# Simultaneous estimation of unknown parameters using a-priori knowledge for the estimation of interfacial heat transfer coefficient during solidification of Sn–5wt%Pb alloy—an ANN-driven Bayesian approach

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## Abstract

The present methodology focuses on model reduction in which the prevalent one-dimensional transient heat conduction equation for a horizontal solidification of Sn–5wt%Pb alloy is replaced with Artificial Neural Network (ANN) in order to estimate the unknown constants present in the interfacial heat transfer coefficient correlation. As a novel approach, ANN-driven forward model is synergistically combined with Bayesian framework and Genetic algorithm to simultaneously estimate the unknown parameters and modelling error. Gaussian noise is then added to the temperature distribution obtained using the forward approach to represent real-time experiments. The hallmark of the present work is to reduce the computational time of both the forward and the inverse methods and to simultaneously estimate the unknown parameters using a-priori engineering knowledge. The results of the present methodology prove that the simultaneous estimation of unknown parameters can be effectively obtained only with the use of Bayesian framework.

## Keywords

Casting GA least squares Bayesian inverse ANN## List of symbols

- \(C_{p}\)
specific heat, (J/(kgK))

- \(f_{s}\)
fraction of solid

- \(h_{a}\)
chill-environmental heat transfer coefficient, \((\hbox {W}/(\hbox {m}^2\hbox {K}))\)

- \(h_{i}\)
interfacial heat transfer coefficient, \((\hbox {W}/(\hbox {m}^2\hbox {K}))\)

*k*thermal conductivity, (W/(mK))

*l*latent heat

*M*number of sensors

- PPDF
posterior probability density function

*t*time, s

- \(T_l\)
liquidus temperature, \(^\circ \hbox {C}\)

- \(T_f\)
fusion temperature, \(^\circ \hbox {C}\)

- \(T_s\)
solidus temperature, \(^\circ \hbox {C}\)

- \(T_C\)
casting surface temperature, \(^\circ \hbox {C}\)

- \(T_M\)
chill surface temperature, \(^\circ \hbox {C}\)

- \(K_p\)
partition coefficient

**T(P)**simulated temperatures, \(^\circ \hbox {C}\)

**Y**simulated measurements, \(^\circ \hbox {C}\)

## Greek

- \(\sigma \)
standard deviation of the prior Gaussian

- \(\rho \)
density, \((\hbox {kg/m}^3)\)

- \(\alpha \)
thermal diffusivity, \( (\hbox {m}^{2}/\hbox {s})\)

- \(\epsilon \)
random variables

- \(\mu \)
mean of prior Gaussian

## Subscripts

*f*fusion

*l*liquidus

*s*solidus

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