Flow and Forced Convection Heat Transfer in Crossflow of Non

Ind. Eng. Chem. Res. 2005, 44, 5815-5827

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Flow and Forced Convection Heat Transfer in Crossflow of Non-Newtonian Fluids over a Circular Cylinder A. A. Soares and J. M. Ferreira Departamento de Fı´sica, Universidade de Tra´ s-os-Montes e Alto Douro, Apartado 1013, 5000-911 Vila Real, Portugal

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

The steady and incompressible flow of power-law type non-Newtonian fluids across an unconfined, heated circular cylinder is investigated numerically to determine the dependence of the individual drag components and of the heat transfer characteristics on power-law index (0.5 e n e 1.4), Prandtl number (1 e Pr e 100), and Reynolds number (5 e Re e 40). The momentum and energy equations are expressed in the stream function/vorticity formulation and are solved using a second-order accurate finite difference method to determine the pressure drag and frictional drag as well as the local and surface-averaged Nusselt numbers and to map the temperature field near the cylinder. The accuracy of the numerical procedure is established using previously available numerical and analytical results for momentum and heat transfer in Newtonian and power-law fluids. The results reported herein provide fundamental knowledge of the flow and heat transfer behavior for the flow of non-Newtonian fluids over a circular cylinder; these results further show that the effect of the power-law index on such behavior is strongly conditioned by the kinematic conditions and less so by the type of thermal boundary condition prescribed at the cylinder surface. 1. Introduction The flow of fluids and forced convection across a heated cylinder has been the subject of considerable research interest because of its relevance in many engineering applications. For instance, a knowledge of the hydrodynamic forces experienced by submerged cylindrical objects such as off-shore pipelines is essential for the design of such structures. On the other hand, because of changing process and climatic conditions, one also needs to determine the rate of heat transfer from such structures. Furthermore, present economic and environmental concerns have raised an interest in methods to determine and control forced convection heat transfer from horizontal cylindrical structures. Additional industrial processes where heat/mass transfer from an isolated cylinder plays an important role include anemometry and chemical or radioactive contamination/purification, glass cooling, plastics and industrial devices, and other processes from turbine blades to electronic circuits. Because of the importance of these applications, there has been a great deal of interest in the flow of Newtonian fluids and the heat transfer across a single cylinder from the experimental, analytical, and numerical standpoints (e.g., see refs 1-8), and excellent reviews are indeed available (e.g., refs 9-12). On the basis of a combination of such analytical/numerical simulations and the available experimental results, it is perhaps fair to say that satisfactory methods are now available which enable the prediction of gross engineering parameters such as * Author to whom correspondence should be addressed. Tel.: 00 91 512 259 7393. Fax: 00 91 512 259 0007/0104. E-mail: [email protected].

pressure and frictional drag, as well as Nusselt and/or Sherwood numbers, for the flow of Newtonian fluids across a single cylinder over most conditions of interest. On the other hand, it is readily acknowledged that many materials (e.g., polymer solutions, melts, muds, emulsions, and suspensions) encountered in chemical and processing applications often exhibit complex nonNewtonian behavior including shear thinning and shear thickening, viscoelasticity, and yield stress.13,14 Moreover, reliable knowledge of the hydrodynamic forces and heat transfer characteristics of submerged cylindrical objects is needed in connection with engineering processes which include the use of wires and thin cylinders as measurement probes and sensors in non-Newtonian flow15-18 and in the design of slurry pipelines where large particles are conveyed in a non-Newtonian vehicle. Additional examples are found in polymer processing operations such as the use of submerged surfaces to form weld lines. Despite such overwhelming importance and frequent occurrence of non-Newtonian behavior, very little work has been reported in the literature on the crossflow and forced convection heat transfer to nonNewtonian fluids from a long, heated cylinder, and the present work aims to fill this gap in the existing literature. It is, however, instructive and useful to briefly summarize the previous scant results available in the literature prior to undertaking the formulation of the present problem. From a theoretical standpoint, because of the rheological complexities of non-Newtonian flow across a long, heated cylinder, most of the work in this field relies on the use of a simple, two-parameter power-law model, which must be combined with governing equations of momentum, continuity, and energy to obtain the pres-

10.1021/ie0500669 CCC: $30.25 © 2005 American Chemical Society Published on Web 06/11/2005

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Figure 1. Real (x, y) plane and the computational (, θ) plane. Variables include the distance (H) from the external boundary to the cylinder surface, the free stream fluid temperature (T∞), and the uniform approach velocity (U∞).

sure and friction drag coefficients and/or heat transfer rate. The use of this simple model enables the degree of complexity of the non-Newtonian effects to be kept at a tractable level, well-suited for the study of the simplest and also the most common type of nonNewtonian behavior, i.e., shear-thinning behavior (n < 1) and shear-thickening behavior (n > 1). Thus, the creeping flow of power-law fluids past an unconfined circular cylinder has been investigated by Tanner,19 who demonstrated that, in the limit of zero Reynolds number, the Stokes paradox is removed for shear-thinning fluids but not for shear-thickening fluids. Subsequently, a similar inference was also reported by Maruˇsic´Paloka.20 Tanner19 also supplemented his analytical results with numerical predictions for values of the power-law index from 0.4 to 0.9; the correspondence between the two results was found to be reasonable. Subsequently, these results have been extended to values of the power-law index as low as n ) 0.2,21 and these new results are consistent with the previous results for spheres and cylinders. Apart from these creeping flow analyses, as far as is known to us, there have been only two studies of power-law fluid flow over a circular cylinder at finite Reynolds numbers.22,23 D’Alessio and Pascal22 numerically investigated the steady power-law flow around a cylinder at Reynolds numbers Re ) 5, 20, and 40 and examined the dependence of critical Reynolds number, wake length, separation angle, and drag coefficient on the value of the power-law index for a fixed blockage ratio β ) 0.037, where β is defined as the ratio between the cylinder diameter and the distance H from the external boundary to the cylinder surface (Figure 1). They reported that a progressive increase in Reynolds number (Re) from 5 to 40 resulted in a decrease in the degree of convergence, which restricted the range of power-law indices for which a fully converged numerical solution was possible from 0.65 to 1.2 at Re ) 5 to 0.95-1.1 at Re ) 40. Subsequently, Chhabra et al.23 extended the work of D’Alessio and Pascal22 to include a wide range of conditions (β ) 0.037, 0.082, and 0.164; Re ) 1, 20, and 40; and 0.2 e n e 1.4). However, in both these studies, only the values of the total drag coefficient were reported, and hence, further work is still needed to determine the effect of the power-law index on the individual pressure and frictional drag coefficients. Coelho and Pinho24,25 undertook a detailed experimental study of the flow characteristics of various vortex shedding regimes for the flow of non-Newtonian fluids around a cylinder at Re ) 50-9000 and reported that shear thinning resulted in decreased cylinder boundary-layer thickness and diffusion length. Aside from these studies, some effort has also been directed at studying the momentum and thermal boundary-layer flows of powerlaw fluids over a circular cylinder, and these have been

reviewed elsewhere.26 Similarly, there have been a few experimental studies dealing with the forced convection heat transfer from a circular cylinder to streaming nonNewtonian fluids,27-32 but most of these relate either to very high values of Reynolds number and/or to the flow of viscoelastic or drag-reducing polymer solutions.33 Admittedly, these studies are not of direct interest here, but these are mentioned here for the sake of completeness. Finally, it is appropriate to mention here that limited results on drag and heat transfer from square cylinders to power-law fluids were reported recently.34 In summary, while scant numerical results are available on the total drag of circular cylinders in powerlaw fluids at finite Reynolds numbers, no information is available on the individual drag components and on the forced convection heat transfer from circular cylinders to power-law fluids. The present work thus aims to fill these gaps in the existing literature and to numerically solve the momentum and heat transfer equations for steady power-law flow across a long, heated circular cylinder over a range of power-law indices (0.5 e n e 1.4), Reynolds numbers 5 e Re e 40, and Prandtl numbers Pr ) 1, 5, 25, 50, and 100, in order to determine the Nusselt number and related parameters, map the isothermal lines, and also extend the work of Chhabra et al.23 and investigate the effect of the power-law index on both the pressure and frictional drag coefficients. A blockage value of β ) 0.037 was selected for this work because it corresponds to asymptotic boundary conditions at 54.6 radii away from the cylinder surface,23 a distance which is regarded to be sufficient to ensure that the flow and heat transfer at this surface are insensitive to boundary effect and, therefore, models the flow of an unbounded fluid across the heated cylinder surface. 2. Basic Equations Consider the steady flow of an incompressible powerlaw fluid with a constant far away streaming velocity (U∞) and temperature (T∞), along the x direction normal to the axis of a circular cylinder (Figure 1a). Since the present study is restricted to long cylinders and flow conditions of Re e 40, the flow across the cylinder is steady and two-dimensional; i.e., all flow variables are independent of the z- coordinate and are, therefore, functions of cylindrical coordinates r and θ alone. Furthermore, the thermophysical properties (density F, heat capacity Cp, thermal conductivity k, and powerlaw parameters K and n) are assumed to be independent of temperature. Under these conditions, the momentum and energy equations are not coupled. The equation of continuity, the r and θ components of the equations of motion, and the thermal energy (in the absence of

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5817

viscous dissipation) in cylindrical coordinates35 can be expressed in their dimensionless stream function/vorticity formulation in terms of the polar coordinates (, θ) with  ) ln(r/a), where a is the radius of the cylinder, as follows,

continuity equation ∂ ∂ψ 1 ∂ ∂ψ e ψ+ )0  ∂ ∂θ ∂θ ∂ e

( )

(

2

(

)

(1)

-momentum component

)(

)

∂ψ ∂ ψ ∂ψ ∂2ψ ∂ψ +ψ +ψ ) + ∂θ ∂∂θ ∂ ∂ ∂θ2 ∂τrθ 1 ∂p 2n - ∂  e (e τrr) + - τθθ (2a) 2 ∂ Re ∂ ∂

-

2

[

]

θ - momentum component

(

) (

) [

∂ψ ∂2ψ ∂ψ ∂2ψ ∂ψ + ψ ) + + ∂θ ∂2 ∂ ∂ ∂∂θ ∂τθθ 1 ∂p 2n -2 ∂ 2 (e τrθ) + e (2b) 2 ∂θ Re ∂ ∂θ 2

(

]

energy equation

)

∂ T ∂ T ∂T eRePr ∂ψ + 2 + + ∂ 2 ∂θ ∂2 ∂θ2 ∂T eRePr ∂ψ eRePr ∂ψ ψ+ +T 1+ ) 0 (3) ∂θ 2 ∂ 2 ∂θ

(

)) (

(

)

(U∞/a)2I2, respectively. In eq 3, the temperature T is scaled in two different ways depending on the thermal boundary condition imposed at the cylinder surface, i.e., at  ) 0 (see Figure 1). The two commonly used thermal boundary conditions at  ) 0 are that of either a constant temperature (Ts) or a constant heat flux (qs). Thus, for the constant temperature boundary condition, the dimensionless temperature T is related to its dimensional counterpart as e-T(Ts - T∞), whereas, for the constant heat flux boundary condition, the dimensionless temperature T is related to its dimensional counterpart as e-Tqsa/k. The Reynolds number is defined as

Re )

Pr )

η ) I2(n-1)/2

(5)

where I2 is the dimensionless second invariant of the rate-of-deformation tensor given as

[(

I2 ) e-2 ψ -

) ( )]

∂2ψ ∂2ψ + 2 ∂2 ∂θ

2

+4

∂2 ψ ∂θ∂

2

(6)

The vorticity in its scaled form is given as

∂ψ ∂2ψ ∂2ψ + 2+2 +ψ+ω)0 2 ∂ ∂ ∂θ

(7)

In eqs 1-7, the dimensionless stream function ψ, vorticity ω, and pressure p are related to their dimensional counterparts as eU∞aψ, e-(U∞/a)ω, and (U∞2F/ 2)p, respectively. The dimensionless components of the extra stress tensor τij and the dimensionless second invariant of the rate of deformation tensor I2 are related to their dimensional counterparts as K(U∞/a)nτij and

( )

CpK U∞ k 2a

n-1

(9)

Eliminating the pressure in eq 2 by the usual method of crossdifferentiation with the introduction of the vorticity ω, followed by some rearrangement, leads to

η

(

)

∂2ω ∂2ω ∂ω ∂ω + 2µ + γω ) F (10a) + 2 + 2λ ∂ ∂θ ∂2 ∂θ

where

(4)

where η is the dimensionless viscosity and ij are the dimensionless components of the rate-of-deformation tensor (e.g., ref 35). The equation for the dimensionless power-law viscosity is

(8)

where K denotes the power-law consistency index and n denotes the power-law index. The Prandtl number is defined as

where p, Re, and Pr are the dimensionless pressure, the Reynolds number, and the Prandlt number, respectively. The dimensionless components of the extra stress tensor for a power-law fluid are written as

τij ) -ηij

F(2a)nU∞2-n K

λ)

Re‚e ∂ψ ∂η - η - n+1 ∂ ∂θ 2

(10b)

µ)

∂η Re‚e ∂ψ + n+1 +ψ ∂θ ∂ 2

(10c)

(

γ)-2

(

F) ψ-

)(

)

Re‚e ∂ψ ∂η +η+ n ∂ 2 ∂θ

)

(10d)

∂2ψ ∂2ψ ∂2η ∂2η ∂η + 2 - 2+2 2 2 ∂ ∂ ∂θ ∂θ ∂ ∂2ψ ∂η ∂2 η 4 (10e) ∂θ∂ ∂θ ∂θ∂

(

)

The exponential scaling for the stream function, the vorticity, and the temperature is appropriate since the stream function is exponentially large far from the cylinder, the vorticity is exponentially small everywhere except in the region of the wake, and the temperature decreases rapidly away from the cylinder. This scaling procedure, used by D’Alessio and Pascal22 and Chhabra et al.,23 is also employed in the present study because it suppressed the numerical instabilities which typically occur for lower values of n. Due to the two-dimensional nature of the flow problem (x-y plane) and because the oncoming flow is in the x direction, it is sufficient to consider the region y g 0 and x2 + y2 g 1 only. Thus, the corresponding region in the (, θ) plane is defined by  g 0 and 0 e θ e π (Figure 1). The physically realistic boundary conditions for this flow are expressed as follows. On the cylinder surface, i.e., at  ) 0, the usual no-slip condition is applied

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∂ψ ∂ψ ) )0 ∂ ∂θ

(11a)

where the dimensionless viscosity in real space (η j ) is given as

η j ) hI2(n-1)/2

which together with eq 7 gives

ψ ) 0 and ω ) -

∂2ψ ∂2

(11b)

The two commonly used thermal boundary conditions at the surface of the solid cylinder are scaled as

T)1

and the dimensionless second invariant of the rate of deformation tensor in real space (Ih2) is obtained through the insertion into eq 6 of ψ ) e-ψ h , where ψ h is the dimensionless stream function in real space. Inserting eq 13 into eq 12a gives

(11c)

CDF ) -

for the constant temperature condition and

2n+1 Re

∫0π (Ih2(n-1)/2 ω))0 sin(θ) dθ

(11d)

for the constant heat flux condition, where the dimensionless temperature (T) used in the present study is related to the previously used34,36,37 dimensionless temperature in real space (T h ) as T ) eT h. On the plane of symmetry at θ ) 0, π, we use the following:

∂T )0 ∂θ

(11e)

Far away from the cylinder surface, for ∞ ) 4, we use the asymptotic approximation for stream function and vorticity given by Chhabra et al.,23

CD - θ e - erf(Q) ψ ≈ sin(θ) + 2 π

(

)

(11f)

p|)0 ) p0 +

CDReI2

2n+1xπ

Qe-Q

2

Q)e

x

Re (1-n)/4 θ I sin n 2 2 2

()

∂T )0 T ) 0 and ∂

2n+1 Re

CDP ) -

∫0π (ηj ω))0 sin(θ) dθ

∫0

∂(η jω j) ∂

)0

dθ

(15)

(16)

Equations 13 and 16 are inserted into eq 15 which is then used to integrate eq 12b by parts, giving

CDP )

2n+1 Re

∫0π [∂∂ (e-hI2(n-1)/2 ω)])0 sin(θ) dθ

(17)

which is rearranged as

CDP )

[

2n+1 Re

- ω) + ∫0π (Ih2(n-1)/2))0 (∂ω ∂ n - 1 1 ∂Ih2 ω 2 hI2 ∂

]

sin(θ) dθ (18)

)0

CD ) CDF + CDP (11h)

(11i)

The system of elliptic partial differential eqs 3, 7, and 10, together with the corresponding boundary conditions (eq 11), has been discretized using the finite difference method. The resulting system of algebraic equations has been solved using an iterative Gauss-Seidel relaxation method. Once the values of ω, ψ, and T are computed in the flow domain, the coefficients of frictional drag CDF, pressure drag CDP, total drag CD, local Nusselt number Nu(θ), and average Nusselt number Nu can be determined from the corresponding equations. The equations for the pressure and frictional drag coefficients are obtained as follows

π

[ ]

The total drag coefficient,

and erf(Q) is the standard error function. The far away stream temperature boundary conditions are

CDF ) -

∫0θ

ω j ) e-ω

(11g)

where CD is the drag coefficient, /2

2n+1 Re

where p0 is the dimensionless stagnation pressure and ω j is the dimensionless vorticity in real space, i.e.,

(1-n)/2

ω≈-

(14)

From ref 38,

∂T )T-1 ∂

ψ)ω)

(13)

(p))0 cos(θ) dθ

(12a) (12b)

(19)

is obtained from eqs 14 and 18. The rate of heat transfer is usually expressed in terms of the Nusselt number. The local Nusselt number on the cylinder surface is defined by36,37

(∂T∂rjh )

Nu(θ) ) -2

rj)1

) -2

(∂T∂ - T)

)0

(20)

for the constant temperature boundary condition (isothermal cylinder surface) and by

Nu(θ) )

h(2a) 2 ) k T

|

)0

(21)

for the constant heat flux boundary condition, where rj ) e and T ) eT h. The surface averaged value of the Nusselt number is given by

Nu )

1 π

∫0π Nu(θ) dθ

(22)

3. Numerical Solution Method The numerical solution procedure used here is an iterative Gauss-Seidel relaxation method already used in an earlier study,23 and to avoid redundancy only the

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5819

main features are included here. For the discretization of eqs 3, 5, 7, and 10, a finite difference method was employed. A second-order upwind differencing technique is used to solve eqs 3 and 10 with one-sided difference approximations to the convective terms of T and ω, whereas for the diffusion terms, the central difference approximation was used. For all other terms in these equations, central difference approximations have been employed, except for the first derivative of ψ in the direction of , which was determined using the secondorder forward difference approximation. This forward difference approximation is necessary to obtain convergence for n e 0.6. Equations 5 and 7 are rewritten as finite difference equations using the central difference of second-order accuracy. The resulting system of equations is solved using a Gauss-Seidel iterative method with an under-relaxation factor of 0.8 to the temperature and vorticity variables. To obtain consistent approximations for ψ, ω, and T, for each iteration a sweep is made through all mesh points and updated values of the drag coefficient CD and the Nusselt number Nu, which correspond to each of the two thermal surface boundary conditions, are determined by numerical integration of eqs 19-21 on the cylinder surface using Simpson’s rule. Note that the solutions of the temperature fields defined by the two thermal surface boundary conditions are obtained simultaneously. Convergence was achieved when, for the same iteration, the variation in CD and Nu values in two successive iterations was less than a preset value of 10-8. To accelerate the convergence of the numerical solution, the Newtonian steady solution values of ψ, ω, and T at every point of the grid were used as the initial guesses for nonNewtonian flow. The outer boundary was positioned at ∞ ) 4, corresponding to asymptotic boundary conditions at a distance of ∼54.6 radii away from the cylinder. For a (N + 1) × (M + 1) computational grid, the spacings in the radial  and angular θ directions are ∞/N and π/M, respectively. For the choice of appropriate grid, additional tests were carried out for the largest Reynolds and Prandtl numbers (Re ) 40 and Pr ) 100) for 0.5 e n e 1.4. Initial tests carried out using three grids (mesh sizes 101 × 51, 201 × 101, and 401 × 101) showed that the gain in accuracy of the drag coefficient and average Nusselt number which resulted from the use of the finest grid (401 × 101) was overall 1, Nu* was a decreasing function of Pr, i.e., for n < 1, the rate of increase of Nu with Pr was larger than that for Newtonian fluids, whereas for n > 1, this rate of increase was smaller than that for Newtonian fluids. The trends seen in Figures 7 and 8 can qualitatively be explained as follows: since both the Reynolds and Prandtl numbers are functions of the kinematic and rheological properties due to the power-law viscosity, defined as K(v/2a)n-1, it is not possible to isolate the role

Figure 8. Effect of the power-law index (n) on the normalized surface-averaged Nusselt number (Nu*) corresponding to various values of Pr, using the constant heat flux boundary condition on the cylinder surface for (a) Re ) 5 and (b) Re ) 40.

of the power-law index on heat transfer. This difficulty, however, can be obviated by using the Reynolds and Peclet (Pe ) Re × Pr) numbers instead of Re and Pr. Figures 7 and 8 relate to the fixed values of Re and show the effect of increasing Prandtl number. It can readily be seen from the definition of Pe that the increasing Pr implies increasing Pe, and for fixed values of F, K, and a, the only way to achieve this is by increasing the velocity. This in turn leads to enhanced levels of shearing (≈ v/2a). For n < 1, this lowers the effective viscosity, which facilitates heat transfer, and of course, the reverse is true for n > 1. Finally, the use of Nu* as a benchmarking criteria to study the effect of nonNewtonian behavior on heat transfer characteristics showed that the choice of the type of thermal boundary condition (constant temperature or constant heat flux) did not produce a significant impact on heat transfer in power-law fluids (Figures 7 and 8). Representative plots of the isotherms for both the constant temperature (top half) and the constant heat flux (bottom half) boundary conditions (Figures 9 and 10) showed that an increase in the Prandtl and/or Reynolds numbers increased the asymmetry and complexity of the aft contours relative to the fore contours

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Figure 9. Effects of n and Pr on the isothermal lines for Re ) 5, using the constant temperature (top) and the constant heat flux (bottom) boundary conditions on the cylinder surface (flow direction ) left to right).

and also compacted the isothermal lines, which resulted in an increased temperature gradient overall. Moreover, since the increase in the Prandtl and/or Reynolds numbers increased the compactness of the isothermals toward the upstream direction (Figures 9 and 10), it also resulted in a more pronounced upstream increase in the local temperature gradient along the cylinder surface, consistent with the corresponding variation in Nusselt number (Figures 3 and 4). The effects of the power-law index on the temperature profiles also became more pronounced at higher Prandtl and/or Reynolds numbers (Figures 9 and 10). For both the constant temperature and constant heat flux boundary conditions, increased shear thinning resulted in an overall increase in the compactness of the isothermals (Figures 9 and 10), which increased the temperature gradient overall. Both the present results for a circular cylinder and those of Gupta et al.34 for the square cylinder showing an increase in the overall temperature gradient with shear thinning are in line with the previously predicted42,43 decrease in boundary layer thickness with increased levels of shear thinning, which has recently been confirmed experimentally for a circular cylinder.25 Before closing the discussion, it is appropriate to mention the two main limitations underlying the results which have been presented in this study. First, it needs to be emphasized here that the governing equations are highly nonlinear due to both the shear dependent

viscosity and the inertial terms. Hence, the assumption of temperature-independent physical properties affords a great simplification, as it does allow the momentum and energy equations to be decoupled. Clearly, in the absence of this assumption, this decoupling is not possible. Besides, it should be borne in mind that these are the very first set of results on this topic, and naturally, these will get superseded by more realistic fluid models and/or by considering temperature-dependent properties and/or by including viscous dissipation effects in future studies. It is suggested that, in the first instance, the effect of the temperature-dependent viscosity could be incorporated by using the same correction as that used for Newtonian fluids. Second, admittedly most non-Newtonian materials exhibit much more complex rheological behavior (especially viscoelasticity) than that captured by the simple power-law fluid model used herein. However, viscoelastic effects are generally significant in transient flows and/or in severely confined flows where extensional deformation can occur. In most other situations, viscous effects dominate the macroscopic flow phenomena such as drag and heat transfer, e.g., see refs 17, 18, and 44. For instance, for flow around a sphere, when both shear thinning and viscoelasticity are present, shear-thinning behavior dominates the drag and convective heat- and mass-transfer behavior.26,45 Therefore, it seems reasonable to start with the

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Figure 10. Effects of the power-law index (n) and Prandtl number (Pr) on the isothermal lines for Re ) 40, using the constant temperature (top) and the constant heat flux (bottom) boundary conditions on the cylinder surface (flow direction ) left to right).

simplest type of non-Newtonian behavior and to build up the level of complexity gradually. 5. Conclusions The non-Newtonian flow across a heated circular cylinder was investigated numerically to determine the hydrodynamic drag components and heat transfer characteristics for a wide range of power-law indices (0.5 e n e 1.4), Reynolds numbers (5 e Re e 40), and Prandtl numbers (1 e Pr e 100). The study of flow parameters showed that the frictional drag component increased with the flow-behavior index n and that the dependence of this component on n increased with Re. For Re e 15, increased shear thinning increased the total drag, whereas for Re g 15, it decreased the total drag. The study of heat transfer showed that the constant heat flux boundary condition was more efficient than the constant temperature condition. For the latter condition, the effect of the power-law index on the local Nusselt number Nu(θ) was less pronounced than that for the constant heat flux condition. A maximum value of Nu(θ) was located close to the front stagnation point, and its peak value was found to increase with shear thinning. At the location where the maximum peaked, the dependence of Nu(θ) on n was found to be strongest. The surface-averaged Nusselt number Nu was found to be a decreasing function of n, although for lower Peclet

numbers such dependence was found to be less pronounced. The effects on isothermal patterns, which result from an increase in Prandtl number and/or Reynolds number, are qualitatively similar to those which result from a decrease in the power-law index. The faster decay in the temperature field for higher Prandtl numbers and a lower power-law index suggests a decrease of the thermal-boundary-layer thickness under the aforementioned conditions. Nomenclature a ) radius of the cylinder (m) CD ) drag coefficient CDF ) friction drag coefficient CDP ) pressure drag coefficient Cp ) heat capacity (J/kg/K) e ) voidage of cylinder assemblage F ) dimensionless function, eq 10e h ) heat transfer coefficient (W/m2 K) H) distance from the external boundary to the cylinder surface (m) I2 ) dimensionless second invariant of the rate-of-deformation tensor hI2 ) dimensionless second invariant of the rate-of-deformation tensor in real space K ) power-law consistency index (Pa sn) k ) thermal conductivity of the fluid (W/m K)

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Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005

L ) wake length (m) n ) power-law index Nu ) average Nusselt number Nu* ) average Nusselt number normalized with the corresponding value for the Newtonian fluid Nu(θ) ) local Nusselt number p ) dimensionless pressure p0 ) dimensionless stagnation pressure Pe ) Peclet number Pr ) Prandtl number r ) cylindrical coordinate (m) rj ) dimensionless radial component in real space qs ) heat flux at the surface of the cylinder (W/m2) Re ) Reynolds number ()F(2a)nU∞2-n/K) U∞ ) uniform approach velocity (m/s) T ) dimensionless temperature T h ) dimensionless temperature in real space Ts ) temperature on the surface of the cylinder (K) T∞ ) free stream fluid temperature (K) vz ) z-component of velocity (m/s) Greek Letters γ ) dimensionless function, eq 10d θ ) angle (radians) θs ) separation angle (degrees) λ ) dimensionless function, eq 10b µ ) dimensionless function, eq 10c ψ ) dimensionless stream function ψ h ) dimensionless stream function in real space η ) dimensionless viscosity η j ) dimensionless viscosity in real space ω ) dimensionless vorticity ω j ) dimensionless vorticity in real space F ) fluid density (kg/m3)  ) dimensionless polar coordinate ()ln(r/a)) ij ) components of rate-of-deformation tensor (s-1) τij ) dimensionless components of extra stress tensor Subscripts θ ) angular component r ) radial component

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Received for review January 18, 2005 Revised manuscript received April 29, 2005 Accepted May 10, 2005 IE0500669