FOREN a Compressed Liquid Density Correlation 5-j.fluidPh

Ž .Fluid Phase Equilibria 168 2000 149–163www.elsevier.nlrlocaterfluid

A compressed liquid density correlation

Kh. Nasrifar, Sh. Ayatollahi, M. Moshfeghian )

Department of Chemical Engineering, Shiraz UniÕersity, Shiraz, Iran

Received 26 August 1999; accepted 30 November 1999

Abstract

A new correlation is developed for calculation of the compressed liquid density of pure compounds andŽ .mixtures. This correlation is used together with the Hankinson–Thomson COSTALD correlation of saturated

liquid density and the Riedel equation for the calculation of vapor pressures. The range of application of thiscorrelation is quite wide; from freezing point temperature to critical point temperature and from saturationpressure to 500 MPa. The average of error for the prediction of the compressed liquid volume of 31 compoundsconsisting of 3324 experimental data points is 0.77% with y0.24% bias from the experimental data. Formixtures, the average of error for the prediction of the compressed liquid volume of 13 mixtures consisting of2101 experimental data points is 1% with y0.22% bias from the experimental data. The comparison with othercorrelations shows that the new correlation is somewhat better and quite reliable to very high pressures. q 2000Elsevier Science B.V. All rights reserved.

Keywords: Correlation; Liquid density; Compressed; Pure; Mixtures

1. Introduction

The compressed liquid density of compounds is an important property for process simulation,pipeline design and liquid metering. Compressed liquid densities are usually calculated usingcorrelations. A good compressed liquid density correlation must be simple and accurate, applicable towide ranges of temperature and pressure for pure compounds and their liquid mixtures. It must also bepredictive so that it can be used when experimental data are not available.

w xYen and Woods 1 developed a generalized compressed liquid density correlation, applicable tothe whole liquid region. This correlation is also applicable to mixtures in addition to the pure

w xcompounds and is fairly accurate. Chueh and Prausnitz 2 introduced a generalized correlation for

) Corresponding author. Tel.: q98-71-30-3071; fax: q98-71-67-2060.Ž .E-mail address: [email protected] M. Moshfeghian .

0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0378-3812 99 00336-2

( )Kh. Nasrifar et al.rFluid Phase Equilibria 168 2000 149–163150

normal compressed liquids. This correlation is simple and applicable to nearly the whole liquid regionand fairly accurate but it has not been extended to mixtures. The compressed liquid density correlation

w xof Brelvi and O’Connell 3,4 is quite accurate. This correlation needs a characteristic volume for eachcompound. In addition, an iterative and integration procedure is required to calculate the compressed

w xliquid density. Thomson et al. 5 developed an excellent compressed liquid density correlation. Thisw xcorrelation was based on the Tait 6 equation. It is applicable to both pure compounds and mixtures.

Although this correlation was optimized for pressures smaller than 68 MPa, it is quite predictive atw xvery high pressures. The Thomson et al. 5 correlation is reliable and it is now widely used.

However, this correlation has a discontinuity at reduced temperatures larger than 0.95 and after thew xdiscontinuity the quality of the correlation is not known. Lee and Liu 7 introduced an equation to

calculate the compressed liquid densities of pure compounds. This model is a modification of thew xsaturated liquid density equation of Spencer and Danner 8 . The model is accurate from low pressures

to very high pressures; however, it requires an iterative procedure. In other words this model isw ximplicit in volume. In addition, the Lee and Liu 7 model has not been extended to mixtures. Aalto et

w x w xal. 9,10 modified the compressed liquid density correlation of Chang and Zhao 11 to improve thecorrelation for a wider range of temperature, from freezing point to critical point temperature.Although this modified correlation is quite accurate for a wide temperature range, the pressure range

Ž .is small up to 20 MPa . Also, the authors asserted that the accuracy of their model gets graduallyworse as pressure increases.

The objective of this work was to develop a new generalized compressed liquid density correlationwith the least limitations. The correlation should be applicable to both pure compounds and theirmixtures. The range of applications should be quite wide, from freezing point temperature to thecritical point temperature and from saturation pressure to 500 MPa. The new correlation should beexplicit in volume so it will not need an iterative procedure to obtain the liquid density. Thecorrelation should be predictive. The ability of the correlation for prediction of the compressed liquiddensity is then compared with the other compressed liquid density correlations.

2. Development of the new correlation

2.1. Saturated liquid density correlation

w xExcept for the correlation of Lee and Liu 7 , the calculation of compressed liquid density is atwo-step procedure for the other previously cited correlations. Using these methods, in the first stepthe saturated liquid density is calculated and in the second step the effect of pressure on the saturatedliquid density is evaluated. There are many accurate saturated liquid density correlations, e.g., the

w xmodified Rackett correlation of Spencer and Danner 8 , and the correlations of Hankinson andw x w x w xThomson 12 , Iglesias-Silva and Hall 13 and Nasrifar and Moshfeghian 14 . Nasrifar and

w xMoshfeghian 15 compared 18 methods for the calculation of the saturated liquid density ofw xrefrigerants and indicated that the Hankinson and Thomson 12 correlation is superior. However, in

w x w xgeneral, the Iglesias-Silva and Hall 13 correlation is better than the Hankinson and Thomson 12w xcorrelation. The Iglesias-Silva and Hall 13 correlation is superior to the Hankinson and Thomson

w x12 correlation for polar compounds and non-hydrocarbons, although for hydrocarbons the Hankinsonw xand Thomson 12 correlation is still superior.

( )Kh. Nasrifar et al.rFluid Phase Equilibria 168 2000 149–163 151

In this work, a two-step procedure is used too. For the calculation of the saturated liquid density,w xthe Hankinson and Thomson 12 correlation is applied. This correlation is quite accurate for reduced

w xtemperatures smaller than 0.95, however, it works well up to the critical point 15 . It was selectedbecause most of experimental data available in the literature are for hydrocarbons. In addition, in thiscorrelation an excellent correlating parameter, the Soave–Redlich–Kwong acentric factor, v , isSRK

used. The parameter v will also be used in formulating of the compressed liquid densitySRK

correlation.

2.2. The correlation of saturation pressure

For developing the compressed liquid density correlation, the vapor pressure of compounds isw xneeded. The generalized Riedel vapor pressure correlation 5,16 is used for this purpose. This

correlation is expressed by:

log P sP Ž0.qv P Ž1. 1Ž .r r SRK r

P Ž0.s5.8031817logT q0.07608141a 2Ž .r r

P Ž1.s4.86601b 3Ž .r

as35y36rT y96.736logT qT 6 4Ž .r r r

bs logT q0.03721754a 5Ž .r

where T is the reduced temperature and P the reduced pressure.r r

2.3. The new compressed liquid density correlation

For the development of a compressed liquid density correlation, the selection of variables plays animportant role. Fig. 1 illustrates the variation of liquid specific volume of n-heptane as a function ofPX with temperature as a parameter. The variable PX is defined by:r r

PX sP yP 6Ž .r r r ,s

Ž .where P is the reduced pressure at saturation P rP . Therefore, the ordinate considers with ther,s s c

saturated specific volume of n-heptane at different temperatures. Fig. 1 also indicates that indepen-dent of the temperature and at high pressures the specific volumes approach an asymptote. Thisconclusion can also be obtained from any equation of state. In fact, equations of state indicate that atinfinite pressure, the molar volume of a compound approaches the molar co-volume. Thus:

n s lim n sb 7Ž .`P™`

Žwhere b is molar co-volume and n is molar volume at infinite pressure practically very high`

. Xpressure . So we can construct the following variable which is a function of P and temperature:r

nyns Xs f P ,T 8Ž . Ž .r rn yn` s

where n is saturated molar volume which in this work is calculated using the Hankinson andsw x Ž . Ž .Thomson 12 correlation. Eq. 8 has a value of 0 at saturation pressure or temperature and a value


ŽFig. 1. Specific volume of n-heptane as a function of excess reduced pressure with temperature as a parameter experimentalw x.data from Ref. 20 .

Ž .of 1 at very high pressure mathematically infinite pressure . Therefore, we must look for anincreasing function of pressure that behaves in this manner. Searching in mathematical handbooksindicates that a tangent hyperbolic function has the stated behavior. So we could have:

nynssC tanhC 9Ž .

n yn` s

where C is a proportionality constant and C an increasing variable which is a function of PX and T .r r

To derive an expression for C , it is supposed that the same expression for C that is valid at lowpressures remains valid at high pressures. In other words, having obtained an expression for C at low

Ž .pressures, it will be used in Eq. 9 . At low pressures when C is small, tanh C can be approximatedŽ . Ž .by C Appendix A . Then Eq. 9 becomes:

nynssCC . 10Ž .

n yn` s

However, we can approximate the molar volume of a liquid at any pressure from the saturated molarvolume using Taylor’s series expansion:

En 1 E2n 2nsn q P yP q P yP q . . . 11Ž . Ž . Ž .s r r ,s r r ,s2ž / ž /EP 2! EPr rs s

Ž .where subscript s indicates that the value must be evaluated at saturation. Eq. 11 is rearranged andrewritten as:

nyns 2sK P yP qK P yP q . . . 12Ž . Ž . Ž .1 r r ,s 2 r r ,sn yn` s


where:

1 E in

iž /i! EPr sK s is1,2,3, . . . 13Ž .in ynŽ .` s

All the derivatives of volume with respect to P are functions of temperature inasmuch as they are atr

saturation. This is easily understood noting that at saturation the degree of freedom is one. In otherw xwords, K is a function of temperature. Using Pade’s approximation 17 , the right-hand side of Eq.´i

Ž .12 can be approximated by a polynomial ratio. Consequently, we arrive at:2 3

nyn AqB P yP qD P yP qE P yPŽ . Ž . Ž .s r r ,s r r ,s r r ,ss 14Ž .2 3

n yn FqG P yP qH P yP q I P yPŽ . Ž . Ž .` s r r ,s r r ,s r r ,s

where A, B, D, E, F, G, H and I are some functions of K , K , . . . . However, our evaluation1 2

during the optimization of the parameters showed that putting DsHs0 has no effect on the qualityŽ .of Eq. 14 and so we will have:

3nyn JqL P yP qM P yPŽ . Ž .s r r ,s r r ,s

sC 15Ž .3n yn FqG P yP q I P yPŽ . Ž .` s r r ,s r r ,s

Ž . Ž .where JsArC, LsBrC and MsErC. Comparing Eq. 15 with Eq. 10 results in:3

JqL P yP qM P yPŽ . Ž .r r ,s r r ,sCs . 16Ž .3

FqG P yP q I P yPŽ . Ž .r r ,s r r ,s

Ž .Eq. 16 is the final expression for C . We have also determined empirically that:1r3 2r3

Js j q j 1yT q j 1yT 17Ž . Ž . Ž .0 i r 2 r

Fs f 1yT 18Ž . Ž .0 r

Csc qc v 19Ž .0 1 SRK

RTcn sV 20Ž .` Pc

VsV qV v 21Ž .0 1 SRK

Ž .The parameters j , j , j , L, M, f , G, I, c , c , V and V are taken as global constants. Eq. 90 1 2 0 0 1 0 1Ž . Ž .together with Eqs. 16 – 21 are the working equations for calculation of the compressed liquid

density of pure compounds. The final values of global constants are given in Table 1.

3. Extension to mixtures

To extend the developed correlation to a mixture, some values of T , P and v for the mixturec c SRK

are required. Because the proposed compressed liquid density correlation is coupled with the saturatedw xliquid density correlation of Hankinson and Thomson 12 , the use of the mixing rules that are used


Table 1Global parameters of the new correlation

y3j 1.3168=100y2j 3.4448=101y2j 5.4131=102y2L 9.6840=10y6M 8.6761=10

f 48.87560

G 0.7185y5I 3.4031=10

c 5.55260

c y2.76591y2V 7.9019=100y2V y2.8431=101

w xfor the Hankinson and Thomson 12 correlation are plausible for the use with the developedcompressed liquid density correlation. Moreover, as will be seen in the Section 4, the predictions of

w xthe compressed liquid density using the mixing rules of Hankinson and Thomson 12 are quite good.The mixing rules used are:

nc nc) )T s x x n T rn 22Ž .Ý Ýc ,m i j i j c , i j m

i j

1r2) ) )n T s n T n T 23Ž .Ž .i j c , i j i c , i j c , j

nc nc nc) ) )2r3 )1r3n s1r4 x n q3 x n x n 24Ž .Ý Ý Ým i i i i i iž / ž /

i i i

nc

v s x v 25Ž .ÝSRK ,m i SRK , ii

and:

P s 0.291y0.080v RT rn ) 26Ž . Ž .c ,m SRK ,m c ,m m

) w xwhere n is characteristic volume and comes from the Hankinson and Thomson 12 saturated liquiddensity correlation. The parameters n ) and v for a large number of compounds were given bySRK

w xHankinson and Thomson 12 .

4. Results

To evaluate the developed correlation and the proposed mixing rules, the relevant global parame-ters of the correlation should be determined. For determining the global parameters of the compressedliquid density correlation, a data bank consisting of 799 experimental data points was built. The data


bank contains only hydrocarbons with T less than 0.95 and pressures less than 500 MPa. Using arw xnonlinear regression package 18 and an objective function defined by:

nps n yncal ,i exp , iF s 27Ž .Ýn

nexp,ii

the 12 global parameters of the correlation were determined. These parameters are reported in TableŽ .1. The percent of average absolute deviation %AAD for determining these parameters was 0.62. The

distribution of errors around experimental data is shown in Fig. 2. The %Dev on the ordinate of Fig. 2is defined by:

n yncal ,i exp , i%Devs 100 28Ž . Ž .

nexp,i

The maximum %Dev for this optimization was 3.2; however, most of the deviations are between y1and 1.

The ability of the proposed correlation to predict compressed liquid density is compared withw x Ž . w x Ž .several correlations. These correlations are Thomson et al. 5 TBH , Yen and Woods 1 YW ,

w x Ž . w x Ž . w x Ž .Chueh and Prausnitz 2 CP , Brelvi and O’Connell 3,4 BO , Chang and Zhao 11 CZ , Lee andw x Ž . w x Ž .Liu 7 LL , and the modified Chang and Zhao correlation by Aalto et al. 9,10 CZA . Tables 2 and

3 illustrate these comparisons for 31 pure compounds composed of paraffins, olefins, halogenatedparaffins, carbon dioxide, nitrogen and ammonia. These tables present the %AAD and %Bias ofdifferent correlations for the calculation of compressed liquid densities. The %Bias is defined by:

nps100 n yncal ,i exp , i%Biass s . 29Ž .Ý

nps nexp,ii

Note that only a portion of hydrocarbon experimental data points cited in Tables 2 and 3 has beenused in the optimization of the parameters of the new correlation and not all of them. Table 2

Fig. 2. Deviation plot of hydrocarbons used for optimization of the global parameters. The total number of experimental dataŽ w x.points is 799 experimental data from Refs. 19,20,22–25 .


Table 2Ability of different correlations in terms of %AADa for prediction of the compressed liquid density of pure compounds

b c d e f g hComponent Pressure range Ref. nps This work YW TBH CP CZA BO CZ LLŽ .MPa

i i bw x Ž . Ž . Ž .CH 0.05–33.5 23,30 26 20 1.79 0.72 2.05 0.67 2.90 1.37 0.87 1.35 1.164w x Ž . Ž . Ž .C H 0.02 23 4 4 0.20 0.20 0.04 0.04 0.04 0.04 0.04 0.04 0.552 6w x Ž . Ž . Ž .C H 0.01–20 20 150 116 1.10 0.84 1.12 0.79 1.16 0.88 0.69 1.26 1.533 8w x Ž . Ž . Ž .n-C H 2–20 20 112 112 0.65 0.65 0.48 0.68 1.20 0.62 0.77 0.75 1.044 10w x Ž . Ž . Ž .i-C H 2–20 20 149 132 0.57 0.38 0.37 0.33 1.52 0.28 0.61 0.61 0.984 10w x Ž . Ž . Ž .n-C H 1–20 20 160 160 0.29 0.29 0.96 0.12 0.30 0.42 0.33 0.31 0.195 12w x Ž . Ž . Ž .n-C H 0.1–503 20,35 207 207 0.95 0.95 2.80 0.60 1.08 1.84 0.76 1.32 1.696 14w x Ž . Ž . Ž .n-C H 5–500 20 60 50 1.23 0.67 4.43 1.05 0.93 7.90 1.03 4.42 10.597 16w x Ž . Ž . Ž .n-C H 2–20 20 152 133 0.36 0.25 1.94 1.67 0.87 0.45 0.75 0.53 0.428 18w x Ž . Ž . Ž .i-C H 0.1–50 20 272 237 1.01 0.73 0.72 0.63 0.86 0.86 0.65 0.96 0.838 18w x Ž . Ž . Ž .n-C H 5–500 20 66 56 1.34 0.79 4.24 1.25 1.03 7.53 0.79 5.06 11.229 20w x Ž . Ž . Ž .n-C H 2–20 20,34 207 207 0.10 0.10 0.36 0.51 0.65 0.59 0.57 0.15 0.3910 22w x Ž . Ž . Ž .n-C H 5–500 20 70 70 0.87 0.87 4.78 1.31 1.00 7.07 1.01 5.37 8.9311 24w x Ž . Ž . Ž .n-C H 5–500 20 70 70 0.83 0.83 4.34 2.10 2.09 7.65 1.21 6.84 8.6113 28w x Ž . Ž . Ž .n-C H 0.1–450 35 27 27 0.75 0.75 3.66 0.23 0.86 4.60 1.38 4.65 1.7816 34w x Ž . Ž . Ž .n-C H 5–500 20 60 60 1.22 1.22 4.49 1.98 2.03 6.99 2.00 7.98 8.6617 36

i i iw x Ž . Ž . Ž .n-C H 5–500 20 50 50 1.62 1.62 4.74 3.27 4.10 8.00 2.32 11.55 8.6920 42w x Ž . Ž . Ž .C H 0.8–36.2 22 198 170 1.28 0.77 1.27 0.45 1.92 1.12 1.50 1.06 1.742 4w x Ž . Ž . Ž .C H 1.4–9.7 24 70 70 0.40 0.40 0.31 0.12 1.66 0.12 0.33 0.42 0.773 6w x Ž . Ž . Ž .1-C H 1.4–69 19 85 65 1.66 0.80 1.11 0.85 1.67 1.32 1.51 1.40 1.164 8w x Ž . Ž . Ž .C H 64–300 25 50 50 1.44 1.44 7.19 0.73 0.45 7.30 0.68 1.68 3.676 6w x Ž . Ž . Ž .CO 1–60 20 306 269 0.61 0.34 0.43 0.50 1.03 0.40 2.65 0.42 0.952w x Ž . Ž . Ž .N 0.4–31.7 23,33 47 45 0.90 0.68 1.24 0.14 1.20 1.13 1.52 0.57 1.652w x Ž . Ž . Ž .NH 0.2–141.8 4 16 16 3.07 3.07 1.85 1.96 2.34 2.34 0.66 1.67 2.353w x Ž . Ž . Ž .CCl FCClF 0.5–3.5 20 110 104 0.61 0.53 0.37 0.16 0.37 0.18 0.21 0.24 0.972 2w x Ž . Ž . Ž .CH CHF 0.7–6.5 21 224 219 0.32 0.30 0.25 1.26 0.73 0.60 0.74 0.57 4.603 2w x Ž . Ž . Ž .CCl FCH 0.5–68 26,27 79 79 0.43 0.43 0.74 0.45 0.26 0.74 0.36 0.40 0.432 3w x Ž . Ž . Ž .CClF CH 1–2 28 6 6 0.34 0.34 0.41 1.43 0.59 1.08 0.96 1.07 1.432 3w x Ž . Ž . Ž .CH F 0.3–20 29 64 60 1.14 1.02 0.49 2.73 1.10 1.22 1.06 0.56 1.212 2w x Ž . Ž . Ž .CHClF 1–6.22 31 212 196 0.58 0.26 0.49 0.25 1.20 0.32 0.36 0.62 0.972w x Ž . Ž . Ž .CHF CHF 0.5–9.3 32 15 15 0.60 0.60 0.38 2.64 0.91 1.29 0.55 2.39 1.212 2

Ž . Ž . Ž .Overall 3324 3075 0.77 0.59 1.40 0.79 1.12 1.62 0.97 1.43 2.24

a Ž . nps < <%AADs 100rnps S n yn rn .i cal, i exp, i exp, ib w xYen and Woods 1 .c w xThomson et al. 5 .d w xChueh and Prausnitz 2 .e w xModified Chang–Zhao by Aalto et al. 9 .f w xBrelvi and O’Connell 3,4 .g w xChang and Zhao 11 .h w xLee and Liu 7 .iThe numbers in the parentheses indicate the case in which T -0.95.r

indicates that the new correlation and the correlations of TBH and BO are better than the othercorrelations with overall %AAD equal to 0.77, 0.79 and 0.97, respectively. The respected overall%Bias presented in Table 3 are y0.24, 0.41 and 0.04, respectively. Note that except for the TBH


Table 3Ability of different correlations in terms of %Biasa for prediction of the compressed liquid density of pure compounds

b c d e f g hComponent Pressure range Ref. nps This work YW TBH CP CZA BO CZ LLŽ .MPa

i i iw x Ž . Ž . Ž .CH 0.05–33.5 23,30 26 20 y0.94 0.12 1.98 0.58 y1.63 1.30 y0.66 1.23 y0.924w x Ž . Ž . Ž .C H 0.02 23 4 4 y0.20 y0.20 y0.02 y0.01 y0.01 y0.01 y0.01 y0.01 0.552 6w x Ž . Ž . Ž .C H 0.01–20 20 150 116 0.26 0.84 1.12 0.79 y0.68 0.86 y0.02 1.15 1.023 8w x Ž . Ž . Ž .n-C H 2–20 20 112 112 0.28 0.28 0.07 0.53 y0.53 0.36 0.09 0.55 0.424 10w x Ž . Ž . Ž .i-C H 2–20 20 149 132 y0.11 0.13 0.31 0.30 y1.32 0.23 0.39 0.29 0.404 10w x Ž . Ž . Ž .n-C H 1–20 20 160 160 y0.29 y0.29 y0.94 0.01 y0.08 y0.20 y0.12 y0.23 y0.175 12w x Ž . Ž . Ž .n-C H 0.1–503 20,35 207 207 y0.95 y0.95 y0.68 y0.56 y1.08 0.62 y0.61 y0.32 y1.686 14w x Ž . Ž . Ž .n-C H 5–500 20 60 50 0.51 0.03 3.49 y0.45 0.80 7.87 0.56 4.22 y10.577 16w x Ž . Ž . Ž .n-C H 2–20 20 152 133 0.08 0.23 y1.87 1.67 0.46 0.29 0.04 0.51 y0.368 18w x Ž . Ž . Ž .i-C H 0.1–50 20 272 237 y0.84 y0.68 0.14 y0.03 0.58 0.71 y0.35 0.59 0.188 18w x Ž . Ž . Ž .n-C H 5–500 20 66 56 0.32 y0.22 2.30 y0.49 0.57 7.24 y0.55 4.59 y11.229 20w x Ž . Ž . Ž .n-C H 2–20 20,34 207 207 0.03 0.03 y0.23 0.51 0.65 0.59 0.57 0.12 y0.3910 22w x Ž . Ž . Ž .n-C H 5–500 20 70 70 0.19 0.19 3.14 0.15 0.79 6.77 y0.97 5.08 y8.3611 24w x Ž . Ž . Ž .n-C H 5–500 20 70 70 0.23 0.23 2.79 1.71 2.09 7.56 y1.00 6.83 y8.4713 28w x Ž . Ž . Ž .n-C H 0.1–450 35 27 27 y0.66 y0.66 3.57 0.07 0.81 4.56 y1.38 4.61 y1.7316 34w x Ž . Ž . Ž .n-C H 5–500 20 60 60 0.07 0.07 2.00 1.55 1.94 6.72 y2.00 7.88 y8.6617 36

i i bw x Ž . Ž . Ž .n-C H 5–500 20 50 50 1.01 1.01 2.43 3.26 4.10 7.94 y2.32 11.55 y8.6420 42w x Ž . Ž . Ž .C H 0.8–36.2 22 198 170 y0.83 y0.25 0.69 y0.27 y1.33 0.47 y1.38 0.31 0.142 4w x Ž . Ž . Ž .C H 1.4–9.7 24 70 70 y0.39 y0.39 0.05 y0.11 y1.66 y0.01 0.30 0.41 y0.253 6w x Ž . Ž . Ž .1yC H 1.4–69 19 85 65 y1.66 y0.80 y0.06 y0.85 y1.43 y0.08 y1.46 y1.26 y1.094 8w x Ž . Ž . Ž .C H 64–300 25 50 50 1.44 1.44 7.19 y0.73 0.45 7.30 0.67 1.68 y1.776 6w x Ž . Ž . Ž .CO 1–60 20 306 269 y0.40 y0.10 0.02 0.48 y0.55 0.25 2.65 0.12 0.322w x Ž . Ž . Ž .N 0.4–31.7 23,33 47 45 y0.36 y0.10 1.21 0.09 y0.92 1.10 y1.51 0.21 1.332w x Ž . Ž . Ž .NH 0.2–141.8 4 16 16 3.07 3.07 1.85 1.96 2.34 2.34 0.52 1.67 2.353w x Ž . Ž . Ž .CCl FCClF 0.5–3.5 20 110 104 y0.61 y0.53 y0.13 0.15 y0.13 0.06 y0.02 0.24 y0.802 2w x Ž . Ž . Ž .CH CHF 0.7–6.5 21 224 219 0.18 0.18 0.07 1.25 y0.12 0.59 0.73 0.56 y4.603 2w x Ž . Ž . Ž .CCl FCH 0.5–68 26,27 79 79 y0.20 y0.20 0.06 y0.39 y0.08 0.53 y0.23 y0.24 0.252 3w x Ž . Ž . Ž .CClF CH 1–2 28 6 6 0.33 0.33 0.41 1.43 0.56 1.08 0.96 1.07 1.432 3w x Ž . Ž . Ž .CH F 0.3–20 29 64 60 0.56 0.66 y0.44 2.71 y0.21 1.10 0.90 0.39 y0.322 2w x Ž . Ž . Ž .CHClF 1–6.22 31 212 196 y0.58 y0.26 y0.24 0.24 y1.06 y0.11 y0.05 0.24 y0.782w x Ž . Ž . Ž .CHF CHF 0.5–9.3 32 15 15 0.17 0.17 y0.08 1.60 0.53 1.29 0.03 2.39 y0.972 2

Ž . Ž . Ž .Overall 3324 3075 y0.24 y0.08 0.40 0.41 y0.17 1.35 0.04 1.03 y1.52

a Ž . npsŽ .%Biass 100rnps S n yn rn .i cal, i exp, i exp, ib w xYen and Woods 1 .c w xThomson et al. 5 .d w xChueh and Prausnitz 2 .e w xModified Chang–Zhao by Aalto et al. 9 .f w xBrelvi and O’Connell 3,4 .g w xChang and Zhao 11 .h w xLee and Liu 7 .iThe numbers in the parentheses indicate the case in which T -0.95.r

correlation, the comparisons are from low temperature to near the critical point and the total numberŽof experimental data points is 3324. However, due to limitation of the TBH correlation the

.discontinuity at Tr)0.95 , the total number of experimental data points used for the TBH correlation

()

Kh.N

asrifaret

al.rF

luidP

haseE

quilibria168

2000149

–163

158

Table 4Ability of different correlations for prediction of the compressed liquid density of mixtures

a bSystem Pressure range Temperature nps Ref. This work TBH CZAŽ . Ž . c dMPa range K %AAD %Bias %AAD %Bias %AAD %Bias

w xCF CHF qCH FCF 0.4–30 280–349 144 36 0.62 y0.62 0.48 0.41 0.18 0.003 2 2 3w xCH F qCH FCF 0.5–3 250–330 36 36 1.67 y1.67 0.92 y0.92 1.28 y1.282 2 2 3w xCH F qCF CHF qCH FCF 0.7–3.2 280–340 53 36 2.02 y2.02 1.12 y1.12 1.53 y1.532 2 3 2 2 3w xCH q n-C H 3.4–31 294–394 174 34 1.47 y1.47 1.17 y1.13 1.01 y1.004 10 22w xC H qC H 1.4–68.9 261–311 170 37 0.39 y0.30 0.53 y0.51 0.68 0.362 6 3 6w xn-C H q n-C H 0.1–430 298–358 104 38 0.61 0.15 1.11 y0.91 2.98 2.9010 22 14 30w xn-C H q n-C H 0.1–331.3 298–358 84 38 0.71 0.51 0.57 y0.57 2.68 2.6612 24 16 34w xn-C H q n-C H q n-C H 0.1–359 298–358 91 38 0.51 0.27 1.05 y1.04 3.56 3.5510 22 14 30 16 34w xn-C H q n-C H 0.1–450 298–373 113 35 0.74 y0.14 1.46 y1.45 4.95 4.9316 34 6 14w xC H qC H 0.6–68.9 311–444 394 39 0.80 y0.75 0.79 y0.74 0.79 y0.033 8 6 6w xC H q1-C H 0.7–68.94 278–344 415 40 1.73 1.29 1.72 1.29 2.35 1.923 8 4 8w xC H qC H 0.6–68.9 261–344 190 41 0.51 y0.40 0.41 y0.41 0.70 0.473 6 3 8w xN qCH 1.3–35.2 92–152 133 42 1.06 y1.04 1.32 y1.32 0.91 y0.332 4

Overall 2101 1.00 y0.22 1.03 y0.35 1.62 0.95

a w xThomson et al. 5 .b w xModified Chang–Zhao by Aalto et al. 10 .c Ž . nps < <%AADs 100rnps S n yn rn .i cal, i exp, i exp, id Ž . npsŽ .%Biass 100rnps S n yn rn .i cal, i exp, i exp, i


ŽFig. 3. Deviation plot for n-undecane: Specific volume percent deviation as a function of pressure experimental data fromw x.Ref. 20 .

was 3075 and the temperature range was for T smaller than 0.95. For the same conditions and ther

same number of experimental data points as used for the TBH correlation, the new correlation gives a%AAD of 0.59 and a %Bias of y0.08. However, the larger errors for the CZA, CZ and the LLcorrelations are due to the limitations of the correlations. They should not be used for the wholerange, although they are quite accurate for their range of applications.

The predictions with our correlation presented in Table 2 show that the maximum error occurs forNH with a %AAD of 3.07. It can be explained noting that NH is a polar liquid and is not suitable3 3

Ž w x.Fig. 4. Deviation plot of N qCH binary mixture as a function of temperature experimental data from Ref. 42 .2 4


w xto be correlated using the acentric factor. In this case, the correlation of Brelvi and O’Connell 3,4 orw xHuang and O’Connell 43 is suitable.

Fig. 3 demonstrates a deviation plot for n-undecane. It shows the %Dev of n-undecane as afunction of pressure at different temperatures. Except for the temperature 573 K, the deviations fromexperimental data points are small from saturation pressure to 500 MPa. Even at the temperature 573K the agreement is quite good for pressures less than 50 MPa. This means that for temperatures nearthe critical point the error increases at high pressures.

The ability of our correlation to predict the compressed liquid density of mixtures is shown inTable 4. The correlation is also compared with the TBH and CZA correlations. The total number ofpoints used for the comparison is 2101 and the maximum pressure is 450 MPa. Table 4 indicates thatthe overall %AAD and %Bias for our correlation are 1.00 and y0.22, for the TBH correlation 1.03and y0.35 and for the CZA correlation 1.62 and 0.95, respectively.

Fig. 4 shows a deviation plot for the mixture of N and CH as a function of temperature. The2 4

pressure range of this system is from 1.28 to 35.2 MPa, the temperature range from 92 to 152 K andthe composition range from 28.36% to 70.60% N . The maximum %Bias is y4.90 and occurs at the2

temperature 152 K; however, the most of deviations are between y2.8 and 0.20.

5. Conclusion

Ž Ž . Ž . Ž ..A generalized compressed liquid density correlation Eq. 9 together with Eqs. 16 – 21 for bothpure compounds and mixtures have been developed, The temperature and pressure range are quitewide; from freezing point to the critical point and from saturation pressure to 500 MPa. Thedeveloped correlation contains 12 global parameters. These parameters have been obtained from aportion of hydrocarbon experimental liquid density data; however, the comparisons show that thecorrelation is quite predictive for other compounds. Using the new correlation, the average of %AADfor predicting the compressed liquid density 3324 experimental data points of 31 compounds is 0.77.For the mixtures, the average of %AAD is 1.00 for 2101 experimental data points of 13 mixtures. Thecomparisons with other correlations show that the developed correlation is superior. It should beemphasized that no adjustable or fitting parameters were used to predict the compressed liquid densityof mixtures. Only the global parameters determined from a few pure compound density data that arereported in Table 1 were used.

6. Nomenclature

Ž 3 .b molar co-volume m rkmolA, B individual constants

Ž .c , c global constants of Eq. 190 1Ž .C a linear function of v as given by Eq. 19SRK

D, E individual constantsŽ .f global constant of Eq. 180


Ž .F a function given by Eq. 18Ž .G global constant of Eq. 16

H a constantŽ .I global constant of Eq. 16Ž .j , j , j global constants of Eq. 170 1 2

Ž .J a function of temperature as defined by Eq. 17Ž .K functions defined by Eq. 13i

Ž .L, M global constants of Eq. 16nc number of compoundsnps number of points

Ž .P pressure MPaPX reduced pressure minus reduced pressure at saturationr

Ž 3 .R gas constant 8314 m Parkmol KŽ .T Temperature K

Ž 3 .n volume m rkmolU Ž 3 .n characteristic volume m rkmol

Ž 3 .n molar volume at infinite pressure m rkmol`

x molar fraction

Greek lettersŽ .a a function given by Eq. 4Ž .b a function given by Eq. 5

F objective functionv acentric factor based on Soave–Redlich–Kwong equation of stateSRK

Ž .V ,V global constants of Eq. 210 1Ž .V a function of v as defined by Eq. 21SRK

Subscriptc criticalcal calculated valueexp experimental valuei, j dummy indexesm mixturer reduceds saturated

Appendix A

Tangent hyperbolic function is expressed by:

eC yeyC

tanhCs A-1Ž .C yCe qe


and the exponential functions eC and eyC are expressed by series:

1 1C 2 3e s1qCq C q C q . . . A-2Ž .

2! 3!

1 1yC 2 3e s1yCq C y C qy . . . A-3Ž .

2! 3!

Ž . Ž .For small values of C , the right-hand sides of Eqs. A-2 and A-3 reduce to the two first terms. So:

eC f1qC A-4Ž .eyC f1yC . A-5Ž .

Ž . Ž . Ž .Substituting Eqs. A-4 and A-5 into A-1 gives approximately:

tanhCfC For small values ofC . A-6Ž . Ž .

References

w x Ž .1 L.C. Yen, S.S. Woods, AIChE J. 12 1966 95–99.w x Ž .2 P.L. Chueh, J.M. Prausnitz, AIChE J. 15 1969 471–472.w x Ž .3 S.W. Brelvi, J.P. O’Connell, AIChE J. 18 1972 1239–1243.w x Ž .4 S.W. Brelvi, J.P. O’Connell, AIChE J. 21 1975 171–173.w x Ž .5 G.H. Thomson, K.R. Brobst, R.W. Hankinson, AIChE J. 28 1982 671–676.w x6 P.G. Tait, Physical and Chemistry of the Voyage of H.M.S. Challenger II: Part IV. S.P. LXI, H.M.S.O, London, 1888.w x Ž .7 H.-Y. Lee, G. Liu, Fluid Phase Equilib. 108 1995 15–25.w x Ž .8 C.F. Spencer, R.P. Danner, J. Chem. Eng.Data 17 1972 236–241.w x Ž .9 M. Aalto, K.I. Keskinen, J. Aittamaa, S. Liukkonen, Fluid Phase Equilib. 114 1996 1–19.

w x Ž .10 M. Aalto, K.I. Keskinen, J. Aittamaa, S. Liukkonen, Fluid Phase Equilib. 114 1996 21–35.w x Ž .11 C.-H. Chang, X. Zhao, Fluid Phase Equilib. 58 1990 231–238.w x Ž .12 R.W. Hankinson, G.H. Thomson, AIChE J. 25 1979 653–663.w x Ž .13 G.A. Iglesias-Silva, K.R. Hall, Fluid Phase Equilib. 131 1997 97–105.w x Ž .14 Kh. Nasrifar, M. Moshfeghian, Fluid Phase Equilib. 153 1998 231–242.w x Ž .15 Kh. Nasrifar, M. Moshfeghian, Fluid Phase Equilib. 158–160 1999 437–445.w x16 R.C. Reid, J.M. Prausnitz, T.K. Sherwood, The Properties of Gases and Liquids, 3rd edn., McGraw-Hill, New York,

1977.w x17 A. Raltson, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.w x18 J.P. Chandler, Department of Computing and Information Sciences, Oklahoma State University, Stillwater, OK 74074,

1975.w x Ž .19 R.H. Olds, B.H. Sage, W.N. Lacey, Ind. Eng. Chem. 38 1946 301–303.w x Ž .20 N.B. Vargaftik, Handbook of Physical Properties of Liquids and Gases Pure Substances and Mixtures , 2nd edn.,

Hemisphere Publication, Washington, 1975, English Translation.w x Ž .21 D.R. Defibaugh, G. Morrison, J. Chem. Eng. Data 41 1996 376–381.w x Ž .22 G.C. Straty, J. Chem. Thermodyn. 12 1980 709–716.w x Ž .23 J.B. Rodosevich, R.C. Miller, AIChE J. 19 1973 729–735.w x Ž .24 W.R. Parrish, J. Chem. Eng. Data 32 1987 311–313.w x Ž .25 M. Gehrig, H. Lentz, J. Chem. Thermodyn. 9 1977 450–455.w x26 H.A. Duarte-Garza, C.-A. Hwang, S.A. Kellerman, R.C. Miller, K.R. Hall, J.C. Holste, K.N. Marsh, B.E. Gammon, J.

Ž .Chem. Eng. Data 42 1997 497–501.w x Ž .27 Y. Maezawa, H. Sato, K. Watanabe, J. Chem. Eng. Data 36 1991 151–155.


w x Ž .28 Y. Maezawa, H. Sato, K. Watanabe, J. Chem. Eng. Data 36 1991 148–150.w x Ž .29 P.F. Malbrunot, P.A. Meunier, G.M. Scatena, W.H. Mears, K.P. Murphy, J.V. Sinka, J. Chem. Eng. Data 13 1968

16–21.w x Ž .30 A.J. Vennix, T.W. Leland Jr., R. Kobayashi, J. Chem. Eng. Data 15 1970 238–243.w x Ž .31 D.R. Defibaugh, G. Morrison, J. Chem. Eng. Data 37 1992 107–110.w x Ž .32 T. Tamatsu, H. Sato, K. Watanabe, J. Chem. Eng. Data 37 1992 216–219.w x Ž .33 G.C. Straty, D.E. Diller, J. Chem. Thermodyn. 12 1980 927–936.w x Ž .34 B.H. Sage, H.M. Lavender, W.N. Lacey, Ind. Eng. Chem. 32 1940 743–747.w x Ž .35 J.H. Dymond, K.J. Young, J.D. Isdale, J. Chem. Thermodyn. 11 1979 887–895.w x Ž .36 J.V. Widiatmo, T. Fujimine, H. Sato, K. Watanabe, J. Chem. Data 42 1997 270–277.w x Ž .37 R.A. McKay, H.H. Reamer, B.H. Sage, W.N. Lacey, Ind. Eng. Chem. 43 1951 2112–2117.w x Ž .38 P.S. Snyder, M.S. Benson, H.S. Huang, J. Winnick, J. Chem. Eng. Data 19 1974 157–161.w x Ž .39 J.W. Glanville, B.H. Sage, W.N. Lacey, Ind. Eng. Chem. 42 1950 508–513.w x Ž .40 G.H. Goff, P.S. Farrington, B.H. Sage, Ind. Eng. Chem. 42 1950 735–743.w x Ž .41 H.H. Reamer, B.H. Sage, Ind. Eng. Chem. 43 1951 1628–1634.w x Ž .42 G.C. Straty, D.E. Diller, J. Chem. Thermodyn. 12 1980 937–953.w x Ž .43 S.H. Huang, J.P. O’Connell, Fluid Phase Equilib. 37 1987 75–84.

Documents

FOREN a Compressed Liquid Density Correlation 5-j.fluidPh