×

Optimal control of the Sabatier process in microchannel reactors. (English) Zbl 1476.80013

Summary: We consider the optimization of a chemical microchannel reactor by means of PDE-constrained optimization techniques, using the example of the Sabatier reaction. To model the chemically reacting flow in the microchannels, we introduce a three- and a one-dimensional model. As these are given by strongly coupled and highly nonlinear systems of partial differential equations (PDEs), we present our software package cashocs which implements the adjoint approach and facilitates the numerical solution of the subsequent optimization problems. We solve a parameter identification problem numerically to determine necessary kinetic parameters for the models from experimental data given in the literature. The obtained results show excellent agreement to the measurements. Finally, we present two optimization problems for optimizing the reactor’s product yield. First, we use a tracking-type cost functional to maximize the reactant conversion, keep the flow rate of the reactor fixed, and use its wall temperature as optimization variable. Second, we consider the wall temperature and the inlet gas velocity as optimization variables, use an objective functional for maximizing the flow rate in the reactor, and ensure the quality of the product by means of a state constraint. The results obtained from solving these problems numerically show great potential for improving the design of the microreactor.

MSC:

80A32 Chemically reacting flows
49M05 Numerical methods based on necessary conditions
35Q35 PDEs in connection with fluid mechanics
65K10 Numerical optimization and variational techniques
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Senderens, JB; Sabatier, P., Nouvelles synthèses du méthane, C R Acad Sci, 82, 514-516 (1902)
[2] Carranza, S.; Makel, DB; Blizman, B.; Ward, BJ; El-Genk, MS; Bragg, MJ, Microchannel reactors for ISRU applications, AIP Conf Proc, 746, 1, 1229-1238 (2005)
[3] Hu, J.; Brooks, KP; Holladay, JD; Howe, DT; Simon, TM, Catalyst development for microchannel reactors for Martian in situ propellant production, Catal Today, 125, 1, 103-110 (2007)
[4] Samplatsky D, Grohs K, Edeen M, Crusan J, Burkey R (2012) Development and integration of the flight Sabatier assembly on the ISS. In: 41st International conference on environmental systems
[5] El Sibai, A.; Rihko Struckmann, LK; Sundmacher, K., Model-based optimal Sabatier reactor design for power-to-gas applications, Energy Technol, 5, 6, 911-921 (2017)
[6] Falbo, L.; Martinelli, M.; Visconti, CG; Lietti, L.; Bassano, C.; Deiana, P., Kinetics of \(CO_2\) methanation on a Ru-based catalyst at process conditions relevant for power-to-gas applications, Appl Catal B, 225, 354-363 (2018)
[7] Spazzafumo, G., Cogeneration of power and substitute of natural gas using electrolytic hydrogen, biomass and high temperature fuel cells, Int J Hydrog Energy, 43, 26, 11811-11819 (2018)
[8] Bailera, M.; Lisbona, P.; Llera, E.; Peña, B.; Romeo, LM, Renewable energy sources and power-to-gas aided cogeneration for non-residential buildings, Energy, 181, 226-238 (2019)
[9] Vogt, C.; Monai, M.; Kramer, GJ; Weckhuysen, BM, The renaissance of the Sabatier reaction and its applications on earth and in space, Nat Catal, 2, 3, 188-197 (2019)
[10] Brooks, KP; Hu, J.; Zhu, H.; Kee, RJ, Methanation of carbon dioxide by hydrogen reduction using the Sabatier process in microchannel reactors, Chem Eng Sci, 62, 4, 1161-1170 (2007)
[11] Engelbrecht, N.; Chiuta, S.; Everson, RC; Neomagus, HWJP; Bessarabov, DG, Experimentation and CFD modelling of a microchannel reactor for carbon dioxide methanation, Chem Eng J, 313, 847-857 (2017)
[12] von Schwerin, M.; Deutschmann, O.; Schulz, V., Process optimization of reactives systems by partially reduced SQP methods, Comput Chem Eng, 24, 1, 89-97 (2000)
[13] Logist, F.; Houska, B.; Diehl, M.; Van Impe, JF, Robust multi-objective optimal control of uncertain (bio)chemical processes, Chem Eng Sci, 66, 20, 4670-4682 (2011)
[14] Benner, P.; Seidel-Morgenstern, A.; Zuyev, A., Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions, Appl Math Model, 69, 287-300 (2019) · Zbl 1461.49047
[15] Burger, M.; Pinnau, R., Fast optimal design of semiconductor devices, SIAM J Appl Math, 64, 1, 108-126 (2003) · Zbl 1056.49035
[16] Hinze, M.; Pinnau, R., An optimal control approach to semiconductor design, Math Models Methods Appl Sci, 12, 1, 89-107 (2002) · Zbl 1173.49302
[17] Thömmes, G.; Pinnau, R.; Seaïd, M.; Götz, G.; Klar, A., Numerical methods and optimal control for glass cooling processes, Transp Theory Stat Phys, 31, 4-6, 513-529 (2002) · Zbl 1011.80003
[18] Pinnau, R.; Thömmes, G., Optimal boundary control of glass cooling processes, Math Methods Appl Sci, 27, 11, 1261-1281 (2004) · Zbl 1049.35044
[19] Blauth, S.; Leithäuser, C.; Pinnau, R., Model hierarchy for the shape optimization of a microchannel cooling system, J Appl Math Mech, 101, 4, e202000166 (2021)
[20] Blauth, S.; Leithäuser, C.; Pinnau, R., Shape sensitivity analysis for a microchannel cooling system, J Math Anal Appl, 492, 2, 124476 (2020) · Zbl 1457.76143
[21] Schmidt, S.; Ilic, C.; Schulz, V.; Gauger, NR, Three-dimensional large-scale aerodynamic shape optimization based on shape calculus, AIAA J, 51, 11, 2615-2627 (2013)
[22] Schmidt, S.; Ilic, C.; Schulz, V.; Gauger, NR, Airfoil design for compressible inviscid flow based on shape calculus, Optim Eng, 12, 3, 349-369 (2011) · Zbl 1284.76330
[23] Leithäuser, C.; Pinnau, R.; Feßler, R., Designing polymer spin packs by tailored shape optimization techniques, Optim Eng, 19, 3, 733-764 (2018) · Zbl 1507.49037
[24] Hohmann, R.; Leithäuser, C., Shape optimization of a polymer distributor using an Eulerian residence time model, SIAM J Sci Comput, 41, 4, B625-B648 (2019) · Zbl 1422.49039
[25] Tegrotenhuis W, King D, Brooks KP, Golladay B, Wegeng R (2002) Optimizing microchannel reactors by trading-off equilibrium and reaction kinetics through temperature management. In: 6th International conference on microreaction technology
[26] Na, J.; Kshetrimayum, KS; Lee, U.; Han, C., Multi-objective optimization of microchannel reactor for Fischer-Tropsch synthesis using computational fluid dynamics and genetic algorithm, Chem Eng J, 313, 1521-1534 (2017)
[27] Jeon, SW; Yoon, WJ; Jeong, MW; Kim, Y., Optimization of a counter-flow microchannel reactor using hydrogen assisted catalytic combustion for steam reforming of methane, Int J Hydrog Energy, 39, 12, 6470-6478 (2014)
[28] Jung, I.; Na, J.; Park, S.; Jeon, J.; Mo, YG; Yi, JY; Chung, JT; Han, C., Optimal design of a large scale Fischer-Tropsch microchannel reactor module using a cell-coupling method, Fuel Process Technol, 159, 448-459 (2017)
[29] Engelbrecht N (2017) Carbon dioxide methanation in a catalytic microchannel reactor. Master’s Thesis, North-West University (South Africa), Potchefstroom Campus
[30] Blauth, S., cashocs: a computational, adjoint-based shape optimization and optimal control software, SoftwareX, 13, 100646 (2021)
[31] Chen, J.; Yang, H.; Wang, N.; Ring, Z.; Dabros, T., Mathematical modeling of monolith catalysts and reactors for gas phase reactions, Appl Catal A, 345, 1, 1-11 (2008)
[32] Zeng, D.; Pan, M.; Tang, Y., Qualitative investigation on effects of manifold shape on methanol steam reforming for hydrogen production, Renew Energy, 39, 1, 313-322 (2012)
[33] Poinsot, T.; Veynante, D., Theoretical and numerical combustion (2005), Philadelphia: RT Edwards, Inc., Philadelphia
[34] Kee, RJ; Coltrin, ME; Glarborg, P., Chemically reacting flow: theory and practice (2005), Hoboken: Wiley, Hoboken
[35] Moioli, E.; Gallandat, N.; Züttel, A., Parametric sensitivity in the Sabatier reaction over \(Ru/Al_2 O_3\)—theoretical determination of the minimal requirements for reactor activation, React Chem Eng, 4, 100-111 (2019)
[36] Mutschler, R.; Moioli, E.; Luo, W.; Gallandat, N.; Züttel, A., \(CO_2\) hydrogenation reaction over pristine Fe Co, Ni, Cu and \(Al_2 O_3\) supported Ru: comparison and determination of the activation energies, J Catal, 366, 139-149 (2018)
[37] Baraj, E.; Vagaskỳ, S.; Hlinčík, T.; Ciahotnỳ, K.; Tekáč, V., Reaction mechanisms of carbon dioxide methanation, Chem Pap, 70, 4, 395-403 (2016)
[38] Lunde, PJ; Kester, FL, Rates of methane formation from carbon dioxide and hydrogen over a ruthenium catalyst, J Catal, 30, 3, 423-429 (1973)
[39] Lunde, PJ; Kester, FL, Carbon dioxide methanation on a ruthenium catalyst, Ind Eng Chem Process Des Dev, 13, 1, 27-33 (1974)
[40] Lunde, PJ, Modeling, simulation, and operation of a Sabatier reactor, Ind Eng Chem Process Des Dev, 13, 3, 226-233 (1974)
[41] Pérez, S.; Del Molino, E.; Barrio, VL, Modeling and testing of a Milli-structured reactor for carbon dioxide methanation, Int J Chem React Eng, 17, 11, 20180238 (2019)
[42] Moioli, E.; Gallandat, N.; Züttel, A., Model based determination of the optimal reactor concept for Sabatier reaction in small-scale applications over \(Ru/Al_2 O_3\), Chem Eng J, 375, 121954 (2019)
[43] Atkins, PW; De Paula, J., Elements of physical chemistry (2017), Oxford: Oxford University Press, Oxford
[44] Bruus, H., Theoretical microfluidics (2007), Oxford: Oxford University Press, Oxford
[45] McBride BJ, Zehe MJ, Gordon S (2002) NASA Glenn coefficients for calculating thermodynamic properties of individual species. Technical report. NASA Glenn Research Center
[46] Svehla RA (1995) Transport coefficients for the NASA Lewis chemical equilibrium program. Technical report. NASA Lewis Research Center
[47] Alnæs MS, Blechta J, Hake J, Johansson A, Kehlet B, Logg A, Richardson C, Ring J, Rognes ME, Wells GN (2015) The FEniCS project version 1.5. Arch Num Software, 3(100)
[48] Logg, A.; Mardal, KA; Wells, GN, Automated solution of differential equations by the finite element method (2012), Heidelberg: Springer, Heidelberg · Zbl 1247.65105
[49] Deuflhard, P., Newton methods for nonlinear problems (2011), Heidelberg: Springer, Heidelberg · Zbl 1226.65043
[50] Balay S, Abhyankar S, Adams MF, Brown J, Brune P, Buschelman K, Dalcin L, Dener A, Eijkhout V, Gropp WD, Karpeyev D, Kaushik D, Knepley MG, May DA, McInnes LC, Mills RT, Munson T, Rupp K, Sanan P, Smith BF, Zampini S, Zhang H, Zhang H (2020) PETSc users manual. Technical Report ANL-95/11-Revision 3.13. Argonne National Laboratory
[51] Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S., Optimization with PDE constraints (2009), New York: Springer, New York · Zbl 1167.49001
[52] Tröltzsch, F., Optimal control of partial differential equations (2010), Providence: American Mathematical Society, Providence · Zbl 1195.49001
[53] Farrell, PE; Ham, DA; Funke, SW; Rognes, ME, Automated derivation of the adjoint of high-level transient finite element programs, SIAM J Sci Comput, 35, 4, C369-C393 (2013) · Zbl 1362.65103
[54] Mitusch, SK; Funke, SW; Dokken, JS, dolfin-adjoint 2018.1: automated adjoints for FEniCS and Firedrake, J Open Source Softw, 4, 38, 1292 (2019)
[55] Rathgeber F, Ham DA, Mitchell L, Lange M, Luporini F, McRae ATT, Bercea G-T, Markall GR, Kelly PHJ (2017) Firedrake: automating the finite element method by composing abstractions. ACM Trans Math Softw 43(3):24 · Zbl 1396.65144
[56] Ham, DA; Mitchell, L.; Paganini, A.; Wechsung, F., Automated shape differentiation in the Unified Form Language, Struct Multidiscip Optim, 60, 5, 1813-1820 (2019)
[57] Rohatgi A (2019) WebPlotDigitizer: Version 4.2
[58] Kelley, CT, Iterative methods for optimization (1999), Philadelphia: Society for Industrial and Applied Mathematics (SIAM), Philadelphia · Zbl 0934.90082
[59] Nocedal, J.; Wright, SJ, Numerical optimization (2006), New York: Springer, New York · Zbl 1104.65059
[60] Andres, M.; Blauth, S.; Leithäuser, C.; Siedow, N., Identification of the blood perfusion rate for laser-induced thermotherapy in the liver, J Math Ind, 10, 17 (2020) · Zbl 1469.92056
[61] Delfour, MC; Zolésio, JP, Shapes and geometries (2011), Philadelphia: Society for Industrial and Applied Mathematics (SIAM), Philadelphia · Zbl 1251.49001
[62] Khan, MG; Fartaj, A., A review on microchannel heat exchangers and potential applications, Int J Energy Res, 35, 7, 553-582 (2011)
[63] Naqiuddin, NH; Saw, LH; Yew, MC; Yusof, F.; Ng, TC; Yew, MK, Overview of micro-channel design for high heat flux application, Renew Sustain Energy Rev, 82, 901-914 (2018)
[64] Zanfir, M.; Baldea, M.; Daoutidis, P., Optimizing the catalyst distribution for countercurrent methane steam reforming in plate reactors, AIChE J, 57, 9, 2518-2528 (2011)
[65] Park S, Jung I, Lee Y, Kshetrimayum KS, Na J, Park S, Shin S, Ha D, Lee Y, Chung J, Lee C-J, Han C (2016) Design of microchannel Fischer-Tropsch reactor using cell-coupling method: effect of flow configurations and distribution. Chem Eng Sci 143:63-75
[66] Engelbrecht, N.; Everson, RC; Bessarabov, D.; Kolb, G., Microchannel reactor heat-exchangers: a review of design strategies for the effective thermal coupling of gas phase reactions, Chem Eng Process, 157, 108164 (2020)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.