Comparison of Mixing Rules for a van der Waal’s Gas ... · scientists are familiar with the van...

Comparison of Mixing Rules for a van der Waal’s Gas Mixture

Gas Phase PVT Properties for the Difluoromethane + Pentafluoroethane (R32 + 125) System : Predicting Pressure Utilizing van der Waal’s Equation of State and

Different Mixing Rules

Molecular Thermodynamics ChE 531

Department of Chemical Engineering University of Tennessee

Knoxville, TN

Submitted by: Austin Newman

12/3/01

Table of Contents I. Introduction 1 II. Experimental Section 1 III. Results 3 IV. Discussion of Results 4 V. References 8 VI. Appendix A – Raw Data 9 VII. Appendix B – Figure 6. (0.5001 Difluoromethane Plots) 16 VIII. Appendix C – Figure 7. (0.6977 Difluoromethane Plots) 19

Austin Newman, CHE 531, University of Tennessee, Fall, 2001

Introduction Many equations of state exist for modeling the behavior of fluids. Frequently, there is a need to model the behavior of gas mixtures. While there is no specific equation of state to model mixtures, there are several different approaches one may take. In this paper, several different “mixing rules” are presented which are used to model a gas mixture. The predicted properties from the equation of state and the mixing rules are compared to actual experimental data with a hope of identifying the best mixing rule for a specific fluid mixture. Experimental Section In order to attempt this experiment, actual PVT data for a gas mixture had to be obtained. PVT data was obtained for a Difluoromethane + Pentafluoroethane system [1]. Data was obtained for two different mixtures, one with 0.5001 mole fraction of Difluoromethane, and another mixture with 0.6977 Difluoromethane. Five sets of constant volume data were available for the 0.5001 mole fraction mixture and eight sets of data were available for the 0.6977 mole fraction mixture. The raw data is included in the appendix of this paper. In order to model the behavior of the fluid properties, an appropriate equation of state was selected. While any equation of state could have been utilized, many engineers and scientists are familiar with the van der Waal’s equation of state, which is why it was selected for this experiment. The van der Waal’s equation of state can be described as:

2Va

bVRTP −−

= (1)

where a and b are constants which depend on the fluid, V is the molar volume, T is the temperature, R is the ideal gas constant, and P is the pressure. The constants a and b may sometimes be found in Chemical Engineering Reference Books. However, a and b are related to the critical properties of all fluids by:

89 cc RTV

a = (2)

3cV

b = (3)

where Vc is the critical volume of the fluid, Tc is the critical temperature and R is the ideal gas constant. As there was great difficulty in obtaining the van der Waal constants for each of these fluids, the relationships in equations (2) and (3) were utilized to approximate the van der

1


Waal’s constants. The critical properties of each of these fluids, along with the van der Waal’s constants are presented in Table 1. Table 1. Properties of Difluoromethane and Pentafluoroethane Property Difluoromethane Pentafluoroethane Tc (K) 351.6 339.41 Vc (cm3 / mol) 58.3 36.39 Pc (bar) 120.8 209.7 a (cm6 bar / mol2) 83.144 83.144 b (cm3 / mol) 3.9728E+06 6.6574E+06 The next step in the experiment was to select mixing rules. Five mixing rules were compared in this experiment. The first mixing rule is described as:

2211 axaxa += (4) and

2211 bxbxb += (5)

where a is the van der Waal’s constant for the gas mixture, x1 is the mole fraction of Difluoromethane, a1 is the van der Waal’s constant, a, for Difluoromethane, x2 is the mole fraction of Pentafluoroethane, and b1 is the van der Waal’s constant, b, for Pentafluoroethane. In each of the following mixing rules, the same convention is followed for the symbols used in the equations. Mixing rule 2 can be described as:

2211 axaxa += (6) and

21bbb = (7)

Mixing rule 3 is somewhat of a combination of mixing rules 1 and 2:

21aaa = (8) and

2211 bxbxb += ( 9) Mixing rule 4 can be shown as:

21aaa = (10) and

21bbb = (11)

2


Mixing rule 5, originally proposed by van der Waal’s is described as [3]:

22221211

21 2 axaaxxaxa ++= (12)

and 2211 bxbxb += (13)

Each of the mixing rules was utilized to model the behavior of the fluid mixture so the “best” mixing rule could be selected for the system described herein. In order to utilize the van der Waal’s equation, temperature and volume were the independent variables and pressure was the dependent variable. Results The performance of each of the mixing rules was analyzed and plotted. A sample plot of this data can be seen in Figure 1. Other figures are included in the Appendix of this report. The error of each of the mixing rules was calculated for each mixture at each volume. The average error for the 0.5001 Difluoromethane mixture is presented in Figure 2. whereas the average error for the 0.6977 Difluoromethane mixture is presented in Figure 3. Upon inspection of Figures 1 and 2 it can easily be seen that mixing rule 1 provided the best model for the data for each of the compositions at each volume. Figure 1. Experimental data and mixing rule behavior for a Difluoromethane / Pentafluoroethane Mixture

Difluoromethane / Pentafluoroethane Mixture(0.5001 Difluoromethane)

25

30

35

40

45

50

320 330 340 350 360 370 380 390

Temperature (K)

Pres

sure

(bar

)

P (actual)P mix 1P mix 2P mix 3P mix 4P mix 5

V = 568 cm^3/mole

3


Figure 2. Percent Error of Mixing Rules for 0.5001 Difluoromethane Mixture

Error Comparison(0.5001 Difluoromethane)

0.00%

5.00%

10.00%

15.00%

20.00%

400 900 1400 1900 2400 2900

Volume (cm^3 / mol)

Ave

rage

Err

or P mix 1P mix 2P mix 3P mix 4P mix 5

Figure 3. Percent Error of Mixing Rules for 0.6977 Difluoromethane Mixture

Error Comparison(0.6977 Difluoromethane)

0.00%

5.00%

10.00%

15.00%

20.00%

350 2350 4350 6350 8350Volume (cm^3 / mol)

Ave

rage

Err

or

P mix 1P mix 2P mix 3P mix 4P mix 5

Discussion of Results From the data presented in the results, it is clearly evident mixing rule 1 was superior in modeling the behavior of the mixture. However, in preparing the numerical computations, several questions were raised. Even though the purpose of this experiment was to determine the “best” mixing rule for a specific mixture of fluids, there are experimental factors identified which may affect the results.

4


In this experiment, the van der Waal’s equation of state was utilized to model the behavior of the gas mixture. The model of the mixtures are dependent on how well the van der Waal’s equation of state models each of the “pure” fluids. Each of the “pure” component data points was compared to the van der Waal model of the data. The average error of the Difluoromethane model was 4% whereas the average error of the Pentafluoroethane model was 8%. For this experiment it would have been desirable to obtain PT data obtained at constant volume. However, the experimental apparatus used to collect the data did not lend the opportunity to collect constant volume data. While the data collected is close to constant volume, it is interesting to note the standard deviation of the volume measurements (presented in Tables 2 and 3). Table 2. Volume data for 0.5001 Difluoromethane mixture Average Volume Standard Deviation 568 cm3 0.5491 cm3 853 cm3 1.0117 cm3 1282 cm3 1.6949 cm3 1927 cm3 2.2441 cm3 2895 cm3 3.7944 cm3 Table 3. Volume data for 0.6977 Difluoromethane mixture Average Volume Standard Deviation 496 cm3 0.4539 cm3 745 cm3 0.7805 cm3 1120 cm3 1.3139 cm3 1683 cm3 2.0264 cm3 2530 cm3 2.8615 cm3 3803 cm3 4.4906 cm3 5715 cm3 6.9268 cm3 8590 cm3 10.7559 cm3 By inspection, of Tables 4 and 5, it can easily be observed that as the volume increased, the standard deviation of the volume increased. While the effects of the volumetric measurements on the model cannot be quantified, it is believed the deviations in the volumetric measurements should be noted. Upon inspection of each of the plots, there were two interesting phenomenon noted across the board for all mixing rules. However, because Mixing Rule 1 incurred the least amount of error, it was examined in detail. In figures 4 and 5 there are two trends in the

5


data, which can be noted by inspection. As the temperature of the gas increases, the percent error in the predicted pressure (compared to the experimental data) drastically decreases. Another trend appearing in the data is the error in the predicted pressure decreases as the volume increases. These two trends appear for all mixing rules which were studied. Figure 4. Error for Mixing Rule 1 for 0.5001 Difluoromethane

Mixing Rule 1 Error(0.5001 Difluoromethane)

0%2%4%6%8%

10%12%14%16%

290 310 330 350 370 390

Temperature (K)

Erro

r

568 cm^3 / mole853 cm^3 / mole1282 cm^3 / mole1927 cm^3 / mole2895 cm^3 / mole

Figure 5. Error for Mixing Rule 1 for 0.6977 Difluoromethane

Mixing Rule 1 Error(0.6977 Difluoromethane)

0%

2%

4%

6%

8%

10%

12%

290 310 330 350 370 390

Temperature (K)

Erro

r

496 cm^3 / mole745 cm^3 / mole1120 cm^3 / mole1683 cm^3 / mole2530 cm^3 / mole3803 cm^3 / mole5715 cm^3 / mole8590 cm^3 / mole

6


Conclusions Several mixing rules were tested utilizing the van der Waal’s equation of state for a Difluoromethane / Pentafluoroethane mixture. Two sets of data were available for this experiment, 0.5001 Difluoromethane mixture and a 0.6977 Difluoromethane mixture. Clearly, mixing rule 1 was superior in predicting the pressure for each of the mixtures. However, there was still an error associated with the predicted values of pressure compared to the experimental values. This model may be useful depending on the needs of the user. At temperatures above 350 K the error associated with this model was less than 5%. As the temperatures increased the error dramatically decreased. If an error of 5% is acceptable, then this model will be sufficient. However, if the user requires greater accuracy, other avenues must be explored. While other mixing rules could be explored, it is believed another equation of state must be utilized. In analyzing the van der Waal’s model for the pure gases there was a significant error demonstrated between the predicted values of pressure and the experimental values. For this specific mixture, a virial type equation of state may be used [1]. Zhang, Sato, and Watanabe were able to model this mixture to an accuracy of ±0.3%. While this may work for this mixture, engineers are continually searching for “general” equations that model a broad range of systems accurately. In the future, it would be interesting to repeat this experiment using a different gas mixture, where each of the pure gases obey the van der Waal’s equation of state. This would eliminate any error introduced by the equation of state. Hypothetically speaking, one would only have to study the experimental error introduced because of:

1. Mixing rules 2. Error from experimental apparatus

After completion of the second experiment, one would have a better feel for the behavior of the mixing rules.

7


References [1] Zhang, Sato, Watanabe, Gas Phase PVT Properties for the Difluoromethane + Pentafluoroethane (R-32 + 125) System, J. Chem. Eng. Data 1996, 41. 1401-1408. [2] Reid, Prausnitz, and Poling, The Properties of Gases and Liquids – 4th Edition, McGraw-Hill, 1987. [3] J. M. Prausnitz, R. N. Lichtenthaler, E. Gomez de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2. ed., Prentice-Hall, 1986.

8


Appendix A Raw Data Difluoromethane T(K) P(kPa) P (bar) rho mol/dm3

290 134.2 1.342 0.0567290 199.9 1.999 0.0853290 296.4 2.964 0.1282290 436 4.36 0.1928290 634.4 6.344 0.2899290 909.6 9.096 0.4359290 1263.9 12.639 0.6555300 186.5 1.865 0.0765300 277.1 2.771 0.115300 409.6 4.096 0.1729300 599.8 5.998 0.2599300 866.5 8.665 0.3909300 1225.3 12.253 0.5877300 1675.7 16.757 0.8837310 179.3 1.793 0.0709310 267 2.67 0.1066310 395.5 3.955 0.1603310 581.6 5.816 0.2411310 845.4 8.454 0.3625310 1207.8 12.078 0.545310 1679.2 16.792 0.8195310 2235.9 22.359 1.2323320 169.1 1.691 0.0646320 252.3 2.523 0.0971320 374.7 3.747 0.146320 553 5.53 0.2196320 808.3 8.083 0.3302320 1164.7 11.647 0.4965320 1642 16.42 0.7465320 2236.9 22.369 1.1225320 2887.3 28.873 1.6878330 153.9 1.539 0.0568330 229.8 2.298 0.0854330 342.2 3.422 0.1284330 506.8 5.068 0.1931330 744.7 7.447 0.2903330 1082.1 10.821 0.4366330 1545.2 15.452 0.6565330 2148.3 21.483 0.9871330 2867.1 28.671 1.4842

9


330 3590.3 35.903 2.2318340 169.3 1.693 0.0606340 252.9 2.529 0.0912340 376.6 3.766 0.1371340 558.1 5.581 0.2062340 821.3 8.213 0.31340 1194 11.94 0.4662340 1706.5 17.065 0.701340 2377.6 23.776 1.0541340 3185 31.85 1.585340 4022.2 40.222 2.3833350 166.3 1.663 0.0578350 248.7 2.487 0.0869350 370.7 3.707 0.1307350 550.2 5.502 0.1965350 811.2 8.112 0.2955350 1184 11.84 0.4444350 1702.9 17.029 0.6682350 2395.4 23.954 1.0047350 3258.8 32.588 1.5108350 4220.2 42.202 2.2716360 431.7 4.317 0.1481360 640.5 6.405 0.2228360 943.6 9.436 0.335360 1376 13.76 0.5037360 1975.5 19.755 0.7573360 2770.9 27.709 1.1388360 3757.5 37.575 1.7123360 4849.5 48.495 2.5747360 5858.1 58.581 3.8713360 6537.2 65.372 5.8211370 150.9 1.509 0.0495370 225.9 2.259 0.0744370 337.6 3.376 0.1118370 502.8 5.028 0.1682370 745.2 7.452 0.2529370 1096.4 10.964 0.3802370 1595.5 15.955 0.5717370 2285.1 22.851 0.8597370 3196.3 31.963 1.2927370 4319.3 43.193 1.9437

Pentafluoroethane T(K) P(kPa) P (bar) rho mol/dm3

350 117.8 1.178 0.0409340 116.1 1.161 0.0415330 131.5 1.315 0.0486320 129.1 1.291 0.0493390 162.4 1.624 0.0505310 146 1.46 0.0578

10


380 181.4 1.814 0.0581360 172.2 1.722 0.0583350 176.3 1.763 0.0615340 173.6 1.736 0.0624300 154.4 1.544 0.0633290 163.6 1.636 0.0698330 196.3 1.963 0.0731370 223.3 2.233 0.0738320 192.6 1.926 0.0741390 243 2.43 0.076310 217.4 2.174 0.0869380 271.1 2.711 0.0873360 257.2 2.572 0.0877350 263 2.63 0.0925340 258.8 2.588 0.0939300 229.3 2.293 0.0952290 242.6 2.426 0.1049330 291.9 2.919 0.1099370 333.2 3.332 0.1109320 286.2 2.862 0.1114390 362.8 3.628 0.1143310 322 3.22 0.1307380 404.1 4.041 0.1313360 382.8 3.828 0.1318350 391 3.91 0.139340 384.2 3.842 0.1412300 338.4 3.384 0.1432290 357.1 3.571 0.1578330 431.8 4.318 0.1653370 494.9 4.949 0.1668320 422.8 4.228 0.1675390 539.7 5.397 0.1719310 473.3 4.733 0.1965380 599.6 5.996 0.1975360 566.8 5.668 0.1982350 577.6 5.776 0.2091340 566.5 5.665 0.2123300 494.8 4.948 0.2154290 519.8 5.198 0.2373330 633.3 6.333 0.2485370 730.8 7.308 0.2507320 618.7 6.187 0.2519390 798.5 7.985 0.2584310 687.3 6.873 0.2954380 883.8 8.838 0.2969360 832.7 8.327 0.298350 845.9 8.459 0.3144340 827.2 8.272 0.3192300 713.5 7.135 0.3238290 743.2 7.432 0.3568

11


330 917.5 9.175 0.3737370 1069 10.69 0.377320 892.7 8.927 0.3787390 1172 11.72 0.3886310 979.8 9.798 0.4442380 1289.7 12.897 0.4465360 1208.9 12.089 0.4482350 1221.9 12.219 0.4727340 1190.2 11.902 0.48300 1005.3 10.053 0.487290 1031.2 10.312 0.5364330 1303 13.03 0.5619370 1542.5 15.425 0.5669320 1261.2 12.612 0.5695390 1699.5 16.995 0.5843310 1355.7 13.557 0.6679380 1854.1 18.541 0.6714360 1724.1 17.241 0.6739350 1728.7 17.287 0.7108340 1672.8 16.728 0.7217300 1363.6 13.636 0.7322330 1796 17.96 0.8449370 2180.8 21.808 0.8525320 1723.6 17.236 0.8563390 2421.7 24.217 0.8785310 1785 17.85 1.0043380 2607.3 26.073 1.0095360 2393 23.93 1.0133350 2369.3 23.693 1.0688340 2267.8 22.678 1.0852330 2356.4 23.564 1.2704370 2991 29.91 1.2818390 3365.4 33.654 1.321380 3553.1 35.531 1.5179350 3096.6 30.966 1.6071340 2908.3 29.083 1.6317330 2859.9 28.599 1.9102

0.5001 Difluoromethane T(K) P(kPa) P (bar) rho mol/dm3

300 363.5 3.635 0.1532300 769.5 7.695 0.3461300 1490.2 14.902 0.7818310 377 3.77 0.1531310 553.7 5.537 0.2302310 803.2 8.032 0.3459310 1142.3 11.423 0.5199310 1580.4 15.804 0.7815310 2079.9 20.799 1.1745320 390.6 3.906 0.1531

12


320 574.7 5.747 0.2301320 835.9 8.359 0.3458320 1196.8 11.968 0.5197320 1667.5 16.675 0.7811320 2227.2 22.272 1.174330 403.8 4.038 0.153330 595.3 5.953 0.2299330 868.9 8.689 0.3456330 1248.7 12.487 0.5194330 1725.5 17.255 0.7807330 2369.8 23.698 1.1734330 3025.8 30.258 1.7636340 417.2 4.172 0.1529340 615.6 6.156 0.2298340 900.2 9.002 0.3454340 1299.9 12.999 0.5192340 1835.4 18.354 0.7803340 2506.9 25.069 1.1728340 3258.5 32.585 1.7627350 430.6 4.306 0.1528350 635.7 6.357 0.2297350 932.5 9.325 0.3453350 1349.5 13.495 0.519350 1917 19.17 0.78350 2640.8 26.408 1.1723350 3485.3 34.853 1.7619360 443.8 4.438 0.1528360 656.1 6.561 0.2296360 964.3 9.643 0.3451360 1400.2 14.002 0.5187360 1997.3 19.973 0.7796360 2774.1 27.741 1.717360 3705.3 37.053 1.76370 457 4.57 0.1527370 676.6 6.766 0.2295370 996.3 9.963 0.3449370 1449.6 14.496 0.5184370 2076.4 20.764 0.7792370 2903.4 29.034 1.711370 3918.9 39.189 1.7602380 94 0.94 0.0299380 140.9 1.409 0.045380 211 2.11 0.0676380 315.3 3.153 0.1015380 469.7 4.697 0.1526380 696.7 6.967 0.2294380 1026.6 10.266 0.3448380 1497.7 14.977 0.5182380 2153.3 21.533 0.7788380 3030 30.3 1.1706

13


380 4131 41.31 1.7593 0.6977 Difluoromethane T(K) P(kPa) P (bar) rho mol/dm3

310 290.7 2.907 0.1166310 429 4.29 0.1753310 628.5 6.285 0.2634310 906.5 9.065 0.3959310 1286.6 12.866 0.595310 1763.8 17.638 0.8943320 300.6 3.006 0.1166320 444.5 4.445 0.1752320 652.7 6.527 0.2633320 946.7 9.467 0.3957320 1348.2 13.482 0.5948320 1864.9 18.649 0.8939320 2463.7 24.637 1.3435330 310.7 3.107 0.1165330 460 4.6 0.1751330 676.6 6.766 0.2632330 984.5 9.845 0.3955330 1408.7 14.087 0.5945330 1963.1 19.631 0.8935330 2631.8 26.318 1.3429330 3311.8 33.118 2.0183340 320.6 3.206 0.1164340 475.3 4.753 0.175340 699.5 6.995 0.263340 1021.8 10.218 0.3954340 1468.1 14.681 0.5942340 2060.2 20.602 0.8931340 2792.7 27.927 1.3423340 3584.2 35.842 2.0174350 330.5 3.305 0.1164350 490.5 4.905 0.1749350 723.9 7.239 0.2629350 1057.7 10.577 0.3952350 1526.4 15.264 0.5939350 2154.9 21.549 0.8926350 2949.5 29.495 1.3416350 3851.5 38.515 2.0164360 340.4 3.404 0.1163360 505.6 5.056 0.1748360 747.1 7.471 0.2628360 1094.8 10.948 0.395360 1581.3 15.813 0.5936

14


360 2246.9 22.469 0.8922360 3102.7 31.027 1.341360 4108.3 41.083 2.0154370 350 3.5 0.1163370 520.6 5.206 0.1748370 770.3 7.703 0.2627370 1131.1 11.311 0.3948370 1638.2 16.382 0.5933370 2338.8 23.388 0.8918370 3253 32.53 1.3403370 4355.8 43.558 2.0144380 161.2 1.612 0.0514380 241.2 2.412 0.0773380 360 3.6 0.1162380 536 5.36 0.1747380 793.8 7.938 0.2625380 1167 11.67 0.3946380 1697 16.97 0.593380 2429 24.29 0.8913380 3401.8 34.018 1.3396380 4600.4 46.004 2.0134

15


Appendix B Figure 6. - 0.5001 Difluoromethane

Difluoromethane / Pentafluoroethane Mixture(0.5001 Difluoromethane)

253035404550

320 330 340 350 360 370 380 390

Temperature (K)

Pres

sure

(bar

)


V = 568 cm^3/mole

Difluoromethane and Pentafluoroethane(0.5001 Difluoromethane)

20

25

30

35

300 320 340 360 380

Temperature (K)

Pres

sure

(bar

)


V = 853 cm^3/mole

16



12141618202224

290 340 390Temperature (K)

Pres

sure

(bar

)


V = 1282 cm^3/mole


10111213141516


Pres

sure

(bar

)


V = 1927

17



10111213141516

290 340 390

Temperature (K)

Pres

sure

(bar

)


V = 1927


6789

1011


Pres

sure

(bar

) P (actual)P mix 1P mix 2P mix 3P mix 4P mix 5V = 2895 cm^3/mole

18


Appendix C Figure 7. - 0.6977 Difluoromethane


303540455055

320 340 360 380Temperature (K)

Pres

sure

(bar

)


V = 496 cm^3/mole


20

25

30

35

40

310 330 350 370 390Temperature (K)

Pres

sure

(bar

)


V = 745 cm^3/mole

19



15171921232527


Pres

sure

(bar

)


V = 1120 cm^3/mole


12131415161718


Pres

sure

(bar

) P (actual)P mix 1P mix 2P mix 3P mix 4P mix 5

V = 1683 cm^3/mole

20



56789

10

300 350Temperature (K)

Pres

sure

(bar

)


V = 3803 cm^3/mole


4

4.5

5

5.5

6


Pres

sure

(bar

)


V = 5715 cm^3/mole

21



1

2

3

4

5


Pres

sure

(bar

)


V = 8590 cm^3/mole

22


23

Comparison of Mixing Rules for a van der Waal’s Gas ... · scientists are familiar with the van...

Documents

Transcript of Comparison of Mixing Rules for a van der Waal’s Gas ... · scientists are familiar with the van...