# One dimensional transient heat flow of a hoop

1. Mar 11, 2009

### crawfs3

1. Problem: derive the energy balance from first principles of a hoop of inner radius ri, outer radius ro. the hoop material has a density of rho (p), heat capacity of c and thermal conductivity k. the center of the hoop has a temperature of T1 and the gas inside the hoop has a convection coefficient of h1. The temperature outside the hoop (not at the outer surface of the hoop) is T2 and the gas outside has a convection coefficient of h2.

2. Attempted solution: since its a hoop, cylindrical coordinates should be used. And since its a one dimensional problem the heat transfer in the y and z components is zero as well as the thermal generation term. I get the following equation: (1/r)d/dr(krdT/dr) = pcdT/dt. This is the equation for the conduction through the hoop and can be solved fully through integration and boundary conditions.

I can derive the temperature profile from the centre of the hoop to the edge of the inner radius using the equivalent thermal circuit but it wants the energy balance so I dont know what to for convection. Please help. Thanks.