Application of Cholette’s model in non-ideal mixing CSTR: a simulation study on dynamic behavior

Chane-Yuan Yang; Yu-Shu Chien; Jun-Hong Chou; Hsing-Ya Li; Chau Wei Hsieh

doi:10.1515/ijcre-2020-0197

Published by De Gruyter February 23, 2021

Application of Cholette’s model in non-ideal mixing CSTR: a simulation study on dynamic behavior

Chane-Yuan Yang , Yu-Shu Chien , Jun-Hong Chou , Hsing-Ya Li and Chau Wei Hsieh

From the journal International Journal of Chemical Reactor Engineering

https://doi.org/10.1515/ijcre-2020-0197

Showing a limited preview of this publication:

Abstract

Non-ideal mixing phenomena are widely found in industrial chemical reactors. In this work we derived the bifurcation formulas for a non-adiabatic CSTR with an irreversible exothermic first order reaction with the non-ideal mixing effect. This is investigated via dynamic behavior simulations based on Chollete’s model. The results show that the non-ideal mixing parameter n (the fraction of the feed entering the perfect mixing zone) determines the variation between six classified regions and dominates the dynamic behavior patterns in the steady-state response diagram. On the other hand, the phase portraits of examples verify the formulas derived in this work. We note that the non-ideal mixing effect has significant importance in CSTR design and control steps. For example, in the safe operating region for an ideal mixing CSTR, non-linear dynamics are obtained by the system under non-ideal mixing conditions (n ≠ 1). The present study has significance and help for chemical reactor design and CSTR control.

Keywords: Chollete’s model; dynamic behavior; non-adiabatic CSTR; non-ideal mixing

Corresponding author: Yu-Shu Chien, Department of Chemical and Material Engineering, National Chin-Yi University of Technology, Taichung, 411, Taiwan, R.O.C, E-mail: yschien@ncut.edu.tw

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix Friedrichs’ theory for bifurcation periodic solutions (Poore 1973)

It is convenient to write Eqs. (4) and (5) in the following form

(A-1)dy‾ds=A‾y‾+μG‾(y‾,μ)

where μ is a small parameter. It is noted that the solutions of the system

(A-2)dy‾ds=A‾y‾

are periodic. This is the case in which the eigenvalues of A‾ are purely imaginary or one and only one of the eigenvalues is zero. Poore (1973) considered only the case where the matrix A‾ was obtained by the linearization about the center and formulated the necessary general theory for the two dimensional autonomous system as follows:

(A-3)dx‾ds=F∧‾(x‾ ,γ)

let aγ∧‾ be defined by F∧‾(aγ∧‾ ,γ)=0 and introduces the following change of variables:

γ=γ0+ϵ,aϵ‾=aγ0+ϵ∧‾,s=T0Tϵt,ϵ=μδ,Tϵ=T0(1+μη),xϵ‾=aϵ‾+μy‾(s,μ),

(A-4)ϵCϵ‾=A‾ϵ−A0‾, C0=dAϵ‾dϵ|ϵ=0,μ2Qϵ‾(y‾,μ)=F‾(aϵ‾+μy‾,ϵ)−μAϵ‾y‾

where T0, δ, η and a0‾are predetermined and μ is an auxiliary parameter. Via the change of variable, the system Eq. (A-3) can be written as

(A-5)dy‾ds=A0y‾+μ{δC(μδ)‾y‾+ηA(μδ)‾y‾+(1+μη)Q‾(μδ)(y‾,μ))}

Consequently, we can construct the following expression.

Theorem A-1

Suppose the two dimensional vector, F‾(x‾,ϵ)∈C2[D×(−ϵ0,ϵ0)], where D is a domain in R² and ε₀ is a positive number. Assume that the equation dx‾dt=F‾(x‾,ϵ) has a constant solution x‾=aϵ‾such that for the value ϵ=0 the eigenvalues of the matrix A0‾=F‾,(a‾0,0)x‾ are purely imaginary of ± iw0 with w0≠0. Furthermore, suppose that the matrix C0‾ does not vanish. There then exists functions η=η(μ)and δ=δ(μ) with ϵ=μδ(μ), Tϵ=T0(1+μη(μ)), η(0)=0, δ(0)=0, and η(μ) and δ(μ)∈C1[0,μ0) for some sufficiently small μ0>0 and a function y‾=(s,μ) with period T0 in s. Assume arbitrarily the prescribed initial value y‾(0,μ)=b0‾ so that

(A-6)xϵ‾=aϵ(μ)‾+μy(T0Tϵ(μ)t,μ)‾

is a solution of the following differential equation

(A-7)dx‾dt=F‾(x‾,ϵ(μ))

In aforementioned theorem, the case Tϵ=T0and ϵ=0 is not excluded. In this case, the point a⁰, which is a center for the linearized system, is also a center for the nonlinear problem. The proof of Theorem one is based on the following fact.

Corollary A-2

Bifurcation from the critic point aγ∧‾ of dx‾dt=F‾∧(x‾,γ)can only occur from those aγ0∧‾ which are the centers in the associated linearized problem or possibly when one and only one of the eigenvalues of the matrix A‾ is zero.

To determine the locations of the solution y‾ and the functions η and δ on μ, we first note that η,δ∈C1[0,μ0). By using η(0)=δ(0)=0, we have

η(μ)=μη1+μdηdμ(θ1,μ) for μ∈[0,μ0) and some 0<θ1<1 andδ(μ)=μδ1+μdδdμ(θ2,μ) for μ∈[0,μ0) and some 0<θ2<1

Note that the magnitudes of μdηdμ(θ1,μ) and μdδdμ(θ2,μ) are o(μ) as μ→0. Since ϵ=μδ(μ)=δ1μ2+o(μ2) as μ→0, the sign of ϵ is determined by the sign of δ1 for sufficiently small μ if δ1≠0. Similarly, the sign of (Tϵ−T0) is determined by the sign of η1. The most important here is that the direction of bifurcation is determined by δ1. Sinceγ−γ0=δ1μ2+o(μ2) as μ→0, the sign of (γ−γ0) is determined by δ1 for sufficiently small μ. If δ1 > 0, a small periodic solution grows from aγ0∧‾ as γ increase beyond γ0. If δ1 < 0, then a small periodic solution grows from aγ0∧‾ as γ decreases below γ0. In this sense the direction of the bifurcation is determined by δ1. To determine δ1 and η1 the following continuity properties of y‾ is needed.

Theorem A-3

Under the continuity assumptions for F‾ in Theorem A-1 and the properties of δ(μ) and η(μ), we have

(A-8)y‾(s,μ)=y0‾(s)+μy1‾(s)+μy−−(s,μ)

where μy−−(s,μ)=0(μ) as μ→0 for s∈[0,∞). The functions y0‾(s), y1‾(s) andy−−(s,μ) are periodic with period T0 for sufficiently small μ. The functions y0‾(s) and y1‾(s) are given by

(A-9)y0‾(s)=Y(s)b0‾

and

(A-10)y1‾(s)=Y(s)∫0sY−1(τ)Q0(y0(τ),0)dτ=∫0sY(s−τ)Q0(y0(τ),0)dτ

where Y(s) is the matrix following the conditions below

(A-11)dYds=A0Y and Y(0)=I

The proof in detail sees in Poore (1973). Using Theorems A-1 and A-3, we obtain the information about δ(μ), η(μ), δ1 and η1. The functions δ(μ) and η(μ) are obtained implicitly from

(A-12)0=η∫0T0Y−1(τ)A‾(ηδ)y‾(τ,μ)dτ+δ∫0T0Y−1(τ)B‾(ηδ)y‾(τ,μ)dτ+∫0T0Y−1(τ)(1+μτ)Q‾(ηδ)y‾((τ,μ),μ)dτ

While δ1 and η1 are determined explicitly from

(A-13)0=η∫0T0Y−1(τ)A‾(ηδ)y‾(τ,μ)dτ+δ∫0T0Y−1(τ)B‾(ηδ)y‾(τ,μ)dτ+∫0T0Y−1(τ)(dQ‾(ηδ)dμy‾((τ,μ),μ))|μ=0dτ

where y‾0is given in Theorem A-3 and Y(s) is the fundamental matrix solution mentioned therein. The detailed proof sees in Poore (1973).

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Received: 2020-10-07

Accepted: 2021-02-03

Published Online: 2021-02-23

Application of Cholette’s model in non-ideal mixing CSTR: a simulation study on dynamic behavior

Abstract

Appendix Friedrichs’ theory for bifurcation periodic solutions (Poore 1973)

Theorem A-1

Corollary A-2

Theorem A-3

References

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