Retrieval of parameters in heat and mass transfer problems using inverse optimization

Abstract

This thesis is aimed at applying the concept of parameter retrieval by optimization techniques to a few engineering systems involving heat and mass transfer. For this, two heat transfer systems involving different types of fins and flat-plate solar collectors have been considered. Furthermore, experimental data based parameter retrieval on a combined heat and mass transfer system concerning forced and induced draft cooling towers is also undertaken. Although the main focus is given on unknown parameter retrieval through inverse analysis, but, many new computational algorithms and empirical correlations for the forward analysis have been also reported from the present research study. The work successfully demonstrates the application of evolutionary optimization algorithms such as the binary-coded Genetic Algorithm (GA) and the real-coded Differential Evolution (DE) for unknown multi-parameter retrievals. Apart from evolutionary methods, a few problems have been also solved with the Golden Section Search Method (GSSM) for single parameter retrieval. Relative merits and limitations of various techniques have been also highlighted. Based on the literature survey, the research work begins with the formulation and the solution of the identified parabolic heat transfer problems on fins involving different levels of nonlinearities, for which either forward and/or inverse analyses were not found. Under this purview, four parabolic heat transfer problems are undertaken according to increase in the difficulty level. Initially for the forward solution, the Adomian Decomposition Method (ADM) is applied for the straight fin, the moving fin and the stepped fin involving all temperature-dependent modes of heat transfer and complex boundary conditions. Furthermore, a comparative study is also conducted on a porous fin. Subsequently, the DE, the GA and the GSSM are applied on these fin problems to inversely predict critical parameters such as the rate of internal heat generation, heat transfer coefficient, fin speed, thermal conductivity. To examine the suitability of optimization technique for single and multi-parameter inverse retrievals, a comparative study is done on the porous fin. Realizing the unsuitability of the GSSM for multiparameter estimation, among the GA and the DE, it is found that the real-coded DE is the most suitable optimization technique for the multiple parameter retrievals. Besides this, the GSSM is found to be the befitting method for single parameter retrieval. After retrieving different parameters, the amount of acceptable simulated measurement errors/noise are also quantified. For the rectangular fin with internal heat generation, satisfactory interpretations are revealed up to 10% simulated measurement error. However, for the same fin without internal heat generation, the reconstructed trends are in satisfactory agreement up to 5% simulated measurement error. Furthermore, for the case of a rectangular moving fin, the allowable/tolerable simulated measurement error is found 7.90% . The concept of heat transfer analysis of fins is next extended to retrieve parameters such as Fourier number, Vernotte number, heat loss coefficient and incident solar heat flux in hyperbolic heat transfer problems of flat-plate solar collectors. The estimations of these parameters lead to determination of the time required to achieve the given temperature, characterization of the heat velocity and the identification of feasible geographical locations. During the appraisal of the acceptable measurement error, it is found that for the solar collector with isothermal boundary condition, the variation between exact and predicted Fourier numbers is 18.33% and 21% for non-Fourier model and Fourier model, respectively. From the multi-parameter retrieval analysis involving the binary-coded GA, it is observed that the reconstructed results obtained with the inversely-estimated parameters are reasonably accurate. It is found that only  0.78% deviation takes place in case of Fourier heat conduction model and  0.44% deviation is observed in case of non- Fourier conduction model. The inverse results using the real-coded DE and the binarycoded GA are compared for solar collector. The maximum acceptable measurement error using the real-coded DE is found to be 5% for adequate reconstruction. In the final chapter, parameter retrieval is accomplished from experiments on a combined heat and mass transfer system involving two different types of cooling tower. In the first case, an induced draft type of cooling tower is considered, whereas, in the other case, a forced draft type of cooling tower is investigated. For the forward analysis, the relevant and required correlations have been developed using the full factorial method-based design of experiments. Thereafter, for single parameter retrieval, the GSSM is used, whereas, for retrieving multiple parameters, the DE is used. Cooling towers’ performance is specified by different performance parameters such as the range, the approach, the effectiveness, the water evaporation rate, the global heat and mass transfer coefficient, the rate of heat transfer and the Merkel number. Among these, the Merkel number is found to be important, because it involves various parameters such as the global heat and mass transfer coefficient, the mass of water, the dimensions (interfacial area per unit volume) and the range within itself. Therefore, it is also known as the tower characteristic ratio. The mass flow rates of air and water are common controlling parameters related to the performance of a cooling tower, but they do not explicitly appear in the expression for the Merkel number. Therefore, the first objective is to propose the relevant correlation for the Merkel number for the present experimental setup as a function of mass flow rates of air and water. These parameters have been finally estimated using the inverse procedure for satisfying a desired Merkel number. At first, the experimental setup of an induced draft cooling tower has been considered which exemplifies a combined heat and mass transfer system. In this study, the correlation of the Merkel number has been proposed with as a function of air mass flow rate. Further, the developed correlation in conjunction with the GSSM has been used to retrieve the unknown controlling parameter (air mass flow rate) for a desired Merkel number of the cooling tower. For a given water flow rate, this case study shows that the present inverse procedure is satisfactory for estimating the required air flow rate to fulfill a desired tower characteristic ratio (Merkel number). It is found that the maximum deviation is 4.4% for the correlation based on the quadratic approximation, whereas, the deviation is 2.8% for the correlation based on the Pade approximation. In the second part of this chapter, a different experimental system involving forced draft cooling tower has been considered for the simultaneously retrieving multiple and feasible combinations of water and air flow rates using the DE-based optimization methodology. In this task, the relevant correlation of the Merkel number as a function of two controlling parameters such as mass flow rate of air and water has been made using bivariate cubic polynomial approximation and bicubic approximations. On comparing the goodness of these two correlations, it is concluded that bivariate cubic polynomial approximation yields better performance than the bicubic approximation. Excellent reconstruction of the Merkel number is obtained using the DE with the aid of the bivariate correlation obtained from the experimental data, with the coefficient of determination of the correlated data against experimental values as 92.05%. It is also found that the present retrieval methodology is an effective approach to estimate unknown controlling parameters for practically satisfying a desired output from a given system.