Minimization of Entropy Generation Rate in Hydrogen Iodide Decomposition Reactor Heated by High-Temperature Helium

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Article Minimization of Entropy Generation Rate in Hydrogen Iodide Decomposition Reactor Heated by High-Temperature Helium

Rui Kong 1, Lingen Chen 2,3,* , Shaojun Xia 1, Penglei Li 1 and Yanlin Ge 2,3

1 College of Power Engineering, Naval University of Engineering, Wuhan 430033, China; [email protected] (R.K.); [email protected] (S.X.); [email protected] (P.L.) 2 Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China; [email protected] 3 School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China * Correspondence: [email protected]; Tel.: +86-27-83615046

Abstract: The thermochemical sulfur-iodine cycle is a potential method for hydrogen production, and the hydrogen iodide (HI) decomposition is the key step to determine the efficiency of hydrogen production in the cycle. To further reduce the irreversibility of various transmission processes in the HI decomposition reaction, a one-dimensional plug flow model of HI decomposition tubular reactor is established, and performance optimization with entropy generate rate minimization (EGRM) in the decomposition reaction system as an optimization goal based on finite-time thermodynamics is carried out. The reference reactor is heated counter-currently by high-temperature helium gas, the optimal reactor and the modified reactor are designed based on the reference reactor design parameters. With the EGRM as the optimization goal, the optimal control method is used to solve the optimal configuration of the reactor under the condition that both the reactant inlet state and hydrogen production rate are fixed, and the optimal value of total EGR in the reactor is reduced by 13.3% compared with the reference value. The reference reactor is improved on the basis of the total EGR in the optimal reactor, two modified reactors with increased length are designed under the condition of changing the helium inlet state. The total EGR of the two modified reactors are the same Citation: Kong, R.; Chen, L.; Xia, S.; as that of the optimal reactor, which are realized by decreasing the helium inlet temperature and Li, P.; Ge, Y. Minimization of Entropy helium inlet flow rate, respectively. The results show that the EGR of heat transfer accounts for a large Generation Rate in Hydrogen Iodide proportion, and the decrease of total EGR is mainly caused by reducing heat transfer irreversibility. Decomposition Reactor Heated by The local total EGR of the optimal reactor distribution is more uniform, which approximately confirms High-Temperature Helium. Entropy the principle of equipartition of entropy production. The EGR distributions of the modified reactors 2021, 23, 82. https://doi.org/ 10.3390/e23010082 are similar to that of the reference reactor, but the reactor length increases significantly, bringing a relatively large pressure drop. The research results have certain guiding significance to the optimum Received: 7 December 2020 design of HI decomposition reactors. Accepted: 5 January 2021 Published: 8 January 2021 Keywords: hydrogen iodide decomposition; tubular reactor; entropy generation rate minimization; finite-time thermodynamics; optimal control theory Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- ms in published maps and institutio- nal affiliations. 1. Introduction Hydrogen energy is a kind of renewable and clean energy, and it is necessary to research and develop hydrogen energy under the background of fossil energy shortage and Copyright: © 2021 by the authors. Li- greenhouse effect. The thermochemical sulfur-iodine (S-I) cycle can couple with a nuclear censee MDPI, Basel, Switzerland. reaction and solar heating [1–3]; then the overall thermal efficiency can reach more than This article is an open access article 50% [4], so it has great development potential. The thermochemical S-I cycle for hydrogen distributed under the terms and con- production was first proposed by the General Atomics Company in the 1970s, and its ditions of the Creative Commons At- reliability was analyzed and demonstrated [5,6]. In the following decades, many related tribution (CC BY) license (https:// researches have been successfully implemented, and the process method of the S-I cycle creativecommons.org/licenses/by/ were further developed and improved [7–9]. 4.0/).

Entropy 2021, 23, 82. https://doi.org/10.3390/e23010082 https://www.mdpi.com/journal/entropy Entropy 2021, 23, x FOR PEER REVIEW 2 of 25

There are three main chemical reactions in the thermochemical S‐I cycle, which are Entropy 2021, 23, 82 the Bunsen reaction, sulfuric acid decomposition and HI decomposition2 of reactions. 25 Figure 1 shows the schematic diagram of the S‐I cycle. For the HI decomposition process, com‐ prehensive researches have been implemented including catalyst performance evaluation There are three main chemical reactions in the thermochemical S-I cycle, which are the [10,11],Bunsen reaction, reaction sulfuric kinetics acid decomposition analysis [12–15] and HI and decomposition reactor reactions.numerical Figure simulation1 shows [16–18]. The conversionthe schematic rate diagram of HI of the decomposition S-I cycle. For the reaction HI decomposition is low, which process, restricts comprehensive the hydrogen yield andresearches thermal have efficiency been implemented of the includingS‐I cycle catalyst system, performance and the choice evaluation of [decomposition10,11], reac- tempera‐ turetion kinetics must analysistake into [12 –account15] and reactor the heat numerical resistance simulation of [the16–18 material]. The conversion and the rate thermal of efficiency HI decomposition reaction is low, which restricts the hydrogen yield and thermal efficiency of ofthe the S-I cycle device. system, The and HI the decomposition choice of decomposition process temperature is considered must take to into be account one of the the most critical stepsheat resistance in hydrogen of the material production and the process thermal efficiency using S of‐I thecycle. device. A large The HI amount decomposition of H2O exists in the process is of considered distilling to HI be onegas of from the most the HIx critical solution, steps in hydrogenthe H2O productionand HI need process to be heated dur‐ ingusing the S-I cycle.reaction A large and amount the unreacted of H2O exists HI in thegas process after ofthe distilling decomposition HI gas from thereaction HIx needs to be circulatedsolution, the Hand2O andreacted HI need again, to be heatedso this during process the reaction requires and a the lot unreacted of heat HI [4]. gas Therefore, after it is nec‐ the decomposition reaction needs to be circulated and reacted again, so this process requires essarya lot of heat to further [4]. Therefore, study it isand necessary analyze to furtherthe thermodynamic study and analyze irreversibility the thermodynamic in the HI decom‐ positionirreversibility process. in the HI decomposition process.

Figure 1. Schematic diagram of the S-I cycle. Figure 1. Schematic diagram of the S‐I cycle. Finite-time thermodynamics (FTT) has made significant progress in physics and engineeringFinite fields‐time since thermodynamics the mid-1970 s [19 (FTT)–33], andhas the made aim issignificant to reduce the progress irreversibility in physics and en‐ of systems under finite time and size constraints. The applications of FTT include the gineeringresearches of fields optimal since performances the mid‐ [197034–50 ]s and [19–33], optimal and configurations the aim is [ to51– reduce68] for various the irreversibility of systemsthermodynamic under processes, finite time devices and and size cycles. constraints. The applications of FTT include the re‐ searchesFTT theory of optimal was also performances applied in chemical [34–50] reaction and processes; optimal see configurations the review article [51–68] [69]. for various thermodynamicSome scholars [70–73 processes,] have conducted devices a lot and of researches cycles. by taking the maximum production rateFTT (PR) theory and minimum was also entropy applied generation in chemical rate (EGR) reaction as optimization processes; targets see the for review article chemical reaction processes. Måsson and Andresen [70] first applied FTT theory to optimize [69].synthetic Some ammonia scholars reaction, [70–73] and the have temperature conducted distribution a lot of curve researches of the reaction by taking mixture the maximum productionwas solved with rate ammonia (PR) and production minimum rate entropy maximization generation (PRM) as rate the optimization(EGR) as optimization goal targets forunder chemical the given reaction inlet status. processes. Chen et al. Måsson [71] established and Andresen the thermodynamic [70] first modelapplied for FTT theory to optimizeremoving CO synthetic2 from acidified ammonia seawater reaction, with hollow and fiberthe temperature membrane contacts, distribution and obtained curve of the reac‐ the optimal configuration of CO2 concentration with the EGRM in the mass transfer pro- tioncess asmixture the optimization was solved objective. with Chen ammonia et al. [72 production] also investigated rate themaximization reaction of carbon (PRM) as the opti‐ mizationdioxide and goal hydrogen under to the synthesize given inlet olefins, status. and analyzed Chen et the al. effect [71] ofestablished reactor structural the thermodynamic modelparameters for onremoving specific EGRs; CO2 thefrom specific acidified EGR could seawater be reduced with byhollow 10.04% fiber and 24.78%membrane contacts, under the optimal catalyst bed density and pipe diameter, respectively. Li et al. [73] studied and obtained the optimal configuration of CO2 concentration with the EGRM in the mass the steam methane reforming reactor with hydrogen PRM as the optimization goal, and transfer process as the optimization objective. Chen et al. [72] also investigated the reac‐ obtained the optimal wall temperature distribution and inlet pressure. The hydrogen PR of tionthe optimal of carbon reactor dioxide increase and by 11.8% hydrogen was compared to synthesize with the referenceolefins, value.and analyzedLi et al. [74 ]the effect of re‐ actor structural parameters on specific EGRs; the specific EGR could be reduced by 10.04% and 24.78% under the optimal catalyst bed density and pipe diameter, respectively. Li et al. [73] studied the steam methane reforming reactor with hydrogen PRM as the optimi‐ zation goal, and obtained the optimal wall temperature distribution and inlet pressure. The hydrogen PR of the optimal reactor increase by 11.8% was compared with the refer‐ ence value. Li et al. [74] further established the model of the steam methane reforming

Entropy 2021, 23, 82 3 of 25

further established the model of the steam methane reforming reactor heated by molten salt, and the total EGR could be reduced by 22% compared with the reference value, by optimizing the mixture and molten salt inlet parameters. Nummedal et al. [75] investigated the steam reforming reaction of methane with EGR minimization (EGRM) as the objective function under the constraint of fixed hydrogen PR, and optimized the gas inlet temperature, the external heat source and the inlet mixture composition by the nonlinear programming method. Wang et al. [76] optimized the decomposition process of sulfuric acid by the non-linear program design method with the goal of SO2 PRM, and the PR was improved by 7% under optimized temperature and pressure profiles. The method of nonlinear programming is to discretize continuous variables, so the results are usually approximate solutions. The optimal control method is relatively complex, but the results are relatively accurate. Johannessen and Kjelstrup [77] first used this method to solve the optimization problem of the sulfur dioxide oxidation reaction, and obtained the optimal heat source temperature distribution curves at different tube lengths with the total EGRM as the optimization goal. The authors Vander Ham et al. [78] studied the decomposition of sulfuric acid in a tubular reactor, obtained the optimal helium temperature distribution under the condition of EGRM, the total EGR is 26% lower than the that of the reference reactor heated by helium counterflow. Zhang et al. [79] investigated olefin synthesis reactors with the optimal control method; the optimal heat reservoir temperature profiles were obtained with EGRM as the optimization target under fixed or free inlet temperature and CO2/H2 ratio. Li et al. [80] analyzed the synthesis of methanol by CO2 and hydrogen under the constraint of fixed methanol PR, and the optimal reactor heat source temperature distribution was obtained with the goal of EGRM; the EGR value in optimal reactor was reduced by 20% compared with the reference reactor using a constant heat source. Kong et al. [81] studied the HI decomposition reactor with a fixed hydrogen PR as the constraint, and the optimal heat source temperature distribution curves were obtained using the optimal control method under the conditions of fixed and free tube length; the total EGR was reduced by 51.3% and 57.6% compared with that in the reference reactor heated with linear heat source temperature, respectively. Zhang et al. [82] used the optimal control method to investigate reverse water gas shift reactors with EGRM as the optimization target, and obtained the optimal configuration of the reactor under different boundary conditions and lengths. On the basis of previous research work, researches involving multiple optimization objectives and multiple reaction processes have been reported. Zhang et al. [83] established the model of reverse water gas reactor heated by high temperature helium, and carried out multi-objective optimization with minimum radial temperature difference and maximum conversion rate. Cao et al. [84] studied the multi-objective optimization of EGR in mass transfer and chemical reaction processes. Sun et al. [85] investigated the sulfuric acid decomposition reaction and carried out multi-objective optimization with minimum EGR and maximum SO2 PR. Zhang et al. [86] established a reaction distillation model of Fetol synthesis and studied the multi-objective optimization problem in the reaction process. Avellaneda et al. [87] carried out multi-objective optimization on the heat and mass transfer process of convective flows in the flow passage. Magnanellia et al. [88] investigated the membrane device to separate carbon dioxide from natural gas, and the research results showed that EGR due to mass transfer in the membrane permeable process could be minimized by controlling the partial pressure and total pressure of the gas components on the permeable side. Kingston et al. [89] studied the air separation process of the packed distillation column and obtained the optimal temperature distribution curve of heat transfer between the column and its surroundings with the aim of EGRM. Korpy´set al. [90] applied an entropy optimization criterion to the methane catalytic combustion process and analyzed the influence of different catalysts on the EGR in the reactor. Kizilova et al. [91] optimized the channel shape of the fuel cell with the minimum EGR caused by viscous flow. Yang et al. [92] studied the heat transfer process and EGR of the two-layer porous media tube, and analyzed the local EGR under different parameters. Li et al. [93] simulated the decomposition process of methane hydrate and analyzed the heat transfer characteristics Entropy 2021, 23, x FOR PEER REVIEW 4 of 25

sition process of methane hydrate and analyzed the heat transfer characteristics and en‐ tropy generation under different pressure conditions. Under the constraint of fixed hy‐ drogen PR, the EGRM was taken as the objective to optimize the HI decomposition reac‐ tion process, and the ultimate goal was to reduce the irreversibility of the reaction process and reduce the energy quality loss. There are three methods to reduce entropy generation in a chemical reactor: optimiz‐ ing the distribution of EGR, increasing the reactor length and improving the catalyst per‐ formance [78]. The first method is to optimize the heat source temperature distribution and adjust the distribution of local EGR; the second method is to increase the heated area and reduce the average heat flux in the heat transfer process; and the third method is to increase the reaction transmission coefficient and increase the chemical reaction rate. Entropy 2021, 23, 82 In this article, the HI decomposition reaction process will4 of 25 be further optimized using FTT theory, in order to explore the potential of reducing total EGR in the reactor. The HI decomposition reactor will be studied using the first two methods to reduce EGR in the and entropy generation under different pressure conditions. Under the constraint of fixed hydrogenreaction PR, system, the EGRM and was the taken influence as the objective of different to optimize design the HI decomposition parameters on reactor performance reactionwill be process, analyzed. and the ultimateThe reference goal was to reactor reduce the is irreversibility heated by of thehigh reaction‐temperature helium gas in a processcounter and‐ reducecurrent the energyway, qualitywhich loss. is closer to the actual industrial reactor. Firstly, the optimal There are three methods to reduce entropy generation in a chemical reactor: opti- mizingcontrol the distributionmethod will of EGR, be increasing adopted the to reactor solve length the andoptimal improving reactor the catalyst performance parameters with performancethe objective [78]. The of firstEGRM method in is the to optimize reaction the heatsystem. source Secondly, temperature distribution by changing the inlet temperature and adjust the distribution of local EGR; the second method is to increase the heated area andand reduce flow the rate average of heat helium flux in thegas heat respectively, transfer process; andtwo the other third method modified is to reactors with increased increaselength the will reaction be transmissiondesigned. coefficient and increase the chemical reaction rate. In this article, the HI decomposition reaction process will be further optimized using FTT theory, in order to explore the potential of reducing total EGR in the reactor. The HI decomposition2. The System reactor Description will be studied using the first two methods to reduce EGR in the reaction system, and the influence of different design parameters on reactor performance will2.1. be Reactor analyzed. Model The reference reactor is heated by high-temperature helium gas in a counter-currentThe chemical way, which reaction is closer to equation the actual industrial of HI decomposition reactor. Firstly, the optimal reaction is as follows [14,15]: control method will be adopted to solve the optimal reactor performance parameters with the objective of EGRM in the reaction system. Secondly, by changing the inlet temperature and flow rate of helium gas respectively,2HI(g) two other H modified22 (g)+I reactors (g) with increasedH=12 length kJ/mol (1) will be designed. where H is enthalpy of reaction. The reaction is endothermic, and the reaction temper‐ 2. The System Description 2.1.ature Reactor is Modelusually controlled between 573 and 823 K. The reactor model has an inner diam‐ The chemical reaction equation of HI decomposition reaction is as follows [14,15]: eter of di and length L, the catalyst particles are considered to be ideally spherical and

uniformly filled2HI in( gthe) Htube,2(g)+ andI2(g) the∆ particleH= 12 kJ/mol diameter is dp The (1) helium temperature, reac‐

wheretion∆ Hmixtureis enthalpy temperature, of reaction. The reaction total is endothermic,pressure, and and the reaction molar temperature flow rate can be expressed as is usually controlled between 573 and 823 K. The reactor model has an inner diameter of d TzTzPzHe (),(),() and Fzk (), respectively, where z representsi the axial position. For the and length L, the catalyst particles are considered to be ideally spherical and uniformly filledfixed in thebed tube, reactor and the particlewith diametergiven parameters, is dp The helium temperature,the axial reactionback mixing mixture in the reactor can be ig‐ temperature,nored, and total the pressure, plug andflow molar model flow rate is applicable can be expressed [15]. as T HeFigure(z), T( z2), showsP(z) the schematic diagram and Fk(z), respectively, where z represents the axial position. For the fixed bed reactor withof the given tubular parameters, plug the axialflow back reactor mixing model. in the reactor Meanwhile, can be ignored, the and reaction the plug is conducted under high flowtemperature model is applicable and low [15]. Figure pressure2 shows operating the schematic condition. diagram of the Therefore, tubular plug the reaction process can be flow reactor model. Meanwhile, the reaction is conducted under high temperature and low pressureassumed operating as follows: condition. Therefore,(1) The thereaction reaction processreaches can a be steady assumed state, as follows: no back mixing in the axial (1)direction; The reaction (2) reaches The a steadyaxial state,and no radial back mixing mass in energy the axial direction; diffusion (2) The can axial be ignored; (3) The reaction andmixture radial mass gas energy is considered diffusion can be to ignored; be an (3) ideal The reaction gas. mixture gas is considered to be an ideal gas.

FigureFigure 2. Schematic 2. Schematic diagram ofdiagram the tubular of plug the ﬂow tubular reactor model.plug flow reactor model.

The parameters of the reference reactor are listed in Table 1. The inlet conditions of the reactor are determined by the process method for HI decomposition in the SI cycle,

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The parameters of the reference reactor are listed in Table1. The inlet conditions of the reactor are determined by the process method for HI decomposition in the SI cycle, and the mixed HI and H2O gas are obtained by reaction distillation directly into the decomposition reactor. The inlet composition, inlet temperature, inlet pressure of mixed gas and reactor size listed in Table1 are all taken from Ref. [ 15]. Nguyen et al. [15] studied the mechanism of HI decomposition and applied the kinetic model to the reactor design, coupled the helium gas-cooled reactor with the S-I cycle, and the high-temperature helium gas was used as the thermal working medium to heat the HI decomposition reactor. The HI and H2O mixture enters the HI decomposition reactor directly from the top of the distiller at a temperature of 468 K and the mixture gas inlet flow rate is 0.6056 mol/s. The high-temperature helium gas flows in the opposite direction at a temperature of 973 K to heat the reaction tube, and the helium inlet flow rate is 3.028 mol/s.

Table 1. The parameters of the reference reactor [15].

Parameters Symbol Value

Inlet temperature of reaction mixture T0 468 K Helium gas inlet temperature THe,in 973 K Overall heat transfer coefficient U 170 W/(K · m2 Mixed gas viscosity µ 3.4 × 10−5 kg/(m · s) Inlet pressure P0 7 bar 3 Catalyst particle density ρc 1250 kg/m Porosity of catalyst bed ε 0.5 −3 Catalyst particle diameter dp 3 × 10 mol/s Helium gas inlet flow rate FHe,in 3.028 mol/s Inlet total flow rate FT,in 0.6056 mol/s Inlet HI molar fraction xHI,in 20.48%

Inlet H2 molar fraction xH2,in 0.01%

Inlet I2 molar fraction xI2,in 0.01%

Inlet H2O molar fraction xH2O,in 79.5% Reactor inner diameter di 0.0762 m Reactor length L 0.982 m

2.2. Conservation Equations The reaction process follows the conservation of energy, mass and momentum. The total heat entering the control unit with an axial direction of dz is used to heat the reaction mixture and to be absorbed by the reaction. The energy conservation equation is [71]:

dT πd q − ρ A (1 − ε)r ∆H/2 = i c c HI (2) dz ∑ kFkCp,k

where q represents heat flux passing the tube wall, q = U(THe − T) when the process obeys Newton’s law of cooling, U, which is the overall heat transfer coefficient, can be approximated 2 as a constant value 170 W/(m · K) [15]. ρc and Ac are the catalyst particle density and cross- sectional area of the reaction tube respectively, ε is the void fraction of catalyst bed, and rHI is the HI decomposition reaction rate, which is strongly dependent on the temperature. Fk and Cp,k are the molar flow rate and molar specific heat capacity for component k, respectively, and the empirical formula of temperature correlation for Cp,k is [94]:

2 3 4 Cp,k = Ak + BkT + CkT + DkT + EkT (3)

where the values of the coefﬁcients Ak,Bk,Ck,Dk and Ek in the formula are listed in Table2. Entropy 2021, 23, 82 6 of 25

The heat released by the high-temperature helium is used to heat the reactor, so the axial temperature distribution of helium gas along the tube can be expressed as [15]:

dT Uπd (T − T) He = i He (4) dz FHeCp,He

Table 2. Thermodynamic parameter values of components [94].

Parameters k = HI k = H2O k = H2 k = I2

Ak 29.770 33.933 25.399 34.151 −3 Bk(×10 ) −7.4945 −8.4186 20.178 13.930 −5 Ck(×10 ) 2.0687 −2.9906 −3.8549 −2.0952 −8 Dk(×10 ) −1.1963 −1.7825 3.1880 1.4362 −12 Ek(×10 ) 2.1010 3.6934 −8.7585 −3.5948 3 Mk × 10 (kg/mol) 127.912 18.015 2.016 253.809

It means that the helium is heated in a counter-current way, so the temperature of helium gas gradually increases along the positive axis. Cp,He is the molar heat capacity of helium, and the value is 20.786 J/(mol · K). The Reynolds number of the reaction mixture ﬂowing in the ﬁxed bed reactor can be expressed as [80]: Rep = Gdp/µ (5)

where G is the mass flow rate per unit area of the reaction mixture, G = Σk(Fk Mk)/Ac, and µ is the viscosity of the reaction mixture. To facilitate the calculation, during the reaction µ is set as a constant value. After preliminary calculation, Rep/(1 − ε) > 500 under the given working parameters, and the pressure drop of the reaction mixture flowing in the tube satisfies the Hick equation [95]: dP (1 − ε)1.2 ρv2 = − −0.2 6.8 3 Rep (6) dz ε dp

where ρ and v are the density ﬂow velocity of the mixed gas, respectively, v = FT RT/(PAc), and R is universal gas constant. The change of gas composition in the reactor is related to the reaction rate. The HI conversion rate (ξ) and molar ﬂow rate (Fk) of each component in the mass conservation equation can be expressed as [82]:

dξ/dz = Acρc(1 − ε)rHI/FHI,in (7)

dFHI/dz = −Acρc(1 − ε)rHI (8)

dFH2 /dz = Acρc(1 − ε)rHI/2 (9)

dFI2 /dz = Acρc(1 − ε)rHI/2 (10)

2.3. Chemical Reaction Rate Shindo et al. [13] deduced the equation for HI decomposition rate based on the assumption that the decomposition of HI on the catalyst surface is a step to determine the reaction rate, and carried out an experimental study on the HI decomposition process with Pt/γ-alumina as catalyst under the conditions of atmospheric pressure. The experimental results showed that the kinetic model can be used to better simulate the reaction process, and the HI decomposition reaction rate equation can be expressed as [13,15]: √ dp pHI − pI pH /Kp r = − HI = k 2 2 (11) HI dt p 2 1 + KI2 pH2 Entropy 2021, 23, 82 7 of 25

1 k = 3.13 × 10 exp[−Ea1 /(RT)] (12) −7 KI2 = 1.80 × 10 exp[−Ea2 /(RT)] (13)

where k and KI2 are the HI decomposition reaction rate constant and iodine adsorption rate

constant, respectively, pHI, pH2 and pI2 are the partial pressures of reaction gas components,

and Ea1 and Ea2 are the activation energies for the decomposition and adsorption reactions,

respectively. When Pt/γ-alumina 1.0 wt% is selected as the catalyst, the values of Ea1 3 3 and Ea2 are 37.1 × 10 and −75.1 × 10 J/mol [15]. Kp and ∆G are the decomposition reaction equilibrium constant and Gibbs free energy changes, and they have the following relationship [96]: −∆G K = exp( ) (14) p RT where ∆G is given by JANAF [97].

2.4. Model Validation Shindo et. al. [13] used a 20 mm tubular ﬁxed-bed reactor for experiments, measured the HI decomposition conversion curves at different temperatures, and the HI solution was passed through the evaporator and entered into the reaction tube for testing during the experiment. Combined with the parameter values in the literature, the numerical calculation results are obtained by applying the established mathematical model of the reactor. Figure3 shows the comparison between the numerical calculation results and the experimental data values at different temperatures. When the decomposition temperature Entropy 2021, 23, x FOR PEER REVIEWis 600 and 700 K, the simulation results are in good agreement with the experimental data, 8 of 25 and the maximum relative percentage error is 5.5%. This indicates that the experimental results can be accurately predicted by the established reactor model.

Figure 3. 3.Comparison Comparison of numericalof numerical simulation simulation values andvalues experimental and experimental data values data at different values at different temperaturestemperatures. 2.5. Entropy Generation Rate 2.5. EntropyAccording Generation to the theory Rate of non-equilibrium thermodynamics, the local EGRs in the reactorAccording are produced to the by threetheory processes of non of‐equilibrium heat transfer, thermodynamics, chemical reaction and the ﬂuid local EGRs in the reactorﬂow [98, 99are], andproduced the local by total three EGR processes can be expressed of heat as: transfer, chemical reaction and fluid flow

[98,99], andσtot the=σ localht + σ crtotal+ σf EGR can be expressed as: 1 1 ∆rG 1 dP (15) =πdiq( tot− ht) + A crcρc(1 f − ε)r(− ) + Acv − T THe T T dz 11rG  1dP (15) =(dqiccc ) A (1)()  r Av  TTHe T Td z

where  ht ,  cr and f represent the local EGRs caused by heat transfer, chemical reac‐ tion and fluid flow, respectively. Each term is the product of flux and force of the corre‐ sponding transport process. The thermodynamic driving force of the heat transfer process is:

111 = (16) TTTHe The chemical driving force of the chemical reaction process is:

0.5 0.5  G ppHI r =lnR 22 (17) TpKHI p

The viscous driving force of fluid flow process is:

1dP   (18) Tzd The total EGR is the local EGR integral in the length of the reaction tube:

L (dSt / d )  d z (19) tot0 tot In addition, according to the entropy balance equation, the total EGR of the reaction system can also be calculated using the following formula [77]:

L qz() (dSt / d ) FS FS d d z (20) tot T,out out T,in in i 0 THe

where Sin and Sout represent the molar entropy flow at the reactor inlet and outlet re‐ spectively, and the last term is the heat entropy flow of high temperature helium gas to

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where σht, σcr and σf represent the local EGRs caused by heat transfer, chemical reaction and fluid flow, respectively. Each term is the product of flux and force of the corresponding transport process. The thermodynamic driving force of the heat transfer process is:

1 1 1 ∆ = − (16) T T THe

The chemical driving force of the chemical reaction process is:

∆ G p 0.5 p 0.5 − r = −R ln H2 I2 (17) T pHIKp

The viscous driving force of ﬂuid ﬂow process is:

1 dP − (18) T dz

The total EGR is the local EGR integral in the length of the reaction tube:

Z L (dS/dt)tot = σtotdz (19) 0 In addition, according to the entropy balance equation, the total EGR of the reaction system can also be calculated using the following formula [77]:

Z L q(z) (dS/dt)tot = FT,outSout − FT,inSin − πdi dz (20) 0 THe

where Sin and Sout represent the molar entropy ﬂow at the reactor inlet and outlet respectively, and the last term is the heat entropy ﬂow of high temperature helium gas to the reaction system. Equation (20) can be used to verify the correctness of Equation (19) in calculating the total EGR, and shows that heating with lower temperature helium can help reduce the total EGR in the reactor under the same import and export conditions.

3. The Optimization Problem In the chemical production process, the reaction product is usually required to meet a certain output, so the H2 PR is kept constant during the optimization process. The optimization problem can be described as solving the optimal THe(z) to minimize the total EGR in the HI decomposition reactor under the constraint of ﬁxed H2 PR. The constraints of the optimization problem can be divided into two categories: one is the energy, momentum and mass balance Equations (2), (6) and (7), which restrict the change of state variables, and another is the inlet and outlet boundary conditions, such as temperature, pressure, and conversion rate at the reactor inlet and outlet.

3.1. The Optimal Control Formulation According to optimal control theory, the Hamilton function H of the optimal control problem is [77,78,100]:

3 H[x(z), λ(z), u(z)] = σtot[x(z), u(z)] + ∑ λi(z) fi[x(z), u(z)] (21) i=1

where x = [T(z), P(z), ξ(z)] is the state variable vector, λ = [λT, λP, λξ] is the covariate variable vector, and fi represents the corresponding conservation equation (Equations (2), (6) and (7)), the control variable is the external helium temperature THe(z) i.e., u(z) = THe(z). Entropy 2021, 23, 82 9 of 25

Based on the Pontryagin minimum value principle [101], the following necessary conditions should be met when the objective function takes the minimum value:

dT/dz = ∂H/∂λT (22)

dP/dz = ∂H/∂λP (23) dξ/dz = ∂H/∂ξ (24)

dλT/dz = −∂H/∂T (25)

dλP/dz = −∂H/∂P (26)

dλξ/dz = −∂H/∂ξ (27) When the Hamiltonian function H takes the extreme value, the control variable satisﬁes the following equation: dH/dTHe = 0 (28) In the Hamiltonian function, H, only the local EGR of heat transfer in Equations (15) and the energy conservation equation in Equation (2) are related to THe. By omitting other terms not related to THe, the Hamiltonian function can be written as:

1 1 πdiq H(THe) = πdiq( − ) + λT (29) T THe ΣkFkCp,k

Substituting q = U(THe − T) into Equations (29) and continuing to omit other items irrelevant to THe yields:

THe T λTTHe H(THe) = − + (30) T THe ΣkFkCp,k

The optimal helium temperature THe(z) can be obtained by solving Equations (28) and (30): !−1/2 λTT THe = T 1 + (31) ΣkFkCp,k

3.2. Boundary Conditions The optimal control boundary conditions can be speciﬁed or are left free. When the state variable is free, the corresponding coordinator variable is 0. According to the second type of boundary condition constraints mentioned earlier, optimal reactor has the same hydrogen production rate, inlet temperature, inlet and outlet pressure as the reference reactor, so the boundary conditions can be summarized as:

ref ξ(0) = 0 and ξ(L) = ξL (32)

ref T(0) =T0 and λT(L) = 0 (33) ref ref P(0) = P0 and P(L) = PL (34) ref ref In the above boundary condition, T0 and P0 are 468 K and 7 bar, respectively. ref ref ξL and PL are obtained by solving the reference reactor.

3.3. Numerical Solution Metho The optimal control problem includes 6 differential equations (Equations (22)–(27)), 1 algebraic equation (Equation (31)), and 6 boundary conditions (Equations (32)–(34)), which can be turned into the boundary value problem of the differential equations, and solved by Matlab solver ‘bvp4c’. The initial value has a great inﬂuence on the calculation result when ‘bvp4c’ is used to solve the optimal problem, the nonlinear programming Entropy 2021, 23, 82 10 of 25

method proposed by Nummedal et. al. [74] is used to discretize various constraint equations and the objective function, and the Matlab solver ‘fmincon’ is used to select the initial 100 grids for calculation to obtain the initial value of discrete points. To meet the calculation accuracy, 3000 grids are selected to solve the function of ‘bvp4c’, the calculation accuracy at this point is greater than 2 × 10−10.

4. Numerical Results and Discussions 4.1. The Reference Reactor Figure4 shows the proﬁles of reaction mixture temperature T and helium temperature THe, the actual HI conversion rate and HI equilibrium conversion rate in the reference reactor. High temperature helium ﬂows in the reverse direction, and its temperature drops almost linearly from 973 to 836 K at the outlet. The reaction mixture temperature increases rapidly at the inlet, then the growth rate gradually slows down, and increases to 856 K at Entropy 2021, 23, x FOR PEER REVIEW 11 of 25 the outlet. The pressure dropped by 0.135 bar. The HI decomposition conversion reaches 0.241 at the outlet; it increases rapidly in the middle position (0.2 < z < 0.6), this indicates a relatively high reaction rate. The difference between the equilibrium conversion rate and the actual conversion rate gradually decreases and remains constant at the end of the reactionthe end tube, of the and reaction this phenomenon tube, and indicates this phenomenon that the reaction indicates driving force that keeps the constant reaction driving force whilekeeps approaching constant while the equilibrium approaching state. the equilibrium state.

Figure 4. 4.The The temperature temperature and hydrogen and hydrogen iodide (HI) iodide conversion (HI) conversion rate in the reference rate in reactor. the reference reactor. Figure5 shows the profiles of local EGRs in the reference reactor. The local total EGR σtot consistsFigure of 5 three shows parts. the The profiles local EGR of local of heat EGRs transfer in isthe relatively reference large, reactor. especially The at local total EGR the inlet, followed by the chemical reaction component, and the fluid flow component is  tot consists of three parts. The local EGR of heat transfer is relatively large, especially at relatively small. The local total EGR gradually decreases with the increase of dimensionless length,the inlet, which followed is consistent by the with chemical the change reaction trend of component, heat transfer and component. the fluid This flow is component is becauserelatively with small. the increase The local of tube total length, EGR the gradually temperature decreases difference betweenwith the helium increase and of dimension‐ mixedless length, gas gradually which decreases, is consistent resulting with in thethe decrease change of trend the heat of transferheat transfer driving force.component. This is Thebecause local EGR withσcr theof chemical increase reaction of tube increases length, first the and temperature then decreases difference gradually from between the helium and reactormixed inlet. gas gradually This is because decreases, the low temperature resulting atin thethe inlet decrease leads to of a smallthe heat reaction transfer rate, driving force. and there is a small chemical driving force near the reaction equilibrium state at the outlet, soThe the local interaction EGR of fluxcr of and chemical force in the reaction chemical increases reaction makes first andσcr reach then the decreases maximum gradually from valuethe reactor in the middle inlet. part.This The is because local EGR theσf of low fluid temperature flow increases at slowly; the inlet this is leads due to to the a small reaction increase of reaction temperature and the acceleration of gas velocity, which leads to the increaserate, and of pressurethere is dropa small rate. chemical driving force near the reaction equilibrium state at the

outlet, so the interaction of flux and force in the chemical reaction makes  cr reach the

maximum value in the middle part. The local EGR f of fluid flow increases slowly; this is due to the increase of reaction temperature and the acceleration of gas velocity, which leads to the increase of pressure drop rate.

Figure 5. The local energy generation rates (EGRs) in the reference reactor.

4.2. The Optimal Reactor The optimal and reference reactors have the same pressure drop, and the relation‐ ships of total and three components EGRs versus the length of the reaction tube are shown

Entropy 2021, 23, x FOR PEER REVIEW 11 of 25

the end of the reaction tube, and this phenomenon indicates that the reaction driving force keeps constant while approaching the equilibrium state.

Figure 4. The temperature and hydrogen iodide (HI) conversion rate in the reference reactor.

Figure 5 shows the profiles of local EGRs in the reference reactor. The local total EGR

 tot consists of three parts. The local EGR of heat transfer is relatively large, especially at the inlet, followed by the chemical reaction component, and the fluid flow component is relatively small. The local total EGR gradually decreases with the increase of dimension‐ less length, which is consistent with the change trend of heat transfer component. This is because with the increase of tube length, the temperature difference between helium and mixed gas gradually decreases, resulting in the decrease of the heat transfer driving force.

The local EGR  cr of chemical reaction increases first and then decreases gradually from the reactor inlet. This is because the low temperature at the inlet leads to a small reaction rate, and there is a small chemical driving force near the reaction equilibrium state at the

outlet, so the interaction of flux and force in the chemical reaction makes  cr reach the

maximum value in the middle part. The local EGR f of fluid flow increases slowly; this Entropy 2021, 23, 82 is due to the increase of reaction temperature and the acceleration11 of 25 of gas velocity, which leads to the increase of pressure drop rate.

Entropy 2021, 23, x FOR PEER REVIEW 12 of 25 Figure 5. The local energy generation rates (EGRs) in the reference reactor. Figure 5. The local energy generation rates (EGRs) in the reference reactor. 4.2. The Optimal Reactor 4.2.The The optimal Optimal and reference Reactor reactors have the same pressure drop, and the relationships of intotal Figure and three 6. components The total EGRsEGR versus of heat the lengthtransfer of the process reaction tube decreases are shown first in Figure and6 .then increases, while theThe totaltotalThe EGR EGRs optimal of heat due transfer andto processchemical reference decreases reaction reactors first and and then have increases,fluid the flow while same are the totalalmostpressure EGRs constants. drop, and When the the relation ‐ due to chemical reaction and fluid flow are almost constants. When the minimum total EGR minimumisships obtained, of total the total corresponding and EGR three is obtained, length components of the reactionthe corresponding EGRs tube is 1.059 versus m. length the length of the of reaction the reaction tube is tube 1.059 are m. shown

Figure 6. 6.Relationships Relationships of the totalof the and total component and component EGRs versus the EGRs reactor versus length L. the reactor length L. Figure7 shows the proﬁles of temperature and HI conversion rate in the optimal reactor.Figure The helium 7 shows gas temperature the profiles increases of from temperature 600 to 1084 K, and and thenHI quicklyconversion drops rate in the optimal reactor.to 858 K at The the outlet.helium The gas temperature temperature of the increases mixed gas increases from 600 gradually to 1084 from K, and the then quickly drops inlet. The difference between temperatures of helium and the mixed gas increases slowly toat ﬁrst,858 andK at then the gradually outlet. The decreases temperature to 0 at the outlet. of the When mixed the heliumgas increases temperature gradually is from the inlet. The difference between temperatures of helium and the mixed gas increases slowly at first, and then gradually decreases to 0 at the outlet. When the helium temperature is near the maximum, the two temperatures have the maximum difference, and the end of the temperature curve is closed due to the free temperature boundary conditions at the outlet. The difference between the equilibrium conversion rate and the actual conversion rate decreases monotonically in the axial direction, the actual conversion rate is close to the equilibrium conversion rate at the outlet.

Figure 7. The temperature and HI conversation rate in the optimal reactor.

Figure 8 shows the local EGRs profiles in the optimal reactor. The local EGR contrib‐ uted by the heat transfer process plays a dominant role; it keeps a certain value at the inlet and remains relatively stable, then drops sharply at the outlet. The local EGR due to the chemical reaction remains stable in the front of the reactor and shows a relatively uniform

Entropy 2021, 23, x FOR PEER REVIEW 12 of 25

in Figure 6. The total EGR of heat transfer process decreases first and then increases, while the total EGRs due to chemical reaction and fluid flow are almost constants. When the minimum total EGR is obtained, the corresponding length of the reaction tube is 1.059 m.

Figure 6. Relationships of the total and component EGRs versus the reactor length L.

Figure 7 shows the profiles of temperature and HI conversion rate in the optimal reactor. The helium gas temperature increases from 600 to 1084 K, and then quickly drops to 858 K at the outlet. The temperature of the mixed gas increases gradually from the inlet. The difference between temperatures of helium and the mixed gas increases slowly at Entropy 2021, 23, 82 12 of 25 first, and then gradually decreases to 0 at the outlet. When the helium temperature is near the maximum, the two temperatures have the maximum difference, and the end of the temperaturenear the maximum, curve the twois closed temperatures due to have the the free maximum temperature difference, boundary and the end conditions of at the outlet. Thethe temperature difference curve between is closed the due toequilibrium the free temperature conversion boundary rate conditions and the at theactual conversion rate decreasesoutlet. The difference monotonically between the in equilibrium the axial conversion direction, rate the and actual the actual conversion conversion rate is close to the rate decreases monotonically in the axial direction, the actual conversion rate is close to the equilibrium conversion conversion rate at therate outlet. at the outlet.

Figure 7. 7.The The temperature temperature and HI conversationand HI conversation rate in the optimal rate reactor. in the optimal reactor. Entropy 2021, 23, x FOR PEER REVIEW 13 of 25 Figure8 shows the local EGRs profiles in the optimal reactor. The local EGR con- tributedFigure by the 8 heat shows transfer the process local plays EGRs a dominant profiles role; in itthe keeps optimal a certain reactor. value at theThe local EGR contrib‐ utedinlet and by remains the heat relatively transfer stable, process then drops plays sharply a dominant at the outlet. role; The it localkeeps EGR a certain due value at the inlet distribution.to the chemical The reaction local remains EGR stabledue to in thefluid front flow of the increases reactor and slowly shows with a relatively increasing tempera‐ anduniform remains distribution. relatively The local stable, EGR due then to fluid drops flow sharply increases at slowly the withoutlet. increasing The local EGR due to the chemicalture.temperature. Local reaction total Local and total remains heat and heattransfer stable transfer EGRs in EGRs the have front have similar similarof the trends;reactor they they and are are shows evenly evenly a relatively distributed uniform indistributed the length in the of length reaction of reaction tube except tube except at the at the inlet inlet and and outlet. outlet.

Figure 8. 8.The The local local EGRs EGRs in the in optimal the optimal reactor. reactor.

4.3. Modified Reactors The design requirements of the modified reactor are as follows: under the constraint of fixed hydrogen PR, the length of the reactor and the inlet conditions of the helium gas are changed to achieve the same total EGR as that of the optimal reactor. The two types of modified reactors designed are realized by changing the inlet temperature and inlet flow rate of helium gas respectively, which are denoted as “Case 1” and “Case 2” reactors; the remaining parameter values in the modified reactor remain the same as reference values. In order to clearly compare the differences between the reactor design parameters, the design parameters of the reference reactor, optimal reactor and modified reactors are listed in Table 3. In the table, the “reference value” (RV) indicates that the parameter is consistent with that of the reference reactor, and the “optimal value” (OV) and the “cal‐ culated value” (CV) indicate that the parameters are obtained by optimization and calcu‐ lation methods, respectively.

Table 3. Comparison of reactor design parameters.

Parameter Ref Opt Case 1 Case 2 T RV OV CV RV He,in F RV ‐ RV CV He,in ()L RV RV RV RV PL() RV RV CV CV TL() RV OV CV CV L RV OV CV CV (dSt / d ) RV OV fixed at OV fixed at OV tot

4.3.1. Case 1 Reactor Figure 9 shows the curve of inlet helium temperature and total EGR changing with reactor length L under the same hydrogen production rate. As the length increases, the corresponding helium inlet temperature decreases, and the total EGR also decreases grad‐ ually. In order to achieve the optimal value of total EGR as that in optimal reactor, the Case 1 reactor length should be extended to 1.291 m and the corresponding helium inlet temperature drops from 973 to 919 K.

Entropy 2021, 23, 82 13 of 25

4.3. Modified Reactors The design requirements of the modified reactor are as follows: under the constraint of fixed hydrogen PR, the length of the reactor and the inlet conditions of the helium gas are changed to achieve the same total EGR as that of the optimal reactor. The two types of modified reactors designed are realized by changing the inlet temperature and inlet flow rate of helium gas respectively, which are denoted as “Case 1” and “Case 2” reactors; the remaining parameter values in the modified reactor remain the same as reference values. In order to clearly compare the differences between the reactor design parameters, the design parameters of the reference reactor, optimal reactor and modified reactors are listed in Table3. In the table, the “reference value” (RV) indicates that the parameter is consistent with that of the reference reactor, and the “optimal value” (OV) and the “calculated value” (CV) indicate that the parameters are obtained by optimization and calculation methods, respectively.

Table 3. Comparison of reactor design parameters.

Parameter Ref Opt Case 1 Case 2

THe,in RV OV CV RV FHe,in RV - RV CV ξ(L) RV RV RV RV P(L) RV RV CV CV T(L) RV OV CV CV L RV OV CV CV (d S/dt)tot RV OV ﬁxed at OV ﬁxed at OV

4.3.1. Case 1 Reactor Figure9 shows the curve of inlet helium temperature and total EGR changing with reactor length L under the same hydrogen production rate. As the length increases, the corresponding helium inlet temperature decreases, and the total EGR also decreases Entropy 2021, 23, x FOR PEER REVIEWgradually. In order to achieve the optimal value of total EGR as that in optimal reactor, 14 of 25 the Case 1 reactor length should be extended to 1.291 m and the corresponding helium inlet temperature drops from 973 to 919 K.

FigureFigure 9. 9.Relationships Relationships of the of helium the helium inlet temperature inlet temperature and total EGR and versus total the EGR reactor versus length theL. reactor length L . Figure 10 shows the variation curves of temperature and HI conversion rate in the Case 1Figure modiﬁed 10 reactor. shows The the outlet variation temperatures curves of of helium temperature and reaction and mixture HI conversion are 784 rate in the andCase 850 1 modified K, respectively. reactor. The actualThe outlet conversion temperatures rate increases of helium rapidly inand the reaction front part, mixture are 784 and 850 K, respectively. The actual conversion rate increases rapidly in the front part, ap‐ proaches close to the equilibrium conversion rate at z=0.6 , and then begins to increase slowly along the equilibrium curve.

Figure 10. The temperature and HI conversion rate in the Case 1 reactor.

Figure 11 shows the profiles of local EGRs in the Case 1 modified reactor. The local EGR contributed by the heat transfer process still accounts for a large proportion, and gradually decreases in the axial direction. The local EGR contributed by the chemical re‐ action first increases to the maximum value and then decreases gradually while the local EGR caused by fluid flow increases slowly. The overall change of local EGRs is similar to that of the reference reactor.

Entropy 2021, 23, x FOR PEER REVIEW 14 of 25

Figure 9. Relationships of the helium inlet temperature and total EGR versus the reactor length L .

Entropy 2021, 23, 82 Figure 10 shows the variation curves of temperature and HI conversion14 of 25 rate in the Case 1 modified reactor. The outlet temperatures of helium and reaction mixture are 784 and 850 K, respectively. The actual conversion rate increases rapidly in the front part, ap‐ proachesapproaches close close to to the the equilibrium equilibrium conversion conversion rate at z =rate0.6 ,at andz=0.6 then begins, and tothen increase begins to increase slowly along along the the equilibrium equilibrium curve. curve.

Figure 10. 10.The The temperature temperature and HI and conversion HI conversion rate in the rate Case in 1 reactor.the Case 1 reactor. Figure 11 shows the profiles of local EGRs in the Case 1 modified reactor. The local EGR contributedFigure by 11 the shows heat transfer the profiles process still of accountslocal EGRs for a in large the proportion, Case 1 modified and gradually reactor. The local EGRdecreases contributed in the axial by direction. the heat The transfer local EGR process contributed still by accounts the chemical for reaction a large first proportion, and Entropy 2021, 23, x FOR PEER REVIEWincreases to the maximum value and then decreases gradually while the local EGR caused 15 of 25 gradually decreases in the axial direction. The local EGR contributed by the chemical re‐ actionby fluid first flow increases increases slowly. to the The maximum overall change value of and local then EGRs decreases is similar to gradually that of the while the local reference reactor. EGR caused by fluid flow increases slowly. The overall change of local EGRs is similar to that of the reference reactor.

Figure 11. 11.The The local local EGRs EGRs in the in Case the 1 Case reactor. 1 reactor. 4.3.2. Case 2 Reactor 4.3.2.Figure Case 12 2 showsReactor the curves of inlet helium flow rate and total EGR changing with reactorFigure length 12L under shows the the same curves hydrogen of PR.inlet The helium corresponding flow rate helium and flow total rate EGR and changing with totalreactor EGR length decrease L when under the the length same increases. hydrogen When PR. the total The EGR corresponding of Case 2 reactor helium is the flow rate and same as the optimal value, the length L of the reactor extends to 1.161 m, and the helium inlettotal flow EGR rate decrease is 1.788 mol/s. when the length increases. When the total EGR of Case 2 reactor is the same as the optimal value, the length L of the reactor extends to 1.161 m, and the helium inlet flow rate is 1.788 mol/s.

Figure 12. Relationships of the helium inlet flow rate and total EGR versus the reactor length L.

Figure 13 shows the variation curves of temperature and HI conversion rate in the Case 2 modified reactor. The helium temperature gradually dropped from 973 to 743 K, the mixture temperature at the outlet is 856 K, and the HI actual conversion rate increases slowly after approaching the equilibrium conversion rate at the dimensionless position z=0.7 .

Entropy 2021, 23, x FOR PEER REVIEW 15 of 25

Figure 11. The local EGRs in the Case 1 reactor.

4.3.2. Case 2 Reactor Figure 12 shows the curves of inlet helium flow rate and total EGR changing with reactor length L under the same hydrogen PR. The corresponding helium flow rate and total EGR decrease when the length increases. When the total EGR of Case 2 reactor is the Entropy 2021, 23, 82 same as the optimal value, the length L of the reactor extends to15 of1.161 25 m, and the helium inlet flow rate is 1.788 mol/s.

Figure 12. 12.Relationships Relationships of the helium of the inlet helium ﬂow rate inlet and flow total EGR rate versus and thetotal reactor EGR length versusL. the reactor length L. Figure 13 shows the variation curves of temperature and HI conversion rate in the Case 2Figure modiﬁed 13 reactor. shows The the helium variation temperature curves gradually of temperature dropped from 973and to HI 743 K,conversion rate in the Entropy 2021, 23, x FOR PEER REVIEWtheCase mixture 2 modified temperature reactor. at the outlet The is helium 856 K, and temperature the HI actual conversion gradually rate dropped increases 16 from of 25 973 to 743 K, slowly after approaching the equilibrium conversion rate at the dimensionless position zthe= 0.7. mixture temperature at the outlet is 856 K, and the HI actual conversion rate increases slowly after approaching the equilibrium conversion rate at the dimensionless position z=0.7 .

Figure 13.13.The The temperature temperature and and HI HI conversion conversion rate inrate the in Case the 2Case reactor. 2 reactor.

FigureFigure 14 14 shows shows the the profiles profiles of localof local EGRs EGRs in the in Case the Case 2 modified 2 modified reactor. reactor. The varia- The varia‐ tions of local EGRs are similar to those of the Case 1 and reference reactor, but with slightly tionsdifferent of local values. EGRs The are local similar EGR of to heat those transfer of the decreases Case 1 and monotonically, reference reactor, and local but EGR with of slightly differentfluid flow values. increases The slowly. local The EGR local of EGR heat of transfer chemical decreases reaction reaches monotonically, its maximum and value local EGR ofat zfluid= 0.2. flow increases slowly. The local EGR of chemical reaction reaches its maximum value at z=0.2 .

Figure 14. The local EGRs in the Case 2 reactor.

4.4. Discussions Table 4 lists the total EGR and component EGR values in the reactors. The total EGR of both optimal and modified reactors decreased by 13.3% compared with the reference value; the decrease of total EGR is mainly caused by the decrease of EGR in the heat trans‐ fer process. Compared with the reference value, the total EGR contributed by the heat transfer process in the three reactors decreased by 14.4%, 15.1% and 14.7%, respectively, and the total EGRs of fluid flow component in two modified reactors increase obviously, which is mainly due to the large pressure drop brought by the increase of tube length. Therefore, the reduction of total EGR in the reaction system can be achieved by changing the temperature distribution of the thermal fluid or by extending the length of the reactor; both methods are achieved by reducing the EGR in the heat transfer process, but extend‐ ing the reactor length usually results in a large pressure loss.

Entropy 2021, 23, x FOR PEER REVIEW 16 of 25

Figure 13. The temperature and HI conversion rate in the Case 2 reactor.

Figure 14 shows the profiles of local EGRs in the Case 2 modified reactor. The varia‐ tions of local EGRs are similar to those of the Case 1 and reference reactor, but with slightly different values. The local EGR of heat transfer decreases monotonically, and local EGR Entropy 2021, 23, 82 of fluid flow increases slowly. The local EGR of chemical reaction reaches 16its of maximum 25 value at z=0.2 .

Figure 14. 14.The The local local EGRs EGRs in thein the Case Case 2 reactor. 2 reactor. 4.4. Discussions 4.4. Discussions Table4 lists the total EGR and component EGR values in the reactors. The total EGR of bothTable optimal 4 lists and the modified total EGR reactors and component decreased by EGR 13.3% values compared in the with reactors. the reference The total EGR value;of both the optimal decrease and of modified total EGR reactors is mainly decreased caused by by the 13.3% decrease compared of EGR inwith the the heat reference transfervalue; the process. decrease Compared of total with EGR the is reference mainly caused value, the by total the EGRdecrease contributed of EGR by in the the heat heat trans‐ transferfer process. process Compared in the three with reactors the reference decreased value, by 14.4%, the 15.1% total andEGR 14.7%, contributed respectively, by the heat andtransfer the total process EGRs in of the fluid three flow reactors component decreased in two modified by 14.4%, reactors 15.1% increase and 14.7%, obviously, respectively, whichand the is total mainly EGRs due of to fluid the large flow pressure component drop in brought two modified by the increase reactors of increase tube length. obviously, Therefore, the reduction of total EGR in the reaction system can be achieved by changing thewhich temperature is mainly distribution due to the of large the thermal pressure fluid drop or by brought extending by the the length increase of the of reactor; tube length. bothTherefore, methods the are reduction achieved byof reducingtotal EGR the in EGR the inreaction the heat system transfer can process, be achieved but extending by changing the reactortemperature length usuallydistribution results of in the a large thermal pressure fluid loss. or by extending the length of the reactor; both methods are achieved by reducing the EGR in the heat transfer process, but extend‐ Tableing the 4. Comparison reactor length of the usually total and results component in a EGRs large in pressure the reactors loss. (J·K− 1·s−1).

Parameter Ref Opt Case 1 Case 2 L (m) 0.982 1.059 1.291 1.165 Heat transfer 3.772 3.228 3.201 3.218 Chemical reaction 0.221 0.216 0.217 0.216 Fluid ﬂow 0.099 0.108 0.132 0.116 Total 4.092 3.550 3.550 3.550

Figure 15 shows the distributions of the local total EGRs in the reactors along the dimensionless axial position. It is obvious that the local total EGR of the optimal reactor is most evenly distributed in the axial position, this distribution can be analyzed and understood by the equipartition of entropy production theory proposed by Johannessen and Kjelstrup [77]; the local EGR tends to be a constant in the middle part in the optimal control system, it changes dramatically at the outlet due to the outlet fixed boundary conditions. The variation trends of local total EGRs in two modified reactors are similar to that in the reference reactor, however, the change of curve is relatively flat, and the corresponding value of the axial position is also relatively small. Entropy 2021, 23, x FOR PEER REVIEW 17 of 25

Table 4. Comparison of the total and component EGRs in the reactors (J∙K−1∙s−1).

Parameter Ref Opt Case 1 Case 2 L (m) 0.982 1.059 1.291 1.165 Heat transfer 3.772 3.228 3.201 3.218 Chemical reaction 0.221 0.216 0.217 0.216 Fluid flow 0.099 0.108 0.132 0.116 Total 4.092 3.550 3.550 3.550 Figure 15 shows the distributions of the local total EGRs in the reactors along the dimensionless axial position. It is obvious that the local total EGR of the optimal reactor is most evenly distributed in the axial position, this distribution can be analyzed and un‐ derstood by the equipartition of entropy production theory proposed by Johannessen and Kjelstrup [77]; the local EGR tends to be a constant in the middle part in the optimal con‐ trol system, it changes dramatically at the outlet due to the outlet fixed boundary condi‐ Entropy 2021, 23, 82 tions. The variation trends of local total EGRs in two modified reactors are similar to17 ofthat 25 in the reference reactor, however, the change of curve is relatively flat, and the corre‐ sponding value of the axial position is also relatively small.

Figure 15. Comparison of the local total EGRs in the reactors.

Vander Ham et al. [[100]100] proposedproposed toto useuse thethe thermodynamicthermodynamic performanceperformance indicatorindicator and equipartition indicator toto evaluateevaluate thethe EGREGR in in chemical chemical reactors, reactors, where where the the equiparti- equipar‐ tion indicator is measured by the change of the local EGR. The standard deviation σ is tition indicator is measured by the change of the local EGR. The standard deviation tot calculated as follows [100]: tot is calculated as follows [100]: s n 1 2 s = ∑ (σz,i − m) (35) n1= n i 1  2 sm()zi, (35) where the discrete n data points were usedn i to1 solve the standard deviation s of the local EGR curve, and m is the average of these data points. The dimensionless coefﬁcient where the discrete n data points were used to solve the standard deviation s of the lo‐ of variation c is the ratio of the standard deviation s to the average m, i.e., c = s/m, cal EGR curve, and m is the average of these data points. The dimensionless coefficient which is used to measure the degree of equalization, so the closer the curve distribution of variation c is the ratio of the standard deviation s to the average m, i.e., csm=/ , which is to the equipartition state, the smaller is the c value. Table5 lists the comparison of isequipartition used to measure indicator the degree in the reactors.of equalization, It can be so seen the closer that the the optimal curve distribution reactor has theis to best the equipartitionequalization characteristics, state, the smaller and is the the Case c value. 1 reactor Table is relatively5 lists the poor.comparison As the of EGR equiparti of heat‐ tiontransfer indicator accounts in the for reactors. a large proportion, It can be seen it is that an important the optimal factor reactor affecting has the the best equipartition equaliza‐ tionindicator characteristics, of local total and EGR. the Case 1 reactor is relatively poor. As the EGR of heat transfer accounts for a large proportion, it is an important factor affecting the equipartition indi‐ Tablecator 5.of Comparisonlocal total EGR. of equipartition indicator in the reactors.

Indicator Ref Opt Case 1 Case 2 c tot 0.8462 0.2167 1.0263 0.7269 cht 0.8715 0.2420 1.0751 0.7492 ccr 1.1050 0.7245 1.3079 1.1172 cf 0.1678 0.1959 0.1669 0.1732

Figure 16 shows the comparison of helium temperature in the reactors along dimensionless axial position. Except for the optimal reactor, the helium temperature in the other three reactors varies monotonously along the axial direction. The helium temperature in the optimal reactor increases gradually from a low temperature value to a maximum value and then drops sharply; the helium ﬂow rate required to achieve the temperature proﬁle is freely variable in the axial direction. Therefore, the results indicate that it is advantageous to use a lower temperature in the reactor inlet to reduce the entropy generation rate. Entropy 2021, 23, x FOR PEER REVIEW 18 of 25

Entropy 2021, 23, x FOR PEER REVIEW 18 of 25

Table 5. Comparison of equipartition indicator in the reactors.

Table 5.Indicato Comparisonr of equipartitionRef indicator inOpt the reactors. Case 1 Case 2 c 0.8462 0.2167 1.0263 0.7269 Indicatotot r Ref Opt Case 1 Case 2 c 0.8715 0.2420 1.0751 0.7492 cht 0.8462 0.2167 1.0263 0.7269 tot c 1.1050 0.7245 1.3079 1.1172 ccr 0.8715 0.2420 1.0751 0.7492 ht c 0.1678 0.1959 0.1669 0.1732 c f 1.1050 0.7245 1.3079 1.1172 cr Figurec 16 shows the comparison0.1678 of helium0.1959 temperature 0.1669in the reactors along0.1732 dimen ‐ f sionless axial position. Except for the optimal reactor, the helium temperature in the other threeFigure reactors 16 variesshows monotonously the comparison along of helium the axial temperature direction. in The the helium reactors temperature along dimen in‐ thesionless optimal axial reactor position. increases Except gradually for the optimal from a reactor,low temperature the helium value temperature to a maximum in the valueother threeand then reactors drops varies sharply; monotonously the helium flowalong rate the requiredaxial direction. to achieve The thehelium temperature temperature profile in theis freely optimal variable reactor in increases the axial gradually direction. from Therefore, a low temperature the results indicate value to thata maximum it is advanta value‐ andgeous then to use drops a lower sharply; temperature the helium in theflow reactor rate required inlet to reduce to achieve the entropy the temperature generation profile rate. Entropy 2021, 23, 82 is freely variable in the axial direction. Therefore, the results indicate18 ofthat 25 it is advanta‐ geous to use a lower temperature in the reactor inlet to reduce the entropy generation rate.

Figure 16. Comparison of the helium gas temperature in the reactors.

FigureFigure 16. 16.Comparison Comparison 17 shows of the of heliumthe the heliumcomparison gas temperature gas temperature of in reaction the reactors. in themixture reactors. temperature in the reactors alongFigure dimensionless 17 shows the comparisonaxial position. of reaction The mixture temperature temperature of reaction in the reactors mixture along in the optimal reactordimensionlessFigure increases 17 axial shows position.slowly the Theat comparison the temperature beginning of reaction reactionand gradually mixture mixture in thesurpass temperature optimal temperature reactor in the curvereactors in alongotherincreases reactorsdimensionless slowly at at thez=0.9 beginning axial. The position. and temperature gradually The surpass temperature variations temperature of of reaction curve reaction in other mixture mixture reactors in in the the modified optimal reactorat z= 0.9 .increases The temperature slowly variations at the ofbeginning reaction mixture and ingradually the modiﬁed surpass reactors temperature and the curve in reactorsreference reactorand the are reference similar. reactor are similar. other reactors at z=0.9 . The temperature variations of reaction mixture in the modified reactors and the reference reactor are similar.

Figure 17. 17.Comparison Comparison of the of reaction the reaction mixture mixture temperature temperature in the reactors. in the reactors.

Figure 18 shows the comparison of thermal driving forces in the reactors along the Figuredimensionless 17. Comparison axial position. of the All reaction the thermal mixture driving temperature forces gradually in the decreasereactors. from the maximum value at the inlet, but the thermal driving forces of the optimal reactor are relatively ﬂat at the middle position, and decrease sharply at the outlet. This is because there is a relatively constant temperature difference between the helium and the mixed gas in the optimal reactor, as shown in Figure7.

Entropy 2021, 23, x FOR PEER REVIEW 19 of 25

Figure 18 shows the comparison of thermal driving forces in the reactors along the dimensionless axial position. All the thermal driving forces gradually decrease from the maximumFigure value 18 shows at the the inlet, comparison but the thermal of thermal driving driving forces forces of the in optimal the reactors reactor along are relthe‐ dimensionlessatively flat at the axial middle position. position, All the and thermal decrease driving sharply forces at the gradually outlet. This decrease is because from there the maximumis a relatively value constant at the inlet,temperature but the differencethermal driving between forces the ofhelium the optimal and the reactor mixed are gas rel in‐ ativelythe optimal flat at reactor, the middle as shown position, in Figure and decrease 7. sharply at the outlet. This is because there Entropy 2021, 23, 82 is a relatively constant temperature difference between the helium and19 of the 25 mixed gas in the optimal reactor, as shown in Figure 7.

Figure 18. Comparison of the thermal driving forces in the reactors.

FigureFigure 18. 18.Comparison Comparison 19 shows of the of the thermal the comparison thermal driving driving forces of in chemical theforces reactors. in thedriving reactors. forces in the reactors along di‐ mensionlessFigure 19 axialshows position. the comparison All the of chemical chemical drivingreaction forces driving in the forces reactors decrease along sharply at the dimensionlessinlet,Figure and the 19 axial chemical shows position. the driving Allcomparison the force chemical of of the reaction chemical optimal driving driving reactor forces forcesis decrease relatively in the sharply highreactors in the along middle di‐ mensionlessatsection. the inlet, This and axialis the because chemical position. the driving Allmixture the force chemical temperature of the optimal reaction reactorin the driving isinlet relatively forcessection high decrease of in the the optimal sharply reactor at the middle section. This is because the mixture temperature in the inlet section of the optimal inlet,is lower, and and the chemicalthe reaction driving is far forceaway of from the optimalthe equilibrium reactor isstate, relatively so a large high chemical in the middle driv‐ section.reactor is lower,This is and because the reaction the ismixture far away temperature from the equilibrium in the state, inlet so section a large chemical of the optimal reactor drivinging force force is isgenerated. generated. is lower, and the reaction is far away from the equilibrium state, so a large chemical driv‐ ing force is generated.

Figure 19. 19.Comparison Comparison of the of chemical the chemical driving driving force in the force reactors. in the reactors.

Figure 20 shows the comparison of the pressures in the reactors along dimensionless Figureaxial position.Figure 19. Comparison 20 Although shows the ofthe the optimal comparison chemical reactor driving lengthof the force increases pressures in the slightly reactors. in the compared reactors with along the dimensionless referenceaxial position. length, the Although pressure dropthe optimal in the tube reactor remains length the same. increases Compared slightly with the refer-compared with the encereference values,Figure length, the 20 lengths shows the of pressurethe two comparison modified drop reactors in of the the increase tube pressures remains by 31.5% in andthe the 18.6%, same. reactors respectively, Compared along dimensionless with the ref‐ and the pressure drops increase by 33.1% and 17.6%, respectively. When the length of the axialerence position. values, Althoughthe lengths the of optimaltwo modified reactor reactors length increasesincrease by slightly 31.5% comparedand 18.6%, with respec the‐ referencemodified reactor length, increases the pressure appropriately, drop heliumin the gastube with remains lower temperature the same. Compared and lower with the ref‐ flowtively, rate and can bethe selected pressure for heating drops toincrease reduce the by total 33.1% EGR and in decomposition 17.6%, respectively. reaction, but When the length erenceof the modifiedvalues, the reactor lengths increases of two modifiedappropriately, reactors helium increase gas withby 31.5% lower and temperature 18.6%, respec and‐ tively, and the pressure drops increase by 33.1% and 17.6%, respectively. When the length of the modified reactor increases appropriately, helium gas with lower temperature and

Entropy 2021, 23, x FOR PEER REVIEW 20 of 25

Entropy 2021, 23, 82 20 of 25

lower flow rate can be selected for heating to reduce the total EGR in decomposition reac‐ tion,the increase but the of reaction increase tube of length reaction will cause tube greater length pressure will cause loss, which greater requires pressure a more loss, which re‐ quirespowerful a compressormore powerful to supercharge. compressor to supercharge.

Figure 20. 20.Comparison Comparison of the of pressure the pressure in the reactors. in the reactors. 5. Conclusions 5. ConclusionsA one-dimensional tubular plug flow reactor model for HI decomposition is estab- lishedA based one‐ ondimensional FTT theory, and tubular the reference plug flow reactor reactor is heated model by helium for countercurrent.HI decomposition is estab‐ lishedBased on based the reference on FTT reactor theory, design and parameters,the reference the reactor optimal reactoris heated and by the helium modified countercurrent. reactors with different parameters are designed by adjusting the distribution of EGR and Basedextending on thethe length reference of the reactor reactor, anddesign the total parameters, EGR of those the reactors optimal are reactor reduced and by the modified reactors13.3% compared with different with the referenceparameters value; are the designed reduction by of totaladjusting EGR is the mainly distribution caused of EGR and extendingby the decrease the oflength EGR in of heat the transfer reactor, process. and the The total optimal EGR reactor of those is solved reactors by the are reduced by 13.3%optimal compared control theory, with and the the distributionreference value; of local totalthe reduction EGR is more of uniform, total EGR the optimal is mainly caused by helium temperature profile is ideal and requires a lower temperature at the reactor inlet. theThe decrease length of the of modifiedEGR in heat reactors transfer increases process. significantly The optimal compared reactor with the is reference solved by the optimal controlreactor length, theory, and and the the local distribution EGR distribution of local remains total similarEGR is to more that ofuniform, the reference the optimal helium temperaturereactor. However, profile the increase is ideal of and length requires will bring a anlower additional temperature pressure drop, at the which reactor inlet. The lengthrequires of a the more modified powerful reactors compressor increases to compensate significantly for the pressure. compared The with method the reference of reactor optimizing the temperature profile of helium can reduce the total EGR without increasing length, and the local EGR distribution remains similar to that of the reference reactor. the pressure drop in the reactor, but achieving this temperature profile in actual production However,practice requires the increase improvement of length of corresponding will bring heating an additional equipment. pressure The results drop, obtained which requires a moreherein powerful can provide compressor some guidelines to compensate for the design for of HI the decomposition pressure. The reactor method structure of optimizing the temperatureparameters. profile of helium can reduce the total EGR without increasing the pressure drop in the reactor, but achieving this temperature profile in actual production practice Author Contributions: Conceptualization, L.C. and S.X.; funding acquisition, S.X.; methodology, requiresR.K. and S.X.; improvement software, R.K., S.X. of and corresponding P.L.; validation, L.C. heating and Y.G.; equipment. writing—original The draft, results R.K. and obtained herein canS.X.; Writing—reviewprovide some and guidelines editing, L.C. for All authors the design have read of andHI agreeddecomposition to the published reactor version structure of param‐ eters.the manuscript. Funding: This work is supported by the National Natural Science Foundation of China (Grant Nos. Author51976235 Contributions: and 51606218) and Conceptualization, Independent Project of NavalL.C. and University S.X.; funding of Engineering acquisition, (No. 20161504). S.X.; methodology, R.K.Acknowledgments: and S.X.; software,The authors R.K., wish S.X. to and thank P.L.; the validation, reviewers for L.C. their and careful, Y.G.; unbiased, writing—original and con- draft, R.K. andstructive S.X.; suggestions, Writing—review which led and to this editing, revised manuscript.L.C. All authors have read and agreed to the published versionConflicts of of Interest:the manuscript.The authors declare no conflict of interest. Funding: This work is supported by the National Natural Science Foundation of China (Grant Nos. 51976235 and 51606218) and Independent Project of Naval University of Engineering (No. 20161504). Acknowledgments: The authors wish to thank the reviewers for their careful, unbiased, and con‐ structive suggestions, which led to this revised manuscript. Conflicts of Interest: The authors declare no conflicts of interest.

Entropy 2021, 23, 82 21 of 25

Nomenclature 2 Ac Cross-section area: m −1 −1 Cp Molar heat capacity, J · mol · K c Dimensionless coefficient of variation d Diameter, m G Mass flow rate per unit area, kg s−1m−2 F Molar flow rate, mol · s−1 H Hamiltonian function of the optimal problem −1 KI2 Adsorption equilibrium constant of iodine, bar Kp Equilibrium constant k Reaction rate constant, mol s−1kg−1 L Length, m M Molar mass, kg mol−1 P Total pressure, bar p Partial pressure of reaction component gas, bar q Heat flow per unit area, W m−2 U Overall heat transfer coefficient, W m−2 K−1 u Control variable v Superficial velocity, m s−1 R Universal gas constant, J mol K−1 Re Reynolds number r Chemical reaction rate, mol s−1 kg−1 S Entropy, J K−1 mol−1 (dS/dt) Entropy generation rate, J K−1 s−1 T Temperature, K x Gas mole fraction x State variable vector z Axial coordinate

Greek Symbol ε Porosity µ Dynamic viscosity, kg m−1 s−1 σ Local entropy generate rate, J K−1 m−1 s−1 ξ Conversion rate ρ Density, kg · m−3 λ Covariate variable vector ∆G Gibbs free energy change of reaction, J mol−1 ∆H Enthalpy of reaction, J mol−1

Subscripts c Catalyst particle i Internal of reactor k Component k in Inlet out Outlet p Particle 0, L Axial position of the reactor

Superscripts ref Reference reactor Entropy 2021, 23, 82 22 of 25

Abbreviations CV Calculated value EGM Entropy generation minimization EGR Entropy generation rate EGRM Entropy generation rate minimization FTT Finite-time thermodynamics HI Hydrogen iodide OV Optimal value PR Production rate PRM Production rate maximization RV Reference value S-I Sulfur-iodine

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