Conjugate Heat and Mass Transfer from a Solid Sphere in the

1112

Ind. Eng. Chem. Res. 1998, 37, 1112-1121

Conjugate Heat and Mass Transfer from a Solid Sphere in the Presence of a Nonisothermal Chemical Reaction Gh. Juncu† Catedra Inginerie Chimica, Politehnica University Bucharest, Polizu 1, 78126 Bucharest, Romania

This paper analyzes the influence of the conductivity ratio, Φλ, the volume heat capacity ratio, Φh, and the diffusivity ratio, ΦD, on the conjugate heat and mass transfer from a particle in the presence of a chemical reaction. The nonisothermal, first-order irreversible chemical reaction takes place in the particle. Both exothermic and endothermic reactions are studied. The creeping flow and the moderate Reynolds number domain are the hydrodynamic regimes considered. For the exothermic reaction, the particle average temperature and concentration and the effectiveness factor are strongly influenced by the variation of the physical properties ratios. For the endothermic reaction, these effects are considerably less significant. The change of the hydrodynamic regime did not lead to any new or spectacular result. The interphase heat- and mass-transfer coefficients are equal and independent of the physical properties ratios. Introduction The conjugate heat and/or mass transfer from a spherical particle is the subject of relatively few theoretical or experimental studies. The general solutions for the case where the transient resistances in both phases are significant were obtained by the following: (a) Plo¨cher and Schmidt-Traub (1972) and Cooper (1977)sthe stagnant phases situation; (b) Abramzon and Borde (1980), Abramzon and Elata (1984), and Juncu (1997)sthe creeping flow regime; (c) Nguyen et al. (1993) and Juncu (1998)sthe moderate Reynolds numbers domain. Due to the goal of this paper, the presentation of the previous contributions in solving the conjugate problem is restricted to the case of the rigid particle. For the same reason, the free convection analyses are not mentioned here. In all previously mentioned studies the chemical reaction is not considered present. To our knowledge, the presence of the chemical reaction in the conjugate transfer was taken into account only by Kleinmann and Reed (1995, 1996). The articles of Kleinmann and Reed (1995,1996) are dedicated only to the case of mass transfer between a sphere and an infinite medium where an isothermal chemical reaction takes place. The main problem analyzed is the validity of the addition rule for mass-transfer resistances function of the diffusivity ratio, the distribution coefficient, and the Damkohler number. The present research is devoted to the conjugate mass and heat transfer between a rigid particle and an infinite convective continuous phase in the presence of a nonisothermal chemical reaction. In our previous papers, Juncu (1996, 1997, 1998), it is shown that the conjugate heat transfer is strongly influenced by the conductivity ratio and the volume heat capacity ratio. The aim of this paper is to extend the analysis on the case when a first-order irreversible nonisothermal chemical reaction takes place in the particle. The exothermic and endothermic chemical reactions are † Phone: +40 1 312 6879. Fax: +40 1 312 6879. E-mail: [email protected].

analyzed in two hydrodynamic regimes: creeping flow and moderate Re number. The quantities of interest are the particle average concentration, the particle average temperature, the effectiveness factor, and the interphase mass- and heat-transfer coefficients. The steady-state results are compared with those provided by the diffusion-reaction model of the steady nonisothermal catalyst particle with external resistances (Doraiswamy and Sharma, 1984). Mathematical Model Consider a rigid particle, with spherical shape and radius a, suspended in an unbounded convective environment with uniform velocity, U. Inside the particle a first-order irreversible nonisothermal chemical reaction takes place for the transferring species. The following assumptions are considered to be valid: (i) the fluid is Newtonian and the flow field is steady, laminar, and axisymmetric; (ii) during the process the physical properties of the phases, the particle shape, and volume remain constant. Under the previously outlined conditions, the dimensionless transient mathematical model equations read as follows: The Continuous Phase: the heat balance equation

Le2

(

)

∂Z2 Peh ∂Z2 Vθ ∂Z2 + VR ) + ∂τ2 2 ∂r r ∂θ ∂Z2 1 ∂ 2 ∂Z2 ∂ 1 r + sin θ (1a) ∂r ∂θ r2 ∂r r2 sin θ ∂θ

(

)

(

)

the mass balance equation

(

)

∂C2 Vθ ∂C2 ∂C2 Pem + VR ) + ∂τ2 2 ∂r r ∂θ ∂C2 1 1 ∂ 2 ∂C2 ∂ r + 2 sin θ (1b) 2 ∂r ∂r ∂θ r r sin θ ∂θ

(

)

The Dispersed Phase:

S0888-5885(97)00429-6 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/07/1998

(

)

Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998 1113

the heat balance equation

Le1

(

with the boundary conditions

)

(

)

∂Z1 ∂Z1 1 ∂ 2 ∂Z1 ∂ 1 ) 2 r + 2 sin θ + ∂τ1 r ∂r ∂r ∂θ r sin θ ∂θ 1 (2a) βφ2C1 exp γ 1 Z1

[(

)]

the mass balance equation

(

)

(

)

∂C1 1 ∂ 2 ∂C1 ∂C1 ∂ 1 ) 2 r + 2 sin θ ∂τ1 r ∂r ∂r ∂θ r sin θ ∂θ 1 φ2C1 exp γ 1 Z1

[(

)]

(2b)

The Initial Conditions:

C1(r,θ,0) ) 0.0, C2(r,θ,0) ) 1.0, Z1(r,θ,0) ) Z2(r,θ,0) ) 1.0 (3) The Boundary Conditions:

C1(0,θ,τ) < ∞,

Z1(0,θ,τ) < ∞

(4)

∂Ci ∂Ci (r,0,τi) ) (r,π,τi) ) 0.0, ∂θ ∂θ ∂Zi ∂Zi (r,0,τi) ) (r,π,τi) ) 0.0, i ) 1, 2 (5) ∂θ ∂θ C2(∞,θ,τ2) ) Z2(∞,θ,τ2) ) 1.0

(6)

Φλ

∂Z1 ∂Z2 (1,θ,τ1) ) (1,θ,τ2); Z1 ) Z2 ∂r ∂r

(7)

ΦD

∂C2 ∂C1 (1,θ,τ1) ) (1,θ,τ2); C1 ) C2 ∂r ∂r

(8)

The mathematical model eqs 1-8 were solved for two hydrodynamic regimes: creeping flow and moderate Re numbers. In creeping flow, the Stokes velocity profiles,

VRe ) (1 - 3/2r-1 + 1/2r-3) cos θ, Vθe ) -(1 - 3/4r-1 - 1/4r-3) sin θ Bird et al. (1960, pp 132-133), were used. At moderate Re numbers the velocities were computed numerically by solving the steady Navier-Stokes equations (Juncu and Mihail, 1990). The steady-state model is obtained by considering the time derivatives in eqs 1 and 2 equal to zero. The steady results of the mathematical model are compared with those provided by the diffusion-reaction model of the steady nonisothermal catalyst particle with external resistances. Using the same dimensionless variables, the model reads as follows: the heat balance equation

(

)

[(

)]

) 0.0 (9a)

[(

)]

) 0.0 (9b)

1 1 d 2 dZ1 r + βφ2C1 exp γ 1 dr Z1 r2 dr the mass balance equation

(

)

1 1 d 2 dC1 r - φ2C1 exp γ 1 dr Z1 r2 dr

dC1 dZ1 (0) ) (0) ) 0.0 dr dr

(10a)

dZ1 (1) ) Bih(1.0 - Z1|r)1), dr dC1 (1) ) Bim(1.0 - C1|r)1) (10b) dr For brevity, model eqs 1-8, and their steady version are referenced as full model, while model eqs 9 and 10 are referenced as simplified model. For a fair comparison between the simplified model and the steady solutions of the full model, a correct correspondence between the different parameters of these models must be established. The reaction parameters, β, γ, and φ are identical for the two models. The heat and mass transport in the continuous phase are expressed in model eqs 9 and 10 by the Biot numbers, Bih and Bim. The main parameter which influences the continuous phase transport eq 1 is the Peclet number. The connection between the Bi numbers and the Pe numbers is established as follows: The Bi numbers are an expressed function of the external Nusselt and Sherwood numbers as

Bih ) ha/λ1 ) (λ2/λ1)(ha/λ2) ) Nuext/Φλ

(11)

Bim ) ka/D1 ) (D2/D1)(ka/D2) ) Shext/ΦD (12) For a given Pe number, Nuext or Shext are computed with the relations presented by Clift et al. (1978) in the paragraphs dedicated to the external problem. For the simplified model, the influence of the physical properties ratio on the conjugate transfer is equivalent to the influence of the Biot numbers, Bih and Bim, on the particle temperature and concentration. The influence of the Bi numbers on the catalyst particle temperature, concentration, and effectiveness factor was analyzed by Carberry (1976). The strategy used here to vary the Bi numbers differs from that employed by Carberry (1976). Quantities of Interest In conjugated heat- and/or mass-transfer studies, the results are usually summarized in terms of (1) the average particle temperature and/or concentration, (2) the local instantaneous Nu number, and (3) the overall instantaneous Nu number. It is obvious that the particle average temperature and concentration, computed by the relations

∫01r2(∫0πZ1 sin θ dθ) dr

(13a)

∫01r2(∫0πC1 sin θ dθ) dr

(13b)

Z h 1 ) 3/2 C h 1 ) 3/2

are quantities which cannot be ignored by any study dedicated to the conjugate transfer phenomena. The rate of heat and mass transfer can be described by the overall heat- and mass-transfer coefficients. Due to the presence of the nonisothermal chemical reaction inside the particle, of more direct practical utility than the overall transfer coefficients, are the “external” transfer

1114 Ind. Eng. Chem. Res., Vol. 37, No. 3, 1998

coefficients (in dimensionless form the external Nu and Sh numbers) which can be viewed as the interphase transfer coefficients. The external Nu and Sh numbers are computed with the relations

∫0π ∂r2|r)1 sin θ dθ ∂Z

Nu2 )

(14a)

Z h S - 1.0

∫0π ∂r2|r)1 sin θ dθ ∂C

Sh2 ) -

(14b)

1.0 - C hS

where the average surface temperature and concentrah s, are given by tion, Z h s and C

∫0πZ1|r)1 sin θ dθ

(15a)

∫0πC1|r)1 sin θ dθ

(15b)

Z h S ) 1/2 C h S ) 1/2

The presence of the chemical reaction leads to some specific quantities of interest. In these systems, the effect of the chemical reaction is taken into consideration, usually by means of two quantities: the enhancement factor or the effectiveness factor. The enhancement factor is usually employed in isothermal gasliquid or liquid-liquid systems. Because the case analyzed here resembles more with the catalyst particle, the effectiveness factor is the parameter selected to monitorize the chemical reaction. The effectiveness factor is given by the relation

(

((

))

∫01r2 ∫0πC1 exp γ 1.0 - 1.0 Z1

η ) 3/2

)

sin θ dθ dr (16)

The relations used to compute the particle average temperature, the particle average concentration, and the effectiveness factor for the simplified model are the onedimensional versions (by considering only the r-dependence) of relations 13a,b and 16. Method of Solution The mathematical models used in this paper were solved numerically. The method selected to solve the transient and the steady full models is multigrid (MG) (the elliptic MG for the steady model and the parabolic MG for the transient model). The elliptic MG and the solving methods of the simplified model are briefly presented in this section. The parabolic MG is presented in the Appendix. All the computations were performed on a HP 9000 715/80 workstation in FORTRAN Double Precision. The Elliptic MG. The continuous-phase equations were discretized on a uniform grid with equal step sizes in both radial and angular directions. This implies the location of ∞ at exp(π), which is a good choice for the Pe number used. In the solid particle, a uniform mesh with unequal step sizes in the radial and angular directions was used. The angular step size in the two phases is equal. The diffusive terms were approximated by central differencing and the convective terms by the exponentially fitted scheme (Hemker, 1977). This choice is imposed by the MG convergence requirements. The nonlinear discrete elliptic equations are solved with the

classical FAS (Brandt, 1977; Hackbusch, 1985), algorithm, suitable for general nonlinear problems. To define the MG iteration step (cycle), the following operators must be specified: the iteration (smoothing) operator, the prolongation operator, and the restriction operator. The smoothing operator used is the alternating line Gauss-Seidel method. The linearization employed is only local and it is in the sense of deferred updating of the exponentials (a Picard-like method). As a whole, the method is a nonlinear iteration belonging to the class of successive substitution methods. The discrete operator is used for the residual restriction and for transferring the coarse grid corrections back to finer grids. The unknowns are transferred on the coarser grid by injection. The nested form of the algorithm is used. Cycles of type V were used with two iterations sweeps performed before and one after the coarse grid correction step. Three levels were used. Tests were made with the coarsest grid having 17 × 17 or 33 × 33 points in each phase. The finest grid has 65 × 65 or 129 × 129 points in each phase. The results obtained showed that the accuracy provided by the 65 × 65 mesh is quite acceptable. The convergence criterion employed was

|resi|2 < 1 × 10-5|res0|2 where |resi|2 means the l2 discrete norm of the residuals after i iterations. The Simplified Model. Three discretization techniques were used to solve the simplified model: the orthogonal collocation on finite elements (Carey and Finlayson, 1975), the spline collocation on finite elements (de Boor, 1978), and the finite differences. The discrete system resulted from the collocation methods was solved with the Newton algorithm. The finite difference approximation was solved by an iterative algorithm (the nonlinear Gauss-Seidel method). The results provided by the three methods are in good agreement. Results The parameters are selected to reflect, as much as possible, the real life systems. The model parameters can be divided into two classes: reaction parameters (β, φ, and γ) and transport parameters (Le1, Le2, Peh, Pem, Φλ, Φh, and ΦD). It is obvious that a complete analysis concerning the influence of all these parameters on the conjugate heat and mass transfer is an impossible task for a single paper. For a conjugate transfer problem the main parameters are those which refer to the physical properties ratio. This paper being focused on the conjugate heat and mass transfer, the key transport parameters are the conductivity ratio, Φλ, the heat capacity ratio, Φh, and the diffusivity ratio, ΦD. The variation interval considered for Φλ and Φh is [0.010, 100.0]. In the previous papers (Juncu, 1997, 1998) dedicated to the influence of the physical properties ratio on the conjugate heat transfer, the values assigned to the Pe numbers belong to the domain of moderate Pe numbers. In this article, the only Peh value considered is Peh ) 100.0. This value can be viewed as typical for the domain of moderate Pe numbers. Two hydrodynamic regimes are considered: creeping flow and moderate Re numbers (Re ) 50.0).


Figure 3. The Φh and Φλ influence on the steady η value for the exothermic reaction in creeping flow. Figure 1. The Φh and Φλ influence on the steady Z h 1 value for the exothermic reaction in creeping flow.

Figure 2. The Φh and Φλ influence on the steady C h 1 value for the exothermic reaction in creeping flow.

For the reaction parameters two sets are selected: one for the exothermic reaction and the other for the endothermic reaction. The parameters values are

the exothermic reaction: β ) 5.0, φ2 ) 0.10, and γ ) 10.0 the endothermic reaction: β ) -5.0, φ2 ) 0.10, and γ ) 16.0 The values were chosen in concordance with the data presented by Froment and Bishoff (1990). To analyze the influence of Φλ and Φh on the conjugate transfer, it is desired to avoid, as much as possible, interferences from the variation of other physical parameters. For this reason, the first results presented in this section were obtained assuming Le1 ) Le2 ) 1.0. The influence of the Le numbers on the conjugate transfer is presented at the end of this section. For Le numbers equal to 1, one obtains ΦD ) ΦR ) Φλ/Φh and Pem ) Peh ) Pe. For the exothermic reaction in creeping flow, the Φλ and Φh influence on the steady values of the particle average temperature, particle average concentration, and effectiveness factor is depicted in Figures 1-3, respectively. The symbols in Figures 1-3 indicate the value of the heat capacity ratio. The agreement between the steady model solution and the long time solution of the transient model is excellent (the relative error is smaller than 1%). The first aspect which must

be mentioned is the absence of a stable steady state at Φλ ) 0.10 and Φh ) 0.010. The time-dependent solution tends to a limit cycle. The system behavior resembles that of a Hopf bifurcation point. The phenomenon is not a consequence of the discretization (it is well-known that the discretization can induce false bifurcation points). On different meshes and with different algorithms the results were the same. Due to the relatively high values of β, γ, and φ, the occurrence of a Hopf bifurcation point may not be considered a surprise. The exothermic reaction is a well-known element of positive feedback which may generate bifurcation. The values depicted in Figures 1-3 are estimations of the unstable steady state. The specific Hopf bifurcation computations were not made in this work. Due to the model’s dimensions, the numerical algorithms are very expensive. Also, the bifurcation behavior of the full model used in this paper is an important problem which deserves a distinct analysis. Figure 1 shows that the Z h 1 - Φλ dependence is monotone; if Φh g 1, the dependence is monotone increasing while, if Φh < 1, the dependence is monotone h 1 is less significant decreasing. The Φh influence on Z at Φλ ) 0.010 but increases with the increase in Φλ. For h 1 versus Φh is monotone a given Φλ, the variation Z increasing. Figure 2 shows that the Φλ and Φh influence on the steady-state values of the particle average concentration is a bit more complicated. At Φh ) 0.01, the dependence C h 1 - Φλ is a curve with a maximum. h 1 - Φλ is monotone For Φh g 0.10, the dependence C h 1 decreases with the decreasing. At a given Φλ, C increase in Φh, if Φh ∈ [0.010, 2.0]. For Φh ∈ [2, 100.0] h 1 increases with the increase in Φh. and Φλ fixed, C Figure 3 shows that for Φh < 1 the dependence η versus Φλ is monotone decreasing. For Φh g 1.0 the curves η - Φλ exhibit a maximum. In the parameter space considered, the steady-state values computed for Nu2 and Sh2 fall in the interval [5.61, 5.89]. These values are close to the value Nuext ) Shext ) 5.38 provided by the external problem. For each pair (Φh, Φλ) the relative difference between Nu and Sh does not exceed 1%. One can state that the steady Nu and Sh values are equal and practically independent of Φh and Φλ. The thermal wake (Abramzon and Elata, 1984) is absent. This result, which confirms the Chilton-Colburn analogy, is completely different from those obtained in Juncu (1997, 1998) for the physical conjugate heat transfer. The main results obtained in Juncu (1997, 1998) can be summarized by (1) the asymptotic Nu number is strongly influenced by


Figure 4. The comparison in terms of Z h 1 between the full and the simplified models; the exothermic reaction in creeping flow.

Figure 5. The comparison in terms of C h 1 between the full and the simplified models; the exothermic reaction in creeping flow.

Figure 6. The comparison in terms of η between the full and the simplified models; the exothermic reaction in creeping flow.

Φh and Φλ; (2) the thermal inversion phenomenon is present, and (3) the thermal wake dimension depends on Φh and Φλ. This comparison leads to the statement that the presence of the chemical reaction radically changes the conjugate heat transfer. The comparison between the full model results and those of the simplified model can be viewed in Figures 4-6. To avoid the agglomeration, only the results obtained at Φh ) 0.010, 1.0, and 100.0 are plotted. The dashed curves in Figures 4-6 are the simplified model results. The solid lines show the full model results. The values assigned to Nuext and Shext, necessary to compute the Bi numbers, are Nuext ) Shext ) 5.38. Figures 4-6 show that, except the Z h 1 values at Φh ) 0.010 and 1.0,

Figure 7. The Φλ influence on the system dynamics for the exothermic reaction in creeping flow; Φh ) 1.0. (a) The time evolution of Z h 1 and C h 1; (b) the η time variation.

significant differences exist between the two models results. At Φh ) 0.010 and Φλ ) 0.10 the steady-state solution is stable. However, these discrepancies are expected. Important, in our opinion, is the fact that the simplified model predicts the same Φh and Φλ influence as the full model. The transient evolution of the particle average temperature, particle average concentration, and effectiveness factor are presented in Figures 7-11. It is obvious that one cannot present the time evolution for all cases for which a steady-state solution was computed. Figures 7 and 8 show the Φλ influence on the system dynamics for two Φh values: Φh ) 1.0 (Figure 7a,b) and Φh ) 0.20 (Figure 8a,b). In Figures 7a and 8a the solid h 1. The symbols lines refer to C h 1 while the dash lines to Z indicate the Φλ value. For a given Φλ, the Φh influence h 1, and η time evolution is presented in Figures on Z h 1, C 9 (Φλ ) 0.010), 10 (Φλ ) 1.0), and 11 (Φλ ) 100.0). The same graphical conventions as in Figures 7 and 8 are used. Figures 7 and 8 show that the time variation of the solutions for Φλ ) 100.0 is different in comparison with that at Φλ ) 0.010 and Φλ ) 1.0. For very high values of Φλ the heat generated by the chemical reaction is mainly dissipated inside the particle and not in the cooling fluid. The temperature gradients inside the particle are small and the particle temperature may be considered uniform. Figures 7-11 also show that, for h 1, C h 1 and η have a similar time a given pair (Φh, Φλ), Z evolution and reach the steady state simultaneously. In Figures 7 and 8, it can be seen that the increase in Φλ reduces the time necessary to reach the steady state. Figures 9-11 show that the time necessary to reach the steady state depends on Φh.


Figure 8. The Φλ influence on the system dynamics for the exothermic reaction in creeping flow; Φh ) 0.2. (a) The time evolution of Z h 1 and C h 1; (b) the η time variation.

All the computations made in creeping flow were replayed at Re ) 50.0. The reason is to analyze the flow separation influence on the conjugate transfer. The presence of the flow separation does not induce any new or spectacular result. All the statements made in the previous paragraphs remain valid. At Φh ) 0.01 and Φλ ) 0.10 the steady-state solution is unstable (a limit h 1, C h 1, cycle exists). The influence of Φh and Φλ on the Z and η stationary values follows the same rules. The comparison with the simplified model leads to the same statements. The Sh2 and Nu2 values can be considered equal and independent of Φh and Φλ. The values obtained belong to the interval [7.10, 7.41]. The external problem provides the values Nuext ) Shext ) 7.29. The exothermic reaction results put in evidence the less significant role of the hydrodynamic regime. For this reason, at the endothermic reaction, only the results obtained at Re ) 50.0 are presented. Figure 12 shows the Φh and Φλ influence on the particle average temperature. It can be seen that the Φh influence is relatively significant only in the domain 1.0-0.010. The results obtained at Φh ) 1.0 and Φh ) 100.0 practically overlap. For all Φh values the dependence Z h 1 - Φλ is monotone decreasing. The same statements can be made for the variations C h 1 versus (Φh, Φλ) (Figure 13) and η versus (Φh, Φλ) (Figure 14). h 1 decreases with the increase in For a given Φλ, Z h 1 and η increase with the increase in Φh. Φh while C The same values of Nu2 and Sh2, as in the case of the exothermic reaction at Re ) 50.0, were obtained. The agreement with the simplified model is better. One

Figure 9. The Φh influence on the system dynamics for the exothermic reaction in creeping flow; Φλ ) 0.010. (a) The time evolution of Z h 1 and C h 1; (b) the η time variation.

h 1, can summarize that the influence of Φh and Φλ on Z C h 1, and η, in the case of the endothermic reaction, is weaker. To analyze the influence of the Lewis numbers on the conjugate transfer, three situations may be considered: (1) Le1 * 1, Le2 ) 1. (2) Le1 ) 1, Le2 * 1. (3) Le1 * 1, Le2 * 1. In all situations, the value of the Peh is considered constant and equal to 100.0. Also, only the exothermic reaction was considered in the numerical tests made. In this section, in a previous paragraph, it was shown that the simplified model predicts qualitatively the same system behavior as the full model. Thus, in a first attempt, one can try to foresee the influence of the Le numbers employing the simplified model. From the beginning, it may be observed that the variation of the Lewis numbers affects only the value of the mass Bi number Bim. For case 1, the relation Le2 ) 1 implies Pem ) Peh. Relation 12 shows that, in this case, Bim depends explicitly only on ΦD ) ΦR/(Le1/Le2) ) ΦR/Le1 * ΦR. This means that different values of ΦR and Le1, which lead to the same value of ΦD, should give the same result. For example, the results obtained at ΦR ) 1 and Le1 ) 0.50 should be identical with those obtained at ΦR ) 2 and Le1 ) 1. The computations made with the full model for Φh ) 0.1, 1.0, and 10.0, Φλ ∈ [0.01, 100.0], Le1 ) 0.5 and 2.0, creeping flow, and Re ) 50 confirmed the previous assumptions.



For case 2, Pem ) PehLe2 * Peh. From relation 12 one obtains

Bim )


Shext Shext Le1 f(Pem) Le1 ) ) ) ΦD ΦR Le2 ΦR Le2 f(PehLe2) Le1 f(PehLe2) ) ΦR Le2 ΦRLe2

The dependence Shext ) f(Pem) may be approximated (Clift et al., 1978) by

Shext ≈ cPedm,

d

Conjugate Heat and Mass Transfer from a Solid Sphere in the

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