Question

Consider steady one-dimensional heat conduction in a plane wall, a long cylinder, and a sphere with constant thermal conductivity and no heat generation. Will the temperature in any of these media vary linearly? Explain.

Short answer

Answer: The temperature varies linearly for a plane wall, but not for a long cylinder or a sphere in the given conditions.

Step-by-step solution

Plane wall

For a plane wall, the heat conduction equation is given by Fourier's Law of heat conduction: q = -k * du/dx where q is the heat flux, k is the constant thermal conductivity, and du/dx represents the temperature gradient along the x-axis. Since we have no heat generation and constant thermal conductivity, this equation simplifies to: q = -k * du/dx = constant To find the temperature distribution, we integrate the equation with respect to x: u(x) = (-q/k) * x + C where C is a constant of integration. Thus, the temperature distribution in a plane wall is a linear function of the spatial coordinate x.

Long cylinder

In a long cylinder, heat conduction is described by the cylindrical heat conduction equation: q = -k * du/dr where q is the heat flux, k is the constant thermal conductivity, and du/dr represents the temperature gradient along the radial coordinate r. Dividing by r, we have: q / r = -k * du/dr q / k = (-du/dr)*r Integrating with respect to r, we find: u(r) = -[(q / k) * ln(r)] + C Since the temperature distribution in a long cylinder depends on the natural logarithm of the radial distance r, the temperature does *not* vary linearly in this case.

Sphere

For a sphere, heat conduction is described by the spherical heat conduction equation: q = -k * du/dr where q is the heat flux, k is the constant thermal conductivity, and du/dr represents the temperature gradient along the radial coordinate r. Now we have: qr^2 = -k * (dr) Integrating with respect to r, we find: u(r) = (q/3k)(R^3 - r^3) + C where R is the radius of the sphere. As the temperature distribution in a sphere depends on the cube of the radial distance r, the temperature does *not* vary linearly in this case either. In summary, for steady one-dimensional heat conduction with constant thermal conductivity and no heat generation, the temperature varies linearly for a plane wall but not for a long cylinder or a sphere.