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Transient heat transfer in a sphere

  • #1
1
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Homework Statement


A metal sphere is subjected to a heat flux, 5000 W/m2. It is originally at 20 C. How long does it take to heat to 90 C?
D = 5 cm
density = 8522 kg/m3
cp = 0.385 kJ/kg-K
k = 104 W/m-k

Homework Equations


rate of heat input = rate of heat accumulation
-k*A*dT/dr = m*cp*dT/dt

The Attempt at a Solution


I've never done one of these where I'm given a heat flux. If this said that the sphere was exposed to a constant external temperature, I would have no problem solving it, because I have those solutions to the differential equations in the book. I'm stuck on trying to solve that differential equation above though. I know that rate of heat input also equals the 5000 W/m2 that the problem statement gives. But if I say that 5000 = m*cp*dT/dt, integrate and solve that for time, it gives me an unreasonably short answer.
 

Answers and Replies

  • #2
19,953
4,107

Homework Statement


A metal sphere is subjected to a heat flux, 5000 W/m2. It is originally at 20 C. How long does it take to heat to 90 C?
D = 5 cm
density = 8522 kg/m3
cp = 0.385 kJ/kg-K
k = 104 W/m-k

Homework Equations


rate of heat input = rate of heat accumulation
-k*A*dT/dr = m*cp*dT/dt

The Attempt at a Solution


I've never done one of these where I'm given a heat flux. If this said that the sphere was exposed to a constant external temperature, I would have no problem solving it, because I have those solutions to the differential equations in the book. I'm stuck on trying to solve that differential equation above though. I know that rate of heat input also equals the 5000 W/m2 that the problem statement gives. But if I say that 5000 = m*cp*dT/dt, integrate and solve that for time, it gives me an unreasonably short answer.
What does the differential equation look like if the surface temperature were constant, rather than the surface heat flux?
 

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