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Investigation into a multiple input/output bifurcated biochemical reaction with substrate inhibition in a real CSTR based on Cholette’s model

  • Chane-Yuan Yang ORCID logo , Ding-Chi Tsai and Yu-Shu Chien EMAIL logo
Published/Copyright: March 28, 2023
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International Journal of Chemical Reactor Engineering
From the journal International Journal of Chemical Reactor Engineering Volume 21 Issue 8

Abstract

CSTR operations entailing high nonlinearity and complexity such as multiple input and output steady-states present a real challenge to chemical engineers and process designers. The input multiplicity in chemical reactions leads to control probems such as process instability and low efficiency. Therefore, it is of critical importance to predict and avoid the multiplicity regions during reactor operation. Since the bifurcation analysis of biochemical processes with nonideal mixing has been carried out by the authors in the previous publication (Yang, C. Y., D. C. Tsai, and Y. S. Chien. 2021. “The Strategy Developed for High Conversion and the Multiplicity Problems of Biochemical Reaction in a Real CSTR with Cholette’s Model.” International Journal of Chemical Reactor Engineering 19: 1245–70), the goal of the present work is to use Sturm’s method, Routh stability criteria and the discriminator roots method with the tangent analysis method to derive the input multiplicity conditions in substrate inhibition in a real CSTR based on Chollete’s model. Four kinetic schemes are used in the analysis as examples to show that all three methods can precisely obtain the bifurcation starting point for the input multiplicity. In addition to the multiple input steady-states, the start-up diagram obtained by the discriminator root method is of critical importance to avoid operating in the input multiplicity regions.

Keywords: bifurcation starting point; Cholette’s model; CSTR; input/output multiplicity
Corresponding author: Yu-Shu Chien, Chemical and Materials Engineering, National Chin-Yi University of Technology, Taichung, 41170, Taiwan, E-mail:

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Sturm’s method

Sturm’s method considers a real algebraic equation, F(x), without multiple roots as Equation (1) (Uspensky 1945). N(x) is the number of sign changes in the function series.

(A-1) F 0 ( x ) = F ( x )
(A-2) F 1 ( x ) = d F ( x ) d x
(A-3) F i ( x ) = − r e m a i n d e r   [ F i − 2 ( x ) F i − 1 ( x ) ] , i ≥ 2
(A-4) F n ( x ) ≠ 0   i s   a   c o n s t a n t

Note that if Equation (A-1) has multiple roots, F n ( x ) is not a constant. Let N (A) and N (B) be the number of sign changes in the sequences of F 0 ( A ) , F 1 ( A ) , … , F n ( A ) and F 0 ( B ) , F 1 ( B ) , … , F n ( B ) , respectively, and N (A)−N (B) is the number of real roots between the upper and lower check points, A and B, respectively. Each multiple root in the interval is counted once merely.

A-1: 3rd-order characteristic equation

For the characteristic equation as a 3rd-order function of D, Equation (1) can be rewritten as

(A-5) F ( D ) = α 3 D 3 + α 2 D 2 + α 1 D + α 0 = 0 , α 3 > 0

Via derivation of the necessary condition of input multiplicity using Sturm’s method, one obtains.

(A-6) F 0 ( D ) = α 3 D 3 + α 2 D 2 + α 1 D + α 0 = 0
(A-7) F 1 ( D ) = 3 α 3 D 2 + 2 α 2 D + α 1 = 0
(A-8) F 2 ( D ) = K 1 D + K 2
(A-9) F 3 ( D ) = K 3

where K 1 , K 2 and K 3 are expressed as

(A-10) { K 1 = 2 α 2 2 9 α 3 − 2 α 1 3 K 2 = α 2 α 1 9 α 3 − α 0 K 3 = ( 2 α 2 − 3 α 3 K 2 K 1 ) ( K 2 K 1 ) − α 1

In the case of N (A)−N (B) > 0 there is a real root (or roots) in the interval [A, B]. The necessary condition for multiplicity can be described as follows,

  1. N (A)−N (B) = 0, the bifurcation points, D − and D + , cannot be found in [A, B], single steady-state.

  2. N (A)−N (B) = 1, D − = D + stands for the input multiplicity bifurcation starting point.

  3. N (A)−N (B) = 2, two distinct real roots D − and D + exist in [A, B], indicating that input multiplicity has occurred.

A-2: 4th-order characteristic equation

The 4th-order characteristic equation is considered as

(A-11) F ( D ) = α 4 D 4 + α 3 D 3 + α 2 D 2 + α 1 D + α 0 = 0 , α 4 > 0

Following Sturm’s algorithm, we obtain the expressions for F i ( D ) and corresponding coefficients as follows

(A-12) F 0 ( D ) = α 4 D 4 + α 3 D 3 + α 2 D 2 + α 1 D + α 0
(A-13) F 1 ( D ) = 4 α 4 D 3 + 3 α 3 D 2 + 2 α 2 D + α 1
(A-14) F 2 ( D ) = K 4 D 2 + K 5 D + K 6
(A-15) F 3 ( D ) = K 7 D + K 8
(A-16) F 4 ( D ) = K 9

where

(A-17) { K 4 = 3 α 3 2 16 α 4 − α 2 2 K 5 = 2 α 3 α 2 16 α 4 − 3 α 1 4 K 6 = α 3 α 1 16 α 4 − α 0 K 7 = 4 α 4 K 6 K 4 + [ ( 3 α 3 − 4 α 4 K 5 ) K 5 K 4 2 ] − 2 α 2 K 8 = ( 3 α 3 K 4 − 4 α 4 K 5 K 4 2 ) K 6 − α 1 K 9 = ( K 5 − K 8 K 4 K 7 2 ) K 8 − K 6

The necessary multiple steady-state condition is the same as that for the 3rd-order characteristic equation.

Appendix B: Routh stability criteria

Consider the characteristic equation that is an Nth-order polynomial as Equation (1), the Routh array is formed as given below:

(B-1) [ α n α n − 2 α n − 4 … α 0 α n − 1 α n − 3 α n − 5 … 0 a 1 a 2 a 3 … 0 b 1 b 2 b 3 … 0 . . . … 0 . . . … 0 c 1 c 2 c 3 … 0 d 1 0 0 … 0 ]

where the coefficients c 1 , c 2 , c 3 , … are in the nth row of the array. a 1 , a 2 , a 3 , b 1 , b 2 , … …. are calculated from the equation.

(B-2) { a 1 = α n − 1 α n − 2 − α n α n − 3 α n − 1 a 2 = α n − 1 α n − 4 − α n α n − 5 α n − 1 b 1 = a 1 α n − 3 − α n − 1 a 2 a 1 b 2 = a 1 α n − 5 − α n − 1 a 3 a 1 . . .

Derivation of the necessary condition for multiplicity using RSC for the 3rd-order characteristic equation is the same as the 4th-order equation; therefore the 4th-order case is taken as an example. The Routh array can be obtained according to Equation (B-2) as

(B-3) [ α 4 α 2 α 0 0 α 3 α 1 0 0 β 1 β 2 0 0 β 3 0 0 0 β 2 0 0 0 ]

where

(B-4) { β 1 = α 3 α 2 − α 4 α 1 α 3 β 2 = α 0 β 3 = β 1 α 1 − β 2 α 3 β 1

By replacing x with x t + t , we obtain the transferred characteristic equation as

F ( x t ) = α 4 ( x t + t ) 4 + α 3 ( x t + t ) 3 + α 2 ( x t + t ) 2 + α 1 ( x t + t ) + α 0
(B-5) = α 4 t x t 4 + α 3 t x t 3 + α 2 t x t 2 + α 1 t x t + α 0 t = 0

where the coefficients are calculated as

(B-6) { α 4 t = α 4 α 3 t = 4 α 4 + α 3 α 2 t = 6 α 4 + 3 α 3 + α 2 α 1 t = 4 α 4 + 3 α 3 + 2 α 2 + α 1 α 0 t = α 4 + α 3 + α 2 + α 1 + α 0

As a result, the Routh array for the transferred characteristic equation can be obtained as

(B-7) [ α 4 t α 2 t α 0 t 0 α 3 t α 1 t 0 0 β 1 t β 2 t 0 0 β 3 t 0 0 0 β 2 t 0 0 0 ]

where the elements are calculated as

(B-8) { β 1 t = α 3 t α 2 t − α 4 t α 1 t α 3 t β 2 t = α 0 t β 3 t = β 1 t α 1 t − β 2 t α 3 t β 1 t

The necessary condition for multiple steady-states is that the number of sign changes in the first column of Equation (3), [ α 4   α 3   β 1 β 3   β 2 ] T , subtracts that in the first column of Equation (8), [ α 4 t   α 3 t   β 1 t β 3 t   β 2 t ] T , equals 2. On the other hand, the bifurcation condition is that there are two equal real roots for the 4th-order characteristic equation in [ D − , D + ] (Chien and Chiu 2009b). In this case, the 1st- and zeroth-order terrms in the transferred characteristic equation, Equation (B-5), vanish. That is,

(B-9) { α 1 t = 0 α 0 t = 0

Substituting Equation (B-9) into Equation (B-8) results in the elements β 2 t and β 3 t becoming zero. As a result, the Rouse array for the starting bifurcation can be written as

(B-10) [ α 4 t α 2 t 0 0 α 3 t 0 0 0 β 1 t 0 0 0 0 0 0 0 0 0 0 0 ]

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Received: 2022-09-06
Accepted: 2023-02-02
Published Online: 2023-03-28

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/ijcre-2022-0176
Keywords for this article
bifurcation starting point; Cholette’s model; CSTR; input/output multiplicity

Articles in the same Issue

  1. Frontmatter
  2. Articles
  3. Enhanced mechanical stirring by eccentric impeller stirring system in zinc hydrometallurgy process for cadmium removal
  4. DEM simulation of biomass pyrolysis in a novel interconnected screw reactor
  5. Numerical and experimental investigations on enhancement mixing performance of multi-blade stirring system for fluids with different viscosities
  6. The technical and economic analysis of processing and conversion of heavy oil cuts to valuable refinery products
  7. Effect of inlet gas velocity on gas-solid fluidization characteristics in fluidized bed
  8. Investigation into a multiple input/output bifurcated biochemical reaction with substrate inhibition in a real CSTR based on Cholette’s model
  9. Performance of photocatalytic oxidation surface with new geometry for indoor environment application: experimental and simulation
  10. Optimization of hydrothermal liquefaction process for bio-oil products from kitchen residue under subcritical conditions
  11. Value-added biochar production from microwave pyrolysis of peanut shell
  12. Short Communications
  13. Environmentally sustainable synthesis of cyclic carbonates from epoxides and CO2 promoted by MCM-41 supported dual imidazolium ionic liquids catalysts
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