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Ind. Eng. Chem. Res. 2007, 46, 7524-7534
Effect of Mass-Transport Limitations on the Performance of a Packed Bed Membrane Reactor for Partial Oxidations. Transport from the Membrane to the Packed Bed M. van Sint Annaland,* U. Ku1rten, and J. A. M. Kuipers Department of Science and Technology, UniVersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
With a packed bed membrane reactor, the product yield can be significantly enhanced for partial oxidation systems, via distributive addition of oxygen to the reaction mixture along the axial coordinate of the reactor, provided that the reaction order in oxygen of the formation rate of the target product is smaller than the reaction order in oxygen of the consecutive reaction toward the waste product. In Part 1, the intrinsic and integral effects of intraparticle diffusion limitations on the performance of the packed bed membrane reactor have been investigated. In this paper, the study is extended by studying the effects of mass-transfer limitations from the membrane wall to the center of the packed bed. Again, first the intrinsic effects are examined, followed by a study of the integral effects over the entire reactor using a two-dimensional reactor model. A modified Thiele modulus φ′′ was defined and it was found that only for φ′′ > 1 radial concentration profiles inside the packed bed does the integral reactor performance deteriorate and that for values 1, radial concentration profiles influence the integral reactor performance. However, for φ′′ < 1, the effect of radial profiles is marginal, and a onedimensional model can sufficiently accurately describe the PBMR performance.
that intraparticle diffusion limitations and external mass-transfer limitations from the interstitial gas to the external surface of the particles can be neglected. The diffusion equation describing the radial transport of oxygen thus reads
D
d2c - (kncn + kmcm) ) 0 2 dx
This can be integrated once (using the substitution p ) dc/dx and the chain rule, so that d2c/dx2 ) dp/dx ) dp/dc dc/dx ) p dp/dc), which yields after substitution of the boundary condition dc/dx(a) ) 0, where a indicates the position where the oxygen gradient becomes zero (for n,m * 0, -1)
dc ) dx 1/2 km 2 1/2 kn (cn+1 - c(a)n+1) + (cm+1 - c(a)m+1) D n+1 m+1 (A.2)
()[
]
or in dimensionless form
[
]
γn+1 - γ(R)n+1 γm+1 - γ(R)m+1 dγ ) x2 φn + pnm dl n+1 m+1
1/2
(A.3)
with
φn ) L
x
km kncn-1 L , pnm ) cm-n ,γ) D kn L c a x , l ) , and R ) (A.4-A.8) cL L L
The relative conversion rate FR,n and selectivity ratio F1-σ have been defined as
∫R1 γn d l FR,n ) 1 (∫R γdl )n
Appendix A Derivation of an asymptotic solution for the relative reactivity and relative selectivity for high values of the modified Thiele moduli. In this appendix, asymptotic solutions are derived for the relative conversion rate FR,n and the relative selectivity F1-σ as defined by eqs 8 and 9. These asymptotic solutions allow a quick estimate of the influence of radial masstransfer limitations from the membrane wall to the center of the packed bed for high values of the modified Thiele modulus φ′′ defined by eq 4. We follow a similar method used by Roberts,8 who investigated the selectivity of porous catalysts for two parallel reactions. Assuming that the catalyst pellet can be represented as a semi-infinite slab of thickness 2L, he derived analytic solutions of the diffusion equation for three sets of reaction orders n ) 1, m ) 0; n ) 2, m ) 0; and n ) 2, m ) 1. His solutions were expressed in cosh functions and incomplete elliptic integrals of the first kind. The method used by Roberts can also be applied here to quantify the effects of radial transport limitations of oxygen from the membrane surface to the center of the packed bed, under the assumption that the hydrocarbons’ concentration gradients are small compared to the oxygen concentration gradient, together with the limiting situation of oxygen depletion in the center of the packed bed. It is assumed that the catalyst bed can be represented by an isothermal and isobaric, semi-infinite slab (of thickness 2L) and
(A.1)
1 - 〈σ〉 ) F1-σ ) 1 - σ(〈c〉)
(A.9)
∫R1γm d l 1 n-m (∫ γ d l ) ∫R1γn d l R
(A.10)
recalling that
∫R1γm d l and σ(〈c〉) ) 〈σ〉 ) 1 - pnm 1 ∫R γn d l
∫R1γ d l )m 1 - pnm 1 (∫R γ d l )n (
(A.11, A.12)
Integration of eq A.3 yields for the limiting case of pnm f 0
∫
1
1 n+1 φn 2
1/2
( ) ∫ γ d l ) φ1 (n +2 1) γ dl ) R
∫
1
R
0
1/2
1 n
R
1 n+1 φn 2
(
∫
x2 (n + 1) φn 3 - n
(A.13)
γ(n-1)/2 dγ )
x2 1 φn (n + 1)1/2
(A.14)
1
0
n
γm d l )
∫
γ(1-n)/2 dγ )
1/2
1
)
1/2
∫
1
0
γ(2m-n-1)/2 dγ )
1/2 x2 (n + 1) φn 2m - n + 1 (A.15)
Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7533
where use has been made of the boundary conditions: γ(R) ) 0 and γ(1) ) 1. Rewriting the Thiele modulus φn in terms of the modified Thiele modulus φ′′
L
φn ) φ′′ L
x
kncn-1 L D
) n+1 m+1 kn〈c〉n-1 + km〈c〉m-1 2 2 D -1/2 n + 1 n-1 m + 1 〈γ〉 + pnm〈γ〉m-1 (A.16) 2 2
x
(
)
which reduces in the limit of pnm f 0 to
φn ) φ′′
(n +2 1)
1/2
〈γ〉(1-n)/2
(A.17)
After substitution of eq A.13, we find
φ ) (φ′′)2/(3-n)
(n +2 1) (3n -+ n1) 1/2
(1-n)/(3-n)
(A.18)
so that
Greek Letters γ ) dimensionless concentration () c/cR) ) porosity ν ) stoichiometric constant F ) density (kg/m3); dimensionless radial coordinate () r/R) σ ) selectivity τ ) tortuosity Φm ) mass flow rate (kg/s) φm,distr ) specific mass flow rate added per volume packed bed (kg/m3 s) ΦV ) volumetric flow rate (m3/s) ω ) mass fraction Subscripts
∫R γ dl ) (φ′′) 1
-2/(3-n)
3 - n -2/(3-n) n+1
(
)
∫R1 γndl ) (φ′′)-2/(3-n)(n3 +- 1n)
(1-n)/(3-n)
∫R1 γm dl ) 2mn-+n1+ 1 (φ′′)-2/(3-n)(n3 +- 1n)
(A.19) (A.20)
(1-n)/(3-n)
(A.21) After substitution in eq A.9 and A.10, we finally yield
∫R1γn dl 3 - n (n+1)/(3-n) FR,n ) 1 ) (φ′′)-2(1-n)/(3-n)( n + 1) (∫R γ dl )n F1-σ )
R ) radius (m) rj′ ) rate of reaction j (mol kgcat-1 s-1) S ) selectivity [-]; reaction source term (kg m-3 s-1) t ) time (s) T ) temperature (K) u ) superficial gas velocity (m/s) V ) gas velocity (m/s) X ) conversion y ) mol fraction z ) axial coordinate (m)
(A.22)
3 - n -2(n-m)/(3-n) n+1 (φ′′)-2(n-m)/(3-n) 2m - n + 1 n+1 (A.23)
(
)
Notation A,P,W ) hydrocarbon reactant, target, and waste product c ) concentration (mol/m3) d ) diameter (m) D ) diffusion or dispersion coefficient (m2/s) FR,n ) relative conversion rate of main reaction (defined in eq 8) F1-σ ) relative selectivity toward the waste product (defined in eq 9) k ) rate constant L ) length of the packed bed (m) m ) reaction order in oxygen of reaction forming the waste product M ) molar mass (kg/mol) mcat ) catalyst mass (kg) n ) reaction order in oxygen of reaction forming the target product nr ) number of reactions p ) pressure (bar) r ) reaction rate (mol m-3 s-1); radial coordinate (m)
0 ) at the reactor inlet/base case A,P,W ) hydrocarbon reactant, target and waste product av ) average b ) bulk of the gas phase cat ) catalyst distr ) distributed eff ) effective m ) molecular M ) at the membrane wall (i.e., at r ) R) p ) particle r ) radial R ) at the membrane wall (i.e., r ) R) t ) tube z ) axial Dimensionless Numbers φ ) Thiele modulus; φ ) L xk1cn-1 b /D φ′ ) modified Thiele modulus )
L x(((n + 1)/2)ν1k1cAbcn-1 + ((m + 1)/2)ν2k2cPbcm-1 ) b b
Deff,p with L ) Rp/3 for spherical catalyst particle φ′′ ) modified Thiele modulus )
L x((n+1)/2)ν1k/1〈c〉n-1+((m+1)/2)ν2k/2〈c〉m-1
Dr with L ) Rt/2 for tubular packed bed pnm ) ratio of reaction rates in the secondary and primary reaction; pnm ) (k2〈cP〉/k1〈cA〉)〈c〉m-n Literature Cited (1) Tsotsas, E.; Schlu¨nder, E. U. Some remarks on channeling and on radial dispersion in packed beds. Chem. Eng. Sci. 1988, 43, 1200. (2) Ku¨rten, U. Ph.D. Thesis, University of Twente, Enschede, The Netherlands, 2003. (3) Ku¨rten, U.; van Sint Annaland, M.; Kuipers, J. A. M. Oxygen distribution in packed-bed membrane reactors for partial oxidations: Effect of the radial porosity profiles on the product selectivity. Ind. Eng. Chem. Res. 2004, 43, 4753.
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Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007
(4) Ku¨rten, U.; van Sint Annaland, M.; Kuipers, J. A. M. Oxygen distribution in packed bed membrane reactors for partial oxidation systems and the effect on the product selectivity. Int. J. Chem. React. Eng. 2004, 2, A24. (5) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The properties of gases and liquids; McGraw-Hill: New York, 1988. (6) Hou, K.; Hughes, R.; Ramos, R.; Mene´ndez, M.; Santamarı´a, J. Simulation of a membrane reactor for the oxidative dehydrogenation of propane, incorporating radial concentration and temperature profiles. Chem. Eng. Sci. 2001, 56, 57.
(7) Ramos, R.; Mene´ndez, M.; Santamarı´a, J. Oxidative dehydrogenation of propane in an inert membrane reactor. Catal. Today 2000, 56, 239. (8) Roberts, G. W. The selectivity of porous catalysts: parallel reactions. Chem. Eng. Sci. 1972, 27, 1409.
ReceiVed for reView May 7, 2007 ReVised manuscript receiVed June 13, 2007 Accepted June 19, 2007 IE070639C
