Open Journal of Optimization 1 1 - ttp://.doi.org/1.36/ojop.1.1 Publised Online December 1 (ttp://www.scirp.org/journal/ojop) Optimization Model of Oil-olume Marking wit Tilted Oil Tank Wei Xie 1 Xiaojing Wang Huize Cui 3 Jun Cen 3 1 Department of Matematics Sicuan University of Science & Engineering Zigong Cina Scool of Computer Science Sicuan University of Science & Engineering Zigong Cina 3 Scool of Automation and Electronic Information Sicuan University of Science & Engineering Zigong Cina Email: xiewei@suse.edu.cn Received September 11 1; revised October 1 1; accepted October 1 ABSTRACT In tis paper te relationsip model between te oil volume and te vertically tilting parameter (α) te orizontally tilting parameter (β) and te displayed eigt of oil ( ) is first constructed wit te tilted oil tank. Ten based on te data of te oil output volume at different time of day an optimization model of oil-volume marking wit tilted oil tank is establised. Finally parameters α =. and β = 3.5 are estimated by using nonlinear least squares metod and te marking number of te tank-volume meter is given. Keywords: Optimization Model; Oil Tank Tilt; Oil-olume Marking Problem; Least Squares Metod; Parameter Estimation 1. Introduction In [1] te oil-volume marking problem wit tilted oil tank is sown and is also a practical one for some oil enterprises. It is well known tat tere are several oil tanks and corresponding oil measuring management system in gas station wic can measure te input or output oil volumes and te eigt of te oil in te tank by means of runoff meters and oil eigt meters. Troug a real-time calculation on te relationsip between te oil eigt and te oil volume we can get te canging conditions of te oil eigt and te oil volume in te tank. Because of te deformation of te ground work te oil tanks ave been tilted after a period of time of operation and it needs to remark te oil-volume meter regularly. Te oil volume stored in an inclined cylinder tank is discussed in [-] and te oil volume stored in an inclined rectangular parallelepiped tank is studied in [5]. In tis paper te oil tank is more sopisticated geometric sape wit a cylinder sape in te middle and two spere-cal top on bot sides. Te studying metod presented in tis paper is: te relationsip between te oil volume and te eigt of te oil is first set up under te situation of no tilting. Secondly based on te symmetry of te oil tank te situations of te vertical tilted tank wit oil on two sides and on only one side and te situations of te orizontal tilted tank will be considered by turning into te corresponding oil volume in te situation of no tilting respectively. Ten te relationsip model between te oil volume and sall be syntesized wen te tank tilts not only vertically but also orizontally. Finally by using non-linear least squares metod parameters and will be estimated and te marking number of te tank-volume meter will be given.. Te Oil olume of te Tank wit No Tilting Te tank consists of two differently saped parts: te main body is a cylinder te two sides are two sperical top. In order to compute te oil volume tey sould be considered respectively (see Figure 1). From Figure 1 we know tat te volume of te sadowed part can be divided into two parts: cylinder part ( ) and sperical body ( ) and C S C L R Rx (.1) Figure 1. Oil tank of no tilting. Copyrigt 1 SciRes.
W. XIE ET AL. 1 were S dy S y H r r R y +arcsin r R y H r H R y d y R H r H (.) is te radius of te sperical body R is te radius of te orizontal round section H is te widt of te sperical body is te oil eigt S y is te random level section of te sperical body. Let denote te oil volume of te tank. Ten it follows from (.1) and (.) tat C S L R R x H r r R y +arcsin r R y H r H R y d y. 3. Te Oil olume of te Tilted Tank (.3) Te tank may tilt vertically or orizontally. Tus we first consider te influence of vertically tilting angle and te orizontally tilting angle to te oil volume meter respectively. Ten we sall syntetically study te relationsip among te oil volume of te tank te oil eigt and te tilted parameter. Based on te known data and te Cartesian coordinate system wit taking te center point of te oil tank as origin it is easy to get te following four equations for boundary contour of te tank: y 1.5 y 1.5 (3.1) y R x3.375 y R x3.375. Under normal circumstances wen te tank tilts vertically wit an angle of (see Figure ) te equation of intersection line P met by te oil surface and vertical section of te tank can be obtained as follows: y x tan 1.5. (3.) 3.1. Situation wit Oil on Bot Sides From Figure 3 (3.1) and (3.) we ave y1 x1 y1 1R x13.375 y1 R tan 1.5 (3.3) were 1 is minimum distance from oil surface to bottom of tank (see Figure 3). Tus (3.3) implies te following relationsip between and : 1 1 1 1 +1.5 R R 3.375 tan wic can be rewritten as 1 1. (3.) Similarly let denote maximum distance from oil surface to bottom of tank (see Figure 3). Ten te relationsip between and is i.e. +1.5 R R 3.375 tan. (3.5) By using te metod of cutting and filling te oil volume of te tank can be sown as follows 11 1 1 (3.6) were ( 1 1i i ) can be calculated from (.3) (3.) and (3.5) as follows Figure. Tilting oil tank. Figure 3. Te situation wit oil on bot sides. Copyrigt 1 SciRes.
W. XIE ET AL. v 1 i i i L R Rx i H r r R y +arcsin r R y H r H R y d y. 3.. Situation wit Oil on Only One Side Similarly let 3 denote maximum distance from oil surface to bottom of tank (see Figure ) wen te tank in bot side as oil on only one side we ave y x y 3 R x+3.375 + y R and te relationsip between tan 1.5 3 3 3 and is +1.5 R R 3.375 tan wic can be rewritten as Based on we ave n 3 3 3. (3.7) y1 x1 tan 1.5 y1 1.5 x1. tan From (.3) and (3.7) now we know 1 L 1.5 3 R d tan R x x 3 L R Rx 3 H r r R y +arcsin r R y H r H R y dy. Tus by using te metod of cutting and filling te oil volume of te tank can be obtained as follows: Figure. Te situation wit oil on only one side. 1 1 3 H r r R y +arcsin r R y H r H R y dy L 1.5 3 R Rx tan 3.3. Horizontally Tilting Model (3.8) Because te main body of te tank is a cylinder (center symmetry) bot te oil marking stylet and te plumb line pass troug te center of its section and te stylet will move wit te lean of te tank wen te tank tilts orizontally wit an angle of. Te relation between te oil marking stylet and te plumb line can be sown in Figure 5. Let denote te real oil eigt of te tank denote te marking eigt on te oil meter. Tus te relationsip between and can been sown as follows: tat is R R cos. (3.9) 3.. Te Tilted Bot ertically and Horizontally Tank Wen te tank is tilted bot vertically and orizontally we replace in (3.) (3.5) and (3.7) by in (3.9) and get ˆ ˆ = k 1 3. (3.1) k k Next we replace k in (3.6) and (3.8) by ˆ in (3.1) for k 1 3 and denote te k Copyrigt 1 SciRes.
W. XIE ET AL. 3 te mean of tat is.577%. By Matlab soft we compare te tested results wit te calculated results after inpouring oil into te tank in Figure 6. Figure 5. Te section sketc map of te orizontally tilted tank. relationsip model between and te oil volume in te situation wit oil on bot sides and wit oil on only one side as follows respectively: j j j 1. (3.11). Confirmation of te Parameter and te Marking Based on te tested data from 1 Contemporary Under-gradute Matmatical Contest in Modeling [1] te oil output i at te different time of day can be obtained. Furter it follows from te relationsip model (3.11) tat we ave 1 j 1. ji j i j i Tus we can construct a optimization model for oil-volume marking wit tilted oil tank as follows: n min S j 1. (.1) i1 By using nonlinear least squares metod and can be estimated. Firstly tirty groups of data are randomly extracted from te first 3 groups of data. Let te step cange of be.5. Ten by (3.1) and (3.) we can calculate intersecting point of line P and te edge of te tank and can fix on j 1 or j in (3.11) based on te intersecting point. Tus by te model (.1) and Matlab soft te minimum relative error between te calculated results and te tested numbers can be confirmed and te tilting parameters. and 3.5 can be computed. Finally we ave te oil volume of te tilted tank (see Table 1). 5. Te Analysis and Test of te Model In order to test te correctness of te model (3.11) te data after inpouring oil into te tank is used to calculate te oil volumes of te corresponding oil eigts wic are analyzed in Excel wit te corresponding tested re sults. Te maximum of absolute value for te relative error of te corresponding calculated results to te tested number is.8% te minimum of tat is.1% and i ji 6. Conclusions Many oil tanks will be tilted after a period of time because of construction operation or te deformation of ground work and te oil volume marking of tank will be canged. A number of autors mainly analyzed ow to calculate te tank volume or studied te factors wic influence te marking and measuring of oil tanks under te normal situation. In fact it is very important to study te recognition of te tilted oil tanks and te marking of oil volume. Te purpose of tis paper is to construct te optimization model of oil-volume marking wit tilted oil tank to estimate te tilting parameters and by using Table 1. Te oil volume of te tilted tank. Heigt/cm alue/l Heigt/cm alue/l Heigt/cm alue/l 1 63. 11 1151.1 1 865.6 178.99 1 3876. 5186.1 3 315.67 13 667. 3 535. 83.3 1 98.5 55677.5 5 675.86 15 36.6 5 5775.1 6 885.9 16 358. 6 5966.7 7 113.1 17 37879.9 7 6189.9 8 1315.9 18 68. 8 6691.5 9 1593.8 19 3369.5 9 6376. 1 1888.1 67.5 3 6331. Figure 6. Grap of oil-volume marking and calculated oil volume. Copyrigt 1 SciRes.
W. XIE ET AL. nonlinear least squares metod and to give te marking number of te tank-volume meter. Moreover te optimization model presented in tis paper can be used to solve te problem of te oil volume marking wen te oil tank is tilted and can also be applied to solve te volume problems of many containers wit various different sapes. 7. Acknowledgements Tis work was supported by te Sicuan Yout Science and Tecnology Foundation (8ZQ6-8) te Open- Foundation of Artificial Intelligence of Key Laboratory of Sicuan Province (9RZ1) and te Scientific Re- searc Fund of Sicuan Provincial Education Department (1ZA136). [] E. Q. Gao and P. Y. Feng Calculation of te Reserve of Horizontal Cylindrical Oil Storage Tank Declined Fiting at Different Liquid Level Journal of Sandong Metallurgy ol. 1 1998 pp. 6-8. [3] T. J. Tian olume Calculation of te Straigt Cylindric Part of Tied Horizontal Tank Journal of Modern Measure and Test ol. 5 1999 pp. 3-36. [] F. J. Sun A Discussion on Some Difficulties in Calibra- tion Calculation of Horizontal Oil Tank olume Journal of Petroleum Products Application Researc ol. 18 No. 5 pp. -. [5] X. G. Pan Measurement and Calculation of Tilt Horizontal Tank olume Journal of Oil & Gas Storage and Transportation ol. 6 No. 6 1987 pp. 7-5. REFERENCES [1] ttp://www.mcm.edu.cn/tml_cn/node/d5ae73f57dea3 8cae73f7635aeee8.tml Copyrigt 1 SciRes.