Pierre KochConsultant

Brussels

A simple calculator program solves problems of partial liquid volumes for a variety of storage and process vessels, including inclined cylindrical vessels and those with conical heads.

Engineers in the oil refining and chemical industries are often confronted with the problem of estimating partial liquid volumes in storage tanks or process vessels.

Formulas are unwieldy tables require interpolation exercises, and programs are available for only limited types of cases.

Such topics as cylindrical vessels have not, to the author's knowledge, been treated in a simple way. Shortcuts and simplifying assumptions are often used. These assumptions introduce errors of unknown magnitude.

Cistern, the calculator program presented here, allows fast and accurate resolution of such problems for a wide range of vessels without user intervention, other than inputting the problem data. Running the program requires no mathematical skills.

Cistern is written for Hewlett-Packard HP 41CV or HP 41CX programmable calculators (or HP 41C with extended memory modules).

### PROBLEMS

Cistern handles three types of problems:

- Calculation of the liquid volume in a specified vessel for a given liquid level
- Calculation of the level reached by a given volume of liquid in a specified vessel
- Calculation of the vessel size required to store a given volume of liquid, for an imposed filling percentage and ratios of dimensions to diameter.

### VESSELS

Cistern accepts both spherical and cylindrical vessels with the following types of heads:

- Type 1: Flat
- Type 2: Torispherical or spherical
- Type 3: Toriconical or conical
- Type 4: Ellipsoidal (particular case, hemispherical).

Fig. 1 shows the pertinent heads and the geometrical parameters that define them. Cylindrical vessels may be horizontal, vertical or inclined on the horizontal by any angle between 0 and 90.

Fig. 2 defines and illustrates the nomenclature used throughout this article.

### MATHEMATICAL FORMULAS

The total volume of the vessel and the partial liquid volume in the vessel are obtained by adding the volumes of, and the partial liquid volume in, the various parts of the vessel.

Explicit formulas exist for the total volumes of spherical, conical, torical, ellipsoidal, and cylindrical parts, as well as for partial liquid volumes in all vertical parts, in horizontal and inclined cylinders, and in horizontal ellipsoidal heads.

For all other types of heads, there are no explicit expressions for the partial liquid volumes.

The volumes must be estimated by integration methods.

A listing of these formulas is given in the accompanying box.

The formula for the partial liquid volume in an inclined cylinder is shown and illustrated in Fig. 3.

Equation 1 (Fig. 3) results from the integration of the wetted cross section, Sx, between x1 and x2.

If the level intersects the lower left side of the cylinder, then x1 = 0. If it intersects the higher right side, then x2 = L.

If the level intersects only the cylindrical part, as shown in Fig. 3, then the formula takes the much simpler form shown in Equation 2 (Fig. 3).

Equation 2 can be derived directly from simple geometrical considerations.

### CISTERN PROGRAM

Fig. 4 is the overall logic diagram for the Cistern program.

A detailed description of the program would be extensive and of little use.

Table 1 shows example data registers for spherical and cylindrical vessels.

### PROCEDURE FOR USE

Cistern can be run with any consistent system of units (m and cu m, ft and cu ft, etc.). Angle of incline, alpha, is specified by the user in degrees (the program converts it to radians for all calculations).

The procedure for running the program follows:

- Clear the calculator memory and partition the memory to have 56 data registers (XEQ SIZE 056).
- Load the program into the calculator.
- Input the problem data into their respective registers, as shown in Table 1.
- Start the run as indicated in Table 2.

The run terminates with a beep, with VVL displayed for Problem 1, and h displaced for Problems 2 and 3. All other calculated data can be found by recalling the appropriate data register, as per Table 1.

When several problems are run successively on the same vessel (not necessarily with the same slope), the second and consecutive runs can be initiated by XEQ I instead of XEQ A or XEQ F, and by XEQ J instead of XEQ B or XEQ G. This will speed up the run.

### EXAMPLE

Design a cylindrical vessel to store 50 cu m of heavy fuel, with a filling percentage of 80%. To ease the removal of settled solids, a slope of 15 on the horizontal is used.

The vessel shall have a lower conical head with d1/D = 0.3, without torical junction, and a higher torispherical head with R2/D = 1 and r2/D = 0.1. In addition, L/D = 4.

This is a "Problem 3" type calculation. Following the instructions in Table 1:

Type of vessel: 3.2 STO 00

L/D = 4 STO 13

d1/D = 0.3 STO 14

r1/D = 0 STO 15

R2/D = 1 STO 16

r2/D = 0.1 STO 17

alpha = 15 STO 18

Take n = 10 STO 20

FP = 80% STO 21

VVL = 50 cu m STO 22

Start the run: XEQ C

The run terminates with a beep with h in display: 3.633 m.

Determine that D = 2.660 m (RCL 01), L = 10.642 m (RCL 02), Vv = 62.5 cu m. (RCL 23), etc.

What is liquid volume for h = 1. 5 m?

Answer: 1.5 STO 19, XEQ I leads to VVL = 9.437 cu m

What is the liquid level height for VVL = 36 cu m?

Answer: 36 STO 22, XEQ J leads to h = 2.893 m.

Editor's note: OGJ subscribers may obtain a free, written (hard) copy of the complete program (1063 entries) by sending a self-addressed, postage paid or stamped envelope to Refining/Petrochemical Editor, Oil & Gas Journal, P.O. Box 1941, Houston, Tex., 77251.

Subscribers outside the U.S. simply send a self-addressed envelope without postage to the same address. This offer will expire July 30, 1992.

*Copyright 1992 Oil & Gas Journal. All Rights Reserved.*