Numerical Study on the Mixed Convection Heat Transfer between a Sphere Particle and High Pressure Water in Pseudocritical Zone

Abstract

Mixed convection heat transfer between supercritical water and particles is a major basic problem in supercritical water fluidized bed reactor, but little work focused on this new area in the past. In this paper, a numerical model fully accounting for thermophysical property variation has been established to investigate heat transfer between supercritical water and a single spherical particle under gravity. Flow field, temperature field and Nusselt number are analyzed based on the simulation results. Results show that buoyancy force has a remarkable effect on flow and heat transfer process. When the direction of gravity and flow are opposite, the gravity enhances the heat transfer before the separation point and inhibits the heat transfer after the separation point. When gravity is incorporated in calculation, a higher temperature gradient and a thinner boundary layer in the vicinity of the particle surface are observed before separation point, and the situations are just the reverse after separation point. Variation of specific heat and conductivity plays a main role in determination of heat transfer coefficient.

1. Introduction

Supercritical water fluidized bed reactor (SCWFBR) is a new promising reactor for gasifying wet biomass to product hydrogen [1]. It is a promising technology because it could avoid plugging effectively and produce hydrogen efficiently [2]. However, determination of heat transfer rate from solid particles to supercritical water (SCW) is a hard work due to complicated gas-solid phase flow and huge property variation of SCW in the reactor. Dong et al. (2012) [3] numerically studied the heat transfer process between bed and wall in SCWFBR. They found that the transfer process in SCWFBR was very different from that in traditional gas-solid fluidized bed. First, the heat transfer in SCWFBR was dominated by the heat transfer of supercritical water. The shape of curves of heat transfer coefficient with bed temperature is similar to the case of supercritical water flow through duct. Second, particle concentration enhanced the heat transfer process, but particle concentration did not greatly affect the heat transfer rate as that in gas-solid fluidized bed. Those differences make the studying the heat transfer processes in SCWFBR become interesting and significant. The reactor always works at a low superficial velocity. So, the mixed convection heat transfer from particles to SCW is obvious, and a simple case for this problem is SCW flowing over a single sphere particle. Little work studied this area in the past.

Mixed convective heat transfer from a sphere in infinite fluid has been studied by many researchers for different interests because it represents a problem related to numerous engineering applications. An exact analytical solution of this problem is hard to achieve due to the nonlinearity in the governing equations [4]. The earliest analytical studies on the combined effect of free and forced convection in assisting mixed convective flow were conducted by Acrivos (1966) (boundary-layer approximation) and Hieber and Gebhart (1969) (matched asymptotic expansion) [5, 6]. Experimental data about mixed convection heat transfer around a sphere in variable fluid were obtained by several researchers by different equipments. Yuge (1960) experimentally studied the mixed convection heat transfer around an isothermal sphere in air flow with a wide range of Reynolds numbers (3.5 ≤ Re ≤ 1.44 × 10⁵) and Grashof numbers (1 ≤ Gr ≤ 10⁵) [7]. Tang and Johnson (1990) obtained various flow patterns of opposing flow by mixed convection experiments with a Grashof number of 9.0 × 10⁵ and Reynolds numbers up to 1100 [8]. Recently, Ziskind et al. (2001) and Mograbi et al. (2002) studied the cross-flow, assisting and opposing flows at low Grashof number and Reynolds number in an electrodynamics chamber (EDC) [9, 10]. Koizumi (2004) studied the time and spatial heat transfer performance around an isothermal sphere in a uniform, downwardly directed air flow by conducting an experiments with Grashof number of Gr = 3.3 × 10⁵ and Reynolds numbers up to 1800 [11]. Later, they (2010) revealed transition process from a steady axisymmetric separated flow to a chaotic flow of an isothermally heated sphere placed in a uniform, downwardly directed flow by experimentation and transient 3D numerical simulation [12].

Due to limitation of experimental and analytical solution, it is necessary to study this problem through numerical simulation. Chen and Mucoglu (1977) numerically investigated the laminar mixed convection around a sphere with constant surface temperature. They found that both the local friction factor and the local Nusselt number increase with increasing buoyancy force for aiding flow and decrease with increasing buoyancy force for opposing flow [13]. In their later work, they presented a similar study for a sphere with a uniform heat flux [14]. Recently, Antar and El-Shaarawi (2002) numerically studied the problem of mixed convection around a liquid sphere in an air stream for aiding and opposing flows [15]. They found that increasing Reynolds number or decreasing the interior-to-exterior viscosity ratio delays the flow separation. The similar conclusions were obtained by later researches. Nazar and Amin (2002) numerically studied the mixed convection flow of an incompressible viscous fluid around a solid sphere with constant surface temperature for both aiding and opposing flow [4]. They found that the aiding flow delays separation of the boundary-layer but the opposing flow brings the separation point closer to the lower stagnation point of the sphere. Later, they extended this problem to micropolarfluid [16]. Kotouč et al. (2008) numerically investigated the issue of loss of axisymmetry in the assisting flow [17]. They found that the acceleration of the flow due to buoyancy considerably stabilized the flow and pushed the onset of instabilities. Bhattacharyya and Singh (2008) studied the aiding flow from an isolated spherical particle with moderate range of Reynolds number (1 ≤ Re ≤ 200) and Grashof number (0 ≤ Gr ≤ 10⁴) [18]. They revealed that the decrease in drag coefficient with increase of Reynolds number is due to the increase of the wake size, and the heat transfer is dominated by the convection effect at higher Reynolds number.

Most of the earlier work on natural or mixed convection was based on the Boussinesq approximation, which treated variable density as a linear relationship with temperature. However, variation of flow thermophysical property becomes very critical in some cases such as heat transfer of supercritical fluid in critical zone. This paper studied the mixed convection around a sphere particle in SCW of pseudocritical zone with moderate range of Reynolds number (5 ≤ Re ≤ 200), which represents supercritical water fluidized bed works at relatively low superficial velocity. The influence of buoyancy force on heat transfer has been investigated. Flow field and heat transfer is analyzed for different values of the Richardson number (Ri = Gr/Re²). Effect of thermophysical property on the flow and heat transfer is studied.

2. Problem Statement and Solution

2.1. Physical Problem

Strictly speaking, the heating of a particle in SCW flow is a transient process. However, as the focus of this study is to understand the effect of flow property variation, the transient particle temperature variations are not considered. Johnson and Patel (1999) numerically and experimentally investigated the flow of incompressible viscous fluid passing a sphere over flow regimes including steady and unsteady laminar flow at Reynolds numbers up to 300 [19]. They found that steady axisymmetric flow maintains at Reynolds numbers up to 200. Although there are no studies verifying weather the results are suitable for supercritical condition, the time average values of drag coefficient and Nusselt number are close to the data obtained in steady assumption with Reynolds number over 200 [13]. Furthermore, based on the results obtained by Antar and El-Shaarawi (2002), Nazar et al. (2002), and Kotouč et al. (2008), buoyancy force keeps the flow steady and symmetry for aiding flow [4, 15, 17]. This work considers that steady, axisymmetric convection heat transport from particle to SCW flow occurs for particles with constant surface temperature at moderate Reynolds numbers. Figure 1 shows the schematics of the SCW flow over a sphere and relative boundary conditions.

Figure 1:

Schematics of a spherical particle in SCW under gravity.

2.2. Mathematical Formulation and Solution

Governing continuum conservation equations for steady flow with variable property over a particle with gravity and negligible viscous dissipation can be given as follows.

Mass conservation is written as

\nabla \cdot ρ \vec{u} = 0 .

(1)

The momentum equation is written as

(\vec{u} \cdot \nabla) ρ \vec{u} = - \nabla p + \nabla (μ \nabla \cdot \vec{u}) + g .

(2)

The energy equation is written as

(\vec{u} \cdot \nabla) ρ h = \nabla \cdot (k \nabla T) .

(3)

The Grashof number is defined as

Gr = \frac{g β ρ^{2} d^{3} Δ T}{μ^{2}} .

(4)

The thermal expansion coefficient of the fluid is defined as

β = - \frac{1}{ρ_{\infty}} (\frac{\partial ρ}{\partial T}) .

(5)

The Nusselt number is defined as

Nu = \frac{d}{2 k_{\infty} (T_{w} - T_{\infty})} \int_{0}^{π} k_{θ} {(\frac{d T}{d r})}_{θ} d θ .

(6)

These equations are solved using a finite volume method. The boundary conditions include specified uniform far field temperature and velocity (free stream), no-slip condition and specified temperature at the particle surface, axisymmetry condition along the centerline, moving wall at the upper boundary (the velocity of wall is the same with free stream), and fully developed exit conditions. The properties of supercritical water are calculated by IAPWS-IF97 equations [20]. The convective terms are discretized using second-order upwind scheme. The SIMPLEC algorithm is applied for the pressure-velocity coupling. Typical values of under-relaxation factors were set from 0.1 to 0.5. A convergence criterion of 10⁻⁶ for each scaled residual component was specified for the relative error between two successive iterations.

2.3. Grid

Figure 1 shows the schematic of the nonuniform grid structure used in this work. It needs a lager domain zone for small Grashof number and low Reynolds number case because of strong viscous reaction and thick boundary layer. The domain size can be determined by the values of R_∞/r. Based on the similar domain grid independence studies conducted by Feng and Michaelides (2000), Dhole et al. (2006), Kumar and Kishore (2009), Dudek et al. (1988), Geoola and Cornish (1981), and Prhashanna and Chhabra (2010) about forced and free convective heat transfer from a sphere, domain grid zone has been determined separately by two flow regimes with different Reynolds number [21 –26]. One is steady flow without wake formation for Re < 20 and the other is 2-D steady flow with wake formation in the range of 20 ≤ Re ≤ 200. In this work, the value of R_∞/r is 150 for Re < 20 and 100 for 20 ≤ Re ≤ 200.

2.4. Validation

There are no any experimental data, simulation results, or theoretical solutions for mixing convective heat transfer from a sphere in supercritical pressure. In order to validate the simulations in this work, Nusselt numbers for constant property fluid passing over a sphere have been compared with the correlations calculated results in the literature. The thermophysical properties of SCW in this paper are calculated by IAPWS-IF97 equations at far field temperature and pressure [20]. The simulations are conducted in consideration of gravity or not referred to Richardson number equaling zero or not. Figure 2(a) shows average Nusselt number predicted by this work for forced convection flow being very close to those correlations proposed by Melissari and Argyropoulos (2005), Whitaker (1972), and McAdams (1954) [27 –29]. Predictions by heat transfer correlations and present numerical studies are accurate with deviation generally within ±10%. In order to validate the model under gravity, average Nusselt numbers are obtained by simulation with Boussinesq approximation (ρ – ρ_∞) = – ρ_∞β(T – T_∞). Boussinesq model assumes that the fluid density is constant in all terms of the momentum equation except the body force term. All properties are evaluated at the fluid film temperature, T_m, defined as (T_w + T_∞)/2. Figure 2(b) shows the comparison of average Nusselt number obtained by simulation with correlations presented by different researchers [30, 31]. It is clear that the trend of the simulation results is consistent with those correlations. The predicted values of Nusselt number by simulation have an agreement with that calculated by Amateo and Tieno's and Churchill's correlations within deviation generally within ±5%. Those comparisons validate the computational model used in this work for constant property fluid. The next section will carefully analyse the effect of variable properties.

Figure 2:

Comparison of average Nusselt number obtained by simulation with correlation: (a) forced convection heat transfer; (b) free convection heat transfer.

3. Results and Discussion

3.1. Flow Field

Effect of gravity on the mixed convection flow field for SCW flowing over a sphere is analyzed in this work. The properties of flow are calculated at the far field temperature in this section. The dimensionless parameters (Nusselt number, Grashof number, Reynolds number, and so on) are all calculated at the temperature in far field.

Figure 3 shows effect of gravity on the flow stream around a sphere in SCW flow in the pseudocritical zone. The general shape of the flow field does not change significantly at rear hemisphere even if the gravity is considered. However, the zone of flow streamlines near the leading hemisphere becomes wide when gravity is incorporated in computed model at Reynolds number of 5. The recirculation zone after sphere becomes smaller under gravity. The rear vortex will vanish if buoyancy force is strong enough. The collapse of rear vortex due to density stratification was also observed by Torres et al. (2000) and Bhattacharyya and Singh (2008) for an assisting convective flow [18, 32]. The buoyancy force is determined by the density difference which results from the nonuniform temperature field. When the direction of gravity is against the flow direction, the buoyancy force assists the convective flow. More fluid is driven by buoyancy force to the leading hemisphere and more fluid is put forward by buoyancy force after rear hemisphere. Another interesting phenomenon is that the divergence in flow stream between Figures 3(a) and 3(b) becomes unconspicuous with an increase in Reynolds number. This is attributed to the fact that the convective effect becomes strong relative to buoyancy force with an increase in Reynolds number.

Figure 3:

Streamlines around spherical particle, P = 23 MPa, T_w = 657 K, and T_∞ = 647 K: (a) without gravity; (b) under gravity.

Figure 4 shows the effect of gravity on surface vorticity magnitude in pseudocritical zone. The surface vorticity increases over the leading hemisphere and then reduces over the rear after it reaches a peak value in both calculations. Obviously, the magnitude of surface vorticity under gravity is much higher than that without gravity. At the same time, the position of surface vorticity maximum value moves toward the back of sphere which is in the same direction of the buoyancy force. When gravity is considered, a thinner boundary layer was formed. With the effect of buoyancy force, fluid around the sphere surface is accelerated. A higher flow velocity near the sphere surface indicates a stronger convective effect, which results in a higher momentum diffusivity and a thinner boundary layer thickness.

Figure 4:

Surface vorticity at different Reynolds number, P = 23 MPa, T_w = 657 K, and T_∞ = 647 K: (a) Re = 5; (b) Re = 50; (c) Re = 150.

Figure 4 further shows a decrease in divergence with a decrease in Richardson number. This is attributed to forced convection which becomes more important in determination of the mixed convection with an increase in Reynolds number. Another noticed phenomenon is that the value of surface vorticity magnitude in recirculation zone under gravity is smaller than that without gravity, as shown in Figure 4(c). It means that the boundary layer thickness in recirculation zone becomes thicker on the effect of buoyancy force. The flow velocity near the sphere surface in recirculation zone is decelerated by buoyancy force because the directions of flow and buoyancy are opposite. A small recirculation zone under gravity is just result from the low fluid inertia force caused by a low fluid velocity.

Figure 5 shows surface pressure variation around the sphere at three conditions: constant property flow, SCW flow, and SCW flow under gravity. The dimensionless pressure was obtained by the free stream momentum ρ_∞U_∞²/2. The trends of pressure variation along sphere for constant property flow and SCW flow are similar to the literature [33]. The highest value of surface pressure is formed at the stagnation point. The dimensionless pressure at the front stagnation point was closer to 1 at Reynolds number of 50 and 150, but nearly 2.5 at Reynolds number of 5. The pressure first decreases and then increases with an increase in streamwise angle. Wen and Jog (2005) [34] obtained similar results when they studied continuum plasma flow over spherical particles. They also found that dimensionless pressure at front stagnation point was closer to 1 at Reynolds number of 50, but greater than 1 at Reynolds number of 20. It is attributed to the effect of strong viscous forces at low Reynolds number. There was no big difference of dimensionless pressure between constant property flow and SCW flow with gravity. So, the variable properties have little influence on the pressure distribution. Besides, the small adverse pressure at low Reynolds number may inhibit the flow separation. Both the gradient and value of dimensionless pressure are elevated when gravity is considered. This results in a small recirculation zone and flow separation delay at the same Reynolds number.

Figure 5:

Dimension pressure distribution at surface of sphere, P = 23 MPa, T_w = 657 K, and T_∞ = 647 K: (a) Re = 5; (b) Re = 50; (c) Re = 150.

3.2. Temperature Field and Local Nusselt Number

Figure 6 shows the effect of gravity on temperature contours in pseudocritical zone. The shapes of temperature contours for two cases are similar. However, when gravity is incorporated in the computations, the temperature gradients become higher before the separation point and lower in the recirculation zone. High temperature gradient results in high heat flux transfer from sphere particle to fluid. Figure 7 shows that the local Nusselt number obtained before the separation point under gravity is higher than that without gravity. However, the local Nusselt number predicted after the separation point becomes lower under gravity. The increase or decrease in the temperature gradient is due to the thinner or thicker thermal boundary layer, which is caused by a higher or lower fluid velocity driven by boundary force. There is a peak of thermal conductivity and specific heat in pseudocritical temperature point. High thermal conductivity results in high heat rate transferring from sphere particle to fluid. At the same time, fluid with higher specific heat is able to store more energy, which makes diffusion of energy confined to a smaller radial distance and results in a higher temperature gradient.

Figure 6:

Temperature contour around sphere particle, P = 23 MPa, T_w = 657 K, and T_∞ = 647 K: (a) without gravity; (b) under gravity.

Figure 7:

Local Nusselt number around spherical particle. P = 23 MPa, T_w = 657 K, and T_∞ = 647 K: (a) Re = 5; (b) Re = 50; (c) Re = 150.

3.3. Nusselt Number

Based on the previous analyses, effect of gravity on flow and temperature field is obvious for SCW flow. Figure 8 further compares the average Nusselt number between forced convection and mixed convection in the ranges of 5 ≤ Re ≤ 200. At low Reynolds number, the gravity assists the convective heat transfer. However, the gravity inhibits the convective heat transfer with the increasing Reynolds number. Based on the analysis in Section 3.2, gravity plays an opposite role in two sides of separation point. Figure 8 further shows this point.

Figure 8:

Effect of property variation on average Nusselt number under gravity.

At the same time, effect of properties on the heat transfer for SCW cannot be ignored in pseudocritical zone. In order to study the roles that thermophysical property plays, the simulations are conducted in this work by varying one or two properties with temperature while the rest of properties are maintained constant as values in the far field. Figure 8 shows the effect of each property on the Nusselt number. Effect of variable fluid density on the flow and temperature contours under gravity includes two parts: the free convection part and the forced convection part. The free convection part comes from the effect of buoyancy force which formed by density stratification. The forced convection part is shown clearly in the literature. For the effect of other properties except density, the main factor which enhances the mix convection heat transfer from particle to flow is the variation of specific heat and thermal conductivity. Effect of variation of viscosity on the average Nusselt number is relatively weak.

The results indicate that effect of variation of specific heat and thermal conductivity on the Nusselt number should be considered when developing a correlation. At a fixed Grashof number, the relationship between Reynolds number and Nusselt number for variable property flow is nearly linear in logarithmic coordinates.

4. Conclusions

The mixed convection heat transfer from sphere particle in SCW was studied by numerical method. A computational model considering the thermal property variation of SCW was developed successfully to describe the flow and heat transfer phenomena. Good agreement on Nusselt number was found between the simulation and correlation for constant property flow. Results show that buoyancy force has remarkable effect on flow and heat transfer process at low Reynolds number. The fluid velocity in mixed convection is influenced by the directions of buoyancy force and flow. When the directions of gravity and flow are opposite, the gravity enhances the heat transfer before the separation point, and inhibits the heat transfer after the separation point. Higher temperature gradient and thinner boundary layer thickness in the vicinity of the particle surface are observed before separation point and the situation is just the reverse after separation point. Effect of variable density on the mixed convection flow includes two parts: the forced convective part and free convective part. Variation of specific heat and conductivity plays a main role in determination of heat transfer coefficient, but variation of viscosity is an ignored factor that influences heat transfer.

Footnotes

Nomenclature

Acknowledgments

This work is currently supported by the National Natural Science Foundation of China through Contract no. 50906069 and Program for New Century Excellent Talents in University (NCET-10-0678).

References

Y. J.

Jin

Guo

L. J.

Zhang

X. M.

Cao

C. Q.

, and Guo

, “Hydrogen production by biomass gasification in supercritical water with a fluidized bed reactor,” International Journal of Hydrogen Energy, vol. 33, no. 21, pp. 6066–6075, 2008.

Zhao

Han

, “Minimum fluidization velocities for supercritical water fluidized bed within the range of 633–693K and 23–27MPa,” International Journal of Multiphase Flow, vol. 49, pp. 78–82, 2013.

Dong

, and Wei

, “Numerical study the heat transfer between bed and wall in supercritical water fluidized bed reactor,” in Proceedings of the Annual Meeting of the Multiphase Flow, Xi'an, China, 2012.

Nazar

Amin

, and Pop

, “On the mixed convection boundary-layer flow about a solid sphere with constant surface temperature,” Arabian Journal for Science and Engineering, vol. 27, no. 2, pp. 117–135, 2002.

Acrivos

, “On the combined effect of forced and free convection heat transfer in laminar boundary layer flows,” Chemical Engineering Science, vol. 21, no. 4, pp. 343–352, 1966.

Hieber

and Gebhart

, “Mixed convection from a sphere at small Reynolds and Grashof numbers,” Journal of Fluid Mechanics, vol. 38, no. 1, pp. 137–159, 1969.

Yuge

, “Experiments on heat transfer from spheres including combined natural and forced convection,” Journal of Heat Transfer, vol. 82, pp. 214–220, 1960.

Tang

and Johnson

A. T.

, “Flow visualization of mixed convection about a sphere,” International Communications in Heat and Mass Transfer, vol. 17, no. 1, pp. 67–77, 1990.

Ziskind

Zhao

Katoshevski

, and Bar-Ziv

, “Experimental study of the forces associated with mixed convection from a heated sphere at small Reynolds and Grashof numbers. Part I: cross-flow,” International Journal of Heat and Mass Transfer, vol. 44, no. 23, pp. 4381–4389, 2001.

10.

Mograbi

Ziskind

Katoshevski

, and Bar-Ziv

, “Experimental study of the forces associated with mixed convection from a heated sphere at small Reynolds and Grashof numbers. Part II: assisting and opposing flows,” International Journal of Heat and Mass Transfer, vol. 45, no. 12, pp. 2423–2430, 2002.

11.

Koizumi

, “Time and spatial heat transfer performance around an isothermally heated sphere placed in a uniform, downwardly directed flow (in relation to the enhancement of latent heat storage rate in a spherical capsule),” Applied Thermal Engineering, vol. 24, no. 17–18, pp. 2583–2600, 2004.

12.

Koizumi

Umemura

Hando

, and Suzuki

, “Heat transfer performance and the transition to chaos of mixed convection around an isothermally heated sphere placed in a uniform, downwardly directed flow,” International Journal of Heat and Mass Transfer, vol. 53, no. 13–14, pp. 2602–2614, 2010.

13.

Chen

T. S.

and Mucoglu

, “Analysis of mixed forced and free convection about a sphere,” International Journal of Heat and Mass Transfer, vol. 20, no. 8, pp. 867–875, 1977

14.

Mucoglu

and Chen

T. S.

, “Mixed convection about a sphere with uniform surface heat flux,” Journal of Heat Transfer, vol. 100, no. 3, pp. 542–544, 1978.

15.

Antar

M. A.

and El-Shaarawi

M. A. I.

, “Mixed convection around a liquid sphere in an air stream,” Heat and Mass Transfer, vol. 38, no. 4–5, pp. 419–424, 2002.

16.

Nazar

Amin

, and Pop

, “Mixed convection boundary layer flow about an isothermal sphere in a micropolar fluid,” International Journal of Thermal Sciences, vol. 42, no. 3, pp. 283–293, 2003.

17.

Kotouč

Bouchet

, and Dušek

, “Loss of axisymmetry in the mixed convection, assisting flow past a heated sphere,” International Journal of Heat and Mass Transfer, vol. 51, no. 11–12, pp. 2686–2700, 2008.

18.

Bhattacharyya

and Singh

, “Mixed convection from an isolated spherical particle,” International Journal of Heat and Mass Transfer, vol. 51, no. 5–6, pp. 1034–1048, 2008.

19.

Johnson

T. A.

and Patel

V. C.

, “Flow past a sphere up to a Reynolds number of 300,” Journal of Fluid Mechanics, vol. 378, pp. 19–70, 1999.

20.

Wagner

and Kretzschmar

H. J.

, International Steam Tables: Properties of Water and Steam Based on the Industrial Formulation IAPWS-IF97: Tables, Algorithms, Diagrams, and CD-ROM Electronic Steam Tables: All of the Equations of IAPWS-IF97 Including a Complete Set of supplementary backward Equations for fast calculations of heat Cycles, Boilers, and Steam Turbines, Springer, 2008.

21.

Feng

Z. G.

and Michaelides

E. E.

, “A numerical study on the transient heat transfer from a sphere at high Reynolds and Peclet numbers,” International Journal of Heat and Mass Transfer, vol. 43, no. 2, pp. 219–229, 2000.

22.

Dhole

S. D.

Chhabra

R. P.

, and Eswaran

, “A numerical study on the forced convection heat transfer from an isothermal and isoflux sphere in the steady symmetric flow regime,” International Journal of Heat and Mass Transfer, vol. 49, no. 5–6, pp. 984–994, 2006.

23.

Kumar

and Kishore

, “2–D newtonian flow past ellipsoidal particles at moderate reynolds numbers,” in Proceedings of the 7th International Conference on CFD in the Minerals and Process Industries, Melbourne, Australia, 2009.

24.

Dudek

D. R.

Fletcher

T. H.

Longwell

J. P.

, and Sarofim

A. F.

, “Natural convection induced drag forces on spheres at low Grashof numbers: comparison of theory with experiment,” International Journal of Heat and Mass Transfer, vol. 31, no. 4, pp. 863–873, 1988.

25.

Geoola

and Cornish

A. R. H.

, “Numerical solution of steady-state free convective heat transfer from a solid sphere,” International Journal of Heat and Mass Transfer, vol. 24, no. 8, pp. 1369–1379, 1981.

26.

Prhashanna

and Chhabra

R. P.

, “Free convection in power-law fluids from a heated sphere,” Chemical Engineering Science, vol. 65, no. 23, pp. 6190–6205, 2010.

27.

Melissari

and Argyropoulos

S. A.

, “Development of a heat transfer dimensionless correlation for spheres immersed in a wide range of Prandtl number fluids,” International Journal of Heat and Mass Transfer, vol. 48, no. 21–22, pp. 4333–4341, 2005.

28.

Whitaker

, “Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and for flow in packed beds and tube bundles,” AIChE Journal, vol. 18, no. 2, pp. 361–371, 1972.

29.

McAdams

W. H.

, Heat Transmission, McGraw-Hill, 1954.

30.

Bejan

and Kraus

A. D.

, Heat Transfer Handbook, Wiley-Interscience, 2003.

31.

Churchill

S. W.

, “Comprehensive, theoretically based, correlating equations for free convection from isothermal spheres,” Chemical Engineering Communications, vol. 24, no. 4–6, pp. 339–352, 1983.

32.

Torres

C. R.

Hanazaki

Ochoa

Castillo

, and Van Woert

, “Flow past a sphere moving vertically in a stratified diffusive fluid,” Journal of Fluid Mechanics, vol. 417, pp. 211–236, 2000.

33.

Clift

Grace

J. R.

, and Weber

M. E.

, Bubbles, Drops, and Particles, Dover, 2005.

34.

Wen

and Jog

M. A.

, “Variable property, steady, axi-symmetric, laminar, continuum plasma flow over spheroidal particles,” International Journal of Heat and Fluid Flow, vol. 26, no. 5, pp. 780–791, 2005.