In this post, I want to share to you how to easily estimate the time needed for the vessel to collapse. The equation involves several simple and basic equation, such as conservation principle and adiabatic compression equation. Let’s dig into the example.

There is a thermally insulated vessel initially at atmospheric pressure and partially filled with water. The vessel is fed with additional water at a constant flow rate of 2 m^{3}/h. The air contained in the vessel headspace is compressed as the additional water is fed. The vessel can withstand a maximum pressure of 3 atm absolute.

Vessel empty volume is 3 m^{3} and initial pressure of vessel is 1 atm absolute. Initial air temperature is 300 K. The ratio of specific heat at constant pressure is 1.4.

Find the air temperature when the vessel collapses and what time needed to make the vessel collapse.

Assume the air is not absorbed by water.

__Solution__

First, we compile the data based on the problem:

- Additional water flow rate = 2 m
^{3}/h - Vessel empty volume = 3 m
^{3} - Initial air temperature = 300 K
- Initial vessel pressure = 1 atm
- Maximum pressure the vessel can withstand = 3 atm
- Gamma (Cp/Cv) = 1.4

Based on mass conservation principle, the liquid volume in the vessel will increase according to the law:

Integrating Equation 1 to give the following equation:

As the water volume increase, the air volume will decrease in the headspace at the same rate:

Based on adiabatic compression:

Based on Equation 4, we got the value of Equation 5 as 300.

For adiabatic transformation:

Based on Equation 6, we got the value of Equation 7 as 4.66.

Then, the final volume of water that causing vessel to collapse is obtained based on Equation 6, which is 1.37 m^{3}.

Use equation below to estimate time needed for the vessel to collapse.

Therefore, the time needed for the vessel to collapse is 0.82 hours.

**Free Spreadsheet**

I share free spreadsheet of the above case, so that you can trace the equation and change the condition. Feel free to download!

I hope you find this post useful.

**Reference:**

Benintendi, Renato. 2021. *Process Safety Calculation *2^{nd} Edition. Elsevier