POLITECNICO DI MILANO Scuola di Ingegneria Industriale e dell’Informazione Corso di Laurea in Ingegneria della Prevenzione e della Sicurezza nell’Industria di Processo UNSTEADY STATE RUNAWAY BOUNDARIES FOR PLUG FLOW REACTORS INVOLVING CATALYTIC OXIDATIONS Relatore: Ing. Sabrina COPELLI Co-relatore: Prof. Renato ROTA Tesi di Laurea di: Sofia CROCI Matr. 833778 Anno accademico 2015/2016 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 2 Ringraziamenti Questo lavoro è la sudata conclusione dei miei cinque anni al Politecnico, di un percorso che sembrava infinito e che invece è volato via in un soffio. Ciò che è accaduto in questi anni non riguarda solo la mia formazione e la mia carriera accademica, ma riguarda un’intera fase della mia vita, in cui ho lottato non solo per costruirmi un futuro professionale, ma anche per costruire me stessa come persona, come donna. Chi mi è stato accanto sa che sono stati anni duri, ma conosce anche la soddisfazione per i risultati che sono riuscita a raggiungere. Quello che forse non tutti conoscono, però, è la gratitudine che ho verso le persone che hanno riempito di gioia la mia vita e hanno reso possibile che io arrivassi fino a questo punto, evitandomi probabilmente un esaurimento nervoso. Tra queste persone ce ne sono alcune che vorrei ringraziare particolarmente: I miei insegnanti e le mie insegnanti, che mi hanno fatto appassionare prima alla matematica, poi alla chimica, e infine al settore a cui ho deciso di dedicare i miei studi e la mia vita: la Maestra Franca e la Maestra Augusta; la Prof.ssa Gioia, la Prof.ssa Doni, la Prof.ssa Rella, la Prof.ssa Bossi e la Prof.ssa Damiano; Francesco Maestri e il Professor Renato Rota. Doveroso ringraziare anche la mia famiglia, il cui supporto logistico, ma soprattutto morale, è stato il pilastro fondamentale che mi ha permesso di arrivare fino in fondo, e la cui pazienza e comprensione è stata degna di Giobbe. Siete stati la condizione necessaria senza la quale probabilmente non ce l’avrei mai fatta. L’elenco degli amici è troppo lungo per poter citare tutti per nome: amici che mi stanno accanto da lunghissimi anni e che mi hanno visto crescere; amici che negli ultimi mesi, nei momenti più difficili, hanno sopportato ore e ore di conversazioni e sfoghi; amici che, indirettamente e magari senza nemmeno accorgersene, mi hanno portato allegria e serenità in innumerevoli occasioni; i ragazzi e le ragazze della Compagnia degli Usberghi e della TMI, con cui ho condiviso esperienze preziosissimi a livello umano. Nella vita di ognuno c’è una persona con cui si crea un legame particolare, che va oltre le definizioni, oltre le distanze, oltre il tempo, che va anche oltre i ringraziamenti all’inizio di una tesi. Io sono molto fortunata, perché di queste persone nella mia vita ce ne sono addirittura due: Ana, la mia vita, la mia sorella, la mia metà; Alan, il mio compagno e il mio sostegno, il sole delle mie giornate. Non ci sono veramente parole per ringraziare a sufficienza tutte le persone che ho nominato, perciò mi limiterò ad aggiungere solo un’ultima cosa: grazie, al parallelepipedo. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 3 Table of Contents Ringraziamenti ..................................................................................................................................... 3 Table of Contents ................................................................................................................................. 4 List of Figures ...................................................................................................................................... 7 List of Tables ..................................................................................................................................... 11 List of Annexes .................................................................................................................................. 12 Abstract .............................................................................................................................................. 13 Runaway termico nel caso di ossidazioni catalitiche condotte in reattori PFR non stazionari .......... 14 Introduction ........................................................................................................................................ 16 1) Plug Flow Reactor Model .............................................................................................................. 19 1.1 Transport Equation for Plug Flow Reactor .............................................................................. 19 1.1.1 Steady State model ............................................................................................................ 19 1.1.2 Unsteady State model ........................................................................................................ 20 1.1.3 Unsteady State model with diffusion ................................................................................ 21 1.1.4 Steady State Model with variable density ......................................................................... 22 1.1.5 Unsteady State model with variable density .................................................................... 23 1.2 Mathematical Model ................................................................................................................ 23 1.2.1 Finite differences method.................................................................................................. 23 1.2.2 Method of Lines ................................................................................................................ 24 1.2.3 Stability ............................................................................................................................. 25 1.2.4 Stiffness ............................................................................................................................. 26 1.2.5 Accuracy and Differentiation Matrix ................................................................................ 26 1.2.6 Boundary conditions ......................................................................................................... 27 2) Thermal Runaway Theories for PFRs ........................................................................................... 29 3) Case study: Naphthalene Oxidation Reaction ............................................................................... 40 3.1 Naphthalene Oxidation Reaction ............................................................................................. 40 3.1.1 Steady State model ............................................................................................................ 41 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 4 3.1.2 Unsteady State model ........................................................................................................ 42 3.1.3 Unsteady State model with diffusion ................................................................................ 42 3.1.4 Steady State model with variable density ......................................................................... 43 3.1.5 Unsteady State model with variable density ..................................................................... 44 3.2 Dimensionless form of transport equations ............................................................................. 44 3.2.1 Steady State model ............................................................................................................ 46 3.2.2 Unsteady State model ........................................................................................................ 46 3.2.3 Unsteady State model with diffusion ................................................................................ 46 3.2.4 Steady State model with variable density ......................................................................... 47 3.2.5 Unsteady State model with variable density .................................................................... 47 3.3 Matlab® Code Setup ................................................................................................................. 48 3.3.1 Steady State Models .......................................................................................................... 48 3.3.2 Unsteady State Models...................................................................................................... 48 4) Runaway Boundaries for Naphthalene Oxidation Reaction .......................................................... 51 4.1 Runaway Boundaries for Steady State model .......................................................................... 51 4.1.1 Parametric sensitivity and critical values .......................................................................... 51 4.1.2 Conversions ....................................................................................................................... 55 4.1.3 Parametric sensitivity with low inlet velocity ................................................................... 57 4.2 Runaway Boundaries for Unsteady State model ...................................................................... 59 4.2.1 Parametric sensitivity and critical values .......................................................................... 59 4.2.2 Conversions ....................................................................................................................... 63 4.2.3 Parametric sensitivity with medium inlet velocity ............................................................ 64 4.2.4 Parametric sensitivity with low inlet velocity ................................................................... 68 4.3 Runaway Boundaries for Unsteady State model with diffusion .............................................. 71 4.3.1 Parametric sensitivity and critical values .......................................................................... 71 4.3.2 Conversions ....................................................................................................................... 75 4.3.3 Parametric sensitivity with medium inlet velocity ............................................................ 77 4.3.4 Parametric sensitivity with low inlet velocity ................................................................... 81 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 5 4.4 Runaway Boundaries for Steady State model with variable density ....................................... 84 4.4.1 Parametric sensitivity and critical values .......................................................................... 84 4.4.2 Conversions ....................................................................................................................... 88 4.5 Runaway Boundaries for Unsteady State model with variable density ................................... 90 4.5.1 Parametric sensitivity and critical values .......................................................................... 90 4.5.2 Conversions ....................................................................................................................... 94 4.5.3 Parametric sensitivity analysis with medium inlet velocity .............................................. 96 4.5.4 Parametric sensitivity analysis with low inlet velocity ................................................... 100 5) Conclusions.................................................................................................................................. 104 5.1 Summary of results ................................................................................................................ 104 5.2 Limitations ............................................................................................................................. 105 5.3 Further developments ............................................................................................................. 105 Nomenclature ................................................................................................................................... 106 Annex 1: An example of Matlab® Code ......................................................................................... 111 References ........................................................................................................................................ 112 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 6 List of Figures Figure 2.1: a) Sensitivity of the maximum temperature rise Tmax as a function of the cooling intensity N; b) first Barkelew diagram showing that all the sensitivity curves have a common envelope; c) second Barkelew diagram for differentiating between sensitive and insensitive operating conditions………………………………………………………………………………...31 Figure 2.2: p-T phase plane, showing p-T trajectories, maxima-curve pm, loci of inflexion points (pi,max and pi,min) and the “simplified” curve pcr................................................................................33 Figure 2.3: S(T,Tin) as a function of the axial coordinate l for different values of the ratio of reactant to coolant heat capacity for a PFR (τPFR)……………………………………………………………37 Figure 3.1: Kinetic scheme of naphthalene oxidation reaction…………………………………..…40 Figure 4.1.1: Tmax as function of Pin, Tin=625K, Tw=625K and v0=1m/s for the steady state model………………………………………………………………………………………………..51 Figure 4.1.2: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=1m/s for the steady state model………………………………………………………………………………………………..52 Figure 4.1.3: Tmax as a function of Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for the steady state model………………………………………………………………………………………………..53 Figure 4.1.4: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for the steady state model………………………………………………………………………………………………..53 Figure 4.1.5: Tmax as a function of Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for the steady state model………………………………………………………………………………………………..54 Figure 4.1.6: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for the steady state model………………………………………………………………………………………………..54 Figure 4.1.7: Steady state conversion along the reactor. Pin=1.8kPa, Tin=625K, Tw=625K and v0=1m/s……………………………………………………………………………………………...55 Figure 4.1.8: Steady state conversion along the reactor. Pin=1.65kPa, Tin=650K, Tw=625K and v0=1m/s……………………………………………………………………………………………...56 Figure 4.1.9: Steady state conversion along the reactor. Pin=1.6kPa, Tin=625K, Tw=625K and v0=0.1m/s……………………………………………………………………………………………57 Figure 4.1.10: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=0.1m/s for steady state model………………………………………………………………………………………………..58 Figure 4.1.11: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=0.1m/s for steady state model………………………………………………………………………………………………..58 Figure 4.1.12: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=0.1m/s for steady state model………………………………………………………………………………………………..59 Figure 4.2.1: Tmax as a function of Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model………………………………………………………………………………………………..60 Figure 4.2.2: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model………………………………………………………………………………………………..60 Figure 4.2.3: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625K and v0=1m/s for unsteady state model……………………………………………………………………………………………......61 Figure 4.2.4: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625K and v0=1m/s for unsteady state model………………………………………………………………………………………………..61 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 7 Figure 4.2.5: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625K and v0=1m/s for unsteady state model………………………………………………………………………………………………..62 Figure 4.2.6: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625K and v0=1m/s for unsteady state model………………………………………………………………………………………………..62 Figure 4.2.7: Unsteady state conversion along the reactor. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=1m/s……………………………………………………………………………………………...63 Figure 4.2.8: Unsteady state conversion along the reactor. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.5 m/s…………………………………………………………………………………………...64 Figure 4.2.9: Unsteady state conversion along the reactor. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.1 m/s…………………………………………………………………………………………...64 Figure 4.2.10: Tmax as a function of Pin, Tin=625K, Tw=625K and v0=0.5m/s for unsteady state model………………………………………………………………………………………………..65 Figure 4.2.11: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=0.5 m/s for unsteady state model………………………………………………………………………………………………..65 Figure 4.2.12: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625K and v0=0.5m/s for unsteady state model………………………………………………………………………………………………..66 Figure 4.2.13: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625K and v0=0.5 m/s for unsteady state model…………………………………………………………………………………………..66 Figure 4.2.14: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625K and v0=0.5 m/s for unsteady state model………………………………………………………………………………………………..67 Figure 4.2.15: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625K and v0=0.5 m/s for unsteady state model…………………………………………………………………………………………..67 Figure 4.2.16: Tmax as function of Pin, Tin=625K, Tw=625K and v0=0.1 m/s for unsteady state model………………………………………………………………………………………………..68 Figure 4.2.17: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=0.1 m/s for unsteady state model………………………………………………………………………………………………..69 Figure 4.2.18: Tmax as a function of Tin, Pin=1.6kPa, Tw=625K and v0=0.1 m/s for unsteady state model………………………………………………………………………………………………..69 Figure 4.2.19: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=0.1 m/s for unsteady state model…………………………………………………………………………………………..70 Figure 4.2.20: Tmax as a function of Tw, Pin=1.6kPa, Tin=625K and v0=0.1 m/s for unsteady state model………………………………………………………………………………………………..70 Figure 4.2.21: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=0.1 m/s for unsteady state model…………………………………………………………………………………………..71 Figure 4.3.1: Tmax as function of Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model with diffusion………………………………………………………………………………………..72 Figure 4.3.2: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model with diffusion………………………………………………………………………………...72 Figure 4.3.3: Tmax as a function of Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for unsteady state model with diffusion………………………………………………………………………………………..73 Figure 4.3.4: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for unsteady state model with diffusion………………………………………………………………………………...73 Figure 4.3.5: Tmax as a function of Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for unsteady state model with diffusion………………………………………………………………………………………..74 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 8 Figure 4.3.6: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for unsteady state model with diffusion………………………………………………………………………………...75 Figure 4.3.7: Unsteady state conversion with diffusion. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=1m/s……………………………………………………………………………………………...76 Figure 4.3.8: Unsteady state conversion with diffusion. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.5 m/s…………………………………………………………………………………………...76 Figure 4.3.9: Unsteady state conversion with diffusion. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.1 m/s…………………………………………………………………………………………...77 Figure 4.3.10: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion………………………………………………………………………………...78 Figure 4.3.11: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion…………………………………………………………………………..78 Figure 4.3.12: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion………………………………………………………………………………...79 Figure 4.3.13: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion…………………………………………………………………………..79 Figure 4.3.14: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.5 m/s for unsteady state model with diffusion………………………………………………………………………………...80 Figure 4.3.15: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625K and v0=0.5 m/s for unsteady state model with diffusion…………………………………………………………………………..80 Figure 4.3.16: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion………………………………………………………………………………...81 Figure 4.3.17: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion…………………………………………………………………………..82 Figure 4.3.18: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion………………………………………………………………………………...82 Figure 4.3.19: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion…………………………………………………………………………..83 Figure 4.3.20: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with diffusion………………………………………………………………………………...83 Figure 4.3.21: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with diffusion…………………………………………………………………………..84 Figure 4.4.1: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=1 m/s for steady state model with variable density……………………………………………………………………………………...85 Figure 4.4.2: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=1 m/s for steady state model with variable density…………………………………………………………………………85 Figure 4.4.3: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for steady state model with variable density………………………………………………………………………………...86 Figure 4.4.4: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for steady state model with variable density…………………………………………………………………………86 Figure 4.4.5: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for steady state model with variable density………………………………………………………………………………...87 Figure 4.4.6: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for steady state model with variable density…………………………………………………………………………88 Figure 4.4.7: Steady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=1 m/s………………………………………………………………………………………..……89 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 9 Figure 4.4.8: Steady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.5 m/s…………………………………………………………………………………………...89 Figure 4.4.9: Steady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.1 m/s…………………………………………………………………………………………...90 Figure 4.5.1: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=1 m/s for unsteady state model with variable density………………………………………………………………………………...91 Figure 4.5.2: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=1 m/s for unsteady state model with variable density…………………………………………………………………………91 Figure 4.5.3: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for unsteady state model with variable density…………………………………………………………………………92 Figure 4.5.4: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for unsteady state model with variable density…………………………………………………………………………92 Figure 4.5.5: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for unsteady state model with variable density…………………………………………………………………………93 Figure 4.5.6: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for unsteady state model with variable density…………………………………………………………………………94 Figure 4.5.7: Unsteady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=1 m/s………………………………………………………………………………………...95 Figure 4.5.8: Unsteady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.5 m/s………………………………………………………………………………………95 Figure 4.5.9: Unsteady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.1 m/s………………………………………………………………………………………96 Figure 4.5.10: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density…………………………………………………………………………97 Figure 4.5.11: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density…………………………………………………………………...97 Figure 4.5.12: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density…………………………………………………………………………98 Figure 4.5.13: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density…………………………………………………………………...98 Figure 4.5.14: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.5 m/s for unsteady state model with variable density…………………………………………………………………………99 Figure 4.5.15: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=0.5 m/s for unsteady state model with variable density…………………………………………………………………...99 Figure 4.5.16: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density………………………………………………………………………..100 Figure 4.5.17: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density……………………………………………………………………………………………...101 Figure 4.5.18: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density………………………………………………………………………..101 Figure 4.5.19: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density………………………………………………………………….102 Figure 4.5.20: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with variable density……………………………………………………………………….102 Figure 4.5.21: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with variable density………………………………………………………………….103 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 10 List of Tables Table 3.1: Data for naphthalene oxidation reaction………………………………………………...41 Table 3.2: Range of values for inlet parameters…………………………………………………….41 Table 3.3: Dimensionless variables…………………………………………………………………45 Table 3.4: Dimensionless parameters……………………………………………………………….45 Table 5.1: Summary of results……………………………………………………………………..104 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 11 List of Annexes Annex 1: An example of Matlab® code…………………………………………………………...111 Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 12 Abstract In chemical industry, one major cause of accidents is thermal runaway. Many industrial processes are based on fast and strongly exothermic reactions. In these systems, a small variation in the system conditions can cause much larger variations in the system behavior, for example a sharp rises in temperature that can lead to serious consequences, from the opening of relief systems to the very explosion of the reactor. For these reasons, fast and strongly exothermic reactions have been the object of a huge number of studies, aimed at identifying safe and unsafe operating conditions for these systems. This work focuses on Plug Flow Reactors (PFR), traditionally used for large, continuous productions, but increasingly employed also to perform typical discontinuous processes. The most used model for PFR is a steady-state model, which neglects the reactor dynamic, i.e. unsteady conditions in the start-up and shut-down operations. This approximation allows treating PFRs with methods very similar to the ones developed for Batch Reactors. In runaway boundaries studies, for example, the parametric sensitivity criterion is often used. In this work, the parametric sensitivity approach has been extended to different PFR models, respectively considering unsteady state conditions, material and thermal diffusion and density variations with temperature. Particularly the removal of the steady state approximation has raised a safety concern, as it produces a change in runaway boundaries towards non-conservative values. Thus, the traditional steady state model may lead to dangerous conclusions in certain cases. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 13 Runaway termico nel caso di ossidazioni catalitiche condotte in reattori PFR non stazionari Il principale scopo dell’industria di processo è di fornire un’ampia gamma di prodotti che possono essere successivamente utilizzati in diversi campi. Di conseguenza, quando si parla di sicurezza nell’industria di processo non si può non parlare dei rischi legati all’utilizzo di un certo prodotto. Infatti, a ogni fase della vita di un prodotto, dalla sua ideazione fino all’eliminazione in forma di rifiuto, è associato un certo grado di rischio, legato alle sue proprietà chimico-fisiche: tossicità, infiammabilità, etc. D’altra parte, la sicurezza di processo, che concerne tutti i rischi associati alla fabbricazione del prodotto, è un altro aspetto chiave dell’industria chimica. Tipici rischi legati ai processi chimici sono runaway termico, esplosioni fisiche, incendi, rilascio di sostanze tossiche o pericolose. In generale, le stesse condizioni operative del processo (temperature, pressioni, portate, logiche di controllo…) possono introdurre instabilità critiche nella miscela reagente e portare a forti deviazioni dalle condizioni di sicurezza. Inoltre, le apparecchiature dell’impianto, incluso il sistema di controllo, possono guastarsi: un esempio molto comune è il guasto del sistema refrigerante nel caso in cui si operi con una reazione esotermica. Infine, i processi chimici sono soggetti all’errore umano. Lo studio di questi elementi costituisce l’enorme campo della “sicurezza di processo”. La valutazione della sicurezza di una reazione chimica, sia nelle normali condizioni operative che in caso di deviazioni dalle stesse, è basata sullo sviluppo di modelli matematici capaci di descrivere tutti i fenomeni chimico-fisici che accadono all’interno di un reattore. Le equazioni di bilancio di massa per ogni specie coinvolta e di bilancio termico globale consentono di considerare gli aspetti cinetici e termici delle reazioni in gioco. Un gran numero di studi è stato dedicato alle reazioni veloci e fortemente esotermiche, che sono una delle principali cause di incidenti nell’industria di processo. I processi chimici che coinvolgono queste reazioni presentano le criticità maggiori, perché possono causare una perdita di controllo termico, che avviene quando la velocità di sviluppo del calore è maggiore della velocità di rimozione da parte del sistema di raffreddamento. Questo fenomeno indesiderato, chiamato “esplosione termica” o “runaway”, può avvenire a causa di un vasto numero di fattori, come il guasto del sistema di controllo, l’inefficienza del sistema di raffreddamento, l’errore umano nel caricamento dei reagenti, etc. Le conseguenze più critiche della perdita di controllo termico si hanno quando la temperatura del reattore si alza fino a raggiungere valori ai quali si scatenano reazioni indesiderate o di decomposizione della miscela reagente. Queste reazioni sono molto più esotermiche e veloci della reazione desiderata e di frequente generano gas incoercibili che possono portare alla pressurizzazione del reattore e quindi al rilascio del contenuto tramite un sistema di emergenza o addirittura all’esplosione fisica del reattore. Essendo le reazioni veloci e fortemente esotermiche tipiche dell’industria farmaceutica e della chimica fine, la letteratura scientifica si è tradizionalmente concentrata sullo studio della stabilità termica dei reattori Batch e Semi-Batch, che operano in modalità discontinua con quantità Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 14 relativamente piccole. Successivamente, i metodi di individuazione delle condizioni di runaway sviluppati per questi reattori sono stati estesi ai reattori continui, come i CSTR e i PFR, normalmente utilizzati per produzioni di larga scala con caratteristiche endotermiche. Recentemente, però, si sta sviluppando la tendenza di trasformare processi tradizionalmente discontinui in processi continui: alcuni processi come le polimerizzazioni in soluzione o emulsione o alcune sintesi tipiche dell’industria farmaceutica iniziano ad essere condotte in reattori continui, ai fini di raggiungere un’intensificazione e un’ottimizzazione, ad esempio una significativa riduzione dei volumi. Perciò, è ragionevole aspettarsi che i reattori CSTR e PFR in futuro saranno impiegati per sintesi di prodotti che coinvolgono reazioni potenzialmente fuggitive. Inoltre, è possibile che questa tendenza porti a utilizzare questi reattori per operazioni di breve termine, o per sintetizzare diversi lotti di prodotti durante l’anno, ciascuno in un periodo più o meno breve. Con riferimento ai PFR, le applicazioni industriali più comuni sono nel campo petrolchimico e nella chimica di base. Alcuni di questi processi, come le ossidazioni catalitiche, coinvolgono reazioni potenzialmente fuggitive. Come già accennato, tutti i criteri per l’individuazione delle cosiddette frontiere di runaway (cioè, il set di parametri critici in corrispondenza dei quali si può perdere il controllo termico del reattore) nei reattori PFR sono stati sviluppati a partire dagli studi condotti sui reattori Batch. Di conseguenza, sono tutti basati su un modello stazionario, che trascura sia il transitorio nella fase di avviamento sia la diffusione termica e materiale e che permette di individuare le frontiere di runaway in modo molto semplice, sia dal punto di vista teorico che computazionale. Tradizionalmente, l’ipotesi di stazionarietà è considerata ragionevole in quanto il transitorio del PFR è molto rapido: solitamente si sviluppa durante il primo passaggio della miscela reagente attraverso il reattore. Sfortunatamente, a causa delle nuove tendenze applicative delineate precedentemente, lo studio della dinamica del PFR, cioè delle operazioni di avviamento e spegnimento del reattore, può diventare essenziale per raggiungere un’ottimizzazione sicura del processo. Lo scopo principale di questo lavoro è di studiare gli effetti della dinamica del reattore sull’individuazione delle frontiere di runaway. Dal momento in cui nessuno dei criteri classici sviluppati per i PFR si può estendere facilmente al caso non stazionario, in questo lavoro è stato utilizzato il criterio della sensitività parametrica, che richiede semplicemente il calcolo di una funzione obiettivo facilmente derivabile dalla soluzione delle equazioni di trasporto del modello. Inoltre, sono stati presi in considerazione altri elementi, come i termini diffusivi e le variazioni della densità con la temperatura, studiando il loro effetto sulle frontiere di runaway. In particolar modo, è stato scoperto che la dinamica del reattore, cioè l’ipotesi di non stazionarietà, ha un effetto significativo sulle frontiere di runaway. Questo modello evidenzia uno spostamento verso valori non conservativi e, perciò, sottolinea l’importanza di considerare anche il transitorio, dal momento in cui un semplice modello stazionario può portare a previsioni pericolose e non conservative. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 15 Introduction The main aim of process industry is to provide a broad range of products which can be successively employed in different ways (e.g., pharmaceuticals, additives, electronic devices, textiles, foods, and so on). Therefore, whenever we talk about chemical safety in process industry, it is not possible to not talk about product safety, that is, the risks linked to the use of a certain product. In fact, each product, between its discovery and elimination, passes through different steps of its history[1]: conception, design, feasibility and market studies, manufacturing, distribution, use, and elimination (in this last step, the product becomes a waste to be disposed of). During all these steps, a certain degree of risk is always associated to the product handling because of its physical-chemical properties: toxicity, eco-toxicity, flammability, explosion, corrosion, etc. On the other hand, process safety, that is all the risks associated with product manufacturing, is another important key aspect in chemical industries. Particularly, risks linked with chemical processes can be thermal (runaway) or physical explosions, fires, and toxic/hazardous releases (of course, all mixed cases are possible). In general, a process often uses operating conditions (temperatures, pressures, flow rates, control strategies, etc...) that, by themselves, may induce critical instabilities in the reacting mixture leading to strong deviations from safety. Moreover, plant equipment, including the control system, may fail: a common example is the refrigerating system breakdown when we are operating with an exothermic reaction. Finally, since chemical processes are work-intensive, they may be subjected to human errors. All these elements, together, constitute what it is called “process safety”. As it can be noticed it is a wide field to be explored. Particularly, the assessment of whether or not a chemical reaction can be carried out in a safe and controlled way under normal as well as upset operating conditions is based on the development of mathematical models capable of describing all the physical-chemical phenomena that occur inside a reactor. In general, a chemical process is always characterized by its stoichiometry and its extent of reaction, independently of the operation mode (e.g. continuous or discontinuous). The rate of change in the extent, however, depends on the reactor type, on the reaction rate and, in the case of heterogeneous reactions, on mass transfer phenomena[2]. All these aspects can be summarized by writing mass balance equations for each independent chemical species involved into the kinetic scheme. On the contrary, the thermal behavior of a chemical reactor depends on the thermodynamics of the process, on the reaction rate but also on how the heat is exchanged with environment or a suitable cooling system. These aspects are described in the overall heat balance equation of the reactor which is that one involved in the assessment of the risk associated with thermal explosions[2]. A huge number of studies has been dedicated to fast and strongly exothermic chemical reactions because they cause a major part of the total number of explosions, fires and hazardous releases occurring in process industries. Chemical processes involving these reactions are the most critical to be carried out because of the possible triggering of a reactor temperature loss of control, that occurs whenever the rate of heat evolution becomes greater than the rate of heat removal provided by the installed cooling equipment. Such an unwanted phenomenon, called “thermal explosion” or Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 16 “runaway”, can take place into every type of reactor (both continuous and discontinuous), operated in whatever temperature control mode (isothermal, isoperibolic, polytropic, etc.), because of a great number of operating and human factors such as control system failure, stirrer breakdown, refrigerating system inefficiency, maintenance, reactants loading errors, dead times enlarging or shortening, etc. The most critical consequences of a loss of thermal control may occur when the reactor temperature increases up to values at which secondary undesired reactions or decompositions of the reacting mixture are triggered. Such reactions, apart from a reduction of the selectivity with respect to the desired product, may lead to a real system thermal loss of control, since they are usually much more fast and exothermic than the desired reaction. Moreover, if a decomposition event is triggered because of the system thermal loss of control, the consequent release of incoercible gases may pressurize the reactor leading to its venting through the installed emergency pressure relief system and, eventually, to its physical explosion. Being fast and strongly exothermic reactions typical of pharmaceutical and fine chemistry industries, all the dedicated scientific literature has been traditionally focused on the study of the thermal stability of Batch Reactors (BRs) (later on Semi-Batch Reactors (SBRs)), which operate in discontinuous conditions and with relatively low volumes of reactants. Particularly, all the runaway detection methods developed for these systems, have been then extended to continuous reactors, i.e. Continuously Stirred Tank Reactor (CSTR) and Plug Flow Reactor (PFR), which were normally employed for endothermic large scale productions (exothermic processes were “exceptions”). But, actually, a growing trend has been developing: changing traditionally discontinuous processes into continuous ones. Therefore, processes that have been always performed in discontinuous reactors, such as solution and emulsion polymerizations or pharmaceutical industry syntheses, start to be carried out in continuous reactors (or, at least, there are documented studies of efforts aimed at using them). The main reason of this choice is not necessarily to increase the production, but mainly to achieve the intensification of the process itself, i.e. a radical volume reduction. Thus, it is reasonable to expect that, in the future, CSTRs and PFRs will be employed to yield an increasingly vast range of products whose synthesis processes involve potentially runaway reactions. There is another considerable consequence of this trend, linked to the different characteristic of these innovative and unusual applications of continuous reactors. Whereas these kinds of reactors are traditionally performed continuously for a whole year, except for extraordinary stops and periodical maintenance, the current trend would possibly lead to short-time operations. As typical in the pharmaceutical field, the very same reactor would be used in the synthesis of different batches of products during the year, each one in a longer or shorter period of time. Referring to PFRs, common industrial applications, in either gas or homogeneous liquid phase systems, are in gasoline production, oil cracking, ammonia synthesis and other processes of oil and basic chemical industry. Some of these processes, as said before, involve potentially runaway reactions, such as catalytic oxidation reactions. As above-mentioned, all the runaway criteria for PFRs have been developed starting from studies upon BRs. As a consequence, all of them are based on a steady state model for describing the functioning of the PFR, which neglects both unsteady conditions in the start-up operation and material and thermal transport by diffusion. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 17 According to this model, the PFR balance equations can be written as: v0 Ci i R l (1.1) (1.2) v0 H T 4U R (T Tw ) l 0 cˆP , mix , g dT 0 cˆP ,mix , g for a single reaction system with constant cooling (the meaning of all the symbols are detailed in the Nomenclature section). It is apparent that these equations are the same equations of a Batch Reactor, with the spatial coordinate z as independent variable instead of time. Thanks to this similarity, it has been possible to treat a PFR with methods derived from those developed for a BR and to identify runaway boundaries in a relatively simple way, from both the theoretical and the computational point of view. The steady state assumption has been traditionally regarded as reasonable because of the extremely rapid transition from the unsteady to the steady state conditions, which in a PFR usually occurs after the first passage of the reaction mixture across the reactor. Unfortunately, with the new trends in the application of continuous reactors discussed above (that is, a PFR can be used for hours instead of one year), the study of the PFR dynamic, i.e. start-up and shut-down operations, may become essential in order to achieve a safe process optimization. The main purpose of this work is to study the effect of the reactor dynamic on the determination of the so-called runaway boundaries (that is, the set of critical parameters in correspondence of which the thermal control of the reactor can be lost). As none of the classical runaway criteria developed for PFRs can be extended easily to an unsteady state model, in this work the generalized parametric sensitivity criterion has been used in order to identify unsteady state runaway boundaries. This choice is completely justified by the fact that the generalized parametric sensitivity criterion requires only the calculation of an object function which can be easily derived from the solution of the unsteady state model. Furthermore, other elements have been taken into account, such as diffusion terms and density variations with temperature, and their effects on runaway boundaries have been investigated. Especially as regards the reactor dynamic, i.e. the unsteady state assumption, it has been found that there is deep effect on runaway boundaries. In fact, this model reveals a change towards non-conservative values and, thus, highlights the importance of taking into account the transition phase, since a simple steady state model is likely to provide dangerous and non-conservatives predictions. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 18 1) Plug Flow Reactor Model 1.1 Transport Equation for Plug Flow Reactor 1.1.1 Steady State model A plug flow reactor (PFR) consists basically of a vessel, i.e., a tube or a duct, through which the reacting fluid flows, in a continuous reaction process. These reactors are normally characterized by a constant cross-sectional area, i.e., constant diameter, through the entire reactor, in a cylindrical geometry. The plug flow is a simple model of the velocity profile of a fluid through a pipe, where the velocity is assumed to be constant across any cross-section perpendicular to the axis of the pipe. As a consequence of this hypothesis, it is assumed that there is no mixing in the direction of the flow and complete mixing in the perpendicular direction. Both molecular diffusion and dispersive mixing processes are neglected. As long as the axial dimension of the reactor is larger than the radial one, which is the case for tubular reactors, all radial variations in the reactor’s variables can be neglected. PFRs are normally handled in steady state continuous operation, i.e., the mass flow rate into the reactor is equal to the mass flow rate out of the reactor, and the temperatures and concentrations at all points in the reactor do not change with time. Thus, the accumulation terms can be neglected. To sum up, the conventional model of Plug Flow is founded on these assumptions: 1) 2) 3) 4) 5) No radial variations in concentrations, temperatures or velocities Perfect mixing in the radial direction No mixing along the axial direction Constant inlet velocity, which is equal to the axial velocity Steady state operation With respect to the reactor operation, it is crucial to define whether a cooling system is present. In the case of exothermic reactions, the cooling system is necessary to remove the heat generated by the reactions. There are many ways to design a heat removal system. In this case, the simplest one is considered: co-current external cooling at constant temperature. A typical example is a coolant fluid which undergoes a phase transition. 6) Constant wall temperature For simplicity’s sake, a single exothermic reaction is considered. The physical properties of the species involved (i.e., reaction mixture density and specific heat) are considered constant. 7) Single exothermic reaction 8) Constant reaction mixture density and specific heat Thus, transport equations for generic species i and for energy can be written as: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 19 v0 dCi iR dl (1.1) v0 H R 4U (T T ) dT w dl 0 cˆP, mix , g dT 0 cˆP ,mix , g (1.2) (1.3) with initial conditions: C l 0 Cin IC: i T l 0 Tin This system is an Ordinary Differential Equations (ODEs) system, in which only the convection term, represented by the first derivative, and the production/consumption term are present. Thus, the system is a spatial one, which means that the system variables are functions of position within the reactor. There is a strong similarity between tubular and well-stirred batch reactors, which are temporal system. In the latter case, for exothermic reactions temperature maximum may occur as the reaction proceeds in time, while in the former at steady state conditions a temperature maximum may be exhibited at some location along the reactor, which is generally referred to as a hot spot. Thus, many criteria for identifying runaway conditions in batch reactors have been developed through an analogy with batch reactors criteria. 1.1.2 Unsteady State model In some cases, unsteady state operation may be used under some circumstances and will prevail during start-up, shut-down, or after a change in one or more of the process variables, due to unexpected failures or setting changes. The proposed model takes into account the accumulation term, i.e., the variations of temperatures and concentrations with time, in order to be able to model also unsteady state operations and especially start-up operations, where a safety concern may arise. All other assumptions are maintained: 1) 2) 3) 4) 5) 6) 7) No radial variations in concentrations, temperatures or velocities Perfect mixing in the radial direction No mixing along the axial direction Constant inlet velocity, which is equal to the axial velocity Constant wall temperature Single exothermic reaction Constant reaction mixture density and specific heat Transport equations for generic species i and for energy can be written as: Ci C v0 i i R t l (1.4) Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 20 H R 4U (T T ) T T v0 w t l 0 cˆP ,mix, g 0 cˆP ,mix , g (1.5) In this case, the system is a Partial Differential Equations (PDEs) system, as both temporal and spatial derivatives are present. Thus, initial conditions and boundary conditions are: C t 0 Cin IC: i T t 0 Tw (1.6) C l 0 Cin BC: i T l 0 Tin (1.7) 1.1.3 Unsteady State model with diffusion Both material and thermal diffusion along the axis of the reactor are usually neglected in PFR models, but they may have an effect on the reactor’s behavior, especially if the values of material and thermal diffusivities are high. The reactor is assumed to perform in unsteady state, whereas all the other assumptions are maintained: 1) 2) 3) 4) 5) 6) No radial variations in concentrations, temperatures or velocities Perfect mixing in the radial direction Constant inlet velocity, which is equal to the axial velocity Constant wall temperature Single exothermic reaction Constant reaction mixture density and specific heat Transport equations for generic species i and for energy can be written as: Ci 2C C Dm 2 i v0 i i R t l l (1.8) H R 4U (T T ) T 2T T 2 v0 w t l l 0 cˆP ,mix , g dT 0 cˆP ,mix , g (1.9) This system is a PDE system with both first-order derivatives, representing convective phenomena, and second-order derivatives, representing diffusive phenomena. Initial and boundary conditions are: C t 0 Cin IC: i T t 0 Tw (1.10) C l 0 Cin Inlet BC: i T l 0 Tin (1.11) Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 21 Ci (l L) 0 l Outlet BC: T ( l L ) 0 l (1.12) Both inlet and outlet boundary conditions are needed in order to solve the second-order PDE system. The conditions at the inlet are constant, whereas the outlet condition expresses the absence of fluxes at the reactor outlet. 1.1.4 Steady State Model with variable density In the previous models, the reaction mixture density has been considered as a constant. This assumption does not take into account variations of density with the reactor temperature which are very likely to occur during operation. In this work, the effect of such variations has been considered by the development and resolution of a more complex model. Thus, the system transport equations have been rewritten under the following assumptions: 1) 2) 3) 4) 5) 6) 7) No radial variations in concentrations, temperatures or velocities Perfect mixing in the radial direction No mixing in the axial direction Constant inlet velocity Constant wall temperature Single exothermic reaction Steady state operation Transport equations for species i and for energy can be written as: (Ci v ) ri l cˆP ,mix , g ˆ mix cˆP ,mix vT H Tv l ˆ mix cˆP ,mix l 4U T Tw R ˆ ˆ d c T mix P , mix (1.13) (1.14) Where the heat capacity of the catalyst has been assumed constant and predominant with respect to the gas phase: ˆ mix cˆ p ,mix 1 ˆ s cˆ p , s const (1.15) (1.16) With initial conditions: C l 0 Cin IC: i T l 0 Tin Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 22 1.1.5 Unsteady State model with variable density In order to develop a full comparison with the constant density case, the variable density model has been considered also for an unsteady state operation, under the same hypothesis: 1) 2) 3) 4) 5) 6) No radial variations in concentrations, temperatures or velocities Perfect mixing in the radial direction No mixing in the axial direction Constant inlet velocity Constant wall temperature Single exothermic reaction The resulting transport equations for species i and energy are: Ci (Ci v) ri t l H cˆ vT T P ,mix , g ( Tv ) ˆ cˆ t ˆ mix cˆP ,mix l l mix P ,mix 4U T Tw R dT ˆ mix cˆP , mix (1.18) (1.19) With the following initial and boundary conditions: C t 0 Cin IC: i T t 0 Tw (1.20) C l 0 Cin BC: i T l 0 Tin (1.21) 1.2 Mathematical Model The simulations contained in this work have been performed with Matlab®. Matlab® indeed has its own PDE solver, called PDEPE, but it has not been possible to use it in this case, because the system is too complex and the solver is not able to reach the convergence in most cases. Thus, the MATMOL code has been implemented. This code has been developed by Wouver, Saucez and Vilas[3] and it is based on MOL (Method of Lines). The main idea underlying this method is to approximate the spatial derivatives using finite differences method, thus obtaining a ODEs system which can be solved through time integration with one of Matlab® ODE solver. 1.2.1 Finite differences method In order to evaluate the spatial derivatives, a spatial grid of N+1 discrete points is built. For simplicity’s sake, the points zi are considered regularly distributed (uniformly spaced) between z0 and zN: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 23 zi z0 iz (1.22) z N z0 N (1.23) (1.24) z i 0,..., N The FD approximation of the nth-order derivative of x(z) in zi, i.e d n x ( zi ) , is based on the Taylor dz n series expansions of x(z) at point zk close to zi. As an example, there are three different ways to approximated the first order derivative of x(z): dx dz dx dz dx dz xi 1 xi z (1.25) xi xi 1 z (1.26) xi 1 xi 1 2z (1.27) zi zi zi The latter expression is a centered approximation and it is the most accurate, as the error decreases quadratically with Δz. The other ones are non-centered and are less accurate, as the error decreases linearly with Δz. Depending on the direction of the “flow of information”, these formulas are termed “upwind” and “downwind” respectively. As for the second order derivatives, they can be approximated with the same method. One example is reported: d 2x dz 2 zi xi 1 2 xi xi 1 z 2 (1.28) 1.2.2 Method of Lines In this study, the PDE system is composed by a set of convection-diffusion-reaction equation, which can be written in the form: xt vxz Dxzz r ( x) (1.29) where xt is the temporal derivative, xz is the first-order spatial derivatives, representing convection, and xzz is the second-order spatial derivatives, representing diffusion, v is the flow velocity, D is the diffusion coefficient and r(x) is the source term, linked to the presence of a chemical reaction producing (or consuming) heat or mass. The PDE is defined in the spatial domain z0<z<zN and must be supplemented of two Boundaries Conditions (BC), at z=z0 and z=zN, as long as an Initial Condition (IC) at t=0. BC can be basically expressed in three ways, depending on the physical nature of the problem: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 24 x z0 , t g 0 (t ) Dirichelet BC (1.30) x ( z , t ) g 0 (t ) z 0 Neumann BC (1.31) x z , t 0 x z0 , t g0 (t ) z 0 Robin BC (1.32) The MOL proceeds as follows: 1) A uniform spatial grid is defined over N intervals (see Eq. 1.22-1.24). 2) The PDE is expressed at each of the grid points, with the exception of points z0 and zN where BCs are imposed: ( xi )t v( xi ) z D( xi ) zz r ( xi ) (1.33) i 0,..., N 1 (1.34) 3) The spatial derivatives are replaced by finite difference formulas, e.g. Eq. 1.25 and Eq. 1.28. This approximation transforms the PDE into a semi-discrete ODEs system. 4) The IC provides the value of the unknown functions x1 (t ), x2 (t ),, xN (t ) at t=0, and the system of ODEs can therefore be integrated using one of Matlab® ODE solver. The choice of spatial discretization and time integrator must be carried with particular attention, as it can deeply affect the stability and the accuracy of the method. In the following, some basic guidelines in order to enhance these features are reported. Nevertheless, a detailed analysis of stability and accuracy of the numerical method used is beyond the purpose of this work. 1.2.3 Stability A numerical method is defined stable if it is able to smother small fluctuations in the input data. In numerical algorithms for differential equations the concern is the growth of round-off errors or initially small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. The stability of the semi-discrete ODEs system depends both on the spatial differentiation scheme and on the time integrator chosen. As regards the spatial differentiation scheme, some practical guidelines can be sketched: 1) First-order spatial derivatives (representing advection or convection) are usually best approximated using upwind FD schemes (physically, useful information on the solution is “sensed” in the upwind direction, from where the solution “flows”). 2) Second-order derivatives (representing diffusion or dispersion) are usually best approximated using centered FD schemes (physically, diffusion has no preferential direction, and information is “sensed” in both directions). Once spatial discretization has been carried out and a stable semi-discrete ODE system has been obtained, a sufficiently small time-step size must be selected in order to remain inside the stability region imposed by the time integrator. This is particularly true when using explicit ODE integrators, Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 25 such as the explicit Euler method. However, implicit time integrators have much larger stability regions, and it can be very advantageous to use one. 1.2.4 Stiffness There is an additional system characteristic, called stiffness, related to the presence of very different time scales, which can make time integration particularly delicate. For instance, a system response to an external perturbation can display small-scales features occurring on periods of a few seconds, and overall transients dominated by large time constants of a few hours. In this case, the limited stability regions of explicit time integrators will impose important restrictions on the time step size, leading to large computational efforts. Again, the use of an implicit ODE solver will be the way around this problem. Matlab® implicit solver ode15s, based on backward differentiation formulas, is optimized to deal with stiff problems and it is very efficient with respect to explicit solvers like ode45. In fact, ode15s can perform calculation with a relatively large time step, requiring much less computational effort to converge. Moreover, its performance are unaffected by the number of grid points or by the discretization scheme chosen. Thus, it is very advantageous and will be used in all the simulations performed in this work. 1.2.5 Accuracy and Differentiation Matrix Accuracy can be influenced at two levels: spatial discretization and time integration. Spatial discretization itself has two tuning features: the number of spatial grid points which determines the fineness or resolution of the spatial grid and the stencil of the FD scheme, i.e., the pattern of points on which the FD scheme is built. Increasing the number of spatial grid points, with a procedure called h refinement, will usually enhance the accuracy of the solution. Increasing the order of the finite difference approximation, with a procedure called p-refinement, can also be beneficial to accuracy. More elaborated discretization schemes can be obtained by combining Taylor series at additional locations zi 1 , zi 2 ,, zi 2 , zi 1 , .As an example, a fourth-order (five-point) based-upwind finite difference approximation for the first derivative is reported: x 6 xi 2 18 xi 1 10 xi 3xi 1 dx i 3 dz zi z (1.35) Such formulas can be used at all the grid points, with the exception of a few points near the boundaries. At those points, alternative formulas have to be developed in order to avoid the introduction of fictitious grid points outside of the spatial domain. In the case of second-order scheme, these latter formulas are derived by considering the Taylor series expansions at the boundary point z0 and zN. For the Matlab® implementation of these formulas the concept of differentiation matrix is useful, so that a spatial derivative can be computed by a simple matrix operation. Differentiation matrices allow higher-order spatial derivatives to be easily computed in either of the following two ways: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 26 1) Stagewise differentiation: xz D1 x xzz D1 xz 2) Direct differentiation: xzz D2 x (1.36) (1.37) (1.38) Where D1 nd D2 are matrices for computing first- and second-order derivatives, respectively. Stagewise differentiation is a simple and very flexible procedure and has been chosen in the simulations performed in this study, although it should be applied with care, as an order of accuracy is lost at each stage. 1.2.6 Boundary conditions An important aspect of the solution of an initial-boundary value problem is the implementation of the boundary conditions. The translation of the boundary conditions into the method of lines is a crucial step, as boundary conditions are a very important aspect of the problem definition. There are different approaches to this implementation. In the following, the mass matrix approach, which has been used in the simulations performed, is recalled. In this approach, boundary conditions can be viewed as algebraic equations. Indeed a Dirichlet condition (Eq. 1.30) can be written as 0 g 0 t x z0 , t g 0 t x0 t (1.39) And a Neumann boundary condition (Eq. 1.31) can be approximated as: 0 x t x0 t x g 0 t z0 , t g 0 t 1 z z (1.40) As an example, the case of a simple diffusion equation supplemented with Neumann and Robin boundary conditions is reported: x 2x t z2 (1.41) IC: x z, 0 1 (1.42) x 0, t 0 z BC: x 1, t x 1, t 0 z (1.43) With 0<z<1 and 0<t<2. The boundary conditions can be expressed as: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 27 00 dx0 (t ) x0 t x1 t dt z (1.44) 00 dxN (t ) xN 1 (1 z ) xN dt z (1.45) The MOL therefore yields the following differential-algebraic system (when using threepoint centered FDs): Where 1 z (1.46) (1.47) This system is in the form Mx f ( x ) , which can be solved using a Differential Algebraic Equation (DAE) solver such as ode15s. Here the RHS f(x) is a linear expression, since the spatial operator representing diffusion is linear, but more general nonlinear terms, such as the reaction term, could appear. The mass matrix M has to be defined and called in the DAE solver in order to integrate the system. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 28 2) Thermal Runaway Theories for PFRs Before the establishment of a rigorous classification and mathematical modeling of chemical reactors, a number of industrial processes (such as synthesis of sulfuric acid, soap and synthetic rubber, production of nitrogen containing compounds, and so on) were designed and put into operation basing only on laboratory experimentations. Moreover, the batch reactor type was the only one practically employed in the industrial syntheses. In the first decades of the nineteenth century, the concepts of unit operations, reactors classification and so on dawned, leading to an ever increasing interesting in the mathematical modeling of chemical processes. Particularly, since the major part of industrial syntheses of interest (e.g., production of sulfuric and nitric acid, nitrations, sulfonations, etc...), were fast and strongly exothermic, there was a growing interest in characterizing the thermal stability of the synthesis reactor (in order to prevent the triggering of runaway phenomena). The first scientists who investigated potentially runaway reactions systematically were Semenov, Todes and Frank-Kamenetzkii[4]. They can be considered as pioneers because they tried to describe in detail both the phenomena of the self-heating and the “explosion-like” temperature rise in a reacting mixture. The criteria derived from their studies are still valid today and constitute the basis of any safety assessment involving chemical reactors. Particularly, according to Semenov theory, cooled batch processes could be operated safe under normal operating conditions (upset operating conditions had not been considered) only if the reactions were moderately exothermic and proceed slowly (a reaction rate may be regarded as slow if its characteristic time is approximately 20 times higher than that of the cooling). Just before and during the years of the Second World War, there was an ever increasing interest in the mathematical modeling of all reactors types (individuated by the studies on their classification) in order to identify the most performing apparatus for a given reaction of interest. Particularly, continuous reactors such as Continuously Stirred Tank Reactors (CSTRs) and Plug Flow Reactors (PFRs), since they guaranteed a major production, were extensively analyzed by mathematicians and physicists concerning all the aspects of their thermal stability. Unfortunately, numerical calculations were still hard to be carried out, therefore, scientists focused their attention on CSTRs rather than PFRs, since they can be described (in their steady-state operation) by algebraic equations. Only in 1956, Bilous and Amundson published a work concerning the stability and sensitivity of a tubular reactor (PFR) [5]. As they noticed, a mathematical approach similar to that one used for CSTRs, that is using the perturbation method, was not feasible because it would have led to write a system of partial differential equations. At that time, calculation of that type was not simple to be carried out, therefore they moved toward a new approach: they introduced, for the first time, the concept of parametric sensitivity. The parametric sensitivity is defined as the system sensitivity behavior with respect to changes in its parameters. By changing these values, the system characteristics can achieve desired or undesired behaviors. When a system operates in the parametrically sensitive region, its performance becomes unreliable and changes with small Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 29 variations in the parameter. In order to detect runaway boundaries, they considered the thermal response of the system, represented by the steady-state temperature profile along the reactor length, as the target behavior to be analyzed as a function of the variation of some model parameters. Their approach was very rigorous since they analyzed both sinusoidal and step parameters variations: anyway, the conclusion was that the two methods led to the same results in terms of reactor parametric sensitivity. Even if they analyzed the variation of the reactor temperature and concentration profile as a function of perturbations of different model parameters, such as reactor wall temperature, inlet reactant concentration, global heat transfer coefficient, etc.., the only conclusion they reached was that a parametric sensitivity region could exist. In 1959, Barkelew[6] tried to find a simple criterion for a PFR, starting from the observation that the equations describing its steady-state behavior are not substantially different with respect to the ones of a BR: simply time had to be replaced by the axial coordinate of the tubular reactor. Particularly, he observed that regions of pronounced parametric sensitivity with respect to the initial temperature and the cooling efficiency could be easily recognizable simply looking at the temperature vs. time (or axial coordinate) profiles and, using such an observation, he was the first who differentiated between sensitive and insensitive operating region of a BR/PFR. Starting from mass and energy balances for a batch reactor, the maximum dimensionless temperature rise, defined by Eq. 2.1 Tmax Tmax Tcool or max Eatt Tmax Tcool n RTcool (2.1) is a function of the dimensionless reacting strength S (Eq. (2.2)), that is a formulation of the adiabatic temperature rise, and the cooling intensity N (Eq. (2.3)), that is an alternative formulation of the classical Semenov number, as it can be seen from Figure 2.1a and 2.1b: S N H C0 c p Tcool Eatt RTcool UA Eatt n 1 c p V k0 exp C0 RTcool (2.2) (2.3) Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 30 a) b) c) Figure 2.1: a) Sensitivity of the maximum temperature rise Tmax as a function of the cooling intensity N; b) first Barkelew diagram showing that all the sensitivity curves have a common envelope; c) second Barkelew diagram for differentiating between sensitive and insensitive operating conditions Referring to Figure 2.1a, it is possible to observe that the maximum reactor temperature with respect to the coolant temperature shows a sharp transition in correspondence of a certain value of the cooling intensity (that is, it exhibits a huge parametric sensitivity behavior). Barkelew created two different types of diagrams in order to differentiate between insensitive and sensitive operating region for a BR. In the first diagram, he represented the thermal behavior of the system (that is, max ) as a function of the cooling intensity and the reaction strength; in the second one, he shows the sets of (S, N) separating the insensitive from the sensitive operating region. Particularly, observing Figure 2.1b, Barkelew showed that all of the curves there represented possessed a common tangential envelope. The points on this envelope could be reported in terms of N/S vs. S couples in order to generate a curve that separated sensitive and insensitive regions of BR operation. All these consideration, as said before, have been easily extended by Barkelew to PFR using the analogy in between time and axial coordinate (initial and inlet concentrations/temperature, etc…). Of course, such an approach has strong limitations that they will be discovered several years later. After the studies carried out by Barkelew, it was clear that a profound analogy in between the mathematical modeling of batch and plug flow reactors could be established. Therefore, since continuous processes had become a sort “must” in order to achieve high productivities for a variety of the chemical compounds, chemical engineers started to refine the modeling for the detection of the runaway boundary for BRs with the main aim of extended the theory to PFRs. In 1964, Dente and Collina[7], referring to the thermal stability of PFRs, published a criterion of Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 31 intrinsic nature that defined the runaway boundary as a critical condition in correspondence of which the temperature vs. axial coordinate profile exhibited a region with positive second-order derivative (that is, concave profile) somewhere before the hot spot. Superimposing that such a region had to have zero amplitude, the following critical conditions arose: d 2 d 3 0 dz 2 dz 3 (2.4) Although developed independently[4], this criterion is perfectly identical to that of Thomas for batch reactors: simply the time is replaced by the axial coordinate. In 1966, Dente and his co-workers published some works[8,9] where they investigated the runaway behavior of a tubular reactor (PFR) in which two parallel or consecutive reactions occurred. The criterion used in their studies was the same that they had previously proposed for single reactions: that is, based on the reactor behavior in the temperature vs. axial coordinate plane (see Eq. 2.4). After the conduction of an extensive set of reactor numerical simulations, it was observed that a runaway of the outlet yield and selectivity occurred whenever a thermal runaway phenomenon was triggered. Some years later, in 1970, van Welsenaere and Froment[10] published a work in which they presented two intrinsic criteria for the identification of the runaway boundary. Particularly, they referred to a gas-solid tubular reactor (PFR) where a single irreversible reaction of the first order occurred (reactor walls are considered at a fixed temperature and densities are independent of the temperature) according to the following equations: dp a A p exp b dz T dT a B p exp b C T Tw dz T (2.5) (2.6) where: A M W PT ˆ S p0 ; B H ˆ S p0 ; cp C 4U ; c p dT l z v (2.7) and initial conditions: p z 0 p0 ; T z 0 T0 Tw (2.8) Apart from these different hypothesis, Eqs. (2.5) and (2.6) presented by Van Welsernaere and Froment can be considered perfectly equivalent to the classical PFR balance equations, when the conversion is replaced by the partial pressure p of the reactant in the gas phase and n is constrained to 1. Using a Runge-Kutta method for the numerical integration of the system describing the reactor, they observed that a pronounced sensitivity of the T vs. z profile, in terms of temperature maxima, Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 32 existed. Therefore, dividing Eq. (2.6) by Eq. (2.5), they obtained a single equation expressing the dependence of the temperature with respect to the partial pressure: that is, a trajectory in the partial pressure vs. temperature plane (see Figure 2.2). dT B C dp A A T Tw a p exp b T (2.9) Figure 2.2. p-T phase plane, showing p-T trajectories, maxima-curve pm, loci of inflexion points (pi,max and pi,min) and the “simplified” curve pcr Varying the values of the wall temperature in a certain range, van Welsenaere and Froment observed that the locus of the partial pressures in correspondence of the temperature maxima defined a curve (pm) which showed a clear maximum (point A in Figure 2.2). The analytical expression of such a curve is easily calculable and can be used to find the critical value of the maximum temperature and pressure for each wall temperature investigated. pm Tm Tw a B exp b C Tm (2.10) van Welsenaere and Froment concluded that the conditions in correspondence of the maximum of the pm vs. Tm curve are considered critical from the safety point of view and can be employed to identify the runaway boundary. It is worth to note that this criterion is based on an intrinsic property of the system. Secondarily, van Welsenaere and Froment focused on the reactor behavior in the T vs. z plane and they concluded that the hot-spot became more and more important (in terms of magnitude) as the inflexion points occurred before the temperature maximum. Such an observation is also based on the conclusions of previous works such as those of Thomas[11] and Dente et al[7]. Differentiating Eq. (2.6) with respect to z and equating it to zero, it is possible to obtain an equation, which is a function of T and p, expressing the locus of the inflexion points in the T vs. z plane. Particularly, investigating the sign of such an equation, it is possible to individuate two limit curves, called pi,min and pi,max, that bound the region of the T vs. z curve where its concavity is downwards (see Figure Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 33 2.2). Whenever an intersection in between a p-T trajectory and pi,min or pi,max occurs, an inflexion point is detected. It is found that the intersection of the p-T trajectories with pi,max always occurs after the intersection with the “pm-curve” and therefore, corresponds to an inflexion point beyond the maximum in the T vs. z curve. Conversely, not all the p-T trajectories have an intersection with pi,min: that is, some T vs. z curves do not have inflexion points before their maximum. As a first approximation, van Welsenaere and Froment assumed that to avoid inflexion points before the maximum of the T vs. z curve, and therefore runaway, the p-T trajectory corresponding to the chosen conditions should not intersect the pi,min curve. But such a condition is too conservative. Following the approach of Thomas[11], they tried to superimpose that the two inflection points were coincident and occurred before the maximum. Even if the mathematical treatment they followed was not completely correct (they referred to as a simplified treatment), they succeed in finding a critical curve, called pcr, which has been reported in Figure 2.2. Consequently, the second criterion of van Welsenaere and Froment can be summarized as follows: “runaway occurs whenever a p-T trajectory intersects the pcr curve”. The critical trajectory corresponding to the runaway boundary is that one that is tangent to the pcr curve. These two runaway criteria are equivalent because they substantially predict the same critical trajectory in the p-T plane. During the remaining years of the ’70, no significant improvements regarding the establishment of new runaway criteria for PFRs were documented. Anyway, it is worth to notice that some studies concerning the coolant flow direction in a PFR were published. Particularly, in 1972, Luss and Medellin[12] observed that the sensitivity behavior of a PFR with counter-current external cooling was quite different from that one described for co-current external cooling. The situation was further complicated by the possible occurrence of multiple steady states, which had not been observed previously for constant and co-current external cooling. Some years later, in 1979, Degnan and Wei[13] observed that the counter-current operating mode of a PFR led to greater possibility of runaway triggering with respect to the co-current one, since near the reactor inlet the temperature difference between the internal reacting mixture and the external coolant generally reached a minimum, while the reaction and the heat generation rates were at their maximum. Anyway they did not propose any runaway criterion. During the first years of the ‘80s, scientists and engineers started to rationalize the approach concerning the detection of the so-called runaway boundaries concentrating their efforts on all the different reactor typologies that were employed in the chemical industry. In order to achieve such a goal, they needed to study in deep all the aspects regarding both the thermal behavior of the different reactors and the relative parametric sensitivity with respect to the variation of a number of operating parameters (that is, refining the mathematical modeling of such systems), together with the eventual steady-state multiplicity conditions. Particularly, in 1981, Soria Lopez and his co-workers[14] published a work on the parametric sensitivity of a fixed bed catalytic reactor (PFR) analyzing the influence of the thermal gradients along the direction of the coolant flow. In order to describe the system, they solved classical PFR balance equations but they replaced the concentration by the partial pressure of the gaseous reactant, as it has been previously done by van Welsenaere and Froment[10] (see Eqs 2.5 and 2.6). At Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 34 that time, most of the runaway criteria discussed for PFRs were based on geometric properties exhibited by the temperature vs. conversion trajectory (reactant conversion was used as independent variable instead of the axial coordinate) corresponding to the solution of the equation resulting from dividing thermal balance equation by mass balance equation. (or Eq. (2.6) by Eq. (2.5)) [10]. It is interesting to observe that the solution of such an equation always predicts the occurrence of a maximum in the reactor temperature profile (that is, the so-called hot-spot always occurs). In the case of batch reactors, this was a safe approach because, as time passes, conversion unavoidably increases from 0 to 1: in the typical case where the coolant temperature does not exceed the initial temperature, a temperature maximum must be always reached. But, for tubular reactors, such an approach ignores the constraint imposed on the system thermal behavior by the finite length of the reactor that also limits conversion: as a matter of fact, when taking the axial coordinate as the independent variable, we may have the situation where no maximum occurs in the temperature profile. Soria Lopez and his co-workers defined such a situation as Pseudo-Adiabatic Operation (PAO), since the global heat transfer coefficient U was not equal to zero (therefore, we cannot talk about adiabatic conditions) but, contextually, the reactor approached a maximum when z∞ (of course such a maximum can correspond to runaway conditions). In other situations, Soria Lopez et al. observed that a Maximum occurred at a Finite Axial Reactor Position and they referred to it as (MFARP). PAO and MFARP operations are two different thermal responses of a PFR under different conditions and their behavior in terms of sensitivity had to be deeply analyzed in order to formulate a correct runaway criterion. Indeed, the criterion developed by Soria Lopez et al. is an extension of the first runaway criterion proposed by van Welsenaere and Froment: simply, Eq. 2.10 for the partial pressure in correspondence of the temperature maximum along the reactor axis (considering the possibility of the occurrence of the PAO region) is changed in: pm Tm Tcool Tm a B exp b C Tm (2.11) where the coolant temperature Tcool(Tm) is that one in correspondence of the maximum temperature Tm. Setting the derivative of such an equation with respect to temperature equal to zero, the maximum of the pm curve can be identified: that is, runaway boundary can be identified. Moreover, Soria Lopez proposed some algebraic simplification in order to calculate easily the maximum of the pm curve that conduced to a numerical error minor than 4% in the estimation of the critical conditions. In 1982, Morbidelli and Varma[15] published the first of a series of studies concerning the concept of parametric sensitivity in chemical reactors as an intrinsic criterion to define runaway boundaries. In this preliminary study, they analyzed the parametric sensitivity of a PFR in order to provide the critical values of the dimensionless adiabatic temperature rise (B) and Stanton number (St) defining both runaway and stable operations for all positive-order exothermic reactions with finite activation energies, and reactor inlet temperatures. In 1985, Morbidelli and Varma[16] published an interesting work which defined the runaway boundary using a generalized criterion based on the normalized parametric sensitivity of the reactor Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 35 temperature maximum θ* (that is, the so-called “objective function”) with respect to any system parameter Φ [17]. The local sensitivity of the dependent variable θ*, with respect to the input parameter Φ, is defined as the first derivative of the dependent variable with respect to the input parameter: s ( *, ) *( ) *( ) *() lim 0 (2.12) Even if higher-order derivatives can be calculated, the treatment to first-order ones is preferred because most applications are based on linear sensitivity analysis. Related to local sensitivity, the normalized sensitivity is defined as: S ( *, ) *( ) s ( *, ) * * (2.13) The original paper developed the criterion referring to batch and plug flow reactors but an extension to CSTRs would be provided in the following year[18]. Particularly, Morbidelli and Varma showed that the critical Semenov number, Ψcr, defined as the value where S(θ*;Φ) is maximum, is the same for any possible choice of the system parameters Φ. This indicates that when the system enters into the parametrically sensitive region, it becomes simultaneously sensitive to all the parameters that affect its behavior[4]. This fact provides great generality to the definition of the parametrically sensitive region, which may then be taken as the boundary between runaway and safe system behavior. It is worth noting that the sign of the normalized objective sensitivity S(θ*,Φ) can be either positive or negative and, such a sign, has a particular meaning: a positive (negative) value indicates that the temperature maximum increases (decreases) as the magnitude of the parameter with respect to the sensitivity is calculated increases. Thus, if the sensitivity is positive, the transition from nonrunaway to runaway behavior occurs as this parameter is increased, while if the sensitivity is negative, the same transition occurs when the corresponding parameter is decreased [4]. Therefore, in order to develop a general definition for the detection of the runaway boundary, it is more convenient to refer to the absolute value of the maximum of the normalized sensitivity function. In 1986, Henning and Perez[19] proposed an intrinsic criterion for PFRs based on the behavior of local sensitivity of reactor temperature (T) to the inlet temperature, Tin, as a function of the axial coordinate l: dS T , Tin dl d dT dl dTin (2.14) (2.15) With the inlet condition: S T , Tin 1 The Henning and Perez criterion defined the critical condition as the first occurrence of a minimum Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 36 on the S(T, Tin) — l curve before the temperature maximum on the T - l trajectory (see Figure 2.3). Figure 2.3. S(T,Tin) as a function of the axial coordinate l for different values of the ratio of reactant to coolant heat capacity for a PFR (τPFR). In order to give an interpretation of this criterion, it is necessary to refer to both the thermal balance equation for PFR and the S(T,Tin)-l critical curve. The first term of the thermal balance equation corresponds to the heat power generated inside the reactor while the second one is the heat power removed by the cooling system. From the inlet up to the minimum point, the S(T,Tin)-l critical trajectory decreases monotonically; this means that the first derivative with respect to inlet temperature for the heat transferred is greater than that for the heat produced. At the minimum point, these derivatives are equal and beyond it they have an opposite behavior. In other words, beyond the minimum, the limiting curve presents a zone where the sensitivity with respect to variations of Tin for the heat generated is greater than that for the heat transferred: that is, critical conditions arises. In 1998, Wu et al.[20] studied the influence of the occurrence of the PAO region on the determination of the runaway boundaries using the generalized parametric sensitivity (objective function: temperature maximum, θ*) but employing two different models for its calculation: the first by solving the dimensionless form of mass and energy balance equations, that is employing the so-called z-MV criterion; the second by solving their ratio, that is employing the so-called x-MV criterion. It is interesting to underline that when the reactor operates in the PAO region, the z-MV criterion is able to determine that the reactor is too short for developing a local temperature maximum (the temperature-axial coordinate profile is a monotonically increasing function) and, therefore, it automatically selects the temperature value at the reactor outlet as θ* in the sensitivity calculation (because it is the maximum along z). On the contrary, when the conversion is taken as the independent variable, the PFR model becomes identical to the BR model; therefore, we implicitly consider the possibility of complete conversion, which can only be assured in reactors of infinite length. It follows that the x-MV criterion does not account for the occurrence of the PAO region, thus providing runaway boundaries that are always more conservative than those predicted by the z- Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 37 MV criterion[4]. In 1999, another sort of revolution occurred: the group of Zaldívar and his co-workers developed the so-called divergence criterion for the detection of the runaway boundaries in BRs[21] and, in 2003, they extended it to whatever type of reactor[22]. Starting from the assumption that the dynamics of a reacting system can be always represented by a system of ordinary differential equations (ODEs), that suitably describes material and energy balances in the general form: dxi Fi x t dt (2.16) They showed that each state of the reacting system corresponded to a unique point onto the state space, which is called state point, and the system state time evolution was represented as a motion along a trajectory. According to the Liouville’s theorem[22], there is a relation between the state space volume of a N-dimensional dynamical system and its divergence. This relation can be expressed as: t V t V 0 exp div F x t 0 dt (2.17) (2.18) Where: N div F x t Fi x t i 1 xi Hence, for a dynamical system, the rate of change of an infinitesimal volume V t , following an orbit x t , is given by the divergence of the flow, which is locally equivalent to the trace of the F x t Jacobian. Even if for dissipative systems, as chemical reactors are, the divergence unavoidably decreases as time approaches , it has been observed[21] that when the system is operated under potentially runaway conditions it exhibits, during a certain period of time, a state space volume expansion. This means that trajectories originating from nearby starting points will diverge and such a feature can be correlated with the parametric sensitivity of the reactor temperature with respect to the system constitutive parameters or initial conditions. Therefore, taking into account such a correlation between parametric sensitivity and the divergence of nearby trajectories, Strozzi et al. defined a general criterion for runaway detection. Particularly, a reactor is operating under runaway conditions when the divergence of the ODEs system describing its thermal behavior becomes positive on a segment of the reaction path. div F x t 0 (2.19) This criterion is so general that can be easily applied to both controlled (there are suitable equations expressing temperature control strategy, e.g. isothermal) and uncontrolled (typically, when the reactor operates in the isoperibolic temperature control mode, coolant temperature is assumed to be Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 38 constant for all process duration) systems independently on the reactor type (CSTR, PFR, BR, SBR). Nevertheless, such a general criterion needs to be subjected to a single constraint in order to be in accordance with both theoretical and experimental evidences on runaway detection. This constraint takes into account the presence of a fast reaction that does not produce directly heat[22]. For describing the kinetics of the reacting systems of interest, usually the minimum number of global reactions (NR) is utilized; therefore, thanks to the stoichiometric constraint, the NC number of moles of each species can be computed from the NR extent-of-reactions, which are easily related to the conversion of some reactants or intermediates. Consequently, the NC material balances can be recast in NR balance equations for the NR extent-of-reactions, one of these being that of the fast reaction that does not produce or consume heat. Particularly, such an extent-of-reaction can generate a strong volume contraction (synthesized in a non-runaway condition according to the divergence criterion) in the N-dimensional state space arising by considering that extent-of-reaction in the divergence computation, but a global expansion (runaway) in the (N-1)-dimensional space arising by not considering that extent-of-reaction into the divergence computation. Therefore, as a general constraint for the runaway criterion based on divergence, all contributions arising from extent-of-reactions that are not related to heat evolutions must be disregarded into the divergence calculation. Such an expedient is necessary in order to avoid evaluation errors due to the presence of strongly negative terms (the divergence is a sum with sign) that are not related to the runaway phenomenon. Apart from this expedient, the divergence criterion can be considered as the actual maximum powerful tool capable of detecting explicitly the runaway boundaries for whatever type of chemical reactor and reaction path. As well as the sensitivity parametric criterion, it is one of the most used nowadays, whereas the other methods here described are interesting mainly from an historical point of view and are rarely employed in current applications. Neither the divergence criterion nor the parametric sensitivity criterion has ever been extended to an unsteady state PFR model. Since the purpose of this work was not to create a new criterion for runaway boundary individuation, the problem of extending one of these criteria has arisen. The parametric sensitivity criterion has been chosen, because of its relative simplicity: being based on first-order derivatives, it has been possible to insert it in the Matlab® code used to integrate the system, as it will be discussed in details in the next section. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 39 3) Case study: Naphthalene Oxidation Reaction 3.1 Naphthalene Oxidation Reaction In this work, the gas-phase naphthalene oxidation to phthalic anhydride is studied. This reaction has already been object of a case study by Morbidelli and Varma[4] and particularly by Sierra Ciro[23], thanks to its simple kinetics and to the availability of data. The phthalic anhydride is a toxic, white crystalline compound used in the manufacture of phthaleins and other dyes, resins, plasticizers and insecticides. It is considered a versatile intermediate in organic chemistry, in part because it is bifunctional and readily available. The primary use is as a chemical intermediate in the production of plasticizers from polyvinyl chloride (PVC). Figure 3.1: Kinetic scheme of naphthalene oxidation reaction Naphthalene oxidation is a highly exothermic reaction, carried out commercially on a V2O5 catalyst in multitubular reactors with external cooling. The kinetic scheme of this reaction is shown in figure 3.1. The reaction rate can be assumed pseudo first-order and given by: E r ' A exp att RT pO pnapht (3.1) This expression is defined on the basis of catalyst weight. Multiplying by the catalyst density: E r A exp att RT pO pnapht ˆ s (3.2) Where pO is the partial pressure of oxygen, which remains constant in the reactor since this reactant is in excess, and pnapht is the partial pressure of naphthalene, which can be expressed in function of naphthalene concentration: pnapht M w PT Cnapht (3.3) Significant data for the naphthalene oxidation reaction are reported in table 3.1. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 40 kmol kg s kPa 2 kJ cˆP ,mix , g 1.044 kgK kJ cˆ p , s 0.95 kgK A 11.16 PT 101.3kPa pO 21.07kPa kJ m s K m2 2.5e 5 s kJ H 1.289e6 kmol 0.99 kg 0 1.293 3 m kg ˆ s 1300 3 m U 9.61e 2 m2 s kJ Eatt 1.134e5 kmol kg M W 29.48 kmol kg M W , s 181.88 kmol Dm 2.0e 6 2 Table 3.1: Data for naphthalene oxidation reaction With reference to reactor inlet conditions, a range of values has been considered, as reported in table 3.2. Pin 1.4 2.2kPa Tin 620 720 K Tw 580 680 K v0 0.1 1 m s Table 3.2: Range of values for inlet parameters The inlet concentration can be computed as: P Cin in RTin (3.4) And so it varies with the inlet parameters. The reactor length L is 2 m and the diameter dT is 0.025 m. Species and energy transport equations for this system according to the various models considered are reported below. 3.1.1 Steady State model Material balance equation for naphthalene: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 41 v0 Cnapht E Aexp att pO pnapth ˆ s RT (3.5) H Aexp Eatt p p ˆ 4U (T T ) T w RT O napth s d cˆ l 0 cˆP,mix , g T 0 P , mix , g (3.6) l Heat balance equation: v0 C l 0 Cin IC: napht T l 0 Tin (3.7) (3.8) H Aexp Eatt p p ˆ 4U (T T ) T T v0 w RT O napth s d cˆ t l 0 cˆP ,mix, g T 0 P , mix , g (3.9) C t 0 Cin IC: napht T t 0 Tw (3.10) C l 0 Cin BC: napht T l 0 Tin (3.11) E Aexp att pO pnapth ˆ s RT (3.12) T 2T T H 4U E 2 v0 Aexp att pO pnapth ˆ s (T Tw ) t l l 0cˆP ,mix , g dT 0cˆP,mix , g RT (3.13) C t 0 Cin IC: napht T t 0 Tw (3.14) 3.1.2 Unsteady State model Material balance equation for naphthalene: Cnapht t v0 Cnapht l E Aexp att pO pnapth ˆ s RT Heat balance equation: 3.1.3 Unsteady State model with diffusion Material balance equation for naphthalene: Cnapht t Dm 2Cnapht l 2 v0 Cnapht l Heat balance equation: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 42 l 0 Cin C Inlet BC: napht T l 0 Tin (3.15) Cnapht (l L) 0 l Outlet BC: T ( l L ) 0 l (3.16) 3.1.4 Steady State model with variable density In order to evaluate the reaction mixture density, the equation of state for ideal gases has been used: P M W RT (3.17) This assumption is reasonable as the reaction mixture consists mostly of air. Thus, the partial pressure of naphthalene can be written as: pnapht RT Cnapht (3.18) In a rigorous approach, the Navier-Stokes equation along the axial direction should be considered in order to link the total pressure and the velocity and solve the system. This can be avoided with some reasonable assumptions: 1) Constant total massive flow rate 2) Constant molecular weight of reaction mixture 3) Constant total pressure The first assumption descends directly from the presence of steady state conditions, while the other two are justified by the large excess of air in the mixture. Thus, the axial velocity can be expressed as a simple function of inlet velocity and temperature variations: T v v0 TIN (3.19) (3.20) The resulting transport equations for naphthalene and for energy are: v T T Cnapht E Cnapht 0 v0 A exp att TIN l TIN l RT cˆP ,mix , g PT M w T H v0 ˆ mix cˆP ,mix RT IN l ˆ mix cˆP ,mix C l 0 Cin IC: napht T l 0 Tin pO ˆ s RT Cnapht 4U E T Tw (3.21) A exp att pO ˆ s RT Cnapht ˆ ˆ RT d c T mix P , mix (3.22) Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 43 3.1.5 Unsteady State model with variable density The assumptions used for the steady state case have been maintained: 1) Constant total massive flow rate 2) Constant molecular weight of reaction mixture 3) Constant total pressure Rigorously, the total massive flow rate is not constant in unsteady state conditions, but potential variations are negligible due to the large excess of air. Thus, transport equations for naphthalene and energy can be written as: Cnapht t Cnapht v0 T T Cnapht E v0 A exp att TIN l TIN l RT cˆ P M T T H P ,mix , g T w v0 t ˆ mix cˆP ,mix RT IN l ˆ mix cˆP ,mix E A exp att RT pO ˆ s RT Cnapht (3.23) 4U T Tw (3.24) pO ˆ s RT Cnapht d ˆ cˆ T mix P , mix C t 0 Cin IC: napht T t 0 Tw (3.25) C l 0 Cin BC: napht T l 0 Tin (3.26) 3.2 Dimensionless form of transport equations Transport equations for heat and mass can be rewritten using dimensionless variables and parameters, respectively reported in tables 3.3 and 3.4. The dimensionless form is typically more suitable than the dimensional one in the frame of parametric sensitivity analysis. Moreover, from the numerical point of view, it eases the system’s convergence. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 44 Dimensionless Definition variable Conversion Cin Cnapht Dimensionless Definition parameter Arrhenius number γ Dimensionless T Tin (3.28) temperature Tin Damköhler number Da Dimensionless T T w w in (3.29) wall Tin temperature Stanton number St 4 U L (3.34) d T 0 cˆP ,mix , g v 0 B H Cin Dm (3.36) Lv0 Dimensionless axial coordinate Cin (3.27) Dimensionless adiabatic temperature rise l z (3.30) L Dimensionless t v 0 (3.31) time L Material diffusion parameter E att (3.32) RTin L A exp γ ρˆ S pO PT M w 0 v0 0 cˆP ,mix , g Tin (3.33) (3.35) Table 3.3: Dimensionless variables Thermal diffusion parameter Lv0 . (3.37) Heat capacity ratio RH 0 cˆP, mix, g ˆ mix cˆP, mix (3.38) Table 3.4: Dimensionless parameters Dimensionless groups defined above have a precise physical meaning. Arrhenius number represents the activation energy of the reaction. Damköhler number represents the ratio between the characteristic process time, which in this case coincides with the characteristic time of convective mass transport, and the characteristic reaction time: it indicates the predominance of either transport or reaction phenomena. Stanton number represents the ratio between the characteristic process time and the characteristic cooling time and thus it is a measure of the cooling system efficiency. The dimensionless adiabatic temperature rise is a measure of the exothermicity of the reaction. Diffusion parameters express the ratio between diffusive phenomena and convective phenomena (i.e., the reciprocal of Péclet numbers). A further dimensionless number is n, which represents the reaction order. It has not been considered as in the examined case the reaction is pseudo first order. It has to be noticed that the expressions reported above are written with reference to the constant density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 45 case, but they differ slightly from the variable density ones, which can be deduced with a simple substitution. 3.2.1 Steady State model Dimensionless material balance equation in terms of naphthalene conversion: d Da exp dz 1 (1 ) (3.39) Dimensionless heat balance equation: 1 d exp B Da dz 1 St (1 ) ( w ) B Da (3.40) z 0 0 IC: z 0 0 (3.41) 3.2.2 Unsteady State model Dimensionless material balance equation in terms of naphthalene conversion: Da exp z 1 1 (3.42) d B Da exp d z 1 (1 ) St ( w ) (3.43) t 0 0 IC: t 0 w (3.44) z 0 0 BC: z 0 0 (3.45) Dimensionless heat balance equation: 3.2.3 Unsteady State model with diffusion Dimensionless material balance equation in terms of naphthalene conversion: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 46 2 2 Da exp z z 1 1 (3.46) (1 ) St ( w ) (3.47) Dimensionless heat balance equation: d 2 2 B Da exp d z z 1 t 0 0 IC: t 0 w (3.48) (3.49) z 1 0 z Outlet BC: z 1 0 z (3.50) z 0 0 Inlet BC: z 0 0 3.2.4 Steady State model with variable density Dimensionless material balance equation in terms of naphthalene conversion: Da exp z 1 1 1 z 1 (3.51) (3.52) (3.53) Dimensionless heat balance equation: RH Da B exp z 1 z 0 0 IC: z 0 0 1 1 St w 3.2.5 Unsteady State model with variable density Dimensionless material balance equation in terms of naphthalene conversion: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 47 1 1 Da exp z 1 z 1 1 (3.54) (3.55) Dimensionless heat balance equation: RH Da B exp z 1 1 1 St w 0 0 IC: 0 w (3.56) z 0 0 Inlet BC: z 0 0 (3.57) 3.3 Matlab® Code Setup 3.3.1 Steady State Models Both steady state model with constant density and with variable density are simple ODEs systems. Matlab® contains a whole series of implicit or explicit ODE solvers with different features. In this case, the implicit solver ode15s has been chosen, as it is very powerful and able to solve also stiff problems. Required inputs for ode15s are the initial conditions of the problem and the interval in which it has to integrate. The independent variable of this problem is the dimensionless length of the reactor, so the integration interval goes from 0 to 1. It is not required to define the integration step as ode15s works with a variable step, choosing it intrinsically while integrating. The obtained outputs are the conversion and temperature profiles along the reactor. 3.3.2 Unsteady State Models In order to solve the PDEs system associated with the unsteady state models, a MatMol code has been implemented. Spatial discretization First of all, it is necessary to define a spatial grid for the discretization of the dependent variables. As discussed before, a thicker grid is able to improve accuracy. On the other hand, increasing the number of points means increasing the time required for the simulation. For the constant density model, two grids have been tested: one with 101 points (step dz=0.02 m) and one with 201 points (step dz=0.01 m). It has been noticed that the time required for the Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 48 simulation is not strongly influenced by the number of points, due to the implicit and powerful solver used. Thus, for accuracy’s sake, the spatial grid of 201 points has been chosen. The variable density model is characterized by a higher complexity and consequently its resolution requires a higher computational effort. In this case, a spatial grid of 201 points is not sufficient to ensure accuracy. Despite the long time required for the simulation, a grid of 501 points has been chosen. Two arrows of discretized dependent variables are obtained: 1 2 i N (3.58) 1 2 i N (3.59) It is possible to write a single arrow which contains both variables, in order to solve the PDEs system as a single PDE: 1 2 i x N 1 2 i N (3.60) The set-up values of this vector are defined in any point by the initial conditions of the problem. Finite Differences Scheme Secondly, finite differences scheme has to be defined. Three different schemes have been tested: a three-points centered scheme, a five-points centered scheme and a seven-points centered scheme. Again, choosing a higher-order differentiation scheme Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 49 may significantly improve accuracy but it means increasing simulation time as well. Stagewise differentiation has been chosen in order to avoid computing a second differentiation matrix for second-order derivatives. With reference to the three-points scheme and the five-points one, slight differences has been detected. Otherwise, using the seven-points scheme, the code is not able to converge. Thus, the fivepoints scheme will be implemented. Boundary conditions With reference to boundary conditions, they have been implemented using a mass matrix, which allows them be included directly in the dependent variables arrow. Time integration Finally, time integration by means of ode15s has to been set up. Initial conditions have been already implemented, while a time interval from 0 to 1 (in dimensionless term) is chosen. This interval is equal to the time in which the reaction mixture is able to travel along the whole reactor up to the outlet. This choice is justified by the PFR behavior: it is indeed well-known that PFR reactors reach steady state conditions right after the first passage. Thus, if some peculiar temporal phenomena are to occur, they will occur before this particular time. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 50 4) Runaway Boundaries for Naphthalene Oxidation Reaction In order to understand how the choice of the reactor model may affect the reactor safety, runaway boundaries have to be computed with the different models considered. The parametric sensitivity criterion has been used in order to compute the sensitivity of the maximum of temperature Tmax with respect to three different input parameters: inlet pressure Pin, inlet temperature Tin, wall temperature Tw. This computation can be simply incorporated in a Matlab® code with a basic for cycle. For each step, the considered parameter is chosen among a defined range, the system is solved for that particular value and the maximum of temperature is found. Eventually, sensitivity can be computed with a simple finite difference method and the plot of Tmax and S(Tmax) can be obtained. The critical values for the different parameters can be identified from the plots. 4.1 Runaway Boundaries for Steady State model 4.1.1 Parametric sensitivity and critical values The plots of Tmax and of S(Tmax, Pin) as a function of Pin (figures 4.1.1. and 4.1.2) are computed in the following conditions: Tin 625K Tw 625K m v0 1 s Figure 4.1.1: Tmax as function of Pin, Tin=625K, Tw=625K and v0=1m/s for the steady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 51 Figure 4.1.2: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=1m/s for the steady state model The critical value of inlet pressure is 1.85 kPa. It can be noticed that this value coincides with the one computed by Morbidelli and Varma [4]. The plots of Tmax and of S(Tmax, Tin) as a function of Tin (figures 4.1.3. and 4.1.4) are computed in the following conditions: Pin 1.6kPa Tw 625K m v0 1 s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 52 Figure 4.1.3: Tmax as a function of Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for the steady state model Figure 4.1.4: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for the steady state model The critical value for the inlet temperature is 661K. The plots of Tmax and of S(Tmax, Tw) as a function of Tw (figures 4.1.5 and 4.1.6) are computed in the following conditions: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 53 Pin 1.6kPa Tin 625K m v0 1 s Figure 4.1.5: Tmax as a function of Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for the steady state model Figure 4.1.6: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for the steady state model The critical value for the wall temperature is 632K. In conclusion, a set of critical values for the three parameters considered has been found: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 54 Pin ,cr 1.85 kPa Tin ,cr 661K Tw,cr 632 K Thus, a safe operation can be achieved by setting these parameters below their critical values. 4.1.2 Conversions With reference to industrial applications, it is important to define whether the safe conditions allow reaching an acceptable conversion values. Conversion profiles along the reactor have been computed in two different safe conditions: in the first case (figure 4.1.7), with a high inlet pressure (1.8 kPa); in the second one (figure 4.1.8), with a high inlet temperature (650 K). Both the cases tend to maximize the reactant conversion; namely, they tried to perform a safe process optimization. Figure 4.1.7: Steady state conversion along the reactor. Pin=1.8kPa, Tin=625K, Tw=625K and v0=1m/s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 55 Figure 4.1.8: Steady state conversion along the reactor. Pin=1.65kPa, Tin=650K, Tw=625K and v0=1m/s In both cases, the reactor is not able to reach total conversion. In order to improve the reactor performance, it is possible to decrease the inlet velocity. A lower inlet velocity means a higher residence time and thus a higher conversion. For example, with an inlet velocity of 0.1 m/s the reactor is able to reach total conversion before the outlet, as shown in figure 4.1.9. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 56 Figure 4.1.9: Steady state conversion along the reactor. Pin=1.6kPa, Tin=625K, Tw=625K and v0=0.1m/s 4.1.3 Parametric sensitivity with low inlet velocity It is necessary to verify whether the proposed solution is acceptable from the safety point of view. Thus, a parametric sensitivity analysis has been performed implementing a lower inlet velocity value, i.e. 0.1 m/s. The sensitivity profiles with respect to the three parameters are reported respectively in figures 4.1.10, 4.1.11, and 4.1.12. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 57 Figure 4.1.10: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=0.1m/s for steady state model Figure 4.1.11: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=0.1m/s for steady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 58 Figure 4.1.12: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=0.1m/s for steady state model The inlet velocity seems to have no effect on runaway boundaries. Consequently, according to this reactor model, the proposed solution can be adopted and a safe optimization of the system has been reached. It is necessary to investigate whether this safe optimization approach is correct. 4.2 Runaway Boundaries for Unsteady State model 4.2.1 Parametric sensitivity and critical values The plots of Tmax and of S(Tmax, Pin) as a function of Pin (figures 4.2.1. and 4.2.2) are computed in the following conditions: Tin 625K Tw 625K m v0 1 s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 59 Figure 4.2.1: Tmax as a function of Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model Figure 4.2.2: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model The critical value of inlet pressure is 1.84 kPa, with a negligible difference with respect to the steady state case. The plots of Tmax and of S(Tmax, Tin) as a function of Tin (figures 4.2.3. and 4.2.4) are computed in the following conditions: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 60 Pin 1.6kPa Tw 625K m v0 1 s Figure 4.2.3: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625K and v0=1m/s for unsteady state model Figure 4.2.4: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625K and v0=1m/s for unsteady state model The critical value of inlet temperature is 657 K, with a slight difference with respect to the steady state case. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 61 The plots of Tmax and of S(Tmax, Tw) as a function of Tw (figures 4.2.5 and 4.2.6) are computed in the following conditions: Pin 1.6kPa Tin 625K m v0 1 s Figure 4.2.5: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625K and v0=1m/s for unsteady state model Figure 4.2.6: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625K and v0=1m/s for unsteady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 62 The critical value of wall temperature is 632 K, without differences with respect to the steady state case. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 1.84 kPa Tin ,cr 657 K Tw,cr 632 K There are small differences between these values and the ones computed with the steady state model. Of course, such differences are not significant from an operative point of view, therefore can be disregarded concluding that the runaway boundary predicted by both models is the same. Conversely, the maximum temperature computed through the unsteady state model is much higher than the one computed through the steady state model: this can lead to some problems from the safety point of view (e.g., issues in material and reactor design). 4.2.2 Conversions Also for the unsteady state model, conversions have been computed as a performance indicator of the system. An example of a conversion profile at finale time along the reactor is reported in figure 4.2.7. Figure 4.2.7: Unsteady state conversion along the reactor. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=1m/s As in the steady state case, high inlet velocity prevents the system from reaching high conversions. The situation can be improved decreasing the inlet velocity. For example, with an inlet velocity of 0.5 m/s the conversion is over 70% and with an inlet velocity of 0.1 m/s the system is able to reach total conversion, as shown respectively in figures 4.2.8 and 4.2.9. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 63 Figure 4.2.8: Unsteady state conversion along the reactor. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.5 m/s Figure 4.2.9: Unsteady state conversion along the reactor. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.1 m/s 4.2.3 Parametric sensitivity with medium inlet velocity In order to verify whether the proposed optimization is safe, parametric sensitivity analyses have been carried out. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 64 First, an inlet velocity of 0.5 m/s has been considered. The sensitivity profiles with respect to the three parameters are reported in figures 4.2.11, 4.2.13 and 4.2.15, as well as the maximum temperature plots (figures 4.2.10,4.2.12 and 4.2.14). Figure 4.2.10: Tmax as a function of Pin, Tin=625K, Tw=625K and v0=0.5m/s for unsteady state model Figure 4.2.11: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=0.5 m/s for unsteady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 65 The critical value of inlet pressure is 1.81 kPa, with a slight difference with respect to the steady state case. Figure 4.2.12: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625K and v0=0.5m/s for unsteady state model Figure 4.2.13: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625K and v0=0.5 m/s for unsteady state model The critical value of inlet temperature is 652 K: differences between the unsteady and the steady state are increasing. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 66 Figure 4.2.14: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625K and v0=0.5 m/s for unsteady state model Figure 4.2.15: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625K and v0=0.5 m/s for unsteady state model The critical value of wall temperature is 631 K, with a negligible difference with respect to the steady state case. In conclusion, a set of critical values for the three parameters considered has been found: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 67 Pin ,cr 1.81 kPa Tin ,cr 652 K Tw,cr 631K As it can be noticed, the differences between the two models are increasing. Although they are still quantitative small, they highlight a qualitative difference in the reactor behavior: in the steady state model, the inlet velocity has no effect on runaway boundary, whereas in the unsteady state model this parameter can change the critical values of the others. Noticeably, decreasing the inlet velocity leads to a change towards lower critical values, i.e. to an unsafe situation. 4.2.4 Parametric sensitivity with low inlet velocity A parametric sensitivity analysis has been performed also with an inlet velocity of 0.1 m/s, which is required to reach total conversion. The sensitivity profiles with respect to the three parameters are reported in figures 4.2.17, 4.2.19 and 4.2.21, as well as the maximum temperature plots (figures 4.2.16, 4.2.18 and 4.2.20). Figure 4.2.16: Tmax as function of Pin, Tin=625K, Tw=625K and v0=0.1 m/s for unsteady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 68 Figure 4.2.17: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=0.1 m/s for unsteady state model The critical value of inlet pressure is 1.66 kPa, confirming the qualitative behavior highlighted in the previous case. Moreover, a quantitative difference between the two models arises in this case. Figure 4.2.18: Tmax as a function of Tin, Pin=1.6kPa, Tw=625K and v0=0.1 m/s for unsteady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 69 Figure 4.2.19: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=0.1 m/s for unsteady state model The critical value for the inlet temperature is 640K, confirming both the qualitative and the quantitative differences. Figure 4.2.20: Tmax as a function of Tw, Pin=1.6kPa, Tin=625K and v0=0.1 m/s for unsteady state model Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 70 Figure 4.2.21: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=0.1 m/s for unsteady state model The critical value for the wall temperature is 627K. The qualitative behavior is confirmed, whereas the quantitative differences are still quite small. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 1.66 kPa Tin ,cr 640 K Tw ,cr 627 K The qualitative behavior highlighted in the previous case has been confirmed. Moreover, with low inlet velocities a quantitative reduction of the critical values can be seen, especially for the inlet pressure and the inlet temperature. Thus, in the case of low inlet velocity, which may be required in industrial applications in order to reach total conversion, the steady state model may lead to unsafe considerations. In practical applications, choosing a value of Pin or Tin lower than the critical one predicted using the steady state model but higher than the one predicted by the unsteady state model would lead to the thermal runaway of the system. Finally, it can be observed that the magnitude of the temperature peaks is much higher in the unsteady state case than in the steady state one. Consequently, the effects of a thermal runaway could be worse than the ones predicted with a steady state model. 4.3 Runaway Boundaries for Unsteady State model with diffusion 4.3.1 Parametric sensitivity and critical values The plots of Tmax and of S(Tmax, Pin) as a function of Pin (figures 4.3.1. and 4.3.2) are computed in the following conditions: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 71 Tin 625K Tw 625K m v0 1 s Figure 4.3.1: Tmax as function of Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model with diffusion Figure 4.3.2: Sensitivity with respect to Pin, Tin=625K, Tw=625K and v0=1m/s for unsteady state model with diffusion The critical value of inlet pressure is 1.85 kPa, the same as the steady state model. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 72 The plots of Tmax and of S(Tmax, Tin) as a function of Tin (figures 4.3.3. and 4.3.4) are computed in the following conditions: Pin 1.6kPa Tw 625K m v0 1 s Figure 4.3.3: Tmax as a function of Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for unsteady state model with diffusion Figure 4.3.4: Sensitivity with respect to Tin, Pin=1.6kPa, Tw=625K and v0=1m/s for unsteady state model with diffusion Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 73 The critical value for the inlet temperature is 657K, the same as the unsteady state model without diffusion and slightly similar to the steady state model. The plots of Tmax and of S(Tmax, Tw) as a function of Tw (figures 4.3.5 and 4.3.6) are computed in the following conditions: Pin 1.6kPa Tin 625K m v0 1 s Figure 4.3.5: Tmax as a function of Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for unsteady state model with diffusion Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 74 Figure 4.3.6: Sensitivity with respect to Tw, Pin=1.6kPa, Tin=625K and v0=1m/s for unsteady state model with diffusion The critical value for the wall temperature is 632K, the same as the two models previously considered. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 1.85 kPa Tin ,cr 657 K Tw,cr 632 K As regards the critical values, there are not significant differences between this model and the previous ones. On the other side, the magnitude of the temperature maximum is higher than the one predicted by the steady state model. 4.3.2 Conversions Conversion profiles along the reactor for different values of inlet velocity are reported in figure 4.3.7, 4.3.8 and 4.3.9. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 75 Figure 4.3.7: Unsteady state conversion with diffusion. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=1m/s Figure 4.3.8: Unsteady state conversion with diffusion. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.5 m/s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 76 Figure 4.3.9: Unsteady state conversion with diffusion. Pin=1.6 kPa, Tin=625K, Tw=625K and v0=0.1 m/s As expected, also in this case decreasing the inlet velocity has a positive effect on the conversion. Thus, sensitivity analysis with medium inlet velocity (i.e., 0.5 m/s) and low inlet velocity (i.e., 0.1 m/s) have been performed. 4.3.3 Parametric sensitivity with medium inlet velocity The plots of temperature maximum and sensitivity as a function of the different parameters are computed with an inlet velocity of 0.5 m/s and are reported below (from figure 4.3.10 to figure 4.3.15). Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 77 Figure 4.3.10: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion Figure 4.3.11: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion The critical value of inlet pressure is 1.81 kPa, slightly smaller than the steady state case but equal to the unsteady state case without diffusion. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 78 Figure 4.3.12: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion Figure 4.3.13: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with diffusion The critical value for the inlet temperature is 652K, equal to the unsteady state case without diffusion and smaller than the steady state case. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 79 Figure 4.3.14: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.5 m/s for unsteady state model with diffusion Figure 4.3.15: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625K and v0=0.5 m/s for unsteady state model with diffusion The critical value for the wall temperature is 631K, as in the case without diffusion. In conclusion, a set of critical values for the three parameters considered has been found: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 80 Pin ,cr 1.85 kPa Tin ,cr 657 K Tw,cr 631 K These results demonstrate the same qualitative behavior of the unsteady state model without diffusion, i.e. the change of critical parameters towards smaller values as the inlet velocity decreases. From a quantitative point of view, the results are almost identical to the ones obtained with the model without diffusion. 4.3.4 Parametric sensitivity with low inlet velocity The plots of temperature maximum and sensitivity as a function of the different parameters are computed with an inlet velocity of 0.1 m/s and reported below, from figure 4.3.16 to figure 4.3.21. Figure 4.3.16: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 81 Figure 4.3.17: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion The critical value of inlet pressure is 1.68 kPa. There is a negligible difference with respect to the unsteady state model without diffusion and a significant difference with respect to the steady state model. Figure 4.3.18: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 82 Figure 4.3.19: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with diffusion The critical value for the inlet temperature is 643 K, slightly similar to the value obtained with the model without diffusion and much different from the value obtained by the steady state one. Figure 4.3.20: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with diffusion Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 83 Figure 4.3.21: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with diffusion The critical value for the wall temperature is 627 K, the same as the unsteady state without diffusion. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 1.68 kPa Tin ,cr 643 K Tw,cr 627 K To sum up, the unsteady state model with diffusion reveals the same qualitative behavior as the model without diffusion and thus it highlights once again the lack of conservativism of the steady state model in the case of low inlet velocity. From a quantitative point of view, the critical values computed through this model are very similar to the ones computed neglecting material and thermal diffusion. Indeed, the differences are not significant and the models can be considered equivalent. Thus, diffusive phenomena seem to have no influence on the runaway boundaries and the presence of the unsteady state term (i.e., temporal derivative) can be regarded as the most influential factor on the reactor’s behavior. 4.4 Runaway Boundaries for Steady State model with variable density 4.4.1 Parametric sensitivity and critical values The plots of Tmax and of S(Tmax, Pin) as a function of Pin (figures 4.4.1. and 4.4.2) are computed in the following conditions: Tin 625K Tw 625K m v0 1 s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 84 Figure 4.4.1: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=1 m/s for steady state model with variable density Figure 4.4.2: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=1 m/s for steady state model with variable density The critical value of inlet pressure is 1.85 kPa, the same as the steady state model with constant density. The plots of Tmax and of S(Tmax, Tin) as a function of Tin (figures 4.4.3. and 4.4.4) are computed in the following conditions: Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 85 Pin 1.6kPa Tw 625K m v0 1 s Figure 4.4.3: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for steady state model with variable density Figure 4.4.4: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for steady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 86 The critical value for the inlet temperature is 661 K, the same as the steady state model with constant density. The plots of Tmax and of S(Tmax, Tw) as a function of Tw (figures 4.4.5 and 4.4.6) are computed in the following conditions: Pin 1.6kPa Tin 625K m v0 1 s Figure 4.4.5: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for steady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 87 Figure 4.4.6: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for steady state model with variable density The critical value for the wall temperature is 632 K, the same as the steady state model with constant density. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 1.85 kPa Tin ,cr 661 K Tw,cr 632 K The critical values are identical to the ones obtained in the case of constant density. 4.4.2 Conversions Conversion profiles along the reactor for different values of inlet velocity are reported in figures 4.4.7, 4.4.8 and 4.4.9. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 88 Figure 4.4.7: Steady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=1 m/s Figure 4.4.8: Steady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.5 m/s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 89 Figure 4.4.9: Steady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.1 m/s The assumption of variable density does affect the behavior of the reactor in terms of conversion. This detailed model indeed considers also the effect of the catalyst, which is very efficient and has a positive effect on the conversion. Nevertheless, a low inlet velocity is still required to reach total conversion before the outlet. The results of sensitivity analysis in the case of lower inlet velocities are not reported, because they are analogous to the one obtained in the case of constant density, i.e., the runaway boundaries are not affected by the inlet velocity. In conclusion, the variable density model shows an effect on reactor performance, i.e. conversions, but no effect on runaway boundaries, which are identical to the ones predicted by the constant density model, regardless of the value of inlet velocity. 4.5 Runaway Boundaries for Unsteady State model with variable density 4.5.1 Parametric sensitivity and critical values The plots of Tmax and of S(Tmax, Pin) as a function of Pin (figures 4.5.1. and 4.5.2) are computed in the following conditions: Tin 625K Tw 625K m v0 1 s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 90 Figure 4.5.1: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=1 m/s for unsteady state model with variable density Figure 4.5.2: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=1 m/s for unsteady state model with variable density The critical value of inlet pressure is 2.67 kPa, much higher than in the steady state case. There is also a difference in the magnitude of the temperature peak, which is higher in the unsteady state case. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 91 The plots of Tmax and of S(Tmax, Tin) as a function of Tin (figures 4.5.3. and 4.5.4) are computed in the following conditions: m Pin 1.6kPa Tw 625K v0 1 s Figure 4.5.3: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for unsteady state model with variable density Figure 4.5.4: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=1 m/s for unsteady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 92 The critical value for the inlet temperature is 664 K, slightly higher than in the steady state case. As regards the magnitudes of the temperature peak and of the sensitivity curve, there is a huge increase with respect to the steady state case. The plots of Tmax and of S(Tmax, Tw) as a function of Tw (figures 4.5.5 and 4.5.6) are computed in the following conditions: Pin 1.6kPa Tin 625K m v0 1 s Figure 4.5.5: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for unsteady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 93 Figure 4.5.6: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=1 m/s for unsteady state model with variable density The critical value for the wall temperature is 643 K, higher than in the steady state case. Also for this parameter, an increase in the magnitudes of the curves is present. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 2.67 kPa Tin ,cr 664 K Tw,cr 643K There are huge differences between the two models. The behavior of the unsteady state system is strongly affected by the assumption of variable density. Consequently, the critical values for the three parameters are higher than the ones predicted by the steady state model and, obviously, higher than the ones predicted by the unsteady state model with constant density. Moreover, a strong effect on the magnitude of the temperature peaks can be seen, as well as on sensitivity values. The parameter that shows this behavior most clearly is the inlet temperature: if it exceeds the critical value, it determines a huge increase on the maximum temperature of the reactor. Thus, a thermal runaway in these conditions is likely to be a catastrophic event. 4.5.2 Conversions Conversion profiles at the final time along the reactor are reported in correspondence with different values of inlet velocity in figures 4.5.7, 4.5.8 and 4.5.9. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 94 Figure 4.5.7: Unsteady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=1 m/s Figure 4.5.8: Unsteady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.5 m/s Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 95 Figure 4.5.9: Unsteady state conversion with variable density. Pin=1.6 kPa, Tin=625 K, Tw=625 K and v0=0.1 m/s In normal operation conditions, there are no differences between the conversion predicted with the steady state model and the ones predicted by the unsteady state one and the same observations are valid. Therefore, sensitivity analyses for medium and low inlet velocity have been performed. 4.5.3 Parametric sensitivity analysis with medium inlet velocity The plots of temperature maximum and sensitivity as a function of the different parameters are computed with an inlet velocity of 0.5 m/s (from figure 4.5.10 to figure 4.5.15). Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 96 Figure 4.5.10: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density Figure 4.5.11: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density The critical value of inlet pressure is 2.02 kPa. The value is still higher than the steady state one, but it has decreased with respect to the case of high inlet velocity. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 97 Figure 4.5.12: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density Figure 4.5.13: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.5 m/s for unsteady state model with variable density The critical value for the inlet temperature is 657 K. The lower inlet velocity has determined a decrease of the critical value, which is now lower than the steady state one. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 98 Figure 4.5.14: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.5 m/s for unsteady state model with variable density Figure 4.5.15: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=0.5 m/s for unsteady state model with variable density The critical value for the wall temperature is 635 K, still higher than the steady state ones but lower with respect to the previous case. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 99 In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 2.02 kPa Tin ,cr 657 K Tw,cr 635 K The qualitative behavior is analogous to the one shown by the unsteady state model with constant density: the critical values of the parameters decreases as the inlet velocity decreases. Nevertheless, the critical values are still higher than the ones predicted by the steady state model, except from the inlet temperature. Also in this case, the magnitudes of the temperature peaks are remarkably high. 4.5.4 Parametric sensitivity analysis with low inlet velocity The plots of temperature maximum and sensitivity as a function of the different parameters are computed with an inlet velocity of 0.1 m/s and are reported below (from figure 4.5.16 to figure 4.5.21). Figure 4.5.16: Tmax as function of Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 100 Figure 4.5.17: Sensitivity with respect to Pin, Tin=625 K, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density The critical value of inlet pressure is 1.77 kPa, lower than the one predicted by the steady state model. In addition, the temperature maximum and the sensitivity value are noticeably higher than the one predicted by the steady state, as it is shown by the figures. Figure 4.5.18: Tmax as a function of Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 101 Figure 4.5.19: Sensitivity with respect to Tin, Pin=1.6 kPa, Tw=625 K and v0=0.1 m/s for unsteady state model with variable density The critical value for the inlet temperature is 645 K, much lower than the steady state one. Also in this case, the magnitudes of the curves are remarkably high. Figure 4.5.20: Tmax as a function of Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with variable density Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 102 Figure 4.5.21: Sensitivity with respect to Tw, Pin=1.6 kPa, Tin=625 K and v0=0.1 m/s for unsteady state model with variable density The critical value for the wall temperature is 629 K, slightly lower than the steady state one. Also in this case, the magnitudes of the curves are remarkably high. In conclusion, a set of critical values for the three parameters considered has been found: Pin ,cr 1.77 kPa Tin ,cr 645 K Tw,cr 629 K In the case of low inlet velocity, the critical values of all the three parameters are lower than the steady state ones. Therefore, the behavior highlighted in the constant density case is confirmed also for the variable density case. This model is able to predict both the effect of variable density, which makes the runaway boundaries move towards higher values, and the effect of decreasing inlet velocity, which makes the runaway boundaries move towards lower values. Although at a first glance this model appears less conservative with respect to the steady state one, in the case of lower inlet velocity it is the opposite and the steady state model can lead to non-conservative conclusions. Such conclusions may be very dangerous, especially because of the huge values of temperature that the system can reach according to the unsteady state model. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 103 5) Conclusions 5.1 Summary of results Table 5.1: Summary of results In summary, this work has succeeded in demonstrating that choosing a steady state model or an unsteady state model can deeply affect the outcomes. First, the steady state model is not able to predict any deviations in the runaway boundaries, neither the ones produced by a change in the value of inlet velocity nor the ones produced by the variable density assumption. The deviations due to lower inlet velocity are of high importance in industrial applications, because it may be necessary to actually reduce the inlet velocity in order to reach high conversions. For low inlet velocity, particularly the steady state model with constant density may lead to nonconservative conclusions, as it predicts higher critical values of the considered parameters with respect to the unsteady state one. The variable density assumptions, instead, tends to move the runaway boundaries towards higher values, i.e., towards a safer situation. Nevertheless, it is important to notice that the steady model is totally unable to predict this sort of effect. Moreover, it may be dangerous to underestimate the risk which can rise in this situation, because for low inlet velocity the runaway boundaries are lower than the steady state ones. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 104 With reference to thermal and material diffusions, they seem not to have a significant influence on runaway boundaries in this kind of system. However, it has to be noticed that the values of thermal and material diffusivities are quite low, thus it is not clear whether they can play a role in a different kind of system. As regards the magnitude of runaway phenomenon, i.e., the temperature maximum values, they appear to be much higher in the unsteady state case than in the steady state one. This can have important effects on the prediction of possible consequences of a thermal runaway in terms of material resistance, structure design and process design. In conclusion, neglecting the unsteady state term, i.e., the temporal derivative, can lead to nonconservative and unsafe estimates of runaway boundaries for both constant density and variable density model. 5.2 Limitations Many limitations from the numerical point of view have been noticed while performing the simulations. The numerical problem faced during this work is indeed very complex and still object of studies in the mathematical and computational field. In particular, the code works properly and stably for the whole range of parameters corresponding to the normal or safe operation. In this range, the code is able to integrate and achieve convergence without any problem or instability. But, as the input parameters exceed the critical one and the system enters a runaway region, the code starts to become unstable, integration requires a longer time and the plot obtained shows oscillating behaviors and sharp peaks. In many cases, the code is actually not able to achieve convergence and the integration is stopped at a time step lower than the one required. For the purpose of this work, this kind of numerical limitations is not very significant, as the objective was to identify where a runaway region starts and not to model the behavior of the system within this region. Thus, observations about critical values of the parameters are solid, although no conclusions about what happen within the runaway region can be drawn. 5.3 Further developments With respect to the computational issues discussed in the previous section, it would be vital to further develop the mathematical and numerical comprehension of this kind of problems, in order to be able not only to identify the runaway boundary but also to undertake a full study of the system’s behavior. For example, it would be interesting to investigate at which stage of the temporal development of the system the runaway phenomenon occurs. As the naphthalene oxidation reaction is a very sample gas phase system, further investigations should be carried out on more complex systems, in order to study the influence of the unsteady state term and of the diffusive terms in different kinds of system (e.g. a liquid phase one). Finally, instrumentation and control issues should be study in depth, in order to investigate the effect of the cooling system (e.g., a co-current heat exchanger) or the implementation of a control system (i.e., number and positions of thermocouples). Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 105 Nomenclature A Heat transfer area of the reactor (associated to the jacket and/or the coil), m2 Pre-exponential factor, kmol/ (kg s kPa2) Factor A, see Eq. (2.7), - a = Eatt/R, activation energy (variant), 1/K B Dimensionless adiabatic temperature increase, - Factor B, see Eq. (2.7) BR Acronym for Batch Reactor b Factor b, see Eq. (2.5), - C Molar concentration, kmol/m3 Factor C, see Eqs. (2.7), - cp Molar heat capacity, kJ/(kmol K) ĉp Massive heat capacity, kJ/(kg K) D Differentiation Matrix Dm Material diffusion coefficient, m2/s Da Damköhler number, - dT Tube diameter, m Eatt Activation energy, kJ/kmol F Jacobian matrix, see Eq. (2.18) FD Acronym for Finite Difference g Example function for BC, see Eq (1.30) ΔH Reaction enthalpy, kJ/kmol k0 Pre-exponential factor, m3(n+m-1)/(kmoln+m-1·s) L PFR total length, m l Axial coordinate, m M Mass matrix Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 106 MFARP Acronym for Maximum occurred at a Finite Axial Reactor Position MW Molecular weight, kg/kmol N Cooling intensity, Eq. (2.3), - n Order of reaction, - P Pressure, Pa PAO Acronym for Pseudo-Adiabatic Operation p Partial Pressure, Pa r Reaction rate referred to the total volume, kmol/(m3·s) Source term r’ Reaction rate referred to the catalyst weight, kmol/(kg·s) R Ideal gas constant = 8.314, kJ/(kmol·K) Production rate, kmol/(m3·s) RH Heat capacity ratio, - S Dimensionless reacting strength, Eq. (2.2), - Normalized parametric sensitivity, - St Stanton number, - s Parametric sensitivity t Time, s T Temperature, K U Overall heat transfer coefficient, kW/(m2·K) V Volume (reactor, liquid, etc..), m3 v Axial velocity, m/s x State variable z Dimensionless axial coordinate, - Subscripts and superscripts * Maximum value in a series Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 107 Cool Coolant cr Critical G Gas phase In Inlet i Inflexion Species i i-th spatial grid point m In correspondence of the maximum MAX or max Maximum value of a quantity or at the maximum value of a quantity MIN or min Minimum value of a quantity or at the minimum value of a quantity mix Reaction mixture N Final spatial grid point napht Naphthalene O Oxygen S Solid phase (i.e., to the catalyst) T Total T Index for the temporal derivative W Wall X Index for first-order spatial derivative xx Index for second-order spatial derivative 0 Initial time, t=0 Initial spatial grid point Greek and other symbols Parameter of Robin BC, see Eq. (1.32) Thermal diffusion coefficient, m2/s Dimensionless activation energy or Arrhenius number, - δ Variation of a quantity Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 108 Material diffusion parameter, - Catalyst porosity, - Η Parameter in MOL method, see eq. (1.47) Θ Dimensionless temperature, - Dimensionless rate of temperature, - Λ Thermal diffusion parameter, - Stoichiometric coefficient (semibatch reactors, paragraph 2.2) ρ Density (molar or massive), kmol/m3 or kg/m3 Dimensionless time, - PFR Ratio of reactant to coolant heat capacity, Figure 2.3, - Φ System parameter Ψ Semenov number, - Χ Molar conversion, - Maximum dimensionless temperature rise, - Acronyms BC Boundary Condition BR Batch Reactor CSTR Continuously Stirred Tank Reactor DAE Differential Algebraic Equation FD Finite Difference IC Initial Condition MFARP Maximum occurred at a Finite Axial Reactor Position MOL Method of Lines MV Morbidelli and Varma method NC Number of components NR Number of reactions ODE Ordinary Differential Equation Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 109 PAO Pseudo-Adiabatic Operation PDE Partial Differential Equation PFR Plug Flow Reactor RHS Right Hand Side SBR Semi-Batch Reactor Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 110 Annex 1: An example of Matlab® Code An example of one of the Matlab® codes used is reported as a separate attachment. Particularly, the syntax for the computation of the sensitivity of the temperature maximum with respect to the inlet pressure in the unsteady state case has been chosen. Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 111 References 1. Stoessel, F., Thermal Safety of Chemical Processes: Risk Assessment and Process Design. Wiley-VCH: 2008. 2. Steinbach, J., Safety Assessment for Chemical Processes. Wiley-VCH: 2008. 3. Vande Wouwer,A.; Saucez,P.; Vilas, C., Simulation of ODE/PDE Models with ® MATLAB ,OCTAVE and SCILAB. Scientific and Engineering Applications. Springer International Publishing Switzerland: 2014 4. Varma, A.; Morbidelli, M.; Wu, H., Parametric Sensitivity in Chemical Systems. Cambridge University Press: 1999. 5. Bilous, O.; Amundson, N. R., Chemical Reactor Stability and Sensitivity II. Effect of Parameters on Sensitivity of Empty Tubular Reactors, AIChE Journal 1956, 2, 117. 6. Barkelew, C. 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M.; Morbidelli, M.; Varma, A., Parametric sensitivity of a CSTR. Chemical Engineering Science 1986, 41, 1647. 19. Henning, G. P.; Perez, G. A. Parametric sensitivity in fixed-bed catalytic reactors. Chemical Engineering Science 1986, 41, 83. 20. Wu, H.; Morbidelli, M.; Varma, A. Pseudo-adiabatic operation and runaway in tubular reactors. AIChE Journal 1998, 44, 1157. 21. Strozzi, F.; Zaldívar, J. M.; Kronberg, A. E.; Westerterp, K. R. On-line runaway detection in batch reactors using chaos theory techniques. AIChE Journal 1999, 45, 2429. 22. Zaldívar, J. M.; Cano, J.; Alós, M. A.; Sempere, J.; Nomen, R.; Lister, D.; Maschio, G.; Obertopp, T.; Gilles, E. D.; Bosch, J.; Strozzi, F. A general criterion to define runaway Sofia Croci - Unsteady State Runaway Boundaries for Plug Flow Reactors involving Catalytic Oxidations Pag. 112 limits in chemical reactors. Journal of Loss Prevention in the Process Industries 2003, 16, 187. 23. 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