Modelling and Dynamic Simulation of One-Dimensional Isothermal Axial Dispersion Tubular Reactors with Power Law and Langmuir-Hinshelwood-Hougen Watson Kinetics

  • Almoruf Olajide Fasola Williams Chemical and Petroleum Engineering Department University of Laagos, Akoka, Lagos, Nigeria
Keywords: Isothermal tubular reactors, power-law kinetics, Langmuir-Hinshelwood-Hougen-Watson kinetics, orthogonal collocation, dynamic simulation


In this paper, the modelling, numerical lumping and simulation of the dynamics of one-dimensional, isothermal axial dispersion tubular reactors for single, irreversible reactions with Power Law (PL) and Langmuir-Hinshelwood-Hougen-Watson (LHHW)-type kinetics are presented. For the PL-type kinetics, first-order and second-order reactions are considered, while Michaelis-Menten and ethylene hydrogenation or enzyme substrate-inhibited reactions are considered for the LHHW-type kinetics. The partial differential equations (PDEs) developed for the one-dimensional, isothermal axial dispersion tubular reactors with both the PL and LHHW-type kinetics are lumped to ordinary differential equations (ODEs) using the global orthogonal collocation technique. For the nominal design/operating parameters considered, using only 3 or 4 collocation points, are found to adequately simulate the dynamic response of the systems. On the other hand, simulations over a range of the design/operating parameters require between 5 to 7 collocations points for better results, especially as the Peclet number for mass transfer is increased from the nominal value to 100. The orthogonal collocation models are used to carry out parametric studies of the dynamic response behaviours of the one-dimensional, isothermal axial dispersion tubular reactors for the four reaction kinetics. For each of the four types of reaction kinetics considered, graphical plots are presented to show the effects of the inlet feed concentration, Peclet number for mass transfer and the Damköhler number on the reactor exit concentration dynamics to step-change in the inlet feed concentration. The internal dynamics of the linear (or linearized) systems are examined by computing the eigenvalues of the linear (or linearized) lumped orthogonal collocation models. The relatively small order of the lumped orthogonal collocation dynamic models make them attractive and useful for dynamic resilience analysis and control system analysis/design studies.


Pereira, Carmo J. & Leib, T.M. (2019). Section 19. Reactors. In: Green, Don W., Southard, M.Z., editor. Perry’s Chemical Engineers’ Handbook; 9th. ed. New York, USA: McGraw-Hill.

Levenspiel, O. (1999). Chemical reaction engineering. 3rd. ed. Wiley. ISBN 9780471254249.

Davis, M.E. & Davis, R.J.; (2003). Fundamentals of chemical reaction engineering, McGraw-Hill. ISBN 0486488551.

Fogler, H.S. (2016). Elements of chemical reaction engineering. 5th ed. Prentice Hall. ISBN 0133887510.

Coker, A. (2001). Modeling of chemical kinetics and reactor design. Gulf Professional Pub. ISBN 9780884154815.

Bahadori, A. (2012). Prediction of axial dispersion in plug-flow reactors using a simple method. Journal of Dispersion Science and Technology, 33(2), 200–205. doi:10.1080/01932691.2011.561159.

Ray, W.H. (1981). Advanced process control, McGraw-Hill Companies.

Butt, J.B. (2000). Reaction kinetics and reactor design, Marcel Dekker. ISBN 9780824777227.

Zhou, W., Hamroun, B., Gorrec, Y.L. & Couenne, F. (2015). Dissipative boundary control systems with application to an isothermal tubular reactor. IFAC-PapersOnLine, 28(13):150–153. doi:10.1016/j.ifacol.2015.10.230.

Li, H.X. & Qi, C. (2010). Modeling of distributed parameter systems for applications - A synthesized review from time-space separation. Journal of Process Control, 20(8), 891–901. doi:10.1016/j.jprocont.2010.06.016.

Finlayson, B.A. (1972). The method of weighted residuals and variational principles: with application in fluid mechanics, heat and mass transfer, Academic Press. ISBN 9780080955964.

Finlayson, B.A. (2003). Nonlinear analysis in chemical engineering, Ravenna Park Pub. ISBN 096317651X.

Villadsen, J., Michelsen, M.L. (1978). Solution of differential equation models by polynomial approximation. Prentice-Hall.

Dochain, D., Babary, J.P., Tali-Maamar, N. (1992). Modelling and adaptive control of nonlinear distributed parameter bioreactors via orthogonal collocation. Automatica, 28(5), 873–883. doi:10.1016/0005-1098(92)90141-2.

G´omez, J.C., Viassolo, D.E. & Junco, S. (1999). Adaptive control of nonlinear distributed parameter systems. International Journal of Modelling and Simulation, 19(2), 194–204. doi:10.1080/02286203.1999.11759971.

Li, M. (2008). Dynamics of axially dispersed reactors. Industrial & Engineering Chemistry Research, 47(14), 4797–4806. doi:10.1021/ie800083e.

Alopaeus, V., Laavi, H. & Aittamaa, J. (2008). A dynamic model for plug flow reactor state profiles. Computers and Chemical Engineering, 32(7), 1494–1506. doi:10.1016/j.compchemeng.2007.06.025.

Petre, E. & Selisteanu, D. (2011). Model approximation and simulations of a class of nonlinear propagation bioprocesses. In: Numerical Analysis - Theory and Application. InTech, 648. doi:10.5772/24131.

Rachidi, S., Karama, A. & Channa, R. (2014). Modelling of a nonlinear distributed parameter bioreactor via Orthogonal Collocation method. In: International Conference on Multimedia Computing and Systems -Proceedings. IEEE Computer Society, 1046– 1050. doi:10.1109/ICMCS.2014.6911140.

Giwa, A. & Giwa, S.O. (2013). Application of Crank-Nicolson finite-difference method to the solution of the dynamic model of a reactor. International Journal of Advanced Scientific and Technical Research, 6(3), 613–623.

Williams, A.O.F. & Adeniyi, V.O. (2018). Design of modified IMC-based PID controllers for isothermal tubular reactors with axial mass dispersion and first-order reaction. NSChE Journal, 33(2), 54–54. URL:

Williams, A. & Adeniyi, V. (1996). A new method for the design of PID-Type controllers. IFAC Proceedings Volumes, 29(1), 6125–6130. doi:10.1016/s1474-6670(17)58663-2.

Conesa, J.A. (2020). Chemical reactors design: Mathematical modeling and applications, Wiley-VCH. ISBN 9783527823383.

Elhajaji, A., Barje, N., Serghini, A., Hilal, K. & Mermri, E.B. (2017). A spline collocation method for integrating a class of chemical reactor equations. International Journal of Nonlinear Analysis and Applications, 8(1), 69–80. doi:10.22075/IJNAA.2017.1653.1436.

Barjes, N., Hajaji, A.E., Serghini, A., Hilal, K. & Mermri, E.B. (2020). A cubic spline collocation method for integrating a class of chemical reactor equations. Investigacion Operacional, 41(1), 54–66.

Dochain, D., Tali-Maamar, N. & Babary, J.P. (1997). On modelling, monitoring and control of fixed bed bioreactors. Computers and Chemical Engineering, 21(11), 1255–1266. doi:10.1016/S0098-1354(96)00370-5.

Cho, Y.S. & Joseph, B. (1983). Reduced-order steady-state and dynamic models for separation processes. Part I. Development of the model reduction procedure. AIChE Journal, 29(2), 261– 269. doi:10.1002/aic.690290213.

Danckwerts, P. (1953). Continuous flow systems: Distribution of residence times. Chemical Engineering Science, 2(1), 1–13. doi:10.1016/0009-2509(53)80001-1.

Sheel, J.G.P. & Crowe, C.M. (1969). Simulation and optimization of an existing ethylbenzene dehydrogenation reactor. The Canadian Journal of Chemical Engineering, 47(2), 183–187. doi:10.1002/cjce.5450470215.

Rase, H.F. (1990). Fixed-Bed Reactor Design and Diagnostics: Gas-Phase Reactions, Elsevier Science. ISBN 9781483162393.

Elnashaie, S.S.E.H. & Elshishini, S.S. (1993). Modelling, simulation, and optimization of industrial fixed bed catalytic reactors, Gordon and Breach Science Publishers. ISBN 2881248837.

Iordanidis, A. A. (2002). Mathematical modeling of catalytic fixed bed reactors (pp. 98-112). Enschede, The Netherlands: Twente University Press. ISBN 9036517524.

Elnashaie, S.S.E.H. & Garhyan, P. (2003). Conservation equations and modeling of chemical and biochemical processes (Google eBook, Marcel Dekker. ISBN 0824709578.

Li, Y., Rangaiah, G.P. & Ray, A.K. (2003). Optimization of styrene reactor design for two objectives using a genetic algorithm. International Journal of Chemical Reactor Engineering, 1(1). doi:10.2202/1542-6580.1013.

Yee, A.K., Ray, A.K. & Rangaiah, G. (2003). Multiobjective optimization of an industrial styrene reactor. Computers & Chemical Engineering, 27(1), 111–130. doi:10.1016/S0098-1354(02)00163-1.

Babu, B., Chakole, P.G. & Syed Mubeen, J. (2005). Multiobjective differential evolution (MODE) for optimization of adiabatic styrene reactor. Chemical Engineering Science, 60(17), 4822–4837. doi:10.1016/J.CES.2005.02.073.

Froment, G.F., De Wilde, J. & Bischoff, K.B. (2011). Chemical Reactor Analysis and Design. Wiley. ISBN 9780470565414.

Srinivasan, A., Depcik, C. (2013).One-dimensional pseudo-homogeneous packed-bed reactor modeling: I. Chemical species equation and effective diffusivity. Chemical Engineering and Technology, 36(1), 22–32. doi:10.1002/ceat.201200458.

Jakobsen, H.A.; (2014). Chemical Reactor Modeling: Multiphase Reactive Flows: Second Edition, Vol. 9783319050. 2nd ed., Springer International Publishing. doi:10.1007/978-3-319-05092-8.

Chabot, G., Guilet, R., Cognet, P., Gourdon, C. (2015). A mathematical modeling of catalytic milli-fixed bed reactor for Fischer-Tropsch synthesis: Influence of tube diameter on Fischer Tropsch selectivity and thermal behavior. Chemical Engineering Science, 127, 72–83. doi:10.1016/j.ces.2015.01.015.

Froment, G.F. (1974). Fixed bed catalytic reactors. technological and fundamental design aspects. Chemie Ingenieur Technik, 46(9), 374–386. doi:10.1002/cite.330460903.

Luyben, W.L.; 2007. Chemical Reactor Design and Control. John Wiley & Sons.

Carberry, J.J.; (1976). Chemical and Catalytic Reaction Engineering. New York: McGraw-Hill. ISBN 0486417360.

Panchenkov, G.M., Lebedev, V.B.; (1976). Chemical Kinetics and Catalysis. Translated from Russian.

Matsuura, T., Kato, M. (1967). Concentration stability of the isothermal reactor. Chemical Engineering Science, 22(2), 171–183. doi:10.1016/0009-2509(67)80009-5.

O’Neill, S., Lilly, M., Rowe, P. (1971). Multiple steady states in continuous flow stirred tank enzyme reactors. Chemical Engineering Science, 26(1), 173–175. doi:10.1016/0009-2509(71)86089-X.

Young, L.C. (2019). Orthogonal collocation revisited. Computer Methods in Applied Mechanics and Engineering, 345, 1033–1076. doi:10.1016/J.CMA.2018.10.019.

Martens, H.R. (1969). A comparative study of digital integration methods. SIMULATION, 12(2), 87–94. doi:10.1177/003754976901200207.

Moler, C., Van Loan, C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM Review, 20(4), 801–836. doi:10.1137/1020098.

Director, S.W., Rohrer, R. (1972). Introduction to Systems Theory. New York, USA: McGraw-Hill.

Williams, A.O.F., Adeniyi, V.O. (2001). Development of some Fortran 77 programs for linear system analysis. NSE Technical Transactions, 36(1), 68–80.

Gerald, C.F. (1978). Applied Numerical Analysis. 2nd. ed.; Reading, MA, Addisson-Wesley.

Lapidus, L.; (1962). Digital Computation for Chemical Engineers. McGraw-Hill.

Georgakis, C., Aris, R., Amundson, N.R. (1977). Studies in the control of tubular reactors-I general considerations. Chemical Engineering Science, 32(11), 1359–1369. doi:10.1016/00092509(77)85032-X.

How to Cite
Williams, A. O. F. (2021). Modelling and Dynamic Simulation of One-Dimensional Isothermal Axial Dispersion Tubular Reactors with Power Law and Langmuir-Hinshelwood-Hougen Watson Kinetics. Journal of Applied Science & Process Engineering, 8(2), 834-858.