2. Benchmark solution#
2.1. Calculation method used for the reference solution#
\(T(x,y,t)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }{A}_{n}\mathrm{sin}\frac{n\pi x}{{L}_{x}}\mathrm{sin}\frac{j\pi y}{{L}_{y}}\mathrm{exp}\left[-(\frac{{\lambda }_{x}{n}^{2}{\pi }^{2}}{{L}_{x}^{2}}+\frac{{\lambda }_{y}{j}^{2}{\pi }^{2}}{{L}_{y}^{2}})t/\rho c\right]\)
where \({A}_{n}=\left[\frac{4({T}_{i})}{{\pi }^{2}jn}[{(-1)}^{n}-1][{(-1)}^{j}-1]-32\right]\frac{5}{9}\) \({T}_{i}=\frac{5}{9}{T}_{0}+32\)
Temperature in \(°C\) to \(t=\mathrm{4320s}\) |
||||||
3.0 |
-17.7778 |
-17.5742 |
-17.3905 |
-17.2448 |
-17.1515 |
-17.1189 |
2.7 |
-17.7778 |
-17.5764 |
-17.3948 |
-17.2507 |
-17.1581 |
-17.1262 |
2.4 |
-17.7778 |
-17.5832 |
-17.4077 |
-17.2684 |
-17.1790 |
-17.1482 |
2.1 |
-17.7778 |
-17.5945 |
-17.4291 |
-17.2979 |
-17.2137 |
-17.1847 |
1.8 |
-17.7778 |
-17.6102 |
-17.4590 |
-17.3391 |
-17.2620 |
-17.2355 |
1.5 |
-17.7778 |
-17.6302 |
-17.4970 |
-17.3914 |
-17.3235 |
-17.3002 |
1.2 |
-17.7778 |
-17.6542 |
-17.5426 |
-17.4541 |
-17.3973 |
-17.3777 |
0.9 |
-17.7778 |
-17.6816 |
-17.5949 |
-17.5261 |
-17.4819 |
-17.4667 |
0.6 |
-17.7778 |
-17.7120 |
-17.6526 |
-17.6056 |
-17.5753 |
-17.5649 |
0.3 |
-17.7778 |
-17.7444 |
-17.7142 |
-17.6903 |
-17.6749 |
-17.6696 |
0.0 |
-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 |
Reference values are obtained with \(n=j=1000\)
2.2. Benchmark results#
\(t=4\mathrm{320s}(\mathrm{1.2hr})\): temperature at the following points:
in \(x=0.6\): for \(y=0.6,1.5,2.4,3.0\)
in \(x=1.5\): for \(y=0.6,1.5,2.4,3.0\)
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.