storage tank design - Society for Industrial and Applied Mathematics
()1991
SIAM Review Vol. 33, No. 2 pp. 271-274, June 1991
Society for Industrial and Applied Mathematics OO5
STORAGE TANK DESIGN * L. J.
GRAY
Abstract. 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.
Key
words, volume integral, symbolic computation
AMS(MOS) subject
classification. 26A63
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.
(o,o,h)
FIG. 1. Problem geometry.
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 L, r, and 0, different circumstances can arise. For example, the elevation angle *Received by the editors February 11, 1990; accepted for publication July 17, 1990. This work was supported by the Y-12 Engineering Division, Martin Marietta Energy Systems, under contract DE-AC05-84OR21400 with the U.S. Department of Energy. Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831. 271
272
CLASSROOM NOTES
shown in Fig. 1 is large enough that the bottom cap of the pipe will be completely covered before the water begins to intersect the top cap. However, for the particular storage tank under consideration the opposite situation holds: 0 is sufficiently small, i.e., L sin(0) _< 2r cos(0), and the water will reach the top cap before the bottom is completely submersed. For computing the volume, it is convenient to employ a coordinate system along the axis of the pipe (Fig. 2). The z h plane defining the water level in the original coordinates becomes the plane sin(0)y + cos(0)z h, and the equation of the pipe is simply (z r) 2 + x 2 r 2, 0 _< y _< L. Figure 2 also shows the three possible volume configurations to be computed.
(sin O)y + (cos e)z=h
L
x
(a) z
z
(b) FiG. 2. The three configurations.
(a) If the water level
SIAM Review Vol. 33, No. 2 pp. 271-274, June 1991
Society for Industrial and Applied Mathematics OO5
STORAGE TANK DESIGN * L. J.
GRAY
Abstract. 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.
Key
words, volume integral, symbolic computation
AMS(MOS) subject
classification. 26A63
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.
(o,o,h)
FIG. 1. Problem geometry.
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 L, r, and 0, different circumstances can arise. For example, the elevation angle *Received by the editors February 11, 1990; accepted for publication July 17, 1990. This work was supported by the Y-12 Engineering Division, Martin Marietta Energy Systems, under contract DE-AC05-84OR21400 with the U.S. Department of Energy. Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831. 271
272
CLASSROOM NOTES
shown in Fig. 1 is large enough that the bottom cap of the pipe will be completely covered before the water begins to intersect the top cap. However, for the particular storage tank under consideration the opposite situation holds: 0 is sufficiently small, i.e., L sin(0) _< 2r cos(0), and the water will reach the top cap before the bottom is completely submersed. For computing the volume, it is convenient to employ a coordinate system along the axis of the pipe (Fig. 2). The z h plane defining the water level in the original coordinates becomes the plane sin(0)y + cos(0)z h, and the equation of the pipe is simply (z r) 2 + x 2 r 2, 0 _< y _< L. Figure 2 also shows the three possible volume configurations to be computed.
(sin O)y + (cos e)z=h
L
x
(a) z
z
(b) FiG. 2. The three configurations.
(a) If the water level
