Teaching arc‐length continuation in the chemical engineering graduate program using MATLAB©

Abstract

We present the solution of six problems drawn from the chemical engineering and applied mathematics fields of study. These problems are solved using the arc‐length continuation method. Among a large number of examples, we have chosen to treat the following case studies because they illustrate basic concepts such as solution multiplicity when dealing with systems of nonlinear algebraic equations or boundary value problems: 1) Plotting an ∞‐shaped curve called the Bernoulli's Lemniscate; 2) Computing the Joule‐Thomson inversion curve for a pure gas using the Soave‐Redlich‐Kwong cubic equation of state; 3) Predicting the Compressibility Factor and Molar Volume versus the Reduced Temperature; 4) Determining region where steady‐state multiplicity is observed in a continuous stirred‐tank reactor (CSTR) with multiple reactions and heat effects; 5) Finding the steady‐state temperature and conversion of a Propylene Glycol Reactor; and 6) Investigating the Frank–Kamenetskii problem arising in the self‐heating of a reactive solid. Reader should be advised that problem 1 may seem slightly mathematical in nature. However, case studies 2–6 could be very useful to instructors involved in teaching not only numerical methods but also chemical thermodynamics and chemical reaction engineering. For one of the above listed case studies, we unveil the MATLAB© code, that we have employed, in the Appendix. In addition, we will emphasize on the versatility of this mathematical software through the extensive usage of the built‐in command fsolve, for the presently undertaken computations. Finally, we conclude this paper by sharing our experience teaching this subject to graduate students at King Fahd University of Petroleum & Minerals.

Nonphysical Behavior in Several Statistical Mechanically Based Equations of State

“…We begin with introducing the general features of bifurcation diagrams generated from equation of state by pseudo arc-length continuation method. ,, The reader may consult our previous work where we described the details of the numerical analysis . A brief summary is given in Appendix B.…”

Simulation‐based equations of state for the Lennard‐Jones fluid: Apparent success and hidden failure

“…One feasible method of generating the bifurcation diagrams is to transform the nonlinear algebraic equation into differential algebraic equations using pseudo arc-length continuation method. 34,42,43 The transformation is obtained by introducing a relationship between the variables of the nonlinear algebraic equation with a variable denoted by arc-length segment(s) using the Pythagorean rule. For instance, we transform the equation of state (f(T, V, P) = 0) into a system of differential algebraic equations given by:…”

“…The technique tracks the evolution of volume roots (V) as temperatures (T) vary at constant pressure (P). The starting point of the evaluation is to construct a V‐T bifurcation diagram at constant pressure using pseudo arc‐length continuation method 42,43 . The study of the V‐T bifurcation diagram is sufficient to discover various types of erroneous behaviors including spurious two‐phase separation regions, unrealistic critical points, unphysical branches, multiple volume roots, and unphysical turning points.…”

“…The starting point of the evaluation is to construct a V-T bifurcation diagram at constant pressure using pseudo arc-length continuation method. 42,43 The study of the V-T bifurcation diagram is sufficient to discover various types of erroneous behaviors including spurious two-phase separation regions, unrealistic critical points, unphysical branches, multiple volume roots, and unphysical turning points. In this work, we study four distinct types of MPS-LJ EOSs: MBWR (with three different parameter sets), Kolafa and Nezbeda, 22 Mecke et al, 23 and Thol et al 21 The three versions of MBWR are Nicolas et al, 17 Johnson et al, 18 and May and Mausbach.…”

Using Mathcad to facilitate the design of chemical reactors involving multiple reactions

“…The traditional way of consolidation or putting into practice the theoretical concepts related to chemical reaction engineering has been accomplished through short‐term laboratory experiences . However, the pedagogy of teaching chemical reaction engineering is continually advancing through the use of new computational tools and, therefore, there is an ongoing effort at many universities to integrate these new tools into the chemical reaction engineering course .…”