2. Benchmark solution

2.1. Calculation method used for the reference solution

\({T}_{(x,y,z,t)}={T}_{w}+\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\sum _{l=1}^{\infty }{a}_{\mathrm{nml}}\mathrm{exp}(-{\kappa }_{\mathrm{mnl}}^{2}\alpha \mathrm{.}t){\mathrm{Tcos}}_{(x,y,z,m,n,l)}\)

with \({\mathrm{Tcos}}_{(x,y,z,m,n,l)}=\mathrm{cos}(\frac{(2m-1)\pi x}{2{L}_{1}})\mathrm{cos}(\frac{(2n-1)\pi y}{2{L}_{2}})\mathrm{cos}(\frac{(2l-1)\pi z}{2{L}_{3}})\)

\({a}_{\mathrm{mnl}}=\frac{64({T}_{0}-{T}_{w})}{{\pi }^{3}(2m-1)(2n-1)(2l-1)}\mathrm{sin}(\frac{(2m-1)\pi }{2})\mathrm{sin}(\frac{(2n-1)\pi }{2})\mathrm{sin}(\frac{(2l-1)\pi }{2})\)

\({\kappa }_{\mathrm{mnl}}={(\frac{(2m-1)\pi }{2{L}_{1}})}^{2}+{(\frac{(2n-1)\pi }{2{L}_{2}})}^{2}+{(\frac{(2l-1)\pi }{2{L}_{3}})}^{2}\)

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

Reference values are obtained with \(m=n=l=100.\)

2.2. Benchmark results

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

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.