Densities of Toluene, Carbon Dioxide, Carbonyl Sulfide, and

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Densities of Toluene, Carbon Dioxide, Carbonyl Sulfide, and Hydrogen Sulfide over a Wide Temperature and Pressure Range in the Sub- and Supercritical State E. Christian Ihmels and Ju 1 rgen Gmehling* Technische Chemie, Carl von Ossietzky Universita¨ t Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany

Densities of toluene, carbon dioxide (CO2), carbonyl sulfide (COS), and hydrogen sulfide (H2S) have been measured with a new computer-controlled high-temperature high-pressure vibrating tube densimeter system (DMA-HDT) for temperatures from 273 K up to 623 K and pressures up to 40 MPa in the sub- and supercritical state. With respect to accuracy, reliability, suitability, and time consumption this system represents an optimum for measuring PFTsproperties in the compressed liquid and supercritical state. Densities of liquid toluene, COS, and H2S as a function of temperature and pressure were correlated with the new three-dimensional density correlation system (TRIDEN). The PFT data from CO2 and the data from COS and H2S in the supercritical state and around the critical point were correlated with a virial-type equation of state. For checking the accuracy and suitability of the vibrating tube densimeter system the experimental densities of toluene and CO2 were compared with the results obtained using reference equations of state. 1. Introduction The reliable knowledge of the PvT behavior of pure compounds and mixtures is of great importance in many fields of research as well as in industrial practice. The densities of fluids as a function of temperature, pressure, and composition are particularly important for the design of industrial plants, pipelines, and pumps. Furthermore, reliable density values are the basis for the development of correlation equations and equations of state. Using equations of state and ideal gas heat capacities, it is possible to calculate phase equilibria and other thermodynamic properties such as enthalpies, entropies, heat capacities, and heats of vaporization at given conditions (temperature, pressure, and composition). These data are needed for solving material and energy balances required for the design and optimization of chemical processes. For this purpose a data bank for pure component thermodynamic and transport properties was developed between 1991 and 2001, which is continuously updated. The main objectives of the pure component database are the determination of recommended values, the fitting of recommended correlation parameters, and the development of improved prediction methods for pure component properties. To accomplish this, the database is thoroughly tested and at the same time data gaps are removed by measurements. For reliable liquid density measurements often vibrating tube densimeters are applied. The vibrating tube method published by Kratky et al.1 is well-known and has been widely applied for more than 30 years in research and development as well as for routine industrial measurements. In this paper the technique is used for the determination of densities in the sub- and * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: ++49-441-7983831. Fax: ++49-441-798-3330. Internet: www.uni-oldenburg.de/tchemie.

supercritical state. In contrast to commercially available vibrating tube densimeters, our prototype can be used over a wider temperature and pressure range. In 1997 the prototype was supplied by “Labor fu¨r Messtechnik Dr. Hans Stabinger” (Graz, Austria). It was designed for temperatures up to 623 K and pressures up to 40 MPa. At first the computer-controlled apparatus was applied for temperatures up to 523 K and pressures up to 10 MPa for the measurement of liquid densities in the subcritical state.2 For the present work the environment of the apparatus was improved and the range of applicability was extended to higher temperatures and pressures. At the same time measurements of compressed supercritical densities were performed. With the computer-controlled measurement system available it is now possible to measure densities in a wide temperature and pressure range in the sub- and supercritical state in a rather short time. For correlating compressed liquid densities the widely used Tait equation3 is employed. This equation needs a reference point for correlating isothermal compressed densities. Mostly the Tait equation is used for data below the normal boiling point with the density at atmospheric pressure as the reference point. Now, we developed a simple and flexible correlation system around the Tait equation for the whole liquid phase up to the critical point. For the correlation of the measured supercritical data a virial-type equation of state was employed. With the correlation equations developed the data can be described within the experimental uncertainties. 2. Experimental Section The density measurement with a vibrating tube is based upon the dependence between the period of oscillation of an unilaterally fixed U-tube and its mass. This mass consists of the U-tube material and the mass respectively the density of fluid filled into the U-tube. The behavior of the vibrating tube can be described by

10.1021/ie001135g CCC: $20.00 © 2001 American Chemical Society Published on Web 09/11/2001

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4471

Figure 1. Schematic diagram of the computer-controlled density measurement unit.

the simple mathematical-physical model of the undamped spring-mass system.1 The characteristic period of oscillation of this model is described by the following equation:

x

τ ) 2π

m0 + VF D

(1)

Rearrangement of the equation and substitution of the mechanical constants lead to the classical equation for vibrating tube densimeters:

F ) Aτ2 - B

(2)

The parameters A and B can be determined by substance calibration measuring the period of oscillation of at least two substances of known density. Unfortunately, the parameters A and B are highly temperature dependent and also pressure dependent. Therefore, the parameters must be determined for each temperature and pressure separately or, like in this work, the classical equation must be expanded with temperatureand pressure-dependent terms. For measurements over such a large range as in this work (273 to 623 K) and up to 40 MPa an extended calibration equation with 14 significant parameters is employed (see eq 3). Using more than 750 data points, a possible over-fitting is prevented. 2.1. Measurement Apparatus. The schematic assembly of the density measurement system developed is shown in Figure 1. The high-pressure high-temperature vibrating tube densimeter (DMA-HDT) is the essential part of the computer-controlled system. The DMA-HDT system consists of the measurement cell and a modified DMA 5000 control unit. The measurement cell contains the vibrating tube (Hastelloy C-276), the sensor and excitation coils, an electronic thermostat with cooling circuit (e.g., for air or water cooling), and two temperature sensors. The period of oscillation measurement and the temperature control is implemented within the DMA-HDT control unit. This control unit is connected to a PC via a serial port (RS232). A target temperature can be sent to the control unit and every second the current temperature and the period of oscillation is controlled by the PC. The vibrating tube unit is connected with highpressure pipes and valves (HIP 30000 psi series from HIP, Erie, PA) to a variable volume cell (dosage pump type 2200-802 from Ruska, Houston, TX). The piston of

the variable volume cell can be moved by a stepping motor (model RSH 125-200-10 from Phytron, Gro¨benzell, Germany) connected to a power/control unit (model ixe alpha-c from Phytron) which is also connected to a PC via a serial port. For the pressure measurement two pressure transducers (model PDCR 911 up to 20 MPa, model PDCR 922 up to 60 MPa, both from Druck, Leicester, England) can be used. A multimeter (model 2000 from Keithley, Cleveland, OH) with serial port is employed for the transformation of the pressure transducer measurement signal. To prevent undesirable temperature- and resulting density gradients in the vibrating tube region of the apparatus, a heating (aluminum block with heating cartridges) around the supply pipes next to the vibrating tube was installed. For the preheating a thermostat unit (model 2416 from Eurotherm, Heppenheim, Germany) is used. For measurements at temperatures from 273 K up to 313 K a thermostat (model RC GCP edition 2000 from Lauda, Lauda-Koenigshofen, Germany) with a water-ethanol mixture is attached to the mentioned cooling circuit of the vibrating tube unit. For evacuating the whole apparatus a vacuum pump (model RZ 2 from Vacuubrand, Wertheim, Germany, with sensor thermovac TM 20 from Leybold, Cologne, Germany) is employed. The vibrating tube unit can be disconnected from the variable volume cell with the help of a three-way valve. In this mode density measurements at atmospheric pressure are possible using small amounts of sample. Density measurements at atmospheric pressure were already presented by Ihmels et al.2 2.2. Measurement Range, Accuracy, and Calibration. The Pt100 temperature sensors installed show a resolution of (3 mK and an accuracy of (30 mK, while the thermostat has a stability of (20 mK. The period of oscillation is measured with a resolution of 1 ns and lies between 2200 and 2600 µs depending on density, temperature, and pressure. The observed reproducibility of the density measurements at atmospheric pressure and temperatures from 298 K up to 373 K is within (0.05 kg/m3. At higher pressures and temperatures hysteresis effects in the vibrating tube material limit the reproducibility to (0.1 kg/m3. The pressure sensors are designed for pressures up to 20 and 60 MPa, respectively, and the accuracy after

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calibration with a dead weight pressure gauge was estimated to be better than (2 and (6 kPa, respectively. The accuracy in the density measurement depends on the accuracy of pressure, temperature, and period of oscillation measurements and on the purity of the reference substances and accuracy of the reference densities used. For the calibration the period of oscillation of the two reference substances water and butane and the period of oscillation in a vacuum were used. Water was measured in the temperature range from 278 to 623 K and butane between 273 and 428 K (both in 5 K steps). Both substances were measured at pressures from ≈0.3 MPa above the vapor pressure up to 40 MPa (in 5 MPa steps). Moreover, the vacuum signal (density ) 0 kg/ m3) was measured between 273 and 623 K. Then, the parameters of a 14-parameter calibration equation (see eq 3) were fitted at the calibration points (temperature, pressure, and period of oscillation) to the reference densities calculated using reference equations of state from Pruss and Wagner4,5 for water and from Younglove and Ely6 for butane and the density zero (in a vacuum) by linear regression. The following equation was used to determine the densities from the measured period of oscillation at a given temperature and pressure. 2

F ) Aτ - B

(3)

with

A)

∑i aiTi + ∑j bjPj + cTP

B)

∑i diTi + ∑j ejPj + fTP

and

with

i ) 0, 1, 2, 3 and j ) 1, 2 For density measurements at atmospheric pressure in the temperature range between 298 and 373 K, a maximum error of (0.1 kg/m3 is obtained. For pressures up to 40 MPa and temperatures from 273 K up to 623 K the total error in the density measurement is estimated to be (0.3 kg/m3. In the moderate range between 298 and 523 K the maximum error is estimated to be (0.2 kg/m3. For the measured liquid densities between 400 and 1500 kg/m3 this leads to relative errors between (0.075 and (0.02%. In the compressed supercritical region only densities above 100 kg/m3 were measured. This leads to maximum relative errors of (0.3%. Because of the strong pressure dependence of the densities near the critical point, higher deviations result in this region. With the accuracy of about (0.006 MPa a maximum error in density of about (0.5% in the supercritical region near the critical pressure and an error of about (2% in the region near the critical point is estimated. 2.3. The Control Program. The control program “Densitas per MotumsDensity Measurement” developed in the first phase of our research work2 was extended for the newly applied hardware (e.g., thermostats, stepping motor control, and power control) and new features were added. With this software fully automized temperature-pressure measurement pro-

Figure 2. Main window of the density measurement control software.

grams can be realized. Figure 2 shows the main window of the program with all current system parameters as well as the measurement program. The current version of the program has an integrated connection to the Dortmund Data Bank (DDB) for pure component properties (DDB-Pure).7 With Antoine parameters from the DDB, dynamic measurement programs with pressures slightly above the vapor pressure can be realized automatically. In addition, the software checks the measurement conditions against the pure component properties (melting point, normal boiling point, critical properties, etc.) and detects possible problems (e.g., whether the final temperature is above the normal boiling point for measurements at atmospheric pressure). While the measurement program is running, all data can be (optionally) stored in an MS-Access database and/or a text file (Excel-CSV format). For a fully automized measurement system, flexible and easy control is required. For this purpose a software package with a service-module program for status and alarm e-mail (e.g., via SMS server to a mobile phone), status-Internet page, and service of a control-client program via the Internet were developed. The controlclient program with the same surface as the main control program enables the operator, for example, in his office to follow the status of the measurements in the laboratory and to interact if necessary. 2.4. Experimental Procedure. For the high-pressure density measurements carried out 75-100 cm3 of degassed liquid or compressed gas are required. Liquids are degassed by vacuum distillation using a Vigreux column with a height of 90 cm. The substances were injected into the evacuated apparatus from a flask (liquid) or a gas cylinder (compressed gas). The measurement programs run from the minimum temperature to the maximum temperature in defined intervals. At each temperature a preset pressure program is executed from minimum to maximum pressure in defined intervals. The control program sets the target temperature and pressure and waits for a defined time after constant system conditions have been reached. After recording

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Figure 3. Densities of toluene at temperatures between 273 and 623 K and pressures above the saturation pressure up to 30 MPa.

and storing the measured values with a specified number of repeats, the program starts the next measurement. After the maximum pressure has been reached, the pressure is reduced to the minimum pressure and the next temperature is set. At the end of the measurement program the temperature and pressure are decreased to the standby values. Complex measurement programs, e.g., with dynamic subcritical measurements (from above the saturation pressure up to 40 MPa) and static supercritical measurements (e.g., between 10 and 40 MPa) as for hydrogen sulfide and carbonyl sulfide, are realized in one run with a measurement program script file (MPS file). A temperature-pressure program between 273 and 623 K and pressures up to 40 MPa with ≈600 measurement points can be realized within 1 week. Atmospheric pressure measurements in a temperature range between 308 and 498 K with 40 experimental require 2 days. 2.5. Chemicals. The compounds used for the measurements are toluene (CH3C6H5, M ) 92.14 g/mol, CAS-RN 108-88-3), carbon dioxide (CO2, M ) 44.01 g/mol, CAS-RN 124-38-9), carbonyl sulfide (COS, M ) 60.07 g/mol, CAS-RN 463-58-1), and hydrogen sulfide (H2S, M ) 34.08 g/mol, CAS-RN 7783-06-4). Toluene (purity 99.99 vol %, checked by gas chromatography) was obtained from Aldrich (Taufkirchen, Germany) and stored over a 3 Å molecular sieve (water content: 7.0 ppm checked by Karl Fischer titration). Carbon dioxide (purity 99.995 vol % checked by gas chromatography) was purchased from Luebke (Oldenburg, Germany). Carbonyl sulfide and hydrogen sulfide (both purity 99.5 vol % checked by gas chromatography) were obtained from Praxair (Oevel, Belgium). 3. Results and Discussion In this work the densities of toluene, CO2, COS, and H2S in the sub- and supercritical state were measured. The measurements of toluene from 273 to 623 K starting above the saturation pressure up to 30 MPa and of CO2 from 283 to 373 K from 10 MPa up to 30 MPa are presented in Figures 3 and 4. The measurements for toluene and CO2 were mainly carried out to test the accuracy and suitability of the apparatus for the measurements of densities at sub- and supercritical conditions. Densities of toluene and CO2 have been measured by different researchers.7 Therefore, these data were compared with the calculated values using the reference equations of state from Goodwin8 and Span and Wag-

Figure 4. Densities of carbon dioxide at temperatures between 283 and 373 K and pressures between 10 and 30 MPa.

Figure 5. Relative deviations between experimental densities at pressures above the saturation pressure up to 30 MPa and the equation of state for toluene from Goodwin8 (Tc ) 591.7 K, Pc ) 4.11 MPa).

Figure 6. Relative deviations between the experimental densities measured in this work and the equation of state for carbon dioxide from Span and Wagner9 (Tc ) 304.1 K, Pc ) 7.38 MPa).

ner,9 respectively. Figure 5 shows the deviations between the experimental densities measured in this work, densities published by other authors,10-12 and the results of the equation of state for toluene from Goodwin. The relatively high deviations in the temperature range from 273 up to 333 K (maximum 0.3% at 273 K) is caused by a weakness of the equation of state for toluene. This is already the case for the data of Akhundov and Abdullaev10 and Straty et al.12, which were used by Goodwin together with other data for fitting the parameters of his equation of state. In Figure 6 the deviations between the experimental densities of CO2 and the equation of state from Span and Wagner9 are presented. In the liquid state and the supercritical state at higher pressures (>20 MPa), the deviations are lower than 0.05 and 0.1%, respectively. However, in the supercritical state the errors reach 0.2% for the densities at pressures below 20 MPa. When the

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parameters can simultaneously be fitted to temperaturedependent experimental density data. With the TRIDEN correlation program the temperature-dependent parameters of the Tait equation can be fitted to isothermal densities (as measured in this work). With the help of these parameters the saturation densities can be obtained at each temperature by extrapolation. The required saturation pressures are calculated with the Wagner equation using our correlation parameters of DDB-Pure7 (fitted to evaluated experimental vapor pressure data from different researchers). The saturation densities are then correlated with the Rackett equation. For nonisothermal data (mainly stored in the DDB-Pure data bank) the Rackett parameters are fitted to experimental saturation densities. Figure 7. Densities of carbonyl sulfide at temperatures from 273 to 623 K and pressures from the saturation pressure up to 40 MPa (above the critical temperature from 10 to 40 MPa).

Modified Rackett equation for saturation density: F0 ) AR/BR[1+(1-(T/CR))

D

R]

(4)

Wagner equation for vapor pressure: ln(P0) ) ln(Pc) + AW(1 - Tr) + BW(1 - Tr)1.5 + CW(1 - Tr)2.5 + DW(1 - Tr)5 Tr

(5) Tait equation for isothermal compressed densities:

[

F ) F0/ 1 - CT ln

Figure 8. Densities of hydrogen sulfide at temperatures from 273 to 548 K and pressures from the saturation pressure up to 40 MPa (above the critical temperature from 10 to 40 MPa).

critical pressure is reached, higher deviations are observed. Span and Wagner estimated uncertainties of 0.03-0.05% in the density for their equation of state. Figures 7 and 8 show the results for COS and H2S in the temperature range from 273 K up to 623 and 548 K, respectively, and pressures up to 40 MPa. The measured PFT data in the liquid phase of toluene, COS, and H2S were correlated with the TRIDEN model. Densities of CO2 and the densities in the supercritical phase of COS and H2S were correlated with a virialtype equation of state. All experimental values are available as Supporting Information. 3.1. Correlation of Liquid PGT data with the TRIDEN Correlation System. For the correlation of the measured compressed liquid densities our new flexible PFT correlation system TRIDEN was employed. TRIDEN stands for the three-dimensional (TRI) correlation of DENsities using the three well-known equations from Tait3 (compressed densities), Rackett13 (saturated densities), and Wagner14 (vapor pressure). The widely used Tait equation3 for isothermal compressed densities was combined with a modified Rackett equation (eq 4) for the liquid saturation densities and the Wagner vapor pressure equation in the 2.5,5-form (eq 5), as reference state (F0 and P0) required for the Tait equation (eq 6). The used Rackett equation is a further modification of the modified form suggested by Spencer and Danner.13 In this Rackett equation all four

(

BT + P BT + P0

)]

(6)

where for the parameter BT the following temperature dependence is used:

T T2 T3 BT ) b0 + b1 + b2 + b3 E E E

()

()

for the parameter CT a linear temperature dependence is used:

(ET)

C T ) c0 + c1

The flexibility of combining different independent equations (for compression, saturation density, and vapor pressure), the adaptability of the number of parameters for the temperature dependence of the Tait parameters BT and CT (e.g., CT may be assumed as a constant for a narrow temperature range), and the possibility of a reliable pressure and temperature extrapolation are the main advantages of this approach. Furthermore, the equations for the saturation density (e.g., polynomial of Tr) or the vapor pressure (e.g., Antoine equation) can be easily exchanged. If no vapor pressure equation is available or applicable, the reference pressure may be set to a constant value, for example, 1 MPa, and the reference density equation describes densities at this pressure. Below the normal boiling point the reference pressure is always 0.1 MPa if a vapor pressure equation is employed. Using these equations, it is possible to correlate the PFT data in the whole liquid state up to the critical point within experimental error. With the developed TRIDEN Excel-Add-In, a number of other properties, for example, isothermal compressibility, thermal expansion coef-

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4475 Table 1. Parameters for the TRIDEN Correlation Model for Toluene, Carbonyl Sulfide, and Hydrogen Sulfide: Temperature Range, Pressure Range, Number of Data Points, Tait Parameters, Rackett Parameters, Wagner Parameters, Critical Temperature, Critical Pressure and Absolute, RMSD, and Relative, RMSDr, Root-Mean-Square Deviations, and the Mean Deviation, bias, as Statistical Values for the TRIDEN Fita

Tmin Tmax Pmin Pmax Fmin Fmax data points c0 (Tait) c1 (Tait) b0 (Tait) b1 (Tait) b2 (Tait) b3 (Tait) E (Tait) A (Rackett) B (Rackett) C (Rackett) D (Rackett) RMSD RMSDr bias A (Wagner) B (Wagner) C (Wagner) D (Wagner) Tc Pc [kPa] a

toluene

carbonyl sulfide

hydrogen sulfide

273 583 0.474 30 466.1 903.6 448 0.079685 -1.45E-06 427.879 -175.867 23.9305 -1.10598 100 81.5414 0.273263 591.750 0.287628 0.5349 0.0770 -0.0176 -7.4070 1.8166 -2.1564 -3.5356 591.7 4113.8

273 368 2 40 699.0 1139.4 242 0.085794 -1.34E-06 350.447 -194.555 32.8760 -1.67972 100 98.4972 0.250994 378.913 0.259821 0.5463 0.05913 -0.0568 -6.8349 2.5199 -2.4992 -1.1595 377.95 6180.83

273 363 2.8 40 539.1 889.5 207 0.085246 -1.17E-06 277.861 -103.188 -0.314774 1.97517 100 77.7065 0.250323 371.408 0.255268 0.5832 0.08199 -0.0503 -5.9367 -0.090618 0.80577 -4.9487 373.2 8936.87

Figure 9. Relative deviations between experimental densities and the TRIDEN correlation for toluene (Tc ) 591.7 K, Pc ) 4.11 MPa).

Figure 10. Relative deviations between experimental densities and the TRIDEN correlation for carbonyl sulfide (Tc ) 377.9 K, Pc ) 6.18 MPa).

Units: K, MPa, and kg/m3.

ficient, or the pressure dependence of the molar heat capacity, can be calculated as well. Besides a deviation plot other statistical values are desirable to evaluate the correlation. The absolute, RMSD (eq 7), and relative, RMSDr (eq 8), root-meansquare deviations and the mean deviation, bias (eq 9), are utilized as statistical values for the TRIDEN fits.

RMSD )

x∑ 1

n

RMSDr ) 100

bias )

1 n

(Fexp - Fcalc)2 [kg/m3]

Figure 11. Relative deviations between experimental densities and the TRIDEN correlation for hydrogen sulfide (Tc ) 373.2 K, Pc ) 8.94 MPa).

(7)

n

x ∑( n

)

Fexp - Fcalc

1

n

Fexp

∑n (Fexp - Fcalc)

2

[kg/m3]

[%]

(8)

(9)

The relative root-mean-square deviation between the DDB-Pure correlation (from literature values) for saturated liquid densities of toluene and the calculation with TRIDEN (extrapolation to the saturation pressure) is 0.09% between 273 and 498 K. The TRIDEN parameters for the Tait equation, the Rackett equation, and the Wagner equation, the temperature and pressure range covered, and additional statistical values are given in Table 1. The units are K, MPa, and kg/m3 with the exception of the critical pressure for the Wagner equation which is used in kPa. For the aspect of a larger range of applicability, all data

were fitted together, except the density data measured between Tc - 10 K up to the critical temperature Tc. These data were omitted from the fitting procedure because of the larger experimental errors near the critical point. However, the wide temperature range covered results in a little lower accuracy of the correlation. In the narrow range from 273 to 373 K a relative root-mean-square deviation of 0.02%, instead of 0.08% for the wider temperature range (from 273 to 583 K) was achieved for toluene. The TRIDEN Excel-Add-In together with special parameters for the narrow range with higher accuracy is available as Supporting Information. In Figures 9-11 the relative deviations between experimental values and the correlation are shown. For toluene, COS, and H2S the deviations are usually within (0.1%. At the highest temperatures larger deviations (near Tc) are observed in all cases. The region below 293 K seems to be problematical for fitting the whole liquid range of toluene. Similar to the equation of state from Goodwin,8 the deviations are about +0.3% at 273 K.

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Table 2. Parameters for the Virial Equation for Carbon Dioxide, Carbonyl Sulfide, and Hydrogen Sulfide: Temperature Range, Pressure Range, Number of Data Points, Equation of State Parameters and Absolute, RMSD, and Relative, RMSDr, Root-Mean-Square Deviations, and the Mean Deviation, bias, of Density and Pressure as Statistical Values for the Fita

Tmin Tmax Pmin Pmax Fmin Fmax data points a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 RMSD (density) RMSDr (density) bias (density) RMSD (pressure) RMSDr (pressure) bias (pressure) a

carbon dioxide 283 373 10 30 4.312 23.157 97 0.0006477 0.32395218 52.05474 -794.87556 -84.209265 1.10 × 10-6 -0.00546645 4.2676144 -7.32 × 10-7 0.00099511 1.55 × 10-7 -0.00012445 2.15 × 10-6 0.01413981 0.10823353 1.91 × 10-5 0.0409715 0.24271685 -0.00014455

carbonyl sulfide 373 623 10 40 2.038 15.801 358 0.00150051 1.0648226 26.900028 -3835.0813 -14.254769 -0.00012753 0.07107329 3.9231224 8.77 × 10-6 -0.00214497 1.76 × 10-7 -0.0003651 1.40 × 10-5 0.00930664 0.17889102 1.00 × 10-6 0.0579453 0.22123425 0.00107488

hydrogen sulfide 368 548 10 40 2.387 21.248 257 0.00113414 0.74666131 26.91403 -2638.9406 -31.778448 -9.43 × 10-5 0.04767593 2.1119278 5.63 × 10-6 -0.00198079 -2.58 × 10-8 -4.69 × 10-5 2.10 × 10-6 0.01268429 0.20776002 -9.84 × 10-6 0.04331493 0.21785414 0.00263373

Figure 12. Relative deviations between experimental densities and the equation of state correlation for carbon dioxide (Tc ) 304.1 K, Pc ) 7.38 MPa).

Figure 13. Relative deviations between experimental densities and the equation of state correlation for carbonyl sulfide (Tc ) 377.95 K, Pc ) 6.18 MPa).

Units: K, MPa, and mol/L.

3.2. Correlation of Compressed Supercritical PGT Data with an Equation of State. A virial-type equation of state (eq 10) was employed for the correlation of the PFT data of CO2. Also, the PFT data of COS and H2S in the supercritical phase and around the critical point were correlated using this equation of state. The equation is a reduced version of the Benedict-Webb-Rubin-type Bender equation of state.15 In this equation the exponential term of the Bender equation was omitted and therefore the number of parameters was reduced from 20 to 13.

Virial-type equation of state: P ) TF‚[R + BF + CF2 + DF3 + EF4 + FF5] with the temperature-dependent functions:

B ) a1 -

a2 a3 a4 a5 T T2 T3 T4

C ) a6 +

a7 a8 + T T2

D ) a9 +

a10 T

E ) a11 +

a12 T

F)

a13 T

(10)

Figure 14. Relative deviations between experimental densities and the equation of state correlation for hydrogen sulfide (Tc ) 373.2 K, Pc ) 8.94 MPa).

The data were correlated by least-squares minimization using the following objective function (eq 11).

S)

∑i [(Fi,exp - Fi,calc)/Fi,exp]2

(11)

Table 2 gives the 13 parameters of the equation of state together with additional statistical values. The absolute, RMSD, and relative, RMSDr, root-mean-square deviations and the mean deviation, bias, were calculated for the density and for the pressure. The units are K, MPa, and mol/L. In Figures 12-14 the relative deviations between the experimental and the correlated values are shown. Deviations between the experimental values and the correlation are within the estimated experimental uncertainties. As expected at the critical point, higher deviations are observed in all cases. Higher deviations at lower pressures and high temperatures are obtained

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4477

for COS and H2S than for CO2. It is strictly recommended the equations only within the temperature and pressure range of correlation be used. For carbon dioxide the isothermal compressibilities (eq 12) calculated with the equation of state were compared with the results obtained using the reference equation of state by Span and Wagner.9

χ)-

1 ∂υ υ ∂P T

( )

(12)

The result was a tolerable relative root-mean-square deviation of 1.46% with a maximum relative deviation of 6.7% near the critical point. 4. Summary and Outlook For the measurement of liquid densities and densities in the compressed supercritical state up to 623 K and 40 MPa, a new computer-controlled high-temperature high-pressure vibrating tube densimeter was developed. The compressed liquid densities and compressed supercritical densities of toluene, carbon dioxide (CO2), carbonyl sulfide (COS), and hydrogen sulfide (H2S) were measured and correlated. Up to now, no density data for COS and H2S covering this wide temperature and pressure range were published. For the correlation of compressed liquid densities for toluene, COS, and H2S, the new correlation system TRIDEN (a combination of well-known equations) was applied. Apparently, vibrating tube densimeters have not yet been used for measurements in the supercritical state. The density measurements are realized fully automized using defined temperature-pressure programs. With respect to accuracy, reliability, suitability, and time consumption, this system represents an optimum for measuring PFT-properties. The system is specialized for the measurement of pure component densities with the integrated connection to the DDB-Pure component properties database. Currently, the system is modified to measure also densities of mixtures for the determination of excess volumes as a function of temperature and pressure. For this purpose the TRIDEN correlation system will be extended for fitting the temperature- and pressure-dependent densities of mixtures and excess volumes, respectively. Acknowledgment The authors would like to thank the “Labor fu¨r Messtechnik Dr. Hans Stabinger” (Graz, Austria) for supplying the DMA-HDT prototype and the “Max BuchnerForschungsstiftung” for financial support of this work. Supporting Information Available: All experimental values are available as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org. Additionally, the mentioned TRIDEN Excel-Add-In (for Excel 97 or higher) may be obtained from the authors. Nomenclature a1...a13 ) parameter of the equation of state a...f ) parameter of the extended vibrating tube equation A, B ) parameter of the basic vibrating tube equation AR...DR ) parameter of the modified Rackett equation AW...DW ) parameter of the Wagner equation BT, bx ) parameter of the Tait equation (MPa)

CT, cx ) parameter of the Tait equation D ) spring constant (N/m) E ) parameter in the Tait equation for reducing the temperature (K) m0 ) mass of the empty U-tube (kg) n ) number of data points P ) total pressure (MPa) P0 ) pressure at reference state (Tait equation) (MPa) Ps ) saturation pressure (MPa) Pc ) critical pressure (MPa) R ) general gas constant (R ) 0.008314472 MPa‚L/mol‚ K) T ) absolute temperature (K) Tc ) critical temperature (K) Tr ) reduced temperature V ) volume of the vibrating tube (m3) π ) constant ()3.14159265) F ) mass density or molar density (EOS) of a fluid (kg/m3 or mol/L) F0 ) density at the reference state (Tait equation) (kg/m3) τ ) period of oscillation of the vibrating tube (µs)

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Received for review December 30, 2000 Revised manuscript received June 22, 2001 Accepted July 11, 2001 IE001135G