Forced Convection Heat Transfer in Power Law Liquids from a Pair of

Ind. Eng. Chem. Res. 2008, 47, 9141–9164

9141

Forced Convection Heat Transfer in Power Law Liquids from a Pair of Cylinders in Tandem Arrangement Rahul C. Patil, Ram P. Bharti,† and Rajendra P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

Forced convection heat transfer characteristics for the flow of incompressible power law fluids over a pair of cylinders (of equal diameters) in tandem arrangement has been studied in the two-dimensional, steady crossflow regime. The field equations have been solved using a finite volume method based solver (FLUENT 6.2) over the ranges of conditions as follows: power law index (0.2 e n e 1.8), Reynolds number (1 e Re e 40), Prandtl number (1 e Pr e 100), and the gap ratio between the two cylinders (2 e G e 10). Extensive results on the isotherm patterns, centerline temperature profiles, and local and average Nusselt numbers have been presented in detail, for the two commonly used thermal boundary conditions, namely, constant temperature or constant heat flux prescribed on the surface of the two cylinders. While the upstream cylinder shows characteristics similar to an isolated cylinder, the downstream cylinder displays a complex dependence on the relevant dimensionless parameters. As expected, at large values of the gap ratio (G), the behavior of the downstream cylinder approaches that of a single cylinder, thereby suggesting no or weak interference between the two cylinders. Depending upon the values of G and n, both the wake interference and power-law rheology influence the heat transfer characteristics to varying extents. Generally, the upstream cylinder shows higher values of the average Nusselt number than the downstream cylinder. On the other hand, the average Nusselt number for both cylinders is seen to be smaller than that for a single cylinder under otherwise identical conditions. With reference to Newtonian fluids, the shear-thinning behavior promotes heat transfer, whereas shear-thickening lowers it. Finally, simple predictive correlations are developed to estimate the value of the average Nusselt number or the j-factor in a new application. 1. Introduction Owing to its fundamental and pragmatic significance, the flow of power-law fluids over a pair of cylinders in tandem arrangement was studied recently in the steady flow regime.1 Extensive numerical results on the detailed and global characteristics including streamline contours, centerline velocity profiles, wake characteristics, surface pressure profiles, and drag coefficients were presented to elucidate the interplay between the power-law rheology (n), the separation between the two cylinders (G), and the Reynolds number (Re). Indeed, depending upon the value of G, the drag on the two cylinders could be quite different from that for a single cylinder. Similarly, wake formation was seen to be somewhat delayed in shear-thinning (n < 1) fluids, whereas it occurred at lower Reynolds numbers in shear-thickening fluids (n > 1) as compared to that in Newtonian fluids. Since these changes in the flow field are likely to impact directly on heat transfer characteristics, this study aims to extend our previous work1 to convective heat transfer over a wide range of Prandtl numbers, as encountered in chemical and processing applications.2–4 Suffice it to add here that, even for the Newtonian fluids, the momentum characteristics have been studied much more rigorously than the corresponding heat/mass transfer problems for this flow configuration. Many substances of multiphase nature and/or of high molecular weight encountered in industrial settings dealing with pulp and paper suspensions, food, polymer melts and their solutions, etc. display shear-thinning and/or shear-thickening behavior.4 Owing to their high viscosity levels, these materials are generally processed at relatively low Reynolds numbers * Corresponding author. Tel.: +91-512-259 7393. Fax: +91-512259 0104. E-mail: [email protected]. † Present address: Department of Chemical & Biomolecular Engineering, University of Melbourne, Parkville 3010, Victoria, Australia.

(laminar flow) as compared to their Newtonian counterparts like air, water, and light-fraction petroleum products. Thus, it makes the voluminous work available with these fluids to be of limited utility when dealing with viscous liquids. Therefore, it seems reasonable to begin with the analysis of purely viscous powerlaw type fluids, and the level of complexity can be built up gradually to accommodate the other non-Newtonian characteristics such as viscoelasticity. As far as is known to us, there has been no prior study on the forced convection heat transfer in incompressible powerlaw liquids from a pair of cylinders in tandem or any other arrangement. This work is, however, concerned with the tandem arrangement. At the outset, it is, however, useful to review the pertinent limited literature for the analogous situation involving Newtonian fluids, for the literature is scant even for heat transfer from single cylinder to power-law fluids, let alone from two cylinders to facilitate the subsequent presentation of the new results. 2. Previous Work As noted earlier, while the bulk of the information available on the flow characteristics of a cylinder in Newtonian fluids has been thoroughly reviewed by Zdravkovich,5,6 the corresponding heat transfer literature has been reviewed by others.7,8 Over the past 10 years or so, limited information has accrued for the flow of power-law fluids over a single cylinder.9–22 The numerical results based on the solution of the complete field equations have been supplemented by the boundary-layer analysis.23 In all these studies, it was implicitly assumed that the 2-D steady flow regime for power-law fluids exists up to the same critical Reynolds number as that for Newtonian fluids. Clearly, this assumption is not justifiable for highly shearthinning and shear-thickening fluids. Indeed, the limiting values

10.1021/ie7017178 CCC: $40.75  2008 American Chemical Society Published on Web 06/18/2008

9142 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008

Figure 1. Schematic representation of (a) the flow over a pair of cylinders in tandem arrangement and (b) an approximation of an unconfined flow (uniform velocity, Uo, and temperature, To) configuration.

of the Reynolds number have been shown to be strongly dependent on the value of the power-law index.20 For instance, the flow ceases to be 2-D and steady at Re ) 33 in a shearthickening fluid with a power-law index of n ) 1.8, whereas this critical value is rather insensitive to the value of the powerlaw index for shear-thinning (n < 1) fluids. In spite of this uncertainty, reliable results are now available for the drag and heat transfer characteristics of power-law fluid flow over a cylinder in the two-dimensional steady symmetric flow regime, embracing the range of conditions as Re e 40 and 0.2 e n e 2. Limited information is also available on the extent of wall effects on drag and heat transfer from a cylinder confined in a planar slit.17,18 In contrast, much less is known about the momentum and heat transfer from two cylinders in tandem arrangement in the steady cross-flow regime (e.g., see papers by Juncu24,25 and Patil et al.1 for a brief review). There seem to be only two studies25,26 related to the steady forced convection heat transfer in Newtonian fluids from two tandem cylinders at relatively low Reynolds numbers. Wong and Chen26 used a finite-element method to solve the full Navier-Stokes and energy equations for laminar mixed convection in air (Pr ) 0.7) from two horizontal cylinders in tandem arrangement over the wide range of Reynolds numbers (5 e Re e 100) and of the Richardson number (0 e Ri e 3) for a constant value of gap ratio, G ) 2. They noted that the flow and heat transfer behavior of the upstream cylinder are close to that of a single cylinder, whereas the downstream cylinder showed a complex dependence on the kinematic parameters like the Reynolds and Richardson numbers. More recently, Juncu25 has numerically solved the transformed vorticity-stream function formulation of the Navier-Stokes equations using a finite-difference method for

Table 1. Effect of Domain Sizes (with grid M1) on the Average Nusselt number (Nu) for Two Tandem Cylinders at Prandtl Number, Pr ) 100 n ) 0.4 Do/D

Nu1

1100 1200 1300

4.0496 4.0493 4.0493

Nu2

n)1 Nu1

Nu2

n ) 1.8 Nu1

Nu2

G ) 2, Re ) 1 (CWT condition) 2.3986 2.3983 2.3984

3.0147 3.0143 3.0140

1.6091 1.6089 1.6087

2.9272 2.9270 2.9269

1.5075 1.5073 1.5073

G ) 2, Re ) 40 (CWT condition) 300 400 500

23.2006 23.1941 23.1915

7.6060 7.6031 7.6041

16.7554 16.7508 16.7482

7.1337 7.1313 7.1300

12.3468a 12.3435a 12.3416a

6.3491a 6.3471a 6.3461a

G ) 2, Re ) 1 (CHF condition) 1100 1200 1300

4.5686 4.5682 4.5682

2.6788 2.6786 2.6786

3.3964 3.3959 3.3955

1.8953 1.8951 1.8949

3.2856 3.2854 3.2852

1.8037 1.8035 1.8035

G ) 2, Re ) 40 (CHF condition) 300 400 500 a

27.3342 27.3261 27.3234

9.6874 9.6826 9.6848

20.0477 20.0422 20.0390

9.0923 9.0893 9.0876

14.7147a 14.7107a 14.7085a

7.7226a 7.7202a 7.7190a

The Nusselt number values are at Re ) 30.

the steady Newtonian flow and heat transfer with two cylinders (of different diameters, D1 and D2) in tandem arrangement (gap ratio G* ) 2L1/D1, where L1 and L2 are the distances of the upstream and downstream cylinders from the origin of coordinate systems. He presented extensive results showing the effects of the Reynolds number (based on D1), Re ) 1-30; of the diameter ratio, D1/D2 ) 0.5, 1, and 2; and of the Prandtl number as Pr ) 0.1, 1, 10, and 100 on the local and average Nusselt

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9143 Table 2. Grid Specifications number of cells in the computational domain (Ncells) δ/D

grid

M1 M2 M3 M4 M5

Do/D ) 400, G ) 2

N

Do/D ) 1200, G ) 2

(a) Details of Grids Used in Grid Independence 200 175852 400 240208 400 279990 400 302388 400 318388

0.010 0.010 0.005 0.004 0.003

Do/D ) 200, G ) ∞

Do/D ) 1200, G ) ∞

160202 192708 221302 245050 268702

221818 254524 290862 309050 332702

Study 213852 278208 317990 340388 356388

(b) Number of grid cells used in final computations for different values of gap ratio (G) with M4 grid gapratio,G 2 3 4 6 10 ∞

Ncells for Do/D ) 400a 302388 430210 379146 383096 352386 245050 (Do/D ) 200)

Ncells for Do/D ) 1200a 340388 468210 417146 421096 390386 309050

N ) number of grid points on the surface of the cylinder; δ/D ) grid spacing in the proximity of the cylinders.

a

Table 3. Effect of Grid Size on the Average Nusselt Number (Nu) for Two Tandem Cylinders at Prandtl Number, Pr ) 100 N ) 0.4

n)1

Do/D

grid

Nu1

Nu2

1200

M1 M2 M3 M4 M5

4.0493 4.0499 4.0483 4.0477 4.0474

2.3983 2.3986 2.3984 2.3982 2.3981

400

M1 M2 M3 M4 M5

23.1941 23.1749 22.9782 22.9485 22.9264

7.6031 7.5290 7.5913 7.5981 7.6045

1200

M1 M2 M3 M4 M5

4.5682 4.5692 4.5661 4.5652 4.5647

2.6786 2.6790 2.6784 2.6781 2.6780

400

M1 M2 M3 M4 M5

27.3261 27.3202 26.8365 26.7664 26.7140

9.6826 9.6549 9.6551 9.6518 9.6504

Nu1

n ) 1.8 Nu2

Nu1

Nu2

1.6089 1.6091 1.6091 1.6090 1.6090

2.9270 2.9272 2.9270 2.9271 2.9269

1.5073 1.5076 1.5076 1.5078 1.5076

7.1313 7.0936 7.1094 7.1110 7.1126

12.3435a 12.3352a 12.3222a 12.3205a 12.3192a

6.3471a 6.3304a 6.3320a 6.3327a 6.3326a

1.8951 1.8953 1.8952 1.8951 1.8951

3.2854 3.2856 3.2851 3.2852 3.2850

1.8035 1.8038 1.8036 1.8038 1.8037

9.0893 9.0550 9.0471 9.0455 9.0448

14.7107a 14.7035a 14.6510a 14.6446a 14.6398a

7.7202a 7.7041a 7.6977a 7.6972a 7.6963a

G ) 2, Re ) 1 (CWT condition) 3.0143 3.0146 3.0142 3.0141 3.0140

G ) 2, Re ) 40 (CWT condition) 16.7508 16.7353 16.6830 16.6763 16.6722

G ) 2, Re ) 1 (CHF condition) 3.3959 3.3963 3.3955 3.3953 3.3952

G ) 2, Re ) 40 (CHF condition) 20.0422 20.0302 19.8668 19.8462 19.8319

The Nusselt number values are at Re ) 30.

a

Table 4. Comparison of the Average Nusselt Number Values for Newtonian (n ) 1) Flow over Two Cylinders in Tandem Arrangement (G ) 2) for CWT Condition Pr

source

Nu1

0.72

present Wong and Chen26

1.2621 1.3350

1

present Juncu25 present Juncu25 present Juncu25

0.6933 0.6988 1.4629 1.4749 3.0141 3.0386

Nu2

Nu1

0.6645 0.6860

1.7254 1.8270

0.4498 0.4525 0.8072 0.8137 1.6090 1.6224

0.9370 0.9434 1.9872 2.0011 4.1041 4.1320

Re ) 5

100

Nu1

0.8397 0.8930

2.3453 2.4490

0.5458 0.5490 1.0230 1.0304 2.0753 2.0884

1.4129 1.4210 2.9968 3.0136 6.2239 6.2572

Re ) 10

Re ) 1

10

Nu2

Nu1

1.0768 1.2050

3.1673 3.3190

0.7270 0.7311 1.4222 1.4307 2.9126 2.9194

2.6225 2.6321 5.5648 5.5835 11.8854 11.9411

Re ) 20

Re ) 2

numbers for the case of two isothermal cylinders. In contrast, as mentioned above, there has been only one study dealing with the flow of power-law fluids over a pair of two cylinders in tandem arrangement,1 embracing the following ranges of

Nu2

Re ) 5

Nu2 Re ) 40 1.3932 1.8670

Re ) 20 1.1878 1.1937 2.3869 2.3917 4.9264 4.8535

conditions: 1 e Re e 40, 0.4 e n e 1.8, and 2 e G e 10. They revealed significant changes in the flow fields and the global flow parameters depending upon the value of the gap ratio (G). Qualitatively, these trends are in line with that reported


Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9145

Figure 2. Influence of the Reynolds and Prandtl numbers and power-law index on the isotherm patterns for gap ratio G ) 2. (a) Temperature contours at Reynolds number Re ) 1, Prandtl number Pr ) 1, three values of the power-law index, n ) 0.4, 1, and 1.8, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement. (b) Temperature contours at Reynolds number Re ) 10, Prandtl number Pr ) 1, three values of the power-law index, n ) 0.4, 1, and 1.8, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement. (c) Temperature contours at Reynolds number Re ) 40 for the power-law index n ) 0.4 and 1.0 and Re ) 30 for n ) 1.8, Prandtl number Pr ) 1, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement. (d) Temperature contours at three values of the Reynolds number Re ) 1, 10, 30, and 40, Prandtl number Pr ) 100, three values of the power-law index, n ) 0.4, 1, and 1.8, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement.

for the flow of Newtonian27 and shear-thinning fluids28 over two spheres. It is reasonable to surmise that these changes in the flow field should also impinge on the resulting temperature field, which in turn will influence the rate of heat transfer from cylinders in power-law fluids.

In summary, there has been, thus, no prior study delineating the role of non-Newtonian characteristics on the forced convection heat transfer from two tandem cylinders immersed in powerlaw liquids. This work aims to fill this gap in the current literature. In particular, the governing equations are solved



Figure 3. Influence of the Reynolds and Prandtl numbers and power-law index on the isotherm patterns for gap ratio G ) 10. (a) Temperature contours at Reynolds number Re ) 1, Prandtl number Pr ) 1, three values of the power-law index, n ) 0.4, 1, and 1.8, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement. (b) Temperature contours at Reynolds number Re ) 10, Prandtl number Pr ) 1, three values of the power-law index, n ) 0.4, 1, and 1.8, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement. (c) Temperature contours at Reynolds number Re ) 40 for the power-law index n ) 0.4 and 1.0 and Re ) 30 for n ) 1.8, Prandtl number Pr ) 1, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandem arrangement. (d) Temperature contours at three values of the Reynolds number Re ) 1, 10, 30, and 40, Prandtl number Pr ) 100, three values of the power-law index, n ) 0.4, 1, and 1.8, and both thermal boundary conditions (CWT and CHF) for two cylinders in a tandam arrangement.

numerically for the two-dimensional (2-D) steady forced convection heat transfer in incompressible power-law fluids from two cylinders over the following ranges of conditions: Reynolds number (1 e Re e 40), Prandtl number (1 e Pr e 100), powerlaw index (0.4 e n e 1.8), and gap ratio (2 e G e 10). 3. Problem Statement and Governing Equations Similar to our recent work, consider the 2-D, steady, crossflow of an incompressible power-law liquid streaming with a uniform velocity (Uo) and temperature (To) over a pair of infinitely long cylinders (of equal diameter, D) in tandem arrangement (gap ratio, G ) L/D, where L is the center-to-center distance), as shown in Figure 1a. While in practice the thermal boundary conditions on the surface of the cylinder can be quite involved, it is customary to consider the two limiting cases, namely, either a constant wall temperature (CWT), Tw (>To), or a constant heat flux (CHF), qw, imposed on the surface of the two cylinders. In order to keep the level of complexity at a tractable level at this stage, the thermophysical properties of the liquid are assumed to be independent of the temperature and the viscous dissipation effects are assumed to be negligible. While these

two assumptions lead to the decoupling of the momentum and thermal energy equations, these also restrict the applicability of the present results to the situations where the temperature difference between the fluid and the cylinder is not too large and for moderate viscosity and/or shearing levels. For both thermal boundary conditions, the maximum temperature difference between the surface of the cylinders and the liquid ∆T ()Tmax - T0) is maintained small (∆T ≈ 2-3 K) so that it is justified to neglect the variation of the physical properties, notably, density and viscosity, with temperature. In the case of the CWT condition, Tmax ) Tw, whereas the value of heat flux (qw) was monitored to keep the maximum value of ∆T within 2-3 K for the CHF condition. The unconfined flow condition is simulated here by enclosing the two cylinders in a large circular outer boundary (of diameter Do), as shown schematically in Figure 1b. The diameter of the outer boundary Do is taken to be sufficiently large to minimize the effects arising from the size of the domain. The continuity and momentum equations in their compact forms are written as follows: Continuity equation ∇ · U ) 0 Momentum equation F(U · ∇ U - f) - ∇ · σ ) 0

(1) (2)


Figure 4. Variation of dimensionless temperature (T*) at the centerline (y ) 0) with the power-law index (n), Prandtl number (Pr), and gap ratio (a) G ) 2, (b) G ) 4, and (c) G ) 10 at Re ) 1. The left two and right two columns are for the CWT and CHF boundary conditions, respectively.

Figure 5. Variation of dimensionless temperature (T*) at the centerline (y ) 0) with the power-law index (n), Prandtl number (Pr), and gap ratio (a) G ) 2, (b) G ) 4, and (c) G ) 10 at Re ) 40 for both thermal boundary conditions. The left two and right two columns are for the CWT and CHF boundary conditions, respectively.


Figure 6. Variation of the local Nusselt number (NuLocal,1) over the surface of the upstream cylinder with the power-law index (n), Prandtl number (Pr), and gap ratio (G) at Re ) 1 for both thermal boundary conditions.

Thermal energy equation Fcp(U · ∇ T) - k∇2T ) 0

(3)

where F, U, T, f, and σ are the density of the power-law fluid, velocity (Ux and Uy components in Cartesian coordinates), temperature, body force, and stress tensor, respectively. The stress tensor, the sum of the isotropic pressure (p), and the deviatoric stress tensor (τ) are given by σ ) -pI + τ

(4)

For incompressible fluids, the extra stress tensor is given by τ ) 2ηε(U)

(5)

where ε(U), the components of the rate-of-strain tensor, is given by ε(U) )

(∇U) + (∇U)T

2

(6)

For a power-law fluid, the viscosity (η) is given by η ) m(I2 ⁄2)(n-1) ⁄ 2 where I2 ) 2(εxx2 + εyy2 + εxy2 + εyx2) (7) where m is the power-law consistency index and n is the powerlaw index of the fluid (n < 1, shear-thinning; n ) 1, Newtonian; and n > 1, shear-thickening) and I2 is the second invariant of


Figure 7. Variation of the local Nusselt number (NuLocal,1) over the surface of the upstream cylinder with the power-law index (n), Prandtl number (Pr), and gap ratio (G) at Re ) 40 (*Re ) 30 for n ) 1.8) for both thermal boundary conditions.

the rate-of-strain tensor (ε). The components of the rate-of-strain tensor (ε) are related to the velocity components and their gradients and are given in standard text books, e.g., see text book by Bird et al.29 Admittedly, the power-law fluid model predicts unrealistic values of viscosity at stagnation points, but these points are geometric singularities with zero surface area. Besides, in spite of these limitations, this simple model has been shown to yield useful results in scores of geometries including cylinders and spheres3 and, thus, has been used here. The appropriate boundary conditions for this flow are written as follows:

•At the inlet boundary: The uniform flow condition is imposed at the inlet. Ux ) Uo Uy ) 0 and T ) To

(8)

•On the surface of the cylinders: The standard no-slip condition is used, and the two cylinders are maintained either at the same constant temperature (Tw) or are subject to a constant heat flux (qw) as follows: Ux ) 0 and Uy ) 0

(9a)

T ) Tw (CWT case) or q ) qw (CHF case)

(9b)


Figure 8. Variation of the local Nusselt number (NuLocal,2) over the surface of the downstream cylinder with the power-law index (n), Prandtl number (Pr), and gap ratio (G) at Re ) 1 for both thermal boundary conditions.

•At the exit boundary: The default “outflow” boundary condition option in FLUENT (a zero diffusion flux for all flow variables) was used in this work. In essence, this choice implies that the conditions at the outflow plane are extrapolated from within the domain without exerting an appreciable influence on the upstream flow conditions. The extrapolation procedure used by FLUENT updates the outflow velocity and the pressure in a manner that is consistent with the fully developed flow assumption, when there is no area change at the outflow boundary. Note that gradients in the cross-stream direction may

still exist at this plane. Also, the use of this condition obviates the need to prescribe a boundary condition for pressure. This is similar to the homogeneous Neumann condition, i.e., ∂Uy ∂Ux ∂T )0 )0 )0 (10) ∂x ∂x ∂x Since it is not known a priori, the numerical computations have been carried out in the full computational domain (Figure 1b), i.e., without assuming the flow to be symmetrical about the midplane. Over the range of conditions studied herein, the


Figure 9. Variation of the local Nusselt number (NuLocal,2) over the surface of the downstream cylinder with the power-law index (n), Prandtl number (Pr), and gap ratio (G) at Re ) 40 (*Re ) 30 for n ) 1.8) for both thermal boundary conditions.

flow field was seen to exhibit symmetry about the midplane, however. The numerical solution of the governing equations (eqs 1-3) in conjunction with the above-noted boundary conditions (eqs 8-10) maps the flow domain in terms of the primitive variables, i.e., velocity (Ux and Uy), pressure (p), and temperature (T) fields. The velocity and pressure fields, in turn, can be used to deduce the local and macroscopic flow characteristics like drag as explained elsewhere,1,15,17,20–22 and the temperature field can be used to evaluate the local Nusselt number as detailed below in brief and described in detail

elsewhere.8,16,18,19 However, at this stage, it is useful to introduce some dimensionless parameters. •The Reynolds (Re) and Prandtl (Pr) numbers for powerlaw fluids are defined as follows: Re )

FDnUo2-n m

and Pr )

( )

mcp Uo k D

(n-1)

(11)

Note that, unlike in the case of Newtonian fluids, the Prandtl number (Pr) for a power-law fluid also depends upon the

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9153 Table 5. Variation of the Surface-Averaged Nusselt Number (Nu1 and Nu2) for Heat Transfer in Power-Law Liquids from a Pair of Unconfined Cylinders in Tandem Arrangement at Pr ) 1 (1 ) upstream, 2 ) downstream) constant wall temperature (CWT) case Re ) 1 G Nu1

Nu2

Re ) 2 Nu1

Nu2

Re ) 5 Nu1

Nu2

constant heat flux (CHF) case

Re ) 10 Nu1

Nu2

Re ) 40

Re ) 1

a

Nu1

Nu2

Nu1

Re ) 2 Nu2

Nu1

Nu2

Re ) 5 Nu1

Nu2

Re ) 10 Nu1

Nu2

Re ) 40a Nu1

Nu2

n ) 0.4 2 0.8108 0.4994 1.0682 0.6115 1.5824 0.8154 2.1806 1.0390 4.2146 1.7620 3 0.8616 0.5238 1.1247 0.6592 1.6398 0.8916 2.2320 1.1468 4.2402 1.9578 4 0.8936 0.5607 1.1544 0.6993 1.6632 0.9559 2.2532 1.2382 4.2430 2.1024 6 0.9258 0.6076 1.1789 0.7662 1.6860 1.0630 2.2801 1.3597 4.2622 2.4135 10 0.9476 0.6760 1.2000 0.8583 1.7117 1.2072 2.3092 1.6083 4.3042 2.9289 ∞ 0.9992 1.2580 1.7634 2.3542 4.3502

0.7854 0.5889 1.0832 0.7378 1.7204 1.0143 2.4732 1.3148 4.9854 2.2185 0.8555 0.6284 1.1651 0.7891 1.7949 1.0883 2.5363 1.4162 5.0174 2.3925 0.9015 0.6542 1.2054 0.8229 1.8217 1.1426 2.5601 1.5004 5.0205 2.5256 0.9460 0.6912 1.2352 0.8746 1.8472 1.2351 2.5912 1.6528 5.0431 2.8928 0.9722 0.7424 1.2592 0.9482 1.8771 1.3645 2.6257 1.8590 5.0926 3.4611 1.0290 1.3265 1.9404 2.6815 5.1471

n ) 0.6 2 0.7282 0.4662 0.9803 0.5682 1.4892 0.7620 2.0618 0.9741 3.9336 1.6511 3 0.7784 0.4936 1.0411 0.6079 1.5539 0.8294 2.1194 1.0742 3.9715 1.8301 4 0.8124 0.5160 1.0760 0.6412 1.5800 0.8870 2.1377 1.1579 3.9750 1.9604 6 0.8512 0.5532 1.1065 0.6976 1.6000 0.9821 2.1575 1.2962 3.9841 2.2197 10 0.8781 0.6091 1.1256 0.7777 1.6197 1.1119 2.1798 1.4865 4.0140 2.6510 ∞ 0.9199 1.1629 1.6538 2.2147 4.0584

0.6868 0.5398 0.9702 0.6791 1.5915 0.9450 2.3066 1.2304 4.6095 2.0750 0.7570 0.5747 1.0586 0.7246 1.6787 1.0123 2.3788 1.3249 4.6554 2.2360 0.8047 0.5969 1.1075 0.7537 1.7099 1.0590 2.3989 1.3969 4.6581 2.3527 0.8589 0.6277 1.1469 0.7964 1.7324 1.1361 2.4216 1.5225 4.6688 2.6386 0.8932 0.6688 1.1687 0.8569 1.7552 1.2476 2.4478 1.7016 4.7041 3.1121 0.9338 1.2109 1.7961 2.4904 4.7569

n ) 1.0 2 0.6933 0.4498 0.9370 0.5458 1.4129 0.7270 1.9306 0.9235 3.5407 1.5404 3 0.7457 0.4748 1.0019 0.5824 1.4842 0.7913 1.9938 1.0202 3.5925 1.7013 4 0.7820 0.4949 1.0400 0.6132 1.5118 0.8462 2.0100 1.0985 3.6034 1.8091 6 0.8245 0.5278 1.0730 0.6648 1.5274 0.9321 2.0210 1.2163 3.6046 2.0066 10 0.8538 0.5774 1.0883 0.7371 1.5388 1.0419 2.0342 1.3682 3.6174 2.3219 ∞ 0.8783 1.1103 1.5625 2.0618 3.6562

0.6463 0.5166 0.9127 0.6483 1.4839 0.8946 2.1226 1.1547 4.0923 1.9116 0.7166 0.5499 1.0060 0.6918 1.5817 0.9600 2.2028 1.2459 4.1518 2.0565 0.7663 0.5708 1.0599 0.7191 1.6159 1.0033 2.2205 1.3092 4.1622 2.1466 0.8256 0.5986 1.1041 0.7573 1.6334 1.0682 2.2327 1.4079 4.1632 2.3457 0.8646 0.6342 1.1219 0.8090 1.6464 1.1568 2.2481 1.5438 4.1784 2.6855 0.8914 1.1465 1.6742 2.2810 4.2245

n ) 1.4 2 0.6976 0.4517 0.9325 0.5449 1.3809 0.7186 1.8566 0.9050 3.2877 1.4826 0.6496 0.5186 0.9052 0.6451 1.4383 0.8779 2.0200 1.1201 3.7635 1.8173 3 0.7527 0.4775 1.0004 0.5824 1.4548 0.7842 1.9213 1.0016 3.3473 1.6345 0.7230 0.5529 1.0018 0.6899 1.5397 0.9447 2.1025 1.2104 3.8301 1.9548 4 0.7906 0.4984 1.0396 0.6144 1.4822 0.8395 1.9369 1.0767 3.3664 1.7256 07746 0.5745 1.0570 0.7181 1.5741 0.9877 2.1198 1.2693 3.8488 2.0253 6 0.8337 0.5327 1.0714 0.6670 1.4949 0.9216 1.9440 1.1834 3.3687 1.8817 0.8347 0.6031 1.1000 0.7564 1.5885 1.0478 2.1274 1.3550 3.8500 2.1697 10 0.8604 0.5828 1.0833 0.7362 1.5021 1.0196 1.9522 1.3136 3.3721 2.1372 0.8707 0.6383 1.1142 0.8045 1.5966 1.1241 2.1368 1.4676 3.8543 2.4402 ∞ 0.8771 1.0987 1.5201 1.9743 3.4027 0.8892 1.1312 1.6174 2.1628 3.8909 n ) 1.8 2 0.7077 0.4565 0.9336 0.5483 1.3643 0.7173 1.8087 0.8964 2.7864 1.3049 3 0.7650 0.4834 1.0062 0.5876 1.4382 0.7848 1.8729 0.9927 2.8440 1.4411 4 0.8040 0.5056 1.0454 0.6212 1.4650 0.8400 1.8894 1.0642 2.8611 1.5243 6 0.8463 0.5419 1.0754 0.6746 1.4766 0.9186 1.8951 1.1629 2.8639 1.6516 10 0.8702 0.5926 1.0854 0.7413 1.4817 1.0091 1.9002 1.2798 2.8655 1.8411 ∞ 0.8830 1.0971 1.4958 1.9179 2.8875 a

0.6598 0.5244 0.9087 0.6479 1.4142 0.8713 1.9541 1.1002 3.1427 1.5906 0.7360 0.5601 1.0073 0.6944 1.5158 0.9393 2.0365 1.1887 3.2086 1.7122 0.7891 0.5828 1.0624 0.7237 1.5497 0.9816 2.0549 1.2440 3.2257 1.7768 0.8483 0.6125 1.1031 0.7622 1.5630 1.0381 2.0609 1.3211 3.2279 1.8870 0.8808 0.6473 1.1152 0.8076 1.5686 1.1070 2.0666 1.4201 3.2298 2.0758 0.8950 1.1281 1.5847 2.0873 3.2560

Re ) 30 at n ) 1.8.

velocity and diameter of the cylinder, in addition to the thermophysical properties. • The local Nusselt number, Nu(θ) or NuLocal, on the surface of the cylinder is evaluated using the temperature field as follows: hD ) Nu(θ) or NuLocal ) k

{

-

∂T ( for CWT condition) ∂ns

1 ( for CHF condition) Tw

(12) 4. Numerical Solution Procedure

where ns (the unit vector normal to the surface of the cylinder) is given as ns )

xex + yey

√x2 + y2

) nxex + nyey

(13)

where ex and ey are the x- and y-components of the unit vector, respectively. The overall surface-average value is obtained as follows: Nu )

1 2π

2π

∫ Nu(θ) dθ

design calculations to estimate the rate of heat transfer from an isothermal cylinder or to estimate the unknown temperature for a given heat flux. Dimensional analysis of the field equations and boundary conditions suggests the average Nusselt number to be a function of the kinematic and geometrical dimensionless numbers, i.e., Nu ) f(Re, Pr, n, G) for a given thermal boundary condition. This functional relationship is developed in this study.

(14)

0

The values of the average Nusselt number (or the average heat transfer coefficient) are needed in process engineering

Since detailed descriptions of the numerical solution procedure are available elsewhere,1,17–22 only the salient features are recapitulated here. In this study, the field equations have been solved using FLUENT (Version 6.2). The unstructured “quadrilateral” cells of nonuniform grid spacing were generated using the commercial grid tool GAMBIT. The 2-D, steady, segregated solver was used to solve the incompressible flow on the collocated grid arrangement. The second-order upwind scheme was used to discretize the convective terms in the momentum and energy equations. The semi-implicit method for the pressure-linked equations (SIMPLE) scheme was used for solving the pressure-velocity decoupling. The “constant density” and “non-Newtonian power-law viscosity” models were used. FLUENT solves the

9154 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 6. Variation of the Surface -Averaged Nusselt Number (Nu1 and Nu2) for Heat Transfer in Power-Law Liquids from a Pair of Unconfined Cylinders in Tandem Arrangement at Pr ) 10 (1 ) Upstream, 2 ) Downstream)a constant wall temperature (CWT) case Re ) 1 G

Nu1

Nu2

Re ) 2 Nu1

Nu2

Re ) 5 Nu1

Nu2

Re ) 10 Nu1

Nu2

constant heat flux (CHF) case Re ) 40

Re ) 1

a

Nu1

Nu2

Nu1

Nu2

Re ) 2 Nu1

Nu2

Re ) 5 Nu1

Nu2

Re ) 10 Nu1

Nu2

Re ) 40a Nu1

Nu2

n ) 0.4 2 1.8622 1.0707 2.4290 1.3341 3.5964 1.7711 5.0008 2.2182 9.9713 3.6815 2.0403 1.2639 2.7191 1.5834 4.1305 2.1425 5.8436 2.789 11.6980 4.5276 3 1.9089 1.1652 2.4733 1.4529 3.6428 1.9377 5.0441 2.4476 10.0994 4.0706 2.0961 1.3591 2.7717 1.7055 4.1891 2.3263 5.9049 2.9813 11.8440 4.8423 4 1.9290 1.2401 2.5013 1.5500 3.6895 2.0989 5.0997 2.6972 10.1099 4.4148 2.1168 1.4289 2.7998 1.8066 4.2396 2.5149 5.9704 3.2884 11.8709 5.1843 6 1.9633 1.3607 2.5502 1.7239 3.7558 2.4190 5.1764 3.1990 10.1478 5.3900 2.1515 1.5438 2.8475 1.9821 4.3093 2.8464 6.0569 3.8197 11.9168 6.4557 10 2.0145 1.5318 2.6113 1.9679 3.8260 2.8202 5.2547 3.7997 10.2557 6.9458 2.2054 1.7070 2.9110 2.2191 4.3874 3.2441 6.1469 4.4285 12.0414 8.0808 ∞ 2.1538 2.7590 3.9582 5.3702 10.3962 2.3647 3.0832 4.5480 6.2858 12.2020 n ) 0.6 2 1.6040 0.9155 2.1548 1.1494 3.2753 1.5708 4.5568 2.0066 3 1.6639 0.9901 2.2146 1.2511 3.3295 1.7285 4.6018 2.2307 4 1.6881 1.0513 2.2413 1.3359 3.3651 1.8703 4.6376 2.4424 6 1.7175 1.1490 2.2818 1.4789 3.4145 2.1285 4.6904 2.8365 10 1.7570 1.2832 2.3282 1.6785 3.4656 2.4686 4.7470 3.3431 ∞ 1.8614 2.4198 3.5485 4.8313

8.8897 3.4153 1.7190 1.0805 2.3856 1.3788 3.7456 1.9252 5.3082 2.4850 10.4670 4.2643 9.0564 3.7453 1.7950 1.1592 2.4565 1.4839 3.8095 2.0935 5.3651 2.7298 10.6310 4.5347 9.1166 3.9951 1.8215 1.2131 2.4838 1.5647 3.8473 2.2439 5.4060 2.9669 10.7123 4.7393 9.1267 4.7342 1.8524 1.2992 2.5246 1.7003 3.8993 2.5018 5.4651 3.3756 10.7275 5.6795 9.1921 5.9477 1.8945 1.4200 2.5730 1.8874 3.9555 2.8355 5.5297 3.8897 10.8060 6.9828 9.3067 2.0121 2.6773 4.0535 5.6302 10.9394

2 1.4629 0.8072 1.9872 1.0230 2.9968 1.4222 4.0910 1.8387 3 1.5371 0.8725 2.0599 1.1199 3.0549 1.5824 4.1329 2.0604 4 1.5676 0.9279 2.0837 1.2016 3.0752 1.7139 4.1488 2.2411 6 1.5917 1.0151 2.1082 1.3279 3.0998 1.9186 4.1733 2.5289 10 1.6148 1.1299 2.1321 1.4915 3.1266 2.1768 4.2046 2.8987 ∞ 1.6645 2.1824 3.1807 4.2669

7.6227 3.1954 1.5374 0.9609 2.1731 1.2397 3.4000 1.7530 4.7338 2.2735 7.8101 3.4998 1.6355 1.0322 2.2604 1.3401 3.4658 1.9152 4.7833 2.4997 7.9506 3.5956 1.6708 1.0789 2.2846 1.4111 3.4869 2.0402 4.8012 2.6835 7.9656 4.0860 1.6960 1.1488 2.3095 1.5205 3.5132 2.2332 4.8287 2.9724 7.9827 4.9064 1.7205 1.2439 2.3354 1.6556 3.5431 2.4810 4.8654 3.3462 8.0650 1.7769 2.3935 3.6065 4.9381

n ) 1.0 8.8922 3.9799 9.1508 4.2776 9.2965 4.2578 9.3216 4.8452 9.3463 5.7657 9.4464

n ) 1.4 2 1.4459 0.7858 1.9418 0.9916 1.3737 2.7921 3.8401 1.7724 3 1.5262 0.8531 2.0194 1.0928 1.5385 2.8476 3.8825 1.9887 4 1.5578 0.9114 2.0403 1.1782 1.6673 2.8531 3.8886 2.1539 6 1.5766 1.0011 2.0539 1.3015 1.8473 2.8592 3.9002 2.396 10 1.5882 1.1101 2.0661 1.4455 2.0604 2.8694 3.9184 2.6873 ∞ 1.6179 2.0995 3.0043 3.9665

6.9167 6.9167 1.5114 0.9391 2.1113 1.2039 3.2352 1.6870 4.4176 2.1726 7.1178 3.4328 1.6176 1.0123 2.2047 1.3073 3.3004 1.8489 4.4652 2.3843 7.2478 3.5331 1.6543 1.0605 2.2259 1.3789 3.3116 1.9654 4.4718 2.5456 7.3652 3.7456 1.6737 1.1297 2.2397 1.4811 3.3245 2.1301 4.4849 2.7804 7.3624 4.3938 1.6861 1.2166 2.2532 1.6052 3.3422 2.3319 4.5059 3.0780 7.4143 1.7169 2.2917 3.3884 4.5627

2 1.4538 0.7854 1.9275 0.9850 2.7921 1.3540 3.6797 1.7403 3 1.5368 0.8571 2.0057 1.0919 2.8476 1.5207 3.7314 1.9426 4 1.5675 0.9196 2.0232 1.1809 2.8531 1.6463 3.7357 2.0949 6 1.5809 1.0131 2.0299 1.3020 2.8592 1.8108 3.7408 2.3064 10 1.5868 1.1173 2.0375 1.4332 2.8694 1.9979 3.7510 2.5605 ∞ 1.6078 2.0621 2.8995 3.7878

5.7215 2.7338 1.5177 0.9395 2.0894 1.1942 3.1362 1.6531 4.2133 2.1133 5.9026 2.9789 1.6268 1.0164 2.1836 1.3015 3.1987 1.8124 4.2685 2.3050 6.0163 3.0665 1.6623 1.0676 2.2013 1.3748 3.2039 1.9236 4.2726 2.4507 6.1044 3.2393 1.6761 1.1386 2.2081 1.4732 3.2105 2.0724 4.2786 2.6586 6.1003 3.7089 1.6825 1.2203 2.2165 1.5848 3.2220 2.2484 4.2905 2.9140 6.1319 1.7063 2.2447 3.2569 4.3341

8.1496 3.8547 8.3178 4.1716 8.4206 4.2091 8.5567 4.3524 8.5577 5.1249 8.6262

n ) 1.8

a

6.6933 3.2984 6.8498 3.5645 6.9462 3.6069 7.0433 3.7283 7.0414 4.2792 7.0853

Re ) 30 at n ) 1.8.

system of algebraic equations using the Gauss-Siedel (GS) point-by-point iterative method in conjunction with the algebraic multigrid (AMG) method solver. The use of AMG scheme can greatly reduce the number of iterations (and, thus, CPU time) required to obtain a converged solution, particularly when the model contains a large number of control volumes. Relative convergence criteria of 10-10 for the continuity and x- and y-components of the velocity and 10-15 for temperature were prescribed in this work. 5. Choice of Numerical Parameters Undoubtedly, the reliability and accuracy of the numerical results is contingent upon a prudent choice of the numerical parameters, namely, optimal domain (value of Do) and grid size. An excessively large value of Do will warrant enormous computational resources, and a small value will unduly influence the results; hence, a prudent choice is vital to the accuracy of the results. Similarly, an optimal grid size should meet two conflicting requirements, namely, it should be fine enough to resolve adequately the flow field, yet it should not be exorbitantly resource-intensive in terms of the memory and CPU time requirements. Since the effects of these parameters (Do and grid size) on drag values for single- and two-cylinder cases have been dealt with in our previous studies,1,15,17,20,22 only the

additional results showing the influence of these parameters on the Nusselt number for two-cylinder assembly are presented here, thereby ensuring that the present results are free from such numerical artifacts. 5.1. Influence of Domain Size on Average Nusselt Number. Following the same approach as that used in our recent studies,1,19,20,22 several values of Do/D ranging from 300 to 1300 were used in this study to examine the role of domain size. As both the hydrodynamic and thermal boundary layers are thick at low Reynolds numbers (Re e 5), a large value of Do/D is required at such low Reynolds numbers1,19,20,22 as compared to that at high Reynolds numbers (Re g 10). Table 1 shows the effect of the domain size (Do/D) on the average Nusselt numbers (Nu1 and Nu2 for the two cylinders) at the maximum value of the Prandtl number, Pr ) 100, used in this work for the extreme values of the gap ratio, the power-law index (n ) 0.4, 1, and 1.8), and the Reynolds number (Re ) 1, 30, and 40) considered in this work. The domain independence study was carried out using grid M1; see Table 2. It is clearly seen that the results change very little (maximum change being 0.012%) as the domain size is increased from 1100 to 1200 and from 1200 to 1300 at Re ) 1. Similarly, a change in domain size from 300 to 400 and 400 to 500 for Re ) 40 (n ) 0.4 and 1) and for Re ) 30 (n ) 1.8) yields very small changes in the value of the

Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9155 Table 7. Variation of the Surface-Averaged Nusselt Number (Nu1 and Nu2) for Heat Transfer in Power-Law Liquids from a Pair of Unconfined Cylinders in Tandem Arrangement at Pr ) 50 (1 ) Upstream, 2 ) Downstream)a constant wall temperature (CWT) case Re ) 1 G Nu1

Nu2

Re ) 2 Nu1

Nu2

Re ) 5 Nu1

Nu2

Re ) 10 Nu1

Nu2

constant heat flux (CHF) case Re ) 40

Re ) 1

a

Nu1

Nu2

Nu1

Nu2

Re ) 2 Nu1

Nu2

Re ) 5 Nu1

Nu2

Re ) 10 Nu1

Nu2

Re ) 40a Nu1

Nu2

n ) 0.4 2 3.2122 1.8868 4.1880 2.3243 6.2373 3.0229 8.7318 3.7185 17.9131 6.0987 3.6127 2.1452 4.7515 2.6580 7.1899 3.5442 10.1936 4.4417 20.9174 7.6841 3 3.2562 2.0343 4.2492 2.5019 6.3288 3.2810 8.8168 4.0895 18.1860 6.7979 3.6660 2.3145 4.8284 2.8852 7.3163 3.9172 10.3295 4.9786 21.1622 8.2195 4 3.3103 2.1644 4.3266 2.6992 6.4353 3.6727 8.9319 4.7062 18.2704 7.2972 3.7182 2.4730 4.9034 3.1330 7.4294 4.3932 10.4656 5.7321 21.3459 8.5415 6 3.3938 2.4238 4.4331 3.0971 6.5679 4.3822 9.0797 5.7898 18.3346 9.3866 3.7930 2.7462 4.9991 3.5438 7.5637 5.1147 10.6294 6.8499 21.4278 11.1929 10 3.4967 2.7673 4.5523 3.5683 6.7011 5.1248 9.2261 6.9114 18.5334 12.6012 3.8913 3.0880 5.1152 4.0114 7.7082 5.8637 10.7947 8.0024 21.6577 14.4777 ∞ 3.7519 4.8222 6.9427 9.4363 18.7934 4.1726 5.4237 7.9989 11.0437 21.9519 n ) 0.6 2 2.7118 1.5466 3.6440 1.9412 5.5660 2.6331 7.7823 3.3312 15.6182 5.6915 3.0348 1.7871 4.1452 2.2730 6.4542 3.1578 3 2.7656 1.6742 3.7121 2.1079 5.6524 2.8917 7.8560 3.7030 15.9344 6.3029 3.0954 1.9289 4.2223 2.4717 6.5606 3.4919 4 2.8052 1.7805 3.7722 2.2685 5.7289 3.2003 7.9278 4.1733 16.1006 6.5736 3.1337 2.0444 4.2809 2.6557 6.6412 3.8450 6 2.8705 1.9727 3.8558 2.5717 5.8238 3.7474 8.0265 4.9995 16.1106 8.0640 3.1951 2.2399 4.3590 2.9634 6.7385 4.3999 10 2.9479 2.2329 3.9431 2.9492 5.9173 4.3678 8.1290 5.9279 16.2224 10.5195 3.2700 2.4943 4.4442 3.3360 6.8390 5.0307 ∞ 3.1359 4.1068 6.0643 8.2776 16.4233 3.4732 4.6258 7.0112

9.1338 4.0463 18.4296 7.2786 9.2388 4.5303 18.6863 7.8179 9.3231 5.0775 18.9313 7.7523 9.4333 5.9339 18.9545 9.6972 9.5492 6.9015 19.0915 12.3002 9.7246 19.3244

n ) 1.0 2 2.4271 1.2981 3.3005 1.6695 4.9924 2.3416 6.8425 3.0275 13.1688 3 2.4991 1.4183 3.3733 1.8391 5.0633 2.6085 6.8857 3.3872 13.5217 4 2.5306 1.5162 3.4127 1.9847 5.1029 2.8596 6.9146 3.7345 13.7557 6 2.5722 1.6736 3.4584 2.2262 5.1481 3.2632 6.9589 4.3090 13.8465 10 2.6138 1.8817 3.5012 2.5232 5.1960 3.7299 7.0143 4.9865 13.8687 ∞ 2.7004 3.5886 5.2897 7.1218 13.9991

5.5517 2.6958 1.5388 3.7460 2.0034 5.7938 2.8463 6.0742 2.7759 1.6681 3.8243 2.1894 5.8771 3.1515 6.1650 2.8061 1.7620 3.8625 2.3374 5.9194 3.4176 6.9023 2.8458 1.9087 3.9074 2.5698 5.9678 3.8211 8.5333 2.8877 2.1019 3.9522 2.8567 6.0207 4.2957 2.9848 4.0524 6.1302

8.0343 3.6922 15.6485 7.0331 8.0972 4.1000 15.9243 7.5641 8.1326 4.4860 16.1168 7.4073 8.1840 5.0851 16.2782 8.2267 8.2480 5.7991 16.3157 10.0683 8.3746 16.4782

n ) 1.4 2 2.3754 1.2385 3.1923 1.5955 4.7254 2.2431 6.3623 2.9086 11.8818 3 2.4545 2.3644 3.2671 1.7737 4.7825 2.5134 6.4189 3.2353 12.2718 4 2.4817 1.4671 3.2946 1.9203 4.8004 2.7398 6.4245 3.5242 12.4877 6 2.5078 1.6208 3.3174 2.1399 4.8215 3.0727 6.4401 3.9715 12.7662 10 2.5271 1.8069 3.3385 2.3912 4.8496 3.4541 6.4680 4.5163 12.7557 ∞ 2.5780 3.3962 4.9181 6.5480 12.8214

5.6045 2.6288 1.4829 3.6129 1.9261 5.4704 2.7220 6.1307 2.7168 1.6155 3.6936 2.1146 5.5390 3.0113 6.3102 2.7427 1.7090 3.7207 2.2557 5.5589 3.2430 6.3973 2.7677 1.8455 3.7439 2.4605 5.5822 3.5748 7.6111 2.7880 2.0134 3.7669 2.7002 5.6140 3.9635 2.8460 3.8332 5.6939

2 2.3724 1.2265 3.1455 1.5730 4.5654 2.1991 6.0641 2.8762 9.7568 3 2.4529 1.3607 3.2172 1.7598 4.6155 2.4641 6.2107 3.1257 10.1193 4 2.4758 1.4698 3.2347 1.9086 4.6205 2.6737 6.2155 3.3781 10.3098 6 2.4900 1.6240 3.2439 2.1129 4.6305 2.9603 6.2177 3.7532 10.5773 10 2.4991 1.7932 3.2570 2.3334 4.6476 3.2891 6.2278 4.2190 10.5636 ∞ 2.5350 3.2996 4.6986 6.2819 10.5977

4.9084 2.6211 1.4730 3.5526 1.8984 5.2709 2.6548 5.3395 2.7105 1.6113 3.6309 2.0898 5.3298 2.9219 5.4923 2.7326 1.7084 3.6487 2.2295 5.3350 3.1347 5.4826 2.7465 1.8425 3.6584 2.4178 5.3462 3.4224 6.3330 2.7566 1.9936 3.6729 2.6279 5.3660 3.7587 2.7975 3.7216 5.4258

7.4472 3.5352 14.1314 7.5157 3.8681 14.4416 7.5230 4.1849 14.5968 7.5419 4.6545 14.9121 7.5752 5.2294 14.9098 7.6702 15.0076

6.9405 7.5216 7.6510 7.3746 8.9266

7.0815 3.4748 11.5641 7.2309 3.7029 11.8577 7.2363 3.9771 11.9989 7.2406 4.3795 12.2908 7.2543 4.8703 12.2823 7.3208 12.3404

5.9584 6.4552 6.5804 6.2630 7.3645

n ) 1.8

a

Re ) 30 at n ) 1.8.

Nusselt number (maximum change being 0.05%). For the limiting case of an isolated cylinder (G ) ∞), the corresponding changes in the average Nusselt number values are also very small (Table 1). It needs to be emphasized here that the extremely small changes seen in the values of the average Nusselt number are accompanied by an enormous increase in the computational time for the extreme conditions. Thus, based on the above study coupled with our previous experience,1,17–22 the domain sizes (Do/D) of 400 (200 for a single cylinder) and 1200 are believed to be adequate in the Reynolds number ranges of 5 < Re e 40 and 1 e Re e 5, respectively, over the ranges of conditions 2 e G e 10 and 0.4 e n e 1.8 considered herein, to obtain the results that are believed to be essentially free from domain effects. 5.2. Influence of Grid Size on Average Nusselt Number. Having fixed the domain size (Table 1), the gridindependence study was carried out using five nonuniform unstructured grids (M1, M2, M3, M4, and M5, as detailed in Table 2) for the same combination of the values of Re, Pr, G, and n as used above in the domain-independence study. A summary of results is presented in Table 3. It can be seen from these results that, in moving from grid M4 to M5, the results show a very small change, with the maximum changes being 0.10% and 0.21% for the CWT and CHF conditions for the two extreme

values of G, respectively. In view of these negligible changes (accompanied by an up to 2-3-fold increase in the computational time), grid M4 is believed to be sufficiently refined to resolve the flow and temperature fields with acceptable levels of accuracy within the range of conditions of interest here. The number of grid cells with grid M4 for different values of the gap ratio (G) is also included in Table 2. Finally, to add further weight to our claim for the accuracy of the results, the present numerical results have been compared with the literature values in the next section. 6. Results and Discussion In this work, the computations have been carried out for the following values of the dimensionless parameters: Reynolds number, Re ) 1, 2, 5, 10, 20, and 40; Prandtl number, Pr ) 1, 10, 50, and 100; power-law index, n ) 0.4, 0.6, 1, 1.4, and 1.8, thereby covering both shear-thinning (n < 1) and shear-thickening (n > 1) fluids; and five values of the gap ratio between the two cylinders, G ) L/D ) 2, 3, 4, 6, and 10. Since, for n ) 1.8, the flow over a single cylinder is known to be symmetric and steady only up to Re ) 30, this is the upper limit for the Reynolds number in this case; whereas for all other values of n, the results are reported up to Re ) 40.20 Finally, the fully

9156 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Table 8. Variation of the Surface-Averaged Nusselt Number (Nu1 and Nu2) for Heat Transfer in Power-Law Liquids from a Pair of Unconfined Cylinders in Tandem Arrangement at Pr ) 100 (1 ) Upstream, 2: Downstream). constant wall temperature (CWT) case Re ) 1

Re ) 2

Re ) 5

constant heat flux (CHF) case

Re ) 10

Re ) 40

Re ) 1

G

Nu1

Nu2

Nu1

Nu2

Nu1

Nu2

Nu1

Nu2

2 3 4 6 10 ∞

4.0477 4.1073 4.1894 4.3050 4.4415 4.7703

2.3982 2.5690 2.7538 3.1276 3.5762

5.2825 5.3734 5.4842 5.6282 5.7847 6.1318

2.9385 3.1484 3.4480 4.0062 4.6102

7.8893 8.0243 8.1695 8.3445 8.5177 8.8280

3.7923 4.1213 4.7165 5.6736 6.6226

11.0694 11.1904 11.3435 11.5364 11.7256 11.9950

4.6351 5.1158 6.0370 7.4907 8.9500

22.9485 7.5981 4.5652 23.2941 8.5423 4.6397 23.4078 9.1386 4.7172 23.4897 12.0320 4.8169 23.7468 16.4639 4.9432 24.0779 5.3015

2 3 4 6 10 ∞

3.3907 3.4515 3.5097 3.5982 3.6994 3.9397

1.9435 2.0981 2.2402 2.5099 2.8496

4.5626 4.6518 4.7355 4.8462 4.9593 5.1681

2.4344 2.6377 2.8675 3.2879 3.7702

6.9885 7.1084 7.2112 7.3350 7.4550 7.6420

3.2886 9.7904 4.1444 3.6187 9.8898 4.6222 4.0693 9.9849 5.3042 4.8059 10.1127 6.4123 5.5929 10.2438 7.6050 10.4323

19.8635 7.1668 3.8296 20.2850 7.9722 3.8974 20.4913 8.2484 3.9525 20.5047 10.2793 4.0329 20.6547 13.6223 4.1276 20.8905 4.3837

2 3 4 6 10 ∞

3.0141 3.0887 3.1310 3.1857 3.2393 3.3489

1.6090 1.7595 1.8851 2.0962 2.3654

4.1041 4.1879 4.2427 4.3023 4.3571 4.4678

2.0753 2.2851 2.4807 2.8071 3.1862

6.2239 6.3138 6.3666 6.4254 6.4866 6.6052

2.9126 3.2459 3.5936 4.1351 4.7275

8.5602 8.5987 8.6344 8.6908 8.7612 8.8968

3.7614 4.2061 4.6871 5.4620 6.3308

16.6763 7.1110 3.3953 17.1372 7.8036 3.4748 17.4213 7.9530 3.5145 17.5736 8.8224 3.5656 17.5992 10.9563 3.6184 17.7374 3.7408

2 3 4 6 10 ∞

2.9388 3.0198 3.0545 3.0879 3.1124 3.1768

1.5265 1.6847 1.8140 2.0143 2.2521

3.9541 4.0381 4.0745 4.1039 4.1308 4.2039

1.9761 2.1953 2.3859 2.6761 2.9963

5.8715 5.9331 5.9558 5.9829 6.0189 6.1056

2.7840 3.1140 3.4152 3.8583 4.3459

7.9495 8.0404 8.0422 8.0559 8.0861 8.1833

3.6205 3.9832 4.3653 4.9705 5.6809

15.0321 15.5451 15.8290 16.2049 16.1971 16.2370

7.2247 7.9219 8.1721 8.1831 9.7522

2 3 4 6 10 ∞

2.9271 3.0086 3.0368 3.0545 3.0659 3.1114

1.5078 1.6755 1.8114 2.0081 2.2216

3.8854 3.9623 3.9845 3.9959 4.0126 4.0667

1.9440 2.1717 2.3606 2.6252 2.9054

5.6627 5.7182 5.7231 5.7348 5.7550 5.8177

2.7280 3.0377 3.3063 3.6825 4.1068

7.5807 7.8255 7.8323 7.8318 7.8407 7.9040

3.6190 3.8536 4.1609 4.6553 5.2729

12.3205 12.7975 13.0431 13.4022 13.3868 13.4073

6.3327 6.9022 7.1025 6.9843 8.0709

Nu1

Nu2

Nu1

Re ) 2

Re ) 5

Re ) 10 Nu1

Nu2

Re ) 40

Nu2

Nu1

Nu2

Nu1

Nu2

Nu1

Nu2

2.6781 2.8996 3.1366 3.5273 3.9763

5.9952 6.1114 6.2166 6.3426 6.4922 6.8867

3.3064 3.6156 3.9890 4.5574 5.1610

9.0820 9.2715 9.4238 9.5988 9.7850 10.1574

4.3907 4.9209 5.6161 6.5822 7.5479

12.8955 5.4820 26.7664 13.0946 6.2411 27.0563 13.2756 7.3222 27.3082 13.4880 8.8123 27.4161 13.7007 10.3268 27.7140 14.0184 28.0863

2.2180 2.4028 2.5685 2.8470 3.1848

5.2188 5.3205 5.4010 5.5022 5.6104 5.8403

2.8175 3.0840 3.3527 3.7800 4.2634

8.1244 8.2755 8.3834 8.5092 8.6375 8.8561

3.9073 4.3702 4.8790 5.6256 6.4376

11.5012 11.6501 11.7631 11.9058 12.0539 12.2758

4.9960 5.6616 6.4413 7.5871 8.8469

23.4490 9.2320 23.7848 9.9510 24.0892 9.6620 24.1220 12.3156 24.2963 15.9095 24.5820

1.8951 2.0626 2.1916 2.3960 2.6541

4.7034 4.7952 4.8466 4.9046 4.9614 5.0878

2.4722 2.7165 2.9239 3.2448 3.6206

7.2657 7.3768 7.4342 7.4972 7.5647 7.7031

3.5163 3.9219 4.2968 4.8442 5.4590

10.0831 10.1510 10.1971 10.2640 10.3461 10.5058

4.5736 5.0913 5.6325 6.4447 7.3754

19.8462 9.0455 20.2010 9.7447 20.4157 9.6433 20.6668 10.4969 20.7130 10.9253 20.8963

1.8198 1.9904 2.1158 2.3020 3.5252

4.5251 4.6167 4.6532 4.6832 4.7124 4.7961

2.3714 2.6144 2.8072 3.0868 3.4022

6.8446 6.9261 6.9527 6.9831 7.0242 7.1256

3.3624 3.7279 4.0474 4.4987 5.0076

9.3479 9.4599 9.4635 9.4811 9.5174 9.6333

4.4180 4.7704 5.2034 5.8428 6.6003

17.9113 8.9602 18.3150 9.7200 18.5210 9.9149 18.9365 9.3449 18.9382 11.4273 19.0273

1.8038 1.9802 2.1087 2.2883 2.4877

4.4400 4.5261 4.5497 4.5619 4.5804 4.6423

2.3332 2.5750 2.7624 3.0162 3.2931

6.5867 6.6564 6.6617 6.6751 6.6989 6.7730

3.2852 3.5968 3.8832 4.2719 4.7166

8.9084 9.1522 9.1623 9.1648 9.1786 9.2581

4.3986 4.5753 4.9195 5.4620 6.1165

14.6446 15.0268 15.2102 15.5953 15.5867 15.6407

n ) 0.4 9.6518 10.4103 10.6679 14.2697 18.8422

n ) 0.6

n ) 1.0

n ) 1.4 3.3027 3.3890 3.4217 3.4536 3.4792 3.5523

n ) 1.8 3.2852 3.3727 3.4000 3.4176 3.4302 3.4818

7.6972 8.3485 8.5261 7.9325 9.3852

Table 9. Empirically Fitted Correlation Coefficient for the Dependence of the Average Nusselt Number (Nu1 and Nu2) and Colburn Heat Transfer Factor (j1 and j2) on the Dimensionless Parameters (Re, Pr, G, and n)a

a b c d

Nu1 (CWT)

Nu1 (CHF)

Nu2 (CWT)

Nu2 (CHF)

j1 (CWT)

j1 (CHF)

j2 (CWT)

j2 (CHF)

0.6285 0.4723 0.2732 0.02674

0.7148 0.4825 0.2728 0.02471

0.2673 0.3836 0.2544 0.3392

0.3282 0.3935 0.2502 0.3015

0.6893 0.5791 0.1700 0.05644

0.7319 0.5482 0.1851 0.0611

0.3345 0.6341 0.2188 0.2344

0.4048 0.6183 0.2145 0.1979

Statistical analysis (total no. of data points for each case ) 600) δmax δavg

34.58 5.66

δ > 20

22

δmax δavg

19.23 5.40

26.82 4.41

36.22 6.47

33.43 6.23

21.19 4.51

22.19 3.97

24.93 5.59

22.93 5.35

No. of data points having deviations 7

19

18

After excluding the data points having δ > 20

a

19.48 4.20

19.92 5.89

19.94 5.71

δ ) percent relative r.m.s. deviation from the numerical data.

converged flow field obtained by Patil et al.1 was used as input for the thermal energy equation. However, prior to presenting the new results, it is appropriate to validate the solution procedure to ascertain the accuracy and reliability of the heat transfer results presented herein. The ranges of parameters selected here are governed in part by the typical values encountered in process engineering applications. For instance, the values of Prandtl number of the order of 20-60 relate to many viscous materials encountered in polymer, food, and pharmaceutical product processing and in petroleum processing applications. Similarly, because of their shear-sensitive nature

and high viscosity, the Reynolds numbers (based on the cylinder diameter) are not very high, being of the order of 20-100. However, because of the growing wakes at Re g 50 or so, the flow field becomes unsteady and asymmetric, and therefore, one needs to solve time-dependent equations. Similarly, the values of G of the order of 2-4 are not uncommon in the process heat exchangers. Finally, many industrially important polymeric systems have value, of n in the range ∼0.3-1, and some of the thick suspensions have values of n in the range 1.2-1.64. Thus, the ranges of parameters selected here are typical of the values encountered in engineering applications.


Figure 10. Dependence of the normalized surface averaged Nusselt number and/or normalized j-factor (Nu1N, Nu2N, and/or j1N, j2N) on the power-law index (n), Prandtl number (Pr), and Reynolds number (Re) at gap ratio G ) 2 for both CWT (1 and 3) and CHF (2 and 4) thermal boundary conditions. The left two and right two columns represent the upstream and downstream cylinders, respectively.

6.1. Validation of Results. Since the detailed validation for the case of a single cylinder is presented elsewhere, Table 4 shows a comparison between the present and literature values of Nusselt number for the case of two cylinders in tandem arrangement. While an excellent correspondence can be seen between the present results and those of Juncu,25 the present values differ from those of Wong and Chen26 by up to about 4-5%. While the exact reasons for this discrepancy are not immediately obvious, it needs to be emphasized here that the domain size used in the present study is much larger than that used by Wong and Chen,26 and this could explain, in part, this difference. For instance, the rectangular domain used by Wong and Chen26 is Lu ) 4.5D, Ld ) 9.5D, and H ) 10D, where Lu, Ld, and H are the upstream and downstream lengths and height

of the computational domain, respectively. The deviations of this order as that seen in Table 4 are not uncommon in such studies and may be attributed to different numerics and solvers, domain, grid, etc. used in these studies. Furthermore, it needs to be emphasized here that, owing to the nonlinear viscous terms, the results for power-law liquids are expected to be intrinsically less accurate than those for Newtonian fluids. In view of the aforementioned considerations, the present results are believed to be reliable to within (2-3%. 6.2. Heat Transfer Results. Some physical insights into the nature of heat transfer can be gained by examining the isotherm (constant-temperature) contours close to the surface of cylinders, centerline temperature variations, and local Nusselt number variation on the surface of the cylinder. From an engineering


Figure 11. Dependence of the normalized surface-averaged Nusselt number and/or normalized j-factor (Nu1N, Nu2N and/or j1N, j2N) on the power-law index (n), Prandtl number (Pr), and Reynolds number (Re) at gap ratio G ) 10 for both CWT (1 and 3) and CHF (2 and 4) thermal boundary conditions. The left two and right two columns represent the upstream and downstream cylinders, respectively.

applications standpoint, it is more appropriate to analyze the functional dependence of the global parameters like the average Nusselt number or the Colburn j-factor on the pertinent dimensionless parameters (Re, n, Pr, and G). As noted earlier, while extensive results have been obtained in this study, only representative results are presented here in the ensuing sections. 6.2.1. Isotherm Profiles. Representative plots showing the dependence of the temperature contours in the vicinity of the two cylinders on the Reynolds number (Re), Prandtl number (Pr), power-law index (n), and gap ratio (G) are presented in Figures 2 and 3. As observed in the local flow patterns,1 at small values of the gap ratio (e.g., G ) 2), a closed stagnant region is formed in between the two cylinders where velocities are

very small. Owing to weak advection, this in turn leads to the formation of a local high-temperature (or low-temperature) region in between the two cylinders, which tends to lower the rate of heat transfer on the downstream side of the upstream cylinder and the upstream side of the downstream cylinder. Furthermore, Figures 2 and 3 suggest the temperature drop between the two cylinders to be maximum for the largest value of the gap ratio (G ) 10), whereas a small temperature drop is observed in the case of G ) 2. This is presumably because the interference between the two cylinders is very small at G ) 10, which gets accentuated at G ) 2. For G ) 10, the isotherm patterns are similar to that for an isolated cylinder.8,14,16,18,19 The crowding of isotherms in the upstream direction increases


Figure 12. Dependence of the normalized surface-averaged Nusselt number and/or normalized j-factor (Nu1SC, Nu2SC and/or j1SC, j2SC) on the power-law index (n), Prandtl number (Pr), and Reynolds number (Re) at gap ratio, G ) 2 for both CWT (1 and 3) and CHF (2 and 4) thermal boundary conditions. The left two and right two columns represent the upstream and downstream cylinders, respectively.

with an increase in the Reynolds and/or Prandtl numbers. Thus, the resulting temperature gradients and, hence, the Nusselt number increase with Re and/or Pr, albeit the dependence on the Reynolds number is somewhat stronger than that on the Prandtl number. The influence of the power-law index (n) on isotherm patterns is seen to be more pronounced at high Reynolds and/or Prandtl numbers for both thermal boundary conditions. As the fluid behavior changes from shear-thickening (n > 1) to shear-thinning (n < 1), an increase in the compactness of the isotherms results in an overall increase in the temperature gradients. This is because the thermal boundary layer is known to be thinner in shear-thinning (n < 1) fluids than that in Newtonian (n ) 1) fluids under otherwise identical conditions. As expected, the constant heat flux (CHF) condition also leads

to the higher rate of heat transfer as compared to that for the constant wall temperature (CWT) condition. 6.2.2. Variation of Dimensionless Temperature (T*) on the Centerline. Figures 4 and 5depict the typical variation of dimensionless temperature (T*) at the centerline (y ) 0) for a range of combinations of the Reynolds and Prandtl numbers, power-law index, and gap ratio for the two thermal boundary conditions. In Figures 4 and 5, the first and third columns show the temperature variation on the centerline connecting the two cylinders, i.e., from the rear stagnation point of the upstream cylinder (XR1) to the front stagnation point of the downstream cylinder (XF2). The second and fourth columns show the corresponding temperature variation from the rear


Figure 13. Dependence of the normalized surface-averaged Nusselt number and/or normalized j-factor (Nu1SC, Nu2SC or j1SC, j2SC) on the power-law index (n), Prandtl number (Pr), and Reynolds number (Re) at gap ratio G ) 10 for both CWT (1 and 3) and CHF (2 and 4) thermal boundary conditions. The left two and right two columns represent the upstream and downstream cylinders, respectively.

stagnation point of the downstream cylinder (XR2) to the distance far from the downstream cylinder in the outflow direction. Over the ranges of conditions, the temperature (T*) profiles show a complex dependence on the Reynolds (Re) and Prandtl (Pr) numbers, gap ratio (G), and power-law index (n). For fixed values of the pertinent dimensionless parameters, the dimensionless temperature (T*) decreases followed by an increase along the centerline in between the two cylinders, i.e., from XR1 to XF2 (see the first and third columns of Figures 4 and 5). Irrespective of the type of thermal boundary condition, at low Reynolds numbers (Re), the increasing values of the Prandtl number (Pr) and/or gap ratio (G) yield qualitatively similar variations of temperature; however, the location of the minimum temperature moves

forward from the midpoint (G/2 distance forward from the upstream cylinder) to XF2. It is probably so because both cylinders are heated, and thus, the stagnation points (XR1 and XF2) of the cylinders are at higher temperature than the fluid in the dead region in between the two cylinders. The temperature decreases along the centerline from the rear stagnation point of the downstream cylinder. As expected, the dimensionless temperature (T*) in the rear of the cylinders decreases with the increasing value of the Prandtl number (Pr). 6.2.3. Local Nusselt Number (NuLocal). Representative variations of the local Nusselt number (NuLocal,1) over the upstream cylinder with the power-law index (n), Reynolds and


Prandtl numbers, and gap ratio (G) and with the thermal boundary conditions are shown in Figures 6 and 7. The corresponding local Nusselt number profiles for the downstream cylinders (NuLocal,2) are shown in Figures 8 and 9. These results show that, for fixed values of Re, n, Pr, and G, the local Nusselt number values for the upstream cylinder are higher than that for the downstream cylinder, i.e., NuLocal,1 > NuLocal,2, albeit both of these values are smaller than that for an isolated cylinder. Likewise, the variation of the local Nusselt number for the upstream cylinder (NuLocal,1) is qualitatively similar to that for an isolated cylinder, e.g., for the extreme values of the Reynolds and Prandtl numbers, the maximum and minimum of the Nusselt number values can be seen at a location (measured from the front stagnation point) other than at θ ) 0o and θ ) 180o, i.e., two distinct peaks are observed for the local Nusselt number under these conditions. Similar dependence is seen for the downstream cylinder, which shows a maximum value of the local Nusselt number at about θ ) 60o at Re ) 40, n ) 0.4, and G ) 2 for both CWT and CHF boundary conditions. All other features of the local heat transfer, especially from the upstream cylinder (NuLocal,1), are seen to be similar to that for a single cylinder,8,14,16,18,19 i.e., it increases with the increasing values of Re and/or Pr. The maximum and minimum values of the local Nusselt number occur close to the front (θ ) 0o) and rear (θ ) 180o) stagnation points, respectively. The location of the minimum value shifts downward with the increasing Re. In shear-thinning (n < 1) fluids, an upstream (downstream to the front stagnation point) shift in the position of the maximum value and a downward shift (upstream to the rear stagnation point) in the location of the minimum value of the local Nusselt number is seen with the increasing values of the Prandtl number. The power-law index (n) appears to influence the maximum values more than the minimum values of the local Nusselt number. In addition, the upstream cylinder (Figures 6 and 7) shows the following peculiar features: the maximum and minimum values of the local Nusselt numbers for the upstream cylinder (NuLocal,1) are almost constant for all values of gap ratio (G). For all values of G and n, the local Nusselt number profile for the upstream cylinder is almost flat at Re ) 1 and Pr ) 1 for the CHF condition. The local heat transfer characteristics from the downstream cylinder (Figures 8 and 9) are also influenced by the Reynolds and/or Prandtl numbers in a fashion similar to that for an isolated cylinder. Because of the wake interference, the local Nusselt number at the front stagnation point of the downstream cylinder is quite different from that seen for an isolated cylinder and for the upstream cylinder in a tandem arrangement. It increases from the front stagnation point (θ ) 0o) and attains its maximum value followed by a decrease along the surface in the rear side of the downstream cylinder. The location of the maximum value shows a downstream shift with the increasing values of Re and/ or G at higher values of Pr. The minimum of the local Nusselt number (NuLocal,2) is close to front stagnation point (θ ) 0o). The local Nusselt number (NuLocal,2) values at the rear stagnation point (θ ) 180o) are somewhat greater than that at the front stagnation point (θ ) 0o), for all values of the Reynolds and Prandtl numbers, power-law index, and gap ratio (except for G > 4) for both thermal boundary conditions. With the increasing value of the gap ratio (G), as expected, the downstream cylinder approaches the heat transfer characteristics of an isolated cylinder. Therefore, the location of the maximum values moves toward the front stagnation point, and that of the minimum values moves toward the rear stagnation point. For instance, an increase in the gap ratio (G) from 2 to 10 moves

the location of the minimum local Nusselt number from θ ) 85o to near the front stagnation point (θ ≈ 0o) at Re ) 1, Pr ) 10, and n ) 0.4 for both thermal boundary conditions. The corresponding locations for Pr ) 100 are seen to be at θ ) 85o and θ ≈ 50o, respectively. Both maximum and minimum values of the local Nusselt number (NuLocal,2) increase with the increasing value of the gap ratio (G). Because of weak advection at Re ) 1 and Pr ) 1, the downstream cylinder also shows almost a flat profile of local Nusselt number for the CHF condition as that seen for the upstream cylinder. Finally, the shear-thinning (n < 1) behavior exerts an appreciable effect on NuLocal,2 as compared to shear-thickening (n > 1) behavior. For instance, the maximum values of NuLocal,2 are ∼14.8, ∼10, and ∼8.5 for n ) 0.4, 1, and 1.8, respectively, at Re ) 10 and Pr ) 100 for the CWT condition. This suggests some gain in heat transfer in shear-thinning fluids and deterioration in shear-thickening fluids. 6.2.4. Average Nusselt Number. Tables 5–8 present the values of the surface-average Nusselt numbers (Nu1 and Nu2) as functions of power-law index (n), Prandtl number (Pr), gap ratio (G), and Reynolds number (Re) considered in this work, for the two thermal boundary conditions. The functional dependence of the average Nusselt number for both cylinders on the dimensionless parameters (n, Pr, Re, and G) is seen to be qualitatively similar to that for a single cylinder. Because of the proximity of the cylinders, the average Nusselt number for both cylinders is smaller than that for a single cylinder, i.e., Nu1(G) < Nu(G ) ∞) and Nu2(G) < Nu(G ) ∞), under otherwise identical conditions. An increase in the value of the Reynolds number (Re) and/or Prandtl number (Pr) and/or a decrease in the power-law index (n) enhances the rate of heat transfer for both thermal boundary conditions. For fixed values of G, Re, and Pr, the average Nusselt number values show an enhancement as the fluid behavior changes from Newtonian (n ) 1) to shear-thinning (n < 1); however, the opposite effect is seen in shear-thickening (n > 1) fluids. For fixed values of Re, n, and G, an increasing value of the Prandtl number (Pr) also enhances the rate of heat transfer, irrespective of the fluid behavior. For fixed values of n and Pr, heat transfer enhances with the increasing value of the gap ratio (G) at low Reynolds numbers (Re). The isoflux (CHF) condition on the surface of the cylinders always results in a larger value of the average Nusselt number than that for the isothermal (CWT) condition; however, the difference between the two values is a function of the Reynolds and Prandtl numbers. Over the ranges of parameters studied herein, as expected, the average Nusselt number for the upstream cylinder (Nu1) is always higher than that for the downstream cylinder (Nu2). For instance, an increase in Prandtl number from 1 to 100 changes the value of the average Nusselt number for the upstream cylinder (Nu1) by a factor of 4.65-5.84, whereas the corresponding change for the downstream cylinder (Nu2) is in the range of 3.3-5.62, respectively. This increase is small in shear-thickening fluids (n > 1). Also, the average Nusselt number values of the upstream cylinder are close to that for a single cylinder as compared to that for the downstream cylinder, thereby suggesting the interference and proximity effects are stronger for the downstream cylinder. At small values of the gap ratio (G) and the Reynolds numbers (Re), the presence of the downstream cylinder shows a significant effect on the average Nusselt number of the upstream cylinder (Nu1) also. For a fixed value of the Reynolds number (Re), the values of Nu1 and Nu2 are always seen to increase with the increasing gap ratio (G), except at Re ) 40 and Pr ) 50 and 100, where different trends are seen possibly because of the strong wake interference effects


at small values of the gap ratio, G ) 2, 3, and 4, as also observed in the flow field.1 Also, the effect of gap ratio (G) is more prominent for the downstream cylinder (Nu2) as compared to that for the upstream cylinder (Nu1). For instance, for fixed values of the Reynolds number (Re ) 1), the Prandtl number (Pr ) 100), and the power-law index (n ) 0.4), the average Nusselt number values, Nu1 and Nu2, for the two cylinders change by 8.9% and ∼33% as the gap ratio (G) is gradually increased from 2 to 10, whereas the corresponding changes are 7.65% and 32.65% for the constant wall flux (CHF) condition. The corresponding changes in the average Nusselt number values at Re ) 40 are (3.4% and 53.9%) and (3.4% and 48.8%) for the CWT and CHF conditions, respectively. The functional dependence of the surface-averaged Nusselt number on the pertinent dimensionless parameters (Re, Pr, G, and n) can be best represented by the following expression: Nu ) aRebGcndPr1⁄3

(15)

The values of the empirically fitted constants (a, b, c, and d) together with their statistical parameters are summarized in Table 9. A statistical analysis revealed that the points relating to certain combinations of the dimensionless parameters (Re, n, Pr, and G), especially for Pr ) 1, Re < 5, and for large values of n (g1), showed rather large deviations. However, if these data points showing deviations larger than 20% are excluded, eq 15 reproduces the remaining data points with an average deviation of 4.2%, which rises to a maximum value of 5.9%. In the engineering literature, it is common to use the so-called Colburn heat transfer factor (j) to correlate heat transfer results, which is defined as j)

Nu RePr1⁄3

(16)

The main virtue of this parameter lies in the fact that it affords the possibility of reconciling the results for a range of Prandtl numbers (Pr) into a single curve. The functional dependence of the present numerical data in terms of the j-factor on the pertinent dimensionless parameters (Re, Pr, n, and G) over the range of conditions considered here can be best represented by the following expression: j)

aGd Rebnc

(17)

The values of the empirically fitted constants (a, b, c, and d) with their statistical analysis with the present numerical data are included in Table 9. While this approach not only reconciles data for a wide range of Prandtl numbers (Pr), the resulting deviations are also somewhat smaller than that associated with eq 15. The errors associated with eqs 15 and 17 seem to be rather large, but these results do indeed embrace wide ranges of the pertinent dimensionless parameters. From an engineering point of view, this level of accuracy is regarded to be acceptable. 6.3. Effect of Power-Law Index (n). In order to elucidate the role of the power-law rheology on the heat transfer characteristics of a pair of cylinders in an explicit manner, the surface-averaged Nusselt number and/or the Colburn j-factor values have been normalized using the corresponding Newtonian values, under otherwise identical conditions, defined as follows: X(n) where X ) Nu1, Nu2, or j1, j2 X(n ) 1) Figures 10 and 11 depict the dependence of the normalized surface-averaged Nusselt number and/or normalized j-factor XN )

(Nu1N, Nu2N and/or j1N, j2N) on the Reynolds number (Re), Prandtl number (Pr), power-law index (n), and gap ratio (G). For fixed values of Re, Pr, and G and for a fixed thermal boundary condition on the surface of the cylinder, an enhancement of heat transfer with the decreasing value of the power-law index (n) is seen; thus, the normalized values of the Nusselt numbers and/or j-factors are seen to be XN > 1 for shear-thinning (n < 1) fluids, whereas the shear-thickening fluids show the opposite trend, i.e., XN < 1. The influence of the power-law index is stronger in the shear-thinning (n < 1) fluids than that in the shear-thickening (n > 1) fluids. It is clearly seen that the use of corresponding results for Newtonian fluids can entail errors up to about 40% depending upon the values of Re, Pr, G, and n. 6.4. Effect of Gap Ratio (G). In order to quantify the extent of interaction between the two cylinders, the Nusselt number and/or Colburn j-factor values are normalized using the corresponding single cylinder (G ) ∞) values, under otherwise identical conditions, defined as follows: X(G) X(G ) ∞) Figures 12 and 13 show the dependence of the normalized average Nusselt number and/or Colburn j-factor (Nu1SC, Nu2SC or j1SC, j2SC) on the Reynolds number (Re), Prandtl number (Pr), power-law index (n), and gap ratio (G). As noted earlier, the average Nusselt number values or j-factor values for both cylinders are smaller than that for a single isolated cylinder, i.e., XSC < 1 under all conditions. These figures also show that, as the gap ratio (G) increases, the normalized values (XSC) increase and approach unity, i.e., both the cylinders approach the single -cylinder limit. For instance, at Re ) 1 and Pr ) 1, as the gap ratio (G) is increased from 2 to 10, the normalized value for isothermal upstream cylinder (Nu1SC) changes (from 0.81 to 0.95), (from 0.79 to 0.97), and (from 0.80 to 0.99) at power-law index n ) 0.4, 1, and 1.8, respectively; the corresponding changes in the normalized value for isothermal downstream cylinder (Nu2SC) are (from 0.5 to 0.68), (from 0.51 to 0.66), and (from 0.52 to 0.67), respectively. Also, the normalized values for the upstream cylinder are closer to that for a single cylinder than that for the downstream cylinder. At small values of the gap ratio (G) and the Reynolds numbers (Re), smaller values of Nu1SC at G ) 2 compared to that at G ) 10 indicate a significant effect on the values of the average Nusselt number of the upstream cylinder (Nu1) due to the presence of the downstream cylinder. The increasing value of the Reynolds number (Re) increases the normalized values for the upstream cylinder (Nu1SC), whereas Nu2SC decreases under otherwise identical conditions. This is presumably due to the increasing wake length of the upstream cylinder. In summary, the heat transfer characteristics of power-law fluids from a pair of cylinders in a tandem arrangement are seen to be influenced in an intricate manner by the value of the Reynolds number (Re), the Prandtl number (Pr), the powerlaw index (n), and the gap ratio (G). At high values of Re, the wake interferences are more prominent when the gap ratio (G) is very small. On the other hand, when the cylinders are sufficiently far away from each other (G f ∞), no wake interference phenomena will occur and both cylinders would act individually as a single cylinder. As Reynolds number is increased, the flow is governed by two nonlinear terms, namely, inertial and viscous, which scale differently with fluid velocity. ˜ on, For instance, the viscous forces approximately scale as U 2 whereas the inertial forces scale as ∼Uo . Thus, keeping everything else fixed, the decreasing value of the power-law index (n)20 suggests diminishing importance of the viscous XSC )


effects for shear-thinning (n < 1) fluids, while the inertial term will still scale as ∝Uo2. On the other hand, viscous effects are likely to grow with the increasing value of the power-law index (n) for a shear-thickening (n > 1) fluid. For the extreme case of n ) 1.8, the viscous term will also scale as ∼Uo1.8, almost identical to the inertial term. It is believed that these different kinds of dependencies on the flow behavior index and velocity are also responsible for the nonmonotonous behavior as seen in this work. Finally, it needs to be emphasized here that, while the tandem arrangement has been considered here, the interactions in the other types of arrangements (like side-by-side and/ or staggered configurations) are also just as important depending upon the lateral gap between the two cylinders. Indeed, a satisfactory understanding of both types of interactions is needed to develop sound design strategies for shell-and-tube heat exchangers, tubular modules of membranes, etc. Similarly, the idealized flow geometry considered can serve us a launching pad to undertake the modeling of multicylinder arrays as encountered in process equipment. Finally, the physical properties (notable viscosity) have been assumed to be independent of temperature. Therefore, the present results should be seen as conservative values of the Nusselt number. In the first instance, one can possibly apply the small correction by multiplying the values of the Nusselt number by the factor (ηb/ηw)0.14 to account for temperature-dependent viscosity. This practice has been found to be satisfactory in the flow of non-Newtonian fluids in other geometries.4 Similarly, while the simple power-law fluid model breaks down in the limit of vanishing small values of shear rate, and therefore, it might be preferable to use a viscosity model that captures the viscosity behavior at low shear rates, e.g., Ellis or Carreau equations. On the other hand, owing to the presence of an additional model parameter, this gives rise to one additional dimensionless group (like Ellis number or Carreau number). Naturally, this adds to the computational efforts as one needs to vary this new dimensionless group to establish its influence on the results. This is in stark contrast to the power-law model, in which case the degree of deviation of fluid behavior from the standard Newtonian fluid behavior is quantified by a single parameter i.e., the value of n. Thus, there exists a delicate balance between capturing the fluid behavior on one hand and obtaining results of engineering utility containing as few parameters as possible on the other. Therefore, the simple power-law model offers an acceptable compromise from a practical standpoint. 7. Concluding Remarks Extensive numerical results on the heat transfer characteristics of steady flow of power-law fluids over a pair of circular cylinders in tandem arrangement have been obtained over the ranges of conditions as follows: 1 e Re e 40, 1 e Pr e 100, 0.4 e n e 1.8, and five values of the gap ratio between the two cylinders (10 g G g 2). The proximity of the cylinders influences the heat transfer from both the downstream side of the upstream cylinder and the upstream side of the downstream cylinder. The wake interference in conjunction with power-law rheology is seen to be a strong influence at high Reynolds numbers, whereas at low Reynolds numbers, the heat transfer is altered by the fluid behavior. The fact that the flow separation is delayed in the shear-thinning fluids versus that in Newtonian and shear-thickening fluids behavior also influences the rate of heat transfer. The maximum temperature gradient is observed in the relatively slow-moving fluid region present in the gap between the two cylinders. The shear-thinning behavior yields some enhancement in heat transfer, whereas shear-thickening

fluid behavior seems to impede it. For all values of the gap ratio, the average Nusselt number values for the upstream cylinder show a dependence on the dimensionless parameters, qualitatively similar to that for a single cylinder. At high values of the gap ratio, the heat transfer characteristics of the downstream cylinder are also qualitatively similar to those for an isolated cylinder. The functional dependence of the numerical results on the dimensionless parameters has also been presented in terms of the average Nusselt number and the Colburn j-factor, which are convenient for process design calculations. Notations cp ) specific heat of the fluid, J/(kg K) D ) diameter of the cylinders, m Do ) outer boundary of the computational domain, m G ) gap between the two cylinders, ()L/D), dimensionless h ) local convective heat transfer coefficient, W/(m2 K) I2 ) second invariant of the rate of the strain tensor, s-2 j ) Colburn factor for heat transfer, dimensionless k ) thermal conductivity of the fluid, W/(m K) L ) center-to-center distance between the cylinders, m m ) power-law consistency index, Pa sn n ) power-law flow behavior index, dimensionless Nu ) surface-averaged Nusselt number, dimensionless NuLocal ) local Nusselt number, dimensionless p ) pressure, Pa qw ) heat flux on the surface of the cylinder, W/m2 Pr ) Prandtl number, dimensionless Re ) Reynolds number, dimensionless Ri ) Richardson number ()Gr/Re2), dimensionless T ) temperature, K To ) temperature of the fluid at the inlet, K Tw ) temperature of the cylinder, K T* ) temperature, dimensionless Uo ) uniform inlet velocity of the fluid, m/s Ux, Uy ) x- and y-components of the velocity, m/s x, y ) streamwise and transverse coordinates, m XN ) normalized average Nusselt number and/or j-factor using the corresponding Newtonian value, dimensionless XSC ) normalized average Nusselt number and/or j-factor using the corresponding value for single cylinder value, dimensionless Greek Symbols η ) viscosity, Pa s θ ) angular displacement from the front stagnation (θ ) 0), degrees F ) density of the fluid, kg/m3 τ ) extra stress tensor, Pa Subscripts 1 ) upstream cylinder 2 ) downstream cylinder, except in I2 o ) free stream w ) surface of the cylinders Superscripts N ) Newtonian value SC ) single circular cylinder

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ReceiVed for reView December 16, 2007 ReVised manuscript receiVed March 18, 2008 Accepted March 24, 2008 IE7017178

Forced Convection Heat Transfer in Power Law Liquids from a Pair of

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