Transient Heat Conduction in Nondeforming Systems

When no deformation occurs, the thermal energy equation reduces to the following form  

Here the source term, S, could represent the rate of energy generated per unit volume due to a phase change, absorbed radiation, or a chemical reaction. For a number of problems encountered in polymer processing this equation takes on two relatively simple forms for planar and cylindrical geometries. For a planar geometry (see Fig. 5.8), Eq. 5.75 becomes 

The solutions of Eqs. 5.76 and 5.77 are subject to various boundary conditions. For a slab of finite thickness (thickness 2b, with the axis at the center of the slab) the boundary conditions are usually given as constant surface temperatures or a step change in the surface temperature due to convection at the free surfaces. Mathematically for the first case the boundary and initial conditions are given as 

In the second case the boundary and initial conditions are given as 

Ta here is the temperature of the cooling fluid. In the case of the cylindrical geometry the corresponding boundary and initial conditions for the constant surface temperature (step change in temperature) or the step change in surface temperature due to convection are written, respectively, as 

---