Volumetric Properties and Solubility Parameters of Cyclohexane +

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Volumetric Properties and Solubility Parameters of Cyclohexane + CO2 Mixtures at High Pressures and their Modeling with the Sanchez-Lacombe Equation of State Michael L Williams, James S Dickmann, John C. Hassler, and Erdogan Kiran Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01287 • Publication Date (Web): 29 Jun 2017 Downloaded from http://pubs.acs.org on July 1, 2017

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Industrial & Engineering Chemistry Research

Volumetric Properties and Solubility Parameters of Cyclohexane + CO2 Mixtures at High Pressures and their Modeling with the Sanchez-Lacombe Equation of State Michael L. Williams, James S. Dickmann, John C. Hassler, Erdogan Kiran* Department of Chemical Engineering Virginia Tech, Blacksburg, VA 24061 Abstract Densities of mixtures of cyclohexane and carbon dioxide containing 0 to 50 wt % CO2 were determined using a variable-volume view-cell at pressures ranging from 5 to 35 MPa and temperatures ranging from 313 to 393 K covering a density range from 0.55 g/cm3 to 0.83 g/cm3 . The data were modeled with the Sanchez-Lacombe (S-L) equation of state. Modeling was conducted both by treating each mixture as a pseudo-pure compound and by using the S-L parameters for the pure components with a composition- and temperature-dependent interaction parameter. Various thermodyna mic properties for the mixtures such as the isothermal compressibility, isobaric expansivity, interna l pressure, excess volumes, and solubility parameters were then evaluated. Mixture excess volumes were also determined, and were modeled with Redlich-Kister type equations. The mixture density data were also modeled using the S-L parameters for pure components and a composition- and temperaturedependent interaction parameter. Isothermal compressibilities ranged from 0.0008 MPa-1 for pure cyclohexane at 313 K and 35 MPa to 0.013 MPa-1 for the mixture containing 50 wt % CO 2 at 393 K and 20 MPa. Isobaric expansivities ranged from 0.0008 K-1 for cyclohexane at 35 MPa and 313 K to 0.0035 K -1 for the 50 wt % CO 2 mixture at 20 MPa and 393 K. Internal pressures ranged from 83.8 MPa for 50 wt % CO2 mixture at 20 MPa and 393 K to 275.5 MPa for cyclohexane at 35 MPa and 313 K. Solubility parameters ranged from 16.6 MPa0.5 for cyclohexane at 35 MPa and 313 K to 9.2 MPa0.5 for 50 wt % CO2 mixture at 20 MPa and 393 K. The mixtures showed a high degree of non-ideality with excess volumes ranging from about −20 cm3 /mol for 50 wt % CO 2 mixture at 20 MPa and 393 K to +2 cm3 /mol also for the 50 wt % CO 2 mixture at 20 MPa, but at 313 K.

______ Corresponding Author. Erdogan Kiran, email: [email protected]

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1. Introduction Compressible fluids such as carbon dioxide and its mixtures with organic solvents have been studied extensively as tunable solvents in various polymer applications ranging from polymer synthesis to polymer modification or recovery1-7 . For example, homogeneous polymerization in carbon dioxide proceeds in a mixture of carbon dioxide and the monomer that has not yet polymer ized until the miscibility boundary is crossed and the newly formed polymer undergoes phase separation. As a second example, in particle formation processes a polymer solution in a traditional organic solvent is injected into carbon dioxide, or vice versa, to induce phase separation. Among the specific mixtures that have been investigated in our laboratory are mixtures of carbon dioxide with ethanol8 , acetone9 , toluene10 , pentane11 , and more recently ethyl acetate12 . Among other mixtures that have been reported in the literature are mixtures of carbon dioxide with various hydrocarbons including but not limited to heptane13 , decane14-16 , undecane17 and tridecane13 . We are now reporting comprehensive data on the density and derived thermodynamic properties of mixtures of carbon dioxide with cyclohexane. Cyclohexane is a well-known solvent for a variety of polymers18-20 . Our interest in its mixtures with carbon dioxide is linked to the potential utility of these mixtures as tunable process or processing fluids for polyolefins20,21 . However, an extensive search of the literature showed that even though there have been several publications on the phase behavior of mixtures of CO 2 and cyclohexane, their modeling and their mixture critical line22-26 , there are essentially no prior studies on their volumetr ic properties. A recent study provides only a comparative evaluation of the modeling of the phase behavior with different cubic equations of state26 . The phase behavior of these mixtures is viewed as being Type I with the vapor-liquid critical curve being continuous between the critical points of the pure components. Critical pressure goes through a maximum around 16 MPa at around 400 K and about 40 wt % CO2 20,26 . The density determinations in the present study have been carried out in a dual-piston variablevolume view-cell in which the positions of two movable pistons are continually monitored to generate nearly continuous density profiles during a pressure scan at a given temperature. The extensive data sets that are obtained provide a unique opportunity for the development of descriptive models for density that can then be used for the evaluation of the derived thermodynamic properties.


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A common approach to modeling density as a function of pressure and temperature is to use the empirical Tait equation27-32 . Even though the Tait equation has been in use for more than 100 years27 and can provide reliable descriptions, full descriptions require a large number of correlation parameters which are merely fitting parameters and lack physical meaning. For example, a Tait equation-based correlation that is provided in the literature for the description of the density of cyclohexane involves six parameters32 . In the present study we use the Sanchez-Lacombe equation of state (S-L EOS) to model the density of cyclohexane, carbon dioxide, and their mixtures. The S-L EOS is a lattice-fluid model which was originally developed for the description of polymer systems33,34 . However, it has been shown to also be effective at describing non-polymeric systems at high pressure35,36 . The basic equation of this lattice fluid model is given by: 1 𝜌̃ 2 + 𝑃̃ + 𝑇̃ (𝑙𝑛(1 − 𝜌̃) + (1 − r ) 𝜌̃) = 0

(1)

where 𝜌̃, 𝑃̃, and 𝑇̃ are the reduced density, pressure, and temperature, respectively, and 𝑟 stands for the number of lattice sites occupied by a molecule. These reduced parameters are further defined by:

𝜌̃ =

𝜌

𝑃̃ =

𝜌∗

𝑃 𝑃∗

𝑇̃ =

𝑇

(2)

𝑇∗

in which 𝜌 ∗ , 𝑃 ∗and 𝑇 ∗ are, respectively, the characteristic density, pressure, and temperature for the given system. The characteristic parameters are defined by: 𝜀∗

𝑃 ∗ = 𝑣∗

𝑇∗ =

𝜀∗ 𝑅

𝑀 𝑃∗

(3)

𝑟 = 𝑅 𝑇 ∗ 𝜌∗

in which 𝜀 ∗ is the interaction energy per mer, 𝑅 is the ideal gas constant, 𝑀 is the molecular weight, and 𝑣 ∗ is the characteristic volume. Once a given system is modeled with the S-L EOS, other thermodynamic properties such as the isothermal compressibility 𝜅, isobaric expansivity 𝛽, and internal pressure 𝜋 can be readily derived using the following equations33,35,37 : 3 ACS Paragon Plus Environment


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1

𝜕𝑉

1

𝑇

1

𝑇

𝜕𝑉

𝑃̃ 𝑣̃2

𝜕𝜌

𝜅 = − 𝑉 (𝜕𝑃 ) = 𝜌 (𝜕𝑃 ) =

1

1 1 ) +( )]−2) 𝑃( 𝑇̃𝑣̃ [( ̃ −1 𝑣 𝑟

1+𝑃̃𝑣̃ 2

𝜕𝜌

𝛽 = 𝑉 (𝜕𝑇 ) = − 𝜌 (𝜕𝑇) = 𝑃

𝑃

𝑇( 𝑇̃𝑣̃ [(

1 1 )+( )]−2) ̃−1 𝑣 𝑟

𝛽

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(4)

(5)

(6)

𝜋 = 𝑇 (𝜅 ) − 𝑃

Another derived property of significance is the solubility parameter, which is of great importance when selecting or designing a solvent system for a polymer. The solubility parameter, 𝜎, is related to the cohesive energy density, which is defined as the internal energy per unit volume through:

𝜎 2 = 𝐶𝐸𝐷 =

𝛥𝑈 𝑉

=

∆𝐻𝑣 −𝑅𝑇

(7)

𝑉

However, information on the solubility parameters of fluids and fluid mixtures at high pressures is lacking. Here, the internal pressure provides a reasonable estimate of the solubility parameter29,38 . Thermodynamically, internal pressure is: 𝜕𝑈

𝜕𝑃

𝛽

𝜋 = (𝜕𝑉 ) 𝑇 = 𝑇(𝜕𝑇 )𝑉 − 𝑃 = 𝑇 (𝜅 ) – 𝑃

(8)

and, thus,

𝛥𝑈

1

𝜕𝑈

1

1

𝜎 = ( 𝑉 )2 ≈ (𝜕𝑉 )2 = 𝜋 2

(9)

This relationship provides a reasonable approximation even though internal pressure is not exactly cohesive energy density, which is the internal energy per unit volume, but rather the change in internal energy upon an infinitesimal change in volume38 . An alternative and equivalent way to express the solubility parameter based on a lattice-fluid model is given by39 : 4 ACS Paragon Plus Environment

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1

𝜌 𝜎 = (𝜌∗ ) (𝑃 ∗ )2

(10)

In this manuscript we report extensive data on the density and the derived thermodyna mics properties of mixtures of CO 2 and cyclohexane such as the isothermal compressibility isobaric expansivity, internal pressure, and excess volume. Additionally, we report for the first time the solubility parameters of these mixtures as a function of pressure from 5 to 35 MPa and as a functio n of composition up to 50 wt % CO 2 over temperatures from 313 to 393 K. The mixtures were modeled using the Sanchez-Lacombe EOS employing two differe nt approaches. Modeling was conducted by either treating each mixture as a pseudo-pure compound, or by using the S-L parameters for the pure components and a composition- and temperature-depende nt interaction parameter. A comparative discussion of the two approaches with respect to compositio na l variation of the characteristic parameters is provided. Mixture excess volumes and their variation with temperature, pressure, and composition were modeled with Redlich-Kister type equations. 2. Experimental Section 2.1. Experimental system Figure 1 is a diagram of the high-pressure variable-volume dual-piston view cell which can be used to determine the density and assess the phase state of fluids and mixtures. The details of the system have been previously described7 . Briefly, it consists of two pairs of sapphire windows; one pair (TLW) is for visual observations or for the measurement of transmitted light intensity, while the other pair (SLW) with a narrow gap between the windows is for the measurement of scattered light intensities. Measurements of the scattered light intensities as a function of the scattering angle and their evolution with time are used for the assessment of the mechanism of phase separation, which is employed when working with polymer solutions; this functionality was not used for these experime nts. The cell is equipped with a magnetically coupled stirring shaft (mixer) for effective mixing of the cell contents. The cell is also equipped with two-variable volume parts (VVP1 and VVP2) which incorporate two movable pistons (mp1 and mp2). The positions of the pistons, and thus the interna l volume of the cell, are altered by the use of external pressure generators (PGN1 and PGN2). 5 ACS Paragon Plus Environment


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The real-time positions (PPS1 and PPS2) of the pistons, and thus the internal volume of the cell, are tracked with two dedicated linear variable differential transformers (LVDT1 and LVDT2). Knowing the initial mass loading of the fluids (charged using a balance with an accuracy of ± 0.005 g) and the internal volume at any given pressure and temperature allows for the assessment of the density of the cell’s contents. The dual-piston arrangement becomes especially useful when mixing very viscous systems (for examples polymer solutions at high concentrations) by moving the whole content of the cell back and forth across the mixer. The cell is heated with cartridge heaters that are uniformly distributed on the main cell body as well as on the variable-volume parts. The external temperature of the cell and the variable-vo lume parts is monitored at multiple locations to ensure uniform heating. The temperature and pressure in the cell are monitored with a Dynisco diaphragm-type pressure transducer that is equipped with an internal J-type thermocouple. The pressure measurements have an accuracy of ± 0.1 MPa, and the temperature measurements have an accuracy of ± 0.5 o C. The piston positions, cell temperature, cell pressure, and transmitted light intensity are all recorded in real time with a dedicated computer.

Figure 1. High-pressure variable-volume dual-piston view-cell7 .

2.2. Materials Cyclohexane with 100% purity was purchased from Sigma-Aldrich, and was used as received. Carbon Dioxide (certified 99.9995% pure) was purchased from Airgas and used without further purification. 2.3. Operational procedure


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Before loading the fluids, the air in the cell is removed by the application of a gentle vacuum. The tubing connections up to the cell are then primed, and the cell is charged with cyclohexane using a HPLC-type liquid pump from a liquid transfer vessel set on a Sartorius balance with 0.01 g readability. The exact mass of cyclohexane loaded to the cell is recorded. The cell is then charged with the target mass of CO 2 from a small pre-pressurized CO2 transfer vessel that is weighed on the same balance. The actual mass loading of CO 2 is recorded. The system is then heated to the target temperature at which the density determinations are to be carried out. Once the target temperature is reached the pistons are moved via the pressure generators to bring the pressure to a value between 5 and 10 MPa, depending on the mass percent of CO 2 in the loading. The mixer is then turned on, and the cell is allowed to equilibrate at that pressure and temperature for half an hour. The pressure is then slowly increased to the desired value and then decreased back to the initial pressure; during this process the piston positions, pressure, and temperature are recorded at a rate of 3 data points per second. Piston position readings are converted to internal volume and then to density at any given time or temperature and pressure. After the system is returned to the starting pressure, the temperature is raised to the next isotherm, in this case 333K, and allowed to equilibrate again. This process is repeated for each isotherm. An example of such data is illustrated in Figure 2 for a mixture of cyclohexane containing 10 wt % CO 2 . The figure shows the change in pressure (initial increase from 4.75 MPa to 35 MPa followed by decrease to 5.35 MPa at a pressure change rate of about 0.15 MPa/s), the associated volume (initial decrease from 98.6 cm3 to 92.2 cm3 , followed by an increase to 97.8 cm3 ), and temperature (which remains essentially constant at 313 K) with time. The figure also shows the change in density with pressure, which is marked as the isotherm at 313 K as the temperature remains essentially constant during the experiment. The density isotherm represents about 1200 data points (about 600 data points in each direction) that appear as continuous curves. The density data show a slight hysteresis between the increasing-volume and decreasingvolume paths of data collection. The effect of this hysteresis is small; for example, the 10% CO2 mixture shown in Figure 2 shows a difference of .04% between the increasing- and decreasing-vo lume paths at 20 MPa. It should be noted that the hysteresis effects can be reduced if the rate of pressure change is reduced. Even though pressure scans in all experiments were conducted in both directions, for consistency, in the remaining part of the manuscript only the data generated along the increasing pressure paths have been used. 7 ACS Paragon Plus Environment


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Density (g/cm3 )

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200 150 100

Decreasing P

0.78

0.76 Increasing P

0.74

50

313 K

313 K

0

0.72 0

100

200

300

400

0

Time (s)

10

20

30

40

Pressure (MPa)

Figure 2. Real time recording of pressure, volume, temperature and the corresponding density isotherm as a function of pressure for 10 wt % CO2 + 90 wt % cyclohexane mixture at 313K.

2.4. Accuracy of the density determinations Figure 3 shows the density isotherms at 313, 333, 353, 373, and 393 K for pure cyclohexa ne and compares the values with those reported in the literature.


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While there is no prior density data on mixtures of cyclohexane and CO 2 for comparison, density

for pure cyclohexane

is available

in the online

NIST database and in several

publications29,32,40,41,42 . Specific data have been reported in the temperature range from 283-323 K up to 80 MPa40 , and in the temperature range from 318 to 413 K up to 65 MPa32 . A more recent publication41 reports density values in the temperature and pressure ranges of 283 to 333 K and 0.1 to 20 MPa. Low pressure (below 8 MPa) density values have been reported at temperatures in the range from 293 to 473 K 42 . Figure 3 shows that the present density values for cyclohexane are slightly higher but in very close agreement (within 1.1%) with the NIST data. At 313 K, the difference is 0.4% at 5 MPa and 0.5% at 35 MPa. At 393 K the difference is slightly larger, being 1 % at 5 MPa and 1.1% at 35 MPa. Density data reported at 313 K by Sun40 and Vega-Maza41 and at 333 K by Amorim32 are in close agreement with the present data at these temperatures. Even though we do not report density data at 303 K, the data by Verdier and Endersen29 is also included in the figure, and is consistent with the direction that is suggested by the present data.

0.8

Density (g/cm3 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.78

303 K

0.76

313 K

0.74

333 K

0.72

353 K 373 K

0.7

0% CO 2

393 K

0.68 0

10

20

30

40

Pressure (MPa) Figure 3. Density isotherms for pure cyclohexane at 313, 333, 353, 373, and 393 K (solid colored curves) and their comparison with density data from NIST (dashed black curves), and density data at 303 K (blue diamonds 29 ), at 313 K (green squares 40 , orange diamonds 41 ) and at 333 K (brown diamonds 32 ).



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2.5 Reproducibility of the density data To ensure the reproducibility and the reliablility of the data, several runs were replicated in their entirity starting with fresh loadings of the view cell. Density data for pure cyclohexane and for a mixture containing 10 wt % CO 2 were generated at all five isotherms. The data were collected over the same timescales (with respect to the rate of change in pressure) for all sets of experiments, and all equilibration times were the same. Mass loading differences were less than 1% in the repeat experiments. The results were highly reproducable. Comparisions of the duplicate runs are provided in the Supporting Information. 3. Results and Discussion Density data were generated for mixtures containing 0, 10, 20, 30 40 and 50 wt % CO 2 at 313, 333, 353, 373 and 393 K. The data were first modeled with the S-L EOS treating each mixture as a pseudo-pure compound. Instead of Tait equation type modeling, these modeled equations were then used to generate derived thermodynamic properties and solubility parameters. The modeling of the mixture density data was also carried out using the S-L EOS parameters for the pure components, CO2 and cyclohexane, and a mixing rule that involves a composition- and temperature- dependent interaction parameter. 3.1. Compositional Dependence of the Density Isotherms Figure 4 shows the compositional dependence of the density for these mixtures at 313, 333, 353, 373 and 393 K. At a given temperature, below a certain pressure, the addition of CO 2 to cyclohexane leads to a reduction of density, which becomes greater with increasing CO 2 addition levels. However, compared to cyclohexane, mixtures containing carbon dioxide show a greater rate of increase in density with pressure, reflecting their higher compressibility. As a result, the density trends are reversed after a crossover point and the mixtures become denser than pure cyclohexane. As shown in the figures, the densities become lower at higher temperatures, and furthermore the crossover pressure shifts to higher pressures at higher temperatures (going from about 19 MPa at 313 K to 25 MPa at 333 K and 34 MPa at 353 K) and occurs at slightly lower densities. It should be noted that at 10 ACS Paragon Plus Environment

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373 and 393 K the crossover pressure is not fully reached by 35 MPa. Such crossover phenomena have been reported for other mixtures of carbon dioxide with organic solvents8,9,11,16,17 .


0.85

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0.8

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0%

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10%

0.7 20%

0.65

30%

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0.6

10%

0.7 0.65 20% 30% 40%

0.6

313 K

50%

0.55 10

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40%

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0.85 0.8

Density (g/cm3 )

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0.75 0.7

0%

10%

0.65

20%

0.6

30% 40%

393 K

50%

0.55 0

10

20

30

40

Pressure (MPa) Figure 4. Density isotherms for mixtures of cyclohexane and CO2 containing 0, 10, 20, 30, 40 and 50 wt % CO2 at 313, 333, 353, 373 and 393 K.

3.2. Modeling the density data with the Sanchez-Lacombe EOS Figure 5 shows the density isotherms at 313, 333, 353, 373, and 393 K for pure cyclohexa ne and mixtures containing 10, 30, and 50 wt % CO 2 . The data for all compositions are presented in the Supporting Information. All the density data generated for each mixture were fitted to the S-L EOS using a least square regression PythonT M program in the pressure range where the mixture is a singlephase solution. The mixtures were first treated as pseudo-pure fluids. The molecular weights for the mixtures used in Equation 3 were calculated from the molecular weights of carbon dioxide (𝑀1 ) and cyclohexane (𝑀2 ) using either the weight fractions (𝑤𝑖) as 1 𝑀𝑚𝑖𝑥

𝑤

𝑤

(11)

= 𝑀1 + 𝑀2 1

2

or, using the mole fractions (𝑥 𝑖) as (12)

𝑀𝑚𝑖𝑥 = 𝑥 1 𝑀1 + 𝑥 2 𝑀2



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which are equivalent. The phase state of the mixtures was assessed by both the transmitted light intensity data and visual observations. Full homogenous conditions were reached at increasingly higher pressures for mixtures with higher CO 2 content. While fully homogenous conditions were attained above 7 MPa for 10% CO 2 , pressures greater than 17 MPa were needed for the mixture containing 50 wt % CO 2 . As noted earlier, the vapor-liquid critical curve for mixtures of carbon dioxide + cyclohexane is reported to be continuous and the critical pressure goes through a maximum around 16 MPa at about 400 K corresponding to about 40 wt % cyclohexane in the mixture20,26 . The S-L model fits that were generated for each system are also included in Figure 5. They are represented as black dotted curves. They cover the full pressure range for pure cyclohexane, but only the fully homogenous solution domains for the other mixtures at each temperature. Figure 6 shows the density data for carbon dioxide from the NIST database and the S-L model fits that were generated in the present study. The S-L EOS parameters for cyclohexane, carbon dioxide and their mixtures when treated as pseudo-pure compounds are given in Table 1. The S-L EOS fits all had correlation coefficients R2 > 0.99.


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40

Pressure (MPa)

Figure 5. Density isotherms at 313, 333, 353, 373, and 393 K for pure cyclohexane and for mixtures containing 10, 30, and 50 wt % CO2 . Figures show the experimental data (solid colored curves) and their respective Sanchez-Lacombe EOS model fits (black dotted curves).



1 313 K

0.9

333 K

0.8

Density (g/cm3 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 47

0.7

353 K 373 K

0.6

393 K

0.5 0.4 0.3 0.2 0.1 0 0

10

20

30

40

Pressure (MPa) Figure 6. Density isotherms at 313, 333, 353, 373, and 393 K for pure CO2 . Figures show the NIST data (solid colored curves) and their respective Sanchez-Lacombe EOS model fits (black dotted curves).

Table 1. S-L EoS parameters for pure cyclohexane, mixtures of cyclohexane and CO 2 , and pure CO2 . In generating these parameters, the mixtures were treated as pseudo-pure fluids.

Mass % CO2

P* (MPa)

T* (K)

ρ* (g/cm3 )

r (# of mers)

ε* (J/mol)

υ* (cm3 /mol)

R2

0

329.74

556.02

0.86392

6.9483

4623.0

14.020

0.998

10

234.38

557.94

0.86733

4.4925

4639.0

19.793

0.996

20

196.62

522.51

0.89252

3.6091

4344.4

22.095

0.997

30

188.25

508.19

0.90874

3.2395

4225.3

22.446

0.996

40

218.40

463.19

0.94775

3.6894

3851.2

17.634

0.998

50

216.63

439.80

0.99121

3.4544

3656.7

16.880

0.999

100

415.74

340.58

1.4093

4.5849

2831.7

6.8112

0.998

The SL-EOS parameters have been previously reported in the literature for both pure cyclohexane18,19 and carbon dioxide43-48 . There are differences between our parameter values and the literature, as well as between the values reported in the literature. Different values that have been reported are listed in Table 2. The differences often reflect the T/P range of the data used to determine 16 ACS Paragon Plus Environment

Page 17 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the parameters. We have compared the predictions of the SL-EOS using the parameters determined in the present study and one set of the parameters given in the literature for cyclohexane 18 and for CO244 with respect to their ability to describe the density data generated in the present study for cyclohexa ne , and the density data for CO 2 given in the NIST database. The results are shown in Figures 7a and 7b. As can be assessed, the parameters determined in the present study describe the density data for cyclohexane and CO 2 better than the parameters that have been published in the literature over the temperature and pressure range covered by the present study. Table 2. Comparison of SL-EOS parameters with values reported in the literature 18,19,33,43-48 .

Cyclohexane

CO2

Reference

P* (MPa)

T* (K)

ρ* (g/cm3 )

Present Study

329.74

556.02

0.86392

Haruki18 ; Sanchez33

383

497

0.902

Sun 19

391.0

517.0

0.917

Present Study

415.74

340.58

1.4093

Kiszka43 ; Cao 47 ; Erkey 48

567

305

1.510

Sato 44 ; Liu 45

720.3

269.5

1.580

Chen 46

611

278.5

1.413



0.81 313

Density (g/cm3 )

0.79

333

0.77

353

0.75

373 393

0.73

0.71 0.69

Cyclohexane 0.67 0

10

20

30

40

Pressure (MPa) Figure 7a. Comparison of experimental density data for cyclohexane obtained in the present study (colored curves) and given in the NIST database (dashed black curves) with S-L EOS predictions using the parameters determined in the present study (filled circles) and given in the literature 18 (open circles) at 313, 333, 353, 373 and 393 K.

1

0.9 0.8

Density (g/cm3 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 47

0.7 0.6 0.5 0.4

0.3 0.2

313 K

0.1

333 K

CO2

0

0

10

20

30

40

Pressure (MPa) Figure 7b. Comparison of the NIST density data for CO2 (colored curves) with S-L EOS predictions using the parameters determined in the present study (filled circles) and given in the literature 44 (open circles) at 313 and 333 K.


Page 19 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.3. Derived Thermodynamic Properties 3.3.1. Isothermal Compressibility Figure 8a shows the isothermal compressibility for pure cyclohexane and for the 50 wt % CO2 mixture at different temperatures as calculated from Equation 4; the data at all compositions is presented in the Supporting Information. At each temperature, as expected the compressibilities decrease with increasing pressure, and at a given pressure they increase with increasing temperature. Figure 8b shows the compressibilities as a function of CO 2 content at two different temperatures, 313 and 393 K. At 313 K the compressibility is around 1 x 10-3 MPa-1 for cyclohexane. It increases almost by a factor of 5 for the mixture containing 50 wt % CO 2 . At 393 K, the compressibility for cyclohexa ne nearly doubles to about 2 x 10-3 MPa-1 . It increases by about a factor of 7 for the mixture containing 50 wt % CO2 . Compressibilities for pure CO 2 are orders of magnitude greater, and were therefore not included in the plots. The literature values for isothermal compressibility for cyclohexane at 303 K range from 1.26 x10-5 MPa-1 to 0.86 x 10 -5 MPa-1 over the range of pressures from 0.1 MPa to 30 MPa29 . At 313 K, literature values range from 1.3 x 10-3 to 0.75 x10-3 over a pressure range of 5 to 65 MPa32 . Sun40 reports tabulated values at 333 K which range from 1.19 x10-3 MPa-1 at 5 MPa to 0.691x10-3 at 65 MPa. Figure 8c compares the present values of compressibility of cyclohexane obtained by S-L modeling of the density data with the values reported in references 32 and 40. There is high degree of similarity. 3.3.2 Isobaric Expansivity Figure 9a shows the isobaric expansivity for cyclohexane and for the 50 wt % CO 2 mixture as a function of temperature at selected pressures as calculated from Equation 5; the data at all compositions is presented in the Supporting Information. The expansivities increase with temperature but decrease with pressure. Figure 9b compares the variation of isobaric expansivities at 20 and 35 MPa for cyclohexane and for mixtures with 10, 20, 30, 40 and 50 wt % CO 2 . As shown the expansivities increase with CO 2 content of the mixture and nearly double in going from cyclohexa ne



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 47

to a mixture containing 50 wt % CO 2 at 313 K, and nearly triple at 393 K. For pure CO 2 , like compressibility, expansivity values are orders of magnitude greater and are not included in the plot. Similar to compressibility, thermal expansivities for cyclohexane has been reported in the literature at 303 K for a pressure range up to 30 MPa over a pressure range from 5 to 65 MPa

32 ,

29 , at

temperatures ranging from 218 to 413 K

and at temperatures ranging from 288 to 323 K over a

pressure range from 0.1 to 85 MPa 40 . At 20 MPa the expansivities are shown to vary from about 1.1 x10-3 K-1 at 315 K to about 1.2 x10-3 K-1 at 415 K

29 .

Figure 9c compares the present result for

cyclohexane with values reported in the literature at 20 MPa over a range of temperatures and at 313 and 333 K at different pressures. The literature values are slightly higher, but they are in the same order of magnitude. 3.3.3. Internal Pressure The internal pressures of the mixtures were evaluated using Equation 6. The results are shown in Figures 10a and 10b. Figure 10a shows that in a given mixture, internal pressure decreases with increasing temperature and increases with increasing pressure. Figure 10b which compares the interna l pressure values at 313 and 393 K for different compositions shows that at a given temperature, interna l pressure decreases with increasing CO 2 content of the mixture with some apparent overlap for the 30 and 40 wt % CO 2 mixtures. The internal pressure data at all compositions is presented in the Supporting Information. Figure 11 shows the internal pressure for CO 2 that was evaluated with the SL-EOS model using the density from the NIST database. The internal pressure, as discussed earlier in the manuscr ipt, provides a reasonable estimate of the solubility parameter for each fluid and mixture, which are discussed in the following section.


0% CO2

393 K

0.0025

Isothermal Compressibility (MPa-1)

0.003

373 K

0.002

353 K

0.0015

333 K 313 K

0.001 0.0005 0 0

10

20

30

0.015

50% CO2

393 K

0.012 373 K

0.009 353 K 333 K

0.006

313 K

0.003

0

40

0

10

Pressure (MPa)

20

30

40

Pressure (MPa)

Figure 8a. Variation of isothermal compressibility with pressure for cyclohexane (0% CO2 ) and for the mixture with 50 wt % CO2 at 313, 333, 353, 373 and 393 K.

0.005 50%

0.004 30%

0.003



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



Page 21 of 47

313 K

40%

20%

0.002

10%

0% CO2

0.001

0 0

10

20

30

40

0.015

50%

393 K

0.012 30%

40%

0.009 20%

0.006 10%

0.003

0% CO2

0 0

Pressure (MPa)

10

20

30

40

Pressure (MPa)

Figure 8b. Comparison of the variation of isothermal compressibility with pressure for cyclohexane (0% CO2 ) and for mixtures containg 10, 20, 30, 40 and 50 % CO2 at 313 and 393 K.


Page 22 of 47

0.002

Cyclohexane 0.0015 333 K

0.001 313 K

0.0005

0 0

10

20

30

40

Pressure (MPa) Figure 8c. Comparison of the isothermal compressibility for pure cyclohexane generated from the S-L model (black solid curves) with data from the literature at 313 K (orange triangles 40 ) and at 333 K (blue squares 32 ).

0.0015

0.004

0% CO2

10 MPa

15 MPa

0.0013

20 MPa 25 MPa 30 MPa

0.0011

35 MPa

0.0009

0.0007

0.0005

Isobaric Expansivity (K-1 )


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



50% CO2

0.0035

20 MPa

0.003

25 MPa

0.0025

30 MPa 35 MPa

0.002 0.0015 0.001 0.0005 0

300

350

400

300

Temperature (K)

350

400

Temperature (K)

Figure 9a. The variation of isobaric expansivity with temperature at selected pressures in cyclohexane (0 wt% CO2 ) and the mixture containing 50 wt % CO2 .


Page 23 of 47

0.0025 50%

20 MPa

0.0035 0.003



0.004

40%

0.0025 30%

0.002

20%

0.0015

10% 0% CO2

0.001 0.0005

35 MPa

50%

0.002 40%

0.0015

30% 20% 10% 0% CO2

0.001

0.0005

0

0 300

350

400

300

350

Temperature (K)

400

Temperature (K)

Figure 9b. Comparison of the variation of isobaric expansivity with temperature at 20 and 35 MPa for cyclohexane and for its mixtures with 10, 20, 30, 40 and 50 wt % CO2 .

0.0015

0.0015


Cyclohexane Isobaric Expansivity (K-1 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.0013

333 K

0.0011

313 K 0.0009

0.0007

Cyclohexane 0.0013

0.0011

0.0009

0.0007

20 MPa

0.0005

0.0005 0

10

20

30

40

300

Pressure (MPa)

320

340

360

380

400

Temperature (K)

Figure 9c. Comparison of the variation of the isobaric expansivity of pure cyclohexane at 313 and 333 K at different pressures (Left), and at 20 MPa at different temperatures (Right) with literature values. Solid black curves (Left) or filled black circles (Right) are values determined from the S-L model in the present study. Blue squares are literature data 32 at 333 K (Left) and at 20.7 MPa (Right), and orange triangles are literature data40 at 333 K (Left) and 20 MPa (Right).



160

300

0% CO2

280 270 260

313 K

250

333 K

240

353 K

230

373 K

220

210

50% CO2

150

Internal Pressure (MPa)


290

140

313 K

130

333 K

120

110

353 K

100

373 K

90

393 K

80 70

393 K

200

60 0

10

20

30

40

0

10

Pressure (MPa)

20

30

40

Pressure (MPa)

Figure 10a. The variation of internal pressure with pressure for cylohexane (0 wt% CO2 ) and for the mixture containing 50 wt % CO2 at different temperatures.

300

300

393 K


313 K

280


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 47

0% CO2

260 240

220 200 180

10%

160 20%

140

40%

30% 50%

120

250 0% CO2

200 150

10%

20%

100

100

30% 40% 50%

50 0

0

10

20

30

40

0

Pressure (MPa)

10

20

30

40

Pressure (MPa)

Figure 10b. Comparison of the internal pressure of cyclohexane and its mixtures with 10, 20, 30, 40 and 50 wt % CO2 at 313 and 393 K.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


200 180


Page 25 of 47

313 K

160 333 K

140 120

353 K

100

373 K

80

393 K

60 40 20 0 0

10

20

30

40

Pressure (MPa) Figure 11. Internal pressure for CO2 generated with the SL-EOS model using the density from the NIST database.

3.3.4 Solubility Parameters The solubility parameter values estimated from Equation 9 as square root of the internal pressure are shown in Figure 12 for cyclohexane and for its mixtures containing 10, 30, and 50 wt % CO2 as a function of pressure at 313, 333, 353, 373 and 393 K; the data for all compositions is presented in the Supporting Information. When used with the characteristic parameters generated by our S-L model, the solubility parameter values estimated by Equation 10 were found to be exactly the same as the values determined by Equation 9. There is no prior data on the solubility parameters for mixtures of cyclohexane with carbon dioxide. There is only one study29 which has reported the solubility parameter for pure cyclohexane at a range of pressures, but only at 303 K. Figure 13 is a comparison of this literature data at 303 K with the present estimations at 313 K and higher temperatures. The plots show that the estimated values are in a reasonable range. Figure 14 shows the solubility parameters estimated for CO 2 using the SL EOS and the density values taken from the NIST database. The solubility parameters increase with increasing pressure at each temperature, and decrease with increasing temperature at each pressure. The solubility parameter for carbon dioxide has been previously calculated and reported in the literature49 using an equation of state of the form 𝑃𝑉 = 𝑧𝑅𝑇. Figure 14 compares these solubility parameters for CO 2 predicted in the 25 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 47

present study using S-L EOS modeling and internal pressure calculations with the values reported in the literature at 313 and 363 K. The 313 K values are close and the match improves at higher pressures. The literature values at 363 K are in between the present values at 353 and 373 K. Figure 15 presents the compositional dependence of the solubility parameters estimated using the S-L EOS evaluated at 20 MPa and 313 and 393 K. They show a gradual decrease in the solubility parameters with increasing CO 2 content. As would be expected, the solubility parameter decreases with increasing CO 2 concentration in the mixture. At 313 K, even though there is a rapid decrease up to about 20 wt % CO 2 , it then appears to level off. At 393 K, there is a continual decrease in the solubility parameter with a further increase in the CO 2 content of the mixture. The solubility parameter can be tuned from a value in the range of 17 MPa 0.5 to about 5 MPa

0.5

by changing the

fluid composition or the T/P conditions. This tunability provides flexibility in bringing about miscibility and/or phase separation in the processing of polymers. Mixtures of cyclohexane and carbon dioxide can be considered for the processing of polyolefins such as polyethylene and its copolymers. The solubility parameter of polyethylene is given in the literature 50 as 16.2 MPa0.5 .


Page 27 of 47

14.5

Solubility Parameter (MPa0.5 )


17 16.5 313 K

16

333 K

15.5 353 K

15

373 K

14.5

393 K

0% CO2

14

13.5

313 K 333 K

353 K

12.5 373 K 393 K

10% CO2

11.5 0

10

20

30

40

0

10

Pressure (MPa)

20

30

40

Pressure (MPa) 13


13


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


12 313 K 333 K

11

353 K 373 K

10

393 K

30% CO2 9

12 313 K

11

333 K 353 K

10 373 K 393 K

9

50% CO2 8

0

10

20

30

40

0

Pressure (MPa)

10

20

30

40

Pressure (MPa)

Figure 12. The variation of the solubility parameter of cyclohexane and its mixtures with 10, 30, and 50 wt % CO2 with pressure at different temperatures.




18 Literature

17.5 17

S-L EOS

303 K

16.5

313 K

16

333 K

15.5

353 K

15

373 K

14.5

393 K

Cyclohexane

14 0

10

20

30

40

Pressure (MPa) Figure 13. Comparison of the solubility parameter of cyclohexane and its variation with pressure reported in the literature29 at 303 K (the uppermost curve) with the values estimated from internal pressure caluclations using the S-L EOS in the present study determined at 313, 333, 353, 373 and 393 K.

14


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 47

12

313 K 333 K

10

373 K

353 K 393 K

8 6 4 2

363 K

0 0

10

20

30

40

Pressure (MPa) Figure 14. The variation of the solubility parameter of pure CO2 estimated by the S-L EOS (solid colored curves) from the NIST density data at 313, 333, 353, 373, and 393 K and their comparison with the literature49 values (dashed black curves) at 333 and 363 K.


Page 29 of 47

17

Solubility Paramter (MPa0.5 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15 313 K

13 11 9

393 K 7

20 MPa 5 0

20

40

60

80

100

wt % CO2 Figure 15. Variation of solubility parameters estimated by S-L EOS with CO2 concentration in the mixture at 313 and 393 K and 20 MPa.

3.3.5 Excess Volumes The excess molar volume (VE) of a mixture represents the difference between the molar volume of the mixture and the sum of the molar volumes of each component at a given T/P condition. It is evaluated using the following equation:

𝑉𝐸 = 𝜌

1

𝑚𝑖𝑥

∑𝑖 (𝑥 𝑖 𝑀𝑖 ) − ∑ (𝑥 𝑖𝑀𝑖 ) 𝜌 𝑖

(13)

𝑖

or, for a binary mixture in expanded form:

𝑉𝐸 =

𝑀𝑚𝑖𝑥 𝜌𝑚𝑖𝑥

−(

𝑥 1 𝑀1 𝜌1

+

𝑥 2 𝑀2 𝜌2

(14)

)

where 𝜌𝑚 is the experimentally determined mixture density, and 𝑥 𝑖 , 𝑀𝑖 , and 𝜌𝑖 are respectively the mole fraction, the molar mass, and the density of the pure components.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 47

The excess volume of each mixture was evaluated using the density values determined in the present study for cyclohexane, and the values given in the NIST database for pure carbon dioxide. Evaluations were performed using the mole fractions corresponding to each mixture. The results are shown in Figure 16 at 313, 333, 373 and 393 K for different pressures. Figure 17 shows the results at 20 and 35 MPa at different temperatures. The data at other temperatures and pressures are provided in the Supporting Information. In these figures, the y-axis (VE) scale has been kept the same to faciliate observation of the trends of the excess volume with pressure at a given temperature, or with temperature at a given pressure. It should be noted that the figures do not include any experimental data in the range from 50 to 100% carbon dioxide. At 313 K, the excess volume is positive at all pressures. At 333 K excess volume remains positive for pressures in the range 25-35 MPa, before becoming negative at 20 MPa. With increasing temperature, it becomes more negative, and is even observed to become negative at higher pressures. At 373 and 393 K, it remains negative at all pressures. It is less negative with increasing pressure, which is further illustrated in Figure 17. The results indicate that the mixtures display greater degree of non-ideality at lower pressures at a given temperature; or at higher temperatures at a given pressure. The excess volume ranges from about +2.4 to −20 cm3 /mol for the mixture with 65 mole % (or 50 wt %) CO 2 . Similar trends are reported for mixtures of CO 2 with other hydrocrabons, such as decane or undecane51 . In mixtures of carbon dioxide with undecane, high negative excess volumes approaching −34 cm3 /mol are reported for the mixtures with high CO 2 content51 . In the literature excess volume is often modeled with a Redlich-Kister type equation51-55 , which can be described by: 𝑉 𝐸 = 𝑥 1 𝑥 2 ∑𝑛𝑖=0 𝐴𝑖 (𝑥1 − 𝑥 2 )𝑖−1

(15)

For binary mixtures, when the summation is limited to the first two terms, it reduces to 𝑉 𝐸 = 𝑥 1 𝑥 2 (𝐴0 + 𝐴1 (𝑥 1 − 𝑥 2 ))

(16)

This can be expanded into a third-order polynomial: 𝑉 𝐸 = (𝐴0 − 𝐴1 )𝑥 1 + (3𝐴1 − 𝐴0 )𝑥 12 + (2𝐴1 )𝑥13

(17) 30

ACS Paragon Plus Environment

Page 31 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Referring to CO 2 as component 1 and cyclohexane as component 2, the present excess volume data were fitted to this third-order polynomial and the parameters Ao and A1 were evaluated using mole fractions describing each composition. They were then fitted to the following secondorder polynomials to describe the temperature dependence of these parameters: 𝐴0 = 𝑎 + 𝑏𝑇 + 𝑐𝑇 2

(18)

𝐴1 = 𝑑 + 𝑒𝑇 + 𝑓𝑇 2

(19)

The coefficients a, b, c, d, e, and f were then fit to second-order polynomials of the form: 𝑖 = 𝑖 0 + 𝑖1 𝑃 + 𝑖 2 𝑃 2

(20)

to describe their variations with pressure (P). Table 3 summarizes these parameter values. The curves that are included in Figures 16 and 17 represent the Redlich-Kister descriptions of the excess volume using these parameters. Table 3. The Redlich-Kister parameters describing excess volume in mixtures of carbon dioxide with cyclohexane.

𝑖0

𝑖1

𝑖2

a

-0.0228

0.0011

-1x10-5

b

11.73

-0.5427

0.0066

c

-1419.1

61.138

-0.6618

d

-0.0116

0.0005

-6x10-6

e

4.9948

-0.1847

0.0016

f

-439.54

8.0548

0.077


5

0

0

-5

-5

VE (cm3 /mol)

5

-10 35 MPa

-15

35 MPa

-15

30 MPa

25 MPa

-20

Page 32 of 47

-10

30 MPa

-20

25 MPa

313 K

20 MPa

333 K

20 MPa

-25

-25 0

0.2

0.4

0.6

0.8

1

0

Mol Fraction CO2 5

5

0

0

-5

-5

-10 35 MPa

-15

0.4

0.6

0.8

1

-10 35 MPa

-15

30 MPa

-20

0.2

Mol Fraction CO2

VE (cm3 /mol)

VE (cm3 /mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

VE (cm3 /mol)


30 MPa

-20

25 MPa

25 MPa

373 K

20 MPa

393 K

20 MPa

-25

-25 0

0.2

0.4

0.6

0.8

1

0

Mol Fraction CO2

0.2

0.4

0.6

0.8

1

Mol Fraction CO2

Figure 16. Comparisons of the variation of molar excess volume with carbon dioxide content at 20, 25, 30, and 35 MPa at 313, 333, 373 and 393 K.


5

5

0

0

-5

-5

VE (cm3 /mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


VE (cm3 /mol)

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-10 313 K

-15

333 K

-10 313 K

-15

333 K

353 K

-20

353 K

-20

373 K

373 K

20 MPa

393 K

35 MPa

393 K

-25

-25 0

0.2

0.4

0.6

0.8

1

0

Mol Fraction CO2

0.2

0.4

0.6

0.8

1

Mol Fraction CO2

Figure 17. Comparisons of the variation of molar excess volume with carbon dioxide content at 313, 333, 353, 373 and 393 K at 20 and 35 MPa .

3.3.6. Predictive modeling with the Sanchez-Lacombe EOS using mixing rules Even though modeling the mixtures as pseudo-pure compounds provides a mechanism to describe the density and to generate the various thermodynamic properties, predictive modeling of the mixture behavior from the pure component parameters is highly desirable. Success in predictive modeling is linked to the success of the mixing rules that can describe the system behavior. However, this often presents challenges when mixtures display high degree of non-ideality. We have explored the predicative modeling of the mixture density data from the pure component S-L EOS parameters (𝜌𝑖∗ , 𝑃𝑖∗, 𝑇𝑖∗ , and 𝑟𝑖 ) for carbon dioxide (1) and cyclohexane (2) (given in Table 1) using the following mixing rules to determine the parameters (𝜌 ∗ , 𝑃 ∗, 𝑇 ∗ , and 𝑟 ) for mixtures. 𝑀 𝑃∗

𝜌 ∗ = 𝑅𝑇 ∗ 𝑟

(21)

∗ 𝑃 ∗ = 𝜙12 𝑃1∗ + 2𝜙1 𝜙2 𝑃12 + 𝜙22 𝑃2∗

(22)

∗ 𝑃12 = √𝑃1∗ 𝑃2∗ (1 − 𝑘12 )

(23)

𝑇 ∗ = 𝑃 ∗ ((

𝜙01 𝑇1∗ 𝑃1∗

)+(

𝜙02 𝑇2∗ 𝑃2∗

(24)

))



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

𝜙10 = 𝜙1 = 1 𝑟

(𝜙1 𝑃1∗ /𝑇1∗ ) ∗ (𝜙1 𝑃1 /𝑇1∗ )+(𝜙2 𝑃2∗ /𝑇2∗ )

(25)

(𝑤1 /𝜌∗1 )

(26)

(𝑤1/𝜌∗1 )+(𝑤2 /𝜌2∗ ) 𝜙0

𝜙0

1

2

(27)

= ( 𝑟1 ) + ( 𝑟2 )

𝑟1 =

Page 34 of 47

𝑀1 𝑃1∗

(28)

𝑅𝑇1∗ 𝜌∗1

where 𝜙1 and 𝜙10 represent the close-packed volume fraction and the average close-packed mervolume fraction, respectively34 , and where 𝑟1 is the number of lattice sites per molecule for component 1 in the pure state, and 𝑘12 is an interaction parameter used to better describe the characteristic pressure 𝑃 ∗ for the mixture. 𝑀 is the mixture molecular weight evaluated by either Equation 11 or 12. In the literature, a range of mixing rules has been previously reported. As summarized by Nicolas36 they include using a quadratic expression for the interaction parameter for mer-mer interaction energy 𝜀 ∗ only; or quadratic expressions for both 𝜀 ∗ and the segment volume 𝑣 ∗ ; or using composition dependent mixing rules involving three binary interaction parameters. In the present model, we have employed only one interaction parameter, 𝑘12 , which was considered to be dependent on both composition and temperature. As can be seen from Equations 21-24, with 𝑘12 being dependent on composition and temperature, the characteristic pressure 𝑃 ∗ will be a function of composition and temperature. Consequently, the characteristic temperature for the mixture, 𝑇 ∗ , will also be dependent on composition and temperature. In the model, the number of occupied lattice sites 𝑟 and the characteristic density, 𝜌 ∗ , of the mixture are considered to be dependent on composition, but independent of temperature. The composition- and temperature- dependent 𝑘12 is expressed as 𝑘12 = (𝑘1𝑎 ∗ 𝜙1 + 𝑘1𝑏 ) ∗ 𝑇 + (𝑘2𝑎 ∗ 𝜙1 + 𝑘2𝑏 )

(29)

where 𝑘1𝑎 , 𝑘1𝑏 , 𝑘2𝑎 , and 𝑘2𝑏 are constants. A PythonT M program was used to determine these coefficients using all the experimental density data (about 7000 data points) generated at all temperatures, pressures, and compositions. From the pure component values for 𝜌𝑖∗ , 𝑃𝑖∗, 𝑇𝑖∗ , and mass fractions 𝑤𝑖, the program evaluates the closed packed volume fractions 𝜙1 (from Equation 26 ) and 34 ACS Paragon Plus Environment

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


then the close-packed mer-volume fractions 𝜙10 (from Equation 25), and 𝑟 (from Equation 27), which allow the formulation of 𝜌 ∗ , 𝑃 ∗, and 𝑇 ∗ for the mixture in terms of 𝑘12 . The coefficients of 𝑘12 which lead to the best fit of the densities as would be predicted by the SL-EOS (Equation 1) to the whole set of the experimental densities are determined. Table 4 shows the coefficients of the interaction parameter that describe the density of the carbon dioxide + cyclohexane mixtures as a function of composition, temperature, and pressure. Once the constants 𝑘1𝑎 , 𝑘1𝑏 , 𝑘2𝑎 , and 𝑘2𝑏 are calculated, a separate PythonT M program is used to calculate the R2 values at each composition to determine if the model represents the experimental data well. Table 5 shows the R2 values for each composition, which are all mostly > 0.99. For comparison, the table also includes the R2 values from modeling the mixtures as pseudo-pure compounds. Also, for ease of comparison, tabulated values of the experimental densities and predictions by pseudo-pure compound assumption and by using mixing rules are given in the Supporting Information at selected pressures and temperatures. With 𝜌 ∗ , 𝑃 ∗, 𝑇 ∗ , and 𝑟 for the pure components and the constants 𝑘1𝑎 , 𝑘1𝑏 , 𝑘2𝑎 , and 𝑘2𝑏 generated, the density of mixtures of carbon dioxide with cyclohexane can be determined with high degree of accuracy by employing the mixing rules described by equations 21-29. Figure 18 shows the comparison of the model predictions of density using the mixing rules with the experimental data. Additional data are given in the Supporting Information. It should be noted that optimization using a constant value for the interaction parameter 𝑘12 or, considering only composition-dependent 𝑘12 (of the form 𝑘12 = 𝑎 + 𝑏𝜙 + 𝑐𝜙2 ), or considering only temperature dependent 𝑘12 (of the form 𝑘12 = 𝑚𝑇 + 𝑛) failed to predict the densities with accuracy. As discussed previously in regards to the excess volume data, these mixtures display high degrees of non-ideality with strong composition and temperature dependence of the mixture behavior. Table 4. Coefficients for the interaction parameter k12 (Eq. 29) for mixtures of CO2 + cyclohexane. k1a

k1b

k2a

k2b

0.004285

-0.00254

-1.9037

1.2709



Table 5. Comparison of the R2 values for the modeling of density by the S-L EOS with mixing rules, or with the assumption that each mixture can be considered a pseudo-pure compound. Mass Fraction CO2

0.1

0.2

0.3

0.4

0.5

R2 - Modeling using mixing rules

0.994

0.989

0.991

0.998

0.998

R2 - Modeling by treating each mixture as a

0.996

0.997

0.991

0.998

0.999

pseudo-pure compound

0.81

0.85

10% CO2

30% CO2

313 K

0.76

373 K 393 K

0.71

Density (g/cm3 )

Density (g/cm3 )

353 K

313 K

0.8

333 K

333 K

353 K

0.75

373 K 393 K

0.7

0.65

0.66

0.6

0

10

20

30

40

0

10

Pressure (MPa)

20

30

40

Pressure (MPa)

0.83

40% CO2

0.83

313 K 333 K 353 K 373 K

0.73

393 K

0.68

0.63

50% CO2

313 K 333 K

0.78

Density (g/cm3 )

0.78

Density (g/cm3 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 47

353 K

0.73

373 K 393 K

0.68 0.63 0.58

0.58

0.53

0

10

20

30

40

0

Pressure (MPa)

10

20

30

40

Pressure (MPa)


Page 37 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 18. Comparisons of the experimental values of density (solid colored curves) with those predicted by the S-L EOS (black dotted curves) using the mixing rules (Equations 21-29) for mixtures with different mass % CO2 .

3.3.7. Comments on the Sanchez-Lacombe EOS parameters of mixtures determined by the pseudo-pure compound versus the mixing-rule approach Figure 19 compares the SL parameters as a function of mole fraction of CO 2 in the mixture as determined by treating each mixture as a pseudo-pure compound, or by using the pure component parameters with the mixing rules described by Equations 21-29 at selected temperatures. For the mixing rule approach, the variation with composition is shown for differe nt temperatures. The effect of temperature is visible in 𝑃 ∗, 𝑇 ∗ and 𝜀 ∗. The values for 𝑣 ∗ do not demonstrate a temperature dependence. The mer-number 𝑟 is independent of temperature, and 𝜌 ∗ does not show any temperature dependence. Overall trends of the parameters obtained from the predictive model are similar to those displayed by the parameters obtained from treating the mixtures as pseudo-pure compounds, except for the characteristic volume 𝑣 ∗ and the 𝑟 parameter describing the number of mers for the mixture. Whereas the mixing rule approach suggests a smooth, almost linear decrease in going from pure cyclohexane to pure carbon dioxide, the pseudo-pure compound approach shows that the characteristic volume goes through a maximum at around 30 mol % CO 2 level. As for 𝑟 values, while the mixing rule approach predicts a smooth decrease in going from cyclohexane to carbon dioxide, the pseudopure compound approach shows that r goes through a minimum at compositions corresponding to the maximum in the characteristic volume. This reverse behavior in 𝑟 is as anticipated since simple manipulations of the relationships in Equation 3 shows that 𝑟 = 𝑀/(𝑣 ∗ 𝜌 ∗ ) and the behavior of 𝑟 must thus closely follow an inverse trend with 𝑣 ∗ . The behavior of 𝑃 ∗ which is equal to 𝜀 ∗ /𝑣 ∗ also should follow the trend in 𝑣 ∗ in an inverse manner which is displayed in 𝑃 ∗ for the pseudo-pure compound approach as well as in the predictive modeling evaluations. It is interesting to note that even though the trends in the behavior in 𝑣 ∗ and 𝑟 that are observed in the pseudo-pure compound approach are not replicated in parameters determined by the predictive modeling, the values that are generated for 𝑃 ∗, 𝑇 ∗ , 𝜌 ∗ and 𝑟 using the mixing rules (Equations 21-29) provide an accurate description of the density isotherms for these mixtures.



450

25

20

350

v* (cm3 /mol)

P* (MPa)

400

300 250

15

10

5

200 150

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

Mol Fraction CO2

0.5

1.0

Mol Fraction CO2

5000

600

4500

500

4000 3500

400

3000

T* (K)

ε* (MPa•cm3 /mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 47

2500 2000

300 200

1500 1000

100

500 0

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

Mol Fraction CO2

0.2

0.4

0.6

0.8

1.0

Mol Fraction CO2


Page 39 of 47

8

1.4

7

r (number of mers)

1.5

1.3

ρ* (g/cm3 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.2 1.1

1 0.9

6 5 4 3 2 1

0.8

0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

Mol Fraction CO2

0.4

0.6

0.8

1.0

Mol Fraction CO2

Figure 19. Variation of the characteristic density (𝜌 ∗ ), characteristic pressure (𝑃 ∗ ), characteristic temperature (𝑇 ∗ ), mernumber (r), interaction energy (𝜀 ∗ ), and the characteristic volume (𝑣 ∗ ) of the Sanchez-Lacombe model for the mixtures with the mole fraction (x) of CO2 in the mixture. Filled blue circles represent the parameters determined by treating the mixtures as pseudo-pure compounds, and the filled squares represent the parameters determined with the mixing rule approach at different temperatures. In plots for 𝑃 ∗ , 𝜀 ∗ , and 𝑇 ∗ , 393 K is at the top of the temperature-dependent trends and 313 K is at the bottom in the descending order (with square symbols representing 393 K – Light Blue; 373 K – Green; 353 K – Grey; 333 K – Yellow; 313 K – Dark Blue).

4. Conclusions The present study provides the first extensive data sets on the density and derived properties for mixtures of carbon dioxide with cyclohexane containing up to 50 wt % carbon dioxide over a wide range of temperatures and pressures. It is shown that the volumetric properties of carbon dioxide, cyclohexane, and their mixtures can be effectively modeled with the Sanchez-Lacombe equation of state if the mixtures are treated as pseudo-pure compounds, or with mixing rules that incorporate a composition- and temperature- dependent interaction parameter to capture the non-ideal behavior of these mixtures. The high degree of non-ideality of these mixtures is displayed by the compositio na l dependence of the excess volume which becomes more negative with increasing temperature or decreasing pressure for a given composition.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 47

The study further shows that the S-L EOS can be used to estimate the solubility parameters of carbon dioxide, cyclohexane, and their mixtures as a function of temperature, pressure and mixture composition. For a given composition the solubility parameters increase with pressure, decrease with temperature, and decrease in a non-linear fashion as composition proceeds from pure cyclohexane to pure carbon dioxide. Supporting Information The Supporting Information document contains: • A comparison of the density data from repeated experimental trials, as discussed in Section 2.5 • Experimental density data and S-L model fits, as presented in Figure 5, with the inclusion of the 20 and 40 wt % CO 2 mixtures • Isothermal compressibility, isobaric expansivity, and internal pressure data, as presented in Figures 8a, 9a, and 10a, with inclusion of the 10, 20, 30, and 40 wt % CO 2 mixtures • Solubility parameter versus pressure, as presented in Figure 12, with inclusion of the 20 and 40 wt % CO 2 mixtures • Excess volume versus mole fraction of CO 2 , as presented in Figures 16 and 17, with inclusion of the 353 K isotherm and the 25 and 30 MPa isobars. • Experimental density data and S-L model fits, as presented in Figure 18, with the inclusion of the 20 wt % CO 2 mixture • Tables of experimental density values, NIST CO 2 density values, and the density predictions made with the Sanchez-Lacombe equation of state, which include predicted values generated by treating the mixtures as pseudo-pure compounds as well as by using the S-L mixing rules. Data is given for the pure components and for mixtures containing 10, 20, 30, 40, and 50 wt % CO 2 .

Acknowledgements This research was in part supported by DuPont.


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Graphical Abstract

0.83

Density (g/cm3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.78

Cyclohexane

0.73

10

0.68 0.63

20

0.58

0.53 0

wt % CO 2 in mixture

30 40 50

333 K 10

20

30

40

Pressure (MPa)


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Volumetric Properties and Solubility Parameters of Cyclohexane +

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