Mathematical Modeling of Isotermal Plug Flow Reactor with Consecutive Reaction Taking into Account the Catalyst Deactivation

Abstract

Background. Mathematical modeling of continuous chemical-technological processes in non-stationary conditions of their implementation is an urgent problem. Solving specific problem numerically on a computer can only provide a formal adequacy of the model to the original. Consequently, the analytical solutions have undeniable advantages over numerical solutions. In the case of heterogeneous catalysis, deactivation of solid catalyst (Kt) takes place with a reduction of the process selectivity, which leads to economic losses. Therefore, the rational (the maximum-beneficial) catalyst lifetime \]{\theta _{\max }} >  > 1\] is the essential part of the problem of industrial Kt selection. 

Objective. The aim of this study is an analytic solution of the problem of operating mode description of the isothermal system  “PFR \[({\tau _L})\] + reaction \[{{\rm A}_1}\xrightarrow[{{\text{Kt}}{\text{,}}\,\,{k_{{\text{d1}}}}}]{{{k_{01}},\,\,{n_1} = 1}}{\alpha _2}{{\rm A}_2}\xrightarrow[{{\text{Kt}}{\text{,}}\,\,{k_{{\text{d2}}}}}]{{{k_{02}},\,\,{n_2} = 1}}{\alpha _3}{{\rm A}_3}\] + Kt \[({k_{{\text{d}}(i)}})\]” under influence of destabilizing factor of deactivation Kt and calculation of rational time \[{\theta _{\max }} = {\tau _{\max }}/{\tau _L}\] its exploitation.

Methods. The modified mathematical model for the calculation of influence deactivation of Kt on the system operation mode is used. Distinctive features of the model are: reactor has a variable length at a constant flow rate and has equal initial and boundary conditions. 

Results. The relative deviations \[|{\varepsilon _{\eta 2}}|\, \sim {k_{{\text{d1}}}}\tau < < 1\] of yield of product \[{\rm A}_2\] and \[{\varepsilon _{s2}} \sim {k_{{\text{d1}}}}\tau\] of the selectivity for conditions of the deactivation of industrial Kt in the linear approximation analytically are calculated. It was found that the magnitudes of \[{\varepsilon _{\eta 2}}\] and \[{\varepsilon _{s2}}\] are determined by the relation \[{\gamma _{\text{d}}}/{\gamma _{0k}}\] of the simplex \[{\gamma _{\text{d}}} = {k_{{\text{d}}2}}/{k_{{\text{d}}1}}\] of rate constants of Kt deactivation and of the simplex \[{\gamma _{0k}} = {k_{01}}/{k_{02}}\] of stages rate constants.

Conclusions. It is proved that with respect to the yield of \[{\rm A}_2\] the self-regulation effect \[({\varepsilon _{\eta 2}} = 0)\] of mode takes place. Nomogram for determining of \[1 < < {\theta _{\max }} < < {({k_{{\text{d1}}}}{\tau _L})^{ - 1}}\] by the maximum-admissible value of \[\varepsilon _{s2\,\max }^{{\text{adm}}} < < 1\] is calculated. For example, at the degree of the conversion \[{x_0} = 75\%\] of reagent \[{\rm A}_1\] and \[\varepsilon _{s2\,\max }^{{\text{adm}}} = 1\% ,\,\,{\gamma _{\text{d}}}/{\gamma _{0k}} = 1\,\,\, \Rightarrow\] \[{\theta _{\max }} \approx 1,2 \cdot {10^3}\,\,({k_{{\text{d1}}}}{\tau _L} = {10^{ - 5}})\]. The rational time \[{\theta _{\max }}\] of exploitation of Kt increases (approximately directly proportional) with a decrease of the simplexes relation \[{\gamma _{\text{d}}}/{\gamma _{0k}}\].