The nonideal mixing effect on the selectivity dynamic of consecutive-parallel reactions in an isothermal continuous stirred tank reactor based on Cholette’s model

Chane-Yuan Yang; Zhengwei Lin; Yu-Shu Chien

doi:10.1515/ijcre-2024-0023

Article

The nonideal mixing effect on the selectivity dynamic of consecutive-parallel reactions in an isothermal continuous stirred tank reactor based on Cholette’s model

Chane-Yuan Yang , Zhengwei Lin and Yu-Shu Chien

Published/Copyright: June 2, 2025

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From the journal International Journal of Chemical Reactor Engineering Volume 23 Issue 4

Abstract

This work presents the mixing conditions and kinetic reaction constant effects on a continuous stirred tank reactor (CSTR) consecutive-parallel reactions selectivity dynamics in an isothermal CSTR based on Cholette’s model. The mixing condition effect is accounted for using the parameter μ = n/m, where n is the flow fraction to the reactor and m is the dead space fraction in the reactor. μ = 1 and μ≠1 represent, respectively, the ideal and nonideal mixing operations. The final desired product yield steady state in the phase plane is first constructed under reaction constant and mixing condition combinations, μ = 0.5, μ = 1 and μ = 2. The simulation results reveal that the larger the μ parameter, the higher the reactant final steady state values, a _s, and the desired product, b _s. For μ = 0.5 < 1, the short circuit flow or channeling effects were less than that of the dead volume in a real CSTR, leading to lower both final steady state values, a _s and b _s, than those under ideal mixing, μ = 1. On the other hand, under the nonideal mixing condition μ = 2 > 1, i.e., the short circuit flow effect or channeling larger than the dead volume in the reactor, resulted in a higher final steady state (a _s,b _s) than those under ideal mixing. A comparative analysis of the mixing condition effects reveals that a higher μ value reduces the transient maximum selectivity regions while increasing the transient minimum selectivity regions. In the selectivity dynamic simulations, the phase plane is divided into a transient extreme yield region and monotonic yield variation region. It is observed, for the first time, that a higher μ value diminishes the transient maximum selectivity region area while expanding the transient minimum selectivity region area in the phase plane. Additionally, the trajectory from the start up to the final steady state is altered due to nonideal mixing. These novel findings, which have not been previously published, hold direct relevance for the design and operation of real reactors. Given that ideal mixing is rarely achievable in practical CSTRs, our results offer valuable insights for optimizing reactor design and start-up policies to enhance the desired product transient yield.

Keywords: consecutive-parallel reaction; selectivity dynamics; nonideal mixing; CSTR; Cholette’s model

Corresponding author: Yu-Shu Chien, Chemical and Materials Engineering, National Chin-Yi University of Technology, Taichung, Taiwan, E-mail: yschien@ncut.edu.tw

Research ethics: Not applicable.
Informed consent: Informed consent was obtained from all individuals included in this study, or their legal guardians or wards.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interests: The authors state no conflict of interest.
Research funding: None declared.
Data availability: The raw data can be obtained on request from the corresponding author.

Appendix

In the work of Varma and DeVera [1]; three distinct cases arose depending on the relative kinetic reaction constant magnitudes. These cases were analyzed separately to determine the specific reactor start-up conditions for consecutive-parallel reactions. The key results throughout the paper are summarized in Table A1.

Table A1:

Varma and DeVera [1] analysis for the consecutive-parallel reaction selectivity dynamic in a CSTR.

Case	Reaction rate constants	Eigenvalue		Characteristic directions		Transient yield region		Monotonic region
Case	Reaction rate constants	λ ₁	λ ₂	z ₁	z ₂	Maximum yield	Minimum yield	Yield increase	Yield decrease
1	k₂ < k₁ + k₃	−(1 + k₂ τ)	−(1 + (k₁ + k₃) τ)	line ASB	line CSD	SCIE	OSD	OSC	DSE
2	k₂ > k₁ + k₃	−(1 + k₂ τ)	−(1 + (k₁ + k₃) τ)	line ASB	line CSD	SAIE	OSBF	OAS	BSEG
3	k₂ = k₁ + k₃	−(1 + k₂ τ)	−(1 + (k₁ + k₃) τ)	line ASB	line ASB^a	SAIE	OSBF	OAS	BSEG

^aIn case 3, because λ₁ = λ ₂, line CSD overlaps the vertical line ASB.

Note that the “Transient yield region” refers to the region where all initial conditions (ICs) lead to maximum or minimum yield for B. The ‘Monotonic region” stands for the region, in which the B yield increases or decreases monotonically over time θ. For Case 1 all ICs lying in the SCIE region give rise to a transient maximum in the B yield, at a point like P ₁ in their approach to the steady state. In Cases 2 and 3, all ICs in the SAIE region result in transient maximum B yields, at a point such as P ₁, in their approach to the steady state S. Moreover, for Case 1, all ICs in the OSD region result in a transient minimum in B yield at point P ₂, and all ICs in the OSBF region result in a transient minimum in B yield at point P ₂.

On the other hand, the B yield increases monotonically over θ for all ICs lying in the OSC region, while all ICs above DSE lead to a monotonic decrease in B yield in Case 1. Similarly, the B yield increases (decreases) monotonically with time θ for all ICs in the OAS (BSEG) regions for Case 2 and Case 3.

Varma and DeVera [1] concluded that for the consecutive-parallel reaction under consideration, given a specific set of reaction and reactor parameters, the transient desired product yield can be effectively enhanced by simply choosing the appropriate reactor start-up policy.

Nomenclature

A: reactant species
a: dimensionless concentration of A
a _max: maximum value of a
a _s: steady state value of a
B: desired reaction product
b: dimensionless concentration of B
b ₀: concentration of species B in the feed
b _max: maximum value of b
b _s: steady state value of b
C: desired reaction product
D: desired reaction product
C _A0: concentration of species A in the feed
C A ′: concentration of species A in the reactor
C _B0: concentration of species B in the feed
C B ′: concentration of species B in the reactor
k _i: reaction constant, i = 1, 2, 3
m: fraction of dead space in CSTR
n: fraction of reactant flow to CSTR
q: reactant volumetric flow rate
r _i,j: dynamic coefficients, i, j = 1, 2, 3
R: dynamic coefficients matrix
S _i,j: final steady state, i, j = 1, 2, 3
t: time
V: CSTR volume
x 1: b − b s
x 2: a − a s
x: vector of x _i, i = 1, 2
z _i: eigenvector, i = 1, 2

Greek symbols

θ: = t/τ, dimensionless time
λ_i: eigenvalue, i = 1, 2
μ: = n/m, mixing condition parameter
τ: = V/q characteristic time of CSTR

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Received: 2024-01-29

Accepted: 2025-05-06

Published Online: 2025-06-02

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Articles in the same Issue

https://doi.org/10.1515/ijcre-2024-0023

Keywords for this article

consecutive-parallel reaction; selectivity dynamics; nonideal mixing; CSTR; Cholette’s model