Question

Consider steady one-dimensional heat conduction through a plane wall, a cylindrical shell, and a spherical shell of uniform thickness with constant thermophysical properties and no thermal energy generation. The geometry in which the variation of temperature in the direction of heat transfer will be linear is (a) plane wall (b) cylindrical shell (c) spherical shell (d) all of them (e) none of them

Short answer

(a) Plane wall (b) Cylindrical shell (c) Spherical shell Answer: (a) Plane wall

Step-by-step solution

Recall the governing equations for heat conduction in different geometries

Recall the heat conduction equation for each of the three geometries: 1. Plane wall: \(\frac{\partial^2T}{\partial x^2} = 0\) 2. Cylindrical shell: \(\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right) = 0\) 3. Spherical shell: \(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right) = 0\) These are the simplified heat conduction equations for each geometry under steady state and one-dimensional conditions.

Solve the heat conduction equation for each geometry

Now, we'll solve the heat conduction equations for each geometry to find the temperature distribution in the direction of heat transfer. 1. Plane wall: Since the second derivative of temperature with respect to x is zero, the temperature will vary linearly with x. Solution: \(T(x) = Ax + B\) 2. Cylindrical shell: Solving the cylindrical equation, we have, \(\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right) = 0 \Rightarrow r\frac{\partial T}{\partial r} = C\) Integrate once, we obtain, \(\int \frac{\partial T}{\partial r} dr = \int \frac{C}{r} dr \Rightarrow T(r) = C \ln(r) + D\) 3. Spherical shell: Solving the spherical equation, we have, \(\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial T}{\partial r}\right) = 0 \Rightarrow r^2\frac{\partial T}{\partial r} = E\) Integrate once, we obtain, \(\int \frac{\partial T}{\partial r} dr = \int \frac{E}{r^2} dr \Rightarrow T(r) = -\frac{E}{r} + F\)

Identify the geometry with linear temperature distribution

Analyzing the temperature distribution solutions for each geometry: 1. Plane wall: \(T(x) = Ax + B\), which is linear with respect to x. 2. Cylindrical shell: \(T(r) = C \ln(r) + D\), which is not linear with respect to r. 3. Spherical shell: \(T(r) = -\frac{E}{r} + F\), which is also not linear with respect to r. From the above solutions, only the plane wall geometry exhibits a linear variation of temperature in the direction of heat transfer. Therefore, the correct answer is: (a) plane wall