Heat Transfer through a lagged sphere
i have the following question:
A storage vessel maybe modeled as a hollow sphere of uniform material. The internal and external radii of r1 and r2 respectively. The temperature of the contents of the sphere is 0in and 0out is the ambient room temperature. The thermal coefficient of the material is ksph. The outside of the sphere is fitted with a uniform layer of lagging, of thickness x, which has a coefficient of thermal conductivity klag. The convective heat transfer coefficient at the inside of the sphere is hin and at the the outside is hout.
I need to show that the rate of heat transfer q through convection and conduction through the lagged sphere can be modeled in the steady state by
q = 4*pi*a^-1(0in - 0out)
a = 1/ksph(1/r1 - 1/r2) + 1/klag(1/r1 - 1/r2 + x) + 1/hin*r1^2 + 1/hout(r2 + x)^2
Im not too sure where to start. Im pretty sure i have to use Fourier's law (d0/dr). I also understand that the 1/r1 - 1/r2 parts relate to the heat loss through the different radii of the sphere itself.
- Do 1/ksph etc come from the differentiation of a logarithm - dx/dy log x
- Also where do the square parts come into it - those relating to 1/hin*r1^2 and 1/hout(r2 + x)^2?
anything to start me off would be good!