A Study on Inverse Problems of Realistic Heat Transfer Systems

Abstract

This thesis aims to contribute towards the optimization of energy requirements. This is achieved by finding the solution of inverse problems in the heat transfer setting. To enhance the performance efficiency of a thermal system, the controlling parameters should be tuned. This is realized for composite walls, where performance is marked through efficiency. Besides, to have deeper information about a practical system, various important thermal parameters are to be estimated. This is procured by performing real-time experiments on fins, where the time-dependent heat flux is retrieved. Marking the presence of tumor through inverse detection of blood perfusion rate demonstrates the versatility of the process. Apart from the retrieval of parameters, the current work focuses on the different techniques for inversion. With the presence of a large number of inversion algorithms for Inverse heat transfer problems (IHTPs) and non-IHTPs, a need for review to have a holistic view is seen. To accomplish the task, an exhaustive literature review, based on regularization techniques and methods to find the solution of inverse problems is performed. With the motivation of selection of the inversion technique best fit for a given problem, the comparison was made for a general inverse problem. Having the exact solution is necessary, to compare the numerical solutions, of a non-linear problem. In this purview, the Adomian decomposition method is established for non-linear heat transfer in double-layered walls. The closed-form of temperature is obtained, with a maximum efficiency of 98.32% for the specific thermal parameters. The contribution of this analysis is seen through its application in an industrial furnace. In the subsequent analysis, the non-suitability of gradient-based algorithms is identified. For the IHTPs, complexity increases due to the presence of several temperature-dependent properties, thus evolutionary algorithms were recognized suitable for the solution of IHTPs. Unlike the steady-state analysis of composite walls, the transient-state retrieval of the functional form of heat flux in pin fin is obtained. An experimental setup with thermocouples of the fin is selected, and experiments are performed to obtain the temperature profile of the fin. A mathematical model with temperature-dependent thermal parameters is solved using Matlab’s pdepe toolbox. GSSM is implemented for retrieval of time-dependent single parameter. To contribute towards the performance of the fin, the effect of measurement errors on the retrieval is depicted. The acceptable measurement errors in temperature are reported as 3%; 2%; and 2% with a maximum error of 6:8%; 9:6%; and 14% respectively for the constant, triangular and realistic case study. Moreover, for the measurement errors due to uncertainty of thermocouples, the temperature profile at V = 80V; 60V; 70V and I = 0:027A; 0:018A; 0:022A is known to have 5%; 6% and 2% reconstruction error. The proposed procedure illustrating an error bound is useful to determine the auto-cut for various devices for their efficient working. After the successful retrieval of a single parameter, multiple critical parameters of the fin are obtained based on sensitivity analysis. For this inverse estimation, an algorithm capable to handle every kind of non-linearity is sought in literature, whose absence laid the foundation of the next objective. The main research question, “Is there a technique that works globally for every inverse problem?”, is asked prior to, “What if the available techniques were not utilized to an extent that they should?” is posed. In lieu of this gap, a general comparative framework is developed, such that an efficient technique is selected, based on the total minimum error. To mitigate the associated ill-posedness, regularization by modification in the objective function is proposed. A suitable objective function out of least squares, Tikhonov/Ridge regularization, lasso estimators and elastic net techniques is selected by comparing the total relative error and the CPU time. Elastic net regularization (0.9) is selected with an error of 0.39 and CPU time of 34s for fin. The regularization parameter (10^-4) is selected based on the minimum total relative error (3.87). TOPSIS analysis is implemented to compare among evolution-based (DE), swarm-based (PSO,WOA), nature-based (WCA, BOA), physics-based (ASO) and hybrid (GWOCS) optimization algorithms for parameter estimation. Inculcating the pdepe-based temperature profile, the best algorithm, along with their performance parameter is obtained in the order BOA (0.77), WOA (0.75) and WCA (0.74), respectively. However, WOA (0.78) works best when experimental temperature profile was utilized, suggesting the robustness of WOA, with elastic net regularization (0:9, 10^-4) for fins. This developed framework could easily be implemented in any other inverse problem for parameter estimation. The proposed procedure designed for comparison enriches the effectiveness in the working of existing inversion methods. To test the applicability of the proposed framework, an inverse bio-medical problem to detect tumor in the human brain is studied. The best inversion algorithm, WOA, obtained previously, together with a hybrid GWOCS are utilized. Pennes model is used for the formulation of heat transfer within the human brain. Using inverse analysis, the unknown blood perfusion rate !i is retrieved in the regularized environment. The current research reported that at positions where tumor cells are present, the temperature rises. Moreover, with an increase in time, temperature of the cancer cells increases, whereas no change in the temperature of tissue without tumor is seen. This observation marks the presence of tumor. Elastic net regularization (0.1, 10^-2) for WOA and (0.9, 10^-4) for GWOCS shows the least RE. After regularization, the perfusion rate is retrieved. A clear distinction of two tumors from the brain tissue is observed. The obtained perfusion rate is used to reconstruct temperature profile. An excellent matching of reconstructed temperature field and the exact field is obtained, even when the forward data contain measurement errors. The obtained set of perfusion rate is appropriate for tumor detection, up to 7% error in the measured data. Thus, the proposed comparative procedure is found suitable for the bio-heat transfer problem. In future, real-time experimental temperature data could be utilized for the detction of tumor through thermal images.