Publication: Estimating a pressure dependent thermal conductivity coefficient with applications in food technology
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2019-06-26
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Taylor and Francis
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Abstract
In this paper we introduce a method to estimate a pressure dependent thermal conductivity coefficient arising in a heat diffusion model with applications in food technology. The strong smoothing effect of the corresponding direct problem renders the inverse problem under consideration severely ill-posed. Thus specially tailored methods are necessary in order to obtain a stable solution. Consequently, we model the uncertainty of the conductivity coefficient as a hierarchical Gaussian Markov random field (GMRF) restricted to uniqueness conditions for the solution of the inverse problem established in Fraguela et al. [1]. Furthermore, we propose a Single Variable Exchange Metropolis-Hastings algorithm (SVEMH) to sample the corresponding conditional probability distribution of the conductivity coefficient given observations of the temperature. Sensitivity analysis of the direct problem suggests that large integration times are necessary to identify the thermal conductivity coefficient. Numerical evidence indicates that a signal to noise ratio of roughly 10 cubed suffices to reliably retrieve the thermal conductivity coefficient.
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“This is an Accepted Manuscript of an article published by Taylor & Francis in Inverse Problems in Science and Engineering on 26-jun-2019, available online: https://www.tandfonline.com/doi/full/10.1080/17415977.2019.1632841.”
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[1] A Fraguela, J A Infante, A M Ramos, and J M Rey. A uniqueness result for the identification of a time-dependent diffusion coefficient. Inverse Problems, 29(12):125009, 2013.
[2] Juan Antonio Infante, Benjamin Ivorra, A M Ramos, and Jose Maria Rey. On the modelling and simulation of high pressure processes and inactivation of enzymes in food engineering. Mathematical Models and Methods in Applied Sciences, 19(12):2203–2229, 2009.
[3] J B Adams. Enzyme inactivation during heat processing of food-stuffs. International journal of food science & technology, 26(1):1–20, 1991.
[4] J P Cardoso and A N Emery. A new model to describe enzyme inactivation. Biotechnology and bioengineering, 20(9):1471–1477, 1978.
[5] Marc Hendrickx, Linda Ludikhuyze, Ilse Van den Broeck, and C Weemaes. Effects of high pressure on enzymes related to food quality. Trends in Food Science & Technology, 9(5):197–203, 1998.
[6] Siegfried Denys, Ann M Van Loey, and Marc E Hendrickx. A modeling approach for evaluating process uniformity during batch high hydrostatic pressure processing: combination of a numerical heat transfer model and enzyme inactivation kinetics. Innovative Food Science & Emerging Technologies, 1(1):5–19, 2000.
[7] L Otero, A M Ramos, C De Elvira, and P D Sanz. A model to design high-pressure processes towards an uniform temperature distribution. Journal of Food Engineering, 78(4):1463–1470, 2007.
[8] Tomas Norton and Da-Wen Sun. Recent advances in the use of high pressure as an effective processing technique in the food industry. Food and Bioprocess Technology, 1(1):2–34, 2008.
[9] L Otero, B Guignon, C Aparicio, and P D Sanz. Modeling thermophysical properties of food under high pressure. Critical reviews in food science and nutrition, 50(4):344–368, 2010.
[10] N. A. S. Smith, S. L. Mitchell, and A. M. Ramos. Analysis and simplification of a mathematical model for high-pressure food processes. Applied Mathematics and Computation, 226:20 – 37, 2014.
[11] Vinicio Serment-Moreno, Gustavo Barbosa-C´anovas, José Antonio Torres, and Jorge Welti-Chanes. High-pressure processing: kinetic models for microbial and enzyme inactivation. Food Engineering Reviews, 6(3):56–88, 2014.
[12] J A Infante, M Molina-Rodr´ıguez, and A M Ramos. On the identification of a thermal expansion coefficient. Inverse Problems in Science and Engineering, 23(8):1405–1424, 2015.
[13] A F Albu and V I Zubov. Identification of thermal conductivity coefficient using a given temperature field. Computational Mathematics and Mathematical Physics, 58(10):1585– 1599, 2018.
[14] Oleg M Alifanov, A V Nenarokomov, S A Budnik, V V Michailov, and V M Ydin. Identification of thermal properties of materials with applications for spacecraft structures. Inverse Problems in Science and Engineering, 12(5):579–594, 2004.
[15] M J Huntul and D Lesnic. An inverse problem of finding the time-dependent thermal conductivity from boundary data. International Communications in Heat and Mass Transfer, 85:147–154, 2017.
[16] P N Vabishchevich and M V Klibanov. Numerical identification of the leading coefficient of a parabolic equation. Differential Equations, 52(7):855–862, 2016.
[17] Dariusz Lydzba, Adrian Rózanski, and Damian Stefaniuk. Equivalent microstructure problem: Mathematical formulation and numerical solution. International Journal of Engineering Science, 123:20–35, 2018.
[18] Jingbo Wang and Nicholas Zabaras. A bayesian inference approach to the inverse heat conduction problem. International Journal of Heat and Mass Transfer, 47(17-18):3927– 3941, 2004.
[19] Jingbo Wang and Nicholas Zabaras. Hierarchical bayesian models for inverse problems in heat conduction. Inverse Problems, 21(1):183, 2004.
[20] Jari P Kaipio and Colin Fox. The bayesian framework for inverse problems in heat transfer. Heat Transfer Engineering, 32(9):718–753, 2011.
[21] Victor Isakov. Inverse problems for partial differential equations, volume 127. Springer, 2006.
[22] Arnold Zellner. Optimal information processing and bayes’s theorem. The American Statistician, 42(4):278–280, 1988.
[23] J Andrés Christen, Colin Fox, et al. A general purpose sampling algorithm for continuous distributions (the t-walk). Bayesian Analysis, 5(2):263–281, 2010.
[24] Paul G Constantine, Carson Kent, and Tan Bui-Thanh. Accelerating markov chain monte carlo with active subspaces. SIAM Journal on Scientific Computing, 38(5):A2779–A2805, 2016.
[25] James Martin, Lucas C Wilcox, Carsten Burstedde, and Omar Ghattas. A stochastic newton mcmc method for large-scale statistical inverse problems with application to seismic inversion. SIAM Journal on Scientific Computing, 34(3):A1460–A1487, 2012.
[26] W Zhang, J Liu, C Cho, and X Han. A fast bayesian approach using adaptive densifying approximation technique accelerated mcmc. Inverse Problems in Science and Engineering, 24(2):247–264, 2016.
[27] Vijay P Singh and Kulwant Singh. Derivation of the gamma distribution by using the principle of maximum entropy (pome) 1. JAWRA Journal of the American Water Resources Association, 21(6):941–952, 1985.
[28] Johnathan M Bardsley. Gaussian markov random field priors for inverse problems. Inverse Problems and Imaging, 7(2):397–416, 2013.
[29] Geoff K Nicholls, Colin Fox, and Alexis Muir Watt. Coupled mcmc with a randomized acceptance probability. arXiv preprint arXiv:1205.6857, 2012