Optimum Size of a Pressure Vessel

by John Duhaime

Submitted : Spring 2009

The objective of this project is to utilize two different equations to find the optimum size of a pressure vessel. This optimum size is determined by finding the optimum volume of a pressure vessel as well as the optimum pressure of the gas that it contains relative to the cost of the vessel. As a vessel becomes larger, more material is needed to create the vessel, which raises the overall cost. This is why gas is often pressurized; it can then be stored in smaller, less expensive vessels. Pressure also affects cost because as the pressure inside of the vessel increases, the walls of the vessel must in turn be made thicker to contain the gas. There is, though, an optimum size of the vessel that minimizes cost with respect to the volume and pressure. To find this optimum size, John must put the first equation (the state of gas equation) in terms of P, pressure, with respect to V, volume. He must then take this result and substitute it into the second equation (the cost equation). To find where cost is at a minimum, he must to differentiate the cost equation with respect V and set the derivative equal to 0. In doing so, he must solve the resulting quadratic to obtain two different values for the volume. To find out which value results in a minimum cost, he must substitute these values into the state of gas equation to solve for P for both values of V. He shall then substitute both sets of P and V into the cost equation to find which cost is lower. The lower cost will be the minimum cost, and the values for corresponding the pressure and volume in the lower cost equation will be the optimum size values. These results show how these equations can be used to minimize costs of storing various gases for many different purposes. These equations can be used to store gases such as helium, oxygen, and nitrogen in the most cost-effective manners. This could greatly reduce the amount of money that companies use in storing different types of gases, as well as enable them to optimize their storage space in doing so.

[ Back ]