Natural convection heat transfer from two vertically aligned circular cylinders in power-law fluids

International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Natural convection heat transfer from two vertically aligned circular cylinders in power-law fluids Radhe Shyam, C. Sasmal 1, R.P. Chhabra ⇑ Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India

a r t i c l e

i n f o

Article history: Received 11 March 2013 Received in revised form 30 April 2013 Accepted 19 May 2013 Available online 15 June 2013 Keywords: Free convection Pair of cylinders Power-law Nusselt number Grashof number Drag

a b s t r a c t In this work, laminar natural convection in power-law liquids from a pair of vertically aligned heated cylinders has been studied numerically over the following ranges of conditions: Prandtl number (0:72 6 Pr 6 102 ), Grashof number (10 6 Gr 6 104 ), power-law index (0:3 6 n 6 1:5) thereby embracing both shear-thinning and shear-thickening fluid behaviour and the center-to-center gap between the two cylinders was varied over the range (2 6 ðS=DÞ 6 20). The heat transfer characteristics of the lower cylinder are little influenced by the presence of the upper heated cylinder in line with the previous studies. On the other hand, the average Nusselt number for the upper cylinder could be higher or lower than that of the lower cylinder depending upon the value of (S/D), Grashof number and power-law index. Overall, all else being equal, shear-thinning behaviour can enhance heat transfer by up to 100% whereas shearthickening has adverse influence on the rate of heat transfer. The present results on Nusselt number and drag coefficient have been correlated using simple analytical forms which permit the estimation of these parameters for the two cylinders in a new application with acceptable levels of accuracy. In addition to the overall heat transfer characteristics, the detailed structure of the velocity and temperature fields is visualised in terms of the streamline and isotherm contours over the preceding ranges of parameters. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Due to its fundamental and pragmatic significance, much has been written about natural convection heat transfer from a circular cylinder in Newtonian fluids like air and water over the past fifty years or so. In free convection, since heat is transported solely by the buoyancy induced flow, the orientation of the cylinder with respect to the direction of gravity and/or the direction of the temperature difference (i.e., whether the cylinder acts as a source or sink) exert varying levels of influence on the overall rate of heat transfer. Thus, for instance, all else being equal, the Nusselt number for a circular cylinder can vary significantly depending upon its orientation, namely, whether it is horizontal, vertical or inclined. Additional complications arise from confinement, viscous-dissipation, temperature dependent viscosity, etc., e.g., see [1]. Suffice it to say here that based on a combination of approximate (boundary layer or asymptotic) treatments, numerical solutions of the governing equations and experimental results, it is now possible to estimate the value of Nusselt number for an unconfined circular ⇑ Corresponding author. Tel.: +91 512 2597393; fax: +91 512 2590104. E-mail address: [email protected] (R.P. Chhabra). Present address: Department of Chemical Engineering, Monash University, Clayton, Australia. 1

0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.05.052

cylinder submerged in quiescent Newtonian media in a new application over wide ranges of Grashof and Prandtl numbers of practical interest [1–5]. Undoubtedly, such single cylinder studies have provided useful insights into the nature of the underlying physical mechanisms, it is readily acknowledged that in most practical applications multiple cylinders arranged in various configurations such as in-line, staggered, vertical or inclined arrays of cylinders are encountered. Typical examples include the cooling of electronic components, pin and compact heat exchangers, etc. [6–8]. In such cases, both the proximity of the other cylinders and/or the confining boundaries influence the velocity and temperature fields around the test cylinder thereby altering the resulting values of the Nusselt number for each cylinder and that of the overall system. Therefore, the next generation of studies in this field have attempted to elucidate the role of interference on heat transfer by considering the case of two or more identical and/or different sizes of heated cylinders arranged in a vertical configuration with varying gap between the two-cylinders. Intuitively, it appears that the Nusselt number for the lower cylinder should be close to that of a single cylinder, as the upper cylinder exerts no influence on the buoyancy-influenced flow around the lower-cylinder except in the case of very small clearance between the two cylinders where the streamlines and isotherm contours interfere with each other even in the weak advection limit. In contrast, the Nusselt number

1128

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Nomenclature C CD C DO CDF CDP CP D D1 FD FDF FDP g Gr h I2 k m n ns nx ny Nul Nu NuO N NP P p Pr R

thermal capacity of fluid (J/kg K) drag coefficient, C d ¼ 2F d =q1 U 2c R (dimensionless) overall drag coefficient of the array, C DO ¼ ðC Du þ C DL Þ=2 (dimensionless) friction component of drag coefficient, C DF ¼ 2F DF =q1 U 2c R (dimensionless) pressure component of drag coefficient, C DP ¼ 2F DP =q1 U 2c R (dimensionless) pressure coefficient (dimensionless) diameter of cylinder (m) diameter of computational domain, m drag force per unit length of the cylinder (N/m) viscous drag force per unit length of cylinder (N/m) pressure drag force per unit length of the cylinder (N/m) acceleration due to gravity (m/s2) Grashof number (dimensionless) heat transfer coefficient (W/m2 K) second invariant of the rate of the strain tensor (s2) thermal conductivity of fluid (W/m K) power-law consistency index (Pa sn) power-law index (dimensionless) unit normal vector on the surface of cylinder (dimensionless) x-component of the unit normal vector (dimensionless) y-component of the unit normal vector (dimensionless) local Nusselt number (dimensionless) average Nusselt number (dimensionless) overall array Nusselt number, NuO ¼ ðNuu þ NuL Þ=2 (dimensionless) total number of cells in computational domain (dimensionless) number of grid points on half of the perimeter of each cylinder (dimensionless) pressure (dimensionless) pressure (Pa) Prandtl number (dimensionless) radius of cylinder (m)

for the upper cylinder is influenced by the lower cylinder on two counts. Firstly, the driving force for heat transfer from the upper cylinder is lower than that for the lower cylinder due to the preheating of the fluid above its free stream value by the lower-cylinder. In the literature, this aspect is known as temperature difference imbalance and many investigators have studied this effect [9]. Secondly, due to the plume rising from the lower cylinder, the upper cylinder experiences effectively a mixed convection regime, i.e., its own buoyancy-induced flow augmented by the plume of the lower cylinder. Naturally, as the distance between the two cylinders increases, the plume expands in the lateral direction and therefore its upward velocity decreases. Therefore, depending upon the relative magnitudes of these two competing phenomena arising from the virtual mixed-convection, the Nusselt number for the upper cylinder may be lower or higher than that of the lower cylinder. Naturally, it is possible to choose values of (S/D), Grashof number and Prandtl number when these two effects will nullify each other and both cylinders will exhibit almost identical values of the Nusselt number. Indeed all these conjectures are borne out by the currently available numerical and experimental studies on natural convection from two cylinders in air and water. Table 1 provides a succinct summary of the range of conditions and the approach used in previous studies dealing with the natural convection from multiple cylinders arranged in vertical configurations. For instance, Park and Chang [9] used the finite difference method

r Ra s S T T⁄

DT UC UX, UY X, Y, Z

distance in the radial direction from the surface of the cylinder (m) Rayleigh number (dimensionless) surface area of each cylinder (m2) center-to-center cylinder spacing (m) temperature of fluid (K) non-dimensional temperature = (T  T1)/(TW  T1) (dimensionless) temperature difference = (TW  T1) (K) characteristic velocity (m/s) X- and Y-components of the velocity (dimensionless) Cartesian coordinates (dimensionless)

Greek symbols b coefficient of thermal volume expansion (K1) h circumferential angle (degree) d distance between two grid points on the surface of the cylinder, (m) e rate of deformation tensor (s1) q density of the fluid (kg/m3) r deviatoric stress tensor (Pa) X composite parameter (dimensionless) W stream function (kg/m) n ratio of lower cylinder to upper cylinder temperature difference = ðDT L =DT u Þ v ratio of upper to lower cylinder average Nusselt number = (Nuu/NuL) Subscripts and superscripts  dimensionless 1 free stream condition i, j X- and Y- co-ordinates l local value L lower cylinder u upper cylinder W cylinder wall condition

to study laminar free convection from two circular cylinders in air. They also examined the influence of the temperature-difference imbalance, defined as n = (DTL/DTu) by considering n = 1 and n = 2. In their work, the ratio v = (Nuu/NuL) varied between 0.8 and 1.1 for n = 1 and the corresponding range for n = 2 was found to be 0.27 to 0.77. Clearly, in the first case, at low Rayleigh numbers, the buoyancy induced current around the lower cylinder is so weak that it hardly contributes anything to the flow around the upper cylinder and the reduced temperature difference (DTu) < (DTL) dominates the overall heat transfer thereby leading to the value of the ratio (Nuu/NuL) smaller than unity. For a fixed value of (S/D), however, the former effect increases with the increasing Rayleigh number thereby leading to values of v > 1, albeit only marginally. Subsequently, Yüncü and Batta [10] have reported qualitatively similar numerical results for free convection from two circular cylinders in air. The ratio of the two Nusselt numbers, v, varied from 0.8 to 1.3 depending upon the value of the Rayleigh number and (S/D). This ratio was found to approach a constant value for the inter-cylinder gap larger than the value of (S/D) > 8 thereby suggesting no interference between the two cylinders. Subsequently, Chouikh et al. [11] have also reported similar numerical results. They found the value of v to increase with Rayleigh number which is in line with the previous numerical studies as well as their limited experimental results [12]. In an extensive numerical study, Corcione [13] has not only reviewed

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

1129

Table 1 Summary of range of conditions spanned by previous studies. Authors

Approach

Operating range of the governing parameters

Park and Chang [9]

Numerical

Yüncü and Batta [10] Chouikh et al. [11] Chouikh et al. [12] Corcione [13] Eckert and Soehngen [14] Marsters [15] Tokura et al. [20] Paykoc et al. [17] Sadeghipour and Asheghi [18] Reymond et al. [21]

Numerical Numerical Experimental Numerical Experimental Experimental Experimental Experimental/Numerical Experimental Experimental

104 6 Ra 6 105 ; 2 6 ðS=DÞ 6 4; n ¼ 1; 2; Pr  0:7 2  104 6 Ra 6 2  105; 2 6 ðS=DÞ 6 10; Pr  0:7 102 6 Ra 6 104; 2 6 ðS=DÞ 6 6; Pr  0.7 Ra = 102 and 104; Pr  0.7 5  102 6 Ra 6 5  105; 2 6 ðS=DÞ 6 50; Pr  0.7 Gr = 14650 and 34300; Pr  0.7 750 6 Gr 6 2000; Pr  0.7 4  104 6 Ra 6 4  105; 1:1 6 ðS=DÞ 6 15; Pr  0.7 1.5  104 6 Ra 6 3  104; Pr  0.7 500 6 Ra 6 7000; 3:5 6 ðS=DÞ 6 30:5; Pr  0.7

Persoons et al. [22]

Experimental

Heo et al. [27] Chae and Chung [29]

Numerical Experimental

the previous pertinent literature for Newtonian fluids, but has also presented extensive results on the detailed velocity and temperature fields and on Nusselt number for laminar free convection in air from vertical arrays consisting of two-to-six cylinders. They also put forward simple empirical correlations for the prediction of the mean Nusselt number for an individual cylinder in the array. The numerical activity in this field has also been complimented by the experimental investigation in this field. In their pioneering work, Eckert and Soehngen [14] studied free convection from three cylinders arranged vertically in air corresponding to three values of the Grashof number as Gr = 14, 650 and 34300. As expected, the Nusselt number for the first cylinder (from bottom) was very close to that of a single cylinder and the Nusselt number for the other cylinders decreased as more and more cylinders were added to the array. Marsters [15], on the other hand, studied the effect of (S/D) and of the number of cylinders in the array on free convection in air in the steady flow regime for the constant heat flux condition prescribed on the surface of each cylinder. They reported the sur-

2  106 6 Ra 6 6  106 ; 1:5 6 ðS=DÞ 6 3; Pr  7 1:8  106 6 Ra 6 5:5  106 ; (S/D) = 2 and 4; Pr  7 1.5  108 6 Ra 6 2.5  1010; 1:02 6 ðS=DÞ 6 9; 0:7 6 Pr 6 2014 7.3  107 6 Ra 6 4.5  1010; 1:02 6 ðS=DÞ 6 9; 2014 6 Pr 6 8334

face temperature of the cylinders to increase from the bottom cylinder upward thereby suggesting decreasing heat transfer due to the reduced temperature gradient. This observation is also consistent with that of Lieberman and Gebhart [16]. Indeed, depending upon the value of (S/D) and Ra or Gr, the average Nusselt number for an individual cylinder in an array could drop to about 50% of the value for a single cylinder at small values of (S/D). In an experimental/numerical study, Paykoc et al. [17] reported the detailed velocity and temperature profiles adjacent to the surface of the two heated cylinders losing heat to quiescent air. While the agreement between the numerical predictions and experimental results was reported to be satisfactory, they found that the Nusselt number for the upper cylinder to approach that of the lower cylinder at about (S/D) P 2 to 2.5 which is in stark contrast to the conclusions reached by Yüncü and Batta [10]. Sadeghipour and Asheghi [18] reported experimental results on the laminar free convection from vertical arrays (made up of 2 to 8 equal diameter cylinders) in air at low Rayleigh numbers. They developed simple correlations

Fig. 1. Schematic representation of the flow and computational domain.

1130

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

thereby enabling the prediction of the average Nusselt number for any cylinder in the array. While these results conform to the expected scaling of Nu  Ra1/4 and are consistent with the literature values e.g., see [19,20], the maximum improvement in the overall heat transfer in a two-cylinder system was of the order of 24% which could be realised at (S/D) P 20. In contrast to the aforementioned studies based on air as the working fluid, Reymond et al. [21] reported extensive experimental results on the local Nusselt number distribution on the surface of single and two circular cylinders for free convection in water. Their key findings can be summarized as follows: for the lower heated and upper unheated case, the Nusselt number for the lower cylinder deviates very little from that of the single cylinder. Of course, when both are heated, the plume rising from the lower cylinder influences the heat transfer characteristics of the upper cylinder in a complex fashion depending upon the value of (S/D) and Ra. As mentioned previously, it denotes a net result of the reduced temperature difference vis-a-vis the virtual forced convection effect which, in turn, is determined by the increasing width of the plume accompanied by its decreasing strength of the buoyancy-induced flow. In a later study, Persoons et al. [22] have reported on the local heat transfer characteristics of a pair of vertically aligned heated cylinders submerged in water and they reported up to 10% enhancement in heat transfer from the upper cylinder and appreciable fluctuations in the value of the local Nusselt number. They also postulated a phenomenological mechanism based on the oscillating nature of the plume formed above the lower cylinder to explain the fluctuating nature of the Nusselt number. However, their surface averaged values of the Nusselt number are in line with the other studies in this field. Free convection from multiple cylinders arrays with and without flow guiding devices in the laminar flow regime has also been investigated by Ashjaee and co-workers [23,24] and others [25] who have also elucidated the influence of the inclination angles of the arrays from the vertical direction. As expected, the use of such flow-guiding plates can enhance the rate of heat transfer by up to 25–30% than that in the absence of such devices. Naturally, this enhancement can be attributed to the forced convection effects introduced by such devices. Of course, this enhancement is accompanied by the concomitant increase in frictional pressure drop across the system. While the assumption of the laminar flow regime is implicit in the foregoing studies, the analogous situation entailing staggered cylinders has been studied experimentally by Sparrow and Boessneck [26] in laminar flow regime and by Heo et al. [27] in turbulent flow regime. The latter authors reported limited numerical results for extremely large values of Prandtl numbers. Their predictions were shown to be consistent with their own experimental observations. For turbulent flow conditions, the ratio, v, of the two Nusselt numbers is nearly equal to unity for about (S/D) P 2. This is also consistent with the trends reported by Toshiyuki et al. [28]. By employing the usual analogy between heat and mass transfer, Chae and Chung [29] in a recent study used the cathodic deposition of copper from an acidic copper sulphate solution onto a vertical array of two cylinders to infer the values of the Nusselt number. While such high values of the Rayleigh number may suggest the occurrence of the turbulent flow regime, but the corresponding values of the Grashof number are likely to be in the range 9  103 to 5  106 which are almost within the laminar flow regime, at least in the case of a single cylinder. This discussion is concluded by briefly mentioning the peripherally related studies. Tokura et al. [20] investigated the effect of confinement on free convection from vertical arrays made up of 2 to 5 cylinders in air. Similarly, D’Orazio and Fontana [30] studied free convection from two vertical arrays of cylinders in air and their results span the center-to-center horizontal and vertical separations between the two arrays 2D to 145D and 4 6 ðS=DÞ 6 12 respectively, albeit the values of the Rayleigh number are extremely

small in their work (2:4 6 Ra 6 11:9). The major thrust of their work was on elucidating the influence of the horizontal and vertical pitches on the overall heat transfer by way of simple correlations. For side-by-side arrangement, the overall Nusselt number was shown to have maximum values at the horizontal separation of about 25–30D irrespective of the value of (S/D). On the other hand, Bejan et al. [31] attempted to develop a relationship between (S/D) and the rate of heat transfer for fixed values of the volume of array and of cylinder diameter to maximise the rate of heat transfer. It is thus fair to say that a reasonable body of literature is available on the free convection from vertical arrays of circular cylinders consisting of varying number of cylinders, albeit most of these relate to the laminar flow regime and for air as the working fluid. For the sake of completeness, the scant literature on natural convection from the vertical arrays of two- and threecylinders especially as applied to subsea pipelines has been recently reviewed by Grafsrønnigen and co-workers [32,33]. In contrast, many fluids of multiphase (foams, emulsions, slurries) and of polymeric nature (polymer melts and solutions) encountered in a broad spectrum of industrial applications exhibit shear-dependent viscosity (shear-thinning and shear-thickening) which is frequently modelled using the simple power-law fluid model [34,35]. Typical examples include polymer and food processing operations, manufacture of personal and health-care products and allied engineering applications [36,37]. Despite their frequent occurrence, the available literature on free convection from variously-shaped heated objects is not only sparse but is also of recent vintage. Early developments in this field are based solely on the application of the approximate boundary layer equations,

Table 2 Comparison of present array average Nusselt number (Nuo) with the numerical work of Corcione [13] in air. S/D

Ra = 103

Ra = 104

Ra = 105

Ref. [13]

Present

Ref. [13]

Present

Ref. [13]

Present

2 3 5 7 9 12

2.4715 2.7116 3.0141 3.2133 3.3621 3.5324

2.5730 2.8476 3.1177 3.2620 3.3558 3.4548

4.2458 4.6582 5.1300 5.1779 5.3726 5.6502

4.4517 4.8704 5.2942 5.5181 5.6519 5.7967

7.8938 8.5224 8.8952 9.5345 9.9295 9.7065

8.0400 8.7239 9.4505 9.8279 10.046 10.285

Table 3 Comparison of the average Nusselt number for the upper cylinder (Nuu) with that of numerical work of Yüncü and Batta [10] in air. Ra

S/D = 2

4

2.0  10 1.0  105 2.0  105

S/D = 5

S/D = 9

Ref. [10]

Present

Ref. [10]

Present

Ref. [10]

Present

4.1615 6.9565 8.5217

4.6355 7.4553 9.0985

5.8385 8.9317 10.571

6.2375 9.3513 11.207

6.5093 9.9379 11.652

6.2806 9.1343 10.786

Table 4 Comparison of the ratio of upstream cylinder average Nusselt number (Nuu) to the single cylinder (Nus) with the experimental work of Sparrow and Niethammer [19] in air. Ra = 2.0  104

Ra = 6.0  104

S/D

Ref.[19]

Present

S/D

Ref. [19]

Present

2 3 5 7 9

0.8273 0.9754 1.1203 1.2160 1.2343

0.8063 0.9595 1.1216 1.2078 1.2596

2 3 5 7 9

0.8421 0.9929 1.1684 1.2421 1.2526

0.8468 1.0001 1.1613 1.2447 1.2949

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152 Table 5 Comparison of present average Nusselt number (Nu) with the numerical work of Paykoc et al. [17] for S/D = 2 in air. Ra

1.5  104 2.0  104 2.5  104 3.0  104

Lower cylinder

Upper cylinder

Ref. [17]

Present

Ref. [17]

Present

5.0264 5.1318 5.4270 5.7221

4.2936 4.6355 4.9198 5.1652

5.6741 5.8938 6.1508 6.4459

5.4184 5.7786 6.0759 6.3313

e.g., see Acrivos and co-workers [38,39], Stewart [40], etc. and this body of literature up to early 1980s dealing with the laminar free convection heat/mass transfer from single spheres, cylinders and plates has been reviewed by Shenoy and Mashelkar [41] whereas a more recent review is available in Ref. [34]. Notwithstanding the utility of such approximate analyses in yielding reliable scaling relationships for skin friction and Nusselt number, these results suffer from two major weaknesses: firstly, the boundary layers are assumed to be thin thereby limiting the applicability of these results to infinitely large values of the Grashof and/or Prandtl num-

1131

bers. Secondly, this analysis entirely neglects the wake region. Each of these assumptions can only be relaxed by seeking numerical solutions of the complete governing differential equations. Indeed reliable numerical results based on the numerical solution of the complete momentum and energy equations for the laminar free convection heat transfer in power-law fluids from a sphere [42], cylinder [5,45], square cylinder [43,44], elliptic cylinder [45], semi-circular cylinder in different orientations [46,47], for instance, have been reported only recently. Broadly, irrespective of the shape of the heated object, shear-thinning behaviour enhances the rate of heat transfer whereas shear-thickening fluid has the opposite influence with reference to the behaviour in Newtonian fluids under otherwise identical conditions. Furthermore, unlike in the case of Newtonian fluids where the mean Nusselt number scales as Ra1/4 in the laminar flow regime, dependence of the Nusselt number on Grashof and Prandtl numbers is strongly modulated by the type of fluid-behaviour, i.e., shear-thinning or shearthickening (namely, the value of power-law index). In addition to the preceding studies on free convection in unconfined powerlaw media, limited results are also available from a square cylinder in confined power-law [48] and a circular cylinder in Bingham flu-

Fig. 2. Comparison of the present temperature profiles with the experimental results of Paykoc et al. [17] (a) Lower cylinder, (b) Upper cylinder at Ra = 1.5  104, S/D = 2 and Pr = 0.71 (symbols for Paykoc et al. and lines for present work).

1132

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

ids [49]. Most of these as well as the other pertinent investigations related to the free-, forced- and mixed- convection regimes from two-dimensional axisymmetric shapes have been summarized in a recent review [50]. In contrast, as far as known to us, in spite of the wide occurrence of shear-thinning and shear-thickening fluid behaviour in scores of industrial settings, there has been no study on the free convection heat transfer from a vertical array of two cylinders immersed in power-law fluids, though the analogous case of two square cylinders has been studied recently [51,52]. In addition to its pragmatic significance, owing to the rapid

spatial decay of the velocity field in shear-thinning fluids, the interference effect may be suppressed or augmented in this case depending upon the gap between the two cylinders. This provides additional motivation for undertaking this investigation. This work aims to fill this gap. In particular, reported herein are new extensive numerical results which elucidate the influence of Grashof number (10 6 Gr 6 104), Prandtl number (0.72 6 Pr 6 100) and power-law index (0.3 6 n 6 1.5) thereby encompassing both shear-thinning (n < 1) and shear-thickening (n > 1) fluid behaviour and of the gap between the two cylinders (2 6 (S/D) 6 20) on the detailed temperature and streamline profiles in close proximity of the two cylinders as well as on the local and mean Nusselt number. Also, the present results have been compared with the previous numerical and experimental results wherever possible

1.2

θ = 90

1

0

θ

U Y / Ra

1/2

0.8

0.6 θ = 135

0

0.4

0.2

Fig. 3. Comparison of the present local values of average Nusselt number (line) for upper cylinder with that of numerical (open symbols) and experimental (filled symbols) work of Paykoc et al. [17] at Ra = 1.5  104 and S/D = 2 with air (Pr = 0.71) as working fluid.

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

r/D Fig. 5. Comparison of present (lines) circumferential velocity profiles with that of Park and Chang [9] shown as symbols at S/D = 3 and Ra = 105 in air for the upper cylinder.

2

Local Nusselt number, Nul

1.6

θ

1

1.4 0.8

1.2 1

S/D = 4

0.6

0.8 0.4

0.6

Local Nusselt number, Nul

1.2

1.8

0.4 0.2 S/D = 2

0.2 0

0

50

100

150

0

θ Fig. 4. Comparison of the present local values of average Nusselt number (line) for the lower cylinder with that of numerical (open symbols) and experimental (filled symbols) work of Paykoc et al. [17] at Ra = 1.5  104 and S/D = 2 with air (Pr = 0.71) as working fluid.

Fig. 6. Comparison of the local Nusselt number distribution (present results shown as lines) with that of Park and Chang [9] (shown as points) at Ra = 105 for the upper cylinder in air.

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

1133

to demonstrate their validity. The paper is concluded by presenting simple expressions which permit the prediction of the Nusselt number and drag values for this system in a new application.

In the above relation, b is the volumetric thermal expansion coefficient defined as:

2. Problem definition and mathematical formulation

For an incompressible fluid, the components of the extra stress tensor, sij are related to the rate of deformation (eij) tensor as follows [53]:

Let us consider the laminar free convection heat transfer from two heated equi-sized vertically aligned long horizontal circular cylinders of diameter, D, immersed in a stagnant and unconfined body of a power-law fluid as shown schematically in Fig. 1. The surface of each cylinder is maintained at a constant temperature of, Tw, whereas the temperature of the surrounding cold fluid is T1, i.e., Tw > T1. The temperature gradient present in the fluid sets up an upward buoyancy induced flow around the two cylinders, which, in turn, leads to heat transfer by free convection. The center-to-center distance between the two cylinders, S, is varied in the range 2 6 (S/D) 6 20. Naturally, since it is not numerically feasible to approach a truly unconfined flow, this idealisation is approached here by enclosing the two cylinders in a cylindrical envelope of fluid (of diameter D1) with its center located at the mid-point of the centreline, as shown in Fig. 1. Of course, a large value of D1 will warrant enormous computational effort without necessarily improving the precision of the results, whereas an unduly small value will disturb the flow and temperature fields themselves. Therefore, a prudent choice is required to keep the computational effort at a reasonable level without incurring significant loss of accuracy of the results. This issue is discussed in detail in a later section. A cartesian coordinate system with its origin situated at the centre of the enclosing fluid envelope is used in this study. The present flow is considered to be incompressible (except for the density term appearing in the y-momentum equation) and steady over the range of conditions spanned herein. The thermo-physical properties of the fluid (density, q; thermal conductivity, k; specific heat, C; power-law constants, m, n, etc.) are assumed to be independent of temperature over the temperature interval, T w 6 T 6 T 1 , except for the density term appearing in the y-component of the momentum equation. Since the cylinders are assumed to be infinitely long in z-direction, it is reasonable to postulate that UZ = 0, and o()/oZ = 0. Furthermore, viscous dissipation effects in the energy equation are also neglected at this stage. Within the framework of these simplifying assumptions, the governing differential equations describing the velocity and temperature fields are written in their dimensionless forms as follows: Continuity equation:

@U X @U Y þ ¼0 @X @Y

 1 @ q q @T T

b¼

sij ¼ 2geij where ði; jÞ ¼ ðX; YÞ

ð6Þ

ð7Þ

For a power-law fluid, the viscosity is given by the expression [53]:

g ¼ ðI2 =2Þð 2 Þ n1

ð8Þ

where n is the power-law index. For Newtonian fluids, n = 1, whereas n < 1 corresponds to shear-thinning and n > 1 denotes shear-thickening fluid behaviour. Finally, in order to complete the problem definition, the following boundary conditions are prescribed.

Fig. 7. Comparison of the present local values of Nusselt number for the upper cylinder (open symbols) with the experimental work of Persoons et al. [22] (filled symbols) in water at S/D = 2.

ð1Þ

x-momentum equation:

    @ðU X U X Þ @ðU X U Y Þ @P 1 @ sXX @ sYX ¼ þ þ pffiffiffiffiffiffi þ @X @Y @X @Y Gr @X

ð2Þ

y-momentum equation:

    @ðU Y U X Þ @ðU Y U Y Þ @P 1 @ sXY @ sYY ¼ þ þ T  þ pffiffiffiffiffiffi þ @X @Y @Y @Y Gr @X

ð3Þ

Energy equation:

UX

@T  @T  1 @2T  @2T  þ UY ¼ þ 1 Þ 2 ð @X @Y @Y 2 Pr  Gr nþ1 @X

! ð4Þ

The linear variation of the fluid density with temperature is treated via the familiar Boussinesq approximation as follows:

q1  q ¼ q1 gðT  T 1 Þ

ð5Þ

Fig. 8. Comparison of the present local values of Nusselt number (open symbols) for the upper cylinder with the experimental work of Reymond et al. [21] (filled symbols) in water at Ra = 2.0  106 and S/D = 2.

1134

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

On the surface of the cylinders (C1, C2): As these are solid surfaces, the usual no-slip boundary condition for flow and a constant temperature condition are used, i.e.,

U X ¼ U Y ¼ 0 and T  ¼ 1

ð9Þ

At the inlet/outlet of the domain: The bottom-half of the computational domain is designated as the ‘‘inlet’’ in the present case. A standard default option in FLUENT known as ‘‘Pressure inlet’’ is imposed at this surface. This assumes a zero total gauge pressure (with respect to the operating pressure which is atmospheric for the present case) and the temperature is equal to that of the far

away fluid, i.e., T⁄ = 0. Similarly, the top-half of the computational domain is assigned as ‘‘outlet’’ and the so-called ‘‘Pressure outlet’’ boundary condition is used at this surface. This option sets the static gauge pressure equal to zero and here, the temperature once again is given by T⁄ = 0. The effect of the far away boundary conditions in free convection has been discussed in Ref. [5]. Axis of symmetry (X = 0): Over the range of values of the Grashof and Prandtl numbers considered herein, the flow is expected to be steady and symmetric about the vertical centreline of the domain (X = 0). Therefore, the computations have been performed only for the right-half domain (X P 0) in order to economize on

Fig. 9. Representative streamlines (right half) and isotherm patterns (left half) at S/D = 2 and n = 0.3.

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

the computational efforts. The symmetry boundary conditions used here are given as follows:

@U Y ¼ 0; @X

U x ¼ 0 and

@T  ¼0 @X

ð10Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In the preceding discussion, D and U C ¼ DgbDT have been used as the characteristic length and velocity scales respectively to nondimensionalize the governing equations and boundary conditions. Thus, q1 U 2C ; m(UC/D)n, etc. have been used to scale the pressure and extra stress components respectively. Temperature is normalised as, T⁄ = (T  T1)/(Tw  T1) which ranges from 0 to 1. Thus, in

1135

the present case for a fixed value of (S/D), the flow and temperature fields are governed by three dimensionless numbers, namely, Grashof number (Gr), Prandtl number (Pr) and power-law index (n). The Grashof and Prandtl numbers are defined here as:  Grashof number:

Gr ¼

q21 Dnþ2 ðgbDTÞ2n m2

ð11Þ

 Prandtl number:

Pr ¼

  2 3ðn1Þ q1 C m ð1þnÞ ð1n D 1þnÞ ðgbDTÞð 2nþ2 Þ k q1

Fig. 10. Representative streamlines (right half) and isotherm patterns (left half) at S/D = 2 and n = 1.5.

ð12Þ

1136

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Note that in the limit of Newtonian fluid behaviour (n = 1), both these definitions reduce to their familiar forms for Newtonian fluids. On the other hand, the effect of the power-law index (n) is seen to be entwined through the definitions of Gr and Pr, in addition to being a dimensionless parameter on its own thereby making the delineation of the role of power-law index in the present case rather difficult. A combined parameter, namely, Rayleigh number, Ra = Gr.Pr is also employed in the literature. In Newtonian fluids, the use of Rayleigh number offers a distinct advantage because the Nusselt number in laminar free convection scales as Nu  Gr1/ 4 1/4 Pr thereby allowing it to be written as Nu  Ra1/4. Owing to

the inherently different scaling of Nu with Gr and Pr in power-law fluids, the use of this definition of the Rayleigh number, Ra, offers no such advantage in the present situation. The numerical solution of the aforementioned governing equations subject to the above-noted boundary conditions maps the flow domain in terms of the primitive variables, i.e., velocity (U x ; U y ), temperature T⁄ and pressure (P). These can be post-processed in order to deduce some global as well as the local flow and heat transfer characteristics. For instance, it is usual to describe the momentum transfer characteristics in terms of streamline contours, surface pressure coefficient and drag coefficient.

Fig. 11. Representative streamlines (right half) and isotherm patterns (left half) at S/D = 6 and n = 0.3.

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Similarly, the heat transfer aspects are often described in terms of isotherm contours, local heat transfer coefficient and average Nusselt number. At this juncture, it is thus appropriate to introduce some of these definitions. Drag coefficient (CD): Due to the velocity gradients prevailing in the close proximity of the heated cylinders, the fluid exerts a net force (in y direction) on each cylinder, described in terms of the familiar drag coefficient. It is made up of two components, namely, pressure drag coefficient and friction drag coefficient defined as follows:

C DP ¼

2F DP

q1 U 2C R

¼

Z

s

C p ny ds

ð13Þ

C DF ¼

2F DF 2 1UC R

q

21n ¼ pffiffiffiffiffiffi Gr

Z

ðsyy ny þ syx nx Þds

1137

ð14Þ

s

and C D ¼ C DP þ C DF

ð15Þ

where Cp is the pressure coefficient defined as follows:

p  p1 CP ¼ 1 q U2 2 1 C

ð16Þ

In Eq. (16), p is the local pressure at a point on the surface of cylinder and p1 is its reference value far away from the cylinder.

Fig. 12. Representative streamlines (right half) and isotherm patterns (left half) at S/D = 6 and n = 1.5.

1138

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Nusselt number (Nu): The local value of the Nusselt number at a point on the surface of the cylinder is calculated by the following expression:

Nul ¼

  hD @T ¼ k @ns

ð17Þ

where ns is the outward drawn unit vector normal to the surface of the cylinder. While the local values of the Nusselt number give an indication of its spatial distribution over the surface of each cylinder, from a practical standpoint, the average value of the Nusselt number is often needed in process design calculations in order to calculate the rate of heat transfer between the cylinder and the surrounding fluid. The surface averaged value of the Nusselt number is calculated by integrating the local values over the whole surface of each cylinder as follows:

Nu ¼

Z

Nul ds

ð18Þ

s

In order to visualise the flow adjacent to each cylinder, streamline contours are examined by using non-dimensional stream function w⁄ defined as follows:

w ¼

w

q1 U C ðRÞ

ð19Þ

where q1 is the reference density far away from the cylinder, UC is the reference velocity and R is the radius of the cylinder.

Finally, it is fair to postulate that both the momentum and heat transfer characteristic parameters are expected to be functions of the Grashof number (Gr), Prandtl number (Pr), power-law index (n), and of course, of the inter cylinder spacing (S/D). These relationships are explored and developed in detail in this work.

3. Numerical solution methodology and choice of computational parameters Since detailed descriptions of the numerical solution methodology employed here are available elsewhere, e.g., see [42–46], these are not repeated here. Broadly, the governing field equations subject to the above-mentioned boundary conditions have been solved numerically using the finite volume based software ANSYS Fluent (version 6.3.26) together with the ANSYS Workbench to mesh the computational domain. The two-dimensional, steady, laminar and pressure based coupled solver was used with second order upwind (SOU) scheme for discretizing the convective terms in the momentum and thermal energy equations. The semi-implicit method for the pressure linked equations (SIMPLE) was used for pressure–velocity coupling and non-Newtonian power-law model was used for specifying the viscosity of the fluid. Naturally, both the velocity and temperature gradients are expected to be steep near the surface of the two cylinders and in the gap in between the two cylinders. Therefore, a very fine

Fig. 13. Representative streamlines (right half) and isotherm patterns (left half) at S/D = 20 and Gr ¼ 104 ; n ¼ 0:3; Pr ¼ 100.

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

mesh is required in these regions to resolve the velocity and temperature fields. On the other hand, the use of such a fine mesh far away from the bluff body would be simply waste of computational resources where gradients and interactions are expected to be negligible. Keeping this in mind, the present computational domain is divided into three sub-regions, namely, region A, B, and C. The region A extends up to a distance of 0.5D radius in all directions in the vicinity of both cylinders whereas the region B spans the distance up to 20–30D in the computational domain. The remaining portion (region C) of the domain extends up to the outer boundary. Each region contains approximately 1/3 of the total number of the computational cells. Regions A and B are mapped using unstructured triangular pave cells whereas region C is meshed using structured quadrilateral map cells. Within each region, the grid is made progressively coarse in the radial direction by using a suitable successive ratio. The solution was initialized by using zero values of the velocity and temperature at each grid point in order to avoid the potential convergence problems. The relative convergence criteria in terms of the residuals of 108 for the momentum equations and 1014 for the thermal energy were prescribed in the present study. These were also found to be adequate to stabilize the values of the drag coefficient and Nusselt number at least up to four significant digits. Much has been written in our recent studies about the effects of the size and shape of the computational domain and of the type of computational mesh [42–47] and hence this discussion is not repeated here. Suffice it to say here that the value of (D1/D) was systematically varied as (D1/D) = 400, 600 and 1200, at the lowest values of the Grashof number, Prandtl number, and n = 1.5 when

1139

the momentum and thermal boundary layers are expected to be very thick. The results for (D1/D) = 600 and (D1/D) = 1200 were found to be virtually indistinguishable from each other at the expense of an appreciable increase in CPU time. Thus, the value of (D1/D) = 600 is considered to be sufficient for the present results to be free from boundary effects. Similarly, three numerical grids were developed and these were tested for the maximum values of the Grashof number (Gr = 104), Prandtl number (Pr = 100) and the minimum value of S/D = 2 and n = 0.3 when the velocity and temperature gradients in the thin boundary layers developed on each cylinder and in the nip region between the two cylinders are expected to be rather steep. A detailed examination of the results revealed that a mesh comprising total number of cells in the half-domain as N = 1,37,930, total number of cells on half-surface of each cylinder Np = 300 and the smallest cell size of d/ D = 0.00523 was sufficient to obtain results which are largely free from such numerical artefacts. Also, the adequacy of these choices is further corroborated by presenting extensive benchmark comparisons with the previous numerical and experimental results in the next section. Such comparisons not only inspire confidence in the reliability of the numerics used here, but also help ascertain the precision of the new results obtained in this work. 4. Results and discussion 4.1. Validation of results While extensive validation for the laminar free convection in Newtonian and power-law fluids from variously shaped single

Fig. 14. Distributions of local Nusselt numbers for upper (lines + symbols) and lower cylinders (lines) at Gr = 10.

1140

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

heated objects including circular and non-circular cylinders, sphere, etc. have been reported elsewhere [42–48], only additional benchmark comparisons involving two circular cylinders are presented here. Table 2 shows a comparison of the present values of the array average Nusselt number with that of Corcione [13] for Pr = 0.70, corresponding to air as the working fluid. Over wide ranging values of parameters, 2 6 ðS=DÞ 6 12 and 103 6 Ra 6 105 , the two results are seen to be within 3–4% of each other. These small differences can probably be ascribed to the inherently different numerics, domain size, mesh quality, etc. used in the two studies and are not uncommon in such studies [54]. Similarly, Table 3 compares the present results with the early results of Yüncü and Batta [10] where the two values are seen to deviate from each other by up to 10–11%. While the exact reasons for such large deviations in this case are not immediately obvious, the following factors must be borne in mind while assessing this comparison. Yüncü and Batta [10] have neither tabulated their numerical values nor have they presented a predictive equation and therefore, their values have been read off from a graph in their paper. Undoubtedly, this will entail errors. Secondly and perhaps more importantly, they have used a relatively coarse mesh (the finest grid spacing being 0.066D and a very small domain given by (D1/ D) = 10 + (S/D) and hence the maximum value of (D1/D) in their study is 20 corresponding to S/D = 10. This value of (D1/D) is too small for the results to be free from boundary effects, especially at low Rayleigh numbers. Tables 4 and 5 show similar comparisons of the present numerical values with the scant experimental re-

sults available in the literature [17,19]. Table 4 contrasts the two sets of results in terms of the ratios of the average Nusselt numbers for the upper cylinder to that of a single cylinder. An excellent correspondence is seen to exist between the two values over the entire range of (S/D) and Rayleigh number spanned in this experimental study [19], but it is limited to a single value of Prandtl number (Pr = 0.71). On the other hand, the agreement is seen to be less good in Table 5 with the experimental results of Paykoc et al. [17], albeit the differences between the predicted and experimental values are well within the experimental uncertainty of 15% inherent in their temperature measurements. It needs to be added here that their results [17] correspond to the case of temperature imbalance of 0.9 between the upper and lower cylinders, albeit they non-dimensionalised the temperature using the value of the lower cylinder. Therefore, the results for the upper cylinder must be divided by 0.9 in order to facilitate a comparison with their results. This factor of about 10–11% should also be applied to the results shown in Table 5, but it is really irrelevant here because they state the experimental uncertainty of their Nusselt number values to be of the order of 15–20%. Now turning our attention to the validation of the detailed temperature and velocity fields, further comparisons with their results [17] are shown in Figs. 2–4 in terms of the local temperature profiles and local Nusselt number distribution along the surface of each cylinder. Thus, overall the present results are seen to be consistent with their detailed measurements [17]. Similarly, Figs. 5 and 6 compare the present predictions of the circumferential velocity and the Nusselt

Fig. 15. Distributions of the local Nusselt numbers for upper (lines + symbols) and lower cylinders (lines) at Gr = 104.

1141

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

(i)

4

6 Pr = Pr = Pr = Pr =

0.71 10 20 50

3

4 2 2

Average Nusselt number, Nu

1

0

0

0.3

0.6

0.9

1.2

1.5

1.8

0

0

0.3

0.6

0.9

1.2

1.5

1.8

0.3

0.6

0.9

1.2

1.5

1.8

Gr = 104 50 50 40 40 30 30 20

20

10

0

10

0

0.3

0.6

0.9

(a)

1.2

1.5

1.8

0

0

Power-law index, n

(b)

Fig. 16. (i) Variation of average Nusselt number with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 2. (a) Lower cylinder; (b) upper cylinder. (ii) Variation of average Nusselt number with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 4. (a) Lower cylinder; (b) upper cylinder. (iii) Variation of average Nusselt number with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 6. (a) Lower cylinder; (b) upper cylinder. (iv) Variation of average Nusselt number with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 20. (a) Lower cylinder; (b) upper cylinder.

number distribution on the surface of the upper cylinder at S/D = 3 and for air. An excellent match is seen to exist here between the present results and the literature values [9]. Fig. 7 shows a comparison of the present numerical values with the experiments of Persoons et al. [22] in water. The present results do seem to capture the rapid decrease in the maximum value of the Nusselt number at the front stagnation point, followed by its near constant value and finally dropping off again towards the rear stagnation point. Bearing in mind the intrinsic difficulties of such measurements, the deviation between the two results are seen to be within the estimated maximum uncertainty of 15–16% in the values of the Nusselt number [22]. Lastly, Fig. 8 contrasts the present results with the experiments of Reymond et al. [21] and once again, the correspondence is seen to be as good as can be expected in this type of work. At this juncture, it is worthwhile to add two more possible reasons for the differences seen in Figs. 2–8. The present numerical predictions are based on the assumption of infinitely long cylinders whereas the length-to-diameter ratios are of the order of 10 in [21,22] and 2–2.5 in [17]. Secondly, the present sim-

ulations also assume the thermo-physical properties of the fluid to be independent of temperature whereas the value of DT (=Tw  T1) = 40 °C was used in [17] and they evaluated the physical properties of air at the mean film temperature. In view of all these factors, the correspondence seen in Figs. 2–8 is regarded to be satisfactory and acceptable. On the other hand, the good correspondence seen in Table 1 coupled with our past experience, the new results reported herein are considered to be reliable to within 2% for the present configuration. 4.2. Streamline and isotherm contours It is customary to visualise the detailed structure of the velocity and temperature fields in terms of the streamline and isotherm contours, especially in the proximity of the two cylinders. Figs. 9– 13 show representative streamline and isotherm patterns for a range of combinations of conditions elucidating the influence of (S/D), Gr, Pr and the value of power-law index (n). At a gross level, the boundary layers are seen to become progressively thinner with

1142

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

(ii)

Gr = 10 5

Pr = Pr = Pr = Pr =

4

0.72 20 50 100

4

Average Nusselt number, Nu

3

2

2

1

0

0

0.5

1

0

1.5

0

0.5

1

1.5

0

0.5

1

1.5

Gr = 10 4 70

50 45

60 40 50

35 30

40

25 30

20 15

20

10 10 5 0

0

0.5

1

(a)

0

1.5

Power-law index, n

(b)

Fig. 16 (continued)

the increasing values of the Grashof number and/or Prandtl number. Furthermore, all else being equal, the boundary layers are seen to be somewhat thinner in shear-thinning fluids (n < 1) than that in Newtonian (n = 1) and thicker in shear-thickening (n > 1) fluids. Further examination of these results reveals the following key trends: At low Grashof numbers, the fluid is entrained mainly from beneath the lower cylinder whereas with the increasing Grashof or Rayleigh number more fluid is entrained from the sides. This can improve the heat transfer from the upper cylinder. The increased strength of the flow can be seen past the upper cylinder due to the plume velocity effects arising from the lower cylinder. However, over the range of conditions spanned here, the flow remains attached to the surface of the lower cylinder which is in line with our previous results for a single cylinder [5]. On the other hand, a small recirculation zone is seen to have formed in shear-thickening fluids (n = 1.5) above the upper cylinder. This is also consistent with the observations of Yüncü and Batta [10] for Newtonian fluids at high Rayleigh numbers. They attributed this effect to the virtual mixed-convection nature of the flow past the upper cylinder. As noted in previous studies, shear-thinning behaviour tends to stabi-

lize the flow thereby deferring the onset of flow separation to high Rayleigh numbers whereas shear-thickening fluid behaviour somewhat advances this transition. Qualitatively, the viscosity of a shear-thinning fluid will be minimum adjacent to the heated cylinder which will increase rapidly (depending upon the value of n) away from the cylinder. This is tantamount to the formation of a virtual wall (fluid body of high viscosity) which is known to have a stabilizing influence on the flow. Of course, a shear-thickening fluid (n > 1) exhibits its maximum viscosity adjacent to the heated cylinders which progressively decreases away from the cylinders. This can be seen as an effectively larger size cylinder immersed in a mobile fluid thereby increasing the local value of the Grashof number and hence leading to the formation of a vortex. Also, the close spacing, e.g., S/D = 2, really influences the flow pattern for the upper cylinder and the heat transfer is affected by the wake. Alternately, one can see the plume rising above the lower cylinder mingle with the boundary layer formed on the upper cylinder. Due to the shear-dependent viscosity of a power-law fluid, the effect of the lower heated cylinder on the structure of the flow and temperature fields around the upper cylinder gets accentuated. Additional

1143

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

(iii)

6

6 Pr = Pr = Pr = Pr =

5

0.72 20 50 100

4

4

Average Nusselt number, Nu

3

2

2

1

0

0

0.5

1

0

1.5

0

0.5

1

1.5

0

0.5

1

1.5

4 Gr = 10 70

50 45

60 40 50

35 30

40

25 30

20 15

20

10 10 5 0

0

0.5

1

(a)

0

1.5

Power-law index, n

(b)

Fig. 16 (continued)

complications arise from the fact that the effective shear rate varies from one point to another along each isotherm thereby yielding local values of the Grashof number and Prandtl number which could appreciably differ from those based on the global values of DT, D, etc. This twist further intensifies the coupling between the velocity and temperature fields in power-law fluids. In view of the preceding discussion, it is perhaps fair to postulate that the Nusselt number should show a positive dependence on the both Grashof and Prandtl numbers which is expected to be stronger on Grashof number (because it depends on the square of viscosity) than that on the Prandtl number. It is also expected that the dependence on Grashof and Prandtl numbers is modulated by the value of power-law index, as predicted by the simplified boundary layer treatments [38–40]. Indeed all these conjectures are borne out by the local Nusselt number results presented in the next section. 4.3. Local Nusselt number distribution At the next level, one can gain further insights by examining the distribution of the Nusselt number along the surface of each cylin-

der. Owing to the symmetry of the flow and temperature fields about the vertical centreline, Figs. 14 and 15 show the results only for half of each cylinder for a range of combinations of the values of power-law index (n = 1.5, 1, 0.3 denoting shear-thickening, Newtonian and shear-thinning behaviour respectively), Prandtl number (0.72 6 Pr 6 100), Grashof number, Gr = 10 and Gr = 104 and for the extreme values of (S/D), namely, 2 and 20. Based on a detailed inspection of these figures and of the other results not shown here, the key trends can be summarized as follows. 4.3.1. Lower cylinder At low values of the Grashof and/or Prandtl number, the Nusselt number varies very little over the surface of the cylinder and this is due to poor advection under these conditions, though the value of the local Nusselt number increases with the increasing Prandtl number and this can safely be attributed to the thinning of the thermal boundary layer and hence the reduced thermal resistance to conduction heat transfer across the boundary layer. For a Newtonian fluid, the viscosity is constant and therefore, each point on the surface of each cylinder corresponds to the same value of the

1144

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

(iv) 6 Pr = Pr = Pr = Pr =

0.71 10 20 50

6

4 4

Average Nusselt number, Nu

2

0

2

0

0.3

0.6

0.9

1.2

1.5

1.8

0

Gr = 10

0

0.3

0.6

0.9

1.2

1.5

1.8

0.3

0.6

0.9

1.2

1.5

1.8

4

80 50 60

40

30

40

20 20 10

0

0

0.3

0.6

0.9

(a)

1.2

1.5

1.8

0

0

Power-law index, n

(b)

Fig. 16 (continued)

Grashof and Prandtl numbers which are identical to their values based on the far away conditions. Thus, the local value of the Nusselt number is determined solely by the temperature gradient normal to the surface of the cylinder. On the other hand, in the case of power-law fluids, the local Nusselt number is still given by the temperature gradient normal to the surface, but each point on the surface of the cylinder corresponds to the local values of the Grashof and Prandtl numbers which can differ from the values given by Eqs. (11) and (12) respectively. Owing to the imposition of the no-slip condition at the solid cylinder, the velocity gradients are expected to be a maximum at the surface of the cylinder which will decay spatially eventually becoming zero far away from the heated cylinder. Similarly, the effective rate of shearing varies from one point to another on the surface of the cylinder, consistent with Eq. (8) where the second invariant of the rate of deformation tensor, I2, is a function of the velocity gradients [53]. Intuitively, one would expect both the temperature and velocity gradients to be maximum at the front stagnation point, at least for the lower cylinder, both of which decrease, albeit at different rates, along the surface towards the rear stagnation point or at least up to the point of flow separation. For Newtonian fluids, the local Nusselt number

of the lower cylinder thus decreases along the surface due to the gradually decreasing temperature gradient along the surface. For power-law fluids, the value of the Nusselt number at a point on the surface is determined by the relative magnitudes of the two effects: viscosity variation and temperature gradient, i.e., the differing rates of the thinning of the momentum and thermal boundary layers. For shear-thickening fluids, these are seen to go hand in hand therefore increasing the rate of decrease with h as can be seen in the results for n = 1.5 (Fig. 15). On the other hand, these two effects are seen to oppose each other thereby leading to the occurrence of the maximum value of the Nusselt number displaced from the front stagnation point, as seen in the results for n = 0.3 (Fig. 14). Finally, suffice it to say here that the results for the lower cylinder are virtually indistinguishable from that for the single cylinder [5,45]. 4.3.2. Upper cylinder As detailed earlier, the heat transfer from the upper cylinder is influenced by the lower heated cylinder on two counts: firstly, by the so called temperature imbalance. For closely placed cylinders and under weak advection conditions (small values of Gr or Pr or

1145

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

(i) 10

8

n = 0.3

9

Lower, Pr = 0.72 Lower, Pr = 20 Lower, Pr = 50 Lower, Pr = 100 Upper, Pr = 0.72 Upper, Pr = 20 Upper, Pr = 50 Upper, Pr = 100

8 7 6

6 5

5

4

Average Nusselt number, Nu

4

3

3

2

2

1

1 0

0

5

10

15

20

0

25

Gr = 10 76

n = 0.3

68 60 52 44 36 28 20 12 4

n = 1.5

7

0

5

10

15

20

25

0

5

10

15

20

25

20

25

4

20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

n = 1.5

0

5

10

15

Inter-cylinder spacing, S/D Fig. 17. (i) Effect of S/D on surface averaged Nusselt number and comparison with single cylinder results [45], arrows- single cylinder, symbols with line – tandem cylinder. (ii) Effect of S/D on total drag coefficient and comparison with single cylinder results [45], arrows- single cylinder, symbols with line – tandem cylinder.

both), the front part of the upper cylinder is influenced by this effect, as is seen in Fig. 14 for S/D = 2 and Gr = 10. The local Nusselt number progressively increases from the front stagnation point up to about the equator (h  90°) irrespective of the value of power-law index, n. Note the much higher peak values in the case of shear-thinning fluids (n = 0.3) than that in Newtonian and shearthickening fluids. This is simply due to the lowering of the effective fluid viscosity in such a highly shear-thinning medium. On the other hand, at S/D = 20 and under weak advection, the preheating effect is virtually irrelevant and the enhancement in heat transfer from the upper cylinder is solely due to the virtual mixed-convection contribution. Indeed, Fig. 14 clearly confirms this assertion for (S/D) = 20 when the curves for the upper cylinder are virtually shifted upward. Also, since the main mode of heat transfer under these conditions is conduction, the liquid viscosity is of little rele-

vance here as can be seen from the rather negligible influence of power-law index, at (S/D) = 20 where supposedly the flow effects are important. With the gradual increase in Grashof number, advection gains strength and more entrainment of cold fluid occurs from the side also. Clearly, this will reduce the preheating effect of the lower cylinder. On the other hand, the velocity effect will enhance the Nusselt number all over the surface, as can be seen in Fig. 15 even for (S/D) = 2. For Newtonian and shear-thickening fluids where the effects arising from the shear-dependent viscosity and decreasing temperature gradient go hand in hand unlike in the case of shear-thinning fluids (n = 0.3) where the location of the peak Nusselt number is shifted closer to the front stagnation point than that at Gr = 10 in Fig. 14. At (S/D) = 20, even greater influence of the plume of the lower cylinder is clearly seen to be present.

1146

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

Total drag coefficient, CD

(ii) 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

30

n = 0.3

28

Lower, Pr = 0.72 Lower, Pr = 20 Lower, Pr = 50 Lower, Pr = 100 Upper, Pr = 0.72 Upper, Pr = 20 Upper, Pr = 50 Upper, Pr = 100

n = 1.5

26 24 22 20 18 16 14 12 10 8 6 4 2

0

5

10

15

20

0

25

Gr = 10 1

0

10

15

20

25

20

25

6

n = 0.3

0.9

n = 1.5

5.5 5

0.8

4.5

0.7

4

0.6

3.5

0.5

3

0.4

2.5 2

0.3

1.5

0.2

1

0.1 0

5

4

0.5 0

5

10

15

20

0

25

0

5

10

15

Inter-cylinder spacing, S/D Fig. 17 (continued)

4.4. Average Nusselt number While the streamline and isotherm profiles and the variation of the local Nusselt number on the surface of each cylinder provide insights at the detailed level, the surface averaged value of the Nusselt number for individual cylinders in an array or for the entire array (simply the mean of the two individual values as the driving force and surface area are equal for both cylinders in the present case) are frequently needed in process design calculations for heat transfer devices. The scaling arguments suggest these values to be a function of four dimensionless parameters: Grashof number (Gr), Prandtl number (Pr), gap between the two cylinders (S/D) and power-law index (n). Fig. 16 shows the combined effects of these factors on the average Nusselt number for each cylinder. An examination of these results shows that, all else being equal, shear-thinning fluid behaviour can augment the value of the Nusselt number by up to 200% under appropriate con-

ditions over and above that observed in Newtonian fluids. As expected, on the other hand, shear-thickening fluid behaviour (n > 1) has adverse influence and indeed the Nusselt number can decrease by up to 20%. Broadly, higher is the value of Grashof number, stronger is the influence of power-law index (n). Aside from this, the Nusselt number exhibits the expected positive dependence on the both Grashof and Prandtl numbers. At low Grashof numbers and small gaps, the preheating effect of the lower cylinder reduces the rate of heat transfer, as is evident in Fig. 16(i) at Gr = 10 and (S/D) = 2, seen to a lesser extent at (S/ D) = 4, Fig. 16(ii). This effect virtually diminishes at (S/D) = 6. In all other cases, the Nusselt number for the upper cylinder is seen to be higher than that for the lower cylinder thereby suggesting that the plume velocity (mixed-convection) effect overshadows the preheating effect. Indeed, the value of the ratios n ranges from about 0.7 to 1.3 which is consistent with the previous numerical and experimental results reported in the literature for Newto-

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

1147

(i)

(b)

(a)

Fig. 18. (i) Variation of total drag coefficient with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 2. (a) Lower cylinder; (b) upper cylinder. (ii). Variation of total drag coefficient with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 20. (a) Lower cylinder; (b) upper cylinder.

nian fluids. Fig. 17(i) shows the influence of the inter-cylinder spacing in a much more explicit manner than that seen in Fig. 16. Interestingly, while in Newtonian fluids, the Nusselt number for the lower cylinder is seen to be pretty close to the single cylinder value in the present study which is in line with the findings of many other workers, it is not so in power-law fluids, as seen here where these values are seen to differ by varying amounts under all conditions. At low Grashof numbers (Gr) while the lower cylinder results are seen to be fairly close to their single cylinder values, these are seen to differ appreciably by up to 7–8% at Gr = 104 and n = 0.3. This is again due to the differing rates of decrease of the temperature gradient and shear rate along the surface of the cylinder in the case of power-law fluids. Finally, it is appropriate to correlate the present numerical results using a simple expression thereby enabling their interpolation for the intermediate values of Gr, Pr, n and (S/D). Following the effectiveness of a composite parameter, X, in correlating the free convection results for a range of 2-D shapes and spheres [47], it will be used here also. It is defined as:

X ¼ Gr1=2ðnþ1Þ Prn=ð3nþ1Þ

ð19Þ

This definition of X, consistent with the boundary layer analysis [38,40], is similar to the Rayleigh number, and in the limiting case of the Newtonian fluid behaviour (n = 1), it leads to X = Ra1/4 which would lead to the scaling of Nu  X reported commonly in the literature in the laminar regime. On the other hand, this parameter also incorporates the fact that the rate of progressive thinning of the thermal boundary layer with respect to the Grashof number and Prandtl number is modulated by the value of the power-law index. It is thus reasonable to postulate the following functional relationship between the average Nusselt number, Nu and the composite parameter X:

Nu ¼ aXb

ð20Þ

The best values of the fitting parameters a and b with the resulting maximum and average deviations for different values of S/D are presented in Table 6. Included here are also the values for the single cylinder results from literature [45]. Altogether

1148

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

(ii)

(a)

(a) Fig. 18 (continued)

there are about 60 data points for each value of S/D. Attention is drawn to the fact that the composite parameter X is seen to play the same role as that of the Rayleigh number in Newtonian media, as is evident from the values of the exponent, b, for the lower cylinder which are nearly equal to unity. On the other hand, the slightly larger values for the upper cylinder can be attributed to the virtual mixed convection flow regime prevailing around the upper cylinder. Furthermore, while the average errors seem to be well below 20%, the maximum errors are seen to be above 100%. Detailed examination of the data showed that most of these instances relate to the lowest value of the Prandtl number used here, i.e., Pr = 0.72, corresponding to air. Strictly speaking, such low values of the Prandtl number are of little relevance in the context of power-law liquids, but these are included here for the sake of completeness and for the purpose of validation. Therefore, if these data points are excluded from the regression, the maximum deviations drop to well below 50% with only about 8 data points (out of about 250) showing deviations larger than 50% with no further discernable trends.

4.5. Drag coefficient The velocity gradients prevailing in the vicinity of the heated cylinders give rise to tangential and normal forces exerted by the fluid on the two cylinders. The component of the resultant force acting in the flow direction is expressed in terms of the usual drag coefficient, Eqs. (13) and (14) respectively. While an extensive literature exists on heat transfer correlations for variously shaped objects, little is available regarding the drag force under these conditions even in Newtonian fluids, let alone in power-law fluids [3,4] except for our recent studies [42–47]. Typical dependence of the total drag coefficient on the Grashof number, Prandtl number, power-law index and (S/D) is shown in Fig. 18. All else being equal, the mixed-convection type flow over the upper cylinder will sharpen the velocity gradients which will, in turn, give rise to greater drag force than that on the lower cylinder, as is borne out by the trends seen in Fig. 18 for both (S/D) = 2 and (S/D) = 20. Under moderate advection conditions (Gr = 104), drag coefficients for both cylinders exhibit dependence on power-law index which is

1149

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

(i)

0.8

0.8

Pr = Pr = Pr = Pr =

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0

0.5

1

1.5

0.4

0

0.5

1

1.5

0.5

1

1.5

Gr = 10 4 0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

C

DP

/ CD

0.72 20 50 100

0.4

0

0.5

1

(a)

1.5

0.4

0

power-law index, n

(b)

Fig. 19. (i) Variation of CDP/CD with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 2. (a) Lower cylinder; (b) upper cylinder. (ii). Variation of CDP/ CD with Grashof number (Gr), Prandtl number (Pr) and power-law index (n) at S/D = 20. (a) Lower cylinder; (b) upper cylinder.

qualitatively similar to that seen for the other 2-D shapes, as discussed in detail elsewhere [46–48]. On the other hand, at low Grashof and/or Prandtl numbers, the drag coefficient shows a complex dependence on power-law index irrespective of the value of (S/D). At this juncture, it is useful to mention here that for a power-law fluid, viscous forces scale as  U nC and inertial forces vary as U 2C . Thus, all else being equal, viscous forces show a much weaker dependence on the velocity in shear-thinning fluids than that in Newtonian and in shear-thickening fluids. This coupled with the fact that the drag of a cylinder is dominated by the form drag suggests that the ratio (CDP/CDF) will be greater in shear-thinning fluids than that in shear-thickening fluids. Indeed, the results shown in Fig. 19 lend support to this conjecture. Furthermore, the effect of power-law index on the ratio (CDP/CD) is seen to diminish with the increasing Prandtl number for both cylinders for (S/D) = 2 and 20. In order to predict the drag coefficient at the intermediate values of the parameters, a simple correlation is developed in terms of X defined in the earlier as follows as:

C D ¼ dXe ðð3n þ 1Þ=4nÞÞf

ð21Þ

The best values of the fitting parameters d, e and f for each value of S/D are reported in Table 7. Since the Rayleigh number plays the same role in free convection as that of the Reynolds number in forced convection, one would expect the value of the exponent e to be close to -1. Indeed, the values shown in Table 7 lend support to this conjecture. Once again, while the average deviations are well below 20%, the maximum deviations in few cases are of the order of 75%. Like in the case of the mean Nusselt number, if the data points pertaining to Pr = 0.72 are excluded, there are only about 10 data points showing errors larger than 50%. Another possible reason is that the composite parameter X is really deduced from the scaling of boundary layer equations which implicitly assume high values of the Grashof number and/or of the Prandtl number (and hence of X) whereas the present work includes values of the Grashof number as low as 10. No further explanation can be offered for the rather large deviations seen in Tables 6 and 7.

1150

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

Gr = 10

(ii)

0.8

0.8

Pr = Pr = Pr = Pr =

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0

0.5

1

0.4

1.5

0

0.5

1

1.5

0.5

1

1.5

Gr = 10 4 1

0.8

C

DP

/ CD

0.72 20 50 100

0.9 0.7 0.8

0.6

0.7

0.6 0.5 0.5

0.4

0

0.5

1

0.4

1.5

(a)

0

Inter-cylinder spacing, S/D

(b)

Fig. 19 (continued)

Table 6 Fitting parameters for average Nusselt number (Nu) in Eq. (20) for different S/D ratios. S/D Lower 2 4 6 20 1 Upper 2 4 6 20 1

a

b

Max. % deviation

Table 7 Fitting parameters for total drag coefficient (CD) in Eq. (21) for different S/D ratios.

Avg. % deviation

S/D

d

e

f

Max. % deviation

Avg. % deviation

Lower 2 4 6 20 1

11.49 14.21 14.70 14.93 44.65

1.025 1.109 1.111 1.113 1.172

0.8410 0.9722 0.9676 0.9683 0.4926

74.25 30.56 65.58 36.04 58.58

14.72 12.45 15.01 15.15 18.53

14.72 22.71 26.73 41.2

0.9429 1.117 1.151 1.296

0.4880 0.3150 0.1563 0.3415

55.14 30.36 35.36 50.04

14.95 8.69 9.27 15.39

0.6902 0.6419 0.6494 0.5896 0.7189

0.9551 0.9811 0.9771 1.004 0.9235

132.8 58.74 58.35 57.47 49.8

15.22 11.87 12.86 13.99 11.45

0.3023 0.4780 0.5651 0.6235

1.188 1.124 1.089 1.093

152.6 43.81 71.49 39.28

17.94 9.89 15.09 11.48

5. Conclusions In this work, the laminar free convection from a pair of two circular cylinders arranged in a vertical configuration in power-law

Upper 2 4 6 20 1

fluids has been studied numerically. The results presented herein span the following ranges of conditions: 10 6 Gr 6 104, 0.72 6 Pr 6 100, 0.3 6 n 6 1.5 and 2 6 (S/D) 6 20 to elucidate the influence of each of them on the streamline and isotherm patterns,

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

distribution of the local Nusselt number on the surface of each cylinder as well as on their average values. The fluid mechanical aspects are characterised in terms of the individual and total drag coefficients as functions of Gr, Pr, n and (S/D). Broadly speaking, the behaviour of the lower cylinder is little influenced by the upper cylinder. On the other hand, the momentum and heat transfer characteristics of the upper cylinder are determined by the two competing processes: preheating effect and the effectively mixed convection effects. Their relative magnitudes are seen to be strongly influenced by the values of the Gr, Pr, (S/D) and n. Thus, the convective heat transfer coefficient of the upper cylinder may be lower or higher than that of the lower cylinder depending upon the relative magnitudes of the two above-noted mechanisms. This ratio is seen to vary from 0.7 to 1.3, which is qualitatively similar to that reported in Newtonian fluids otherwise under identical conditions. On the other hand, all else being equal, shear-thinning behaviour (n < 1) is seen to promote the rate of heat transfer; however, shear-thickening behaviour has an adverse influence on the overall rate of heat transfer. Extensive comparisons with the previous numerical and experimental results (limited to Newtonian fluids) lend credibility to the reliability of the present new results for power-law fluids. Based on the present numerical data, reliable simple predictive expressions have been developed which permit the estimation of the Nusselt number and of drag coefficient in a new application. References [1] R.M. Fand, E.W. Morris, M. Lum, Natural convection heat transfer from horizontal cylinders to air, water and silicone oils for Rayleigh numbers between 3  102 and 2  107, Int. J. Heat Mass Transfer 20 (1977) 1173–1184. [2] V.T. Morgan, The overall convective heat transfer from smooth circular cylinders, Adv. Heat Transfer 11 (1975) 199–264. [3] O.G. Martynenko, P.P. Khramstov, Free Convective Heat Transfer, Springer, New York, 2005. [4] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancy-Induced Flows and Transport, Hemisphere Publishing, New York, 1988. [5] A. Prhashanna, R.P. Chhabra, Laminar natural convection from a horizontal cylinder in power-law fluids, Ind. Eng. Chem. Res. 50 (2011) 2424–2440. [6] D.S. Steinberg, Cooling Techniques for Electronic Equipment, second ed., Wiley, New York, 1991. [7] J.E. Hesselgreaves, Compact Heat Exchangers, Pergamon Press, Oxford, 2001. [8] S. Kakac, H. Liu, Heat Exchangers: Selection, Rating and Thermal Design, second ed., CRC Press, Boca Raton, FL, 2002. [9] S.K. Park, K.S. Chang, Numerical study on interactive laminar natural convection from a pair of vertically separated horizontal cylinders, Numer. Heat Transfer 14 (1988) 61–74. [10] H. Yüncü, A. Batta, Effect of vertical separation distance on laminar natural convective heat transfer over two vertically spaced equi-temperature horizontal cylinders, Appl. Sci. Res. 52 (1994) 259–277. [11] R. Chouikh, A. Guizani, M. Maâlej, A. Belghith, Numerical study of the laminar natural convection flow around an array of two horizontal isothermal cylinders, Int. Commun. Heat Mass Transfer 26 (1999) 329–338. [12] R. Chouikh, A. Guizani, A. El Cafsi, M. Maalej, A. Belghith, Experimental study of the natural convection flow around an array of heated horizontal cylinders, Renew. Energy 21 (2000) 65–78. [13] M. Corcione, Correlating equations for free convection heat transfer from horizontal isothermal cylinders set in a vertical array, Int. J. Heat Mass Transfer 48 (2005) 3660–3673. [14] E.RG. Eckert, E.E. Soehngen, Studies on heat transfer with laminar free convection with the Zehnder–Mach interferometer, Tech Rep. 5747, USAF Air Material Command, 1948. [15] G.F. Marsters, Arrays of heated horizontal cylinders in natural convection, Int. J. Heat Mass Transfer 15 (1972) 921–933. [16] J. Lieberman, B. Gebhart, Interactions in natural convection from an array of heated elements, experimental, Int. J. Heat Mass Transfer 12 (1969) 1387– 1396. [17] E. Paykoc, H. Yüncü, M. Bezzazog˘lu, Laminar natural convective heat transfer over two vertically spaced isothermal horizontal cylinders, Exp. Therm. Fluid Sci. 4 (1991) 362–368. [18] M.S. Sadeghipour, M. Asheghi, Free convection heat transfer from arrays of vertically separated horizontal cylinders at low Rayleigh numbers, Int. J. Heat Mass Transfer 37 (1994) 103–109. [19] E.M. Sparrow, J.E. Niethammer, Effect of vertical separation distance and cylinder-to-cylinder temperature imbalance on natural convection for a pair of horizontal cylinders, J. Heat Transfer 103 (1981) 638–644.

1151

[20] I. Tokura, H. Saito, K. Kishinami, K. Muramoto, An experimental study of free convection heat transfer from a horizontal cylinder in a vertical array set in free space between parallel walls, J. Heat Transfer 105 (1983) 102–107. [21] O. Reymond, D.B. Murray, T.S. O’Donovan, Natural convection heat transfer from two horizontal cylinders, Exp. Therm. Fluid Sci. 32 (2008) 1702–1709. [22] T. Persoons, I.M. O’Gorman, D.B. Donoghue, G. Byrne, D.B. Murray, Natural convection heat transfer and fluid dynamics for a pair of vertically aligned isothermal horizontal cylinders, Int. J. Heat Mass Transfer 54 (2011) 5163– 5172. [23] M. Ashjaee, T. Yousefi, S. Farahmand, E. Chavoshi, Enhancement of free convection heat transfer from a vertical array of isothermal cylinders by flow diverters, Heat Transfer Eng. 30 (2009) 197–206. [24] M. Ashjaee, T. Yousefi, Experimental study of free convection heat transfer from horizontal isothermal cylinders arranged in vertical and inclined arrays, Heat Transfer Eng. 28 (2007) 460–471. [25] R. Karvinen, T. Kauramäki, Effect of orientation on natural convection of a cylinder array in water, Int. Commun. Heat Mass Transfer 13 (1986) 155–161. [26] E.M. Sparrow, D.S. Boessneck, Effect of transverse misalignment on natural convection from a pair of parallel, vertically stacked horizontal cylinders, J. Heat Transfer 105 (1983) 241–247. [27] J.H. Heo, M.S. Chae, B.J. Chung, Influences of vertical and horizontal pitches on the natural convection of two staggered cylinders, Int. J. Heat Mass Transfer 57 (2013) 1–8. [28] T. Misumi, K. Suzuki, K. Kitamura, Fluid flow and heat transfer of natural convection around large horizontal cylinders: experiments with air, Heat Transfer Asian Res. 32 (2003) 293–305. [29] M.S. Chae, B.J. Chung, Effect of pitch-to-diameter ratio on the natural convection heat transfer of two vertically aligned horizontal cylinders, Chem. Eng. Sci. 66 (2011) 5321–5329. [30] A. D’Orazio, L. Fontana, Experimental study of free convection from a pair of vertical arrays of horizontal cylinders at very low Rayleigh numbers, Int. J. Heat Mass Transfer 53 (2010) 3131–3142. [31] A. Bejan, A.J. Fowler, G. Stanescu, The optimal spacing between horizontal cylinders in a fixed volume cooled by natural convection, Int. J. Heat Mass Transfer 38 (1995) 2047–2055. [32] S. Grafsrønningen, A. Jensen, Natural convection heat transfer from two horizontal cylinders at high Rayleigh numbers, Int. J. Heat Mass Transfer 55 (2012) 5552–5564. [33] S. Grafsrønningen, A. Jensen, Natural convection heat transfer from three vertically arranged horizontal cylinders with dissimilar separation distance at moderately high Rayleigh numbers, Int. J. Heat Mass Transfer 57 (2013) 519– 527. [34] R.P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, second ed., CRC Press, Boca Raton, FL, 2006. [35] R.P. Chhabra, J.F. Richardson, Non-Newtonian Flow and Applied Rheology: Engineering Applications, second ed., Butterworth-Heinemann, Oxford, UK, 2008. [36] Z. Berk, Food Process Engineering and Technology, Academic Press, New York, 2008. [37] J. Welti-Chanes, J.F. Vélez-Ruiz, G.V. Barbosa-Cánovas, Transport Phenomena in Food Processing, CRC Press, Boca Raton, FL, 2003. [38] A. Acrivos, A theoretical analysis of laminar natural convection heat transfer to non-Newtonian fluids, AIChE J. 6 (1960) 584–590. [39] M.J. Shah, E.E. Petersen, A. Acrivos, Heat transfer from a cylinder to a powerlaw non-Newtonian fluid, AIChE J. 8 (1962) 542–549. [40] W.E. Stewart, Asymptotic calculation of free convection in laminar three dimensional systems, Int. J. Heat Mass Transfer 14 (1971) 1013–1031. [41] A.V. Shenoy, R.A. Mashelkar, Thermal convection in non-Newtonian fluids, Adv. Heat Transfer 15 (1982) 143–225. [42] A. Prhashanna, R.P. Chhabra, Free convection in power-law fluids from a heated sphere, Chem. Eng. Sci. 65 (2010) 6190–6205. [43] C. Sasmal, R.P. Chhabra, Laminar natural convection from a heated square cylinder immersed in power-law liquids, J. Non Newtonian Fluid Mech. 166 (2011) 811–830. [44] C. Sasmal, R.P. Chhabra, Effect of orientation on laminar natural convection from a heated square cylinder in power-law liquids, Int. J. Therm. Sci. 57 (2012) 112–125. [45] C. Sasmal, R.P. Chhabra, Effect of aspect ratio on natural convection in powerlaw liquids from a heated horizontal elliptic cylinder, Int. J. Heat Mass Transfer 55 (2012) 4886–4899. [46] A. Chandra, R.P. Chhabra, Laminar free convection from a horizontal semicircular cylinder to power-law fluids, Int. J. Heat Mass Transfer 55 (2012) 2934–2944. [47] A.K. Tiwari, R.P. Chhabra, Laminar natural convection in power-law liquids from a heated semi-circular cylinder with its flat side oriented downward, Int. J. Heat Mass Transfer 58 (2013) 553–567. [48] M. Sairamu, R.P. Chhabra, Natural convection in power-law fluids from a tilted square in an enclosure, Int. J. Heat Mass Transfer 56 (2013) 319–339. [49] M. Sairamu, N. Nirmalkar, R.P. Chhabra, Natural convection from a circular cylinder in confined Bingham plastic fluids, Int. J. Heat Mass Transfer 60 (2013) 567–581. [50] R.P. Chhabra, Fluid flow and heat transfer from circular and non-circular cylinders submerged in non-Newtonian liquids, Adv. Heat Transfer 43 (2011) 289–417.

1152

R. Shyam et al. / International Journal of Heat and Mass Transfer 64 (2013) 1127–1152

[51] R. Shyam, R.P. Chhabra, Effect of Prandtl number on heat transfer from tandem square cylinders immersed in power-law fluids in the low Reynolds number regime, Int. J. Heat Mass Transfer 57 (2013) 742–755. [52] R. Shyam, R.P. Chhabra, Natural convection in power-law fluids from two square cylinders in tandem arrangement at moderate Grashof numbers, Heat Mass Transfer 49 (2013) 843–867.

[53] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Fluid Dynamics, second ed., vol. 1, Wiley, New York, 1987. [54] P.J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, NM, USA, 1998.