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Transient heat transfer in a sphere

  1. Apr 29, 2016 #1
    1. The problem statement, all variables and given/known data
    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

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

    3. 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.
  2. jcsd
  3. Apr 29, 2016 #2
    What does the differential equation look like if the surface temperature were constant, rather than the surface heat flux?
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