[advances in heat transfer] advances in heat transfer volume 12 volume 12 || heat transfer in flows...

Heat Transfer in Flows with Drag Reduction

YONA DIMANT and MICHAEL POREH

Faculty of Civil Engineering, Technion, Israel Institute of Technology, Hai fa, Israel

I. Introduction . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . 11. Models of Flows with Drag Reduction

111. Previous Heat Transfer Models IV. Analysis of Heat Transfer . . . . . . . . . . .

A. Analysis of Thermally Developed Flows . . . . . B. The Influence of Temperature-Dependent Fluid Properties

D. The Entrance Region in the Constant Temperature Mode . . . . . . . .

VI. Conclusions . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

C. The Entrance Region in the Constant Flux Mode . . .

V. Comparison with Experimental Data

. . . . . 17

. . . . . 19

. . . . . 86

. . . . . 88

. . . . . 89

. . . . . 91

. . . . . 95

. . . . . 98

. . . . . l o o

. . . . . 1 0 9

. . . . . 110

. . . . . 1 1 1

I. Introduction

The fascinating effects of minute quantities of long chain polymer molecules on the frictional drag of turbulent flows have stimulated considerable research of the phenomenon known as drag reduction or the Toms effect. Drag reduction is usually defined as

at the same Reynolds number. Value of DR above 75% have been recorded in dilute solutions of poly(ethy1ene oxide), polyacrylamides, Guar gum, and other polymers.

A large number of mechanisms of drag reduction have been proposed, but it appears from the literature that no rigorous theory that can fully explain the phenomenon and predict its characteristics has been formulated C1-41. On the other hand, considerable data have been collected that made it possible

I1

DR = 1 -fsolutionlfsolvent

78 YONA DIMANT AND MICHAEL POREH

to formulate phenomenological models and semiempirical correlations, similar to those employed in the description of turbulent Newtonian flows, for flows with drag reduction.

Naturally, primary attention has been given to the effect of the polymers on the turbulent motion and the momentum transfer which determines the drag reduction, but interest in diffusion and heat transfer in these flows has also been large. More than a dozen experimental investigations and a similar number of theoretical and semiempirical models of heat transfer in flows with drag reduction have been presented in the past decade. In general, the proposed models resemble classical heat transfer models which had been used success- fully for predicting heat transfer in various turbulent flows. All the models indicate that drag reduction is associated with a reduction in the heat transfer between the wall and the fluid, but large differences exist between some of the models. As shown by Smith el al. [S], models that correlate satisfactorily some sets of data fail to agree with other sets. Moreover, measurements that appear to be under similar conditions yielded discordant records of heat transfer rates.

Some of the scatter in the experimental data is undoubtedly due to shear and thermal degradation of the polymers, which can change the properties of the solutions along the test section. However, consistent differences between sets of measurements in several carefully conducted experiments, which cannot be attributed to degradation effects alone, have not been explained and have therefore raised doubts as to whether the classical analogy between momentum and heat transfer can be applied in a universal form to flows with drag reduction.

We shall review in this work the models proposed to describe both momentum and heat transfer in flows with drag reduction and examine their limitations. It will be shown that in flows with drag reduction one, in general, has to distinguish between the constant wall temperature and the constant heat flux modes of heat transfer, which has not been done in any of the previous works.

Analysis of the heat transfer in established conditions and in the thermal entrance region and analysis of the effect of temperature-dependent fluid properties for the two modes of heat transfer are then presented. The analysis is based on a simple phenomenological model for describing the velocity profiles in flows with drag reduction which uses Van Driest’s mixing length expression with a variable damping parameter [4]. The analysis yields an implicit correlation between the heat transfer rate and the friction not contain- ing any undetermined coefficients. It shows that several effects that have so far been casually neglected, assuming a similarity to turbulent Newtonian flows, are quite significant in flows with drag reduction.

Taking these effects into consideration, the experimental data are re- analyzed and are found to be in good agreement with the proposed model.

FLOWS WITH DRAG REDUCTION 79

11. Models of Flows with Drag Reduction

The earliest descriptions of flows with drag reduction were proposed by Meyer [6 ] and Elata et al. [7], who employed a two-layer model to describe the velocity profile; a viscous sublayer where

u+ = y+ (1)

(2)

and a log region where

U' = 5.75 logy' + 5.5 + Au' The term Au', which describes the upward shift of the log profile in the conventional law of the wall representation, was empirically related to the shear velocity and polymer characteristics by the equations

Au' = u log(V*/V:); V* > V: (3a)

Au' = 0; v* I v: (3b)

where V: is the shear velocity at the onset of drag reduction and LY is a concentration dependent parameter.

Integration of u' over the area of the pipe yields an expression for the friction coefficient$ At high Reynolds numbers and small to moderate values of u', the contribution of the sublayer to the integral of u' is negligible, yielding for V* > V: the equation

llfl'' = 4.0 log(Re f I/') - 0.4 + Aui/& (4) Plotted in Prantdl coordinates, f - 1 / 2 versus Ref''', Eq. (4) gives straight

lines that intersect the Newtonian line (u = 0) at (Ref''2),,. Dimant [8] proposed that Eqs. (3a) and (3b) be replaced by one equation:

AU+ = (44) iogp +(v*/v~",)~-J ( 5 )

which, as shown in Fig. 1 [8a] gives a better description of the data. Both Eqs. (3) and ( 5 ) seem to fail at very large shear rates.

A three-layer model describing the velocity profile in drag reducing fluids has been proposed by Virk [9]. Virk suggested that drag reducing polymers create a new intermediate layer, between the viscous sublayer and the log region, which he termed the elastic sublayer. The velocities in this layer, according to Virk, can be described by a universal law :

u' = 11.7 lny' - 17.0; yv' < y + < ye' (6)

where y,' is given by the intersection of Eqs. (1) and (6) and ye+ is given by the intersection of Eqs. (2) and (6). The relation between Au' and the thickness of the elastic sublayer is thus given by

Au' = 9.2 ln(y,+/yy+) (7)


FIG. 1. Typical friction curves near the onset of drag reduction. Data from Wells and Spangler [8a].

Except for small values of R+ and large values of Au+, the contributions of both the viscous and elastic layers to the integral of u+ are small (see Table 3, Virk [9]). The details of the sublayer do not affect the calculation of the friction coefficient, and Virk’s model gives the same results as the two-layer models. Virk termed this case the “polymeric regime.”

In the other extreme case, when the elastic sublayer becomes large, the contribution of the log region to the integral of u+ is negligible and the friction coefficient is described by a universal law obtained by integration of (6) :

(8) Equation (8), termed the “maximum drag reduction asymptote,” describes reasonably well the maximum records of drag reduction obtained at small values of R+ in many investigations.

A very similar, but slightly more complicated three-layer model has been offered independently by Tomita [lo].

Several investigators [4, 11, 123 have lately proposed similar models that give smooth velocity distributions. The authors [4] have proposed a model based on Van Driest’s expression for the mixing length:

where A + = 26 and k = 0.4 in Newtonian fluids.

velocity gradient and shear stress can be obtained:

l/f112 = 19.0 log(Ref’”) - 32.4

I = ky[I-exp(-y+/A+)] (9)

Using this expression, the following relations between the dimensionless


where T + = T/T, and z, = P V * ~ . In Van Driest’s original solution, the velocity profile is calculated using the constant shear stress approximation, 7 = T, or 7’ = 1, which gives:

duo+ 2 -= dy +

1 + { 1 + 4k2y+’ [ 1 - exp ( - y+ /A+) ] ’ } ‘ I 2

Integration of Eq. (11) gives uo+ = y + at small values of y+, and u+ = ( l / k ) In y + + B ( A + ) at large values ofy’, similarly to Eq. (2). It was therefore natural to exmine whether Van Driest’s mixing length model could be used to describe the entire velocity profile in flows with polymers. Such a model can be used only if one does not adopt the constant shear stress approximation. This approximation gives, for small R + / A + , a large region where u,,’ = y + . As illustrated in Fig. 2, the velocities there might exceed those in a fully

PARABOLIC PROFILE

FIG. 2. The solution uo + based on the constant shear approximation for large and small R + / A + .

laminar flow, which are given by the parabolic equation

u+ = y’(l -y+/2R+) (12)

This limitation has been overlooked by previous investigators. Van Driest’s model, as well as the original Prandtl model Z = ky, fails to

describe faithfully the velocity profile in the central region of the pipe. Various methods for correcting this deficiency have been proposed. Some investigators use a constant eddy viscosity model in this region, whereas others modify the velocity profile by using the “law of the wake.” The differences between the velocity profiles obtained by the various methods are not large and the choice of a particular method is largely a matter of taste. The authors suggest modi- fying the velocity profile ul+ using the law of the wake by writing that

u+ = u l + + u2+ (13)


where u1 + is the solution of Eq. (lo), with t+ = 1 - y+/R+ for pipe flows. The limit of Eq. (lo), with T+ = 1 -y+/R+ for small values of R+/A+ is

du,+/dy+ = 1 - y + / R + (14)

which gives the correct parabolic equation for laminar pipe flows. This result implies that uZ+, which is zero near the wall, has to vanish identically for small values of R+/A+. In other words, the deviation from the "law of the wall" has to decrease with drag reduction.

This condition is satisfied by the following equation proposed for u2+ :

n 2k

UZ' = - [l -cos(ny++lR+)][l -exp(-2R+/A+)] (15)

where ll = 0.67 is a universal constant for pipe flows. The value of ll has been determined so that the value of the Newtonian friction factor at Re = 5 x lo5 would satisfy Eq. (4) with ci = 0. One notes that for large values of R+/A+, which is always the case if A + = 26, the exponential term in Eq. (15) vanishes and uz+ becomes identical to Coles' wake function.

Velocity profiles calculated using this model are compared with pipe flow measurements in Figs. 3-7 [9, 12a]. Friction coefficients for various values of A + obtained by numerical integration of u+ are plotted in Prandtl coordinates in Fig. 8. Poreh and Grunblatt [13] have extended this model to flat plate flows with polymer solutions using for z+ the distribution

z+ = exp[-3.45(~~+/6+)~]

which appears to be a satisfactory approximation for both laminar and turbulent boundary layers with zero pressure gradient. Figure 9 compares the velocity profiles calculated by Poreh and Grunblatt, using II = 0.8 1, with the measurements of Kumor and Sylvester [14].

It must be realized that the value of the parameters appearing in these models, namely the parameter A + in the present model and the parameter Au+ in Virk's model, are determined by the friction measurements. Un- doubtedly, it is desired to find a correlation between these parameters and the polymer properties. Virk has proposed such a correlation on the basis of friction measurements in pipe flows in the polymeric regime. Since in this region the present model is practically identical to Virk's model, one can use his correlation to calculate the dependence of A + on the polymer properties. Such a calculation gives [4]

(16) A u ' I a N 40 log(A+/A,+ +4.0) - 28

where A,+ = 26. The relation between A + and the polymer properties can


be approximated therefore by

A+/A,+ = 5.0~1 +(v*/v341~/40 - 4.0 (17)

where V: is the shear velocity at the onset of drag reduction and A = c r / ~ is Virk's parameter given by

A = KN3/2(.N~/M)1/Z (18)

N being the number of chain links in the polymer macromolecules, JV Avogadro's number, M molecular weight, and c the conventional mass con-

50

4 0

U+

30

20

10

01. , , ,1 ' " ' , , , , ' ' ' " e c i c l ' ' I 10 lo* f 10'

FIG. 3. Measured and calculated velocity profiles. Data from Virk [9].

I * , , , , , , , I


84 YONA DIMANT AND MICHAEL P o r n



centration in grams of polymer per gram of solution. K is found to be a proportionality constant which equals (2.3 f0.8) over the range 5 x lo3 < N < 0.5 x 10'. Recent experimental data [14aJ have shown, however, that the applicability of Virk's correlation (1 8) is limited. For this reason we have refrained from using Eqs. (17) and (1 8). The following analysis of heat transfer is valid for any functional dependence of A+ on the polymer properties and shear.


40

30

U+

20

10

0 5 10' 10' 3110'

FIG. 7. Measured and calculated velocity profiles. Data from Rollin and Seyer [%I.

FIG. 8. Friction factors for different values of A'


1 I I 1 I l l I I I I I I I I I I 1 I I 1 1 1 1 "I 25

0 I I I 1 1 1 1 I I 1 1 1 1 1 1 I I i I 1 1 1 1 lo00

Y + 2 lo 100

FIG. 9. Velocity distribution in a boundary layer with zero pressure gradient [13].

HI. Previous Heat Transfer Models

Almost all the phenomenological heat (or mass) transfer models use eddy viscosity and eddy diffusivity transport equations for describing the shear stress and the heat flux :

7 = p(v+E)du/dy (19) = - ( ( K + P C ~ E ~ ) dT/dy (20)

where E is the eddy viscosity, and E,, the eddy diffusivity. Using dimensionless parameters, Eqs. (19) and (20) give for constant fluid properties

dT' q' du' -= - Preff - dy' 2' dy'

where the effective Prandtl number is defined as 1 + E / V Preff = 1/Pr + E / V Pr,

Pr, = E/E,, is the turbulent Prandtl number and T' = (Tw-T)CprW/q,V*. Equation (21) enables one to calculate the temperature distribution and the


bulk temperature which is defined as

The Nusselt number and the Stanton number are directly related to the bulk temperature by the expressions

and

Such a procedure requires, however, that Pr, and q+/r+ be known. None of the previous studies of heat transfer in flows with drag reduction uses the exact distribution of 4'12'. Instead, approximations that had been proposed earlier for Newtonian turbulent flows were adopted.

Howard and McGrary [ 151 employed a two-layer model for the temperature profile using q+/r+ = 1, E = 0 in the sublayer and Pr,, = Pr, = 0.62 in the turbulent region.

The effect of the buffer zone (y,' -= y+ < yz+) was first considered by Poreh and Paz [16], who proposed to describe the velocity profile in the buffer zone by a log law

where y,+ = 0.43yj+ and yj+ is defined by the intersection of Eqs. (1) and (2). In order to calculate the temperature profile the assumptions r+ = q+ = 1 and Pr, = 1 were adopted. In the sublayer E = 0 was used, whereas the molecular diffusion in the turbulent core was neglected. The value of Pre, in the buffer zone was calculated from Eq. (22).

Howard [17] used a different three-layer model based on the velocity distribution proposed by Granville [18]. The sublayer was assumed to extend to y,+ = 5.0+Au+, whereas the velocity profile in the outer region and the transition were described by

(27)

Nu = 2PrRf/Tb+ (24)

St = cf/2)1'2/Tb' (25)

u+ = Y, + ln(y+/y,+> + Y l + (26)

U + = 2.5 In(y+-J) + 5.5 + Au+

The value of J was determined by the matching condition at y l + . The value of Pr,, in the outer region was calculated using Pr, = 0.62. Within the inner layer, E = 0 was assumed.

All the above-mentioned models assumed that E = 0 in the viscous sublayer. It is well known that the flow in the viscous sublayer is not really Iaminar and although the turbulent level there is small, it cannot be neglected when the temperature, or the diffusion, sublayer is much smaller than the viscous sublayer. These models cannot be accurate therefore for large Pr or Sc numbers, particularly in flows with large drag reduction having relatively large viscous sublayers.


This severe limitation is not present in the model of Levich [19], originally proposed for calculating the diffusion of mass in Newtonian flows with large Schmidt numbers, which was adopted by Rubin [20] to calculate heat transfer in polymer solutions. Rubin [20] employs a two-layer model to describe the concentration or the temperature profile. Following Levich, the thickness of the diffusion sublayer, where E = 0, was assumed to decrease with the Prandtl or Schmidt numbers. It is found, however, that the model is not very accurate for small Pr or Sc numbers.

A different approach, based on the analysis of Reichardt, has been used by Gupta et al. [2l] and Wells [22]. If one assumes y+ = z+ and uses a constant value of Pree in the outer region, the following expression for the Stanton number can be obtained by integration of Eq. (21) :

Prt (fl2) St = CD + (Pr/Prt- l)Cf/Z)'/'H

where H is the following function of the Pr number:

0 is the centerline-to-average velocity ratio and Y is the analogous temperature ratio. Friend and Metzner [23] determined experimentally the unknown function H for purely viscous fluids and found that

H = /3pre1l3 (30)

where B = 11.8. Both Gupta et al. [21] and Wells [22] assumed that these relations are valid in dilute polymer solutions. Gupta et al. 1211 took Y = 1 and proposed that 1.1 8 < @ < 2.0. Wells [22] assumed that fl = 1 .02yj, Y = 1, and CD = 1.2. He found that

where yj+ is the edge of the viscous sublayer as defined earlier. The assumption that H i s the same with and without polymers is not fully justified by either author. Moreover, it is noted that these models give at very large Sc numbers St a S C - ~ / ~ rather than S C - ~ / ~ , as observed experimentally [24] (see Fig. 20).

IV. Analysis of Heat Transfer

In view of the inherent limitations of the previously described heat transfer models and the difficulties in correlating all the experimental data, which have been fully described by Smith et al. [S], it is proposed to analyze the heat


transfer using the eddy diffusivities calculated from the mixing length model of the authors described earlier, which gives continuous velocity distributions in good agreement with experimental data. The advantages of such a model are that it does not neglect the eddy diffusivity in the sublayer and that it can be extended to nonisothermal flows and to developing thermal boundary layers.

We have seen earlier that the constant shear stress approximation, z+ = 1, is not satisfactory in flows with large drag reduction. Similarly, the use of the approximation q+ = z+ cannot be justified in flows with a relatively thick viscous sublayer. This observation implies, as we shall see later, that it is necessary to distinguish between flows with “constant wall temperature” (CT) and flows with “constant heat flux” from the wall to the flow (CF), which have different distributions of 4’.

We shall consider only hydrodynamically fully developed flows in which u = u(r) only. The energy equation in such flows can be written as

p c , ~ aqax = (l/r) a(rq)/ar (32)

where q, the heat flux, is related to the temperature gradient aT/ar by Eq. (20). The heat transfer in the CT and CF modes will be analyzed for the thermally fully developed flows as well as for the entrance region. The effect of the temperature dependent fluid properties on thermally developed flows will also be calculated.

A. ANALYSIS OF THERMALLY DEVELOPED FLOWS

The proposed model is based on the eddy viscosity and diffusivity equations (1 9) and (20) and on the use of Eq. (21) to relate the temperature to the velocity distribution which is determined by the model described earlier. It is assumed in this analysis that Pr, = 1, and thus E,, can be calculated from the mixing length model proposed earlier.

In thermally fully developed flows the following expression is satisfied

A(=) = 0 dx Tw-Tb (33)

When the wall heat flux qw is constant, the following conditions have to be satisfied :

aT/ax = dTw/dx = dTb/dx = 2g,/pCpUbR (34)

Using these conditions, Eq. (32) can be integrated to give

(35) 1 - (4/Re)SY,’ u+(l - y + / R + ) dy’

1 - y’/R” q+ =


1.10

Now, for constant fluid properties,

I I I I I I l l I

P r = l Pr = 10 --- -\

q+ = (l/Pr+&/v Pr,) dT+/dy+ (36)

and thus for constant qw

(37) dT+

dY + (1 -y+/R+)ED+ 1 - (4/Re)K u+(l - y + / R + ) dy+ -- -

where

1 1 2 + 2 du+ ED+ = -+---k y [ l - e x p ( - y + / ~ + ) ] 2 -

Pr Pr, dY +

Different results are obtained in pipes with constant wall temperature T, = constant (CT). In this case Eq. (33) gives

aT T,-T dTb _ - ax T,-T, dx

where

dTb/dx = %w/Pcp ub

Thus, integration of Eq. (32) gives for constant T,

(39)

d T + 1 -(4/Re)fi+ u+( l - y+ /R+) (T+/T ,+) dy+

dY + ( 1 -y+/R+)E,+

where ED+ is given by Eq. (38). The difference between Eqs. (41) and (37) is obvious.

(41) -- -

lo4 Re lo5

FIG. 10. Comparison of the Nusselt number at constant heat flux (NuCF) and at constant wall temperature (Nun) modes.


The numerical solution of the “constant flux” problem is a straightforward procedure since d T + / d y + in this case [Eq. (37)] is completely determined by the velocity field. The solution of the “constant temperature” problem, on the other hand, is more complicated because of the dependence of q+ on the temperature distribution. The authors have solved this problem numerically using an iteration procedure in which the temperature distribution T + for constant flux is used as the first approximation [25].

The calculated values of the heat transfer coefficients NuCT (constant flux) and Nk- (constant wall temperature) are compared in Fig. 10 for the cases Pr = 1 and Pr = 10, A + = 26, and A + = 350. Obviously, the difference between the two modes is very small in the Newtonian case (A+ = 26); however, it is not always negligible in flows with large A + , small Reynolds numbers, and small Pr numbers.

B. THE INFLUENCE OF TEMPERATURE-DEPENDENT FLUID PROPERTIES

All the theoretical models discussed so far assume constant fluid properties. Analysis of the effect of the temperature-dependent fluid properties in flows with drag reduction is essential for future engineering applications as well as for meaningful comparison between the theoretical results and the available experimental data.

For most liquids, the specific heat and the thermal conductivity are relatively independent of temperature; however, the viscosity decreases remarkably with temperature. The general effect of the variation of the fluid properties is to change both the velocity and temperature profiles, yielding different friction and heat transfer coefficients than would be obtained if properties were constant.

Two methods for evaluating this effect are widely used. In the reference temperature method, a characteristic temperature is chosen at which the properties appearing in the nondimensional groups (Re, Pr, Nu, etc.) should be evaluated. Thus the friction factor and heat transfer coefficients can be calculated using correlations relevant to constant properties behavior. A more convenient method is the property ratio method in which all variables are evaluated at the bulk temperature, and the effects of the variable liquid properties are lumped into a function of the ratio pW/&, which denotes the ratio of the viscosity evaluated at surface temperature to the viscosity evaluated at the bulk temperature [26]. It was found that in such a scheme the following power laws offer an excellent approximation for flow at given Reynolds and Prandtl numbers :


where the subscript cp refers to the appropriate constant-properties solution or very small temperature differences.

When using the property ratio method, attention must be paid to the following problem. When a temperature difference is applied to an experimental system, both the shear stress and the discharge can change, depending on the boundary conditions of the experiment. Different values of f/&, and Nu/Nu,, are obtained, at the same bulk-to-wall temperature difference, when the Reynolds number remains constant (constant Re) and when the shear stress remains constant (constant R+) ; it is necessary to distinguish between these two cases.

It is possible to calculate the effect of the temperature-dependent fluid properties using a procedure similar to that proposed by Deissler [26a]. In liquids with variable properties the equations for the shear and heat flux are

where y + = yv*/vb.

model, can be calculated by Using these equations, the velocity distribution, according to the present

(46) 22 + -- - dul+

dy+ v/vb f {(V/Vb)z+4kzy+z[1 -eXp( -y+Vb/A+V)]z~+}”z

The temperature profile can be obtained by integration of Eq. (37), provided that the value of u+ appearing in these equations is calculated by Eq. (46), and ED+ is calculated using

&/vb = k2y+’[1 - e ~ p ( - - y + v ~ / A + v ) ] ~ d u + / d y + (47)

In order to solve these equations for a given temperature ratio Tw/Tb, the dependence of the viscosity on the temperature T for the particular fluid has to be determined. A good approximation is the relation [26]

PIPW = TWIT (48)

where Tis in Fahrenheit degrees. The same approximation appears to be valid in polymer solutions as well [27]. Using this approximation and defining

P = 9wv*lcpTw~w Eq. (48) becomes

V 1

(49)


0 20 40 60 80 0.7

Tw-Tb(OF 1 FIG. 11. The calculated and measured effect of wall-to-bulk temperature differences on

the Nusselt number and friction coefficient in water (constant Re).

where Tb and T, are in Fahrenheit degrees. Equations (46) and (37) were solved in this work using the following iteration scheme [25]. The constant properties solution is taken as a first approximation. The velocity distribution is then calculated, using Eqs. (46) and (50), and is then used to evaluate the next approximation.

The authors have used this scheme to evaluate the dependence of Nu/Nuc, and flf,, in Newtonian fluids ( A + = 26) for the constant heat flux problem. The numerical calculations for Pr, = 8 and Tb = 60°F are compared with the data of Allen and Eckert [28] in Fig. 11. The agreement with the data in Newtonian flows is excellent for Nu/Nu,, and good for f / j& .

Calculations for various values of A + > 26 show that the power law approximations, Eqs. (42) and (43), are also adequate in drag reducing flows. We have plotted in Fig. 12 the variation of the exponents rn and n in the power laws with A + for two values of R'. Both exponents increase with A. Consistent results are obtained at all Prandtl numbers, as shown in Fig. 13 for p , /p , = 0.5.

One sees from these figures that the effect of the temperature difference on the heat transfer is about the same with and without polymers. On the other hand, the change in the friction factor f due to a given temperature difference, is larger in flows with drag reduction.

One may thus conclude that smaller bulk-to-wall temperature differences are required for flows with drag reduction to be considered as isothermal.

94

- 0.30

-a32

YONA DIMANT AND MICHAEL POREH

n- - 0.38 m--- /

/ /

- - // -

1 I I I I l l 1 I I I .

/ - 0.34

- -

Prbulk

FIG. 13. The dependence of Nu/Nu., and f/j& on the Prandtl number Cu.,/,ub = 0.5 and constant R+).


C. THE ENTRANCE REGION IN THE CONSTANT FLUX MODE

We shall consider now the development of the thermal boundary layer in a fully developed velocity field in a pipe, namely u = u(r ) only. For the constant heat flux mode (CF), the dimensionless variables x+ = x/R, 5 = y/R, and

e+ = (T-T,)~c,,v*/~, (51)

where T, is the bulk temperature at x = 0, will be used. The differential equation for the dimensionless temperature difference 6+, derived from Eq. (32), is

and the corresponding boundary conditions are

aO+(x+,i)/a( = 0, ae+(x+,o)/a( = - R + P ~ , O+(o,c) = o (53)

8+ 8+

TRANCE R E G I O ~

FIG. 14. Schematic description of the constant heat flux mode.

A schematic description of the expected variation of the various variables

Now the variable O + can be expressed as for the CF mode is given in Fig. 14.

8+ = OF+ + 0,' (54) where OF+(x+c) is the dimensionless temperature at the thermally fully developed region, namely large x+. It is clear from Fig. 14 that

TF - T,'= (dTb/dx)X + (T-Tb) (55)

Multiplying this expression by pCpV*/qw and using the relation dTb/dx =


2qw/pCpVbR [Eq. (34)], the definition of T + and the relation f = 2ubZ/v*', one finds that

OF+ = 2cf/2)1'2x+ + (Tb+- T + )

OF+ = 2 St, Tb+x+ + (Tb+ - T + )

(56)

(57)

Using Eq. (25) the above equation can also be written as

The new variable 01+ satisfies the differential equation (52) and the following boundary conditions

ae,+(x+,i)/ag = 0, ae,+(x+,o)/ag = 0, e,+(o,O = --eF+(O,[)

(58) Separating 8,' into an x+ dependent and 5 dependent functions and sub- stituting in Eq. (52), one finds that

m

i = 2 e,+ = C piFi(()exp(-ilix+) (59)

where li and Fi([) are the eigenvalues and the eigenfunctions of the following Sturm-Liouville system

dFldr = 0 at 5 = 0 and [ = 1

The values of /Ii are determined from the initial conditions for 8,' which, in view of the orthonormality of the eigenfunctions Fi, can be written as

Bi = R'~Fi(~)U'(l-r)S,+(O,i) dl (61)

The above solution determines the Stanton number at each x, which can be written as

Sparrow [29] computed, with a desk calculator, the first five values of Fi(0) and li for several pairs of Pr numbers using Deissler's expressions for U + and ED+ in turbulent Newtonian flows. Tabulated values appear in Kays [26]. The authors have prepared a computer program that calculates, using Galerkin's method [30], the numerical values of li and Fi(l) in a 30-term approximation to the series [25]. As an auxiliary function in Galerkin's method, the function c$k = Jo [bk (1 - r)'] was used where bk , k = 2, . . . ,30, satisfy Jl(bk) = 0. The 30-term solution gives accurate temperature distributions of O+(x+, [) and St,, even for small values of x/D, of the order of unity, for any values of Re, Pr, and drag reduction (or A').


1.15

-Allen 8 Eckert

Present work 1.10 - - StX st,

1.05

1.00

0 10 20 30

x l D I

FIG. 15. Variation of the St number in the entrance region for Newtonian flows, CF mode. Figure reproduced from Allen and Ekkert [28].

The calculated solution for the Newtonian case (Pr = 7, Re = 50,000, A + = 26) is compared in Fig. 15 with the solution of Sparrow, another solution of Deissler [26a] who used an integral method, and the experimental data of Allen and Eckert [28] and Hartnett [31] as presented in Allen and Eckert [28]. The agreement with the experimental data is very good.

The calculated effect of drag reduction on the variation of the Stanton number in the entrance region is demonstrated in Fig. 16. One sees that drag reduction has a drastic effect on the development of the thermal boundary layer and that the entrance length is largely increased when the friction coefficient is reduced. The development of the thermal boundary layer, for a given drag reduction, depends on the Reynolds number as shown in Fig. 17. In the same figure we have also plotted the variation of St,/St, for Re = 50,000 and Pr = 50, which shows that the effect of the Pr number on the thermal development is smaller than that of the Re number.

2.5

Drag reduclion=78% 2.0

- Stx St,

1.5

I .o

0 LO 50 lo xlD’O

FIG. 16. Calculated variation of St, at different values of drag reduction, CF mode.


FIG. 17. Calculated variation of the St, at different Re and Pr reduction, CF mode.

numbers for 75% drag

D. THE ENTRANCE REGION IN THE CONSTANT TEMPERATURE MODE

dimensionless temperature difference In the case of constant wall temperature it is convenient to define the

8 = (TV--n/(Kv-n Now 8 is also a solution of Eq. (52) but it satisfies different boundary conditions :

aO(x+,l)/ay = 0, 8(X+,O) = 0, 8(0,[) = 1 (63)

A schematic description of the expected variation of 8, T,, and Tb for this case is shown in Fig. 18. The solution of 8 can also be expressed as

m

8 = yiGi([) exp(-bix+) i = l

where bi and Gi(C) are the eigenvalues and eigenfunctions of the corresponding Sturm-Liouville system. Note that 6, # 0 in this case. The values of yi are determined by the initial value of 8 so that

The Stanton number in this case is given by

2 (@/a0 I ( = 0 St = Pr Re 8,

FLOWS WITH DRAG REDUCTION

T

99

,I Twzconst.

(_ENTRANCE REGION- 1 FIG. 18. Schematic description of the constant wall temperature mode.

One can obtain &,(x+) by integration of qw(x+) from x+ = 0 to x+, which gives

The local Stanton number is thus given by

The value of St at the fully developed region is obtained by letting x+ --* a, which gives

St" = 412 (68)

The algorithm for solving this problem using Galerkin's method with the auxiliary functions 4 k = J~ [a,(l -()'I, where a, satisfies Jo(ak) = 0, is also given in Dimant [25]. More terms are required in this case to obtain an accurate solution at small values of x /D. It is estimated that 30 terms give reasonable accuracy, of the order of 10% for x / D = 3.

Calculated distribution of St,/St, in the CT and the CF modes are shown in Fig. 19 for 70 and 78% drag reduction. The effect of drag reduction on the development of the thermal boundary layer in the constant wall temperature mode is found to be much smaller than in the constant heat flux mode, but in


X l D

FIG. 19. Comparison of the St number in the entrance region in the CT and CF modes.

this case too one finds larger entrance lengths than in the Newtonian case. As expected, the difference between the two modes of heat transfer in the New- tonian case is negligible.

V. Comparison with Experimental Data

The analysis presented earlier has revealed several important features of flows with drag reduction that had not been recognized before. It has shown that it is necessary to distinguish between the constant wall temperature (CT) and the constant heat flux (CF) modes of heat transfer, that the effect of the wall-to-bulk temperature differences on the flow is larger than in turbulent Newtonian flows, and that the length of the thermal entrance region increases considerably in flows with drag reduction, particularly in the CF mode. A meaningful comparison between theoretical models and experimental data is therefore possible only if these findings are properly taken into consideration. In addition it is required to minimize the effect of shear and thermal degradation of the polymer solutions.

These requirements impose difficult constraints which were not always satisfied. To eliminate degradation, for example, short test sections have been used. On the other hand it has been shown that in many cases the entrance region extends beyond x / D = 80.

Special care has to be taken when comparing experimental data with the predictions of theoretical models for isothermal flows since very small wall-to- bulk temperature differences are required in flows with drag reduction for eliminating the effect of the temperature-dependent fluid properties. All the proposed models for constant fluid properties (cp) give implicit relations :

which are independent of the polymer properties and parameters that describe


them, like A + in our model. In order to evaluate such models, the values of St,, , calculated with the appropriate implicit relation and the measured values of Re, .Pr, and5 are compared with the measured values of the Stanton number Stex,. When Tw/Tb > 1.1, a large difference between the calculated and measured values is expected, due to the change of the fluid viscosity near the wall. It has been shown, however, that the effect of the variable fluid properties on the St number is not as large as their effect on the friction coefficient f. Thus, if one wishes to evaluate St,, , it is preferable to measure the value off separately at zero heat flux. Closer simulation of the isothermal conditions is therefore achieved in the experiments where f and St are not measured simultaneously. It should also be noted that i f f is measured separately at zero heat flux (ZHF), it has to be measured at the same value of the bulk temperature, otherwise the Reynolds number corresponding to f and St would not be the same. A comparison between the theoretically calculated St,, and the measured St is not justified when TWITb > 1.1, and one should use the present model for variable fluid properties (vp) [Eqs. (37) and (46)] which provides an implicit relation

St = F",(Re, Pr,f; Tw/Tb) (70)

In this model all the fluid properties are evaluated at the bulk temperature. It will be shown that only a few of the measurements in flows with drag

reduction satisfy the above requirement and it is therefore desired to compare the various models with measurements in turbulent Newtonian flows at different Pr numbers as well. Such a comparison provides a sensitive measure of the ability of the phenomenological models to describe the transport processes in the various parts of the boundary layer. Since polymer additives increase the thickness of the viscous sublayer, it is expected that proposed models for flow with drag reduction should give accurate predictions of heat transfer rates over a wide range of Prandtl numbers in Newtonian fluids.

Heat and mass transfer measurements in ten experimental studies in turbulent Newtonian flows were compiled by Deissler [24] for comparison with his phenomenological model. Deissler used a two-layer model that cannot be modified directly to describe the effect of polymers as the Van Driest model. However, the two models predict almost identical heat transfer rates in Newtonian fluids. Figure 20 compares the Stanton numbers predicted by the present model (with A + = 26), with the models of von Karman [16], Levich [19], Friend and Metzner [23], which were modified to account for drag reduction by Poreh and Paz [16], Rubin [20], and Wells [22]. It is seen from this figure that the effect of the Pr or Sc number is excellently described by the present model. The crude von Karman model appears to be accurate only for 0.5 < Pr < 5, whereas the model of Levich is adequate only for Pr > 5. The model of Friend and Metzner is fairly accurate up to Pr numbers


lo3

- - - - LEVlCH (RUBIN I

st

lo''

I 10 lo2 I o3 Pr , Sc

FIG. 20. Calculated and measured values of St numbers at different Pr or Sc numbers for Newtonian fluids.

of the order of 100. At higher Pr numbers it deviates from the data, and as noted earlier it indicates a power law dependence of St a Pr-2/3 rather than St a Pr-3/4 as suggested by the model of Deissler, the present model, and the model of Levich, and as confirmed by experimental data.

The measurements of heat transfer in flows with drag reduction obtained in the nine experimental studies listed in Table I [S, 21,27, 32-40] have been examined by the authors. Different polymers and concentrations over a wide range of Reynolds and Prandtl numbers were used, giving small and large values of drag reduction. Some solutions were quite concentrated and revealed other non-Newtonian phenomena besides drag reduction, such as shear dependent viscosity. The friction and heat transfer were measured simultaneously in some experiments and separately in others. Both modes of heat transfer were used. Unfortunately not all the data presented in these works could be used in our work since the values of the Pr numbers were not always reported. It is evident from Table I that the values of T,/T, in most studies have been large, and the effect of the variable fluid properties cannot be neglected, particularly in those studies where f and St were measured simultaneously. It is also obvious from Table I that several measurements that have been considered to be at the fully developed region had actually been taken in the entrance region.

The closest to ideal, constant fluid properties conditions are the measurements of Debrule [39], where the friction has been measured at zero heat flux

TABLE I EXPERIMENTAL STUDIES OF HEAT TRANSFER IN FLOWS WITH DRAG REDUCTION

Polymer: Pipe Heat concentration diameter Reynolds Prandtl Tw Tb transfer Friction

Study (PPm) (in.) numbers numbers ( O F ) (OF) X / D mode mode"

Marrucci and Astarita [32,331

Gupta 1341 Gupta et al. [21] McNaIly [35]

ET597: 600, 1000 0.472

ET597: 100,500,4500 0.745

WSR301: 2, 10,20 0.78

Smith et al. [5]

Corman [36]

Khabakhpasheva et al. [37,381

Monti [27]

Debrule [39]

Yo0 [40]

WSR301: 10 0.117 WSR N3000: 1000 Guar gum: 50,200,600, 0.62

1200,2400 0.92 Polyox: 70 0.39 Polyacrylamide 120 ET597: 250, 750, 1O00, 0.423

2000,4000,6000 Polyox: 10,50 0.377

Seperan AP-30: 0.87 100,500,1000,1500

WSR-301: 1000,2000

Re=6x103 t 6 x lo4

Re=7x102 t9 .2 x 104

+ 2.5 x 105

t 2 . 5 x 104

Ref = 2.5 x lo4

Reb = 5 x lo2

Re' = 7 x lo3 + 9 x lo4

Reb = lo3 t 3 . i x 104

Re'=2x103 t 6 x lo4

Reb = 2X lo4 t 2 x 1 0 5

t 104 Re, = lo3

?

Prb=7+12: 80 t90

Prf = 2.5 t 2.8

Prb = 6 t 9

?

Prb = 6.5 + 9.0

?

Pr, = 4.4 f 10

Pr, = 6 t 8 0 6-80

? 100 CT ?

103 t 182 190 t 199 98 f 102

?

- 93

75 2- CF t 9 5 40

-140 52 CT

78 18 CF - 80 70 78, CF t90 116 -84 60 CF

? ? -40 CF

Tb+5 48 40 CF t 103

75 105* 5' CF +80 t115 t l l O

ZHF

ZHF

ZHF

SIM

SIM

?

?

ZHF

ZH F

ZHF, SIM friction is measured at zero h d t flux (ZHF) or simultaneously with St (SIM). 'Reported only for measurements in the entrance region.


R. 10'

(a1

I

FIG. 21. Comparison of calculated values of St and the data of Debrule [39] (set 7, aged solution).


and the heat transfer has been measured maintaining Tw/Tb < 1.1. It has been, therefore, decided to use only Debrule’s measurements for comparison with the isothermal models of Wells [22], Rubin [20], and the one presented in this work.

As shown in Fig. 21 the measurements of Debrule in water and in 10 ppm polyox solutions, both fresh and aged, are in very good agreement with the present model. The predictions based on the models of Wells and Rubin, on the other hand, deviate consistently from these data. The 50 ppm data do not follow consistently any of the models. The measurements at Pr = 4.38 and 10.3 are still in good agreement with the present model. However, a large deviation is observed at the intermediate value Pr = 6.16, which could not be explained.

McNally [35] has measured heat transfer and friction in polyox solutions. The friction data were measured at zero heat flux at Tb = 77’F, whereas a bulk temperature Tb N 140’F and film temperature Tf N 177’F were recorded at the measuring station during the heat transfer measurements. When presented a s j = St Pr;/’ versus Re,, namely the fluid properties were evaluated at the film temperature, the onset Reynolds number for the heat transfer was threefold larger than the corresponding one for friction. McNally has therefore offered a correction factor C = 2.76, and plotted j,,,, =jC0*’ versus Recorr = Re,C. The corrected values of j were found to be close to f/2, as suggested by the Colbrun analogy.

It appears that the need for the above correction is due to the evaluation of the fluid properties at different temperatures. Note that the ratio of the viscosities of water at 170 and 77’F is about 2.2. The authors have replotted the heat transfer measurements of McNally versus the Reynolds number evaluated at the bulk temperature Tb = 140’F, using the measured values of the friction coefficients at the same Reynolds number, and compared the data with the values obtained by the present model for Tb/Tw = 0.715. As seen from Fig. 22 good agreement is found between the theory and the measurements.

Measurements in the constant flux mode throughout the entrance region have been reported by Gupta [34] and by Yo0 [40]. Comparison of the reported measured values of St,/St, with the theoretically calculated values of St,/St, for constant fluid properties are shown in Fig. 23. The absolute values of Nu, reported by Gupta for another case where the entrance region was longer than the test section are compared in Fig. 24 with theoretically calculated values. The theoretical curve in this figure was calculated assuming that

Although this assumption can be used only at very small wall-to-bulk


2.1,jL + 20p.p.m.

I I l l l l I I I

lo5 5 . d Re

FIG. 22. Comparison between measured values of St by McNally [35] and the calculated values with the present model.

FIG. 23. Comparison of calculated and measured values of St, in the entrance region, CF mode.

temperature differences, good agreement is found between the calculations and the data measured at T,/T, = 0.6.

No measurements in the entrance region at the constant wall temperature mode were found in the literature.

The measurements of Smith et al. [ 5 ] at x / D = 18, in loo0 ppm solution of N3000 and in 10 ppm of a W301 solution, and the measurements of Gupta [34] at x / D = 40.3, in 4500 ppm solution of ET597, were found to be in the entrance region. To demonstrate the effects of the wall-to-bulk temperature difference and the entrance region on the data, we have plotted in Fig. 25 the experimental values of St versus three sets of calculated values of St for :

FLOWS WITH DRAG REDUCTION

zoorl

107

I 50

FIG. 24. Comparison of calculated and measured values of Nu, in the entrance region, CF mode.

I 8

4 0 6

/ z f10% 3 v c 0 u 0, 4- /o- - /

/ 4

0 0 - /

/

2 - '/

-

0~ I 1 1 I I 0 1 2 3 4 5

Stexp'104

M'

- 0

st exp"10 6 7 8 5 9 10

FIG. 25. Comparison of the measured values of St and calculated values of St with and without the entrance length and variable fluid properties effects.


(1) constant fluid properties and x / D = 00 ; (2) variable fluid properties and x / D = co ; (3) variable fluid properties and the correct value of x/d.

The results clearly demonstrate the magnitude of the errors introduced when the effects of the variable fluid properties and the entrance region are neglected.

All the available measurements from studies listed in Table I, except for the one set of measurements of Debrule (Pr = 6.16, C = 50 ppm) discussed earlier, are compared with the theoretically calculated values by the present model in Fig. 26. Both the effect of T,/T,.and the effect of the entrance region have been included in the calculations. It should be noted that the Pr number in most data varies from one point to the other and it is therefore not possible to plot the data as in Figs. 21 and 24.

The only data set that consistently falls above the line Stexp/Sttheo = 1.1 5 is from Khabakhpasheva [37]. Now, it was not clearly stated in his work

FIG. 26. Comparison of measured and calculated St number with the present model. (Run 9, Fig. 21c omitted.)


whether the friction coefficients had been measured at zero heat flux or not. It has been assumed, for the purpose of calculating St, that the friction coefficients have been measured simultaneously with St. If it were assumed that f had been measured at zero heat flux, the differences between the calculated and experimental values would have been considerably smaller.

VI. Conclusions

A phenomenological model for calculating heat transfer in flows with drag reduction has been presented. The model is based on a mean velocity closure that uses Van Driest’s mixing length expression with a variable damping parameter and the classical Reynolds-Prandtl analogy between momentum and heat transfer. It does not adopt, however, the commonly used constant shear stress and constant flux approximations.

The model has been used to analyze the heat transfer in pipe flows both in the thermally established region and in the entrance region. The effect of the temperature-dependent fluid properties has also been analyzed. Algorithms providing implicit relations between the heat transfer, the friction, the properties of the fluid and the flow, which do not include any empirical coefficients, were derived for each case.

The analysis has revealed several important features of the transport processes in flows with drag reduction that are not found in turbulent Newtonian flows :

(a) Large differences between the constant wall temperature and constant

(b) Long thermal entrance regions which depend on the drag reduction,

(c) A larger effect of the temperature dependent fluid properties.

heat flux modes of heat transfer.

the Reynolds number, and the mode of heat transfer.

Although the exact mechanism of drag reduction is not fully understood, the physical explanation of the differences between heat transfer in turbulent flows with and without drag reduction is clear. Drag reduction is due to the reduction of the turbulent exchange of momentum near the wall, This reduction is accompanied by a comparable reduction in the turbulent transfer of heat and mass. When the relative transfer by the vertical turbulent fluctuations in the wall region is reduced, features that characterize heat transfer in laminar flows are observed. It is well known that in laminar flows it is necessary to distinguish between the two modes of heat transfer, that the effect of the temperature dependent fluid properties is larger than in turbulent flows, and that the entrance region is longer, particularly in the constant heat flux mode, and that its length is Reynolds number dependent [26].


Previous models of heat transfer in flows of drag reduction were based on approximations adequate for turbulent Newtonian flows only, and have, therefore, failed to describe these effects.

It has also been shown that these features of heat transfer in flows with drag reduction have not been fully taken into consideration in the planning of experimental studies and in the analysis of data. When properly analyzed, the large reported differences between data measured at various experiments are reduced, and overall good agreement is found with the theoretical calculations that assume an analogy between heat and momentum transfer.

Thus, although the proposed model does not explain the physical mechanisms of the phenomenon of drag reduction, it provides a better understanding of the observed features, as well as a valuable tool for analyzing and predicting transport processes in flows with drag reduction.

ACKNOWLEDGMENTS

This study is based in part on the D.Sc. Dissertation of Y. Dimant written under the supervision of Prof. M. Poreh. The work was supported by a grant from the United States- Israel Binational Science Foundation (BSF), Jerusalem, Israel.

NOMENCLATURE

Damping parameter in the mixing length equation (9)

Polymer concentration Specific heat Constant wall heat flux mode Constant wall temperature mode Drag reduction,

Pipe diameter Dimensionless effective eddy dif-

Fanning friction factor,

Eigenfunctions in the CF entrance

Eigenfunctions in the CT entrance

Bessel function of order zero Bessel function of order one Constant, Eq. (9) Constant, Eq. (18) Mixing length, Eq. (9) Molecular weight Number of polymer chain links Avogadro's number

D R = 1 -f.olutionlfmlvcnt

fusivity, Eq. (38)

f = 2rw/pub2

region solution

region solution

Nu

Pr prt

Pr.a

4+ 4

Re Re' R r R+

s c St

SIM

T Tb Tw T +

Nusselt number,

Prandtl number, Pr = pc, v / K

Turbulent Prandtl number, Pr, =

Effective Prandtl number, Eq. (22) Heat flux Dimensionless heat flux, q+ = q/qw Reynolds number, Re = u b Dlv Generalized Reynolds number Pipe radius Cross-stream coordinate Dimensionless pipe radius, R+ =

Schmidt number Stanton number,

Denotes simultaneous measurements of the friction factor and Stanton number

NU = 2Rqw/~(Tw--Tb)

&/&b

RV*/v

S t = q w / P c p u b ( T w - T b )

Temperature Bulk temperature Wall temperature Dimensionless temperature differ-

ence, T + = (T, - T ) Cp rwlqw V*

FLoWS WITH DRAG REDUCTION 111

U + V* V2

X

xi

Y

Y +

ZHF

a

B 81

Y l

61

6+

A A U +

.!?b

.!?

K

1,

CC

Dimensionless velocity, u+ = u/V* Shear velocity, V* = (7,&)’” Critical shear velocity at the onset

of drag reduction; Eq. (3) Streamwise coordinate Dimensionless streamwise coordi-

Cross stream coordinate measured

Dimensionless cross-stream coor-

Denotes friction measurement per-

Concentration dependent parameter

Parameter defined in Eq. (49) Constants corresponding to the

eigenfunctions in Eq. (61) Constants corresponding to the

eigenfunctions in Eq. (64) Eigenvalues in the CT entrance

region solution Dimensionless boundary layer thick-

ness for external flows Virk‘s polymer parameter, Eq. (18) Defined in Eq. (2) Eddy viscosity Eddy diffusivity Thermal conductivity Eigenvalues of the CF entrance

Viscosity

nate, x + = x /R

from the wall

dinate, y + = yV*/v

formed at zero heat flux

in Eq. (3)

region solution

Kinematic viscosity Constant in Eq. (15) Density Shear stress Dimensionless shear stress,

Dimensionless temperature differ-




Dimensionless cross-stream coordi-

7’ = 7/7.,

ence, Eq. (51)

ence, Eq. (51)

ence, Eq. (54)

ence, Eq. (62)

nate, C = yIR = y + / R +

SUBSCRIPTS

Property evaluated at bulk tempera-

Constant fluid properties Denotes conditions at x = 0 Experimentally measured value Property evaluated at film tempera-

Thermally established property Theoretically calculated value Denotes variable fluid properties Property evaluated at the wall Property at evaluated at the station x Property evaluated at x 00

ture

ture, Tf = (Tw + Tb)/2

REFERENCES

1. J. W. Hoyt, The effect of additives on fluid friction. J. Basic Eng. 94,258 (1972). 2. J. L. Lumley, Drag reduction by additives. Annu. Rev. Fluid Mech. 1, 367 (1969). 3. F. H. Bark, E. J. Hinch, and M. T. Landahl, Drag reduction in turbulent flow due to

additives: A report on Euromech 52 flow. J. Fluid Mech. 68, Part 1, 129-138 (1975). 4. M. Poreh and Y. Dimant, Velocity distribution and friction factors in flows with drag

reduction. Proc. Znt. Symp. Nav. Hydrodyn., 9fh, p. 1305 (1972); also Technion, I.I.T., Civ. Eng. Publ. No. 175 (1972).

5. K. A. Smith, P. S. Keuroghlian, P. S. Virk. and E. W. Merrill, Heat transfer to drag reducing polymer solutions. AZChEJ. 15,294 (1969).

6. W. A. Meyer, A correlation of the frictional characteristics for turbulent flow of dilute non-Newtonian fluids in pipes. AZChE J. 12, 522-525 (1 966).

7. C. Elata, J. Lehrer, and A. Kahanovitz, Turbulent shear flow of polymer solution. Zsr. J. Technol. 4, 84 (1 966).

8. Y. Dimant, “Transport of Momentum Heat and Mass in Turbulent Flow of Polymer Solutions,” D.Sc Thesis. Technion, Haifa, Israel, 1974 (in Hebrew).


8a. C. S. Wells and J. G. Spangler, Injection of drag reducing fluid into turbulent pipe flow of a Newtonian fluid. Phys. Fluids 10, No. 9,1890 (1967).

9. P. S. Virk, An elastic sublayer model for drag reduction by dilute polymer solutions of linear macromolecules. J. Fluid Mech. 45, Part 3,417-440 (1971).

10. Y. Tomita, Pipe flows of dilute aqueous polymer solutions. Bull. JSME 13, No. 61 (1970).

11. N. G. Vasetskaya and V. A. Ioselevich, “Semi-empirical Turbulence Theory for Dilute Polymer Solutions” (transl. from An SSR Mekhanika Zhidkosti i Gaza, Vol. 3, No. 2, 1970). Plenum (Fluid Dynamics Consultants Bureau), New York, 1972.

12. D. B. Spalding. A model and calculation procedure for the friction and heat transfer behavior of dilute polymer solution in turbulent pipe flow. In “Progress in Heat Trans- fer,” Vol. 5. Pergamon, Oxford, 1972.

12a. A. Rollin and F. A. Seyer, “Velocity Measurements in Turbulent Flow of Dilute Viscoeleastic Solutions.” Dept. Chem. 8c Petrol. Eng., University of Alberta, Edmonton, Alberta, 1971.

13. M. Poreh and L. Grunblatt, “Phenomenological models of boundary layer flows with drag reduction,” Publ. No. 196. Technion, I.L.T., Haifa, Israel, 1973.

14. M. S. Kumor and D. W. Sylvester, Effect of drag reducing polymer on the turbulent boundary layer. AIChE Symp. Ser. 69, No. 130 (1973).

14a. C. Wang, Correlation of the friction factor for turbulent pipe flow of dilute polymer solutions. 2nd. Eng. Chem., Fundam 11, No. 4 (1972).

15. R. G. Howard and D. M. McGrary, “The Correlation between Heat and Momentum Transfer for Solutions of Drag Reducing Agents,” Report 3232. Naval Ship Res., Annapolis, Maryland, 197 1.

16. M. Poreh and U. Paz, Turbulent heat transfer to dilute polymer solutions. Int. J. Heat Mass Transfer 11, No. 5,805-812 (1968).

17. R. G. Howard, “Heat and Momentum Transfer in Drag Reducing Solutions,” Report 32260. Naval Ship Res., Annapolis, Maryland, 1970.

18. P. S. Granville, “The Frictional Resistance and Velocity Similarity of Drag Reducing Dilute Polymer Solutions,” Report 2502. Naval Ship Res., Washington, D.C., 1966.

19. V. G. Levich, “Physicochemical Hydrodynamics,” pp. 139-1 57. Prentice-Hall, Engle- wood Cliffs, New Jersey, 1962.

20. H. Rubin, Scaling of heat transfer to dilute polymer solutions. J. Hydronaut. 9, 147-150 (1971).

21. M. K. Gupta, A. B. Metzner, and J. P. Hartnett, Turbulent heat transfer characteristics of viscoelastic fluids. Znt. J . Heat Mass Transfer 10,1211-1224 (1967).

22. C. S. Wells, Turbulent heat transfer in drag reducing fluids. AZChE J. 14, No. 3,406410 (1968).

23. W. L. Friend and A. B. Metzner, Turbulent heat transfer inside tubes and the analogy among heat, mass and momentum transfer. AZChEJ. 4, No. 4,393402 (1958).

24. R. G. Deissler, Turbulent heat transfer in smooth passages. In “Turbulent Flows and Heat Transfer” (C. C. Lin, ed.), p. 288. Princeton Univ. Press, Princeton, New Jersey, 1959.

25. Y. Dimant, “Computer Program for Calculating Momentum, Heat and Mass Transfer in Dilute Polymer Solutions,” Publ. No. 204. Technion, I.I.T., Civ. Eng., Haifa, Israel, 1975.

26. W. M. Kays, “Convective Heat and Mass Transfer,” Ser. Mech. Eng. McGraw-Hill, New York, 1966.

26a. R. G. Deissler, Turbulent heat transfer and friction in the entrance regions of smooth passages. Tram. ASME 77, 1221 (1955).


27. R. Monti, Heat transfer in drag reduction solutions. In “Progress in Heat Transfer,” Vol. 5, p. 239. Pergamon, Oxford, 1972.

28. R. W. Allen and E. R. G. Eckert, Friction and heat transfer measurements to turbulent pipe flow of water (Pr = 7) and 8 at uniform wall heat flux. J. Heat Transfer 86,301 (1 964).

29. E. M. Sparrow, T. M. Hallman, and R. Siegal, Turbulent heat transfer in the thermal entrance region of a pipe with uniform heat flux. App. Sci. Res., Sect. A 7 , 37 (1957).

30. L. V. Kantorovich and V. I. Krylov, “Approximate Methods of Higher Analysis” (transl. by C. D. Benster). Wiley (Interscience), New York, 1964.

31. J. P. Hartnett, Experimental determination of the thermal entrance length for flow of water and of oil in circular pipes. Trans ASME 77, 1211 (1955).

32. G. Marrucci and G. Astarita, Turbulent heat transfer in viscoelastic fluids. I d . Eng. Chem., Fundam. 6,470 (1967).

33. G. Astarita and G. Marrucci, Heat transfer in viscoelastic liquids in turbulent flow. Znt. Congr. Znd. Chem. 36th, 1966, p. 243 (1966).

34. M. K. Gupta, “Turbulent Heat Transfer Characteristics of Viscoelastic Fluids,” M.Sc. Thesis. University of Delaware, Newark, 1966.

35. W. A. McNally, “Heat and Momentum Transport in Dilute Poly (enthylene oxide) Solutions,” TR No. 44, Naval Underwater Weapons Research and Engineering Station, Newport, Rhode Island, 1968.

36. J. C. Corman, “Experimental Study of Heat Transfer in Viscoelastic Fluids,” Rep. 68-C-206. General Electric Research and Development Center, 1968; also in Znd. Eng. Chem., Process. Des. Dev. 9,254 (1970).

37. E. M. Khabakhpasheva, V. I. Popov, and B. V. Pereplitsa, Heat transfer in viscoelastic fluids. Zn “Heat Transfer 1970,” Vol. IV, p. Rh 2. 1970.

38. E. M. Khabakhpasheva and B. V. Pereplitsa, Turbulent heat transfer in weak polymeric solutions. Heat Transfer-Sou. Res. 5, No. 4 (1973).

39. P. M. Debrule, “Friction and Heat Transfer Coefficient in Smooth and Rough Pipes with Dilute Polymer Solutions,” Ph.D. Thesis. California Institute of Technology, Pasadena, 1972; also in Znt. J. Heat Mass Transfer 17, 529-540 (1974).

40. S. S. Yoo, “Heat Transfer and Friction Factors for Non-Newtonian Fluids in Turbulent Pipe Flow,” Ph.D. Thesis. University of Illinois, Chicago, 1974.

[advances in heat transfer] advances in heat transfer volume 12 volume 12 || heat transfer in flows...

Documents