperature profile in a fin is shown in Fig. 4.12. The details of calculating the temperature distribution are quite complex and will not be given here; the most comprehensive reference on this subject is the book, "Extended Surface Heat Transfer" by Kern and Kraus (3). The results depend upon a number of parameters, including fin geometry (shape, height, and thickness), fin material, and outside fluid temperature and heat transfer coefficient. It is also necessary to make a number of assumptions; for example, most analyses assume that the outside fluid has a constant bulk temperature and a constant heat transfer coefficient at all points on the fin surface. This is known not to be true, but the real state of affairs is not well understood and would introduce great complexity into the analysis if one tried to be completely rigorous. As a practical matter, the results obtained from the simplified analysis seem to be consistent with experience and lead to acceptable designs. The subject of fin efficiency was discussed in Chapter I, and curves for fin efficiency and fin resistance were given for low-finned Trufin. Since the values for high fin were not given, the method of obtaining the values will be repeated. 2. Fin Efficiency and Resistance. The fin efficiency, Φ, is the ratio of the total heat transferred from the real fin in a given situation to the total heat that would be transferred if the fin were isothermal at its base temperature. For the kinds of fins that are considered here, a good equation to use over the range of interest is:

r

o

ddm

31

12

+

=Φ (4.1)

where

215

YkRh

Hm

wfoo

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=1

2 (4.2)

Equation (4.1) is actually based upon fins of uniform thickness, whereas the fins on Wolverine high-finned Trufin are actually slightly thicker at the base and thinner at the tips. The error is small and in fact the Wolverine fins are slightly more efficient than this equation indicates. The geometrical variables are defined in Fig. 4.2 and ho and Rfo are respectively the actual convective heat transfer coefficient and the actual fouling resistance on the fin side, based on the fin area. To gain an appreciation of the probable magnitude of Φ in a typical problem, consider the following example: Type H/R tube, 3003 aluminum:

dr = 1.00 in. do = 1.875 in. H = 0.437 in. s = 0.076 in. Y = 0.015 in. kw = 110 Btu/hr ft2°F ho = 10 Btu/hr ft2°F Rfo = 0.0

Then:

439.0

12015.0110

101

212437.0

2

2=

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

°⎟⎟⎠

⎞⎜⎜⎝

⎛ °⎟⎠⎞

⎜⎝⎛=

ftFfthr

BtuBtu

Ffthrftm

919.0

000.1875.1

3)439.0(1

12

=

+

=Φ , i.e., 91.9% fin efficiency

There are small differences between the nominal dimensions and the actual dimensions, and some variation from lot to lot in the latter. See Section 6 for details. Nominal dimensions will be used in the examples in this Section. As we will observe later, this efficiency is, if anything, biased towards the low side of most applications. Cop per fins have a higher thermal conductivity and would give a higher Φ. (Copper nickel (90/10) would give Φ = 0.730 under otherwise identical conditions, but is not commonly used for high-finned tubes.) Thicker fins (our example used the thinnest available) would give higher efficiencies. The film heat transfer coefficient was typical of atmospheric air under nominal operating conditions; an extreme value of 20 Btu/hr ft°F would give Φ = 0.862. A quantity somewhat more directly useful in design calculations is the "Fin Resistance", Rfin, defined as:

216

⎥⎦

⎤⎢⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Φ+

Φ−= fo

oAAfin R

hA

fin

root

11 (4.3)

where Aroot is the surface area of a unit length of plain(unfinned) tube between the fins and Afin is the heat transfer area of all of the fins on a unit length of tube. Continuing with the example above, we can compute the value of Rfin as follows:

lengthofftftftin

infins

ftAroot /219.012076.0.12

.11

121 2=⎟

⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛= π

( ) ( )[ ] lengthofftftftin

finsides

infinsftA fin /62.3

122.

1114411875.1

42222 =

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛×⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛−=

π

BtuFfthr

FfthrBtuR fin

°=

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

°⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+

−=

2

262.3219.0

00827.00/101

919.0919.01

which corresponds to an effective heat transfer coefficient for the fins only of 121 Btu/hr ft2°F. This may be compared to a typical value of h,, for air-cooled exchangers of 10 Btu/hr ft2°F, which indicates that the fin resistance is only a small part of the total resistance to heat transfer. The fin resistance then can be directly incorporated into the equation for the overall heat transfer coefficient as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

Δ+++=

i

o

ii

ofi

m

o

wfinfo

oo AA

hAA

RAA

kxRR

hU111 (4.4)

The value of Rfin may be calculated for any desired case by using Eqns. 4.1, 4.2 and 4.3. 4.2.2. Effect of Fouling on High-Finned Trufin As a matter of consistency and principle, the analysis to this point has steadfastly incorporated the term Rfo, the resistance due to fouling on the finned surface. As a matter of fact, fouling on high-finned Trufin with air on the fins is seldom a serious problem, unless there is extensive deposition of material as from massive corrosion (indicating a poor material choice) or a heavy dust storm or ingestion of debris. In the latter cases, continued operation is out of the question, and there is no alternative but to shut down and remove the obstructions. Under normal conditions, the continuous movement of air past the surface tends to minimize deposition of sand and dust, and such deposits as may form can usually be removed by occasionally running a compressed air jet over the surface. Accordingly, Rfo is usually taken as zero for high finned Trufin applications.

217

4.2.3. Contact Resistance in Bimetallic Tubes In Type L/C Trufin, there is an internal liner of a metal other than the 3003 aluminum of the outer tube and fins. The two metals will sometimes be in imperfect contact with one another, leading to an additional resistance to the flow of heat. Generally at low temperatures of the metal-to-metal interface, the liner is exerting a positive pressure upon the aluminum finned tube. But as the tube temperature rises, the aluminum expands more rapidly than the liner and a definite gap develops. The gap is filled with air, introducing a substantial additional resistance to the flow of heat. There have been several studies, both experimental and analytical, made of this problem and the results have been surveyed by Kulkarni and Young (4). This paper and its references should be consulted for details and predictive methods, but it is desirable to summarize here the main findings:

1. At the fabrication temperature of approximately 70°F, there is a positive contact pressure of about 400 psi for a stainless steel liner inside aluminum. Presumably a similar value would exist for other liner metals.

2. This results in a contact resistance of about 0.00005 hr ft2°F/Btu, based upon the contact surface.

This is negligible for any practical application.

3. At the point of zero contact pressure (which occurs at a bond temperature of about 200-215°F in the steel/aluminum case), the bond resistance has been measured to be about 0.0002 hr ft2°F/Btu. This is still negligible for most applications.

4. At tube side fluid temperatures of 1000°F and air side temperatures of 200°F, the bond

resistance is computed to increase to values as high as 0.003 hr ft2°F/Btu (based on contact area) at air side coefficients of 5 Btu/hr ft2°F (based on fin area) and 0.002 hr ft2°F/Btu for air side coefficients of 10 Btu/hr ft2°F. When the corresponding area ratios (say between 1: 10 and 1:20) are taken into account, bond resistance is seen to be about 10-25 percent of the total resistance to heat transfer and definitely needs to be considered in the design. However, it would not seem that a very detailed calculation of the effect is in order unless many such high temperature cases are to be handled.

The complete formulation of the overall heat transfer coefficient calculation for the bimetallic tube with contact resistance is then:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ+++

=

i

o

ii

ofi

m

o

w

w

b

ob

m

o

w

wfinfo

o AA

hAA

RAA

kx

AA

RAA

kx

RRh

U11

1

21

0 (4.5)

where 1⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ

mw

owAkAx

is the wall resistance for the fin metal root, Rb is the bond resistance based

upon the bond contact area Ab, 2⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ

mw

owAkAx

is the wall resistance for the liner tube, and the other

terms have their usual meaning.

218

4.3. Heat Transfer and Pressure Drop in High-Finned Trufin Tube Banks

4.3.1. Heat Transfer Coefficients in Crossflow 1. Briggs and Young Correlation. A number of correlations have been published for heat transfer during flow across banks of finned tubes. The number and range of variables are so large that it would be surprising if a relatively simple correlation would be generally applicable. More complex correlations require correspondingly greater data sets with particular emphasis upon wide ranges of variables and multivariate interactions over these ranges, and the published data generally do not meet these criteria. Therefore, the published correlations must be used with great care to ensure that they are applicable in the range of interest. Within the above caution, one of the best published correlations for high finned Trufin tubes is due to Briggs and Young (5).

12.02.03/1

68.0max Pr134.0

−−

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

sY

sHVd

kdh

airair

airr

air

roμρ

(4.6)

This correlation represents data for root diameters from 0.44 in. to 1.61 in. and fin heights from 0.056 in. to 0.652 in. Fin spacings ranged from 0.035 in. to 0. 117 in. The tubes were in equilateral triangular pitch tube banks with pitches up to 4.5 in. Vmax is the maximum air-side velocity going through the finned tube bank and the other quantities have their usual meaning. Fin spacing, s, is related to the number of fins per inch Nf by the equation:

YN

sf

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

1 (4.7)

It will be useful here to demonstrate the use of this correlation and definitions of the terms by carrying out a typical calculation. 2. Example: Calculate the air side heat transfer coefficient for the case of an aluminum Trufin Type H/R tube with a 3/4 in. root diameter with 9 fins/in. with a mean fin thickness of 0.019. The tubes are in an equilateral triangular layout with a 1 7/8 in. pitch. Air is flowing at 70°F with a face velocity of 600 ft/min. Solution:

The geometrical parameters are as follows:

dr = 0.750 in.

H = 21 (1.625 – 0.75) in. = 0.438 in.

Y = 0.019 in.

219

s = (1/9 – 0.019) in. = 0.092 in.

Vmax: The problem was stated in terms of the "face velocity", i.e., the average velocity of the air approaching the first row of tubes. To convert this to Vmax we must first find the free flow area between two tubes at the point of closest approach, per foot of tube. Since the centers of adjacent tubes are 1.875 in. apart and the root diameter is 0.75 in., there is a 1.125 in. clearance between the root diameters of the tubes giving a clearance area of 12(1.125) = 13.50 in.2/ft. From this clearance area must be subtracted the area blocked by the fins on each tube, which per foot is

2(9 fin/in.)(12 in./ft.)(0.019 in./fin)(0.438) = 1.80 in2/ft.

Thus the free flow area between tubes per foot of length is 11.70 in.2 The "face area" corresponding to these same two adjacent tubes per foot of length is simply the center-to-center distance between the tubes – the pitch – or

(1.875 in.)(12 in./ft.) = 22.50 in2/ft.

The air flowing at 600 ft/min. approaching the face must accelerate to

min/115070.1150.22600 ft=⎟

⎠⎞

⎜⎝⎛

to flow between the tubes and this is the value of Vmax. The physical properties of air at 70°F are:

k = 0.0150 Btu/hr ft°F ρ = 0.0765 Ibm/ft3 μ = 0.439 lb,m/ft hr

cp = 0.240 Btu/Ibm°F

Pr = ( ) 702.00150.0

0439.0240.0=

Finally we may calculate ho:

( )( ) ( )( )( )( ) ( )12.02.0

3/168.0

092.0019.0

092.0438.0702.0

0439.06011500765.012/750.0

12/750.00150.0134.0 −−

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛×⎥⎦

⎤⎢⎣⎡=oh

= 0.0322(432)(0.889)(0.732)(1.208)

ho = 10.9 Btu/hr ft2°F

This is a very typical value of the air side coefficient, and it may be compared to the relatively very much higher values for tube-side (~1000 for water, 2000 + for condensing steam, ~300 for a medium organic

220

liquid, ~100 for a heavy organic liquid). This comparison illustrates immediately the importance of high-finned Trufin in air cooled service. 4.3.2. Mean Temperature Difference in Crossflow 1. The LMTD. It was pointed out in Chapter 1, the great simplification introduced into heat exchanger design by the Mean Temperature Difference (MTD) concept. In its simplest practical form, assuming countercurrent flow, constant overall heat transfer coefficient, etc., the correct definition of the MTD is the Logarithmic Mean Temperature Difference (LMTD) defined as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−−=

)()(

)()(

io

oi

iooi

tTtT

n

tTtTLMTD

l

(4.8)

where Ti and To are the hot fluid inlet and outlet temperatures and ti and to are the cold fluid inlet and outlet temperatures, respectively. With this concept one may write the basic design equation (if all of the conditions are satisfied) as

)(LMTDUQA

oo = (4.9)

where Ao and Uo are referred to the same reference area, usually the total outside heat transfer area of the heat exchanger. 2. MTD for Crossflow. In air-cooled heat exchangers, the air and the process fluid are in crossflow to one another, not in countercurrent flow as assumed in the LMTD derivation. In this case, it is necessary to apply a correction factor F which may be obtained by mathematical analysis. The Eq. (4.9) may be written as:

)(LMTDFUQA

oo = (4.10)

where, it is important to note, the LMTD is calculated as in Eq. (4.8). The correction factor F is a function of the parameters and

ii

iotTtt

P−−

= (4.11)

and

io

oittTT

R−−

= (4.12)

F is plotted in Fig. 4.13 for crossflow with one tube-side pass, i.e., the tube side fluid flows in parallel through all the tubes. It is assumed that an equal amount of fluid flows through each tube. F is plotted in Fig. 4.14 for crossflow with two tube-side passes, with an equal number of tubes in each pass. There may be more than one row of tubes in each pass. Note that the tube-side flow goes through

221

the uppermost tubes first, and that the overall flow pattern is moving towards the tube side fluid being in countercurrent flow to the air flow. In fact, for three or more passes, the overall flow pattern is so close to countercurrent that F can be taken to equal to 1.00 with very small error.

222

4.3.3. Pressure Drop in Crossflow 1. Robinson and Briggs Correlation. Correlations for pressure drop across banks of finned tubes are subject to the same cautions as used for heat transfer, and in fact pressure drop is subject to even greater uncertainty. One of the better correlations in the open literature is due to Robinson and Briggs (6):

93.032.0max47.9

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

r

t

air

airrr d

PVdf

μρ

(4.13)

where fr is the friction factor and Pt is the transverse pitch between adjacent tubes in the same row. The friction factor is defined as

2max2 Vn

gpf

air

cairr

ρ

Δ= (4.14)

where ΔPair is the pressure drop across the tube bank (in say lbf/ft2) and n is the number of tube rows in the bank. This correlation represents data for tube banks with root diameters from 0.734 to 1.61 in., fin diameters from 1.561 in. to 2.750 in., and pitches from 1.687 to 4.500 in.

223

Most of the data are for equilateral triangular arrangements. However, two tube banks with isosceles triangular arrangements were tested, and it was found that the data for these two banks could be adequately cor related if the additional factor

52.0

⎟⎟⎠

⎞⎜⎜⎝

⎛PPt

were included on the right-hand side of Eq. (4.13). In the above expression, P is the "longitudinal" pitch of the tube bank, defined as the distance between the centers of adjacent tubes in different rows, measured along the diagonal. For an equilateral arrangement, Pt = P. Since one of the isosceles tube banks tested was close to a staggered square arrangement there is some reason to believe that this correlation will prove adequate for predicting pressure drop in such geometries also. 2. Example. Using the same tube and arrangement of the previous example and the same air flow rate, calculate the pressure drop across a tube bank of five rows of tubes.

Solution: The geometrical and flow parameters (including Reynolds number) are unchanged. Then the friction factor fr is, by Eq. (4.13):

( )( )( )( ) ( )( ) 0.232 0.4260.05759.47 750.0875.1

0.4396011500.07650.750/1247.9

93.032.0==⎟

⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡=

−−

rf

and the pressure drop is, by Eq. (4.14):

( )( )( )( ) 22

2

2m

23m ./0141.0/03.2

minsec60

sec

lb 32.2

ft/min 1150/ftlb 0.0765520.232inlbftlb

lb

ftp ff

f

air ==

⎟⎠⎞

⎜⎝⎛⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=Δ

It is customary in stating the air-side pressure drop in air-cooled heat transfer equipment to quote it in "inches of water", that is, as the number of inches of height of a column of water at the earth's sur face that would be supported by the pressure difference. The pressure drop is converted from lbf/ ft2 to inches of water by multiplying the reciprocal of the density of water (commonly taken as 62.4 lbm/ft3), by (12 in./ft), and by gc/g which has a numerical value of unity but converts the units properly. So, for this case:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Δ=Δ

gg

ft

lbpOHofp c

OH

fairin

2

122. ρ

(4.15)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

f

m

m

fin lb

lbftin

ftlbft

lbOHofp 1

.12

/4.62103.2

322.

OHinOHofpin 22. .39.0=Δ

224

This value is well within the capability of fans that would be used in this service. A pressure drop of 1/2 in. of water is a common design value. 4.3.4. Other Air-Side Pressure Effects There can be other sources of pressure loss on the air-side of an air-cooled exchanger, such as the louver system (even when it is open), fan guards, structural elements, plenums and shrouds. Few data or correlations are generally available in the literature to permit accurate evaluations of these effects. Fortunately, the losses should be very small and a rough estimate of their magnitude is usually sufficient, mainly as reassurance that they will not be major features in limiting the performance of the units. The following is a useful rule of thumb procedure for any flow situation in which the major pressure effect is from a sudden acceleration of the flow followed by a sudden deceleration. Calculate the increase in air velocity that must be achieved in order for the air to flow through the restriction (say the open louver system, for example). Then allow twice the velocity head calculated from this velocity increase as the irrecoverable pressure loss. Thus, suppose the flow in the previous example had to accelerate from 800 feet/minute to 900 feet/minute to pass through a fan guard. The velocity head represented by that velocity changeis:

( )c

airg

VVP

2

21

22 −

=Δρ

( ) ( ) ( )[ ]2

2f

m

223m

minsec60

seclb

ftlb32.22

/min800 - ft/min 900 /ftlb 0.0765

⎟⎠⎞

⎜⎝⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

OHin.011.0in.

lb109.3

ft

lb056.0 22

f42f =×== −

Doubling this loss gives 0.02 in. H2O as the probable maximum effect. This is a sufficiently small loss (compared to 0.39 in. of H2O across the tube bank) that it would not cause a major problem in the design or operation. If the pressure drop thus calculated turns out to be substantial by this estimate, manufacturers' data or a much more detailed calculation of the effect is required. Finally, if the discharge air plume is to be accelerated strongly in a smoothly converging duct in order to cause it to travel far above the unit before it mixes with the surrounding air, the additional pressure drop required for acceleration can be conservatively taken to be one velocity head based upon the desired discharge velocity.

225

4.4. Preliminary Design Procedures 4.4.1. Principles of the Design Process The essential feature of most design problems involving air coolers is that a certain thermal change must be made on the process stream, using air which can only undergo limited temperature and pressure changes. The air-side heat transfer process is usually controlling in the heat removal process, and the limited pressure drop possible with the air restricts the values of velocities (and, therefore, heat transfer coefficients) to a very narrow range. The first problem of design is to select the general features of the heat exchanger configuration. The second problem is to calculate whether or not the configuration selected will transfer the required amount of heat within the pressure drop limitations. There are basically three possible outcomes to the second problem: 1. The configuration selected will transfer the heat, using (but not exceeding) the available pressure

drops. This then represents the desired design. 2. The configuration will not transfer the heat specified unless the air-side pressure drop is exceeded. In

this case, it is necessary to select a different (larger) design and return to the heat transfer and pressure drop calculations, until the first outcome above is achieved.

3. The configuration will transfer the heat specified, but does not use the pressure drop available. In this

case, the exchanger is too large and can be reduced in size, saving money, until the first outcome is realized.

The above is a gross over-simplification of the design problem, but it illustrates the essentials of the process and the criteria by which success is measured. This section will concentrate upon the first problem – the selection of a design whose major features are fairly close to the final design. In fact, for many purposes, such as plant capital cost estimates or preliminary plant layout, the first-cut design may be sufficient. 4.4.2. Selection of Preliminary Design Parameters 1. Selection of Tube. Finned surface tubes are mainly of interest when the fluid on the fin side has a much lower coefficient than on the tube-side. To a very rough approximation, the tube should be chosen so that

hiAI ≈ hoAo (4.16) or in terms of the area ratios available,

o

ihh

AiAo

≈ (4.17)

226

Since, as we have seen, ho for typical air-cooler applications is about 10 Btu/hr ft2°F, we may make a rough tube selection by choosing one with an Ao/Ai ratio approximately one-tenth the numerical value of hi when the latter is given in Btu/hr ft2°F. Thus, if experience had taught that hi was about 100 Btu/hr ft2°F (typical of cooling a medium weight organic liquid in the lower turbulent flow regime), then one would consider choosing a tube with Ao/Ai ≈ 10, of which there are many available. However, many streams to be cooled (such as water or condensing steam) give an hi ≥ 1000 Btu/hr ft2°F, and there simply are no tubes available that have the corresponding area ratios. In those cases, one simply selects one of the higher ratio tube configurations, taking into account material, availability, prices, and other less tangible considerations. Process stream considerations do affect the choice of tube also. For low flow rates, and for liquids generally, the smaller diameter tubes are generally preferable so that tube-side velocities can be kept up to ensure that the flow is in the turbulent regime and that fouling is minimized. Turbulent flow is generally preferred, partly because of the better heat transfer coefficient, but also to reduce the possibility of a flow maldistribution among the tubes. For high tube-side flow rates, or for gases and condensing vapors, or where tube-side pressure drop is limited, larger diameter tubes are generally preferable. No absolute rules can be written - each case must be considered on its own merits in terms of operability and cost. 2. Selection of a Tube Layout. It has been emphasized above that staggered tube layouts - usually equilateral triangular, less commonly other triangular or rotated square - must be used to minimize bypassing in tube banks. Within this limit, however, and for a given tube, there is the question of what pitch to use. There must be some minimum clearance - on the order of 3/16 to ¼ inch - between fin tips to prevent fin-to-fin impact (with noise and mechanical damage resulting) during operation. Larger clearances are possible and perhaps desirable, since pressure drop decreases more rapidly than heat transfer coefficient as the pitch increases and the velocity decreases. Inevitably, however, the result is to increase the size and cost of the exchanger. Therefore, the usual practice is to put the tubes as close together as possible. 3. Selection of Design Air Temperatures. The selection of the inlet and exit air temperatures are both matters of concern to the designer, though the considerations in their respective selection are quite different. The inlet air temperature at any given moment at a given location is set by nature, but there is usually a substantial range of air temperatures over the course of a year or even a day. For most plant locations, the data are available to plot the percentage of hours in the year when the air temperature will exceed a given value. The designer (or usually, the process engineer) must then decide which temperature to choose in terms of the fraction of time that the heat exchanger will be nominally under-designed and incapable of handling the design heat load at the process stream temperatures specified. Thus, the designer may elect the 5 percent level – that air temperature that is exceeded only 5 percent of the hours of the year. In principle, this means the exchanger will fail to meet demand 5 percent of the year and will be over-designed the other 95 percent. In practice, it is not nearly that simple. Uncertainties at many other points in the design process, including heat transfer coefficients, process stream conditions, changes in the operation of the plant, fouling transients, etc., plus the provision of process

227

control and flexibility in other parts of the process greatly affect even the criterion of what constitutes failure to meet process requirements. Therefore the selection of an inlet temperature loses much of the central importance that has been assigned to it in some past discussions, and the major concern is to select one near but below the maximum air temperature likely to be encountered. Then the exchanger is designed with an eye upon flexibility to application and operation in the particular circumstances of the given problem. It should be noted that problems are as likely to arise from the fact that the air temperature is colder than the design value most of the time. Thus the process stream may be overcooled, leading to freeze-up or, in a reflux condenser, overloading a column or its reboiler. These problems should be anticipated by the process engineer, rather than the exchanger designer, but someone needs to make provision for controlling the air flow rate or other variables to avoid the worst consequences. More often than not, choice of the design inlet air temperature fixes the minimum process fluid exit temperature. This is because it is not generally economically justified to cool the tube-side fluid to a temperature lower than about 20°F hotter than the inlet air; stated another way, the temperature approach at the cold end is generally chosen to be at least 20°F. However, approaches as low as 10°F have been specified, leading to a substantial increase in area. Larger approaches can be used, of course, if there is no need to cool the process fluid so far. The exit air temperature must also be chosen by the designer, but the considerations limiting this choice are set by the process rather than the climate. That is, the exit air temperature must always be less than the inlet temperature of the process stream. If there are several tube-side passes and if the other assumptions underlying the validity and the logarithmic mean temperature difference derivation are satisfied, then in theory one can obtain a workable design for any exit air temperature less than the inlet process fluid temperature. In practice, this is never pushed to the limit, first, because there are too many things that can cause the theoretical model to be violated, and second, because it simply leads to an ex-cessively large and costly exchanger. The approach temperature limit sometimes occurs between the exit air and the inlet process fluid. In that case, the approach is rarely less than 20'°F and more commonly 40'°F. Where only one or two tube-side passes are involved, it is important to calculate F, the MTD correction factor, for the temperatures chosen and make sure that it has a good value (F > 0.8) and that it is not on or near the steep portion of the curve. If these conditions are not satisfied, it is necessary to back off on the temperatures specified and pay the penalty in process efficiency. If the assumptions underlying the F-LMTD derivation are not satisfied, then a much more careful analysis of the approach temperature selection and the MTD evaluation are required. 4.4.3. Fundamental Limitations Controlling Air-Cooled Heat Exchanger Design 1. The Nature of the Problem. There are two or three aspects of air-cooled heat exchanger design which are in some sense in competition. On the one hand, there is the limited capacity of the air to absorb heat; this we may term the "thermodynamic limitation." On the other hand, there is the limited rate at which heat can be transferred to the air; this we may term the "rate limitation." The thermodynamic limitation calls for moving very large quantities of air across the exchanger, accepting the maximum possible change in the air temperature. Given the low pressure drops acceptable, however, this large air flow must be accommodated by using a large face area and shallow depth in the exchanger.

228

The "rate limitation" requires that the temperature difference between the streams be kept as great as possible (which implies small temperature changes in the air) and that the velocities be kept as high as possible, again within the fan capabilities. Both limitations have in common the desire to use high air velocities to overcome them, and the mutual limit upon this we could call the "pumping limitation." In every problem, these three limitations must be balanced, but the point of balance is very dependent upon the particular features of each problem. In this section, we present the common design standards that have evolved and deduce from them a procedure for selecting a preliminary design. 2. The "Pumping Limitation. " The fans in use on air cooled heat exchangers give a maximum practical pressure drop of one inch of water (5.20 lbf/ft2 = 0.0361 lbf/in2). However, the usual design range is from 0.3 inches of water to 0.7 inches, with 0.5 inches being a convenient design target. This converts very approximately into two somewhat more convenient design quantities. The first is in terms of the mass velocity passing through the minimum flow area, defined in terms already used a, ρairVmax, Typical values of ρairVmax as a function of the number of rows of tubes are shown in Table 4. 1. The second design quantity commonly cited is the face velocity, the average air velocity approaching the face of the tube bank, which as a function of the number of rows is given in Table 4.2.

Table 4.1 Table 4.2 Typical Mass Velocities for Air-Cooler Design Typical Face Velocities for Air-Cooler Design

n, No. of Rows

of Tubes ρairVmax lbm/hr ft2 n, No. of Rows

of Tubes Vface ft/min

3 4 5 6 8

5000-6000 5000 4500 4000 3500

3 4 5 6 8

900 800 700 600 500

These are, of course, only representative values, and the comparability of these values between the two tables is only approximate. They are, however, very useful for preliminary estimates. 3. The "Thermodynamic Limitation. " The thermodynamic limitation is nothing more than a heat balance and thus exists in all heat exchangers. But it is particularly critical in air-cooled exchangers because of the low mass rate at which air may be blown across the tube bank. Thus, if the total duty of the exchanger is Q Btu/hr, and if the air inlet and exit temperatures are ti and to, respectively, then the mass flow rate of air required is

)(, ioairpair ttc

Qw−

= (4.18)

229

For all practical purposes, the value of cp,air may be taken as constant at 0.24 Btu/lb°F. The mass flow rate of air may be related to the face velocity Vface and face area Aface by

faceair

airface V

wA

ρ= (4.19)

where ρair is evaluated at air inlet temperature. Care must be taken of course to keep the units consistent. But as noted in the discussion of the pumping limitation, there is a fairly narrow range of values of Vface that can be provided in an air-cooled exchanger, and this is inversely related to the number of tube rows. By combining Eqs. (4.18) and (4.19).

( )faceairioairp

face

VttcQ

TA

ρ)(1

, −= (4.20)

where the notation (Aface)T indicates that this is the face area required by the "thermodynamic limitation". From Table 4.2 we may find typical design values for Vface as a function of n from the pumping limitation. Substituting these values into Eq. (4.20), noting the ft/min units used on Vface, allows us to calculate the face area required per unit of heat to be transferred as a function of n given the values of ti and to.

Example: Assume that 100,000 lb/hr of water is to be cooled from 180°F to 120°F, using air available at 90°F (ρair = 0.0737 lbm/ft3). If the air is heated to 140°F, what face area is required on a thermodynamic basis as a function of the number of tube rows, using the values from Table 4.2: Solution: From Eq. (4.20),

( ) ( )( ) hrBtuft

VVFQ

A

ftm

mface

hrft

faceftlb

FlbBtu

Tface

/01885.0

600737.09014024.0

1)( 2

minmin

min/3

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

⎟⎠⎞

⎜⎝⎛°−⎟

⎠⎞⎜

⎝⎛

=

°

Since Q = (100,000 lbm/hr)(1 Btu/(lbm°F))(180 – 120) °F = 6x106 Btu/hr, we may also calculate the face area required, the results are:

(Aface)T ft2 (Aface)T n

Vface, ft/min Q Btu/hr ft2

3 4 5 6 8

900 800 700 600 500

2.09x10-5

2.36x10-5

2.69x10-5

3.14x10-5

3.77x10-5

126 141 162 189 226

Alternatively, had we specified a tube and tube layout we could have carried out a similar set of calculations based upon Table 4. 1. This would probably be a better criterion to use for actual design, but it is not quite so convenient as the procedure given. Before developing the implications of the thermodynamic limitation further, let us take a look at the heat transfer rate limitation.

230

4. The "Rate Limitation. " The basic equation for the rate of heat transfer is:

)(MTDUQA

oo = (4.21)

where Ao and Uo must be on the same area basis, usually the total outside area of the finned tube. If reasonable estimates of Uo and MTD can be made quickly, the Ao is easily found. This heat transfer area can be put on a comparable basis with the "Thermodynamic Limitation" if it is translated into the face area required by the equation:

( )*HT

oHTface

An

AA = (4.22)

where n is the number of tube rows, and A*

HT is the finned tube heat transfer area per square foot of face area and per row. A*

HT must be determined for each choice of tube and layout; for the case calculated in the previous example,

( ) ( )[ ] ( )( ) ftftftin

finsidesinlengthofftA ft

ininfins

fin /45.2129/.144

/2.75.0in. 1.625

4/ 2.

.2222 =

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

π

( )( )( ) ftftinft

inftftin

inlengthofftA ft

ininfins

root /163.0.12

1.019.01291

/.12.75.0

/ 2.. =⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎟⎠

⎞⎜⎜⎝

⎛= π

Total heat transfer area per foot of tube = 2.61 ft2

With a 1.875 in. pitch, each foot of width of tube bank has 40.6.875.1

.12=

inin

tubes per row, so

A*

HT = (6.40 tubes/ ft-row)(2.61 ft2/ft) = 16.70 ft2/ft2 of face, per row of tubes By combining Eqs. (4.21) and (4.22) we may obtain the equation for the face area required by the rate limitation:

( ))(

1* MTDUAnQ

A

oHT

HTface= (4.23)

A necessary condition of a final design is that the left hand sides of Eqs. (4.20) and (4.23) be equal. It is

immediately observable that, other things being equal, Q

A Tface )( increases with increasing number of

rows of tubes n, whereas Q

A HTface )( decreases with n increasing. Therefore, there is some cross-over

point for any given design problem, and if this point can be approximately located by rapid calculations, the main features of the preliminary configuration can be set. In order to do that, it is necessary to estimate values of Uo and MTD.

231

As noted previously, the overall heat transfer coefficient is largely controlled in air-cooled exchangers by the air side coefficient. In the example given, the air-side coefficient was found to be 10.9 Btu/hr ft2°F at a face velocity of 600 ft/min and a maximum mass velocity of (1150 ft/min)(60 min/hr)(0.0765 lbm/ft) = 5280 Ibm/ft2 hr. Corresponding values of h. would be 14.4 Btu/hr ft2 °F at a face velocity of 900 ft/min and 8.3 Btu/hr ft2 °F at a face velocity of 400 ft/min. These values may be compared to typical values for the other terms given in Eq. (4.5) as follows:

Btu/hr ft2°F

hi, Based on inside tube area Viscous liquid 50

High pressure gas 75 Medium liquids 150 Light liquids 250 Water 1200 Condensing organic vapors 300 Condensing steam 2000

1/Rfi, Based on Heavy fouling 100 inside tube area Moderate fouling 500

Light fouling 2000

w

w

xkΔ

(Stainless steel liner, based on liner area) 2000

w

w

xkΔ

(Aluminum tube, based on root area) 15,000

1/Rc (Maximum contact resistance,

based on contact area) 300 1/Rfin (Maximum resistance for aluminum

fins, based on fin area) 125 From these values, we can see that the air-side resistance can vary from almost 100 percent of the total resistance (for, e.g., condensing steam) to about half of the total (for a viscous liquid, assuming that the tube has been chosen following the hoAo ≈ hiAi criterion). Thus, for preliminary calculations, it is a reasonable approximation to estimate Uo = 10 Btu/hr ft2°F, based on total outside finned tube area, shading this to values as low as 5 Btu/hr ft2°F where low air velocities and/or low intube heat transfer coefficients are involved. Alternatively, for high tube-side coefficients and high air velocities (corresponding to shallow tube banks), the overall coefficient may be estimated as high as 12 Btu/hr ft2°F, based on total outside tube area. To carry out the rate calculations, it is also necessary to have an estimate of the mean temperature difference. For preliminary estimation purposes, in cases where very close temperature approaches are not contemplated, it is often sufficient to use the arithmetic mean temperature difference:

232

AMTD = 2

1 [(Ti – to) + (To – ti])] (4.24) The AMTD is always equal to or greater than the LMTD, the difference depending upon the ratio (Ti – to)/(To – ti). When this ratio is close to unity, AMTD ≈ LMTD; as the ratio departs further from unity, the discrepancy between AMTD and LMTD becomes greater. Additionally, for one or two tube-side passes, the configuration correction factor F must be used to convert the LMTD to the MTD. For all practical air-cooler designs,*1.0 ≤ F ≤ 0.8, so a value of F = 0.9 is a good estimate. *If F < 0.8 for a given problem, using Figs.. (4.13) or (4.14), it is probably necessary to change design temperatures or number of tube-side passes to ensure a good design.

Example. Continue using the previous example, cooling water from 180°F to 120°F using air available at 90°F. Assume also that Wolverine H/R Trufln with dr = 3/4 in. and do = 1 5/8 in., on a 1 7/8 in. equilateral triangular pitch is used; thus A*

HT = 16.70 ft2/ft2 of face area per row. If we start by estimating an exit air temperature of 140°F as before, the AMTD is quickly found to be:

AMTD = 21 [(180 - 140) + (120 - 90)] = 35°F

The LMTD is 34.8°F. We may also wish to check F at this point:

556.09018090140

=−−

=P

20.190140

120180=

−−

=R

and from Fig. 4.13 or 4.14 we get for F a value of about 0.82 if there is one tube-side pass and about 0.91 for two tube-side passes.

We may now set up a table for HT

face

Q

A⎟⎟⎠

⎞⎜⎜⎝

⎛similar to that for

T

face

Q

A⎟⎟⎠

⎞⎜⎜⎝

⎛ in the previous section:

Comparison of this table with the previous one indicates that the two closely correspond at the point of an air cooler with six rows of tubes and a face area of 190 ft2 or in round numbers, a unit 20 feet long and 10 feet wide. The expectation that the fan requirements are probably within design range is due to the fact that we have used typical face velocities. The total heat transfer area (including fins) required is (6

rows)(10 feet wide) ⎟⎟⎠

⎞⎜⎜⎝

⎛pitchinftin

.875.1/.12

(20 feet long)(2.61 ft2/ft of tube) = 20,000 ft2, or a total of 7680 ft of

tube. The next step is to verify the heat transfer coefficients and pressure drop in each side, and verify that two tube rows/pass are necessary and sufficient to maintain good tube-side velocity. From the air-side pressure drop calculation, a fan and driver specification may be obtained. There will almost certainly be some modifications in the approximate design obtained here, but this gives the designer a good place to start.

233

(Aface)HT ft2 (Aface)HT

n No. of Tube Side Passes

F

MTD °F

Vface ft/min

Uo Q Btu/hr ft2

3 4 5 6 8

1 2 2 3 4

0.82 0.91 0.91 1.00 1.00

28.5 31.7 31.7 34.8 34.8

900 800 700 600 500

12 11 10 9 8

5.84x10-5

4.29x10-5

3.78x10-5

3.19x10-5

2.69x10-5

350 258 227 191 161

234

4.5. Final Design The steps for the preliminary design of an air-cooled heat exchanger were given. This procedure can be followed regardless of the nature of the heat transfer inside of the tubes by making a reasonable initial estimate of the coefficients. The next step is to calculate all the coefficients and pressure drops, using appropriate correlations, to verify the design meets the exchanger requirements. It is likely that some adjustments will have to be made in the physical arrangement but with several iterations, a suitable design is usually obtained. Whether the tube-side heat transfer is single-phase or two-phase, the coefficients are generally much larger than the air-side and do not become controlling. Thus, while some of the tube-side correlations, in particular the two-phase equations, are not exact, in accuracy is generally not a serious problem for air-cooled heat exchangers. Attention must be given to the pressure drops so that they are within design limits. This is more important in two-phase flow because of the large volumes of vapor and generally results in larger diameters of Trufin being selected. For single-phase flow, efforts should be made to keep the fluid in the turbulent flow regime. Correlations for the tube-side heat transfer and pressure drop are found in Chapter 2 for sensible heat transfer, Chapter 3 for condensing and Chapter 5 for boiling heat transfer.

235

NOMENCLATURE A Surface area for heat transfer. Ao and Ai are the corresponding values for the

outside and inside surface, respectively, and Am denotes the logarithmic mean of Ao and Ai. Afin is the total heat transfer area/ft for the fins on a tube, and Aroot is the area/ft of the bare tube remaining between the fins. Ab is the bond contact area (per foot of length) in a bimetallic tube. A*HT is the total outsideheat transfer area of a bank of finned tubes per square foot of face area per row.

ft2

Aface Face area, or plain area of a finned tube heat exchanger. This is the total

flow area of the air approaching the tube bank, (Aface)HT is the face area re-quired in a given exchanger by purely heat transfer considerations; (Aface)T is the face area required by purely thermodynamic considerations.

ft2

AMTD Arithmetic mean temperature difference defined by Eq. (4.24). °F cp Specific heat of the flowing fluid. Btu/Ibm°F d Diameter of a tube. do and di are the outside and inside diameters

respectively, and dm denotes the logarithmic mean. Dr, is the root diameter of a finned tube. dfin is the outside diameter of the fin.

in. or ft.

F Correction factor for the logarithmic mean temperature difference (LMTD) to

make it applicable to heat exchangers in which the flow is not entirely countercurrent or cocurrent.

dimensionless

fr The friction factor for tube banks, defined by Eq. (4.14). dimensionless g Gravitational acceleration at Earth's surface. 4.17xl08 ft/hr2 gc Gravitational conversion constant. 4.17xl08lbmft/lbf hr2 H Fin height from root to tip. in. h Film heat transfer coefficient. ho and hi are the values for the outside and the

inside of the heat transfer surface, respectively. hf is an equivalent heat transfer coefficient for any fouling that may be present, equal to the reciprocal of the fouling resistance.

Btu/hr ft2°F

k Thermal conductivity of a material. kw refers to the wall material, while kair, kl,

kv, and kg refer to air, the liquid phase, the vapor, and gas, respectively. Btu/hr ft2(°F/ft)

L Length, usually of a tube. ft. LMTD Logarithmic mean temperature difference, defined by Eq. (4.8) °F MTD True mean temperature difference, F (LMTD) °F

236

m Quantity characterizing fin geometry and properties, defined by Eq. (4.2). dimensionless Nf Number of fins per inch. (in.)-1 n Number of rows of tubes in a tube bank, measured in the direction of flow. dimensionless P Parameter in MTD calculations, defined by Eq. (4.11). dimensionless P Longitudinal tube pitch: distance between adjacent tubes in different rows,

measured along the diagonal. in.

Pt Transverse tube pitch, distance between adjacent tubes in the same row in

a tube bank. in.

Pr Prandtl number of a fluid defined as (cpμ/k). Subscripts “air",”l”,”v", and "g"

refer to air, liquid, vapor, and gas phases, respectively. dimensionless

p Pressure of a liquid. lbf/in2 absolute Δp Pressure drop for flow of a fluid through a given path. The subscript "air"

refers to the pressure drop across the tube bank on the air-side. lbf/in.2

Q heat flow rate. Btu/hr R Parameter in MTD calculations, defined by Eq. (4.12) dimensionless Rb Bond resistance based on bond contact area. hr ft2°F/Btu Rf Resistance to heat transfer due to fouling. Rfo and Rfi are fouling resistances

on the outside and inside of a heat transfer surface, respectively. hr ft2°F/Btu

Rfin Resistance to heat transfer in a fin, given by Eq. (4.3). hr ft2°F/Btu Rw Resistance to heat transfer due to wall conduction. hr ft2°F/Btu r Radius of a tube. ro and ri are the outside and inside radii respectively; rm is

the logarithmic mean of ro and ri. r' is the outside radius of the inner tube and the inside radius of the outer tube in a bimetallic tube.

in. or ft.

s Distance between fins, surface to surface. in. or ft. T, t Temperatures. Both symbols (usually subscripted) are used more or less

interchangeably and for this reason every temperature must be carefully defined for each particular discussion. Usually, capital letters refer to the hot fluid and lower case to the cold fluid, but sometimes capitals refer to the outside fluid and lower case to the inside. Ti and ti usually refer to the inlet temperatures of the two streams and To and to to the outlet temperatures.

°F

Uo Overall heat transfer coefficient for heat transfer between two fluids

separated by a finned surface, referenced to the outside (finned) surface Btu/hr ft2°F

237

area Ao. U+ is the combined heat transfer coefficient for the wall and fin resistance, the coolant and any dirt films. U' is the combined heat transfer coefficient for the condensate film, tube side fouling, wall and fin resistance and air film coefficient.

V Mean velocity of a flowing fluid. For tube banks, Vmax is calculated as the

mean velocity at the point where the tubes are closest together. Vface is the air velocity approaching the face of the tube bank.

ft/sec

W, w Mass flow rates of the fluids in a heat exchanger. Ibm/hr x Usually, a length variable, especially when it appears as Δxw, the wall

thickness of a tube. ft.

Y Thickness of a fin. in. or ft. Greek Φ Fin efficiency: the ratio of the total heat transferred from a real fin to that

transferred if the fin were isothermal at-its base temperature. dimensionless

μ Viscosity of a fluid. μair refers to air. Ibm/ft3 ρ Density of a fluid. airρ refers to air. Ibm/ft3

238

BIBLIOGRAPHY 1. Weierman, C., Taborek, J., and Marner, W.J., "Comparison of the Performance of Inline and

Staggered Banks of Tubes with Segmented Fins," Paper presented at 15th National Heat Transfer Conference, San Francisco, August 10-13, 1975. (In press.)

2. Bell, K. J., and Kegler, W.H., "Analysis of Bypass Flow Effects in Tube Heat Banks and Heat

Exchangers," Paper presented at 15th National Heat Transfer Conference, San Francisco, August 10-13, 1975. (In press.)

3. Kern, D. Q., and Kraus, A. D., "Extended Surface Heat Transfer", McGraw-Hill Book Company, New

York (1972). 4. Kulkarni, M. V., and Young, E. H., Chem. Eng. Prog. 62, No. 7, 68 (1966). 5. Briggs, D. E., and Young, E. H., Chem. Eng. Prog. Symp. Series No. 41, "Heat Transfer - Houston",

59, 1, (1963). 6. Robinson, K. K., and Briggs, D. E., Chem. Eng. Prog. Symp. Series No. 64, "Heat Transfer - Los

Angeles", 62, 177 (1965).

239

5.1. Trufin in Boiling Heat Transfer Boiling is the formation of vapor bubbles at the heating surface. These bubbles form at nucleation sites whose number and location depend upon the surface roughness or cavities, fluid properties, and operating conditions. The boiling heat transfer coefficient is very sensitive to the temperature difference between the surface and the liquid. In addition, the heat transfer coefficient is affected by the local vapor-liquid mixture ratios and velocities, which are a function of the vaporizer design and operating con-ditions. The complex interaction of all these variables makes the accurate prediction of a boiling coefficient virtually impossible, but in large commercial vaporizers the two-phase flow heat transfer becomes controlling and reduces the number of variables. In this section these variables will be discussed and some references given with the aim of providing the engineer an understanding of these factors that affect design of vaporizers. Also some design principles to allow him to produce a vaporizer design will be given. In general, the philosophy of design is that of a designer of process vaporizers. 5.1.1. Pool Boiling Curve If a heating surface is immersed in a pool of liquid that is at the boiling point and the surface temperature is slowly increased, then a plot of the heat flux and the derived heat transfer coefficient versus the temperature difference between the heating surface and the liquid boiling point results in a curve as shown in Fig. 5.1. For the present we are considering a single component liquid; mixtures will be discussed later. Up to the point A or A', heat transfer occurs by natural convection and no bubbles are seen. The liquid pool is superheated and evaporation occurs at the liquid-vapor interface. At point A or A', the local superheat is sufficient to activate nucleation sites on the heating surface and vapor bubbles are formed. The very rapid, almost explosive, formation of the bubbles causes a very strong local velocity within the liquid film and increases heat transfer. In the region from A to B (A' to B') more bub6le nucleation sites are activated and this is the region of nucleate boiling. At point B on the heat flux curve (defined as the critical temperature difference also called the departure from nucleate boiling, DNB, or the bum-out point) the heat flux decreases with a further increase of the surface temperature. Note that point B does not correspond to point B' on the coefficient curve but is at the ΔT where the slope of

240

the h vs. ΔT curve is -1. Several phenomena are occurring as one approaches point B and passes it into the B to C region. The numerous nuclei and the rapid evolution of vapor prevent the liquid from approaching the surface and thus starve the surface of liquid, which was defined by Zuber (1) as a hydrodynamic crisis phenomenon. However, just beyond B after a short transition zone, film boiling oc-curs. Happel and Stephan (2) have also observed and reported the formation of continuous vapor films well before the minimum heat flux, point C. In film boiling a continuous layer of vapor covers the heating surface and keeps the liquid from contacting the surface. The insulating effect of the vapor reduces the rate of heat transfer and the coefficient. As the temperature difference increases, the vapor film becomes thicker and eventually reaches a maximum thickness somewhere near point C' and then the coefficient slowly increases due to the effect of radiation and perhaps further convection effects within the vapor film. The transition from nucleate to film boiling involves a zone where the rapid vapor evolution blankets the tube with a rough vapor-liquid interface that pulsates and occasionally collapses thus wetting the tube. However, as the ΔT is further increased this film becomes smooth and stable but the heat flux is less. The extent of this transition seems to depend upon its definition. Many references define the transition to be between the maximum, B, and the minimum heat flux, C, points as shown by the upper arrows in Figure 5.1. However, some experiments (2,3) have seen stable films well before the minimum flux, C, is reached. Film boiling appears to be closely related to the Leidenfrost effect. This phenomenon was first described by Leidenfrost in 1756 and bears his name. He noted that when liquids were spilled or placed on very hot surfaces, drops were formed which did not contact the surface but floated above the surface and slowly evaporated. However, when the surface temperature was reduced below a certain temperature the drops contacted the surface and rapidly evaporated. The Leidenfrost phenomenon has undergone several periods of intense experimentation and neglect but references to the early literature are found in [references 3 and 4 and more recent work in references 5, 6 and 7]. As noted by Drew and Mueller (3) the temperature differences for the Leidenfrost effect and the boiling critical temperature difference seemed to be closely related. Hence, the actual maximum flux, point B, may be governed by both the hydrodynamic and the film boiling effects. However, film boiling can occur without ever entering the nucleate boiling region as for example in the quenching of metals. From a practical standpoint only the A-B portion of the curve is of interest as operation in the B-C region results in excessive surface temperatures. However, there are occasions when film boiling is unavoidable as in the vaporization of low boiling liquids or in cryogenic vaporizers. The deliberate use of film boiling in attempting to reduce fouling or corrosion has been suggested but is impractical due to variation of operating conditions during start up or shut down and the fluctuations of the film in the transition region. The start up procedure of a vaporizer can affect its subsequent operation whenever the temperature of the heating medium would correspond to a temperature difference greater than the critical ΔT or point B. If the heating source is applied before any liquid is in the vaporizer, then the tube surface temperature, Tw, will reach the medium (steam) temperature, Ts, since no heat transfer is occurring. Then when the liquid is fed to the vaporizer, boiling would begin in the film or B-C region; i.e., (Tw – Tsat,) > ΔTc. However, if the vaporizer is full of liquid when the steam is turned on, then the wall temperature starts at the liquid temperature and rises on the A-B portion of the curve until at equilibrium the wall temperature, Tw, corresponds to a temperature less than point B and the temperature difference, Ts – Tw, results from the heating medium resistance, fouling resistance, and tube wall resistance. Hence, whenever Ts > (Tsat + ΔTc) then the liquid should be in the vaporizer for a cold start up. In normal operation the wall temperature, Tw, would be less than Tsat + ΔTc due to other resistances.

241

5.1.2. Nucleation Nucleation can be either homogenous (occurring within the liquid) or heterogeneous (occurring at a liquid-solid interface). The nucleation phenomenon has been extensively studied both theoretically and experimentally and even a brief review of all the factors involved is beyond the scope of this manual. Cole (8) has an excellent review of nucleation and considerable information is also available in (9, 10, 11). Very briefly, due to the surface tension forces we have across the interface of a spherical bubble

csat rPP /2σ=− l (5.1) where Psat is the saturation pressure of the vapor and the liquid pressure, o the surface tension and rlP c the radius of curvature of the bubble. Through the vapor pressure curve for a fluid, these pressures can be related to a superheat in the liquid in order to maintain a bubble of radius, rc, at equilibrium. The superheat for homogenous nucleation is very large and is difficult to obtain in the presence of surfaces. In thermodynamic equilibrium theory the isotherms on a P-V diagram go through a minimum, (∂P/ ∂V)T = 0, at a liquid superheat which is taken as a homogenous nucleation temperature. In kinetic theory there is a probability that a sufficient number of molecules with greater than average energy can join to form a cluster with an equilibrium radius. The resulting kinetic theory equation for superheat is

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=−

)/(316 3

hJnkTnkTvT

TTv

vsatsat

ll

ll σπρρ

ρλ

(5.2)

n this equation k is the Boltzman constant and h the Plank constant. The heterogeneous nucleation requires less superheat and is in addition a function of the angle of contact between the vapor and solid. Cole (8) has shown the above equation can be modified to

2/13

)/(3)(16

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=−

hJnkTnkTfvT

TTv

vsatsat

ll

ll θσπρρ

ρλ

(5.3)

where the function of f(θ) is a factor involving the bubble contact angle θ. This contact angle is affected by the wettability of the surface, shape of the surfaces (pores etc.) local temperature gradients, etc. Although an excellent understanding of the factors involved in nucleation has been developed it is of little use in the design of vaporizers. This is due to the inability to manufacture and know in advance all the surface characteristics, to control the changes in surface during operation due to corrosion and fouling, and the effect of dissolved gases, mixtures, and solids upon the physical properties of the liquid especially with the local temperature and concentration gradients that will exist around a developing bubble. Further, the effectiveness of nucleation in improving heat transfer decreases as two-phase flow becomes the dominant factor. Figure 5.2 shows how sensitive the boiling flux is to surface conditions and how this surface can change during operation.

242

Improving the nucleation characteristics of a surface has the effect of moving the curves of Figure 5.1 to the left but has little effect on the maximum heat flux at point B. The net result is a higher heat transfer at a given temperature difference or for a given heat flux, a lower temperature difference. These results become important when the available temperature differences are small or become very important as in cryogenic services where power consumption is closely related to ΔT. Special proprietary surfaces have been developed and are commercially available (12), as well as mechanically formed surfaces including Trufin. These surfaces can substantially reduce the superheat required for nucleation and increase the heat transfer coefficients by factors of 3 to 10. The relative effectiveness of these surfaces can change for different fluids depending upon how well the specific surface characteristics (pore size and distribution, surface wettability, etc.) can be matched to the fluid characteristics. However, these surfaces must be used with care so that their effectiveness is not destroyed by fouling, corrosion, or accumulation of high boiling residues. 5.1.3. Nucleate Boiling Curve The nucleate boiling curve is considered to start at point A of Figure 5.1 and ends at point B. This portion of the boiling curve has a very steep slope ranging from 2 to 4 and for the heat transfer coefficient, h = a ΔTm. Observations of boiling shows that as the temperature difference, ΔT, is increased more nuclei are activated; however, eventually when the nuclei spacing is less than the bubble diameter the effectiveness of additional nuclei should diminish. So far a theoretical proof of why the exponent, in, should be so high has not been made. Studies of bubble diameters and frequencies, heat transfer under the developing bubble and the pumping action of the bubbles in carrying away the superheated liquid from the film layers (9,10) have only partially explained the heat transfer in nucleate boiling.

243

The position of the A-B section of the boiling curve can be shifted to the left or right by changes in the surface characteristics, surface tension, pressure, dissolved gases or solids, or high boiling components in a mixture. As shown in Figure 5.2 any changes in surfaces affecting the nucleation properties of surfaces, changes the A-B curve, and the effect of pressure on the nucleate curves is shown in Figure 5.3. In binary mixtures where both components can be vaporized, the boiling curves for mixtures usually lie between those of the pure components. With mixtures, the effect of mass diffusion, local concentration gradients caused by the greater evaporation rate of the more volatile component and the resultant effects on the physical properties of the mixture as well as changes in interface saturation temperatures during bubble growth all influence the boiling curves. If one of the components has a very high boiling point so that it is essentially non-volatile, then the effect of increasing its concentration is to shift the curve A-B to the right reducing the coefficient. Further, the accumulation of the high boiler in the nuclei cavities can cause these to become inactive; hence, when certain special surfaces are used their effectiveness can be greatly reduced depending upon the ability of the circulating liquid to wash out the concentrated high boilers from these cavities. 5.1.4. Maximum or Critical Heat Flux In Figure 5.1 point B is the maximum heat flux for the nucleate boiling regime. Theoretically higher fluxes can be obtained by proceeding along the C-D portion, the film boiling regime, of the curve to very high temperature dif-ferences. The above statements apply to single tube pool boiling only as maximum fluxes for boiling inside tubes or bundles are also experienced but are due to hydrodynamic conditions. Several explanations have been made for the maximum heat flux, such as close packed nuclei forcing the liquid away from the surface and the non-wetting of the surface (the Leidenfrost effect). It is possible that both ex-planations may be simultaneously involved. The maximum flux is a function of pressure and also goes through a maximum as was first shown by Cichelli and Bonilla (13) in Figure 5.4. Based on models assuming force balances (14) the following equation for non-metallic liquids was derived with the constant determined empirically:

( )4/1

2max 18.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

v

cv

v

ggq

ρ

ρρσλρ

l (5.4)

However since many physical properties can be related to the critical pressures, Pc, Mostinski (15) derived the following simple expression

244

9.035.0max )1()(803 rr

cPP

Pq

−= (5.5)

The above equation implies the maximum flux goes to zero at the critical pressure; nevertheless, experiments (16) show there is still a similar effect at supercritical pressures due in part to local density differences as a result of local temperature variations. The maximum flux is somewhat influenced by the heating surface orientation and shape; also some surface effects are seen but in general small changes in surface characteristics do not greatly affect the maximum. Mixtures and the presence of non-volatile liquids can also affect the maximum flux. However, these effects are unpredictable and are neglected in equipment design. 5.1.5. Film Boiling In fully developed film boiling the vapor blankets the heating surface in a smooth continuous film except where the generated vapor escapes from the film in very large bubbles. If the heating surface is vertical and extends through the liquid level, the vapor can escape from the ends of the annular spaces and bubbles may not be generated. There is a transitional region between nucleate and film boiling where the surface is essentially enveloped by the vapor but the interface is rough, tends to fluctuate, and the liquid may occasionally touch the surface for a brief period. This transition occurs in the B-C portion of the curve of Figure 5.1. Depending upon the investigator's definition of film boiling, it can be defined as starting in the B-C portion of the curve or starting at the minimum point C. In fully developed film boiling no effect of surface finish is seen, Figure 5.5. The effect of mixtures, especially those containing non-volatile liquids, does not seem to have been published. In solutions containing dissolved solids, it is difficult to get to the fully developed film region due to rapid fouling of the surface. While film boiling heat transfer is reasonably predictable (18) and it may appear to have some advantages (e.g., corrosion, or fouling by dissolved solids), it is largely avoided because of the high temperature differentials involved. With the high costs of energy, such wastage of temperature potential is uneconomical. Further, the implied advantages are likely to be illusionary. However, there are occasional instances, such as in cryogenic services or certain low boiling chemicals, where film boiling may be unavoidable due to plant constraints on heating sources.

245

5.1.6. Boiling Inside Tubes Although boiling inside tubes may also be in nucleate or film boiling regimes, there are additional factors involved because the vapor and liquid must travel together through the tube. Both heat transfer and pressure drop are, therefore, affected by the pattern of the resulting two phase flow, which because of the evaporation of liquid, changes along the tube. Depending upon the fraction vaporized several different flow regimes are possible, Figure 5.6. For total evaporation in a circulating system the pressure drop through the tube causes an increase in the local boiling point with reference to the pressure existing at the tube outlet; hence, there will be a liquid heating zone at the inlet. When the local tube surface temperature is sufficiently superheated with reference to the local pressure then bubbles will form at the tube surface nuclei and the regimes governed by the nucleate boiling coefficients. As the two-phase mixture accelerates, the two-phase heat transfer can dominate. Depending on tube orientation, a two layer segregated regime (horizontal tubes) or a slug flow (vertical) regime can form which upon further evaporation will develop into an annular flow regime. When the vapor fraction is very large, greater than 60% by weight, a mist flow regime develops and the surface becomes dry and heat transfer is again a convective form but now forced convection to a vapor. The vapor tends to superheat and then evaporate the entrained droplets. If the tube wall temperatures are high enough, a film boiling regime may also occur. Note that in this case an inverted annular flow takes place with an annular vapor film surrounding a liquid core. The overall performance of a tube may also show a maximum heat flux effect similar to the A-B-C portion of the boiling curve in Figure 5.1. However, the maximum flux may be caused by: (a) film boiling, (b) high vapor fractions producing the dry wall regime, or (c) a hydrodynamic instability resulting in surging and unsteady flow through the tube. This latter, (c), effect is a result of the characteristics of the two-phase flow pressure drop curves as a function of the vapor fraction showing maximum and minimum points. Consequentially the instability limit, (c), is a function of the pressure drops in the entire flow circuit loop. 5.1.7. Subcooling and Agitation All of the above discussion for both pool and in-tube boiling was based on the liquid feed being at the vapor saturation temperature.

246

If the liquid pool is subcooled, the heat transfer to the liquid will be a natural convection coefficient if the tube wall temperature is too low to activate the nuclei. Once the nuclei are activated, heat transfer is high as in nucleate boiling but the bubbles rapidly collapse as they penetrate the liquid film on the surface or after the bubbles depart from the surface. This is defined as incipient boiling. Heat transfer coefficients, however, quickly approach those for saturation nucleate boiling.

Agitation in pool boiling (9) effects are shown in Figure 5.7. For in-tube forced convection, heat transfer follows the convective curve until nucleation begins and the nucleate boiling curve is followed, Figure 5.8. This figure also shows that surface finish has very little effect in the coefficients.

247

5.2. Vaporizers - Types and Usage 5.2.1. General Vaporizers are constructed in numerous designs and operated in many modes. Depending upon the service application the design, construction, inspection, testing, operation, and maintenance are governed by various specifications, and by codes (national, state, and local government regulatory agencies.) The ultimate objective of all these rules is to insure the safety of the operators and community during the operation of the vaporizer. It is beyond the scope of this manual to discuss all these facets of vaporizers. Instead we will limit our discussion to the types of vaporizers frequently used in the process industries, specifically shell and tube exchangers. These can be broadly classified as those where the vaporization occurs (a) inside the tubes, or (b) outside the tubes. The selection of the vaporizer type depends upon the evaluation of many factors such as; (1) the purpose of vaporization - to generate a vapor or to cool the heating medium, (2) the boiling fluid; single or multicomponent, and does it contain non-volatile liquids or dissolved solids, (3) the type of heating: liquid, gas, radiant, or electric, (4) the fouling characteristics and blow down requirements, (5) the operating pressures; this is especially important under vacuum or near critical pressure. In the following discussion tubes can be plain, finned, or enhanced to improve the nucleation and boiling coefficients, and/or to improve the heating media coefficients. The characteristics and application of Trufin tubes will be discussed later. 5.2.2. Boiling Outside Tubes (a) Kettle Reboilers. A kettle reboiler installation is shown in Figure 5.9 in simple line form. Here a horizontal U-bundle is placed at the bottom of an oversized shell. The liquid level over the bundle is controlled by means of a baffle. Excess liquid (bottoms or blow-down) overflows the baffle into the end section where the level is controlled by means of a level controller. The space above the baffle-liquid level is used to disengage the vapor from the splash and spray above the bundle. The recirculating liquid returns to the bundle in the eccentric annular space between the shell and the tube bundle. One or more (depending upon bundle length) vapor nozzles are used to remove the vapors. A U-tube bundle is frequently used thus eliminating the need for an internal floating head which could have gasket problems; also there are no stresses between tubes and shell. If a high velocity for liquid heating media is required additional passes can be formed. Advantages are: insensitive to hydrodynamics and therefore reliable and easy to size. High heat fluxes are possible, can operate at low,ΔT, can handle high vaporization up to 80%, simple piping, unlimited area. Disadvantages are: all the dirt collects here and non-volatiles accumulate unless an adequate

248

draw-off is maintained; shell side is difficult to clean; difficult to determine the degree of mixing and, thus, determine the correct AT for wide boiling range liquids; the oversize shell is expensive.

Another version of the kettle reboiler, Figure 5.10, is a fixed tube sheet shell and tube exchanger with a full bundle but operated with a liquid level control on the shell side liquid, These types are used in large refrigeration and air conditioning plants. Here the refrigerant is clean, or at most contains traces of oils, and is completely evaporated. The tubes in the vapor space serve to dry and superheat the vapors for use in the compressors. A small draw off is occasionally used to keep the oil content low in the shell. (b) Column Internal Reboiler. Figure 5.11 shows a U-tube bundle inserted into the side of a column. It acts the same as a kettle reboiler but doesn't have the shell and connecting piping. There are fewer hydraulic problems as improper location and improper sizing of the feed and vapor lines in a kettle reboiler can cause operating problems. The disadvantages are the limited amount of surface area that can be installed, requires a large flange and internal supports, tubes are short hence a costly bundle, and the column must be shut down in order to clean as no alternate operation is possible. (c) Horizontal Thermosyphon Reboilers. A regular baffled shell and tube exchanger of the TEMA "X”, “G”, “H", OR "E" type is used as shown in Figure 5.12 with boiling occurring in the shell-side. By piping arrangement, a driving force for circulation is established by the density differences between the liquid in the column and the two-phase mixture in the exit piping. The heating medium flows in the tubes in single or multiple passes. Advantages are: Higher circulation rate can give a better ΔT than a kettle reboiler; column skirt height is less than for a vertical thermosyphon; high velocity and low exit vapor fractions decrease the effect of residual high boilers and reduce fouling, unlimited area, can handle a fouling heating liquid. Disadvantages are: fouling on shell-side; baffling and tube supports may create vapor blankets and localized dryout; multiple nozzles and complicated piping is required; baffles may be needed to prevent

249

flashing of low boilers near the inlets and concentration of heavies at the exchanger ends; very little design information is available and hydrodynamic problems are not well known or defined; requires large plot area, and high structural costs. (d) Vertical Shell-Side Thermosyphon. The vertical shell-side boiling exchanger is an unusual arrangement not used much in connection with columns but is often encountered in packed catalyst tube reactors. Here, because of the packing and the need for close temperature control, a vertical packed tube reactor of an arrangement shown in Figure 5.13 is used. Sometimes the catalyst bed is on the upper tube sheet when a fast reaction quench is required. Circulation is obtained through the use of an elevated steam drum. Depending upon the temperature conditions at the upper tube sheet, special problems may occur due to the tendency of the steam to form a vapor blanket underneath the tube sheet. 5.2.3. Boiling Inside Tubes (a) Vertical Thermosyphon. A single pass TEMA "E" type shell is used as shown in Figure 5.14. General characteristics of this vaporizer are a large exit pipe with a cross sectional area about equal to the total cross sectional area of the tubes and arranged to minimize the vertical distance between the top tube sheet and the column nozzle. The liquid level in the column is usually kept at the top tube sheet level in order to provide for maximum circulation. Better heat transfer coefficients are obtained if this level is dropped to 1/3 to 1/2 of the tube length; however, circulation is reduced and fouling may increase. For vacuum service where the hydrostatic head can significantly affect the local boil-ing point with a low ΔT, a lower tube submergence is often used. The driving head for circulation is the density difference between the liquid in the column and the two phase mixture in the tubes. These vaporizers frequently are constructed with 3/4 to 1 in. tubes, 8 to 12 ft. long. The exit vapor weight fraction for hydrocarbons ranges from 0.1 to 0.35 and for water about 0.02 to 0.1. Advantages are: circulation is relatively high and

250

tends to minimize fouling; tube-side fouling is easier to clean; the "E" shell and connecting piping is relatively inexpensive, easily supported, and compact. The disadvantages are: requires more head room and column skirt height; maximum heat flux may be lower than kettle reboilers due to instability; the hydrostatic head effect on the boiling point may be a problem at low AT and/or vacuum service, maximum reboiler area is limited, and limited to about 30% vaporization. Experience has shown that the instability encountered at high fluxes is very sensitive to the size, hence, pressure drop in the exit piping. However, stability can be regained by increasing the pressure drop in the liquid recirculation line by means of restrictions or valves. (b) Vertical Long Tube Evaporators. In these evaporators the arrangement is almost the same as in Figure 5.14 except that the recirculation line maybe omitted. Basically they are an once-through evaporator with the feed rate separately controlled. The weight fraction vapor can be very high, approaching 100%, depending upon boiling range and fouling characteristics of the feed. Recirculation, if any, is done by mixing the recir-culated liquid with the fresh feed before pumping the mixture to the evaporator. The effective submergence is very low so that an annular climbing film or mist flow regime exists in the major portion of the tube. These evaporators can operate at relatively low pressures (down to 2 in. Hg abs.) and can handle viscous, wide boiling range mixtures; however, the ΔT used is moderate, about 15 °F. Although smaller tubes have been used many of these evaporators have tube diameters of 2 in. and tube lengths of 20 or more feet. (c) Forced Circulation Evaporators. These may be either vertical or horizontal exchangers and may be of multi-pass construction. Very little vaporization occurs in these tubes especially before the last pass. Essentially these units are treated as a heater and the liquid is flashed after passing through a restriction and then the separated liquid is returned to the pump suction.

251

Advantages are: high velocity and no vaporization reduces fouling; high heat transfer rates; can be used for very low absolute pressures as hydrostatic head effects on boiling points is avoided, mandatory for viscous bottoms, can use standard exchangers, piping is smaller. Disadvantage is the cost of pumps and the power for their operation. 5.2.4. Other Types of Evaporators (a) Fired Vaporizers. In direct fired vaporizers such as steam generators and tube stills the major heat transfer problem is with the heat source and these are specialized designs with proprietary design methods and are beyond the scope of this manual. (b) Film Vaporizers. In falling film vaporizers a thin liquid film flows down the inside of vertical tubes, Figure 5.15, or can fall down across a bank of horizontal tubes. These films are thin so that the heat transfer coefficient is high and bubble formation is very low or non-existent. Since the pressure drops are low and the hydrostatic head is negligible, these vaporizers are well suited to operation at very low pressures as well as having high coefficients at low temperature differences. The hold-up is low and therefore can handle temperature sensitive liquids; also a stripping gas can be used to aid in reducing the temperature level. The disadvantages are: the difficulty of distributing the liquid uniformly on the tubes; the need to maintain an adequate film on all the surfaces; and the possibility to form dry patches when certain physical conditions are exceeded. In Figure 5.15, the tubes are extended above the tube sheet forming a suitable depth pool to aid in uniformly distributing the liquid. A primary overflow weir aids in distributing the feed around the bundle periphery. Each tube may be fabricated as in Figure 5.16 into additional distributing devices. Uniformity of distribution is very essential for vertical tube units. On horizontal tubes the initial uniformity of distribution is not critical and simple overflow or spray devices are used. In horizontal units the bundles are rectangular, with staggered tube pitches, and relatively deep bundles. The major development of horizontal falling film vaporizers was government sponsored research by the Office of Saline Water and the DOE project Ocean Thermal Energy Conversion. Reports of various conferences of these projects contain references and data concerning these vaporizers. In mechanically agitated film vaporizers a film is formed inside a large tube or vessel which has a close fitting agitator, or in some designs a scraper blade, rotating at speeds high enough to keep the liquid on

252

the walls and spread out in a film. Again these vaporizers are suitable for very low pressure (1 mm Hg) operation. The agitator performs several functions: it forms the liquid film; it prevents dry spots, provided sufficient liquid is fed; it agitates the film thus resulting in high heat and mass transfer rates; and fouling is reduced. The disadvantages are: power consumption for mechanical drives; problems with bearings and seals; very expensive design due to close tolerance required; and size and capacity is limited thus necessitating multiple units for large evaporation loads.

253

5.3. Boiling Heat Transfer The prediction of the heat transfer coefficient to boiling liquids is subject to large errors due to the inability to specify, manufacture, and maintain the nucleation characteristics of surfaces, as was illustrated by Figures 5.2 and 5.5. However, the boiling heat transfer coefficient is only part of the overall coefficient and the effect of these large errors is reduced in the overall design but must be considered in selection of safety factors or the selection of operating parameters; e.g., using only a fraction of the available steam pressure. Boiling heat transfer has been studied extensively and good summaries of these researches are found in (9, 10, 21, 22). The design procedures and equations are different for boiling outside of tubes or pool boiling and for boiling inside of tubes. A single tube boiling heat flux curve is a basic starting point and the development of this curve is as follows. A natural convection coefficient equation is used to generate the natural convection heat flux curve by

qnc = hncΔT (5.6) and plotted in Figure 5.17 as line OA. Then a nucleate boiling heat flux curve is calculated and plotted as line AB. The intersection of these lines is point A. A calculation of the maximum flux determines point B on the nucleate curve. Usually these two curves are sufficient but if the critical ΔT at point B can be exceeded, then the minimum flux and the corresponding temperature difference is calculated and establishes point C and the film boiling flux curve CD drawn. No predictive equations exist for the intermediate region (BC) but a straight line is drawn between these points. What we now have is an approximation of the curve in Figure 5.1 by a series of straight lines. The real curve (as in Figure 5.1) has a smooth transition between these straight lines but in design these transition curves are ignored and results in a slight conservatism. 5.3.1. Pool Boiling - Single Tube (a) Natural Convection When the wall temperatures are too low to initiate nucleate boiling the heat transfer coefficient is based on the liquid natural convection coefficient where the ΔT is based on the difference between the wall and the liquid temperatures. For horizontal tubes the following equation is used

254

25.

2

2353.0

ll ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ Δ=

k

cTgdk

dh poonc μ

μ

βρ (5.7)

(b) Nucleate Boiling Although several attempts (23, 24) of theoretical type equations utilizing the fluid properties have been proposed, they are impractical because of the required physical property data (often unavailable for the designer's problem), their complicated evaluation, and their inherent uncertainty due to the surface conditions. A simpler approach, widely used by designers, is based on the work of Borishanski (25) who utilized the law of corresponding states and, as modified by Mostinski (15) and Collier (26), is given as

hnb = A* q0.7F(Pr) (5.8) where A* is a constant evaluated at a reference reduced pressure of Pr = 0.0294 and F(Pr) is a function of reduced pressure as shown in Figure 5.18.

A* = 0.00658 69.0cP (5.8a)

F(Pr)1 = 1.8 + 4 + 10 (5.9) 17.0

rP 2.1rP 10

rP However, dropping the last two terms is a safe design (26) for pool boiling hence

255

F(Pr)2 = 1.8 (5.10) 17.0rP

However, for refrigerants (R11, R12, R113, R115, etc.), F(Pr) should be evaluated as

F(Pr)3 = 0.7 + 2 Pr[4 + 1/(1 – Pr)] (5.11) and the values of A* for these refrigerants given in the square brackets in Table 5.1 be used in equation 5.8. (26) It should be emphasized that the above equation was developed for single component liquids and that the system pressure, tube surface condition, presence of non-condensable gases, hystersis of the boiling curve, size and orientation of the surface, subcooling, wettability, and gravitation acceleration are some of the variables that can affect the result. (c) Critical or Maximum Heat Flux. Cichelli and Bonilla (13) found that the maximum flux was a function of reduced pressure, Pr = P/Pc, and that this curve also had a maximum at a reduced pressure of 0.3, Figure 5.4. Based on the assumption that the maximum flux was limited by the hydrodynamic flow pattern at the surface, Zuber (24), developed the following theoretical equation which seems to correlate the data and with a minor adjustment of the theoretical constant, π / 24, is

4/1

2)(

18.0⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

v

vcvcr

ggq

ρ

ρρσλρ l (5.12)

This equation was for flat plates and some effect of geometry is found which shows the constant ranging from 0.12 to 0.2 (27) depending upon a dimensionless parameter

2/1)(⎥⎦

⎤⎢⎣

⎡ −σρρ

c

vg

gL l

where L is radius or a length of plate. Curves are given for several different shapes, sphere, plates, and cylinders, Figure 5.19. In addition, other factors such as liquid viscosity, subcooling, and surface conditions can affect the values given by equation 5.12. However, equation 5.12 is generally used in commercial design as there are other contributing factors in an actual exchanger that influence the maximum. (d) Minimum Film Boiling Flux. The minimum heat flux, qf, for film boiling, point C in Figure 5.17, occurs when the minimum rate of vapor formation to sustain a stable vapor film is reached. The Zuber theory for the minimum heat flux in film boiling was improved by Berghmans (28) who considered second order perturbations and included the

256

effect of vapor film thickness into the analysis. From an analysis by Berenson (29) for flat plates and the maximum flux equation 5.12, we find that

2/1

09.018.0

⎥⎦

⎤⎢⎣

⎡ +=

v

v

mf

crqq

ρρρl (5.13)

and that the corresponding ΔTmin is

3/12/13/2

min )()()(

127.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−⎥⎦

⎤⎢⎣

⎡+−

=Δvc

v

v

c

v

v

v

vgg

ggk

Tρρ

μρρ

σρρρρρ λ

lll

l (5.14)

A straight line between points B and C in Figure 5.17 then is used for the so-called transition region. Through point C a film boiling flux curve CD can be drawn using the heat flux calculated from the film boiling coefficients. (e) Film Boiling Heat Flux. At high temperature differences a continuous vapor film covers the surface and analytic analysis has followed an analogy to film condensation. For large horizontal plates

[ ]

4/13

2/

)(425.0

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

Δ

−=

πμ

λρρρ

cv

evvf LT

gkh

l (5.15)

where Lc is the shortest unstable wave length for the Taylor instability given by

[ ]2/1

2 ⎥⎦

⎤⎢⎣

⎡−

=v

cc g

gL

ρρσ

πl

(5.16)

and λe is an effective latent heat including the superheat effect

λe = λ [1 + 0.4(cpv ΔT/λ)] (5.17) For tubes the equations are

4/1'3 )(]/069.59.0[

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

Δ

−+=

cv

evvcf TL

gkdLh

μ

λρρρ l (5.18)

but here

257

λe’ = λ [1 + 0.34(cpv ΔT/λ)]2 (5.19) These equations give the conduction heat transfer but in addition at these temperature levels radiation becomes important. Hence, the total film coefficient as suggested by Bromley (30) is

hft = hf + 0.75 hr (5.20) where hr is the coefficient for radiation transfer assuming the liquid is a black body and radiation is between infinite parallel plates. The flux is then calculated by equation 5.6. 5.3.2. Single Tube in Cross Flow The effect of cross flow on the boiling coefficients is shown in Figure 5.20. Basically the forced convection coefficient may exceed the nucleate boiling coefficient at low temperature differences but at higher ΔT the nucleate boiling coefficient becomes dominant. The simple rule is to use the highest coefficient. The critical or maximum heat flux is also affected by cross flow velocity but the magnitude of this effect is a function of tube size and seems to disappear when ap-proaching industrially used dimensions as shown in Figure 5.21. In the film boiling region equation 5.18 will apply for low cross flow velocities but for higher velocities and for less than 45°C (81°F) subcooling the following equation applies

2/1

7.2 ⎥⎦

⎤⎢⎣

⎡Δ

= ∞

sato

vvc Td

kVh

λρ (5.21)

5.3.3. Boiling on Outside of Tubes in a Bundle In spite of the wide use of horizontal tube bundles there is very little data and only elementary suggestions for predicting its heat transfer. The very early work of Abbot and Comley (31) showed the coefficients for a

258

bundle and a single tube were essentially the same. Other reports by Palen et al., (32) showed that bundles may perform better than single tubes due to the circulation induced through the bundle. Although circulation models are being developed, very little has been published on these models. As shown in Collier (26) and in Figure 5.22 the coefficients vary in a haphazard fashion throughout the bundle but, in general, increase from the bottom to top. Palen (34) recommends using

hb = hnbl Fb Fm + hnc (5.22) where Fb = 1.5, hnc = natural convection coefficient, and hnbl = the single tube nucleate coefficient. Fm is the mixture correction. The 1.5 factor is a conservative approximation as it could range up to 3 depending on a bundle layout, size, and heat flux. There is a maximum flux for a bundle that is different, and lower, than the single tube maximum flux. Based on some plant experience Palen and Small (35) proposed a model assuming a vapor blanketing effect. They developed a correction term, Φb, which is used to multiply the qmax as calculated by equation 5.12 and corrected for mixture effects, eqn. 5.38. Their result can be simplified to

1.1)/(2.2

tpBo

tp

s

Bb

LDd

LK

ALD

=⎟⎟⎠

⎞⎜⎜⎝

⎛=Φ

π (5.23)

where K = 4.12 for square pitch

K = 3.56 for triangular pitch (Φb)min = 0.1

This result is reported to be conservative by Palen et al. (32). 5.3.4. Boiling Inside Tubes As shown in Figure 5.6 vaporization inside tubes involves a number of different flow regimes each of which requires a different evaluation of the coefficient plus a local temperature difference which in turn requires corresponding pressure drop calculations. Since most are of a natural circulation design, only the available liquid head is known thus resulting in a trial and error series of calculations to determine the feed rate per tube. The calculated procedure is, thus, too tedious for hand calculation and computers are utilized. However, the computer programs are also complex and expensive to develop and, therefore, become proprietary. We list below the various equations used for the design of vertical in tube vaporizers. Although horizontal in-tube vaporization is also used, the heat transfer equations and methods are the same but the flow pattern now includes a stratified two-layer region. The major difference between horizontal and vertical in-tube vaporization is the definition of and the flow criteria used to define the limits of each regime. The appropriate heat transfer equation is then used for each regime.

259

(a) Single Phase Liquid Region. In a circulating vaporizer the temperature of the liquid entering the tube is below the local boiling point due to the effect of the hydrostatic head on the saturation temperature. This liquid zone extends to the point where the temperature has increased and the local pressure decreased such that the local saturation point has been reached. Actually some further superheat is required to initiate nucleation. The liquid zone heat transfer coefficients are calculated from

1.

2

2325.43.33.

PrPr

17.0⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ Δ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

l

ll

l

l

ll μ

βρμμ

Tgdk

Gdkdh i

w

iic (5.24)

for L/di > 50 and diG/μ < 2000. For turbulent flow and diG/μ > 10,000 use

3/18.

023.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

l

l

ll k

cGdkdh piic μ

μ (5.25)

and interpolate between these two equations on a Re number basis for 2000 < diG/μ < 10,000. (b) Boiling Region. The boiling region can be further subdivided into a sub cooled boiling, saturated boiling, and two-phase boiling regions with predictive equations for each (9,26). Another approach taken by Chen (36) is to combine the saturated and two-phase regions into one, with an equation combining the convective and nucleate boiling mechanisms

hb = s hnb + hcb (5.26) where hb = the boiling coefficient

hnb = the nucleate boiling coefficient hcb = the convective coefficient s = Chen suppression factor

The convective coefficient is a function of the Martinelli two-phase flow parameter, Xtt, and the Chen correlation using this factor is

chttc

cb Fxfhh

== )( (5.27)

73.0

213.0135.2 ⎥⎦

⎤⎢⎣

⎡+=

ttch x

F (5.28)

( )11.057.0

1⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= −

v

vxx

ttxμμ

ρρ l

l

(5.29)

260

x = weight fraction of vapor hc = liquid phase heat transfer coefficient based on the amount of liquid present, Equation 5.25.

The nucleate boiling coefficient, hnb, is determined as

hnb = hnbl Fm (5.30) where Fm is a correction applied for mixtures (discussed later) and hnbl is the coefficient determined from equation 5.8. The suppression factor, s, is determined as follows:

1. Calculate liquid phase ll μ/Re Gdi= 2. Calculate two-phase Retp = (5.31) 25.1Re chFl

3. Calculate s= 1/[.1 + 2.53(10-6) ] 17.1Re tp

Now equation 5.26 can be solved for hb. The subcooled boiling coefficient can be obtained by again using equation 5.26 but with s = (ΔTb/ΔTo) where ΔTb is the temperature difference between the tube wall and the saturation temperature of the liquid at the given local pressure and ΔTo is the difference between the tube wall and subcooled bulk temperature. Instead of the convective coefficient, hnbl, the liquid coefficient (eqn. 5.24 or 5.25) is used. The nucleate coefficient, hnbl, is obtained from the transformed equation 5.8 as

hnbl = 5.43(10-8)( cP )2.3(ΔT)2.33[F(P)]3.33 (5.32) and equation 5.32 changed to

hnb = hnbl Fm (5.33) where Fm is from equation 5.38. (c) Mist Flow. In mist flow the small amount of remaining liquid is en trained as droplets and the tube wall is essentially dry. The coefficient drops rapidly and approaches that of heat transfer to gas. In this regime sensible heat is transferred to the gas which in turn transfers some of the heat to the droplets until they are completely evaporated after which only sensible heat transfer to gas occurs. The main problem is the determination of the vapor temperature, hence, temperature difference. Two extreme conditions are: (1) no heat is transferred to the droplets hence the vapor temperature rises rapidly; and (2) heat is rapidly transferred to the droplets until they disappear and during this evaporation phase the vapor is at saturation temperature. Condition I is approached at low pressures and velocities and condition 2 at high pressure and velocities. The actual case is somewhere between 1 and 2. Some attempts to develop empirical and theory based equations are reported (26) but the range of data seem too limited. We would recommend to use an equation like 5.25 based on gas properties and then make an engineering judgment guess of the fraction of the sensible heat transferred to the gas that is used up as latent heat for the evaporation of drops. The

261

resulting effect on vapor temperature could be used to calculate an LMTD for the mist region and with the calculated gas coefficient used to determine the heat flux. The mist region can be determined from a Fair (37) map or from the simple equation derived from this map

Gmm = 1.8(106) Xtt lb/hr ft2 (5.34) where Gmm is the maximum mass velocity before mist flow begins. (d) Film Boiling. This type of boiling should be avoided, if possible, due to control problems, possible fouling, and lack of data on pressure drop calculations. But if the temperature difference is high enough over the entire tube length, then the heat transfer coefficient can be calculated by the Glickstein and Whiteside (38) correlation

Nu = 0.106 Re0.64 Pr0.4 (ρb/ρv)0.5 (5.35) where the bulk average density on a no slip basis is

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 11

vb x

ρρ

ρρ ll (5.36)

Properties in eqn. 5.35 are based on the liquid. However, the main problem is to determine the mass velocity, G. In film boiling inside a tube we have a core of liquid surrounded by an annular layer of gas which is of very low viscosity. No data in the open literature exist for this case and, thus, determining the circulation rate is a real problem. An alternative estimate of the film coefficient could be made based on pool boiling correlations. (e) Maximum Heat Flux for Stable Operation. In vertical tube thermosyphons there are several limits to operation such as surging, critical heat flux, and mist flow. The surging instability occurs as ΔT is increased beyond a limit but this surging is dependent upon the hydraulic layout and is controllable by the physical arrangement. Blumenkrantz and Taborek (39) discuss the phenomenon. The surging can be controlled by increasing the frictional resistance in the inlet piping. Usual recommended design for thermosyphons has the outlet pipe cross-section area equal to the total cross section area of the tubes. The inlet liquid line is usually smaller and in the range of 25 to 50% of the outlet pipe area. As the temperature difference increases, the evaporation rate of a given tube will increase, pass through a maximum, and then decrease. This was investigated by Lee (40) and confirmed by Palen et.al. (41) and both presented correlations for this effect. The Palen correlation is preferred due to its simplicity

qmax = 16066(di/L)0.35(Pc)0.61 (Pr)0.25( 1 – Pd) (5.37)

262

5.3.5. Boiling of Mixtures When a mixture is boiled the heat transfer coefficient predictions are further complicated by the effect of the local changes in composition, which in turn affect physical properties and boiling points. Some understanding of the factors involved result from studies on boiling of binary mixtures. When a binary mixture is boiled, the vapor generated is richer in the lower boiling point component and a plot of the boiling points of the liquid compositions and the dew points of the resulting vapor compositions is shown in Figure 5.23. As the binary mixture, y1 is heated the first vapor bubble forms at the temperature called the bubble point and that vapor has the composition of Y2 as shown in the figure. A plot of these bubble points versus composition gives the lower (liquid) curve. In cooling a binary vapor of composition y, the first drop of condensate forms at the temperature called the dew point and a plot of dew points is the vapor curve. Temperatures between the dew point and bubble point correspond to a mixture of vapor and liquid each of different composition but whose sum total composition is y1. These curves in Figure 5.23 are generated under the special condition of equilibrium between the vapor and liquid. Boiling is a non-equilibrium process; however, the formation of bubbles at the surface depletes the liquid film of the low boiling component and the remaining liquid has a higher local boiling point. Thus the effective ΔT from the surface to the liquid film boiling point is less than the apparent ΔT and the resulting calculated coefficient is lower than the actual coefficient. Further, the bulk liquid temperature is taken as the equilibrium boiling point of the mixture and ignores any superheat in the bulk liquid. (a) Mixture Heat Transfer Coefficients. Early experiments on mixtures showed, as in Figure 5.24, that the heat fluxes lie between the values for the pure components. The heat transfer coefficients, based on the apparent ΔT, are always less than the pure component coefficients and the minimum values occur at the concentrations where there is the greatest separation between the vapor and liquid lines (42) as illustrated in Figures 5.25 and 5.26.

263

While most mixtures follow the curves as in Figure 5.23 where one component is more volatile over the entire concentration range, there are other systems where one component is more volatile over only a portion of the concentration range and less volatile over the remaining portion. These systems form azeotropes where the azeotropic composition is one where the composition of the vapor and liquid are identical. This azeotropic mixture boils as though it were a single component liquid. Examples of such systems (42) are shown in Figures 5.27 and 5.28. These figures show a minimum boiling point azeotropic mixture but maximum boiling point azeotropic mixtures also exist. Recent papers by Stephan (42) and by Thomas (43) give short reviews of some current theories for heat transfer to boiling mixtures. When we have multi-component mixtures the theories become too complicated for design purposes. A simple empirical approach to calculating mixture boiling coefficients is based on the paper of Palen and Small (35) which was later confirmed as suitable for equipment design in (32) and recommended by Palen (34). Here a mixture correction factor, Fm, is used to modify the calculated coefficient of the volatile component, hnbl, determined from equation 5.32 so that the mixture coefficient = hnblFm where

Fm = exp(-0.015 BR) (5.38) where BR = boiling range, dew point-bubble point, °F. with a lower limit of Fm = 0. 1. This relation is shown in Figure 5.29. This equation is recommended (34) as a reasonable approach for multicomponent systems. This empirical equation is based on the boiling range which is the spread between the vapor and liquid curves as shown in Figures 5.23, 5.25, and 5.26 and is close to some of the theoretical methods (42). (b) Mixture Maximum Heat Flux. Although studies of maximum heat flux for mixtures have shown some instances, Figure 5.30, where the maximum flux of a mixture can be greater than the maximum flux of the components, (44) there is some disagreement

264

on the cause for some of the higher values and some question on the adequacy of some theories, see e.g., (44,45,46) for details. For design purposes it is recommended (34) that equation 5.5 be used with the critical pressure of the mixture based on the molar average. This will give maximum fluxes lying between those of the components and will be conservative.

265

5.4. Falling Film Heat Transfer Falling film vaporizers are characterized by having thin films in contact with the heat transfer surfaces; therefore, the heat transfer equations are basically the same as the condensation equations, the only difference being the direction of heat flow. Hence, we give below only a few basic equations and refer you to section 3 for the influence of shear, mass transfer, etc. The above statements assume no nucleate boiling occurs. Due to the high coefficients and low ΔT used in these units, nucleation is normally not experienced; however, the effect of nucleation is to increase the rate of heat transfer, hence, this is a conservative assumption. There is one special characteristic that appears in the vaporizers and that is the dry patch phenomenon. A dry patch can form due to (1) insufficient liquid to wet the surface, and (2) too high a surface temperature and heat flux causing the liquid to form rivulets thus resulting in dry spots. The dry spots have very low heat fluxes and thus reduces capacity and should be avoided. We will discuss the dry spot effects later. 5.4.1. Vertical In-Tube Vaporizer Uniformity of liquid distribution to all tubes is essential for in-tube falling films. The type of distributor, Figure 5.16, used on each tube affects the allowable hydraulic gradient for the flow across the tube sheet; e.g., a simple overflow weir is very sensitive to the hydraulic gradient while a slotted tube is less sensitive. Distributors are not a commercially available item and are engineered for each application. Although the feed to the vaporizers may be at or very close to the saturation temperature, nevertheless, a preheat section is required to establish the temperature gradient within the film. In the laminar region the coefficient is (47)

h = 4.71 k/(3Γ μ/g ρ2)1/3 (5.39) and for the turbulent region the recommended (48) equation is

34.04.03

3/1

23

2 4)10(7.5 ⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ Γ=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −

kc

gkh μ

μρ

μ (5.40)

For surface evaporation the respective equations for local heat transfer coefficients are (48): Laminar flow

22.03/1

23

2 4821.0−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ Γ⎟⎟⎠

⎞⎜⎜⎝

⎛=

μρ

μ

gkh (5.41)

Turbulent flow

( ) 65.04.03/1

23

23 4108.3 ⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ Γ⎟⎟⎠

⎞⎜⎜⎝

⎛=

−−

kc

gkh μ

μρ

μ (5.42)

266

The laminar equation 5.41 included the effect of waves and ripples. The transition Reynolds number is just the value from the intersection of these equations, not the transition of flow regimes, and is

06.158004 −

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Γk

c

trans

μμ

(5.43)

The above equations provide the local heat transfer coefficients which are used in stepping through the vaporizer. All the fluid properties are evaluated for the liquid phase. This stepping process will also account for the temperature changes and thus give an integrated UΔT value. For mixtures there may be additional mass transfer resistances in the liquid and gas films and techniques of handling these complications are the same as for condensation, therefore, refer to section 3 for further information. 5.4.2. Horizontal Shell-Side Vaporizer Distribution is less critical with horizontal units since drippage between tube rows soon overpowers the initial distribution. While there is still much to be resolved regarding the effect of tube spacing on the effect of drippage and vapor velocities on the motion of drops within the tube bundles, the following equations can be used for approximating the heat transfer, where all the fluid properties are evaluated for the liquid phase. For the preheat zone:

25.12.

2

2336.21.3/1

23

2

PrPr439.0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ Γ=⎟

⎟⎠

⎞⎜⎜⎝

⎛−

w

ogdk

c

gkh

μ

ρμμρ

μ (5.44)

For the evaporation zone (50):

66.024.03/1

23

2 418.0 ⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ Γ=⎟

⎟⎠

⎞⎜⎜⎝

⎛

kc

gkh μ

μρ

μ (5.45)

Horizontal units are used with finned tubes or enhancements on both inside and outside of the tubes to improve heat transfer. See the government literature associated with the development of sea water desalinization and ocean thermal energy conversion for further information. 5.4.3. Dry Spots - Film Breakdown Although theories have been developed to explain the dry spot or film breakdown due to either minimum flows or high fluxes, the agreement with experimental data is only fair, see (9, 47) for further references. The best suggestion is to have a minimum flow at the bottom of the tube or bundle, for water, above 150 lb/hr ft and to not exceed a film ΔT of 18°F (10°C) The requirement for minimum wetting rate means a minimum amount of recirculation may be necessary and it is best to assure the recirculation is sufficient for adequate wetting. When handling mixtures this recirculated liquid can affect the concentrations and boiling points of the feed.

267

5.5. Special Surfaces Commercially available special surfaces used to enhance the boiling side heat transfer are finned surfaces, special plated surfaces, and porous surfaces produced by electroplating, sintering, or machining. The Trufin tubes, used for boiling and condensation enhancement, usually have 16 to 40 fins/inch, approximately 1/16 inch high and resemble a screw thread. The basic purpose is to increase the surface per unit length that is exposed to the boiling liquid and an area ratio increase of 2.2 to 6.7 is obtained. The fins are formed by an extrusion process and are available in most metals. The finning process also changes the surface nucleation characteristics and an improved surface factor is reported by Palen et al. (49). Although the heat transfer coefficients are high the fin efficiency is still high because of the very short fin length. For the high conductivity metals, such as copper, the fin efficiency is almost 100% but for low conductivity metals such as stainless steel the efficiency may drop into the 70-80% range. Surface enhancement is not a universally acceptable solution to the improved performance of reboilers. Careful consideration must be given to the particular operating conditions. Some of the factors to be evaluated are as follows. (1) These are expensive surfaces and cost comparisons should be made. (2) These surfaces are available only for a limited number of metals and the corrosion requirements of the system need evaluation. (3) The boiling range and fouling or corrosive characteristics of the liquids could significantly affect the final performance. Implied here is the ability to clean these surfaces. (4) The performance of a tube bundle can be significantly different from the performance of a single tube as described by Palen et al. (49). Here the relative effect of nucleate boiling to two-phase convection heat transfer needs to be determined. Further, apparent comparisons of tube bundle performance vs. single tubes needs a careful consideration of the effect of bundle layout, tube pitch, etc. on the circulation rate, hence two phase heat transfer, about which we are only beginning to understand. (5) The improvement of the boiling coefficient may not improve overall performance if the heating medium, fouling, or tube wall coefficients are limiting. The major application of enhanced surfaces is in boiling clean liquids at low temperature differences. Trufin tubes depend more on surface increase and seem less subject to problems of fouling and wide boiling range liquids. The maximum heat flux appears to equal that of a plain tube based on the projected area. Enhanced surfaces find industrial applications for two reasons: (a) For a given temperature difference the heat duty will be two or three times higher than for plain tubes at low ΔT. This can result in smaller reboilers with savings in space and weight. At high ΔT the relative performance of enhanced vs. plain tubes is less. (b) For a given duty the required temperature difference will be smaller than for plain tubes. This is of great importance when the cost of other equipment, such as compressors, and operating costs are considered. Heat transfer performance for some commercially available enhanced surface tubes are described by Yilmaz (50). 5.5.1. Boiling on Fins One of the problems of boiling from fins is the determination of fin efficiency. As the nucleate boiling coefficient is strongly dependent upon the temperature difference, which in a fin is varying along its length, the calculation of a fin efficiency requires a stepwise computer program. Fin efficiency calculations with a linear variation along the length were derived by Han (51) and Chen (52). A closer approach to

268

boiling conditions was made by Cumo (53) who made a numerical solution for the case where the heat flux was proportional to the third power of the local wall to liquid temperature difference. Haley and Westwater (54) solved a one dimensional general conduction equation. The effect of fin clearances was studied by Westwater (55) who found that bubble size was a factor but a 1/16-in clearance at atmospheric pressure was sufficient to avoid interference. 5.5.2. Mean Temperature Difference In all boiling processes the liquid is superheated with reference to the vapor saturation temperature but the effect of superheat on the temperature difference calculation is neglected. The amount of superheat cannot be predicted and is usually small compared to the ΔTsat; thus this effect is ignored. For in-tube boiling the heat transfer coefficient calculational procedure requires the division of the tube into zones for each flow regime which in turn requires calculating the heat transferred zone by zone and thus the temperature difference is also included in the calculations. The net result is no separate calculation of a mean temperature difference is made for in-tube vaporization. For boiling outside of tubes the determination of a mean temperature difference is an arbitrary choice based on the amount of subcooling, the boiling range, and to some extent the geometry of the reboiler. The problem is the degree of mixing that occurs in the shell which depends on the circulation rate. Very little is presently known about shell-side circulation and methods for estimating the circulation rates are only in early stages of development. As shown by Palen et al. (32) in Figure 5.31 the bulk temperature within the tube bundle varies with length and the effective ΔT differs substantially from the assumed ΔT or the ΔT based on exit temperatures; thus depending upon the assumptions made, several methods of calculating the mean temperature difference can be used. Some of these choices are: (1) For a pure component or a narrow boiling range mixture, the vapor saturation temperature is used, and if a single component condensing vapor is the heating medium, the temperature difference is this difference. If the heating medium is transferring sensible heat, then a log mean of the vapor saturation temperature and the heating medium terminal temperatures is used. (2) For a wide boiling range mixture or where the sensible heat load is a substantial fraction of the total heat load, a counterflow log mean ΔT gives optimistic results (32, 34) and an LMTD based on exit vapor temperature is recommended. (3) When the effect of static head on the boiling point is significant; e.g., in large bundles and/or in vacuum service, then the boiling temperature should be based on the mean pressure in the bundle including the imposed static head. (4) When a horizontal thermosyphon reboiler is used, a counterflow LMTD is very optimistic and use of a cocurrent flow LMTD is suggested. Since experimentally measured boiling coefficients, and any resulting correlations, are dependent on the temperature differences used to calculate these coefficients from the basic data, then the same LMTD method should be used in the design of reboilers when using these data or correlations. However, the

269

generalized correlations given above have such a spread of data that the temperature difference determination is a minor factor in the data spread and the MTD suggestions in the above paragraph should be followed.

270

5.6. Pressure Drop The total pressure, ΔPt, across a system consists of three components: (a) a static pressure difference, ΔPs, due to the density and elevation of the fluid, (b) a pressure differential, ΔPm, due to the change of momentum, and (c) a pressure differential due to frictional losses, ΔPf. That is

ΔPt = ΔPs + ΔPm + ΔPf (5.46) In a boiling system we are dealing with a vapor-liquid mixture and, in the evaporating zone, the relative quantity of liquid and vapor are changing. Therefore, these component pressure differentials must be determined for the two-phase mixture existing at each point and then integrated over the system. The two-phase flow adds many complications to the calculation of pressure drop. As shown in Figure 5.6 the flow pattern can vary substantially from inlet to outlet. Although both the liquid and vapor are traveling together in the same direction they usually travel at different speeds as a result of slippage between them. The local flow pattern and flow rates can fluctuate resulting in a fluctuating pressure drop. Thus the ability to calculate the pressure drop is much poorer than for a single phase system. In spite of hundreds of researches on two-phase flows a ± 30% accuracy of a predicted pressure drop is considered excellent, a ± 50% a very good prediction, and a ± 100% error is very probable. Therefore, any vaporizer design should incorporate enough of a safety factor to allow for the uncertainties in calculating pressure drops and the corresponding flow rates as well as the potential effect on heat transfer coefficients. 5.6.1. Tube-Side Pressure Drop The method of calculations given below is based on the Lockhart-Martinelli analysis which is a separated flow model; i.e., the flow rates of the vapor and liquid are based on the same pressure gradient. This seems to be the best current general model in the literature, although proprietary improvements have been made. (a) Static Head Loss. The static head loss is very important in vertical units when the heat flux is low and when in the bubble flow regime. Here

∫=Δ θρ sindHggP tpc

s (5.47)

for vertical units sin θ = 1. tpρ can vary with height, H, and is also affected by slip. Here

tpρ = Rv vρ + (1 – Rv) lρ (5.48) and where the volume fraction of vapor is based on the Martinelli relationship

Rv = 1 – 1/Φltt = 1 – Rl (5.49) where Φϑtt, is defined in (c).

271

(b) Momentum Head Loss. The momentum loss is easily determined from the inlet and outlet conditions for either each incremental step or on the overall system.

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−−

−⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−−

+=Δ1

22

2

222

)1()1(

)1()1(

vvvvvvc

tm R

xR

xR

xR

xgG

Pρρρρ ll

(5.50)

where x is local weight fraction of vapor. (c) Friction Head Loss. The friction head loss equations are in two forms either liquid or vapor based. Both will give the same result (both phases turbulent) but the liquid form is better when > 4000 and vapor form for < 4000. The single phase pressure drops are

lRe lRe

)2/1()1()/(4 22

lll ρcti gxGdLfP −=Δ (5.51)

)2/1()/(4 22vctivv gxGdLfP ρ=Δ (5.52)

and the two-phase pressure drops are

ll PP ttf ΔΦ=Δ 2 (5.53)

vvttf PP ΔΦ=Δ 2 (5.54) Here

22 )/(1/201 tttttt xx ++=Φl (5.55)

22 )/(1201 ttttvtt xx ++=Φ (5.56) where

11.057.01

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

=v

vtt x

xxμμ

ρρ l

l

(5.29)

All of the above equations are applied in a stepwise manner along the tube and the final overall pressure drop is the sum of the drops across each step. In a thermosyphon reboiler the above calculated pressure drop must match the available driving head which is total available head minus the sum of the recirculating liquid line frictional and momentum losses.

272

5.6.2. Shell-Side Pressure Drop The calculation of pressure drop for two-phase flow across the tube bundles is done about the same as for flow inside tubes. However, there is much less data for these flows especially in the high liquid fraction ratios en countered in boiling. Most of the experimental data is in the high vapor fraction region as in condensers. Ishihara et al. (56) reviewed the available data and correlations (57, 58, 59, 60) for shell-side flow and concluded that these correlations well represented the author's data but occasionally failed when compared against all the data. Ishihara et al. proposed that the Martinelli separated flow method be used and these equations are given below. Figure 5.32 shows the data compared to this method. The agreement is good in the high vapor fraction region but scatters more in the low vapor fraction region. Further improvements are claimed by the authors but are proprietary and unpublished. The static head and momentum losses are calculated as in the above equations; however, the mass velocity, Gt, in equation 5.50 is based on the minimum flow area bet-ween the tubes. The friction head loss equations are those for flow across tube banks where the maximum mass velocity is based on the minimum flow area between the tubes, hence

)2/1()1(4 22max lll ρct gxGNfP −=Δ (5.58)

)2/1(4 22

max vctvv gxGNfP ρ=Δ (5.59) and

22 )(/81 tttttt xx ++=Φl (5.60)

22 )(81 ttttvtt xx ++=Φ (5.61) Equations 5.53, 5.54, and 5.29 are unchanged.

273

The above equations can be used to develop circulation models for shell-side boiling. In a kettle reboiler with a cylindrical tube bundle, additional problems arise as to how to consider the flow through the bundle as both the height and flow area varies as one proceeds vertically through the bundle. There are also problems in how to calculate the downleg flow in the area between the shell and bundle outside diameter. A further complication is an allowance for a lower density due to some bubble entrainment in the recirculation stream. All these problems need further investigation plus some data on low vapor fraction two-phase flow pressure drops before shell-side circulation prediction methods can be useful. Fair and Klip (61) proposed a shell-side circulation model and included the effect of flow across bundles of varying width and depth. They used the Grant and Chisholm (62) correlation equations. The analysis appears promising but needs to be compared to a wider range of reboiler designs together with a need for specific experiments to attempt circulation measurements in kettle reboilers.

274

5.7. Fouling The contamination of a surface (fouling) on which a liquid is boiling can produce unexpected results. For instance, on smooth surfaces having few nucleation sites the initial surface contamination can increase the number of nucleation sites and increase the boiling coefficient sufficiently to overcome the thermal resistance of the fouling layer. However, this effect is dependent upon a number of special conditions and exists only during the initial stages of fouling. Later as the fouling layer builds up, its thermal resistance will reduce the overall coefficient and heat flux. The strong dependence of the nucleate boiling coefficient on the film temperature difference can magnify the influence of a fouling coefficient when the overall temperature difference is fixed. This effect is the same regardless of the source of the increased resistance, be it fouling, change in tube wall coefficient due to change in thickness or thermal conductivity, or changes in the heating medium coefficient. However, at constant flux the increase in overall temperature difference due to an increased resistance is the normal expected value of just an additional ΔT of the fouling layer. At a constant overall temperature difference any change in fouling, wall, or heating medium resistances will change the temperature distribution and result in a change in the boiling coefficient. Usually a trial and error calculation is required; however, in Figure 5.33 we have a generalized graphical solution. Here equation 5.8 is rearranged to

q = [A* F(P)]3.33 ΔT3.33 = B ΔT3.33 (5.62) In Figure 5.33 we plot q/B versus ΔT with curves of RoB as parameters where Ro is the total of all resistances other than the boiling resistance. The curve RoB = 0 represents the boiling curve. The following examples illustrate the use of this figure. For the examples we will use B = 2 and ΔT = 15. (a) If there are no other resistances then RoB = 0 and the flux is q = 8250 x 2 = 16,500 and hb = 1100. (b) If a resistance of 0.001 were added then RoB = 0.001 x 2 = 0.002 and at ΔT = 15, q = 2250 x 2 = 4500. At this flux, q/B = 2250, we find from curve RoB = 0 that the boiling film ΔT is 10 thus the boiling coefficient, hb, was reduced to 450. To have maintained the same flux as in (a), that is q/B = 8250, we find from curve RoB = 0.002 that the temperature difference would have to be 31.5. (c) Suppose the resistance of 0.001 in (b) represented the wall and heating medium resistances in the clean condition. Now suppose an additional resistance of 0.001 was added due to fouling, change of tube material, or a different heating medium coefficient. Now RoB = (0.001+ 0.001) x 2 = 0.004. From the figure at ΔT = 15 we find the flux would now be q = 1500 x 2 = 3000 and that the boiling film ΔT = 9 (from RoB = 0 curve). Fouling coefficients as currently used are vague and unreliable. The purpose of a fouling coefficient is to permit operation of the equipment for a reasonable length of time before shutting down and cleaning. However, the time interval associated with a fouling coefficient is unspecified and the term reasonable depends upon circumstances. For instance, in crystallizing evaporators a shut down once a shift or once a day for washing out the evaporators is reasonable but in a reboiler where polymerization fouls the tubes and requires a physical cleaning such cycles are unreasonable while a several month cleaning cycle should include evaluating the costs of cleaning, lost production, and cost of exchanger surface to determine the optimum cycle. Unfortunately present knowledge of fouling is insufficient to permit a

275

prediction of fouling rates and even a basic understanding of the effect of velocity, temperatures, or compositions on fouling rates is lacking. For additional information see references (63) through (67).

Fouling under boiling conditions could be different than fouling under convective heat transfer and depends upon the type of boiling. For instance, in the early transition and film boiling regimes, B-C portion of the boiling curve in Figure 5. 1, fouling can be very rapid due to the alternate wetting and drying of the surface. There is little published data on fouling in vaporizers and most values lack documentation as to

276

the operating conditions and cycle time. However, the following values can be used as a guide to making a guess of the values to use in your designs. Boiling Side

Cl-C8 normal hydrocarbons 0 - 0.001 °F ft2 hr/Btu

Heavier normal hydrocarbons 0.001-0.003 Diolefins and polymerizing hydrocarbons 0.003-0.005

Heating Side Condensing steam 0-0.0005

Condensing organics 0.0005-0.001

Sensible heating, organic liquids 0.0005-0.002 Also the TEMA standards (68) can be consulted for further values. Excessive conservatism for fouling allowances may result in oversizing the reboiler such that under clean or initial operation problems may arise in control of the boil up, problems in condensate removal because of low pressure, economic penalties for unnecessary surface, the possibility that reduced circulation may permit more rapid fouling, and for a fixed ΔT that the film boiling region might be reached. Therefore, it is best to make realistic estimates of the fouling avoiding over-conservatism and to evaluate the performance of the reboiler under the full range of operating conditions (from reduced production when clean to full production when fouled). Fouling rates are affected by the type of boiling surface. For instance, low-finned tubes have performed well in fouling conditions (69, 70). These articles report instances where the substitution of low-finned tubes for plain tubes resulted in lower rates of fouling and reduced cleaning times.

277

5.8. Design Procedures 5.8.1. Selection of Reboiler Type The following factors must be considered when the selection of a reboiler type is made.

1. Cleanability (Fouling) -Tube-side is easier to clean than shellside. 2. Corrosion - corrosion or process cleanliness may dictate the use of expensive alloys; therefore, these fluids are placed inside tubes in order to save the cost of an alloy shell. 3. Pressure - high pressure fluids are placed on tube side to avoid the expense of thick walled shells. For very low pressures (vacuum) other factors involved in the selection of reboiler type determines the tube-side fluid. 4. Temperatures - very hot fluids are placed inside tube to reduce shell costs. The lower stress limits at high temperatures affect shell design the same as high pressures. 5. Heating medium requirements may be more important than the boiling liquid requirements. 6. Boiling fluid characteristics: Temperature sensitive liquids require low holdup design. Boiling range and mixture concentration together with available ΔT affect circulation requirements to avoid stagnation. Foaming can be better handled inside tubes. 7. Temperature difference and type of boiling (film or nucleate) affects the selection. 8. Space constraints; e.g., if head room is limited then vertical units would be inappropriate or the limitation of space for internal reboilers. 9. Enhanced surfaces are suitable only for some types.

5.8.2. Pool Type Reboilers The usual pool type (kettle) reboilers, Figure 5.9, have submerged tube bundles and a vapor disengaging space. Here the vapor leaves the reboiler at the saturation temperature and may have some entrained liquid. If a dry or superheated vapor is required as; e.g., feed to a compressor, then additional separators are used or tubes placed in the vapor space (Figure 5.10) to dry and superheat the vapor. Typically the bundle diameter is about 6007o of the shell diameter and liquid level is sufficient to just submerge the bundle. The actual shell diameter is determined by the amount of acceptable entrainment and the corresponding vapor velocity. An empirical equation used to determine this velocity is

5.0

5 )(10(86.62290

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−=

−v

vVLρρ

σρl

(5.63)

where VL is vapor load lb/hr ft3 (vapor rate divided by volume of vapor space), σ is surface tension lb/ft. Due to swelling of the liquid volume during boiling and to some foaming the surface area on which the

278

above velocities are based is several (3-5) inches above the static level and the vapor volume is then calculated on this chord length. Entrainment is also effected by the vapor horizontal velocity in flowing to the vapor nozzles. As a rule of thumb the number of vapor nozzles are

Nn ≈ L/(5 Ds) (5.64) and an equal number of feed nozzles. The tube diameters range from 5/8 to 1 inch with 3/4 and 1 inch being most common. Tube length is determined by the allowable pressure drop for the heating medium or space limitations. U-tubes are used to avoid an internal gasket but U-tubes are difficult to clean in the bend ends. Straight tubes with fixed tube sheets can be used if mechanical tube side cleaning is required. Tube pattern depends upon the need for cleaning, a square pitch for fouling conditions, a triangular pitch for clean liquids. There is little agreement on the pitch ratios as both close, 1.2, and wide, 1.5 – 2.0, ratios are used. The best pitch ratio is a function of the amount of circulation and fraction vaporized, the effect of circulation on the coefficient, and the boiling range. Insufficient data exists to analyze the interaction of these factors and cur rent theory on circulation is in a developing stage (61). For pure single component liquids, as long as the tubes are wet, the heat transfer will be high, but multicomponent liquids will require adequate circulation to reduce the effect of boiling point elevation due to stripping of the light components. Also the relative contribution of nucleate and forced convection coefficients to the overall boiling coefficient will affect its response (slope of curve) to temperature difference. The maximum flux of the bundle (see eqn. 5.23) is affected by vapor blanketing. Sometimes, and based on the designer's intuition, vapor lanes are provided in large bundles to help improve circulation and vapor removal. The liquid level can be controlled by external level controllers which measure the equivalent static level but the actual level can be higher because of the lower density of the vapor-liquid mixture in the reboiler. An overflow baffle in the end of the reboiler is used when there is a continuous bottoms withdrawal to control the kettle concentrations. This baffled zone allows separation of vapor and liquid and is sized to provide a sufficient (several minutes) holdup suitable for a bottoms pump control. This baffle then sets the level of the dynamic mix in the bundle. If the vapor release volume is marginal then a small (a few inches) change in liquid level can rapidly increase the entrainment such that the resulting increased two-phase flow pressure drop in the vapor line can reduce the feed rate especially if a gravity feed from a column is involved. These upsets seriously affect column performance. 5.8.3. In-tube or Thermosyphon Reboilers The in-tube reboilers are vertical units with a condensing heat medium on the shell-side. The horizontal thermosyphons have a process stream or heating medium on the tube-side. Both thermosyphons depend upon a natural circulation induced by the density difference in the feed and exit streams. The vertical in-tube unit may, however, be operated as a once through vaporizer in which case the feed rate is controlled externally. In this discussion we limit ourselves to the thermosyphon action reboilers. For in-tube vaporizers tube diameters range from 1 to 2 inches and tube lengths from 8 to 20 ft. although usually the shorter tubes are preferred where head room is limited. The horizontal vaporizers can use longer tubes. The horizontal thermosyphon reboilers appear to have been developed in the petroleum industry and information and data on their design and performance are unavailable. However, the basic approach is to estimate the fraction vaporized, then determine the circulation rate from the piping layout and estimated exchanger pressure drop, and finally calculate the heat transfer rates using either convective liquid flow

279

equations or, if desired, corrected for two-phase flow. The calculations are repeated and adjusted until the vapor fraction calculated and estimated agree within an acceptable limit. The fraction vaporized may be as large as 25%. For the in-tube vaporizers a flow rate per tube is chosen and the calculations for heat transfer and pressure drop made for the chosen tube size and length. The total number of tubes for the required service can then be determined. The pressure drop in the piping is calculated and the total available head compared to the pressure drop. The flow rates are adjusted until a satisfactory agreement of the available and used pressures are obtained or a valve (or restriction) adjusted to balance the pressures. In the application of reboilers to distillation columns the choice of reboiler type and the piping arrangement can affect the performance of the reboiler. Jacobs (71) describes various feed arrangements and column internal baffling. Considering only the simplest arrangements Figure 5.34 shows the column feed, G, mixes with the liquid discharge, L, from the reboiler and the mixture, F, is the feed to the reboiler. The effect of this arrangement is it produces the lowest AT in the reboiler, sacrifices a stripping plate, and is poor for thermal fouling because of a long holdup at high temperature. It is, however, a very simple arrangement. Figure 5.35 shows a gross bottom feed system where all the column downflow, G, flows directly through the reboiler in a once through manner. This results in the best average reboiler ΔT, has low holdup and, therefore, best for fouling liquids but because of the once through operation and to obtain the best reboiler performance this arrangement is limited to a maximum boil up of 30% of the feed, F. A combination of the two above methods is the mixed bottoms feed system shown in Figure 5.36. The average ΔT will be slightly less than the gross bottom feed (Fig. 5.35) but it is satisfactory for most fouling services and is a flexible system. The column baffling is more complex.

280

281

282

5.9. Special Considerations As the boil up rate is very sensitive to the ∆T it is necessary to determine the range of the ∆T to provide for the anticipated boil up range under both clean and fouled conditions. This ∆T range should be within the controllable limits of the heating media. The boil up is basically controlled by varying the heating medium temperatures. For a condensing media a pressure control is used and for a sensible heat source media a by-pass control is used but here both temperature difference and heat transfer coefficient of the heating medium changes. For a condensing media, a problem arises when boiling at low loads with a clean vaporizer that the required pressure may be so low (or a vacuum) that condensate removal becomes difficult. For a sensible heat source, the amount of by-pass may so reduce the flow in the vaporizer that problems in distribution or accelerated fouling may occur. The vaporizer should have liquid in it and be started up at the lowest ∆T possible. It is possible in those cases of large fouling allowances, excessive conservatism in design, or underestimated coefficients that the design ∆T can be large enough for a clean reboiler to get into the film boiling regime. The design flux may also be low enough so that it can be met in the transition or film boiling modes. Under these conditions the effect of ∆T is reversed from that of nucleate boiling thus the controls will be ineffective and the rate of fouling may be increased. If a condensing medium pressure becomes too low at low loads or clean condition, then a partial flooding of the surface should allow increasing the pressure into a controllable range. The surface should be flooded a fixed amount. Trying to control the boil up by varying the amount of flooding is difficult due to the slow response. With partial flooding consider the effect of stresses especially in horizontal units where the stresses are different in the flooded and unflooded tubes. Wide boiling range mixtures, where the boiling point of the heavy component exceeds the heating medium temperature, require reboilers in which circulation occurs even in the convective heat transfer region otherwise the light component will be stripped out and stagnation will occur. Operation near the critical pressure have several problems. The density difference between vapor and liquid is low so the driving head for circulation is low. The maximum heat flux and the critical ∆T are also low. Operating under film boiling conditions might be considered as a possible solution. Low pressure (vacuum) operation requires careful analysis of the effect of static head on the boiling point and the possible temperature pinch thus caused at the top of the liquid zone. This boiling point elevation also results in much greater liquid preheat lengths and also reduces the available head for circulation. The large density differences cause higher acceleration pressure losses which result in reduced circulation. Low pressures require larger cavities for nucleation thus suppressing nucleate boiling coefficients, but the use of enhanced surfaces can be effective. Low ∆T operation (8°F) may sometimes be necessary because of process economics. Here the problem is nucleation and the use of enhanced surfaces is required. However, prediction of heat transfer coefficients is very uncertain and experimental tests on the specific surface liquid combination are recommended. Boiling curves developed from a small single tube pool boiling apparatus will provide a guide to the basic heat transfer coefficients which can be modified as necessary from the above discussed methods.

283

Very high ∆T is often encountered when vaporizing a low boiling point liquid; e.g., ammonia, with low pressure steam and here another problem of potential freezing of the heating medium must be considered. An available high temperature medium is another cause for high ∆T. At high ∆T the three limitations of film boiling, mist flow, and instability must be evaluated by the methods given above. Instability can be controlled within limits by throttling of the recirculation line or can be avoided by a shellside reboiler. Mist flow only represents a low heat transfer coefficient. Film boiling is a possible mode of operation if the potential for transition boiling is avoided where the control criteria are reversed. A possible solution to transition and fihn boiling is the use of medium- to high-finned tubes where the temperature drop along the fins is such that nucleate boiling occurs near the fin tips. Boiling on fins has been tested in the laboratory but no publication of a commercial application has been found. 5.9.1. Examples of Design Problems To design a heat exchanger requires that one first specify essentially all the dimensions of an exchanger and then one can proceed with the heat transfer calculations to determine whether the assumed design will give the desired performance and, if not, then the initial design is modified and the calculations repeated until an acceptable match of design and performance is obtained. A design problem is, therefore, a series of rating problems. In the rating of an exchanger its dimensions are known and only the heat transfer calculations are required; however, for vaporizers these calculations still involve considerable iterative trials to converge on an answer. It is obvious that the better the initial guess matches the final design the less the amount of calculation; hence, here is where experience is beneficial to the designer. In an attempt to aid a novice designer each of the examples below show in the initial steps one way of getting an initial design. In the second example, 5 steps are also used, however, in step 4 the guess of fraction vaporized is based on experience or as in this case prior knowledge of the experimental value.

5.10. Example of Design Problems for Trufin in Boiling Heat Transfer

5.10.1. Design Example - Kettle Reboiler Size a kettle reboiler to transfer 43.3(106) Btu/hr to vaporize a hydrocarbon mixture at 170 psia using steam available at 395°F. The critical pressure of this liquid is 434 psia and it has a boiling range of 60°F. The boiling temperature is 330°F. Design the reboiler using 3/4-in. OD tubes on 1.125-in square pitch. We will estimate the latent heat as 144 Btu/Ibm and liquid density as 41 lbm/ft3. Step 1. Calculate or estimate heating medium, tube wall, and fouling coefficients. For this example (and in order to compare to a test unit) the steam coefficient is 2000 and the tube wall is 4800. This reboiler was claimed to be clean; hence,

fwo

o Rhh

R ++=11

Ro = 1/2000 + 1/4800 = 0.000708

Step 2. Calculate the mixture correction factor, Fm from eq. 5.38.

Fm = exp(– 00.015 x 60) = 0.41 Step 3. Calculate B and RoB and find q. From eqns. 5.8a, 5.10 and 5.62.

A* = 0.00658(434).69 = 0.435

F(P)2 = 1.8 ( ) 17.434170 = 1.535

B = [(0.435)(1.535)]3.33 = 0.26

Correcting B for the mixture, use fig. 5.29 at BR of 60°F,

B = 0.26 x 0.41 = 0.1066 hence

RoB = 0. 1066 x 0.000708 = 7.5(10-5) At ΔT=65 Figure 5.33 gives q/B=280,000 hence

q = 0.1066 x 280,000 = 29,848 Btu/hr ft2

Step 4. Calculate single tube maximum q1, eq. 5.5

284

q1max = 803(434)(170/434).35 (1 – 170/434).9 = 160,488 Btu/hr ft2

Step 5. Preliminary estimate of bundle size For a bundle

qb = q1max Φb where

Φb = 2.2(πDBL/AB s). If we approximate

Φ = 2.2Ψ by letting Ψ be (for square pitch)

oB

t

pdLD

B

s

BdD

pLDA

LD

t

oB πππ

ππ

2

4

4

2

2=

×=

Now let

max1

242.2

qq

dDp b

oB

tb =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=Φ

π

ftdq

qPD

ob

tB 118.2

)12/75.0)()(848,29()488.160()12/125.1)(4)(2.2()4)(2.2( 2

max12

===∴ππ

As the above approximation ignores the additional effect of circulation on the boiling coefficient, DB = 2 ft. B

Step 6. Calculate bundle maximum flux, eqn, 5.23 For U-tube on this pitch a total of 180 U-tubes or 360 ends will form a 2 foot diameter. For one foot of bundle length

0889.0)12/75(.)360(

)1)(2(===Ψ

πππ

s

BA

LD

Φb = 2.2Ψ = (2.2)(0.0889) = .1956

maximum bundle flux

q = Φbq1max

285

q = 0. 1956 x 160,488 = 31,392 Btu/hr ft2

Step 7 Calculate the bundle heat transfer For a 2 ft bundle assume q = 28,600 Btu/hr ft2 and calculate heat transfer coefficients based on this flux and the values obtained in steps 3 and 5. From eqn. 5.8 calculate hnbl

hnbl = (0.435)(1.535)(28,600)0.7 = 878.3 Btu/hr ft2°F Step 8. Calculate natural convection coefficient, eqn 5.7 We have insufficient information to calculate this coefficient but we will assume it is 40 Btu/hr ft2°F. Step 9. Calculate bundle coefficient, eqn. 5.22

hb = 878.3 x 0.41 x 1.5 + 40 = 580.1 Btu/hr ft2°F

U = 1/(115 80.1 + 0.000708) = 411.2 Btu/hr ft2°F

q=UΔT

q = 411.2 x 65 = 26,730 Btu/hr ft2°F The measured coefficient for this reboiler (72) was 440 Btu/hr ft2°F or 7% higher. Step 10. Check bundle design. Step 9 heat flux (26,730) is less than the maximum allowed bundle flux of step 6 (31,392) hence OK. Since Φb in step 6 is greater than 0. 1 no vapor lanes or larger pitches are required; therefore, bundle is OK. Step 11. Size the bundle.

Required length = 1963.360730,2661043××

× = 22.8 ft

This length checks with the test unit length of 23 ft. Step 12. Check for entrainment. Number of vapor nozzles per eqn. 5.64

Nn = 2523× = 2.3 round up to 3

Vapor per nozzle

286

Wn = 3144000,300,43× = 100,231 lbm/hr

Entrainment limit, eq. 5.63

VL = 2290 X 1.7255.

725.1415

⎥⎦⎤

⎢⎣⎡

− = 1409 lbm/hr ft3

(Note dynes/cm = [Ibf/ft] / 6.86 x 10-5) Therefore the vapor volume/nozzle = 100,231/1409 = 71.1 ft3. If the shell is 25 ft long then the cross section area for vapor above the liquid level is 71.1/8.33 = 8.537 ft2. The shell diameter is then determined from tables of segmental areas; however, for first approximation assume a liquid level at the center line then

Ds = (2 x 8.537 x 4/π)0.5 = 4.66 ft This is a large shell compared to the bundle diameter; therefore, consider the use of entrainment separation devices. 5.10.2. In-Tube Thermosyphon - Example Problem Size a vertical thermosyphon vaporizer to transfer 1,483,000 Btu/hr to an organic liquid with the following properties: boiling point @ 17 psia = 185.5°F, = 0.45, latent heat= 154.8 Btu/lb, lpc lμ = 0.96 lb/ft. hr, μ v

= 0.0208 lb/ft. hr, k = 0.086 Btu/hr ft. °F, and densities lb/ft3 liquid = 44.8, vapor = 0. 18 1, cP = 593.9 psia. Heating medium is steam at 217.4°F. Use 1-in. 12 BWG carbon steel tubes 8 ft. long. For this problem assumes no other fouling is present. This example is based on a test by Johnson (73). Boiling point elevation for 8 ft static head is 9°F. The heat source is steam condensing on the outside of the tubes with a coefficient of 1000. Step 1. Calculate Ro

Rw = )891)(.30()1)(12/109.0( = 0.00035

Ro = 1000

1 + 0.00034 = 0.00135

Step 2 Calculate the maximum limiting flux using eqn. 5.37

qmax = 16066 ( )35.2

812/782.

⎥⎥⎦

⎤

⎢⎢⎣

⎡(593.9).61

25.

9.59317

⎟⎠⎞

⎜⎝⎛

(1 – .0286) = 22,548 Btu/hr ft2

287

288

This is a high flux and would require a 22548 x.00135 = 30.4 temperature drop across the steam tube wall. As only 217.4 – 185.5 = 31.9°F is available it is obvious the operation is well below the maximum. Step 3. Determining a boiling flux Calculate a nucleate boiling flux using Figure 5.33 Here

B = [0.00658(593.9).69(1.8)(17 / 593.9).17]3.33 = 0.1214 (5.62) hence

RoB = 0.00135 x 0.1214 = 0.00016 For ΔT = 31.9° from the figure we should calculate

q = 44,000 x 0.1214 = 5342 Btu/hr ft2

This flux represents only the nucleate boiling coefficient and this is a lower limit. To include a two-phase convective effect assume a 50% increase in the boiling side. Hence, from the above flux and ΔT get U (167.4), subtract the Ro (.00135) resistances to get the boiling coefficient (216.4) increase the nucleate coefficient by the assumed ratio (= 324.6), then recalculate the new overall coefficient (225.7) and heat flux (7200). Step 4. Determining the recirculation rate.

Vapor per tube = 8 x 0.2618 x 7200 / 154.8 = 97.4 lb/hr Now one has to assume the fraction vaporized. We will short cut this trial and error by assuming the experimental value of 9%. Therefore, the feed rate/tube = 97.4/.09 = 1082 lb/hr. Step 5. Calculate basic values needed to check pressure drop, circulation rate, and preheat zone.

Gt = 1082 / (π x (.782)2 / [4 x 1441) = 324,404 lb/ft2 hr

V = 324,404 / (3600 x 44.8) = 2.01 ft/sec

Re = .782 x 324,404 / (12 x .96) = 22,021 From friction factor charts f = 0.0075 Hence in the liquid zone the head loss per foot of tube is by eqn. 5.51

ΔH = (4 x .0075 x 12 / .782) x 2.012 / 64.4 = 0.029 ft/ft Using an average vaporization of 9/2 = 4.5% we can calculate Xtt, (eqn. 5.29)

398.10208.096.0

8.44181.0

045.0045.1X

11.057.0

tt =⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ −

=

289

Next get (eqn. 5.55) 2

ttΦl

2ttΦl = 1 + 20 / 1.398 + (1 / 1.398)2 = 15.82

The two-phase AH based on average liquid content of 0.955 is

ΔH = 15.82 x .029 (0.955)2 = 0.42 ft/ft The two-phase density due to slip is (eqn. 5.48 and 5.49)

Rv = 82.15/11− = 0. 749

ρtp = (.749 x .181) + [(1 – .749) x 44.8] = 11.38 lb/ft3

The boiling zone static head loss is

ΔH = 11.38/44.8 = 0.254 ft/ft Using eqn. 5.50 for PΔm

Gt = 324,404/3600 = 90.11 lb/ft2 sec

( ) ( ) ( ) ft751.0lb/ft64.33749.181.

09.251.8.44

09.12.32

11.90ΔP 2222

m ==⎟⎟⎠

⎞⎜⎜⎝

⎛

×+

×−

=

Heat transfer in preheat zone; eqn. 5.25

( ) ⎟⎠⎞

⎜⎝⎛ ××

⎟⎠⎞

⎜⎝⎛ ×

=782.

782.12086.086.

96.45.22021023.0h3/1

8. = 121.1 Btu/hr ft2 °F on outside area

Therefore

U = 1 / (1 / 121.1 + .00135) = 104. 1 Btu/hr ft2 °F Using a ΔT = 31°F the temperature rise in preheat zone is

45.1012312618.1.104

×××

= 1.86 °F/ft

Step 6. Estimating preheat and boiling lengths. Assume preheat zone = 3 ft Friction loss in preheat zone = 3 x .029 = 0.087 ft

290

Effective submergence at this point = total head (8) – friction loss (.087) – preheat zone (3) = 4.91 ft liquid which is equivalent to a boiling point elevation of

(4.91/8) x 9 = 5.53 °F Length required for this temperature rise is 5.53/1.74 = 3.18 ft. Close enough. Check on circulation and pressure drops Available head = 8 ft liquid neglecting liquid line losses Overall momentum loss = .751 ft Friction losses

boiling zone 5 x .42 2.100 preheat zone .087

Static heads

boiling zone 5 x .254 1.270 preheat zone 3.000

7.21ft Considering there is some losses in the liquid recirculating line the above agreement is close enough. Step 7. Calculate heat transfer in boiling zone From eqn. 5.8

hnbl = 0.00658(593.9).69(7200).7[1.8(17 / 593.9).17 = 266.2 x .782 / 1 = 208. 1 Btu/hr ft2 °F on OD area

From eqn. 5.28

226.2213.0398.1135.2F

73.0

ch =⎟⎠⎞

⎜⎝⎛ +=

Determines from eqn. 5.31

Retp = 22,021 x 2.2261.25 = 59,874

s = 1 / {1 + [2.53(10-6) x (59,874)1.17]} = 0.504 From eqn. 5.27

hcb = 121.1 x 2.226 = 269.6 Btu/hr ft2 °F on an outside area basis From eqn. 5.26

291

hb = (.504)(208.1) + 268.6 = 374.5

Adding the steam and wall resistance to obtain U for the boiling section

U = 1 / [(1 / 374.5) + 0.00135] = 249 Step 8. Calculate average coefficient for tube and area An average coefficient for the preheat and boiling zone is

Uav = (3 x 104.1 + 5 x 249.0)/8 = 194.5 Btu/hr ft2 °F Required area = 1,483,000/194.5 x 31.9 = 239 ft2 vs. 201 ft2 in the test vaporizer. Thus, this simplified calculation came within 19% of predicting the test results which is acceptable. In design case after calculating the required area (239 ft2) a safety factor should be added to allow for the error spread in all the involved equations. Also fouling should be considered and should be included in the term Ro term. We did not include fouling in this example since we were trying to compare the calculation method with data obtained in a clean vaporizer. 5.10.3. Boiling Outside Trufin Tubes - Example Problem To illustrate the value of and methods of calculation for Trufin tubes in boiling, a comparison of the performance of a plain surface and finned surface tube will be made. The plain tube is 0.75 and o.d., 18 B.W.G. wall and 90/10 Cu-Ni. The Trufin is Wolverine Cat. No. 65-265049-53. This tube has a surface area of 0.640 ft2/ft with an Ao/Ai ratio of 4.61, a fin height of 0.057 and width of 0.012 inches. There are 26 fins per inch. The tubes are heated with steam having a coefficient of 2000. A pure hydrocarbon having a critical pressure of 489 psia will be boiled at 100 psia with an overall temperature difference of 10'F. The bundle factor, Fb, is 1.5 and the surface factor, Fs, for this temperature is 1.0 for the plain tube and 1.5 for the Trufin tube. Evaluation of the Plain Tube Performance 1. Calculate Ro.

where Ro = wall resistance + tube-side resistance

( )( )( ) 000162.652.29

75.12/049.R wall ==

hwall = 6174

( ) 00074.652.2000

75.6174

1R ο =+=

2. Calculate the single tube boiling coefficient using eq. 5.32

hnbl = (5.43)(10-8)(489)2.3[1.8(100 / 489)0.17]3.33 ΔT2.3 = 0.24 ΔT2.3

292

assuming the maximum possible ΔT of 10°F

hnbl = (0.24)(10)2.3 = 47.9

3. Calculate the bundle boiling coefficient, overall U, and the heat flux then check the assumed ΔT. Assume a natural convection coefficient, hnv = 40, and using the bundle factor of 1.5 in eq. 5.22.

hb = (47.9)(1.5) + 40 = 111. 8

U0 = 1 / (1 / 111.8 + .00074) = 103.2

the available boiling ΔT is then

ΔTb = 10 – (10)(.00074)(103.2) = 9.2°F

This is not close enough to the assumed value of 10 so repeat steps 2 and 3. 2’ Assume ΔTb = 9.2

hnbI = (0.24)(9.2)2.33 = 42.25 3' hb = (42.25)(1.5) + 40 = 103.4

U0 = 1 / [(1 / 103.4) + .00074] = 96 4. Calculate available boiling ΔT.

ΔTb = 10 – (10)(.00074)(96) = 9.29°F

q = UΔT = (96)(10) = 960 Btu/hr ft2 (outside area) Evaluation of the Trufin Tube Performance 1. Calculate Ro

The inside area basis will be used

( )( )( )( ) 0.00013

579.2953.12/049.R wall ==

Ro (wall + steam resistance) = 0.00013 + 1/2000 = 0.00063

2. Calculate the boiling coefficient using eq. 5.32 with a surface factor of 1.5

hnbl = (1.5)(0.24) ΔT2.33 = 0.36 ΔT2.33

assume a boiling ΔT of 8°F

hnbl = (0.36)(8)2.33 = 45.8

293

using eq. 5.22 with Fb = 1.5 and hc =30

hb = (45.8)(1.5) + 30 = 98.7 3. Adjust for fin efficiency.

Figure 5.37 is used. This was derived for the case boiling liquids on fins where h = bΔT2.

using the assumed ΔT of 8 and hb = 98.7

b = 98.7 / (8)2 = 1.542

the abscissa for fig 5.37 is then

( )( )( )( ) 320.812/018.029

542.1212057. =×

an efficiency of 87% is read and

hb = (98.7)(.87) = 85.9 on an outside area basis

On an inside area basis;

hb (85.9)(4.61) = 396

U = 1/ (1/396 + .00063) = 317

q = UΔT = (317)(10) = 3170 Btu/hr ft2 (inside basis)

Check assumed value of boiling ΔT of 8°F.

ΔT (wall + steam) = (0.00063)(3170) = 2.0

ΔTboiling = 10 – 2 = 8°F

This checks with assumed value. If not then, repeat steps 2 and 3 with a new value.

Comparison of Performance Since the area per foot of the two tubes are different, comparison will be made on a per foot of length basis. 1. For plain tube

q/foot = (960)(.1963) = 188.5 Btu/hr-foot length 2. For Trufin

q/foot =(3170)(.640/4.61) = 440.1 Btu/hr-foot length

Therefore the performance ratio of Trufin to plain is: 440.1 / 188.5 = 2.3

294

Table 5.1

Simple dimensional equation for nucleate pooling boiling heat transfer (after Borishanski)

Liquid Pressure

range atm. A*

from exp A*

Eqn 5.9 Critical

pressure atm. No. in

Fig 5.18

Water Water Water Water Water Water

Pentane

Heptane (80%) n-heptane Benzene Benzene Diphenyl

Methanol Ethanol Ethanol Butanol

R11 R12 R12 R13

R13B1 R22 R113 R115

RC318

Methylene chloride

Ammonia Methane

1 – 70

1 – 196 0.09 – 1 1 – 72.5 1 – 170 1 – 5.25

1 – 28.6

0.45 – 14.8 0.45 – 14.8

1 – 44.4 0.9 – 20.7

0.9 – 8

0.08 – 1.39 1 – 20.7 1 – 59

0.17 – 1.38

1 – 3 1 – 4.9

6 – 40.5 2.8 – 10.5 17 – 39

0.4 – 2.15 1 – 3 8 – 31

3.6 – 27

1 – 4.5 1 – 8 1 – 42

1.61 1.58 2.28 1.76 1.75 2.26

.429 .464 .642 .417 .520 .441

(.272) .720 1.019 (.173)

.768 [.681]

.956 1.37 [1.01]

.705 1.744 [.976] [.941]

.488 1.49 [.934]

1.23 [.984]

(.752) 1.54 1.06

1.66 1.66 1.66 1.66 1.66 1.66

.449 .381 .381 .588 .583 .425

.815 .701 .701 .547

.539 .516 .516 .496 .508 .586 .453 .425

.394

.677 1.039 .563

216.9 216.9 216.9 216.9 216.9 216.9

32.8 25.9 25.9 48.1 48.1 30.4

78.0 62.6 62.6 43.8

42.9 40.3 40.3 37.9 39.1 48.4 33.4 30.6

27.3

59.6 110.8 45.6

1 2 3 4 5 6

7 8 9 11 -- --

13 10 12 14

-- 15 -- -- -- -- -- --

--

-- -- --

Values shown in round brackets ( ) are uncertain. Values shown in brackets [ ] relate to the use of Equations 5.11 for F(P).

295

NOMENCLATURE A* Constant defined in equation 5.9. dimensionless As Surface area. ft2 B Constant defined in equation 5.62. dimensionless BR Boiling range, dew point-bubble point. °F cp Specific heat, for liquid and clpc pv, for vapor Btu/lbm °F

d Tube diameter, do for outside and di for inside. ft. Dp Diameter of tube bundle. ft. Ds Shell diameter. ft. Fb Tube bundle correction factor. dimensionless Fcb Chen Factor. dimensionless Fm Mixture correction factor. dimensionless f Friction factor. dimensionless G Mass velocity. Ibm/ft2 hr Gt Mass velocity based on total flow. Ibm/ft2 hr Gtmax Total mass velocity based on minimum cross flow area. Ibm/ft2 hr Gmm Mass velocity at beginning of mist flow. Ibm/ft2 hr g Gravitational constant. ft/hr2

gc Conversion constant. Ibm ft/lbf hr2

H Height. ft Hl

Height of liquid zone. ft ΔH Head loss per foot of tube. ft/ft h Film heat transfer coefficient; hb = boiling, hc = convective, hf film, =

liquid, hlh

r = radiation, hcb = convective boiling, hft = film total, hnb = nucleate boiling, hnbl = single tube nucleate boiling.

Btu/hr ft2 °F

296

K Constant in equation 5.23. dimensionless k Thermal conductivity. Btu/hr ft2 °F L Length. ft Lc Minimum unstable wave length. ft m Exponent. dimensionless N Number of tube rows. dimensionless Nn Number of vapor nozzles. dimensionless Nu Nusselt number. dimensionless P Pressure. lbf/ft2

cP Critical pressure. lbf/in2

Pr Reduced pressure = P/PC. dimensionless Pr Prandtl number. dimensionless Psat Saturation pressure at plane interface. lbf/ft2 pt Transverse tube pitch. ft ΔP Pressure drop; ΔPT = total, ΔPs =static, ΔPm = momcntum, ΔPf = friction. lbf/ft2

q Heat flux; qmax = maximum, qmf = minimum film, qnc = natural convection, qcr

= critical. Btu/hr ft2

Re Reynolds number. dimensionless Rl, Rv Volume fraction of liquid, vapor. dimensionless Ro Sum of thermal resistances other than the boiling resistance. hr ft2 °F/Btu rc Radius of bubble. ft s Chen suppression factor. T Temperature; Ts = steam, Tw = wall, Tsat = saturation. °F ΔT Temperature difference; ΔTb = tube wall-saturation, ΔTc = critical, ΔTO = tube

waIl-bulk liquid, ΔTmin = difference at minimum film boiling coefficient. °F

V Velocity. ft/hr

297

V∞ Velocity approaching tube. ft/hr VL Vapor load. lbm/hr ft3 Xtt Martinelli parameter, equation 5.29. x Weight fraction of vapor. y Mole fraction low boiling component in liquid. GREEK β Coefficient of thermal expansion. 1/°R Γ Flow rate per unit length. Ibm/hr ft λ Latent heat; λe, λ’ = effective latent heats see eqn. 5.17, 5.19. Btu/Ibm

μ Dynamic viscosity; lμ = liquid, vμ = vapor lb./ft hr

ρ Density; ρl = liquid, ρv = vapor, ρb = bulk average, ρtp = two-phase. σ Surface tension. lbf/ft v Specific volume change liquid-vapor. ft3/lbm Φb Bundle maximum flux correction factor. dimensionless

2vtt

2tt Φ,Φl Martinelli two phase factors. dimensionless

298

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