A plane wall of thickness 0.1 m and thermal conductivity 25 W/m·K having uniform volumetric heat generation of 0.3 MW/m3 is insulated on one side, while the other side is exposed to a fluid at 32°C. The convection heat transfer coefficient between the wall and the fluid is 400 W/m2·K. Determine the maximum temperature in the wall.

T = 167 ° C

Explanation:

To solve the question we have the following known variables

Type of surface = plane wall ,

Thermal conductivity k = 25.0 W/m·K,  

Thickness L = 0.1 m,

Heat generation rate q' = 0.300 MW/m³,

Heat transfer coefficient hc = 400 W/m² ·K,

Ambient temperature T∞ = 32.0 °C

We are to determine the maximum temperature in the wall

Assumptions for the calculation are as follows

Therefore by the one dimensional conduction equation we have

During steady state

= 0 which gives

From which we have

Considering the boundary condition at x =0 where there is no heat loss

  = 0 also at the other end of the plane wall we have

hc (T - T∞) at point x = L

Integrating the equation we have

from which C₁ is evaluated from the first boundary condition thus

0 =  from which C₁ = 0

From the second integration we have

From which we can solve for C₂ by substituting the T and the first derivative into the second boundary condition s follows

→ C₂ =

T(x) = and T(x) = T∞ +

∴ Tmax → when x = 0 = T∞ +

Substituting the values we get

T = 167 ° C

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