A Solution to Identification of Oil Storage Tank Displacement and Calibration Table of Tank Capacity Abstract When studying the problem of underground oil storage tank displacement, we use micro-element method and the principle of equal effects to establish a mathematical model about the identification of oil storage tank displacement and the calibration table of tank capacity. As a result, we achieve the tank dip and the calibration table of tank capacity. For the first problem, we work out the volume of each part by using integration after cutting the tilted tank into two regular parts. Then we can get the relationship between the heights and the reserves in the tilted tank, we can also wok out the calibration table of tank capacity. Since the expression of the relationship between the heights and the reserves is so complex, we use a simply multinomial (a unitary function) instead the complex expression considering the engineering calculation. For the second problem, the tank transverse displacement does not have effects on the oil shape in the tank because the body shape of the oil storage tank is cylinder, so the process which be simultaneous vertical tilt in the horizontal deflection is equivalent to two processes, as if the horizontal deflection occurs after the tank has been tilted in the vertical direction. After using the method of segmentation, combination and calculus, we present a approximate calculation expression based on compensation principle; but the expression just points the relation between oil reserves and the height in the vertical tilt direction. Then finding out the relation between the final oil level height and the height in the horizontal deflection direction, which is on the basis of their geometrical relationship, we get the final expression between the oil reserves and the final height, the longitudinal and transverse gradient deflection angle. Afterwards the standard library table can be established to contain both vertical and horizontal angle as projects, the relationship between the oil reserves and the final oil level height based the expression. According to the standard function library table, we find that the vertical angle is about 5° and the horizontal angle is about 7°when compared the curve fitted by the given data with the functions in the standard table. At last, we work out calibration table of tank capacity relation to oil height in between 10cm. Key Words: segmentation and combination; micro-element method; method of equal effect; compensation principle; standard function library table
1 Introduction Generally, gas stations store fuel by several underground oil storage tanks, matching with “an oil height measurement management system”. The system can do real-time calculation using the predetermined calibration table of tank capacity, based on the data of the oil height and in/out oil quantity which is measured by flow meter and oil level indicator. As a result, the relation between the fuel reserves and the tank oil level height could be worked out. However, due to the deformation of foundation leaded by some factors, the tank position will ensue longitudinal tilt and lateral deflection (hereinafter are called displacements) after using the oil tank a period of time, leading the change of calibration table of tank capacity. According to relevant requirements, the capacity table has been required periodic re-calibration. Thus, to create mathematical models is the way to solve the problem of identification of oil storage tank displacement and calibration table of tank capacity. (
  1) In order to master holding tank of tank capacity charts after modification, to have initial test on a small elliptic storage tank (two end of elliptic cylinder), study separately tanks without modification and inclination for longitudinal displacement. According to the experimental data, please create a mathematical model researches the influence of capacity table because of tanks displacements, and give a tank capacity table calibration after displacement between in the oil level height 1cm. (
  2) For the actual storage tank (subject for cylinder and ends for the ball crown body), trying to establish (horizontal displacement and the longitudinal change occurred in the meantime) calibration tank capacity table by creating mathematical model that oil reserves and the quantity that oil level height as well as the modification parameters (the longitudinal tilt angle α and the lateral deflection angle β) between the general relationship. Please use the actual data testing in the process of oil inlet/outlet after storage tank displacement, then determine into parameters according to the mathematical model, and give a tank capacity table calibration after displacement between in the oil level height 1cm at the same time. Furthermore, please use the attached actual testing data to analyze and inspect the validity of the model and the method’s reliability. 2 Assumptions and Conditions (
  1) Regardless of the oil tank’s thickness and deformation caused by pressure, in the whole process only foundation deformation cause position changes, such as the longitudinal and transverse offset tilt; (
  2) In the course of analysis, ignoring the volume of other attachments in the oil tank, and considering only one fuel sending unit is there; (
  3) The longitudinal tilt only occurs in one end. 3 Notation and Definitions S : Oil storage tank cross sectional area; V : The volume of oil storage tank;
h : The oil level height measured by the fuel sending unit in the oil storage tank;
L : The length of the oil storage tank; D : The ball diameter of the ball crown body corresponding spherical tank at both
1037 ds; α : The longitudinal tilt angle;
β :The lateral displacement angle.
4 Model Analyses For the first problem, based on the elliptic oil storage tank's size and shape, we can build a space coordinate on the premise of basic assumptions. Then the relation between the oil reserves and the oil level height measured by the fuel sending unit could be respectively found out after considering two conditions of without modification and longitudinal displacement in this space coordinates. With the experimental data of two conditions to validate the model, the original one can be improved and optimized through the analysis about the data relationship. For the second problem,
5 General model and Solution
  5.1 The mathematical model of calibration tank capacity table after elliptic oil tank tanks tilt
  1.1 Establish space Cartesian coordinate system Since the small elliptic cylinder storage tank’s shape is similar to cylinder, their only difference is that any tank cross section is oval, and elliptic is a perfectly symmetrical graphics with its center as the origin. If establishing the right-angle coordinates on the tank’s cross section, the long axis of the oval can be taken as x axis and the short axis can be taken as y axis while the long and short axes are symmetric about the origin. We set the oval long axis length for 2a , the oval short axis length for 2b . At last, the axis along with the oil tank’s cylinder shaft direction and through the center origin will be taken as z axis, setting the length of z axis for L . Now two plane coordinate systems xoy and yoz are given here. In cross section, the plane coordinate system of xoy is established as in figure 2: the elliptical long axis a =
  0.6 m and the short axis b =
  0.89 m respectively.
Figs. 1 xoy coordinate system on cross section of small elliptic oil tank.
Both crew-cut ends of storage tank are parallel, so are tangent plants of the tank bottom, furthermore both sides with the following are vertical, so positive section of the small ellipse tank is a rectangle. We can establish yoz coordinate system on the rectangle and the length of the storage is L , L =
  2.45m , as follows figure
Figs. 2 yoz coordinate system on the positive section of small elliptic oil tank.
The complete spaces coordinate as follows the figs. 3 shows.
Figs. 3 The complete spaces coordinate system in the tank.

  1.2 The establishment of mathematic model
  1) The relation between storage quantity and oil level height without modification
Under this situation, the oil plane and the storage tank fluctuation cut-side are always parallel, in addition that shape of the oil stored in storage tank is a rule cram (namely its volume expression can be similar to the simple cylinder), and setting height for oil is h shown in figs.
  4. Oil cross-sectional area is partial elliptic area S (h) in the tank; the length of the oil body is L , the oil volume is Vbb , obtained Vbb = S (h) L
Now seek the cross-sectional area S ( h) : For storage liquefied petroleum gas tanks, national relevant specification: in loading and unloading temperature difference shall not exceed 30 ℃, the biggest filling quantity is 85% of the total volume. So tank oil body (namely the oil tank shape) of cross section only is partial ellipses (within the prescribed scope, no matter how much oil storage quantity is), as figs.4 shows.
Figure 4 elliptic graphics In the figs. 4, the dotted line is for oil plane and the part below dotted line is for the object part, the elliptic area S ( h) . The intersection point of dotted with y axis is point H , whose distance to intersection of short axis and negative axis namely is the oil height h . Setting the elliptical equation as:
x2 y2 + =1 a2 b2
According to the micro element method, partial elliptical surface area S ( h) can be given as follows:
S ( h) = ∫
2a 2 2a ? h 2 b2 h b2 ? b ? y 2 dy = b ? h 2 + arcsin + π ? ? ?b b b ?2 2 b 4 ?
(1 )
Thus, the relation between storage quantity (i.e. volume) and oil level height without modification as follows:
Vbb = SL =
2a ? h 2 b2 h b2 ? b ? h 2 + arcsin + π ? L ? b ?2 2 b 4 ?
(2 )
  2).tanks occur when oil displacement and storage quantity of high relations: At this time of storage tank left side slopes downward, oil noodles no longer and tanks with the following parallel, tank storage form no longer rules easy, volume calculation become complicated. Now put the oil body split into two parts volume calculation, still from positive diagram to analysis, as shown in figure
Figure 4 displacements small elliptic oiltank shaft section From figure 4 cross-sectional perspective, divided into two parts of the area of the oil body respectively, as shown in figure 5 shows, right, left cross-court shaded area for cross-court shaded area for.
Figure 5 section projection drawing In figure 4, oil noodles, along in the plane of the parallel to the tank bottom the oil body cut into two parts, from positive projection perspective, is divided into two parts, make represent part volume for that represent part, storage quantity namely volume oil volume
V = V1 + V2
Currently in use in reservoirs in the right wall wasn't highly notes for (figure
  4) instead of oil shown in measuring high float oil operations, and high and high exists between an equation relations, find out the relationship between again after conversion, and finally arrive at the relational expression. Volume calculations are as follows: Similar to a split the shape of the body, and -axes pointed face of cross-sectional area of parallel along the direction decrease gradually, using the micro element method to cross section (both for
part of the ellipsoid) in the direction of integral come to this part of volume, range is, cross section of sapce still with (
  1) type,
2a 2 b ? y 2 dy b 2a ? h1 + l tan α b2 h + l tan α h1 2 b2 h? ) b 2 ? ( h1 + l tan α ) 2 + arcsin 1 = ? b ? h12 ? arcsin 1 ? ?( b ? 2 2 b 2 2 b? S1 = ∫
h1 +l tan α
V1 = ∫ S1dl
0 L ? 2 a ? h + l tan α b2 h + l tan α h1 2 b2 h ?? ? ? = ∫ ? ?( 1 ? b ? h12 ? arcsin 1 ? ? dl ) b 2 ? (h1 + l tan α )2 + arcsin 1 0 b ? 2 2 b 2 2 b ?? ? ? ?
Volume calculations are as follows: This part of the oil storage is the shape of the rules, can directly by the bottom area with high volume, get multiplied by (
  1) type method to calculate the available:
2a 2 2a ? h1 2 b2 h1 b 2 ? 2 2 S2 = ∫ b ? y dy = b ? h1 + arcsin + π ? ? ?b b b ?2 2 b 4 ?
V2 = S2 L =
2a ? h1 2 b2 h b2 ? b ? h12 + arcsin 1 + π ? L ? b ?2 b 4 ? 2
Comprehensive get total volume
V = V1 + V2
L ? 2 a ? h + l tan α h + l tan α h1 2 h ?? b2 b2 ? ? = ∫ ? ?( 1 ? b ? h12 ? arcsin 1 ? ? dl ) b 2 ? (h1 + l tan α )2 + arcsin 1 0 2 2 b 2 2 b ?? ?b ? ? ?
2a ? h1 2 b2 h b2 ? b ? h12 + arcsin 1 + π ? L ? b ?2 2 b 4 ?
And whatever variant or not, oil float always perpendicular to the tank bottom, when it measured for its high oil and oil tank, the contact point to the underside of the vertical distance, the diagram below point six shown.
Figure 6 displacements small elliptic oiltank shaft section Represents the position of oil float (fixed), said oil noodles, namely for oil float test oil high, be high, make for, and perpendicular to the above relations get by
h = h1 + ?h
But questions is given in oil float position data, the oil from the distance is float
  2.05 m, too
?h =
  2.05 tan α
Oil for high expression
h1 = h ?
  2.05 tan α
Will cover the total volume in expression, get the oil when high and oil displacement quantity of equation
V = V1 + V2
L ?



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