7. Long Fin

\[\theta = C_1e^{-mx} + C_2e^{mx} \tag*{(1)}\] For a sufficiently long fin, it is reasonable to assume that the temperature of the fin tip approaches \(T_\infty\).

Boundary Conditions (BC): \[\begin{aligned} {3} \text{BC-1} \qquad \theta &= \theta_o = (T_o-T_\infty) & \qquad \text{at} \quad x&=0 \\ \text{BC-2} \qquad \theta &= 0 \qquad & \text{at} \quad x&\rightarrow \infty \end{aligned}\] Application of BC-2 to Eqn.(1) gives \[\begin{aligned} 0 &= C_1e^{-\infty m} + C_2e^{\infty m} \\ &= 0 + C_2C_3 \\ \Longrightarrow \quad C_2 &=0 \end{aligned}\] And, from BC-1 to Eqn.(1) we get \[\theta_o = C_1 e^{-0m} \qquad \Longrightarrow \quad C_1=\theta_o\]


Hence, \[\theta = \theta_o e^{-mx} \qquad \Longrightarrow \quad \frac{\theta}{\theta_o} = e^{-mx}\] i.e., \[\frac{\theta}{\theta_o} = \frac{T-T_\infty}{T_o-T_\infty} = e^{-mx}\] Heat flow through the fin (\(Q\)) is given by \[Q = \int_{x=0}^L hP\theta dx\] or, \[Q = -Ak\left.\frac{d\theta}{dx}\right|_{x=0}\] From any of the above two equations, we get \(Q = Akm\theta_o\).

i..e, \[\boxed{Q = \theta_o\sqrt{PhkA}}\]