# Lumped system analysis in transient heat conduction

I had this question where I had a sphere and a cylinder with given dimensions and propreties ($\rho$, C and k). He also gave me initial temperatures, then both of them dipped in a bath of water of given temperature but unknown h (convective coefficient). Then he gave me the temperature of the sphere after 2 mins and wants the temperature of the cylinder after 5 mins.

My first thought was to get h, which is relatively easy but i had a problem choosing between 2 methods: first one was to assume it was a lumped system (the sphere) and use the exp(Bi*F) rule where the h will be the only unknown, or I can use the Heisler Charts where I have the temperature ratio and the Fourier number and I can use them to get the 1/Bi.

In both cases the h will result in a lumped system in the cylinder, but the final temperatures in each case is different ( 5 degrees different), so it's pretty obvious that one of them is true.

My professor solved using the assumption, but when I asked him he said that would work too but never said which one is more right, so is there any way I can test for the lumped system assumption with an unknown h?

A pic of the question is attached (question number 6) 