Ind. Eng. Chem. Res. 1992,31, 727-732
727
Influence of Rheology on Laminar Heat Transfer to Viscoelastic Fluids in a Rectangular Channel Chunbo Xie and J a m e s
P.Hartnett*
Energy Resources Center, University of Illinois at Chicago, Chicago, Illinois 60680
Experimental studies of the laminar pressure drop and heat-transfer behavior of two types of aqueous polymer solutions were carried out in a 2 1 rectangular channel. The fluids studied were lo00 wppm of neutralized Carbopol934 in deionized water and 1000 wppm of Separan AP-273 in tap water. Three different thermal boundary conditions were studied. The experimental friction factors for the two polymer solutions agree with the value predicted for a purely viscous power law fluid. The measured Nusselt values for the two polymer solutions were considerably higher than the corresponding values for a power law fluid and higher than the experimental values for water. It is postulated that these high heat-transfer values are the result of secondary flows which arise from normal stress differences imposed on the boundaries of viscoelastic fluids in laminar flow through noncircular geometries. In addition, it is hypothesized that under laminar flow conditions the low frequency oscillatory behavior determines the relative elasticity, which in turn influences the heat-transfer performance of such fluids. Introduction The fluid mechanics and heat-transfer behavior of non-Newtonian fluids, particularly viscoelastic fluids, has been of special interest in recent years because of the wide use of such fluids in the chemical, pharmaceutical, and food industries. Throughout these industries non-Newtonian fluids undergo fluid mechanical and heat-transfer processes. Aqueous polymer solutions represent a large family of viscoelastic non-Newtonian fluids. Most of these polymer solutions, such as carboxymethyl cellulose, poly(ethylene oxide) and polyacrylamide experience substantial decreases in heat transfer and pressure drop as compared to Newtonian fluids under condition of turbulent channel flow (Cho and Hartnett, 1982). In contrast, aqueous solutions of poly(acry1ic acid) (Carbopol) do not show the anticipated heat-transfer and pressure drop reduction in turbulent channel flow but rather behave like clay slurries which are generally considered to be purely viscous. The semiempirical predictions (Friend and Metzner, 1955; Dodge and Metzner, 1959) are in good agreement with the turbulent heat transfer and pressure drop measurements of clay slurries and aqueous Carbopol solutions. Measurements of the fully established pressure drop and the local heat transfer under laminar circular tube flow conditions reveal that all aqueous polymer solutions, including Carbopol, are in excellent agreement with the analytical predictions using the purely viscous power law model (Cho and Hartnett, 1982). This demonstrates that elasticity plays no role in the case of the circular tube geometry. In the case where aqueous polymer solutions flow through noncircular channels under laminar conditions, it has been predicted that secondary flows would occur (Green and Rivlin, 1956; Wheeler and Wider, 1966) reflecting the fact that the stresses acting on orthogonal faces of a fluid element are not equal in viscoelastic fluids. Measurements taken with aqueous polyacrylamide solutions in laminar flow through a 2:l rectangular channel revealed dramatic increases in heat transfer (of the order of 100-200%) as compared with the analytical predictions for a purely viscous power law fluid (Kostic, 1984). At the same time the measured pressure drop was in good agreement with the purely viscous prediction. Thus Kostic concluded that the predicted secondary flows caused by the normal stress differences have little influence on the dimensionless pressure drop but dramatically increase the 0888-5885192f 2631-0727$03.00JO
heat transfer as compared to a purely viscous fluid. Against this background, laminar flow measurements of heat transfer and pressure drop in a 2:l rectangular duct were carried out for aqueous Carbopol solutions that had earlier revealed anomalous behavior under turbulent flow conditions. Surprisingly, the Carbopol solutions demonstrated the same general performance as reported for the Separan solution (Hartnett and Xie, 1989). In the case of a loo0 wppm Carbopol solution (using deionized water and neutralizing the solution to a pH of 7.0 by adding sodium hydroxide) the heat transfer could be predicted by the equation
Nu = 6.0Re0.2
(1) This equation was valid for the three boundary conditions studied: (1)Hl(1L) with upper wall heated alone, (2) Hl(1L) with lower wall heated alone, and (3) Hl(2L) with both upper and lower walls heated. No influence of the Rayleigh number was found, suggesting that the secondary flow associated with the elastic nature of the fluid overwhelmed any free-convection effects. In the present study a detailed comparison of the pressure drop and heat-transfer behavior of 1000 wppm Carbopol and loo0 wppm Separan was carried out, along with measurements of the steady shear viscosity and the fluid elasticity. The objective of the study is to obtain sufficient experimental data to be able to predict the performance of these fluids. In addition, the measurements should contribute to an understanding of the behavior of these special fluids.
Experimental Apparatus The heat-transfer experiments were carried out in a 2 cm X 1cm rectangular duct with three possible thermal boundary conditions: (1) top wall heated, other walls adiabatic; (2) lower wall heated, other walls adiabatic; (3) both top and bottom walls heated with side walls adiabatic. The flow system, which is show in Figure 1,consists of a 400-gal reservoir, two auxiliary plastic tanks of 55 gal each, a Moyno positive displacement pump, a test section, a stainless steel heat exchanger, and a weighing tank to collect the solutions. The test section is 640 cm long and its hydraulic diameter is 1.2 cm. The top and bottom walls of the test section are fabricated of stainless steel and the side walls are made of 1.27-cm-thick polycarbonate sheet with a thermal conductivity of 0.23 W/(m K). Heating is 0 1992 American Chemical Society
728 Ind.
Eng. Chem. Res., VoL 31, No. 3,1992
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accomplished hy passing electric current through one or d both of the stainleas steel walls. Pressure taps are l along the length of the test section and the local pressure drop, under conditions of fully developed flow, are measured with a manometer. The outaide wall temperature distribution along the top and bottom walls was measured by 42 thermocouples on each side installed along the length of the duct. In addition, the side wall temperature was monitored by six thermocouples while three single thermocouples measured the fluid inlet and exit tempera!mea and the ambient mom temperature. All thermocouples were T type copper constantan. The entire test section is insulated by a 30.5 cm X 30.5 cm plywood box filled with Styrofoam beads having an effective conductivity of 0.036 W/(m K). The heat loss to the environment is considerably less than 1% of the total heat input.
Test Fluids Two different polymer solutions were used in the heabtransfer study (1) polyamylamide (Separan 273 from Dow Chemical Company) with a concentration of lo00 wppm in Chicago tap water, the chemistry of which can be found in an earlier publication (Kwack and Hartnett, 1980); (2) poly(acry1ic acid) (Carbopol 934 from B.F. Goodrich Co.) with a concentration of lo00 wppm in deionized water. The aqueous Carbopol polymer solution was neutralized to a pH value of 7.0 hy adding sodium hydroxide. The chemical composition of the two basic polymers is given on Figure 2. According to the manufacturers' catalogues exposure to high shear can degrade Separan by rupture of the polymer chain (Dow, 1988), while Carbopol934 resin possesses high shear resistance and maximum thermal stability (B.F. Goodrich, 1989). To check the stability of the fluids, both steady shear apparent viscosity and dynamic rheology measurements were taken after each heat-transfer run using a capillary tube vis-
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Figure 3. Viseosity of Separan AP-273 in tap water (loo0wppm).
cometer in our laboratory and a Rheometrics fluid spectrometer a t Argonne National Laboratory.
Results and Discussion Rheology Results. In light of the above-cited coneern a b u t polymer degradation resulting from exposure of the aqueous polymer solutions to the high shearing rates encountered during their circulation in the flow loop, the steady shear apparent viscosity and the dynamic (cscillatory) rheology measurements were made before the initial heat-transfer run and after each subsequent run. Each heabtransfer run involved the establishment of a fixed flow rate and a fixed heat input, followed by monitoring to determine when steady state had occurred, and finally, measurement of all relevant data to evaluate the heab transfer performance. Following each run a sample of the heat-transfer fluid was taken from Tank #2 (Figure 1)for rheological measurements. Figure 3 shows the measured steady shear apparent viscosities of the Separan AP273 solution as a function of shear rate with the run number (Le., hours of circulating the fluid) as a parameter. It can be seen that relatively little change in the viscosity occurs above a shear rate of lo2s-l. However, at lower shearing rates the viscosity is seen to undergo dramatic decreases with increasing time of circulation of the fluid. This observed decrease in the low shear rate viscosity reflects the breaking of the polymer
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 729 MI,.. "l
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Figure 4. Oaeillatory phase shift of Separan AP-273 in tap water (loo0wpm).
Figure 6. Oscillatory p h w shift of Carbwl934 in deionizsd water (loo0wppm). Friction Factor 01 Polymer Solutions in 2 1 Duct
e -Sparan Ap 273 in T.W. 1000 wppm 0-Carbop01834 in D.W.1000 wppm Temperature
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Figure 5. Viscosity of Carbopol 934 in deionized water (loo0 wppm).
bonds caused by the shearing stresses encountered during the flow through the circulating pump, around bends, and throughout the flow loop. This can be seen even more clearly on Figure 4, which presents the phase shift angle, 6,as a function of the frequency of oscillation, w. In this measurement, taken with the test fluid placed in a cone and plate rheometer, one of the surfaces undergoes small-amplitude willatmy motion and the other surface responds with an oscillatory motion. The phase angle between the shear strain input and shear stress output is a measure of the fluid elasticity. If the phase shift is zero (i.e., no phase shift) the material being tested is elastic while a 90° phase shift corresponds to a purely viscous fluid. Thus we see in Figure 4 that the phase shift, 8, incTeases at all frequencies as the number of heattransfer runs increase (i.e., as the hours of circulation increase). This is clear evidence of the lm of elasticity of the Separan solution. In contrast, the aqueous Carbopol solutions were clearly more stable, showing very little evidence of polymer degradation even after being circulated for over 40 h. This is evidenced in Figure 5, which shows the steady shear v b m i t y as a function of shear rate, and in Figure 6,which presents the phase shift as a function of frequency of oscillation. Figure 6 also reveals that aqueous Carbopol solutions are viscoelastic. Pressure Drop Results. The dimensionleas pressure drop data under fully establiahed laminar flow conditions in the 2 1 rectangular duct were measured simultaneously with the heat-transfer experimenta and are shown on
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Figure 1 ' . Friction factor of aqueous Separan and Carbopol solutions in a 21 rectangular duct (loo0 wppm).
Figure 7 for both the Separan and the Carbpol solutions. The solid line represents the predicted value for a purely viscous power law fluid, given by the relationship f = 16/Re*, where Re* is the Kozicki generalized Reynolds number (Kozicki et al. 1966):
Re* = pU'"Dh"/[8n-1K(0.7278+ 0.2439/n)"]
(2)
It is seen that the measured values are in good agreement with the purely viscous analysis. Heat-Transfer Results. The heattransfer performance of the lo00 wppm Separan solution is shown on Figure 8 for the case where the upper wall is heated while the other three walls are adiabatic. Earlier results with water revealed that free-convection effects are negligible for upper wall heating (Hartnett and Xie, 1989) and this should also apply to the aqueous polymer solution. The local Nusselt number is expected to be a function of the Graetz number and the dimensionleas elasticity. Instead the heat-transfer performance is dominated by the Rey-
730 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Id
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nolds number as evidenced by the experimental results shown on Figure 8. The experimental measurements at Re = 100 are much lower than the measurements taken at Re = 500, suggesting that the secondary flows increase with increasing Reynolds number. The observed decrease in the Nusselt number with hours of circulation at a fixed Reynolds number is evidence of a decrease in the fluid elasticity caused by the rupture of the molecular bonds of the polymer as the fluid is exposed to shearing stresses. This is consistent with the rheological measurements shown on Figures 3 and 4. Figure 8 also shows the predicted local Nusselt number for a Newtonian fluid with a Prandtl number equal to 10 for the simultaneous development of the thermal and hydrodynamic boundary layers with the thermal boundary condition being constant wall heat flux axially and constant wall temperature peripherally on all four walls, the socalled Hl(4) boundary condition (Wibulswas, 1966). Also, shown for reference is the limiting Nusselt number for the case where only one of the longer walls is heated, the Hl(1L) boundary condition (Shah and London, 1978). A purely viscous power law fluid having a value of n equal to 0.6 would yield Nusselt values approximately 10% higher than those of a Newtonian fluid. A comparison of the experimental results for the Separan solution with the values predicted for a Newtonian fluid or for a purely viscous fluid reveals that the measured heat-transfer coefficients for the Separan solution are much higher. This behavior is consistent with earlier reported results (Kostic, 1984) and is ascribed to secondary flows resulting from normal stress differences acting on the fluid boundary. The experimental heat-transfer data for the neutralized aqueous Carbopol solution with upper wall heating are shown on Figure 9, where it may be seen that the Nusselt number at a fixed Reynolds number is independent of the hours of circulating the fluid. This reflects the stability of the Carbopol solution and is consistent with the rheological measurement shown on Figures 5 and 6. The experimental Nusselt values for the Carbopol solutions are considerably higher than the predicted values for a power law fluid. As in the case of Separan, this higher performance reflects the secondary flows which occur when a viscoelastic fluid flows in laminar motion through a rectangular channel. Figure 10 compares the experimental heat transfer results for water, the aqueous Carbopol solution, and the aqueous Separan solution. The water results are in good agreement with the predicted Newtonian forced-convection values and as expected reveal little influence of natural
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convection for the downward-facing heated wall. The Nusselt values for the two polymer solutions are much higher than the values for water. Furthermore, the Carbopol solution shows Nusselt values approximately twice that found for the Separan solutions, suggesting that the Carbopol fluid is more elastic than the Separan fluid. Turning to Figures 4 and 6, it can be seen that the Separan solutions show lower values of the phase shift, 6, than the Carbopol solutions at high frequencies; that is, at high frequencies the Separan solution is more elastic than the Carbopol solutions. However, this situation is reversed a t low frequencies with the Carbopol solutions being more elastic. When these observations are coupled with the measured heat transfer performance of these two fluids, it is tentatively concluded that the low-frequency elastic behavior plays the dominant role in determining the laminar heat transfer performance. In particular, it is suggested that the limiting value of the phase shift, 6, as the frequency, w, goes to zero is the appropriate measure of fluid elasticity under laminar flow conditions. Figure 11presents the experimental Nusselt values for the case where the lower wall is heated with the other three walls being adiabatic. The influence of natural convection on the results for water can be clearly seen by comparing Figures 10 and 11. In contrast, the Nusselt values for the Carbopol solution showed no influence of free convection, giving evidence that the secondary flows resulting from the normal force differences acting on the fluid boundaries dominate the free-convection flow. For the aqueous Sep-
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 731 ld
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aran solution some influence of free convection can be seen as the dimensionleas heat transfer in Figure 11is somewhat higher than the values shown on Figure 10. This behavior is also consistent with the conclusion that the Separan solution is not as elastic as the Carbopol. The results for the case where both the upper and lower walls are simultaneously heated are shown on Figure 12. Also shown are the results of Wibulswas for Hl(4) and the limiting values for Hl(2L) (Shah and London, 1978). In general, these measurements are in good agreement with the results obtained by superimposing the values found on Figures 10 and 11. The water results are somewhat higher than the values predicted for pure forced convection for the Hl(2L) boundary condition, reflecting the influence of free convection. The heat-transfer measurements for both polymer solutions were much higher than those found for water, and in all cases the Carbopol results were much higher than those of the Separan solution. Some words of caution are in order in interpreting these experimental results. It is clear that the heat-transfer coefficient is higher for the polymer solutions than for the water. However, it should be noted that the pressure drop is proportional to the fluid viscosity when compared at a fixed flow rate. Taking this into account, it follows that the pressure drop is an order of magnitude higher for the polymer solutions than for water when compared at the same flow rate. Thus the heat transfer per unit pressure drop (Le., per unit pumping power) is much lower for the polymer solutions than for water.
Conclusion The measured friction factors for aqueous polymer solutions in laminar flow through rectangular channels are in good agreement with the values predicted for purely viscous power law fluids. The Separan solutions showed substantial degradation (i.e., rupturing of polymer bonds) as they circulated through the flow loop, whereas the Carbopol solutions revealed no such degradation. The Nusselt numbers for the aqueous polymer solutions were considerably higher than the values found for water, reflecting the secondary flows associated with the flow of viscoelastic fluids through rectangular passages. The aqueous Carbopol solutions revealed much higher values of the Nusselt number than found for the Separan solutions, suggesting that the Carbopol is more elastic than the Separan. This is consistent with the low frequency oscillation results which showed Carbopol to be more elastic than Separan. It is hypothesized that the low frequency oscillatory behavior determines the relative elastic behavior of fluids under conditions of laminar flow in rectangular channels. Acknowledgment We acknowledge the financial support of the Engineering Division of the Office of Basic Energy Sciences of the U.S.Department of Energy under Grant No. ER13311. Thanks are also given to the Materials Division of Argonne National Laboratory for the convenience of using their Rheometrics equipment.
Nomenclature Gz = Graetz number = mC / k x Nu = Nusselt number = h&/k NuLl= Nusselt number for lower wall in the case of Hl(1L) boundary condition Nuul= Nusselt number for upper wall in the case of Hl(1L) boundary condition Nu,,, = Nuseelt number for mean value in the case of Hl(2L) boundary condition Pr = Prandtl number = $,/k Ra, = Ftayleigh number = gj3q'Dh4p2Cp/k2T Re = Reynolds number = puDh/? Re* = Kozicki generalized Reynolds number, eq 2 C , = specific heat of fluid, J/(kg.K) Dh = hydraulic diameter of the duct, m f = fanning friction factor = ~ / ( 1 / 2 p L F ) g = acceleration due to earth gravity, m/s2 h = local convective heat transfer coefficient = qf'/(T,-Tb), W/(K.m2) Hl(1L) = thermal boundary conditions representing constant wall heat flux axially and constant temperature peripherally with one longer wall heated Hl(2L) = thermal boundary condition representing constant wall heat flux axially and constant temperature peripherally with two longer walls heated Hl(4) = thermal boundary condition representing constant wall heat flux axially and constant temperature peripherally with all four walls heated K = consistency index in power law fluid model, T = K y " , N.s"/m2 m = mass flow rate, kg/s n = power law index in power law fluid model, T = K y " q f f = heat flux per unit heating area, W/m2 T, = local wall temperature of the duct, "C Tb= local fluid bulk temperature, "C U = mean velocity in axial direction, m/s wppm = weight parta per million x = axial location from the duct entrance, m
Ind. Eng. Chem. Res. 1992,31,732-735
732 Greek Letters
6 = volumetric coefficient of thermal expansion, 1/K 6 = phase shift, deg 9 = shear rate, l/s r) = viscosity of Newtonian fluid or apparent viscosity of non-Newtonian fluid, N.s/m2=1000 CP w = angular frequency, rad/s p = density of fluid, kg/m3 T = shearing stress, N/m2 Registry No. Carbopol934,9007-16-3; Separan 273,3722428-5.
Literature Cited Cho, Y. I.; Hartnett, J. P. Non-Newtonian Fluids in Circular Pipe Flow. Adu. Heat Transfer 1982,15,59-141. Dodge, D. W.; Metzner, A. B. Turbulent Flow of Non-Newtonian Systems. AZChE J. 1959,5,(2), 189-204. Dow Chemical Company. Let Separan Settle Your Liquid-Solids System; Design Products Department: Midland, MI, 1988. Friend, W. L.; Metzner, A. B. Turbulent Heat Transfer inside Tubes and the Analogy Among Heat, Mass, and Momentum Transfer. AZChE J . 1955,4 (4). B.F. Goodrich Chemical Group. Carbopol Water Soluble Resins; The B.F. Goodrich Company, Speciality Polymer and Chemicals Division: Cleveland, OH, 1989.
Green, A. E.; Rivlin, R. S. Steady Flow of Non-Newtonian Fluids through Tubes. Q. Appl. Math. 1956,15,299-308. Hartnett, J. P.; Xie, C. Influence of Polymer Concentration on Laminar Heat Transfer to Aqueous Carbopol Solutions in a 2:l Rectangular Duct. AZChE Symposium Series; AIChE: Philadelphia, 1989;Vol. 85 (No. 269),pp 454-459. Kostic, M. Heat Transfer and Hydrodynamics of Water and Viscoelastic Fluid Flow in a Rectangular Duct. Ph.D. Thesis, University of Illinois at Chicago, 1984. K o k k i , W.; Chou, C. H.; Tiu, C. Non-Newtonian Flow in Ducts of Arbitrary Cross-sectional Shape. Chem. Eng. Sci. 1966, 21, 665-679. Kwack, E. Y.; Hartnett, J. P.; Cho, Y. I. Chemical Effects in the Flow of Dilute Polymer Solutions. Lett. Heat Mass Transfer 1980,7, 1-6. Shah, R. K.; London, A. L. Laminar Flow Forced Convection in Ducts; Academic Press: New York, 1978. Wheeler, J. A.; Wider, E. H. Steady Flow of Non-Newtonian Fluids in a Square Duct. Trans. SOC.Rheol. 1966,10,353-367. Wibulswas, P. Laminar Flow Heat Transfer in Non-circular Ducts. Ph.D. Dissertation, Department of Mechanical Engineering, University of London, 1966.
Received for review January 11, 1991 Revised manuscript receiued April 10, 1991 Accepted May 3, 1991
Energy Cost of Intracellular Organization Grace H. Okamoto* and Edwin N. Lightfoot Department of Chemical Engineering, University of Wisconsin, 1415 Johnson Dr., Madison, Wisconsin 53706
Intracellular diffusional processes are examined by use of a simple model selected to highlight the role of Brownian motion in determining cellular anatomy. Limiting calculations of energy consumption show that mechanical transport of formed elements costs very little, but it is prohibitively expensive for small metabolites. Biologicai systems have evolved efficient means to maintain spatial organization of intracellular elements in the presence of randomizing diffusive forces.
Introduction The purpose of this paper is to consider the role of intracellular diffusion, in an attempt to assess the impact on intracellular energy consumption. The underlying rationale for this analysis is the belief that the evolution and metabolic behavior of biological structures at the cellular level have been heavily influenced by mass-transport considerations. We shall see that, in the absence of evolutionary adaptation, such transport can be too slow for optimum operation and that the cell has reacted with remarkable effectiveness to overcome this transport constraint. Understanding these constraints and adaptations requires an effective knowledge of pertinent time constants and a careful study of intracellular anatomy and physiology. In fact, the study of intracellular transport is particularly instructive from a modeling standpoint due to the complexity of the cell interior and because intracellular mass transport was only recently recognized as a significant problem. We begin briefly with a bit of historical perspective in the form of Figure 1: a continuous stirred tank reactor (CSTR) model of a typical mammalian cell. A CSTR model assumes that the cell is just an unorganized collection of metabolites and enzymes. Since the cell is seldom more than about 10 pm across, diffusional processes are expected to be very rapid, and the unwary investigator
* Author
to whom correspondence should be addressed.
Table I. Intracellular Diffusion Coefficients Adapted after Mastro et al.. (1984)" radius, i O W , cm2/s a H 2 0 l A compound Mr scab ~ 3 e e l h 170 PCAOL 3.2 64 33 1.9 2.5 sorbitol 1.9 182 94 50 methylene blue 2.6 320 3.7 40 15 2.6 324 sucrose 4.4 52 20 8 648 eosin 6.0 40 5.0 3600 12.0 18 3.5 5.0 dextran 3.0 5.0 inulin 5500 13.0 15 9.2 2.5 dextran loo00 23.3 3.7 24000 35.5 6.3 1.5 4.2 dextran 5.3 0.03 167.0 43000 23.2 actin 6.9 0.10 71.0 bovine serum albumin 68000 36.0 "Literature references for individual materials will be found in the original table.
wodd be tempted to assume that the cell interior is always well mixed. For a typical cell with a radius of 5 pm (L= 5 pm), the diffusional times can be estimated with the following equation.
Tn = (2.5 X
%
L2/3)
cm2)/31
(1)
When considering a typical globular protein in water, the diffusivity (3))is approximately 7 X cm2/s, and the time to diffuse across the cell (Tn) is less than ' / 2 s.
0888-5885/92/2631-0732$03.00/00 1992 American Chemical Society
