Flow of Heat in a Sphere

Consider a hollow sphere with the internal radius r1, external radius r2, having constant coefficient of thermal conductivity λ. Internal surface of the sphere is held at temperature T1. A heat exchange with the environment having temperature T2 takes place on the external surface. Intensity of convective heat transfer is characterized by the heat transfer coefficient β.

Analytical solution of the problem has the form:

For numerical calculation consider 1/ 8 th part of the hollow sphere (see figure). On the lateral edges we specify symmetry conditions (the heat flux across the lateral edges is equal to 0).
Let us use the following data: internal radius of the sphere r1 = 150 mm external radius of the sphere r2 = 250 mm. Coefficient of thermal conductivity λ of the material of the sphere is equal to 47 W / m.K.
The temperature T1 on the internal surface of the sphere is 373.15 K (or 100 oC). The temperature of ambient environment T2 is equal to 298.15 K (or 25 oC), heat transfer coefficient β is equal to 100 W / (m2 . oC) .

After carrying out calculation with the help of AutoFEM, the following results are obtained:

Table 1. Parameters of finite element mesh

Table 2. Result "Temperature" at r= (3 r1+r2) / 4 = 0.175 m

Table 3.Result "Temperature" at r = (r1+r2) / 2 = 0.200 m

Table 4. Result "Temperature" at r = (r1+ 3 r2) / 4 = 0.225 m

Table 5. Result "Temperature" at r = r2 = 0.250 m

 

 

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

 

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