2. Benchmark solution

2.1. Calculation method used for the reference solution

\(T(x,y,z,t)={T}_{0}+{\mathrm{2q}}_{w}\frac{\sqrt{\alpha t}}{\lambda }(A+B+C)\) with:

\(A=\sum _{m=0}^{\infty }\left[\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{1}+x}{2\sqrt{\alpha t}}\right]+\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{1}-x}{2\sqrt{\alpha t}}\right]\right]\)

\(B=\sum _{m=0}^{\infty }\left[\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{2}+y}{2\sqrt{\alpha t}}\right]+\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{2}-y}{2\sqrt{\alpha t}}\right]\right]\)

\(C=\sum _{m=0}^{\infty }\left[\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{3}+z}{2\sqrt{\alpha t}}\right]+\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{3}-z}{2\sqrt{\alpha t}}\right]\right]\)

\(\alpha =\frac{\lambda }{\rho {C}_{p}}\)

Reference values are obtained with \(m=1000.\)

2.2. Benchmark results

Temperature at points: \(O(\mathrm{0,0}\mathrm{,0})\), \(H(0.5\mathrm{,0}.8\mathrm{,1}\mathrm{.})\) and \(C(1.\mathrm{,1}.6\mathrm{,2}\mathrm{.})\)

2.3. Uncertainty about the solution

Analytical solution.

2.4. Bibliographical references

• M.J. Chang, L.C. Chow, W.S. Chang, « Improved alternating direction implicit for solving transient three dimensional heat diffusion problems », Numerical Heat Transfer, vol. 19, vol. 19, pp 69-84, 1991.