{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T21:06:45Z","timestamp":1780520805177,"version":"3.54.1"},"reference-count":68,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,28]],"date-time":"2021-12-28T00:00:00Z","timestamp":1640649600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51976235 and 51606218"],"award-info":[{"award-number":["51976235 and 51606218"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The exothermic reactor for ammonia synthesis is a primary device determining the performance of the energy storage system. The Braun-type ammonia synthesis reactor is used as the exothermic reactor to improve the heat release rate. Due to the entirely different usage scenarios and design objectives, its parameters need to be redesigned and optimized. Based on finite-time thermodynamics, a one-dimensional model is established to analyze the effects of inlet gas molar flow rate, hydrogen\u2013nitrogen ratio, reactor length and inlet temperature on the total entropy generation rate and the total exothermic rate of the reactor. It\u2019s found that the total exothermic rate mainly depends on the inlet molar flow rate. Furthermore, considering the minimum total entropy generation rate and maximum total exothermic rate, the NSGA-II algorithm is applied to optimize seven reactor parameters including the inlet molar flow rate, lengths and temperatures of the three reactors. Lastly, the optimized reactor is obtained from the Pareto front using three fuzzy decision methods and deviation index. Compared with the reference reactor, the total exothermic rate of the optimized reactor is improved by 12.6% while the total entropy generation rate is reduced by 3.4%. The results in this paper can provide some guidance for the optimal design and application of exothermic reactors in practical engineering.<\/jats:p>","DOI":"10.3390\/e24010052","type":"journal-article","created":{"date-parts":[[2021,12,28]],"date-time":"2021-12-28T06:55:03Z","timestamp":1640674503000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Multi-Objective Optimization of Braun-Type Exothermic Reactor for Ammonia Synthesis"],"prefix":"10.3390","volume":"24","author":[{"given":"Tianchao","family":"Xie","sequence":"first","affiliation":[{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0950-3686","authenticated-orcid":false,"given":"Shaojun","family":"Xia","sequence":"additional","affiliation":[{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chao","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Power Engineering, Naval University of Engineering, Wuhan 430033, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,28]]},"reference":[{"key":"ref_1","first-page":"271","article-title":"High temperature solar tests with 1kWchem ammonia reactor","volume":"36","author":"Luzzi","year":"1995","journal-title":"Fuel Energy Abstr."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0038-092X(98)00108-X","article-title":"Techno-economic analysis of a 10 MW solar thermal power plant using ammonia-based thermochemical energy storage","volume":"66","author":"Luzzi","year":"1999","journal-title":"Sol. Energy"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/0038-092X(96)00291-5","article-title":"Endothermic reactors for an ammonia based thermochemical solar energy storage and transport system","volume":"56","author":"Lovegrove","year":"1996","journal-title":"Sol. Energy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/j.solener.2003.07.020","article-title":"Developing ammonia based thermochemical energy storage for dish power plants","volume":"76","author":"Lovegrove","year":"2004","journal-title":"Sol. Energy"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1115\/1.1352737","article-title":"Maximizing thermal power output of an ammonia synthesis reactor for a solar thermochemical energy storage system","volume":"123","author":"Kreetz","year":"2001","journal-title":"J. Energy Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/S0038-092X(02)00024-5","article-title":"Exergy analysis of an ammonia synthesis reactor in a solar thermochemical power system","volume":"73","author":"Kreetz","year":"2002","journal-title":"Sol. Energy"},{"key":"ref_7","first-page":"21","article-title":"Numerical simulation of ammonia-based exothermic reactor used in solar thermal power generation","volume":"36","author":"Long","year":"2008","journal-title":"J. South China Univ. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1016\/j.solener.2017.11.064","article-title":"Design and optimization of an ammonia synthesis system for ammonia-based solar thermochemical energy storage","volume":"159","author":"Chen","year":"2018","journal-title":"Sol. Energy"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.solener.2017.06.049","article-title":"Modeling of ammonia synthesis to produce supercritical steam for solar thermochemical energy storage","volume":"155","author":"Chen","year":"2017","journal-title":"Sol. Energy"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40807-016-0034-4","article-title":"A mathematical model for ammonia solar and synthesis reactors","volume":"3","author":"Abdiwe","year":"2016","journal-title":"Renew. Wind. Water Sol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1016\/j.energy.2017.06.157","article-title":"Modeling and optimization of an industrial ammonia synthesis unit: An exergy approach","volume":"137","year":"2017","journal-title":"Energy"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.energy.2016.05.096","article-title":"On the efficiency, exergy costs and CO2 emission cost allocation for an integrated syngas and ammonia production plant","volume":"117","year":"2016","journal-title":"Energy"},{"key":"ref_13","first-page":"6","article-title":"Optimal reactor length of an auto-thermal ammonia synthesis reactor","volume":"10","author":"Ksasy","year":"2010","journal-title":"Int. J. Elec. Comput. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1016\/j.compchemeng.2004.11.010","article-title":"Optimal design of an auto-thermal ammonia synthesis reactor","volume":"29","author":"Babu","year":"2005","journal-title":"Comput. Chem. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1360\/N092018-00220","article-title":"Processes in generalized thermodynamic dynamic-optimization of irreversible processes","volume":"49","author":"Chen","year":"2019","journal-title":"Sci. Sin. Tech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1360\/N092018-00220","article-title":"Processes in generalized thermodynamic dynamic-optimization of irreversible cycles","volume":"49","author":"Chen","year":"2019","journal-title":"Sci. Sin. Tech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1115\/1.3450705","article-title":"The concept of irreversibility in heat exchanger design: Counterflow heat exchangers for gas-to-gas applications","volume":"99","author":"Bejan","year":"1977","journal-title":"J. Heat Trans."},{"key":"ref_18","first-page":"239","article-title":"Notes on the history of the method of entropy generation minimization (finite time thermodynamics)","volume":"21","author":"Bejan","year":"1996","journal-title":"J. Non-Equilib. Thermody."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chen, L.G., Feng, H.J., and Ge, Y.L. (2020). Power and efficiency optimization for open combined regenerative Brayton and inverse Brayton cycles with regeneration before the inverse cycle. Entropy, 22.","DOI":"10.3390\/e22060677"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Chen, L.G., Ge, Y.L., Liu, C., Feng, H.J., and Lorenzini, G. (2020). Performance of universal reciprocating heat-engine cycle with variable specific heats ratio of working fluid. Entropy, 22.","DOI":"10.3390\/e22040397"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Goupil, C., and Herbert, E. (2019). Adapted or adaptable: How to manage entropy production?. Entropy, 22.","DOI":"10.20944\/preprints201912.0272.v1"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tsirlin, A., and Gagarina, L. (2020). Finite-time thermodynamics in economics. Entropy, 22.","DOI":"10.3390\/e22080891"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yasunaga, T., and Ikegami, Y. (2020). Finite-time thermodynamic model for evaluating heat engines in ocean thermal energy conversion. Entropy, 22.","DOI":"10.3390\/e22020211"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Rogolino, P., and Cimmelli, V.A. (2020). Thermoelectric efficiency of Silicon\u2013Germanium alloys in finite-time thermodynamics. Entropy, 22.","DOI":"10.3390\/e22101116"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2425","DOI":"10.1016\/j.energy.2004.03.034","article-title":"Distribution of heat exchange in optimum diabatic distillation columns","volume":"29","author":"Koeijer","year":"2004","journal-title":"Energy"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1016\/S0009-2509(02)00629-2","article-title":"Transport equations for distillation of ethanol and water from the entropy production rate","volume":"58","author":"Kjelstrup","year":"2003","journal-title":"Chem. Eng. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Andresen, B., and Essex, C. (2017). Eigentimes and very slow processes. Entropy, 19.","DOI":"10.3390\/e19090492"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Andresen, B., and Essex, C. (2020). Thermodynamics at very long time and space scales. Entropy, 22.","DOI":"10.3390\/e22101090"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6304","DOI":"10.3390\/e17096304","article-title":"Metrics and energy landscapes in irreversible thermodynamics","volume":"17","author":"Andresen","year":"2015","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1044","DOI":"10.1021\/ie020319n","article-title":"Minimizing the entropy production rate of an exothermic reactor with a constant heat-transfer coefficient:\u2009 The ammonia reaction","volume":"42","author":"Nummedal","year":"2003","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1021\/i200032a010","article-title":"Optimal temperature profile for an ammonia reactor","volume":"25","author":"Maansson","year":"1986","journal-title":"Ind. Eng. Chem. Process Des. Dev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5360","DOI":"10.1002\/er.5286","article-title":"Optimal design and operation of ammonia decomposition reactors","volume":"44","author":"Badescu","year":"2020","journal-title":"Int. J. Energy Res."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1016\/j.energy.2003.11.002","article-title":"The second law optimal path of a four-bed SO2 converter with five heat exchangers","volume":"29","author":"Johannessen","year":"2004","journal-title":"Energy"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Kong, R., Chen, L., Xia, S., Li, P., and Ge, Y. (2021). Minimization of entropy generation rate in hydrogen iodide decomposition reactor heated by high-temperature helium. Entropy, 23.","DOI":"10.3390\/e23010082"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"114436","DOI":"10.1016\/j.enconman.2021.114436","article-title":"Performance analysis of hydrogen iodide decomposition membrane reactor under different sweep modes","volume":"244","author":"Kong","year":"2021","journal-title":"Energy Convers. Manag."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/j.egyr.2020.03.011","article-title":"Entropy generation rate minimization for steam methane reforming reactor heated by molten salt","volume":"6","author":"Li","year":"2020","journal-title":"Energy Reps."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Li, P., Chen, L., Xia, S., and Zhang, L. (2019). Entropy generation rate minimization for methanol synthesis via a CO2 hydrogenation reactor. Entropy, 21.","DOI":"10.3390\/e21020174"},{"key":"ref_38","unstructured":"Elias, I.F. (2014). Thermodynamics with Chemical Engineering Applications, Cambridge University Press."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2073","DOI":"10.1051\/e3sconf\/202126702073","article-title":"Multi-objective performance optimization of ammonia decomposition thermal storage reactor","volume":"267","author":"Xie","year":"2021","journal-title":"E3S Web Conf."},{"key":"ref_40","first-page":"9","article-title":"Maximum hydrogen production rate optimization for tubular steam methane reforming reactor","volume":"17","author":"Li","year":"2019","journal-title":"Int. J. Chem. React. Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1021\/i160035a032","article-title":"Pressure drop in packed beds of spheres","volume":"9","author":"Hicks","year":"1970","journal-title":"Ind. Eng. Chem. Fundam."},{"key":"ref_42","unstructured":"Groot, S.R., and Mazur, P. (1984). Non-Equilibrium Thermodynamics, Dover."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., Bedeaux, D., Johannessen, E., and Gross, G. (2010). Non-Equilibrium Thermodynamics for Engineers, World Scientific.","DOI":"10.1142\/7869"},{"key":"ref_44","unstructured":"Kubie, J., Tariq, M., and Grassie, T. (2012). Heat Transfer: A Problem-Solving Approach, Routledge."},{"key":"ref_45","unstructured":"Robert, A.G. (2012). Fluid Mechanics, Dover Publications."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE T Evolut. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1109\/MCI.2017.2742868","article-title":"PlatEMO: A MATLAB platform for evolutionary multi-objective optimization [Educational Forum]","volume":"12","author":"Tian","year":"2017","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Zhang, R.Z. (2018). A study of industrial structure optimization under economy, employment and environment constraints based on MOEA. International Conference on Bio-Inspired Computing: Theories and Applications, Springer.","DOI":"10.1007\/978-981-13-2826-8_11"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1063\/1.4757266","article-title":"Bayesian uncertainty quantification and propagation in molecular dynamics simulations: A high performance computing framework","volume":"137","author":"Angelikopoulos","year":"2012","journal-title":"J. Chem. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"094104","DOI":"10.1103\/PhysRevB.89.094104","article-title":"Combinatorial screening for new materials in unconstrained composition space with machine learning","volume":"89","author":"Meredig","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"110200","DOI":"10.1016\/j.nucengdes.2019.110200","article-title":"Validation and uncertainty quantification of multiphase-CFD solvers: A data-driven Bayesian framework supported by high-resolution experiments","volume":"354","author":"Liu","year":"2019","journal-title":"Nucl. Eng. Des."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"107636","DOI":"10.1016\/j.ress.2021.107636","article-title":"Uncertainty quantification for Multiphase-CFD simulations of bubbly flows: A machine learning-based Bayesian approach supported by high-resolution experiments","volume":"212","author":"Liu","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1080\/00295639.2018.1512790","article-title":"Validation and uncertainty quantification for wall boiling closure relations in multiphase-CFD solver","volume":"93","author":"Liu","year":"2019","journal-title":"Nucl. Sci. Eng."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.applthermaleng.2018.08.041","article-title":"Data-driven modeling for boiling heat transfer: Using deep neural networks and high-fidelity simulation results","volume":"144","author":"Liu","year":"2018","journal-title":"Appl. Therm. Eng."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1016\/j.enconman.2013.06.031","article-title":"A novel machine learning approach for estimation of electricity demand: An empirical evidence from Thailand","volume":"74","author":"Mostafavi","year":"2013","journal-title":"Energy Convers. Manag."},{"key":"ref_56","first-page":"163","article-title":"How to represent crystal structures for machine learning: Towards fast prediction of electronic properties","volume":"89","author":"Glawe","year":"2013","journal-title":"Phys. Rev. B"},{"key":"ref_57","first-page":"2460","article-title":"Bayesian bounds for parameter estimation and nonlinear filtering\/tracking","volume":"123","author":"Trees","year":"2008","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"96405","DOI":"10.1103\/PhysRevLett.114.096405","article-title":"Molecular Dynamics with On-the-Fly Machine Learning of Quantum-Mechanical Forces","volume":"114","author":"Li","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","first-page":"29","article-title":"Introduction to the structure of a Braun Process ammonia synthesis tower","volume":"4","author":"Zheng","year":"1991","journal-title":"Chem. Equip. Des."},{"key":"ref_60","unstructured":"Chen, Q.Q. (2011). A Simulation of the Ammonia Synthesis Reactor and Process, East China University of Science and Technology."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0165-0114(97)00377-1","article-title":"Extensions of the TOPSIS for group decision-making under fuzzy environment","volume":"114","author":"Chen","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0377-2217(03)00020-1","article-title":"Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS","volume":"156","author":"Opricovic","year":"2004","journal-title":"Eur. J. Oper. Res."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1016\/j.mcm.2006.03.023","article-title":"An extension of TOPSIS for group decision making","volume":"45","author":"Shih","year":"2007","journal-title":"Math. Comput. Model."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1016\/j.jcss.2005.11.001","article-title":"Fuzzy LINMAP method for multiattribute decision making under fuzzy environments","volume":"72","author":"Xia","year":"2006","journal-title":"J. Comput. Ssyt. Sci."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.knosys.2013.02.012","article-title":"An interval-valued intuitionistic fuzzy LINMAP method with inclusion comparison possibilities and hybrid averaging operations for multiple criteria group decision making","volume":"45","author":"Chen","year":"2013","journal-title":"Knowl-Based Syst."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1016\/j.mcm.2005.09.012","article-title":"Application of shannon\u2019s entropy to classify emergent behaviors in a simulation of laser dynamics","volume":"42","author":"Guisado","year":"2005","journal-title":"Math. Comput. Model."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"53","DOI":"10.3390\/e12010053","article-title":"Imprecise Shannon\u2019s Entropy and Multi Attribute Decision Making","volume":"12","author":"Lotfi","year":"2010","journal-title":"Entropy"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/j.ecolind.2018.02.015","article-title":"Integrated flood vulnerability assessment approach based on TOPSIS and Shannon entropy methods","volume":"89","author":"Yang","year":"2018","journal-title":"Ecol. Indic."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/1\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:54:37Z","timestamp":1760169277000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/1\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,28]]},"references-count":68,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["e24010052"],"URL":"https:\/\/doi.org\/10.3390\/e24010052","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,28]]}}}