Chemical Engineering Journal 171 (2011) 655–668 Contents lists available at ScienceDirect Chemical Engineering Journal j ourna l ho mepage: www.elsevier.com/locate/cej Multiplicities of temperature wave trains in periodically forced networks of catalytic reactors for reversible exothermic reactions Erasmo Mancusi a,1 , Pietro Altimari b,∗ , Lucia Russo c,∗ , Silvestro Crescitelli d a Facoltà di Ingegneria, Università del Sannio, Piazza Roma, 82100 Benevento, Italy b Dipartimento di Ingegneria Chimica Alimentare Università di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA), Italy c CNR, Istituto delle Ricerche sulla Combustione, P.le Tecchio 80, Napoli 80125, Italy d Dipartimento d’Ingegneria Chimica Università “Federico II” Piazzale Tecchio 80, I-80125 Napoli, Italy a r t i c l e i n f o Article history: Received 10 January 2011 Received in revised form 4 April 2011 Accepted 13 April 2011 Keywords: Loop reactor Process intensification Pattern formation Waves train Periodically forced network Bifurcation analysis Multistability a b s t r a c t Networks of catalytic reactors with periodically switched inlet and outlet sections offer a competitive technological solution to the operation of reversible exothermic reactions. Traditionally, this operation mode is implemented by periodically shifting inlet and outlet sections so as to jump a single reactor unit in the flow direction. Here, a network of four catalytic reactors carrying on the methanol synthesis process is considered and the effect of varying the number (n s ) of reactor units jumped by inlet and outlet sections on network stability and performance is investigated. Increasing n s , a greater variety of periodic regimes giving rise to trains of temperature waves characterized by spatial periodicity are detected as the switching velocity varies. These regimes well reproduce the inter-stage cooling effect of multistage fixed bed reactors and, hence, guarantee in general large conversion values. Moreover, an intriguing coexistence between T-periodic and multi-periodic temperature wave trains is revealed, T being the period needed for the system to recover its initial configuration. A T-periodic symmetric wave train characterized by k waves always coexists with a number of k - 1 stable symmetric kT-periodic regimes, except when symmetry breaking is encountered. The k - 1 coexisting regimes correspond to wave trains with a number of waves ranging between 1 and k - 1. Bifurcational analysis is performed to characterize the stability range of periodic regimes and to systematically analyze multiplicities and bifurcations as the switching velocity is varied and at different n s . © 2011 Elsevier B.V. All rights reserved. 1. Introduction Networks (NTWs) of catalytic reactors with periodically switched inlet and outlet sections have been proved to offer an effective technological solution to achieve autothermal operation of exothermic catalytic processes [1]. The periodic variation of the NTW feeding sequence enables to trap an exothermic reaction front within the bed ensuring the possibility to operate at high conver- sion regimes even with streams characterized by very low adiabatic temperature rise. Since the NTW keeps unchanged the flow direc- tion, it also prevents the emission of unconverted reactants caused by the inversion of the flow in the RFR [1,2]. Moreover, the for- mation of a declining temperature zone close to the NTW outlet ∗ Corresponding authors. E-mail addresses: paltimar@unina.it (P. Altimari), lucrusso@unina.it (L. Russo). 1 Erasmo Mancusi is spending a period as Visiting Professor at the Universidade Federal de Santa Catarina. The actual adress is Departamento de Engenharia Química e Engenharia de Alimentos, Universidade Federal de Santa Catarina, Laboratório de Simulac ¸ ão Numérica de Sistemas Químicos, LABSIN, Campus Universitário Cx. P. 476, 88.040-900 Florianópolis (SC), Brazil. section makes this solution competitive as reversible exothermic reactions are considered, guaranteeing a significant increase in the average conversion due to achievement of more favorable ther- modynamic equilibrium conditions [3,4]. NTWs of reactors are periodically forced systems which exhibits spatio-temporal sym- metries [5]. As a consequence, if T is the system period, that is the time after which the NTW recovers the same feeding sequence, the expected regimes of the forced NTW are symmetric T-periodic. Under symmetric regime, each reactor of the NTW exhibits the same spatial profile only shifted of a time corresponding to the switch period. When the forced NTW is operated at switching to thermal veloc- ity ratios around unity, these regimes are characterized by a single travelling temperature waves [6]. The switching velocity range over which the NTW can sustain such regimes becomes progres- sively smaller as the adiabatic temperature rise and/or the feed temperature are decreased. Outside of this range, transitions to multi-periodic, quasi-periodic and chaotic solutions as well sym- metry breaking bifurcations can likely occur [7,8]. However, symmetric T-periodic regimes have been also shown to arise at switching to thermal front velocity ratios greater than 1385-8947/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cej.2011.04.026