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# Calculate the Wetted Surface Area of Pressure Vessels

By Babak Firoozi, Oilcor |

**This article provides a comparative study of two approaches to protect ASME semi-elliptical pressure tanks from overpressure in the event of a fire**

Calculating the wetted surface area is a crucial step to protect American Society of Mechanical Engineers’ (ASME) pressure vessels from overpressure in the event of a fire. The historical approach relies on an analytical approximation, using a linear equation to calculate partial surface area of 2:1 semi-elliptical (SE) heads that has been published [1]. This article presents a rigorous numeric analytical method and compares partial surface area calculations of 2:1 SE heads using both methods.

## Pressure relief from fire

The loss of fluids from enclosed ASME pressure vessels as a result of overpressure due to fire is a major safety concern at chemical process industries (CPI) facilities. Such an event can not only lead to worker injury and damage to mechanical assets, but has the potential for catastrophic losses to the facility, the surrounding community and the environment, as well. Fire is often the cause of the greatest relief load through a pressure safety valve (PSV), and thus becomes a critical aspect of appropriately sizing the PSV.

Fluids in a vessel absorb the heat of a fire through the wetted surface area (Figure 1). The fluid’s increase in volume due to vaporization then reaches a pressure that exceeds the pressure capacity of the fixed vessel volume. The expanded fluid must be relieved through the PSV to avoid rupturing the pressure vessel.

Per American Petroleum Inst. (API) Standard 521, the head absorption from fire is calculated by Q = C _{1} A _{WS} ^{0.82}, in which C _{1} is a constant and A _{ws} is the wetted surface area [2]. The absorbed energy across the wetted surface area provides heat required to boil the liquid.

## Surface area calculations

Heads of ASME pressure vessels are typically in the shape of an oblate spheroid at a ratio of 2:1 (Figures 1 and 2). An analytic approximation for wetted surface area of both 2:1 SE heads has been presented as S=2.168Dh.

The shape of ellipsoids is well defined by the general equation

x ^{2} /a ^{2} +y ^{2} /b ^{2} +z ^{2} /c ^{2} = 1

and its total surface area is defined by Knud Thomsen’s formula [1]:

Where p = ln(3)/ln(2).

An ellipsoid’s partial surface area is not defined but can be calculated by numeric analysis as the sum of finite segments of frustums along any axis. Table 1 shows the equations used for this calculation. A frustum between each elliptical xy-plane of any shaped ellipsoid (for example, an egg) along its z-axis is calculated using the following Equation 3:

A 2:1 SE head is an oblate spheroid (Figure 1), of which c = a and b = a/2. The calculation of each frustrum from a _{i+1} to a _{i} then reduces to:

Per the Pythagorean theorem, the semi-major axis (a) of each successive xy plane is

(Figure 2), along the height of the z-axis [3].

It is helpful to use a standard spreadhseet or other software because 100 or more frustums should be calculated for the best accuracy.

## Liquid volume

The liquid volume of a vessel may be required to determine the total relief load (total liquid inventory) during a fire. The liquid volume of a partially filled 2:1 SE head is calculated by

which is derived from the volume calculation of a spherical cap [3]. By rigorous numeric analysis, the volume is calculated as a sum of all zones of each cross-sectional x plane. As shown in Figure 3, the volume calculated by both methods are the same.

## Comparison

Compare the calculation of the wetted surface area of a 2:1 SE head by analytic approximation (S=2.168Dh) against a rigorous numeric analysis (Figures 4 and 5). The analytic measure is a linear function of liquid level h. The rigorous numeric analysis results in a non-linear function of h.

The numeric method’s algorithm divides the head into a minimum of 100 xy-planes. The sum of finite segments of frustums between each xy-plane along the height of the z-axis (liquid level h) is the wetted surface area of 0 to h.

By Figure 4 and Figure 5, we observe that the maximum difference of wetted surface area calculated by analytic approximation versus rigorous numeric analysis is less than 3% at about 75% liquid level. Typically fire relief load is calculated at 70–80% liquid level.

## Discussion

A linear equation S = 2.168Dh gives a very close approximation of the wetted surface area of a 2:1 SE head as compared to a rigorous numeric analysis. Of course, the linear equation is much easier to apply; however, for the engineer experienced with common spreadsheet software, the rigorous numeric analysis gives an exact result.

## References

1. Firoozi, B., Wetted surface area calculations for fire-relief sizing in ASME pressure vessels, Hydrocarbon Processing, August 2015.

2. API Standard 521, “Pressure-relieving and Depressuring Systems,” 6th Ed., January 2014.

3. Kern, W.F., Bland, J.R., “Solid Mensuration with Proofs,” 2nd Ed., Wiley, New York, 1948.

4. Szirtes, T., “Applied Dimensional Analysis and Modeling,” Elseiver Press, New York, 2006.

Author

Babak Firoozi is a chemical engineer working in the Energy & Chemicals Div. of Oilcor, Inc. (Email: babak.firoozi@oilcor.com). He has several years of process engineering experience in the oil-and-gas industry, primarily petroleum refining and landfill gas processes. He holds a B.E. in chemical engineering from the University of Baja California, Mexico, and an M.S. in chemical engineering from California State University. He is a registered professional engineer (PE) in the state of California.

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