Theses and Dissertations Collection

My thesis focuses on the evolution of the solid-liquid interface during melting and

solidication in a material with constant internal heat generation and prescribed heat

ux

at the boundary in a cylinder. A phase change process in which a material transitions

between two phases, solid and liquid, is known as a Stefan moving boundary problem.

We assume that the internal heat generation is constant and the same in both phases.

The material properties in both phases are also taken to be constant and equal. We assume

that the heat is transferred only by conduction and we neglect convection in the liquid phase.

In addition, we assume that there is a sharp interface between two phases where the phase

changes at a single melting temperature and there is no mushy zone at the interface. The

presence of the internal heat generation makes the problem nonhomogeneous. Starting from

the heat conduction equation, the approach nds steady-state and transient temperature

solutions in each phase and employs the separation of variables technique to nd transient

solutions. A nonlinear rst-order dierential equation with Fourier-Bessel series terms is

derived for the time-dependent motion of the interface. Analytic solutions for temperature

proles in each phase are derived. We do not introduce the Stefan number since there is only

one xed temperature: melting temperature. Instead we introduce dimensionless heat

ux

at the boundary. The initial value problem is solved numerically, and solutions compared to

the previously derived quasi-steady ones. It is shown that when the internal heat generation

and the heat

ux at the boundary have a close range of values, it takes longer for the front

to reach steady state than when the values are farther apart. As the dierence between

the internal heat generation and the

ux increases, the transient solution becomes more

dominant and the numerical solution of the phase change front does not reach steady-state

before the outer boundary or centerline is reached. This shows that the prescribed heat

ux can be used as a parameter that controls the motion of the interface. The problem has

applications for a nuclear fuel rod during meltdown.