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Ind. Eng. Chem. Res. 2010, 49, 8114–8124
Effectiveness Factors for Partially Wetted Catalysts Arie Jan van Houwelingen,†,‡ Schalk Kok,§ and Willie Nicol*,† Department of Chemical Engineering, UniVersity of Pretoria, Pretoria, 0002, South Africa, and AdVanced Mathematical Modeling, CSIR Modeling and Digital Science, P. O. Box 395, Pretoria 0001, South Africa
The finite element method (FEM) is used to solve the diffusion-reaction equation for spherical particles that are partially wetted by the liquid reagent in a trickle-bed reactor. Boundary conditions are specified using true wetting geometries from photographs of spherical particles that were partially wetted under trickle-flow conditions. Three types of reactions were investigated: completely liquid-limited reactions, completely gaslimited reactions, and reactions that are dependent on both the gas and the liquid reagent, following elementary kinetics. The study is limited to spheres, since wetting geometries were only available for these particles. Both monodispersed and eggshell catalysts were simulated. On the basis of the results, existing models for trickle-bed pellet efficiency factors were verified and models are proposed for the prediction of reaction rates in a partially wetted eggshell and monodispersed catalysts for gas-limited reactions, liquid-limited reactions, and elementary reactions for which both gas- and liquid-reagent concentrations are of importance. 1. Introduction The estimation of catalyst effectiveness factors, defined as the ratio between the observed reaction rate in a pellet and the reaction rate in the absence of mass-transfer resistances, plays an important role in chemical reactor engineering and is governed by solution to the steady-state reaction-diffusion equation within the catalyst: ∇2C - φ2y(C) ) 0
(1)
where φ describes the ratio of reaction to diffusion and y(C) is the reaction rate as a function of reagent concentration(s), C. The effectiveness factor can be evaluated by the integral of the reagent concentrations over the catalyst volume or the reagent fluxes over the catalyst external area. Analytical solutions for eq 1 are generally only available for first- and zero-order reactions in pellets with well-defined geometries such as the one-dimensional geometries of a sphere, a semiinfinite cylinder, and a semiinfinite slab.1 Unique boundary conditions are also required, where the boundary condition at the external surface is the same over the whole surface. In trickle-bed reactors, which are commonly used in industry to process gaseous and liquid reagents over a packed catalyst bed, the reaction-diffusion problem generally does not conform to these requirements: At least two reagents (a gas and a liquid) react in a trickle-bed reactor, and the kinetic expression is probably not simply first order in one reagent. Also, boundary conditions for partially wetted particles are mixed; that is, the boundary condition for the wetted surface differs from that for the dry surface. Except for specific (theoretical) wetting and particle geometries, mixed boundary conditions will increase the dimensionality of eq 1. To simplify effectiveness factor derivations, the reaction in a trickle-bed reactor is usually classified as being either gas- or liquid-limited; i.e., the concentration of either the liquid or the gas reagent respectively is constant throughout the catalyst * To whom correspondence should be addressed. E-mail: willie.nicol@ up.ac.za. † University of Pretoria. ‡ Current address: Sasol Technology Research and Development, Klasie Havanga Rd., P. O. Box 1, Sasolburg 1947, South Africa. § CSIR Modeling and Digital Science.
particle so that pseudo-first-order kinetics can be assumed for the limiting reagent. A reaction is usually classified as gas- or liquid-limited on the basis of γ, where γ)
DBCB,bulk DAC*A
(2)
where DA and DB are the effective diffusivities of the gas and liquid reagents, respectively. C*A and CB,bulk are the saturated gas concentration and the liquid reagent concentration in the bulk liquid, respectively. A reaction is said to be gas-limited if γ . 1 and liquid-limited if γ , 1. Liquid-limited reactions are modeled differently from gas-limited reactions in trickle beds with partially wetted catalysts. In this paper, the finite element method (FEM) is used to model pellet efficiency factors for partially wetted spherical particles. Boundary conditions are specified according to wetting geometries that were experimentally determined, using a colorimetric method described in a previous paper.2 Results from this numerical modeling exercise are used to evaluate existing models for effectiveness factors for liquid- and gas-limited reactions in partially wetted particles. Investigations that deal with this question already exist3,4 but not where true partial wetting geometries are used. To reconcile models for gas- and liquid-limited reactions, the reaction rA ) RrB ) -RkrCACB taking place in a partially wetted catalyst is also investigated. A numerical study of this reaction was also performed by Yentekakis and Vayenas,5 but no suggestions were made for an easy-to-use expression to estimate the effectiveness factor for such reactions. Such an expression is developed in this paper and is validated with numerical simulations. Many trickle-bed reactors make use of eggshell catalysts. Though it is rather easy to derive analytical expressions for a completely wetted eggshell catalyst with first-order reaction kinetics, the role of partial wetting may differ significantly for eggshell catalysts than for monodispersed catalysts. The effect of partial wetting on the performance of this type of catalyst is therefore also investigated. All work is limited to isothermal conditions and constant reagent diffusivity.
10.1021/ie9017176 2010 American Chemical Society Published on Web 07/23/2010
Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010
2. Numerical Method 2.1. First-Order Reaction, -r ) krC. Models for gas- and liquid-limited reactions in partially wetted catalysts assume a constant concentration of the nonlimiting reagent and first-order kinetics for the limiting reagent so that the reaction can be written as -r ) krC. In dimensionless form, the equation for steady-state diffusion combined with such a reaction within a catalyst sphere is given by ∇2c - φ2c ) 0 c)
C ; Cbulk
φ ) rp
(4)
∑ [( ∫ [B] [B] dV + ∫ φ [N] [N] dV + ∫ Bi [N] [N] dS){c}] ) ∑ ( ∫ Bi [N] el
T
V
T
S
T
el
el
S
el
dS) (5)
el
T
A
2
T
V
T
B
el
T
B
el
[ ] ∂Ra ∂Ra ∂{a} ∂{b} ∂Rb ∂Rb ∂{a} ∂{b}
RkrCB,bulk DA
(6)
B
{ } { } {∆a} {∆b}
)-
Ra Rb
(10)
The method terminates when |Ra,b| < ε, where ε is some specified tolerance. The matrix on the left-hand side of eq 10, the consistent tangent, is assembled from submatrices that are given by ∂Ra ∂{a} ∂Ra ∂{b} ∂Rb ∂{a} ∂Ra ∂{a}
) KB + φA2Kb + BiAKN ) φA2Ka
(11) ) φB2Kb ) KB + φB2Ka + BiBKN
where Ka, Kb, KB, KA, and KN are given by
KB Here [B] contains the spatial derivatives of the shape functions [N], Biel is the Biot number of the particular element under consideration, and {c} is the concentration at each node in the mesh. The stiffness matrix [K] and vector [F] are functions of the geometry and known constants only, so that the nodal concentrations {c} are solved from a system of linear eqs 5. The code is implemented in Matlab, and a direct sparse solver is used to compute {c}. 2.2. Reactions of the Form rA ) rrB ) -rkrCACB. In dimensionless form, the reaction-diffusion equations for the reaction rA ) RrB ) -RkrCACB in a spherical pellet are as follows:
S
Since both equations contain the terms {a} and {b}, the equations cannot be solved in the same fashion as eq 5, and the concentration profiles were solved for using the Newton-Raphson iterative scheme for coupled nonlinear systems.8 This method iteratively improves the estimates for {a} and {b} by computing updates {∆a} and {∆b} from
∑ ( ∫ [N] [N]{a}[N] dV) ) ∑ ( ∫ [N] [N]{b}[N] dV) ) ∑ ( ∫ [B] [B] dV) ) ∑ ( ∫ [N] [N] dS) ) ∑ ( ∫ [N] dS) T
V
T
el
KN
V
T
el
KA
V
T
el
A
∑ ( ∫ [B] [B] dV + ∫ φ [N] [N]{a}[N] dV + ∫ Bi [N] [N] dS){b} - ∑ ( ∫ Bi [N] dS) ) {0} (9)
el
φA ) rp
S
el
Kb
∇2a - φA2ab ) 0;
T
T
S
Ka )
which is of the form
(7)
A
el
T
Since the boundary conditions are geometrically complex, involving both Biw and Bid, the solution of eqs 3 and 4 need to be solved numerically. The FEM is a mature numerical technique that is ideally suited for this type of partial differential equation.6 The application of FEM to solve the diffusionreaction equation for partially wetted catalysts was first described by Mills et al.,7 where it was compared to other approximation methods. The FEM method was recommended for nonlinear rate equations. In FEM, the control volume is divided into small volume elements V and associated surfaces S. Concentrations {c} on the nodes of these elements are interpolated to provide the concentration field c ) [N]{c}, where [N] contains interpolation shape functions. Quadratic shape functions were used to interpolate the nodal concentrations. These were found to be more suited to the typical concentration profiles than linear interpolation. For a first-order reaction coupled with diffusion, the standard Bubnov-Galerkin finite element method yields the equation: 2
2
T
V
S
∇c · nj ) Biw(1 - c) at the wetted external surface ∇c · nj ) Bid(1 - c) at the dry external surface
T
krC*A DB
∑ ( ∫ [B] [B] dV + ∫ φ [N] [N]{b}[N] dV + ∫ Bi [N] [N] dS){a} - ∑ ( ∫ Bi [N] dS) ) {0} (8) el
with the following boundary conditions:
V
Ra )
Rb ) kr D
φB ) rp
The FEM equations for this reaction are as follows:
(3)
∇2b - φB2ab ) 0;
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(12)
S
T
el
S
This scheme is quadratically convergent and generally requires less than 10 iterations to converge to the solution. Note however that this solution strategy requires the solution of a linear system of twice the size of that for a first-order reaction when the same FEM grid is used and is therefore more computationally intensive. As before, a direct sparse solver is used to solve eq 10. 2.3. Meshing. A tetrahedral mesher for Matlab that was developed by Persson and Strang9 was used to generate threedimensional meshes of a sphere. In total, six different meshes
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for a spherical eggshell catalyst: A1eφλ + A2e-φλ , Feλe1 c(λ) ) λ (14) c(λ) ) c(F), λ e F φ φ(1-2F) φ - 1/F -1 φ 2φ A1 ) e + e ; A2 ) e - A1e φ + 1/F
(
Figure 1. Finite element meshes for (a) a first-order reaction in a monodispersed particle; (b) a first-order reaction in an eggshell particle; (c) a reaction of the form -rA ) RkrCACB in a monodispersed particle; (d) the same reaction in an eggshell particle where F ) 0.9.
)
Here, λ is the dimensionless radial profile, and F ) rs /rp where rs is the inner dimension of the catalyst shell. Solutions are accurate for φ e 30, as is shown in Figure 3. For the reaction -rA ) -rB ) krCACB the FEM solutions can be verified as follows. (a) For symmetric boundary conditions, φA . φB and no external mass-transfer resistances, the concentration profile of reagent A should be given by eq 13. (b) For symmetric boundary conditions, negligible external mass-transfer resistances, and any value of φA′ and φB′ , the following relationship should hold true: b)
were used for the investigation: Simulations of monodispersed and eggshell particles were performed with different meshes, since the shell had to be well-defined for the simulation of eggshells. For a grid size of n nodes, the solution of eq 5 requires the solution of an n × n system, whereas the iterative solution of eqs 8 and 9 requires the (iterative) solution of a 2n × 2n system. Therefore, computational limitations required the meshes for the reaction rA ) RrB ) -RkrCACB to be coarser than the corresponding meshes for first-order reactions. Meshes were created for both reactions, monodispersed particles, and eggshell particles with inner to outer shell diameter ration’s of F ) 0.9 and F ) 0.5. Cross-sections of the meshes for the monodispersed particles and eggshell particles with F ) 0.9 are shown in Figure 1. Boundary conditions are based on the wetting geometries that were obtained by a colorimetric method that was reported in previous work.2 This method yields photographs of particles that were partially wetted by liquid doped with a colorant under trickle-flow conditions. The translation of a two-dimensional photograph onto the FEM grid can be done with simple geometrical calculations: Each pixel on the photograph was translated to the corresponding three-dimensional Cartesian coordinate on the surface of a unit sphere. Each surface triangle on the FEM grid was then assigned a Biot number for wet or dry surface external mass transfer, based on the coordinates of its centerpoint and the coordinates of the pixels in the photograph. For the liquid reagent, BiB,d ) 0, and for the gas reagent BiA,d . BiA,w. The process involves some interpolation, since the set of pixel coordinates and node point coordinates are not the same. The method is illustrated in Figure 2. For each particle, photographs of two opposing hemispheres were used to obtain a complete description of the wetting geometry. In total, 15 particles with different wetting geometries were simulated with the FEM, with wetting efficiencies ranging from f ) 0.26 to f ) 1. 2.4. Solution Accuracy. The FEM solution for a first-order elementary reaction can be verified with the following analytical expressions for the concentration profile in the absence of external mass-transfer resistances: for a monodispersed catalyst sphere: sinh(φλ) c(λ) ) sinh(φ)
(13)
a - 1 + γ′ γ′
(15)
where γ′ ) (φA′ /φB′ )2 and a and b are the dimensionless concentrations of A and B. Unlike φA and φB, which are based on bulk concentrations, φA′ and φB′ are based on reagent surface concentrations. The derivation for this relationship is shown in Appendix A. Numerical and analytical solutions of concentration profiles for the reaction -rA ) RkrCACB and the maximum simulated Thiele modulus are shown in Figure 4. 3. Monodispersed Particles Preliminary results of the FEM simulations showed that the behavior of partially wetted monodispersed catalysts differs significantly from that of eggshell catalysts, so that these will be discussed separately. In this section, the behavior of partially wetted monodispersed catalysts and different reaction expressions is described, based on FEM results. 3.1. Theory. To obtain an understanding of the behavior of partially wetted monodispersed catalysts for different reaction cases, it is first necessary to understand the relevant existing theories. 3.1.1. Geometry. For the theoretical particle geometries of a semiinfinite slab, a semiinfinite cylinder, and a sphere, a firstorder reaction, and a single boundary condition at the external
Figure 2. Translation of experimental wetting geometry photographs to boundary conditions for the FEM simulations. (a) Graphical representation of the boundary conditions. Black indicates wetted surface external mass transfer, and white indicates dry surface external mass transfer. (b) Original photograph of the partially wetted particle (f ) 0.74).
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Figure 3. Numerical solutions for (a) a monodispersed catalyst and (b) an eggshell catalyst with F ) 0.9, for Φ ) 30 and no external mass transfer tested against the analytical solutions given in eqs 13 and 14.
Figure 4. Numerical and analytical solutions for the reaction -rA ) RkrCACB for the case where φA . φB ) 30: (a) Monodispersed particle; (b) eggshell particle with F ) 0.9.
surface, eq 1 is one-dimensional; and exact analytical expressions for the pellet efficiency can be derived: semiinfinite slab: tanh φG ; η) φG semiinfinite cylinder: 2I1(φC) ; η) φCI0(φC)
φG ) L
VR kr ) D Sx
2VR kr ) D Sx
φC ) rC
kr D
(16)
(17)
kr D
sphere: η)
3 (φ coth φ - 1); φ2
φ ) rp
3VR kr ) D Sx
kr (18) D
Aris10 showed that the efficiencies of all the above geometries have more or less the same functionality with the modulus of a slab, φG ) L(kr/D)1/2 (termed the generalized modulus), where
L is the ratio between the particle volume and external area. He therefore proposed to use eq 16 to approximate pellet efficiency factors, regardless of the particle shape. The maximum error encountered when modeling a sphere as a slab is approximately 10% at intermediate φ. In the limits of L f 0 and L f ∞, efficiency factors for spheres, cylinders, and slabs are exactly the same for the same φG. The idea of a generalized modulus forms the basis for modeling partially wetted particles under liquid-limited reaction conditions. The expressions so far are for pellet efficiency factors and do not take external mass-transfer resistances into account. To obtain an overall efficiency factor, one can make use of the following relationship: η0 ) η
(
)
CS φG2η VRkc )η 1+ , where Bi'' ) (19) Cbulk Bi'' SxD
This can be derived from the following equality:
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Sxkc(Cbulk - Cs) ) VRηkrCs
(20)
3.1.2. Modeling of Partial Wetting. Effectiveness factors for particles in trickle-bed reactors are complicated by the fact that the boundary conditions for eq 1 are mixed: Due to incomplete wetting, two boundary conditions must be satisfied, one for the wetted and one for the dry surface. Even for the 1-D geometries of a sphere, a slab, and a cylinder, the diffusionreaction problem obtains extra dimensions, though some geometries have previously been suggested where partial wetting can be specified without increasing the dimensionality of the problem.11,12 For liquid-limited reactions, Dudukovic13 made use of the work of Aris10 to derive an expression for the catalyst efficiency of partially wetted particles for liquid-limited reactions, realizing that only the wetted area can supply the reagent so that Sx ) Spf: η)
tanh φG , φG
where φG )
Vp fSp
kr D
(21)
Equation 21 suggests that partial wetting affects the effective geometry of a particle due to its effect on the external area available for reagent supply. It is important for the rest of this paper to realize that if all shapes can be approximated for a slab of thickness L ) VR/Sx, then it should also be possible to approximate all shapes with a sphere of radius rp ) 3VR/Sx, so that eq 21 becomes η)
φ 3f2 φ coth -1, 2 f f φ
[
() ]
where φ )
3Vp Sp
kr D
(22)
The overall efficiency that takes external mass-transfer limitations into account can easily be modeled, making use of eq 19, and taking into account that only the wetted area is available for external mass transfer,
0 is only the same as the liquid-limited description of eq 23 at large moduli. Note that eq 25 describes gas-liquid-particle surface mass transfer as one step, so that Biw is a combined Biot number for gas-liquid and liquid-solid mass transfer. This can be done when the overall rate of liquid-solid and gas-liquid mass transfer is the same. For high conversions of the liquid reagent and negligible expense of the gas, this is a reasonable assumption. Valerius et al.12 suggested a particle efficiency model which can take wetting efficiency into account and can be used for gas-limited and liquid-limited reactions. The model is based on a hollow cylinder geometry where the outer surface area represents the wetted surface and the inner surface the dry surface of a partially wetted pellet and can therefore only be used for f > 0.5. This geometry is used in a later paper15 to simplify the numerical calculations for complex kinetic expressions by transforming a 3-D problem to a 2-D problem, rather than to obtain analytical expressions for pellet efficiency factors. Only the analytical expression for a liquid-limited reaction was reported and will be verified in this section. 3.1.2. Kinetics. Exact explicit expressions of pellet efficiency factors only exist for simple kinetic expressions such as zeroand first-order kinetics.1 Large amounts of literature that present methods of approximating effectiveness factors for arbitrary kinetics are therefore available. Arguably the most important among these is that of Bischoff,16 who suggested a general modulus for kinetics of any form. This modulus has more or less the same effect on pellet efficiency independent of the kinetic expression. To understand how the Bischoff modulus can be used, a short version of the derivation of this modulus is now shown: d2C diffusion-reaction equation dx2 dC C(0) ) Cs ; boundary conditions dx x)L
r(C) ) D
(26)
slab geometry: η0 )
tanh φG φG(1 + (φG tanh φG)/Biw′ f)
sphere geometry: φ φ φ φ - 1 /φ2 1 + f coth - 1 /Biw η0 ) 3f2 coth f f f f
( [
( ) ] )(
[
(23)
dC , dx
d d )p dx dC
then
∴r(C) )
D d 2 (p ) 2 dC
(27)
-1
() ] )
(24) The above approach cannot be followed for a gas-limited reaction, since the gas enters via both the wetted and the dry surface area. The most widely accepted model for a gaslimited reaction in a partially wetted catalyst was provided by Ramachandran and Smith.14 The model is based on infinite slab geometry and the assumption that the reagent entering through the dry and the wetted parts of the slab can be treated separately and do not interact throughout the slab volume: slab geometry: η0 )
let p )
f tanh φG (1 - f) tanh φG + φG(1 + (φG tanh φG)/Biw′ ) φG(1 + (φG tanh φG)/Bid′ ) (25)
When compared to eq 19, it is clear that this expression is analogous to that of a slab where the entire surface is at a surface j s, where C j s ) fCs,w + (1 - f)Cs,d. Clearly, the concentration C modeling of gas-limited reactions is completely different from that of liquid-limited reactions: the limit of eq 25 where Bid f
integrate from p ) 0 (at x ) L) to p: p 2 C d(p2) ) r(β) dβ 0 D CL C 2 dC )p)( C r(β) dβ)1/2 ∴ L dx D
∫
∫
(28)
∫
The effectiveness factor can be evaluated using the flux of reagent into the slab:
η)
∫
C dC -SpD( C r(β) dβ)1/2 dx x)0 L ) Vpr(Cs) Vpr(Cs)
-SpD
(29)
It is well-known that, for simple-order reactions in a slab, the pellet efficiency factor-Thiele modulus curve has the relationship η ) φG-1 when φG . 1. At such high Thiele moduli, L or r is very large and CL f Ceq, where Ceq is the concentration where r(Ceq) f 0. For example Ceq ) 0 when a reaction is irreversible and involves only one reagent. To have the same behavior for an arbitrary reaction at a large Bischoff modulus (large L and/or fast reaction), this modulus can be defined as
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Figure 5. Particle efficiency versus generalized modulus for partially wetted particles and liquid-limited reaction conditions for real wetting geometries. The parity plots show the prediction performance for the discussed models for liquid-limited particle efficiency.
φT′′ )
Vpr(Cs) 1 ) ( ηCeq Sp√2D
∫
Cs
Ceq
r(β) dβ)-1/2
(30)
where the original derivation was done for a slab (to be used with eq 13) one can, for the same reason as discussed earlier, also define a “spherical” Bischoff modulus that can be used with eq 15: φT′′ )
3Vpr(Cs) 1 ( ) ηCeq Sp√2D
∫
Cs
Ceq
r(β) dβ)-1/2
(31)
3.2. Verification of Existing Models. Expressions for efficiency factors of partially wetted catalysts are only available for completely liquid-limited and completely gas-limited reactions where only one of the reagents plays a role. The accuracy of these models can now be evaluated with the FEM models for true wetting geometries. In this section, the Bischoff approximation for the reaction -rA ) -rB ) RkrCACB is also derived and verified for fully wetted particles.
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3.2.1. Liquid-Limited Reactions. Monodispersed particle efficiencies of all the photographed particles shown in Figure 2 were calculated for 0.1 e φ e 30 in the absence of external mass-transfer resistances. Results are shown in Figure 5 as a function of the generalized modulus approach for partially wetted particles.13 It is clear from the figure that this approach yields rather good results, as can also be inspected by the parity plot in the top left corner. The cylinder shell model12 performs better but is limited to f > 0.5. The treatment of external mass transfer for liquid-limited reactions is analytically correct and need not be verified. 3.2.2. Gas-Limited Reactions. The most important model for the evaluation of partially wetted particle efficiency under gas-limited conditions is the “weighting method”14 (eq 22). This weighting method should be accurate if the difference between dry and wet surface concentrations is small, so that unsymmetrical boundary conditions do not have a major influence on the concentration profiles in the slab.14 Usually, Bid is very high, and it was therefore investigated for which values of Biw the weighting method would still be accurate when external masstransfer limitations on the dry part of the catalyst were negligible. Results are shown as parity plots in Figure 6. Liquid-solid mass-transfer Biot numbers are typically larger than 10 for trickle-bed reactor applications as is also the case for gas-liquid mass transfer.17 It can be concluded that eq 25 is accurate for realistic values of Biw. The only significant source of prediction error is the fact that slab geometry is used to model other pellet geometries (spherical in this work), as can be seen from the subplot in the bottom right corner. The equivalent of eq 25 for a sphere is η0 )
3f(φ coth φ - 1) + φ2(1 + (φ coth φ - 1)/Biw) 3(1 - f)(φ coth φ - 1) (32) 2 φ (1 + (φ coth φ - 1)/Bid)
Though the original derivation of eq 25 was only valid for the slab geometry defined in that work,14 Goto et al.4 have shown that one can also use eq 32, as can also be seen from the FEM results.
Figure 6. Parity plots of pellet efficiency factors as calculated with the weighting model vs FEM results for different extents of mass-transfer resistances over the wetted part of the catalyst if gas-solid mass transfer is negligible. In the last subplot, the parity between spherical and slab efficiency at the same generalized modulus is also shown.
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Figure 7. Pellet efficiency factors as a function of the Bischoff modulus for the reaction rA ) RkrCACB, fully wetted particles with no external mass-transfer resistances. Also shown is the η-φ relationship for a first-order reaction in a sphere.
Figure 8. Performance of the unified model. (a) Liquid-limited reaction, with some external mass-transfer resistances for the gaseous reagent over the wetted surface (BiA,w ) 10, BiA,d f ∞, BiB,w f ∞); (b) reaction that is not classified as either gas- or liquid-limited with varying rates of external mass transfer for the gaseous reagent (γ ) 1, BiA,d f ∞, BiB,w f ∞); (c) gas-limited reaction (BiA,w ) 10, BiA,d f ∞, BiB,w f ∞); (d) arbitrarily chosen conditions (φA ) 3, γ ) 5, BiA,w ) 5, BiA,d ) 50, BiB,w ) 10). All Thiele moduli in a-c are as defined in the legend of a.
3.2.3. Bischoff Modulus. For a completely wetted particle, the Bischoff modulus (based on surface concentrations) for the reaction -rA ) RkrCACB can be derived as follows, starting at the definition of the Bischoff modulus: φT )
3Vpr(Cs) Sp√2D
(
∫
Cs
Ceq
r(β) dβ)-1/2
(33)
This definition allows for the integration of a reaction rate which is described in terms of one reagent only. If γ > 1, CAeq ) 0 and CBeq is unknown, so that CB should be written in terms of CA. This can be done by using eq 15, which relates the concentrations of A and B in the pellet:
φT ) ) )
3Vpr(CA,s, CB,s) Sp√2DA rpRkrCA,sCB,s
√2DA
rp√RkrCB,s
√2DA
(
) φA′ 1 -
(
∫
CA,s
0
[ ∫ Rk C 1
0
(∫
1
0
1 3γ′
r(CA) dCA))-1/2 2
r A,s
( a - γ′1 + γ′ ) da]
-1/2
CB,sa
)
a2 - a + γ′a da γ′
-1/2
-1/2
)
(34) When γ′ < 1, CBeq ) 0 and the above derivation should be performed in terms of CB to obtain
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Figure 9. Predictions of the FEM data in Figure 8 by traditional models for liquid- and gas-limited reactions.
Figure 10. Performance of the generalized modulus approach for liquid-limited reactions in partially wetted spherical eggshell catalysts, where F ) 0.9.
(
φT ) φB′ 1 -
γ′ 3
-1/2
)
(35)
In this derivation a and b are dimensionless concentrations based on surface concentrations. Either eq 34 or 35 can be used when γ′ ) 1, since Ceq will then be zero for both reagents. Pellet effectiveness factors for completely wetted particles, -rA ) RkrCACB, and negligible external mass transfer were calculated using FEM. These are shown as a function of the abovederived Bischoff modulus in Figure 7. Clearly, the Bischoff modulus for this reaction can be used for a good approximation of pellet efficiency. 3.3. Unified Model for rA ) rrB ) rkrCACB. The previous section has shown that the existing models for liquid- and gaslimited reactions are satisfactory for true wetting geometries. These models are specific to either liquid- or gas-limited reactions but provide useful descriptions of the effect of partial wetting on the behavior of the liquid and gaseous reagents, which can be summarized as follows:
(a) For liquid-limited reactions, the generalized modulus approach13 is sufficiently accurate while very simple to use. According to this approach, partial wetting affects the effective geometry (diffusion path length) of the limiting reagent when a reaction is liquid-limited. For a monodispersed catalyst pellet, the Thiele modulus of a partially wetted particle can be corrected for partial wetting by adjusting the effective geometry so that φcorr ) φ/f. Where in the previous discussion the Thiele modulus-efficiency relationship of a slab was used for any geometry, it is also possible to use that of a sphere. Overall pellet efficiency can be evaluated by taking external masstransfer resistances into account, as is shown in eq 19. Mathematically, the approach to liquid-limited reactions can be written as η0 ) η(φB′′/f)
CB,s CB,bulk
(36)
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(b) Partial wetting affects the average external surface concentration of the limiting reagent when the reaction is gas-limited. Correcting overall pellet efficiency with the average external surface concentration weighted according to the fractional wetting yields good results for realistic rates of external mass transfer. It is preferable to use the model of Ramachandran and Smith14 adapted to the Thiele modulus-pellet efficiency relationship specific to the particle geometry. η0 ) η(φA′′)
fCA,s | w + (1 - f)CA,s | d CA,bulk
(37)
Combining the above approaches with the Bischoff modulus derived for the reaction -rA ) RkrCACB, the following unified model for this reaction is suggested to predict the efficiency of partially wetted particles over the whole γ-range: η0 )
j A,sCB,s 3C φT2C* ACB,bulk
( (
2
φT ) φA′ 1 -
φA′ ) rp
(
φB′
2
3φA′
2
φT ) φB′ 1 -
φA′
2
3φB′
) )
RkrCB,s ; DA
if φA′ g φB′ if φB′ g φA′
φB′ )
rp f
j A,s krC DB
2 j A,s φA′ (φT coth φT - 1) C )f 1+ C* φ 2Bi A
( (
(1 - f) 1 +
Figure 11. Particle efficiencies as a function of the modulus defined in eq 11 for eggshell particles with F ) 0.5 and 0.9.
(φT coth φT - 1)
T
A,w
)
-1
2
φA′ (φT coth φT - 1) φT2BiA,d 2
φ′Bf(φT coth φT - 1) CB,s ) 1+ CB,bulk φ 2Bi T
B,w
(38)
+
) )
-1
-1
The unified model is reported here in terms of spherical particles. For other shapes, the applicable η-φ relationship and Biot number should be used. The principles remain the same: the average surface concentration of the gas and the modulus for the liquid component are affected by fractional wetting according to the traditional approaches.12,14 The external surface concentration of the liquid reagent is calculated according to eq 19. The model uses a Bischoff modulus that was derived for fully wetted particles. The results are good for a wide γ-range, as is shown in Figure 8, especially when compared to predictions of the traditional liquid-limited and gas-limited models shown in Figure 9.
10. A possible reason why the generalized modulus approach performs so poorly for partially wetted eggshell catalysts is that, within such catalysts, zones can exist where diffusion without reaction can take place and the particle can therefore not be directly related to a slab with diffusion and reaction throughout its volume. Due to symmetry, such zones will not exist in a fully wetted particle and the generalized modulus approach is still accurate at high wetting efficiencies. A modulus that will be able to describe partially wetted spherical eggshell catalyst will have the following characteristics: (a) If completely wetted, the modulus should be equal to the generalized modulus, as can also be seen from the data for particles with a high fractional wetting in Figure 10. (b) The generalized modulus approach works well for monodispersed catalysts. Therefore, the new modulus should also be equal to the generalized modulus when F ) 0. (c) The modulus should be higher than the generalized modulus for a partially wetted particle, increasingly so at low fractional wettings and a thin outer shell. On the basis of the above, the following modulus is proposed for spherical eggshell particles:
4. Eggshell Particles The most obvious difference between eggshell and monodispersed catalyst spheres is that the former tend to behave according to slab geometry as the shell thickness is decreased. This clearly shows in the behavior of such particles under gaslimited conditions: for the investigated eggshell catalysts, eq 22, which was developed for a semiinfinite slab, predicts all FEM data within 5% for Biw g 5. Diffusion limitations of the liquid reagent in partially wetted spherical eggshell catalysts are however not predicted well by the traditional generalized modulus approach, and partial wetting has a stronger detrimental effect on these catalysts than in monodispersed spheres, as is shown in Figure
This modulus should be used with the modulus-efficiency relationship of a semiinfinite slab and correlates the FEM results for particles with F ) 0, 0.5, and 0.9 sufficiently, as can be seen from Figures 5 and 11. To model eggshell particles in which resistance to internal and external mass transfer of both liquid and gas reacting according to the rate expression -rA ) -RrB ) RkrCACB occur, eq 11 can be used in the unified model to describe liquid-diffusional effects. Since eq 11 describes a modulus
Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010
8123
Figure 12. Overall efficiency factor for the reaction -rA ) -RrB ) RkrCACB in an eggshell catalyst with F ) 0.9. (a) Liquid-limited reaction (BiA,w ) 10, BiA,d f ∞, BiB,w f ∞); (b) reaction with γ ) 1, BiA,d f ∞, and BiB,w f ∞; (c) gas-limited reaction (BiA,w ) 10, BiA,d f ∞, BiB,w f ∞). In all cases, Φi ) 10, where i refers to the limiting reagent.
for slab geometry, the unified model should be used accordingly: η0 )
j A,sCB,s C tanh φT φTC* A CB,bulk
( ) ( )
Appendix A. Derivation of Equation 15 For a sphere with symmetric boundary conditions, the diffusionreaction equations for an elementary A + B f C reaction are d2a 2 da - φA2ab ) 0 + F dλ dλ2 d2b 2 db - φB2ab ) 0 + 2 F dλ dλ RCB,bulkDB φA 2 γ) ) C*D φB A A
2
φT ) φA′ 1 φT ) φB′ 1 φA′ )
φB′
φA′
2
3φB′
rp(1 - F3) 3
(
if φA′ g φB′
2
3φA′
RkrCB,s ; DA
2 j A,s φA′ tanh φT C )f 1+ C* φTBiA,w ′ A
(
(
2
( )
if φB′ g φA′
2
)
)
-1
φB′ )
rp(1 - F3)[1 + F(1 - f)] 3f
(
2
φA′ tanh φT +(1 - f) 1 + φTBiA,d
CB,s φB′ f tanh φT ) 1+ CB,bulk φTBiB,w
)
-1
)
j A,s kr C DB
(
(40)
Predicted overall efficiencies for eggshell particles with F ) 0.9 are compared to FEM simulation results in Figure 12. 5. Conclusions Using FEM simulations, it was shown that the traditional approaches for particle efficiency modeling of partially wetted particles under liquid-limited13 and gas-limited conditions14 are sufficiently accurate for true wetting geometries. On the basis of these descriptions and the Bischoff modulus for general reaction kinetics,16 a simple unified pellet efficiency model was proposed for partially wetted particles where both the liquid and gas concentrations determine the reaction rate, following elementary kinetics. This model was also validated with FEM simulations. Lastly, a correction factor is proposed for the modeling of liquid-limited reactions in eggshell particles as it was shown that the model of Dudukovic13 does not describe these sufficiently. It is shown that this correction factor can also be used in the proposed unified model for reactions of the form -rA ) -RrB ) RkrCACB.
)
d2b d2a 2 da 2 db ) γ + + 2 2 λ dλ λ dλ dλ dλ 2 d 2d (a - γb) ∴ 2 (a - γb) ) λ dλ dλ ∴
-1
Let θ ) a - γb Then d 2θ 2 dθ )0 + 2 λ dλ dλ It can be shown that, for y ) λθ, d 2y )0 dλ2 ∴y(λ) ) k1λ + k2 and θ(λ) ) k1 + Since θ is finite at λ ) 0,
k2 λ
θ(λ) ) k1 At λ ) 1, a ) 1 and b ) 1 (no external mass-transfer resistances).
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Ind. Eng. Chem. Res., Vol. 49, No. 17, 2010
∴k1 ) 1 - γ ) θ ∴a - γb ) 1 - γ a-1+γ ∴b ) γ The same relationship can be derived for semiinfinite slab geometry.
φ ) Thiele modulus for a sphere (based on bulk concentrations for higher order reactions) φ′ ) Thiele modulus for a sphere based on surface concentrations φ′′ ) Thiele modulus based on surface concentrations, not specific to any geometry φE ) eggshell modulus, defined in eq 11 φG ) generalized modulus, based on slab geometry φT ) Bischoff modulus Subscripts
Nomenclature a ) dimensionless concentration of liquid reagent, CB/CB,bulk b ) dimensionless concentration of gaseous reagent, CA/C*A c ) dimensionless concentration of limiting reagent Bi ) Biot number for a sphere, Bi ) rpkc/D Bi′ ) general Biot number based on slab geometry, Bi′ ) Vpkc/ (SpD) Bi′′ ) Biot number defined in eq 19, applicable to any geometry C* ) saturation concentration, mol/m3 Ceq ) concentration of (limiting) reagent for which the reaction rate is zero Ci,bulk ) concentration of reagent i in the bulk liquid, mol/m3 Ci ) concentration of reagent i, mol/m3 Ci,s ) surface concentration of reagent i, mol/m3 D ) effective diffusivity, m2/s f ) fractional wetting Ii(j) ) Bessel function of the second kind of order i, evaluated at j kr ) reaction rate constant based on catalyst volume, (SI) units depend on rate expression kc ) external mass-transfer coefficient based on Sp, m/s L ) slab thickness or characteristic length, m nj ) unit vector normal to particle external surface ri ) rate of reaction of reagent i, mol/s rC ) radius of cylindrical particle, m rp ) radius of spherical particle, m S ) external surface of surface element used in FEM Sp ) total pellet external area, m2 Sx ) external surface area through which reagent can enter the pellet, m2 V ) volume of element used in FEM Vp ) total pellet volume, m3 VR ) pellet volume in which reaction takes place, m3 X ) dimensionless diffusion distance in a slab
Greek Letters R ) reaction stoichiometry, R ) rA/rB γ ) (CB,bulkDB)/(C*ADA); γ ) φA2/φB2 for the reaction rA ) RrB ) -RkrCACB γ′ ) same as γ, but based on surface concentrations rather than bulk concentrations η ) efficiency factor based on surface concentrations η0 ) efficiency factor based on bulk concentrations λ ) dimensionless radial position in a sphere, λ ) r/rp F ) ratio of inner shell radius to particle radius
A ) refers to gaseous reagent B ) refers to liquid reagent d ) refers to dry surface el ) refers to volume or surface element in FEM modeling w ) refers to wetted surface
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ReceiVed for reView October 31, 2009 ReVised manuscript receiVed February 24, 2010 Accepted June 15, 2010 IE9017176
