MATHEMATICAL MODELLING OF SOLIDIFICATION PROCESSES

Abstract

Heat transfer phenomena involving phase change also known as moving boundary problem are associated with many practical application like metal casting, environmental engineering, thermal energy storage system, aerodynamic ablation, thawing of foodstuff, cryopreservation and cryosurgery etc. Cryosurgery is use of extremely low temperature with an instrument called as cryoprobe to damage all cancer cells while sparing adjacent healthy tissues. During phase change, interface between the frozen and unfrozen phase is moving with time and the boundary conditions at this interface require specific treatment. Except initial and boundary conditions, two more conditions are needed on the moving boundary, one to determine the boundary itself and another to complete the solution of the heat equation in each region. These problems are non-linear due to the existence of a moving boundary between the two phases associated with the release of latent heat. Neither position nor the velocity of the interface can be predicted in advance. Mathematical analysis becomes yet more complicated, when the physical properties of the system are temperature dependent. Phase change problems have a limited number of analytical solutions. These solutions are limited for one dimensional, infinite or semi-infinite region with simple boundary conditions. Numerical methods appear to offer a more practical approach for solving phase change problems. Various methods have been proposed for the numerical solution of phase change problems based on front tracking, front fixing and fixed domain approach. Numerical methods based on enthalpy effective heat capacity formulation are most popular methods to solve phase change problems, The present thesis entitled Mathematical modelling of solidification processes deals with some phase change problems in biology and alloy. Numerical solutions are obtained using finite difference method. The whole work is presented in the form of six chapters, as follows Chapter 1 is introductory in nature and gives a brief account for solidification process, its mathematical formulation and numerical solution in biology and alloy. At the end of the chapter, summary of the whole work embodied in the thesis is given. In chapter 2, the effect of cryoprobe diameter, cryoprobe temperature and heat generation due to metabolism and blood, perfusion, on phase change heat transfer process during cryosurgery in lung tumor has been analyzed numerically. Results show that (i) increase in cryoprobe diameter, (ii) decrease in cryoprobe freezing temperature, lead to increase in minimum temperature, freezing rate, freezing area in tissue and decrease in tumor freezing time. Further decrease in minimum temperature, cooling rate, freezing area and an increase in time to freeze tumor has been observed with the presence of heat source term compared to the case where heat source term was not included. In chapter 3, Pennes bioheat Equation is used to find transient temperature profile, freezing and thawing interface during combined cryosurgery and hyperthermia treatment of lung. Three blood re-flow patterns when (i) blood vessel take very short time to resume its function on thawing (Case 1), (ii) blood vessel are completely destroyed (Case 2) and (iii) blood vessel need a time delay to resume on thawing (Case 3), the frozen tissue, were also taken into account [Zhang, 2002]. The temperature raising in tissue has been found highest ii for case 1 and least for case 2. Temperature profiles and phase change interfaces are obtained for all cases. Information obtained are beneficial to know whether the tumor has been damaged or not, and to minimize the damage to neighboring healthy lung tissue by over-freezing and overheating, and hence to optimize the treatment planning. In chapter 4, transient temperature profile and position of phase change interfaces are obtained numerically, which is important to apply cryosurgery precisely. Informations from this study are significant for the operation of a successful cryosurgical treatment and can also be applied to cryopreserved living organ. In chapter 5, a mathematical model has been developed to study the phase change phenomena during freezing and thawing process in biological tissues considered as porous media. Numerical simulation is used to study the effect of porosity, on the motion of freezing and thawing front and transient temperature distribution in tissue. It is observed that porosity has significant effect on transient temp profile and phase change interfaces, further decrease in freezing and heating rate has been found with increased value of porosity. In chapter 6, transient heat transfer analysis has been done to study the effect of volumetric heat generation on one-dimensional solidification in finite media with convective cooling. The whole process is divided in four different stages [Katiyar, 1989]. It is found that motion of freezing interface slows down with increase in heat generation rate, while it accelerates with respect to increased rate of convective cooling. iii