Effect of Diffusion Limitations on the Selectivity of Parallel Heterogeneous Reactions
Abstract Generated abstract
This paper examines how external diffusion limitations influence the selectivity of parallel heterogeneous reactions, particularly when the competing reactions have different kinetic orders. Using a macrokinetic formulation that relates surface reaction rates to near-surface reactant concentrations, diffusion-rate constants, flow velocity, and temperature, the authors derive expressions for selectivity in kinetic and external-diffusion regimes. The analysis indicates that, unlike parallel reactions of identical order, reactions of different orders can show selectivity changes caused by mass-transfer conditions, with the effect increasing as the difference in reaction order grows. A conditional gas-phase estimate suggests that for reactions differing in order by one, a thirtyfold change in flow velocity may change selectivity by about 30 percent.
Full Text
EFFECT OF DIFFUSION COMPLICATIONS ON THE SELECTIVITY OF PARALLEL HETEROGENEOUS REACTIONS
S. I. SMOLYANINOV, V. M. MIRONOV
(Submitted by the organic section of the KhTF seminar)
The macrokinetic region in which parallel heterogeneous reactions proceed can determine their selectivity.
In the kinetic region, the selectivity of such reactions depends only on the external conditions affecting changes in the rates of the reactions being compared. These include temperature, pressure, and the concentration of the reactants.
In the region of internal diffusion, the question of the dependence of the selectivity of parallel reactions on the conditions of diffusive transport of reactants was analyzed by E. Wheeler [1]. He showed that changing the diffusion conditions of reactants for reactions of the type
\[ A \to B \]
\[ A \to C \]
affects the selectivity of these reactions only when their kinetic orders differ; that is, diffusion complications inside the catalyst pores more strongly retard the rate of the slower reaction.
In the external-diffusion region, the dependence of process selectivity on the diffusion conditions of reactants has been determined only as applied to first-order reactions [2, 3, 4, 5, 6]. In this case, the selectivity of the process is in no way determined by mass-transfer conditions, since here the rates of the parallel reactions are determined to the same extent by diffusion of the reactants.
With regard to the extent to which the selectivity of parallel reactions of different orders changes with the rate of diffusion of the reactants, we have found no studies, apart from a brief remark by W. A. Reuter and C. C. Gaughman [4], in which the supposition was expressed that this influence is insignificant.
Let us attempt to evaluate how the selectivity of parallel heterogeneous reactions of different kinetic orders changes with the conditions of external diffusion, which is determined mainly by the rate of reactant flux. We write the equations of the investigated, simultaneously occurring, parallel heterogeneous reactions in the following general form:
\[ A_o \xrightarrow{\beta_A} mA_{\text{п}} \xrightarrow{k_A} B, \]
\[ C_o \xrightarrow{\beta_C} nC_{\text{п}} \xrightarrow{k_C} D. \]
$A_o$ and $C_o$ are the initial components of the reaction in the bulk;
$A_{\mathrm{p}}$ and $C_{\mathrm{p}}$ are the same components at the phase-interface surface;
$\beta_A$ and $\beta_C$ are the diffusion-rate constants of the initial substances;
$k_A$ and $k_C$ are the rate constants of the surface reactions;
$m$ and $n$ are the kinetic orders of the reactions under study.
Let us assume that the rate of sorption–desorption of the reaction components is much higher than the rate of the chemical reaction itself. If the reaction rates depend only on the concentrations of the initial substances (the reaction is practically irreversible), then we may write:
\[ W_A = k_A [A_{\mathrm{p}}]^m, \]
\[ W_C = k_C [C_{\mathrm{p}}]^n, \]
where $W_A$ and $W_C$ are the rates of the surface reactions;
$[A_{\mathrm{p}}]$ and $[C_{\mathrm{p}}]$ are the surface concentrations of the reactants.
Defining selectivity by the ratio of the true rates of the reactions being compared, we obtain:
\[ \xi = \frac{W_A}{W_C} = \frac{k_A}{k_C} \cdot \frac{[A_{\mathrm{p}}]^m}{[C_{\mathrm{p}}]^n}, \]
from which it is clear that the selectivity of the process in this case is determined both by the ratio of the rate constants of the reactions being compared and by the ratio of the concentrations of the initial substances at the phase-interface surface. And since the reaction-rate constants are determined only by the process temperature, while the surface concentrations are proportional to the diffusion-rate constants of the corresponding reactants, and also
\[ \beta_i = f(D_i;\, v), \]
where $D_i$ is the diffusion coefficient of the $i$-th component;
$v$ is the velocity of the reactant flow, then
\[ \xi = \varphi\left(T;\, \frac{D_A}{D_C};\, v^a\right); \]
$a$ is a constant depending on the orders of the reactions being compared and on the hydrodynamic regime in the apparatus.
The expression obtained is characteristic of the region of external diffusion and should hold even if it is assumed that the concentrations of the initial substances on the active surface are much smaller than those in the reaction volume.
In the kinetic region,
\[ [A_{\mathrm{p}}] \simeq [A_o] \quad \text{and} \quad [C_{\mathrm{p}}] \simeq [C_o], \]
therefore
\[ \xi = \varphi(T;\, [A_o];\, [C_o]), \]
which corresponds to the statement made earlier. In the most frequently encountered case
\[ A_o \begin{cases} \xrightarrow{\beta_A}\; m A_{\mathrm{p}} \xrightarrow{k_1} B,\\ \xrightarrow{\beta_A}\; n A_{\mathrm{p}} \xrightarrow{k_2} C, \end{cases} \]
according to the above, we have:
\[ \xi = \frac{k_1}{k_2}\cdot [A_{\mathrm{p}}]^{m-n}. \]
Reactions of a more complex type can also lead to a similar type of dependence, for example,
\[ A_0 \begin{array}{c} \nearrow mA_{\mathrm{p}} + B_{\mathrm{p}} \to C \\ \searrow nA_{\mathrm{p}} + B_{\mathrm{p}} \to E \end{array} , \]
that is, differing in order with respect to only one component. Analogously to the preceding, in the external-diffusion region
\[ \xi = \varphi[T; v^{b(m-n)}], \]
where \(b\) is a constant determined by the hydrodynamic regime in the apparatus. In the kinetic region, for this type of reaction,
\[ \xi = \varphi(T). \]
In the case where the orders of the reactions being compared are identical, that is, \(m=n\), then
\[ v^{b(n-n)} = 1, \]
and in the region of external diffusion
\[ \xi = \varphi(T), \]
which indicates the independence of the selectivity of parallel reactions in this particular case from diffusion conditions and its invariance upon transition from one limiting region to another. In particular, for \(m=1\) and \(n=1\) an analogous conclusion was drawn in the works cited above [2, 3, 4, 5, 6].
For gas reactions the selectivity of the process may be expressed as follows:
\[ \xi = \frac{k_1'}{k_2'} \cdot p_{A_{\mathrm{p}}}^{\,m-n}. \]
\(p_{A_{\mathrm{p}}}\) is the near-surface partial pressure of component \(A\).
\[ p_{A_{\mathrm{p}}} = p_A - \Delta p_A, \qquad \text{where} \]
\(p_A\) is the partial pressure of component \(A\) in the bulk;
\(\Delta p_A\) is the difference between the partial pressures of component \(A\) in the bulk and at the catalytic surface.
\[ k_1' = \frac{k_1}{(RT)^m} = \mathrm{const}; \qquad k_2' = \frac{k_2}{(RT)^n} = \mathrm{const}. \]
Having calculated \(\Delta p_A\) by the formula proposed by G. K. Boreskov [7], it can be shown, on a conditional model, in a first approximation, that the selectivity of two parallel reactions proceeding in the diffusion region and differing in order by unity changes by 30% when the flow velocity changes by a factor of 30.
Thus, on the basis of the foregoing, it may be concluded that the selectivity of two parallel reactions of different kinetic orders proceeding in the region of external diffusion depends on the conditions of diffusive transport of the reactants and is greater the larger the difference between the orders of the reactions being compared.
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