Ind. Eng. Chem. Res. 1988,27, 536-540
536
Spijker, R. UKF Internal Report YBO-RP-83-248, 1983; UKF, IJuiden. Spijker, R. UKF, IJmuiden, personal communication, 1985. Technische Keramiek; Technieuws Washington W-84-09; Ministerie van Economische Zaken, Direktie R en 0: 's Gravenhage, 1984. Teslenko, V. V.; Rakov, E. G. Khim. Prom. 1981, 12, 744.
Wertz, D. L.; Cook, G . A. J. Sol, Chem. 1986, 14(1), 41. Weteringe, K. "The Utilization of Phosphogypsum". Proc. of the Fertilizer Society of London No. 208, 1982. Received for review April 21, 1987 Accepted September 29, 1987
COMMUNICATIONS Prediction of Thermophysical and Thermodynamic Properties of Trichlorofluoromethane and Chlorotrifluoromethane in the Single- and Two-Phase Region Using the BACK Equation of State T h e parameters of the Boublik-Alder-Chen-Kreglewski (BACK) equation of state have been calculated from vapor pressure and saturated liquid density data for trichlorofluoromethane and chlorotrifluoromethane. The predictions of other thennophysical properties from the BACK equation are compared with experimental data. The predictions of second virial coefficients are excellent, while predictions of P-V-T data in the one-phase region are good, with a maximum deviation of less than &4% in the volume. It has been shown that the Boublik-Alder-Chen-Kreglewski (BACK) equation of state (EOS) is one of the most reliable equations for predicting the thermodynamic properties of pure compounds and mixtures (see Kreglewski and Hall (1983), Simmick et al. (1979), Boublik (1983), Machat and Boublik (1985a,b), and Chandnani et al. (1984)). In this paper, the parameters of the BACK equation of state have been calculated from vapor pressure and saturated liquid density. The predictions of the BACK equation for other properties agree well with experimental values for both trichlorofluoromethane (Freon R11) and chlorotrifluoromethane (Freon R13) The BACK equation of state is briefly described in the next section. For further details see Kreglewski (1984).
RACK Equation of State The Boublik-Alder-Chen-Kreglewski equation of state was developed by Chen and Kreglewski (1977). It has been found that it is accurate in predicting phase equilibria for both pure fluids and fluid mixtures and that, in practice, only a few constants are required for each substance. These constants can be related to the properties of the molecules. The BACK equation of state has the form PV/RT = Z = Z h + Z a (1) where the compressibility factor 2 is the sum of a repulsive term Zh and an attractive term Za. Chen and Kreglewski used the hard-sphere equation of Boublik (1975) for the repulsive term Z h, 1 + (3a - 2)y + ( 3 2 - 3a + 1)y' - a 9 3 Z h
=
(1 - Y)3
(2)
where a is a characteristic constant of the substance which depends on the nonsphericity of the molecule. For spherical molecules, a = 1,and for nonspherical molecules a > 1. y is the ratio of the volume of the molecules to the total volume: y = 0.74048V0/V (3) 0888-5885/88/ 2627-0536$01.50/0
where V ois the close packed volume of the molecular hard cores given by
Vo = VW[l- C exp(-3uo/kT)l3
(4)
where u O is a characteristic energy of the fluid, VWis the value of Vo a t 0 K, and C is a constant which has been found to be equal 0.12 for most compounds. The attractive term is represented by a polynomial expansion of Alder et al. (1972):
where u is a characteristic energy which for spherical molecules is independent of the temperature. For nonspherical molecules u = .o(
1+
&)
where q > 0. The first approximation to q is given by q / k T c = 0 . 5 0 5 ~+ 0.702~'
(7)
where Tc is the critical temperature and w is the Pitzer acentric factor. D,, are 24 universal constants originally obtained by Alder et al. (1972) from molecular dynamics results for a fluid with a square-well potential but redetermined by Chen and Kregewski using argon data. The latter set of values were used in this work. In application, the four BACK equation of state parameters which are characteristic of an individual compound, VW,a,u and q , must be evaluated from experimental data. Experimental Data A literature survey and evaluation of the experimental data was carried out for vapor pressure, liquid and vapor densities a t saturation, P-V-T data, and the critical properties. Table I summarizes the data available for vapor pressure, for saturated density, and for P-V-T data. 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 537 Table I. References to Vapor Pressure, Saturated Liquid Density, and P-V-T
property vapor pressure . .
compd R11
authors Benning and McHarnes Osborn et al. Fernandez-Fassnacht and Rio Riedel Albright and Martin Sinka et al. Perel'shtein Oguchi et al. Fernandez-Fassnacht and Rio Benning and McHarnes Chavez et al. Albright and Martin Sinka et el. Perel'shtein Oguchi et al. Rivikin and Kremenvskaya Albright and Martin Michels et al. Sinka et al. Oguchi et al. Geller et al. Takahashi et al. Castro-Gomez
R13
saturated density
R11 R13
P-V-T
R11 R13
date 1940a 1941 1984 1938 1952 1970 1975 1975b 1985 1940b 1986 1952 1970 1975 1975b 1975 1952 1966 1970 1975a 1979 1985 1987
no. of points 7 8 32 30 7 10 18 29 41 14 31 11
5 18 6 164 45 288 33 204 68 170 34
Table 11. Characteristic Constants
conqx~ ~ M / c m 3mol-' R11 52.385 R13 38.708
c
a
0.12 0.12
1.0398 1.0257
Polk 435.31 275.99
9lk 60.0 38.5
The reduced temperature and pressure ranges are given for each set of data. The tabulated values published by ASHRAE (1981) for R11 and by Du Pont (1978) for R13 for vapor pressure and saturated liquid density were compared with the experimental data and were found to be in a good agreement and hence were used in determining the BACK EOS parameters. Determination of the Characteristic Constants. The Pitzer acentric factor was used to determine an initial value for q from eq 7. From values of vapor pressure and the liquid volume a t saturation as a function of the temperature and the critical data, the other three parameters were adjusted by using a nonlinear least-squares fit. The parameters obtained were then used to calculate the vapor pressure from the BACK equation by determining the liquid and vapor volumes for which both the pressure and chemical potentials of each phase were the same a t a specified temperature (see below). The mean deviation between the experimental and calculated values of the vapor pressure was then determined. This procedure was repeated using different values for q to give a new set of Voo,a,and u o until a minimum in the vapor pressure residual was obtained. Figure 1 shows the variation of the residual vapor pressure with q / k for R11 and for R13. For both fluids a distinct minimum was observed. Table I1 gives the characteristic constants of R11 and R13. Determination of the Properties along the Saturation Curve. For a specified temperature the saturated liquid volume V,' and the correspondingvapor volume V/, the vapor pressure Po,and the thermodynamic properties of both the liquid and vapor a t saturation were calculated from the BACK equation of state by solving
P,' = Pug
(8)
G,' = G,g (9) where P,' and P,g are the vapor pressure of the saturated liquid and vapor, respectively, and G,' and Gag are the
6'o
reduced pressure range 0.009-0.957
reduced temp range 0.52-0.99 0.50-0.62 0.46-0.61 0.44-0.99 0.73-0.99 0.67-0.94 0.44-0.99 0.77-1.00 0.62-0.90 0.52-0.98 0.34-0.98 0.43-0.99 0.76-0.94 0.44-0.99 0.93-0.995 0.58-1.004 0.719-1.236 1.07-1.40 0.992-1.59 0.904-1.23 0.325-0.981 0.904-1.24 0.99-1.32
0.001-0.020
0.0004-0.019 0.0002-0.946 0.026-0.946 0.065-0.888 0.0002-0.946 0.155-1.00 0.022-0.508
0.182-4.48 0.074-1.08 1.06-12.2 0.385-1.95 0.430-2.75 1.29-12.9 0.0262-0.269 0.966-16.6
r-
001
'
'
'
'
60
' 65
es
EO
v/k
0.01 25
"
"
"
'
'
30
"
'
36
I
d k
Figure 1. Variation of percentage deviation of vapor pressure with 91 k .
Gibbs free energies of the liquid and vapor, respectively, given by
G' RT
A' RT
- = -(T,V)
+ 2 - 1 - In 2
where Ar - = Jvm(Z- 1) dV/V RT
Results and Discussion Along the Saturation Curve. The properties of both liquid and vapor at equilibrium calculated from the BACK
538 Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988
a) R11 4 [ 055
"
"
085
"
"
"
0 75
"
I
'
0 85
-4
0 85
055
Reduced Temperature
066
0 85
0 76
0 85
Reduced Temperature
4
b) R13
I 0
>
-L
cr z
I
r
J
-2 -
bl R13 -41
"
066
'
I
"
065
0 76
0 86
0 96
Reduced Temperature
Figure 2. Percentage deviation from BACK equation: vapor pressure. (a) ( 0 )Benning and McHarnes (1940); (X) Osborn et al. (1941); (0) ASHRAE (1969); (A)ASHRAE (1981); (+) FernandezAlbright and Martin (1952); (A) Fassnacht and Rio (1984). (b) (0) Oguchi et al. (197513); (+) Perel'shtein (1975); (X) Fernandez-Fassnacht and Rio (1985). Table 111. Second Virial Coefficients T/K R11, B/cm3 mol-' T/K -1106.7 180 260 -927.6 200 280 220 -801.0 298.15 240 -684.3 320 260 -594.6 340 280 -523.9 360 -465.6 298.15 380 -416.6 320 400 -375.1 340 420 -339.4 360 440 380 -308.5 460 -281.4 400 480 -257.6 450 500 -236.4 500 520 -208.7 550 550 -171.0 600 600 -141.0 650 650 -116.5 700 700 -96.2 750 750 -79.0 800 800 850 -64.3 850 -51.6 900 900 -40.5 950 950 1000 1000 -30.8
R13, B/cm3 mol-l -667.4 -523.2 -422.5 -349.2 -293.8 -250.7 -219.1 -188.1 -164.7 -145.0 -128.1 -113.5 -84.3 -62.4 -45.3 -31.7 -20.6 -11.3 -3.5 3.1 8.9 13.9 18.3 22.2
equation of state for R11 and R13 have been calculated (see supplementary material). Figures 2-4 show the percentage deviation between the predicted and the experimental values for vapor pressure, liquid volume a t saturation, and vapor volume a t saturation for R11 and R13. The percentage difference in the vapor pressure does not Table IV. Calculated comDd R11 R13
Critical Constants TJK % dev 472.5 -0.29 -0.65 304.0
exceed f3% over the reduced temperature range 0.55-1.0 for both R11 and R13, while the percentage difference in the density is f0.2% over the range 0.55-0.99 in case of the saturated liquid and f 4 % for the saturated vapor. Thermodynamic Properties in the One-Phase Region. Pressure-Volume-Temperature. The P-V-T properties have been calculated from 260 to lo00 K for R11 and from 180 to lo00 K for R13. In both cases the pressure range is from 1to 5000 bar (see supplementary material). To compare the predicted P-V-T data with the experimental values, the difference between the experimental values and the values calculated by using the BACK equation of state was plotted, as percentage volume deviation, against the reduced pressure at different temperatures, Figure 5. In the case of R13, the maximum volume deviation is less than 4% a t about 400 K ( T , = 1.32) and 400 bar (P, = 16). On the other hand, there are no available data for comparison for R11 higher than 473 K (T, = 1.004) and pressure a t about 175 bar (P, = 4). At this condition the deviation is less than 2.4%. Second Virial Coefficients. Figure 6 compares the calculated second viral coefficients with experimental values over a temperature range for both R11 and R13. Numerical values are given in Table 111. The predicted values show excellent agreement with experimental values, particularly as only the vapor pressure and the saturated density results were used to determine the parameters. The Critical Point. The critical pressure, temperature, and volume were calculated from BACK EOS for both R11 and R13. Table IV gives the calculated values and the
P,/ bar
% dev
44.95 40.41
-1.95 -4.4
V,/cm3 mol-' 247.5 177.0
% dev
0.24 2.1
2.. 0.283 0.283
Ind. Eng. Chem. Res., Vol. 27, No. 3, 1988 539 -200
- 4 L ,
"
"
'
'
"
"
"
"
'
1
"
"
0.76
0.86
0.66
,
0 06
0.86
Reduced Temperature
Reduced Temperature 0 , -100
4 1
4 U 0
._ 5
-
-200 -300 -
L
E
-400 -
0
m v)
-..o}
b) A13 0)
-4
,
,
'
,
066
1
,
,
'
,
8
'
,
'
,
0 76
066
8
.
,
,
'
-?OO I
1
g '
b) R13 '
06
0 86
0 86
-600 -
Figure 4. Percentage deviation from BACK equation: saturated vapor molar volume. (a) (+) Benning and McHarnes (1940a); (A) ASHRAE (1969);(0) ASHRAE (1981). (b) (X) Albright and Martin (1952);(A)Oguchi et al. (1975b);(+) Perel'shtein (1975);(0) Du Pont (1978).
- 0 2 1 ,' 00
'
'
'
'
'
'
10
'
'
'
'
'
'
20
'
'
'
30
'
0
tf "
00
' ' 60
"
" ' 10 0
'
'
'
,
'
'
60
40
~
20
Conclusions The BACK equation of state is an adequate equation for predicting vapor-liquid equilibrium and thermodynamic properties along the saturation curve as well as P-V-T data in the one-phase region for both R11 and R13. One advantage of this equation is that the parameters can be determined from the minimum of experimental data. For the compounds studied here, only the liquid density and the saturation vapor pressure were used to obtain the parameters. These properties are usually known with reasonable accuracy for many compounds where no other data exist. The major limitation in using the BACk equation in ita present form is that the predictions below a reduced temperature of 0.55 show increasing deviations from experimental values. Registry No. Cl,FC, 75-69-4;ClF&, 75-72-9.
b) R13 ~
16
Figure 6. Comparison of second virial coefficient calculated by BACK equation with experimental values. (a) (-) This work; (0) Hirschfelder et al. (1942);(+) Hajjar and MacWood (1968);(A)Kunz and Kapner (1969);(X) Sutter and Cole (1970). (b) (-) This work; (0) Michels et al. (1966);(V)Hajjar and MacWood (1968);(A)Kunz and Kapner (1969);(X) Haworth and Sutton (1970);(+) Sutter and Cole (1970);(0)Schramm and Gehramann (1980).
Reduced Pressure
'
10
Reduced Temperature
Reduced Temperature
~ 16.0
"
~
~
~
'
~
20 0
Reduced Pressure
Figure 5. Percentage deviation from BACK equation: fluid molar 0.64,(0) volume. (a) Rivikin and Kremenvskaya (1975), T, = (0) 0.69,(A)0.71,(+) 0.73,(X) 0.79,(0) 0.83,(V)0.87,( 0 )0.94,(X) 0.96, (0) 1.004. (b) Michels et al. (1966),T,= (+) 1.32,(0) 1.15;Geller et al. (1979),T,= (X) 0.90;Castro-Gomez (1987),T,= (A)1.32,(0) 1.15,(0)0.90.
percentage deviations from the selected experimental values.
'
Supplementary Material Available: Values for vapor pressures, molar volumes, enthalpies, and heat capacities for liquid and vapor at saturation (Table 5) for R11 (range 260-470 K at 10 K intervals) and (Table 6) for R13 (range 160-300 K at 10 K intervals) and values for the compressibility factors, fugacity coefficients, and residual thermodynamic properties of the real fluid (Table 7) for R11 and (Table 8) for R13 (10 pages). Ordering information is given on any current masthead page.
Literature Cited Albright, L. F.; Martin, J. J. Ind. Eng. Chem. 1952,44, 188. Alder, B. J.; Young, D. A,; Mark, M. A. J. Chem. Phys. 1972,56, 3013. ASHRAE Thermodyanics Properties of Refrigerants; Am. SOC. Heat. Refrig. Air-Cond. Eng.: New York, 1969. ASHRAE Handbook: Fundamentals; Am. SOC.Heat. Refrig. AirCond. Eng.: Atlanta, 1981.
540 Ind.
Eng. Chem. Res., Vol. 27, No. 3, 1988
Benning, A. F.; McHarnes, R. C. Znd. Eng. Chem. 1940a, 32, 497. Benning, A. F.; McHarnes, R. C. Znd. Eng. Chem. 1940b, 32, 814. Boublik, T. J. Chem. Phys. 1975,63,4084. Boublik, T. Collect. Czech. Chem. Commun. 1983, 48, 2713. Castro-Gomez, R. C. Ph.D. Dissertation, Texas A&M University, College Station, 1987. Chandnani, P. P.; Chakma, A.; Lielmezs, J. Thermdyn. Chem. Acta. 1984, 82, 263. Chavez, M., Hablchayn, P.; Maass, G.; Tsumura, R. J . Chem. Eng. Data 1986, 31, 218. Chen, S.; Kreglewski, A. Ber. Bunsenges Phys. 1977, 81, 1048. Du Pont “Freon” Technical Bulletin T-l3A, 1978. Fernandez-Fassnacht, E.; Rio, F. D. J . Chem. Thermodyn. 1984,16, 1003. Fernandez-Fassnacht, E.; Rio, F. D. Cryogenics 1985, 25, 204. Geller, V. Z.; Porichanskii, E. G.; Svetlichnyi, P. I.; El’kin, Yu. G . Kholod. Tekh. Tekhnol. 1979, 29, 43. Hajjar, R. F.; MacWood, G. E. J . Chem. Phys. 1968,49, 4567. Haworth, W. S.; Sutton, L. E. Trans. Faraday SOC.1970,67, 2907. Hirschfelder, J. 0.;McClure, F. T.; Weeks, I. F. J. Chem. Phys. 1942, 10, 201. Kreglewski, A. Equilibrium Properties of Fluids and Fluid Mixtures; Texas A&M University Press: College Station, 1984. Kreglewski, A,; Hall, K. R. Fluid Phase Equilib. 1983, 15, 11. Kunz, R. G.; Kapner, R. S. J. Chem. Eng. Data 1969,14, 190. Machat, V.; Boublik, T. Fluid Phase Equilib. 1985a, 21, 1. Machat, V.; Boublik, T. Fluid Phase Equilib. 1985b, 21, 11. Michels, A.; Wassernaar, T.; Wolker, G. J.; Prins, Chr.; Klundert, L. v. d. J . Chem. Eng. Data 1966, 11, 449. Oguchi, K.; Tanishita, I.; Watanabe, K.; Yamaguchi, T.; Sasayama, A. Bull. JSME 1975a, 18 (No. 126), 1448.
Oguchi, K.; Tanishita, I.; Watanabe, K.; Yamaguchi, T.; Sasayama, A. Bull. JSME 1975b, 18 (No. 126), 1456. Osborn, D. W.; Garner, C. S.; Docscher, R. N.; Yost, D. M. J . Am. Chem. SOC.1941,63, 3496. Perel’shtein, I. I. Thermophysical Properties of Matter and Substances; Rabinovich, V. A., Ed.; Amerind Pub.: New Delhi, 1975; Eng. Transl., Vol. 4, p 68. Riedel, L. 2.Ges. Kaelte-Znd. 1938, 45, 1938. Rivikin, S. L.; Kremenvskaya, E. A. Thermophysical Properties of Matter and Substances; Rabinovich, V. A,, Ed.; Amerind Pub.: New Delhi, 1975; Eng. Transl., Vol. 4, p 68. Schramm, B.; Gehramann, R. The Virial Coefficients of Pure Gases and Mixtures; Dymond, J. H., Smith, E. B., Eds.; Clarendon: Oxford, 1980; p 18. Simmick, J. J.; Lin, H. M.; Chao, K. C. Equation of State in Engineering and Research; Advances in Chemistry Series No. 182; Chao, K. C., Robinson, R. L., Eds.; Wiley: New York, 1979; p 209. Sinka, J. V.; Rosental, E.; Dixon, R. P. J. Chem. Eng. Data 1970,15, 73. Sutter, H.; Cole, R. H. J. Chem. Phys. 1970, 52, 132. Takahashi, M.; Takahashi, S.; Iwasaki, H. J . Chem. Eng. Data 1985, 30, 10.
Naila M. Gadalla, Kenneth N. Marsh* Thermodynamics Research Center Texas A&M University College Station, Texas 77843 Received for review May 29, 1987 Revised manuscript received November 3, 1987 Accepted November 10, 1987
