Spherical Storage Tank Development Through Mathematical Modeling of Constituent Sections

A sphere is the optimal geometry for a closed pressure vessel in the sense of being the most structurally efficient shape. A cylindrical vessel is somewhat less efficient for two reasons: (1) the wall stresses vary with direction, (2) closure by end caps can alter significantly the ideal membrane state, requiring additional local reinforcements. However the cylindrical shape may be more convenient to fabricate and transport.

Spherical storage tanks are preferred for storage of high pressure fluids. A spherical tank is considered stronger than its counterparts such as the common fixed roof tank, open top tank, and floating roof tank [1-3]. The even distribution of stresses on the sphere's surfaces, both internally and externally, generally means that there are no weak points.   Pressure inside a true spherical tank is known to be identical on every axis. Common storage tanks are comprised of numerous components or pieces of metal that are welded or bolted together, in the field or in the shop.  Welds and seams are generally accepted as weak points in high pressure scenarios. The spherical shape creates great strength in resisting these pressures and offers the least amount of exterior surface, which reduces the transfer of warmer ambient temperatures on the overall volume. Spherical storage tanks are more expensive to fabricate than the other common types, and become more economically feasible as the tank design gets larger. Vertical cylindrical tanks resting on the ground are however sometimes used in lieu of spherical tanks on account of their low manufacturing cost and higher capacities [1].

Spherical tank finds great use in applications involving pressure vessels and pressure vessels have been in wide use for many years in petroleum, chemical, military industries and also in nuclear power plants [4, 5]. Other recorded uses of spherical tanks include storing of various liquids ranging from non-flammable liquids to dangerous flammable or toxic chemicals with explosive nature and are installed almost in each sector of contemporary industry like nuclear, energy, chemical etc. [3]. During ordinary operation, the liquid storage. They are known to be usually subjected to high pressures and temperatures which may be constant or cycling.

3.1 Stress assumptions in spherical vessels

The following assumptions are made when analyzing the stresses in Spherical Vessels

1. All shear stresses are zero:

$\tau_{\mathrm{r\upsilon}}=\tau_{\mathrm{\upsilon r}}=0, \tau_{\mathrm{r} \theta}=\tau_{\theta \mathrm{r}}=0$ and $\tau_{\theta \mathrm{\upsilon}}=\tau_{\mathrm{\upsilon\theta}}=0$

2. The normal stress σ_rr varies from zero on the outside free surface to the negative of the pressure p on the inside surface. Again there is the need to neglect this value when compared to the other normal stresses and justify this assumption a posteriori.

The force generated on the shell (the surface of revolution) due to axially symmetrical load can be represented by meridional force (N_φ) and circumferential force (N_Ɵ) [6].

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Figure 2. Tensile stresses σ in a spherical pressure vessel [11]

A more general approach at finding the Tensile Stress σ is by cutting the sphere into two hemispheres [10, 11] as shown in Figure 2 which can be used to show that

$\sigma=\frac{P r}{2 t}$   (1)

where σ represents the tensile stresses in the wall of the vessel, the fluid pressure is p and t is the wall thickness

Any section that passes through the center of the sphere yields the same result.

3.2 Sections development

Mathematical modeling of spherical tank as previously developed was derived based on the structure and the output transfer function where the spherical tank system was identified as a nonlinear complex structure [12]. The framework developed in this work is however simplified and has been successfully used to develop some prototypes. Another previous Mathematical model of spherical screw lines developed also involved the use of relatively complex single-blade helicoid equation together with the equation of a curve of crossing of surfaces of the sphere and helicoid [13, 14].

In this work however, the approach employed in developing the spherical tank involved dividing the sphere into two hemispheres and subsequent division of each of the hemispheres into sections. The material used is mild steel and a simplified procedure was established for the section/profile development and the number of sections needed to obtain a near-perfect spherical shape established. Each of the sections developed is as represented by the profile PCD in Figures 3 and 4.

The procedure for the sectional development (Figures 3 and 4) involves the following

1. Divide the sphere into two hemispheres
2. If the number of sections/profiles each hemisphere will be divided into is represented by n₁, then the arc length CD in Figure 3 will be the ratio of the circumference of one of the great circles by n₁. That is, length of each of the base of the divided spherical section is

$X_{0}=^{2 \pi R} / n_{1}$   (2)

3. If the angle subtended by the arc CD at the centre E of the sphere is α, then $\alpha=\frac{360}{n 1}$; (See Figure 6)
4. The length of arc PQ (Figures 3c and 4) is $L=\frac{90}{360} 2 \pi R=\frac{\pi R}{2}$
5. The development of the sectional profile PCD in Figure 3 with the arc length PQ transforming into the height of the bisecting line PQ of magnitude $L=\frac{\pi R}{2}$ is shown in Figure 5
6. In contrast to the basic mass-balance equation used by others [1, 4, 5, 16] to analyze the minor circles in a hemispherical shape, the method employed here is a simple geometrical analysis of the shape. The arc length AB of one of the minor circles of the sphere in Figure 3a is represented by x₅ in the development shown in Figure 5 with a corresponding minor radius of r₅ and a vertical height h₅ from the sphere horizontal diametral plane as illustrated in Figure 4
7. The height PE in Figure 3a represents the radius R of the sphere and can as well be divided into n₂ number of sections with n₂ representing the number of minor circles in each hemisphere as illustrated in Figures 4 and 5
8. The angle each arc of the minor circles such as arc AB in Figure 3a subtends at a distance of radius R from the centre E of the sphere is represented by β as shown in Figure 4.
9. Analyses of such sectors of the minor circles described in (viii) above are as shown in Figures 7 to 9 with the accompanying Mathematical formulations.

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Figure 3. Sectional development

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Figure 4. Hemisphere used for mathematical analysis

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Figure 5. Sectional divisions along minor circles

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Figure 6. Major circle sectional analysis

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Figure 7. First minor circle sectional analysis

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Figure 8. Fifth minor circle sectional analysis

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Figure 9. n₂th minor circle sectional analysis

From Figure 6 and using the Cosine Rule for the cord length C₀ is

$C_{0}^{2}=R^{2}+R^{2}-2 . R . R \cos \alpha$

$C_{0}=\sqrt{2 R^{2}(1-\cos \alpha)}$

$C_{0}=\sqrt{2 R^{2}(1-\operatorname{Cos}(360 / n 1))}$  (3)

Length of the arc CD from Figures 5 and 6 is

$x_{0}=\frac{2 \pi R}{n 1}=\frac{\alpha}{360} 2 \pi R$ (4)

Using Figures 7 to 9;

From the Cosine Rule, Cord length C₁ corresponding to arc length x₁ is

$C_{1}^{2}=R^{2}+R^{2}-2 . R . R \operatorname{Cos} \beta_{1}$

$C_{1}=\sqrt{2 R^{2}\left(1-\operatorname{Cos} \beta_{1}\right)}$   (5)

where β₁=90/n₂ (see Figure 4)

From Figure 7, using Pythagoras theorem and taking h1 = h = R/n₂; $d_{1}=\sqrt{C_{1}^{2}-h^{2}}$; $r_{1}=R-d_{1}$ (see Figure 8).

Similarly from Figure 8,

$C_{5}=\sqrt{2 R^{2}\left(1-\cos \beta_{5}\right)}$

where $\beta_{5}=5 \beta_{1}=5 * 90 / n_{2}$ and $d_{5}=\sqrt{C_{5}^{2}-h_{5}^{2}}=\sqrt{C_{5}^{2}-(5 h)^{2}}$; $r_{5}=R-d_{5}$

For n₂ number of divisions which is equivalent to the number of minor circles obtained for each of the two hemispheres of the spherical shape of interest,

$\begin{aligned} d_{n 2}=\sqrt{C_{n 2}^{2}-h_{n 2}^{2}} &=\sqrt{C_{n 2}^{2}-\left(n_{2} . h\right)^{2}} \\ r_{n 2} &=R-d_{n 2}=R-R=0 \end{aligned}$     (6)

since $d_{n 2}=R$ (7)

Eq. (7) implies that there is no minor circle at the tip of the sphere.

With reference to Eq. (6);

Since $d_{n 2}=R$; together with n₂ and h are known

Eq. (6) becomes

$R=\sqrt{C_{n 2}^{2}-\left(n_{2} \cdot h\right)^{2}} \quad$ or

At $x_{o}=\frac{2 \pi R}{n 1} ; \mathrm{d}_{\mathrm{o}}=0 ; \mathrm{h}_{\mathrm{o}}=0$

This paper has developed mathematical and procedural frameworks that will facilitate the development of spherical storage tanks. The effect of the number constituent sections on the sphericalness obtainable was also illustrated by developing a 225-litre capacity spherical storage tank using ten and twenty sections respectively. It is hoped that the framework developed will encourage the development and usage of spherical tanks when they are needed and not approximate them to cylindrical tanks with hemispherical caps.