Research Article Measurement and Prediction Method of Compressibility Factor at High Temperature and High Pressure

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1 Mathematical Problems i Egieerig Volume 16, Article ID , 8 pages Research Article Measuremet ad Predictio Method of Compressibility Factor at High Temperature ad High Pressure Xiaoxu Zhu, Bochao Xu, ad Zhoghe Ha DepartmetofPowerEgieerig,NorthChiaElectricPowerUiversity,Baodig713,Chia Correspodece should be addressed to Xiaoxu Zhu; zhuxiaoxu@hotmail.com Received 3 March 16; Accepted 14 April 16 Academic Editor: Giuseppe Vairo Copyright 16 Xiaoxu Zhu et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. I order to get the compressibility factor Z of workig fluid uder differet coditios, experimetal measuremet method of Z uder high pressure ad high temperature ad data miig method were studied i this paper. Experimetal measuremet method based o real gas state equatio ad predictio method based o Least Squares Support Vector Machie were proposed. First, a experimetal method for measurig Z at high temperature ad high pressure was desiged; i this method the temperature, pressure, ad desity (mass ad volume) of correspodig state were measured ad substituted ito the actual gas equatio of state, ad the Z ca be calculated. Meawhile, i order to obtai cotiuous value i T-p plae, Squares Support Vector Machies are itroduced to establish the predictio model of Z. Take Hexamethyldisiloxae, for example; the experimetal data of Z was obtaied usig the experimetal method. Meawhile the predictio model of Z, which ca be used as calculatio fuctio of Z, was established based o those experimetal data, ad the Z (T:5K K, p:1.3mpa 2.25 MPa) was calculated by usig this calculatio fuctio. By compariso with this published data, it was foud that the average relative error was 2.14%. 1. Itroductio Compressibility factor (Z-factor) [1 3] is a importat parameter of thermal physical properties ad is ofte used to calculate other physical parameters. Z-factor chages with temperature ad pressure. I egieerig calculatio, the Zfactor values uder differet (T, p) are eeded. However, for the ew refrigerat or differet batches, these data are ot complete, especially at high temperature ad high pressure. Curretly the method based o the law of correspodig states is a commo mea [4, 5]. However, this approach is a empirical method. At high temperature ad high pressure regio, sometimes the error of this method will be serious. Therefore, it is very ecessary to devise a method to measure ad calibrate Z-factor i the regio of high temperature ad high pressure. It is direct to calculate Z-factor by actual gas state equatio [6]. It ca be kow from the equatio that Z-factor ad desity (mass m ad volume V)arecorrespodigwhe temperature T ad pressure p were give. The, Z-factor ca be solved by T, p, m,adv.soitisthepremisetoobtaim ad V value uder the coditio of (T, p). Basedotheaboveaalysis,ameasuremetmethodof Z-factor is desiged based o actual gas state equatio. I this experimet, a give quality of workig fluid is heated i a cotaier, whose volume is costat, ad the chages of T ad p are recorded, ad m ad V at differet (T, p) aregot;fiallyz-factor ca be obtaied correspodigly. This method is simple, ad the precisio ca be guarateed at the high temperature ad high pressure regio, eve at supercritical regio. However, Z-factor obtaied by experimet is discrete i particular (T, p) status, ad the eeds of egieerig applicatios caot be met. I geeral, empirical formula established by the experimetal data is used to meet the practical egieerig applicatio. Ad the accuracy of the empiricalformulaisthekey.ithispaper,z-factor predictio methodbasedoleastsquaressupportvectormachieis proposed.ithismethod,lssvmisusedtoreplacethe traditioal empirical formula to establish predictio fuctio i T-p plae. I this paper, Hexamethyldisiloxae (MM) was take, for example, by usig this method; Z-factor at differet (T, p) was measured, ad Z-factor predictio fuctio of MM

2 2 Mathematical Problems i Egieerig was established. Z-factor values of MM (T: 5K K, p: 1.3 MPa 2.25 MPa) ad the Z-p graph were obtaied by the predictio fuctio. By compariso with the published data, it was foud that the average relative error was 2.14%. The effectiveess of this method ad precisio of the fuctio were verified. 2. Desig Desity Measuremet Method 2.1. The Scheme of Experimet. By the real gas state equatio [7], Itcabegotthat pv = ZRT. (1) Z= MVp mrt = Mp ρrt. (2) It ca be see from formula (2), as log as the correspodig temperature, pressure, ad desity (mass ad volume) ca be measured, that the correspodig Z-factor ca be calculated. I the regio of high temperature ad high pressure, especially i a supercritical state, because of sealig, the chage of mass ad volume teds to brig measuremet error. So i this experimet, the mass ad volume are fixed. I this method a certai mass workig fluid is added to a cotaier, whose volume is costat. The workig fluid i the cotaier will be at superheated or supercritical state if the cotaier is heated cotiuously. Durig the heatig process, the chages of T ad p should be recorded correspodigly. Further aalysis showed that, beside the measurig workig fluid, there is also air i the cotaier. So p is the total pressure of measurig workig fluid ad air, but ot the actual pressure of this measured medium. For the ideal gas, the pressure of compoet ca be solved by Dalto law of partial pressure [8, 9]: p i p = i =x i, (3) p i =x i p, where p is total pressure, p i is partial pressure of the ith compoet, is total molar mass, i is molar mass of the ith compoet, ad x i is the molar mass fractio of the ith compoet. However, this law cosiders that Z-factor is costat equal to 1. For the actual gas, Z-factor is chaged with temperature ad pressure. So the Z-factor of each compoet i each state must be got before calculatig the partial pressure of actual gas. Obviously, this cotradicts with the purpose of this paper. Therefore, i order to elimiate the ifluece of other compoets (air) o the pressure measuremet of workig fluid,thecotaiermustbevacuumedbeforebeigheated. The p is the actual pressure of workig fluid. Furthermore the correspodig desity ρ of (T, p)poitscabegot. The correspodig Z-factor ca be obtaied by substitutig T, p,adρ to formula (2). Temperature cotroller Filler Heatig device Pressure vessels Vacuum T sesor p sesor Figure 1: Experimetal apparatus. Computer 2.2. Experimetal Devices. The whole experimet requires the followig devices: the measured medium, pressure vessels, temperature sesors, pressure sesors, electroic scale, heatig device, temperature cotrol device, vacuum pump, ad so o. The experimetal structure was show i Figure 1. As show i Figure 1, the temperature cotrol device heats the cotaier through adjustig the temperature of heatig equipmet. The temperature ad pressure of the cotaier are measured by the temperature ad pressure sesors. 3. Experimet 3.1. Experimetal Material. As a importat orgaic workig medium, MM is icreasigly applied to ORC system with its excellet thermodyamic properties uder the state of high temperature ad high pressure [1, 11]. Take MM, for example; this paper used the experimetal method to measure its desity i the state of high temperature ad high pressure (T: 5 K 85 K, p: 1.2 MPa 2.1 MPa) ad calculated the correspodig Z-factor. Ktypethermocouplesadpressuregaugewereusedto measure the temperature ad pressure, the use of electroic scales measurig refrigerat mass. At the same time, it used resistace box as the heatig device ad used temperature cotroller to adjust the heatig temperature. Pressure vessel, whose volume is costat of 184 ml, was made of stailess steel Error Aalysis. First of all, the experimetal ucertaity should be aalyzed. The accuracy of K type thermocouple is.75%, the accuracy of pressure gauge is.4%, ad the electroic precisio is 3 kg ±.1 g, so the ucertaity of this experimetal system isestimatedtobe.87%. Atthesametime,therelativeerroradthemearelative error were calculated by the followig formula: δ i = δ= 1 Z i Z is Z is 1, Z i Z is Z is 1, where Z i is measuremet (computig) value ad Z is is published values. (4)

3 Mathematical Problems i Egieerig 3 Table 1: Z-factor obtaied by experimet (ρ = kg/m 3 ) Aalysis of the Experimetal Process ad Results. First ofall,weadded52.5g,54.4g,58.4g,64.8g,ad71.2gmm to each pressure vessel, ad the correspodig desities were kg/m 3, 5.18 kg/m 3, kg/m 3, kg/m 3,ad kg/m 3. Heat ad record T ad p after overheatig state. Measuremet ad calculatio results of Z-factor are show i Tables 1 5. This paper uses the PEFPROP property search software to query Z of MM, ad the error aalyses are show i Tables 1 5. It ca be see from Tables 1 5, i this experimet, that the relative error is small, ad the average relative error is about 2.25%. For the supercritical state is difficult to measure, the result is also very good. Beside the fact that the measuremet of T ad p may lead to errors, o the other had, the tightess of experimet device, which results i a small amout of leakage i the heatig process, will lead to the chages of desity ad cause the error of Z-factor. But the effect is small, which ca be ehaced by the sealig meas. 4. The Predictio Model of Z-Factor Based o LSSVM Z-factor of refrigerats ca be obtaied accurately by this experimetal method. Ad the precisio is high eve at high temperature ad high pressure coditios. However, Z-factor Table 2: Z-factor obtaied by experimet (ρ = 5.18 kg/m 3 ) obtaied by the experimet method above is discrete. I practice, all Z-factors o T-p plae are eeded. Moreover,all the experimetal data are o the isodese lie. That is difficult for the practical applicatio. Obviously, it is extremely difficult, eve impossible, to get all Z-factors o T-p plae by usig the experimet oly. So this paper proposesa predictiomethodof Z-factor based o LSSVM (Z-LSSVM). Ad a calculatio fuctio of Z-factor o the plae is established; meawhile Z-factorofMMo T-p plae had bee got by usig this fuctio Least Squares Support Vector Machie. SVR [12, 13] theory was first proposed by Vapik i the 199s. SVR is a machie learig techique based o the structural risk miimizatio priciple, which solves the problem of local miima ad oliear problem. O the base of SVR, the Least Square Support Vector Machie was proposed by Suykes [14, 15] to improve the solutio speed ad covergece precisio. For a give traiig set, T = {(x i,y i ),2,3,...,}, x i R, y i R. (5) Solvig for x ad y mappig fuctio f(x), y=f(x) =ω T φ (x) +b. (6)

4 4 Mathematical Problems i Egieerig Table 3: Z-factor obtaied by experimet (ρ =53.87 kg/m 3 ) Table 4: Z-factor obtaied by experimet (ρ =59.78 kg/m 3 ) I the formula, ω is the weight vector, b is the offset, ad φ(x) is a mappig fuctio. The equatio ca be used to estimate the ukow oliear fuctios. Based o the priciple of structural risk miimizatio, costructig, ad solvig the optimizatio problem: Mi J (ω, e, b) = 1 2 ωt ω+ 1 2 γ e 2 i s.t. y i =ω T φ(x i )+b+e i,,2,..., e i,,2,...,l. I the formula, γ is the pealty coefficiet, which maily cotrols the puishmet degree, ad J (ω, e, b) istheloss fuctio. I order to solve ω ad e, the Lagrage method is itroduced. The defiitio of Lagrage fuctio is as follows: L(ω,b,e,α,γ)= 1 2 ωt ω+ 1 2 γ e 2 i α i {(ω T φ(x i ))+b+e i y i }. I the formula, α i isthelagragemultiplier. (7) (8) Optimizig the formula accordig to Karush-Kuh- Tucker (KKT), L ω = ω= L b = α i =, L = e i γδ i e i =α i, L = α i (ω T φ(x i ))+b+e i y i =, 1 T V [ 1 [ V Ω+ I ] [ b α ]=[ y ]. λ] α i φ(x i ), (9)

5 Mathematical Problems i Egieerig 5 Table 5: Z-factor obtaied by experimet (ρ =65.68 kg/m 3 ) as Amog them, y=[y 1,...,y ], 1 V = [1,...,1], α=[α 1,...,α ]. (1) The trasformatio by kerel fuctio ca be expressed Ω ij =φ(x i ) T φ(x j )=K(x i,x j ). (11) The fittig fuctio LSSVM ca be expressed as f (x) = α i k (x, x i ) +b. (12) 4.2. The Predictio Method of Z-Factor Based o LSSVM. Predictio method is very importat ad must have the followig characteristics: (1) the ability to solve the small-scale sample problem [16]: the data obtaied by experimet is limited, so the method should fid out the variatio from smallscalesampleset; (2) the ability o oliear problem [16]: Z-factor chages with T ad p, ad its relatioship with T ad p is oliear. So, the predictio model must have the ability to mie the oliear relatioship. Throughtheaboveaalysis,LSSVMisasuitablemethod for the predictio of Z-factor. The predictio of Z-factor is to establish the fuctio of Z-factor with T ad p: Z=f(T,p). (13) Accordig to the experimet above, groupsofz-factor ad (T, p)ca be got,amely, groups of traiig samples, S = {(x 1,Z 1 ),(x 2,Z 2 ),...,(x,z )}, (14) where x i =[T i,p i ]. Substitutig (X i,z i ) ito type (7) ad solvig the optimizatio problem i accordace with the formulas (8)-(9), the the optimal solutio of Lagrage multiplier is obtaied: Fially the mappig fuctio is got: where ω= b=z j α=(α 1,α 2,...,α ) T. (15) Z=ω φ(t, p) +b, (16) α i φ(t i,p i ), α i K((T i,p i ) (T j,p j )) + e j. (17) 4.3. Predictio Results ad Error Aalysis of the MM Z- Factor. ThispapertakesMMasaexample,adZ-factor predictio model of MM is established. The experimetal data of Tables 1 5 were used as the traiig samples of LSSVM predictio model. Through the model optimizatio, the Zfactor predictio fuctio of MM ca be got fially: Z=Z(T, p) = 137 α i K ((T, p), (T i,p i )) (18) Lagrage multiplier of the predictio fuctio is show i Table 6. I order to verify the accuracy of Z-LSSVM fuctio, Zfactor o the T-p plae (T = K K, p=1.3mpa 2.25 MPa) was calculated by this fuctio ad compared with the published data. Predictio ad compariso results are show i Tables Figures 2 6 are Z-p diagram of MM ad relative error i K, 65 K, 7 K, 75 K, ad K. It ca be see from Tables 7 11 ad Figures 2 6 that the predicted results are ideal, ad the average relative error is about 2.14%. The maximum relative error i supercritical regio is withi 5%. The precisio of Z-LSSVM model was verified.

6 6 Mathematical Problems i Egieerig Table 6: Lagrage multiplier ofz-lssvm fuctio for MM. α 1 14 α α α α 57 7 α α α α α α i : Lagrage multiplier. Table 7: Z-factor calculated by Z-LSSVM ( K). p/mpa Z i Z is δ/% Table 8: Z-factor calculated by Z-LSSVM (65 K). p/mpa Z i Z is δ/% Table 9: Z-factor calculated by Z-LSSVM (7 K). P/MPa Z i Z is δ/% Figure 2: The Z-p diagram ad the relative error of MM ( K). 5. Coclusio The Z-factor is a importat parameter i the calculatio of thermodyamic egieerig. Accurate measuremet ad 1

7 Mathematical Problems i Egieerig 7 Table 1: Z-factor calculated by Z-LSSVM (75 K). p/mpa Z i Z is δ/% Table 11: Z-factor calculated by Z-LSSVM ( K). p/mpa Z i Z is δ/% calculatio of compressibility factor is sigificat. The measuremet at high temperature ad high pressure is the focus of the study. Durig the study, the followig coclusios ca be obtaied: (1) Based o the actual gas equatio of state, the experimetal method to measure Z-factor at high temperature ad pressure state is preseted. Z-factor value i superheated ad supercritical regio ca be accurately measured by this method, ad the method isrelativelysimpleadeasytooperate. (2) Z-factor predictio method based o LSSVM has bee proposed. With this method, the Z-factor predictio model of MM has bee established, ad the empirical formula of Z-factor is obtaied. The experimets show that the average relative error of Figure 3: The Z-p diagram ad the relative error of MM (65 K) Figure 4: The Z-p diagram ad the relative error of MM (7 K) Figure 5: The Z-p diagram ad the relative error of MM (75 K). the predictio model is 2.14% ad the precisio of Z- LSSVM method has bee verified. (3) The method proposed i this paper is uiversal ad ca be widely used for other orgaic workig fluids. Besides, it should also have the same accuracy to measure ad predict the desity of orgaic workig fluids

8 8 Mathematical Problems i Egieerig Figure 6: The Z-p diagramadtherelativeerrorofmm(k). Competig Iterests The authors declare that they have o competig iterests. Ackowledgmets This work was sposored by the Fudametal Research Fuds for the Cetral Uiversities (15MS12). 1 [1] R. Abbas, A. Schedema, C. Ihmels, S. Eders, ad J. Gmehlig, Measuremet of thermophysical pure compoet properties for a few siloxaes used as workig fluids for orgaic rakie cycles, Idustrial ad Egieerig Chemistry Research, vol.5,o.16,pp ,11. [11] R. Abbas, E. C. Ihmels, S. Eders, ad J. Gmehlig, Measuremet of trasport properties for selected siloxaes ad their mixtures used as workig fluids for orgaic rakie cycles, Idustrial & Egieerig Chemistry Research,vol.5,o.14,pp , 11. [12] V. Vapik, A overview of statistical learig theory, IEEE Trasatio o Neural Network, vol. 1, o. 5, pp , [13] F. Ruimig, Theory ad Applicatio Aalysis of Support Vector Machie, Electric Power Press, Beijig, Chia, 7 (Chiese). [14] J. A. K. Suykes ad J. Vadewalle, Least squares support vector machie classifiers, Neural Processig Letters,vol.9,o. 3,pp.293 3,1999. [15]X.Wag,J.Che,C.Liu,adF.Pa, Hybridmodeligof peicilli fermetatio process based o least square support vector machie, Chemical Egieerig Research ad Desig, vol. 88, o. 4, pp , 1. [16] Z.-H. Ha ad X.-X. Zhu, Selectio of traiig sample legth i support vector regressio based o iformatio etropy, Proceedigs of the Chiese Society of Electrical Egieerig, vol. 3, o., pp , 1. Refereces [1] N. Azizi, R. Behbahai, ad M. A. Isazadeh, A efficiet correlatio for calculatig compressibility factor of atural gases, Natural Gas Chemistry, vol. 19, o. 6, pp , 1. [2] A. Kamari, A. Hemmati-Sarapardeh, S.-M. Mirabbasi, M. Nikookar, ad A. H. Mohammadi, Predictio of sour gas compressibility factor usig a itelliget approach, Fuel Processig Techology,vol.116,pp.9 216,13. [3] A. Chamkalai, S. Zedehboudi, R. Chamkalai, A. Lohi, A. Elkamel, ad I. Chatzis, Utilizatio of support vector machie to calculate gas compressibility factor, Fluid Phase Equilibria, vol.358,pp.189 2,13. [4] H.W.Xiag,The Correspodig States Priciple ad Its Practice, Elsevier Sciece & Techology, 5. [5] M. A. Mahmoud, Developmet of a ew correlatio of gas compressibility factor (Z-factor) for high pressure gas reservoirs, i Proceedigs of the North Africa Techical Coferece & Exhibitio,Cairo,Egypt,April13. [6] P. M. Drachuk ad J. H. Abou-Kassem, Calculatio of Z factors for atural gases usig equatios of state, Caadia Petroleum Techology,vol.14,o.3,pp.34 36,1975. [7] J. R. Travis, D. Piccioi Koch, J. Xiao, ad Z. Xu, Real-gas equatios-of-state for the GASFLOW CFD code, Iteratioal Hydroge Eergy,vol.38,o.19,pp ,13. [8] F.R.Hilgema,G.Bertrad,adB.Wilso, UsigDalto slaw of partial pressures to determie the vapor pressure of a volatile liquid, Chemical Educatio, vol. 84, o. 3, pp , 7. [9] C. Jiamig ad L. Gebao, Advaced Egieerig Thermodyamics, Pekig Uiversity Press, 1.

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