, 43:62 | Cite as

A Bayesian inference approach: estimation of heat flux from fin for perturbed temperature data

  • Harsha Kumar
  • Gnanasekaran Nagarajan


This paper reports the estimation of the unknown boundary heat flux from a fin using the Bayesian inference method. The setup consists of a rectangular mild steel fin of dimensions 250×150×6 mm3 and an aluminium base plate of dimensions 250×150×8 mm3. The fin is subjected to constant heat flux at the base and the fin setup is modelled using ANSYS14.5. The problem considered is a conjugate heat transfer from the fin, and the Navier–Stokes equation is solved to obtain the flow parameters. Grid independence study is carried out to fix the number of grids for the study considered. To reduce the computational cost, computational fluid dynamics (CFD) is replaced with artificial neural network (ANN) as the forward model. The Markov Chain Monte Carlo (MCMC) powered by Metropolis–Hastings sampling algorithm along with the Bayesian framework is used to explore the estimation space. The sensitivity analysis of the estimated temperature with respect to the unknown parameter is discussed to know the dependency of the temperature with the parameter. This paper signifies the effect of a prior model on the execution of the inverse algorithm at different noise levels. The unknown heat flux is estimated for the surrogated temperature and the estimates are reported as mean, Maximum a Posteriori (MAP) and standard deviation. The effect of a-priori information on the estimated parameter is also addressed. The standard deviation in the estimation process is referred to as the uncertainty associated with the estimated parameters.


Mild steel fin heat flux ANN Bayesian inference MCMC standard deviation 



asymptotic computational fluid dynamics


artificial neural network


heat transfer coefficient, W/m2 K


thermal conductivity of the fin material, W/m K


height of the fin, m


perimeter of the fin, m


reference heat flux, W/m2


flux input, W/m2


reference temperature, K


temperature for given value of heat flux, K


non-dimensional length


distance from the base of fin, m

Greek symbols


non-dimensional temperature


\( 1/T \), thermal expansion coefficient K−1


density, kg/m3


kinematic viscosity, m2/s


  1. 1.
    Beck J V, Blackwell B and Clair C S 1985 Inverse heat conduction: ill-posed problems. New York: WileyzbMATHGoogle Scholar
  2. 2.
    Ozisik M N and Orlande H R B 2000 Inverse heat transfer: fundamentals and applications. New York: Taylor and FrancisGoogle Scholar
  3. 3.
    Mota C A A, Orlande H R B, Carvalho M O M D, Kolehaminen V and J P Kaipio 2010 Bayesian estimation of temperature-dependent thermophysical properties and transient boundary heat flux. Heat Transfer Eng. 31:570–580CrossRefGoogle Scholar
  4. 4.
    Wang J and Zabaras N 2004 A Bayesian inference approach to the inverse heat conduction problem. Int. J. Heat Mass Transfer 47: 3927–3941CrossRefzbMATHGoogle Scholar
  5. 5.
    Giralomi M 2008 Bayesian inference for differential equations. Theor. Comput. Sci. 408: 4–16MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liang Yan, Fenglian Yang and Chuli Fu 2009 A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems. J. Comput. Appl. Math. 231:840–850MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Parthasarathy S and Balaji C 2008 Estimation of parameters in multi-mode heat transfer problems using Bayesian inference-effect of noise and a prior. Int. J. Heat Mass Transfer 51: 2313–2334CrossRefzbMATHGoogle Scholar
  8. 8.
    Gnanasekaran N and Balaji C 2011 A Bayesian approach for the simultaneous estimation of surface heat transfer coefficient and thermal conductivity from steady state experiments on fins. Int. J. Heat Mass Transfer 54: 3060–3068CrossRefzbMATHGoogle Scholar
  9. 9.
    Konda Reddy B, Gnanasekaran N and Balaji C 2012 Estimation of thermo-physical and transport properties with Bayesian inference using transient liquid crystal thermography experiments. J. Physics: Conference Series 395: 012082Google Scholar
  10. 10.
    Cheung S H and Beck J L 2009 Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters. J. Eng. Mech. 135: 243–255CrossRefGoogle Scholar
  11. 11.
    Gnanasekaran N and Balaji C 2013 Markov Chain Monte Carlo (MCMC) approach for the determination of thermal diffusivity using transient fin heat transfer experiments. Int. J. Therm. Sci. 63: 46–54CrossRefGoogle Scholar
  12. 12.
    Balaji C. and Tamanna Padhi 2010 A new ANN driven MCMC method for multi-parameter estimation in two-dimensional conduction with heat generation. Int. J. Heat Mass Transfer 53: 5440–5455CrossRefzbMATHGoogle Scholar
  13. 13.
    Deng S and Hwang Y 2006 Applying neural networks to the solution of forward and inverse heat conduction problems. Int. J. Heat Mass Transfer 49: 4732–4750CrossRefzbMATHGoogle Scholar
  14. 14.
    Ghadimi B, Kowsary F and Khorami M 2015 Heat flux on-line estimation in a locomotive brake disc using artificial neural networks. Int. J. Therm. Sci. 90: 203–213CrossRefGoogle Scholar
  15. 15.
    Somasundharam S and Reddy K S 2016 Inverse estimation of thermal properties using Bayesian inference and three different sampling techniques. Inverse Probl. Sci. Eng. zbMATHGoogle Scholar
  16. 16.
    Cole K D, Tarawneh C and Wilson B 2009 Analysis of flux-base fins for estimation of heat transfer coefficient. Int. J. Heat Mass Transfer 52: 92–99CrossRefzbMATHGoogle Scholar
  17. 17.
    Cole K D, Beck J V, Keith A Woodbury and Filippo de Monte 2014 Intrinsic verification and a heat conduction database. Int. J. Therm. Sci. 78: 36–47CrossRefGoogle Scholar
  18. 18.
    Prietob D, Asensioa M I, Ferraguta L and Cascón J M 2015 Sensitivity analysis and parameter adjustment in a simplified physical wildland firemodel. Adv. Eng. Softw. 90: 98–106CrossRefGoogle Scholar
  19. 19.
    Pereyra S, Lombera G, Frontini G and Urquiza S A 2014 Sensitivity analysis and parameter estimation of heat transfer and material flow models in friction stir welding. Mater. Res. 17(2): 397–404CrossRefGoogle Scholar
  20. 20.
    Cheng-Hung Huang, Li-Chun Jan, Rui Li and Albert J Shih 2007 A three-dimensional inverse problem in estimating the applied heat flux of a titanium drilling – theoretical and experimental studies. Int. J. Heat Mass Transfer 50: 3265–3277CrossRefzbMATHGoogle Scholar
  21. 21.
    Huang C H and Chen W C 2000 A three-dimensional inverse forced convection problem in estimating surface heat flux by conjugate gradient method. Int. J. Heat Mass Transfer 43(17): 3171–3181CrossRefzbMATHGoogle Scholar
  22. 22.
    Chen T C, Liu C C, Jang H Y and Tuan P C 2007 Inverse estimation of heat flux and temperature in multi-layer gun barrel. Int. J. Heat Mass Transfer 50(11–12): 2060–2068CrossRefzbMATHGoogle Scholar
  23. 23.
    Wikstrom P, Blasiak W and Berntsson F 2007 Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method. Appl. Therm. Eng. 27(14–15): 2463–2472CrossRefGoogle Scholar
  24. 24.
    Cui Miao, Yang Kai, Liu Yun-fei and Gao Xiao-wei 2012 Inverse estimation of transient heat flux to slab surface. J. Iron Steel Res. Int. 19(11): 13–18CrossRefGoogle Scholar
  25. 25.
    Man Young Kim 2007 A heat transfer model for the analysis of transient heating of the slab in a direct-fired walking beam type reheating furnace. Int. J. Heat Mass Transfer 50: 3740–3748CrossRefzbMATHGoogle Scholar
  26. 26.
    Incropera F P and De Witt D P 2002 Fundamentals of heat and mass transfer. 5th edn. New York, NY: John Wiley & SonsGoogle Scholar
  27. 27.
    Wang J and Zabaras N 2005 Hierarchical Bayesian models for inverse problems in heat conduction. Inverse Probl. 21: 183–206MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNational Institute of Technology KarnatakaMangaloreIndia

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