368 Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 3, 1978
COMMUNICATIONS
Effectiveness Factors for Oxidation Kinetics
Effectiveness factors 7 were evaluated for a type of intrinsic kinetics employed for the liquid-phase oxidation of organic compounds with porous catalyst particles. The results were obtained by conventional numerical solution of the differential mass balance equation and include the high Thiele number region where the concentration of limiting reactant is zero in the central core of the particle. Since this numerical solution requires extensive calculations, effectiveness factors were also obtained by approximation methods. The Bischoff method proved to be useful for obtaining approximate results (within 10% of the values of the more exact numerical solution). Orthogonal collocation, with two collocation points and using the reaction-zone concept, gave very good results (within 0.5%) and required but 1% of the computer time needed for the numerical solution. At high concentrations of oxidizable organic, the kinetics reduce to half order in oxygen concentration. Effectiveness factors were also evaluated in this region since results for half-order kinetics at large values of the Thiele modulus apparently have not been published.
Oxidation kinetics of liquid organic compounds often follow a half-order dependency on oxygen and a Langmuir-type dependency on concentration of organic (Goto et al., 1977). For example, this type of kinetics has been found to represent rates of oxidation of dilute, aqueous solutions of acetic acid using iron oxide (Levec and Smith, 1976) and copper-manganese-lanthanum oxide (Levec et al., 1976) catalysts in water-purification reactors. In these reactions where the pores of the catalyst particles are filled with liquid, the low diffusivities can cause significant diffusion resistance, even for small particles. Yet the available literature presenting effectiveness factors for complex kinetics (for example, Chu and Hougen, 1962;Roberts and Satterfield, 1965,1966; Schneider and Mitschka, 1965a,b, 1966) does not treat the oxidation case. Surprisingly, the solution for half-order kinetics, corresponding to large values of kzCA in eq 1,does not seem to be available except for relatively low values of the Thiele modulus, 4 h . The thesis of Volkman (1967) gives results when the reactant concentration, CB, is finite throughout the catalyst particle (&, < 6.5), but no information is available when CB 0 at a finite position within the particle. The objective of this paper is to provide effectiveness factors, 1, for liquid-phase oxidation kinetics, or for any irreversible reaction whose intrinsic kinetics follows the equation klCACB”2 RB = (1) 1 kzCA
-
+
and where B is the limiting reactant. The nonlinearity of eq 1requires that a numerical solution for 7 be used. Since such solutions necessitate considerable computer time, the accuracy of approximation methods was also evaluated. While oxidation in dilute aqueous solutions is likely to be isothermal, oxidation of pure liquid organics can lead to intraparticle temperature gradients. Hence, effectiveness factors are also presented for the nonisothermal case. Numerical Solution For an isothermal, spherical catalyst particle with a constant effective diffusivity @,&, the conservation equation for B within the particle may be written d2CB 2 d C ~ RB -++-p-=o dr2 r d r (De)~
If the stoichiometry of the reaction is A
+ bB
-
products
(3)
the conservation equation for A is d2CA 2dCA ---+--dr2 r d r
RB
p-=O
b(De)A
(4)
where the boundary conditions require that the concentration gradients of A and B are zero at the particle center and that ~ the particle radius. the concentrations are C A ,and ~ C B , at Then eq 2 and 4 can be solved to give the following relation between CA and CB
Equation 5 can be used with eq 1to express R B in terms of CB. Then eq 1 and 2 can be combined to yield one differential equation involving only CB and r as variables. From the solution of this single, differential equation, the effectiveness factor is obtained from the expression
(13)
( R B )is~ the reaction rate, from eq 1, evaluated a t outer surface conditions (Intermediate equations are in the Supplementary Material. See paragraph a t the end of the paper regarding this material.) The two-point boundary value problem described by the differential equation and boundary conditions was solved numerically using a fourth-order Runge-Kutta method (Carrahan et al., 1969). The solution procedure is available (Supplementary Material). Illustrative results are shown in Figures 1 and 2, which display the effectiveness factor as a function of the Thiele modulus 4 for various values of the dimensionless parameters a , @,and y (defined in Nomenclature). As a increases, the reaction kinetics beome independent of CAand half order in CB, with an equivalent rate constant of kllk2. Hence, the values of 7 in Figure 1 should approach the effectiveness factors for half-order kinetics as a becomes large. This test of the numerical solution was carried out by solving the same equations as before except that eq 1 is re-
0019-7882/78/1117-0368$01.00/0 0 1978 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 17,No. 3, 1978
369
1.0
0.8
F
.
0.6
