The distribution of sensor points

Fig 2 is the coordinate system. Vertical tanks are generally cylindrical, and their cross-sections are generally circular. Therefore, our methods and tests are all carried out with circular cross-sections. Vertical tank is divided into upper part and bottom part. The bottom part of vertical tank is usually irregular because of ground subsidence or other reasons, and the capacity of bottom part is obtained by volumetric method or other methods. The bottom of the vertical tank generally only occupies little share of the entire storage tank, and the bottom volume is only used when the liquid is first fed or when the tank is cleared. In the daily custody transfer, the capacity table on the upper part of the vertical tank is generally used for calculation, and the amount of liquid in each custody transfer is generally not less than two meters in the height of the vertical tank. Due to the deformation of the bottom plate and other reasons, it is difficult to measure the bottom volume of the vertical tank very accurately by the geometric measurement method and the laser scanner method, and more attention is paid to the measurement of the upper part of the storage tank. Thus, this paper focuses on acquiring the capacity of upper part. The height of bottom part is H_bottom, and the height of upper part is H_upper. The inner radius of the vertical tank is R. The capacity of the vertical tank is synthesized from the upper part and the bottom part, shown in Eq 1.

Fig 2

The coordinate system.

Fig 3 shows the distribution of sensor points. Sensor points are equiangularly arranged on different layers. The number of layers is N_layer and the number of sensor points on each layer is N_mono. The projection of the sensor points of each layer on O-XY surface are coincidence. The interval between layers is h_layer and the total number of sensor points is N_sensor. Thus,

Fig 3

Sensor points distribution.

Criterion of the Monte Carlo test

Criterion of the Monte Carlo test is used to decide whether a sample point falls in the vertical tank. As the vertical tank is convex and the sensor points are located on the inner surface of the tank, sample points in the enclosed area formed by all sensor points must be in the tank, shown in Fig 4. If sample point P_sample locates in the enclosed area formed by all sensor points, we have a, b and c, make

Fig 4

The enclosed area formed by all sensor points.

(a) 3D illustration of the enclosed space; (b) The projection of the enclosed space on O-XY surface.

If we cannot find a, b and c let P_sample satisfies Eq 4, it is still uncertain whether P_sample falls in the vertical tank or not. The subsequent calculation will be performed.

Fig 5 is the positional relationship of P_sample,P_i1,j1, and P_i2,j2. If P_sample is between P_i1,j1, and P_i2,j2(i.e. P_sample falls within the shaded area in Fig 4), we have

Fig 5

Positional relationship of P_sample, P_i1,j1, and P_i2,j2.

After finding out two sensor points making distance is the smallest, for example, P_i1,j1 and P_i2,j2 satisfy Eqs 5, 6, and 7, we construct an ellipsoid with P_i1,j1 and P_i2,j2 as the endpoints, shown in Fig 6. Point A and B are the foci of the ellipse. Thus, $$ \overrightarrow{{\text{P}}_{i1,j1}\text{A}}=\lambda \cdot \overrightarrow{{\text{P}}_{i1,j1}{\text{P}}_{i2,j2}}$$, $$ \overrightarrow{{\text{P}}_{i1,j1}\text{B}}=\left(1-\lambda \right)\cdot \overrightarrow{{\text{P}}_{i1,j1}{\text{P}}_{i2,j2}}$$, where λ is a scale factor, 0 < λ < 1.

As $$ \overrightarrow{\text{OA}}=\overrightarrow{{\text{OP}}_{i1,j1}}+\overrightarrow{{\text{P}}_{i1,j1}\text{A}}=\overrightarrow{{\text{OP}}_{i1,j1}}+\lambda \cdot \overrightarrow{{\text{P}}_{i1,j1}{\text{P}}_{i2,j2}}$$ and $$ \overrightarrow{\text{OB}}=\overrightarrow{{\text{OP}}_{i1,j1}}+\overrightarrow{{\text{P}}_{i1,j1}\text{B}}=\overrightarrow{{\text{OP}}_{i1,j1}}+\left(1-\lambda \right)\cdot \overrightarrow{{\text{P}}_{i1,j1}{\text{P}}_{i2,j2}}$$, we have the coordinates of Point A are x_A = x_i1,j1+λ · (x_i2,j2 –x_i1,j1), y_A = y_i1,j1+λ · (y_i2,j2 –y_i1,j1), z_A = z_i1,j1+λ · (z_i2,j2 –z_i1,j1). The coordinates of Point B are x_B = x_i1,j1+(1-λ) · (x_i2,j2 –x_i1,j1), y_B = y_i1,j1+(1-λ) · (y_i2,j2 –y_i1,j1), z_B = z_i1,j1+(1-λ) · (z_i2,j2 –z_i1,j1). Then the ellipsoid equation is

Fig 6

The ellipsoid with P_i1,j1, and P_i2,j2 as the endpoints.

In Fig 6, the distance from O to P_i1,j1 and P_i2,j2 along the surface of the tank are $$ l_{i1,j1}=\Vert \overrightarrow{\text{OC}}\Vert +\Vert \overrightarrow{{\text{CP}}_{i1,j1}}\Vert $$ and $$ l_{i2,j2}=\Vert \overrightarrow{\text{OD}}\Vert +\Vert \overrightarrow{{\text{DP}}_{i2,j2}}\Vert $$. The distance from P_i1,j1 to P_i2,j2 along the surface of the tank is defined as l_{i1,j1↔i2,j2}. Since the surface continuity of the vertical tank, approximate solution to get l_{i1,j1↔i2,j2} is

Solving C_half = l_{i1,j1↔i2,j2}, we can get λ and the ellipsoid equation, i.e. Eq 8.

According to the definition of an ellipsoid, if the sum of the distance from P_sample to A and B is shorter than L_minor, P_sample is in the ellipsoid. That is, if

In summary, if a sample point P_sample falls in the vertical tank, we have

The coordinates of P_sample satisfy Eq 1.

If the coordinates of P_sample do not satisfy Eq 1, they should satisfy Eq 11.

The vertical tank capacity

When conducting the Monte Carlo tests, sample points are randomly generated in the smallest cuboid that contains the vertical tank. If x_min = min{x_i,j}, y_min = min{y_i,j}, z_min = min{z_i,j}, x_max = max{x_i,j}, y_max = max{y_i,j}, z_max = max{z_i,j}, where 1≤ i ≤ N_layer, 1≤ j ≤ N_mono, x_i,j, y_i,j and z_i,j are the coordinates of sensor points, the coordinates of sample points would be x_min≤x≤x_max, y_min≤y≤y_max, z_min≤z≤z_max.

After the Monte Carlo tests, if the number of sample points satisfying the criterions of the Monte Carlo Test is N_IN, and totally N_sample sample points are generated, the capacity of the upper part would be

The Monte Carlo test

Fig 7 is the vertical tank model of the Monte Carlo test. The radius of vertical tank is R = 1.300[m], and the upper part height of the vertical tank is H_upper = 8.476[m]. The maximum capacity of upper part is π · R² · H_upper ≈ 45.000[m³]. Sensor points are distributed in ten layers (N_layer = 10). Each layer has ten sensor points uniformly distributed equiangularly (N_mono = 10). Thus, N_sensor = N_mono · N_layer = 100, i.e., there are 100 sensor points in total. Considering the size of the vertical tank, we initially set up 100 sensor points. The influence of the number of sensor points on the measurement results will be further studied in the follow-up work. Currently, 100 sensor points are used to explore the feasibility of the proposed method. The difference of heights between each layer is h_layer = 0.8476[m]. Sample points are generated in Cuboid L–W–H_upper, whose L = 2.600[m], W = 2.600[m] and H_upper = 8.476[m]. Thus, x_min = -1.300[m], y_min = -1.300[m], z_min = 0, x_max = 1.300[m], y_max = 1.300[m], z_max = 8.4776[m], and the coordinates of sample points satisfy -1.300[m]≤x≤1.300[m], -1.300[m]≤y≤1.300[m], 0≤z≤8.476[m].

Fig 7

The vertical tank model.

The Monte Carlo tests are performed on Matlab, and the sample points are generated with Matlab’s built-in random function ─ unifrnd (). Totally N_sample = 10⁶ sample points are generated in each test. A total of 10 tests were conducted.

In order to discuss the measurement results of vertical tanks of different volumes, the absolute error of capacity per unit volume is calculated as

Fig 8 is the absolute error of capacity per unit volume of Test 1~10. It shows that there is a significant linear relationship between ε_k and h_k in each test. The slopes of the fitting results of Test 1~10 are all around 0.0252.

Fig 8

The absolute error of capacity per unit volume of Test 1~10.

To further investigate the relationship between the absolute error of capacity per unit volume and the size of the vertical tank, another sixty tests are carried out for different heights and radii of tank.

Fig 9 is the absolute error of capacity per unit volume with different heights of vertical tank. The heights of vertical tank are 8.476 [m], 6.781 [m], 4.238 [m], and 2.543 [m]. Each height is tested 10 times and the inner radius of vertical tank is 1.300 [m] in each test. Fig 8 shows the average value of ε_k. From the fitting results, even though H_upper has changed, there is still a significant linear relationship between ε_k and h_k. The slopes of the fitting results are different. Interestingly, we find that 0.02524 × 8.476 ≈ 0.214, 0.03154 × 6.781 ≈ 0.214, 0.05047 × 4.238 ≈ 0.214, and 0.08415 × 2.543 ≈ 0.214. This shows that the product of the slope and H_upper is a constant value, which 0.214.

Fig 9

The absolute error of capacity per unit volume with different heights of vertical tank.

Fig 10 is the absolute error of capacity per unit volume with different radii of vertical tank. The radii of vertical tank are 1.300 [m], 1.040 [m], 0.910 [m], and 0.780 [m]. Each radius is tested 10 times and the height of vertical tank is 8.476 [m] in each test. Fig 10 shows the average value of ε_k. It can be seen from Fig 10 that there is a significant linear relationship between ε_k and h_k, and the slope hardly changes due to changes in radius.

Fig 10

The absolute error of capacity per unit volume with different radii of vertical tank.

Based on the above analysis of the influence of H_upper and R on ε_k, in the linear relationship between ε_k and h_k, R has little effect on the slope, and the product of H_upper and the slope is a constant, which is 0.214. Therefore, the relationship between ε_k and h_k can be written as

Vertical tank test and result analysis

Fig 11 are devices used for vertical tank testing, including a vertical tank [13] and a metal tank [14]. The inner diameter of the vertical tank is R = 0.299[m]. One hundred sensor point labels are evenly affixed on the inner wall of the vertical tank, a total of 10 layers, each with 10 points. The distance between the lowest sensor point and the highest sensor point is H_upper = 0.513[m]. The distance between the sensor points of each layer is h_layer = 0.057[m]. The capacity of bottom part is Q_bottom = 0.0175[m³]. A ruler is attached to the inner wall of the vertical tank for reading the liquid level with a resolution of 0.001[m]. The nominal capacity of the metal tank is 0.02[m³], indicating that it can accurately hold 0.02[m³] of liquid. The liquid in the metal tank can be transferred to the vertical tank through the plastic hose. The liquid used in the test is water. Since the nature of water is relatively stable, compared to other media, such as oil, the volume of water is less affected by temperature and volatility, so we used water to verify the accuracy of the proposed method to measure the capacity of vertical tanks.

Fig 11

Vertical tank test devices.

Fig 12 is the sequence of vertical tank test, including two parts: vertical tank test and Monte Carlo test. In the vertical tank test, the first step is to pour Q_bottom = 0.0175[m³] of water into the tank, and record the liquid level as $$ {h^{\prime }}_0$$ at this time. Then 0.02[m³] of water is poured into the tank one by one, and the corresponding liquid levels are recorded. The recorded liquid levels are $$ {h^{\prime }}_1$$, $$ {h^{\prime }}_2$$, $$ {h^{\prime }}_3$$, $$ {h^{\prime }}_4$$, $$ {h^{\prime }}_5$$, $$ {h^{\prime }}_6$$, and $$ {h^{\prime }}_7$$ in sequence. A total of 5 tests are carried out. In order to prevent the difference of Q_bottom from affecting the liquid level of the upper part of vertical tank, the liquid levels when the water volume in the vertical tank is 0.0375[m³], 0.0575[m³], 0.0775[m³], 0.0975[m³], 0.1174[m³], 0.1375[m³] and 0.1575[m³] are $$ {h^{\prime }}_1-{h^{\prime }}_0$$, $$ {h^{\prime }}_2-{h^{\prime }}_0$$, $$ {h^{\prime }}_3-{h^{\prime }}_0$$, $$ {h^{\prime }}_4-{h^{\prime }}_0$$, $$ {h^{\prime }}_5-{h^{\prime }}_0$$, $$ {h^{\prime }}_6-{h^{\prime }}_0$$ and $$ {h^{\prime }}_7-{h^{\prime }}_0$$ respectively. Finally, the relationship between the capacity and liquid level of the vertical tank, which is $$ {Q^{\prime }}_k-h_k$$, is obtained by linear fitting [15]. The $$ {Q^{\prime }}_k-h_k$$ obtained from the vertical tank test is used as the exact value to verify the effectiveness of the method proposed in this paper.

Fig 12

Vertical tank test sequence.

In the Monte Carlo test, a total of 10 tests are performed, and the parameters of each test are shown in Fig 12. Based on the average value of 10 tests, $$ {\stackrel{⌢}{Q}}_k-h_k$$ is calculated using Eqs 13 and 19.

Table 1 is the results of vertical tank tests. Based on the average values, the relationship between the capacity and liquid level of the vertical tank is obtained through linear fitting, shown in Fig 13.

Fig 13

The relationship between the capacity and liquid level of the vertical tank.

Table 1

Results of vertical tank tests.

	$$ [m]	$$ [m]	$$ [m]	$$ [m]	$$ [m]	$$ [m]	$$ [m]
Test1	0.072	0.142	0.213	0.285	0.356	0.427	0.498
Test2	0.071	0.142	0.213	0.284	0.355	0.426	0.497
Test3	0.071	0.142	0.213	0.284	0.356	0.426	0.497
Test4	0.071	0.142	0.213	0.284	0.355	0.426	0.497
Test5	0.071	0.143	0.214	0.285	0.356	0.427	0.497
Average	0.071	0.142	0.213	0.284	0.356	0.427	0.497

Fig 14 is the absolute error distribution of the measurement results obtained through the Monte Carlo method, e.g. $$ {\stackrel{⌢}{Q}}_k-{Q^{\prime }}_k$$. In Fig 14, the black mark is the average value of the absolute error of the 10 tests. The red area represents the standard deviation of the absolute error of the 10 tests. The maximum absolute error of using Monte Carlo method to measure the vertical tank capacity does not exceed ±0.0003[m³].

Fig 14

Absolute error distribution.

Fig 15 is the relative error shrinkage curve, e.g. $$ \left|{\stackrel{⌢}{Q}}_k-{Q^{\prime }}_k\right|/{Q^{\prime }}_k$$. The relative error finally converges to less than 0.18%.

Fig 15

Relative error shrinkage curve.