Cmjv03i03p0344 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

Finite Element Analysis of Mixed Convective Heat and Mass Transfer Flow Through a Conentric Annulus with Heat Sources and Radiation D. CHITTI BABU1 and D. R. V. PRASADA RAO2 1

Department of Mathematics, Government College (A), Rajahmundry, A.P., INDIA. 2 Department of Mathematics, S.K. University, Anantapur, A.P., INDIA (Received on: May 31, 2012) ABSTRACT We investigate the free and forced convection flow through a porous medium in a co-axial cylindrical duct with radiation effect where the boundaries are maintained at constant temperature and concentration. The Brinkman-Forchhimer extended Darcy equations which takes into account the boundary and inertia effects are used in the governing linear momentum equations. The effect of density variation is confined to the buoyancy term under Boussinesq approximation,the momentum, energy and diffusion equations are coupled equations. In order to get a better insight into this complex problem, we make use of Galerkin finite element analysis with quadratic polynomial approximations. The behaviour of velocity, temperature and concentration is analyzed at different axial positions. The shear stress and the rate of heat and mass transfer have also been obtained for variations in the governing parameters. Keywords: Convective heat transfer, mass transfer, Radiation

1. INTRODUCTION Transport phenomena involving the combined influence of thermal and concentration buoyancy are often encountered in many engineering systems

and natural environments. There are many applications of such transport processes in the industry notably in chemical distilleries, heat exchangers, solar energy collectors and thermal protection systems. In all such classes of flows, the driving force is

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

provided by a combination of thermal and chemical diffusion effects. In atmosphere flows, thermal convection of the earth by sunlight is affected by differences in water vapour concentration.In particular, design engineers require relationships between heat transfer geometry and boundary conditions which can be utilized in cost-benefit analysis to determine the amount of insulation that will yield the maximum investment. It is known that canisters filled with radioactive rays be buried in earth so as to isolate them from human population ans is of interest to find the surface temperature of these canisters.This phenomenon is ideal to the study of convection flow in a porous medium contained in a cylindrical annulus6,7,8. Free convection in a vertical prous annulus has been extensively studied by Prasad6, Prasad and Kulacki7 and Prasad et al.8 both theoretically and experimentally. Chen and Yuh3 have investigated the heat and mass transfer characteristics of natural convection flow along a vertical cylinder under the combined buoyancy effects of thermal and species diffusion. Antonio2 has investigated the laminar flow, hat transfer in a vertical cylindrical duct by taking into account both viscous dissipation and the effect of buoyancy. The limiting case of fully developed natural convection in porous annuli is solved analytically for steady and transient cases by Sharawi and Al-Nimir9 and Al-Nimir(1), Philip5 has obtained solutions for the annular porous media valid for low modified Reynolds number. Sudheer Kumar et al.10 have studied the effect of radiation on natural convection over a vertical cylinder in a porous media. Padmavathi4 has analyzed the convective hat

transfer in a cylindrical annulus by using finite element method. In this paper we discuss the free and forced convection flow through a porous medium in a co-axial cylindrical duct with radiation effect where the boundaries are maintained at constant temperature and concentration.The effect of density variation is confined to the buoyancy term under Boussinesq approximation, the momentum, energy and diffusion equations are coupled equations. In order to get a better insight into this complex problem, we make use of Galerkin finite element analysis with quadratic polynomial approximations. The Galerkin finite element analysis has two important features. The first is that that approximation solution is written directly as a linear combination of approximation functions with unknown nodal values as coefficients. The behaviour of velocity, temperature and concentration is analysed at different axial positions. The shear stress and the rate of heat and mass transfer have also been obtained for variations in the governing parameters. Rate of heat and mass transfer have also been obtained for variations in the governing parameters. 2. FORMULATION OF THE PROBLEM We consider the free and forced convection flow in a vertical circular annulus through a porous medium whose walls are maintained at a constant temperature and concentration. The flow, temperature and concentration in the fluid are assumed to be fully developed. The Boussenissq approximation is invoked so that the density variation is confined to the thermal and molecular buoyancy forces with radiation effect. The Brinkmanâ&#x20AC;&#x201C;Forchhimerâ&#x20AC;&#x201C;

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

Extended Darcy model which accounts for the inertia and boundary effects have been used for the momentum equation in the porous region. The momentum, energy and diffusion equations are coupled and non– linear. Also the flow is unidirectional along the axial direction of the cylindrical annulus. Making use of the above assumptions the governing equations are

− ∂p µ  ∂ 2 u 1 ∂u  µ pδF 2  − u − +  2 + u δ  ∂r ∂z r ∂r  k k + ρgβ (T − T0 ) + ρgβ ∗ (C − C 0 ) = 0 (2.1)  ∂ 2T 1 ∂T  ∂T ∂  + Q − (q r ) ρ c pu = λ  2 + ∂Z r ∂r  ∂y  ∂r (2.2)

 ∂ C 1 ∂C  ∂C  = D1  2 + ∂Z r ∂r   ∂r 2

(2.3)

where u is the axial velocity in the porous region. T, C are the temperature and concentration of the fluid, k is the permeability of porous medium, F is a function that depends on Reynolds number, the microstructure of the porous medium and D1 is the molecular diffusivity, β is the coefficient of volume expansion, cP is the specific heat, ρ is density and g is gravity.The relevant boundary conditions are u = 0,T = Ti,C = Ci at r = a u = 0,T = T0,C = C0 at r=a + s

(2.4)

Introducing Rosseland approximation for Radiative heat flux qr =

− 4 T ∂ T4 3βr ∂ u r

We now define the following non – dimensional variables.

z ∗ r ∗ a paδ , r = , u = u , D∗ = , a a r ρ r2 s T − T0 C − C0 θ∗ = , s∗ = , C ∗ = a C1 − C0 Ti − T0 z∗ =

Introducing those non–dimensional variables, the governing equations in the non-dimensional form are (on removing the stars) −1 ∂ 2 u 1 ∂u π + δ ( D )u + + = (2.5) ∂r 2 r ∂r δ 2 A u 2 − δG (θ + NC )

∂ 2θ 1 ∂θ + = P1 N1u − α ∂r 2 r ∂r

(2.6)

∂ 2 C 1 ∂C + = Sc N 2 u ∂r 2 r ∂r

(2.7)

The corresponding non – dimensional boundary conditions are u = 0, θ = 1, c = 1 at r = 1

(2.8)

u = 0, θ = 0, c = 0 at r=1+s

(2.9)

3. FINITE ELEMENT ANALYSIS The finite element analysis with quadratic polynomial approximation functions is carried out along the radial distance across the circular duet. The behavior of the velocity, temperature and concentration profiles has been discussed computationally for different variations in

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

governing parameters. The Galerkin method has been adopted in the variational formulation in each element to obtain the global coupled matrices for the velocity, temperature and concentration in course of the definite element analysis. Choose an arbitrary element ek and c let uk, θk and ck be the values of u, θ and in the element ek.We define the error residuals as d  du k r dr  dr

(

  + δG θ k + Nc k  −1 k 2 δ (0 ) r u − δ A r (u k ) 2 E pk =

(3.1)

d  dθ k Eθk =  r dr  dr

  − r P1 N1u k + α 

(3.2)

d  dC k Ec =  r dr  dr

  − r ScN 2u k 

(3.3)

u k = u1k ψ 1k + u3k ψ 1k + u3k ψ 3k

(3.4)

A 4X 4 = B 4

(3.5)

The global matrix for u is A5X 5 =B5

(3.6)

Solving these coupled global matrices for temperature, concentration and velocity (3.4) – (3.6) respectively and using the iteration procedure we determine the unknown global nodes through which the temperature, concentration and velocity at different radial intervals at any arbitrary arial cross sections are obtained. The respective expressions are given by

θ (r ) = ψ 11 θ11 + ψ 211 θ12 + ψ 31 θ13

θ k = θ1k ψ 1k + θ 2k ψ 2k + θ 3k ψ 3k

1 ≤ r ≤ 1 + s ∗ 0.2 =ψ 12 θ13 + ψ 22 θ14 + ψ 32 θ15

C k = C1k ψ 1k + C2k ψ 2k + C3k ψ 3k

ψ1k , ψ 2k , ψ 3k ...........

A 3X 3 = B 3 The global matrix for C is

4. SOLUTION OF THE PROBLEM

where uk, θk and ck are the values of u, θ and c in the arbitrary element ek. These are expressed as linear combinations in terms of respective local nodal values.

where

order 2n+1. The ultimate coupled global matrices are solved to determine the unknown global nodal values of the velocity, temperature and concentration influid region. In solving these global matrices an iteration procedure has been adopted to include the boundary and effects in the porous medium. For computational purpose we choose five elements in flow region. The global matrix for θ is

etc

1 + s ∗ 0.2 ≤ r ≤ 1 + s ∗ 0.4 =ψ 13 θ15 + ψ 23 θ16 + ψ 33 θ17 are

Lagrange’s quadratic polynomials. Incase we choose n quadratic elements, then the global matrices are of

1 + s ∗ 0.4 ≤ r ≤ 1 + s ∗ 0.6 =ψ 14 θ17 + ψ 24 θ18 + ψ 34 θ19 1 + s ∗ 0.6 ≤ r ≤ 1 + s ∗ 0.8

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

=ψ 115 θ19 + ψ 25 θ 20 + ψ 35 θ 21

1 + s∗ 0.8 ≤ r ≤ 1 + s u (r ) = ψ 11 u11 + ψ 211 u12 + ψ 31 u13

1 + s ∗ 0.8 ≤ r ≤ 1 + s C (r ) =ψ 11 c11 + ψ 211 c12 + ψ 31 c13

1 ≤ r ≤ 1 + s∗ 0.2 =ψ 12 u13 + ψ 22 u14 + ψ 32 u15

1 ≤ r ≤ 1 + s ∗ 0.2 =ψ 12 c13 + ψ 22 c14 + ψ 32 c15

1 + s ∗ 0.2 ≤ r ≤ 1 + s ∗ 0.4 =ψ 13 u15 + ψ 23 u16 + ψ 33 u17

1 + s ∗ 0.2 ≤ r ≤ 1 + s ∗ 0.4 =ψ 13 c15 + ψ 23 c16 + ψ 33 c17

1 + s ∗ 0.4 ≤ r ≤ 1 + s ∗ 0.6 =ψ 14 u17 + ψ 24 u18 + ψ 34 u19

1 + s ∗ 0.4 ≤ r ≤ 1 + s ∗ 0.6 4 4 4 = ψ1 c17 + ψ 2 c18 + ψ3 c19

1 + s ∗ 0.6 ≤ r ≤ 1 + s ∗ 0.8 =ψ 115 u19 + ψ 25 u20 + ψ 35 u 21

1 + s ∗ 0.6 ≤ r ≤ 1 + s ∗ 0.8 =ψ 115 c19 + ψ 25 c20 + ψ 35 c21

1 + s ∗ 0.8 ≤ r ≤ 1 + s

0.4

0.18

0.3

0.16 0.14

0.2 I

0.1 u

II III

0 1

1.2

1.4

1.6

1.8

-0.1

0.12

0.1

0.08

III

0.06 0.04

-0.2

0.02 -0.3

-0.4

3x10

5x10

8x10

-3x10

1.4

1.6

1.8

−1

Fig.1 Variation of u with G I II III IV 3

1.2

Fig. 2 Variation of u with D I II III

V 3

-5x103

0.25

−1

5x103

103

2x103

0.4 0.3

0.2 I

0.2

0.15 u

III IV

0.1

u0.1

III

1.2

1.4

1.6

1.8

-0.1

0.05

-0.2 0 1

1.2

1.4

1.6

1.8

Fig. 3 Variation of u with α I II III IV 4 6 -4 -6

-0.3

Fig. 4 Variation of u with N I II III IV -0.5 -0.8 1 2

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012) 1.5

0.25

0.2

1 I II

0.15 u

III

I II

0.5

III IV

0.1

0.05

1.2

1.4

1.6

1.8

-0.5

0 1

1.2

1.4

1.6

1.8

-1

Fig. 5 Variation of u with Sc I II III IV Sc 2.01 0.24 0.6 1.3

Fig.6 Variation of θ with G I II III IV V 3x103 5x103 8x103 -3x103 -5x103

1.2

0.8

III

0.6

III

0.4

0.4 0.2

0.2 0 1

0 1

1.2

1.4

1.6

1.8

Fig. 7 Variation of θ with D I II III D

−1

5x103

103

1.2

1.4

-0.2

−1

1.6

1.8

Fig. 8 Variation of θ with α I II III IV

2x103

-4

-6

1.2

1 0.8

0.8

I II

0.6 θ

III

0.4

I II

0.6

III

0.4 0.2 0.2 0 1 -0.2

1.2

1.4

1.6

1.8

Fig. 9 Variation of θ with N -0.5 -0.8 1 2

0 1

1.2

1.4

1.6

1.8

Fig. 10 Variation of θ with Sc 2.01 0.24 0.6 1.3

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012) 1.2

1.2

1 I

0.8

C 0.6

III

0.8

0.6

III

0.4

0.2

0.2 0

0 1

1.2

1.4

1.6

1.8

Fig.11 Variation of C with G I II III IV 3x103

5x103

8x103

1.2

1.4

1.6

1.8

Fig. 12 Variation of C with D I II III

-3x103 -5x103

−1

5x103

103

−1

2x103

1.2

1 I

0.8

C 0.6

III

0.6

0.8

III

0.4

0.2

1.2

1.4

1.6

1.8

Fig. 13 Variation of C with α I II III IV 4 6 -4 -6

1.2

1.4

1.6

1.8

Fig. 14 Variation of C with N I II III IV -0.5 -0.8 1 2

1.2 1 I

0.8

III

0.6

0.4 0.2 0 1

1.2

1.4

1.6

1.8

Fig. 15 Variation of C with Sc I II III IV 2.01 0.24 0.6 1.3

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D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

5. DISCUSSION OF THE RESULTS In this analysis, we investigate the convective heat and mass transfer flow of a viscous fluid in the annular region with radiation within two concentric cylinders which are maintained at constant temperature and concentrations. By using Galerkin finite element analysis with quadratic polynomials, the velocity, temperature and concentration have been analysed for different values of the governing parameters G(Grashoff number), D −1 (Darcy parameter), α (heat source parameter, N(Radiation parameter), Sc(Schmidt number) respectively. In this analysis, the actual axial flow is in vertically downward direction and hence u>0 correspondence to the reversal flow. The actual axial velocity u is shown in figs 1-5 for different sets of parameters. Fig.1 represents the variation of u with G. It is found that the velocity u is positive for G>0 and negative for G<0 thereby exhibiting a reversal flow in the entire flow region for G>0. u experiences an enhancement with

u with maximum attained at r=1.5. From fig.2, we observe that the variation of u is remarkably appreciable for higher values of D −1 . Lesser the permeability of the porous medium larger u and for further lowering of the permeability higher u in the region 1.4 ≥ r ≤ 1.6 and smaller u in the region 1.7 to 1.9. The variation of u with α reduces when α<0 and the actual velocity experiences an enhancement with an increase in α (fig.3). The variation of u radiation parameter reveals that u exhibts a reversal flow in the entire region for N<-0.8

and no such flow exists anywhere in the region for any value of N (fig.4).Fig.5 represents the variation of u with Sc fixing the other parameters. It is found that lesser the molecular diffusivity smaller the axial velocity in the entire flow field. Figs.6-10 represent the variation of temperature θ with different governing parameters. We follow the convention that the temperature is positive or negative according as the actual temperature is greater/lesser than the temperature on the inner cylinder. It is found that the temperature is negative for G<0 and positive for G>0. The variation in the temperature is more predominant in the vicinity of the boundaries r=1 and 2. It is found that he actual temperature depreciates in the vicinity of inner cylinder r=1 and enhances in the vicinity of outer cylinder r=2 with G>0(fig.6). From fig.7, we notice that lesser the permeability of the porous medium larger the actual temperature. The presence of heat sources/sinks on θ is exhibited in fig.8 For α>0, θ is negative and for α<0,θ is positive. An increase in the strength of heat source reduces the actual temperature in the entire flow region while it enhances with α (<0). From fig.9, we observe that the actual temperature experiences an enhancement with N<0 while in the opposite direction it experiences a depreciation in the flow region. From fig.10, we notice that lesser the molecular diffusivity larger the actual temperature in the flow phenomena. The non-dimensional concentration dstribution (C) is exhibited in figs 11-15 for different parameters. As in the case of temperature distribution the concentration is also exhibits remarkable change in the vicinity of the boundaries r=1 and r=2. It is found that the

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

actual concentration depreciates with G . The depreciation in the actual concentration with G>0 is remarkably appreciable than in the case of G<0(fig.11). Lesser the permeability of the porous medium lesser the actual concentration everywhere in the region(fig.12). The influence of heat sources on the concentration is shown in fig.13. It is found that the actual concentration experiences a depreciation for α>0 and an enhancement for α<0. The variation of C with N is shown in fig.14.The variation of concentration experiences depreciation when N ≤ -0.8 and enhances when N>1. The variation of C with Sc indicates that actual concentration enhances in the flow region(fig.15). The shear stress( τ ) on the inner cylinder r=1 and the outer cylinder r=2 is shown in tables 1-4 for different values of governing parameters. It is observed that the shear stresses at r=1 and r=2 enhances in magnitude with increase in G .Lesser the permeability of the porous medium larger τ at both the cylinders. The variation of

τ with Sc reveals that

the lesser the molecular diffusivity smaller the magnitude at both the boundaries. For different values of N, the shear stress at r=1 depreciates and enhances at r=2.

The Nusselt number(Nu) which measures the rate of heat transfer at the inner and outer cylinder is shown in tables 5-8. It is observed that the rate of heat transfer depreciates in magnitude with increase in G at both the boundaries.Lesser the permeability of the porous medium smaller Nu at r=1 and r=2.Also lesser the molecular diffusivity larger the rate of heat transfer at both the cylinders. For the variation in radiation parameterN, Nu depreciates for increase in N while for the N<0, it enhances at both the boundaries. The Sherwood number(Sh) which measure the rate of mass transfer at r=1 and r=2 is shown in tables 9-12 for different parameters. It is noticed that the rate of mass transfer is positive for all variations. The rate of mass transfer enhances at r=1 and reduces at r=2 with increae in G>0 while an increase in G<0 reduces at r=1 and enhances at r=2. The variation of Sh with D −1 shows that the lesser the permeability of the porous medium lesser Sh at r=1 and larger at r=2. Also lesser the molecular diffusivity larger Sh at r=1 and smaller Sh at r=2. The variation of Sh with N shows that, the rate of mass transfer reduces at r=1 and enhances at r=2 (tables 9-12).

Table-1 Shear stress( τ ) at r=1 D

−1

III

VII

0.02375

0.07194

-0.02463

-0.07323

0.02379

0.02373

0.02378

3x10

0.02434

0.07370

-0.02523

-0.07503

0.02438

0.02431

0.24368

5x10 G

0.02495

0.07554

-0.02586

-0.07690

0.02499

0.02493

0.02497

Sc N

10 1.3 1

3x10 1.3 1

-1x10 1.3 1

352

-3x10 1.3 1

10 2.01 1

10 0.24 1

10 0.6 1

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

353

D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012) Table-2 Shear stress( τ ) at r=1

−1

III

0.04851

-0.01345

-0.02091

3x10

0.049704

-0.01378

-0.02142

5x10 G

0.050946

-0.01412

-0.02195

Sc N

10 1.3 -0.5

10 1.3 2

10 1.3 -0.8

Table-3 Shear stress( τ ) at r=2 D

−1

III

-0.03267

-0.09864

0.03334

0.099408

VII

-0.0265

-0.032705

-0.03269

3x10

-0.03315

-0.100084

0.03383

0.100864

-0.03312

-0.03319

-0.033171

5x10 G

-0.03364

-0.101579

0.3434

0.102379

-0.033623

-0.03367

-0.033666

Sc N

3x10 1.3 1

10 1.3 1

-1x10 1.3 1

-3x10 1.3 1

10 2.01 1

10 0.24 1

Table-4 Shear stress( τ ) at r=2 D

−1

III

-0.06652

0.0181609

0.08336

3x10

-0.06745

0.018426

0.028751

5x10 G

-0.06850

0.018703

0.029182

Sc N

10 1.3 2

10 1.3 -0.5

10 0.6 1

10 1.3 -0.8

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

354

D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012) Table-5 Nusselt Number(Nu) at r=1

−1

III

VII

-0.03401

-0.04584

-0.02209

-0.010135

-0.033908

-0.034133

-0.034089

3x10

-0.03415

-0.04630

-0.021987

-0.009655

-0.034062

-0.034284

-0.034240

5x10 G

-0.03431

-0.04678

-0.02144

-0.009144

-0.03422

-0.034441

-0.034398

3x10 1.3 1

10 1.3 1

Sc N

-1x10 1.3 1

-3x10 1.3 1

10 2.01 1

10 0.24 1

10 0.6 1

Table-6 Nusselt Number(Nu) at r=1 D

−1

III

-0.04006

-0.02579

-0.02384

3x10

-0.04035

-0.02571

-0.02373

5x10 G

-0.04067

-0.02568

-0.02358

10 1.3 2

Sc N

10 1.3 -0.5

10 1.3 -0.8

Table-7 Nusselt Number(Nu) at r=2 D

−1

III

VII

-1.82754

-1.9606

-1.69438

-1.56112

-1.82667

-1.82877

-1.82836

3x10

-1.82851

-1.96354

-1.69338

-1.55814

-1.82766

-1.82973

-1.82932

5x10 G

-1.82952

-1.9666

-1.69234

-1.55505

-1.82868

-1.83073

-1.83032

Sc N

10 1.3 1

3x10 1.3 1

-1x10 1.3 1

-3x10 1.3 1

10 2.01 1

10 0.24 1

10 0.6 1

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

355

D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012) Table 8 Nusselt Number(Nu) at r=2

−1

III

-1.8952

-1.73662

-1.71473

3x10

-1.8972

-1.73612

-1.71394

5x10 G

-1.89927

-1.73561

-1.71311

10 1.3 2

Sc N

10 1.3 -0.5

10 1.3 -0.8

Table-9 Sherwood number at r=1 D

−1

III

VII

1.43043

1.43625

1.42448

1.4185

1.42792

1.43395

1.43278

3x10

1.4305

1.43658

1.4244

1.4182

1.42800

1.43397

1.43815

5x10 G

1.43059

1.43682

1.42432

1.4180

1.42822

1.43398

1.43285

Sc N

10 1.3 1

3x10 1.3 1

-1x10 1.3 1

-3x10 1.3 1

10 2.01 1

10 0.24 1

10 0.6 1

Table-10 Sherwood number at r=1

−1

III

1.43221

1.45401

1.44972

3x10

1.43225

1.45415

1.44988

5x10 G

1.43228

1.45431

1.45004

Sc N

10 1.3 2

10 1.3 -0.5

10 1.3 -0.8

Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)

D. Chitti Babu, et al., J. Comp. & Math. Sci. Vol.3 (3), 344-357 (2012)

356

Table-11 Sherwood number at r=2 D

â&#x2C6;&#x2019;1

III

VII

0.72517

0.713664

0.736687

0.748199

0.728136

0.72094

0.722357

3x10

0.72510

0.713443

0.736761

0.748419

0.728022

0.72093

0.722324

5x10 G

0.72502

0.713219

0.736838

0.748643

0.727904

0.720918

0.722291

Sc N

10 1.3 1

3x10 1.3 1

-1x10 1.3 1

-3x10 1.3 1

10 2.01 1

10 0.24 1

10 0.6 1

Table-12 Sherwood number at r=2 D

â&#x2C6;&#x2019;1

III

0.721203

0.692004

0.698774

3x10

0.721161

0.691883

0.698603

5x10 N

0.72118

0.691754

0.698485

-0.5

-0.8

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