Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor

Abstract

Background. The urgency of mathematical modeling of continuous chemical-technological processes in non-stationary con­ditions caused by the actions of various destabilizing factors is undoubted. In this case, analytical solutions have undeniable advantages over numerical ones, since they make it possible to clarify the nature of the cause-effect relationships in the properties of the modeling object under consideration and, as a practical result, to give physically grounded recommendations for increasing the efficiency of its functioning. For catalytic processes, the reason for non-stationarity is the deactivation of the solid catalyst (Kt). This leads to a decrease of the conversion degree \[{x_ \bullet } = 1 - {c_{1 \bullet }}\]  of the A₁ reagent, and hence to a negative increase of its concentration \[{c_{1 \bullet }}\] in the reaction mixture at the outlet of the plug flow reactor (PFR) of length \[{L_{0 \bullet }},\] which leads to economic losses. Therefore, a rational (maximally expedient) catalyst lifetime \[{\theta _{\max }} > > 1\] has fundamental importance and is a significant part of the individual problem of selecting Kt.

Objective. The aim of the paper is analytical calculation of the maximally expedient lifetime \[{\theta _{\max }} = {\tau _{\max }}/{\tau _{L \bullet }}\] of industrial Kt at the point \[{x_{0 \bullet }}\] of maximum of the nominal yield \[{\eta _{02 \bullet }} \equiv \eta _{02}^{\max }\]  of the product A₂ for the isothermal system “PFR at the optimum resi­den­ce time \[{\tau _{L \bullet }} = {L_{0 \bullet }}/{u_0}\] of the reactants + consecutive catalytic reaction \[\mathrm{A}_{1}\xrightarrow[\mathrm{Kt},k_{\mathrm{d}1}]{k_{01},n_{1}=1}\alpha _{2}\mathrm{A}_{2}\xrightarrow[\mathrm{Kt},k_{\mathrm{d}2}]{k_{02},n_{2}=1}\alpha _{3}\mathrm{A}_{3}"\] under the influence of the Kt deactivation destabilizing factor.

Methods. A linearized mathematical model in the form of a system of ordinary differential equations of characteristics for calculating the relatively small Kt deactivation influence on the system operating mode stationarity has been used.

Results. In the case of the first-order reaction for the conditions of the industrial Kt deactivation, the relative deviations \[\left | \varepsilon _{x\bullet } \right |=\left | (x_{\bullet }/x_{0\bullet })-1 \right |\sim k_{\mathrm{d}1}\tau _{\mathrm{max}}< < 1\] of the degree of A₁ conversion, the relative deviations \[|{\varepsilon _{\eta 2\bullet }}|\, \sim {k_{{\text{d1}}}}\tau_{max}\] of the A₂ yield and the relative deviations \[\varepsilon _{s2\bullet }\sim {k_{{\text{d1}}}}\tau_{max}\] of selectivity \[{s_{2 \bullet }} = {\eta _{2 \bullet }}/{x_ \bullet }\] from the nominal values are analytically calculated in the linear appro­ximation. It is established that the magnitudes \[{\varepsilon _{x \bullet }} < 0,\] \[|{\varepsilon _{\eta 2 \bullet }}|\geqslant 0\] and \[{\varepsilon _{s2 \bullet }} > 0\] are determined by the simplex \[\gamma _{0k}=k_{01}/k_{02}\]  of the nominal rate constants and by the simplex \[\gamma _{\mathrm{d}}=k_{\mathrm{d}2}/k_{\mathrm{d}1}\] of the Kt deactivation rate constants of the reaction stages.

Conclusions. It is proved that with respect to the yield of A₂ product, there is a self-regulation effect \[({\varepsilon _{\eta 2 \bullet }} \approx 0)\] of the stationary mode at the condition of the equality \[\gamma _{\mathrm{d}}=1\] of the Kt deactivation rate constants. A nomogram for determining \[1 < <\theta _{max}< < (k_{\mathrm{d}1}\tau _{L\bullet })^{-1}\] on the maximum admissible value of \[\left | \varepsilon _{x\bullet } \right |_{max}^{\mathrm{adm}}< < 1.\] is calculated. For example, at a nominal degree of conversion \[{x_{0 \bullet }} = 75\% \Leftrightarrow {\gamma _{0k}} = 2\] of A₁ reagent and \[\left | \varepsilon _{x\bullet } \right |_{max}^{\mathrm{adm}}= 5\%\Rightarrow \theta _{max}\approx 1,1\cdot 10^{3}(k_{\mathrm{d}1}\tau _{L\bullet }=10^{-4}).\] It is shown that the rational catalyst lifetime \[{\theta _{\max }}\] is inversely proportional to the complex \[k_{\mathrm{d}1}\tau _{L\bullet }\] of the Kt deactivation rate constant of the first stage.