How Do Pressure and Temperature Changes Affect Stresses in a Pressure Vessel?

[rock.freak667]

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?

 

[KataKoniK]

Thank you for sharing your problem with us. I am a scientist with expertise in thermodynamics and heat transfer, and I would be happy to help you with your question.

Firstly, you are on the right track by using the hoop stress equation to calculate the stresses due to pressure changes. This is a good starting point for analyzing the vessel's structural response to the changes in pressure.

For the thermal stresses, as you have correctly pointed out, you will need to consider the heat conduction equation. However, I would recommend using the full 3-D heat conduction equation, as it will give you a more accurate representation of the thermal stresses within the vessel. The simplified 1-D approach may not capture all the complexities of the temperature distribution within the vessel.

To solve the 3-D heat conduction equation, you will need to use numerical methods such as finite element analysis (FEA) or finite difference method (FDM). These methods will allow you to model the temperature distribution within the vessel and calculate the thermal stresses at different time intervals.

You mentioned that your data collection against time does not fit any equation trendline. This could be due to the non-linear nature of the heat conduction equation and the complexity of the vessel geometry. In this case, it would be challenging to solve the equation analytically. Therefore, using numerical methods would be the most suitable approach.

In terms of simplifying the situation, you can make some assumptions to simplify the FEA or FDM model. For example, you can assume that the vessel is axisymmetric, which means that the temperature distribution and stresses will only vary in the radial direction. This will reduce the complexity of the model and make it easier to solve.

In summary, to accurately determine how the stresses vary within the vessel due to changes in pressure and temperature, I would recommend using the full 3-D heat conduction equation and solving it using numerical methods such as FEA or FDM. This will give you a more precise understanding of the thermal stresses within the vessel. I hope this helps, and please feel free to ask any further questions. Good luck with your analysis!