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The bifurcation behavior of tubular reactors
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“By applying first-principle modelling through relevant mass balance relations on an infinitesimally small section of the reactor, the dynamics of the reactant concentration can be described by the PDE given in Equation ( 1), belonging to the class of second order parabolic PDEs commonly used to characterize diffusion-convection-reaction systems [25] in chemical engineering.”
From Optimal control of axial dispersion tubular reactors with recycle: Addressing state‐delay through transport PDEs (2025) · cites this paper as [25]
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Cites
Bifurcation, perturbation of simple eigenvalues, itand linearized stability
Crandall, M. G., & Rabinowitz, P. H. (1973) · Archive for Rational Mechanics and Analysis · doi:10.1007/bf00282325 · Cited by 830
On the dynamic behavior of continuous stirred tank reactors
Uppal, A., Ray, W. H., & Poore, A. B. (1974) · Chemical Engineering Science · doi:10.1016/0009-2509(74)80089-8 · Cited by 737
Self-citationThe QR Transformation A Unitary Analogue to the LR Transformation--Part 1
Francis, J. G. F. (1961) · The Computer Journal · doi:10.1093/comjnl/4.3.265 · Cited by 696
Solution of boundary-value problems by orthogonal collocation
Villadsen, J. V., & Stewart, W. E. (1967) · Chemical Engineering Science · doi:10.1016/0009-2509(67)80074-5 · Cited by 673
Francis, J. G. F. (1962) · The Computer Journal · doi:10.1093/comjnl/4.4.332 · Cited by 384
Orthogonal collocation on finite elements
Carey, G. F., & Finlayson, B. A. (1975) · Chemical Engineering Science · doi:10.1016/0009-2509(75)80031-5 · Cited by 338
Uppal, A., Ray, W. H., & Poore, A. B. (1976) · Chemical Engineering Science · doi:10.1016/0009-2509(76)85058-0 · Cited by 284
Self-citationThe Hopf Bifurcation Theorem in infinite dimensions
Crandall, M. G., & Rabinowitz, P. H. (1977) · Archive for Rational Mechanics and Analysis · doi:10.1007/bf00280827 · Cited by 181
On the theory and application of the Hopf-Friedrichs bifurcation theory
Poore, A. B. (1976) · Archive for Rational Mechanics and Analysis · doi:10.1007/bf00248886 · Cited by 132
Hlaváček, V., & Hofmann, H. (1970) · Chemical Engineering Science · doi:10.1016/0009-2509(70)85030-8 · Cited by 119
An efficient general purpose method for the intergration of stiff ordinary differential equations
Michelsen, M. L. (1976) · AIChE Journal · doi:10.1002/aic.690220327 · Cited by 114
Multiplicity, Stability, and Sensitivity of States in Chemically Reacting Systems—A Review
Schmitz, R. A. (1975) · Advances in chemistry series · doi:10.1021/ba-1975-0148.ch007 · Cited by 113
Padberg, G., & Wicke, E. (1967) · Chemical Engineering Science · doi:10.1016/0009-2509(67)80167-2 · Cited by 112
Hegedus, L. L., Oh, S. H., & Baron, K. (1977) · AIChE Journal · doi:10.1002/aic.690230503 · Cited by 107
ORTHOGONAL COLLOCATION IN CHEMICAL REACTION ENGINEERING
Finlayson, B. A. (1974) · Catalysis Reviews · doi:10.1080/01614947408079627 · Cited by 96
Ignition/extinction phenomena in a wall cooled fixed bed reactor
Kalthoff, O., & VORTMEYER, D. (1980) · Chemical Engineering Science · doi:10.1016/0009-2509(80)80056-x · Cited by 72
Hlaváček, V., & Hofmann, H. (1970) · Chemical Engineering Science · doi:10.1016/0009-2509(70)85031-x · Cited by 70
Fixed Bed Catalytic Reactors. Technological and Fundamental Design Aspects
Froment, G. F. (1974) · Chemie Ingenieur Technik · doi:10.1002/cite.330460903 · Cited by 69
Studies in the control of tubular reactors-I General Considerations
Georgakis, C., Aris, R., & Amundson, N. R. (1977) · Chemical Engineering Science · doi:10.1016/0009-2509(77)85032-x · Cited by 67
Some further observations on tubular reactor stability
Amundson, N. R. (1965) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.5450430201 · Cited by 64
An experimental study of steady state multiplicity in a loop reactor
Root, R. B., & Schmitz, R. A. (1969) · AIChE Journal · doi:10.1002/aic.690150509 · Cited by 53
Creeping Profiles in Catalytic Fixed Bed Reactors. Continuous Models
Rhee, H., Lewis, R. P., & Amundson, N. R. (1974) · Industrial & Engineering Chemistry Fundamentals · doi:10.1021/i160052a004 · Cited by 50
Fieguth, P., & Wicke, E. (1971) · Chemie Ingenieur Technik · doi:10.1002/cite.330431004 · Cited by 48
Varma, A., & Amundson, N. R. (1973) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.5450510211 · Cited by 46
Hysteresis and Periodic Activity Behavior in Catalytic Chemical Reaction Systems
Hlaváček, V., & Votruba, J. (1979) · Advances in catalysis · doi:10.1016/s0360-0564(08)60054-1 · Cited by 45
Instabile Reaktionszustände bei der heterogenen Katalyse
Wicke, E. (1974) · Chemie Ingenieur Technik · doi:10.1002/cite.330460902 · Cited by 45
The behaviour of an adiabatic fixed bed reactor for the oxidation of carbon monoxide—II
Sharma, C. S., & Hughes, R. (1979) · Chemical Engineering Science · doi:10.1016/0009-2509(79)85107-6 · Cited by 44
Moving reaction zones in fixed bed reactors under the influence of various parameters
VORTMEYER, D., & Jahnel, W. (1972) · Chemical Engineering Science · doi:10.1016/0009-2509(72)80041-1 · Cited by 44
The behaviour of an adiabatic fixed bed reactor for the oxidation of carbon monoxide—I
Sharma, C. S., & Hughes, R. (1979) · Chemical Engineering Science · doi:10.1016/0009-2509(79)85106-4 · Cited by 40
Some general observations on tubular reactor stability
Luss, D., & Amundson, N. R. (1967) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.5450450603 · Cited by 39
The influence of the lewis number on the dynamics of chemically reacting systems
Ray, W. H., & Hastings, S. P. (1980) · Chemical Engineering Science · doi:10.1016/0009-2509(80)80007-8 · Cited by 37
Self-citationHlaváček, V., Hofmann, H., & Kubı́ček, M. (1971) · Chemical Engineering Science · doi:10.1016/0009-2509(71)86052-9 · Cited by 37
Hlaváček, V., & Hofmann, H. (1970) · Chemical Engineering Science · doi:10.1016/0009-2509(70)85074-6 · Cited by 34
Creeping reaction zone in a catalytic, fixed-bed reactor: a cell model approach
Rhee, H., Foley, D., & Amundson, N. R. (1973) · Chemical Engineering Science · doi:10.1016/0009-2509(73)80060-0 · Cited by 31
Varma, A., & Amundson, N. R. (1972) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.5450500406 · Cited by 30
Simon, B. H., & VORTMEYER, D. (1978) · Chemical Engineering Science · doi:10.1016/0009-2509(78)85076-3 · Cited by 30
Tubular reactor steady state and stability characteristics. I. Effect of axial mixing
McGowin, C. R., & Perlmutter, D. D. (1971) · AIChE Journal · doi:10.1002/aic.690170412 · Cited by 29
Experimental Study of Multiple Steady States in Adiabatic Catalytic Systems
Hlaváček, V., & Votruba, J. (1974) · Advances in chemistry series · doi:10.1021/ba-1974-0133.ch041 · Cited by 28
Puszynski, J. A., & Hlaváček, V. (1980) · Chemical Engineering Science · doi:10.1016/0009-2509(80)85013-5 · Cited by 27
The non‐adiabatic tubular reactor: Stability considerations
Varma, A., & Amundson, N. R. (1973) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.5450510411 · Cited by 26
Quasi-stationäres verhalten von wandernden brennzonen
Gilles, E. D. (1974) · Chemical Engineering Science · doi:10.1016/0009-2509(74)80120-x · Cited by 25
Ampaya, J. P., & Rinker, R. G. (1977) · Chemical Engineering Science · doi:10.1016/0009-2509(77)85027-6 · Cited by 24
Effects of catalyst particle size on multiple steady states
OH, S., Baron, K., Sloan, E. D., & Hegedus, L. L. (1979) · Journal of Catalysis · doi:10.1016/s0021-9517(79)80031-7 · Cited by 24
Simulation von wandernden Reaktionszonen eines Festbettreaktors im Digitalrechner
VORTMEYER, D., & Jahnel, W. (1971) · Chemie Ingenieur Technik · doi:10.1002/cite.330430713 · Cited by 23
Modelling of chemical reactors-XXXII
Kubíčekt†, M., Hofmann, H., & Hlaváček, V. (1979) · Chemical Engineering Science · doi:10.1016/0009-2509(79)85104-0 · Cited by 22
Modeling of chemical reactors—XXX
Sinkule, J., Votruba, J., Hlaváček, V., & Hofmann, H. (1976) · Chemical Engineering Science · doi:10.1016/0009-2509(76)85004-x · Cited by 21
Kubı́ček, M., & Hlaváček, V. (1974) · Chemical Engineering Science · doi:10.1016/0009-2509(74)87027-2 · Cited by 21
Hlaváček, V., Hofmann, H., Votruba, J., & Kubı́ček, M. (1973) · Chemical Engineering Science · doi:10.1016/0009-2509(73)85073-0 · Cited by 19
Mathematical models of fixed bed reactors
Amundson, N. R. (1970) · Berichte der Bunsengesellschaft für physikalische Chemie · doi:10.1002/bbpc.19700740205 · Cited by 18
Algorithm for Evaluation of Complex Bifurcation Points in Ordinary Differential Equations
Kubı́ček, M. (1980) · SIAM Journal on Applied Mathematics · doi:10.1137/0138008 · Cited by 18
MATHEMATICAL MODELING OF HETEROGENEOUS CATALYTIC REACTORS
Karanth, N. G., & Hughes, R. (1974) · Catalysis Reviews · doi:10.1080/01614947408075373 · Cited by 18
Tubular Chemical Reactors: The “Lumping Approximation” and Bifurcation of Oscillatory States
Cohen, D. S., & Poore, A. B. (1974) · SIAM Journal on Applied Mathematics · doi:10.1137/0127032 · Cited by 18
Michelsen, M. L. (1977) · The Chemical Engineering Journal · doi:10.1016/0300-9467(77)85005-3 · Cited by 18
Exponential collocation of stiff reactor models
Guertin, E. W., Sørensen, J. P., & Stewart, W. E. (1977) · Computers & Chemical Engineering · doi:10.1016/0098-1354(77)85005-9 · Cited by 17
Nishimura, Y., & Matsubara, M. (1969) · Chemical Engineering Science · doi:10.1016/0009-2509(69)85071-2 · Cited by 17
Stability and optimisation of a methanol converter
Stephens, A. D. (1975) · Chemical Engineering Science · doi:10.1016/0009-2509(75)85111-6 · Cited by 16
Wicke, E., & Padberg, G. (1968) · Chemie Ingenieur Technik · doi:10.1002/cite.330402102 · Cited by 16
Stability of high dimensional nonlinear systems using Krasovskii's theorem
Berger, A. J., & Lapidus, L. (1969) · AIChE Journal · doi:10.1002/aic.690150209 · Cited by 16
A comparison of techniques for local stability analysis of tubular reactor systems
McGowin, C. R., & Perlmutter, D. D. (1971) · The Chemical Engineering Journal · doi:10.1016/0300-9467(71)80006-0 · Cited by 15
Autothermal reactor with internal heat exchange—II
Ampaya, J. P., & Rinker, R. G. (1978) · Chemical Engineering Science · doi:10.1016/0009-2509(78)80048-7 · Cited by 14
Local stability of tubular reactors
Clough, D. E., & Ramirez, W. F. (1972) · AIChE Journal · doi:10.1002/aic.690180223 · Cited by 14
Global asymptotic stability in distributed parameter systems: comparison function approach
Varma, A., & Amundson, N. R. (1972) · Chemical Engineering Science · doi:10.1016/0009-2509(72)80006-x · Cited by 13
An experimental study of unstable states in a loop reactor
Root, R. B., & Schmitz, R. A. (1970) · AIChE Journal · doi:10.1002/aic.690160308 · Cited by 13
Experimental observation of multiple steady-states for diluted and blended catalyst beds
Votruba, J., Hlaváček, V., & Sinkule, J. (1976) · Chemical Engineering Science · doi:10.1016/0009-2509(76)87050-9 · Cited by 12
Studies in the control of tubular reactors—III stabilization by observer design
Georgakis, C., Aris, R., & Amundson, N. R. (1977) · Chemical Engineering Science · doi:10.1016/0009-2509(77)85034-3 · Cited by 11
Mikuš, O., Puszynski, J. A., & Hlaváček, V. (1979) · Chemical Engineering Science · doi:10.1016/0009-2509(79)85082-4 · Cited by 10
Convergence of the Galerkin Method for Nonlinear Problems Involving Chemical Reaction
Finlayson, B. A. (1971) · SIAM Journal on Numerical Analysis · doi:10.1137/0708032 · Cited by 9
Studies in approximation methods—III
Birnbaum, I., & Lapidus, L. (1978) · Chemical Engineering Science · doi:10.1016/0009-2509(78)80003-7 · Cited by 9
Vergleich von kontinuierlichen Modellen zur Beschreibung des dynamischen
Lübeck, B. (1974) · The Chemical Engineering Journal · doi:10.1016/0300-9467(74)80023-7 · Cited by 8
Studies in the control of tubular reactors—II Stabilization by modal control
Georgakis, C., Aris, R., & Amudson, N. R. (1977) · Chemical Engineering Science · doi:10.1016/0009-2509(77)85033-1 · Cited by 8
Gupalo, Y. P., & Ryazantsev, Y. S. (1969) · Chemical Engineering Science · doi:10.1016/0009-2509(69)85072-4 · Cited by 7
Kubı́ček, M. (1979) · Chemical Engineering Science · doi:10.1016/0009-2509(79)80015-9 · Cited by 7
On gas-liquid mass transfer in packed trickle beds
Patwardhan, V. S. (1979) · Chemical Engineering Science · doi:10.1016/0009-2509(79)85083-6 · Cited by 7
Kubı́ček, M., & Hlaváček, V. (1971) · Chemical Engineering Science · doi:10.1016/0009-2509(71)83007-5 · Cited by 7
The stability of fixed bed catalytic reactors
McGreavy, C., & Soliman, M. A. (1973) · Chemical Engineering Science · doi:10.1016/0009-2509(73)85145-0 · Cited by 6
Hlaváček, V., Mikuš, O., Jíra, E., & Pour, V. (1980) · Chemical Engineering Science · doi:10.1016/0009-2509(80)80095-9 · Cited by 6
Experimental observations of multiple steady states in an adiabatical catalytic reactor
Hlaváček, V., Votruba, J., & Hofmann, H. (1976) · Chemical Engineering Science · doi:10.1016/0009-2509(76)85032-4 · Cited by 5
Stephanopoulos, G., & Schuelke, L. M. (1976) · AIChE Journal · doi:10.1002/aic.690220507 · Cited by 5
Liou, C., Lim, H., & Weigand, W. A. (1974) · Chemical Engineering Science · doi:10.1016/0009-2509(74)80186-7 · Cited by 4
Untersuchungen zur Instabilität isothermer heterogener Festbettreaktoren
Renken, A., Müller, M., & Helmrich, H. (1975) · Chemie Ingenieur Technik · doi:10.1002/cite.330472413 · Cited by 4
Liou, C., Lim, H., & Weigand, W. A. (1972) · AIChE Journal · doi:10.1002/aic.690180536 · Cited by 4
Modeling and stability of catalytic fixed bed reactors
Hlaváček, V. (1970) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.5450480602 · Cited by 3
McGreavy, C., & Soliman, M. A. (1973) · AIChE Journal · doi:10.1002/aic.690190125 · Cited by 3
On the use of Krasovskii's theorem for stability analysis
Berger, A. J., & Lapidus, L. (1968) · AIChE Journal · doi:10.1002/aic.690140229 · Cited by 3
An experimental study of instability in a non-adiabatic multi-reaction tubular reactor
Schleppy, R., & Shah, Y. T. (1977) · Chemical Engineering Science · doi:10.1016/0009-2509(77)80074-2 · Cited by 2
Stewart, W. E., Liou, C., Lim, H., Weigand, W. A., Berger, A. J., & HÁLA, E. (1972) · AIChE Journal · doi:10.1002/aic.690180443 · Cited by 2
Research works printed but not linked: 11
- Gilles, 1976, Proc. 4th Int. Symp. Chem. React. Engng Heidelberg, 459
- Hlaváček, 1978, Advan. Catal., 27, 59
- Butakov, 1973, Dokl. Akad. Nauk. SSSR, 209, 643
- Hlaváček, 1974, Int. J. Chem. Engng, 14, 461
- Butakov, 1978, Dokl. Akad, Nauk SSSR, 238, 166
- Oh, 1978, am. Chem. Soc. Symp. Ser., 65, 461
- Paspek, 1980, A.I.Ch.E.J.
- Vortmeyer, 1961, Z. Electrochem., 65, 282
- McGinnis, 1965, Chem. Engng Prog. Symp. Ser., 55, 2
- Sørensen, 1973, A.I.Ch.E.J., 969, 1286
- Poore, 1980, A.I.Ch.E. Annual Meeting
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Tezduyar, T. E., & Park, Y. J. (1986) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/0045-7825(86)90003-4 · Cited by 269
Olefin/paraffin separations by adsorption: π‐Complexation vs. kinetic separation
Rege, S. U., Padin, J., & Yang, R. T. (1998) · AIChE Journal · doi:10.1002/aic.690440405 · Cited by 225
Multiplicities and instabilities in chemically reacting systems — a review
Razón, L. F., & Schmitz, R. A. (1987) · Chemical Engineering Science · doi:10.1016/0009-2509(87)80055-6 · Cited by 193
Meurer, T., & Kugi, A. (2009) · Automatica · doi:10.1016/j.automatica.2009.01.006 · Cited by 184
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Metadata
- Authors
- Jensen, K. F., & Ray, W. H.
- Affiliations
- University of Wisconsin–Madison (US)
- Corresponding author
- Ray, W.Harmon
- ORCID
- Jensen, Klavs F.
- Venue
- Chemical Engineering Science
- Publisher
- Elsevier BV
- Published
- 1982
- Volume
- 37
- Issue
- 2
- Pages
- 199-222
- ISSN
- 0009-2509 (ISSN-L)
- OpenAlex
- W2065214383
- Cited by
- 180
- References
- 86
- Citations by year
- 2026: 2 · 2025: 5 · 2024: 5 · 2022: 3 · 2021: 4 · 2020: 4 · 2019: 5 · 2018: 3 · 2017: 4 · 2016: 3 · 2015: 6 · 2014: 1 · 2013: 7 · 2012: 2
- FWCI
- 5.55
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Chemical Engineering Science (publishedVersion)
- Defense Technical Information Center (DTIC) (submittedVersion)
- Topics
- Physical Sciences › Engineering › Computational Mechanics › Heat and Mass Transfer in Porous Media
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Dynamics and Pattern Formation
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
- Keywords
- tubular reactor, bifurcation analysis, Hopf bifurcation, steady-state profiles, axial dispersion, first-order irreversible reaction, multiple steady states, periodic solutions, Lewis number, Peclet number, fixed-bed reactor, reaction-diffusion systems, two-phase model
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