Storage Tank Design

A calculus problem arising in the operation of a storage tank for a chemical processing plant is described. Constructing an appropriate gauge for the storage tank requires computing the volume of water residing inside a tilted cylindrical pipe. The following calculus problem arose in the engineering design of a storage tank facility, and is possibly more interesting and challenging for students than the usual textbook exercises on computing volumes. It is also a good example for illustrating the usefulness of symbolic computation programs such as Maple [1]. Although the laborious part of this problem, evaluating the integral expressions, can be easily handled by computer, the student must still be able to do the fun (or possibly from their point of view, the hard) part of choosing the coordinate system and setting up the integrals. Consider a cylindrical pipe of radius r and length L which has been elevated an angle 0 from the horizontal (see Fig. 1); the pipe is capped at both ends. This is the basic geometry of a storage tank to be used in a chemical processing plant, and the operator of the facility must be able to continuously monitor the volume of water contained in the tank. As the simplest measurement that can be made is to observe the water level height h, designing a gauge requires knowing the volume of water V in the pipe as a function h. Depending upon the values of the parameters