# Multiple steady states in two-phase reactors under boiling conditions

TL;DR: In this article, the authors analyzed the nonlinear behavior of two-phase reactors under boiling conditions and identified three necessary conditions for the existence of multiple steady states: the reactant A has to be the light-boiling component, the difference in boiling point temperatures between the reaction A and the product B is sufficiently large, and the order of the reaction B is less than some physical parameter α.

Abstract: In this paper, we analyze the nonlinear behavior of two-phase reactors under boiling conditions. First we focus on a simple n th-order reaction of the form A → B , which allows a rigorous analytical treatment. Three necessary conditions for the existence of multiple steady states have been identified: the reactant A has to be the light-boiling component, the difference in boiling point temperatures between the reactant A and the product B has to be sufficiently large, and the order of the reaction has to be less than some physical parameter α . This parameter α can be interpreted as a measure for the phase-equilibrium-driven self-inhibition of the reaction mechanism. Thus, we have found an elegant explanation for the occurrence of multiplicities. Analytical and therefore general quantitative criteria identifying the regions of multiplicity for the model system are presented. Practical relevance of our results is demonstrated by means of two examples, the Monsanto process for the production of acetic acid and the ethylene glycol reactive distillation system.

##### Citations

More filters

••

[...]

TL;DR: In this article, a single loop pulsating heat pipe exhibits multiple operational quasi-steady states and a temporal scaling analysis is presented to estimate the order of magnitude of the equilibrium frequency of phase change and ensuing oscillations.

Abstract: The novel fact that, keeping all the operating and boundary conditions fixed, a single loop pulsating heat pipe exhibits multiple operational quasi-steady states is reported in this paper. For a specified heat power input level and volumetric filling ratio, continuous online measurements of static pressure and temperature at crucial locations, along with flow visualization, have been carried out for more than twelve hours per experimental run of device operation. Four distinct quasi-steady states have been observed in these experimental runs. Each quasi-steady state is characterized by a unique specific two-phase flow pattern and corresponding effective device conductance, revealing the strong thermo-hydrodynamic coupling guiding the thermal performance. The quasi-steady state corresponding to best thermal performance consists of continuous unidirectional flow circulations, while the state corresponding to poor thermal performance is characterized by the intermittent bidirectional flow reversals. A temporal scaling analysis is presented to estimate the order of magnitude of the equilibrium frequency of phase change and ensuing oscillations. These order-of-magnitude estimates closely match with the experimentally observed frequencies. The spectral contents of each quasi-steady state are analyzed and it is found that dominant frequencies of flow oscillations are in the range of 0.1 to 3.0 Hz with each quasi-steady state exhibiting a characteristic power spectrum. This provides the necessary velocity scaling estimates, primary information needed for any progress in design of pulsating heat pipes.

127 citations

••

[...]

TL;DR: In this article, the impact of input multiplicity on the control of a simulated industrial scale methyl acetate reactive distillation (RD) column is studied and a new metric, rangeability, is defined to quantify the severity of inputs multiplicity in a steady-state input-output (IO) relation.

Abstract: The impact of steady-state multiplicities on the control of a simulated industrial scale methyl acetate reactive distillation (RD) column is studied. At a fixed reflux rate, output multiplicity, with multiple output values for the same reboiler duty, causes the column to drift to an undesirable steady-state under open loop operation. The same is avoided for a fixed reflux ratio policy. Input multiplicity, where multiple input values give the same output, leads to “wrong” control action under feedback control severely compromising control system robustness. A new metric, rangeability, is defined to quantify the severity of input multiplicity in a steady-state input–output (IO) relation. Rangeability is used in conjunction with conventional sensitivity analysis for the design of robust control structures for the RD column. Results for the two synthesized control structures show that controlling the most sensitive reactive tray temperature results in poor robustness due to low rangeability causing “wrong” control action for large disturbances. Controlling a reactive tray temperature with acceptable sensitivity but larger rangeability gives better robustness. It is also shown that controlling the difference in the temperature of two suitably chosen reactive trays further improves robustness of both the structures as input multiplicity is avoided. The article brings out the importance of IO relations for control system design and understanding the complex dynamic behavior of RD systems.

35 citations

••

[...]

TL;DR: In this paper, a case study for dynamic simulation of a reactive distillation (RD) process for the production of TAME is presented, where nonlinear dynamic effects such as oscillations, multiple steady states, and internal state multiplicity have been observed under certain conditions.

Abstract: The liquid phase synthesis of octane-enhancing ethers like methyl tert-butyl ether (MTBE) or tert-amyl methyl ether (TAME) can be advantageously performed in a reactive distillation (RD) processes with ion-exchange resin catalyst. In the present paper, a case study for dynamic simulation of a RD process for the production of TAME is presented. Nonlinear dynamic effects such as oscillations, multiple steady states, and internal state multiplicity have been observed under certain conditions. Feed condition and Damkohler number are the important parameters that have influence on the existence of these effects. The presence of these effects has been confirmed through independent bifurcation analysis. The influence of various modeling parameters, reaction kinetics, and phase equilibrium models on this observation is studied.

20 citations

••

[...]

TL;DR: In this article, a relatively simple method to identify multiplicity in reactive distillation due to the interaction of reaction and distillation is proposed, where the nonlinear generation term and linear removal term, f

Abstract: We propose a relatively simple method to identify multiplicity in reactive distillation due to the interaction of reaction and distillation. The nonlinear generation term and linear removal term, f...

12 citations

••

[...]

TL;DR: In this article, temperature control of a two-stage continuous bulk styrene polymerization process is developed to predict the performance of auto-refrigerated CSTR and tubular reactors.

Abstract: Dynamic simulation and control of a two-stage continuous bulk styrene polymerization process is developed to predict the performance of auto-refrigerated CSTR and tubular reactors. The tubular reactor is subdivided into three temperature-control jacket zones. In this paper temperature control of auto-refrigerated continuous stirred tank reactor and tubular reactor are carried out, simultaneously. Two strategies are proposed for the control of tubular reactor. At the first strategy the controlled variable is jacket temperature and in the second strategy the controlled variable is the reactor temperature at the exit of each section. The set points for polymer grade transition are obtained using optimization of reactors temperatures via genetic algorithm (GA). Simulation results show that both of the control strategies are successful but second strategy has better performance in the control of polymer properties in the presence of disturbance and model mismatch.

10 citations

### Cites methods from "Multiple steady states in two-phase..."

[...]

##### References

More filters

•

[...]

TL;DR: An overview of Chemical Reaction Engineering is presented, followed by an introduction to Reactor Design, and a discussion of the Dispersion Model.

Abstract: Partial table of contents: Overview of Chemical Reaction Engineering. HOMOGENEOUS REACTIONS IN IDEAL REACTORS. Introduction to Reactor Design. Design for Single Reactions. Design for Parallel Reactions. Potpourri of Multiple Reactions. NON IDEAL FLOW. Compartment Models. The Dispersion Model. The Tank--in--Series Model. REACTIONS CATALYZED BY SOLIDS. Solid Catalyzed Reactions. The Packed Bed Catalytic Reactor. Deactivating Catalysts. HETEROGENEOUS REACTIONS. Fluid--Fluid Reactions: Kinetics. Fluid--Particle Reactions: Design. BIOCHEMICAL REACTIONS. Enzyme Fermentation. Substrate Limiting Microbial Fermentation. Product Limiting Microbial Fermentation. Appendix. Index.

8,133 citations

•

[...]

01 Jan 1969

TL;DR: In this article, the authors introduce the notion of uniformity of intensive potentials as a criterion of phase equilibrium, and propose a model for solubilities of solids in liquid mixtures.

Abstract: 1. The Phase Equilibrium Problem. 2. Classical Thermodynamics of Phase Equilibria. 3. Thermodynamic Properties from Volumetric Data. 4. Intermolecular Forces, Corresponding States and Osmotic Systems. 5. Fugacities in Gas Mixtures. 6. Fugacities in Liquid Mixtures: Excess Functions. 7. Fugacities in Liquid Mixtures: Models and Theories of Solutions. 8. Polymers: Solutions, Blends, Membranes, and Gels. 9. Electrolyte Solutions. 10. Solubilities of Gases in Liquids. 11. Solubilities of Solids in Liquids. 12. High-Pressure Phase Equilibria. Appendix A. Uniformity of Intensive Potentials as a Criterion of Phase Equilibrium. Appendix B. A Brief Introduction to Statistical Thermodynamics. Appendix C. Virial Coefficients for Quantum Gases. Appendix D. The Gibbs-Duhem Equation. Appendix E. Liquid-Liquid Equilibria in Binary and Multicomponent Systems. Appendix F. Estimation of Activity Coefficients. Appendix G. A General Theorem for Mixtures with Associating or Solvating Molecules. Appendix H. Brief Introduction to Perturbation Theory of Dense Fluids. Appendix I. The Ion-Interaction Model of Pitzer for Multielectrolyte Solutions. Appendix J. Conversion Factors and Constants. Index.

4,477 citations

••

[...]

01 Jan 1985

TL;DR: Singularities and groups in bifurcation theory as mentioned in this paper have been used to solve the problem of finding a group of singularities in a set of problems with multiple solutions.

Abstract: This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob- lems and we hope to convert the reader to this view In this preface we will discuss what we feel are the strengths of the singularity theory approach This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V I Arnold In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence In this volume our emphasis is on singularity theory, with group theory playing a subordinate role In Volume II the emphasis will be more balanced Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation As we use the term, bifurcation theory is the study of equations with multiple solutions

3,127 citations

•

[...]

27 Apr 1988

TL;DR: The Canonical Formalism Statistical Mechanics in the Entropy Representation as discussed by the authors is a generalization of statistical mechanics in the Helmholtz Representation, and it has been applied to general systems.

Abstract: GENERAL PRINCIPLES OF CLASSICAL THERMODYNAMICS. The Problem and the Postulates. The Conditions of Equilibrium. Some Formal Relationships, and Sample Systems. Reversible Processes and the Maximum Work Theorem. Alternative Formulations and Legendre Transformations. The Extremum Principle in the Legendre Transformed Representations. Maxwell Relations. Stability of Thermodynamic Systems. First--Order Phase Transitions. Critical Phenomena. The Nernst Postulate. Summary of Principles for General Systems. Properties of Materials. Irreversible Thermodynamics. STATISTICAL MECHANICS. Statistical Mechanics in the Entropy Representation: The Microanonical Formalism. The Canonical Formalism Statistical Mechanics in Helmholtz Representation. Entropy and Disorder Generalized Canonical Formulations. Quantum Fluids. Fluctuations. Variational Properties, Perturbation Expansions, and Mean Field Theory. FOUNDATIONS. Postlude: Symmetry and the Conceptual Foundations of Thermostatistics. Appendices. General References. Index.

2,854 citations

•

[...]

01 Jan 1960

TL;DR: The Canonical Formalism Statistical Mechanics in the Entropy Representation as mentioned in this paper is a generalization of statistical mechanics in the Helmholtz Representation, and it has been applied to general systems.

Abstract: GENERAL PRINCIPLES OF CLASSICAL THERMODYNAMICS. The Problem and the Postulates. The Conditions of Equilibrium. Some Formal Relationships, and Sample Systems. Reversible Processes and the Maximum Work Theorem. Alternative Formulations and Legendre Transformations. The Extremum Principle in the Legendre Transformed Representations. Maxwell Relations. Stability of Thermodynamic Systems. First--Order Phase Transitions. Critical Phenomena. The Nernst Postulate. Summary of Principles for General Systems. Properties of Materials. Irreversible Thermodynamics. STATISTICAL MECHANICS. Statistical Mechanics in the Entropy Representation: The Microanonical Formalism. The Canonical Formalism Statistical Mechanics in Helmholtz Representation. Entropy and Disorder Generalized Canonical Formulations. Quantum Fluids. Fluctuations. Variational Properties, Perturbation Expansions, and Mean Field Theory. FOUNDATIONS. Postlude: Symmetry and the Conceptual Foundations of Thermostatistics. Appendices. General References. Index.

2,478 citations