A hollow cylinder is heated on the outside in such a way that the cylinder is maintained at a uniform temperature. The outside of the cylinder is otherwise insulated, so the energy must be transferred by radiation from the inside of the cylinder through the cylinder ends. The system is in a vacuum

Question

A hollow cylinder is heated on the outside in such a way that the cylinder is maintained at a uniform temperature. The outside of the cylinder is otherwise insulated, so the energy must be transferred by radiation from the inside of the cylinder through the cylinder ends. The system is in a vacuum, so radiation is the only mode of energy transfer. As shown in Figure 5.10, there is a disk centered on the cylinder axis and facing normal to the ends of the cylinder. The disk is exposed on both sides, so the side facing away from the cylinder radiates to the surroundings at [latex]T_e[/latex]. The other side receives radiation from the inside of the cylinder. Provide equations to compute the disk temperature, [latex]T_d,[/latex] if the inside of the cylinder is at a uniform temperature [latex]T_c[/latex].

Accepted Answer

The heat loss from the surface of the disk facing away from the cylinder can be considered as radiation into a large environment at [latex]T_e.[/latex] Hence from Equation 5.24, the net energy flux leaving this surface is

[latex]Q_1=A_1\epsilon _1(T_1)\sigma (T_1^4-T_2^4) [/latex] (5.24)

[latex]q_{d,e}=\epsilon _{d,e} \sigma (T_d^4-T_e^4).[/latex] The exchange with the cylinder is analyzed as a three-surface enclosure consisting of the inside of the cylinder, the side of the disk facing the cylinder, and the open area between the disk and the cylinder (a frustum of a cone) combined with the open area at the left side of the cylinder (this is a first approximation without subdividing the cylinder and disk areas to achieve greater accuracy). In the configuration-factor designations, the subscript e designates the combination of the two open areas at [latex]T_e.[/latex]Since the energy lost to the environment is not asked for, only two equations are written, one for the inside of the cylinder and one for the surface of the disk facing the cylinder. The [latex]q_{d,c}=-\epsilon _{d,e} \sigma (T_d^4-T_e^4).[/latex] is the radiative heat flux supplied to the left surface of the disk. From Equation 5.29 the two equations are

[latex]q_c\left\lgroup\frac{1}{\epsilon _c}-F_{c-c}\frac{1-\epsilon _c}{\epsilon _c} \right\rgroup-q_{d,c} F_{c-d}\frac{1-\epsilon _{d,c}}{\epsilon _{d,c}} =F_{c,d}\sigma (T_c^4-T_d^4)+F_{c-e}\sigma (T_c^4-T_e^4)[/latex] (5.31)

[latex]q_{c} F_{d-c}\frac{1-\epsilon _{c}}{\epsilon _{c}} +q_{d}\frac{1}{\epsilon _{d,c}} =F_{d-c}\sigma (T_d^4-T_c^4)+F_{d-e}\sigma (T_d^4-T_e^4)[/latex] (5.32)

The [latex]q_c[/latex] is eliminated from Equations 5.31 and 5.32; this yields an analytical relation for [latex]T_d. [/latex]

Question 5.12: A hollow cylinder is heated on the outside in such a way tha...

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