1、5、R-球罐体的半径6、k-相同储油量时未变位和纵向变位油面高度差与变位角度的比例系数五、模型的建立与求解5.1 储油罐储油量与油面高度之间的数学模型 通过对问题的分析,我们试图找到罐体无变位时储油量与油面高度和罐体变位时储油量与油面高度的一种函数关系。为此,我们将附件1中的数据录入到计算机中,并进行曲线拟合,分别建立不同的回归方程,结果发现二次曲线的拟合度最高(见图1)。无变位数据散点图变位后数据散点图图1然后我们把进油和出油这两种不同的情形的数据录入到一个图形中,发现图形基本吻合,因此,我们可以用进油的数据得出储油量和油面高度的曲线方程: 无变位时:V=-4.822e-004*h2+4.4
2、98*h-515.4 变位时:V=-5.287e-005*h2+4.255*h-806 根据表达式,我们画出无变位和变位时曲线方程的图形(见图2),图2 进油实验散点图比较由图形可以看出,当罐容值相同时,变位后的油面高度偏高,并且当油面高度趋近于零时,储油量为负值。这是因为油管下方距罐底还有一定距离。当油面低于油管下部时,外界无法获取油,因此认定此时油罐已空,而事实上罐内还存有油。 我们根据无变位时储油量与油面高度的关系式标注倾斜角为=4.1。纵向变位时油面高度间隔为1cm的罐容表标定值。表1 高度间隔为1cm的罐容表标定值油位高度(mm)变位前标定值(L)变位后标定值(L)-515.406-
3、805.97810-470.469-763.42920-425.629-720.89230-380.885-678.36440-336.238-635.84850-291.687-593.34260-247.232-550.84670-202.874-508.36180-158.613-465.88790-114.448-423.423100-70.3789-380.97110-26.4067-338.52712017.4691-296.09513061.2485-253.673140104.931-211.262150148.518-168.862160192.008-126.4721702
4、35.401-84.093180278.699-41.7244190321.8990.633672200365.00442.9812210408.01185.3181220450.923127.644230493.738169.96240536.456212.265250579.078254.56260621.604296.844270664.033339.118280706.366381.381290748.602423.633300790.742465.875310832.785508.106320874.732550.326330916.583592.537340958.337634.7
5、36350999.994676.9253601041.56719.1033701083.02761.2713801124.39803.4283901165.66845.5754001206.84887.7114101247.91929.8364201288.9971.9514301329.781014.064401370.571056.154501411.271098.234601451.861140.314701492.361182.374801532.771224.424901573.071266.465001613.291308.495101653.41350.515201693.421
6、392.525301733.341434.525401773.171476.515501812.891518.485601852.531560.455701892.061602.415801931.51644.355901970.851686.296002010.091728.216102049.241770.136202088.31812.036302127.251853.926402166.111895.86502204.881937.686602243.551979.546702282.122021.396802320.592063.236902358.972105.067002397.
7、252146.887102435.442188.687202473.532230.487302511.522272.277402549.422314.057502587.222355.817602624.922397.577702662.532439.317802700.042481.057902737.452522.778002774.772564.488102811.992606.188202849.122647.888302886.152689.568402923.082731.238502959.912772.898602996.652814.548703033.292856.1888
8、03069.842897.818903106.292939.429003142.642981.039103178.93022.639203215.063064.229303251.133105.799403287.093147.369503322.963188.919603358.743230.459703394.423271.9998034303313.519903465.483355.0210003500.873396.5210103536.173438.0210203571.363479.510303606.463520.9710403641.463562.4310503676.3736
9、03.8710603711.183645.3110703745.93686.7410803780.513728.1610903815.033769.5611003849.463810.9611103883.793852.3411203918.023893.7211303952.153935.0811403986.193976.4411504020.134017.7811604053.984059.1111704087.734100.4311804121.384141.7511904154.944183.0512004188.44224.34 然后我们对这两个曲线方程进行分析,试图找到在相同储油
10、量的情况下,无变位时油面高度和有变位时油面高度以及倾斜角之间的关系。 我们求出21组不同储油量情形下对应的油面高度(见表2),对数据进行分析,发现高度差基本为一定值,h=65.7。很明显,这是因为倾斜角的影响,即h =k。因此可以得到比例系数k=16.02 。表2 储油量相同时油面高度比较 储油量(L)无变位油面高度(mm)变位油面高度(mm)平均高度差h(mm)312187.826465.7462222.7299612257.9335762293.3912329.14061062365.14411212401.54771362438.25131512475.15491662512.55841
11、812550.11962588.16562112626.56922262665.27282365.83691.97522467.06718.47792617.06758.08132718.83785.18372868.83825.48763018.83866.29093168.83907.39455.2 罐体变位后标定罐容表的数学模型我们先看只发生纵向倾斜(倾斜角为)的情况。由上面的分析,我们可以得到在相同的储油量时纵向倾斜时油面高度h纵与未变位时油面高度h未的关系式:h纵-h未=h 又h = k, 又k在上面已经求出,k=16.02 我们再在发生纵向倾斜的基础上再发生横向倾斜(倾斜角为)的情
12、况。由分析知,我们可以得到 由和 得 又由于,(-1.5z h=159.02 176.14 192.59 208.50 223.93 238.97 253.66 268.04 282.16 296.03 309.69 323.15 336.44 349.57 362.56 375.42 388.16 400.79 413.32 425.76 438.12 450.40 462.62 474.78 486.89 498.95 510.97 522.95 534.90 546.82 558.72 570.61 582.48 594.35 606.22 618.09 629.96 641.85 65
13、3.75 665.67 677.63 678.54 690.53 690.82 702.85 714.91 727.03 739.19 751.42 763.70 764.16 776.53 788.99 801.54 814.19 826.95 839.83 852.84 866.00 879.32 892.82 892.84 906.53 920.45 934.61 949.05 963.80 978.91 994.43 1010.43 1026.99 1044.25 1062.37 1081.59 1102.33 1125.32 1152.36 1193.49 ; v=312 362 4
14、12 462 512 562 612 662 712 762 812 862 912 962 1012 1062 1112 1162 1212 1262 1312 1362 1412 1462 1512 1562 1612 1662 1712 1762 1812 1862 1912 1962 2012 2062 2112 2162 2212 2262 2312 2315.83 2365.83 2367.06 2417.06 2467.06 2517.06 2567.06 2617.06 2666.98 2668.83 2718.83 2768.83 2818.83 2868.83 2918.8
15、3 2968.83 3018.83 3068.83 3118.83 3168.83 3168.91 3218.91 3268.91 3318.91 3368.91 3418.91 3468.91 3518.91 3568.91 3618.91 3668.91 3718.91 3768.91 3818.91 3868.91 3918.91 3968.91 ; h1=142.62 160.48 177.54 193.94 209.81 225.21 240.21 254.88 269.24 283.33 297.18 310.82 324.27 337.55 350.67 363.64 376.4
16、9 389.22 401.84 414.36 426.80 439.15 451.43 463.65 475.80 487.90 499.96 511.97 523.95 535.90 547.82 559.72 571.61 583.48 595.35 607.21 619.08 630.96 642.84 654.75 666.68 678.63 690.61 702.64 714.70 726.81 738.98 751.21 763.51 775.88 788.33 800.87 813.52 826.27 839.14 852.15 865.30 878.61 892.10 905.
17、78 919.69 933.84 948.26 962.99 978.08 993.57 1009.54 1026.08 1043.29 1061.36 1080.51 1101.15 1123.99 1150.72 ; v1=314.72 364.72 414.72 464.72 514.72 564.72 614.72 664.72 714.72 764.72 814.72 864.72 914.72 964.72 1014.72 1064.72 1114.72 1164.72 1214.72 1264.72 1314.72 1364.72 1414.72 1464.72 1514.72 1564.72 1614.72 1664.72 1714.72 1764.72 1814.72 1864.72 1914.72 1964.72 2014.72 2064.72 2114.72 2164.72 2214.72 2264.72 2314.72 2364.72 2414.72 2464.72 2514.72 2564.72 2614.72 266
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