Achievements and future prospects in the scientific field of mathematical modeling of the catalytic reforming of gasoline
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DOI:
https://doi.org/10.32523/2616-6771-2025-151-2-56-86Keywords:
catalytic reforming, mathematical model, bifunctional catalyst, reactor, deactivation, coke formation, organochlorine compounds, optimisation, raw material flowsAbstract
Catalytic reforming of gasoline is a pivotal industrial process among those aimed at enhancing the octane rating of motor fuels - products that are projected to maintain high demand in the foreseeable future. Extensive research has been devoted to improving the design of reforming processes, reactor configurations, and catalyst formulations. Given the complexity of catalytic reforming - which involves a multicomponent feedstock and product mixture, a bifunctional catalyst system, numerous parallel and sequential reactions, and catalyst deactivation mechanisms - mathematical modeling remains the principal tool for process investigation and optimization. This paper provides a concise review of the evolution of kinetic modeling approaches for catalytic reforming, beginning with foundational work from 1959. Particular attention is given to the research conducted at the National Research Tomsk Polytechnic University, where a well-established scientific school has focused on developing mathematical models for petrochemical and refining processes. Using the catalytic reforming of gasoline as a case study, the paper outlines a methodological approach for constructing non-stationary models and describes the key principles underlying their development. The modeling results presented demonstrate the potential for optimizing reactor design, maintaining the balance between the catalyst’s acidic and metallic functions, and minimizing coke formation on the catalyst surface. The results of improvement of the mathematical model of catalytic reforming with a stationary catalyst bed taking into account the involvement of additional feed streams are presented.
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References
Abramin, A.L. (2010). Improvement of industrial gasoline reforming processes with a moving-bed catalyst using mathematical modeling [Sovershenstvovanie promyshlennykh protsessov riforminga benzinov s dvizhushhimsja sloem katalizatora metodom matematicheskogo modelirovanija in Russian]. Tomsk Polytechnic University, Tomsk. https://earchive.tpu.ru/handle/11683/6546
Ali, S.A., Siddiqui, M.A., Ali, M.A. (2005). Parametric study of catalytic reforming process. Reaction Kinetics and Catalysis Letters 87(1), 199–206. https://doi.org/10.1007/s11144-006-0001-y
Ancheyta-Juárez, J., Villafuerte-Macías, E. (2000). Kinetic modeling of naphtha catalytic reforming reactions. Energy and Fuels 14(5), 1032–1037. https://doi.org/10.1021/ef0000274
Ancheyta, J. (2011). Modeling and Simulation of Catalytic Reactors for Petroleum Refining. Willey & Sons. https://doi.org/10.1002/9780470933565
Antos, G.J., Aitani, A.M. (Eds.). (2004). Catalytic Naphtha Reforming, Revised and Expanded. CRC Press. https://doi.org/10.1201/9780203913505
Arab Aboosadi, Z., Jahanmiri, A.H., Rahimpour, M.R. (2011). Optimization of tri-reformer reactor to produce synthesis gas for methanol production using differential evolution (DE) method. Applied Energy 88(8), 2691–2701. https://doi.org/10.1016/j.apenergy.2011.02.017
Arani, H.M., Shirvani, M., Safdarian, K., Dorostkar, E. (2009). Lumping procedure for a kinetic model of catalytic naphtha reforming. Brazilian Journal of Chemical Engineering 26(4), 723–732. https://doi.org/10.1590/S0104-66322009000400011
Barbier, J., Corro, G., Zhang, Y., Bournonville, J.P., Franck, J. P. (1985). Coke formation on platinum-alumina catalyst of wide varying dispersion. Applied Catalysis 13(2), 245–255. https://doi.org/10.1016/S0166-9834(00)81143-4
Barbier, J., Marecot, P., Martin, N., Elassal, L., Maurel, R. (1980). Selective poisoning by coke formation on pt/Al2O3. Studies in Surface Science and Catalysis 6(C), 53–62. https://doi.org/10.1016/S0167-2991(08)65218-0
Bartholomew, C.H. (2001). Mechanisms of catalyst deactivation. Applied Catalysis A: General 212(1–2), 17–60. https://doi.org/10.1016/S0926-860X(00)00843-7
Belyi, A.S. (2005). Reforming catalysts of the PR family: Scientific foundations and technological advancement. Kinetics and Catalysis 46(5), 684–692. https://doi.org/10.1007/s10975-005-0123-7
Benitez, V., Boutzeloit, M., Mazzieri, V.A., Especel, C., Epron, F., Vera, C.R., Marécot, P., Pieck, C.L. (2007). Preparation of trimetallic Pt-Re-Ge/Al2O3 and Pt-Ir-Ge/Al2O3 naphtha reforming catalysts by surface redox reaction. Applied Catalysis A: General 319, 210–217. https://doi.org/10.1016/j.apcata.2006.12.006
Benitez, V.M., Pieck, C.L. (2010). Influence of indium content on the properties of Pt-Re/Al2O3 naphtha reforming catalysts. Catalysis Letters 136(1–2), 45–51. https://doi.org/10.1007/s10562-009-0202-x
Bishara, A., Stanislaus, A., Hussain, S. (1984). Effect of feed composition and operating conditions on catalyst deactivation and on product yield and quality during naphtha catalytic reforming. Applied Catalysis 13(1), 113–125. https://doi.org/10.1016/S0166-9834(00)83332-1
Chen, Z., Yan, Y., Elnashaie, S.S.E.H. (2004). Catalyst deactivation and engineering control for steam reforming of higher hydrocarbons in a novel membrane reformer. Chemical Engineering Science 59(10), 1965–1978. https://doi.org/10.1016/j.ces.2004.01.046
Choudhary, V.R., Mulla, S.A.R., Rane, V.H. (2000). Coupling of exothermic and endothermic reactions in oxidative conversion of ethane to ethylene over alkaline earth promoted La2O3 catalysts in presence of limited O2. Applied Energy 66(1), 51–62. https://doi.org/10.1016/S0306-2619(99)00040-9
Ciapetta, F.G., Wallace, D.N. (1972). Catalytic naphtha reforming. Catalysis Reviews 5(1), 67–158. https://doi.org/10.1080/01614947208076866
Delmon, B., Yates, J.T. (1987). Studies in surface science and catalysis. Studies in Surface Science and Catalysis 31(C), XI–XII. https://doi.org/10.1016/S0167-2991(08)65388-4
Demirbas, A. (2011). Competitive liquid biofuels from biomass. In Applied Energy 88(1), 17–28. https://doi.org/10.1016/j.apenergy.2010.07.016
Ding, M., Hayakawa, T., Zeng, C., Jin, Y., Zhang, Q., Wang, T., Ma, L., Yoneyama, Y., Tsubaki, N. (2013). Direct conversion of liquid natural gas (LNG) to syngas and ethylene using non-equilibrium pulsed discharge. Applied Energy 104, 777–782. https://doi.org/10.1016/j.apenergy.2012.12.017
Dyusembaeva, A.A., Vershinin, V.I. (2019). Modeling of Catalytic Reforming: Effect of Kinetic Parameters on the Expected Composition of Products. Kinetics and Catalysis 60(1), 106–111. https://doi.org/10.1134/S002315841901004X
Elizalde, I., Ancheyta, J. (2015). Dynamic modeling and simulation of a naphtha catalytic reforming reactor. Applied Mathematical Modelling 39(2), 764–775. https://doi.org/10.1016/j.apm.2014.07.013
Figoli, N.S., Beltramini, J.N., Barra, A.F., Martinelli, E.E., Sad, M.R., Parera, J.M. (1982). Influence of total pressure and hydrogen: hydrocarbon ratio on coke formation over naphtha-reforming catalyst. ACS Symposium Series 239–252. https://doi.org/10.1021/bk-1983-0202.ch012
Galushin, S.A. (2004). Modeling of transient processes on the surface of platinum-containing catalysts in industrial gasoline reforming reactors [Modelirovanie nestatsionarnykh protsessov na poverkhnosti platinosoderzhashhikh katalizatorov v promyshlennykh reaktorakh ustanovok riforminga benzinov in Russian]. Tomsk Polytechnic University, Tomsk. http://earchive.tpu.ru/handle/11683/6149
García-Dopico, M., García, A., Santos García, A. (2006). Modelling coke formation and deactivation in a FCCU. Applied Catalysis A: General 303(2), 245–250. https://doi.org/10.1016/j.apcata.2006.02.026
Gyngazova, M.S. (2011). Modeling the operation of gasoline reforming reactors with continuous catalyst regeneration, taking coke formation into account [Modelirovanie raboty reaktorov protsessa riforminga benzinov s nepreryvnoy regeneratsiej katalizatora s uchetom koksoobrazovanija in Russian]. Tomsk Polytechnic University, Tomsk. http://earchive.tpu.ru/handle/11683/6682
Gyngazova, M.S., Kravtsov, A.V., Ivanchina, E.D., Korolenko, M.V., Chekantsev, N.V. (2011). Reactor modeling and simulation of moving-bed catalytic reforming process. Chemical Engineering Journal 176–177, 134–143. https://doi.org/10.1016/j.cej.2011.09.128
Gyngazova, M.S., Kravtsov, A.V., Ivanchina, E.D., Korolenko, M.V., Uvarkina, D.D. (2010). Kinetic model of the catalytic reforming of gasolines in moving-bed reactors. Catalysis in Industry 2(4), 374–380. https://doi.org/10.1134/S2070050410040124
Hamied, R., Sukkar, K.A., Raouf, S. (2022). Modeling, Kinetic and Experimental optimization of Reforming Unit for Al- Doura Heavy Naphtha over bi and Tri-metallic Catalysts. Iraqi Journal of Oil and Gas Research (IJOGR) 2(1), 108–121. https://doi.org/10.55699/ijogr.2022.0201.1020
Hongjun, Z., Mingliang, S., Huixin, W., Zeji, L., Hongbo, J. (2010). Modeling and simulation of moving bed reactor for catalytic naphtha reforming. Petroleum Science and Technology 28(7), 667–676. https://doi.org/10.1080/10916460902804598
Hou, W., Su, H., Hu, Y., Chu, J. (2006). Lumped kinetics model and its on-line application to commercial catalytic naphtha reforming process. CIESC Journal 57(7), 1605.
Hu, Y., Xu, W., Su, H., Chu, J. (2004). A dynamic model for naphtha catalytic reformers. Proceedings of the IEEE International Conference on Control Applications 1, 159–164. https://doi.org/10.1109/cca.2004.1387204
Hui, L., Yunchang, D., Jiongliang, Y., Zeyun, W. (1995). Study of the electrochemical performance of nickel hydroxide. Journal of Power Sources 57(1–2), 137–140. https://doi.org/10.1016/0378-7753(95)02266-X
Iranshahi, D., Pourazadi, E., Paymooni, K., Rahimpour, M.R. (2012). A novel dynamic membrane reactor concept with radial-flow pattern for reacting material and axial-flow pattern for sweeping gas in catalytic naphtha reformers. AIChE Journal 58(4), 1230–1247. https://doi.org/10.1002/aic.12664
Ivanchina, E.D. (2002). Improving industrial hydrocarbon feedstock processing technology using platinum catalysts based on an unsteady-state model [Sovershenstvovanie promyshlennoj tekhnologii pererabotki uglevodorodnogo syr’ya s ispol’zovaniem platinovykh katalizatorov na osnove nestatsionarnoy modeli in Russian]. Doctoral dissertation, Tomsk Polytechnic University, Tomsk. https://dissercat.com/content/sovershenstvovanie-promyshlennoi-tekhnologii-pererabotki-uglevodorodnogo-syrya-s-ispolzovani
Khobragade, M., Majhi, S., Pant, K.K. (2012). Effect of K and CeO2 promoters on the activity of Co/SiO2 catalyst for liquid fuel production from syngas. Applied Energy 94, 385–394. https://doi.org/10.1016/j.apenergy.2012.02.002
Kmak, W.S., Stuckey, A.N. (1973). Powerforming process studies with a kinetic simulation model. AIChE National Meeting 56a.
Koksharov, A.G. (2023). Increasing the efficiency of gasoline reforming technology by reducing coke formation intensity using a mathematical model [Povyshenie effektivnosti tekhnologii riforminga benzinov putem snizheniya intensivnosti protsessa koksoobrazovaniya s ispol’zovaniem matematicheskoy modeli in Russian]. Tomsk Polytechnic University, Tomsk. http://earchive.tpu.ru/handle/11683/74989
Koksharov, A.G., Ivanchina, E.D., Faleev, S.A., Fedyushkin, A.I. (2015). The way of increasing resource efficiency of naphtha reforming under conditions of catalyst acid and metal activity balance by mathematical modeling method. Procedia Engineering 113, 1–7. https://doi.org/10.1016/j.proeng.2015.07.276
Konstantinovich Z.I. (2016). Optimization of catalyst regeneration processes for reforming, dehydrogenation, and hydrotreating in circulation loop units [Optimizatsiya protsessov regeneratsii katalizatorov riforminga, degidrirovaniya, gidroochistki v apparatakh tsirkulyatsionnykh konturov in Russian]. https://dissercat.com/content/optimizatsiya-protsessov-regeneratsii-katalizatorovriforminga-degidrirovaniya-gidroochistki
Kostenko, A.V. (2006). Improvement of the design and efficiency of the reactor block in the catalytic reforming process of hydrocarbon feedstock [Sovershenstvovanie konstruktsii i povyshenie effektivnosti raboty reaktornogo bloka protsessa kataliticheskogo riforminga uglevodorodnogo syr’ya in Russian]. Tomsk Polytechnic University, Tomsk. https://dissercat.com/content/sovershenstvovanie-konstruktsii-i-povyshenie-effektivnosti-raboty-reaktornogo-bloka-protsess
Krane, H.G. (1959). Reactions in catalytic reforming of naphthas. Proceeding of the V World Petroleum Congress. https://hero.epa.gov/hero/index.cfm/reference/details/reference_id/8435336
Lid, T., Skogestad, S. (2008). Data reconciliation and optimal operation of a catalytic naphtha reformer. Modeling, Identification and Control 29(4), 117–129. https://doi.org/10.4173/mic.2008.4.1
Marin, G.B., Froment, G.F., Lerou, J.J., De Backer, W. (1983). Simulation of a catalytic naphtha reforming unit. EFCE Publication Series (European Federation of Chemical Engineering) 27.
Mazzieri, V.A., Pieck, C.L., Vera, C.R., Yori, J.C., Grau, J.M. (2009). Effect of Ge content on the metal and acid properties of Pt-Re-Ge/Al2O3-Cl catalysts for naphtha reforming. Applied Catalysis A: General 353(1), 93–100. https://doi.org/10.1016/j.apcata.2008.10.024
Namioka, T., Saito, A., Inoue, Y., Park, Y., Min, T.J., Roh, S.A., Yoshikawa, K. (2011). Hydrogen-rich gas production from waste plastics by pyrolysis and low-temperature steam reforming over a ruthenium catalyst. Applied Energy 88(6), 2019–2026. https://doi.org/10.1016/j.apenergy.2010.12.053
Ostrovsky, N.M., Sokolov, V.P., Aksenova, N.V., Lukyanov, B.N. (1989). Kinetics of gasoline fraction reforming and mathematical model of the process [Kinetika riforminga benzinovykh fraktsii i matematicheskaya model’ protsessa in Russian]. Proceedings of Khimreaktor 10, Tolyatti.
Padmavathi, G., Chaudhuri, K.K. (1997). Modelling and Simulation of Commercial Catalytic Naphtha Reformers. Canadian Journal of Chemical Engineering 75(5), 930–937. https://doi.org/10.1002/cjce.5450750513
Pchelintseva, I.V. (2019). Regularities of the catalytic conversion of hydrocarbons in the gasoline reforming process under reduced pressure [Zakonomernosti kataliticheskogo prevrashhenija uglevodorodov v protsesse riforminga benzinov pri snizhenii davlenija in Russian]. Tomsk Polytechnic University, Tomsk. http://earchive.tpu.ru/handle/11683/56166
Pereira, C.S.M., Silva, V.M.T.M., Pinho, S.P., Rodrigues, A.E. (2010). Batch and continuous studies for ethyl lactate synthesis in a pervaporation membrane reactor. Journal of Membrane Science 361(1–2), 43–55. https://doi.org/10.1016/j.memsci.2010.06.014
Petrova, D.A., Gushchin, P.A., Ivanov, E.V., Lyubimenko, V.A., Kolesnikov, I.M. (2021). Modelling Industrial Catalytic Reforming of Lowoctane Gasoline. Chemistry and Technology of Fuels and Oils 57(1), 143–159. https://doi.org/10.1007/s10553-021-01234-x
Poluboyartsev, D.S. (2007). Selection and evaluation of the efficiency of Pt catalysts for the gasoline reforming process using a modeling system [Vybor i otsenka effektivnosti Pt-katalizatorov protsessa riforminga benzinov s primeneniem modeliruyushchey sistemy in Russian]. Author’s abstract of Cand. Tech. Sci. dissertation. Tomsk Polytechnic University, Tomsk. https://dissercat.com/content/vybor-i-otsenka-effektivnosti-pt-katalizatorov-protsessa-riforminga-benzinov-s-primeneniem-m
Pujadó, P.R., Rabó, J.A., Antos, G.J., Gembicki, S.A. (1992). Industrial catalytic applications of molecular sieves. Catalysis Today 13(1), 113–141. https://doi.org/10.1016/0920-5861(92)80191-O
Rahimpour, M.R., Bahmanpour, A.M. (2011). Optimization of hydrogen production via coupling of the Fischer-Tropsch synthesis reaction and dehydrogenation of cyclohexane in GTL technology. Applied Energy 88(6), 2027–2036. https://doi.org/10.1016/j.apenergy.2010.12.065
Rahimpour, M.R., Esmaili, S., Bagheri Ghalehghazi, N. (2003). A kinetic and deactivation model for industrial catalytic naphtha reforming. Iranian Journal of Science and Technology, Transaction B: Technology 27(2), 279–290.
Rahimpour, M.R., Jafari, M., Iranshahi, D. (2013). Progress in catalytic naphtha reforming process: A review. In Applied Energy 109, 79–93. https://doi.org/10.1016/j.apenergy.2013.03.080
Rahimpour, M.R., Rahmani, F., Bayat, M. (2010). Contribution to emission reduction of CO2 by a fluidized-bed membrane dual-type reactor in methanol synthesis process. Chemical Engineering and Processing: Process Intensification 49(6), 589–598. https://doi.org/10.1016/j.cep.2010.05.004
Ramage, M.P., Graziani, K.R., Krambeck, F.J. (1980). Development of mobil’s kinetic reforming model. Chemical Engineering Science 35(1–2), 41–48. https://doi.org/10.1016/0009-2509(80)80068-6
Ren, X.H., Bertmer, M., Stapf, S., Demco, D.E., Blümich, B., Kern, C., Jess, A. (2002). Deactivation and regeneration of a naphtha reforming catalyst. Applied Catalysis A: General 228(1–2), 39–52. https://doi.org/10.1016/S0926-860X(01)00958-9
Reutova, O.A., Iriskina, O.V. (2000). Model’ reaktora riforminga. I. kineticheskaya model’ dlya polifunktsional’nogo katalizatora [Reforming reactor model. I. Kinetic model for a multifunctional catalyst]. Matematicheskie Struktury i Modelirovanie [Mathematical structures and modeling] 1(5). https://cyberleninka.ru/article/n/model-reaktora-riforminga-i-kineticheskaya-model-dlya-polifunktsionalnogo-katalizatora
Roddy, D.J. (2012). Development of a CO2 network for industrial emissions. Applied Energy 91(1), 459–465. https://doi.org/10.1016/j.apenergy.2011.10.016
Rodríguez, M.A., Ancheyta, J. (2011). Detailed description of kinetic and reactor modeling for naphtha catalytic reforming. Fuel 90(12), 3492–3508. https://doi.org/10.1016/j.fuel.2011.05.022
Sharova, E.S. (2010). Improving the efficiency of the reactor unit of the gasoline reforming process with a fixed granular layer katalizatora [Povyshenie effektivnosti raboty reaktornogo uzla protsessa riforminga benzinov s nepodvizhnym zernistym sloem katalizatora in Russian]. https://dissercat.com/content/povyshenie-effektivnosti-raboty-reaktornogo-uzla-protsessa-riforminga-benzinov-s-nepodvizhny
Sharova, E.S. (2010). Improving the efficiency of the reactor unit in the gasoline reforming process with a fixed granular catalyst bed [Povyshenie effektivnosti raboty reaktornogo uzla protsessa riforminga benzinov s nepodvizhnym zernistym sloem katalizatora in Russian]. Tomsk Polytechnic University, Tomsk. https://dissercat.com/content/povyshenie-effektivnosti-raboty-reaktornogo-uzla-protsessa-riforminga-benzinov-s-nepodvizhny
Smith, R. (1959). Kinetic analysis of naphtha reforming with platinum catalyst. Chem Eng Prog 76–80.
Stijepovic, M.Z., Vojvodic-Ostojic, A., Milenkovic, I., Linke, P. (2009). Development of a kinetic model for catalytic reforming of naphtha and parameter estimation using industrial plant data. Energy and Fuels 23(2), 979–983. https://doi.org/10.1021/ef800771x
Taghvaei, H., Shirazi, M.M., Hooshmand, N., Rahimpour, M.R., Jahanmiri, A. (2012). Experimental investigation of hydrogen production through heavy naphtha cracking in pulsed DBD reactor. Applied Energy 98, 3–10. https://doi.org/10.1016/j.apenergy.2012.02.005
Taskar, U., Riggs, J.B. (1997). Modeling and Optimization of a Semiregenerative Catalytic Naphtha Reformer. AIChE Journal 43(3), 740–753. https://doi.org/10.1002/aic.690430319
Teixeira, M., Madeira, L.M., Sousa, J.M., Mendes, A. (2010). Modeling of a catalytic membrane reactor for CO removal from hydrogen streams - A theoretical study. International Journal of Hydrogen Energy 35(20), 11505–11513. https://doi.org/10.1016/j.ijhydene.2010.04.101
Vagizovna P.I. (2019). Regularities of hydrocarbon catalytic transformation in gasoline reforming at reduced pressure [Zakonomernosti kataliticheskogo prevrashcheniya uglevodorodov v protsesse riforminga benzinov pri snizhenii davleniya in Russian]. https://dissercat.com/content/zakonomernosti-kataliticheskogo-prevrashcheniya-uglevodorodov-v-protsesse-riforminga-benzino
Wei, J., Prater, C.D. (1962). The Structure and Analysis of Complex Reaction Systems. Advances in Catalysis 13(C), 203–392. https://doi.org/10.1016/S0360-0564(08)60289-8
Yusuf, A.Z., Aderemi, B.O., Patel, R., Mujtaba, I.M. (2019). Study of industrial naphtha catalytic reforming reactions via modelling and simulation. Processes 7(4). https://doi.org/10.3390/pr7040192
Yusuf, A.Z., John, Y.M., Aderemi, B.O., Patel, R., Mujtaba, I.M. (2019). Modelling, simulation and sensitivity analysis of naphtha catalytic reforming reactions. Computers and Chemical Engineering 130. https://doi.org/10.1016/j.compchemeng.2019.106531
Zagoruiko, A.N., Belyi, A.S., Smolikov, M.D. (2021). Thermodynamically consistent kinetic model for the naphtha reforming process. Industrial and Engineering Chemistry Research 60(18), 6627–6638. https://doi.org/10.1021/acs.iecr.0c05653
Zagoruiko, A.N., Belyi, A.S., Smolikov, M.D., Noskov, A.S. (2014). Unsteady-state kinetic simulation of naphtha reforming and coke combustion processes in the fixed and moving catalyst beds. Catalysis Today 220–222, 168–177. https://doi.org/10.1016/j.cattod.2013.07.016
Zaynullin, R.Z., Koledina, K.F., Gubaydullin, I.M., Akhmetov, A.F., Koledin, S.N. (2020). Kinetic Model of Catalytic Gasoline Reforming with Consideration for Changes in the Reaction Volume and Thermodynamic Parameters. Kinetics and Catalysis 61(4), 613–622. https://doi.org/10.1134/S002315842004014X
Zhorov, Y.M., Kartashev, Y.N., Panchenkov, G.M., Tatarintseva, G.M. (1980). Mathematical model of platforming under stationary conditions with allowance for isomerization reactions. Chemistry and Technology of Fuels and Oils 16(7), 429–432. https://doi.org/10.1007/BF00726749
Zhu, X., Li, Q., He, Y., Cong, Y., Yang, W. (2010). Oxygen permeation and partial oxidation of methane in dual-phase membrane reactors. Journal of Membrane Science 360(1–2), 454–460. https://doi.org/10.1016/j.memsci.2010.05.044
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