Hello, I really need help with this heat transfer question. The professor wrote

Question

Hello, I really need help with this heat transfer question. The professor wrote it himself so it is an "original" question.Here it is:

The value of thermal conductivity (k) is theoretically a constant. In practice, however, it may be a (relatively weak) function of temperature. Assume that k increases linearly with distance into the wall, i.e.:

k = k0 [1 + 1.8z]

a) Write an energy balance for a differential thickness of the wall. Integrate the equation to yield an expression for the temperature profile in the wall.

b) If k0 = 5.2 W/m-K, calculate a value of the heat flux (q”) through the wall.

c) If k were assumed constant (and equal to k0), what would be the value of the heat flux?

Any help provided will be really helpful thank you!

Explanation / Answer

1) equation of conductivity: kdT/dz=q

dT/q = dz/k = k0 ( 1+1.8z)^-1 dz

integrating...

(T2-T1)/q= k0 (0.555(ln(z+0.5555) +2.19722) this is the energy balance

2) (100-20)/q = 5.2( 0.555(ln(0.05+0.555) +2.19722)

q= 84.9249 W/m^2

3) k=5.2

dT/q =k dz

q= k dT / dz = 5.2 (100-20) / 0.05 = 832 W/m^2

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