Numerical evaluation of the stability of stationary points of index-2 differential-algebraic equations: Applications to reactive flash and reactive distillation systems

Harney, David Anthony; Mills, Thomas; Book, Neil L.

doi:10.1016/j.compchemeng.2012.09.021

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…The topological form of the bifurcation shapes of this work is of type " " such as typically displayed in continuous stirred tank chemical reactor and reactive flash. This is supported by other studies (Rodríguez et al [7], Ruiz et al [8], Alvarez-Ramirez [27], Jaime-Leal et al [9], and Harney et al [26]).…”

Section: Resultssupporting

confidence: 80%

“…Jaime-Leal et al [9] introduced a new approach and conditions to identify input multiplicity in reactive flash, based on the application of reaction-invariant composition variables. Finally, Harney et al [26] demonstrate that dynamic behavior of reactive flash and reactive distillation represented by index-2 system of differential algebraic equations (DAEs) can be reduced to an ordinary differential equations system (ODE) by single differentiation; the resulting Jacobian matrix will have a null eigenvalue at every steady state with multiplicity of at least the dimension y. These null eigenvalues must be accounted for when determining the stability of a steady state.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Biodiesel Production by Reactive Flash: A Numerical Simulation

Regalado-Méndez

Skogestad

Natividad

et al. 2016

International Journal of Chemical Engineering

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Reactive flash (RF) in biodiesel production has been studied in order to investigate steady-state multiplicities, singularities, and effect of biodiesel quality when the RF system approaches to bubble point. The RF was modeled by an index-2 system of differential algebraic equations, the vapor split (ϕ) was computed by modified Rachford-Rice equation and modified Raoult’s law computed bubble point, and the continuation analysis was tracked on MATCONT. Results of this study show the existence of turning points, leading to a unique bubble point manifold,(xBiodiesel,T)=(0.46,478.41 K), which is a globally stable flashing operation. Also, the results of the simulation in MATLAB® of the dynamic behavior of the RF show that conversion of triglycerides reaches 97% for a residence time of 5.8 minutes and a methanol to triglyceride molar flow ratio of 5 : 1.

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Section: Resultssupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Biodiesel Production by Reactive Flash: A Numerical Simulation

Regalado-Méndez

Skogestad

Natividad

et al. 2016

International Journal of Chemical Engineering

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“…Hence, the interconnection among the subsystems in ( 29) is power-preserving. 7 The system conformed by ( 29) and ( 31) is an index-2 differential algebraic system in Hessenberg form (see, e.g., [49]). Based on [49], the time derivative of the algebraic constraint (29c) can be computed once to explicitly obtain z K in terms of xE and x∆ as follows:…”

Section: B Interconnection Structurementioning

confidence: 99%

Port-Hamiltonian Modeling of Hydraulics in 4th Generation District Heating Networks

Strehle

Machado

Cucuzzella

et al. 2022

2022 IEEE 61st Conference on Decision and Control (CDC)

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A fundamental precondition for the secure and efficient operation of district heating networks (DHNs) is a stable hydraulic behavior. However, the ongoing transition towards a sustainable heat supply, especially the rising integration of distributed heat sources and the increasingly meshed topologies, introduce complex and potentially destabilizing hydraulic dynamics. In this work, we propose a unifying, passivity-based framework which guarantees asymptotic stability of any forced hydraulic DHN equilibrium while allowing for meshed, timevarying topologies and different, dynamically interacting distributed heat sources. To establish the desired hydraulic equilibria, we propose decentralized, passivity-based pressure and volume flow rate controllers for the pumps and valves in the actuated DHN subsystems. In particular, we leverage the equilibriumindependent passivity (EIP) properties of the DHN subsystems, the skew-symmetric nature of their interconnections, and LaSalle's Invariance principle to assess asymptotic stability in a modular manner. The obtained results hold for the state-of-theart as well as future DHN generations featuring, for example, multiple distributed heat sources, asymmetric pipe networks, and multiple temperature layers. We verify our findings by means of simulations. * * F. Strehle and J. Machado contributed equally.* * * The work of J.

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“…On the basis of the ideas of Chang and Sahinidis and Harney et al, we developed a new method to addresses the robust steady-state optimization of index-2 DAE systems under parametric uncertainty. Because the Lyapunov linearization theorem cannot be applied to DAE systems, the matrix pencil and Routh–Hurwitz test are used to formulate the stability constraints.…”

Section: Introductionmentioning

confidence: 99%

“…A high-index problem can be caused by the wrong choice of input variables, internal state constraint equations (e.g., reaction equilibrium and phase equilibrium), or submodels interconnection. 18 Recently, Harney et al 19 used the matrix pencil of the linearized system to directly evaluate the stability of the stationary points of index-2 DAE systems and performed bifurcation analyses of reactive flash and reactive distillation systems. They also showed that the index-reduced system by differentiation is not guaranteed to possess the same stability characteristics as that of the original system.…”

Section: Introductionmentioning

confidence: 99%

Robust Optimization of Index-2 Differential Algebraic Equations with Guaranteed Stability under Parametric Uncertainty: Application to a Reactor–Separator–Recycle Process

Xia

Zhao

2015

Ind. Eng. Chem. Res.

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In this article, we propose a method for steady-state optimal design of index-2 differential algebraic systems under parametric uncertainty. We use the matrix pencil to evaluate directly the stability of index-2 differential algebraic equations and formulate stability constraints using the Routh–Hurwitz test. The underlying mathematical problem is difficult to solve because it involves infinite stability constraints. We developed an algorithm where an infinite number of constraints can be implemented as several relaxation problems that are solved iteratively. Additionally, the simulation result under parametric uncertainty is used to estimate the bound of the state perturbations rather than assumptions based on experience that may lead to overly conservative or not implementable designs. To illustrate the method, we apply it to a reactor–separator–recycle process and obtain the robustly stable design.

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Numerical evaluation of the stability of stationary points of index-2 differential-algebraic equations: Applications to reactive flash and reactive distillation systems

Cited by 6 publications

References 16 publications

Biodiesel Production by Reactive Flash: A Numerical Simulation

Biodiesel Production by Reactive Flash: A Numerical Simulation

Port-Hamiltonian Modeling of Hydraulics in 4th Generation District Heating Networks

Robust Optimization of Index-2 Differential Algebraic Equations with Guaranteed Stability under Parametric Uncertainty: Application to a Reactor–Separator–Recycle Process

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