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