2. Benchmark solution

2.1. Calculation method used for the reference solution

A semi-analytical solution involving error functions is available:

\(\mathit{erf}(x)\mathrm{=}\frac{2}{\sqrt{\pi }}{\mathrm{\int }}_{0}^{x}{e}^{\mathrm{-}{t}^{2}}\mathit{dt}\) and \(\mathrm{erfc}(x)=\frac{2}{\sqrt{\pi }}{\int }_{x}^{+\infty }{e}^{-{t}^{2}}\mathrm{dt}\)

This solution is valid for a semi-infinite medium, so it can only be used in a limited range of variation in the time variable.

Either

the position of the solid/liquid interface. Let’s say \({s}_{t}=\frac{L}{\sqrt{{t}_{\mathrm{total}}}}\) and \(\lambda =\frac{{s}_{t}}{2\sqrt{{d}_{s}}}\) where \({d}_{s}\) and \({d}_{l}\) designate the diffusivity of solid and liquid media \(({d}_{s}=\frac{{k}_{s}}{{c}_{s}},{d}_{l}=\frac{{k}_{l}}{{c}_{l}})\). The solution to the heat equation is of the form:

\({T}_{s}(x,t)={T}_{0}+\frac{{T}_{m}-{T}_{0}}{\mathrm{erf}(\lambda )}\mathrm{erf}(\frac{x}{2\sqrt{{d}_{s}t}})\) if \(x\le {x}_{t}\)

\({T}_{l}(x,t)={T}_{i}+\frac{{T}_{m}-{T}_{i}}{\mathrm{erfc}(\lambda \sqrt{\frac{{d}_{s}}{{d}_{l}}})}\mathrm{erfc}(\frac{x}{2\sqrt{{d}_{l}t}})\) if \(x\ge {x}_{t}\)

The data of \({t}_{\mathrm{total}}\) is enough to define the solution, so we set \({t}_{\mathrm{total}}=420.\)

2.2. Benchmark results

TEMPS : Absciss	0.5	1.0	1.0	1.5	2.5	2.5	3.0
.000
.005	682.43	661.33	661.33	647.50	638.74	632.69	628.20
.010	726.05	705.75	705.75	692.06	682.43	675.24	669.63
.015	738.11	728.70	728.70	718.44	709.60	702.23	696.06
.020	739.86	737.22	737.22	731.99	726.05	720.27	714.94
.025		739.50	737.56	737.56	734.47	730.81	727.00
.030		739.93	739.39	739.39	738.11	736.20	733.88
.035		739.99	739.88	739.88	739.45	738.61	737.40
.040			739.98	739.86	739.86	739.55	739.00
.045				739.97	739.87	739.65
.050					739.97	739.89
.055						739.97
.060
.065
.070
.075
.080
.085
.090
.095
.100

TEMPS : Absciss	3.5	4.0	4.0	5.0	5.5	5.5	6.0
.000
.005	624.68	621.84	621.84	619.48	617.48	615.25	614.25
.010	665.09	661.33	661.33	657.43	653.65	650.37	647.49
.015	690.83	686.33	686.33	682.43	678.99	675.95	673.22
.020	710.11	705.75	705.75	701.81	698.25	709.92	692.06
.025	723.23	719.60	719.60	716.17	712.95	720.89	707.09
.030	731.34	728.70	728.70	726.05	723.43	728.48	718.44
.035	735.89	734.18	734.18	732.34	730.43	733.42	726.53
.040	738.21	737.22	737.22	736.07	734.79	736.44	731.99
.045	739.29	738.77	738.77	738.11	737.33	738.18	735.47
.050	739.74	739.50	739.50	739.15	738.71	739.12	737.56
.055	739.91	739.81	739.81	739.65	739.42	739.60	738.75
.060	739.97	739.93	739.93	739.86	739.75	739.83	739.39
.065	739.99	739.98	739.98	739.95	739.90	739.93	739.72
.070		739.99	739.98	739.98	739.96	739.97	739.88
.075				739.99	739.99	739.95
.080						739.98
.085						739.99
.090
.095
.100

(In \(°C\), according to the abscissa in meters and the time in seconds).

Note:

We limit ourselves to variations during the first 6 seconds, beyond 10 seconds the limit condition at the end \(x=1\) is no longer assured.

2.3. Uncertainty about the solution

Unknown, due to evaluation of error functions.

2.4. Bibliographical references

• Mr. Necati Özisik - Heat Conduction - Chapter 10: Phase-change problems example 10-3 - John Wiley & Sons.