2. Benchmark solution

2.1. Calculation method used for the reference solution

\(T(x,y,t)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }{B}_{n}\mathrm{cos}\frac{(\mathrm{2n}-1)\pi x}{2{L}_{x}}\mathrm{sin}\frac{j\pi y}{{L}_{y}}\mathrm{exp}\left[-(\frac{{\lambda }_{x}{(\mathrm{2n}-1)}^{2}{\pi }^{2}}{4{L}_{x}^{2}}+\frac{{\lambda }_{y}{j}^{2}{\pi }^{2}}{{L}_{y}^{2}})t\right]\)

where \({B}_{n}=\left[\frac{8{T}_{i}}{{\pi }^{2}j(\mathrm{2n}-1)}{(-1)}^{n+2}[{(-1)}^{j}-1]-32\right]\frac{5}{9}\) \({T}_{i}=\frac{5}{9}{T}_{0}+32\)

	Temperature in \(°F\) to \(t=1.2\mathrm{hr}(\mathrm{4320s})\)
2.7	—15,6151	—15,6480	—15,7455	—15,9049	—16, 1211	—16,3876	—16,6964	— 17,0381	—17,4022	—17,7778
2.4	— 15,6462	—15,6786	—15,7748	—15,9318	—16, 1449	—16,4076	—16.7120	—17,0487	—17,4076	—17,7778
2.1	—15,7391	—15,7700	—15,8620	—16.0122	—16,2160	—16,4673	—16,7584	—17,0805	— 17.4238	—17,7778
1.8	—15,8921	—15,9208	—16.0058	—16, 1447	—16,3333	—16,5657	—16,8349	—17,1328	—17,4503	—17,7778
1.5	—16, 1025	—16,1279	—16,2035	—16, 3269	—16,4944	—16,7009	—16,9401	—17,2048	—17,4869	—17,7778
1.2	—16,3655	—16,3869	—16,4506	—16,5547	—16,6959	—16,8700	—17.0716	—17.2947	—17.5325	—17,7778
0.9	—16,6744	—16,6911	—16,7409	—16.8222	—16,9325	— 17,0685	— 17,2261	—17,4004	—17,5862	—17,7778
0.6	— 17,0203	—17,0318	—17.0660	—17,1218	—17, 1975	— 17,2909	—17,3991	—17.5187	— 17.6462	—17,7778
0.3	—17,3923	—17,3982	—17.4156	—17.4440	—17.4825	— 17.5300	—17,5851	— 17.6459	—17.7108	—17,7778
0.0	—17,7778	—17,7778	—17,7778	—17,7778	—17,7778	—17,7778	—17,7778	—17,7778	—17,7778	—17,7778
\(Y\uparrow\) \(X\to\)	0.0	0.3	0.6	0.9	1.2	1.5	1.8	2.1	2.4	2.7

Reference values are obtained with \(n=j=1000\)

2.2. Benchmark results

\(t=\mathrm{1.2hr}(\mathrm{4320s})\): temperature at the following points:

• in \(x=0.0\): for \(y=0.6,1.5,2.7\),

• in \(x=0.9\): for \(y=0.6,1.5,2.7\),

• in \(x=1.8\): for \(y=0.6,1.5,2.7\).

2.3. Uncertainty about the solution

Analytical solution.

2.4. Bibliographical references

• J.C. Bruch Jr., G. Zyroloski, “Transient two-dimensional heat conduction problems solved by the finite element method”, Int. J. num. Meth. Engng, vol. 8, no. 3, pp 481-494, 1974.