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

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## Abstract

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 mm^{3} and an aluminium base plate of dimensions 250×150×8 mm^{3}. 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.

## Keywords

Mild steel fin heat flux ANN Bayesian inference MCMC standard deviation## Nomenclature

- ACFD
asymptotic computational fluid dynamics

- ANN
artificial neural network

- h
heat transfer coefficient, W/m

^{2}K- k
thermal conductivity of the fin material, W/m K

- L
height of the fin, m

- p
perimeter of the fin, m

- q
_{ref} reference heat flux, W/m

^{2}- q
flux input, W/m

^{2}- T
_{ref} reference temperature, K

- T
_{new} temperature for given value of heat flux, K

- X
non-dimensional length

- x
distance from the base of fin, m

## Greek symbols

- θ
non-dimensional temperature

- β
\( 1/T \), thermal expansion coefficient K

^{−1}- ρ
density, kg/m

^{3}- ν
kinematic viscosity, m

^{2}/s

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