A hot reservoir with a temperature of 700 K is 0.87 m away from a cold reservoir

Question

A hot reservoir with a temperature of 700 K is 0.87 m away from a cold reservoir with a temperature of 311 K. The two reservoirs are insulated from each other except for a rod of brass (k = 109 W/m-K) that has a cross-sectional area of 0.087 m2. The entire system is allowed to reach a steady-state condition.

2. Assume that you need to transfer 5885060 J of energy in ten minutes between the two reservoirs. To enhance the rate of energy transfer, a steel rod (k = 43 W/m-K) of the same length is added between the two reservoirs. What should the cross-sectional area of the steel rod be in order to achieve the proper rate of energy transfer?

Explanation / Answer

Energy transfer rate=-k × A × dT/dx

Energy transfer (E) = -k × A × dT/dx × t

cross-sectional area of the steel rod A = E /  -k ×dT/dx × t
k=thermal conductivity=(43 + 109)W/m.K = 152 W/m.K
A=cross sectional area=?
dt=temperature difference=(311-700)K = -389K
dx=distance between the reservoirs=0.87m
Time=10minutes=10 × 60=600 seconds

cross-sectional area of the steel rod A = E /  -k ×dT/dx × t

A = 5885060 J /( -152w/m-K * -389K * 600s)

A = 0.165 m2

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