Optimization Model of Oil-Volume Marking with Tilted Oil Tank

In this paper, the relationship model between the oil volume and the vertically tilting parameter (α), the horizontally tilting parameter (β) and the displayed height of oil (h) is first constructed with the tilted oil tank. Then, based on the data of the oil output volume at different time of day, an optimization model of oil-volume marking with tilted oil tank is established. Finally, parameters α = 2.2 ̊ and β = 3.05 ̊ are estimated by using nonlinear least squares method and the marking number of the tank-volume meter is given.


Introduction
In [1], the oil-volume marking problem with tilted oil tank is shown and is also a practical one for some oil enterprises.
It is well known that there are several oil tanks and corresponding "oil measuring management system" in gas station, which can measure the input or output oil volumes and the height of the oil in the tank by means of runoff meters and oil height meters.Through a real-time calculation on the relationship between the oil height and the oil volume, we can get the changing conditions of the oil height and the oil volume in the tank.Because of the deformation of the ground work, the oil tanks have been tilted after a period of time of operation, and it needs to remark the oil-volume meter regularly.
The oil volume stored in an inclined cylinder tank is discussed in [2][3][4] and the oil volume stored in an inclined rectangular parallelepiped tank is studied in [5].In this paper, the oil tank is more sophisticated geometric shape with a cylinder shape in the middle and two sphere-cal top on both sides.The studying method presented in this paper is: the relationship between the oil volume and the height of the oil is first set up under the situation of no tilting.Secondly, based on the symmetry of the oil tank, the situations of the vertical tilted tank with oil on two sides and on only one side, and the situations of the horizontal tilted tank will be considered by turning into the corresponding oil volume in the situation of no tilting, respectively.Then, the relationship model between the oil volume and shall be synthesized when the tank tilts not only vertically but also horizontally.Finally, by using non-linear least squares method, parameters  * , ,h     and  will be estimated and the marking number of the tank-volume meter will be given.

The Oil Volume of the Tank with No Tilting
The tank consists of two differently shaped parts: the main body is a cylinder, the two sides are two spherical top.In order to compute the oil volume, they should be considered respectively (see Figure 1).
From Figure 1, we know that the volume of the shadowed part can be divided into two parts: cylinder part ( ) and spherical body ( ), and Copyright © 2012 SciRes. where is the radius of the spherical body, is the radius of the horizontal round section, R H is the width of the spherical body, is the oil height, h y S is the random level section of the spherical body.Let h denote the oil volume of the tank.Then, it follows from (2.1) and (2.2) that

The Oil Volume of the Tilted Tank
The tank may tilt vertically or horizontally.Thus, we first consider the influence of vertically tilting angle  and the horizontally tilting angle  to the oil volume meter, respectively.Then, we shall synthetically study the relationship among the oil volume of the tank, the oil height and the tilted parameter   ,   .Based on the known data and the Cartesian coordinate system with taking the center point of the oil tank as origin, it is easy to get the following four equations for boundary contour of the tank: 1.5, 1.5, 3.375 , 3.375 .
Under normal circumstances, when the tank tilts vertically with an angle of  (see Figure 2), the equation of intersection line met by the oil surface and vertical section of the tank can be obtained as follows:

Situation with Oil on Both Sides
From Figure 3, (3.1) and (3.2), we have where 1 is minimum distance from oil surface to bottom of tank (see Figure 3).Thus, (3.3) implies the following relationship between and which can be rewritten as (3.4) Similarly, let 2 denote maximum distance from oil surface to bottom of tank (see Figure 3).Then the relationship between and (3.5) By using the method of cutting and filling, the oil volume of the tank can be shown as follows ) can be calculated from (2.3), (3.4) and (3.5) as follows

Situation with Oil on Only One Side
Similarly, let 3 denote maximum distance from oil surface to bottom of tank (see Figure 4), when the tank in both side has oil on only one side, we have 2 tan 1.5, , +3.375 + , and the relationship between and which can be rewritten as 3) and (3.7), now we know Thus, by using the method of cutting and filling, the oil volume of the tank can be obtained as follows:

Horizontally Tilting Model
Because the main body of the tank is a cylinder (center symmetry), both the oil marking stylet and the plumb line pass through the center of its section and the stylet will move with the lean of the tank when the tank tilts horizontally with an angle of  .The relation between the oil marking stylet and the plumb line can be shown in Figure 5.Let 4 denote the real oil height of the tank, denote the marking height on the oil meter.Thus, the relationship between and , can been shown as follows: (3.9)

The Tilted Both Vertically and Horizontally Tank
When the tank is tilted both vertically and horizontally, we replace in (3., , , ,  Thus, we can construct a optimization model for oil-volume marking with tilted oil tank as follows: can be computed.Finally, we have the oil volume of the tilted tank (see Table 1).

The Analysis and Test of the Model
In order to test the correctness of the model (3.11), the data after inpouring oil into the tank is used to calculate the oil volumes of the corresponding oil heights, which are analyzed in Excel with the corresponding tested re sults.The maximum of absolute value for the relative the mean of that is 0.577%.By Matlab soft, we compare the tested results with the calculated results after inpouring oil into the tank in Figure 6.

Conclusions
error of the corresponding calculated results to the tested number is 2.822%, the minimum of that is 0.001% and Many oil tanks will be tilted after a period of time beon, operation or the deformation of cause of constructi ground work, and the oil volume marking of tank will be changed.A number of authors mainly analyzed how to calculate the tank volume, or studied the factors which influence the marking and measuring of oil tanks under the normal situation.In fact, it is very important to study the recognition of the tilted oil tanks and the marking of oil volume.
The purpose of this paper is to construct the optimization model of oil-volume marking with tilted oil tank, to estimate the tilting parameters  and  by using

Figure 1 .
Figure 1.Oil tank of no tilting.

Figure 3 .
Figure 3.The situation with oil on both sides.

Figure 4 .
Figure 4.The situation with oil on only one side.

Figure 5 . 4 .
Figure 5.The section sketch map of the horizontally tilted tank.relationship model between   * , ,h  and the oil volume in the situation with oil on both sides and with oil on only one side as follows, respectively:  * , , j j V V h    , .(3.11) 1, 2 j 

. 1 )
By using nonlinear least squares method,  and  can be estimated. Firstly, thirty groups of data are randomly extracted from the first 302 groups of data.Let the step change of ,   be 0.05.Then, by (3.1) and (3.2), we can calculate intersecting point of line P and the edge of the tank and can fix on 1 j  or 2 j  in (3.11) based on the intersecting point.Thus, by the model (4.1) and Matlab soft, the minimum relative error between the calculated results and the tested numbers can be confirmed, and the tilting parameters 2

Figure 6 .
Figure 6.Graph of oil-volume marking and calculated oil volume.