Answer:
T=128°C
Explanation:
Given Data:
thickness = 7.5 cm ;
internal heat generation rate =
= 105W/m³ ;
First we have the equation of conduction for steady state withh heat generation as:
k
+
= 0
by re-arranging it we get
= -
/k
Here,
is the rate of internal heat generation,
k is thermal conductivity constant
dT is temperature differential
dx is differential along x-axis
Now if we integrate both sides w.r.t x, one derivative is removed from left side of equation and and a variable is multiplied to the term on the right side and a unknown integrating constant C is added.
= - (
/k)*x + C -------------------- equation (1)
Now if we apply Dirichlet's first boundary equation at x = 0, dT/dx = 0
we get,
= - (
/k)*x + C
0 = - (
/k)*0 + C
C = 0
Now put C = 0 in equation 1
we get,
k
= -
*x ------------------ equation (2)
Now integrate it w.r.t x
T= - (
/2k)*x² + C₁ ------------------ equation (3)
Now apply Dirichlet's second boundary condition on equation 2, i.e, x=L, k
= h⁻(T-T₀), equation 2 becomes
h⁻(T-T₀) = L
------------------ equation (4)
Put value of T from equation 3 in equation 4
h⁻(- (
/2k)*L² + C₁-T₀) = L
By rearranging it we get,
C₁= T₀ +
L[(L/2k)+(1/h⁻)] ------------------ equation (5)
Now putting value of C₁ from equation 5 in equation 3, we get
T = T₀ + (
/2k)[L² + 2kL/h⁻]
Now by putting the given values of T₀ = 90°C,
= 10 W/m², k = 12 W/mK, L = 0.075m, h⁻ = 500 W/m²K
we get, T= 128°C