Inverse Estimation of Multi- Parameters Using Bayesian Framework Combined with Evolutionary Algorithms for Heat Transfer Problems

Abstract

This thesis focuses on the estimation of unknown parameters using various inverse methods for the heat transfer problems. The first class of problem elaborately discusses about the estimation of interfacial heat transfer coefficients during the solidification of casting. To accomplish this, a prevalent one dimensional transient horizontal directional solidification of Sn-5%wtPb alloy with temperature dependent thermophysical properties and latent heat is considered to be the mathematical model/forward model and numerically solved using Explicit Finite Difference Method to obtain temperature distribution from the known boundary and initial conditions. The temperatures from the forward model is validated with the literature and an absolute error of 5% from the actual measurements was observed. In order to mimic the real time experiments, the temperatures are added with σ=0.01Tmax, σ=0.02Tmax and σ=0.03Tmax Gaussian white noise (simulated measurements) and compared with two different objective functions: (i) Least Squares and (ii) Bayesian Framework. Meantime, to expedite the solution of the inverse problem, the numerical model is then replaced with Artificial Neural Network (ANN), which acts as a fast forward model to estimate the unknown constants present in the correlation of interfacial heat transfer coefficient. A total of 473 data sets of inputs and corresponding outputs were used to create a trained artificial neural network which produced temperatures with an accuracy less than 0.1◦C temperature difference from the exact temperature data. Genetic Algorithm (GA) was implemented as an inverse method and it was found that ANN-GA-Bayesian framework was more effective compared to ordinary least squares for noise added data with an overall average error of 2%. Furthermore, an extended study on the advantage of Bayesian framework for the estimation of multi-parameters during Al-4.5wt%Cu alloy solidification is also discussed in detail. The main aim is to retrieve more information with less available simulated measurements. A sensitivity analysis is performed to understand the dependency of the unknown parameters like modeling error, latent heat and heat transfer coefficient parameters on the solution. It showed that the values of constants of the IHTC correlation and latent heat affect the temperature distribution in casting significantly. For iiithe solution of inverse estimation, the use of two different metaheuristic algorithms (i) Genetic Algorithm (GA) and (ii) Particle Swarm Optimization (PSO) is illustrated. A careful examination of the mentioned algorithms is performed to fix the algorithm parameters. The results revealed that PSO combined with Bayesian framework provides a better computational solution compared to GA-Bayesian with an overall absolute error less than 6%. Also, the study on the effect of multiple sensors revealed that using two sensor the average % error for the estimation of a ,b and latent heat was 0.247, 0.3 and 0.45 respectively and suggesting that two sensors were sufficient for the present analysis. The second class of problem is extended to retrieve the unknown heat flux and heat transfer coefficient for a 3-D steady state conjugate fin heat transfer problem. A mild steel fin with dimensions 150x250x6 mm3 is placed centrally on to an aluminium base of dimensions 150x250x8 mm3 and experiments are conducted for different heat flux values of 305, 544, 853 and 1232 W/m2 and corresponding temperature distribution along the vertical fin is recorded. Navier-Stokes equation is solved to obtain the necessary temperature distribution of the fin. Heat flux with the range between 305W/m2 and 3300 W/m2 and its corresponding temperature distribution of the fin is obtained using commercial software. A total of 24 Computational Fluid Dynamics (CFD) simulations are performed to create a neural network model that can surrogate the forward problem in order to expedite the computational process. The estimation of the heat flux and heat transfer coefficient using GA, PSO and PSO- Broyden Fletcher Goldfarb Shanno (BFGS) is carried out for both simulated and experimental data. A detailed comparison study on the effect of algorithm parameters on the solution is demonstrated in order to examine the performance of the algorithms. For simulated temperature measurements, all the mentioned algorithms proved to be effective but PSO-BFGS estimated the heat flux with an absolute % error of 0.86 and heat transfer coefficient with 0.105% for experimental temperatures. The results show that the PSO-BFGS method outperforms GA and PSO and is observed to be a formidable approach in the estimation of the unknown parameters