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## Homework Statement

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).

## Homework Equations

Hoop stress

3-D heat conduction equation

## 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

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?

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