1. The problem statement, all variables and given/known data Problem is described as follows, I have a pressure vessel which basically consists of a hollow cylindrical body with two hemispherical shells attached to both ends. Within the vessel, there is a gas flowing from top to bottom. Over a period of one week, the vessel was subjected to changes in both temperatures and pressures (inlet and outlet). I need to get how the stresses varied within the vessel (due to pressure and temp). 2. Relevant equations Hoop stress 3-D heat conduction equation 3. The attempt at a solution Both hemispheres and cylinders are thin, so at each time interval hence change in pressure I can calculate hoop stress using σ=PD/2t. However the thermal stress becomes a bit confusing as I am not sure how to model it/solve it. I am simplifying the situation by ignoring the convective element of the fluid flowing and concentrating on conduction. The heat equation is as follows [Broken] I can simplify my situation by converting the problem to 1-D such that my temperature function T will just be of t and r i.e. T = T(r,t). My main issue is determining how to get ∂T/∂t. Plotting my data collection against time doesn't really fit any equation trendline and just looks a bit erratic. or do I assume T(r,t)=X(r)Y(t) and solve the PDE using separation of variables which if I remember correctly will eventually give me a Fourier Series which might complicate my situation. Is there any way to make this easier to do by hand rather than an FEA simulation?