Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor
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 pos...
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| Format: | Article |
| Language: | English |
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Igor Sikorsky Kyiv Polytechnic Institute
2018-03-01
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| Series: | Наукові вісті Національного технічного університету України "Київський політехнічний інститут" |
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| Online Access: | http://bulletin.kpi.ua/article/view/105885 |
| _version_ | 1818610537835003904 |
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| author | Igor D. Lucheyko Igor I. Lucheyko Roman V. Kotsiurko |
| author_facet | Igor D. Lucheyko Igor I. Lucheyko Roman V. Kotsiurko |
| author_sort | Igor D. Lucheyko |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-16T15:16:01Z |
| format | Article |
| id | doaj.art-67830e379f8e40e2a9c6d060f999ffeb |
| institution | Directory Open Access Journal |
| issn | 1810-0546 2519-8890 |
| language | English |
| last_indexed | 2024-12-16T15:16:01Z |
| publishDate | 2018-03-01 |
| publisher | Igor Sikorsky Kyiv Polytechnic Institute |
| record_format | Article |
| series | Наукові вісті Національного технічного університету України "Київський політехнічний інститут" |
| spelling | doaj.art-67830e379f8e40e2a9c6d060f999ffeb2022-12-21T22:26:48ZengIgor Sikorsky Kyiv Polytechnic InstituteНаукові вісті Національного технічного університету України "Київський політехнічний інститут"1810-05462519-88902018-03-0101889610.20535/1810-0546.2018.1.105885105885Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow ReactorIgor D. Lucheyko0Igor I. Lucheyko1Roman V. Kotsiurko2Ternopil Ivan Puluj National Technical UniversityTernopil Ivan Puluj National Technical UniversityTernopil Ivan Puluj National Technical UniversityBackground. 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.http://bulletin.kpi.ua/article/view/105885Mathematical modelingPlug flow reactorConsecutive irreversible reactionDeactivation of solid catalystCatalyst lifetime |
| spellingShingle | Igor D. Lucheyko Igor I. Lucheyko Roman V. Kotsiurko Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor Наукові вісті Національного технічного університету України "Київський політехнічний інститут" Mathematical modeling Plug flow reactor Consecutive irreversible reaction Deactivation of solid catalyst Catalyst lifetime |
| title | Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor |
| title_full | Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor |
| title_fullStr | Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor |
| title_full_unstemmed | Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor |
| title_short | Analytical Determination of Rational Catalyst Lifetime for Consecutive Reaction in Isothermal Plug Flow Reactor |
| title_sort | analytical determination of rational catalyst lifetime for consecutive reaction in isothermal plug flow reactor |
| topic | Mathematical modeling Plug flow reactor Consecutive irreversible reaction Deactivation of solid catalyst Catalyst lifetime |
| url | http://bulletin.kpi.ua/article/view/105885 |
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