Natural Convection Heat Transfer of Non-Newtonian Power-Law Fluids from a Pair of Two Attached Horizontal Cylinders

Transcription

1 Natural Convection Heat Transfer of Non-Newtonian Power-Law Fluids from a Pair of Two Attached Horizontal Cylinders Subhasisa Rath *1, Sukanta K. Dash 2 * 1 School of Energy Science & Engineering, Indian Institute of Technology Kharagpur, India Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, India Abstract: Laminar natural convection of unconfined Power-law fluids from two attached horizontal cylinders has been investigated numerically to elucidate the non-newtonian behavior of the fluid over a wide range of pertinent kinetics: Grashof number (10 Gr 10 ), Prandtl number (0.71 Pr 100), and Power-law index (0.2 n 1.8). The heat transfer characteristics are elucidated in terms of isotherms, local Nusselt number (Nu) distributions and average Nusselt number values whereas the flow characteristics are interpreted in terms of streamlines, local distribution of the pressure drag and skinfriction drag coefficients along with the total drag coefficient values. Under identical conditions, the average Nusselt number shows a positive dependence on both Grashof and Prandtl number whereas it shows an adverse dependence on Power-law index. Overall, shear-thickening (n > 1) fluid behavior impedes the convection heat transfer whereas shear-thinning (n < 1) fluid behavior promotes it with reference to the generalized Newtonian fluids (n = 1). The heat transfer from an individual cylinder in attached condition has been degraded compared to the case of a single horizontal cylinder. Furthermore, the heat transfer rate of the top cylinder is reduced by 29-41% relative to that of the bottom cylinder due to preheating of the thermal plume. Keywords: Numerical heat transfer; Natural convection; Non-Newtonian fluid; Power-law fluid; Horizontal cylinder; Correlation 1. Introduction: The momentum and heat transfer characteristics in natural convection heat transfer have emerging potential applications within the domain of transport phenomena in both Newtonian and non-newtonian fluid mediums. Due to the temperature dependent buoyancy flow, the coupled momentum and energy equations are to be solved concurrently which brings analytical and computational challenges in natural convection. Owing to the pragmatic significance, over the last fifty years or so, extensive research has been done on free convection from heated cylinders of different shapes in confined as well as an unconfined Newtonian fluid medium like water and air [1-6]. In addition to the aforementioned research, 5 * Corresponding Author: address: subhasisa.rath@gmail.com (S. Rath)

2 in numerous process industries including food, agricultural, pharmaceutical, polymer, biomedical, biochemical, beverage etc., many structured fluids exhibit shear-thinning or shear-thickening non- Newtonian behavior under pertinent industrial settings. Heating and/or cooling of such fluids are commonly encountered by natural convection heat transfer. Typical examples of process engineering applications include thermal processing of food-stuffs, fruit yogurts, some packaged ready to eat foods, reheating of polymer melts or solutions, particulate suspensions containing pulps and paper fibers, chemical treatment in multiphase reactors including slurry reactors, packed and fluidized-bed reactors etc. [7-12]. Unlike Newtonian fluids, the apparent viscosity of a Power-law fluid exhibits a measurable variation with shear rate in the physical domain. Thus it is exemplary to mention here that the coupled momentum and heat transfer characteristics (or the coupled velocity and thermal fields) are further accentuated subject to non-newtonian fluids. Extensive research on natural convection from heated cylinders of different cross-sections like circular [13], semi-circular [14,15], square [16], tilted square [17], and elliptical [18] in an unconfined power-law medium has been investigated numerically in recent years. In the aforementioned literature [13-18] on Power-law fluids, the heat transfer characteristics were represented in terms of local Nusselt number distributions along the surface of the heated object and the average Nusselt number. Under identical conditions, it has been found that the shear-thinning behavior of the power-law fluid enhances the heat transfer whereas shear-thickening behaviour diminishes the heat transfer with reference to that of the generalized Newtonian fluids, albeit different cross-sections and/or orientations of the heated object has a significant influence on the rate of heat transfer. The preceding research on free convection is restricted to a single heated cylinder in Newtonian or non-newtonian fluid mediums. Undoubtedly, appreciable physical insights into the underlying physical phenomena have been found from such research on single cylinder over a wide range of Grashof and Prandtl numbers of industrial interests [19]. However, it has been readily acknowledged that free convection from various configurations of multiple cylinders has significant practical applications including compact heat exchangers, multi-pin extended surfaces, cooling of electronic equipment, etc. [20-23]. Experimental and numerical study on natural convection heat transfer from two vertically staggered cylinders were conducted by Heo et al. [24] and it has been reported that due to the presence of the bottom cylinder, the heat transfer from the top cylinder was reduced, exhibiting strong interaction and preheating of the thermal plume, which is commonly known as temperature difference imbalance in the literature. This effect of temperature difference imbalance has also been reported in the numerical investigation of Park and Chang [25]. Cianfrini et al. [26] numerically studied the steady-state natural 2

3 convection heat transfer from a heated cylinder in the presence of a downstream cylinder. Both numerical and experimental investigations on free convection from two horizontal cylinders arranged vertically have been performed by Chae and Chung [27]. It has been reported in their research that with an increase in pitch to diameter ratio, the ratio of Nusselt numbers of upper to lower cylinder (Nuu / Nul) gradually increases. Despite the arrangement of horizontal cylinders in a vertical array, a scant research has also been done for cylinders arranged in horizontal array [28-31]. It has been reported in the preceding investigations that owing to the development of chimney effect, the convection heat transfer increases with decrease in the horizontal spacing between the cylinders and corresponding to the optimum spacing, it reaches a maximum value. At the optimal spacing between the cylinders, the two thermal plumes just come in contact adjacent to the cylinders and with further decrease in the horizontal spacing, the two thermal plumes tend to merge and come out as a single plume resulting in a decrease in the rate of heat transfer. Notwithstanding the preceding investigations [22-31] in Newtonian fluids, subsequently, this physical mechanism of free convection from multiple cylinders has been extended to non-newtonian fluids and a scant research is now available in Power-law fluids. Shyam at al. [32] numerically studied the free convection from two vertically aligned circular cylinders in power-law fluids. In their study, the simulations were conducted for a wide range of pertinent kinetics for both shear-thinning and shearthickening Power-law fluid behaviors along with varying vertical spacing between the two cylinders in the range (2 (S/D) 20). Shyam and Chhabra [33] numerically investigated the laminar natural convection from two vertically arranged square cylinders at moderate Grashof number in the range (10 Gr 1000) in both shear-thinning as well as shear-thickening Power-law fluids (0.4 n 1.8). Simulations were conducted by varying the inter-cylinder spacing in the range, 2 L/d 6 to elucidate its effect on flow and thermal fields. To the best of our knowledge, no prior literature has been seen to be present on free convection from multiple attached-cylinders in non-newtonian fluids, albeit limited literature are available in Newtonian medium like air, [34-36]. Most recently, Rath and Dash [34] numerically studied both laminar and turbulent natural convection from different stacks of horizontal cylinders in air. Attached cylinders of three, six and ten in numbers having the same diameter were arranged in a triangular manner to form different stacks for the study of heat transfer. Liu et al. [35] numerically investigated the natural convection in laminar regime from two heated circular cylinders attached horizontally. Computational study on free convection from two vertically attached horizontal cylinders was done by Liu et al. [36]. 3

4 Owing to the interaction of the plumes and vortex formations, the heat transfer from individual cylinder was found to decrease compared to that of a single cylinder. The resulting heat transfer is attributed by two competing mechanisms: decreasing momentum of the flow due to preheating of the plume by the bottom cylinder (creates temperature difference imbalance) and the mixed convection behavior experienced at the top cylinder due to the plume developed by the bottom cylinder. A summary of the numerical study on natural convection from multiple horizontal cylinders is presented in Table 1. Table 1 Summary of the literature on natural convection from multiple horizontal cylinders (numerical study) Authors Working fluid No. of cylinders Arrangement Operating conditions Chae and Chung [27] Air 2 Vertical Ra ; 2014 Pr 3883;1.02 P / D 9 Park and Chang [25] Air 2 Vertical Ra 10 ;2 S/ D 4 Lu et al. [22] Molten Salts 2-8 Vertical Ra 510 ;2 S/ D 10 Persoons et al. [23] Water 2 Vertical Ra ; M. Corcione [28] Air 1-4 Vertical-Horizontal (Array) 2 S / D Ra 10 ;1.4 W / D 24 2 H / D 12 Stafford and Egan [30] Air 2 Horizontal & Ra 10 ;0 S/ D 12 Vertical Shyam at al. [33] Power-law 2 Vertical 4 2 S / D 20;10 Gr 10 ; (Square cylinders) 0.72 Pr 100;0.3 n 1.5 Shyam and Chhabra [32] Power-law 2 Vertical 3 10 Gr 10 ;0.71 Pr 50; 0.4 n 1.8;2 L / D 6 Liu et al. [35] Air 2 Horizontal 5 10 Ra 10 Liu et al. [36] Air 2 Vertical 5 10 Ra 10 Despite the previous investigations, the literature on non-newtonian fluids lacks a comprehensive study on free convection from different configurations of a pair of attached cylinders. Hence, the present study is motivated towards the computational study of laminar free convection from a pair of two attached horizontal cylinders arranged in both vertical and horizontal direction to address the effect of physical configurations and the role of power-law rheology on momentum and heat transfer characteristics. The applications of such geometrical configurations can be found in polymer processing industries where attached heated cylinders of different shapes are used to form weld lines through natural convection heat transfer. The reported results herein include the streamlines and isotherm contours in close proximity of 4

5 the attached cylinders, local and average values of Nusselt number, distribution of local pressure drag and friction drag coefficients along with the total drag coefficients values. Finally, the paper is concluded with the development of some Nusselt number correlations for individual cylinder as well as for the overall configurations of the attached cylinders, which can be beneficial to practicing engineers and academic researchers. 2. Problem Formulation 2.1. Physical description To perform the numerical investigations on external free convection from two attached horizontal heated cylinders, it is customary to immerse the cylinders in an extensive unconfined fluid domain of sufficiently large size. Fig. 1 shows the schematic representation of the physical problem, where a pair of two equi-diameter horizontal cylinders, attached together in two different configurations such as case- I (vertically attached) and case- II (horizontally attached) are placed in a two-dimensional computational domain. The cylinders are considered to be very long in the z-direction. Hence, 2D simulation supports it, as the variation of any field property would be negligible in the z-direction. Gravity is set up in the negative Y-direction. For the Cartesian coordinate system, the origin is set exactly where the two cylinders come in contact. The heated cylinders are maintained at a fixed temperature of Tw which are immersed in a surrounding quiescent non-newtonian power-law fluid maintained at temperature T (Tw > T ). The presence of temperature gradient leads to thermal expansion of the fluid in the domain and as a result, the fluid near the heated cylinders will be lighter compared to the fluid far away from the cylinders. Owing to the density gradient, an upward flow or buoyancy will set up adjacent to the cylinders which in turn, results in free convection of fluid in the domain. The objective is to assess the flow and thermal field around the attached cylinders and hence predict the Nusselt number and drag coefficients for it. 5

6 2.2. Mathematical Modeling Assumptions: The flow is assumed to be steady, laminar, buoyancy-driven, incompressible, twodimensional, and symmetric about the vertical centreline. The shear-dependent viscosity of the fluid has been captured by using the non-newtonian power-law model. The thermo-physical properties (except density) of the fluid are assumed to be independent of temperature and evaluated at mean film temperature (Tm). The temperature dependent density is taken into consideration by employing Boussinesq approximation as Eq. (1): T T (1) It is worthwhile to mention here that to justify the assumption of the use of constant thermo-physical properties at mean film temperature (Tm) for non-newtonian fluids, the maximum temperature difference (a) (b) Fig. 1. Schematic representation of the physical problem, (a) case- I (vertically attached cylinders) and (b) case- II (horizontally attached cylinders) T Tw T should never exceed 5K in the system [37-39]. The maximum temperature difference for all the simulations in the present study are set to T 1K which is quite reasonable to assume the thermal expansion coefficient as a constant value as expressed in Eq. (2): 1 1 T T P m (2) 6

7 Furthermore, in the present study, the radiative heat transfer is assumed to be negligible since temperature being very small and the viscous dissipation effect is also neglected owing to the maximum value of Brinkman number, Br << Governing Equations: With reference to the aforementioned suitable assumptions, the coupled flow and thermal fields can be written with the governing conservation equations (continuity, momentum, and energy) in their non-dimensional form as follows: Continuity: X- Momentum: Y- Momentum: Thermal Energy: U X X U Y Y 0 UXUX UXUY P 1 XX YX X Y X Gr X Y UYUX UYUY P 1 XY YY X Y Y Gr X Y 2 2 U U 1 X Y 1/(n1) 2 2 X Y Gr Pr X Y For an incompressible fluid, the deviatoric stress tensor ( deformation tensor ( ij ), as given by Bird at al. [40]: The rate of deformation tensor or strain rate tensor ( field [40] and can be written as: ij ij (3) (4) (5) (6) ij ) is directly associated with the rate of 2 (7) 1 U U i j ij 2 j i ij ) is in turn associated with the velocity gradient The non-newtonian viscosity for a power-law fluid is interlinked with the second invariant of the strain rate tensor as given by Bird at al [40], which in turn defined as: (8) Where the second invariant I 2 (n1)/2 I m 2 2 of the strain rate tensor is defined as: (9) I. (10) 2 ij ji i j 7

8 All the non-dimensional variables appearing in Eqs. (3-6) are defined as follows: x y u u p T T X,Y, U, U,P, L L u u T T x y X Y 2 c c uc w (11) Where, L is the characteristic length scale (defined as; L = D) and velocity (defined as: uc g TL ) used for the present simulation. u c is the characteristic reference In the present study, the apparent viscosity of a non-newtonian fluid is approximated by using the Ostwald de Waele power-law viscosity model [40] as follows: n1 m (12) Boundary Conditions: To solve the governing conservation Eqs. (3-6), some suitable physically realistic boundary conditions are imposed to the computational domain, which can be summarized in mathematical form as follows. On the cylinder surface: The surface of the cylinders are assigned as wall boundary with isothermal condition. No slip and no penetration boundary conditions are imposed on these walls, i.e.: U U 0 and 1 (13) X Y At the outer boundary: The outer surface of the flow domain is designated as pressure outlet boundary where pressure is set to be the atmospheric pressure (zero static gauge pressure). Pressure outlet boundary would allow the fluid to either go out of the domain or come into the domain depending upon the inside pressure condition in the domain. The temperature of any back-flow would be same as the surroundings quiescent fluid temperature, i.e.: P P 0 (gauge); 0 (back-flow temperature) (14) At the vertical axis( X 0 ) : Owing to the appearance of steady and symmetric flow about the central vertical axis (at X = 0) of the computational domain, the computations are conducted only with a half computational domain (X 0). Hence, a planar symmetry boundary is imposed at this vertical centreline Heat transfer and fluid parameters U X X Y UX 0, 0 and 0 The numerical solution represents the flow domain in terms of some field variables, i.e., velocity, temperature and pressure. Evidently, in the present study, the coupled momentum and thermal fields are (15) 8

9 influenced by three dimensionless parameters, i.e., Grashof number (Gr), Prandtl number (Pr) and powerlaw index (n) which are defined as follows for non-newtonian power-law fluids. Grashof number: 2 n2 2 n D gt Gr (16) 2 m 2 1n C 1 n 3 Prandtl number: n1 p m 1n Pr D gtd 2 n1 K Moreover, the power-law index, in its own right, is a non-dimensional parameter but owing to its appearance in both Grashof and Prandtl numbers, it increases the level of difficulties in particularising its role on the numerical results. In the present numerical study, the power-law index, Grashof number, and Prandtl number are considered for a wide range of pertinent industrial conditions such as; (0.2 n 1.8), 5 (10 Gr 10 ), and (0.71 Pr 100), respectively. Where, n < 1 indicates the shear-thinning fluid behavior, n > 1 indicates the shear-thickening fluid behavior and n = 1 corresponds to the generalized Newtonian fluid behavior. In the present study, the numerical results are summarized in 4 of this paper, in terms of streamline and isotherm contours, Nusselt numbers, and drag coefficients as a function of Grashof number, Prandtl number, and power-law index in the aforementioned ranges. Thus, it is prudent to describe some of these parameters here. Drag coefficient (Cd): It is associated with the total drag force, which is the resultant force on the cylinders in the direction of buoyancy due to the gradients of velocity appearing in the vicinity of the cylinders. The resulting drag force is the sum of the normal pressure force and shearing force. Hence, the total drag coefficient (Cd) has two components such as pressure drag coefficient (Cpr) and skin friction drag coefficient (Csf), which are due to pressure difference in the direction of flow and shear stress, respectively. Where, (17) Cd Cpr Csf (18) F C C n ds (19) d,pr pr 2 s pr, ˆ y 1/ 2ucD P P The local pressure drag coefficient: Cpr, 2 1/ 2u c and d,f sf 2 s sf, 1/ 2ucD (20) F C C n ˆ ds (21) 9

10 The local skin friction coefficient: C sf, 2 1/ 2 uc 1n 2 U Gr ˆn Nusselt number (Nu): The convective heat transfer is associated with the non-dimensional parameter called Nusselt number which is the ratio of convective to conductive heat transfer at the surface. The average Nusselt number is numerically calculated by taking the area weighted average or integrating the values of local Nusselt numbers over the cylinder surface as written below: Where the local Nusselt number value is calculated as follows: Nu wall (22) Nu Nu l ds (23) l s hd l (24) K nˆ s Overall Nusselt number (NuO): It is the average Nusselt number of the overall pair of attached cylinders as follows: 2 for case I : Nu Nu Nu / (25) O T B f or case II : Nu O Nu L Nu R (26) 3. Numerical Procedure The aforementioned governing Eqs. (3-6) along with the imposed boundary conditions (13-15) have been solved numerically by using a finite volume method (FVM) based algebraic multi-grid (AMG) solver of the commercial software package ANSYS-Fluent-v15. For coupling the pressure-velocity terms, the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm has been implemented since it was found to be the most stable one. PRESTO (PREssure STaggering Option) scheme was employed for pressure discretization which calculates the pressure at the face center and avoids the interpolation errors and hence, gives more accurate results. Second Order Upwind (SOU) scheme was applied for discretization of the convective terms. Furthermore, for a smooth convergence, the scaled residuals or the convergence criterions were set to a very low value (unlike as for Newtonian fluids) such as: for energy equation, it was set to and for continuity and momentum equations, it has been set to These criterion values are more adequate for the present study since the monitor of Nusselt number and drag coefficient was consistent up to five significant digits under these criterions. In addition, the non- Newtonian power-law model and Boussinesq approximation were used for capturing the shear-dependent viscosity and temperature dependent density, respectively. It is customary to mention here that the use of any specific values of thermo-physical properties has no importance since the results are reported here in 10

11 their non-dimensional form. The under-relaxation factors used for the present simulations are given in Table 2. Table 2 Under-relaxation factors: Pressure Density Body forces Momentum Energy Choice of computational domain and grid distributions: To conduct the numerical simulations for a purely unconfined flow problem is quite impossible. Hence, a sufficiently large cylindrical fluid domain of diameter D has been considered for the present study as shown in Fig.1. The flow is expected to be steady and symmetric flow about the vertical centreline (X = 0) of the computational domain. Hence, symmetry condition has been taken in the present simulations and computation has been performed for half computational domain only (X 0) as shown in Fig. 2(a) which would minimize the computational cells and resources to a great extent. It is a common intuition that near the heated cylinders the temperature and velocity gradients will vary appreciably whereas, the gradient of field properties will vary moderately at far away from the cylinders. Hence, the variation of temperature and velocity distributions were captured by providing very fine cells over the cylinders using inflation technique (first layer thickness = 0.01D and growth rate = 1.2) for first 20 cells from the cylinder wall. In contrast, it is just an unnecessary computational effort to use fine mesh in a region far away from the cylinders. Keeping this in mind, the computational domain is split into two regions. The Region-I (about 10D from the origin) where relatively moderate size unstructured quadrilateral pave cells are provided and Region-II (rest of the domain) where relatively courser quadrilateral mapped cells are provided as shown in Fig. 2(b) & 2(c) Domain independence test: The size of the computational domain would affect the numerical results to some extent. Hence, a domain-independent study is carried out such that it would not affect the flow field and also keep the requisite computational resources at an optimum level at the same time. For the present study, domainindependent test was carried out by varying the domain size, D in the range spanning from 50D to 800D at a lowest value of Prandtl number (Pr = 0.71) where the boundary layers are expected to be quite thick and at two extreme values of power-law index (n = 0.2 and n = 1.8) as shown in Table 3 for case- I (vertically attached cylinders). Domain-independent study has been done for individual Grashof numbers 11

12 and the domain size is fixed to 600D, 400D and 200D for Gr = 10, Gr = 10 3 and Gr = 10 5, respectively which are found to be sufficient for the present study since, the relative variation in average Nusselt number beyond this size were found to be less than 1%. Domain independence test for case- II (horizontally attached cylinders) has also been carried out with the same preceding conditions as in case- I and exactly same aforementioned domain sizes were found for case- II. (a) (b) (c) Fig. 2. Schematic representation of grid distributions in half computational domain; (a) overall grid structure, (b) blown-up view near the cylinders for case- I, and (c) blown-up view near the cylinders for case- II Table 3 Domain independence test (half domain): Variation of overall average Nusselt number with domain size for case- I Domain, D Gr = 10, Pr = 0.71 Gr = 10 3, Pr = 0.71 Gr = 10 5, Pr = 0.71 n = 0.2 n = 1.8 n = 0.2 n = 1.8 n = 0.2 n = D D D D D D Grid independence test: Grid independence study has also been carried out to ensure the numerical results remain to be independent of the computational cell size. In the present study, grid independence test is carried out by 12

13 using boundary adaption technique at the highest Prandtl number (Pr = 100) where the boundary layers are expected to be quite thin over the cylinders and at two extreme values of power-law index (n = 0.2 and n = 1.8) as shown in Table 4 for case- I (vertically attached cylinders). After getting a converged solution using the initial mesh (no adaption 3.1 ), successive boundary adaption of 10 number of cells has been implemented on the cylinder walls and symmetry boundaries. The schematics of successive adaptions are shown in Fig. 3. From Table 4 it can be seen that the second adaption for Gr = 10 and third adaption for Gr = 10 5 are found to be grid independent for the present computations since the relative variation in average Nusselt number with its succeeding adaption was less than 1%. Hence, second adaption for all the simulations under Gr = 10 and third adaption for all the simulations under Gr = 10 3 and Gr = 10 5 are carried out for both case- I and II before reporting any result. (a) (b) (c) (d) Fig. 3. Boundary adaption grids over the cylinder surface; (a) no-adaption, (b) 1 st adaption, (c) 2 nd adaption, and (d) 3 rd adaption Table 4 Grid independence test (half-domain): Variation of overall average Nusselt number with grid size for case- I Gr = 10, Pr = 100 Gr = 10 5, Pr = 100 Boundary No. of n = 0.2 n = 1.8 No. of n = 0.2 n = 1.8 N Adaption P N P cells Nu Nu cells Nu Nu No st nd rd th

14 4. Results and Discussions In the present study, numerical simulations of natural convection from two attached horizontal cylinders in non-newtonian power-law fluids have been conducted over a wide range of pertinent conditions: 5 10 Gr 10, 0.71 Pr 100 and 0.2 n 1.8. Overall 270 numerical simulations have been performed to elucidate both momentum and heat transfer characteristics around the two attached cylinders. The choice of the above-mentioned ranges of parameters are not so arbitrary, these are governed by some practical behavior of real fluids as well as numerical limitations. The range of power-law index, n has been taken in such a way that it exhibits both shear-thinning and shear-thickening fluid behavior since the values of power-law index for most of the polymeric melts and solutions are as low as 0.5 to 0.3 and for most of the starch and corn-flour solutions are as high as 1.5 to 1.7 [7,8]. Most of the power-law fluids are found to be more viscous compared to any Newtonian fluid. Hence, the extreme value of Grashof number is expected to be low (about 10 5 ) for non-newtonian power-law fluids compared to its counterpart Newtonian fluids like air and water (laminar up to Gr = 10 7 or 10 8 ). Furthermore, the Prandtl number for most of the realistic industrial fluids has been found to be as large as 500 or even more [8,12]. However, with a high value of Prandtl number, the temperature and velocity gradients are expected to be extremely thin and ultra-fine computational grids are required to resolve any gradients of property within this boundary layer which demand expensive computational resources. With these aforementioned considerations, the flow characteristics are elucidated in terms of streamlines and total drag coefficients along with local pressure drag and skin-friction drag coefficient distribution over the cylinders whereas the heat transfer characteristics are described by isotherm contours, average Nusselt number, and local Nusselt number distribution over the cylinders. However, before reporting the present numerical results, it is customary to validate the accuracy and reliability of the implemented numerical methodology with prior literature Validation of the numerical methodology The present numerical methodology has been validated with the numerical results of Prhashanna and Chhabra [13] for an unconfined natural convection heat transfer from a single horizontal cylinder in non-newtonian power-law fluids as shown in Fig. 4. The local Nusselt number distribution over the cylinder is shown in Fig. 4(a) at Gr =10 3 and n = 1.8 for two different Prandtl number values of 7 and 50. Furthermore, the average surface Nusselt numbers with respect to the power-law index, n are shown in Fig. 4(b) and 4(c) for Gr = 10 and 10 3 respectively, for three different Pr of 7, 20,and 50. It can be seen 14

15 from Fig. 4 that the present simulation results for a single cylinder show a very good agreement with the numerical results of Prhashanna and Chhabra [13]. In addition, the present numerical results for a pair of attached horizontal cylinders has been validated with the numerical work of Liu at al. [35,36] in air for Pr=0.71 and n=1. Table 5 shows the comparison of an overall average Nusselt number between the present work and the numerical work of Liu at al. [35,36] for both vertically as well as horizontally attached cylinders where the relative errors were found to be less than 6%. Hence, it can be concluded here that the numerical methodology used for the present simulations is accurate and reliable and eventually may give us satisfactory results for natural convection heat transfer in non-newtonian power-law fluids. (a) (b) (c) Fig. 4. Comparison of local and average Nusselt numbers of a single cylinder with the numerical results of Prhashanna & Chhabra [13], (a) distribution of local Nusselt number over the cylinder surface, (b) average Nusselt number as a function of power-law index at Gr = 10 and (c) average Nusselt number as a function of power-law index (n) at Gr =

16 Table 5 Comparison of the overall average Nusselt number of the present work with the numerical work of Liu at al. [35,36] for attached cylinders in air (Pr = 0.71 and n = 1) Ra Vertically attached Horizontally attached Liu at al. [36] Present work Liu at al. [35] Present work Streamline and isotherm contours The thermal and flow fields are visualized in terms of streamlines and isotherm contours in the close proximity of the pair of attached heated cylinders. Fig. 5 and 6 shows the representative normalized isotherm (left half) and streamline patterns (right half) for some combination of Gr = 10, 10 3 & 10 5, Pr = 0.71 & 100 and n = 0.2 (shear-thinning) & 1.8 (shear-thickening) for case- I (vertically attached cylinders) and case- II (horizontally attached cylinders), respectively. Newtonian behavior (n = 1) of the fluid is not presented here for the sake of brevity. Owing to the thermal expansion or density gradient, the fluid is dragged towards the heated cylinders and a buoyancy-driven flow is set up thereby resulting in the formation of a plume adjacent to and above the cylinders. The fluid inertia or the strength of the buoyancy is so weak at low Grashof number (Gr = 10) that the buoyancy flow remains attached to the surface of the cylinders. In addition, at low Grashof number, flow due to buoyancy levitates from beneath of the attached cylinders and flows vertically upward by sliding over the cylinder surfaces whereas, the strength of the buoyancy increases with increase in the Grashof number and a significant amount of fluid is drawn from the sides of the attached cylinders. Irrespective of the power-law index, the momentum and thermal boundary layers are quite thick at a low value of Grashof number and/or Prandtl number and gradually becoming thinner with increase in the strength of the buoyancy or with an increase in Grashof number or Prandtl number. This increasing strength of the buoyancy enhances the convection owing to a thinner thermal boundary layer. Furthermore, it can be anticipated from the preceding discussion that Nusselt number would show a positive dependence on both Grashof number and Prandtl number, albeit this dependence of heat transfer is more prominent on Grashof number compared to Prandtl number, as evident in Fig. 5 and 6. 16

17 n = 1.8 Pr = 100 Pr = 100 Pr = 0.71 Pr = 0.71 Gr = 10 Gr = 10 3 Gr =10 5 n = 1.8 n = 0.2 n = 0.2 (a) (b) (c) (d) Fig. 5. Normalized isotherms (left half) & streamlines (right half) for case- I (vertically attached cylinders) 17

18 n = 1.8 Pr = 100 Pr = 100 Pr = 0.71 Pr = 0.71 Gr = 10 Gr = 10 3 Gr =10 5 n = 1.8 n = 0.2 n = 0.2 (a) (b) (c) (d) Fig. 6. Normalized isotherms (left half) & streamlines (right half) for case- II (horizontally attached cylinders) 18

19 Qualitatively, in the underlying physics of natural convection, the maximum value of the effective shear-rate is expected near the isothermal cylinders which gradually diminishes away from the cylinders and vanishes to zero where the power-law fluid is almost stationary. Thus, the shear-dependent viscosity is minimum near the cylinders for a shear-thinning fluid which progressively increases and reaches to a maximum value away from the cylinders. Thus, an imaginary cavity like fluid structure is formed, which is virtually bounded by the surrounding highly viscous fluid. Hence, the shear-thinning fluid always tries to stabilize the flow and consequently eliminates the possibility of flow separations and vortex formations as much as possible. In contrast, a completely contradictory behavior can be seen in shear-thickening fluids. Furthermore, it is worthwhile to notice in Fig. 5 that for case-i (vertically attached cylinders), two recirculation zones are formed in both sides to the contact point of the two cylinders and the size of these vortices progressively reduces with increasing value of the Grashof number or Prandtl number and with decreasing value of the power-law index. The contact point of the two cylinders and the stagnation point or the flow separation point on the surface of the cylinders are accountable for the vortex formations. The precise justifications for this observation can be explained qualitatively as: with increase in Grashof number, more fluid entrains from the side and penetrate towards the contact point, hence the stagnation point shifts towards the contact point of the attached cylinders, the flow separation delays and the size of the vortices decreases. Owing to the thinning of boundary layers and increase in velocity gradient, the flow separation becomes slower and the size of the vortices reduces with an increase in Prandtl number. As discussed in the preceding paragraph, the shear-thinning (n < 1) fluid stabilizes the flow, hence with decrease in power-law index, the size of the vortices gradually decreases. In contrast, for case-ii (horizontally attached cylinders), two recirculation zones are developed at the top (back region) and below (front region) the contact point as shown in Fig. 6. Unlike the previous case (case-i), the size of these vortices gradually increases with increasing value of the Grashof number whereas it still follows the same behavior and the size progressively reduces with increase in Prandtl number, but the power-law index plays a vital role here for the size of the vortices. At low Grashof number (Gr = 10), the size of the vortices increases with decrease in power-law index whereas, at high Grashof number, it decreases with decrease in power-law index. The exact justifications can be mentioned herein as: at low Grashof number, fluid rises from beneath the two cylinders and the stagnation point of the front vortex moves towards the contact point. Hence, small size vortices are seen to form whereas, when Grashof number increases, more fluid entrains from the sides and the front stagnation point gradually 19

20 moves downstream and the back stagnation point gradually moves upstream. Hence, the flow separation advances and gradually the size of the vortices increases. At a low Grashof number (Gr = 10), the shearrate at the bottom of the cylinders is quite low which exhibits a little higher apparent viscosity for the shear-thinning fluids. This destabilizes the flow even as compared to the shear-thickening fluids in this particular case of the present study. Hence, at low Grashof number (Gr = 10), the size of the vortices increases with decrease in power-law index whereas, at high Grashof number, the effect of power-law index on the growth of the vortices is similar to that of the previous case (case-i). These streamline patterns are completely different from that of a single cylinder or two horizontal cylinders with spacing, where the vortex formations are rarely seen. Hence, the momentum and heat transfer characteristics are expected to be different prominently for the attached cylinders. Moreover, under identical conditions, the boundary layers are found to be marginally thin in shearthinning fluid behavior and thick for shear-thickening fluid behavior with reference to the generalized Newtonian fluids. Thus, one can anticipate an augmentation in heat transfer in shear-thinning fluids and a depletion in shear-thickening fluids except for a particular case (case-ii, Gr = 10) in the present study where shear-thinning fluid shows thick boundary layers and shear-thickening fluid shows thin boundary layers. Hence, a completely antithetical heat transfer characteristic can be expected for this case. Finally, it is worth to mention here that at a low value of Grashof number and Prandtl number (at low Rayleigh number), the mode of heat transfer is primarily by conduction and hence, the effect of the power-law index has a very negligible influence on the isotherms as shown in Fig. 5 and 6. In addition, the recirculation zones (vortices) have relatively less influence on the thermal field (isotherms) at low and moderate Grashof numbers whereas, at high Grashof number (Gr = 10 5 ), it may have a significant influence on the isotherms in some cases, as reported in the literature [16,41]. This effect is also found in the present study in Fig. 6(c) and 6(d) in case-ii (horizontally attached cylinders), where temperature inversions can be seen in the isotherm contours just above the contact point at Gr = 10 5 and Pr = 100. The effect of this trend would be discussed in the next section under local Nusselt number distributions Local Nusselt number distribution Notwithstanding the fact that the heated attached cylinders are maintained at an isothermal condition but the temperature gradient varies over the surface of the cylinders. Hence, it is customary to elucidate the local heat transfer characteristics in terms of distribution of the local Nusselt numbers over the attached horizontal cylinders, which in turn can provide further physical insights into the flow and thermal fields. Owing to the symmetry of the flow and thermal plume about the vertical centreline of the two attached 20

21 cylinders, the representative results of the local Nusselt number distribution are shown in Fig. 7 and 8 only for half of each cylinder in case-i (vertically attached) and only for the right cylinder in case-ii (horizontal cylinder), respectively annotating the combined effects over the range of conditions, Grashof number (10 Gr 10 5 ), Prandtl number (0.71 Pr 100) and Power-law index (n = 0.2, 1 and 1.8; manifesting shearthinning, Newtonian and shear-thickening fluid behavior, respectively). For a Newtonian fluid, the local value of the Nusselt number is solely governed by the temperature gradient normal to the surface of the cylinders since the viscosity of the fluid is constant throughout the computational domain. Conversely, for a non-newtonian Power-law fluid, in addition to the temperature gradient, the local value of the Nusselt number is also governed by the local apparent viscosity (which changes progressively with shear-rate) of the fluid on the surface of the cylinders where the local values of the Grashof number and Prandtl number are expected to be different from those at the far field conditions, owing to the appearance of the power-law index in the definition of both Grashof number and Prandtl number as given by Eqs. (16) and (17), respectively. In addition, the temperature gradient is further influenced by the relative local magnitude of the two causes such as the temperature difference between the cylinder surface and the neighboring fluid along with the velocity gradient at the surface of the cylinders. It can be seen from Fig. 7 and 8 that at a low value of Grashof number or Prandtl number, the local Nusselt number changes very moderately with the circumferential angle (θ) over the surface of the cylinders, owing to the poor advection or dominating effect of conduction. Irrespective of the power-law index, the value of the local Nusselt number increases progressively with increasing Grashof number and/or Prandtl number, attributed to the thinning of the thermal boundary layer, in turn, increases the temperature gradient which was already anticipated from the isotherm contours in Fig. 5 and 6 in the preceding section. The overall trends and the key findings from this analysis are summarized for case-i and case-ii separately as follows: Case-I (vertically attached cylinders): Irrespective of the power-law index, the local Nusselt number values of the top cylinder are found to be much lower compared to that of the bottom cylinder due to the interaction of their plumes and temperature imbalance or preheating of the fluid coming from the bottom cylinder as shown in Fig. 7. For Newtonian (n = 1) and shear-thickening (n > 1) fluids, the maximum value of the local Nusselt number is seen to be at the front stagnation point of the bottom cylinder (θ = 0 0 ), which monotonically decreases with the circumferential angle (θ) over the surface of the bottom cylinder to a zero value at approximately θ = , owing to gradual decrease in temperature gradient over the surface and remain zero in the 21

22 region θ = to its rear stagnation point at (θ = ). This may be attributed to the additional heating effect of the thermal plume by the top heated cylinder before reaching to the rear stagnation point (θ = ) such that the temperature gradient reduces significantly at this region. For the top cylinder, the value of local Nusselt number is seen to be zero in the region from its front stagnation point (θ = ) to around θ = , owing to the reduced temperature gradient as discussed above. Beyond this region, the local Nusselt number increases monotonically with θ (due to fluid being drawn from the side of the cylinder) and reaches to a maximum value at approximately θ = depending on Gr and Pr, while with further increase in θ, the local Nusselt number progressively decreases to a finite value at the rear stagnation point (θ = ). The rate of decrease in the local Nusselt number over the surface of the cylinder is more prominent in shear-thickening fluid compared to that of a Newtonian fluid, albeit they follow the same pattern. This can be attributed to an increase in the apparent viscosity of a shear-thickening fluid with increasing shear-rate, resulting in a thicker boundary layer over the cylinders [13]. In contrast, for shear-thinning (n < 1) fluids, the maximum value of the local Nusselt number is no longer seen at the front stagnation point (θ = 0 0 ), it has been shifted to approximately θ = downstream depending on Grashof number and/or Prandtl number as shown in Fig. 7, indistinguishable from that seen for a single cylinder [13]. This behavior can be explained qualitatively by considering the magnitudes of two competing mechanisms such as (i) decrease in the temperature gradient along the surface of the cylinders, which impedes the heat transfer and (ii) reduction in apparent viscosity with an increase in shear-rate over the cylinders, which enhances the heat transfer. However, beyond θ = , it follows the same pattern as Newtonian and shear-thickening fluid, though with little deviation in the rate of increase or decrease of the local values of Nusselt number, due to the lower viscosity of the fluid. It can also be noticed in Fig. 7 that the zero value region (θ = to ), where the value of local Nusselt number is zero, gradually decreases with increase in Grashof number, due to the delay of the flow separation or gradual decrease in the size of the vortices as explained in the preceding section for streamlines patterns. Overall, all else being equal, the value of the local Nusselt numbers are found to be higher in shear-thinning fluids than that of shear-thickening fluids with reference to that of a Newtonian fluid. 22

23 (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 7. Distribution of Local Nusselt number along the surface of the cylinders for case- I (vertically attached cylinders) Case-II (horizontally attached cylinders): Unlike the previous case, here all the fluid behaviors (shear-thinning, Newtonian and shearthickening) have almost followed the same pattern over the cylinder surface as seen in Fig. 8. The maximum value of local Nusselt number is seen to be in a region approximately θ = depending on Gr, Pr and n. From the starting point (θ = 0 0 or ), the local Nusselt number monotonically decreases over the cylinder surface to a zero value at approximately θ = , due to gradual decrease in temperature gradient over the surface and remain zero in the region from θ = to the contact 23

24 point of the two cylinder (θ = ). This may be attributed to the merging of two thermal plumes, leading to the formation of flow separation and vortex formations near the contact point, such that the temperature gradient reduces significantly at this region. Up to the contact point (θ = ), the value of Nusselt number is still found to be zero up to a region around θ = , owing to the formation of vortices at bottom of the contact point. Beyond this region, the value of local Nusselt number increases monotonically with θ, due to increase in thermal gradient with a favorable pressure gradient over the surface and attains the maximum value at approximately θ = It can be seen that with increase in Grashof number and/or Prandtl number, the location of maximum Nusselt number shifts downstream along the surface of the cylinder, owing to the downstream movement of the flow separation point. Furthermore, the rate of decrease or increase in the local Nusselt number is more prominent in shear-thickening fluid than that of a shear-thinning fluid, since in shear-thickening fluid, the effective viscosity increases with increase in shear-rate and vice-versa whereas completely opposite behaviour is seen for the shear-thinning fluid. Unlike the previous case, the zero value region (θ = to ) gradually increases with increase in Grashof number, due to early flow separation or gradual increase in the size of the vortices. Owing to the thick boundary layer for shear-thinning fluids compared to Newtonian and shear-thickening fluids at Gr = 10 as already been seen in Fig. 6 of 4.2, it is clear from Fig. 8 (a), (b) and (c) that the value of local Nusselt numbers are also marginally lower in shear-thinning fluids compared to Newtonian and shear-thickening fluids. In addition, the effect of temperature inversion as explained in the preceding section in isotherm contours can be clearly seen in Fig. 8 at high Grashof number of Due to a temperature inversion, the local value of Nusselt number suddenly changes from its regular pattern and marginally a higher value is found as expected from the isotherms. It can also be noticed that the effect of temperature inversion is prominent at high Grashof number (Gr = 10 5 ) and Prandtl number of 10 onwards (Pr 10) for shear-thinning fluids, Pr 20 for Newtonian fluids and Pr 50 for shear-thickening fluids. Apparently, in the present study, the local heat transfer characteristics are distinguishable from that of a single horizontal cylinder [13] or a pair of two cylinders in a vertical array with spacing [32]. Hence, it is fair to postulate from these conjectures that the global characteristics would also predict differently from those of a single or two cylinders with spacing, which is discussed in the next section as average surface Nusselt number. 24

25 (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 8. Distribution of Local Nusselt number along the surface of the cylinders for case- II (horizontally attached cylinders) 4.4. Average Nusselt number It is customary to figure out the global heat transfer characteristics in terms of average surface Nusselt number as expressed in Eq. (23), which has great importance in process engineering and industrial design calculations. The dependence of average surface Nusselt number on Grashof number, Prandtl number and power-law index are shown in Fig. 9 and 10 for case-i (vertically attached cylinders) and case-ii (horizontally attached cylinders), respectively. For case-i, the average Nusselt numbers are expected to be different for both top and bottom cylinder, hence these are shown separately in Fig. 9, whereas for case-ii, both left and right cylinder would show the same value of the average Nusselt number 25

26 as shown in Fig. 10. As anticipated from the local Nusselt number distribution, the average Nusselt numbers are found to be lower for the top cylinder compared to that of the bottom cylinder in case-i, due to preheating of the plume by the bottom cylinder. Irrespective of the power-law index, the average Nusselt number increases with increase in Grashof number and/or Prandtl number, owing to progressive thinning of the thermal boundary layer. Conversely, with increase in the power-law index from shear-thinning (n < 1) to shear-thickening (n > 1), the value of average Nusselt number progressively decreases except a particular case at Gr = 10 in case-ii (horizontally attached cylinders) as postulated from preceding discussions in 4.2 and 4.3. This is due to the slower reduction in apparent viscosity over the cylinder surface and a little higher viscosity at the beneath of the two horizontally attached cylinders, which leads to thickening of boundary layers. (a) (b) (c) (d) (e) (f) Fig. 9. Dependence of average Nusselt number on power-law index, Grashof number and Prandtl number for case- I (vertically attached cylinders) 26

27 (a) (b) (c) Fig. 10. Dependence of average Nusselt number on power-law index, Grashof number and Prandtl number for case- II (horizontally attached cylinders) Furthermore, at low Grashof number (Gr = 10), conductive heat transfer dominates over advection hence, the variation of average Nusselt number with power-law index is seen to be moderate. Overall, the average Nusselt number shows a positive dependence on both Grashof and Prandtl numbers, whereas it shows an adverse dependence on Power-law index A comparison with a single cylinder It is clear from the preceding discussions that compared to a single cylinder, the momentum and heat transfer characteristics for a pair of attached cylinders are completely different. Hence, in the present study, an attempt has been made to compare the present results of case-i (vertically attached cylinders) and case-ii (horizontally attached cylinders) with a single horizontal cylinder in unconfined power-law fluids under the same pertinent conditions. Fig. 11 shows the ratio of the overall Nusselt number of the combined cylinders to the average Nusselt number of a single cylinder as a function of power-law index, Prandtl number and Grashof number. Owing to the interaction of the plumes and formation of vortices near the contact point, overall Nusselt number for both case-i and case-ii are found to be lower than that of a single horizontal cylinder. Furthermore, under identical conditions, the overall Nusselt number for case-i is higher than case-ii. This may be due to the fact that for vertically attached cylinders, the top cylinder is experienced a mixed convection effect by the presence of the bottom cylinder, which accelerate the advection and as a result, the heat transfer from the top cylinder increases significantly. This effect is completely absent in case-ii, where the bottom of the two horizontally attached cylinders experiences a poor movement of the fluid due to lower favorable pressure gradient. It can be seen that with increase in Grashof number, the value of the overall Nusselt number increases for both the cases compared to that of 27