Calculating fluid volumes in partially filled tanks can be challenging. This article presents formulas that allow rapid and accurate computation of fluid volumes in partially filled vertical cylindrical tanks with concave bottoms, and in vertical tanks with identical top and bottom closed heads. These formulas supplement ones previously published for determining fluid volumes in partially filled horizontal and vertical cylindrical tanks of other commonly encountered shapes .
- When the fluid height in the closed-head vertical tank is below any portion of the top head, use the formulas for open-head tanks.
- When the fluid height is anywhere within the top head, compute the total tank volume by doubling the value computed using the open-head vertical tank formulas with a fluid height equal to half the total tank height. Then, subtract from that value the air space volume computed using the open-head vertical tank formulas with h = air space height.
Figure 3 is a diagram of a closed-head vertical cylindrical tank with applicable parameters. To simplify the illustration, the figure only shows convex and concave ellipsoidal heads, although the heads may be conical, ellipsoidal, spherical or torispherical. In the case of torispherical heads, a = a1 + a2 for convex heads and a = -(a1 + a2) for concave heads. Figure 3 shows h(v) = fluid height in a convex-bottom tank and h(c) = fluid height in a concave-bottom tank. In the fluid volume equations for these tanks, however, only h is used for fluid height; therefore, the user must make the correct association to the tank bottom configuration under consideration.
Figure 3. Only convex and concave ellipsoidal heads are shown on the vertical cylindrical tank, although the heads may also be conical, spherical or torispherical.
Equations for calculating the volumes of closed-head vertical cylindrical tanks where the top and bottom heads are identical in type and concavity are provided at left.
Vopen-head is the volume calculated by the formulas for open-head vertical tanks provided earlier in this article for concave-bottom tanks and in the previous article for convex-bottom tanks.
If the top and bottom heads of closed-head vertical cylindrical tanks differ in type or in concavity, the same technique described above can be used to compute the fluid volume for any level within the tank. However, careful manipulation of the formulas is necessary to account correctly for the difference in head types.
To compute the fluid volume in closed-head vertical elliptical tanks, the same formulas presented in the earlier article can be used to correct the volumes calculated using the formula here for closed-head vertical cylindrical tanks.
The following examples will serve to check that you are properly applying the principles and equations above and from the earlier article. In all cases, L = 120 in. and D = 13 in., while a = 33 in. for convex heads and a = -33 in. for concave heads. For torispherical heads, f = 0.71 and k = 0.081. All fluid heights extend into the top heads of the tanks. Tank details are as indicated in Table 2, where v = convex and c = concave.
Equations for Torispherical Concave Bottoms
Dan Jones is a senior process chemist for Stockhausen Louisiana LLC, Garyville, La.
Let's first consider vertical cylindrical tanks with concave bottoms. For open-top vertical tanks, Figures 1 and 2 describe possible bottom configurations and define the variables used in the equations. The value for "a," the distance that the bottom extends into the tank, must always be a negative number (or zero, in the case of a flat-bottom tank), i.e., a 0, to use the formulas in this article.
Figure 1 shows the parameters for conical, ellipsoidal and spherical concave bottoms. For the formula to apply, the ellipsoidal bottom must be exactly half of an ellipsoid of revolution. Likewise, the spherical bottom cannot protrude upward into the cylindrical portion of the tank a distance greater than the radius of the cylindrical portion of the tank.
Three Types of Concave Bottoms
Figure 1. Vertical cylindrical tanks with conical, ellipsoidal and spherical concave bottoms use the same variables in their formulas.
Figure 2 details the parameters for cylindrical vertical tanks with concave torispherical bottoms. For torispherical-bottom tanks, the dish parameter, f, must exceed 0.5, i.e., f > 0.5, and the knuckle parameter, k, must be greater than or equal to zero but less than or equal to 0.5, i.e., 0 k 0.5. The parameter "a" is not a formula input for a torispherical bottom; it can be calculated via a = -(a1 + a2).
Torispherical Concave Bottom
Figure 2. Variables such as the dish, f, and knuckle, k, parameters must be used for vertical cylindrical tanks with torispherical concave bottoms.
The equations for fluid volumes in vertical cylindrical tanks with concave bottoms are shown on p. 30. The volume of a flat-bottom vertical cylindrical tank may be found using any of these equations and setting a = 0. Radian angular measure must be used for trigonometric functions.
To check that you are correctly applying the equations, calculate the volumes in gallons of the following cylindrical tanks with concave conical, ellipsoidal and spherical bottoms for D = 113 in. and a = -33 in. and for a torispherical bottom where f = 0.71 and k = 0.081, for fluid heights of 15 in., 25 in. and 50 in. The answers are given in Table 1.
For the torispherical case, the calculated a = -(a1 + a2) = -27.22 in.
Fluid volumes in vertical elliptical tanks with any of the concave bottoms covered in this article may be calculated using the method described in a previous article  for correcting cylindrical volumes to elliptical volumes.
Concave Bottom Equations Address Three Types
Closed-head tanksCalculation of fluid volumes in closed-head vertical cylindrical tanks requires using the formulas presented in the earlier article  for convex bottoms and here for concave bottoms. Follow this procedure: