Multi-objective optimization for helium-heated reverse water gas shift reactor by using NSGA-II

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Multi-objective optimization for helium-heated reverse water gas shift reactor by using NSGA-II Lei Zhang a,b,c, Lingen Chen a,b,∗, Shaojun Xia a,b, Yanlin Ge a,b, Chao Wang b, Huijun Feng a,b a

Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan, 430205, P.R. China School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan, 430205, P.R. China c Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, P.R. China b

a r t i c l e

i n f o

Article history: Received 28 November 2018 Revised 18 October 2019 Accepted 8 November 2019 Available online xxx Keywords: Reverse water gas shift High-temperature helium Finite-time thermodynamics Multi-objective optimization Two-dimensional pseudo-homogeneous model Generalized thermodynamic optimization

a b s t r a c t Thermodynamic performance of helium-heated reverse water gas shift (RWGS) reactor is investigated by using the theory of finite-time thermodynamics. Taking into account both the radial temperature gradients and the diffusion-reaction phenomenon inside the catalytic pellets, a comprehensive twodimensional pseudo-homogeneous mathematical model is established to represent the reactor. By using numerical calculations, the thermodynamic performance of reactors in the given conditions is analyzed, the influences of the effective parameters on reactor performance are also examined. Eventually, from the perspective of heat management and production improvement, a multi-objective optimization (MOO) procedure based on the non-dominated sorting genetic algorithm (NSGA-II) is applied to investigate the best working parameters considering the minimum radial temperature difference and maximum conversion rate as optimization objective functions. The results show that significant radial temperature gradients and the resulting radial gradients of apparent reaction rate can be observed. The thermodynamic performance of the counter-flow reactor is superior to that of the parallel-flow pattern. The MOO is an effective technique for selecting the best working parameters to reduce the radial temperature difference and improve the conversion rate simultaneously. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Producing sustainable liquid fuels by capturing carbon dioxide (CO2 ) and generating hydrogen (H2 ) which are available at or near the point of use is of interesting for both military and nonmilitary applications [1-3]. Technologies related to CO2 chemical utilization [4-6] on land have been applied in fuel synthesis for a sea-based operation [7-11]. Extraction of CO2 and H2 from seawater and catalytic conversion to synthesize fuels is a novel way to achieve littoral land-based or even shipboard production of liquid fuels which is of importance to the Navy scenarios [7]. The Navy’s ability to produce different fuel types for military devices including warships and carrier-based aircrafts at or near the point of use is a revolution of oversea combat capabilities and energy security. It can significantly enhance the safety of the Navy logistics and offers operational advantages by increasing the flexibility of refueling [12]. The U.S. Naval Research Laboratory has devoted to developing novel technologies for recovering CO2 and H2 from seawater by the method of electrochemical acidification [13-16],

∗

Corresponding author. E-mail address: [email protected] (L. Chen).

and brand new catalysts for CO2 hydrogenation have been developed and tested [7-11,17,18]. Technologies currently exist on land is to capture CO2 directly from air or from flue gases exhausting by heavy industries, which are impractical for a littoral operation due to the requirements of flexibility and high efficiency. The ideal carbon source and hydrogen source for sea-based fuel synthesis would be carbon dioxide and hydrogen recovering from seawater which are at the point of use in theater. The concentration of CO2 in the atmosphere is 0.7 mg/l on a weight/volume basis, and that in the world’s oceans is approximately 100 mg/l which is about 140 times greater than that can be extracted in air [14]. NRL broke through the challenges of recovering carbon dioxide by extracting CO2 from the acidic seawater in the electrochemical acidification cell [14-20]. The extraction rate of CO2 up to 98% was achieved by using the hollow fiber membrane contactor in an experimental prototype [19, 20]. Under the circumstance of producing hydrogen by electrolysis, a theoretical energy efficiency of about 60% was determined by an energy balance analysis [2]. The technical feasibility of the new littoral energy conversion processes combined with nuclear energy and ocean thermal energy conversion was discussed [2] and the technology was verified at the bench-scale by a successful flight of model plane which used synthetic fuel. NRL demonstrated that the synthetic fuel price at sea was competitive

https://doi.org/10.1016/j.ijheatmasstransfer.2019.119025 0017-9310/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Schematic diagram of the HTGR fuel synthesis system.

Fig. 2. Schematic diagram of the helium-heated RWGS reactor (parallel-flow operation).

compared with the fuel price with the addition of the logistic costs [2, 12]. Meanwhile, the remote military installations at Diego Garcia and Djibouti were used as economic examples of sea-based fuel synthesis from seawater and nuclear energy [7]. From the perspective of energy conversion, one of the scientific and technical challenges is the requirements of large energy input because of the overall unfavorable energy balance of the sea-based operation for fuel synthesis [12]. Renewable primary energy such as ocean thermal, solar, wind and nuclear energy are proposed to be the energy sources for the liquid fuel production process in the remote littoral regions [2, 7]. Furthermore, from the perspective of environmental protection, the combination of renewable energy supply units (CO2 -free) and fuel synthesis process are tremendously beneficial for the reduction of CO2 emissions because the process is CO2 neutral and the liquid fuel products are sulfur and nitrogen-free which are environmentally friendly [1]. The number and distribution of ocean thermal, solar and wind resources are extremely dependent on the usage scenarios, it may be difficult to ensure the long-term stability for the Navy operations. Ship-based or littoral land-based nuclear platforms are expected to be superior energy sources to maintain the continual and long-term seabased operations, especially the reduction of capital costs and the energy conversion efficiency can be enhanced by the use of more advanced nuclear reactors [21]. The modular high temperature gas cooled reactor (HTGR), one of the candidates for Generation IV innovative reactors, is an ideal energy source for mass sea-based fuel synthesis process because of its inherent passive safety, high ther-

mal efficiency, highly modular feature and the availability of providing high-temperature process heat (900∼950 °C) for other industrial units [22-25]. Reverse water gas shift (RWGS) reaction is one of important steps in the indirect fuel synthesis routes from CO2 hydrogenation [1, 18]. Carbon monoxide (CO), which can be produced from RWGS reaction, has a lower energy barrier than thermodynamic stable CO2 . Therefore, CO can be used more flexibly in the downstream Fischer-Tropsch reactors [26, 27]. RWGS reaction is an endothermic process, favorable high-temperature conditions can be achieved by heating up the reactors using external heat reservoir. The modular HGTRs, especially constructed in the navy vessels and littoral land-based nuclear platforms [28-33], is a suitable heat source for the RWGS reactors, and its conceptual design is shown in Fig. 1. In addition, the net amount of CO2 emissions of sea-based fuel production process driven by HTGR can be reduced considerably because nuclear energy is CO2 -free, and the compact configuration of modular HTGR has nature advantages for mobile navy applications [21]. In HTGR fuel synthesis system, the thermal performance of the RWGS reactors directly influence the system safety, efficiency and economy [34, 35]. Conceptual and technological designs for the HTGR process heat application have been investigated [23-25, 34-37]. Nevertheless, it is difficult to investigate the heat and mass phenomena in the reactor system by the method of experiments, due to the complexity and high cost of the experiment systems. The thermal design and optimization by numerical simulation and modern thermodynamic optimization theory is an effec-

Please cite this article as: L. Zhang, L. Chen and S. Xia et al., Multi-objective optimization for helium-heated reverse water gas shift reactor by using NSGA-II, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119025

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Fig. 3. Variations of the temperatures and heat flux along the dimensionless axial coordinate z∗ for the parallel-flow reactor.

Fig. 4. Variations of the temperatures and heat flux along the dimensionless axial coordinate z∗ for the counter-flow reactor.

tive method to understand and improve the thermodynamic performance of the real-world units. Finite-time thermodynamics (FTT) or thermodynamic optimization theory [38-48] is a multidisciplinary field, which combines fundamental sciences such as thermodynamics, heat transfer, fluid mechanics and chemical reaction kinetics. FTT can solve the optimal performance and optimal configuration problems of different energy conversion devices and systems, which finite-time and/or finite-size constraints are commonly ignored by classical thermodynamics. Great efforts have been put forth to investigate the optimal performances and optimal configurations of different chemical reaction processes based on FTT theory. In terms of optimizations with maximum production rate as objective function, Månson and Andresen [49] studied the FTT problem of maximum production rate for ammonia synthesis reactor with the aid of optimal control

theory. Piña et al. [50] obtained the optimal heat-flux path along the steam methane reforming (SMR) reactors with the constraints of constant heat duty and maximum methane conversion rate as objective function. Pantoleontos et al. [51] obtained the optimal path of the exterior wall temperature for SMR reactors by optimizing H2 production rate. Bak et al. [52] and Chen et al. [53] studied the optimal concentration path of the consecutive chemical reaction (A  B  C) [52] and a more generalized one (xA  yB  zC) [53] respectively, which can be conducted to obtain the maximum yield of B. Wang et al. [54] investigated the problem of maximum production rate for sulphuric acid decomposition process, and the optimal inlet conditions and optimal path of the exterior wall temperature were obtained. Besides the objective of maximizing production rate, some researchers also studied the chemical reaction and mass transfer process using the minimum entropy generation

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2. System description The endothermic RWGS reaction can produce CO from CO2 [27]:

CO2 + H2  CO+H2 Or H > 0

Fig. 5. Variations of the temperature differences T along the dimensionless axial coordinate z∗ .

rate or specific entropy generation rate as objective functions [5565], including methanol synthesis reactor [55], ammonia synthesis reactor [56], sulfur dioxide oxidation reactor [57], SMR reactor [58, 59], sulphuric acid decomposition reactor [60], CO2 hydrogenation to light olefins reactor [61], dimethyl ether synthesis reactors [62], RWGS reactors [63], first-order consecutive reaction (A  B  C) process [64] and extraction process of CO2 from acidic seawater by using hollow fiber membrane contactor [65]. In engineering, various performance indicators, which are commonly conflicting, should be considered together by designers. Therefore, solving multi-objective issues has considerable theoretical significances and practical application values [66]. Multiobjective optimization (MOO) method based on evolutionary algorithms has been widely employed in various energy conversion devices and systems [67-71]. The non-dominated sorting genetic algorithm (NSGA-II) is obtained by combining the genetic algorithm with the method of Pareto, it retains the global searching ability of genetic algorithm in single objective optimization, and extends the genetic algorithm to the field of MOO by utilizing the method of Pareto. Ahmadi et al. [67-69] applied NSGA-II to investigate the multi-objective optimization problems of combined cycle power plants [67], irreversible Carnot refrigerator [68] and Atkinson engine [69]. Arora et al. [70] performed multi-objective and multi-parameter optimization of two-stage thermoelectric generator through NSGA-II. Shahhosseini et al. [71] obtained the optimal CO2 injection flow rate of SMR reactors for the downstream methanol reactors, with NSGA-II and decision making methods. According to a thorough open literature survey, RWGS reactor performances have only been demonstrate at the bench-scale [72], the conceptual design of the RWGS reactor coupling with HTGR at the industry-scale is firstly proposed in the present study. In additions, researches on the thermal design and optimization of RWGS reactors by MOO have been not yet published. The present work is devoted to establishing a comprehensive FTT model of heliumheated RWGS reactor based on the assumption of two-dimensional pseudo-homogeneous modeling, with considering the irreversibilities of heat transfer, viscous flow and chemical reactions. Performance analyses and parameter study are conducted to investigate the thermodynamic optimization potential and the influences of working parameters on performance indicators. A MOO procedure considering heat management and production improvement aspects is also reported.

(1)

From point of view of thermodynamics, high-temperature conditions, which are favorable to relieve equilibrium restrictions for the endothermic reactions, not only offer the ability to increase reaction rate, but also inhibit the exothermic side reaction of methanation [73]. In order to achieve the technical requirements of industrial applications, a kind of high active, selective and temperature-stable Pt-based RWGS catalysts is designed by authors’ group. The measured data from the selectivity and active tests at high temperature (60 0∼70 0 °C) reveals that a stable high CO selectivity (around 99%) can be achieved by using this novel Ptbased catalyst [63]. Therefore, side reactions of methanation and coke formation are ignored. Under the typical operating conditions of high temperatures, all compositions are in gaseous states and the ideal gas assumption can be safely used. As shown in Fig. 2, the helium-heated RWGS reactor consists of two concentric tubes which length is defined as L. The inner tube is uniformly filled with spherical catalyst pellets as the porous medium reaction channel. The high-temperature hot gaseous helium from HTGR flows through an annular heating section to provide heat to the endothermic RWGS reaction in the inner tube. The flow of the hot gaseous helium through the outer tube can be both parallel and counter compared to the feed gasses. The different diameters and the different temperatures used in modeling are illustrated in the schematic sketch of the cross section. The diameters of the annular section, outer and inner wall of the reactor tube are denoted as da , do and di , respectively. T and Ta represent the temperatures of reaction mixtures and hot gaseous helium. The inner and outer wall temperatures of the reactor tube are defined as Tiw and Tow . The complete mathematical model consists of three parts: the reactor tube, the catalyst pellets and the annular heating section. 2.1. Reactor tube model A suitable reactor model gives consideration to both the accuracy and the converge efficiency [74]. Reasonable and necessary assumptions are thus made to help get accurate and stable simulation results. Some simplifications based on two-dimensional pseudo-homogeneous modeling assumptions are considered as follows: (1) Steady-state operations; (2) Under the circumstance of high gas velocity (1.5 m/s) and small catalyst particles (6 mm), the heat and mass transfers between the solid pellet phase and the gas bulk are strong enough. Therefore, the heat and mass transfers between the catalyst pellets and the fluid could be ignored safely in engineering: Pseudo-homogeneous reactor model is applied [74, 75]; (3) A tubular flow reactor is adopted herein, and the ratio of N = L/di ≈ 50. Thus, the RWGS reactor can be assumed to be presented as a plug flow one with the high Reynolds number, namely, neglecting the axial heat and mass dispersion effects [74, 75]; (4) The pressure drop in the present RWGS reactor is small (less than 0.2 MPa). It implies that the effects of pressure change on the reactor performance are small [76]. Thus, radial pressure gradients are neglected in the present model to simplify the calculation procedure under the conditions of ensuring the accuracy of the results. The state of the reaction mixtures is characterized by the temperature T, pressure P and conversion rate ξ . The profiles of T and

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Fig. 6. Variations of the apparent reaction rate rapp , effectiveness factor η and temperature T along the dimensionless radial coordinate R∗ at the axial position z∗ = 0.76.

Table 1 Kinetic parameter values [63].

2.1.2. Conservation equations The energy conservation equation is expressed as [77]

aH2 O

bCO2

A/mol/(s · g · MPa)

E/kJ/mol

65

7.4

2.324

28.91

  ∂T λer ∂ 2 T 1 ∂ T ρ r Hrη = + − b ∂ z Gc p ∂ R2 R ∂ R Gc p

ξ vary along the radial and axial coordinate R and z, respectively. The trajectory of P changes along the axial coordinate z only. The conversion rate is defined as

  ξ (z, R ) = FCO2 ,in − FCO2 (z, R ) /FCO2 ,in

(2)

where Fk,in and Fk (z, R) represent the molar flow of component k at the inlet and the position of coordinate (z, R). Subscript “in” means the inlet state. The molar flow rate of component k is therefore as follows

Fk = Fk,in + FCO2 ,in vk ξ

(3)

where vk is the stoichiometric coefficient for component k. 2.1.1. Chemical reaction rate The kinetic equations can quantitatively characterize the effects of some effective working parameters (such as temperature and concentration) on the chemical reaction rate. A general RWGS rate equation based on the earlier work of Refs. [9-11, 63] is used

r=k

PCO2 PH2 − PCO PH2 O /K PCO + aH2 O PH2 O + bCO2 PCO2

(4)

where Pk is the partial pressure of component k. aH2 O and bCO2 are the inhibition coefficients representing the inhibition effects of H2 O and CO2 , respectively. The equilibrium constant K, which can be calculated from standard Gibbs energy, is introduced to quantify the chemical equilibrium limitations. k is the rate constant that follows the Arrhenius equation

k = Aexp[−E /(Rg T )]

(5)

where Rg is the gas constant, kinetic constants of the preexponential factor A and the activation energy E can be obtained by fitting the measured data in the testing platforms from the authors’ research group. Table 1 gives the values of the parameters in the rate equation.

(6)

where λer is the effective radial thermal conductivity, G = ρv and cp are mass flow velocity and specific heat capacity of the reaction mixtures. ρ and v are the density and velocity of the reaction mixtures. ρ b is the catalytic bed density, r, r H and η are the intrinsic reaction rate, enthalpy of reaction and the effectiveness factor, respectively. The mass conservation equation is written as [77]

  ∂ξ Der ∂ 2 ξ 1 ∂ξ ρ Ac r η = + + b ∂z v ∂R R ∂R FCO2 ,in

(7)

where Der is the effective radial diffusivity, Ac is the cross-sectional area of the inner tube of the chemical reactor. For larger Reynolds numbers (Re/(1 − ε ) > 500), momentum balance is thus modeled by Hick’s equation [78] 1. 2

dP (1 − ε ) = −6.8 dz ε3

Re−0.2

ρv2 dp

(8)

where ɛ and Re = vρ d p /μm are the void fraction and the Reynolds numbers for the catalytic bed. dp is the diameter of the catalytic pellets, and μm is the viscosity of reaction mixtures. Boundary conditions for conservation equations are written as

At z = 0, 0 ≤ R ≤ Ri T = Tin ; ξ = 0;P = Pin

(9)

At R = 0, 0 ≤ z ≤ L

∂T ∂ξ = 0; =0 ∂R ∂R

(10)

At R = Ri , 0 ≤ z ≤ L

∂ξ =0 ∂R

(11)

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Fig. 7. Variations of the chemical reaction rate rp along the dimensionless radial coordinate ζ ∗ for different radial positions inside the tube, at the axial position z∗ = 0.76.

Fig. 8. Variations of CO2 partial pressure Pp,CO2 along the dimensionless radial coordinate ζ ∗ for different radial positions inside the tube, at the axial position z∗ = 0.76.

2.3. Annular heating section model

∂T λer = hi (Tiw − TR=Ri ) ∂R

(12)

where Ri is the inner radius of the reactor tube, hi is the convective heat-transfer coefficient inside the reactor tube.

The assumptions of the negligible radial temperature gradients and pressure drop are made to model the annular heating section model. The energy conservation equation of the hot gaseous helium can be written as

2.2. Catalyst pellets

dTa l π do q = dz FaC p,a

The reactions take place at the active sites of the solid catalyst pellets, where the reaction rate can be affected by the gradients of temperature and concentrations. The main transport mechanism of reaction components is diffusion inside the catalyst pellets. The large thermal conductivity of metallic catalysts can used as a validity to assume the non-existing temperature gradients inside the pellets. An isothermal spherical modeling assumptions are thus made to model the catalyst pellets. The mass conservation equations are written as [78]

106 × Dem,k



Rg T

∂ 2 Pp,k 2 ∂ Pp,k + ζ ∂ζ ∂ζ 2



where q is the heat flux transferred between the hot fluid in the annular heating section and the reaction mixtures in the reactor tube. Fa and Cp, a are the molar flow rate and molar heat capacity of the hot fluid, respectively. Compared to the flow of the reaction mixtures in the reactor tube, the flow pattern of the hot fluid in the annular heating section can be parallel- and counter-flow, which are denoted by

l = −1 for parallel − flow l = 1 for counter − flow

(18)

2.4. Radial energy conservation equations

+ ρ p νk r p = 0 , k =

CO2 , H2 , CO, H2 O

(17)

(13)

where subscript “p” denotes the states inside the pellets. Dem,k and Pp, k are the effective diffusivity and partial pressure of component k, respectively. ρ p is the density of the catalyst pellet, and ζ is the radial coordinate for the catalyst pellets. Boundary conditions are expressed as

The radial wall temperatures of Tiw and Tow can be obtained by solving the radial energy conservation equations through the entire axial positions. At the inner wall of the reactor tube R = Ri , the energy conservation equation is written as

hi (Tiw − TR=Ri ) =

λw (Tow − Tiw ) Ri ln (Ro/Ri )

(19)

Pp,k = Pk , ζ = R p

(14)

where Ro is the outer radius of the reactor tube, and λw is the thermal conductivity of the tube metal. At the outer wall of the reactor tube R = Ro, the energy conservation equation is written as

∂ Pp,k = 0, ζ = 0 ∂ζ

(15)

λw (Tow − Tiw ) = ho (Ta − Tow ) Ro ln (Ro / Ri )

where ho is the convective heat-transfer coefficient outside the reactor tube.

where Rp is the radius of the catalyst pellets. The effectiveness factor is calculated as follows

η=

 Vp 0

r p dV Vp r

where Vp is the volume of the catalyst pellet.

(20)

(16)

2.5. Modeling parameters The modeling parameters used herein are divided into structure and working parameters as well as physical-chemical and transfer

Please cite this article as: L. Zhang, L. Chen and S. Xia et al., Multi-objective optimization for helium-heated reverse water gas shift reactor by using NSGA-II, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119025

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Fig. 9. Influence of the inlet helium temperature Ta, in on the temperature difference T and mean conversion rate ξ¯ .

Fig. 10. Influence of the inlet molar flow rate of helium Fa, in on the temperature difference T and mean conversion rate ξ¯ .

parameters. Structure and working parameters are set according to the engineering standards. The physical-chemical and transfer parameters are calculated based on the classical equations and correlations, which are presented in Appendix A and B. All the thermodynamic data needed in model simulations are obtained from Ref. [79]. 3. Numerical solution and optimization procedure 3.1. Numerical solution method The solution of the present mathematical model, where most of the physical-chemical and transfer parameters are depend on the state variables at every position through the system, belongs to a hybrid problem consisting of partial differential equations (PDEs), ordinary differential equations (ODEs) and nonlinear algebraic equations. Due to the complexity of the problem, analytical solution cannot be obtained and a numerical solution routine must

be proposed. According to the flow arrangement of the hot gaseous helium in the annular heating section, the solution of the model is divided into the initial value problem (parallel-flow reactor) and boundary value problem (counter-flow reactor). PDEs are solved by the method of lines (MOL). By MOL, the radial independent variables are discretized by using the method of finite differences with Nr − 1 elements (Nr nodes), and the axial independent variables are remained. Therefore, the original PDEs can be transformed into ODEs that can be integrated by the classical Runge-Kutta method along the axial direction with Na =50 0 0 nodes. The boundary value problem of PDEs is solved by the shooting method. The catalytic pellet model, which is a boundary value problem of ODEs, is solved by “bvp4c” solver in MATLAB, with N p =30 0 0 nodes along the catalyst radial direction. The simulation in all present cases have been running using Nr = 6. With a finer grid of Nr = 21, the computation results of outlet mean conversion rate differ by about 0.0098. Compared with a mesh size (Nr = 51) for which results can be con-

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Fig. 11. Influence of the inlet temperature of feed gases Tin on the temperature difference T and mean conversion rate ξ¯ .

Fig. 12. Influence of the inlet total pressure of feed gases Pin on the temperature difference T and mean conversion rate ξ¯ .

sidered to be “exact”, the Nr = 6 case has about 1.2% error in the outlet mean conversion rate.

3.2. Optimization formulation MOO is an effective tool to find the optimal solutions with multiple objective functions, which are commonly conflicting. Because of the non-existing global optimal solution that can maximize or minimize all objective functions simultaneously, the Pareto optimal set, which are the set of the non-dominated solution, are used to represent the optimal compromise among decision variables. The Pareto optimal frontier, which is the relevant objective functions to the Pareto optimal set, are obtained by using NSGA-II. The optimization procedure is implemented in a MATLAB routine based on the MOO solver “gamultiobj”. In this paper, the values of population size, Pareto fraction and generation are set to be 200, 0.35 and 200, respectively.

The main goal of the chemical reactors is to achieve the production target in a safe mode. For the endothermic RWGS reactor, high temperatures are favorable for the production improvement. However, the severe temperature gradients caused by hightemperature operation may affect the production indictor and safe operation by the resulting catalyst deactivations, thermal stresses and thermal cracks in the tube [35]. Therefore, heat management inside the reactors are considered together with the production improvement. For this study, the minimum radial temperature difference and maximum outlet mean conversion rate are taken as the objective functions, which can be written as

min maxT max ξ¯out

(21)

where maxT = max(TR=Ri − TR=0 ) denotes the maximum allowable radial temperature difference through the reactor tube, subscript “out” denotes the outlet states.

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Fig. 13. Influence of the inlet total molar flow rate of feed gases FT, in on the temperature difference T and mean conversion rate T.

Table 2 Comparisons of outlet mean conversion rate ξ¯out between experimental data and simulation results [63]. Temperature/K

Gas hourly space velocity/h − 1

Experimental data/-

Simulation results/-

873 923 973 873 873 873

12,000 12,000 12,000 18,000 24,000 30,000

0.530 0.575 0.612 0.519 0.497 0.490

0.567 0.607 0.647 0.566 0.562 0.555

Five independent parameters, including inlet helium temperature Ta, in , inlet helium molar flow rate Fa, in , inlet temperature, and inlet total molar flow rate and inlet total pressure of reaction mixtures Tin , FT, in and Pin , are chosen as the decision variables. The constraints used in MOO are given as follows

20 ≤ Fa,in (mol /s ) ≤ 60 1173 ≤ Ta,in (K ) ≤ 1373 773 ≤ Tin (K ) ≤ 973 3 ≤ FT,in (mol /s ) ≤ 7 1.5 ≤ Pin (MPa ) ≤ 4.5

(22)

Firstly, for the given values of Ta, in , Fa, in , Tin , FT, in and Pin , the temperature difference T = (TR=Ri − TR=0 ) at different axial positions are calculated, and the maximum value of the temperature difference T is determined. Secondly, all of the values of Ta, in , Fa, in , Tin , FT, in and Pin changes, and the corresponding different values of maxT = max(TR=Ri − TR=0 ) are further calculated. Finally, the minimum value of maxT = max(TR=Ri − TR=0 ) is chosen, i.e. the min maxT is solved, and the corresponding values of Ta, in , Fa, in , Tin , FT, in and Pin are determined. 4. Results and discussions 4.1. Model validation Experiments in industrial scale are the most suitable measures to verify the accuracy of the present model. However, due to the large and complexity experiments for helium-heated RWGS reactor, it is difficult and high-cost to investigate the heat and mass

phenomena in the reactor system by the method of experimental measurements. In the stage of conceptual research, experimental data from lab scale reactor can be applied to demonstrate the validity of the principle parts in the system. The research on lab scale mainly focuses on the mechanism and the reaction process of chemical reaction, and belongs to those of intrinsic chemical reaction kinetics excluding effects of all physical transfer processes at the micro level, which is independent of the shape and size of the equipment. The research on industrial scale needs to consider effects of macro physical processes such as heat transfer, mass transfer and fluid flow on chemical reaction, and belongs to those of the chemical reaction kinetics including effects of all physical transfer processes at the macro level. It mainly focuses on the quantitative description of chemical reaction rate law, which is related to the specific shape and size of the equipment. Due to the influences of physical processes such as heat transfer, mass transfer and fluid flow, the experimental data obtained from the equipment at different scales are generally different, while they coincides with each other only under the ideal conditions of physical processes. The former is the basis of the latter. It is impossible to get the experimental data on industrial scale, and this paper uses the experimental data on lab scale to fit that on industrial scale, so the accuracy of the reaction kinetic model is ensured as much as possible under the existing conditions. More authentic and accurate information about industrial systems may help us improve the model and correct the simulation results in the future work. Therefore, a lab scale experimental reactor established by the research group of the authors [63] is used to verify the accuracy of the present two-

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Table 3 Prescribed parameters for the model simulation. Parameters

Notation

Value

Catalyst porosity Catalyst tortuosity Catalyst pellet diameter Thermal conductivity of catalysts Void fraction of catalyst bed Catalyst pellet diameter Inlet total molar flow rate of feed gas Catalyst bed density Inner diameter of the reactor tube Outer diameter of the reactor tube Outer diameter of the annulus Thermal conductivity of the tube wall Reactor length Inlet temperature of feed gas Inlet total pressure of feed gas Inlet CO2 molar fraction Inlet H2 molar fraction Inlet CO molar fraction Inlet H2 O molar fraction Inlet molar flow of hot gaseous helium Inlet temperature of hot gaseous helium

θ τ

0.3 4 20 nm 0.5 W•m−1 •K−1 0.65 0.006 m 5 mol•s−1 1020 kg•m−3 0.1016 m 0.1322 m 0.1722 m 100 W•m−1 •K−1 5m 873 K 3 MPa 0.495 0.495 0.005 0.005 40 mol•s−1 1273 K

dpore

λc

ɛ dp FT, in

ρb di do da

λw L Tin Pin yCO2 yH2 yCO yH2 O Fa, in Ta, in

dimensional pseudo-homogeneous model. The schematic diagram of the lab-scale fixed-bed reactor was shown in Fig. 1 in Ref. [63]. Table 2 lists the comparisons between experimental data obtained from the small scale reactor system and simulation results with a mesh size of Nr = 6. The inlet molar ratio of H2 and CO2 is 2.5 and the inlet pressure is atmospheric in all experimental cases. It can be immediately observed that the errors between experiments and simulations do not exceed 15%, which can be considered as an acceptable agreement. It can also be seen from Fig. 3 in Ref. [63]. 4.2. Model analysis The model simulations in a given operation condition are conducted to investigate the performances and thermal effects of the helium-heated RWGS reactors. Table 3 lists all the structure and operation parameters used herein. Figs. 3 and 4 illustrate the profiles of temperatures and heat fluxes for the parallel- and counterflow operations, respectively. Table 4 lists the performance comparisons for different flow modes, where T¯ is the radial mean temperature, the directions of the arrows denote the flow directions of the fluids inside and outside the reactor tubes. As shown in Figs. 3 and 4, more uniform axial distributions of the temperature difference between hot gaseous helium and outer wall of the tube (Ta − Tow ), the temperature difference between outer wall and inner wall (Tow − Tiw ) and the heat flux are observed in the counterflow reactor, compared to the parallel-flow one. The profiles of the radial mean temperature in both reactors show slight decreases near the inlet, which is caused by the endothermic RWGS reactions. The areas under the curves of the heat fluxes give graphical description of the heat duty of per unit area listed in Table 4. As shown in Table 4, the outlet mean conversion rate of the counterflow reactor (ξ¯L = 0.440) is slightly larger than that of the parallelflow reactor (ξ¯L = 0.437), which is achieved by a larger heat duty. Moreover, the outlet helium temperature in a counter-flow reactor is higher than that of a parallel-flow reactor. It denotes that the heat-transfer efficiency in a counter-flow reactor is superior to that in a parallel-flow reactor. The pressure drop is almost identical in both reactors. Fig. 5 shows the profiles of the temperature differences T = TR=Ri − TR=0 along the dimensionless axial coordinate z∗ . As shown in Fig. 5, the profile of the temperature difference in parallel-flow

Fig. 14. Pareto optimal frontiers for the reactor.

reactor shows a sharp increase up to the maximum near the inlet, and then is followed by a rapid decrease that is due to the lower local heat flux and reaction rate. The temperature difference in counter-flow reactor shows a relatively smooth trend, it increases rapidly near the inlet, and then increase quite slowly to the maximum which is less than the counterpart in the parallel-flow reactor. Based on the results above, the counter-flow reactor has a superior thermal performance and production capacity. Therefore, only counter-flow reactor is chosen for further parametric analyses. Fig. 6 shows the radial variations of the apparent chemical reaction rate (rapp = r · η), effective factor η and temperature T of the reaction mixtures at the position z∗ = 0.76 (the position of max T). The counterparts in the parallel-flow show the similar trends. As shown in Fig. 6, the effectiveness factor decreases slightly along the dimensionless radial coordinate R∗ , and the values of the effective factor along the tube are less than 0.1, which denotes that the large mass diffusion resistance exists in the catalyst pellets. The profile of the apparent chemical reaction rate increases significantly, which is caused by the radial temperature

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11

Table 4 Performance comparisons of different reactors.

Outlet mean conversion ξ¯out /− Outlet temperature of helium Ta, out /K Heat duty per unit area Q/kW · m-2 Pressure drop P/MPa

Parallel-flow reactor

Counter-flow reactor

0.437 1130 75 0.170

0.440 1273 80 0.1682

Fig. 15. Comparisons of different cases without optimization, with single-objective and multi-objective optimizations for the reactor.

gradient. The observed radial gradient of the apparent chemical reaction rate implies the existence of the poor catalyst utilization near the center line of the reactor tube [77]. Figs. 7 and 8 show the profiles of the chemical reaction rate rp and CO2 partial pressure Pp,CO2 inside the pellets for different radial

positions at the axial position z∗ = 0.76. As shown in Figs. 7 and 8, the active layer of about 10% depth can be observed obviously, which means that RWGS reactions are in equilibrium after 10% of the radius. An alternative shape of eggshell is more favorable than the spherical shape used herein. The use of eggshell catalyst pel-

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L. Zhang, L. Chen and S. Xia et al. / International Journal of Heat and Mass Transfer xxx (xxxx) xxx Table 5 Comparisons of the outlet conditions and computation time for calculated and constant effectiveness factors. Simulation case

Outlet mean temperature T¯out /K

Outlet mean conversion ξ¯out /−

Computation time / min

Parallel-flow operation using calculated η Parallel-flow operation using constant η (= 0.05 ) Counter-flow operation using calculated η Counter-flow operation using constant η (= 0.05 )

997.3 997.2 1012.9 1012.8

0.437 0.435 0.440 0.437

2.8 0.05 12 0.25

Table 6 Values of the optimal decision variables in point C.

Parallel-flow operation Counter-flow operation

Fa,in /mol · s−1

Ta, in /K

Tin /K

FT,in /mol · s−1

Pin /MPa

29.4 38.19

1239 1229

973 973

5.74 5.52

4.09 4.47

lets is a trade-off between a good internal mass transfer and a low pressure drop in the tube [80]. The chemical reaction rate rp near the tube wall is relatively higher due to the higher temperature. 4.3. Parameter studies The influences of the decision variables on the temperature difference T and mean conversion rate ξ¯ of the counter-flow reactor are examined. Previous to this, the validity of using the constant effectiveness factor is evaluated. Table 5 lists the simulation results with calculated and constant effective factors, for both parallel- and counter-flow operations. It can be observed that simplification of using constant effectiveness factors (η = 0.05) gives almost identical results and overlapping profiles of the state variables (not shown in the paper) compared with the full model, as it reduces the computation time by a factor of around 50. Therefore, it is an acceptable simplification for the following simulations and optimizations with constant effective factors. Figs. 9 and 10 show the influences of inlet helium temperature Ta, in and molar flow rate Fa, in on the profiles of temperature difference T and mean conversion rate ξ¯ along the tube. Different values of the inlet helium molar flow rates denote different temperature profiles of the heat source, i.e. the larger the inlet helium molar flow rate is, the higher the average heat source temperature is, which affects the chemical reaction by heat transfer finally. The decision variables increase along the directions of arrows. As shown in the Figs. 9 and 10, by increasing the inlet helium temperature, the resulting larger temperature difference T leads to higher heat supplies, which is the main reason for the increase of the mean conversion rate in Fig. 9. Meanwhile, larger inlet molar flow rate of helium in the annular heating section means that greater heat capacity of heat carriers can be used for endothermic RWGS reactions. It implies that a trade-off between the production improvement and heat management exists, by choosing a set of appropriate working parameters. The effects of inlet temperature Tin , pressure Pin and total molar flow rate FT, in on the profiles of temperature difference T and mean conversion rate ξ¯ along the tube are shown in Figs. 11, 12 and 13. As illustrated in Fig. 11, by raising the inlet temperature of feed gasses, the temperature difference T decreases while mean conversion rate ξ¯ increases, which means that the performance of the reactor is better. This can be justified that the increment of inlet temperature Tin increases the convective heat transfer coefficient inside the reactor tube (hi ), which can result in a better heat transfer performance and break the restriction of chemical equilibrium. Fig. 12 shows that the temperature difference T and mean conversion rate ξ¯ increase with the increment of the inlet pressure Pin as higher chemical reaction rate caused by higher Pin leads to a higher production rate. However, the resulting response of changing Pin is not significant compared with other decision variables. As

shown in Fig. 13, an increase in inlet total molar flow rate FT, in decreases both the temperature difference T and mean conversion rate ξ¯ . The increment of inlet total molar flow rate FT, in denotes the increase of production intensity, which results in the decrease of residence time of gas mixtures. 4.4. Results of multi-objective optimization Multi-objective optimization is performed for both the paralleland counter-flow operations to find the optimal solution points as Pareto optimal frontiers, which are illustrated in Fig. 14, respectively. The endpoints in Pareto optimal frontiers (point A and B) can be considered as single-objective optimal points (point A: maximum ξ¯out ; point B: minimum max (T)). The ideal and nonideal points are hypothetical points where both objective functions reach their upper and lower bounds. Many decision-making methods, including TOPSIS and LINMAP methods [70], are employed to find the optimal solution with the aid of the ideal and non-ideal points which are not located in the Pareto optimal frontier. The Pareto optimal frontiers herein cannot balance the two objective functions very well, i.e. a small change in outlet mean conversion rate ξ¯out may cause a large variation in maximum temperature difference max (T). The ideal and non-ideal points are not recommended to be utilized for decision making in the present study. Each point in the Pareto optimal frontier can be selected as the final optimal solution based on the preferences and criteria of the decision-maker. Therefore, point C is selected as one of the optimal solutions. From the perspective of engineering, the designer can choose the point according to the technical, economic and environmental requirements or the designer experience. Point C in this paper is a relative superior point obtained according to the designer experience. Fig. 15 shows comparisons of the maximum temperature difference and outlet mean conversion rate for the base case (without optimization), singleobjective optimization cases, and MOO case. As for parallel operation, the outlet mean conversion rate increases by 14.8% with raising the maximum temperature difference by 26.5% (15.1% and 21% increments for ξ¯out and max (T) respectively, in counter-flow operation) under the single-objective optimization for maximum ξ¯out . The application of MOO in parallel-flow reactor can simultaneously achieve 5.1% increment and 37% decrement (6.6% increment and 31% decrement in counter-flow operation) for ξ¯out and max (T) respectively. Optimizing the maximum temperature difference of the parallel reactor individually decrease max (T) and ξ¯out by 60.5% and 0.9% respectively. It is interesting that max (T) is reduced by 50.9% with raising ξ¯out by 1.1% for optimizing the maximum temperature difference of the counter-flow reactor. In other words, the selected optimal solution from MOO can achieve a satisfactory trade-off between the two conflicting objectives as shown in Fig. 15. Table 6 lists the optimal deci-

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sion variables in point C for both parallel- and counter-flow operations. It can be observed that the optimal inlet temperature of feed gasses attains its upper bound for both different operations and values of other decision variables distribute in the given ranges.

13

expressions in temperature [79]. n is the number of components. Aij is an interaction parameter for components i and j, which is expressed as [81]



Aij =

1+



μi / μj



0 . 5

Mi /Mj

8 1 + Mi /Mj

5. Conclusions

0.25 2

(A.2)



In the present study, a conceptual model of RWGS reactor coupling with HTGR is proposed, and FTT theory is applied to implement the thermal design and optimization of the system. Analyses and optimizations are conducted from the viewpoint of production improvement and heat management. Strong radial temperature gradients in the reactor tube and diffusions in catalyst pellets are observed and analyzed in given working conditions. Furthermore, the influences of some working parameters on performance indicators are investigated in detail. Finally, optimal working parameters are gathered as Pareto optimal frontiers to select the appropriate optimal solution according to the preferences and criteria of the decision makers. The results show that the performance of counter-flow reactor, with more moderate temperature gradients and better heattransfer performance, is superior to that of parallel-flow reactor. Strong radial temperature gradients result in distinct gradients of chemical reaction rate, which implies that the performance improvements can be conducted in viewpoint of reducing temperature gradients and optimizing the structure and distribution of catalyst pellets. Chemical equilibrium is reached near the surface of pellets because of a large mass transfer resistance. Therefore, catalyst structure of eggshell is recommended to be employed in RWGS reactors. The study of parametric effects reveals that raising the inlet temperature of feed gasses is an effective measure to reduce radial temperature difference and improve outlet conversion rate simultaneously. In addition, the results of MOO are compared with the optimal solutions obtained by the method of single-objective optimization, indicating that MOO can balance conflicting indictors well and find an appropriate trade-off among various decision variables. For a sea-based helium-heated RWGS reactor which energy supply may be nuclear energy or renewable energy, the crucial point or first priority are safety and production capacity. The principle aspects about production improvement and heat management have been investigated in the present study. Besides the points mentioned herein, one can build more plentiful optimization models by taking energy efficiency as one of the objective functions.

where Mi is the molar mass of component i. The thermal conductivity of reaction mixtures λg, m can be calculated by the following expression [81]

Acknowledgments

Appendix B. Calculation of heat and mass transfer parameters

This paper is supported by the National Natural Science Foundation of P. R. China (Project Nos. 51606218and 51576207), Hubei Provincial Natural Science Foundation of China(Project No. 2018CFB708), Self-Topic Project of Naval University of Engineering (Project No. 20161504). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.

The effective radial thermal conductivity λer is estimated by considering the stagnant and flow effect, which is calculated by the following formula [84]

λg,m =

n 

i=1

yi λg,i n j=1 y j Aij

(A.3)

where λg, i is the thermal conductivity of component i, which can be obtained by the polynomial expressions in temperature as well [79]. The interaction parameter Aij is calculated approximately by Eq. (A.2). Both molecular diffusion and Knudsen diffusion are considered in the pores of catalyst pellets. The effective diffusion coefficient for component i can be calculated as follows [81]

Dem,i =

θ τ



1 1 /DM ,i + 1 /DK,i



(A.4)

where θ and τ are porosity and tortuosity of catalysts. DM, i and DK, i are molecular and Knudsen diffusion coefficients, respectively. The molecular diffusion coefficient for each gas is calculated based on the Wike model [81]

DM ,i =

n  DM,ij yj

(A.5)

j=1 j=k

where DM, ij is the binary diffusion coefficient for components i and j, which is obtained by the method of Fuller [82] 1/2

DM,ij =

10−7 T 1.75 (1/Mi + 1/Mj )

(A.6)

2

(9.8692P )(V¯i1/3 + V¯j1/3 )

where V¯i represents the diffusion volumes for component i. Knudsen diffusion coefficient is estimated by the following formula [80]



DK,i = 0.0485dpore

T Mi

(A.7)

Where dpore is the catalyst pore diameter. The porosity θ , tortuosity τ and pore diameter dpore are estimated based on Ref. [83].

λer λ0 Re · Pr = er + λg,m λg,m 6.3 + 34.7d p /di

(B.1)

Appendix A. Calculation of physic-chemical parameters

where Pr is the Prandtl number in the reactor tube. λ0er is the stagnant effective radial thermal conductivity, which can be expressed as follows [85]

The viscosity of reaction mixtures μm is obtained as follows [81]

λ0er 1−ε =ε+ λg,m 0.22ε 2 + 2λg,m /(3λc )

μm =

n  i=1

y i μi n j=1 yj Aij

(A.1)

where yi is the molar fraction of component i, μi is the viscosity of the pure component i, which can be calculated from polynomial

(B.2)

where λc is the thermal conductivity of catalysts. The convective heat transfer coefficient inside the reactor tube hi is calculated according to [86]



hi = 1 − 1 .5 ( di /d p )

−1.5

 λg,m dp

1/3

Re0.59 Pr

(B.3)

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The convective heat transfer coefficient outside the reactor tube ho is expressed as [87]

ho =

  2/3 λg,a ( f /8 )Rea Pra d 1+( h) F  2/3 L dh 1 + 12.7 f /8(Pra − 1 )

(B.4)

where Rea and Pra are Reynolds and Prandtl number in the annular heating section, respectively. f is the friction factor, F is the correction factor for the boundary conditions, λg, a is the thermal conductivity of heating fluid, and dh is the hydraulic diameter which can be expressed as follows

dh = da − do

(B.5)

The friction factor f is calculated by the following correlation [87]

f = (1.8log10 Rea − 1.5 )−2

(B.6)

For the boundary condition of heat transfer from the inner tube with insulating outer tube, the correction factor F is given as follows [87] −0.16

F = 0.86(do/da )

(B.7)

The effective radial diffusivity Der can be estimated by the correlation reported in the Ref. [81]

Der =

[m5G;November 22, 2019;3:34]



d p /v 2

10 1 + 19.4(d p /di )



(B.8)

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Please cite this article as: L. Zhang, L. Chen and S. Xia et al., Multi-objective optimization for helium-heated reverse water gas shift reactor by using NSGA-II, International Journal of Heat and Mass Transfer, https://doi.org/10.1016/j.ijheatmasstransfer.2019.119025