A Bayesian approach for the simultaneous estimation of surface heat transfer coefficient and thermal conductivity from steady state experiments on fins

This paper explores the potential of the Bayesian approach to estimate multiple parameters from simple, inexpensive experiments on natural convection heat transfer from a vertical rectangular fin. A vertical fin of rectangular cross-section is placed on two heated horizontal aluminium blocks of size 250 × 75 × 10 (all in mm) that act as a horizontal base and hold the extended surface made up of mild steel of size 250 × 160 × 4 (all in mm). A heater is placed below the aluminium block and to restrict the heat flow, the bottom side of the heater is insulated with glass wool. Steady state experiments are carried out to obtain the temperature distribution for different levels of heating and temperatures on the fin are recorded with K-type thermocouples. Using data from these experiments, two critical parameters namely, the average heat transfer coefficient and the thermal conductivity of the extended surface are first individually and later simultaneously obtained using the Bayesian approach thereby obviating the need for sophisticated equipment. The above two parameters are correlated thereby making their simultaneous estimation very challenging. The Markov Chain Monte Carlo (MCMC) is used for the estimation without and with subjective priors on the two parameters. The uncertainties are obtained explicitly in the form of standard deviation. The addition of subjective priors is the hallmark of the Bayesian approach as it reduces the standard deviation of the estimates. This considerably helps regularizing the ill-posedness and becomes a necessity for estimating correlated parameters. A discussion on the optimum number of temperature measurements needed for estimating the parameters with a given accuracy using the Bayesian method is also presented.