Momentum and Heat Transfer Phenomena of Confined Spheroid

Article pubs.acs.org/IECR

Momentum and Heat Transfer Phenomena of Confined Spheroid Particles in Power-Law Liquids C. Rajasekhar Reddy and Nanda Kishore* Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India ABSTRACT: Effects of wall confinement and the power-law fluid behavior index on momentum and heat transfer phenomena of confined spheroid particles are numerically investigated over wide ranges of conditions such as the Reynolds number, Re, 1− 100; Prandtl number, Pr, 1−1000; particle aspect ratio, e, 0.5−2.5; wall confinement factor, λ, 2−5; and power-law fluid behavior index, n, 0.4−1.8. Some of the key findings include that the size of the recirculation wake at the rear end of the particle increases with the increasing particle aspect ratio and/or Reynolds number and/or wall factor. The total drag coefficient (Cd) decreases with the increasing wall factor and/or Reynolds number and/or the decreasing power-law index. Regardless of values of the Reynolds and Prandtl numbers and regardless of the shape of the spheroid particles, the average Nusselt number increases with the decreasing power-law index and/or with the decreasing wall factor.

1. INTRODUCTION The resistance to flow of solid particles in viscous fluids and associated heat transfer between solid and liquid phases is of fundamental importance due to their overwhelming applications in chemical, biochemical, and many other processing industries. Some important applications include fixed and fluidized bed reactors, mineral slurries, coal-oil slurry transport, wastewater treatment, heterogeneous catalytic reactors, thermal treatment of several food products, processing of pharmaceutical products, etc. Therefore, in the design of such solid−liquid contacting equipment, process engineers often require reliable information on the settling velocity (or drag force) and/or the rate of heat transfer (or Nusselt number). If the design information is already available via other sources, then also many operational conditions can be calculated or modified (depending on the nature of applications) from the information of drag and Nusselt numbers. This information on the drag and Nusselt numbers can be conveniently expressed as functions of the Reynolds number, Prandtl number, particle aspect ratio, wall confinement, and model parameters of non-Newtonian fluids. However, in many of the aforementioned applications, often large numbers of irregular particles interact with the surrounding fluid, with neighboring particles, and with the confining container wall. Despite this fact, unfortunately most of the solid−liquid contacting equipment is designed on the basis of the drag and Nusselt numbers of unconfined spherical particles in Newtonian fluids in the limit of Re → 0 and Pe → ∞. Therefore, there is a strong prerequisite to incorporate the effects of size and shape of particles along with the rheological parameters of the liquid/suspension in the design of such contacting equipment. However, incorporating all details (such as number of particles, irregular shape of particles, wall confinement, rheology of continuous phase, etc.) together is a very cumbersome task; hence researchers have tended toward developing correction factors (due to shape, wall confinement, and rheological parameters) to the existing theoretical results.1,2 Thus, in the present numerical work, the spheroid shape of the particles is considered which is found to be a very good approximation for nonspherical particles.2 These spheroid © 2013 American Chemical Society

particles can be described by the aspect ratio (e), which can cover both plate-like (e < 1) and cylinder-like (e > 1) shapes including spheres with e = 1. Furthermore, the effects of wall confinement are incorporated in the present study by locating these particles in tubes of different diameters. On the other hand, many of the industrial fluids in chemical, pharmaceutical, food, polymer, and other process industries generally obey power-law type non-Newtonian characteristics.3,4 Hence, in the present simulations, for the continuous phase, a shear-thinning and shear-thickening type power-law fluid viscosity model has been adopted. Finally, the aim of this work is to report numerical results on the drag and heat transfer characteristics of confined spheroids in power-law fluids over a wide range of pertinent parameters.

2. PREVIOUS WORK The creeping motion (Re → 0) past spheroid particles perhaps first investigated by Oberbeck,5 and this approach was extended to investigate the hydrodynamics around a variety of axially symmetric bluff bodies by Payne and Pell.6 Subsequently, many semianalytical and/or seminumerical results7−12 have been published in the open literature for the case of flow past spheroid particles at small but finite Reynolds numbers and have presented the drag on spheroid particles as a function of shape correction factors to the existing classical Stokes solution for spheres. Haider and Levenspiel13 developed explicit equations for drag coefficients and terminal velocities of falling spheroidal particles and validated them with other existing equations. Militzer et al.14 compiled the existing experimental and theoretical results on the drag of unconfined spheroids and proposed a correlation which is valid over wide ranges of Reynolds number and particle aspect ratios. However, both at low and high Reynolds numbers, their correlation overestimates Received: Revised: Accepted: Published: 989

September 30, 2013 December 9, 2013 December 18, 2013 December 18, 2013 dx.doi.org/10.1021/ie4032428 | Ind. Eng. Chem. Res. 2014, 53, 989−998

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematic representation of fluid and heat flow around confined spheroids.

the drag of a spherical particle (e = 1). Chhabra et al.15 has also evaluated several experimental results on nonspherical but regular particles settling in viscous fluids. Wang et al.16 conducted experiments on spheroids settling with facets facing upward and downward up to Reynolds numbers of the order of 105. They reported that the drag of both facing upward and downward spheroids are larger than those of spheres. TranCong et al.17 experimentally investigated the drag coefficients of various irregular particles and proposed a correlation as a function of the Reynolds number, ratio of surface equivalent sphere diameter to nominal diameter, and particle circularity. Recently, Pinar et al.18 experimentally studied the hydrodynamic interactions between side-by-side spheres at large Reynolds numbers by employing dye visualization and particle image velocimetry. In the past few decades, numerous researchers numerically investigated the transport phenomena with unconfined spheroid particles in Newtonian fluids by solving the complete Navier−Stokes equations in wide ranges of low to moderate Reynolds numbers and particle aspect ratios.19−25 The axisymmetric viscous flow past spheres and ellipsoids of an axis ratio of 2:1 has been analytically and numerically investigated by Chang et al.23 using the method of matched asymptotic expansion and a deterministic vortex method in the range of Reynolds numbers of 100−3000. The interfacial transport characteristics of spheroid bubbles and spheroid solid particles have been numerically investigated by Li et al.24 using the SIMPLE algorithm implemented on a nonorthogonal boundary fitted staggered grid in the range of Reynolds numbers up to 100. They have reported extensive results on recirculation wake structures and rate of mass transfer; however, little information is provided on the drag behavior of spheroid bubbles and particles. Perhaps Tripathi et al.26 were the first to numerically study the drag experienced by the spheroidal particles in shear-thinning fluids at low to moderate Reynolds numbers, and they found that the drag on spheroids due to shear-thinning fluids is larger compared to Newtonian fluids for moderate values of the Reynolds number. Further, Tripathi and Chhabra27 extended their study for shearthickening fluid flow past unconfined spheroid particles.

On the other hand, the literature is limited pertaining to the heat transfer phenomena of spheroid particles in Newtonian and power-law fluids. An approximate series solution in powers of time for transient heat flow from spheroid particles has been reported by Normiton and Blackwell.28 The rates of heat and mass transfer from spheroid particles of aspect ratio 0.2 to surrounding air was numerically investigated at intermediate Reynolds numbers.29 The forced convection mass transfer from thin oblate (thin ice bar) to surrounding air was numerically investigated at Reynolds numbers up to 20 by Pitter et al.30 The semiempirical correlations for forced convective heat and mass transfer from spheroid particles to surrounding fluids with Prandtl numbers less than unity have been presented by Yovanovich.31 The laminar axisymmetric thermal flow from oblate spheroids (of aspect ratio 0.2−1) to surrounding air and water are numerically investigated at intermediate Reynolds numbers.32 Zheng and List33 have conducted wind tunnel experiments to investigate latitudinal and overall convective heat transfer from rotating spheres and spheroids with nonuniform surface temperature at very high Reynolds numbers. Wen and Jog34 have numerically obtained the effects of variable thermal properties on the flow field and heat transfer around spheroid particles. Juncu35 has numerically investigated the forced convective heat transfer from spheroid particles at intermediate Reynolds numbers and for Pr = 1 and 10. Recently, Richter and Nikrityuk36 numerically studied the drag and heat transfer effects on fixed spherical, cuboidal, and ellipsoidal particles in Newtonian fluids and concluded that apart from Reynolds number three geometrical parameters influence the particle−fluid interactions, namely, normalized longitudinal length, sphericity, and crosswise sphericity. In their subsequent study,37 the effect of orientation of ellipsoid particles on their drag and Nusselt numbers has been investigated using ANSYS Fluent and proposed correlations for drag and Nusselt numbers of ellipsoids. Kishore and Gu38 numerically investigated the drag and heat transfer phenomena of unconfined spheroid particles and proposed correlations for total drag coefficients and average Nusselt numbers of unconfined spheroid particles. Kishore and Gu39,40 extended their study to confined spheroids and developed empirical 990

dx.doi.org/10.1021/ie4032428 | Ind. Eng. Chem. Res. 2014, 53, 989−998


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velocity components. The appropriate momentum boundary conditions for this flow problem are the standard no-slip velocity on the surface of the spheroid particle and along the confining wall; the velocity inlet and pressure outlet at the inlet and outlet, respectively, and the standard axisymmetry condition along the central axis of the confining tube. In order to solve the energy equation, the uniform inlet temperature T0 and constant temperature Ts along the surface of the spheroid and the axisymmetry condition along the central axis are used. The governing conservation eqs 1−3 along with these boundary conditions are numerically solved for velocity, pressure, and temperature distributions in the entire computational domain using ANSYS Fluent. The fully converged velocity and temperature fields are further used to evaluate the streamlines, isotherm contours, drag coefficients, and surface and average Nusselt numbers as functions of the pertinent dimensionless numbers. The dimensionless parameters for power-law fluids those appear in this work are defined as follows: Reynolds number:

correlations for drag coefficients and average Nusselt numbers of confined spheroids in Newtonian fluids over a wide range of conditions. However, to the best of the authors’ knowledge, no literature is available on power-law fluid flow past and heat transfer from confined spheroids even in the limiting conditions, let alone in an intermediate range of conditions. Thus this work is endeavored to fill this gap in the literature over the range of conditions as of Re, 1−100; Pr, 1−1000; e, 0.5−2.5; λ, 2−5; and n, 0.4−1.8.

3. PROBLEM STATEMENT AND MATHEMATICAL FORMULATION As shown in Figure 1, consider that a spheroid particle (heated at temperature Ts) is fixed along the central axis of a tube and the particle is located at an upstream distance Lu from the inlet and at a downstream distance Ld from the outlet. The length of the tube is L (=Lu + Ld) and the diameter is Dt. The ratio of the diameter of the tube and the polar diameter of the particle (2b) is defined as the wall factor (λ). An incompressible power-law liquid is assumed to flow in the tube with a uniform velocity V and a temperature T0 at the inlet. The fluid experiences resistance force (drag force) because of the presence of spheroid particle at an upstream distance of Lu and, because of convection effect, a recirculation wake would appear in the rear end of the particle. The heat transfer is taking place from the heated spheroid surface to the surrounding flowing power-law liquids. Furthermore, both the fluid resistance and the rate of heat transfer are being affected by the confining tube wall and the non-Newtonian characteristics of the surrounding fluid. Therefore, this work is to find the effects of the wall factor and the power-law behavior index on the resistance force (drag) and the rate of heat transfer (Nusselt number) as functions of the Reynolds and/or Prandtl numbers and particle aspect ratio. In order to study the effect of the wall retardation, the value of the wall factor has been varied by changing the diameter of the tube by keeping the polar diameter of the particle fixed for a given spheroid. Further the velocity, pressure, and temperature fields around confined spheroid particles can be obtained by numerically solving the continuity, momentum, and energy equations of the following forms.43 continuity equation: (1) ∇·V = 0

λ=

(6)

n−1 ⎛ Cpm ⎞⎛ V ⎞ ⎜ ⎟ Pr = ⎜ ⎟⎜ ⎟ ⎝ k ⎠⎝ deq ⎠

(7)

Peclet number: (8)

Pe = Re·Pr total drag coefficient: Cd =

Fd 1 A p 2 ρV 2

(

)

= Cdp + Cdf (9)

Nusselt number:

Nu =

(2)

hdeq k

(10)

where deq is the sphere volume equivalent diameter of the spheroid particle; Ap is the projected area of the particle; Fd is the drag force exerted by the fluid on the particle; Cd is the total drag coefficient; Cdp and Cdf are the pressure and friction components of the total drag coefficient, respectively; and Cp, k, and h are the specific heat, thermal conductivity, and convective heat transfer coefficient, respectively.

(3)

where ρ is the density of the fluid, αf is the thermal diffusivity, and τ is the extra stress tensor for an incompressible fluid defined as43 τ = 2ηε. The viscosity equation for a power-law fluid can be written as1,3,43 ⎛ I ⎞(n − 1)/2 η = m⎜ 2 ⎟ ⎝2⎠

Dt 2b

Prandtl number:

energy equation: (V ·∇T ) = αf ∇2 T

(5)

m

wall factor:

momentum equation: ρ[(V ·∇)V ] = −∇P + ∇·τ

deq nV 2 − nρ

Re =

4. NUMERICAL METHODOLOGY The velocity, pressure, and temperature profiles in the entire computational domain are obtained by numerically solving the continuity and momentum equations as discussed elsewhere;38−42 hence only salient features are presented herein. The segregated approach has been used to solve eqs 1−3; i.e., first continuity and momentum equations are solved using the two-dimensional axisymmetric solver option in ANSYS Fluent

(4)

where the power-law fluid consistency index (m) and the power-law behavior index (n) are assumed to be independent of the temperature in the present range of investigation. The rate of the deformation tensor (ε) and the second invariant of the rate of the deformation tensor (I2) are related to the 991



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Table 1. Entry and Exit Effects on Cd of Confined Spheroids at Re = 1 e = 0.5

e = 2.5

λ=2

λ=5

λ=2

λ=5

Lu

Ld

n = 0.4

n = 1.8

n = 0.4

n = 1.8

n = 0.4

n = 1.8

n = 0.4

n = 1.8

35 50 65

35 50 65

59.008 58.628 58.811

2624.606 2593.776 2627.416

32.251 34.283 34.315

236.199 236.667 236.883

113.293 113.202 113.164

1248.179 1246.696 1248.376

90.418 84.794 84.838

182.109 182.364 182.351

to obtain fully converged velocity and pressure fields in the computational domain, and then the so-obtained velocity profile is used as input to solve the energy equation. The momentum transfer in the computational domain has been obtained using the semi-implicit method for pressure linked equations (SIMPLE) algorithm along with quadratic upstream interpolation for the convective kinematics (QUICK) scheme for convective terms. The residual tolerances for continuity and momentum equations are allowed to reach 10−8 in order to achieve fully converged velocity and pressure profiles. Then the fully converged velocity profile is used as input to solve the energy equation using the same approach with a residual tolerance for temperature profile of 10−12. The drag coefficients and Nusselt numbers along with other near-surface kinematics are evaluated by using these fully converged velocity and temperature profiles as discussed in section 3. However, prior to reporting new results, it is customary to check for numerical artifacts such as entry/exit and grid effects of the numerical solver and its validity. 4.1. Entry and Exit Effects. The upstream and downstream distances from the particle to inlet and outlet, respectively, often affect the velocity and pressure profiles, and hence optimum values of these distances are obtained through numerical experiments so as to neutralize their affect. Thus computational experiments were carried out for different values of Lu and Ld for extreme values of the wall factor, powerlaw index, and aspect ratio of a spheroid particle at Re = 1 and are presented in Table 1. It can be seen from Table 1 that, except for Lu = Ld = 35, all other values of Lu and Ld produce almost identical results; however, a larger value, i.e., Lu = Ld = 65, is chosen for all other computations. 4.2. Grid Independence Study. Detailed grid independence is carried out for extreme values of the power-law index, wall factor, and aspect ratio of the spheroid particle at Re = 100 and is shown in Table 2. The drag coefficients obtained by using different grids are almost identical for all combinations of these parameters. However, the finer grid 3 is chosen for λ = 2 and the moderately fine grid 2 is used for λ = 3, 5. Thus obtained optimum upstream and downstream distances and grids are used to solve the energy equation as well.

Table 2. Effect of Grid on Cd of Confined Spheroids at Re = 100 Cd

no. of nodes grid

spheroid surf.

grid 1 grid 2 grid 3

100 150 200


100 150 200


100 150 200


100 150 200

Lu e= 300 350 400 e= 300 350 400 e= 300 350 400 e= 300 350 400

Ld 0.5, λ = 300 350 400 0.5, λ = 300 350 400 2.5, λ = 300 350 400 2.5, λ = 300 350 400

tube wall

n = 0.4

n = 1.8

850 900 950

0.861 0.851 0.848

26.529 26.732 26.729

850 900 950

0.578 0.574 0.572

5.195 5.217 5.222

850 900 950

4.281 4.832 5.037

19.896 19.953 19.975

850 900 950

2.759 2.862 2.870

8.104 8.107 8.109

2

5

2

5

where;38−42 hence they are not repeated herein. However, in this work, additional validation of present Cd values of a spheroid of aspect ratio e = 0.5 in Newtonian liquids is shown with those of Richter and Nikrityuk36 in Table 3 and the Table 3. Comparison of Present Values of Cd of a Spheroid of e = 0.5 in Newtonian Fluids Re

Richter and Nikrityuk36

present

10 25 50 75 100

3.897 2.035 1.293 1.005 0.846

3.885 2.032 1.292 1.005 0.850

agreement between the two sets of results is excellent. Thus, on the basis of our previous experience38−42 and present validation, it is safe to conclude that the present solver is reliable and accurate to develop new results on the momentum and heat transfer characteristics of confined spheroid particles in power-law fluids. 5.2. Flow Phenomena. Figure 2 shows the streamline patterns of a shear-thinning fluid (n = 0.4) flow past confined spheroids of aspect ratio e = 0.5 (Figure 2a−c) and e = 2 (Figure 2d−f) at Re = 100 (upper half) and Re = 20 (lower half). For all values of the wall factor, for e = 0.5, there is no recirculation wake in the rear end at Re = 20; however, as the Reynolds number increases to Re = 100, a small recirculation wake has appeared because of the increased contribution of the convection over viscous forces. On the other hand, in the case of e = 2, the recirculation wake is observed for both Re = 20 and Re = 100 for all values of the wall factor. Regardless of the value

5. RESULTS AND DISCUSSION In order to delineate the drag and heat transfer behavior of confined spheroid particles in power-law fluids, the following range of conditions are considered: particle aspect ratio, e = 0.5, 1, 1.5, 2, 2.5; Reynolds number, Re = 1, 5, 10, 20, 50, 100; Prandtl number, Pr = 1, 10, 100, 100; wall factor, λ = 2, 3, 5; and power-law index, n = 0.4, 0.6, 0.8, 1, 1.4, 1.8. 5.1. Validation. The present numerical solver is extensively validated by comparing the present values of drag coefficients and Nusselt numbers with existing literature values for the cases of unconfined spheres and spheroids, and of confined spheres in Newtonian and power-law liquids and presented else992



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Figure 2. Streamlines of a shear-thinning fluid (n = 0.4) flow past confined spheroids of (a−c) e = 0.5 and (d−f) e = 2 at Re = 100 (upper half) and Re = 20 (lower half).

Figure 3. Streamlines of a shear-thickening fluid (n = 1.8) flow past confined spheroids of (a−c) e = 0.5 and (d−f) e = 2 at Re = 100 (upper half) and Re = 20 (lower half).

and power-law index; hence, they are not repeated. Furthermore, unlike in Newtonian fluids, for shear-thinning fluids, as the rate of shear increases the apparent viscosity decreases, and thus the rate of settling of the spheroids will be faster as compared to Newtonian fluids; the opposite trend is true for shear-thickening fluids. In other words, the shearthinning type power-law viscosity behavior reduces the viscous forces; hence, for fixed values of the Reynolds number and the wall factor, the length of the recirculation wake behind the spheroid increases as compared to the case of Newtonian fluid while the opposite trend is true for shear-thickening fluids. Finally, the size of the recirculation wake behind the spheroid increases with the increasing Reynolds number and/or with the increasing particle aspect ratio and/or with the increasing wall factor and/or with the decreasing power-law index. The physical significance of these results indicates that, at a given Reynolds number for a given particle, the settling velocity is higher in shear-thinning fluids followed by Newtonian and shear-thickening fluids.

of the particle aspect ratio, with the increasing wall factor, i.e., with decreasing retardation forces, the size of the recirculation wake increases because the convective forces gradually increase with the decreasing confinement. Figure 3 shows the streamline patterns of a shear-thickening fluid (n = 1.8) flow past spheroid particles of aspect ratio e = 0.5 (Figure 3a−c) and e = 2 (Figure 3d−f) at Re = 100 (upper half) and Re = 20 (lower half) for different values of the wall factor. In the case of shearthickening fluid flow past spheroids of e = 0.5 (Figure 3a−c), there is no recirculation wake even at Re = 100, whereas for spheroids of e = 2 (Figure 3d−f) the trends are similar to the case of shear-thinning fluids. However, the size of the recirculation wake has substantially reduced for shearthickening fluids as compared to shear-thinning fluids because of the suppression of convection forces due to increased dilatant nature of the fluid. In the case of shear-thickening fluids as well, irrespective of the value of the particle aspect ratio, the size of the recirculation wake increases with the increasing wall factor. Qualitatively similar trends are seen for other combinations of the Reynolds number, aspect ratio, wall factor, 993



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Figure 4. Cd values of confined spheroidal particles of (a−c) e = 0.5 and (d−f) e = 2 for different values of power-law index.

Figure 5. Ratio between pressure and friction drag coefficients of confined spheroidal particles of (a−c) e = 0.5 and (d−f) e = 2 for different values of power-law index.

shear-thinning fluids are smaller than those of Newtonian fluids. The opposite trend is true for shear-thickening fluids. Thus, in a physical sense, spheroid particles can easily settle in shear-thinning fluids (n < 1) filled in tubes of larger diameters, whereas settling of same size spheroid particles receive higher resistance to settling in shear-thickening fluids (n > 1) filled in narrow tubes. Figure 5 represents the combined effects of the Reynolds number, power-law index, and wall factor on the relative contributions of pressure and friction drag coefficients of spheroid particles of aspect ratio e = 0.5 (Figure 5a−c) and e = 2 (Figure 5d−f). As the value of the Reynolds number increases, for n = 0.4, the ratio between two individual drag coefficients gradually increases for all values of e and λ.

5.3. Drag Phenomena. Figure 4 shows the total drag coefficients of confined spheroids of aspect ratio e = 0.5 (Figure 4a−c) and e = 2 (Figure 4d−f) in power-law fluids as functions of the Reynolds number and the wall factor. Regardless of the shape of the spheroid particle, for fixed values of the Reynolds number and the power-law index, the drag coefficient decreases with increasing wall factor because of decreasing retardation effects. Regardless of values of the wall factor and the powerlaw index, as the value of the Reynolds number increases the drag coefficient decreases for spheroid particles of e < 1 and e > 1. For fixed values of the Reynolds number and the wall factor, the drag value increases with increasing power-law index; i.e., for a given combination of Re and λ, the drag coefficients of 994



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Figure 6. Temperature contours around confined spheroids of (a−c) e = 0.5 and (d−f) e = 2.0 in shear-thinning fluids of n = 0.4 at Re = 100 for Pr = 1000 (upper half) and Pr = 10 (lower half).

Figure 7. Temperature contours around confined spheroids of (a−c) e = 0.5 and (d−f) e = 2.0 in shear-thickening fluids of n = 1.8 at Re = 100 for Pr = 1000 (upper half) and Pr = 10 (lower half).

ranges of conditions as 1 ≤ Re ≤ 100, 0.4 ≤ n ≤ 1.8, 2 ≤ λ ≤ 5, and 0.5 ≤ e ≤ 2.5:

However, increase in the drag ratio is more significant for the case of spheroids of e > 1 when compared to spheroid particles of e < 1. The slope of the increment of this drag ratio decreases with increasing power-law index; moreover, it becomes almost independent of the Reynolds number for shear-thickening fluids. The physical senses of these results indicate that spheroids of e < 1 can more easily settle in shear-thinning fluids than in shear-thickening fluids, whereas the spheroid particles of e > 1 experience more resistance from the fluid. Finally, an attempt has been made to fit the numerical results of Cd of confined spheroid particles in the form of an empirical correlation and the following form is found to be suitable in the B=1+

Cd =

22n + 1(n + 2) (1 + 0.157Re 0.687)A(Re , n , λ) Re B(Re , n , λ)

(11)

where A=1+

4.94Re−0.34(n + 0.6)0.82 1 + 1.2n(λ − 2.08)

(12)

and

0.44(e − 0.4)Re 0.27[1 − (n − 0.48)Re−0.1] 1 + (λ − 2.82)[− 0.36(1 − 0.13 ln Re) + 0.07(λ − 3)(1 − 0.18 ln Re)]

995

(13)



Article

Figure 8. Nusselt number distribution along the surface of confined spheroids of (a−c) e = 0.5 and (d−f) e = 2.0 in power-law fluids at Re = 20 and Pr = 100.

Figure 9. Average Nusselt numbers of confined spheroids of (a−c) e = 0.5 and (d−f) e = 2.0 in power-law fluids at Re = 20.

Equation 11 reproduces the present numerical Cd values (720 data points) with an average relative difference of ±10.38%, which rises to a maximum of ±59.38% for extreme conditions. Furthermore, eq 11 reduces to the well-known Schiller and Naumann44 correlation for total drag coefficients of solid spheres settling in unbounded (large values of λ) Newtonian fluids (n = 1). 5.4. Isotherm Contours. Figure 6 shows the isotherm contours around confined spheroids of aspect ratio e = 0.5 (Figure 6a−c) and e = 2 (Figure 6d−f) in shear-thinning fluids of n = 0.4 at Re = 100 and Pr = 1000 (upper half) and Pr = 10 (lower half). For spheroids of aspect ratio e = 0.5 (Figure 6a−c)

and for all values of the wall factor, at Pr = 1000, the isotherm contours at the rear end are being sucked toward the particle because of small recirculation wake formation at Re = 100. However, for Pr = 10, only small amounts of contours are being sucked toward the rear of the particle while the majority of them are being convected in the flow direction. Hence, the thermal boundary layer is thinner for Pr = 1000 and, as expected, the rate of heat transfer increases. Further, by increasing the value of the wall factor, because of decreased retardation forces, the boundary layer has gradually become thicker. For the case of spheroids of aspect ratio e = 2 (Figure 6d−f), regardless of values of the wall factor, for both Pr = 10 996



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6. CONCLUSIONS The power-law fluid flow past and heat transfer from confined spheroid particles is numerically investigated over wide ranges of conditions. The recirculation wake length behind confined spheroid particles increases with increasing Reynolds number and/or wall factor and/or aspect ratio and/or with decreasing power-law index. Further, these recirculation lengths are prominent in spheroids of e > 1 compared to spheroid particles of e < 1. The total drag coefficient on spheroids decreases with the increasing Reynolds numbers and/or wall factor and/or with the decreasing power-law index and/or particle aspect ratio. The effect of Reynolds number on the relative contributions of pressure and friction drag coefficients is large for shear-thinning fluids. For all values of the Reynolds and/or Prandtl numbers, regardless of the shape of the spheroid particles, with the decreasing power-law index and/or with the decreasing wall factor, the average Nusselt number increases.

and Pr = 1000, the formation of recirculation at the rear end is very strong and its length is very large compared to spheroids of e = 0.5. Thus the boundary layer has become thinner for spheroids of e > 1 to cause an increase in the rate of heat transfer. Figure 7 shows the isotherm contours around confined spheroids in shear-thickening fluids of n = 1.8 at Re = 100 and Pr = 1000 and Pr = 10, and qualitatively similar trends are seen as in the case of Figure 6. However, because of reduced recirculation wake for shear-thickening fluids, the thermal boundary layer is thick enough to cause a decrease in the overall rate of heat transfer. 5.5. Nusselt Numbers. Figure 8 shows the surface Nusselt number distribution for spheroids with e = 0.5 (Figure 8a−c) and spheroids with e = 2 (Figure 8d−f) for different values of the wall factor and power-law index. For the case of spheroids of e = 0.5, the surface Nusselt number decreases from the front stagnation point to the rear stagnation point for both shearthinning and shear-thickening fluids. The surface Nusselt numbers do not change much with change in either powerlaw index or wall factor; however, for spheroids of e = 2, discernible trends are observed. For shear-thinning fluids, the surface Nusselt number of spheroids of e = 2 increases from its value at the front stagnation point to θ ≈ 60−70°, then again decreases up to θ ≈ 120°, and finally forms a secondary wake in the rear end because of the recirculation wake seen in the case of momentum transfer. For shear-thickening fluids, for the surface Nusselt numbers in the fore aft of the spheroid of e = 2, a reverse trend is seen as compared to shear-thinning fluids. Further, in the rear of the spheroid of e = 2, the size of the secondary wake decreases with increasing power-law index and wall factor. Furthermore, as the value of the wall factor increases, the trends are qualitatively similar; however, the magnitude of the Nusselt number decreases. Figure 9 shows the combined effects of the wall factor and power-law index on the average Nusselt number of confined spheroids of aspect ratio e = 0.5 (Figure 9a−c) and e = 2 (Figure 9d−f) at Re = 20. The characteristic trend of increasing average Nusselt number with the increasing Peclet number has been observed for all values of the particle aspect ratio, wall factor, and power-law index. The average Nusselt number increases with the decreasing powerlaw index and/or with the decreasing wall factor and/or with the increasing particle aspect ratio. Thus, in a physical sense, the rate of heat transfer from prolate spheroids to shearthinning fluids is higher compared to Newtonian and shearthickening fluids. Finally, on the basis of present numerical results (2880 data points), the following form of empirical correlation for the average Nusselt number of confined spheroids in power-law liquids has been found to be suitable over the range of conditions as 1 ≤ Re ≤ 100, 1 ≤ Pr ≤ 1000, 0.4 ≤ n ≤ 1.8, 2 ≤ λ ≤ 5, and 0.5 ≤ e ≤ 2.5:

■

*E-mail: [email protected]. Tel.: +91-361-2582276. Notes

The authors declare no competing financial interest.

■

Nuavg = [2.4Pe 0.3Re 0.02 − α(n − 1)Pe 0.4]{e 0.7}

NOMENCLATURE a = equatorial radius of the spheroid particle, m Ap = projected area of the particle, m2 b = polar radius of the spheroid particle, m Cd = total drag coefficient, dimensionless Cdf = friction drag coefficient, dimensionless Cdp = pressure drag coefficient, dimensionless Cp = specific heat capacity, J kg−1 K−1 Dt = diameter of the tube, m e = aspect ratio, dimensionless Fd = drag force, N h = convective heat transfer coefficient, W m−2 K−1 k = thermal conductivity of fluid, W m−1 K−1 I2 = second invariant of rate of deformation tensor, s−2 L = length of the tube, dimensionless Ld = downstream distance, dimensionless Lu = upstream distance, dimensionless m = power-law fluid consistency index, Pa·sn n = power-law fluid behavior index, dimensionless Nu = Nusselt number, dimensionless P = pressure, Pa Pe = Peclet number, dimensionless Pr = Prandtl number, dimensionless Re = Reynolds number, dimensionless T = temperature, K T0 = surrounding fluid temperature, K Ts = particle surface temperature, K V = velocity vector, m/s

Greek Symbols

− (λ − 2)[0.15Pe 0.38 − (λ − 3)0.025Pe 0.46] {1.2 e − 0.1[e + 1]}

AUTHOR INFORMATION

Corresponding Author

(14)

This correlation reproduces the present numerical results within a relative average error of ±10.35%, which rises to the maximum of ±56.81% for extreme values of Peclet numbers for spheroids of aspect ratio 2.5. Furthermore, for e = 1, eq 14 reduces to our previous results on average Nusselt numbers of confined spheres in power-law liquids.42

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η = viscosity of the fluid, kg/m·s λ = wall factor, dimensionless ρ = density of the fluid, kg/m3 τ = extra stress tensor, Pa ε = rate of deformation, s−1

REFERENCES

(1) Chhabra, R. P. Bubbles, Drops and Particles in Non-Newtonian Fluids; CRC: Boca Raton, FL, 2006.

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Article

(2) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic: New York, 1978. (3) Chhabra, R. P.; Richardson, J. F. Non-Newtonian Flow and Applied Rheology, 2nd ed.; Butterworth-Heinemann: Oxford, U.K., 2008. (4) Macosko, C. W. Rheology: Principles, Measurements and Applications; Wiley-VCH: New York, 1994. (5) Oberbeck, A. Ü ber die stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung. J. Reine Angew. Math. 1876, 81, 62−80. (6) Payne, L. E.; Pell, W. H. The Stokes flow problem for a class of axially symmetric bodies. J. Fluid Mech. 1960, 7, 529−549. (7) Breach, D. R. Slow flow past ellipsoids of revolution. J. Fluid Mech. 1961, 10, 306−314. (8) Aoi, T. The steady flow of viscous fluid past a fixed spheroidal obstacle at small Reynolds numbers. J. Phys. Soc. Jpn. 1955, 10, 119− 129. (9) Chwang, A. T.; Wu, T. Y. Hydrodynamics of low-Reynoldsnumber flow. Part 4: Translation of spheroids. J. Fluid Mech. 1976, 75, 677−689. (10) Sugihara-Seki, M. The motion of an ellipsoid in tube flow at low Reynolds numbers. J. Fluid Mech. 1996, 324, 287−308. (11) Alassar, R. S.; Badr, H. M. Steady flow past an oblate spheroid at small Reynolds numbers. J. Eng. Math. 1999, 36, 277−287. (12) Blaser, S. Forces on the surface of small ellipsoidal particles immersed in a linear flow field. Chem. Eng. Sci. 2002, 57, 515−526. (13) Haider, A.; Levenspiel, O. Drag coefficient and terminal velocity of spherical and non-spherical particles. Powder Technol. 1989, 58, 63− 70. (14) Militzer, J.; Kan, J. M.; Hamdullahpur, F.; Amyotte, P. R.; Al Taweel, A. M. Drag coefficient for axisymmetric flow around individual spheroid particles. Powder Technol. 1989, 57, 193−195. (15) Chhabra, R. P.; Agarwal, L.; Sinha, N. K. Drag on non-spherical particles: An evaluation of available methods. Powder Technol. 1999, 101, 288−295. (16) Wang, J.; Qi, H.; You, C. Experimental study of sedimentation characteristics of spheroid particles. Particuology 2009, 7, 264−268. (17) Tran-Cong, S.; Gay, M.; Michaelides, E. E. Drag coefficients of irregularly shaped particles. Powder Technol. 2004, 139, 21−32. (18) Pinar, E.; Sahin, B.; Ozgoren, M.; Akilli, H. Experimental study of flow structures around side-by-side spheres. Ind. Eng. Chem. Res. 2013, 52, 14492−14503, DOI: 10.1021/ie4022732. (19) Rimon, Y.; Lugt, H. J. Laminar flow past oblate spheroids of various thickness. Phys. Fluids 1969, 12, 2465−2472. (20) Masliyah, J. H.; Epstein, N. Numerical study of steady flow past spheroids. J. Fluid Mech. 1970, 44, 493−512. (21) Comer, J. K.; Kleinstreuer, C. A numerical investigation of laminar flow past non-spherical solids and droplets. J. Fluids Eng. 1995, 117, 170−175. (22) Pitter, R. L.; Pruppacher, H. R.; Hamielec, A. E. A numerical study of viscous flow past a thin oblate spheroid at low and intermediate Reynolds numbers. J. Atmos. Sci. 1973, 30, 125−134. (23) Chang, C. C.; Liou, B. H.; Chern, R. L. An analytical and numerical study of axisymmetric flow around spheroids. J. Fluid Mech. 1992, 234, 219−246. (24) Li, W. Z.; Yan, Y. Y.; Smith, J. M. A numerical study of the interfacial transport characteristics outside spheroidal bubbles and solids. Int. J. Multiphase Flow 2003, 29, 435−460. (25) Zamyshlyaev, A. A.; Shrager, G. R. Fluid flows past spheroids at moderate Reynolds numbers. Fluid Dyn. 2004, 39, 376−383. (26) Tripathi, A.; Chhabra, R. P.; Sundararajan, T. Power law fluid flow over spheroid particles. Ind. Eng. Chem. Res. 1994, 33, 403−410. (27) Tripathi, A.; Chhabra, R. P. Drag on spheroidal particles in dilatant fluids. AIChE J. 1995, 41, 728−731. (28) Norminton, E. J.; Blackwell, J. H. Transient heat flow from constant temperature spheroids and the thin circular disk. Q. J. Mech. Appl. Math. 1964, XVII, 65−72. (29) Masliyah, J. H.; Epstein, N. Numerical solution of heat and mass transfer from spheroids in steady axi-symmetric flow. Prog. Heat Mass Transfer 1972, 6, 613−632.

(30) Pitter, R. L.; Pruppacher, H. R.; Hamielec, A. E. A numerical study of the effect of forced convection on mass transfer from a thin oblate spheroid of ice in air. J. Atmos. Sci. 1974, 31, 1058−1066. (31) Yovanovich, M. M. General expression for forced convection heat and mass transfer from isopotential spheroids. AIAA 26th Aerospace Sciences Meeting, Reno, NV; American Institute of Aeronautics and Astronautics: Reston, VA, 1988; paper 0743, pp 11−14. (32) Comer, J. K.; Kleinstreuer, C. Computational analysis of convection heat transfer to non-spherical particles. Int. J. Heat Mass Transfer 1995, 38, 3171−3180. (33) Zheng, G.; List, R. Convective heat transfer of rotating spheres and spheroids with non-uniform surface temperatures. Int. J. Heat Mass Transfer 1996, 39, 1815−1826. (34) Wen, Y.; Jog, M. A. Variable property, steady, axi-symmetric, laminar, continuum plasma flow over spheroid particles. Int. J. Heat Fluid Flow 2005, 26, 780−791. (35) Juncu, G. Unsteady heat transfer from an oblate/prolate spheroid. Int. J. Heat Mass Transfer 2010, 53, 3483−3494. (36) Richter, A.; Nikrityuk, P. A. Drag forces and heat transfer coefficients for spherical, cuboidal and ellipsoidal particles in cross flow at sub critical Reynolds numbers. Int. J. Heat Mass Transfer 2012, 55, 1343−1354. (37) Richter, A.; Nikrityuk, P. A. New correlations for heat and fluid flow past ellipsoidal and cubic particles at different angles of attack. Powder Technol. 2013, 249, 463−474. (38) Kishore, N.; Gu, S. Momentum and heat transfer phenomena of spheroid particles at moderate Reynolds and Prandtl numbers. Int. J. Heat Mass Transfer 2011, 54, 2595−2601. (39) Kishore, N.; Gu, S. Wall effects on flow and drag phenomena of spheroid particles at moderate Reynolds numbers. Ind. Eng. Chem. Res. 2010, 49, 9486−9495. (40) Kishore, N.; Gu, S. Effect of blockage on heat transfer phenomena of spheroid particles at moderate Reynolds and Prandtl numbers. Chem. Eng. Technol. 2011, 34, 1551−1558. (41) Reddy, C. R.; Kishore, N. Wall retardation effects on flow and drag phenomena of confined spherical particles in shear-thickening fluids. Ind. Eng. Chem. Res. 2012, 51, 16755−16762. (42) Reddy, C. R.; Kishore, N. Effects of wall confinement and power-law fluid viscosity on Nusselt number of confined sphere. Chem. Eng. Technol. 2013, 36, 1568−1576. (43) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley; New York, 2002. (44) Schiller, L.; Naumann, A. Z. Ü ber die grundlegenden Berechungen bei der Schwerkraflaufbereitung. Z. Ver. Dtsch. Ing. 1933, 77, 318−320.

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Momentum and Heat Transfer Phenomena of Confined Spheroid

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