Ind. Eng. Chem. Res. 2003, 42, 6977-6985
6977
Solubility of Ethylene in Mixtures of Toluene, Norbornene, and Cyclic Olefin Copolymer at Various Temperatures and Pressures Liang-sun Lee,*,† Ruey-fu Shih,‡ Hsin-jung Ou,† and Tien-san Lee‡ Department of Chemical and Materials Engineering, National Central University, Chungli 32054, Taiwan, and Union Chemical Laboratories, Industrial Technology Research Institute, Hsinchu 300, Taiwan
In this study, the solubilities of ethylene in mixtures of toluene, norbornene, and cyclic olefin copolymer (COC) were measured at various temperatures (between 323.15 and 423.15 K), pressures (between 5 and 25 bar), concentrations of norbornene (between 0 and 85 wt %), and concentrations of COC (between 0 and 40 wt %). The experiments were conducted by the pressure decaying method using a newly designed apparatus. The experimental results show that the solubility of ethylene increases with increasing system pressure but decreases with increasing system temperature in the above mixture. Another interesting observation is that the solubility decreases when the concentration of reaction product, COC, is increased. Thus, in addition to temperature and pressure, the COC concentration affects the reaction extent, and beyond a certain COC concentration, further reaction favoring COC generation is impossible because of the opposite effect on ethylene solubility. Also, in this study, the experimental solubility data were expressed in vapor-liquid equilibrium relationship and correlated by bubble-pressure calculations with the Peng-Robinson equation of state (PR EOS; Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59) incorporating the modified van der Waals one-fluid (vdW-1) mixing rules and the Zhong-Masuoka (Z-M; Zhong C.; Masuoka, H. Fluid Phase Equilib. 1996, 123, 59) mixing rules, including the consideration of binary interaction parameters. The average absolute deviation percentages (AAD) of the correlation are less than 3.0%, except that of pressure with the Z-M mixing rules. Introduction In the 1950s, polycycloolefins were produced from cycloolefins with the Ziegler catalyst with the defect of inconsistent physical properties. Later, metallocene and methylaluminoxane catalysts were used instead of the Ziegler catalyst to produce different polymers of polycycloolefins from norbornene (the cyclic olefin monomer, e.g., bicyclo[2,2,1]-2-heptene). However, this new product still suffers the defect of having a very high glass transition temperature, causing some problems in practical industrial applications. An improved process, in which an ethyl group is attached to norbornene to produce better-quality polymers, called cyclic olefin copolymer (COCs), was therefore developed. These products consist of a rigid part made up of the polycycloolefin group and a flexible part made up of the attached linear ethyl group, so that they have good physical properties of low dielectric constants, lower glass transition temperatures, opaqueness, solvent resistance, and biocompatibility. They can be used as electronic materials, photoelectric materials, and materials for medical product containers. Bergstro¨m and Seppa¨la¨3 and Ruchatz and Fink4-6 studied the effects of the reaction temperature, pressure, concentration of norbornene, and catalyst on the properties of the resulting COC. They found that the number of cyclic groups and attached linear ethyl groups on the structure of the COC are affected by the reaction * To whom correspondence should be addressed. Tel.: +8863-425-0224. Fax: +886-3-425-2296. E-mail: t3100206@ ncu.edu.tw. † National Central University. ‡ Industrial Technology Research Institute.
conditions and will determine the properties of the COC. However, one might think that, in addition to the reaction temperature and pressure, the concentration of dissolved ethylene, i.e., the solubility of ethylene, in the reaction solution would first determine the extent of polymerization and then further affect the properties of the polymer product, the COC. Determination of the solubility of ethylene in the reaction solution is the first step in understanding this reaction system and was the main concern of this study. In recent years, the solubilities of gases in polymer solutions have been studied and reported by many researchers. Bogdanovic et al. (1990)7 studied the solubility of ethylene in polyethylene (PE) in the temperature range from 383.15 to 583.15 K and pressure range from 200 to 1000 bar. They found that the solubility of ethylene did not behave consistently at different pressures. One can consider the effect of temperature and pressure on the solubility of ethylene through the expression
dseth )
( ) ∂seth ∂P
T
dP +
( ) ∂seth ∂T
P
dT
(1)
The derivatives represent the isothermal and isobaric solubility coefficients, respectively, of ethylene in PE. Bogdanovic et al.7 explained that the solubility of ethylene decreases with temperature at low pressure, i.e., (∂seth/∂T)P is negative, whereas it increases with temperature at higher pressure, i.e., (∂seth/∂T)P is positive, and (∂seth/∂T)P is 0 at a certain temperature called the inversion point. This inversion phenomenon was shown in the solubility vs pressure diagram of ethylene, where two isotherms intersect at a specific pressure.
10.1021/ie030412r CCC: $25.00 © 2003 American Chemical Society Published on Web 11/12/2003
6978 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
Sato et al.8 (1996) measured the solubilities of nitrogen and carbon dioxide in polystyrene (PS) and reported that the solubility of carbon dioxide decreases with increasing temperature but the solubility of nitrogen increases with increasing temperature (in the range from 373.2 to 453.2 K). This is the opposite solubility effect that occurs for gases with low critical temperatures such as helium, hydrogen, and nitrogen. In 1999, Sato et al.9 studied the solubilities of nitrogen and carbon dioxide in polypropylene (PP) and high-density polyethylene (HDPE), as well as that of nitrogen in PS (in a temperature range lower than that used in their work in 19968). They observed that the solubility behaviors of nitrogen and carbon dioxide in PP and HDPE are consistent with the behaviors of these gases in PS reported in 1996.8 However, a different behavior was observed for nitrogen dissolution in PS in the temperature range from 313.2 to 353.2 K, where the nitrogen solubility decreases with increasing temperature. This observation is completely contrary to that made in 1996.8 Sato et al.9 concluded that the lowest solubility of nitrogen in PS happens at 350.2 K and that the solubility will increase with either increasing or decreasing temperature. The solubility data obtained by the above researchers were of binary systems that had fewer degrees of freedom than the present quaternary system. However, their works are valuable for reference. It is worth mentioning that the experimental results in the present study are for a system using metallocene as a catalyst that generates a different COC than the system using the Ziegler catalyst. Experimental Work Chemicals. The chemicals ethylene, toluene (99.5 mass %), and norbornene (>99.0 mass %) were purchased from Merck and Aldrich companies, respectively. The COC (provided by Union Chemical Laboratories, Taiwan) has a mass-average molecular weight, Mw, of 83 249; a number-average molecular weight, Mn, of 34 817; and a glass transition temperature, Tg, of 408.15 K. Carbon dioxide, ethylene, and helium gases were purchased from Chien-jen Co. (Taiwan). The purity of the carbon dioxide and ethylene was >99.5%, and that of helium was >99.99%. All chemicals were used without further purification. Apparatus and Experimental Procedure. The flow and semiflow types of apparatuses usually used for high-pressure vapor-liquid equilibrium systems are not suitable for the present mixture. The apparatus for present study is a static equilibrium system designed similarly to that used by Sato et al.;8 it is shown schematically in Figure 1. The major components of this apparatus include a high-pressure ethylene gas cylinder, a gas storage cell (Whitey Co., model 316L-50DF4-150) with a maximum operating pressure of 5000 psi, an equilibrium cell (Jerguson Gauge 11-T-20) with a scaled side-view glass and with a maximum operating pressure of 1300 psi at 150°C. A gas chromatograph (GC) for composition analysis was included on-line with the equilibrium cell. Two temperature-controlled thermostats with an uncertainty of (0.1 K were used. The accessory components of this apparatus include a vacuum pump, connecting lines, needle valves, check valves, and temperature control systems. During experiments, the temperatures were measured with mercury thermometers (Amarell Co.) with an uncertainty of (0.1 K. The pressures were measured with pressure gauges (Max-
Figure 1. Apparatus used for the present solubility study: A, high-pressure gas cylinder; B, gas cleaner; C, gas storage tank; D, safety valve; E, single-way valve; F, liquid injector; G, equilibrium cell; H, vacuum pump; GC, gas chromatograph; T1 and T2, temperature meters; P1 and P2, pressure meters.
thermo Co.) with an uncertainty of (0.1 psi. In this study, the composition of the gas mixture was analyzed with a GC (Taiwan Gas Chromatography Co., model 8900) with a TCD detector. The column used for present study was 80/100 Porapak Q (2.5 m × 1/8 in.). The operating conditions were as follows: injection temperature ) 110 °C, oven temperature ) 70 °C, detector temperature ) 120 °C, detector current ) 60 mA, and carrier gas ) helium with a flow rate of 1.8 mL‚min-1. It is worth mentioning that the accuracy of the compositions reported in this study depends on the uncertainties in the weights of the chemicals; the uncertainties in the volumes of the storage cell, equilibrium cell, and sampling loop; the uncertainties in the readings from the prescaled side-view glass of the equilibrium cell; and the uncertainties in the densities and compressibility factors calculated from the equation of state. In this study, the uncertainties in the above factors were considered to determine the final significant digits of all experimental data. For brevity, the methods used to determine these uncertainties are omitted here. The experiments were accomplished through the following careful operation steps: (1) Leakage from any connection or valve was carefully examined and prevented before each experimental run. (2) The whole apparatus was evacuated with a vacuum pump. (3) A sample of the designated feed composition (ternary mixture of toluene, norbornene, and COC) was inserted into the equilibrium cell through a liquid injector. Note that the constituent chemicals had been weighed with an electronic balance (Ohaus Analytycal Plus Co.) with an uncertainty of (0.1 mg. (4) Ethylene gas from the ethylene cylinder was introduced into the gas storage cell through a needle valve. After a static state had been reached, the temperature (T1,I) and pressure (P1,I) of the storage cell were recorded. Note that the storage cell was equipped with a safety valve (High-Pressure Equipment Inc.) for safety reasons. (5) The check valve (Swagelok Co.) between the storage cell and the equilibrium cell was then opened to allow a certain quantity of ethylene gas to flow into the equilibrium cell. The check valve was then closed when the equilibrium cell pressure reached the desired pressure, and the new static temperature (T1,F) and pressure (P1,F) of the storage cell were recorded. (6) The time-dependent pressure (P2) of the equilibrium cell at the desired temperature was recorded, as the pressure of the equilibrium cell decreased very slowly as a result of the dissolution of ethylene into the liquid mixture. (7) After the equilibrium cell pressure had remained constant for more than 30 min, equilibrium was considered to have been reached, and the final pressure of the equilibrium
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6979 Table 1. Experimental and Literature VLE Data for the Dioxide and Toluene System at 352.60 K Ng and Robinson10 this work
P (bar)
xCO2
3.8 14.0 30.8 1.6 4.0 9.7 12.0 14.6 17.6 20.9 23.6 29.8
0.0200 0.0765 0.1720 0.0049 0.0186 0.0480 0.0640 0.0785 0.0994 0.1189 0.1337 0.1672
Figure 2. P-x diagram of the carbon dioxide (1) + toluene (2) binary system at 352.60 K: ], experimental data; b, data from Ng and Robinson.10
cell was recorded. Steps 3-7 were then repeated to generate a set of isothermal P-x-y data. (8) To obtain another data set for a new isotherm, the designated temperature was changed, and steps 1-7 were then repeated. The calculation of the solubility of ethylene in the liquid mixture from the collected data is explained in a later paragraph. Before the experiments for the present system were conducted, an experimental test was run on the solubility of carbon dioxide in toluene at 352.60 K to ensure the reliability of this apparatus and experimental skill. The results of the test experiment were compared to the literature data of Ng and Robinson reported in 197810 and given in Table 1, where more new experimental data for this mixture are also provided. However, Table 1 does not obviously show how well these two data sets match, as no comparison can be made directly between experiments run under identical experimental conditions. Thus, Figure 2 was drawn and obviously shows the good agreement between the test data and the data of Ng and Robinson.10 Calibration Curves for Ethylene, Toluene, and Norbornene. To obtain the compositions of each phase of the present system, calibration curves for ethylene, toluene, and norbornene were first prepared. For the ethylene calibration curve, ethylene gas was introduced into the equilibrium cell at the desired temperature and
left undisturbed until the cell pressure did not vary. Then, the sampling valve was opened, and the volume of ethylene gas flowing into the GC was recorded. The number of moles of ethylene flowing into the GC can be calculated using the density of ethylene estimated by an equation of state (the PR EOS1 was used here) at the specific temperature and pressure. The data on peak area analyzed by the GC and the number of moles of ethylene were used to construct the calibration curve for ethylene. The calibration curve of toluene was constructed by injection of a specific volume of toluene directly into GC to obtain the peak area. The number of moles of toluene at the temperature of interest was calculated with the known density of toluene. The data on the peak area and number of moles of toluene were then used to construct the GC calibration curve for toluene. The norbornene calibration curve was constructed in a different way by dissolving norbornene in a cosolvent, toluene, because norbornene is a solid at room temperature. A mixture of known composition of toluene and norbornene was prepared, and a 0.1-µL sample was taken and injected into the GC for analysis. The peak area ratio and the known mole fraction of norbornene were used to construct the norbornene calibration curve. Each point of three calibration curves was determined by averaging three closest data points from the GC analysis. Solubility Calculation. For the present system, the solubility of ethylene in toluene + norbornene + COC mixture was computed by material balance before and after an experimental run. Let Wtotal, Wtol, Wnor, and WCOC represent the carefully weighed masses of the liquid mixture, toluene, norbornene, and COC, respectively. The amount of ethylene flowing from the storage cell to the equilibrium cell, Weth, was calculated using the equation
(
Weth ) MWeth
VsP1,F VsP1,I Z1,IRT1,I Z1,FRT1,F
)
(2)
where MWeth is the molecular weight of ethylene, Vs is the volume of the storage cell, and Z is the compressibility factor of ethylene calculated by the PR EOS.1 The subscripts I and F represent the initial and final conditions, respectively, of each experiment. During each experiment, the amount of ethylene introduced from the storage cell to the equilibrium cell did not completely dissolve in the liquid mixture. However, this amount is equal to the sum of the ethylene existing in the vapor phase (superscript v) and in the liquid phase (superscript l) in the equilibrium cell. Thus
Weth ) Wveth + Wleth
(3)
The mass of ethylene in the vapor phase was calculated from the equation of state
Wveth
(
) MWeth
V vP2,F Z2,FRT2,F
)
(4)
where V v is the volume of the equilibrium gas phase in the equilibrium cell. This volume can be read from the prescaled side-view glass of the equilibrium cell. The compressibility factor Z2,F at the equilibrium conditions T2,F and P2,F was also calculated by the PR EOS.1 For
6980 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 Table 2. Experimental VLE Data for the Ethylene (1) + Toluene (2) + Norbornene (3) + COC (4) Quaternary System temp (K)
initial tol/nor/COC wt ratio
323.15
40/56/4
x1
x2
x3
x4
y1
y2
y3
5.1 9.9 14.6 19.2 23.8 6.4 10.8 14.6 19.1 25.2 5.7 9.4 16.0 20.6 23.3 6.1 10.7 15.4 20.5 23.7 5.6 9.9 13.7 19.4 25.6 7.0 10.5 14.8 19.9 25.3 6.8 10.7 15.4 20.0 23.9
1.60 3.21 5.02 6.98 9.15 1.57 2.99 4.36 5.90 8.36 1.12 2.30 3.76 5.18 6.61 1.11 2.24 3.33 4.65 5.63 0.88 1.69 2.51 3.80 5.30 0.55 1.18 1.79 2.88 3.85 0.47 1.00 1.71 2.50 3.13
0.0526 0.1001 0.1481 0.1948 0.2406 0.0583 0.1057 0.1471 0.1894 0.2486 0.0494 0.0958 0.1479 0.1928 0.2339 0.0371 0.0719 0.1032 0.1385 0.1630 0.0337 0.0627 0.0904 0.1306 0.1734 0.0185 0.0393 0.0583 0.0906 0.1177 0.0181 0.0382 0.0632 0.0866 0.1102
0.3995 0.3794 0.3592 0.3395 0.3202 0.3968 0.3768 0.3594 0.3416 0.3166 0.4003 0.3807 0.3588 0.3399 0.3227 0.4147 0.3997 0.3863 0.3710 0.3605 0.4090 0.3968 0.3850 0.3680 0.3499 0.4149 0.4061 0.3980 0.3844 0.3729 0.4140 0.4055 0.3950 0.3851 0.3752
0.5478 0.5204 0.4926 0.4656 0.4391 0.5444 0.5171 0.4931 0.4686 0.4344 0.5493 0.5225 0.4924 0.4665 0.4426 0.5481 0.5283 0.5104 0.4904 0.4764 0.5568 0.5400 0.5241 0.5010 0.4763 0.5665 0.5545 0.5436 0.5249 0.5093 0.5674 0.5558 0.5413 0.5278 0.5141
0.0001 0.0001 0.0001 0.0001 0.0001 0.0005 0.0004 0.0004 0.0004 0.0004 0.0010 0.0010 0.0009 0.0008 0.0008 0.0001 0.0001 0.0001 0.0001 0.0001 0.0005 0.0005 0.0005 0.0004 0.0004 0.0001 0.0001 0.0001 0.0001 0.0001 0.0005 0.0005 0.0005 0.0005 0.0005
0.9625 0.9767 0.9860 0.9876 0.9893 0.9660 0.9852 0.9867 0.9900 0.9908 0.9622 0.9757 0.9855 0.9886 0.9902 0.8305 0.9012 0.9257 0.9411 0.9476 0.8305 0.8919 0.9210 0.9402 0.9500 0.4610 0.6322 0.7084 0.7833 0.8162 0.4725 0.6479 0.7411 0.7878 0.8170
0.0088 0.0057 0.0037 0.0030 0.0029 0.0093 0.0034 0.0033 0.0021 0.0024 0.0122 0.0072 0.0043 0.0032 0.0032 0.0681 0.0276 0.0208 0.0167 0.0153 0.0762 0.0614 0.0371 0.0258 0.0238 0.1940 0.1223 0.0942 0.0692 0.0469 0.1784 0.1113 0.0820 0.0655 0.0559
0.0287 0.0176 0.0103 0.0094 0.0078 0.0247 0.0114 0.0100 0.0079 0.0068 0.0256 0.0171 0.0102 0.0082 0.0066 0.1014 0.0712 0.0535 0.0422 0.0371 0.0933 0.0467 0.0419 0.0340 0.0262 0.3450 0.2455 0.1974 0.1475 0.1369 0.3491 0.2408 0.1769 0.1467 0.1271
30/42/28
40/56/4
35/49/16
423.15
40/56/4
35/49/16
the present study, it was assumed that the traces of toluene and norbornene in the gas phase were negligible and would not affect the computation of the compressibility factor of ethylene. This assumption significantly simplifies the computation. The solubility of ethylene in the liquid mixture was then calculated as
Seth )
vapor phase
solubility × 102 (geth/gmixture)
35/49/16
373.15
liquid phase
pressure (bar)
Wleth Wltol + Wlnor + WlCOC
(5)
and the mole fraction of ethylene in the liquid phase was calculated as xeth ) Wleth/MWeth Wleth/MWeth + Wlnor/MWnor + WlCOC/MWCOC + WItol/MWtol
(6) where MWCOC is the number-average molecular weight of the COC. The mole fractions of other components in the liquid phase can be calculated according to the same method as used for ethylene.
of the longer time required to reach equilibrium. In addition, leakage might occur, particularly at higher temperatures, which cause nonuniform expansion at the connections of the lines. Following the experimental procedure described in the preceding section with careful technique, 35 experimental data points were obtained, as reported in Table 2, where the original liquid mixtures containing toluene, norbornene, and COC of the designated compositions are also included. This table shows that the solubility of ethylene in the liquid mixture increases with increasing pressure but decreases with increasing temperature. The effect of the liquid composition on the ethylene solubility is not so obviously indicated by the numerical values listed in table, as no experimental data have been obtained at the identical experimental conditions for comparison. However, the solubility could still be arranged qualitatively in decreasing order by the fraction weight ratio of toluene/norbornene/COC as 40/56/4 > 35/49/16 > 30/42/28. Thus, ethylene has a higher solubility in a solution with a higher concentration of toluene and norbornene at the early stage of reaction and a lower solubility when more COC is present in solution phase. The experimental data were plotted in Figures 3-5, where the solubilities of ethylene in the liquid mixture at different temperatures are clearly shown.
Experimental Results The solubilities of ethylene in the liquid mixture were measured at temperatures of 323.15, 373.15, and 423.15 K and pressures ranging from 5 to 25 bar. The present experiment is more difficult than VLE measurements by a flow or semiflow approach at high pressure because
Experimental Data Reduction Phase Equilibrium Correlation. The fugacity coefficient-activity coefficient (φ-γ) approach to vaporliquid equilibrium correlations is suitable for systems at low or moderate pressure, as the solution model is
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6981
Figure 3. Solubility of ethylene in toluene, norbornene, and COC mixtures at 323.15 K: 9, 2, b, f, experimental data; -, - - -, -‚-. -‚‚-, calculated using the PR EOS with the vdW-1 mixing rules (k12 ) 0.571, k13 ) 0.581, k14 ) 0.881, k23 ) -0.329, k24 ) 1.803, k34 ) 2.831) for different initial toluene/norbornene/COC weight ratios of 42/58/0, 40/56/4, 35/49/16, and 30/42/28, respectively.
Figure 5. Solubility of ethylene in toluene, norbornene, and COC mixture at 423.15 K: 9, 2, b, experimental data; -, - - -, -‚-, calculated by using the PR EOS with the vdW-1 mixing rules (k12 ) 0.391, k13 ) 0.493, k14 ) 1.175, k23 ) 0.074, k24 ) 1.839, k34 ) 2.904) for different initial toluene/norbornene/COC weight ratios of 42/58/0, 40/56/4, and 35/49/16, respectively.
At equilibrium, the vapor fugacity and the liquid fugacity are equal, as also are the temperatures and pressures in the two phases, and this equality can be expressed in terms of the fugacity coefficients as
yiφˆ vi ) xiφˆ li
i ) 1, 2, ..., n
(7)
where the fugacity coefficient is calculated according to the following equation
ln φˆ i )
Figure 4. Solubility of ethylene in toluene, norbornene, and COC mixtures at 373.15 K: 9, 2, b, experimental data; -, - - -, -‚-, calculated using the PR EOS with the vdW-1 mixing rules (k12 ) 0.449, k13 ) 0.536, k14 ) 0.671, k23 ) -0.168, k24 ) 5.917, k34 ) 1.728) for different initial toluene/norbornene/COC weight ratios of 42/58/0, 40/56/4, and 35/49/16, respectively.
usually inadequate at high pressure because of the inherent disadvantage of choosing a pseudo-liquid fugacity as the standard state of a constituent component. Instead, the fugacity coefficient-fugacity coefficient (φφ) approach is suitable for systems at high pressure. For the present system, the constituent component, ethylene gas, has a critical temperature lower than the present experimental temperature and a critical pressure higher than the present experimental pressure. Thus, the φ-φ method is applied here.
∫0p
1 RT
[( ) ∂nv ∂ni
-
T,P,nj*i
]
RT dP P
(8)
where the partial differential of volume with respect to component i in the above equation is calculated with an equation of state (EOS). The literature works related to the present study include that of Sato et al.,9 who applied the EOS of Sanchez and Lacome11 to correlate the experimental solubility data of nitrogen in polypropylene (PP), carbon dioxide in high-density polyethylene (HDPE), and nitrogen in polystyrene (PS). Liu and Wong12 applied the Schotte EOS13 to the system of ethylene dissolved in polyethylene (PE). They reported better results obtained with the vdW-1 mixing rules than with the Wong-Sandler14 (W-S) mixing rules. In the present study, the Peng-Robinson equation of state1 suitable for systems at high pressure is applied. This EOS is given in the Appendix for reference. The fugacity coefficient of component i in a mixture can be derived from the original PR EOS and eq 8 after some mathematical manipulations and expressed as
ln φˆ i )
b′m (Z - 1) - ln(Z - Bm) bm b′m Z + 2.414Bm Am a′m 1 ln (9) Z - 0.414Bm 2x2B n am bm
(
m
) |
|
6982 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
where am and bm are the energy and covolume parameters of the mixture calculated by the mixing rules. The other terms in the above equation are as follows
Am )
amP
bmP Bm ) RT
, (RT)2
[ ]
∂(n2am) a′m ) ∂ni
[ ]
∂(nbm) , b′m ) ∂ni T,v,nj*i
(10)
T,nv,nj*i
∑i ∑j zizjaij
bm )
∑i zibi
(11)
and
aij ) xaiaj
bij
xbibj
(1 - Lij)
bi + bj bij ) 2
(12)
where Lij denotes the binary interaction parameter of the vdW one-fluid mixing rules. Note that a volume correction factor was introduced into aij by Saravia et al.15 in these modified mixing rules. The terms a′m and b′m in eq 9 in the vdW one-fluid mixing rules are expressed as
∑j njaij
a′m ) 2
b′m ) bi
component
MW
Tc (K)
Pc (bar)
acentric factor
CO2a C2H4a toluene C7H8a norbornene, C7H10b
44.010 28.054 92.141 94.156
304.1 282.4 591.8 583.0
73.8 50.4 41.0 39.3
0.239 0.089 0.263 0.159
a Data from Reid et al.20 b Data from the databank of Aspen Plus.21
To estimate the energy and covolume parameters and the terms given in eq 10 for the mixture, an appropriate set of mixing rules is required, and the selection of the mixing rule is as important as the choice of a suitable EOS for VLE correlations, as illustrated by Liu and Wong.12 They used the Wong-Sandler14 (W-S) and the van der Waals one-fluid (vdW-1) mixing rules for the system of ethylene dissolved in polyethylene (PE) and reported that the results obtained with the vdW-1 mixing rules were reasonable and better than those obtained with the W-S mixing rules14 for the Schotte EOS. The mixing rules used in this study are the ZhongMasuoka (Z-M)2 and the modified vdW-1 of Saravia et al.15 for polymer solutions. Other mixing rules such as the Wong-Sandler mixing rules used by Liu and Wang12 were not considered because they often require a physical parameter, such as the volume fraction, that is difficult to determine for the polymer component, COC, in the present mixture. The modified vdW-1 mixing rules were used here because of their simplicity and the reasonable final results obtained for the present mixture. For the correlation of VLE of polymer systems, the Z-M mixing rules were used by Zhong and Masuoka2 and found to give better results than the vdW1, vdW-2, and W-S mixing rules.14 Louli and Tassios16 (2000) also obtained similar results for the VLE correlation of polymer mixtures. The modified vdW one-fluid mixing rules are expressed as
am )
Table 3. Physical Properties of the Compounds
(13) (14)
Zhong and Masuoka proposed their mixing rules suitable for cubic equations of state in 1996. Their consideration was based on the assumptions that, at very high pressure, the repulsive term tends to zero and the contribution of the attractive term is much less than that of the repulsive term and can be neglected. Thus, the excess Helmholtz free energy at high pressure is negligible. The mixing rules proposed by Zhong and Masuoka2 are
D am ) RTQ 1-D bm )
(15)
Q 1-D
(16)
where
D)
ai
∑i zib RT,
Q)
i
(b - RTa )
) ij
(
bi -
(
a
)
∑ ∑zizj b - RT ij
) (
(17)
)
ai aj + bj RT RT (1 - kij) (18) 2
After some mathematical manipulations, the terms a′m and b′m for the Z-M mixing rule2 and for the fugacity coefficient estimation can be obtained as
a′m ) nRT
b′m )
1
(
[∑( )
1-D
zj b -
2
j
)
aibm + Db′m biRT a
RT
ij
-
Q
(19)
(
1-D
1-
ai
)]
biRT
(20)
where kij denotes the binary interaction parameter of the Z-M mixing rules. The binary interaction parameters of the Z-M mixing rules, kij, and of the mixing rules of van der Waals, Lij, are determined in the section on the VLE correlation. Estimations of Energy and Covolume Parameters of the COC Polymer. For thermodynamic property estimations of a mixture with an equation of state, the molecular weights, critical properties, and acentric factors of the constituent components are required prior to determination of the energy and covolume parameters of the equation of state. The above physical property data of ethylene, toluene, and norbornene are given in Table 3. However, it is very difficult to obtain the PVT data for the COC polymer either from the literature or from the experiments required to determine the energy and covolume parameters. To estimate these two parameters for the COC polymer, we used the molar volume of the COC polymer estimated with the extended group contribution volume of
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6983
The performance of the above minimization will generate pair interaction parameters among all of the constituent components. For the present quaternary system, there are six pairs of binary interaction parameters to be determined from the experimental VLE data given in Table 2. In general, the optimum binary interaction parameters are generated from the experimental data involving all of the components appearing in the objective function, eq 22. However, during data correlation, the number of binary parameters is sometimes reduced to make it easier to determine all of the binary parameters. Thus, for the present system, the pair interaction parameters of ethylene, benzene, and norbornene (pairs L12, L13, and L23 or k12, k13, and k23) were determined from the VLE data of the ethylene, benzene, and norbornene ternary system estimated in our other report,19 which are quoted in Table 5. In this table, the average absolute deviations (AADs) of the system pressure and the composition of the gas phase are also included to indicate that the original correlation of the ternary system was reasonable. Thus, there were only three binary interaction parameters to be determined: L14, L24, and L34 for the modified vdW-1 mixing rules and k14, k24, and k34 for the Z-M mixing rules2 The optimum values of these parameters are given in Table 6. All of the binary interaction parameters given in Tables 5 and 6 are random and show no dependency on temperature and pressure; moreover, all have absolute magnitudes less than 1 for the modified vdW-1 mixing rules. This observation can be explained by the fact that the constituent components in the mixture are nonpolar or weakly polar compounds that have small values of pair interaction parameters. During the correlation, we noticed that some of the binary interaction parameters, kij, for the Z-M mixing rules were larger than 1 and possibly changed the sign of am from positive to negative, i.e., from an attractive contribution to a repulsive contribution. However, we carefully checked the final numerical values of am for the present case and observed that am is always positive and should not cause any unreasonable results. During the correlation,
Table 4. Energy and Covolume Parameters for the PR EOS for the Polymer COC a (m3/gmol)2‚bar
b (m3/gmol)
AADa (%)
8.6913 × 10-3
2.7460 × 10-2
0.24
COC a
AAD (%) )
1 N
∑|
|
vcal - vGCVOL × 100%. vGCVOL
liquid (GCVOL) method of Tasibanogiannis et al.17 at different temperatures as the true value. In addition, the vapor pressure was adopted as the constant value of 10-7 MPa used by Orbey and Sandler18 for polymers. Then, the energy and covolume parameters of the PR EOS for the COC were determined by minimizing the following objective function
[ ∑(
min OBJ ) min
)]
100 N |vGCVOL - vcal| N
vGCVOL
1
(21)
where N is the number of volumes at different temperatures and vcal is the volume calculated using the PR EOS containing the parameters to be determined. The optimal energy and covolume parameters obtained from this regression are given in the Table 4. VLE Correlation. The vapor-liquid equilibrium calculation of a mixture is usually obtained through the bubble point or dew point of the pressure or temperature. For the present isotherm solubility measuring system, the bubble pressure calculation would be suitable for our purposes. The objective function defined for the present correlation is expressed as
min OBJ ) min
[ (
100 N |Pexp - Pcal|
∑ N 1
∑ |yi,exp - yi,cal| i)1 k
+
Pexp
])
(22)
where N is the number of experiments and k is the number of components.
Table 5. Interaction Parameters and Deviations Obtained Using the PR EOS with the vdW-1 and Z-M Mixing Rules for the VLE of the Ethylene(1) + Toluene(2) + Norbornene (3) Ternary System from Lee et al.19 vdW-1 mixing rules
a
Z-M mixing rules
temp (K)
L12/L13/L23
AAD P (%)a
AAD y × 102 b
k12/k13/k23
AAD P (%)a
AAD y × 102 b
323.15 373.15 423.15
0.120/0.149/0.012 0.077/0.129/-0.035 0.092/0.129/0.013
2.63 1.42 2.46
1.07 1.20 1.93
0.571/0.581/-0.329 0.449/0.536/-0.168 0.391/0.493/0.074
3.09 1.52 2.73
0.43 1.05 2.13
AAD P (%) )
∑| N
|
Pcal - Pexp
1
Pexp
b
× 100%.
AAD y )
1 N
∑[∑|y
i,cal
- yi,exp|
i
]
Table 6. Interaction Parameters and Deviations Obtained Using the PR EOS with the vdW-1 and Z-M Mixing Rules for the VLE of the Ethylene (1) + Toluene (2) + Norbornene (3) + COC(4) Quaternary Systema vdW-1 mixing rules
a
b
temp (K) 323.15 373.15 423.15
Z-M mixing rules
L14/L24/L34
AAD P (%)b
AAD y × 102 c
k14/k24 / k34
AAD P (%)b
AAD y × 102 c
0.800/0.184/-0.085 0.738/0.368/-0.402 0.694/-0.497/-0.184
2.88 1.35 2.08
0.85 1.11 1.53
0.881/1.803/2.831 0.670/5.917/1.728 1.175/1.839/2.904
4.02 1.57 2.10
0.36 1.49 1.73
L12, L13, L23 and k12, k13, k23 are from the ethylene (1) + toluene (2) + norbornene (3) ternary system (Table 5).
AAD P (%) )
∑| N 1
|
Pcal - Pexp Pexp
× 100%.
c
AAD y )
1 N
∑[∑|y
i,cal
i
- yi,exp|
]
6984 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
we also tried to put constraints to confine the values of kij in the range between 0 and 1, but doing so resulted in much worse correlation results than those given in Table 6. Thus, for the present case, the fact that the values of kij given in Table 6 are larger than 1 is tolerable given that the sign of am is not changed and is still theoretically reasonable. The correlation results listed in Table 6 show that the AADs of the system pressure and of the gas-phase composition are less than 3.0%, except for the AAD of pressure by the Z-M mixing rules. Upon comparison of correlation results from the vdW-1 and Z-M mixing rules, it can be observed that both mixing rules are suitable for correlating this quaternary system and the modified vdW-1mixing rules are slightly better than the Z-M mixing rules for the present system. The experimental results and the results correlated with the vdW-1 mixing rules are also plotted in Figures 3-5 for temperatures of 323.15, 373.15, and 423.15 K, respectively. These figures show the good agreement between the experimental and correlated data. The slightly worse results obtained with the Z-M mixing rules are omitted here. Conclusion The isothermal solubilities of ethylene in mixtures of toluene, norbornene, and COC were experimentally determined in the temperature range between 323.15 and 423.15 K, the pressure range between 5 and 25 bar, the norbornene concentration range between 0 and 85 wt %, and the COC concentration range between 0 and 40 wt %. The experiments were carried out with a newly designed apparatus by the pressure decaying method and the experimental data were useful in designing and operating the process producing the COC. The experimental results showed that the thermodynamic behavior of this system is similar to the recognized behavior of the dissolution of a gas in a liquid, i.e., the solubility of ethylene in the present mixture increases with increasing system pressure but decreases with increasing system temperature. The experimental results showed that the ethylene solubility decreases in the mixture when the concentration of COC is increased. Thus, there is a critical COC concentration at a certain operating temperature and pressure beyond which further reaction favoring COC generation is not possible. The experimental solubility data were correlated and expressed in the vapor-liquid equilibrium relationship fairly well with the PR EOS1 incorporating the modified vdW-1 and the Zhong-Masuoka2 mixing rules with the consideration of optimum binary interaction parameters. The correlation results showed that the AADs of system pressure and the gas-phase composition were less than 3.0%, except for the AAD of the pressure obtained with the Z-M mixing rules. Upon comparison of the correlation results from the vdW-1 and Z-M mixing rules, it was observed that both mixing rules were suitable for representing this quaternary system and the vdW-1 mixing rules were slightly better than the Z-M mixing rules for the present system. Acknowledgment The authors acknowledge the financial support of this research from Union Chemical Laboratories of Industrial Technology Research Institute of Taiwan.
Nomenclature A ) GC peak area Am ) term defined in eq 10 ai ) energy parameter of PR EOS am ) term defined in eq 15 a′m ) term defined in eq 13 or 19 Bm ) term defined in eq 10 bi ) covolume parameter of PR EOS bm ) term defined in eq 15 b′m ) term defined in eq 14 or 20 D ) defined in eq 17 f ) fugacity G ) Gibbs free energy kij ) binary interaction parameter of the Zhong-Masuoka mixing rules Lij ) binary interaction parameter of the van der Waals one-fluid mixing rules Mw ) mass-average molecular weight Mn number-average molecular weight N ) number of data points n ) number of moles OBJ ) objective function P ) system pressure, bar Q ) defined in eq 17 R ) ideal gas constant S ) solubility T ) temperature, K V ) molar volume Vs ) volume of storage cell V v ) volume of gas phase at equilibrium state W ) mass x ) liquid mole fraction y ) vapor mole fraction Z ) compressibility factor Greek Characters µ ) chemical potential φ ) fugacity coefficient ω ) acentric factor Superscripts o ) reference state l ) liquid phase s ) saturated state v ) vapor ∧ ) identity in a mixture Subscripts cal ) calculated value COC ) COC polymer eth ) ethylene exp ) experimental value GCVOL ) value estimated by the GCVOL method i ) component i j ) component j m ) mixture nor ) norbornene tol ) toluene
Appendix The Peng-Robinson equation of state was proposed in the form
P)
a(T) RT v - b v(v + b) + b(v - b)
(A-1)
where the repulsive term was retained as in the original van der Waals form whereas the attractive term was modified by Peng and Robinson1 to
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6985
a(T) ) a(Tc) R(Tr,ω)
(A-2)
The covolume and attractive parameters at critical conditions were proposed by Peng and Robinson as
R2Tc2 a(Tc) ) 0.457 24 Pc
(A-3)
RTc b(T) ) 0.077 880 Pc
(A-4)
R1/2 ) 1 + κ[1 - (T/Tc)1/2}
(A-5)
and
with
κ ) 0.374 64 + 1.542 26ω - 0.269 92ω2 (A-6) Literature Cited (1) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (2) Zhong C.; Masuoka, H. A New Mixing Rule for Cubic Equation of State and its Application to Vapor-Liquid Equilibria of Polymer Solutions. Fluid Phase Equilib. 1996, 123, 59. (3) Bergstro¨m, C. H.; Seppa¨la¨, J. V. Effects of Polymerization Conditions When Making Norbornene-Ethylene Copolymers Using the Metallocene Catalyst Ethylene Bis(indenyl) Zirconium Dichloride and MAO to Obtain High Glass Transition Temperature. J. Appl. Polym. Sci. 1997, 63, 1063. (4) Ruchatz, D.; Fink, G. Ethene-Norbornene Copolymer Using Homogeneous Metallocene and Half-Sandwich Catatlyst: Kinetics and Relationships between Catalyst Structure and Polymer Structure. 2. Comparative Study of Different Metallocene and Half-Sandwich Methylaluminoxane Catalysts and Analysis of the Copolymer by 13C Nuclear Magnetic Resonance Spectroscopy. Macromolecules 1998, 31, 4674. (5) Ruchatz, D.; Fink, G. Ethene-Norbornene Copolymer Using Homogeneous Metallocene and Half-Sandwich Catatlyst: Kinetics and Relationships between Catalyst Structure and Polymer Structure. 3. Copolymerization Parameters and Copolymerization Diagrams. Macromolecules 1998, 31, 4681. (6) Ruchatz, D.; Fink, G. Ethene-Norbornene Copolymer Using Homogeneous Metallocene and Half-Sandwich Catatlyst: Kinetics and Relationships between Catalyst Structure and Polymer Structure. 4. Development of Molecular Weight. Macromolecules 1998, 31, 4684.
(7) Bogdanovic´, V. Zˇ .; Tasic´, A. Zˇ .; Djordjevic´, B. D. Inversion Phenomena of Ethylene Solubility in Polyethylene. J. Appl. Polym. Sci. 1990, 41, 3091. (8) Sato, Y.; Yurugi, M.; Fujiwara, K.; Takishima, S.; Masuoka, H. Solubilities of Carbon Dioxide and Nitrogen in Polystyrene under High Temperature and Pressure. Fluid Phase Equilib. 1996, 125, 129. (9) Sato, Y.; Fujiwara, K.; Takikawa, T.; Sumarno,; Takishima, S.; Masuoka, H. Solubilities and Diffusion Coefficients of Carbon Dioxide and Nitrogen in Polypropylene, High-Density Polyethylene, and Polystyrene under High Pressures and Temperatures. Fluid Phase Equilib. 1999, 162, 261. (10) Ng, H. J.; Robinson, R. B. Equilibrium Phase Properties of the Toluene-Carbon Dioxide Sytem. J. Chem. Eng. Data 1978, 23, 325. (11) Sanchez, I. C.; Lacombe, R. H. An elementary Molecular Theory of Classical Fluids. Pure Fluid. J. Phys. Chem. 1976, 80, 2352. (12) Liu, J. L.; Wong, D. S. H. Application of Wong-Sandler Mixing Rules to Polymer Solutions. Fluid Phase Equilib. 1996, 117, 92. (13) Schotte, W. Vapor-Liquid Equilibrium Calculations for Polymer Solutions. Ind. Eng. Chem. Process Des. 1982, 21, 289. (14) Wong, D. S. H.; Sandler, S. I. A Theoretically Correct Mixing Rule for Cubic Equation of State. AIChE J. 1992, 38, 671. (15) Saravia, A.; Kontogeorgis, G. M.; Harismiadis, V. I.; Fredenslund, A.; Tassios, D. Application of van der Waals Equation of State to Polymers IV: Correlation and Prediction of Lower Critical Soluiton Temperatures for Polymer Soluitons. Fluid Phase Equilib. 1996, 115, 73. (16) Louli, V.; Tassios, D. Vapor-Liquid Equilibrium in PolymerSolvent Systems With a Cubic Equation of State. Fluid Phase Equilib. 2000, 168, 165. (17) Tasibanogiannis, I. N.; Kalospiros, K. S.; Tassios, D. Extension of the GCVOL Method and Appliciation to Some Complex Compounds. Ind. Eng. Chem. Res. 1994, 33, 1643. (18) Orbey, N.; Sandler, S. I. Vapor-Liquid Equilibrium of Polymer Solutions Using a Cubic Equation of State. AIChE J. 1994, 40, 1203. (19) Lee, L. S.; Shih, R. F.; Ou, H. J.; Lee,T. S. Gas-Liquid Equilibria of Binary and Teranry Mixtures of Ethylene, Tolune, and Norbornene, manuscript in preparation. (20) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987; Appendix A. (21) ASPEN PLUS Reference Manual, version 10.1; AspenTech: Cambridge, MA, 2000.
Received for review May 9, 2003 Revised manuscript received September 23, 2003 Accepted October 9, 2003 IE030412R
