Using computational singular perturbation as a diagnostic tool in ODE and DAE systems: a case study in heterogeneous catalysis

Abstract

We have extended the computational singular perturbation (CSP) method to differential algebraic equation (DAE) systems and demonstrated its application in a heterogeneous-catalysis problem. The extended method obtains the CSP basis vectors for DAEs from a reduced Jacobian matrix that takes the algebraic constraints into account. We use a canonical problem in heterogeneous catalysis, the transient continuous stirred tank reactor (T-CSTR), for illustration. The T-CSTR problem is modelled fundamentally as an ordinary differential equation (ODE) system, but it can be transformed to a DAE system if one approximates typically fast surface processes using algebraic constraints for the surface species. We demonstrate the application of CSP analysis for both ODE and DAE constructions of a T-CSTR problem, illustrating the dynamical response of the system in each case. We also highlight the utility of the analysis in commenting on the quality of any particular DAE approximation built using the quasi-steady state approximation (QSSA), relative to the ODE reference case.

Keywords:

• CSP
• ODE
• DAE
• QSSA
• T-CSTR
• heterogeneous catalysis

Acknowledgments

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy's National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the US Department of Energy or the United States Government.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This assumption can be relaxed using the implicit function theorem [61,62].

2 We denote surface species with an asterisk superscript, as in C${}^{\ast }$, and an empty surface site with $\ast$.

3 For example, the gas term in the temperature equation has $V_i^{\mathrm{\prime }}:=-\frac{1}{\rho c_p}\left(\sum _{k=1}^{K_g}{h}_k{W}_k{\nu }_{ki}\right),i=1,\dots ,L_{\mathrm{g}}$

Additional information

Funding

This work was supported by the U.S. Deparment of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division, as part of the Computational Chemistry Sciences Program (Award Number: 0000232253).