Ind. Eng. Chem. Res. 2004, 43, 4753-4760
4753
Oxygen Distribution in Packed-Bed Membrane Reactors for Partial Oxidations: Effect of the Radial Porosity Profiles on the Product Selectivity U. Ku 1 rten, M. van Sint Annaland,* and J. A. M. Kuipers Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
A two-dimensional, pseudohomogeneous reactor model was presented to describe the radial and axial concentration profiles in a packed-bed membrane reactor and the local velocity field while accounting for the influences due to the distributive membrane flow and the radial porosity profile. The effect of the radial porosity profile on the model results has been evaluated for cases with a low tube-to-particle diameter ratio (dt/dp) with insufficient radial transport resulting in strong radial oxygen concentration profiles. From these calculations, it was concluded that the overall effect of the porosity profile on the conversion-selectivity plot is unexpectedly small. If the implementation of a radial porosity profile is not required by other considerations, e.g., for an accurate description of temperature effects, then the effect of the porosity profile can safely be neglected. If a radial porosity profile is implemented, however, the effect of the porosity profile on the dispersion coefficient should be taken into account to achieve a proper description of the radial transport. 1. Introduction Packed-bed membrane reactors (PBMRs) can be applied to improve the selectivity toward the target product in partial oxidation systems. Via a membrane, which can be permselective or not, oxygen is fed distributively to the catalyst bed and the reaction mixture. From the membrane surface, the oxygen permeates into the packed bed, while it is consumed. If the chemical consumption rate is high compared to the dispersion rate, concentration gradients will emerge, which may affect the performance of the PBMR. An important design criterion (in view of operating costs) of tubular packed-bed reactors is the maximum allowable pressure drop. To reduce the pressure drop, the size of the catalyst particles is chosen to be as large as possible but where internal transport limitations can just be avoided. Consequently, low aspect ratios of the tube-to-particle diameter (dt/dp), and thus a strong porosity distribution with increased porosity near the membrane wall, are to be expected for industrial-scale PBMRs. Because of the increased porosity near the wall, where the oxygen concentration is maximal, the catalyst concentration, and thus the activity of the packed bed, is decreased and the axial velocity is increased, causing a bypass flow. The present paper discusses the effects of the porosity distribution on the performance of the PBMR in terms of the conversion and selectivity for a simple, but representative, consecutive reaction scheme, from which it can be assessed whether the porosity distribution needs to be accounted for in a two-dimensional PBMR model. If the effect of the radial porosity profile would be small, the required reactor model can be considerably simplified, using an a priori calculated velocity profile (see ref 1). * To whom correspondence should be addressed. Tel.: (+31) 53 4894478. Fax: (+31) 53 4892882. E-mail:
[email protected].
Before the effect of the porosity distribution is discussed, the selected reaction system, the reactor model, and the model parameters used in this study are described. 2. Selection of the Reaction System Selectivity improvements in a PBMR for partial oxidation systems can be achieved because of the reduced level of the oxygen concentration for a reaction system, which exhibits a lower reaction order in oxygen for the reaction forming the target product than for the formation of waste products. In this paper, the following simple, but representative, consecutive reaction scheme will be considered (with m > n):
A + ν1O2 f P
r1 ) k1pApO2n
(1)
P + ν2O2 f W
r2 ) k2pPpO2m
(2)
where A represents the hydrocarbon reactant, P the target product, W the waste product, νi the stoichiometric constants, and ki the reaction rate constants (Table 1). The reaction rates of both reactions are chosen to be first order in the hydrocarbons, and the reaction orders of oxygen for the target product formation and for the waste product formation are indicated with n and m, respectively. 3. Two-Dimensional, Pseudohomogeneous PBMR Model In a PBMR with large particles relative to the tube diameter, a bypass flow may emerge near the tubular membrane wall. Oxygen is added to the packed bed in this region of increased bed porosity and increased axial velocity, reducing the residence time of oxygen in the catalyst bed. To evaluate the extent of this effect, the description of the two-dimensional flow field was in-
10.1021/ie030771m CCC: $27.50 © 2004 American Chemical Society Published on Web 07/08/2004
4754 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 Table 1. Parameters Selected for the Model Kinetics rate constants: n ) 1.0/m ) 2.0
0.150 mol/(g s bar1+n) 4.500 mol/(g s bar1+m) 1 1
k1 k2 ν1 ν2
stoichiometric coefficients
cluded in the PBMR model. The developed PBMR model was based on the following assumptions: 1. Pseudohomogeneous Reactor Model. In the model, mass-transfer limitations from the gas bulk to the catalyst surface and inside the catalyst particle have been ignored in order to avoid obscuring the effects of the porosity profile. The effects of these mass-transfer limitations have also been studied2 and will be reported elsewhere. The heterogeneous reaction rates can thus be described as a direct function of the gas-phase concentrations. Furthermore, the structure of the packing is characterized by the bed porosity, which can, however, be a function of the radial position in the tube. The bed porosity is considered here as a locally averaged value. Also axisymmetry is assumed for the PBMR. 2. Diffusive and Dispersive Transport Described with a Standard Dispersion Model (SDM). It is assumed that gas-phase mass transport in the PBMR can be modeled with a two-dimensional model with convective flow with superimposed radial and axial dispersion. First, the flow model (gas-phase total continuity and Navier-Stokes equations) is described, where after the component mass balances are given. 3.1. Flow Model. The two-dimensional PBMR model calculates the gas-phase velocity field by solving the total continuity and momentum equations for the gas phase given respectively by
desired because of the large differences in time scales on which the hydrodynamic phenomena and chemical reactions take place. Because the steady-state solution is of main interest in this work, this decoupling allows one to use different time steps and scales, speeding up the calculations tremendously. The total continuity and Navier-Stokes equations are solved with a finite-difference technique on a staggered computational mesh using a first-order time discretization and implicit treatment of the pressure gradient and linearized implicit treatment of the drag forces. The implicit treatment of the pressure gradient term requires solution of a pressure correction equation (Poisson equation) derived from the mass defect of the gas-phase continuity equation. Details on the numerical technique can be found in ref 4. The convection terms have been discretized using a first-order upwind scheme, while the dispersion terms have been discretized with standard second-order finite-difference representations. The computational scheme is thus as follows: For a new time step, first the density field is calculated from the old pressure and concentration field data using the ideal gas law. Subsequently, the velocity field is calculated using the discretized momentum equations, followed by the calculation of the new pressure field using the pressure correction equation. Then, the density field is updated using the equation of state, and the iteration loop is repeated until all variables have converged. Because of the explicit treatment of the convection and dispersion terms in the momentum equation, the maximum allowable time step is determined by the following two stability criteria:
Viscosity stability criterion:
∂(Fg) + ∇(Fgvj ) ) 0 ∂t
µg∆t/Fg∆x2 < 0.5
(3)
(8)
Courant-Friedrichs-Lewy stability criterion (Courant condition):
and
∂ (F vj ) + ∇(Fgvj vj ) ) - ∇p - βFgvj - ∇(τjjg) + Fggj ∂t g (4) where
Friction coefficient: β ) 150
(1 - )2 µg 3
2
Fgdp
(Ergun’s equation) with |vj | )
1 - |vj | (5) 3 dp
+ 1.75
xvr2+vz2
Newtonian fluid: jτg ) - λg - 2µg (∇vj )Ih - µg[(∇vj ) + (∇vj )T] (6) 3
(
)
λg is the contribution of the bulk viscosity of the gas to the normal stress, which can be neglected for lowdensity gases.3
Ideal gas law: Fg ) pMg/RTg
(7)
Although the physical properties (especially density and viscosity) are determined by the local composition, which are affected by chemical reactions, the component mass balances are solved sequentially after having solved the flow model. This decoupling is possible and
x( ) ( ) |vr|∆t ∆r
2
+
|vz|∆t ∆z
2
