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

Igor D. Lucheyko, Igor I. Lucheyko, Roman V. Kotsiurko

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.


Keywords


Mathematical modeling; Plug flow reactor; Consecutive irreversible reaction; Deactivation of solid catalyst; Catalyst lifetime

References


A.Yu. Zakgeym, Introduction to Modeling of Chemical-Technological Processes. Moscow, Russia: Khimiya, 1982 (in Russian).

K.R. Westerterp et al., Chemical Reactor Design and Operation. New York: Wiley, 1987. doi: 10.1002/aic.690320124

Yu.M. Zhorov, Kinetics of Industrial Organic Reactions. Moscow, Russia: Khimiya, 1989 (in Russian).

M.E. Davis and R.J. Davis, Fundamentals of Chemical Reaction Engineering. Boston: McGraw-Hill, 2003.

D.V. Evdokimov et al., “Analysis of development tendencies of modern mathematical and numerical modelling”, Visnyk DNU, Ser. Modeliuvannia, no. 8, pp. 3–17, 2009 (in Russian). doi: 10.15421/140901

U. Mann, Principles of Chemical Reactor Analysis and Design. Hoboken, New Jersey: John Wiley & Sons, 2009. doi: 10.1002/ 9780470385821

Catalysis: From Principles to Applications. M. Beller et al., eds. New York: Wiley-VCH, 2012. doi: 10.1002/anie.201210089

J.B. Rawlings and J.G. Ekerdt, Chemical Reactor Analysis and Design Fundamentals. Madison, Wisconsin: Nob Hill Publishing, LLC, 2012.

A. Rasmuson et al., Mathematical Modeling in Chemical Engineering. Cambridge, UK: MPG Print-group Ltd., 2014. doi: 10.1002/cite.201590043

N.N. Kulov and L.S. Gordeev, “Mathematical modeling in chemical engineering and biotechnology”, Theor. Found. Chem. Eng., vol. 48, no. 3, pp. 243–248, 2014. doi: 10.1134/S0040579514030099

A.K. Verma, Process Modelling and Simulation in Chemical, Biochemical and Environmental Engineering. Boca Raton: CRC Press, 2014. doi: 10.1002/cite.201690094

L.K Doraiswamy and D. Uner, Chemical Reaction Engineering. Beyond the Fundamentals. Boca Raton, London, New York: CRC Press, 2014.

S.R. Upreti, Process Modeling and Simulation for Chemical Engineers: Theory and Practice. New York: Wiley, 2017. doi: 10.1002/9781118914670.ch8

I.D. Lucheyko, “Deactivation of catalyst in system “consecutive reaction A1 ® a2A2 ® a3A3 + catalyst + mixing continuous reactor”, Naukovi Visti NTUU KPI, no. 6, pp. 145–151, 2012 (in Ukrainian).

I.D. Lucheyko, “Mathematical modeling system “mixing continuous reactor + reaction A1 ®a2A2”at conditions of catalyst deactivation”, Izvestiya Vuzov. Khimiya i Khim. Tekhnologiya, vol. 57, no. 12, pp. 88–92, 2014 (in Russian).

I.D. Lucheyko, “Criterion Damkohler as parameter defining mathematical description of non-stationary operating mode of CSTR”, in Proc. XX Mendeleev Congress on General and Applied Chemistry, vol. 3, Ekaterinburg, Russia, 2016.

I.D. Lucheyko, “Mathematical modeling of isothermal plug flow reactor with consecutive reaction taking into account the catalyst deactivation”, Naukovi Visti NTUU KPI, no. 5, pp. 106–115, 2016 (in Ukrainian). doi: 10.20535/1810-0546.2016.5.71920


GOST Style Citations






DOI: https://doi.org/10.20535/1810-0546.2018.1.105885

Refbacks

  • There are currently no refbacks.




Copyright (c) 2018 Igor Sikorsky Kyiv Polytechnic Institute

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.