球体の発熱による温度分布-熱源が一様ではなく，中心に局在している場合-32

球の内側（非発熱源）の熱拡散方程式

\(\Large T_{in} = -\frac{C_2}{r} + C_3 \)

\(\Large \frac{dT_{in}}{dr} \vert_{r=R} = \frac{C_2}{r^2} \vert_{r=R} = \frac{C_2}{R^2} = - \frac{p \ H^3}{3 \kappa_{in} R^2} \)

\(\Large C_2 = - \frac{p H^3}{3 \kappa_{in}} \)

\(\Large T_{in} = \frac{p H^3}{3 \kappa_{in}} \frac{1}{r} + C_3 \)

\(\Large T_{in} (R) = T_{m} (R) \)

より，

\(\Large \frac{p \ H^3}{3 \kappa_{in}} \frac{1}{R} + C_3 = \frac{p \ H^3}{3} \left[ \frac{1}{\kappa_m R} + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \)

\(\Large \begin{align*} C_3 &= \frac{p H^3}{3} \left[ -\frac{1}{ \kappa_{in} R} + \frac{1}{\kappa_m R} + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \\
&= \frac{p H^3}{3} \left[ \frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \end{align*} \)

\(\Large \begin{align*} T_{in} &= \frac{p H^3}{3 \kappa_{in}} \frac{1}{r} +\frac{p H^3}{3} \left[ \frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \\
&= \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} r}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \end{align*} \)

球の内側（発熱源）の熱拡散方程式

\(\Large T_h = -\frac{1}{6} \frac{p}{\kappa_h} r^2 + C_1 \)

\(\Large T_{h} (H) = T_{in} (H) \)

より，

\(\Large -\frac{1}{6} \frac{p}{\kappa_h} H^2 + C_1 = \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} H}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \)

\(\Large \begin{align*} C_1 &= \frac{1}{6} \frac{p}{\kappa_h} H^2 + \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} H}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \\
&= \frac{pH^2}{6} \left( \frac{1}{\kappa_h} + \frac{2}{\kappa_{in}} \right) + \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} H}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \end{align*} \)

\(\Large T_h = -\frac{1}{6} \frac{p}{\kappa_h} r^2 + \frac{pH^2}{6} \left( \frac{1}{\kappa_h} + \frac{2}{\kappa_{in}} \right) + \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} H}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \)

となります．

まとめ

\(\Large T_{out} = \frac{p \ H^3}{3 \kappa_{out}} \frac{1}{r} + R.T \)

\(\Large T_m = \frac{p H^3}{3} \left[ \frac{1}{\kappa_m r} + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \)

\(\Large T_{in} = \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} r}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \)

\(\Large T_h = -\frac{1}{6} \frac{p}{\kappa_h} r^2 + \frac{pH^2}{6} \left( \frac{1}{\kappa_h} + \frac{2}{\kappa_{in}} \right) + \frac{p H^3}{3} \left[ \frac{1}{\kappa_{in} H}+\frac{1}{R} \left( \frac{1}{\kappa_m}-\frac{1}{ \kappa_{in}} \right) + \frac{1}{R+M} \left( \frac{1}{\kappa_{out}} - \frac{1}{\kappa_m} \right) \right] + R.T \)

となります．

次に，具体的な値を入れてみましょう．