Multi-objective optimization method for enhancing chemical reaction process

Accepted Manuscript Multi-objective Optimization Method for Enhancing Chemical Reaction Process Xuepu Cao, Shengkun Jia, Yiqing Luo, Xigang Yuan, Zhiwen Qi, Kuo-Tsong Yu PII: DOI: Reference:

S0009-2509(18)30697-3 https://doi.org/10.1016/j.ces.2018.09.048 CES 14524

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

22 May 2018 6 September 2018 26 September 2018

Please cite this article as: X. Cao, S. Jia, Y. Luo, X. Yuan, Z. Qi, K-T. Yu, Multi-objective Optimization Method for Enhancing Chemical Reaction Process, Chemical Engineering Science (2018), doi: https://doi.org/10.1016/j.ces. 2018.09.048

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Multi-objective Optimization Method for Enhancing Chemical Reaction Process Xuepu Cao a, Shengkun Jia a, Yiqing Luo a, Xigang Yuan a,b, Zhiwen Qi c, Kuo-Tsong Yu a,b a

School of Chemical Engineering and Technology, Tianjin University, 300072, Tianjin, China

b

State Key Laboratory of Chemical Engineering, Tianjin University, 300072, Tianjin, China

c

State Key Laboratory of Chemical Engineering, East China University of Science and Technology, 200237,

Shanghai, China

Abstract：An approach to enhancing the chemical reaction process based on flow pattern construction was proposed. We demonstrated that the entropy generation due to mass transfer, chemical reaction and the viscous dissipation tend to their extremums when the performance of the reaction is enhanced. Based on this finding, a multi-objective optimization problem was formulated. The weight coefficients of individual objectives (various entropy generations) were derived according to their relative importance. The final objective function was formulated by the linear combination of the entropy generations of mass transfer and chemical reaction. By minimization of the objective function, subject to a constraint of fixed viscous dissipation, through calculus of variations, a velocity field and associate body force field can be obtained, by which the reaction process was effectively enhanced. The flow pattern obtained by optimization calculation can be used to guide the design of the internal structures in the reactors. Keywords: Reaction process enhancement; Multi-objective optimization; Weight coefficient; Entropy generation; Viscous dissipation Nomenclature ai Mole concentration, ci Mass concentration, c Total molar concentration, C0, L1,L2,L3 Lagrange multiplier D Diffusion coefficient, Additional volume force per unit volume, F

rG

Gibbs free energy,

J k L

Objective function Chemical reaction rate constant, Lagrangian function

 Corresponding author. E-mail address: [email protected] (Xigang Yuan).

M Molecular weight, N Correction factor P Pressure, R Universal gas constant, S Entropy generation, T Temperature, Velocity vector, U x Cartesian coordinates, y Cartesian coordinates, w weight coefficient Greek symbols  Viscosity,  Stoichiometric coefficient

i 

  

 

Chemical potential for species , Density, Local entropy generation, Viscous dissipation, Viscous dissipation, Eigenvalue of the matrix Rate of the chemical reaction

Superscripts 

Standard condition

Subscripts A B C BA CA i R 

Related to component A Related to component B Related to component C Physical parameters for component B relative to component A Physical parameters for component C relative to component A Related to component, i Related to reaction Related to viscous dissipation

1 Introduction Reactors are widely used in chemical or biochemical process industries for a variety of applications, such as chemical absorption, organic synthesis (Albini and Fagnoni, 2009), combustion process (Ghermay et al., 2011), catalytic reaction (Albini and Fagnoni, 2009; Martín et al., 2016; Müller et al., 2005), biodegradation process (Gavrilescu and Chisti, 2005; Jia et al., 2010), etc. Fluid dynamics and heat and mass transfer affect the conversion of reactions through the kinetics. As the reaction process is usually complicated, the optimal design of reactors or systematic approach to enhancing the reaction processes to increase the conversion and the yield remain challenging. The better understanding of the process is of great importance. In chemical reactors, mass transfer in the forms of fluid convection as well as component diffusion is essential in some cases to guarantee the conversion. Attempts have been made to

enhance this kind of mass transfer or chemical reaction process, for example, using “Lateral Sweep Vortex Generators” (Aravind and Deepu, 2017), adding corrugated fluidic networks in redox flow batteries (Lisboa et al., 2017), adding carbon nanotube nanofluids (Yazid et al., 2017), adding ribs plate for photocatalytic oxidation reactor (Chen and Meng, 2008). In addition to these passive measures, there are many active measures for chemical process enhancement, such as, flow rate control during vanadium flow battery operations (Tang et al., 2014; Zheng et al., 2016) and imposing oscillatory frequency in grooved channels for pulsatile flow (Nishimura et al., 2000). All these measures above used in different chemical process are trying to optimize the relationship between the mass transport enhancement and the energy consumption. These methods can be used to enhance the chemical reaction process, as they can alter the fluid flow field, change the temperature distribution and the concentration distribution of the species, and ultimately affect the chemical reaction kinetics. However, these measures enhance the process at expense of mechanical power, as the energy input by either the augmented pressure of the feed supply to the reactor with additional baffles or the increased agitation power is not recoverable but dissipated in the viscose fluid flow. Therefore, a reasonable design of a reactor depends on a proper trade-off between enhancement extent of the reaction process and the operating cost for the mechanical energy input. Along this line of design, a fundamental question to a designer retains: what is the best flow field to enhance the process at a given mechanical energy input. Actually, this question is usually answered empirically by designers’ experiences. The same issue has been addressed extensively in the studies of convective heat transfer processes by minimizing the entropy production (the irreversibility) of process (Bejan, 1995, 1997). And, a number of investigations on the entropy generation as a design tool in other transport or chemical reaction processes can be found (Bejan and Tondeur, 1998; Sciacovelli et al., 2015; Tondeur and Kvaalen, 2007). Li et al. (2014) developed a physically feasible geometry configuration by minimizing the entropy generation due to heat transfer to improve the heat transfer process in a solar receiver. Jia et al. (2016) achieved an optimal velocity field by taking the extremum of the entropy generation due to mass transfer to enhance the mass transfer process. Johannessen and Kjelstrup (2004) presented a way to minimize the entropy production rate in plug flow reactors for SO2 oxidation. Their study showed that they reduced the entropy production rate through regulating the reactor length and the temperature of the cooling/heating medium, under the certain production of chemicals. Gutierrez and Méndez (2012) developed an optimization method based on entropy generation minimization to improve the design and operation of the reactor of methane thermal cracking. Kjelstrup et al. (2000) analyzed the entropy production of the reaction that produced methanol from the mixed gas of carbon dioxide and hydrogen, and showed that it was possible to accomplish considerable reductions of entropy production due to heat transfer. Nummedal et al. (2003) minimized the entropy production rate of the ammonia reactor, subject to a fixed ammonia production. Achievements have been made in the previous works, but optimal designs are mainly based on an overall entropy generation of the process as the objective. In fact, the various entropy generations and their respective effect on the chemical reaction process are different, even in opposite directions. So it is essential to split the overall entropy generation according to the individual contributions: fluid friction, heat transfer, mass transfer and reaction. This classification can help designers identify the contributions of various entropy generations to the process: which

ones affect the most the irreversibility (Arjmandi and Amani, 2015; Bidi et al., 2010; Lior et al., 2006; Rana et al., 2014) and how much these ones are more important than the others. As a result, the multi-objective optimization problem needs to be considered to enhance the chemical reaction process performance. The main purpose of the present paper is to propose a method for optimal design of a reaction process, in which the entropy generations due to the mass transfer, chemical reaction and viscous dissipation are considered, assuming that the heat effect can be neglected. The various entropy generations are used as objective functions with the viscous dissipation as a constraint. By solving these optimization problems through calculus of variations, the dependences of the optimal velocity fields and the associated entropy generations on the viscous dissipation are obtained. The weight coefficients for each entropy generation are calculated by using the principal eigenvector of the positive pairwise comparison matrix, which is obtained by comparing the relative importance of the different entropy generations. The multi-objective problem is thus converted to a single objective problem by linear combination of the various entropy generations with the weight coefficients. The problem is solved for an optimal velocity field by which the reaction process is effectively enhanced. Two different types of reactors with one-order irreversible reaction were numerically studied to illustrate and demonstrate the applicability of the proposed approach.

2 Entropy generation analysis 2.1 Physical model The reactor model is a two-dimensional rectangular channel with the length of 30 mm and the height of 5 mm, as shown in Fig. 1.

Fluid out (A, C)

Fluid in (A)

Mass flux (B) Fig. 1 The rectangular channel reactor model The reactant A flows into the channel from the left (inlet). The reactant B flows into the channel from a segment of 10 mm in width at the bottom at a constant flux, 0.01 m-2·s-1. The product C together with reactant A which is not converted in the reaction flows out from the right (outlet). The reaction in this model may correspond to the acid gases absorption process which has been studied by many investigators because of theoretical interest as well as industrial importance (Hikita et al., 1976; Kloosterman et al., 1987). The velocity of the fluid is uniformly distributed with a value of 0.001 m/s at the entrance. The corresponding value of Reynolds number based on the height of channel at the entrance is 5. At the outlet, the boundary condition is outflow. At the wall, no-slip boundary condition is applied. The reaction is described as follows:

A B C

(1)

The following assumptions are adopted for this model. i. The chemical reaction process is in steady-state with fluid flow. ii. Reactant A is excess compared to reactant B, and the product C can be regarded as a dilute solute in the reactant A. The diffusion between the components B and C is neglected. iii. The heat of chemical reaction is neglected. As the reactant A is excess, the chemical reaction given by Eq. (1) can be known as an irreversible pseudo-first-order reaction, where the reaction rate depends only on the concentration of reactant B. For the chemical reaction in this two-dimensional channel reactor, the governing equations are as follows: Continuity equation:

  U  0

(2)

  U  U  P  2U  F

(3)

Momentum equation:

Species conservation equation:

 U cB     DBAcB   M B

(4)

 U cC     DCAcC    M C

(5)

The ω is the reaction rate as follows:

k

 cB

(6)

MB

For a two-dimension problem, if the body force, F, is known, Eqs. (2), (3), (4) and (5) contain five unknowns, ux, uy, cB, cC, P, the same as the number of equations. It is clear that for different values of F, the solutions of the equation set for velocity field and concentration field are different, which may affect the reaction process, and hence lead to the different flux of product C at the exit. For our problem in this paper, the body force F is taken as unknown, and to be optimized so that the production rate of the product C is maximized. The extremum principle of entropy generation in the chemical reaction process is employed to solve the problem for the optimal body force F.

2.2 Entropy generation variation analysis For the reaction system in this work, the irreversibility mainly occurs due to mass transfer, chemical reaction, and the viscous effects of the fluid. The entropy generation rate can be expressed as the sum of contributions of these effects, and thus, the volumetric entropy generation rate, σ, at each point can be calculated as follows (Carrington and Sun, 1991; Hirschfelder et al., 1964; Jia et al., 2016; Teng et al., 1998):



 2 RDBA

M B M AcB 1  cB  c

 cB   2

 2 RDCA

M C M AcC 1  cC  c

 cC 

2

1 1   r G  Ф T T

(7)

The first two terms of the right hand side of the Eq. (7) are the entropy generation due to the mass transfer of reactant, B, and product, C, respectively. The third term is the entropy generation due to the chemical reaction, and the last term is the entropy generation due to viscous effects of the fluid. The c represents the total molar concentration, Φ is the viscous dissipation, which is given by: 2 2   u 2  u y   ux u y   x Ф   2         2 x   y   y x    

(8)

Gibbs free energy of the reaction, △rG, is defined as: K

 r G   vi i

(9)

i 1

where μi is the chemical potential of i species, and it is expressed based on the chemical potential at the standard condition, μiΘ:

i  i  RT lna i

(10)

Gibbs free energy of the reaction is the driving force of the chemical reaction and decides the direction and extent of the chemical reaction. The absolute value of it can be interpreted as the "distance" deviating from the chemical equilibrium state. The increase of the "distance" will increase the entropy production due to the reaction. So, in order to enhance the chemical reaction process, the reaction rate and the Gibbs free energy of the reaction should be increased, and this lead to the increase of the entropy production due to the chemical reaction. For a process shown in Fig. 1, where reactant B comes into the fluid system with a constant mass flux, the increasing of mixing effect of the reactants can enhance the process. And increasing the mixing effect leads to the decrease of the entropy generation due to mass transfer of the reactants (Jia et al., 2016). In a viscous fluid flow, the viscosity of the fluid will take energy from the motion of the fluid (kinetic energy) and transform it into internal energy of the fluid. This process is irreversible and is referred to as viscous dissipation. And thus, the viscous dissipation determines the mechanical energy input to the process. More mechanical energy input can increase the mixing effect of the reactants and thus enhance the chemical reaction process. However, the velocity field of the fluid corresponding to a given mechanical energy input may not be unique, and thus there must be a best velocity field to give the maximum enhancement of the process. Based on the analysis above, the chemical reaction process can be enhanced by maximizing the entropy generation due to the chemical reaction and minimizing the entropy generation due to the mass transfer of reactant B for a given value of viscous dissipation. So the process enhancement can be transformed into the resolution of a multi-objective optimization problem for the optimal velocity field and the associate body force filed. In the following sections, the extremums of individual entropy generation due to mass transfer of reactant B, mass transfer of product C, and the chemical reaction, with a given viscous dissipation, are performed respectively, and their variations with respect to the process enhanced degree, i.e. the flux of product C at the exit of the reactor are investigated.

2.2.1 Maximization entropy generation of reaction The entropy generation due to the reaction in the chemical reaction process is used as the

objective function firstly. The objective function is given by  1  max J R      r G d    T 

(11)

△rG is a function of the concentration cB and cC which are affected by the flow field. The chemical reaction process must satisfy the continuity equation, Eq. (2), momentum equations, Eq. (3), and species conservation equations, Eqs. (4) and (5). The viscous dissipation given in Eq. (12) is to be constrained as a constant.

 d   

 const

(12)

Then, the problem becomes the maximization of the objective function with differential and integral constraints. To solve the problem, the Lagrange function which includes the objective function and constraint equations is formulated by using the Lagrange multiplier method (Finlayson, 2013) as follows:  1    r G +C0  L1   U cB     DBAcB  + M B    T LR    d     L2   U cC     DCAcC    M C   L3   U      

(13)

where C0 is constant, L1, L2 and L3 are variables which vary with position. The extremum of the functional can be found by using the Euler-Lagrange equations in calculus of variations (Finlayson, 2013). The variational of Eq. (13) with respect to L1, L2 and L3 are respectively for species conservation Eqs. (4), (5) and continuity Eq. (2). The variational of Eq. (13) with respect to cB, cC is as follows:

 U L1    DBA  L1     kL1 

 kM C MB

k M BT

rG 

 kR 

NcC 1  2cB  cC   ln  M B  cB (1  cB  cC ) 1  cB  cC 

(14)

L2

 U L2    DCA  L2  

 kR MB

1  cB cC 1  cB  cC 

The variational of Eq. (13) with respect to U is: 1 1 1 2 U  L1cB  L2cC + L3  0 2C0 2C0 2C0

(15)

(16)

Combining Eq. (16) with the momentum Eq. (3) gives the follow relations:

L3  2C0 P F

(17)

1 1 L1cB  L2cC +   U  U 2C0 2C0

  U  U  P  2 U 

1 1 L1cB  L2cC +   U  U 2C0 2C0

(18) (19)

Then the equation set composed of the continuity equation Eq. (2), the modified momentum Eq. (19), species conservation Eqs. (4)，(5) and Eqs. (14)，(15), (18) can be solved for the velocity field, the concentration field, the pressure and body force F. The boundary condition of Eq. (14) for the Lagrange multiplier, L1, depends on the constant mass flux boundary condition, and it is given by

L1 cB  R  n M A M B cB (1 - cB )c n

(20)

The boundary condition of Eq. (15) for Lagrange multiplier, L2, under the isoconcentration boundary condition, is given by L2  0 (21)

2.2.2 Minimization entropy generation of mass transfer of reactant B In this section, the entropy generation due to mass transfer of reactant B is used as the objective function which is given by   2 RDBA 2 min J B    cB  d    M M c  B A B 1  cB  c 

(22)

Similar to the Section 2.2.1, for solving the optimization problem, the Lagrange function is constructed, as given by Eq. (23), including the objective function, Eq. (22), and constraint equations, Eqs. (2), (4), (5) and (12).    2 RDBA 2  cB  +C0  L1   U cB     DBAcB  + M B    LB    M B M AcB 1  cB  c d     L   U c     D c    M   L    U   C CA C C 3  2 

(23)

The variational of Eq. (23) with respect to cB, cC gives the following relations:

 U L1    DBA  L1  

 2 RDBA 

 cB   cB        kL1  M B M Ac  c 1  c c 1  c     B  B B    B 

 U L2    DBA  L2 

(24)

(25)

The variational of Eq. (23) with respect to U gives the similar results as Eqs. (16), (17), (18) and (19). The solution of Eq. (24) for L1 depends on the boundary condition given by Eq. (20). And the Eq. (25) for L2, the boundary condition is given by Eq. (21).

2.2.3 Extremum entropy generation of mass transfer of product C In this section, the entropy generation due to mass transfer of product C is used as the objective function given by   2 RDCA 2 J C     cC  d   M M c  C A C 1  cC  c 

(26)

The Lagrange function is also constructed as Eq. (27), including the objective function, Eq. (26) and constraint equations, Eqs. (2), (4), (5) and (12).    2 RDCA 2  cC  +C0  L1   U cB     DBAcB  + M B    LC    M C M AcC 1  cC  c d     L   U c     D c    M   L    U   C CA C C 3  2 

The variational of Eq. (27) with respect to cB, cC is as follows:  kM C  U L1    DBA  L1    kL1  L2 MB

(27)

(28)

 U L2    DCA  L2  

 2 RDCA  

cC   cC      M C M Ac   cC 1  cC   cC 1  cC  

(29)

The variational of Eq. (27) with respect to U also gives the similar relations as Eqs. (16), (17), (18) and (19).The boundary condition of Eq. (28) for L1 is given by Eq. (20). And the boundary condition of Eq. (29) for L2 is given by Eq. (21).

3 Results and discussion The CFD (Computational Fluid Dynamics) software FLUENT 14.5™ is utilized in the solution of the optimization problems. A pressure based solver is used for the calculation with a laminar viscous model. The second-order upwind scheme is used to discretize the governing equations and the ‘‘SIMPLE” algorithm is employed to solve the pressure and velocity. The UDS (user defined scalar) package provided in FLUENT 14.5™ is used for solving equations, (4), (5), (14), (15), (24), (25), (28) and (29). The UDF (user defined function) is used for adding the external body force F to the momentum Eq. (19). For the two-dimension optimization problem in this work, the equation set to be solved contains ten unknowns, ux, uy, cB, cC, P, Fx, Fy, L1, L2, L3, the same number as the equations. The grid with a number of 240,000 meshes was selected after checking the balance between the stability of the numerical solution and the computational time. The simulation was carried out on a Dell work station with 24 parallel processors of 3.00 GHz Intel Xeon E5-2687W. On average, the computation time was about 12 hours for an optimal calculation under a given value of viscous dissipation.

3.1 Individual objectives optimization 3.1.1 Simulation results without optimization For comparison, the chemical reaction process without optimization is firstly simulated by solving equations (2), (3), (4) and (5) with F = 0 for the rectangular channel reactor model. The results of the velocity contour, streamline and the concentration distribution of the product C are shown respectively in Fig. 2 and Fig. 3. As shown in Fig. 2, the flow is normal through the reactor with straight streamlines. The maximal concentration of product C is 0.0736, the flux of product C at the exit is 5.043×10-3 kg/s.

Fig. 2 Velocity magnitude (m/s) and streamlines in rectangular channel reactor with F = 0

Fig. 3 Concentration distribution of product C in rectangular channel reactor with F = 0

3.1.2 Maximization entropy generation due to chemical reaction The chemical reaction process with maximization of entropy generation of reaction is calculated by solving the set of equations (2), (4), (5), (14), (15) and (19). The optimized results with C0 = -770 are shown in the Fig. 4 and Fig. 5.

Fig. 4 Velocity magnitude (m/s) and streamlines by maximization entropy generation of reaction with C0 =-770

Fig. 5 Concentration distribution of product C by maximization entropy generation of reaction with C0 =-770 Compared with Fig. 2, an irregular clockwise eddy appears at the bottom where the reactant B flows into the rectangular channel as shown in Fig. 4. The extent of the reaction process enhancement in this process can be measured by the flux of products C at the exit. Comparing Fig. 5 with Fig. 3, the concentration of product C obtained by the optimization computing is higher than that by normal simulation. The maximal concentration of product C increases from 0.0736 to 0.201 and the flux of product C at the exit increases from 5.043×10-3 kg/s to 5.411×10-3 kg/s.

14 0.570 13 0.565 12 0.560 11 0.555 10 0.550

9

0.545 5.00

Ψ Viscous dissipation ×108 W

SR The reaction entropy generation W/K

15 0.575

5.05

5.10

5.15

5.20

5.25

5.30

5.35

5.40

8 5.45

Flux of product C at the exit ×103 kg/s

Fig. 6 Variations of the entropy generation due to the reaction, SR, and viscous dissipation, Ψ, with respect to the flux of product C at the exit by maximization entropy generation of reaction The Lagrange multiplier C0 is a constant and determines the viscous dissipation proportion in the Lagrange function, as shown in Eq. (13). As a term in the Lagrange function, a smaller value of C0 means a bigger value of the viscous dissipation, Φ. According to Eq. (18), the constant C0 determines the external body force field F. So, the optimal external body force field F corresponding to Fig. 4 and Fig. 5 can be obtained by solving Eq. (18) with C0 = -770. In other words, if this optimal body force field F is added to the system, the reaction process can be enhanced. By performing the optimization with different values of C0, different value of the viscous dissipation and external body force field F can be obtained. Fig. 6 shows the variations of the entropy generation due to the reaction, and viscous dissipation, and the flux of product C at the exit, with respect to different values of C0. Fig. 6 demonstrates that entropy generation due to the reaction accompanied by the viscous dissipation is increased with the increase of the flux of products C at the exit. It indicates that the reaction process can be enhanced at the cost of increased viscous dissipation (more mechanical energy input). It should be noted that the optimal results obtained by the calculus of variations method can only be guaranteed as the local optima. Although a global optimum cannot be ensured, the objective for the optimization (i.e. the outlet flux of product C) is effectively improved for different given viscous dissipation restrictions as shown in Fig. 6.

3.1.3 Minimization entropy generation due to mass transfer of reactant B The chemical reaction process with minimization of entropy generation due to mass transfer of reactant B is calculated by solving the set of Eqs. (2), (4), (5), (19), (24) and (25). The optimized results with C0 = 200 are shown in the Fig. 7 and Fig. 8.

Fig. 7 Velocity magnitude (m/s) and streamlines by minimization entropy generation of mass transfer of reactant B with C0 =200

12.0 11.5

7.2

11.0 7.0 10.5 6.8 10.0 6.6 9.5 6.4

Ψ Viscous dissipation ×108 W

7.4

of reactant B ×102 W/K

SB Mass transfer entropy generation

Fig. 8 Concentration distribution of product C by minimization entropy generation of mass transfer of reactant B with C0 =200 Compared with Fig. 2, an irregular clockwise eddy appears at the bottom of the rectangular channel as shown in Fig. 7. And by comparing Fig. 8 with Fig. 3, the maximal concentration of product C increases from 0.0736 to 0.104 and the flux of product C at the exit increases from 5.043×10-3 kg/s to 5.181×10-3 kg/s.

9.0

6.2 5.04

5.06

5.08

5.10

5.12

5.14

5.16

5.18

8.5 5.20

Flux of product C at the exit ×103 kg/s

Fig. 9 Variations of the entropy generation due to mass transfer of reactant B, SB, and viscous dissipation, Ψ, with respect to the flux of product C at the exit by minimization of entropy generation due to mass transfer of reactant B A series of different values of the viscous dissipation and the corresponding entropy generation due to mass transfer of reactant B, with respect to the flux of product C at the exit, have been obtained, as shown in Fig. 9. It indicates that the entropy generation due to mass transfer of reactant B is reduced and the viscous dissipation is increased, with the increase of the flux of products C at the exit. The variation of the entropy generation due to mass transfer of reactant B with respect to the flux of product C at the exit is opposite to that of the entropy generation due to

the chemical reaction, which agrees with the results analyzed in Section 2.2. This tendency can be interpreted as that, for the reactant B with a constant mass flux boundary condition, the decrease of entropy generation due to mass transfer of reactant leads to the increase of the mixing effect of the reactant, which favors evidently the reaction.

3.1.4 Maximization entropy generation due to mass transfer of product C The chemical reaction process with extremum of entropy generation due to mass transfer of product C is calculated by solving the set of Eqs. (2), (4), (5), (19), (28) and (29). The optimized results with C0 = -700 are shown in the Fig. 10 and Fig. 11.

Fig. 10 Velocity magnitude (m/s) and streamlines by extremum entropy generation of mass transfer of product C with C0 =-700

Fig. 11 Concentration distribution of product C by extremum entropy generation of mass transfer of product C with C0 =-700 Comparing with Fig. 2, an irregular clockwise eddy also appears at the bottom of the reactor as shown in Fig. 10. Compared Fig. 11 with Fig. 3, the maximal concentration of product C is also increased, from 0.0736 to 0.191, and the flux of product C at the exit increases from 5.043×10-3 kg/s to 5.439×10-3 kg/s.

14

10 13 9 8

12

7

11

6 10 5

Ψ Viscous dissipation ×108 W

15

11

of product C ×102 W/K

SC Mass transfer entropy generation

12

9

4 3

8 5.0

5.1

5.2

5.3

5.4

5.5

Flux of product C at the exit ×103 kg/s

Fig. 12 Variations of the entropy generation due to mass transfer of product C, SC, and viscous dissipation, Ψ, with respect to the flux of product C at the exit by extremum of entropy generation

due to mass transfer of product C The variation of entropy generation due to mass transfer of product C and the viscous dissipation, with respect to the flux of product C at the exit was obtained as shown in Fig. 12. It indicates that both the entropy generation due to mass transfer of product C and the viscous dissipation are increased, with the increase of the flux of products C at the exit. Note that the variation tendency of entropy generation due to mass transfer of product C with respect to the viscous dissipation is similar to that of entropy generation due to the reaction. This suggests that the extremum of the entropy generation due to mass transfer of product C is the maximum which has been used as the objective function (Finlayson, 2013). The same conclusion has been obtained in our previous study (Jia et al., 2016). An explanation is that, the increase of entropy generation due to mass transfer of product C means that the process of mass transfer of the product occurs under higher concentration gradient, which can enhance the diffusion of the product going away from the high concentration sites, and is thus favorable to the reaction process.

3.2 Multi-objective optimization All of the entropy generation should be taken into account at the same time for higher process enhancement extent. So, the enhancement of this chemical reaction process converts into solving a multi-objective optimization problem. There are many different methods for multi-objective optimization, and the methods can be classified in many ways according to different criteria (Miettinen, 1999). The weighting method is used in the present work, as it is an often and simple used multi-objective scalarization method in practice though it has drawbacks (Miettinen, 1999). Many methods applied in chemical engineering (Logist et al., 2012; Miettinen, 1999; Mitra and Majumder, 2011) can mitigate the intrinsic drawbacks of the weighting method, but to some extent increase the difficulties of variational calculation. The relative importance of various entropy generations was studied firstly to derive their weight coefficients. Pairwise comparisons play an important role in assessing the relative importance in decision theory, because of ease of understanding and application (Saaty, 1977; Takeda and Yu, 1995). In this paper, three objectives are included, which are the entropy generation due to the reaction, the entropy generation due to mass transfer of reactant B and the entropy generation due to mass transfer of the product C, under a given viscous dissipation. A series of data about each entropy generation variation with respect to the viscous dissipation has been obtained, as shown in Fig. 13. 0.13 0.58 0.12 0.11 0.56 0.10 0.55 0.09 0.54 0.08 0.53 0.07 0.52 0.06

SR Reaction

0.51

Entropy generation W/K

Entropy generation W/K

0.57

SB Mass transfer of reactant B 0.05 SC Mass transfer of product C

0.50

0.04

8

9

10

11

12

13

14

Viscous dissipation ×108 W

Fig. 13 Variation of entropy generations with respect to the viscous dissipation

The standard deviation of the entropy generations in a certain variation range of viscous dissipation is used as the criterion to judge their relative importance. The pairwise comparison matrix is as follows:  a11 A   a21  a31

a12 a22 a32

a13   1 0.11 0.45    a23   9.09 1 4.12  a33   2.21 0.24 1 

(30)

where a12 is an estimate of importance of objective 1(entropy generation due to mass transfer of reactant B) relative to objective 2(entropy generation due to mass transfer of product C), a13 is an estimate of importance of objective 1(entropy generation due to mass transfer of reactant B) relative to objective 3(entropy generation due to the reaction), a23 is an estimate of importance of objective 2(entropy generation due to mass transfer of product C) relative to objective 3(entropy generation due to the reaction), and a11= 1, a22= 1, a33 = 1. The Eq. (31) is used to calculate the weight coefficients (Saaty, 1977; Takeda and Yu, 1995).

Aw  max w w   wB , wC , wR    0.081, 0.740, 0.179  T

(31) T

(32)

where, λmax is the greatest eigenvalue of the matrix A, w is the normalization weight vector. Then, the weight coefficients, wB, wC, wR, for entropy generation due to mass transfer of reactant B, mass transfer of product C, and the reaction, respectively, are obtained. Based on the analysis and calculation in Section 2 and Section 3.1, the linear combination of entropy generation due to mass transfer of reactant B, mass transfer of product C and the reaction using the weight coefficients in Eq. (32), is used as the final objective function, which is given by    2 RDCA  2 RDBA 1 2 2 min J     cB   9.1   cC  +2.2   r G d   M M c 1  c c M M c 1  c c T B C A C  C  B A B 

(33)

For solving the optimization problem to get the optimal velocity field, the concentration field and the optimal body force field F, the Lagrange function is constructed as Eq. (34), which includes the objective function, Eq. (33) and constraint equations, Eqs. (2), (4), (5) and (12).    2 RDCA  2 RDBA 1 2 2  cB   9.1*  cC  +2.2*  r G   M M c 1  c c M M c 1  c c T B C A C  C  B A B  L    d   +C   L   U c     D c  + M  1 B BA B B  0    L   U c     D c    M   L    U   C CA C C 3  2 

(34)

The variational of Eq. (34) with respect to c1, c2 is as follows:  U L1    DBA  L1  

 2 RDBA  

c B   cB  k rG     2.2*  M B M Ac   cB 1  cB   cB 1  cB   M BT

 kR 

NcC 1  2cB  cC   kM C 2.2*  L2 ln    kL1  M B  cB (1  cB  cC ) 1  cB  cC  MB

 U L2    DCA  L2   9.1*

 2 RDCA 

 cC   cC  1  cB  kR      2.2*  M C M Ac  M B cC 1  cB  cC    cC 1  cC   cC 1  cC   

(35)

(36)

The variational of Eq. (34) with respect to U also gives the similar relations, Eqs. (16), (17),

(18) and (19).The boundary condition of Eq. (35) for L1 is Eq. (20). The boundary condition of Eq. (36) for L2 is Eq. (21). The reaction process is optimized by solving the set of equations (2), (4), (5), (19), (35) and (36). The optimized results with C0 = 5900 are shown in the Fig. 14 and Fig. 15.

Fig. 14 Velocity magnitude (m/s) and streamlines by optimization with C0 =5900

6.6 11

6.4 6.2

10

6.0

9

9 8 7 6 5

0.570

0.565

0.560

0.555

0.550

W/K

6.8 12

10

0.575

SR The reaction entropy generation

7.0 13

11

of product C ×102 W/K

7.2

14

12

SC Mass transfer entropy generation

7.4

of reactant B ×102 W/K

Ψ Viscous dissipation ×108 W

15

SB Mass transfer entropy generation

Fig. 15 Concentration distribution of product C by optimization with C0 =5900 Compared with Fig. 2, the optimal flow field is obtained, in which a big irregular clockwise eddy appears at the bottom where the reactant B flows into the rectangular channel as shown in Fig. 14. And by comparing Fig. 15 with Fig. 3, the maximal concentration of product C increases from 0.0736 to 0.198 and the flux of product C at the exit increases from 5.043×10-3 kg/s to 5.442×10-3 kg/s.

5.8 8

4 5.0

5.1

5.2

5.3

5.4

0.545

5.5

Flux of product C at the exit ×103 kg/s

Fig. 16 Variations of the entropy generation due to the reaction, SR, mass transfer of reactant B, SB, mass transfer of product C, SC, and viscous dissipation, Ψ, with respect to the flux of product C at the exit Fig.16 indicates that the entropy generation due to mass transfer of reactant B is decreased; the entropy generation due to mass transfer of product C, the entropy generation due to the reaction and the viscous dissipation are increased, with the increase of the flux of products C at the exit. The variation tendency of these parameters is consistent with the results in Section 3.1.

The linear combination of the three entropy generations is used as an objective, as shown in Eq. (33), which is optimized under fixed viscous dissipation. The minimum of this objective is corresponding to the maximum of the flux of product C at the exit. The outlet flux of product C at any given value of viscous dissipation in Fig. 16 is the highest one under such viscous dissipation. This implies that no matter what kinds of measures are taken to enhance the chemical reaction process by changing the velocity field, the outlet flux of product C cannot exceed the corresponding value in Fig. 16. The results in Fig. 16 can then be regarded as thermodynamic limit for chemical reaction process enhancement. The points correspond to the different values of C0 i.e., different values of external body force field F and associated flow field. It can be regarded as the best flow field with the same mechanical energy input, and can be used as the guidance for the engineers to do optimal design. For proving the feasibility and reliability of the method of deriving the weight coefficients for the individual objectives, the linear combination of entropy generation due to mass transfer of reactant B, mass transfer of product C and the reaction with uniform weight coefficients is used as the objective function, as Eq. (37). Similar to the problem with the weight coefficients obtained by the proposed method, the chemical reaction process optimization with uniform weight coefficients is also calculated.    2 RDCA  2 RDBA 1 2 2 J     cB   cC  +  r G d     M M c M C M AcC 1  cC  c T  B A B 1  cB  c 

(37)

Fig.17 shows that the flux of product C at the exit with calculated weight coefficients is higher than that with uniform weight coefficients, at the same viscous dissipation. The results indicate that the calculation method of weight coefficients is valid in the presented problem. Flux of product C at the exit ×103 kg/s

5.46 5.44 5.42 5.40 5.38 5.36

w1 : w2 : wR=1.0 : 9.1 : 2.2

5.34

w1 : w2 : wR=1 : 1 : 1 5.32 5.30 10

11

12

13

14

15

8

Total viscous dissipation ×10 W

Fig. 17 Optimization results comparison with different weight ratios

3.3 Square reactor model To prove the optimization method is valid for different type of reactors, another simulation is carried out as shown in Fig.18. The reactor model is square, with 10 mm long. The reactant A enters from the top of the reactor, and the product C together with the reactant A which is not converted in the reaction flow out at the bottom of the reactor on both sides through 1 mm wide outlets. The reactant B flows into the reactor at a constant flux, 0.01 m-2·s-1. At the entrance (top

side of the square), the velocity of the fluid is uniformly distributed with a value of 0.001 m/s. The fluid is in laminar flow. Fluid in (A)

Release source (B)

Fluid out (A, C)

Fluid out (A, C)

Fig. 18 The square reactor model The chemical reaction process without optimization was firstly simulated. And the results of the velocity contour, streamline and the concentration distribution of the product C are shown respectively in Fig. 19 and Fig. 20.

Fig. 19 Velocity magnitude (m/s) and streamlines in square reactor with F = 0

Fig. 20 Concentration distribution of product C in square reactor with F = 0 Then, similar to rectangular channel reactor model case in Section 2, the chemical reaction process is optimized. The optimized results with C0 = 3800 are shown in the Fig. 21 and Fig. 22.

Fig. 21 Velocity magnitude (m/s) and streamlines in square reactor by optimization with C0 =3800

Fig. 22 Concentration distribution of product C in square reactor by optimization with C0 =3800 Compared with Fig. 19, four axisymmetric eddies appears in the middle of the reaction region where the reactants B flow into the square reactor as shown in Fig. 21. And by comparing Fig. 22 with Fig. 20, the concentration of product C of the optimization computing is higher than

7.0 6.5

0.095

6.0 5.5

0.090

5.0 4.5

0.085

4.0

14 12 10 8 6

0.75

0.70

0.65

0.60

SR The reaction entropy generation

0.100

16

W/K

7.5

18

0.80

of product C ×102 W/K

0.105

8.0

SC Mass transfer entropy generation

Φ Viscous dissipation ×107 W

8.5

20

of reactant B W/K

9.0

SB Mass transfer entropy generation

that obtained by normal simulation. The maximal concentration of product C increases from 0.105 to 0.202 and the flux of product C at the exits increases from 5.044×10-3 kg/s to 7.266×10-3 kg/s.

0.55

4 5.0

5.5

6.0

6.5

7.0

7.5

Flux of product C at the exits ×103 kg/s

Fig. 23 Variations of the entropy generation due to the reaction, SR, mass transfer of reactant B, SB, mass transfer of product C, SC, and viscous dissipation, Ψ, with respect to the flux of product C at the exits Fig. 23 indicates that the entropy generation due to mass transfer of reactant B is reduced; the entropy generation due to mass transfer of product C, the entropy generation due to the reaction and the viscous dissipation are increased, with the increase of the flux of products C at the exits. The variation tendency of the entropy generation due to the reaction, mass transfer of reactant B, mass transfer of product C, and viscous dissipation, with respect to the flux of product C at the exits are similar to that in the rectangular channel reactor case in Fig. 1. This suggests that the method proposed for chemical reaction process enhancement is also valid in this type of reactor. Fig. 23 also demonstrates the approximate thermodynamic limit of this reaction process enhancement. The optimal flow field obtained by the methods proposed in this paper can be used to guide the design of more efficient reactors, as illustrated by Cao et al. (2018), where a micro-combustor for methane/air burning process was improved by adding proper type of baffles based on optimal velocity field for higher efficiency. The same principle has been used in high temperature heat transfer process, and an application can be found in Li et al. (2014), where an optimal distribution of porous media was designed to enhance the heat transfer in solar receiver.

4 Conclusions In the present work, an approach to reaction process enhancement was proposed based on flow pattern construction. A multi-objective optimization problem was formulated based on analysis of variation tendency of entropy generations including chemical reaction entropy generation and mass transfer entropy generation, with respect to the viscous dissipation. The objective function was constructed by the combination of these various entropy generations. By minimization of the objective function, subject to a constraint of fixed viscous dissipation, through the calculus of variations, a velocity field and the associate body force field were obtained, by which the reaction process was effectively enhanced. And it should be noted that the method proposed in this paper can be applied to other chemical reaction processes, not only for the one-order irreversible reaction.

We demonstrated in the present work that the entropy generation due to the chemical reaction and the mass transfer of the product are increased, with the increase of the viscous dissipation and the output flux of the product. While, the entropy generation due to the mass transfer of the reactant is decreased with the increase of the viscous dissipation. We obtained the weight coefficients of these various entropy generations according to their relative importance, with which the optimal results are better than that with uniform weight coefficients. The approximate thermodynamic limit was obtained for the process enhancement by improving the fluid flow in the reactor at a given mechanical energy input (viscous dissipation). It should be noted that the multi-objective optimization performed in this paper is based on the assumption that all model parameters of the chemical reaction system under consideration are constant. However, the real situations of chemical reaction process are different and involve uncertainty due to many factors (Mitra, 2009; Vallerio et al., 2015). The model parameters in our work, such as, the inlet velocity and mass flux of reactant may have uncertainty with the variation in the process and environmental data. The results in this paper are of importance for further understanding the reaction process for enhancement. The solution of the optimal function gives the best flow field in the reactors. However, the methods of how to obtain this flow field are not addressed in this paper. In fact, the method proposed in this paper can provide design engineers with a direction to an optimal design of the internal structures in the reactors. The designers can calculate the sizes, angles, and positions etc. of the devices, such as baffles, porous mediums and stirrers, inside the reactor to get a flow pattern approaching to the optimal one for higher performance. Acknowledgements Support by National Natural Science Foundation of China (Grant 91434204) is acknowledged. References Albini, A., Fagnoni, M., 2009. Handbook of Synthetic Photochemistry. Journal of the American Chemical Society 132, 16726–16726. Aravind, G.P., Deepu, M., 2017. Numerical study on convective mass transfer enhancement by lateral sweep vortex generators. International Journal of Heat and Mass Transfer 115, 809-825. Arjmandi, H.R., Amani, E., 2015. A numerical investigation of the entropy generation in and thermodynamic optimization of a combustion chamber. Energy 81, 706-718. Bejan, A., 1995. Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, pp. 174-177. Bejan, A., 1997. Advanced engineering thermodynamics / A. Bejan. Bejan, A., Tondeur, D., 1998. Equipartition, optimal allocation, and the constructal approach to predicting organization in nature. Revue Générale De Thermique 37, 165-180. Bidi, M., Nobari, M.R.H., Avval, M.S., 2010. A numerical evaluation of combustion in porous media by EGM (Entropy Generation Minimization). Energy 35, 3483-3500. Cao, X., Jia, S., Luo, Y., Yuan, X., Yu, K.T., 2018. Optimal design of transport and reaction pattern in premixed methane-air micro-combustor. Proceedings of the 13th International Symposium on Process Systems Engineering – PSE 2018. Carrington, C.G., Sun, Z.F., 1991. Second law analysis of combined heat and mass transfer phenomena.

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Highlights

   

Various entropy generations have different effects on chemical reaction process. The weighted sum of various entropy generations is optimized. Optimal velocity field and the associate body force field are obtained. The results can be regarded as the thermodynamic limit to the process enhancement.