Analytical solution of non-Fourier heat conduction in a 3-D hollow sphere under time-space varying boundary conditions

In the current research, a comprehensive analytical technique is presented to evaluate the non-Fourier thermal behavior of a 3-D hollow sphere subjected to arbitrarily-chosen space and time dependent boundary conditions. The transient hyperbolic thermal diffusion equation is driven based upon the Cattaneo-Vernotte (C-V) model and nondimensionalized using proper dimensionless parameters. The conventional procedure of separation of variables is applied for solving the 3-D hyperbolic heat conduction equation with general boundary conditions. In order to handle the time dependency of the boundary conditions, Duhamel's theorem is employed. Furthermore, for the purpose of demonstrating the applicability of the obtained general solution, two particular cases with different time-space varying boundary conditions are considered. Subsequently, their respective non-Fourier thermal characteristics are elaborately discussed and compared with the Fourier case. The quantitative analysis is carried out, including the profiles of the time-dependent temperature and 3-D distributions of temperature at different time frames. Eventually, the influences of the controlling factors such as Fourier number and Vernotte number on the temperature field distributions within a hollow sphere for both cases are assessed. The findings reveal that the lag time in the hyperbolic thermal propagation diminishes with a decrement of Vernotte number, and it asymptotically vanishes for the Fourier case. Also, the number and severity of the jump points that occurred in the non-Fourier cases decrease by increasing the Fourier number, and these points finally vanish at particular Fourier numbers.