Effect of Blockage on Heat Transfer from a Sphere in Power-Law Fluids

Ind. Eng. Chem. Res. 2010, 49, 3849–3861

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Effect of Blockage on Heat Transfer from a Sphere in Power-Law Fluids Daoyun Song and Rakesh K. Gupta Department of Chemical Engineering, West Virginia UniVersity, P.O. Box 6102, Morgantown, West Virginia 26506

R. P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India 208016

This work endeavors to elucidate the influence of confining walls on the convective heat transfer from a sphere to power-law fluids. The governing equations (mass, momentum, and thermal energy) have been solved numerically over the following ranges of conditions: power-law index, 0.2-1, i.e., only for shear-thinning fluid behavior; sphere Reynolds number, 5-100; sphere-to-tube-diameter ratio, 0-0.5; and Prandtl number, 1-100. Extensive results of the local and surface averaged values of the Nusselt number are presented herein to delineate the influence of each of the aforementioned parameters on the rate of heat transfer from a sphere. Broadly speaking, the Nusselt number shows positive dependence on both the Reynolds and Prandtl numbers. All else being equal, shear-thinning fluid behavior is seen to facilitate heat transfer with reference to that in Newtonian fluids. Indeed, it is possible to augment the rate of heat transfer by up to 60-70% under appropriate conditions. However, the imposition of confining walls is seen to limit the enhancement in heat transfer, especially at low Reynolds and/or Prandtl numbers. Therefore, the severity of confinement together with the values of the Reynolds and Prandtl numbers influences the value of the Nusselt number in an intricate manner. Introduction In a recent paper,1 the severity of wall effects on the terminal velocity (expressed in terms of the usual drag coefficient) of a sphere settling at the axis of a cylindrical tube filled with powerlaw fluids was examined. Depending upon the values of the diameter ratio, Reynolds number, and power-law index, the confining walls were shown to exert varying levels of extra retardation force, thereby slowing the steady descent of the sphere. Naturally, all else being equal, the higher the value of the sphere-to-tube-diameter ratio (λ) is, the more severe the wall effect is. Qualitatively, the increase in the drag force exerted by the fluid is attributed to the sharpening of the velocity gradients near the sphere and at the walls, to the backflow of the fluid displaced by the sedimenting sphere, and to the wake modifications due to the presence of the confining walls. The numerical predictions were shown to be in fair agreement with the literature data on wall effects on spheres falling in Newtonian and power-law fluids up to about λ e 0.5 and Reynolds number (based on sphere diameter) values up to 100. Intuitively, one would expect a similar enhancement in the rate of convective heat transfer from a sphere to power-law fluids when the temperatures of the sphere and of the ambient power-law fluid (contained in a cylindrical tube) differ from each other. This assertion is based on the fact that, akin to the velocity gradient, there must be the sharpening of the temperature gradient close to the surface of the sphere in a confined medium as opposed to the one in an unconfined medium. This in turn should result in enhanced values of the convective heat transfer coefficient. This work sets out to elucidate this phenomenon for a range of values of the Prandtl number, Reynolds number, sphere-to-tubediameter ratio, and power-law index. However, before embarking upon the detailed presentation of the new results obtained in this study, it is useful and instructive to provide here a brief * To whom correspondence should be addressed. Tel.: +91 512 2597393. Fax: +91 512 2590104. E-mail: [email protected].

account of the previous scant literature available on heat transfer from a confined sphere, even for Newtonian fluids. Previous Work The flow of fluids and heat transfer from spherical particles is encountered in a range of industrial applications including during continuous aseptic processing of variously shaped food particles in water and in non-Newtonian polymer solutions,2-5 processing of suspensions in tubular heat exchangers in mineral, chemical, pharmaceutical, and petroleum reservoir engineering applications,6-8 non-Newtonian liquid-solid fluidized beds,9 use of electric and magnetic fields to enhance the rates of heat/ mass transport in narrow channels, etc. Aside from the aforementioned wide-ranging potential industrial applications, model studies based on a single sphere configuration are also germane to the development of appropriate frameworks for the analysis of multiparticle systems as encountered in real-life applications. For instance, it is not uncommon to use the results for a single sphere as a basis to estimate the rates of heat and mass transfer in fixed and fluidized beds.10-12 Despite such an overwhelming pragmatic significance, very little information is available on the influence of confining walls on the rate of heat transfer from a sphere even in Newtonian fluids, let alone in power-law fluids.7,13-15 For instance, Perkins and Leppert16,17 studied the effect of confinement on the rate of heat transfer from a cylinder in air and water in the (pipe) Reynolds number range 2000 to 1.2 × 105 for a range of blockage ratios varying from 0.2 to 0.42, and they reported enhancements in the value of the Nusselt number of up to 50-60% with reference to the unconfined cylinder at the same values of the Reynolds and Prandtl numbers. Similarly, the preliminary findings of Sastry et al.18 on the liquid-solid heat transfer in the context of food processing suggested varying levels of enhancement in the convective heat transfer coefficients. Indeed, Sastry et al.18 reported up to 100% increase in the heat transfer coefficients in water when the particle-to-tube-diameter ratio was increased

10.1021/ie901524h  2010 American Chemical Society Published on Web 03/08/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010

from 0.26 to ∼0.62, over the particle Reynolds number in the range 3600-27 000. However, as far as is known to us, there have been very few analytical/numerical studies elucidating the influences of walls on heat transfer from a confined sphere. Maheshwari et al.19 studied heat transfer from a single sphere and from an in-line array of three spheres in a tube for a range of sphere-to-tube-diameter ratios (0.1-0.5), for a range of Reynolds numbers (1-100), and for two values of the Prandtl number pertaining to air and water, though there have been numerous experimental and numerical studies on the prediction of wall effects on a falling sphere.7,20 Depending upon the combination of the values of Re, Pr, and λ, they reported up to 20% enhancement in the value of the average Nusselt number. This is in stark contrast to the extensivesexperimental, analytical, numericalsliterature on heat transfer from an unconfined sphere, e.g., see refs 13 and 15, and for recent numerical studies on heat transfer from a sphere, see refs 21-25. On the other hand, there have been even fewer studies on heat transfer from a sphere in power-law fluids and most of these have been reviewed recently.7,26 A cursory inspection of the available literature clearly shows that most of the early studies are based on the boundary layer approximation together with the thin boundary layer assumption (large Peclet and/or Prandtl numbers); e.g., see Kawase and Ulbrecht.27 Only recently, Dhole et al.26 have numerically solved the momentum and thermal energy equations to elucidate the role of powerlaw rheology on heat transfer from an unconfined sphere. Their numerical results were shown to be in line with the scant experimental data on mass transfer from spheres to power-law fluids. Broadly, shear-thinning behavior was seen to facilitate heat transfer and shear-thickening behavior impedes it. This finding is also consistent with the trends reported for circular,28,29 elliptical,30 and square cylinders31,32 submerged in free streams of power-law fluids. However, Dhole et al.26 encountered severe convergence difficulties for values of the power-law index smaller than ∼0.5-0.6 and they circumvented this difficulty by using a coarse mesh. The lack of this information is particularly acute because many fluids of industrial significance7,8 display values of the power-law index as low as 0.2. It is also appropriate to add here that, as noted elsewhere,1,7,33 indeed there have been only a few studies even on wall effects experienced by a sphere or a cylinder translating in nonNewtonian fluids. From the foregoing discussion, it is thus abundantly clear that indeed very little is known about the role of confining walls on the phenomenon of convective heat transfer from a sphere even in Newtonian fluids. Furthermore, owing to the severe numerical difficulties, currently available literature on heat transfer from an unconfined sphere immersed in power-law fluids is also restricted to only a moderate degree of shearthinning fluid behavior, n > ∼0.5-0.6. This work aims to alleviate this situation. In particular, reported herein are the values of the local and mean Nusselt numbers for a sphere submerged in power-law fluids over the following ranges of conditions: Reynolds number, 5-100; blockage ratio, 0-0.5; power-law index, 0.2-1; and Peclet number, e4000. The range of Reynolds numbers considered here is well within the steady axisymmetric flow regime for the flow of Newtonian fluids past a sphere,34 and since no such information is available for powerlaw fluids, in the first instance, this criterion is used here also for power-law fluids. Problem Description and Formulation Consider a sphere of diameter d located at the axis of a cylindrical tube of diameter D that is infinitely long (in the

Figure 1. Schematics of the flow configuration together with the cylindrical coordinate system (r, z).

z-direction) so that the end effects are negligible. The case of a sphere falling in a stationary power-law fluid is mimicked here by moving the fluid and the tube with a constant velocity Vo over a fixed (stationary) sphere, as shown schematically in Figure 1. The surface of the sphere is maintained at a constant temperature Tw, and the oncoming fluid is at a temperature of To. The thermophysical properties of the fluid (density, F; thermal conductivity, k; heat capacity, Cp; power-law constants, m and n) are assumed to be temperature-independent, and the viscous dissipation is also assumed to be negligible. While this simplifying assumption leads to the decoupling of the momentum and thermal energy equations, it also restricts the applicability of the present results to situations wherein the temperature difference between the fluid and sphere is small. Within the framework of the foregoing assumptions, the continuity, momentum, and thermal energy equations in their compact forms are written as follows: continuity: momentum:

∇·V ) 0

(1)

F(V · ∇V - f) - ∇ · σ ) 0

(2)

FCp(V · ∇T) - k∇2T ) 0

(3)

thermal energy:

where V, f, σ, and T are the velocity vector, body force, stress tensor, and temperature of the fluid, respectively. The stress tensor (σ) is the sum of the isotropic pressures (p) and the deviatoric component (τ), which is linked to the rate of deformation tensor (ε) as τ ) 2ηε

(4)

∇V + (∇V)T 2

(5)

where ε)

Finally, for a power-law fluid, the viscosity η is related to the second invariant of the rate of deformation tensor, I2, as η ) m(I2 /2)(n-1)/2

(6)


In turn, I2 is related to the velocity components and their derivatives; e.g., see Bird et al.35 In eq 6, m and n are the two power-law constants. Evidently, n < 1 denotes the so-called shear-thinning behavior and n ) 1 corresponds to the standard Newtonian fluid behavior. The physically realistic boundary conditions for this flow are written as follows: • On the surface of the sphere, no-slip condition and constant temperature conditions are implemented, i.e., Vr ) 0, Vz ) 0, and T ) Tw. • At the inlet, uniform flow in the z-direction and uniform temperature are prescribed, i.e., Vr ) 0, Vz ) Vo, and T ) To. • On the tube wall, no-slip and adiabatic conditions are used, i.e., Vr ) 0, Vz ) Vo, and ∂T/∂r ) 0. • At the exit, both pressure and the axial temperature gradient are set to zero, i.e., ∂T/∂z ) 0. Note that the imposition of zero pressure at the exit boundary is not necessary and one can set a constant value, as only the pressure gradient needs to be evaluated. However, knowledge of this predetermined value is required to define the dimensionless pressure coefficient as explained in ref 1. The numerical solution of the governing equations (eqs 1-3) subject to the above-noted boundary conditions maps the flow domain in terms of the velocity components, pressure, and temperature fields. These, in turn, can be used to evaluate the global characteristics such as the individual and total drag coefficients, the wake parameters, and the Nusselt number. Since the flow characteristics have been dealt with in detail elsewhere,1 the heat transfer characteristics are treated here. For this purpose, it is convenient to introduce a dimensionless temperature, T*, defined as T - To Tw - To

T* )

(7)

One can now define a local heat transfer coefficient on the surface of the sphere as h(Tw - To) ) -k

∂T ∂ns

|

(8) surface

which when nondimensionalized yields the following definition of the Nusselt number: Nuθ )

∂T* hd )k ∂ns

|

(9) surface

In turn, the surface averaged Nusselt number is obtained as Nu )

1 2

∫

π

0

Nuθ sin θ dθ

(10)

The scaling of the governing equations and of the boundary conditions suggests the local and average Nusselt numbers to be functions of the following four dimensionless parameters: Reynolds number: Re )

FVo2-ndn m

Prandtl number: Pr ) diameter ratio: power-law index:

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( )

Cpm Vo k d

n-1

λ ) d/D n

This functional relationship is explored in this work. In addition, some further insights are also provided by examining the role of confining boundaries on the temperature distribution close to the falling sphere. Numerical Solution Procedure and Choice of Numerical Parameters In this study, the governing equations together with the abovementioned boundary conditions were solved using COMSOL Multiphysics (version 3.5a). Since a detailed description of the domain and grid selection is available elsewhere,1 it is not repeated here in detail. In brief, the flow geometry was drawn using the built-in CAD tools, and the quadrilateral elements of the nonuniform grid were also generated using the built-in meshing function of COMSOL Multiphysics. In each element, the second-order Lagrange elements were used to approximate the velocity components while linear elements were used to approximate the pressure values in order to handle the velocity pressure coupling, i.e., the scheme Lagrange -P2P1 in COMSOL. The constant density and power-law viscosity modules have been used to input the values of the physical properties depending upon the desired values of the Prandtl and Reynolds numbers. Initially, the linear solver Direct (PARDISO, which stands for parallel sparse direct linear solver and is used for symmetric and nonsymmetric systems) was used, and if it proved unsuccessful to achieve the desired level of convergence, only then it was switched to Direct (UMFPACK, which stands for unsymmetric-pattern multifrontal package for sparse LU factorization and is generally employed to solve asymmetric sets of equations). The relative convergence criterion for each variable was set to 10-6. Furthermore, simulations were deemed to have converged only when the drag and Nusselt number values had stabilized at least up to five significant digits. For fixed values of the input parameters (Re, Pr, n, λ), the solution was always initiated by using the corresponding Newtonian flow and temperature flow fields to accelerate the rate of convergence. Similarly, the unconfined sphere case was simulated by considering a sphere-in-sphere configuration wherein the radius of the enclosing sphere is 500R. In other words, no further exploration was done in this regard and the grid and domain used in our recent study1 were considered to be satisfactory. Results and Discussion In this study, extensive numerical results encompass wide ranges of conditions as 0.2 e n e 1, 5 e Re e 100, 0 e λ e 0.5, and Pr e 100. However, the range of the Prandtl number is somewhat dependent upon the corresponding value of the Reynolds number, for the maximum value of the Peclet number ()Re · Pr) in this work is restricted to 4000. The maximum value of the Peclet number is dictated by the fact that increasingly fine mesh is required to adequately resolve the boundary layers at high Peclet numbers, but the range of Prandtl numbers covered here is believed to be sufficient to delineate the functional dependence of the Nusselt number on the Prandtl number. However, before embarking upon the presentation of

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Table 1. Surface Average Values of Nusselt Number in Newtonian Fluids value of Nu Re

Pr

ref 22

ref 25

present work

5

1 5 50 100 1 5 10 20 1 5 10 20

2.93 4.346 8.300 10.22 5.963 9.542 11.783 14.606 7.718 12.543 15.564 19.370

3.032 4.326 8.077 9.93 5.963 9.505 11.707 14.478 7.749 12.895 16.210 20.429

3.025 4.314 8.026 9.802 5.963 9.498 11.636 14.168 7.683 12.551 15.376 18.487

50

100

Table 2. Comparison of Average Values of Nusselt Number in Power-Law Fluids value of Nu n 0.6

Re

Pr

present work

ref 26

5

1 10 50 100 1 5 10 20 1 5 10 20 1 10 50 100 1 5 10 20 1 5 10 20

3.155 5.735 9.307 11.511 6.559 10.950 13.610 16.711 8.611 14.726 18.231 21.984 3.078 5.379 8.53 10.471 6.227 10.112 12.458 15.211 8.10 13.477 16.569 19.927

3.139 5.667 9.063 11.183 6.512 10.778 13.374 16.482 8.556 14.652 18.46 22.62 3.081 5.389 8.554 10.551 6.234 10.140 12.556 15.59 8.11 13.567 16.97 20.65

50

100

0.8

5

50

100

the new results, it is instructive to benchmark the present results with the literature values to demonstrate the adequacy of the domain size and grid used herein. Validation of Results. For Newtonian fluids, the present values of the mean Nusselt number for an unconfined sphere are seen to be in excellent agreement with the literature values (Table 1). The present results are seen to be within 5% of the literature values, except for one result at Re ) 100 and Pr ) 20. Table 2 shows a similar comparison for heat transfer in power-law fluids from an unconfined sphere for a range of values of Re, Pr, and n. Once again, the correspondence between the two sets of independent results is seen to be excellent; the two values seldom differ by more than 2-3%. Finally, the present results are compared with those of Maheshwari et al.19 for two values of the blockage ratio, namely, λ ) 0.1 and λ ) 0.5, for two values of the Prandtl number corresponding to air and water (Table 3). Needless to say, again, an excellent match is seen in Table 3. At first sight, a value of Nu < 2, as seen here for λ ) 0.5, Pr ) 0.74, and Re ) 1, might seem counterintuitive, but it can be explained qualitatively as follows: the limiting value of Nu ) 2 corresponds to conduction from an isothermal sphere submerged in an infinite body of stagnant fluid (maintained at a

Table 3. Effect of Blockage on Average Nusselt Number λ ) 0.1

λ ) 0.5

Pr

Re

present work

ref 19

present work

ref 19

0.74

1 10 50 100 1 10 50 100

2.247 3.374 5.494 7.035 3.172 5.895 10.544 14.082

2.249 3.378 5.503 7.05 3.172 5.894 10.548 14.092

0.708 3.787 6.38 8.073 3.7 7.583 12.658 16.532

0.712 3.795 6.391 8.084 3.697 7.584 12.66 16.527

7

constant temperature). In this case, the faraway boundary condition (at r f ∞) in terms of a known temperature or zero temperature gradient (in radial direction) does not influence the final result. Limited simulations performed in this study also corroborate this assertion for λ e ∼0.1 or so. However, the faraway boundary condition begins to influence the results as the value of λ is gradually increased. In this case, there will not be any heat transfer possible at steady state if zero temperature gradient is applied at this boundary. This corresponds to Nu ) 0.The present case of λ ) 0.5, Pr ) 0.74, and Re ) 1 falls between these two limits in the sense that the wall is adiabatic, but there is a weak convection effect at this value of the Reynolds number. The bulk flow does allow removal of some heat from the sphere, though the wall does not. Therefore, one can expect values of the Nusselt number to be lower than the conduction limit of Nu ) 2. Indeed, a similar gradual reduction in the value of heat transfer is seen in the results reported in ref 19 at Re ) 1 with the increasing value of λ. In addition to the aforementioned comparisons for the mean Nusselt number, the present values of the Nusselt number at the front stagnation point were compared with that of Dhole et al.26 over the overlapping ranges of conditions; the two values are almost indistinguishable from each other up to about Re ) 50 whereas the two values differ by 5-6% for Re > 50. It needs to be emphasized here that the deviations of such magnitude as seen here in Tables 1-3 are not at all uncommon in such numerical studies due to the differences arising from grid and domain effects, solution procedure, convergence criterion, etc.36 Based on the aforementioned comparisons and that reported elsewhere,1 the present results on heat transfer are believed to be reliable to within 2-3%. Local Nusselt Number. Figures 2-9 show the variation of the local Nusselt number on the surface of the sphere for a range of combinations of values of Re, Pr, n, and λ. A detailed examination of these figures (and of the results not presented here for the sake of brevity) suggests the following overall trends. (1) For a Newtonian fluid (n ) 1), as there is no mechanism for change in its viscosity due to the varying rate of deformation on the surface of the sphere, the Nusselt number is maximum at the front stagnation point (as the temperature gradient is maximum here) and it gradually deceases as one traverses along the surface of the sphere toward the rear stagnation point, as is seen for Re ) 5 (see Figure 2) and Re ) 10 (see Figure 3). Such a monotonic decrease in the value of the Nusselt number from the front stagnation point all the way to the rear stagnation point occurs as long as the flow remains attached to the surface of the sphere. The flow over a sphere begins to separate at about Re ) 20-25 for λ ) 0, and beyond this value of the Reynolds number, the Nusselt number decreases from its maximum value at the front stagnation point, reaching its minimum value at the point of separation (θs ≈ 130°) before turning upward, as is seen for Re ) 50 (see Figure 4) and Re ) 100 (Figure 5). For fixed values of Re and λ, an increase in the value of the Prandtl


Figure 2. (a) Local Nusselt number variation on the surface of the sphere for λ ) 0, Re ) 5, and n ) 1.0. (b) Local Nusselt number variation on the surface of the sphere for λ ) 0.2, Re ) 5, and n ) 1.0. (c) Local Nusselt number variation on the surface of the sphere for λ ) 0.5, Re ) 5, and n ) 1.0.

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number shifts the curve upward due to the thinning of the thermal boundary layer, thereby augmenting the rate of heat transfer. Similarly, all else being equal, as the value of λ is progressively increased the value of the Nusselt number increases due to the sharpening of the temperature gradients in the vicinity of the heated sphere. (2) In shear-thinning fluids and for an unconfined case (λ ) 0), the local Nusselt number progressively decreases along the surface of the sphere up to a certain value of the Prandtl number. Once the Prandtl number exceeds this critical value, the Nusselt number is no longer maximum at the front stagnation point (refer to Figures 6 and 7). The value of the Prandtl number delineating these two regimes is strongly dependent on the values of the Reynolds number, the power-law index, and the sphere-to-tubediameter ratio. For instance, for λ ) 0 and n ) 0.6, the critical value of the Prandtl number ranges from Pr ∼ 10 to Pr ∼ 1 as the value of the Reynolds number is progressively increased from Re ) 5 to 100 as shown in Figures 6a and 7a. The corresponding values for n ) 0.3 are Pr ∼ 5 to Pr < ∼1 seen in Figures 8a and 9a. Broadly, the lower the value of n and/or larger the Reynolds number, the smaller the value of the Prandtl number separating the two types of behaviors. The effect gets accentuated in the presence of confining walls. There are two competing mechanisms responsible for this behavior. First, since the velocity field decays very quickly in shear-thinning fluids,1 there is therefore only a small region in the vicinity of the sphere where the fluid is subject to intense shearing. This is almost equivalent to viscoplastic behavior wherein when the rate of deformation approaches zero, the apparent viscosity rises very rapidly. Indeed, the lower the value of n, the stronger this effect, as can be seen from the results for n ) 0.3 included in Figures 8 and 9. The net result is that the fluid in the vicinity of the sphere is subject to intense shearing, thereby lowering its effective viscosity, which facilitates heat transfer. This is responsible for the increase in the value of the Nusselt number. This increase is somewhat offset by the fact that the temperature gradient decreases as one traverses along the surface of the sphere away from the front stagnation point. Thus the maximum value (away from the front stagnation points) seen in Figures 6-9 is a net result of these two competing mechanisms. One can also interpret this competing process in terms of the different rates of thinning of the momentum and thermal boundary layers for power-law fluids. Furthermore, it is also well recognized that the confining walls influence the flow field much more significantly than the temperature field,13 even in Newtonian fluids. Consequently, the enhancement in heat transfer is nowhere near as much as the extra drag force on a sphere caused by the confining walls. As will be seen in the next section, the average Nusselt number results lend further support to this assertion. Average Nusselt Number. While the detailed distribution of the Nusselt number on the surface of the sphere provides useful insights, the surface average values are frequently required for process design calculations. Figures 10 and 11 show representative results elucidating the influence of λ, Pr, Re, and n on the average value of the Nusselt number. A detailed examination of these results shows that the mean Nusselt number increases with the increasing Reynolds number, Prandtl number, and the decreasing value of the power-law index. Irrespective of the values of λ and Re, the dependence of the average Nusselt number is very weak on the flow behavior index, n, at Pr ) 1. This is obviously due to weak advection at Prandtl numbers of ∼1, and the chief mode of heat transfer is conduction under these conditions; therefore the flow characteristics of the fluid are largely irrelevant under these

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Figure 10. (a) Dependence of average Nusselt number on power-law index and Prandtl number for Re ) 5 and λ ) 0. (b) Dependence of average Nusselt number on power-law index and Prandtl number for Re ) 5 and λ ) 0.2. (c) Dependence of average Nusselt number on power-law index and Prandtl number for Re ) 5 and λ ) 0.5.

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Figure 12. (a) Effect of power-law index and blockage ratio on normalized Nusselt number at Re ) 5 and Pr ) 1. (b) Effect of power-law index and blockage ratio on normalized Nusselt number at Re ) 5 and Pr ) 100.

Figure 11. (a) Dependence of average Nusselt number on power-law index and Prandtl number for Re ) 100 and λ ) 0. (b) Dependence of average Nusselt number on power-law index and Prandtl number for Re ) 100 and λ ) 0.2. (c) Dependence of average Nusselt number on power-law index and Prandtl number for Re ) 100 and λ ) 0.5.

conditions. However, as the value of the Prandtl number is gradually increased, all else being equal, it is possible to achieve enhancements in the rate of heat transfer of up to 50-55% as the fluid behavior index changes from n ) 1 to n ) 0.2. This is simply due to the lowering of the effective viscosity of the fluid with the decreasing value of the fluid behavior index. A detailed inspection of the surface averaged values suggests the scaling Nu ∝ ∼Pr1/3, except for the results corresponding to Re ) 5. This finding is also consistent with that reported for Newtonian fluids. Furthermore, since the velocity field has been shown to decay much more rapidly in shear-thinning fluids than that in Newtonian fluids, it is plausible that the flow attains free stream conditions in the lateral direction well before reaching the confining wall. Since, for Pr > 1, the thermal boundary layer is expected to be thinner than the momentum boundary layer, the temperature field is therefore likely to decay over even a shorter (lateral) distance. Under these conditions, the temperature gradient in shear-thinning fluids is not influenced by the confining wall to the same extent as in Newtonian fluids. Therefore, the enhancement in heat transfer is likely to be lower in shear-thinning fluids than that in Newtonian fluids.


Figure 13. (a) Effect of power-law index and blockage ratio on normalized Nusselt number at Re ) 50 and Pr ) 1. (b) Effect of power-law index and blockage ratio on normalized Nusselt number at Re ) 50 and Pr ) 80.

Indeed Figures 12-14 confirm this expectation for all values of λ in which the Nusselt number (normalized with respect to the corresponding value for Newtonian fluids) is plotted against the power-law index for scores of values of the Reynolds number, Prandtl number, and λ. The key trends borne out by these results can be summarized as follows. At low values of Re and Pr or of both, the imposition of walls seems to have a detrimental effect on the rate of heat transfer as the value of the normalized Nusselt number is maximum for an unconfined sphere (λ ) 0) as compared to that for a confined sphere. It, however, needs to be emphasized here that shear-thinning fluid behavior still promotes heat transfer, albeit not as much as in the case of an unconfined sphere. For instance, for Re ) 5, Pr ) 100, and n ) 0.2 (see Figure 12b), one can still expect an increase of about ∼35% in the Nusselt number as compared to that in a Newtonian fluid at λ ) 0.5, but this is roughly only about 70% of the value predicted for an unconfined sphere at the same values of the Reynolds and Prandtl numbers and power-law index. On the other hand, as the value of the Reynolds number or Prandtl number or both is progressively increased, the degree of enhancement in heat transfer increases. Also, at Re ) 10

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Figure 14. (a) Effect of power-law index and blockage ratio on normalized Nusselt number at Re ) 100 and Pr ) 1. (b) Effect of powerlaw index and blockage ratio on normalized Nusselt number at Re ) 100 and Pr ) 40.

and Pr ) 100 (not shown), one begins to see that the normalized Nusselt number for λ ) 0.1 exceeds the unconfined case values, and this effect is seen in Figure 13b, where for all values of λ the confinement leads to greater enhancements in heat transfer than that for an unconfined sphere. The same trend can be clearly seen in Figure 14b at Re ) 100 and Pr ) 40. The switchover seen in these figures is a clear indication that the net rate of heat transfer is determined by two competing mechanisms, namely, the rapid decay of the flow which may or may not extend up to the confining wall and therefore the confining wall may not contribute to the sharpening of the temperature gradients. On the other hand, because the boundary layers (momentum and thermal) are known to be thinner in shear-thinning fluids than in Newtonian fluids, one would expect to see some enhancement in convection, but naturally this mechanism will contribute appreciably only at moderate values of the Reynolds and Prandtl numbers. Thus the former mechanism dominates at low Re and Pr values, while the latter gains importance at intermediate Reynolds and Prandtl numbers.

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In summary, depending upon the values of the Reynolds number, Prandtl number, power-law index, and blockage ratio, it is possible to realize enhancement in heat transfer by up to about 60-70% with respect to that in Newtonian fluid under otherwise identical conditions. This can lead to improved energy utilization and process efficiency. Finally, before leaving this section, it is worthwhile to reiterate here that the present analysis does not account for the temperature dependence of the thermophysical properties, notably viscosity, of the fluid. The available limited literature suggests that this correction is small for Newtonian fluids (e.g., see the correlation due to Whitaker37), and therefore, in the first instance, the same correction can be used for power-law fluids also until this effect is delineated for power-law fluids. Conclusions In this work, the role of confinement has been studied on the rate of convective heat transfer from a sphere sedimenting in power-law fluids in cylindrical tubes over wide ranges of pertinent variables, 5 e Re e 100, 0.2 e n e 1, and 0 e λ e 0.5, and for varying ranges of the Prandtl number subject to the limitation of the maximum Peclet number e4000. Broadly speaking, irrespective of the values of λ and n, the Nusselt number shows positive dependence on both Reynolds and Prandtl numbers. Similarly, all else being equal, shearthinning behavior facilitates heat transfer (owing to the lowering of the effective viscosity) compared to that in Newtonian fluids. Indeed, it is possible to achieve enhancement up to about 60-70% in heat transfer. On the other hand, while the imposition of walls does lead to an increase in the value of the convective coefficient, the degree of increase exceeds the unconfined case values only at moderate values of the Reynolds and Prandtl numbers. Finally, the smaller the value of the power-law index, the greater the enhancement in heat transfer. This work clearly shows that the rate of heat transfer from a sphere immersed in confined power-law fluids is determined by an intricate interplay between the physical and kinematic variables. Acknowledgment R.P.C. is grateful to IIT, Kanpur, and WVU, Morgantown, for providing partial travel support to the United States, thereby facilitating this collaboration. Notation Cp ) heat capacity, J/kg K d ) sphere diameter, m D ) tube diameter, m f ) body force, N I2 ) second invariant of the rate of deformation tensor, dimensionless h ) heat transfer coefficient, W/m2 K k ) thermal conductivity of fluid, W/m K Ld ) downstream distance, dimensionless Lu ) upstream distance, dimensionless m ) power-law consistency index, Pa · sn n ) power-law index, dimensionless Nu ) Nusselt number, dimensionless p ) isotropic pressure, Pa Pr ) Prandtl number, dimensionless R ) radius of sphere, m R∞ ) radius of spherical domain, m Re ) Reynolds number, dimensionless

r, z ) coordinates, m T ) fluid temperature, K To ) free stream fluid temperature, K Tw ) constant temperature at the surface of the sphere, K T* ) dimensionless temperature V ) velocity vector, m/s Vo ) free stream velocity, m/s Greek Symbols R ) thermal diffusivity, m2/s ε ) rate of strain tensor, s-1 η ) viscosity, Pa · s λ ) diameter ratio ()d/D), dimensionless θ ) angular position on the surface of the sphere measured from the front stagnation point, deg F ) fluid density, kg/m3 τ ) extra stress tensor, Pa σ ) stress tensor, Pa

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ReceiVed for reView September 29, 2009 ReVised manuscript receiVed February 15, 2010 Accepted February 22, 2010 IE901524H

Effect of Blockage on Heat Transfer from a Sphere in Power-Law Fluids

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