Question

A plane wall with constant properties is initially at a uniform temperature To. Suddenly, the surface at x = L is exposed to a convection process with a fluid at T[infinity](>To) having a convection coefficient h. Also, suddenly the wall experiences a uniform internal volumetric heating ˙ q q˙ that is sufficiently large to induce a maximum steady-state temperature within the wall, which exceeds that of the fluid. The boundary at x = 0 remains at To.

Answer 1

Answer:

The distribution is as depicted in the attached figure.

Explanation:

From the given data

• The plane wall is initially with constant properties is initially at a uniform temperature, To.
• Suddenly the surface x=L is exposed to convection process such that T∞>To.
• The other surface x=0 is maintained at To

• Uniform volumetric heating q' such that the steady state temperature exceeds T∞.

Assumptions which are valid are

1. There is only conduction in 1-D.
2. The system bears constant properties.
3. The volumetric heat generation is uniform

From the given data, the condition are as follows

Initial Condition

At t≤0

$T(x,0)=T_o$

This indicates that initially the temperature distribution was independent of x and is indicated as a straight line.

Boundary Conditions

At x=0

$T(0,t)=T_o$

This indicates that the temperature on the x=0 plane will be equal to To which will rise further due to the volumetric heat generation.

At x=L

$-k\frac{\partial T}{\partial x}]_{x=L}=h[T(L,t)-T_{\infty}]$

This indicates that at the time t, the rate of conduction and the rate of convection will be equal at x=L.

The temperature distribution along with the schematics are given in the attached figure.

Further the heat flux is inferred from the temperature distribution using the Fourier law and is also as in the attached figure.

It is important to note that as T(x,∞)>T∞ and T∞>To thus the heat on both the boundaries will flow away from the wall.