Laminar Natural Convection from a Horizontal Cylinder in Power-Law

ARTICLE pubs.acs.org/IECR

Laminar Natural Convection from a Horizontal Cylinder in Power-Law Fluids A. Prhashanna and R. P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India 208016 ABSTRACT: In this work, free convective heat transfer from a horizontal cylinder immersed in quiescent power-law fluids has been investigated numerically in the laminar flow regime. The governing differential equations (continuity, momentum, and energy) have been solved over the following ranges of conditions: Grashof number, 10 e Gr e 105; power-law index, 0.3 e n e 1.8; Prandtl number, 0.72 e Pr e 100. The flow and heat transfer characteristics have been visualized in terms of streamlines and isotherm contours which help delineate the regions of high/low temperature. As expected, the value of the local Nusselt number decreases from its maximum value at the front stagnation point along the surface of cylinder, as the flow remains attached to the surface of the cylinder over the range of conditions covered in this study. Finally, the surface-averaged Nusselt number shows positive dependence on both Grashof and Prandtl numbers. All else being equal, shear-thinning behavior enhances the rate of heat transfer with reference to its value in Newtonian fluids. Shear-thickening behavior, however, has an adverse influence on heat transfer. The paper is concluded by presenting comparisons with the previous approximate analysis and scant experimental data available in the literature.

’ INTRODUCTION Over the past 50 years or so, natural convection from a horizontal cylinder submerged in quiescent media has received considerable attention in the literature, especially in Newtonian fluids. The interest in such model flows stems from both theoretical, such as to gain useful insights into the nature of the underlying transfer processes, as well as pragmatic considerations, such as that reliable values of heat transfer coefficients are often required in process engineering calculations. Consequently, over the years, an extensive body of knowledge has accrued on the heat transfer characteristics of a horizontal cylinder in the free convection regime as far as the Newtonian fluids are concerned, albeit there is a preponderance of studies pertaining to air and water. It hardly needs to be emphasized here that in most practical situations, free (or natural) convection always contributes, how so ever small, to the overall rate of heat transfer from a cylinder to the surrounding fluid medium or vice versa. This contribution progressively increases with the decreasing celerity of the imposed flow, i.e., with the decreasing value of the Reynolds number. Unlike in the case of a sphere, additional complications arise in this case depending upon the orientation of the cylinder, e.g., whether the cylinder is oriented horizontal, vertical, or inclined with reference to the direction of gravity. This is so simply due to the fact that the flow patterns induced by the temperature-dependent density are strongly influenced by the orientation of the cylinder, which, in turn, influence the rate of heat transfer to varying extents depending upon the orientation and kinematic conditions. Indeed, the rate of heat transfer (or Nusselt number) varies significantly from one orientation to another under otherwise identical conditions. All in all, based on a combination of approximate analytical (thin boundary layer approximation (without taking into account the curvature effects)), numerical solutions of the governing field equations and experimental results, suffice it to say here that it is now possible to estimate r 2011 American Chemical Society

the value of the Nusselt number in the free convection regime in Newtonian media over wide ranges of the governing parameters, namely, Prandtl and Grashof or Rayleigh numbers for a horizontal cylinder in quiescent Newtonian fluids. In contrast, it is readily conceded that many materials (notably of polymeric and multiphase nature, for instance) display a range of rheological complexities including shear-dependent viscosity, viscoelasticity, yield stress, etc.1 This class of materials is frequently encountered in polymeric, food, pharmaceutical, and personal care product manufacturing sectors.1-3 In spite of their such overwhelming pragmatic significance in scores of modern processing applications, very little is known about the nature of free convection in such media, especially from a horizontal cylinder.4,5 This work aims to alleviate this situation. Furthermore, among the various nonlinear characteristics, perhaps the so-called shear-thinning and shear-thickening behavior denote the most common types of non-Newtonian characteristics encountered in diverse industrial settings. This type of behavior is conveniently approximated by the two-parameter power-law model. Also, in order to keep the level of complexity at a tractable level, it seems reasonable to begin with this type of fluid behavior and the level of complexity can gradually be built up to incorporate the other non-Newtonian aspects of fluid behavior. Thus, this work is concerned with prediction of the rate of heat transfer from a heated horizontal cylinder immersed in stagnant power-law fluids over wide ranges of the pertinent kinematic and physical parameters in the laminar natural convection regime. It is, however, deemed useful and instructive to briefly summarize the current state of the art on natural convection from a horizontal cylinder in Newtonian media followed by the scant analogous literature for Received: September 20, 2010 Accepted: December 17, 2010 Revised: November 11, 2010 Published: January 12, 2011 2424

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Industrial & Engineering Chemistry Research non-Newtonian fluids which, in turn, facilitates the presentation and discussion of the new results obtained in this work.

’ PREVIOUS WORK From a theoretical standpoint, the governing differential equations describing the free convective transport of heat (or mass) from a cylinder are coupled via the body force term in the momentum equations, thereby excluding the possibility of rigorous analytical solutions, even for Newtonian fluids. Early attempts in this field are based on the boundary layer approximation together with the negligible curvature effects. Obviously, such analyses not only are restricted to high values of Prandtl number and/or of Grashof number but also do not capture the wake region of the cylinder. In other words, the assumption of high Rayleigh numbers is implicit in all such studies wherein the boundary layer is thin and it is thus justified to neglect the curvature effects. Similarly, one can approach the other limit of vanishingly small values of the Rayleigh number, and this case is well suited to the matched asymptotic analysis. The studies of Narain,6 Yao et al.,7 Farouk and Guceri,8 and Merk and Prins,9 for instance, exemplify the utility of this approach. Obviously, neither of these approximations is applicable for intermediate values of the Rayleigh number.10,11 Furthermore, most of the aforementioned studies pertain to the case when the surface of the submerged cylinder is maintained at a constant temperature. Wilks12 examined the analogous situation for the other commonly employed boundary condition, i.e., a constant heat flux being prescribed on the surface of the cylinder. In particular, he delineated the choice of transformation and of an appropriate independent variable to employ the boundary layer approximation for these conditions. Despite these limitations, the aforementioned analyses have proved to be of value in delineating the correct scaling of the Nusselt number with the pertinent governing parameters, namely, Prandtl number and Grashof number or Rayleigh number, at least in the limits of low/high Rayleigh numbers. This, in turn, aided by scores of experimental studies, has led to the development of widely accepted predictive equations spanning the full range of flow regimes as well as wide ranges of Prandtl and Grashof numbers.13-17 In particular, Fand et al.14 critically examined the various functional forms of the dependence of Nusselt number on Rayleigh number and Prandtl number. On the basis of their experimental data encompassing 3 orders of magnitude variation in Prandtl number (0.7 e Pr e 3090) and 5 orders of magnitude variation in Rayleigh number (250 e Ra e 1.8 107), they developed reliable empirical expressions for Nusselt number for a horizontal cylinder. Similarly, Fand and Bruker18 assessed the role of viscous dissipation in free convection by introducing the so-called Gebhart number, akin to the well-known Eckert number in forced convection. They identified conditions when the viscous dissipation effects cannot be ignored. It is thus imperative that one must resort to the numerical solution of the full governing equations to examine the role of intermediate values of the Rayleigh number in free convective transport. Kuehn and Goldstein11 used a finite difference method to solve the Navier-Stokes equations together with the energy equation for free convection heat transfer from a horizontal cylinder over the range of Rayleigh number as 1 e Ra e 107. While they stated their results to be consistent with the literature data, more importantly their results highlighted the inadequacy of the boundary layer approximations for intermediate values of the Rayleigh number. Later, Moon et al.19 numerically analyzed heat transfer in the mixed convection

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regime from a horizontal cylinder and also confirmed the findings of Kuehn and Goldstein11 in the limit of free convection. They also examined the role of the size of the domain on the results. Subsequently, Saitoh et al.20 reported highly accurate numerical results for Nusselt number in the free convection regime for a horizontal cylinder in air over the range 103 e Ra e 105. Indeed, these results have been frequently used to carry out benchmark comparisons by subsequent researchers in this field to establish the reliability of their results. Furthermore, they also examined the influence of the type of boundary condition prescribed on the outer boundary on the heat transfer results. The commonly used two conditions are that of inflow and outflow, such as that used by Kuehn and Goldstein,11 Zeitoun and Ali,21 and Moon et al.,19 and of treating the outer boundary as the solid boundary, such as that explored by Saitoh et al.20 Intuitively, from a theoretical stand point, far away from the cylinder (r f ¥) it is expected that the buoyancy-induced flow would have subsided, i.e., velocity components would approach zero and the temperature will approach its free stream value. However, numerical solutions impose these free stream conditions at finite distance (r = r¥). Intuitively, it thus appears that the influence of the outer boundary conditions should progressively vanish as the value of r¥ is gradually increased. Naturally, it stands to reason that the thicker the boundary layer (small Rayleigh number or Grashof number), the larger the value of r¥ needed to eliminate the influence of the outer boundary conditions. The study of Saitoh et al.20 recommends the results to be free from this artifact for D¥ g 700D. Wang et al.22 also demonstrated that the inflow/ outflow type of boundary conditions are only appropriate for the steady flow regime. Wang et al.22,23 reported transient and steady-state values of temperature profiles and Nusselt number for free convection from a horizontal cylinder in air for both types of the boundary conditions, namely, constant temperature or constant heat flux imposed on the surface of the cylinder. Similarly, in a recent study, Haldar et al.24 elucidated the contribution of fins to the overall rate of heat transfer from a cylinder in air. Their limited results for a cylinder without fins are in good agreement with the values of Saitoh et al.,20 thereby lending credibility to their results for the case of a cylinder fitted with fins. The bulk of the literature on convective heat transfer from a horizontal cylinder in Newtonian media has been reviewed by Morgan25 and in several other books.26-28 Suffice it to add here that it is now possible to estimate reliable values of the heat transfer coefficient for a horizontal circular cylinder in the free convective regime in Newtonian fluids over conditions of practical interest (10-11 e Ra e 1012), encompassing both laminar and turbulent flow regions. In contrast, very few studies are available on free convection in non-Newtonian fluids in general and from a heated horizontal cylinder in particular.29-31 Acrivos32 was seemingly the first to tackle this problem for free convection from axisymmetric-shaped objects immersed in stagnant power-law fluids. He employed the thin boundary layer approximation which is obviously applicable at large values of the Prandtl and/or Rayleigh numbers. By choosing an appropriate shape factor, his analysis can be used to infer the values of Nusselt number for a sphere, or a plate, or a horizontal cylinder, etc. Subsequent more detailed numerical and experimental studies reveal that this approach is satisfactory for a sphere or plate at large values of the Prandtl or Rayleigh number.33-36 Chang et al.37 exploited the similarity between the energy equations for the forced and free convection to introduce a hypothetical outer field stream function. This coupled with the approach of Kim 2425

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Industrial & Engineering Chemistry Research et al.38 led them to obtain a series solution for free convection from a horizontal cylinder in power-law fluids. Their predictions for a vertical plate (where curvature effects are absent) are consistent with those of Acrivos.32 Lyons et al.39 merely demonstrated that the dosing of water with high molecular weight polyethylene oxide led to a significant reduction in the rate of heat transfer from thin cylinders in the free convection regime. While they did not report detailed rheological properties of their test fluids, the deterioration in heat transfer can safely be attributed to the increase in viscosity (lowering of Grashof number) due to polymer addition. Gentry and Wollersheim40 presented detailed temperature profiles and Nusselt number values for heated cylinders immersed in stagnant polymer solutions (modeled as power-law fluids). However, their polymer solutions were only weakly shear thinning as the power-law index varied from n = 0.8 to n = 1 in their study. By using dimensional considerations, Ng and Hartnett41 showed these data to be consistent with the scaling suggested by the boundary layer analysis of Acrivos.32 Subsequently, they argued that owing to extremely high values of viscosity of their polymer solutions (∼0.001 e Ra e ∼100) there was virtually no fluid motion at all in the fluid adjacent to the immersed heated cylinder, and indeed, their experimental values of the average Nusselt number were close to those based on the assumption of heat transfer by pure conduction only.42 Their subsequent experimental studies43,44 elucidated the role of curvature effects (by using very thin wires such that the boundary layer thickness would be comparable to the radius of the wire) and the viscoelasticity of fluids. The resulting values of the Rayleigh number were too small ( T¥). Owing to the thermal expansion, the fluid adjacent to the cylinder will be lighter than the fluid far away from it. Such a density gradient, in turn, will set up an upward flow near the cylinder, thereby resulting in heat transfer by free convection. Depending upon the temperature dependence of the density and the temperature difference between the cylinder and the fluid, the resulting flow may be steady or time-dependent, laminar, or turbulent. Since no prior information is available in the literature on this aspect, the flow is assumed to be two-dimensional, laminar, and symmetric with reference to the centerline of the cylinder in the direction of the gravity over the range of conditions encompassed here. This flow regime is likely to happen when the value of the Reynolds number based on the gravity-induced velocity is small. Since the non-Newtonian fluids tend to be generally far more viscous than their Newtonian counterparts, this is a reasonable assumption, at least to begin with. The thermophysical properties except for the density term in the ymomentum equation are assumed to be temperature independent, and the viscous dissipation is also assumed to be negligible here. The variation of density with temperature is assumed to be linear and approximated here by the usual Boussinesq approximation, as done elsewhere.33,34 Under these conditions, the flow and temperature fields are governed by the continuity, momentum, and thermal energy equations written in their compact forms as follows. • continuity equation DUx DUy þ ¼ 0 Dx Dy

ð1Þ

• x component of momentum equation F¥

DUx DP Dτxx Dτyx ¼ - þ þ Dx Dx Dt Dy

ð2Þ

• y component of momentum equation F¥

DUy Dτxy Dτyy DP ¼ - þ gβðT - T¥ ÞF¥ þ þ Dy Dt Dx Dy

ð3Þ

• energy equation DT ¼ Rr2 T Dt 2426

ð4Þ


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In eq 3, the coefficient of thermal expansion, β, is defined as 1 DF β ¼ F DT T

Evidently, owing to the variation of density with temperature, the momentum and energy equations are coupled via the body force term in the y component of the momentum equations. Furthermore, the flow past a cylinder is known to exhibit several flow regimes like fully attached, separated and steady, unsteady, etc. with the increasing value of the Reynolds number (based on the characteristic velocity due to densityinduced flow). In the present work, the flow is not assumed a priori to be steady, and therefore, transient terms are retained here in both the momentum and the energy equations. If the flow turns out to be steady, the solution will ultimately approach steady state, which was found to be the case over the range of conditions encompassed in this study. For a power-law fluid, the components of the deviatoric stress tensor, τ, are related to the second invariant of the rate of deformation tensor as τ ¼ 2mð2I2 Þn - 1=2 εðuÞ

ð5Þ

where ε(u), the rate of deformation tensor, is related to the velocity field as i 1h ð6Þ εðuÞ ¼ rU þ ðrUÞT 2 and I2, in turn, is related to the velocity components and velocity gradients, and these expressions are available in standard texts, such as that by Bird et al.54 In eq 5, m and n are the power-law consistency and flow behavior index, respectively. Evidently, n < 1 corresponds to shear-thinning behavior, n = 1 indicates the standard Newtonian flow behavior, and n > 1 denotes the shear-thickening behavior. As noted earlier, obviously, it is not possible to solve an unconfined flow numerically, and hence, it is necessary to impose an artificial domain with suitable boundary conditions which adequately mimick the unconfined flow situation. In this case, a concentric cylindrical envelope of fluid of diameter D¥ has been used which encloses the heated cylinder. The diameter of the envelope of the fluid (D¥) is chosen such that it does not unduly affect the flow and at the same time keeps the required computational effort at a modest level. The physically realistic boundary conditions for this flow are written as follows. • That of no-slip and constant temperature on the surface of the cylinder, i.e., at r = R Ux ¼ 0, Uy ¼ 0, and T ¼ Tw

ð7Þ

• At the outer boundary, i.e., r = R¥, the Neumann-type boundary condition, i.e., normal gradients of all variables except temperature, are set equal to zero. To ensure zero radial mass flux, the normal (radial) velocity is also prescribed to be zero. In mathematical terms DUθ ¼ 0 and Ur ¼ 0 ð8Þ T ¼ T¥ ; Dr As noted earlier, much confusion exists regarding the type of temperature boundary condition at the outer boundary. Some20

Figure 1. (a) Schematics of the flow. (b) Computational domain.

have treated it as a solid boundary (Ux = 0, Uy = 0, and T = T¥), while others11,19,21 have treated it as an inflow/outflow boundary, i.e., T = T¥, prescribed over the inflow part and ∂T/∂r = 0 implemented on the outflow segment of the domain. However, as mentioned earlier, these effects are expected to be significant only when relatively short domains are used, as is the case in the study of Kuehn and Goldstein11and of Moon et al.19 for instance. In view of the relatively high value of Rayleigh number and the fact that the value of D¥ is also significantly large (as will be seen later), these boundary conditions are used here, albeit the effect of outer boundary conditions is likely to be minimal in view of the large value of D¥. Indeed, limited simulations have been carried out in the present study using both types of conditions, i.e., T = T¥ or ∂T/∂r = 0, and as will be seen later, the resulting values of drag coefficient and Nusselt number were virtually indistinguishable from each other, albeit the use of ∂T/∂r = 0 condition accelerates the approach to convergence, thereby reducing the 2427



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Table 1. Domain Independence Tests at Gr = 10 and Pr = 0.72 CD

CDP

Nu

domain size (D¥/D)

n = 0.3

n=1

n = 1.8

n = 0.3

n=1

n = 1.8

n = 0.3

n=1

n = 1.8

1000

7.151

10.525

6.807

4.682

5.400

3.067

2.152

1.594

2.088

1200

7.167

10.526

6.807

4.695

5.400

3.067

2.156

1.594

2.088

Table 2. Grid independence tests Gr = 103, Pr = 100 CD

CDP

Nu

grid

total no. of cells

n = 0.3

n=1

n = 1.8

n = 0.3

n=1

n = 1.8

n = 0.3

n=1

n = 1.8

G1

25 600

0.5505

0.5497

0.7246

0.3734

0.2815

0.3503

27.817

16.561

12.477

G2 G3

55 680 121 600

0.5483 0.5481

0.5535 0.5547

0.7277 0.7345

0.3662 0.3743

0.2941 0.3005

0.3611 0.3725

27.583 27.541

16.444 16.400

12.439 12.408

Gr = 105, Pr = 50 CD

CDP

Nu

n=1

n = 1.8

n=1

n = 1.8

n=1

n = 1.8

G2

55 680

0.1934

0.3655

0.0992

0.1849

41.803

22.207

G3

121 600

0.1961

0.3674

0.1061

0.1905

41.225

22.143

G4

200 000

0.1964

0.3670

0.1078

0.1914

41.148

22.143

CPU time .This lends support to the assertion of Saitoh et al.20 that the influence of the outer boundary condition for temperature on the flow and temperature fields in the vicinity of cylinder diminishes with the increasing value of D¥. Suffice it to add here that similar boundary conditions have also been used by others.55-58 The aforementioned governing eqs 1-6 along with the boundary conditions outlined in eqs 7 and 8 were solved using FLUENT. The numerical solution maps the flow domain in terms of the primitive variables, namely, pressure, temperature, and velocities for a range of thermophysical properties, albeit the absolute values of these properties like F, m, n, Cp, k, T0, and Tw, etc., are of no consequence because the final results are presented in terms of dimensionless quantities. The scaling considerations of the governing equations and boundary conditions suggest this flow to be governed by the following five dimensionless groups: drag coefficient, Nusselt number, Grashof number, Prandtl number, and power-law index. At this juncture, it is appropriate to introduce definitions of these and some of the other pertinent parameters. Drag Coefficient (CD). The upward flow induced by the buoyancy forces gives rise to the shearing and normal forces, thereby resulting in a net drag force acting on the cylinder in the y direction. This is usually expressed in terms of a drag coefficient (CD) which is made up of two components, i.e., due to shearing (CDF) and the normal (CDP) forces. These, in turn, are defined as follows Z FDF 21 - n ¼ pffiffiffiffiffi τ 3 ns dS CDF ¼ 1 ð9Þ Gr s FUc2 D 2 Z FDP ð10Þ CDP ¼ 1 ¼ Cp ny dS s FUc2 D 2

where Cp is the pressure coefficient defined as Ρ0 - Ρ¥ Cp ¼ 1 ð11Þ FUc2 2 Here, Ρ0 is the local pressure at a point on the surface of the cylinder and Ρ¥ is the reference pressure far away from the surface of the cylinder. Here, Uc is the reference velocity defined as Uc = (RgβΔT)1/2 and, of course, the total drag coefficient, CD, is simply the sum of these two components. Nusselt Number (Nu). It gives the nondimensional rate of heat transfer. The local Nusselt number at a point on the surface of the cylinder is defined as hD Dξ ¼ ð12Þ N uθ ¼ k Dns surf ace where ns is the outward drawn unit normal vector on the surface of the cylinder. However, in process design calculations, one frequently requires the overall mean value of the Nusselt number to calculate the rate of heat transfer between the cylinder and the fluid or to estimate one of the temperatures if heat flux is known. The average Nusselt number is obtained simply by integrating the local values of the Nusselt number over the surface of the cylinder as follows Z 1 Nuθ dS ð13Þ Nu ¼ S s The variables appearing in the governing equations and boundary conditions are rendered dimensionless using D, Uc, D/Uc, and FUc2 as scaling variables for length, velocity, time, and pressure, respectively. The temperature difference is nondimensionalized as ξ = (T - T¥)/(Tw - T¥). The scaling of these 2428



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equations suggests that the flow and temperature fields are governed by Grashof number (Gr), Prandtl number (Pr), and the power-law index (n). These are defined here as follows Gr ¼ 2=ð1 þ nÞ

Pr ¼

FCp m k F

F2 R nþ2 ðgβΔTÞ2 - n m2

ð14Þ

ðRÞð1 - nÞ=ð1 þ nÞ ðRgβΔTÞ3ðn - 1Þ=2ðn þ 1Þ

of iterations required to achieve the preset level of convergence, thereby reducing the CPU time, particularly when the model equations contain a large number of control volumes. The flow geometry, i.e., cylinder-in-cylinder, was created in GAMBIT, and the computational domain (Figure 1) was meshed using a grid of unstructured quadrilateral cells with nonuniform spacing. Since the flow is symmetric (about the y axis), the solution was sought

ð15Þ Evidently, both the drag coefficient (CD) and the Nusselt number (Nu) are expected to be functions of the Grashof number (Gr), Prandtl number (Pr), and power-law index (n). These relationships are explored and developed in this work. It is interesting to note here that unlike in the case of Newtonian fluids, the definitions of Grashof and Prandtl numbers are entwined via the power-law constants (m, n), and it is therefore not at all possible to delineate the influence of the power-law index in this case in an unambiguous manner.

’ SOLUTION PROCEDURE The present work has been carried out using FLUENT (6.3.26), which uses the finite volume formulation of the governing equations. FLUENT solves the system of algebraic equations using the Gauss-Siedel (G-S) point-by-point iterative method in conjunction with the algebraic multigrid (AMG) method. The use of the AMG scheme greatly reduces the number Table 3. Comparison of Nusselt Number Values with the Literature Results for Newtonian Fluids Nu

Pr

Kuehn and Goldstein11

Wang et al.22

Saitoh et al.20

present work

125

0.7

3.09

3.06

3.024

3.021

1250

0.7

4.94

4.86

4.826

4.805

Ra

Figure 2. Temperature profiles at various angles in the vicinity of the cylinder at Ra = 12500, Pr = 0.7, and n = 1.

Table 5. Effect of the Outer Temperature Boundary Condition case I (T = T¥) n

case II (∂T/∂r = 0)

Gr

Pr

CDP

CDF

Nu

CDP

CDF

5

Nu

12 500

0.7

8

7.937

1.8

10

50

0.1907

0.1763

22.11

0.1904

0.1768

22.14

1250

1

5.06

4.971

1

105

50

0.1058

0.0899

41.23

0.1060

0.0899

41.225

1250

5

5.66

5.552

0.6

103

100

0.2124

0.2908

22.37

0.2116

0.2903

22.34

1250

10

5.81

5.723

0.3

103

100

0.3649

0.5477

27.47

0.3658

0.5482

27.50

7.97

7.898

Table 4. Comparison of Local Nusselt Number Values at Different Angles with the Literature Results for Newtonian Fluids (Pr = 0.7) Nuθ θ = 0

θ = 30

θ = 60

θ = 90

θ = 120

θ = 150

θ = 180

present Saitoh et al.20

3.806 3.813

3.766 3.772

3.635 3.64

3.375 3.374

2.863 2.866

1.978 1.975

1.217 1.218


3.89

3.85

3.72

3.45

2.93

2.01

1.22

Wang et al.22

3.86

3.82

3.7

3.45

2.93

1.98

1.20

present

5.949

5.892

5.713

5.387

4.757

3.312

1.533

Saitoh et al.20

5.995

5.935

5.75

5.41

4.764

3.308

1.534


6.24

6.19

6.01

5.64

4.82

3.14

1.46

Wang et al.22

6.03

5.98

5.8

5.56

4.87

3.32

1.5

present Saitoh et al.20

9.747 9.675

9.646 9.557

9.337 9.278

8.814 8.765

7.976 7.946

5.912 5.891

1.983 1.987


10.15

10.03

9.65

9.02

7.91

5.29

1.72

9.8

9.69

9.48

8.9

8

5.8

1.94

Ra 103

104

105

source

Wang et al.22

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Figure 3. Streamline patterns in the vicinity of the cylinder n = 0.3.

only in one-half of the domain (x g 0). The grid near the surface of the cylinder was sufficiently fine to resolve the steep gradients of the flow and temperature fields within the relatively thin momentum and thermal boundary layers. The grid used in the present study was made finer at the surface of the cylinder and relatively coarse far away by using the continuous stretching option using the successive ratio parameter in GAMBIT. A successive ratio of 1.05 was used in the radial direction, while the number of cells is maintained constant with a uniform spacing in the angular direction. Furthermore, the unsteady, laminar, pressure-based 2-D solver was used with QUICK scheme for handling the convective terms in the momentum and energy equations. The time marching numerical scheme was used with an initial velocity profile of zero velocity everywhere (stationary fluid) to initiate the calculations even for the steady-state case to avoid potential convergence problems. As noted earlier, the

density is assumed to be a function of temperature and is imposed in the solver by using the usual Boussinesq approximation. The semi-implicit method for pressure-linked equations (SIMPLE) scheme was used for pressure-velocity coupling, and the nonNewtonian power-law model was used for estimating the relevant value of viscosity corresponding to the desired value of the Grashof or Prandtl number or both. In the present study, the iterations were stopped when the values of the drag coefficient had reached a steady state, i.e., drag did not change with time and/or stabilized up to four significant digits for a period of 200 time units. For a particular time step, a convergence criterion of 10-6 was used for continuity. In turn, this led to the corresponding residuals for the x and y momentum and energy equations being on the order of 10-10. However, it is appropriate to add here that convergence became increasingly difficult with the increasing value of the Grashof number and/or decreasing value 2430



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Figure 4. Streamline patterns in the vicinity of the cylinder at n = 1.8.

of the power-law index. Broadly, the lower the value of the power-law index, the smaller the limiting value of the Grashof number above which convergence could not be attained to the desired level. This limited the maximum value of the Grashof number achievable depending upon the value of the power-law index. For instance, this was found to be Gr = 103 for n = 0.3, and beyond this value, it was not possible to satisfy the same convergence criterion. On the other hand, in view of the fact that most shear-thinning fluids (like polymer melts and solutions) tend to be far more viscous than their Newtonian counterparts the resulting values of the Grashof number are typically much smaller than that obtained in air and water. Despite this limitation, the ranges of conditions covered herein are believed to be sufficiently wide to delineate the scaling of the Nusselt number on the Grashof number, Prandtl number, and powerlaw index.

’ CHOICE OF NUMERICAL PARAMETERS It is needless to emphasize here that the accuracy and reliability of the numerical results are contingent upon a prudent choice of an optimal grid and domain size. In this study, the computational domain is characterized by the value of the diameter, D¥, of the enclosing cylindrical envelope of the fluid with the heated cylinder located at its center. Intuitively, it appears that the smaller the value of the Grashof number and/or Prandtl number, the larger the domain needed to carry out the numerical simulations in order to minimize the effect of domain size on the flow characteristics. This is obviously so because the boundary layer will be thick under these conditions and the buoyancyinduced currents are expected to decay rather slowly. Therefore, in this work, the domain-independence test has been carried out for Gr = 10, Pr = 0.72, and for the extreme values of powerlaw index, i.e., n = 0.3 and 1.8. The value of (D¥/D) was 2431



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Figure 5. Temperature contours in the vicinity of the cylinder at n = 0.3.

systematically varied from 400 to 1200, and a domain of D¥/D = 1200 was found to be appropriate to obtain results which are largely free from domain effects. Table 1 gives a summary of the present results, lending support to this assertion where the values of both pressure and total drag coefficients and Nusselt number are seen to be virtually indistinguishable from each other for D¥/D = 1000 and 1200. Having fixed the domain size as D¥/D = 1200, an optimal grid should be sufficiently fine enough to resolve the steep gradients close to the cylinder and, of coarse, without necessitating excessive computational resources. This objective was realized by stretching the grid continuously in the radial direction while keeping the equal spacing in the angular direction. The effect of grid spacing was checked for the conditions when the boundary layer would be extremely thin, i.e., for Gr = 103, Pr = 50, and for Gr = 105 (only for shear-thickening fluids) and for three values of the power-law index n = 0.3, 1, and 1.8 (Table 2). A thorough

examination of these values suggests that grid G2 (grid spacing close to the surface of the cylinder, δ/D = 0.005) is adequate, because as one moves from grid G2 to G3, the values change by less than 1% whereas the CPU time increased many fold. It is obvious that for the case of Gr = 105, one needs even finer grids. Further grid-independence tests were carried for Gr = 105, Pr = 50, and power-law index n = 1 and 1.8. The results shown in Table 2 suggest that grid G3 (grid spacing close to the surface of the cylinder, δ/D = 0.0025) is appropriate in this case. Finally, even though the range of conditions dealt with in the present work lie well within the steady-state flow regime, the unsteady solver was used in this work to avoid potential convergence problems. Thus, the time step used for time integration does not have any special significance in the context of final results except for the fact that it can accelerate or slow down the approach to the desired convergence level. Broadly, an adaptive time step procedure was used. It ensures the convergence at each 2432



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Figure 6. Variation of the local Nusselt number over the surface of the cylinder at n = 1.8.

time step to the desired level of criterion within the maximum number of iterations for a given time step. Therefore, in summary, the results reported herein are based on the following choices of numerical parameters: D¥/D = 1200, grid G2 for all cases except for Gr = 105 where grid G3 has been used. It needs to be emphasized here that the domain and grid spacing (near the cylinder) used here are way beyond that suggested by Saitoh et al.,20 and the domain used here is much larger than that used in previous studies.11,20 Finally, the adequacy of these choices is further demonstrated in the next section by presenting benchmark comparisons between the present and literature values for a few well-known cases.

’ RESULTS AND DISCUSSION In this work, extensive numerical results have been obtained over the following ranges of conditions: 0.3 e n e 1.8, 10 e Gr e 105, and 0.72 e Pr e 100. A comment about the choice of these ranges is also in order. The values of n are such that these embrace both shear-thinning and shear-thickening fluid behavior.

While most polymeric systems (solutions and melts) exhibit shear-thinning behavior, the widely used starch and corn flour suspensions in food-related applications display shear-thickening behavior.1,3 The upper limits of the Grashof number and Prandtl number are dictated by the fact that an increasingly refined mesh is needed to resolve the extremely thin boundary layers at high values of the Grashof number, Gr, and Prandtl number, Pr, or both. Not only does this inevitably lead to acute convergence problems but also these computations become prohibitively computational resource intensive. However, the ranges of Gr and Pr covered here are not only sufficient to delineate the scaling of the Nusselt number with Prandtl and Grashof numbers but also similar to that encountered in food-processing applications.19 Thus, the ranges of parameters employed here are guided in part by potential engineering applications and in part by numerical limitations. Before presenting the new results obtained in this study, it is important to demonstrate the reliability and accuracy of the present results by comparing them with prior results available in the literature. 2433


Industrial & Engineering Chemistry Research 1. Validation of Results. The numerical solution methodology used herein has been validated by comparing the present values with the scant literature available on the free convection heat transfer from a horizontal cylinder in Newtonian fluids (Table 3). Since none of the previous studies have reported the values of drag coefficient, only the values of Nusselt number are included in Table 3. The present results are seen to be within (2% of that due to Kuehn and Goldstein,11 within (1% of that due to Wang et al.,22 and within (0.5% of that due to Saitoh et al.20 It is needless to say here that it is not at all uncommon to encounter differences of this magnitude among different numerical studies employing differing domains, grids, and solution methodologies, etc.59 Next, Table 4 reports a typical comparison in terms of the local Nusselt number at various points on the surface of the cylinder. Once again, the present results are seen to be remarkably close to that of Saitoh et al.20 and others. However, it is worthwhile to point out here that the values of Kuehn and Goldstein11 at θ = 150 do differ slightly (by up to 15%) from the other values, especially at Ra = 105. Saitoh et al.20 attributed this to the inadequacy of grid spacing and domain used by Kuhen and Goldstein.11 In fact, this is a more stringent check for the reliability and accuracy of the present results than that seen in Table 3 in terms of the surface-averaged values of the Nusselt number. Finally, to add further weight to our claim regarding the accuracy of the present results, Figure 2 compares the temperature distribution in air in the vicinity of a heated cylinder with that of Kuehn and Goldstein11 for Ra = 12 500. Note that in this figure only the radial distance, r, has been nondimensionalized using Ra1/4/D. Once again, the correspondence is seen to be excellent. Lastly, before embarking upon the study of free convection from a horizontal cylinder in power-law fluids, the benchmark problem of the lid-driven laminar flow of Newtonian and power-law fluids in a square cavity was also investigated. The resulting values of the center line velocities were found to be within (2% of that reported by Ghia et al.60 for Newtonian fluids and by Neofytou61 for power-law fluids. Such close correspondence lends further credibility to the numerical solution methodology employed in this work. Finally, as noted earlier, the influence of the type of temperature boundary condition at the outer boundary, namely, T = T¥ (case I) and ∂T/∂r = 0 (case II), is explored here. Table 5 summarizes the results for the two cases. Clearly, the two values are virtually indistinguishable from each other, and the difference is well below 0.5%. Thus, in summary, based on the aforementioned comparisons, not only are the new results reported herein believed to be reliable to within 1-2% but also the influence of the temperature boundary condition at the outer boundary is expected to be negligible, at least over the ranges of conditions spanned here. 2. Streamlines and Temperature Profiles. Figures 3 and 4 show the representative streamline plots for a power-law index of n = 0.3 and 1.8 and for extreme values of the Grashof number, Gr, and the Prandtl number, Pr. Since in the present case Tw > T¥, the fluid close to the cylinder is lighter than that faraway, it is clearly seen that the fluid is drawn toward the heated cylinder and an upward current is set up along the surface of the cylinder. A steady buoyant plume and a stagnation ring are seen to form above the cylinder which grows in size with decreasing values of Grashof number and/or Prandtl number and with increasing power-law index. This is due to the thickening of the momentum and thermal boundary layers with the decreasing value of Grashof number and/or Prandtl number and with the increasing value of the power-law index (i.e., as the fluid behavior transits from shear-thinning to shear-thickening via the standard Newtonian

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Figure 7. Variation of the local Nusselt number over the surface of the cylinder at n = 0.3.

fluid behavior). Conversely, this behavior can be attributed to the fact that a shear-thinning fluid exhibits the minimum effective viscosity near the surface of the cylinder, which gradually increases due to the weakening of the density-induced flow current. As expected, a completely opposite behavior is seen in shear-thickening fluids, and hence, the flow field would decay rather slowly in this case than that in shear-thinning fluid. Figure 5 shows the influence of the power-law index, Grashof number, and Prandtl number on temperature contours for the same values of Gr and Pr for n = 0.3. Qualitatively similar isotherm contours are obtained for Newtonian and shear-thickening fluids and thus are not included here for the sake of brevity. Suffice it to add here that the thermal boundary layer extends up to x ≈ 10D only and beyond this point T ≈ T¥ even for Ra = 7.2, the lowest value used here. Irrespective of the value of the powerlaw index, it is observed that the thermal boundary layer gradually thins with increasing values of the Grashof number and/or Prandtl number or both. Also, the thermal boundary layer is seen to be thinner in shear-thinning fluids than that in shearthickening fluids under otherwise identical conditions. 1 2434



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Figure 8. Dependence of average Nusselt number on Grashof number, Prandtl number, and power-law index.

Therefore, one can expect to see some enhancement in the rate of heat transfer in shear-thinning fluids as compared to that in Newtonian fluids. Conversely, some reduction in the rate of heat transfer can be anticipated in shear-thickening fluids. This expectation is indeed borne out by the Nusselt number results presented in the next section. 3. Local Nusselt Number on the Surface of the Cylinder. The local value of the Nusselt number varies along the surface of the cylinder. At a given point (θ) on the surface of the cylinder, the value of the Nusselt number is governed by the value of the temperature gradient which, in turn, is influenced by the local velocity field in an intricate manner. The value of the local velocity gradient together with the value of n determines the local value of the fluid viscosity which influences the value of the Nusselt number due to its occurrence in the definition of both Grashof and Prandtl numbers. Figures 6 and 7 show representative results elucidating the combined effects of n, Pr, and Gr on the Nusselt number distribution on the surface of the cylinder for a shear-thinning (n = 0.3) and shear-thickening fluid (n = 1.8). A detailed examination of these figures suggests the following overall trends.

For Newtonian fluids (results not shown here), it was seen that the local Nusselt number decreases from its maximum value at the front stagnation point (θ = 0) to the rear stagnation point (θ = 180). This is simply due to the decreasing temperature gradient along the surface of the cylinder from θ = 0 to 180. Since the flow remains attached to the surface of the cylinder in ranges of the conditions studied herein, the local Nusselt number continually decreases from the front to the rear stagnation point. Also, irrespective of the type of fluid behavior (i.e., the value of n), it is seen that the local Nusselt number shows positive dependence on the Grashof number and Prandtl number. Thus, at Gr = 10, the buoyancy-induced flow current is so weak that the main mode of heat transfer is by conduction, and hence, the flow properties of the fluid are of little consequence and the local Nusselt number varies very little over the surface of the cylinder. For shear-thickening fluids (n > 1, Figure 6), the local Nusselt number decreases from θ = 0 to 180 monotonically as discussed above for Newtonian fluids but with a greater slope. This is due to the combined effects of the following two mechanisms: first, the temperature gradient decreases from θ = 0 to 180 which reduces the rate of heat transfer. Second, the 2435


Industrial & Engineering Chemistry Research effective viscosity of the fluid increases in shear-thickening fluids with the increasing shear rate which also suggests the thickening of the boundary layer as well as poor circulation of the fluid over the cylinder. All these factors contribute together, thereby leading to the steep decrease in the local value of the Nusselt number and thus impeding the overall rate of heat transfer as compared to that in Newtonian fluids. Indeed, this behavior is also qualitatively consistent with that seen for a sphere in free convection34 as well as in forced convection regimes.62 For shear-thinning fluids (n < 1, Figure 7), the Nusselt number progressively decreases along the surface of the cylinder from θ = 0 to 180 up to a certain Rayleigh number. Once the Rayleigh number exceeds a critical value, the Nusselt number no longer exhibits its maximum value at θ = 0 as is seen in Figure 7, similar to that seen for a sphere also.34 This can be explained, at least qualitatively, as follows. First, the temperature gradient decreases along the surface of the cylinder. Second, in the case of shearthinning fluids, the effective viscosity of the fluid is lowered in the region of intense shear close to the surface which enhances the rate of heat transfer. The maximum value (away from θ = 0), seen in Figure 7, is the net result of these two mechanisms. However, the magnitudes of these two opposing mechanisms depend on the values of Gr, Pr, and n in an intricate manner, e.g., see the definitions of Gr and Pr given in eqs 14 and 15. For instance, for n = 0.3 the critical value of Rayleigh number is seen to be approximately Ra ≈ 200 for the shifting of the location of the maximum value away from θ = 0. Aside from the complex behavior seen above, the Nusselt number shows a positive dependence on Grashof and Prandtl numbers irrespective of the type of fluid behavior, i.e., shear thinning or shear thickening. 4. Average Nusselt Number. Though the local Nusselt number distributions on the surface of the cylinder and temperature contours provide valuable insights, the average Nusselt number is what is often required in process design calculations. Figure 8 shows representative plots elucidating the influence of the Grashof number, Gr, Prandtl number, Pr, and flow behavior index, n on the area-averaged value of the Nusselt number. As expected, the Nusselt number is seen to be a weak function of the power-law index at low Rayleigh numbers which can safely be ascribed to weak advection under these conditions. However, as the value of the Rayleigh number is gradually increased and flow becomes stronger, the power-law index begins to exert an increasing influence on the value of the Nusselt number. Broadly, all else being equal, there is an enhancement in the rate of heat transfer in shear-thinning fluids as compared to that in Newtonian fluids while it deteriorates somewhat in shear-thickening fluids. This is solely due to the shear dependence of the effective viscosity of the fluid. It is reduced for shear-thinning fluids, whereas it rises in shear-thickening fluids with the rate of fluid deformation caused by the buoyancy-induced flow. While the actual extent of enhancement (or deterioration) in the rate of heat transfer is strongly influenced by the values of the Grashof number, Prandtl number, and power-law index, enhancements on the order of 80-90% are possible under appropriate conditions in shear-thinning fluids. Broadly, the higher the value of the Grashof number or Prandtl number or both, the greater the enhancement in heat transfer. Finally, the present results have been correlated using the following simple form h i0:89 ð16Þ Nu ¼ 1:19 Gr 1=2ðn þ 1Þ Pr ðn=ð3n þ 1ÞÞ

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Figure 9. Parity plot between the present results and predictions of eq 16.

Equation 16 correlates 80 numerical data points with an average error of 6.8%, which reaches the maximum value of 21%. Figure 9 shows the parity plot between the actual numerical values and the predictions of eq 16. 5. Drag Coefficient. Figure 10 shows the functional dependence of the drag coefficient on the Grashof number, Prandtl number, and power-law index. The drag coefficient shows an inverse dependence on the Grashof and Prandtl numbers, which is similar to its dependence on the Reynolds number in the case of imposed flow. The effect of the power-law index on the drag coefficient is evident only at high Rayleigh numbers (when there is a significant velocity), and it is more pronounced in the case of shear-thickening fluids than that in shear-thinning fluids. This is presumably due to the differences arising from the scaling of the viscous and buoyancy effects on the characteristic velocity and power-law index. In essence, at low values of the Grashof and/or Prandtl numbers or both, the resulting velocity gradients are not very steep and therefore the resulting stress levels are low. The effect of the power-law index gets accentuated by the thinning of the boundary layer and sharpening of the velocity gradient at high values of Gr, Pr, or both. This, in turn, leads to higher values of drag as seen in Figure 10. 6. Comparison with Other Studies. It is also worthwhile to contrast the present results with the previously available scant analytical and experimental results. As noted earlier, Acrivos32 presented a boundary layer approximation which is applicable at large values of Grashof number and/or Prandtl number. Table 6 shows a typical comparison for two values of powerlaw index. Excellent correspondence is seen to exist for Gr(1/(2nþ1))Pr(n/(3nþ1)) > ∼10, as asserted by Acrivos.32 Below this value, the two results begin to veer away from each other, at least for non-Newtonian fluids. It is useful to recall here that the Grashof number is proportional to (effective viscosity)-2, and therefore, the Grashof number is likely to be small in situations involving viscous non-Newtonian fluids. Of course, some of this effect will be countered, to some extent, by the corresponding high value of the Prandtl number which is proportional to viscosity. Therefore, the present results for low Grashof numbers indeed supplement the high Grashof number predictions of 2436



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Figure 10. Dependence of drag coefficient on Grashof number, Prandtl number, and power-law index.

Table 6. Comparison of Present Values of Nusselt Number with That of Acrivos32 Nu n 1.5

1.0

(1/2(nþ1))

Gr

(n/(3nþ1))

32

Unfortunately, they have not provided sufficient details for their results to be recalculated in the form required here, and thus, it is not possible to present detailed comparisons with their results. However, suffice to say here that their results are within (10% of the predictions of eq 16 which is regarded to be satisfactory. Finally, Ng and Hartnett42,44 rearranged the results of Acrivos32 in the following form

Pr

Acrivos

1.45

1.30

5.56

5.01

5.91

13.98

12.58

13.58

29.06

26.16

26.80

where the modified Rayleigh number, RaN, is defined as follows

1.64 5.18

1.38 4.35

1.79 4.30

16.38

13.76

11.86

47.29

39.72

41.23

ðFgβΔTÞD2nþ1 ð18Þ mRn Note that the three definitions of the Rayleigh number, namely, Ra, RaD, and RaN, used here and in the literature only differ through a function of power-law index, n. Ng and Hartnett42,44 further argued that eq 17 correctly predicts the limiting case of Nu Ra(1/4) for Newtonian fluids. Since the present results are in fair agreement with the boundary layer analysis of Acrivos32 at high values of Grashof and/or Prandtl numbers, it suggests these to be consistent with eq 17 also.

present 1.67

Acrivos.32 Gentry and Wollersheim40 reported free convection heat transfer results for a horizontal cylinder immersed in carbopal solutions which exhibited mild shear-thinning behavior. They found their results to be in line with the values for Newtonian fluids, though their results deviate increasingly from the line as the value of the Rayleigh number is gradually decreased.

1=3n þ 1

Nu ¼ CRaN

ð17Þ

RaN ¼

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Table 7. Comparison of the Present Values of Nusselt Number with the Predictions of eq 19 in Newtonian Media Nu Pr 0.72

7

20

50

100

RaD

present

eq 19

57.6

1.79

1.29

5.76 103

4.30

4.07

5.76 105 560

11.86 3.13

12.86 2.53

5.6 104

8.24

7.99

5.6 106

24.10

25.3

1.6 103

4.03

3.45

1.6 105

10.90

10.91

1.6 107

32.30

34.50

4 103

5.00

4.53

4 105 4 107

13.80 41.23

14.30 45.30

8 103

5.90

5.56

8 105

16.43

17.60

Finally, the present results are compared with the experimental data on the Nusselt number in Newtonian fluids. Using scores of silicone oils of different viscosities, air, and water, Fand et al.14 presented the following expression applicable over wide ranges of conditions as 0.7 e Pr e 3090 and 250 e RaD e 1.8 107 0:047 Nu ¼ 0:474Ra0:25 D Pr

ð19Þ

Table 7 shows a comparison between the present numerical results and the predictions of eq 19. A detailed examination reveals that barring the two cases for RaD = 57.6 and 560, the two values are typically within 10% of each other, which is also on the order of the uncertainty associated with eq 19. The differences of this order are not at all uncommon and mainly arise from the variation of the physical properties with temperature. For instance, the value of ΔT ranges from ∼1 to 110 C in the experiments of Fand et al.,14 whereas it was typically 5-10 C in the present work. Besides the first data point in Table 7 is clearly outside the range of validity in eq 19. Therefore, all in all, the correspondence seen in here is regarded to be satisfactory and acceptable. Before leaving this section, it is also useful to draw an analogy between the role of shear-thinning or shear-thickening behavior, on one hand, and that of the direction of temperature difference, on the other,34 i.e., whether T¥ < Tw or T¥ > Tw. For a shearthinning fluid, the effective viscosity of the fluid will be lowest close to the cylinder (because the rate of deformation is a maximum here) and will progressively increase away from the cylinder. This is equivalent to the case of a heated cylinder losing heat to a cold quiescent medium, i.e., Tw > T¥ for Newtonian fluids as the viscosity will be minimum adjacent to the surface of the cylinder (maximum temperature). One can similarly demonstrate the corresponding analogy for a cold cylinder receiving heat from the hot ambient fluid, i.e., Tw < T¥ for a shear-thickening fluid. Some of these ideas are well supported by the limited numerical results63 wherein the effect of temperature-dependent viscosity on heat transfer from a cylinder in Newtonian fluids has been studied.

’ CONCLUSIONS In this work, the effect of Grashof number, Prandtl number, and power-law index has been studied on the rate of heat transfer

from a heated horizontal cylinder immersed in stagnant powerlaw fluids for wide ranges of conditions: 0.3 e n e 1.8, 10 e Gr e 105, and 0.72 e Pr e 100. Extensive results on the streamline and isotherm contours and local Nusselt number are reported along with the dependence of the average Nusselt number and drag coefficient on Grashof number, Prandtl number, and power-law index. The present results are found to be in good agreement with the previous approximate analytical studies available in the literature for a high value of Grashof and Prandtl numbers when the boundary layer is very thin and curvature effects are negligible. Broadly, shear-thinning fluid behavior enhances the rate of heat transfer, while the shear-thickening behavior has deleterious effect on it. Indeed, under appropriate conditions, it is possible to realize augmentation in the rate of heat transfer by up to 70-80% in shear-thinning fluids. This can certainly result in significant energy savings in process applications. The present predictions are also in line with the scant experimental data available in the literature on free convective heat transfer from a horizontal cylinder in mildly shearthinning fluids and Newtonian fluids at high Prandtl numbers.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: 0091512 2597393. Fax: 0091512 2590104. E-mail: chhabra@ iitk.ac.in.

’ NOMENCLATURE CD = drag coefficient, CD = FD/((1/2)(FU2c D)), dimensionless CDF = frictional component of drag coefficient, CDF = FDF/((1/2)(FU2c D)), dimensionless CDP = pressure component of drag coefficient, CDP = FDP/((1/2)(FU2c D)), dimensionless Cp = pressure coefficient, dimensionless D = diameter of cylinder, m D¥ = diameter of the outer domain, m FD = drag force per unit length of cylinder, N 3 m-1 FDF = frictional component of drag force per unit length of cylinder, N 3 m-1 FDP = pressure component of drag force per unit length of cylinder, N 3 m-1 g = acceleration due to gravity, m 3 s-2 Gr = Grashof number, dimensionless GrD = Grashof number based on diameter, dimensionless h = heat transfer coefficient, W 3 m-2 3 K-1 I2 = second invariant of the rate of the strain tensor, s-2 k = thermal conductivity of fluid, W 3 m-1 3 K-1 m = power-law consistency index, Pa 3 sn n = power-law index, dimensionless Nu = average Nusselt number, dimensionless Nuθ = local Nusselt number, dimensionless Ρ = pressure, Pa Pr = Prandtl number, dimensionless PrD = Prandtl number based on diameter, dimensionless r = radial distance, r = (x2 þ y2)1/2, m R = radius of cylinder, m Ra = Rayleigh number (= GrPr), dimensionless RaD = Rayleigh number based on diameter, dimensionless RaN = modified definition of the Rayleigh number, eq 18, dimensionless T = temperature of fluid, K Tw = cylinder surface temperature, K T¥ = ambient fluid temperature, K 2438


Industrial & Engineering Chemistry Research Uc = characteristic velocity, m 3 s-1 Ux,Uy = x and y components of the velocity, m 3 s-1 x,y = Cartesian coordinates, m

’ GREEK SYMBOLS R = thermal diffusivity, m2 3 s-1 β = coefficient of volumetric expansion, K-1 ε = component of the rate of the strain tensor, s-1 η = viscosity, Pa 3 s θ = position on the surface of the cylinder, degree ξ = nondimensional temperature, ξ = (T - T¥)/(Tw - T¥) F = density of the fluid, kg 3 m-3 τ = extra stress tensor, Pa ’ SUBSCRIPTS ¥ = ambient condition w = cylinder surface condition ’ REFERENCES (1) Chhabra, R. P.; Richardson, J. F. Non-Newtonian Flow and Applied Rheology, 2nd ed.; Butterworth-Heinemann: Oxford, 2008. (2) Berk, Z. Food Process Engineering and Technology; Academic Press: New York, 2008. (3) Chanes, J. W.; Velez-Ruiz, J. F. Transport Phenomena in Food Processing; CRC Press: Boca Raton, 2003. (4) Rohsenow, W. M.; Hartnett, J. P.; Cho, Y. I. Handbook of Heat Transfer; 3rd ed.; McGraw Hill: New York, 1998. (5) Ramaswamy, H. S.; Zareifard, M. R., Dimensionless Correlations for Forced Convection Heat Transfer to Spherical Particles under Tube Flow Heating Conditions. In Transport Phenomena in Food Processing; Chanes, J. W., Velez-Ruiz, J. F., Eds.; CRC Press: Boca Raton, FL, 2003; Chapter 32. (6) Narain, J. P. Free and Forced Convective Heat Transfer from Slender Cylinders. Lett. Heat Mass Transfer 1976, 3, 21–30. (7) Yao, L. S.; Catton, I.; McDonough, J. M. Free-Forced Convection from a Heated Longitudinal Horizontal Cylinder. Num. Heat Transfer 1978, 1, 255–266. (8) Farouk, B.; Guceri, S. I. Natural and Mixed Convection Heat transfer around a Horizontal Cylinder within Confining Walls. Num. Heat Transfer 1982, 5, 329–341. (9) Merk, H. J.; Prins, J. A. Thermal Convection in Laminar Boundary Layers, Parts I, II and III. Appl. Sci. Res. 1953-54, 4, 11–24.Appl. Sci. Res. 1953-54, 4, 195-206. Appl. Sci. Res. 1953-54, 4, 207-224. (10) Peterka, J. A.; Richardson, P. D. Natural Convection from a Horizontal Cylinder at Moderate Grashof numbers. Int. J. Heat Mass Transfer 1969, 12, 749–752. (11) Kuehn, T. N.; Goldstein, R. J. Numerical Solution to the Navier-Stokes Equations for Laminar Natural Convection about a Horizontal Isothermal Circular Cylinder. Int. J. Heat Mass Transfer 1980, 23, 971–979. (12) Wilks, G. External Natural Convection about Two-Dimensional Bodies with Constant Heat Flux. Int. J. Heat Mass Transfer 1972, 15, 351–354. (13) Churchill, S. W.; Chu, H. H. S. Correlating Equations for Laminar and Turbulent free Convection from a Horizontal Cylinder. Int. J. Heat Mass Transfer 1975, 18, 1049–1053. (14) Fand, R. M.; Morris, E. W.; Lum, M. Natural Convection from Horizontal Cylinders to Air, Water and Silicone Oils for Rayleigh numbers between 300 and 2 107. Int. J. Heat Mass Transfer 1977, 20, 1173–1184. (15) Kitamura, K.; Kami-iwa, F.; Misumi, T. Heat Transfer and Fluid Flow of Natural Convection around Large Horizontal Cylinders. Int. J. Heat Mass Transfer 1999, 42, 4093–4106. (16) Fujii, T.; Takeuchi, M.; Fujii, M.; Suzaki, K.; Uehara, H. Experiments on Natural Convection Heat Transfer from the Outer

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Laminar Natural Convection from a Horizontal Cylinder in Power-Law

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