Free convection from a heated circular cylinder in confined power-law fluids

International Journal of Thermal Sciences 74 (2013) 156e173

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Free convection from a heated circular cylinder in confined power-law fluids Radhe Shyam, M. Sairamu, N. Nirmalkar, R.P. Chhabra* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 May 2012 Received in revised form 14 June 2013 Accepted 14 June 2013 Available online 24 July 2013

In this work, free convection heat transfer from a horizontal cylinder (isothermal or dissipating heat at a constant rate) placed on the horizontal axis of a square duct filled with an incompressible power-law fluid has been investigated. On the other hand, along the vertical centre line, three relative positions of the heated cylinder are considered, namely, at the centre, close to the upper wall and close to the bottom wall to elucidate the effect of the symmetric and asymmetric confinement. The governing differential equations, namely, continuity, momentum and energy have been solved numerically to elucidate the effect of the pertinent kinematic (Rayleigh number, Ra; Prandtl number, Pr; power-law index, n) and geometric parameters, especially the relative positioning of the heated cylinder with reference to the bottom wall. Overall, the present results span the wide range of conditions as 102
Ra
106; 0.71
Pr
100; 0.2
n
1.8 and 0.25
b1
0.75, and the flow is believed to be laminar and steady over this range of Rayleigh numbers. Overall, the average Nusselt number shows positive dependence on Grashof number and Prandtl number. The overall heat transfer also decreases as the cylinder gradually approaches the upper adiabatic wall. The streamline and isotherm contours reveal interesting flow patterns which show a rather strong dependence on the value of b1. Ó 2013 Elsevier Masson SAS. All rights reserved.

Keywords: Natural convection Cylinder Power-law fluid Rayleigh number Prandtl number Nusselt number

1. Introduction Due to the frequent occurrence of non-Newtonian fluid behaviour in wide ranging industrial applications, considerable research efforts have been devoted to understanding the convective transport of heat and mass in the free convection regime in such fluids in recent years. Typical examples include the re-heating of processed foodstuffs [1], melting of polymeric pellets and chips, polymer processing operations, flow and heat transfer in fibrous beds, etc. The role of shear-thinning and shear-thickening fluid viscosity on the laminar free convection from a sphere [2], cylinder [3], a square cylinder [4,5], elliptic cylinders [6] and semi-circular cylinders [7,8] in an unconfined power-law medium has been numerically investigated recently. Irrespective of the body shape, all else being equal, shear-thinning fluid behaviour was shown to enhance the rate of heat transfer whereas shear-thickening viscosity impedes it, albeit the degree of enhancement varies from one shape to another. However, the preceding studies are restricted to the case of the constant surface temperature condition prescribed on the surface of the heated object. Furthermore, the aforementioned numerical

* Corresponding author. Tel.: þ91 512 2597393; fax: þ91 512 2590104. E-mail address: [email protected] (R.P. Chhabra). 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2013.06.005

studies for finite values of the Grashof and Prandtl numbers also lend support to the scaling relationships deduced from the boundary layer analyses which are applicable in the limits of large values of the Grashof number and/or of Prandtl number or both [9,10]. In addition to the foregoing pragmatic significance of such results in process engineering calculations, such model configurations also constitute an important class of problems within the domain of transport phenomena. It is worthwhile to point out here that in the case of Newtonian fluids, viscosity is constant everywhere in the flow domain (provided it is assumed to be independent of temperature) whereas, on the other hand, it shows spatial variation in the case of power-law fluids even when its temperature-dependence is neglected. Thus, the coupling between the velocity and temperature fields is further accentuated in the case of non-Newtonian fluids. Similarly, when free convection occurs in a confined fluid medium, the type of temperature boundary conditions prescribed on the boundary walls as well as the size and shape of the enclosure exert an appreciable influence on the flow and temperature fields prevailing in the enclosure, even in the case of Newtonian fluids like air and water. In turn, these changes also manifest in the resulting values of the average Nusselt number. For instance, Larson et al. [11] studied numerically and experimentally free convection from a horizontal circular cylinder enclosed in a rectangular enclosure up to Rayleigh number values of 109 and

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

was studied numerically. Subsequently, this work has been extended to Bingham plastic fluids which introduced additional complexities due to the formation of the yielded and unyielded regions in the domain [27]. Hence, only scant results are available for two-dimensional axisymmetric shapes like square, elliptical and semi-circular cross-section cylinders [4,6e8,28]. It is worthwhile to add here that the analogous problem of forced convection heat/mass transfer in power-law fluids from variously shaped objects has begun to receive systematic attention only over the past ten years or so [29e35], as is revealed by a recent review on this topic [36]. The present study is aimed at extending our recent study [24] to the case of a circular cylinder otherwise under identical conditions. In particular, the governing partial differential equations (mass, momentum and energy) have been solved numerically for the case of a circular cylinder maintained either at a constant temperature (Case I or CWT) or dissipating heat at a constant rate (Case II or CHF) to the ambient power-law fluid confined coaxially in a long square duct, with three relative positions along the y-axis. The numerical results reported herein span the range of conditions as follows: Rayleigh number, 102
Ra
106; Prandtl number, 0.71
Pr
100 and power-law index, 0.2
n
1.8. Furthermore, the top and bottom walls of the square duct are treated as adiabatic whereas the two side walls are maintained at a constant temperature which is lower than that of the heated cylinder. Finally, while the influence of the relative positioning of the circular cylinder with reference to the centre of the enclosure was studied for three configurations, the overall aspect ratio of the system defined as b2 ¼ R/L was maintained fixed at 0.2 in this study, in line with the previous study for Newtonian fluids [37]. 2. Problem statement and governing equations Let us consider a circular cylinder (of radius, R, and infinitely long in z-direction) heated to maintain its surface either at a constant temperature Ts or it dissipates heat at a constant rate, i.e., constant heat flux (qs). It is immersed in a stagnant power-law medium (at a temperature of Tf < Ts) confined in a duct of square cross-section (side, L) as shown schematically in Fig. 1. Irrespective of the boundary condition prescribed on the surface of the heated cylinder, due to the temperature difference present in the system, density gradient in the fluid induces an upward flow thereby transferring heat from the cylinder to the fluid by free convection. Numerous possibilities exist in terms of the boundary conditions including isothermal, adiabatic or mixed type that one can impose on the confining walls. Following our recent study [24] and the

Adiabatic wall

Isothermal wall, Tf

g Ts or qs y

R

x

Centre line

l

Adiabatic wall L Fig. 1. Schematic of the flow configuration.

Isothermal wall, Tf

Prandtl number of unity. They reported the velocity and temperature fields to be stable and steady up to such high values of the Rayleigh number. Similarly, Elepano and Oosthuizen [12] studied the two-dimensional laminar free convection from an isothermally heated cylinder placed inside a rectangular enclosure with the two side and bottom walls maintained adiabatic and the top wall being isothermal at a lower temperature than the cylinder itself. Based on their numerical results for air (Pr ¼ 0.7), they found that the mean Nusselt number was influenced primarily by the value of the Rayleigh number while the positioning of the cylinder with reference to the top wall had negligible influence on heat transfer up to Ra
105. In an attempt to develop a benchmark study for validating the efficacy of numerical solution methodology, Demirdzic et al. [13] provided accurate numerical results for the case of free convection from an isothermal circular cylinder enclosed in a square duct with its horizontal walls being adiabatic and the vertical side walls of isothermal nature. They reported detailed results for two values of the Prandtl number, Pr ¼ 0.1 and Pr ¼ 10 at a fixed value of the Rayleigh number, Ra ¼ 106. The effect of the relevant geometrical factors like the aspect ratio of the enclosure, ratio of the size of the cylinder and the enclosure, and the relative positioning of the heated cylinder with respect to that of the enclosure has been studied numerically by Ghaddar [14] for air as the working fluid (Pr ¼ 0.71) for the case of a cylinder dissipating heat at a constant rate. The case of an isothermal cylinder enclosed in an enclosure of cold (isothermal) walls has been investigated by Moukalled and Acharya [15] for air as the working fluid spanning the range of Ra
107. Their results clearly show a positive dependence of the aspect ratio (cylinder diameter/side of square enclosure) on the average Nusselt number. Cesini et al. [16] reported an extensive numerical-cum-experimental study on the temperature and Nusselt number distribution for an isothermal cylinder enclosed in a cavity of varying widths. Their experimental results span a rather narrow range of Rayleigh number (1.3  103
Ra
3.4  103) and these are also limited to Pr ¼ 0.71 or so, albeit their numerical predictions encompass the values of Rayleigh number up to about w105. Other similar studies relating to different geometries, or the values of the Rayleigh number based on a wide ranging numerical techniques including finite volume method [17,18], differential quadrature method [19], immersed boundary method [20,21], etc. have been reported in the literature. Suffice it to say here that most of the aforementioned studies show good agreement with each other. In contrast, the studies involving water as the working fluid (Pr ¼ 7e8) are indeed scarce [22,23]. Thus, all in all, the flow and temperature patterns in buoyancy induced flow in an enclosure are governed by an intricate interplay between the shape and geometry of the heated object as well as that of the enclosure, the values of Rayleigh and Prandtl numbers and the type of thermal boundary conditions prescribed on the surface of the heated object and the walls of the enclosure. In summary, a reasonable body of information is now available on the free convection from a circular and a square cylinder in Newtonian fluids in such enclosures. Depending upon the geometry and boundary conditions, the rate of free convection may increase or decrease with reference to that in an unconfined medium as summarized in a recent study by Sairamu and Chhabra [24]. Also, it is appropriate to add here that even in unconfined Newtonian fluids, the bulk of the available literature pertains to the case of a sphere, followed by that for a circular cylinder and a plate in various orientations [2,3,25,26]. In contrast, much less is known about the analogous flow with power-law type non-Newtonian fluids. As far as known to us, there has been only one such study with a tilted square cylinder immersed in power-law fluids in a square duct [24]. In this work, the effect of power-law rheology on the rate of heat transfer from a heated square bar in a square box filled with a power-law medium

157

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R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

other prior studies involving Newtonian fluids [37,38], in this work the top and bottom horizontal walls are assumed to be adiabatic whereas the side (vertical) walls of the enclosure are maintained at a uniform temperature of Tf. However, the two commonly used conditions, namely, constant temperature (Case I) and constant heat flux (Case II) were prescribed on the surface of the cylinder in order to delineate the role of boundary condition on heat transfer. Furthermore, intuitively it appears that the buoyancy-induced flow would be strongly influenced by the positioning of the heated cylinder in the confining duct. In this work, three cases, namely, close to the top surface, symmetric positioning at the centre and close to the bottom surface have been considered, albeit in each case the cylinder was located on the x ¼ 0 plane (Fig. 1). As the cylinder is infinitely long in the z-direction, the flow is assumed to be two-dimensional, i.e., Vz ¼ 0 and there are no gradients in the z-direction, i.e., v()/vz ¼ 0. Furthermore, over the range of conditions spanned here, the flow is expected to be laminar and steady. This assertion is based on the findings of previous workers in Newtonian fluids, e.g., see Ref. [11] wherein the flow was found to be stable and steady up to Rayleigh number values of 109 which is three orders of magnitude higher than the maximum value of the Rayleigh number of 106 used in the present study. In the absence of corresponding information for power-law fluids, the assumption of steady flow regime is reasonable in the present case, at least as a first approximation. As will be seen later, this conjecture is borne out by the limited time-dependent simulations performed in this study and are discussed in a later section. Next, the thermo-physical properties of the fluid (density r, thermal conductivity, k, heat capacity, c, power-law constants, m, n) are assumed to be temperature-independent except for the fluid density appearing in the body force term in the y-component of the momentum equation. Hence the results presented herein are applicable to situations wherein the maximum value of DT ¼ (Ts  Tf) is small so that one can evaluate these properties at the mean film temperature (Ts þ Tf)/2. Finally, the fluid is also assumed to be incompressible except for the body force term (in the y-momentum equation) which is written using the Boussinesq approximation for the variation of density with temperature. Within the framework of these assumptions, the coupled velocity and temperature fields are governed by the continuity, momentum and thermal energy equations (written in their dimensionless forms) as follows:

In free convection, fluid flow is caused entirely by the buoyancy force, thus the equilibrium of inertial and buoyancy forces can be written as follows:

rf Vc2 R

wrf gbDT

Therefore, from Eq. (5) the characteristic velocity scale is obpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RgbDT for case I and Vc ¼ R g bqs =k for case II. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Therefore, in Eqs. (1)e(4), the velocity is scaled by Rg bDT or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R g bqs =k for Case I and Case II respectively and the linear distance is scaled using the radius of the cylinder, R. The temperature is scaled as q ¼ (T  Tf)/(Ts  Tf) or q ¼ (T  Tf)/(qsR/k) depending upon the boundary condition prescribed on the surface of the cylinder. Over the narrow range of temperature variation entailed here, the Boussinesq approximation is adequate to capture the variation of density with temperature as follows: tained as Vc ¼

  r  rf ¼ b Tf  T rf

1 vr b¼  rf vT

(7) T

For an incompressible power-law fluid, the components of the extra stress tensor are related to the rate of deformation tensor as:

sij ¼ 2heij where ði; jÞ ¼ ðx; yÞ

 x-momentum:

(2)

 y-momentum:



    v Vy Vx v Vy Vy vsxy vsyy vp 1 þ þ ¼  þ q þ pffiffiffiffiffiffi vy vy vx vy Gr vx

(3)

 Thermal energy equation:

vq vq 1 Vx þ Vy ¼ 1 vx vy Pr$Grðnþ1Þ

v2 q v2 q þ vx2 vy2

(8)

and the components of the rate of deformation tensor, eij are given as, see Bird et al. [39]:

(1)


   vðVx Vx Þ v Vx Vy vp 1 vsxx vsyx þ þ ¼  þ pffiffiffiffiffiffi vy vx vx vy Gr vx

(6)

where rf is the density of the fluid at the reference temperatureTf; b is the volumetric expansion coefficient at constant temperature and it is defined as:

 Continuity:

V$V ¼ 0

(5)

! (4) Fig. 2. Zoomed view of mesh in the vicinity of cylinder.

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173 Table 1 Comparison of the present results obtained from the steady and transient solutions. n

Nuavg (b1 ¼ 0.25) Steady solution

Transient solution

Ra ¼ 106, Pr ¼ 100 (CWT) 0.2 13.824 13.810 1 17.835 17.832 1.8 10.462 10.456

8.9507 13.445 8.2917

8.9321 13.432 8.3125

Ra ¼ 106, Pr ¼ 0.71 (CWT) 0.2 61.350 61.323 1 16.445 16.502 1.8 5.7188 5.7272

40.659 12.856 4.6991

40.675 12.863 4.7025

Ra ¼ 106, Pr ¼ 100 (CHF) 0.2 6.2374 6.2410 1 12.055 12.123 1.8 10.186 10.210

5.1793 10.509 8.0200

5.1802 10.509 8.0210

Ra ¼ 106, Pr ¼ 0.71 (CHF) 0.2 16.395 16.402 1 11.276 11.291 1.8 4.8094 4.8251

33.310 26.828 25.738

33.326 26.840 25.752

eij ¼

1 vVi vVj þ vi 2 vj

definitions of the Grashof and Prandtl numbers are different for the two cases.

Nuavg (b1 ¼ 0.75) Steady solution



159

Transient solution

Case I: For constant wall temperature (CWT):

Gr ¼

r2f ðgbDT=RÞ2n R4

(12)

m2

rf C m Pr ¼ k rf

!

2 ðnþ1Þ

2ðn1Þ ðnþ1Þ



. 3ðn1Þ 2ðnþ1Þ g bDT R

R

(13)

Case II: For the constant heat flux (CHF):

Gr ¼

 (9)

r2f ðgbqs =kÞ2n R4

(14)

m2

rf C m Pr ¼ k rf

!

2 ðnþ1Þ

2ðn1Þ ðnþ1Þ

. 3ðn1Þ 2ðnþ1Þ g bqs k



R

(15)

Owing to the non-viscometric flow conditions in the present case, it is appropriate to write the non-dimensional viscosity h in terms of the second invariant of the rate of deformation tensor as:

The contemporary literature on free convection employs another dimensionless group, namely, Rayleigh number, Ra, which is given simply by the product of the Grashof number and Prandtl number, i.e.,

h ¼ ðI2 =2Þðn1Þ=2

Ra ¼ Gr$Pr

(10)

The stress components and viscosity have been rendered pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionless using mð RgbDT =RÞn or mð g bqs =kÞn and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi mð RgbDT =RÞn1 or mð g bqs =kÞn1 respectively. The second invariant I2 itself is given by Bird et al. [39]:

I2 ¼

XX i

eij $eji

(11)

j

Finally, no-slip (Vx ¼ 0, Vy ¼ 0) boundary condition is implemented on all four walls of the confining duct and on the surface of the cylinder. While q ¼ 0 is prescribed on the vertical side walls, the condition of (vq/vy) ¼ 0 is employed on the top and bottom horizontal walls. For the case of isothermal cylinder (Case I), q ¼ 1 is used and when the cylinder is dissipating heat at a constant rate (Case II), the corresponding boundary condition on the surface of the cylinder is given by vq/vns ¼ 1 where ns is the outward drawn unit vector normal to the surface of the cylinder. The two dimensionless groups appearing in Eqs. (2)e(4), namely, Grashof number, Gr and Prandtl number, Pr are defined here. Since the characteristic temperature difference in the two cases is given as (Ts  Tf) and (qsR/k) respectively, the resulting

(16)

However, out of these three parameters, Gr, Pr and Ra, only two are independent non-dimensional groups which characterize this flow, in addition to the power-law index (n) which is a dimensionless group in its own right. Following the prior literature studies, the Rayleigh and Prandtl numbers are employed in this work to present and discuss the results here. At this juncture, it is worthwhile to make two observations: firstly, in the limit of n / 1, Eqs. (12)e(15) reduce to their expected forms for Newtonian fluids. Secondly, unlike in the case of Newtonian fluids wherein the Prandtl number is a physical property group, in the present case, it shows complex dependence on the size of the cylinder and the characteristic temperature difference. This indeed makes the delineation of the role of power-law index on the resulting velocity and temperature fields far from being straightforward. Thus, for a fixed geometric configuration, while the local Nusselt number will show spatial variation along the surface of the cylinder, the average Nusselt number is expected to be a function of three dimensionless groups, namely, Ra, Pr and n for a given set of boundary conditions prescribed on the surface of the heated cylinder and on the walls. This work endeavours to explore and develop this relationship.

Table 2 Comparison between the present and literature values of the average Nusselt number (Nuavg) for CWT case for Pr ¼ 0.7 and b1 ¼ 0.5 (walls of enclosure at constant temperature). Ra

Nuavg

b2 ¼ 0.1 104 105 106 107

b2 ¼ 0.2

Present

Moukalled and Acharya [15]

Present

Moukalled and Acharya [15]

Shu and Zhu [19]

Ren et al. [21]

Hussain and Hussein [18]

6.5845 12.0130 19.4297 32.2696

6.5913 12.1738 19.4367 32.6384

5.1335 7.8104 14.1690 23.8943

5.3016 8.0853 14.9196 25.1313

5.1568 7.7351 14.1652 e

5.0310 7.6970 13.6018 e

5.4189 8.1631 14.9430 24.9873

160

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

Table 3 Comparison between the present values and that of Kim et al. [20] of the average Nusselt number (Nuavg) for CWT case for Pr ¼ 0.7 (isothermal walls of the enclosure). Ra

Nuavg Kim et al. [20]

Present

8.1905 8.6191 11.3333 16.7619

8.2172 8.6517 11.3668 16.7599

5.0925 5.1428 7.7619 14.1429

5.0408 5.1336 7.8104 14.1691

8.1905 8.2857 10.0476 14.3810

8.2170 8.3441 10.1084 14.4296

b1 ¼ 0.25 103 104 105 106

b1 ¼ 0.5 103 104 105 106

b1 ¼ 0.75

103 104 105 106

For fixed values of Ra, Pr, n and b1, the numerical solution of Eqs. (1)e(4) subject to the aforementioned boundary conditions maps the flow domain in terms of the primitive variables Vi(x,y), p(x,y) and q(x,y) which, in turn, can be post processed to evaluate the secondary variables of interest like isotherm contours, streamlines, local Nusselt number and the average Nusselt number. At this juncture, it is thus appropriate to introduce their definitions. By writing an energy balance on the surface of the cylinder, it is easily seen that the local Nusselt number, Nu, is given by:

hd vq for CWT ðCase IÞ ¼  k vns hd 1 Nu ¼ ¼ for CHF ðCase IIÞ q k

Nu ¼

(17a) (17b)

The surface averaged value Nuavg is obtained simply by integrating the local values over the surface of the cylinder.

Fig. 3. Comparison of the present (shown as lines) and Moukalled and Acharya [15] values of the dimensionless centre line vertical velocity for Pr ¼ 0.7, b1 ¼ 0.5 and b2 ¼ 0.1 (CWT case). (B Ra ¼ 104, - Ra ¼ 105, , Ra ¼ 106, C Ra ¼ 107).

section have been solved numerically using the finite volume solver FLUENT (version 6.3.26). Since detailed descriptions of the solution procedure are available elsewhere [4,24], it is not repeated here. Similarly, since the reliability and accuracy of the numerical results is strongly influenced by the type of mesh used, extensive experimentation has been carried out to select an optimal grid. Needless to say here that since the velocity and temperature gradients are expected to be steepest close to the surface of the heated cylinder and the boundary walls, non-uniform mesh was used in these regions to meet the conflicting requirements of resolving the velocity and temperature fields adequately without necessitating expansive

3. Numerical solution scheme and choice of parameters In this work, the governing differential equations subject to the aforementioned boundary conditions outlined in the preceding Table 4 Comparison between the present values and that of Hussain and Hussein [18] of the average Nusselt number (Nuavg) for CHF case for Pr ¼ 0.7 (isothermal walls of the enclosure). Ra

Nuavg Hussain and Hussein [18]

Present

7.8823 8.1069 9.9679 13.3690

7.9360 8.0102 9.2901 12.1380

5.0588 5.0909 6.4706 10.7380

5.0399 5.0487 5.6637 9.2546

7.8823 7.8182 8.6203 11.6684

7.9262 7.9017 8.1657 10.5549

b1 ¼ 0.25 103 104 105 106

b1 ¼ 0.5 103 104 105 106

b1 ¼ 0.75 103 104 105 106

Fig. 4. Comparison of the present (shown as lines) and Cesini et al. [16] values of the Local Nusselt number for Pr ¼ 0.71, b1 ¼ 0.5, W* ¼ 2.1, H* ¼ 4.07 (CWT case). (B Ra ¼ 1.3  103, , Ra ¼ 2.4  103, 7 Ra ¼ 3.4  103) filled e Numerical values, unfilled e Experimental values.

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173 Table 5 Comparison between the present values and that of Cesini et al. [16] of the average Nusselt number (Nuavg) for Pr ¼ 0.71, b1 ¼ 0.5, W* ¼ 2.1, H* ¼ 4.07(CWT case). Ra

1.3  103 2.4  103 3.4  103

Nuavg Present

Cesini et al. [16] Numerical

Experimental

2.39 2.63 2.79

2.36 2.61 2.77

2.46 2.80 3.07

computational resources. Following the approach employed in the recent work by Sairamu and Chhabra [24], three grids were used for this purpose. Each grid was characterized in terms of the number of control volumes on the surface of the cylinder (Np) and the total number of cells (N) in the entire computational domain, and the relative minimum grid spacing (d/R) on the surface of the cylinder. Since the boundary layer thickness is expected to be minimum corresponding to the maximum values of Rayleigh number (Ra), Prandtl number (Pr) and minimum value of power-law index (n) [40], grid independence tests have been carried out at Ra ¼ 106, Pr ¼ 100, and n ¼ 0.2 for each value of b1 used in this work. Irrespective of the type of boundary condition, i.e., CWT or CHF, prescribed on the surface of the cylinder, a grid with Np ¼ 160 and d/ R ¼ 0.0039 was found to be satisfactory. On the other hand, the value of N (total number of cells) varied as 46,316, 34,624 and 46,286 for b1 ¼ 0.25, 0.5 and 0.75 respectively. A zoomed view of the present mesh distribution is shown in Fig. 2. For asymmetric

161

placement of the cylinder, i.e., b1 ¼ 0.25 and 0.75, more control volumes were employed to achieve the desired level of accuracy. Finally, over the range of conditions spanned here, the buoyancy induced flow is expected to be steady and laminar (and hence symmetric about y-axis), therefore the computations have been carried out only in half-domain, i.e., x  0 in order to economize on the computational effort. Furthermore, the validity of the steady flow assumption has been investigated here by performing a few full domain time-dependent computations for scores of values of the Rayleigh number, Prandtl number and the two boundary conditions, as shown in Table 1. Clearly, the average Nusselt number values computed by using the transient simulations are seen to be close to the values returned by the steady state solution. This justifies the assumption of the steady flow over the range of conditions spanned here. The adequacy of the selected solution mesh, convergence criterion and the other numerical aspects is further demonstrated in the next section where the present results are benchmarked against the literature values. 4. Results and discussion As noted earlier, extensive numerical results on the temperature field and Nusselt number have been obtained over the following ranges of conditions: 102
Ra
106; 0.71
Pr
100; 0.2
n
1.8 for the values of b1 ¼ 0.25, 0.5 and 0.75. Some discussion about the range of values of power-law index (n), Prandtl number (0.71e100) and Rayleigh number (102e106) used here is in order. Most polymeric melts and solutions exhibit values of the power-law index as low as 0.2e0.4 [41,42]. On the

β1 = 0.25

β1 = 0.5

20

24

20

Local Nusselt number, Nu

Local Nusselt number, Nu

a b c

16

12

8

4

0

16

12

8

4

0 a

b

c

b

c

β1 = 0.75

24

Local Nusselt number, Nu

a

Distance along the cylinder

Distance along the cylinder

20

16

12

8

4

0

a

b

c

Distance along the cylinder Fig. 5. Comparison of the present (shown as lines) and Kim et al. [20] values of the local Nusselt number on the surface of the cylinder for Pr ¼ 0.71 for CWT case. (B Ra ¼ 103, 4 5 6 - Ra ¼ 10 , , Ra ¼ 10 , C Ra ¼ 10 ).

162

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

other hand, typical starch-in-water and corn flour-in-water dispersions used in food processing applications are characterized by the values of the power-law index, n ¼ 1.4e1.6 or so [1]. Similarly, most power-law fluids tend to be far more viscous than their Newtonian counterparts like air and water and therefore, the resulting values of the Grashof number are not expected to be as high as in the case of air and water, as can be ascertained from the available scant experimental results for an unconfined sphere and cylinder [2,3]. Similarly, many such fluids exhibit a value of the Prandtl number as high as 100 or even greater [41,42]. It is readily conceded that extremely fine computational meshes are needed to resolve the extremely thin boundary layers under these conditions. On the other hand, more than two orders of magnitude variation in the value of the Prandtl number is regarded to be sufficient for the purpose of delineating the scaling of the Nusselt number with Prandtl number. While admittedly, no power-law fluid corresponds to Pr ¼ 0.71, these results are used here for the purpose of validation and thus these are included here. In summary, the range of parameters spanned here is based on a combination of practical and numerical considerations. However, prior to undertaking the detailed presentation and discussion of the new results obtained in this study, it is desirable to benchmark the efficiency of the present numerical solution methodology to ascertain the reliability and precision of the new results.

4.1. Validation of results As noted earlier, many investigators have reported numerical results for the problem studied herein with air as the working fluid (Pr ¼ 0.7e0.71). These are used here to establish the reliability and precision of the new results for power-law fluids reported herein. Tables 2e4 show comparisons with prior results culled from various sources and spanning wide ranges of geometrical parameters (b1, b2), value of the Rayleigh number (based on L as the characteristic linear dimension) and for both CHF and CWT conditions imposed on the surface of the cylinder. An inspection of Table 2 reveals near perfect correspondence between the present and literature values of the average Nusselt number, Nuavg ¼ hL/k for a fixed value of b1 ¼ 0.5. Table 3 shows a similar comparison for three different values of b1 and match is seen to be excellent once again. Table 4 shows a validation for the case of a cylinder dissipating heat (case II) at a constant rate to air in an isothermal enclosure. Up to Ra ¼ 104, the two values are within 1% of each other, except for the case of Ra ¼ 106 where the two values differ from each other by 14%. While the exact reasons for such a large discrepancies are not immediately obvious, a detailed examination of isotherm contours at Ra ¼ 105 and Ra ¼ 106 presented by Hussain and Hussein [18] indicates inadequate resolution of the steep velocity and temperature gradients in thin boundary layers at such high values of the Rayleigh number. It is also appropriate to add

Fig. 6. Representative streamline (left half) and isotherm (right half) profiles for CWT at b1 ¼ 0.5.

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

here that the numerical mesh used in the present work consists of almost twice as many control volumes as that in their study and hence the present predictions are regarded to be more reliable at high Rayleigh numbers. However, a few simulations were performed with a numerical mesh similar to the one used in their study, the gap between the two predictions decreased a little, but still cannot explain the discrepancy of 14%. The only other difference is in the type of cells used in the two studies. No more explanation can be given for this case at this stage. On the other hand, the validation shown in Tables 2e5 is regarded to be sufficient to demonstrate the reliability of the new results reported herein. Finally, Fig. 3 shows a comparison between the present values of the dimensionless Vy (scaled using m/rL) at the centerline of the enclosure with that of Moukalled and Acharya [15]. The two predictions are seen to be in near perfect agreement. In our view, the close match seen in Fig. 3 constitutes a much more stringent test of the reliability of the present predictions than that seen in Tables 2e4. This section is concluded by showing a comparison of the present numerical results with the experimental results of Cesini et al. [16] in Fig. 4 and in Table 5. While the present numerical predictions are in agreement with their results, the match with the experimental results is comparable in the two cases. Fig. 5 shows representative comparisons between their and present results in terms of the local Nusselt number. Unfortunately, no

163

experimental results are available for power-law fluids in the configuration studied here whereas the comparisons for an unconfined cylinder are reported elsewhere [3]. Based on the aforementioned extensive comparisons, the new results for power-law fluids are considered to be reliable within w2%. 4.2. Streamline and isotherm patterns Since in the present case, the flow is induced by temperaturedependent density, the geometric configuration of the heated cylinder exerts an appreciable influence on the detailed structure of the velocity and temperature fields. Thus, for instance, if the heated cylinder is placed close to the upper adiabatic wall, the bulk of the liquid beneath the heated cylinder participates very little in the overall heat transfer process. On the other hand, if the heated cylinder is close to the bottom adiabatic surface, the bulk of the fluid present above it partakes in heat transfer by free convection. This aspect is captured here by varying the value of b1. However, this effect is further compounded by the fact that the rate of deformation progressively decreases away from the heated cylinder which is accompanied by progressively increasing effective viscosity for a shear-thinning fluid (n < 1) which in turn adversely influences the rate of heat transfer. Therefore, the velocity and temperature fields in this case are determined by a

Fig. 7. Representative streamline (left half) and isotherm (right half) profiles for CWT at b1 ¼ 0.25.

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complex interplay between the geometry and the values of the Rayleigh number, Prandtl number and power-law index. Figs. 6e8 show representative streamline (left half) and isotherm (right half) patterns for a range of combinations of the Rayleigh number (Ra), Prandtl number (Pr), power-law index (n) and the relative position of the cylinder along the y-axis. We begin with the symmetric case when the centre of the heated cylinder coincides with the centre of the enclosing duct (Fig. 6). At low values of the Rayleigh number (Ra ¼ 100), the power-law index (n) is seen to have very little influence on the temperature field. This is simply so due to weak advection under these conditions and heat transfer occurs primarily by conduction therefore the fluid viscosity is largely irrelevant under these conditions. For conduction, one would expect the isotherms to be concentric circles as is borne out by the isotherms in the close proximity of the heated cylinder and parallel to the isothermal side walls. Furthermore, at such low values of the Rayleigh number, one can also expect the flow to be symmetric about x ¼ 0 and y ¼ 0 planes as is evidenced in the streamline patterns seen at Ra ¼ 100. The double-celled flow structures seen in the streamline patterns are qualitatively similar to that observed by others for a square and circular cylinder in Newtonian fluids [20,37] and for a square cylinder in power-law fluids [24]. The power-law index (n) is seen to exert very little influence on the size of the recirculating regions. However, as the

value of the Rayleigh number is increased to Ra ¼ 106, buoyancy induced flow gains momentum and therefore, the role of conduction progressively diminishes resulting in significant bending of isotherms as can be seen in Fig. 6. More fluid is seen to be entrained from below and sides and with the progressive thinning of temperature boundary layer (crowding of isotherms close to the surface of the cylinder), a plume is formed above the cylinder. As the heated fluid rises above and due to the adiabatic top wall, it flows along the cold side walls thereby leading to the development of boundary layers along the vertical side walls. Under these conditions, the power-law index is seen to exert significant influence on the streamline patterns, as can be seen in Fig. 6 for n ¼ 0.2, n ¼ 1 and n ¼ 1.8. Firstly, in highly shear-thinning fluids (n ¼ 0.2) and in Newtonian fluids, significant flow is seen to be present above the cylinder leaving the fluid relatively calm beneath the cylinder, albeit the flow beneath the cylinder is influenced significantly by the growing boundary layers on the side walls. The double-celled structures are distorted and/or merged and shift upward irrespective of the type of fluid behaviour. Similar merging occurs in Newtonian and shear-thinning fluids at Pr ¼ 100 where further bending and crowding of isotherms and streamlines is seen to occur due to increased advection. These trends are consistent with that reported previously in the literature for Newtonian fluids.

Fig. 8. Representative streamline (left half) and isotherm (right half) profiles for CWT at b1 ¼ 0.75.

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

Now turning our attention to Fig. 7 which corresponds to the cases of b1 ¼ 0.25, i.e., the heated cylinder is close to the bottom adiabatic wall. While at low Rayleigh numbers, the isotherms resemble the ones corresponding to conduction regardless of the value of power-law index. Similarly, the flow pattern is singlecelled in each case which is due to the loss of symmetry about the y ¼ 0 plane. Naturally, in this case, the limited amount of fluid beneath the heated cylinder is entrained which partakes in the process of heat transfer. Apart from this overall behaviour, the thermal boundary layer is seen to thin with the increasing value of the Rayleigh number or Prandtl number or both. The boundary layers on the side walls also show qualitatively similar trend as is evident by the crowding of isotherms or of streamlines close to the side walls. On the other hand when the heated cylinder is close to the top adiabatic wall, i.e., b1 ¼ 0.75 (Fig. 8), due to the loss of symmetry about y ¼ 0 plane, the flow field consists of single-celled recirculating region except at high Rayleigh and/or Prandtl numbers which show stratification of cold and hot fluids beneath and above the cylinder. The lack of contribution of the fluid beneath the heated cylinder to the overall heat transfer is also evident from the shape of the isotherms seen at Ra ¼ 106 and Pr ¼ 100, where the bulk of the temperature gradient is confined to the thin boundary layer. Qualitatively similar streamlines and isotherm contours were obtained for Case II, i.e., when the condition of constant heat flux is

165

prescribed on the surface of the cylinder instead of the constant wall temperature and thus only representative results for three values of b1 but at fixed values of Ra ¼ 106 and Pr ¼ 100 are shown in Fig. 9. Heated fluid layers are confined to the proximity of the heated cylinder thereby suggesting some reduction in the overall heat transfer. For the asymmetric cases of b1 ¼ 0.25 and b1 ¼ 0.75, reduced gaps between the adiabatic walls and heated cylinder restrict the contact between the surrounding fluid and the heated cylinder as compared to the symmetric case of b1 ¼ 0.5. The aforementioned discussion of the flow patterns and isotherm contours suggests that the rate of heat transfer from the heated cylinder should show positive dependence on the Rayleigh number and roughly an inverse dependence on the value of b1. However, it is much less evident to postulate a priori the influence of the power-law index. Firstly, the power-law index (n) appears in the definitions of the Rayleigh and Prandtl numbers in an intricate manner in addition to being a dimensionless parameter in its own right. Undoubtedly, all else being equal, the boundary layers both on the cylinder and side walls tend to be thinner in shear-thinning fluids than that in Newtonian fluids and vice versa in shearthickening fluids [40,41]. This would suggest an enhancement in heat transfer in shear-thinning fluids and similarly some reduction in shear-thickening fluids. This assertion is based, however, on the assumption that the boundary layers develop in unconfined flows,

Fig. 9. Representative streamline (left half) and isotherm (right half) profiles for CHF at Ra ¼ 106, Pr ¼ 100.

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which is not the case here. Also, it is plausible that the boundary layers on the side walls and that on the cylinder may interact with each other especially at small values of the Rayleigh number or of Prandtl number or both. This effect is expected to diminish with the increasing values of Ra and/or Pr or both. On the other hand, while at low Rayleigh numbers, the fluid from below the cylinder is entrained whereas fluid from side is increasingly entrained with increasing values of the Rayleigh number. Similarly, the role of Pr is also far from obvious. Indeed all these conjectures are borne out by the local Nusselt number results presented in the ensuing sections. 4.3. Local Nusselt number distribution on the surface of the cylinder At the outset, it is useful to recall here that the free convection heat transfer is strongly influenced by the buoyancy-induced vepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi locity which is of the order of Rg bDT for case I as noted earlier. This velocity plays the same role here as that of the imposed

velocity in the case of forced convection. For Newtonian fluids, the Grashof number is directly proportional to (gbDT) whereas the Prandtl number is nearly independent of this parameter, at least over short temperature intervals, when the change in the value of Prandtl number on account of temperature variation is negligible. On the other hand, for power-law fluids, the effective viscosity is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by mðg_ Þn1 where g_ is of the order of g bDT=R. Thus, for a pffiffiffiffiffiffiffiffiffiffiffiffi cylinder of given size, g_ f g bDT and hence this parameter appears in the definitions of both Ra and Pr thereby accentuating the role of power-law index. For instance, Rafðg bDTÞf1 ðnÞ and Prfðg bDTÞf2 ðnÞ when f1(n) and f2(n) are functions of n in accordance with Eqs. (12)e(15). For n ¼ 1, Ra f (gbDT) and Pr is independent of this parameter, which is consistent with the expected behaviour for Newtonian fluids. For n ¼ 0.2, the corresponding values are f1(0.2) ¼ 0.8 and f2(0.2) ¼ 0.5. Thus, the role of (gbDT) is weakened as far as the Rayleigh number is concerned, but the Prandtl number shows inverse dependence on this parameter. At the other extreme

Ra = 102 2.5

3 nn = 0.2 nn = 0.4 nn = 0.7 nn = 1 nn = 1.5 nn = 1.8

2.5

2

2.9

a b

2.4

2

c 1.9 1.5

1.5

1.4 1

1

0.5

0.9

a

b

c

0.5

a

b

c

0.4

Distance along the cylinder

Distance along the cylinder

a

c

b

Distance along the cylinder

Local Nusselt number, Nu

Ra = 104 8

7.5

7.8

6

6.5

6

5.2 4.5 4

3.9 3 2.6

2

0

1.5

a

b

c

0

1.3

a

b

c

0

Distance along the cylinder

Distance along the cylinder

a

c

b

Distance along the cylinder

Ra = 106 40

32

30

30

25

25

20

20

15

15

10

10

5

5

c

24

16

8

0

a

b

c

Distance along the cylinder

Pr = 10

0

a

b

Distance along the cylinder

Pr = 50

c

0

a

b

Distance along the cylinder

Pr = 100

Fig. 10. Local Nusselt number (Nu) distribution on the surface of the cylinder for b1 ¼ 0.5 (CWT).

c

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

of n ¼ 1.8, f1(n) ¼ 0.63 and f2(n) ¼ 0.43, therefore in this case the dependence is further weakened, but at least both Gr and Pr show positive dependence on this parameter. Thus it appears that for a fixed value of power-law index (n), the Nusselt number should show a positive dependence on the Rayleigh number. Since the effect of Prandtl number is already incorporated in the definition of the Rayleigh number, whether the Nusselt number shows any additional dependence on Prandtl number or not is immediately not obvious. On the other hand, for fixed values of Ra and/or Pr, the dependence of the Nusselt number on power-law index is likely to be non-monotonic and/or modulated by the values of Ra or Pr or both. This functional relationship is further complicated by the geometrical aspects of this configuration. This complex dependence is borne out by the results shown in Figs. 10e13 for the CWT and CHF boundary conditions respectively. A detailed examination of these results reveals the following overall trends: With the

167

gradually increasing Rayleigh number, advection gains strength and therefore, the influence of power-law index begins to manifest. At low Prandtl numbers, such as Pr ¼ 0.71 (not shown here), increasing degree of enhancement was observed in heat transfer in shear-thinning fluids whereas shear-thickening fluid behaviour is seen to suppress it, though at Ra ¼ 102, the effect of power-law index is seen to be marginal due to the dominance of conduction. On the other hand, for a fixed value of Rayleigh number, as the value of the Prandtl number is progressively increased, this is accompanied by concomitant decrease in the value of the Grashof number and hence the Nusselt number decreases. In other words, the relative contribution of conduction increases at the expense of decreasing convection. For a Newtonian fluid, this is due to the thickening of the boundary layer which presents more resistance to heat transfer. Furthermore, the fact that the Nusselt number scales as Nu f Ra1/4 implies similar roles of Grashof and Prandtl numbers

Fig. 11. Local Nusselt number (Nu) distribution on the surface of the cylinder for b1 ¼ 0.25 (CWT).

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in Newtonian fluids. On the other hand, in addition to the complex dependence of power-law index, n, via the definitions of Grashof and Prandtl numbers, the available boundary layer analysis [9] and recent numerical results [3] show that the dependence of the Nusselt number on the Grashof number becomes stronger whereas it weakens on the Prandtl number with the decreasing value of power-law index. On the other hand, while the Nusselt number shows a weaker dependence on Grashof number, but it still scales as Nu f Pr0.25. However these scalings are expected to be more appropriate at large values of the Grashof and Prandtl numbers, an assumption inherent in boundary layer treatments. Also, the effective shear rate is expected to be a maximum on the surface of the cylinder and/or at the solid walls and it progressively decreases away from it. This would mean that a shear-thinning fluid would exhibit minimum viscosity in this region and it will quickly increase depending upon the value of n until it hits again the high shear region close to the walls. It is thus conceivable that depending upon

3

the value of n, these two fluid-like regions are separated by a highly viscous mass in between thereby impeding the rate of heat transfer in a highly shear-thinning fluids as seen at Ra ¼ 106 at Pr ¼ 50 and Pr ¼ 100. Of course, by the same reasoning, some enhancement can be anticipated in shear-thickening fluids, as is evident in Fig. 10 at Ra ¼ 102 and Ra ¼ 104 at Pr ¼ 50 and Pr ¼ 100. Figs. 11 and 12 show the corresponding results for b1 ¼ 0.25 and b1 ¼ 0.75 respectively. At low Prandtl numbers, some further enhancement in heat transfer is realized for b1 ¼ 0.25 which is primarily due to the corresponding high value of the Grashof number. Due to the narrow gap between the bottom wall and the cylinder in this case, certainly more fluid is seen to partake in the process of heat transfer, though the role of power-law index is seen to be much more involved here than that in the case of b1 ¼ 0.5. Finally, for the symmetric case of b1 ¼ 0.5, under the conditions of moderate advection, e.g., Ra ¼ 104 and 106, the Nusselt number progressively decreases from its maximum value at the front

Ra = 102

2.4

2.9

2.4 1.9

c

2

1.9 1.4 1.4 1

0.9

0

a

b

c

0.4

0.9

a

b

c

0.4

Distance along the cylinder

Distance along the cylinder

a

b

c

Distance along the cylinder

Ra = 104

Local Nusselt number, Nu

9

7.5

7 6

6

5

6

4.5

4 3

3

3

2 1.5 1 0

a

b

c

0

a

b

c

0

Distance along the cylinder

Distance along the cylinder

b

c

Distance along the cylinder

Ra = 106

30

40

a

30

25

25

20

20

15

15

10

10

5

5

30

20

10

0

a

b

c

Distance along the cylinder

Pr = 10

0

a

b

Distance along the cylinder

Pr = 50

c

0

a

b

Distance along the cylinder

Pr = 100

Fig. 12. Local Nusselt number (Nu) distribution on the surface of the cylinder for b1 ¼ 0.75 (CWT).

c

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

stagnation point, albeit in highly shear-thinning fluids at low Prandtl number (n ¼ 0.2, Pr ¼ 0.71 for instance) or in moderately shear-thinning fluids and at high Prandtl numbers (n ¼ 0.7, Pr ¼ 50), the maximum value of the Nusselt number occurs at a position intermediate between the front stagnation point and the equator (point b). Similar trends have also been reported for free convection from unconfined spheres and cylinders [2,3], forced convection from a sphere [29], etc. Basically, the occurrence of the maxima in Nusselt number at positions other than the front stagnation point is ascribed to two competing mechanisms: firstly, while the variation of the shear rate along the surface of the cylinder leads to the increasing or decreasing local viscosity of the fluid (depending upon the value of power-law index) and this, in turn, alters the local values of the Prandtl number (proportional to viscosity) and Grashof number (inversely proportional to the square of viscosity). Secondly, the temperature gradient progressively decreases from its maximum value at front stagnation point.

169

Thus, in a given situation, if the local value of the Prandtl number increases, it will facilitate heat transfer. This increase will partially be offset by the corresponding decrease in the local value of the Grashof number which, in turn, will translate into reduced heat transfer. All else being equal, since the rate of change of viscosity with shear rate is influenced solely by the value of the power-law index, the shift in the position of the maximum Nusselt number is generally seen at small values of n and this critical value of the power-law index keeps rising with the increasing Prandtl number. On the other hand, shear-thickening fluids exhibit the trends which are qualitatively similar to that seen in Newtonian fluids over most range of conditions. These trends are seen to be accentuated in the case of b1 ¼ 0.25 (Fig. 11) and somewhat suppressed for b1 ¼ 0.75 (Fig. 12). Due to the lack of participation of the fluid beneath the cylinder in the case of b1 ¼ 0.75, heat transfer is impeded with reference to the symmetric case of b1 ¼ 0.5 otherwise under identical conditions. This section is concluded by showing

Ra = 102 2.1

2.6

3

2.3

2.5

2

2

1.7

1.5

1.4

1

a b

1.8

c 1.5

1.2

n n n n n n

0.9

0.6

a

= 0.2 = 0.4 = 0.7 = 1.0 = 1.5 = 1.8

b

c

1.1

Local Nusselt number, Nu

Distance along the cylinder

a

b

c

0.5

Distance along the cylinder

4

b

c

Distance along the cylinder

Ra = 104

6.5

5

a

6

5.5

5

4.5

4

3.5

3

2.5

2

1.5

1

3

2

1

0

a

b

c

0.5

Distance along the cylinder

a

b

0

c

Distance along the cylinder

Ra = 106

12

15

15

8

10

10

4

5

5

b

c

Distance along the cylinder

β1 = 0.25

0

a

b

Distance along the cylinder

β1 = 0.5

c

20

20

a

b

Distance along the cylinder

16

0

a

c

0

a

b

Distance along the cylinder

β1 = 0.75

Fig. 13. Local Nusselt number (Nu) distribution on the surface of the cylinder at Pr ¼ 100 (CHF).

c

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R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

representative results for the constant heat flux condition in Fig. 13. Broadly, the Nusselt number is seen to be maximum at the front stagnation point for the symmetric case of b1 ¼ 0.5 as long as there is some degree of advection. Naturally, the case of Pr ¼ 100 and Ra ¼ 102 corresponds to Gr ¼ 1 and under these conditions, heat transfer will occur primarily by conduction as is seen in Fig. 13 for b1 ¼ 0.5. However, there appears to be some evidence of convection in shear-thickening fluids (n ¼ 1.5 and n ¼ 1.8). This counterintuitive behaviour can be explained, at least qualitatively, as follows: a shear-thickening fluid will exhibit its maximum viscosity on the surface of the cylinder and therefore, one can postulate that the heated circular cylinder is encapsulated in a highly viscous fluid which is tantamount to an effectively large virtual object immersed in low viscosity fluid which will naturally lead to some enhancement in heat transfer as seen in Figs. 10e13. For a fixed Rayleigh number, as the value of the Prandtl number is progressively decreased which is accompanied by the concomitant increase in

the value of the Grashof number which, in turn, promotes heat transfer. Finally, suffice it to say here, the local Nusselt number is always found to be lower for the CHF boundary condition than that for CWT condition at fixed values of Gr, Pr and n. 4.4. Average Nusselt number While the streamline and isotherm contours provide visual representation of the flow and temperature fields help delineate the local “hot/cold” region which may be relevant during the processing of temperature-sensitive materials, process engineering and design calculations necessitate reliable knowledge of the mean heat transfer coefficient, i.e., Nusselt number, Nuavg. It stands to reason that for a given set of boundary conditions, the average Nusselt number would be a function of the Rayleigh number (Ra), Prandtl number (Pr), power-law index (n) and the value of b1. Broadly, one would expect a positive correlation between Nuavg and

Ra = 102 1.85

1.65

1.95 Pr = 10 Pr = 50 Pr = 100 Pr

1.7

1.85

1.55 1.75

1.55

1.45 1.65

1.4

1.35 1.55

1.25

0

0.4

0.8

1.2

1.6

2

1.45

0

Average Nusselt number, Nuavg

0.4

0.8

1.2

1.6

2

1.25

0

0.4

0.8

1.2

1.6

2

Power-law index, n

Power-law index, n

Power-law index, n

Ra = 104

9.5

8.5

6

7

5

5.5

4

4

3

2.5

2

8 6.5 5 3.5 2 0.5

0

0.4

0.8

1.2

1.6

2

1

0

0.4

0.8

1.2

1.6

2

1

0

0.4

0.8

1.2

1.6

2

Power-law index, n

Power-law index, n

Power-law index, n

Ra = 106 30

20

10

0

0

0.5

1

1.5

Power-law index, n

β1 = 0.25

2

30

30

25

25

20

20

15

15

10

10

5

5

0

0

0.5

1

1.5

Power-law index, n

β1 = 0.5

2

0

0

0.5

1

1.5

Power-law index, n

β1 = 0.75

Fig. 14. Variation of mean Nusselt number with Rayleigh number, Prandtl number, power-law index and b1 (CWT).

2

R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

171

two-dimensional shapes submerged in power-law fluids in mixed, forced and free convection regimes [36,40]. Conversely, at Ra ¼ 100, the value of b1 exerts much more influence than the power-law index or the Prandtl number. Fig. 15 shows the corresponding results for the constant heat flux condition and qualitatively similar trends are seen here as far as the influence of Ra, Pr, n and b1 on the average Nusselt number is concerned. Thus, in summary the free convection heat transfer from an isothermal or isoflux cylinder in confined power-law fluids shows a complex dependence on the kinematic (Ra, Pr, n) and geometric parameters (b1, b2). In this work, the effect of b1 has been studied in detail at a fixed value of b2. Similarly, the enclosure is made up of isothermal side walls and adiabatic top and bottom walls. The available scant studies suggest the boundary conditions also to exert varying levels of influence on the rate of heat transfer from the cylinder. Further studies in this field will hopefully address these as well as the other related aspects of this problem.

Ra and an inverse dependence on b1. This functional relationship is examined in detail in this section. Fig. 14 shows representative results spanning wide ranges of conditions which elucidate the influence of Ra, Pr, n and b1 on the average Nusselt number. The broad trends can be summarized as follows: due to the progressive thinning of the thermal boundary layer with the increasing Rayleigh number, the thermal resistance to heat transfer decreases and hence, all else being equal, the average Nusselt number is seen to increase with Rayleigh number. Naturally, as noted earlier, the Nusselt number is seen to decrease with the increasing value of b1 as can be seen in Fig. 14. Further examination of these results shows that shear-thinning behaviour facilitates heat transfer with reference to the corresponding value in Newtonian fluid and of course, shear-thickening is seen to have a deleterious effect on heat transfer. This effect becomes increasingly significant with the increasing Rayleigh number. This finding is consistent with the general trend observed with variously shaped

Ra = 102 1.7

2.2

1.8

Pr = 10 Pr = 50 Pr = 100

1.6

1.7 2

1.5

1.6

1.4

1.8

1.5

1.3

1.4 1.6

1.2 1.1

1.3

0

0.4

0.8

1.2

1.6

2

1.4

0

0.8

1.2

1.6

2

1.2

0

Power-law index, n

Power-law index, n

Average Nusselt number, Nuavg

0.4

0.4

0.8

1.2

1.6

2

Power-law index, n

Ra = 10 4

5

4

6

4.5

5

3.8

4

3.1

3

2.4

2

1.7

3

2

1

0

0.4

0.8

1.2

1.6

2

1

0

0.4

0.8

1.2

1.6

2

1

0

Power-law index, n

Power-law index, n

0.4

0.8

1.2

1.6

2

Power-law index, n

Ra = 106 20

20

25

20

15

15

15

10

10 10

5

0

5

5

0

0.5

1

1.5

Power-law index, n

β1 = 0.25

2

0

0

0.5

1

1.5

Power-law index, n

β1 = 0.5

2

0

0

0.5

1

1.5

Power-law index, n β1 = 0.75

Fig. 15. Variation of mean Nusselt number with Rayleigh number, Prandtl number, power-law index and b1 (CHF).

2

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R. Shyam et al. / International Journal of Thermal Sciences 74 (2013) 156e173

Finally, before concluding this section, it is worthwhile to make an observation regarding the characteristic velocity scale used in this work to render the governing partial differential equations dimensionless. The characteristic velocity scale given by Eq. (5) obviously implicitly assumes the dominance of inertial effects which is certainly pertinent at large values of Grashof or Rayleigh numbers. On the other hand, at low Grashof or Rayleigh numbers, viscous forces are dominant rather than the inertial forces. It is thus conceivable to deduce a characteristic velocity scale for this flow by equating the viscous and buoyancy forces which leads to the representative velocity being given by {rfgbDTRnþ1/m}1/n. However, it can easily be shown that the definition of the Grashof number based on this velocity scale is simply related to the definition used in this work, Eq. (12), by Gr1/n. The fact that the power-law index, n, is dimensionless on its own makes it possible to convert results from one definition of the Grashof number to the other. In nut shell, for a given geometric configuration (values of b1 and b2) and fixed boundary conditions, heat transfer characteristics in the present case are governed only by three independent dimensionless groups, i.e., Pr, Gr and n or combinations thereof. 5. Conclusions This work elucidates the influence of Rayleigh number, Prandtl number, power-law index and of the relative position of a heated cylinder (isothermal or isoflux) submerged in a power-law fluid in a long square duct. The range of variables encompassed here (102
Ra
106; 0.71
Pr
100; 0.2
n
1.8 and 0.25
b1
0.75) is such that the flow field is expected to be two-dimensional, steady and symmetric about the vertical centerline. Under these conditions, the flow field exhibits doubled-cell recirculating regions at low Rayleigh numbers when the cylinder and duct are coaxial. However, single recirculating region is formed with the increasing Rayleigh number and/or for the asymmetric location of the heated cylinder. The local and surface average Nusselt number for the cylinder show a positive dependence on Rayleigh number and it also decreases as the cylinder is progressively moved towards the top wall. This is so due to the lack of participation of the body of cold fluid beneath the cylinder. On the other hand, all else being equal, shear-thinning fluid behaviour promotes heat transfer whereas shear-thickening impedes it, especially at high Rayleigh numbers. References [1] Z. Berk, Food Process Engineering and Technology, Academic Press, New York, 2008. [2] A. Prhashanna, R.P. Chhabra, Free convection in power-law fluids from a heated sphere, Chem. Eng. Sci. 65 (2010) 6190e6205. [3] A. Prhashanna, R.P. Chhabra, Laminar natural convection from a horizontal cylinder in power-law fluids, Ind. Eng. Chem. Res. 50 (2011) 2424e2440. [4] 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) 811e830. [5] 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) 112e125. [6] C. Sasmal, R.P. Chhabra, Effect of aspect ratio on natural convection in powerlaw fluids from a heated horizontal elliptic cylinder, Int. J. Heat Mass Transfer 55 (2012) 4886e4899. [7] A. Chandra, R.P. Chhabra, Laminar free convection from a horizontal semicircular cylinder to power-law fluids, Int. J. Heat Mass Transfer 55 (2012) 2934e2944. [8] 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) 553e567. [9] A. Acrivos, A theoretical analysis of laminar natural convection heat transfer to non-Newtonian fluids, AIChE J. 6 (1960) 584e590. [10] W.E. Stewart, Asymptotic calculation of free convection in laminar threedimensional systems, Int. J. Heat Mass Transfer 14 (1971) 1013e1031.

[11] D.W. Larson, D.K. Gartling, W.P. Schimmel, Natural convection studies in nuclear spent-fuel shipping casks: computation and experiment, J. Energy 2 (1978) 147e154. [12] A.R. Elepano, P.H. Oosthuizen, Free convective heat transfer from a heated cylinder in an enclosure with a cooled upper surface, in: Proceedings of the 1st International Conference on Advanced Computational Methods in Heat Transfer, Computational Mechanics Publication, Southampton, Boston, vol. 2, 1990, pp. 99e109. [13] I. Demirdzic, Z. Lilek, M. Peric, Fluid flow and heat transfer test problems solutions for non-orthogonal grids: bench-mark solutions, Int. J. Numer. Methods Fluids 15 (1992) 329e354. [14] N.K. Ghaddar, Natural convection heat transfer between a uniformly heated cylindrical element and its rectangular enclosure, Int. J. Heat Mass Transfer 35 (1992) 2321e2334. [15] F. Moukalled, S. Acharya, Natural convection in the annulus between concentric horizontal circular and square cylinders, J. Thermophys. 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Heat Mass Transfer 56 (2013) 319e339. [25] W.M. Lewandowski, Natural convection heat transfer from plates of finite dimensions, Int. J. Heat Mass Transfer 34 (1991) 875e885. [26] O.G. Martynenko, P.P. Khramstov, Free Convective Heat Transfer, Springer, Berlin, 2005. [27] 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) 567e581. [28] C.-Y. Cheng, Natural convection boundary layer on a horizontal elliptical cylinder with constant heat flux and internal heat generation, Int. Commun. Heat Mass Transfer 36 (2009) 1025e1029. [29] S.D. Dhole, R.P. Chhabra, V. Eswaran, Forced convection heat transfer from a sphere to non-Newtonian power-law fluids, AIChE J. 52 (2006) 3658e 3667. [30] R.P. Bharti, R.P. Chhabra, V. Eswaran, Steady forced convection heat transfer from a heated circular cylinder to power-law fluids, Int. J. Heat Mass Transfer 50 (2007) 977e990. [31] A.A. Soares, J.M. Ferreira, R.P. Chhabra, Flow and forced convection heat transfer in cross flow of non-Newtonian fluids over a circular cylinder, Ind. Eng. Chem. Res. 44 (2005) 5815e5827. [32] A.K. Sahu, R.P. Chhabra, V. Eswaran, Forced convection heat transfer from a heated square cylinder to power-law fluids in the unsteady flow regime, Numer. Heat Transfer, Part A 56 (2009) 109e131. [33] M. Bouaziz, S. Kessentini, S. Turki, Numerical prediction of flow and heat transfer of power-law fluids in a plane channel with a built-in heated square cylinder, Int. J. Heat Mass Transfer 53 (2010) 5420e5429. [34] R.P. Bharti, P. Sivakumar, R.P. Chhabra, Forced convection heat transfer from an elliptical cylinder to power-law fluids, Int. J. Heat Mass Transfer 51 (2008) 1838e1853. [35] A. Chandra, R.P. Chhabra, Flow over and forced convection heat transfer in Newtonian fluids from a semi-circular cylinder, Int. J. Heat Mass Transfer 54 (2011) 225e241. [36] R.P. 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Nomenclature C: thermal heat capacity of fluid, J kg1 K1 e: rate of the deformation tensor, s1 g: acceleration due to gravity, m s2 Gr: Grashof number, dimensionless h: heat transfer coefficient, W m2 K1 H*: ratio of the height of cavity to the diameter of cavity used in Cesini et al. [16]. I2: second invariant of the rate of the strain tensor, s2 k: thermal conductivity of fluid, W m1 K1 l: position of the centre of the heated cylinder from the bottom wall, m m: power-law consistency index, Pa sn n: power-law index, dimensionless ns: unit normal vector Nu: local Nusselt number, dimensionless Nuavg: average Nusselt number, dimensionless N: total number of cells in entire computational domain NP: number of grid points on half of the circumference of the cylinder p: pressure, dimensionless Pr: Prandtl number, dimensionless

173

qs: uniform heat flux on the surface of the cylinder, W m2 R: radius of the cylinder, m Ra: Rayleigh number, dimensionless T: temperature of fluid, K Tf: side wall temperature (¼reference temperature), K Ts: cylinder surface temperature, K DT: temperature difference ¼ (Ts  Tf), K Vc: characteristic velocity scale for free convection, m s1 Vx,Vy: x- and y-components of the velocity, dimensionless W*: ratio of the width of cavity to the diameter of cavity used in Cesini et al. [16]. x,y: Cartesian co-ordinates, dimensionless Greek symbols

b: coefficient of volume expansion, K1 b1: dimensionless distance of inner cylinder from bottom wall (¼l/L), dimensionless b2: ratio of radius of inner cylinder to side wall (¼R/L), dimensionless g_ : shear rate, s1 h: power-law viscosity, dimensionless q: non-dimensional temperature, q ¼ (T  Tf)/(Ts  Tf) or (T  Tf)/(qsR/k) m: Newtonian viscosity, Pa s r: density of the fluid, kg m3 rf: fluid density at reference temperature, Tf, kg m3 s: extra stress tensor, Pa Subscripts avg: average value f: fluid property s: cylinder surface condition Abbreviations CHF: constant heat flux CWT: constant wall temperature