儲(chǔ)油罐的變位識(shí)別與罐容表標(biāo)定數(shù)模競(jìng)賽論文

儲(chǔ)油罐的變位識(shí)別與罐容表標(biāo)定2010高教社杯全國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽承諾書(shū)我們仔細(xì)閱讀了中國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽的競(jìng)賽規(guī)則.我們完全明白，在競(jìng)賽開(kāi)始后參賽隊(duì)員不能以任何方式（包括電話、電子郵件、網(wǎng)上咨詢等）與隊(duì)外的任何人（包括指導(dǎo)教師）研究、討論與賽題有關(guān)的問(wèn)題。我們知道，抄襲別人的成果是違反競(jìng)賽規(guī)則的,如果引用別人的成果或其他公開(kāi)的資料（包括網(wǎng)上查到的資料），必須按照規(guī)定的參考文獻(xiàn)的表述方式在正文引用處和參考文獻(xiàn)中明確列出。我們鄭重承諾，嚴(yán)格遵守競(jìng)賽規(guī)則，以保證競(jìng)賽的公正、公平性。如有違反競(jìng)賽規(guī)則的行為，我們將受到嚴(yán)肅處理。我們參賽選擇的題號(hào)是（從A/B/C/D中選擇一項(xiàng)填寫(xiě)）： 我們的參賽報(bào)名號(hào)為（如果賽區(qū)設(shè)置報(bào)名號(hào)的話）：所屬學(xué)校（請(qǐng)?zhí)顚?xiě)完整的全名）：參賽隊(duì)員(打印并簽名)：12.3.指導(dǎo)教師或指導(dǎo)教師組負(fù)責(zé)人(打印并簽名)：日期：2010年9月賽區(qū)評(píng)閱編號(hào)（由賽區(qū)組委會(huì)評(píng)閱前進(jìn)行編號(hào)）：2008高教社杯全國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽編號(hào)專用頁(yè)賽區(qū)評(píng)閱編號(hào)（由賽區(qū)組委會(huì)評(píng)閱前進(jìn)行編號(hào)）：賽區(qū)評(píng)閱記錄（可供賽區(qū)評(píng)閱時(shí)使用）：評(píng)閱人評(píng)分備注全國(guó)統(tǒng)一編號(hào)（由賽區(qū)組委會(huì)送交全國(guó)前編號(hào)）：全國(guó)評(píng)閱編號(hào)（由全國(guó)組委會(huì)評(píng)閱前進(jìn)行編號(hào)PAGE50儲(chǔ)油罐的變位識(shí)別與罐容表標(biāo)定摘要本文是與儲(chǔ)油罐的變位識(shí)別與罐容表標(biāo)定相關(guān)的數(shù)學(xué)問(wèn)題。問(wèn)題一中，我們運(yùn)用高等數(shù)學(xué)中的微元的思想，分別建立了儲(chǔ)油罐在變前位、變位后的數(shù)學(xué)模型，并對(duì)其求解，得出變位前與變位后，在相同油高下，間隔為1cm的罐容表標(biāo)定值（詳見(jiàn)39頁(yè)）。油體高度變位后的體積變位前的體積121.6016.1718222.2024.4621………1093658.103786.66201103687.703809.9516通過(guò)對(duì)變位前、變位后標(biāo)定值的對(duì)比，在時(shí)得出傾斜對(duì)罐容表的影響為：傾斜之后的油罐，油體體積的標(biāo)定值在任意高度下都要比傾斜之前標(biāo)定體積要小。當(dāng)儲(chǔ)油罐內(nèi)裝油過(guò)多或過(guò)少時(shí)，傾斜對(duì)油體體積的標(biāo)定值影響較?。划?dāng)儲(chǔ)油罐內(nèi)油面深度接近罐體中線時(shí)影響最大。問(wèn)題二中，儲(chǔ)油罐發(fā)生縱向變位的同時(shí)還發(fā)生橫向變位。通過(guò)幾何分析找出傾角、偏角與油體高度間的函數(shù)關(guān)系。按照問(wèn)題一中的分析方式，利用微元法得到體積與深度、縱向傾角和橫向偏角的關(guān)系式，但方程比較復(fù)雜，不能夠得到解析解。因此我們采用近似算法，找出相鄰油層間的體積，利用最優(yōu)化理論得到、的最優(yōu)解，，再將代入上述關(guān)系式中求解，即得到罐體變位后油位高度間隔為的罐容表標(biāo)定值。(詳見(jiàn)41頁(yè))高度標(biāo)定值高度標(biāo)定值403.681836…26058.4239505.415106…27060.2533607.358329…28061.851我們用和高度算出體積，計(jì)算出理論出油體積，與實(shí)際出油體積作對(duì)比，算出相對(duì)誤差均值在0.1%左右，證明我們模型的準(zhǔn)確性較高。最后，我們用蒙特卡洛法對(duì)模型中的高度進(jìn)行擾動(dòng)時(shí)，發(fā)現(xiàn)對(duì)高度添加一個(gè)服從正態(tài)分布隨機(jī)擾動(dòng)值時(shí)產(chǎn)生的相對(duì)誤差與的值產(chǎn)生的相對(duì)誤差在一個(gè)數(shù)量級(jí)。從而證明了我們所建立的模型具有較好穩(wěn)定性。一、問(wèn)題重述（略）二、問(wèn)題分析加油站的地下儲(chǔ)油罐通過(guò)預(yù)先標(biāo)定的罐容表（即罐內(nèi)油位高度與儲(chǔ)油量的對(duì)應(yīng)關(guān)系）進(jìn)行實(shí)時(shí)計(jì)算，以得到罐內(nèi)油位高和儲(chǔ)油量的變化情況。罐體發(fā)生變位后會(huì)發(fā)生縱向傾角和橫向偏角，影響油浮子的位置，致使讀出的油位高度與實(shí)際高度不符，從而導(dǎo)致罐容表發(fā)生改變。通過(guò)對(duì)附表1數(shù)據(jù)的分析，可以看出進(jìn)油曲線和出油曲線完全重合，即沒(méi)有漏油存在，且流量計(jì)讀數(shù)完全正確。我們不考慮溫度、壓力及不計(jì)浮盤(pán)上下浮動(dòng)對(duì)罐內(nèi)油量體積的影響，但要考慮到油罐內(nèi)有注油管道和出油管道（未達(dá)油罐底部）以及油位探測(cè)管道（焊接至油罐底部），會(huì)占一部分體積?？紤]到在實(shí)際情況中，壓力及溫度對(duì)油罐變形的影響，為確保安全儲(chǔ)油罐不可能完全裝滿，也不可能在油量很少的情況下供油，且儲(chǔ)油罐的變位不可能很大，縱向傾角和橫向偏角都在小范圍內(nèi)。簡(jiǎn)化的儲(chǔ)油罐如下圖，我們只考慮在油量在陰影部位的情況。簡(jiǎn)化的儲(chǔ)油罐2.1對(duì)問(wèn)題一的分析：我們首先找出油罐傾斜前、傾斜后體積與傾角，高度間的函數(shù)關(guān)系，然后通過(guò)互相對(duì)比得出傾角對(duì)罐容表的影響程度。未傾斜的油罐：如圖：對(duì)于橫截面為橢圓形的儲(chǔ)油罐，封閉的罐體內(nèi)油面為水平面，在統(tǒng)一的坐標(biāo)系下，油罐內(nèi)處處油體深度相等，且處處橫截面面積相等。只需計(jì)算出任一截面面積，并對(duì)其長(zhǎng)度方向積分即可求出體積。通過(guò)得到的體積與高度間的關(guān)系式，即可得到任意高度時(shí)儲(chǔ)油罐中的理論儲(chǔ)油量。但實(shí)際中儲(chǔ)油罐內(nèi)含有輸油、進(jìn)油管道等其他影響油體體積的因素。其標(biāo)定值可能不等于理論值。我們通過(guò)修正得到它們之間的差值，從而找到正確的標(biāo)定值。傾斜的油罐：罐體縱向傾斜時(shí)，油深不再是處處相等，各截面面積也此時(shí)傾角、油體高度對(duì)體積都有影響。我們打算通過(guò)微元法建立他們的函數(shù)關(guān)系，對(duì)其進(jìn)行積分求解。同樣找出修正值，從而得出油罐傾斜時(shí)的正確標(biāo)定值。2.2對(duì)問(wèn)題二的分析：附件2中給出的數(shù)據(jù)是在傾角和偏角同時(shí)影響下的實(shí)際檢測(cè)數(shù)據(jù)。同時(shí)考慮傾角和偏角對(duì)罐容表標(biāo)定的影響，為了簡(jiǎn)化數(shù)學(xué)建模的求解，我們分為兩步求解，為此我們首先考慮橫向偏轉(zhuǎn)的情況，找出偏轉(zhuǎn)之后油體的實(shí)際高度與讀數(shù)高度、偏轉(zhuǎn)角之間的函數(shù)關(guān)系。將讀數(shù)高度用偏轉(zhuǎn)角和油體實(shí)際高度的函數(shù)表達(dá)，即建立：這樣既可將偏轉(zhuǎn)后實(shí)際油體高度轉(zhuǎn)為讀數(shù)高度，再考慮縱向變位時(shí)，就可將問(wèn)題轉(zhuǎn)化為問(wèn)題一的情況。從而得到的關(guān)系式。但問(wèn)題二中的儲(chǔ)油罐的主體是圓柱體，兩端為球冠體。這就使得建立積分方程比較復(fù)雜，無(wú)法得出解析解。我們采用兩種方法來(lái)解決這個(gè)問(wèn)題。對(duì)積分方程進(jìn)行數(shù)值解；采用近似法求得兩油層間體積的近似解，步驟如下：1）首先找出液面所在的截面面積與高度、變位角度之間的函數(shù)關(guān)系；2）取上下兩層距離為的兩油層，相鄰高度的體積可以用一個(gè)近似柱體來(lái)代替，其計(jì)算公式為：表示上層油面面積，表示下層油面面積，表示兩油層間的間距3）由題中所給數(shù)據(jù)，得到相鄰高度間的采集出油量，用最小二乘法思想建立優(yōu)化模型，可得到的估計(jì)值。把高度h、代入積分函數(shù)，便可以算出任意高度體積。從而給出油位高度間隔為的罐容表標(biāo)定值。相比之下，第二種方法比第一種方法更易編程實(shí)現(xiàn)，因此我們采用第二種方式進(jìn)行建模。三、模型假設(shè)1、流量計(jì)所測(cè)數(shù)據(jù)準(zhǔn)確；2、儲(chǔ)油罐為剛體，即油罐不會(huì)變形3、忽略因溫度變化而對(duì)油體容積的影響；4、不計(jì)油品的靜壓力作用而引起得油體的膨脹和收縮的容積；5、油浮子與標(biāo)尺為點(diǎn)接觸，不會(huì)因?yàn)橛透∽佑绊懹腕w體積；6、罐體發(fā)生變位時(shí)縱向傾角的取值范圍為，橫向偏角的取值范圍為四、符號(hào)說(shuō)明出油罐內(nèi)油體的讀數(shù)高度油體的實(shí)際高度橢圓的長(zhǎng)半軸橢圓的短半軸儲(chǔ)油罐的長(zhǎng)度按照讀數(shù)高度得出的油罐內(nèi)油體體積縱向傾角橫向偏角橢圓橫截面面積兩相鄰為的油層間的體積上層油面面積下層油面面積兩油層間的間距真實(shí)體積部分變量在使用時(shí)說(shuō)明五、模型的建立與求解5.1問(wèn)題（一）微分方程模型5.1.1油罐未變位時(shí)的模型根據(jù)對(duì)前面對(duì)問(wèn)題一的分析，首先建立了未變位時(shí)，儲(chǔ)油罐內(nèi)油量與油位高度的數(shù)學(xué)程模型[1]。建立如圖5.1的坐標(biāo)系，設(shè)油罐內(nèi)液面高度為，為恒定值；橢圓的長(zhǎng)半軸為,短半軸為，xyh簡(jiǎn)化小橢圓儲(chǔ)油罐Axyh簡(jiǎn)化小橢圓儲(chǔ)油罐AAA－A圖5.1橢圓軌跡方程為：(5.1)對(duì)半弓橫截面（陰影部分）進(jìn)行積分得出面積：(5.2)由（1）式得出為：(5.3)將(3)式代入（2）式得：(5.4)積分后可得：(5.5)面積再乘以長(zhǎng)度即為體積： (5.6)其中，半弓橢圓橫截面面積，表示油罐長(zhǎng)度，表示油罐內(nèi)油體體積。代入已知值以及變值，用求解得出結(jié)果部分見(jiàn)表1，詳見(jiàn)附錄1。累加進(jìn)油量/L實(shí)際累加體積/L油位高度/mm油高/m計(jì)算體積/m3實(shí)際體積/m3差值50312159.020.159020.322882590.3120.01088259100362176.140.176140.3746330340.3620.012633034…365739191152.361.152364.0556137683.918910.136703768370739691193.491.193494.1073621283.968910.138452128累加出油量/L實(shí)際累加體積/L油位高度/mm油高/m計(jì)算體積/m3實(shí)際體積/m3差值52.7239161150.721.150724.0527979523.916190.136607952102.738661123.991.123994.0010492213.866190.134859221…3653316.2160.480.160480.327208490.316190.011018493703266.2142.620.142620.2754733340.266190.009283334表1注：實(shí)際累加體積是在罐內(nèi)油量初值的基礎(chǔ)上累次加上每次進(jìn)油量得出，計(jì)算體積是按照上述函數(shù)式（5）代值求解的。不考慮其他任何影響因素的情況下，油深應(yīng)該與高度的關(guān)系為（6）式所得積分值。則計(jì)算體積應(yīng)該與實(shí)際體積相等，沒(méi)有差值產(chǎn)生。但由結(jié)果可見(jiàn)計(jì)算體積與實(shí)際體積之間有差值。對(duì)題中所給附件1中數(shù)據(jù)分析可知，儲(chǔ)油罐在13:20以前一直不間斷進(jìn)油，儲(chǔ)存油量到達(dá)最高值，之后一直出油，油量減到最低值。差值按照先增后減的趨勢(shì)排列。我們考慮差值與油深間存在著對(duì)應(yīng)關(guān)系，利用擬合函數(shù)找出它們間的關(guān)系式：（5.7）并求出相關(guān)系數(shù)，相關(guān)系數(shù)越接近1，說(shuō)明線性關(guān)系很強(qiáng)。將計(jì)算值減去差值得出正確值，該值與真實(shí)值十分貼近。這也驗(yàn)證了我們前面的分析，即不可忽視油罐內(nèi)的注油管道和出油管道（未達(dá)油罐底部）以及油位探測(cè)管道（焊接至油罐底部）占去的體積。圖5.2出油量和進(jìn)油量誤差排序后的擬合曲線圖5.2出油量和進(jìn)油量誤差排序后的擬合曲線算出的油體體積與真實(shí)值不符，它們之間有一個(gè)差值即真實(shí)體積=計(jì)算體積-差值，聯(lián)系（6）（7）式：根據(jù)此式可計(jì)算出未變位時(shí)，油位高度間隔為的罐容表標(biāo)定值（見(jiàn)表2詳見(jiàn)附錄4）：h（m）V/m3v（L升）0.010.01617216.171750.020.02446224.462110.030.03554535.54459………1.073.7370553737.0551.083.7623433762.3431.093.7866623786.6621.13.8099523809.952表2未傾斜時(shí)罐容表標(biāo)定值5.1.2油罐發(fā)生縱向變位時(shí)的模型：圖5.3建立如圖5.3的三維坐標(biāo)。設(shè)通過(guò)油位計(jì)讀出的油深為；某一微小截面處油深為，值不恒定，隨著油深改變；已知橢圓的長(zhǎng)半軸為；短半軸為；油罐逆時(shí)針變位縱向傾角。如圖5.3在區(qū)間內(nèi)任意去某一處微小段，同時(shí)用與底面平行的平面與液體面相截，得到圖中陰影部分微元面積元的體積：(5.8)由上述（5）中推出的弓形面積共識(shí)式可知：(5.9),,,之間的幾何關(guān)系見(jiàn)下圖,為已知數(shù)據(jù)：ffdZh’h(5.10)聯(lián)立（7）（8）（9）式得：(5.11)對(duì)上式進(jìn)行積分:(5.12)再代入各項(xiàng)已知數(shù)據(jù)，利用MATLAB求出結(jié)果如下表。由于篇幅有限，在此僅列出部分結(jié)果（見(jiàn)表3），詳見(jiàn)附錄5。累加進(jìn)油量（升）油位高度實(shí)際體積計(jì)算體積實(shí)際與計(jì)算體積的差值747.86411.29962.861.01000.0472797.86423.451012.861.05830.0455……………3297.731034.243512.733.5694574790.05673299.741035.363514.743.5732283220.0585累加出油量（升）油位高度實(shí)際體積計(jì)算體積實(shí)際與計(jì)算體積的差值501020.653464.743.5231136560.05837371001007.733414.743.4780872070.0633472……………2500425.831014.741.0678373580.05309742550411.73964.741.0117861410.0470461表3由（11）式就可得出傾角為時(shí)，小橢圓儲(chǔ)油罐在不同高度下的計(jì)算體積。同樣，油罐傾斜時(shí)的計(jì)算體積與實(shí)際體積間仍有差值，差值=計(jì)算體積-實(shí)際體積。實(shí)際體積是在罐內(nèi)油量初值的基礎(chǔ)上累次加上每次進(jìn)油量得出，計(jì)算體積將值代入函數(shù)式（11）中求出的。同樣按照5.1.1的方法將差值按照先增后減的趨勢(shì)排列（見(jiàn)附錄6），油罐傾斜時(shí)的差值與讀數(shù)高度同樣存在對(duì)應(yīng)關(guān)系。利用MATLAB擬合工具箱CFTOOL對(duì)差值和高度進(jìn)行分析（程序代碼見(jiàn)附錄7），得出差值與油面深度呈非線性關(guān)系為（見(jiàn)圖5.4）：（5.13）,擬合度好。圖5.4圖5.4利用函數(shù)式（11）計(jì)算出的油體體積與真實(shí)值不符，它們之間有一個(gè)差值即：真實(shí)體積=計(jì)算體積-差值（5.14）按照題目要求即可得到油位高度間隔1cm的罐容表標(biāo)定值（見(jiàn)表3，詳見(jiàn)附錄8）：h(cm)h（m）v（m3）v(L升)180.180.2136213.6190.190.2378237.8200.200.2630263.0…………1071.073.59653596.51081.083.62773627.71091.093.65813658.11101.103.68773687.7表3傾斜之后的罐容表標(biāo)定值說(shuō)明：標(biāo)定高度范圍在之間。原因：由前面的假設(shè)，縱向傾角和橫向偏角都,討論極限情況，即油體液面剛好位于Ⅰ線處，假設(shè)這時(shí)的傾角為最大值。幾何關(guān)系見(jiàn)圖5.5ⅠffⅠ圖5.5為已知數(shù)據(jù)，則則油面深度的標(biāo)定區(qū)間為。5.1.3傾斜前后罐容表標(biāo)定值比較（詳細(xì)見(jiàn)附錄8）：油體高度未傾斜時(shí)罐容表標(biāo)定值傾斜之后的罐容表標(biāo)定值同等高度下傾斜與未傾斜油量差值）0.0116.1717521.6-5.40.0224.4621335.5445923.811.7…………1.083762.3433627.7134.61.093786.6623658.1128.61.103809.9523687.7122.3利用Excel對(duì)同等高度下變位前后的油體體積差值進(jìn)行處理，并生成圖形：圖5.6（實(shí)線為變位后的體積，虛線為變位前的體積）圖5.7由數(shù)據(jù)和圖5.6可以看出傾斜之后的儲(chǔ)油罐，油體體積的標(biāo)定值在任意同等高度下都要比傾斜之前標(biāo)定體積要小。。從圖5.7可以清楚的看到，當(dāng)儲(chǔ)油罐內(nèi)裝油過(guò)多或過(guò)少時(shí)，傾斜對(duì)油體體積的標(biāo)定值影響較小；當(dāng)儲(chǔ)油罐內(nèi)油面深度接近罐體中線時(shí)影響最大。5.2問(wèn)題（二）數(shù)學(xué)模型ZZ5.2．1微元法求出函數(shù)關(guān)系如圖：在此問(wèn)題中，油罐的形狀可以看成是兩個(gè)圓的部分體積與圓柱體體積之和組成，因此可以按液面在水平面的投影分為三個(gè)區(qū)間處理。第Ⅰ部分（由液面圍成的左邊圓球部分）建立如圖坐標(biāo)系，以為坐標(biāo)遠(yuǎn)點(diǎn)，同樣采用微元法求解體積，由公式（5）知道弓形面積仍然是：其中為圓柱體的半徑值，為弓形截面水深高度，兩者均是變值；定性分析，在第一部分的區(qū)間內(nèi)，隨著Z值的增大，R增大，有勾股定理得：其r為圓球體的半徑值，是已知值；與測(cè)量水深的之間有幾何知識(shí)推導(dǎo)出；綜上所敘，可以的到第一部分的水容量的幾分方程式：第Ⅱ部分（由液面圍成的圓柱體部分）第二部分以為圓心，分析方式與問(wèn)題以中發(fā)生變位時(shí)相同；為一確定值（圓筒的半徑）第Ⅲ部分（由液面圍成的右邊圓球部分）第三部分以為圓心，向Z軸負(fù)方向積分，整個(gè)推到過(guò)程和第一部分相似，以下部分分析省略；則（5.2.1）從而得到5.2．2優(yōu)化模型求我們打算用優(yōu)化模型求解參數(shù)，這需要體積關(guān)于的顯示函數(shù)關(guān)系式，這由于上面對(duì)體積的積分沒(méi)有解析解，無(wú)法得到他們的具體函數(shù)關(guān)系，我們用其他方法得到體積和的函數(shù)關(guān)系。數(shù)據(jù)中給出的每次出油高度變化很小，可以把減少的體積當(dāng)做一層薄油膜，油膜的體積即出油量，我們可以用柱體來(lái)替代油膜，從而用柱體體積近似出油量，這里我們的柱體面積也是高度的函數(shù)，所以油膜體積可以用高度、傾角和偏角表示出來(lái)。這首先找出油面面積與各參數(shù)之間的函數(shù)關(guān)系：當(dāng)罐體被油面截開(kāi)后的形狀為兩端為圓形的一部分，中間為橢圓形的一部分。則油面面積（5.2.2）圖5.9A求出的面積：A由圖5.9，幾何關(guān)系如下：由勾股定理得：由投影關(guān)系知：由弓形面積＝圓弧面積－三角形面積得:(5.2.3)同理 求出的面積：(5.2.4)下面求解的面積：OOYXabh為如圖橢圓面積的一部分；橢圓面積則 橢圓長(zhǎng)軸：橢圓短半軸：由圖中陰影部分面積對(duì)Y軸積分得面積同理得到面積(5.2.5)其中聯(lián)立(5.2.2)(5.2.3)(5.2.4)（5.2.5）得到油面面積與高度、縱向傾角和橫向偏角的函數(shù)關(guān)系式：即優(yōu)化模型：由于可以把油膜近似為柱體，和對(duì)面積表達(dá)式的推算。我們用最小二乘法思想，建立優(yōu)化模型，確定使近似出的體積和實(shí)際的體積差值平方和最小。模型如下：其中：表示第個(gè)油膜的真實(shí)體積，即出油量可以從數(shù)據(jù)中得到表示第個(gè)油膜的下表面面積，可以參照上面的推算表示第個(gè)油膜的上表面面積，即第個(gè)油膜的下表面面積表示第個(gè)油膜的真實(shí)高度表示第個(gè)油膜的真實(shí)高度用求解模型，確定變位參數(shù)、。解得。將該值代入（５.２.１）積分表達(dá)式，利用對(duì)體積積分，便可以得到高度的體積，計(jì)算罐體變位后油位高度間隔為的罐容表標(biāo)定值（見(jiàn)附錄9）。六、模型準(zhǔn)確性分析我們對(duì)第二問(wèn)的模型進(jìn)行檢驗(yàn)，算出參數(shù)、，帶入積分函數(shù)對(duì)積分算出體積，體積做差，便得出油膜體積，和真實(shí)的出油量做比較，算出相對(duì)誤差來(lái)驗(yàn)證模型的準(zhǔn)確性。數(shù)據(jù)如下：實(shí)際出油計(jì)算體積相對(duì)誤差實(shí)際出油計(jì)算體積相對(duì)誤差149.09148.762230.2198%238.33237.762960.2379%68.4568.5247540.1092%42.9243.2011430.6550%199.27197.402550.9371%171.34170.55720.4569%70.0570.4974430.6387%212.34212.28370.0265%136.36135.219920.8361%92.3892.3345280.0492%232.74233.170590.1850%243.85243.057540.3250%107.97108.863570.8276%206.69208.203280.7321%49.2448.8350410.8224%224.5223.432820.4754%229.93229.29010.2783%169.26169.639330.2241%181.7180.035710.9160%220.09218.761630.6036%238.52238.757460.0996%117.54117.487760.0444%131.79132.08750.2257%93.4494.0749180.6795%174.69175.871080.6761%114.46114.220460.2093%由數(shù)據(jù)計(jì)算得相對(duì)誤差平均值小于，說(shuō)明模型準(zhǔn)確性比較高。七、模型的穩(wěn)定性分析我們利用蒙特卡洛法進(jìn)行模型的穩(wěn)定性檢驗(yàn)，當(dāng)對(duì)高度施加一個(gè)微小擾動(dòng)時(shí)，即，檢驗(yàn)此時(shí)的變化情況，如其波動(dòng)很小，則模型穩(wěn)定。我們進(jìn)行了多組檢驗(yàn)，發(fā)現(xiàn)誤差隨服從正態(tài)分布,。將代入函數(shù)式，得出的值，比較擾動(dòng)前后值的變化情況：原值相對(duì)誤差率相對(duì)誤差率第一組2.05691.48430.00%0.00%第二組2.05341.47020.17%0.95%第三組2.05541.48010.07%0.28%由數(shù)據(jù)可知穩(wěn)定性較好。擾動(dòng)后對(duì)油體體積的影響見(jiàn)附錄10。八、模型的優(yōu)缺點(diǎn)分析模型的優(yōu)點(diǎn)：1、問(wèn)題一中，我們采用橫截面為弓形面的微元法，對(duì)整個(gè)積分區(qū)間進(jìn)行一次積分，該方法簡(jiǎn)單一算，而且很好的運(yùn)用到第二問(wèn)中；2、問(wèn)題二中，我們?cè)诓捎梦⒃ń鉀Q問(wèn)題時(shí)發(fā)現(xiàn)不能找到原函數(shù)的解析式，為了采用優(yōu)化模型求的、，我們?cè)谇篌w積時(shí)采用了近似方法，從而是、的表達(dá)列式為顯示形式。模型的缺點(diǎn)：1．在對(duì)、的值的確定是，用很薄的梯形圓柱面的體積看成兩液面見(jiàn)體積，產(chǎn)生了誤差。2.實(shí)際中，儲(chǔ)油罐內(nèi)油體體積與溫度、壓力等因素有關(guān)，但我們建模時(shí)未予考慮。九、模型的推廣在問(wèn)題而中我們建立優(yōu)化模型，其中，將兩層液面間所夾體積看成是梯形柱面體積產(chǎn)生誤差，我們可以用微積分的方法找到它的精確近似解，從而減少誤差。即。七、參考文獻(xiàn)[1]高恩強(qiáng)豐培云，臥式傾斜安裝圓柱體油罐不同液面高度時(shí)貯油量的計(jì)算，[2]樊映川,等編.高等數(shù)學(xué)講義下冊(cè),北京:高等教育出版社,1993:62.附錄1 問(wèn)題（一）中，罐體未變位時(shí)的高度與體積值累加進(jìn)油量/L實(shí)際累加體積/L油位高度/mm油高/m計(jì)算體積/m3實(shí)際體積/m3差值50312159.020.159020.322882590.3120.01088259100362176.140.176140.3746330340.3620.012633034150412192.590.192590.4263649440.4120.014364944200462208.500.20850.4781317650.4620.016131765250512223.930.223930.529851880.5120.01785188300562238.970.238970.5816057660.5620.019605766350612253.660.253660.6333520370.6120.021352037400662268.040.268040.6850810480.6620.023081048450712282.160.282160.7368467550.7120.024846755500762296.030.296030.7885776580.7620.026577658550812309.690.309690.8403288240.8120.028328824600862323.150.323150.892056110.8620.03005611650912336.440.336440.9438023350.9120.031802335700962349.570.349570.9955422760.9620.0335422767501012362.560.362561.0472971141.0120.0352971148001062375.420.375421.0990547911.0620.0370547918501112388.160.388161.1508080961.1120.0388080969001162400.790.400791.202554141.1620.040554149501212413.320.413321.2542938711.2120.04229387110001262425.760.425761.3060316551.2620.04403165510501312438.120.438121.3577748871.3120.04577488711001362450.400.45041.4094914081.3620.04749140811501412462.620.462621.4612354451.4120.04923544512001462474.780.474781.5129789061.4620.05097890612501512486.890.486891.5647375991.5120.05273759913001562498.950.498951.6164859881.5620.05448598813501612510.970.510971.6682425061.6120.05624250614001662522.950.522951.7199836111.6620.05798361114501712534.900.53491.7717297481.7120.05972974815001762546.820.546821.8234588661.7620.06145886615501812558.720.558721.8751928761.8120.06319287616001862570.610.570611.9269543171.8620.06495431716501912582.480.582481.9786789731.9120.06667897317001962594.350.594352.0304335481.9620.06843354817502012606.220.606222.0821977682.0120.07019776818002062618.090.618092.1339513712.0620.07195137118502112629.960.629962.1856740792.1120.07367407919002162641.850.641852.2374325912.1620.07543259119502212653.750.653752.2891626842.2120.07716268420002262665.670.665672.3408868872.2620.07888688720502312677.630.677632.3926703372.3120.08067033720542316678.540.678542.3966050962.315830.08077509621042366690.530.690532.4483710132.365830.08254101321052367690.820.690822.4496211782.367060.08256117821552417702.850.702852.5013953292.417060.08433532922052467714.910.714912.553114852.467060.0860548522552517727.030.727032.6048838042.517060.08782380423052567739.190.739192.6565911262.567060.08953112623552617751.420.751422.7083374992.617060.09127749924052667763.700.76372.7600100822.666980.09303008224072669764.160.764162.7619398162.668830.09310981624572719776.530.776532.8136645192.718830.09483451925072769788.990.788992.8654193242.768830.09658932425572819801.540.801542.9171692792.818830.09833927926072869814.190.814192.9689183042.868830.10008830426572919826.950.826953.0206668582.918830.10183685827072969839.830.839833.0724115042.968830.10358150427573019852.840.852843.1241444273.018830.10531442728073069866.000.8663.1758919843.068830.10706198428573119879.320.879323.2276344323.118830.10880443229073169892.820.892823.2793836373.168830.11055363729073169892.840.892843.2794597643.168910.11054976429573219906.530.906533.3311793833.218910.11226938330073269920.450.920453.3829373763.268910.11402737630573319934.610.934613.4346743433.318910.11576434331073369949.050.949053.4864252783.368910.11751527831573419963.800.96383.53816723.418910.119257232073469978.910.978913.5899206123.468910.12101061232573519994.430.994433.6416697423.518910.122759742330735691010.431.010433.6934184863.568910.124508486335736191026.991.026993.7451380653.618910.126228065340736691044.251.044253.7968908033.668910.127980803345737191062.371.062373.8486493093.718910.129739309350737691081.591.081593.9003852743.768910.131475274355738191102.331.102333.9521385153.818910.133228515360738691125.321.125324.0038632053.868910.134953205365739191152.361.152364.0556137683.918910.136703768370739691193.491.193494.1073621283.968910.138452128累加出油量/L體積油位高度/mm油高m體積m3實(shí)際體積差值52.7239161150.721.150724.0527979523.916190.136607952102.738661123.991.123994.0010492213.866190.134859221152.738161101.151.101153.94931663.816190.1331266202.737661080.511.080513.8975703723.766190.131380372252.737161061.361.061363.8458377813.716190.129647781302.736661043.291.043293.7940731313.666190.127883131352.736161026.081.026083.7423470073.616190.126157007402.735661009.541.009543.6905846413.566190.124394641452.73516993.570.993573.6388412173.516190.122651217502.73466978.080.978083.587112053.466190.12092205552.73416962.990.962993.5353563743.416190.119166374602.73366948.260.948263.4836214523.366190.117431452652.73316933.840.933843.4318856023.316190.115695602702.73266919.690.919693.3801339863.266190.113943986752.73216905.780.905783.3283664693.216190.112176469802.73166892.100.89213.2766419853.166190.110451985852.73116878.610.878613.2248931413.116190.108703141902.73066865.300.86533.1731547963.066190.106964796952.73016852.150.852153.1214147563.016190.10522475610032966839.140.839143.0696525022.966190.10346250210532916826.270.826273.0179210172.916190.10173101711032866813.520.813522.9661883312.866190.09999833111532816800.870.800872.9144165952.816190.09822659512032766788.330.788332.8626870512.766190.09649705112532716775.880.775882.8109548622.716190.09476486213032666763.510.763512.759212892.666190.0930228913532616751.210.751212.7074512912.616190.09126129114032566738.980.738982.6557002612.566190.08951026114532516726.810.726812.6039460952.516190.08775609515032466714.700.71472.5522159612.466190.08602596115532416702.640.702642.5004930462.416190.08430304616032366690.610.690612.4487158962.366190.08252589616532316678.630.678632.3969942052.316190.08080420517032266666.680.666682.3452646252.266190.07907462517532216654.750.654752.2935058212.216190.07731582118032166642.840.642842.2417392152.166190.07554921518532116630.960.630962.1900294552.116190.07383945519032066619.080.619082.1382666882.066190.07207668819532016607.210.607212.0865148882.016190.07032488820031966595.350.595352.0347943871.966190.06860438720531916583.480.583481.9830382181.916190.06684821821031866571.610.571611.9313102581.866190.06512025821531816559.720.559721.8795437911.816190.06335379122031766547.820.547821.8278030241.766190.06161302422531716535.900.53591.7760653971.716190.05987539723031666523.950.523951.7243089721.666190.05811897223531616511.970.511971.6725557731.616190.05636577324031566499.960.499961.6208283141.566190.05463831424531516487.900.48791.5690639431.516190.05287394325031466475.800.47581.5173299791.466190.05113997925531416463.650.463651.4656088511.416190.04941885126031366451.430.451431.4138423661.366190.04765236626531316439.150.439151.3621012211.316190.04591122127031266426.800.42681.3103728321.266190.04418283227531216414.360.414361.2586054841.216190.04241548428031166401.840.401841.2068748441.166190.04068484428531116389.220.389221.1551345861.116190.03894458629031066376.490.376491.1033835351.066190.03719353529531016363.640.363641.0516243291.016190.0354343293003966.2350.670.350670.9999035630.966190.0337135633053916.2337.550.337550.9481532050.916190.0319632053103866.2324.270.324270.8963917650.866190.0302017653153816.2310.820.310820.8446440670.816190.0284540673203766.2297.180.297180.7929043960.766190.0267143963253716.2283.330.283330.7411775050.716190.0249875053303666.2269.240.269240.6894437950.666190.0232537953353616.2254.880.254880.6377003980.616190.0215103983403566.2240.210.240210.5859291020.566190.0197391023453516.2225.210.225210.5342061820.516190.0180161823503466.2209.810.209810.4824663040.466190.0162763043553416.2193.940.193940.4306930950.416190.0145030953603366.2177.540.177540.3789614050.366190.0127714053653316.2160.480.160480.327208490.316190.011018493703266.2142.620.142620.2754733340.266190.009283334附錄2出油量與進(jìn)油量差值排序排序后排序后高度h:差值：高度差值0.159020.01088260.142620.00928330.176140.0126330.159020.01088260.192590.01436490.160480.01101850.20850.01613180.176140.0126330.223930.01785190.177540.01277140.238970.01960580.192590.01436490.253660.0213520.193940.01450310.268040.0230810.20850.01613180.282160.02484680.209810.01627630.296030.02657770.223930.01785190.309690.02832880.225210.01801620.323150.03005610.238970.01960580.336440.03180230.240210.01973910.349570.03354230.253660.0213520.362560.03529710.254880.02151040.375420.03705480.268040.0230810.388160.03880810.269240.02325380.400790.04055410.282160.02484680.413320.04229390.283330.02498750.425760.04403170.296030.02657770.438120.04577490.297180.02671440.45040.04749140.309690.02832880.462620.04923540.310820.02845410.474780.05097890.323150.03005610.486890.05273760.324270.03020180.498950.0544860.336440.03180230.510970.05624250.337550.03196320.522950.05798360.349570.03354230.53490.05972970.350670.03371360.546820.06145890.362560.03529710.558720.06319290.363640.03543430.570610.06495430.375420.03705480.582480.0666790.376490.03719350.594350.06843350.388160.03880810.606220.07019780.389220.03894460.618090.07195140.400790.04055410.629960.07367410.401840.04068480.641850.07543260.413320.04229390.653750.07716270.414360.04241550.665670.07888690.425760.04403170.677630.08067030.42680.04418280.678540.08077510.438120.04577490.690530.0825410.439150.04591120.690820.08256120.45040.04749140.702850.08433530.451430.04765240.714910.08605490.462620.04923540.727030.08782380.463650.04941890.739190.08953110.474780.05097890.751420.09127750.47580.051140.76370.09303010.486890.05273760.764160.09310980.48790.05287390.776530.09483450.498950.0544860.788990.09658930.499960.05463830.801540.09833930.510970.05624250.814190.10008830.511970.05636580.826950.10183690.522950.05798360.839830.10358150.523950.0581190.852840.10531440.53490.05972970.8660.1070620.53590.05987540.879320.10880440.546820.06145890.892820.11055360.547820.0616130.892840.11054980.558720.06319290.906530.11226940.559720.06335380.920450.11402740.570610.06495430.934610.11576430.571610.06512030.949050.11751530.582480.0666790.96380.11925720.583480.06684820.978910.12101060.594350.06843350.994430.12275970.595350.06860441.010430.12450850.606220.07019781.026990.12622810.607210.07032491.044250.12798080.618090.07195141.062370.12973930.619080.07207671.081590.13147530.629960.07367411.102330.13322850.630960.07383951.125320.13495320.641850.07543261.152360.13670380.642840.07554921.193490.13845210.653750.07716271.150720.1366080.654750.07731581.123990.13485920.665670.07888691.101150.13312660.666680.07907461.080510.13138040.677630.08067031.061360.12964780.678540.08077511.043290.12788310.678630.08080421.026080.1261570.690530.08252591.009540.12439460.690610.0825410.993570.12265120.690820.08256120.978080.12092210.702640.0843030.962990.11916640.702850.08433530.948260.11743150.71470.0860260.933840.11569560.714910.08605490.919690.1139440.726810.08775610.905780.11217650.727030.08782380.89210.1104520.738980.08951030.878610.10870310.739190.08953110.86530.10696480.751210.09126130.852150.10522480.751420.09127750.839140.10346250.763510.09302290.826270.1017310.76370.09303010.813520.09999830.764160.09310980.800870.09822660.775880.09476490.788330.09649710.776530.09483450.775880.09476490.788330.09649710.763510.09302290.788990.09658930.751210.09126130.800870.09822660.738980.08951030.801540.09833930.726810.08775610.813520.09999830.71470.0860260.814190.10008830.702640.0843030.826270.1017310.690610.08252590.826950.10183690.678630.08080420.839140.10346250.666680.07907460.839830.10358150.654750.07731580.852150.10522480.642840.07554920.852840.10531440.630960.07383950.86530.10696480.619080.07207670.8660.1070620.607210.07032490.878610.10870310.595350.06860440.879320.10880440.583480.06684820.89210.1104520.571610.06512030.892820.11054980.559720.06335380.892840.11055360.547820.0616130.905780.11217650.53590.05987540.906530.11226940.523950.0581190.919690.1139440.511970.05636580.920450.11402740.499960.05463830.933840.11569560.48790.05287390.934610.11576430.47580.051140.948260.11743150.463650.04941890.949050.11751530.451430.04765240.962990.11916640.439150.04591120.96380.11925720.42680.04418280.978080.12092210.414360.04241550.978910.12101060.401840.04068480.993570.12265120.389220.03894460.994430.12275970.376490.03719351.009540.12439460.363640.03543431.010430.12450850.350670.03371361.026080.1261570.337550.03196321.026990.12622810.324270.03020181.043290.12788310.310820.02845411.044250.12798080.297180.02671441.061360.12964780.283330.02498751.062370.12973930.269240.02325381.080510.13138040.254880.02151041.081590.13147530.240210.01973911.101150.13312660.225210.01801621.102330.13322850.209810.01627631.123990.13485920.193940.01450311.125320.13495320.177540.01277141.150720.1366080.160480.01101851.152360.13670380.142620.00928331.193490.1384521附錄3第一次差值數(shù)據(jù)擬合LinearmodelPoly1:f(x)=p1*x+p2Coefficients(with95%confidencebounds):p1=0.1353(0.1342,0.1364)p2=-0.01223(-0.01302,-0.01144)Goodnessoffit:SSE:0.0005653R-square:0.9973AdjustedR-square:0.9973RMSE:0.001941附錄4未變位時(shí)，油位高度間隔為的罐容表標(biāo)定值hi（m）vi(M3)vi（L）0.010.01617216.171750.020.02446224.462110.030.03554535.544590.040.04885548.855120.050.06406364.063260.060.08094380.94310.070.09932799.326580.080.119082119.08170.090.140101140.1010.10.162294162.2940.110.185584185.58370.120.209903209.90250.130.235191235.1910.140.261396261.39580.150.288469288.46890.160.316366316.36640.170.345048345.04820.180.374477374.47710.190.404619404.61870.20.435441435.44060.210.466913466.91290.220.499007499.00690.230.531696531.69590.240.564954564.95430.250.598758598.7580.260.633084633.08370.270.667909667.90940.280.703214703.21390.290.738977738.97660.30.775178775.1780.310.811799811.79910.320.848822848.82160.330.886228886.22760.340.924923.99990.350.962122962.12170.361.0005771000.5770.371.0393491039.3490.381.0784231078.4230.391.1177831117.7830.41.1574151157.4150.411.1973041197.3040.421.2374361237.4360.431.2777961277.7960.441.3183711318.3710.451.3591471359.1470.461.4001111400.1110.471.4412491441.2490.481.4825491482.5490.491.5239971523.9970.51.565581565.580.511.6072871607.2870.521.6491031649.1030.531.6910181691.0180.541.7330181733.0180.551.775091775.090.561.8172241817.2240.571.8594061859.4060.581.9016251901.6250.591.9438681943.8680.61.9861231986.1230.612.0283782028.3780.622.0706212070.6210.632.1128392112.8390.642.1550222155.0220.652.1971552197.1550.662.2392282239.2280.672.2812282281.2280.682.3231422323.1420.692.3649592364.9590.72.4066652406.6650.712.4482492448.2490.722.4896972489.6970.732.5309962530.9960.742.5721352572.1350.752.6130982613.0980.762.6538752653.8750.772.694452694.450.782.734812734.810.792.7749412774.9410.82.814832814.830.812.8544622854.4620.822.8938232893.8230.832.9328972932.8970.842.9716692971.6690.853.0101243010.1240.863.0482463048.2460.873.0860183086.0180.883.1234243123.4240.893.1604473160.4470.93.1970683197.0680.913.2332693233.2690.923.2690323269.0320.933.3043363304.3360.943.3391623339.1620.953.3734883373.4880.963.4072913407.2910.973.440553440.550.983.4732393473.2390.993.5053333505.33313.5368053536.8051.013.5676273567.6271.023.5977693597.7691.033.6271973627.1971.043.6558793655.8791.053.6837773683.7771.063.710853710.851.073.7370553737.0551.083.7623433762.3431.093.7866623786.6621.13.8099523809.952附錄5問(wèn)題（一）中，罐體變位后的高度與體積值累加進(jìn)油量/L油位高度/mm高度m體積L體積m3體積計(jì)算差值747.86411.290.41129962.860.962861.01000.0472797.86423.450.423451012.861.012861.05830.0455847.86438.330.438331062.861.062861.11800.0552897.86450.540.450541112.861.112861.16750.0547947.86463.900.46391162.861.162861.22210.0593997.86477.740.477741212.861.212861.27920.06631047.86489.370.489371262.861.262861.32750.06461097.79502.560.502561312.791.312791.38260.06981147.79514.690.514691362.791.362791.43360.07081197.73526.840.526841412.731.412731.48500.07221247.73538.880.538881462.731.462731.53610.07341297.73551.960.551961512.731.512731.59190.07911347.73564.400.56441562.731.562731.64510.08241397.73576.560.576561612.731.612731.69740.08471447.73588.740.588741662.731.662731.74990.08721497.73599.560.599561712.731.712731.79660.08391547.73611.620.611621762.731.762731.84880.08611597.73623.440.623441812.731.812731.90000.08731647.73635.580.635581862.731.862731.95270.09001697.73646.280.646281912.731.912731.99920.08651747.73658.590.658591962.731.962732.05270.09001797.73670.220.670222012.732.012732.1032310790.09051847.73680.630.680632062.732.062732.1484490190.08571897.73693.030.693032112.732.112732.2022725250.08951947.73704.670.704672162.732.162732.2527394330.09001997.73716.450.716452212.732.212732.3037360960.09102047.73727.660.727662262.732.262732.3521746010.08942097.73739.390.739392312.732.312732.4027460540.09002147.73750.900.75092362.732.362732.4522364460.08952197.73761.550.761552412.732.412732.4978951990.08522247.73773.430.773432462.732.462732.5486560440.08592297.73785.390.785392512.732.512732.5995552140.08682347.73796.040.796042562.732.562732.6446893730.08202397.73808.270.808272612.732.612732.6962772680.08352447.73820.800.82082662.732.662732.7488374490.08612497.73832.800.83282712.732.712732.7988716260.08612547.73844.470.844472762.732.762732.8472225330.08452597.73856.290.856292812.732.812732.8958626850.08312647.73867.600.86762862.732.862732.9420691220.07932697.73880.060.880062912.732.912732.9925684490.07982747.73892.920.892922962.732.962733.0442130030.08152797.73904.340.904343012.733.012733.0896427980.07692847.73917.340.917343062.733.062733.140831720.07812897.73929.900.92993112.733.112733.1897229260.07702947.73941.420.941423162.733.162733.2340478730.07132997.73954.600.95463212.733.212733.284115830.07143047.73968.090.968093262.733.262733.3346067540.07193097.73980.140.980143312.733.312733.3790235960.06633147.73992.410.992413362.733.362733.4235491330.06083197.7