Comparison of two models of the dimer–dimer reaction on composite catalyst

Abstract. In this paper two mathematical models of the dimer–dimer reaction on supported catalysts are compared. The PDE model describe the bulk diffusion of both reactants and product, adsorption and desorption, the surface diffusion of adsorbates and intermediate. The ODE model is derived from the first, assuming that all materials are well-mixed. The catalytic reactivity, calculated using the PDE and ODE models, is compared for two variants of reactants adsorption and variuos bulk and surface diffusion.


Introduction
Adsorption is an essential first step in heterogeneous catalysis when a molecule in the gas phase or in solution binds to atoms on the solid surface. The molecule that is binding is called the adsorbate and the surface to which it binds is the adsorbent. In heterogeneous catalysis the reactants diffuse to the catalyst surface and adsorb onto it via the formation of chemical bonds. A part of the adsorbates split from the adsorbent. The product molecules desorb and diffuse away from the catalyst surface.
The surface of catalyst can consist of small active catalyst particles affixed on solid material which is inactive in reaction but active in adsorption process. The spillover, when adsorbates can migrate from the catalyst particle to support and from the support to catalyst particle, has an important influence on catalytic reactions proceeding on supported catalyst. Parts of the catalyst surface that are inactive in the surface reaction are active for adsorption and desorption process. Spillover increases or decreases the concentrations of adsorbates or product molecules on active parts of the catalyst surface [1,2,5].
The dimer-dimer heterogeneous surface reaction proceeding on supported catalyst was examined in [5] using a rather complicated mathematical model of parabolic PDEs. One part of partial differential equations was solved in 2D domain. Another part of PDEs was determined on a boundary of the domain. Two-molecular reaction with spillover effect was investigated in [4] assuming that materials are well-mixed.
The purpose of this study was to compare the results obtained using two models: (a) the PDE model given in [5], which takes into account the bulk diffusion of both reactants and the reaction product and the surface diffusion of the adsorbed molecules of each reactant; (b) the simplified model based on ODEs if all species are well-mixed.
The dimer-dimer heterogeneous surface reaction that occurs on a composite catalyst, , is examined, provided that the reactants and reaction product are well-mixed. The following is the reaction scheme, where S is the vacant adsorption site, A 2 and B 2 are reactants, and P = A 2 B is the product of the reaction of A 2 and B 2 : Here AS, BS and ABS are adsorbates of reactants A, B and intermediate AB. Let reactants A, B and product P fill in the domain Ω with surface ∂Ω =S ∪S. The catalyst surfaceS is composed from two active and inactive in reaction regions S 1 and S 2 . Let k 1i , k 2i be the adsorption and k −1i , k −2i the desorption rates constants for inactive site (i = 1) and for active one (i = 2) of reactants A and B. k 3 is the reaction between adsorbates AS and BS rate constant, k 4 is the reaction between adsorbate AS and intermediate ABS rate constant, and k 5 is the conversion rate constant of ABS into product P .
The mathematical model of the surface reaction proceeding on a composite catalyst is written using the Langmuir-Hinshelwood kinetics and the surface diffusion mechanism based on the particle jumping into a nearest vacant adsorption site [3].
Assume that s 1 , s 2 are densities of the inactive and active in reaction adsorption sites of strips S 1 and S 2 . In case where densities s 1 , s 2 depend only on variable x 1 and the initial concentrations of reactants are constants the 3D in space problem can be reduced in the 2D problem (see [5]). Let x * ∈ (0, 1) be the catalyst-support interface then S 2 = [0, x * ) and S 1 = (x * , 1] are domains consisting of the active and inactive in reaction adsorption sites.
Assume that u 12 , u 22 , u 32 are densities of the active in the surface reaction sites occupied by the adsorbed molecules of reactants A, B and particles of intermediate AB. Similarly, u 11 , u 21 , u 31 denote densities of particles of adsorbates AS, BS and ABS bound to the inactive in reaction sites. Differencies s 1 − u 11 − u 21 − u 31 and s 2 − u 12 − u 22 − u 32 are densities of the vacant adsorption sites of strips S 1 and S 2 .
Let λ 2,11 , λ 2,21 and λ 2,31 be the constants of the jump rates via the catalystsupport interface x * of particles of AS, BS and ABS from the inactive position x * + 0 into the nearest-neighbour free active site x * − 0. Similarly, λ 1,12 , λ 1,22 and λ 1,32 present the constants of the jump rates via x * of particles of adsorbates AS, BS and ABS from the active position x * − 0 into the nearest-neighbour vacant inactive site x * + 0.
A simplified mathematical model based on the ODEs is derived from the PDE model given in [5] using the averaging procedure. Concentrations a(t), b(t) and p(t) of reactants A 2 , B 2 and their reaction product P = A 2 B, respectively, are solutions P. Katauskis of the system: System (1) is solved together with the following equations to find the unknown values of densities u ij :  From (1)-(7) it follows two mass conservation laws: In this paper the turn-over rate -the catalyst surface S 2 specific conversion rate of the reactants molecules into product molecules -is studied. This time function is defined by the formula z = k 5 u 2 32 + k 4 u 12 u 32 /s 2 .

Numerical results
An implicit difference scheme based on the alternating direction method was used to solve the PDE problem. To get the numerical solution of systems (1)-(7) the standard MATLAB ODE solver ode45 was applied. The following values of parameters were used in calculations: k ij = 0.017, k −ij = 0.0017, i, j = 1, 2; k 3 = 0.1; k 4 = 0.1; k 5 = 0.1; λ j+1,i j = λ j,i j+1 = 0.1, i = 1, 2, 3, j = 1; a 0 1 = a 0 2 = 1. Assume that κ B =: κ a = κ b = κ p and if values of κ ij do not depend on the values of indices we use κ S for the PDE model from [5]. In the calculations laws (8) were satisfied with 10 −6 accuracy for both models.
Calculation the values of the turnover rate z(t) for t ∈ [0, 1500] s using the PDE model requires approximately 180 times more the computer time than calculation these values when the ODE model is applied.
Two variants of reactants adsorption are studied: both reactant can adsorb on the active in reaction catalyst surface and inactive in reaction support; both reactants adsorb only on the support. Fig. 1(a) demonstrates the influence of the variation of the bulk diffusion coefficients of both reactants and product in the case where the reactants A and B can adsorb only on the support, i.e. k 12 = k 22 = 0. The surface reaction occurs only due to spillover. The turn-over rate z(t) grows faster, reaches a large maximum, and approaches the curve z(t) calculated using the ODE model if the diffusion coefficients κ B increase. The maximal values of z(t) are given in Table 1. We observe that curves 1 and 2 practically not differ from the ODE line for t > 300 if κ B = 0.1 or 0.01 and κ S = 0.1. The catalytic reactivity calculated by the PDE and ODE models differs significantly if the bulk diffusion is small (curve 3 with κ B = 0.001). The maximum values of z(t), corresponding to the bulk diffusion coefficients 0.1, 0.01, 0.001, are 95.96, 92.65 and 64.71 percent of the maximum z(t) determined by the ODE model.
The effect of the surface diffusivity κ S of adsorbates AS, BS and particles of intermediate ABS on the turn-over rate is more noticeable (see Fig. 1(a) and Table 1) although the qualitative behavior of z(t) is similar. Plots in Fig. 1(a) show noticeable differences between maximal values of z(t) corresponding to the surface diffusivity 0.01, 0.005 and the fixed bulk diffusivity κ B = 0.1. In this case the maximum values Table 1. Maximal values of the turn-over rate z(t) of PDE and ODE models in case k 12 = k 22 = 0, k 11 = k 21 = 0.017.  of z(t), corresponding to the surface diffusion coefficients 0.01 and 0.005, equal to 75.37 and 63.6 percent of the maximum z(t) determined by the ODE model. We observe a qualitatively similar dependence of the catalytic reactivity on the bulk diffusion of reactants and the product, and the surface diffusion of adsorbates and intermediate when the reactants can adsorb on both active and inactive intervals ( Fig. 1(b)). The maximal values of z(t) are written in Table 2. In this case, the differences between curves calculated using the PDE and ODE models are smaller. Now the maximum values of z(t) are 97.61, 95.23, 73.74 percent for the bulk diffusivity 0.1, 0.01, 0.001, and 90.19, 83.02 percent for the surface diffusivity 0.01 and 0.005 of the maximum z(t) determined by the ODE model.