The method of lines with numerical differentiation of the sequential temperature–time histories for a facile solution of 1D inverse heat conduction problems
 Antonio Campo,
 Mohammad R. Salimpour,
 John Ho
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
This work addresses an inverse heat conduction problem (IHCP) in a large planar slab receiving a certain heat flux at one surface while the other surface is thermally insulated. The two different heating conditions to be studied are: (1) constant heat flux and (2) a timedependent triangular heat flux. For the IHCP, the temperature–time variations at the insulated surface of the large planar slab are obtained with a single temperature sensor. The two temperature–time histories are generated from the solutions of the direct heat conduction problem. The central objective of the paper is to implement the method of lines for the descriptive 1D heat equation combined with numerical differentiation of: (1) the “measured” temperature–time history at the insulated surface and (2) the temperature–time history at other subsurface locations. In the end, it is confirmed that excellent predictions of the temperature–time variations at the directly heated surface are obtainable for the two dissimilar heating conditions. This is accomplished with small systems of firstorder differentialdifference equations, one with two equations and the other with four equations.
 Arpaci, VS (1966) Conduction heat transfer. AddisonWesley, Reading
 Poulikakos, D (1994) Conduction heat transfer. Prentice Hall, Upper Saddle River
 Beck, JV, Blackwell, B, St, CR (1985) Clair, inverse heat conduction: illposed problems. Wiley Interscience, New York
 Özisik, MN, Orlande, HRB (2000) Inverse heat transfer. Taylor & Francis, New York
 Burgrraf, OR (1964) An exact solution of the inverse problem in heat conduction theory and applications. J Heat Transf 86C: pp. 373382 CrossRef
 Murio, DA (1993) The mollification method and the numerical solution of illposed problems. Wiley, New York CrossRef
 Beck, JV, Blackwell, B Inverse problems. In: Minkowycz, WJ, Sparrow, EM, Schneider, GE, Pletcher, RH eds. (1988) Handbook of numerical heat transfer, chapter 19. Wiley, New York
 Maciag, A, AlKhatib, JM (2000) Stability of solutions of the overdetermined inverse heat conduction problems when discretized with respect to time. Int J Numer Meth Heat Fluid Flow 10: pp. 228244 CrossRef
 Ling, X, Atluri, SN (2006) Stability analysis for inverse heat conduction problems. Comput Model Eng Sci 13: pp. 219228
 Stolz, G (1960) Numerical solutions to an inverse problem of heat conduction for simple shapes. J Heat Transf 82: pp. 2026 CrossRef
 Hore PS, Kruttz GW, Schoenhals RJ (1977) Application of the finite element method to the inverse heat conduction problem. ASME Paper, No. 77WA/TM4
 Deng, S, Hwang, Y (2006) Applying neural networks to the solution of forward and inverse heat conduction problems. Int J Heat Mass Transf 49: pp. 47324750 CrossRef
 Shidfar, A, Molabahrami, A (2010) A weighted algorithm based on the homotopy analysis method: application to inverse heat conduction problems. Commun Nonlinear Sci Numer Simul 15: pp. 29082915 CrossRef
 Singh, KM, Tanaka, M (2001) Dual reciprocity boundary element analysis of inverse heat conduction problems. Comput Methods Appl Mech Eng 190: pp. 52835295 CrossRef
 Vakili, S, Gadala, MS (2009) Effectiveness and efficiency of particle swarm optimization technique in inverse heat conduction analysis. Numer Heat Transf Part B Fundam 56: pp. 119141 CrossRef
 Narayanan, VAB, Zabaras, N (2004) Stochastic inverse heat conduction using a spectral approach. Int J Numer Meth Eng 60: pp. 15691593 CrossRef
 Feng, ZC, Chen, JK, Zhang, Y (2010) Realtime solution of heat conduction in a finite slab for inverse analysis. Int J Therm Sci 49: pp. 762768 CrossRef
 Zueco, J, Alhama, F, GonzálezFernández, CF (2004) Numerical inverse problem of determining wall heat flux. Heat Mass Transf 41: pp. 411418 CrossRef
 Eldén, L (1995) Numerical solution of the sideways heat equation by difference approximation in time. Inverse Prob 11: pp. 913923 CrossRef
 Taler, J, Duda, P (2001) Solution of nonlinear inverse heat conduction problems using the method of lines. Heat Mass Transf 37: pp. 147155 CrossRef
 Liskovets OA (1965) The method of lines, review (in Russian). Differenzial’nie Uravneniya 1:1662–1668. English translation: Differential Equations 1:1308–1323
 Sarmin, EN, Chudov, LA (1963) On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method. USSR Comput Math Math Phys 3: pp. 15371543 CrossRef
 Zafarullah, A (1970) Application of the method of lines to parabolic partial differential equations with error estimates. J Assoc Comput Mach 17: pp. 294302 CrossRef
 Verwer, JG, SanzSerna, JM (1984) Convergence of method of lines approximations to partial differential equations. Computing 33: pp. 297313 CrossRef
 Wouwer, AV, Saucez, P, Schiesser, E (2001) Adaptive method of lines. Chapman & Hall/CRC, Boca Raton CrossRef
 Campo, A, Salazar, AJ (1996) Matching solutions for unsteady conduction in simple bodies with surface heat fluxes. J Thermophys Heat Transf 10: pp. 699701 CrossRef
 www.mathworks.com
 Chapra, SC, Canale, RP (2009) Numerical methods for engineers. McGrawHill, New York
 www.office.microsoft.com
 Title
 The method of lines with numerical differentiation of the sequential temperature–time histories for a facile solution of 1D inverse heat conduction problems
 Journal

Heat and Mass Transfer
Volume 49, Issue 3 , pp 369379
 Cover Date
 20130301
 DOI
 10.1007/s0023101210885
 Print ISSN
 09477411
 Online ISSN
 14321181
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Antonio Campo ^{(1)}
 Mohammad R. Salimpour ^{(2)}
 John Ho ^{(3)}
 Author Affiliations

 1. Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX, 78249, USA
 2. Department of Mechanical Engineering, Isfahan University of Technology, 8415683111, Isfahan, Iran
 3. Department of Mechanical Engineering, University of Vermont, Burlington, VT, 05405, USA