韓 勇, 郭向利, 龍新平

(1. 中國工程物理研究院化工材料研究所, 四川 綿陽 621999; 2. 中國工程物理研究院, 四川 綿陽 621999)

1 引 言

炸藥爆轟時(shí),爆轟產(chǎn)物處于高溫高壓狀態(tài),爆轟產(chǎn)物壓力由數(shù)十吉帕逐漸衰減至兆帕量級(jí),溫度也由數(shù)千開爾文逐漸衰減至1000K甚至更低,建立能夠準(zhǔn)確描述爆轟產(chǎn)物氣體高溫、高壓的狀態(tài)方程對(duì)有效表征爆轟產(chǎn)物狀態(tài)變化過程具有十分重要的意義。

目前,實(shí)現(xiàn)爆轟產(chǎn)物氣體所處的高溫、高壓試驗(yàn)條件十分困難,關(guān)于高溫、高壓氣體狀態(tài)的實(shí)驗(yàn)數(shù)據(jù)很少。分子動(dòng)力學(xué)(MD)/蒙特卡洛方法(MC)是研究爆轟產(chǎn)物氣體高溫高壓熱力學(xué)性質(zhì)的有力工具,它們通過有限的分子數(shù)目,結(jié)合邊界的約束處理,能夠準(zhǔn)確計(jì)算氣體的宏觀熱力學(xué)性質(zhì)。然而MD方法和MC方法計(jì)算耗時(shí),且無法直接應(yīng)用。解析形式的狀態(tài)方程則可以直接應(yīng)用于實(shí)際問題,形成子程序直接納入相關(guān)程序中應(yīng)用。在描述爆轟產(chǎn)物氣體方面,已有BKW[1]、VLW[2]、LJD[3]、JCZ[4]等解析狀態(tài)方程形式。其中,VLW狀態(tài)方程為我國吳雄教授借鑒相似理論,認(rèn)為各階維里系數(shù)在高溫下是相似的,高階維里系數(shù)可以通過二階維里系數(shù)求得,進(jìn)而將維里物態(tài)方程以一種簡(jiǎn)化形式寫出,形成了VLW爆轟產(chǎn)物狀態(tài)方程[5-6],其成功應(yīng)用于炸藥爆轟性能的理論計(jì)算。與BKW狀態(tài)方程相比,VLW狀態(tài)方程在描述爆轟環(huán)境下氣體高溫高壓熱力學(xué)狀態(tài)時(shí)有了一定改善。然而VLW狀態(tài)方程形式仍然存在一些不足,VLW狀態(tài)方程不能有效描述爆轟產(chǎn)物氣體組分的高溫高壓熱力學(xué)狀態(tài),從而使其所采用的爆轟產(chǎn)物勢(shì)參數(shù)缺乏必要的物理基礎(chǔ)支撐。作者[7]前期應(yīng)用VLW狀態(tài)方程計(jì)算H2O沖擊Hugoniot曲線的結(jié)果表明,即使在優(yōu)化后的勢(shì)參數(shù)條件下,VLW狀態(tài)方程仍然不能在寬的壓力范圍有效描述H2O的高溫高壓狀態(tài)。

針對(duì)VLW狀態(tài)方程存在的上述不足,本研究提出了一種描述爆轟環(huán)境下高溫、高壓氣體的對(duì)比態(tài)維里型狀態(tài)方程VHL(Viral-Han-Long),該狀態(tài)方程基于LJ勢(shì)能函數(shù),通過對(duì)炸藥爆轟產(chǎn)物中的重要?dú)怏w組分CO2高溫高壓熱力學(xué)狀態(tài)的描述,表明對(duì)比態(tài)VHL狀態(tài)方程能夠有效地描述爆轟環(huán)境下CO2氣體的壓力、體積和溫度(pVT)熱力學(xué)關(guān)系。

2 對(duì)比態(tài)VHL狀態(tài)方程

理論上,任何氣體的狀態(tài)方程,都可以用維里形式描述。然而,實(shí)踐中隨著維里系數(shù)階數(shù)的提高,計(jì)算的復(fù)雜性迅速增大。Barker等[8]基于LJ勢(shì)能函數(shù),針對(duì)無量綱第三階、第四階、第五階系數(shù),進(jìn)行了精確的理論計(jì)算,其以表格形式列出了特定溫度時(shí)的系數(shù)值,不適宜實(shí)際使用。在前期研究中,作者等[9]提出用一種簡(jiǎn)化維里型狀態(tài)方程形式描述高溫甲烷氣體的熱力學(xué)pVT狀態(tài)。為有效描述爆轟環(huán)境下其它氣體產(chǎn)物的pVT熱力學(xué)關(guān)系,本研究提出了一種基于對(duì)比態(tài)原理的維里型狀態(tài)方程形式VHL,表達(dá)式見(1)。

(1)

C*=c1T*c2+c3T*c4+c5T*c6+c7T*c8+c9T*c10

D*=d1T*d2+d3T*d4+d5T*d6+d7T*d8+d9T*d10

E*=e1T*e2+e3T*e4

b0=2πNσ3/3

式中,p為氣體實(shí)際壓力,GPa;V為摩爾體積,cm3·mol-1;T為溫度,K;R為摩爾氣體常數(shù)8.3145,J·mol-1·K-1;N為阿佛加德羅常數(shù)6.022045×1023mol-1,第二階無量綱維里系數(shù)B*采用變步長(zhǎng)辛卜生求積法近似計(jì)算獲得[10];C*、D*分別為第三階和第四階無量綱維里系數(shù),c1～c10、d1～d10為常數(shù),通過該表達(dá)式所得第三階和第四階無量綱維里系數(shù)值與理論值十分吻合; 對(duì)于第五階以上的無量綱維里系數(shù),則采用組合函數(shù)表示,其系數(shù)值e1～e4、a、b、f由無極性分子CH4的熱力學(xué)數(shù)據(jù)[11-12](溫度1000 K以上的112組pVT數(shù)據(jù))確定; 各階維里系數(shù)的擬合常數(shù)獲得方式見文獻(xiàn)[9]所述,具體數(shù)值見表1。T*為無量綱溫度,bCH4為67.21,w為對(duì)比態(tài)參量,VHL狀態(tài)方程通過參量w實(shí)現(xiàn)其它氣體與甲烷狀態(tài)方程的對(duì)比。ε、σ為L(zhǎng)J勢(shì)參數(shù)。

表1VHL狀態(tài)方程中各階維里系數(shù)擬合常數(shù)值

Table1Fitting constant values of each order viral coefficient in VHL Equation of State(EOS)

coefficientvaluecoefficientvaluecoefficientvaluec1-0.96665 d1-2.31154 e12.15701c2-4.51039 d2-7.49608 e2-2.52736c3-2.58733 d3-2.56864 e30.27080c4-0.85487 d4-1.14947 e4-1.00190c52.08033 d52.09534 a-0.00243c6-0.51631 d6-0.84757 b0.00018c72.02825 d72.81486 f3.60800c8-2.15543 d8-4.44480c9-0.12489 d9-0.30669c10-7.65161 d10-11.36945

3 CO2氣體高溫高壓熱力學(xué)狀態(tài)的計(jì)算

由于液態(tài)CO2樣品制備條件苛刻、難度大,其高溫、高壓的基礎(chǔ)實(shí)驗(yàn)數(shù)據(jù)較少。1990年初,Schott等[13]才發(fā)表了用化爆技術(shù)在5～30 GPa區(qū)域獲得的一組試驗(yàn)點(diǎn)。Nellis等[14]用二級(jí)輕氣炮技術(shù)在25～70 GPa區(qū)域獲得了一組試驗(yàn)點(diǎn)。劉福生等[15]利用二級(jí)輕氣炮作沖擊加載手段,獲得了CO2在20～60 GPa區(qū)域六個(gè)Hugoniot數(shù)據(jù)點(diǎn)。實(shí)驗(yàn)數(shù)據(jù)點(diǎn)偏少,且缺乏溫度的直接測(cè)量數(shù)據(jù)。因此,本研究以Belonoshko等[12]的分子動(dòng)力學(xué)計(jì)算數(shù)據(jù)作為CO2的基礎(chǔ)數(shù)據(jù),溫度范圍為718～4978 K,壓力范圍為0.5116～111.078 GPa,基于VHL狀態(tài)方程,應(yīng)用復(fù)形調(diào)優(yōu)法[16]優(yōu)化了CO2的LJ勢(shì)參數(shù),勢(shì)參數(shù)值與文獻(xiàn)值比較見表2所示。采用VHL、VLW狀態(tài)方程計(jì)算結(jié)果和分子動(dòng)力學(xué)計(jì)算值比較見表3所示,不同溫度或壓力下CO2體積計(jì)算偏差如圖1、圖2所示。由表3、圖1和圖2可得,采用VHL狀態(tài)方程計(jì)算得CO2體積平均絕對(duì)偏差為0.971%,最大偏差為4.04%, VHL狀態(tài)方程計(jì)算所得體積偏差與壓力和溫度參量無明顯的相關(guān)性。VLW狀態(tài)方程計(jì)算所得體積偏差則與溫度具有明顯的相關(guān)性,在較低溫度下,計(jì)算所得體積偏差較大，最大偏差87.149%,隨著溫度的升高,計(jì)算體積偏差逐漸減小,但仍普遍高于VHL狀態(tài)方程計(jì)算結(jié)果,采用VLW狀態(tài)方程計(jì)算所得平均絕對(duì)偏差20.2%。

表2本研究所采用CO2勢(shì)參數(shù)值與文獻(xiàn)值比較

Table2Comparison of the potential parameter values of CO2used in this paper and literature ones

(ε/k)/Kb0/mL·mol-1reference247.063.37[17]205.085.05[5]181.867.30thispaper

圖1不同壓力下VHL狀態(tài)方程、VLW狀態(tài)方程預(yù)測(cè)CO2的體積誤差百分比

Fig.1Error percentage of volume predicted by VHL EOS and VLW EOS at different pressure

圖2不同溫度下VHL狀態(tài)方程、VLW狀態(tài)方程預(yù)測(cè)CO2的體積誤差百分比

Fig.2Error percentage of volume predicted by VHL EOS and VLW EOS at different temperature

為驗(yàn)證對(duì)比態(tài)VHL狀態(tài)方程在更低壓力下的有效性,同時(shí)與VLW狀態(tài)方程計(jì)算結(jié)果比較,本研究引

用NIST數(shù)據(jù)庫的數(shù)據(jù)[18],對(duì)CO2在1000 K,20～800 MPa的熱力學(xué)狀態(tài)進(jìn)行了計(jì)算,結(jié)果見表4所示。采用對(duì)比態(tài)VHL狀態(tài)方程計(jì)算得CO2體積絕對(duì)平均偏差為0.698%,采用VLW狀態(tài)方程計(jì)算所得體積絕對(duì)平均偏差為11.988%。在固定溫度1000 K條件下,隨著壓力的增加,VLW計(jì)算所得體積偏差逐漸增大,在壓力為800 MPa時(shí),體積偏差最大達(dá)-19.771%。其原因可能與VLW狀態(tài)方程形式的高階維里系數(shù)過度簡(jiǎn)化有關(guān),隨著壓力增大,描述多個(gè)氣體分子同時(shí)相互作用的高階維里系數(shù)的準(zhǔn)確性要求提高,而VLW狀態(tài)方程中,除第二階維里系數(shù)與理論值相符合外,隨著維里系數(shù)階級(jí)的增大,其與理論值的差距也逐漸增大,故CO2的體積計(jì)算結(jié)果偏差隨壓力增大而增大。而本文所提出的對(duì)比態(tài)VHL狀態(tài)方程的第三階、第四階維里系數(shù)均與理論值吻合,高階維里系數(shù)則通過甲烷高溫高壓熱力學(xué)狀態(tài)數(shù)據(jù)優(yōu)化獲得,其具有扎實(shí)物理基礎(chǔ),因此,該狀態(tài)方程能夠很好描述CO2高溫狀態(tài)下較低壓力范圍內(nèi)的熱力學(xué)狀態(tài)。

表3VHL、VLW狀態(tài)方程計(jì)算CO2高溫pVT關(guān)系

Table3ThepVTrelation of CO2at high temperature calculated by the means of VHL EOS and VLW EOS

No. T/K p/GPaV/cm3·mol-1MD[12]VHLVLWerror/% VHL VLW1718.80.51164040.9118.132.275-54.6752798.80.58294040.3125.830.775-35.4253920.20.63964040.7331.431.825-21.4254963.90.68354040.3632.260.900-19.350511130.77324040.4335.221.075-11.950612300.84604040.4436.721.100-8.200712960.92084039.8536.74-0.375-8.1508720.60.77453535.9514.892.714-57.4579774.70.84463535.5320.111.514-42.54310902.00.94643535.4625.661.314-26.68611805.71.4803030.3617.631.200-41.23312999.41.7433029.9922.55-0.033-24.8331311901.9913029.8024.87-0.667-17.1001415092.2903030.0427.480.133-8.4001516752.4393030.1728.400.567-5.3331618072.6683029.8128.49-0.633-5.0331719452.8323029.7528.86-0.833-3.80018773.82.1682727.2613.790.368-49.22719887.32.3622727.0217.49-0.515-35.6042010162.4882727.1120.00-0.184-26.3622111683.6992524.7119.42-1.160-22.3202219834.9522524.7723.39-0.920-6.4402327196.0672524.7224.72-1.120-1.1202436377.0652525.0426.030.1604.1202544197.9142525.1826.720.7206.8802650608.0822525.8727.813.48011.240

Table3continued

No. T/K p/GPaV/cm3·mol-1MD[12]VHLVLWerror/% VHL VLW27700.84.4822222.662.7994.040-87.14928812.94.7212222.4511.923.076-45.27129913.44.9552222.3114.032.433-35.5833010245.2902222.1215.431.561-29.1553112187.9302020.1915.350.950-23.25032186410.522019.5617.35-2.200-13.25033262812.182019.6318.86-1.850-5.70034432115.052019.9220.70-0.4003.50035497816.362019.8821.00-0.6005.00036803.211.351818.238.881.278-50.68937122812.471818.0013.310.000-26.05638164113.561818.0015.070.000-16.27839197714.521817.9915.94-0.056-11.44440241415.641818.0216.770.111-6.83341275616.741817.9617.17-0.222-4.61142316117.561818.0417.680.222-1.77843346918.241818.0617.990.333-0.05644783.019.471616.147.190.875-55.08745118620.621615.8811.14-0.750-30.37546164521.921615.8812.91-0.750-19.31347200622.941615.9213.79-0.500-13.81348231824.111615.9014.31-0.625-10.56249277025.071616.0015.010.000-6.18750307226.321615.9415.27-0.375-4.56351338427.041615.9915.59-0.063-2.56252395627.991616.1216.160.7501.00053803.726.221515.097.010.600-53.28054119527.391514.8510.27-1.000-31.53355160828.631514.8311.77-1.133-21.53356206929.971514.8712.81-0.867-14.60057240030.921514.9113.36-0.600-10.93358271131.951514.9213.77-0.533-8.20059307532.891514.9714.19-0.200-5.40060340133.671515.0114.520.067-3.20061393834.741515.1015.010.6670.06762787.235.671414.126.170.857-55.90063121937.021413.849.51-1.143-32.07164165938.361413.8110.91-1.357-22.07165205039.581413.8311.71-1.214-16.35766234040.501413.8512.17-1.071-13.07167279341.971413.8912.74-0.786-9.00068319042.681413.9613.19-0.286-5.78669340144.181413.9013.29-0.714-5.07170789.649.451313.145.701.077-56.12371119250.771312.868.57-1.077-34.05472159452.071312.819.80-1.462-24.64673198453.321312.8210.57-1.385-18.69274231354.381312.8411.07-1.231-14.84675288656.211312.8711.74-1.000-9.69276307957.431312.8511.89-1.154-8.53877355158.001312.9312.34-0.538-5.07778395658.711312.9912.67-0.077-2.53879782.069.771212.215.111.750-57.425

Table3continued

No. T/K p/GPaV/cm3·mol-1MD[12]VHLVLWerror/% VHL VLW80120871.241211.917.86-0.750-34.52581161372.641211.848.94-1.333-25.50882199173.981211.839.61-1.417-19.91783240775.371211.8410.17-1.333-15.25084285476.901211.8610.64-1.167-11.33385323878.241211.8710.97-1.083-8.58386349879.161211.8811.16-1.000-7.00087807.0100.71111.264.962.364-54.882881209102.21110.987.12-0.182-35.264891599103.61110.908.06-0.909-26.755901971104.91110.888.66-1.091-21.273912416106.61110.879.20-1.182-16.400922866108.11110.889.63-1.091-12.500933012109.21110.879.73-1.182-11.545943326109.61110.899.98-1.000-9.236953959111.11110.9210.41-0.727-5.364

Note: error=100×(VEOS-VMD)/VMD

表41000 K時(shí)CO2pVT關(guān)系的VHL、VLW狀態(tài)方程計(jì)算值及與NIST數(shù)據(jù)庫數(shù)據(jù)的比較

Table4Comparison of the calculated values of CO2pVTrelation at 1000 K by VHL EOS and VLW EOS and the data of CO2in NIST database

No.p/GPa V/cm3·mol-1 NIST[18] VLW VHLerror/% VLW VHL10.02434.60436.9437.400.5290.64820.04228.77230.1231.500.5811.18730.06161.37161.6163.700.1431.47240.08128.35127.5130.30-0.6621.52750.10108.89107.0110.50-1.7361.43860.1296.07393.3597.31-2.8341.28970.1486.96983.5787.96-3.9081.13580.1680.15376.1980.94-4.9440.98590.1874.84270.4175.48-5.9220.847100.2070.57865.7671.08-6.8260.716110.2267.06961.9167.47-7.6920.595120.2464.12458.6864.43-8.4900.482130.2661.61355.9261.84-9.2400.374140.2859.44253.5359.60-9.9460.270150.3057.54251.4457.64-10.6040.172160.3255.86249.5855.91-11.2460.079170.3454.36447.9354.36-11.835-0.009180.3653.01746.4552.97-12.387-0.092190.3851.79845.1151.71-12.912-0.170200.4050.68743.8950.56-13.410-0.242210.4249.66942.7749.52-13.890-0.309220.4448.73241.7548.55-14.327-0.371230.4647.86640.8047.66-14.762-0.430240.4847.06239.9246.83-15.176-0.484250.5046.31239.1146.07-15.551-0.533260.5245.61138.3545.35-15.919-0.578270.5444.95437.6444.67-16.270-0.621280.5644.33636.9744.04-16.614-0.660290.5843.75436.3543.45-16.922-0.697300.6043.20435.7642.89-17.230-0.732310.6242.68335.2042.36-17.532-0.764

Table4Continued

No.p/GPaV/cm3·mol-1NIST[18]VLWVHLerror/%VLWVHL320.6442.18934.6741.85-17.822-0.794330.6641.71934.1741.38-18.095-0.822340.6841.27233.7040.92-18.347-0.848350.7040.84533.2440.49-18.619-0.872360.7240.43832.8140.08-18.863-0.896370.7440.04932.4039.68-19.099-0.920380.7639.67632.0139.30-19.322-0.941390.7839.31931.6338.94-19.555-0.963400.8038.97631.2738.59-19.771-0.983

Note: error=100×(VEOS-VNIST)/VNIST.

4 結(jié) 論

本研究提出了一種基于LJ勢(shì)能函數(shù)的對(duì)比態(tài)維里型狀態(tài)方程VHL用于描述爆轟環(huán)境下CO2的高溫高壓熱力學(xué)狀態(tài)。采用VHL狀態(tài)方程計(jì)算得CO2體積平均絕對(duì)偏差為0.971%,最大偏差為4.04%,采用VLW狀態(tài)方程計(jì)算所得平均絕對(duì)偏差20.2%,最大偏差87.149%。因此,在計(jì)算CO2高溫、中高壓熱力學(xué)狀態(tài)時(shí),VHL狀態(tài)方程的計(jì)算準(zhǔn)確性得到了大幅度提高。VHL狀態(tài)方程計(jì)算所得體積偏差與壓力和溫度參量無明顯的相關(guān)性; VLW狀態(tài)方程計(jì)算所得體積偏差則與溫度具有明顯的相關(guān)性,隨著溫度的升高,計(jì)算體積偏差逐漸減小。

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