Question

Consider a long pipe of inner radius \(r_{1}\), outer radius \(r_{2}\), and thermal conductivity \(k\). The outer surface of the pipe is subjected to convection to a medium at \(T_{\infty}\) with a heat transfer coefficient of \(h\), but the direction of heat transfer is not known. Express the convection boundary condition on the outer surface of the pipe.

Short answer

Answer: The convection boundary condition on the outer surface of the pipe can be expressed as: \(-k \frac{d T}{d r}\Big|_{r=r_2} = h (T_s - T_{\infty})\).

Step-by-step solution

Understand the conduction and convection processes

In this scenario, heat is transferred from the pipe to the surroundings in two ways: 1. Conduction: This is the process by which heat is transferred within the pipe through its walls. It depends on the temperature gradient and the pipe's thermal conductivity. 2. Convection: This is the process by which heat is transferred from the pipe's outer surface to the surrounding medium. It depends on the heat transfer coefficient and the temperature difference between the outer surface and the surrounding medium.

Write the heat transfer equation for conduction

Using Fourier's Law of conduction, the heat transfer rate due to conduction (\(q_c\)) can be given as: $$q_c = -k \frac{d T}{d r}$$ Where \(k\) is the thermal conductivity of the pipe and \(\frac{d T}{d r}\) is the temperature gradient across the pipe wall.

Write the heat transfer equation for convection

Using Newton's Law of cooling, the heat transfer rate due to convection (\(q_h\)) can be given as: $$q_h = h (T_s - T_{\infty})$$ Where \(h\) is the heat transfer coefficient, \(T_s\) is the temperature at the outer surface of the pipe, and \(T_{\infty}\) is the temperature of the surrounding medium.

Express the convection boundary condition on the outer surface of the pipe

At the outer surface (i.e., \(r=r_2\)), the heat transfer rate due to conduction should be equal to the heat transfer rate due to convection. Thus, the convection boundary condition can be given as: $$-k \frac{d T}{d r}\Big|_{r=r_2} = h (T_s - T_{\infty})$$ This equation represents the convection boundary condition on the outer surface of the pipe.