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## Prevent tank-bottom failure through reliability analysis

Apply service-life methods to optimize inspections and maintenance for storage units

Keywords: [tank bottom] [reliability] [ probability] [storage units] [maintenance] [ defects] [MTBF] [life] [mean time between failures]

Most oil storage tank failures result from defects like corrosion pits and flaws on the bottom plates.1 Since oil storage tanks in the petroleum and petrochemical industries are usually made from welded steel plates, defects in the tanks should be detectable during the welding process or in storage service phases.2 But even with efforts at advance flaw detection, there is still much complexity and randomness associated with tank-bottom corrosion. This makes it necessary to use probability methods in the reliability analysis and life/mean time between failures (MTBF) calculation for tank bottoms with corrosion pits.

#### Probability model.

The structure of a steel tank bottom can be considered as a series system from the point of view of the failure mechanism, which means that all plates must be in a normal state in order for the bottom to function properly. The tank fails when any one of its bottom plates has the first perfected pit. This model is called a series model or weakest link model.

Then, for a tank bottom made of m plates with the reliability Ri(t) for the ith plate, its reliability, RT(t), is the probability that all plates simultaneously survive to time t and can be expressed under the independence assumption as

(1)

The failure or hazard rate for a series system can also be expressed by an exponential distribution, where lT(t) is the failure rate of the tank bottom, li(t) is the failure rate of the ith plate.3

(2)

In another way, the reliability of a tank mostly depends on the corrosive statuses on its bottom plates. When one plate has defects, the applied stresses or corrosives will increase the size of these defects and, ultimately, failure occurs when the size of any one defect in the plate reaches a critical value. Usually, the defect growth is the main cause of failure and a defect with the least resistance to the applied stress or corrosives will be the first to fail. In this case, the reliability of the tank bottom of a steel tank will be

(3)

where Ri is the reliability of the ith plate of the tank bottom. It shows that it is an extreme value distribution.

Therefore, the reliability of a steel tank could be calculated by combining an extreme value distribution and an exponential distribution in the weakest-link or series model.

#### Calculating tank-bottom reliability.

According to extreme value theory, the tank bottom will fail as soon as only one defect penetrates the plate of the tank bottom. When there are n corrosive pits or defects in a tank bottom, the corrosive-resistance function can be expressed as

(4)

where Z is the surplus thickness of any one of the tank-bottom plates, Tb is the original thickness of the bottom plates, ai is the depth of the ith corrosive pit or defect, t is the time before failure of the tank-bottom, ti is the time to failure of the ith defect.

Applying the concept of smallest extreme value distribution,3 the probability of failure for the tank-bottom is

(5)

where F(z) or F(t) is the cumulative distribution of defects. For simplicity, assume that n r ∞ and then:

(6)

It is evident that the time of the plate penetration is proportional to the difference between the plate thickness and the initial depth of defects and there is:

(7)

where Tb is the thickness of tank-bottom plates, mm; k is the corrosion rate of the tank bottom, mm/year and ai is the original depth of the ith defect.

Assume that the probability density function for the depth of corrosive defects be

(8)

where a is the depth of defects, mm, and ā is the average depth of all defects in a tank bottom, mm.

The probability for the failure of the corrosive defects is

(9)

Thus, the reliability of a tank bottom is

(10)

The previous formula shows that the reliability of a tank-bottom has a correlation with several factors—such as the number and mean depth of defects, the rate of corrosion and the original thickness of the bottom plates—and that it declines with the tank’s service time or MTBF in a double exponential function.

#### Analyzing tank bottom reliability.

The reliability of a tank bottom can be shown and perceived more directly by visualizing its calculation in MATLAB.4 By substituting the different values of the number and the mean depth of defects, the corrosion rate and the plate thickness of the tank bottom into Eq.10, the reliabilities of the tank bottom with very different states of defects and plates can be calculated and shown in Fig.1 through Fig. 4. These figures clearly show that reliability drops rapidly with the increase of the defect-number and the corrosive rate as well as the mean depth of defects. This is especially true when the service time is more than 20 years (as shown in Fig.1, Fig. 2 and Fig. 3). It should be noted that an increase of the bottom plate’s thickness of just 1 millimeter or 0.5 millimeter will greatly upgrade the reliability as well as prolong the service time or MTBF of a tank. From the curves in Fig. 4, it can be seen that an increase or decrease of 0.5 millimeter of the bottom plates’ thickness makes the reliability vary greatly at the same service time of the tanks.

 Fig. 1. Reliability examined with different   numbers of defects in a tank bottom.

 Fig. 2. Reliability considered with different   tank-bottom corrosion rates.

 Fig. 3. Reliability examined with different   average depth of the defects in a tank bottom.

 Fig. 4. Reliability comparison of different   thicknesses for tank-bottom plates.

#### Life/MTBF for steel storage tanks.

The life/MTBF of a steel tank can be calculated by rearranging Eq. 10, which results in

(11)

The previous equation indicates that the life or MTBF for a tank bottom is a complicated function of reliability, which is affected by the number and depth of defects and the thickness of the bottom plate, while there is a simple linear relation between the life or MTBF and the corrosive rate of the tank bottom.

Ordinarily, the thickness of tank-bottom-plates can be designed and the reliability predicted in advance. The corrosion rate of a tank bottom can be evaluated by its surrounding conditions, such as the tank-base humidity and the water content of oil stored. Thus, the life of a tank bottom will be decided mainly on the number of defects, which can be investigated in periods during the inspection of tanks.

According to the sizes of most oil storage tanks in the industry, set the reliability at 95% and the bottom plate’s thicknesses and corrosive parameters into two groups: one with thinner bottom plates and another with thicker ones, which respectively are:

Group A: Tb = 5 mm, 6 mm, 7 mm; k = 0.10, 0.15 mm / a; a = 1.0 mm;
Group B: Tb = 8 mm, 10 mm, 12 mm; k = 0.15, 0.2 mm / a; a = 2.0 mm.

With this established, the life or MTBF of these tank-bottoms can be calculated and visualized with different corrosive parameters as shown in Fig. 5 and Fig. 6. The curves show the MTBF descends at the initial of the defects arising in the tank-bottom plates with whatever thicknesses much more than at the later time. It is during the initial arising of several defects in the bottom-plates that for the number of defects increase by one or two will cause MTBF to reduce greatly.

 Fig. 5. MTBF of tank bottoms with thinner   plates.

 Fig. 6. MTBF of tank bottoms with thicker   plates.

Further, the case that the life/MTBF of a tank-bottom change with its reliability required can be demonstrated in the following examples of calculations. Take two examples: one is the tank with Tb = 5 mm, k = 0.1 mm / year, ā = 0.5 mm and another is the tank with Tb = 10 mm, k = 0.15 mm / year, ā = 1.5 mm. The life/MTBF function of the tank-bottoms related with variables of the reliability and the number of defects can be visualized in Fig. 7.

 Fig. 7. The tank-bottom life/MTBF with   different numbers of defects.

The previous calculation results show that the life or MTBF of the tank bottom with a different number of defects will be very different and will decrease with the increasing number of corrosive pits. The higher reliability choice is the selection of thicker bottom plates. This is a very effective method to increase the life/MTBF and the reliability as well as the resistance to corrosion. At the same time, we can also take other measures to protect the bottom plates from being corroded, such as coating them and keeping them away from water. In fact, this calculation and analysis are much more important for planning the inspection and maintenance of a tank for safe storage than for knowing its accurate life/MTBF.

#### Bringing it all together.

There is a functional relation between the life/MTBF and reliability of a steel storage tank with defects in its bottom plate, which is also dependent on the parameters of tank-bottom structure and corrosion (such as the corrosion rate and the thickness of the tank-bottom plate). So we can increase the life/MTBF and reliability of steel storage tanks by selecting thicker bottom plates and strengthening the resistance to corrosion. The other key component is to reduce and/or eliminate the factors that cause or accelerate bottom corrosion pits and defects.

The MTBF of a steel storage tank will depend mainly on the defects in its bottom. There seems to be no difference regarding the size of the tank-bottom area in the previous life calculation. However, usually there are more defects in a larger tank bottom. So, a larger tank should be designed with the thicker bottom plates at the same corrosive environment and reliability level. The larger tanks should also be inspected in shorter periods than the smaller ones in order to maintain proper reliability.

Through the analysis of life/MTBF and reliability, a more reasonable plan for the future inspection and maintenance of a tank bottom can be made. This will greatly increase cost efficiency and avoid both unneeded work and losses that come from not finding the defects that can lead to failure in time. HP

LITERATURE CITED

1 Ji-Yi, F., Analysis on a thousand cases of accidents in oil depots, Sino Petrochem Publishing House, pp. 228 and 355, Beijing, 2005.
2 Guang-chen, G. and Z. Zhang, Design and management of petroleum depots, Petroleum University Publishing House, pp. 221-227, Dongying, Shandong, 1991.
3 Shu-Ho, D. and M. Wang, Reliability analysis in engineering applications, Van Nostrand Reinhold, pp. 32–35, pp. 359–361, New York, 1991.
4 Cheng, W., et al, MATLAB 5.3 essentials and programming with advanced application, China Machine Press, pp. 82–87.

 The authors Lichuan Liu is a professor in the petroleum engineering department of Logistic Engineering University in China. She engages in research on petroleum storage and transportation system design and reliability. Tianqi Liu is a professor in the electric engineering school of Sichuan University in China. She engages in research on electricity power system design and reliability.

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