where Ste is the Stefan number defined as Ste = [c.sub.u]([T.sub.ph] - [T.sub.0])/l.
In this model, the Stefan number is taken small compared to the unity.
The latent heat required for the phase change completion from the initial state, [L.sup.*], is measured here by the Stefan number
Ste = [C.sub.[infinity]]([T.sub.[infinity]] - [T.sub.f])/[L.sup.*].
The governing equations of the problem in enthalpy form are nondimensioned as equations set (1) using following nondimensional distance, time, temperature, enthalpy, Stefan number and boundary heat flux
The proposed approach is tested over a range of Stefan numbers and it is observed that the method is independent of the Stefan number.
Table 1 Comparison of RMS of computed heat flux for different Stefan numbers RMS Error (%) Stefan Number [X.sub.m] = 0.1 [X.sub.m] = 0.3 [X.sub.m] = 0.5 0.5 2.33 4.39 6.14 1 2.37 4.37 6.52 1.5 2.32 4.39 6.14 2 2.32 4.44 7.83 RMS Error (%) Stefan Number [X.sub.m] = 0.7 [X.sub.m] = 0.9 0.5 8.05 10.15 1 8.82 10.22 1.5 8.05 10.16 2 9.14 10.56 Table 2 Comparison of objective function and RMS of computed heat flux for different [sigma] Triangular Heat Flux [sigma] Objective Function RMS Error (%) 0 9.88E-07 0.056 0.02 2.28E-04 0.363 0.05 1.3E-03 0.58 0.1 4.9E-03 0.62 Step heat flux [sigma] Objective Function RMS Error (%) 0 9.65E-06 2.32 0.02 2.20E-04 2.70 0.05 1.3E-03 3.32 0.1 5E-03 3.62
The Stefan number St, which is the ratio of heat released during crystallization and the heat capacity of the body, is a constant for all four cases discussed above.
The Stefan number is the ratio of the latent heat to the product of the specific heat and imposed temperature difference.
where De is the thermal Deborah number (De = [[alpha]t.sub.c]/[D.sup.2]) and St is the Stefan number (St = [H.sub.f]/(Cp([T.sub.i] - [T.sub.[infinity]]))).