6.4 VLE data regression
6.4.2 Modelling results for the 2-propanol (1) + 4-methyl-2-pentanone (2) system
112
Figure 6-16: Temperature dependence of the NRTL-HOC model combination parameters for 1-propanol (1) + 4-methyl-2-pentanone (2) system. (■), g21-g12; (♦) g12-g21
6.4.2 Modelling results for the 2-propanol (1) + 4-methyl-2-pentanone (2) system
113 values of 0.006, 0.022 and 0.053 respectively.
In summary, it can be seen that the ∆y1avg values obtained for each data set are lower than 0.01 indicating that at each isotherm the VLE data measured pass the first criterion of the point test (Van Ness, 1973). As observed in Figures (6-19) the y residuals plot for isotherm 323.15K with the NRTL-HOC and NRTL-NTH model combinations show important negative bias and failed in the random scattering about the x-axis this was due to the systematic error in the calibration of the temperature probe and the pressure transducer. Also comparing the RMSD δln(γ1/γ2) values (the highest value being 0.066) obtained for the data sets with the consistency index table of Van Ness (Van Ness, 1995) as discussed earlier shows that the measured data passed the direct test.
Generally, all the model combinations fit the data set at the three different isotherms well as shown in the Figures below and the P-x-y plots presented in Appendix A. For the 323.15 K isotherm, Figures (6-17) and (6-18) give the comparison of the model fits to the experimental data. The graphical representation of the point test and direct test is shown in Figures (6-19 and 6-20). The P-x-y plots showing the comparison of the activity coefficient models for the three isotherms are shown in Appendix A. The point test and direct test plots for this system are presented in Appendix A.
114
Table 6-5: Model results obtained for the 2-propanol (1) + 4-methyl-2-pentanone (2) system
model T/ K aij aji bij bji AAD
∆P/kPa AAD
(∆y) RMSD δln(γ1/γ2)
NRTL-HOC 323.15 11.6 -3.98 -3564.3 1360.34 0.011 0.006 0.050 338.15 8.41 6.21 -2688.7 -1986.5 0.024 0.006 0.053 353.15 -12.4 -0.40 4555.28 152.99 0.019 0.006 0.053 NRTL-NTH 323.15 13.11 -0.11 -4040.3 112.47 0.010 0.006 0.053 338.15 8.34 6.71 -2690.1 -2142.7 0.025 0.007 0.059 353.15 12.25 0.22 -4137.8 -49.82 0.024 0.008 0.066 WILSON-HOC 323.15 -0.07 -0.2 -81.24 -113.96 0.010 0.006 0.051 338.15 -2.51 -3.44 731.15 1025.66 0.023 0.006 0.055 353.15 1.77 3.11 -676.05 -1253.6 0.022 0.006 0.053 UNIQUAC-HOC 323.15 46.44 -155 0.00 0.00 0.01 0.006 0.051 338.15 0.00 0.00 65.59 -172.11 0.023 0.006 0.056 353.15 -0.07 0.09 79.59 -173.69 0.022 0.006 0.053
u(T) = 0.06 K bij and aij are the binary interaction parameters for the activity coefficient models in ASPEN, (τij = aij + bij/T), αij (non-randomness parameter) set to 0.3 for the NRTL-HOC and NRTL-NTH model combinations, RMSD δln(γ1/γ2) (the Root Mean Square deviation for the direct test)
115
Figure 6-17: Fit of the NRTL-HOC and the NRTL-NTH models to the P-x-y plot of the 2-propanol (1) + methyl isobutyl (2) system at 323.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
Figure 6-18: Fit of the NRTL-HOC and NRTL-NTH model combinations to the x-y plot of the 2-propanol (1) + methyl isobutyl ketone (2) system at 323.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
7,0 9,0 11,0 13,0 15,0 17,0 19,0 21,0 23,0 25,0
0,0 0,2 0,4 0,6 0,8 1,0
P/ kPa
x1,y1
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
0,0 0,2 0,4 0,6 0,8 1,0
y1
x1
116
Figure 6-19: Point test (varying EOS): ∆y1 for the 2-propanol (1) + methyl isobutyl ketone (2) system at 323.15K: (◦) NRTL-HOC, (∆) NRTL-NTH
Figure 6-20: Direct test (varying EOS): δln(γ1/γ2) for the 2-propanol (1) + 4-methyl-2- pentanone (2) system at 323.15K: (◦) NRTL-HOC, (∆) NRTL-NTH
-0,025 -0,020 -0,015 -0,010 -0,005 0,000 0,005 0,010 0,015 0,020
0,0 0,2 0,4 0,6 0,8 1,0
y1exp-y1calc
x1
-0,11 -0,09 -0,07 -0,05 -0,03 -0,01 0,01 0,03
0,0 0,2 0,4 0,6 0,8 1,0
δ ln(γ1/γ2)
x1
117
Figure 6-21: Fit of the NRTL-HOC and NRTL-NTH model combinations to the p-x-y plot of the 2-propanol (1) + 4-methyl-2-pentanone (2) system at 338.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
Figure 6-22: Fit of the NRTL-HOC and NRTL-NTH model combinations to the x-y plot of the 2-propanol (1) + 4-methyl-2-pentanone (2) system at 338.15 K for the combined method: (•) this work, (—) NRTL-HOC, (- - - -) NRTL-NTH
15 20 25 30 35 40 45 50 55
0,0 0,2 0,4 0,6 0,8 1,0
P/ kPa
x1,y1
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
0,0 0,2 0,4 0,6 0,8 1,0
y1
x1
118
Figure 6-23: Fit of the NRTL-HOC and NRTL-NTH model combinations to the p-x-y plot of the 2-propanol (1) + 4-methyl-2-pentanone (2) system at 353.15 K for the combined method: (•) this work, (—) NRTL-HOC, (- - - -) NRTL-NTH
Figure 6-24: Fit of the NRTL-HOC and NRTL-NTH model combinations to the x-y plot of the 2-propanol (1) + 4-methyl-2-pentanone (2) system at 353.15 K for the combined method: (•) this work, (—) NRTL-HOC, (- - - -) NRTL-NTH
25 35 45 55 65 75 85 95 105
0,0 0,2 0,4 0,6 0,8 1,0
P/ kPa
x1,y1
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
0,0 0,2 0,4 0,6 0,8 1,0
y1
x1
119
Figure 6-25: Comparison between the experimentally determined liquid-phase activity coefficients and those calculated from the NRTL-HOC model combination for 2- propanol (1) + 4-methyl-2-pentanone (2) system at 323.15 K, 338.15 K and 353.15 K.
323.15 K (―), 338.15 K (…) and 353.15 K (- - -) for NRTL-HOC model combination;
323.15 K (○), 338.15 K (□) and 353.15 K (Δ), for experimental γ1; 323.15 K (●), 338.15 K (■) and 353.15 K (▲), for experimental γ2
A representative plot of the comparison between the calculated liquid phase activity coefficients and the values obtained from the NRTL-HOC model combination is shown in Figure 6-25. From the figure, it shows that the NRTL-HOC model combination fitted the data set well.
-0,1 0,1 0,3 0,5 0,7 0,9
0 0,2 0,4 0,6 0,8 1
ln(γ1,γ2)
x1
120
Figure 6-26: Temperature dependence of the NRTL-HOC model combination parameters for 2-propanol (1) + 4-methyl-2-pentanone (2) system. (■) g21-g12; (♦) g12-g21
6.4.3 Modelling results obtained for the 2-pentanone (1) + methylpropan-1-ol system Three isothermal VLE data sets were measured for this system and the data sets were regressed with the combined method approach following the procedure explained in section 6-4. Table 6-6 summarizes the results obtained for the temperature dependent binary interaction parameters, the ∆yavg, ∆Pavg and the RMSD δln (γ1/γ2) for the three isotherms.
Similarly to the previously discussed systems, four model combinations were used for the regression of the data sets in order to obtain the best fit as a model combination may fit a data set well than the others. The model combinations are: the NRTL-HOC, the NRTL-NTH, the WILSON-HOC and the UNIQUAC-HOC model combinations. The model variations were done to determine of the best fit model combinations.
Varying the equation of state for the first isotherm (343.15 K) i.e. using the NRTL-HOC and NRTL-NTH model combination, the NRTL-NTH combination gave the best fit for the VLE data set based on the ∆yavg values, although both gave the same value for the ∆Pavg. The WILSON-HOC and UNIQUAC-HOC gave the best fit when the activity coefficient models were varied for this isotherm; both have the same values for ∆yavg and ∆Pavg.
∆g12= 0.9853T2- 659.81T + 111791
∆g21= -0.972T2+ 634.49T - 102695 0
400 800 1200 1600 2000
320 325 330 335 340 345 350 355
(g12-g22)/Jmol-1,(g21-g11)/Jmol-1
Temperature/K
121
For the 358.15 K, the NRTL-NTH model combination gave the best fit based on ∆Pavg when the equations of state models were varied. The NRTL-HOC and the WILSON-HOC gave the same values for ∆yavg and ∆Pavg but the UNIQUAC-HOC has the best fit based on ∆Pavg when activity coefficient models were varied.
The NRTL-HOC model combination fitted the data set for the 363.15 K both when the equations of state models were varied and when the activity coefficient models were varied.
The experimental VLE data sets were subjected to thermodynamic consistency testing with the point test (Van Ness, 1973) and direct test (Van Ness, 1975) in order to verify the quality of the measured VLE data. For the point test, the ∆yavg values calculated for each data set were compared with the value of 0.01 (suggested by Danner and Gess, 1995) and for the direct test, the RMSD δln (γ1/γ2) values calculated also for each data set were compared with the consistency index table of Van Ness (Van Ness, 1995). The y and P residuals for the three isotherms scatter well about the x-axis. The measured VLE data for the three isotherms pass the two thermodynamic consistency tests indicating that the newly measured VLE data are thermodynamically consistent. The P-x-y plots showing the comparison of the model fit to experimental data are shown in Appendix A. Plots for the point and direct test are shown in Appendix A. The graphical representations of the comparison of the model fit to the experimental VLE data are shown in Figures 6-28 to 6-35.
122
Table 6-6: modelling results obtained for the 2-pentanone (1) + 2-methylpropan-1-ol system
model T/ K aij aji bij bji AAD
∆P/kPa
AAD (∆y)
RMSD δln(γ1/γ2) NRTL-HOC 343.15 -1.97 -11.3 631.67 4072.9 0.006 0.005 0.036
358.15 0.73 8.48 -299.5 -2863 0.015 0.005 0.031 363.15 2.53 10.34 -990.6 -3546 0.026 0.005 0.035 NRTL-NTH 343.15 -2.49 -11.7 803.3 4245.3 0.006 0.004 0.033 358.15 1.27 8.86 -504.4 -2991 0.014 0.005 0.032 363.15 3.10 10.76 -1211 -3687 0.026 0.005 0.036 WILSON-HOC 343.15 0.00 0.00 -163.4 -2.186 0.006 0.005 0.035 358.15 0.00 0.00 -132.0 -0.87 0.015 0.005 0.033 363.15 0.00 0.00 -160.6 26.29 0.026 0.005 0.037 UNIQUAC-HOC 343.15 0.00 0.00 12.75 -61.95 0.006 0.005 0.035 358.15 0.00 0.00 2.509 -42.07 0.014 0.005 0.033 363.15 0.00 0.00 28.27 -69.09 0.026 0.005 0.037
u(T) = 0.06 K bij and aij are the binary interaction parameters for the activity coefficient models in ASPEN, (τij = aij + bij/T), αij (non-randomness parameter) set to 0.3 for the NRTL-HOC and NRTL-NTH model combinations, RMSD δln(γ1/γ2) (the Root Mean Square deviation for the direct test)
Table 6-7: Best model for the new isotherms measured in this work
Systems Temperature (K) Best model
1-propanol (1) + 4-methyl-2-pentanone 338.15 NRTL-HOC
353.15 WILSON-HOC
368.15 NRTL-HOC
2-propanol (1) + 4-methyl-2-pentanone 323.15 NRTL-HOC
338.15 WILSON-HOC
353.15 WILSON-HOC
2-pentanone (1) + 2-methylpropan-1-ol 343.15 NRTL-HOC
358.15 NRTL-HOC
363.15 NRTL-HOC
123
Figure 6-27: Fit of the NRTL-HOC and NRTL-NTH model combinations to the P-x-y plot of the 2-pentanone (1) + 2-methylpropan-1-ol (2) system at 343.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
Figure 6-28: Fit of the NRTL-HOC and NRTL-NTH model combinations to the x-y plot of the 2-pentanone (1) + 2-methylpropan-1-ol (2) system at 343.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
19 21 23 25 27 29 31 33 35 37
0 0,2 0,4 0,6 0,8 1
P/ kPa
x1,y1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 0,2 0,4 0,6 0,8 1
y1
x1
124
Figure 6-29: Point test (varying EOS): ∆y1 for the 2-pentanone (1) + 2-methylpropan-1- ol (2) system at 343.15 K: (◦) NRTL-HOC, (∆) NRTL-NTH
Figure 6-30: Direct test (varying EOS): δln(γ1/γ2) for the 2-pentanone (1) + 2- methylpropan-1-ol (2) system at 343.15 K: (◦) NRTL-HOC, (∆) NRTL-NTH
-0,014 -0,012 -0,010 -0,008 -0,006 -0,004 -0,002 0,000 0,002 0,004
0 0,2 0,4 0,6 0,8 1
y1exp-y1calc
x1
-0,07 -0,06 -0,05 -0,04 -0,03 -0,02 -0,01 0,00 0,01 0,02
0 0,2 0,4 0,6 0,8 1
δln(γ1/γ2)
x1
125
Figure 6-31: Fit of the NRTL-HOC and NRTL-NTH model combinations to the P-x-y plot of the 2-pentanone (1) + 2-methylpropan-1-ol (2) system at 358.15 K for the combined method: (•) this work, (—) NRTL-HOC, (- - - -) NRTL-NTH
Figure 6-32: Fit of the NRTL-HOC and NRTL-NTH model combinations to the x-y plot of the 2-pentanone (1) + 2-methylpropan-1-ol (2) system at 358.15 K for the combined method: (•) this work, (—) NRTL-HOC, (- - - -) NRTL-NTH
40 42 44 46 48 50 52 54 56 58 60
0 0,2 0,4 0,6 0,8 1
P/ kPa
x1,y1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 0,2 0,4 0,6 0,8 1
y1
x1
126
Figure 6-33: Fit of the NRTL-HOC and NRTL-NTH model combinations to the P-x-y plot of the 2-pentanone (1) + 2-methylpropan-1-ol (2) system at 363.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
Figure 6-34: Fit of the NRTL-HOC and NRTL-NTH model combinations to the x-y plot of the 2-pentanone (1) + 2-methylpropan-1-ol (2) system at 363.15 K for the combined method: (•) this work, (— ) NRTL-HOC, (- - - -) NRTL-NTH
50 55 60 65 70 75
0 0,2 0,4 0,6 0,8 1
P/ kPa
x1,y1
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 0,2 0,4 0,6 0,8 1
y1
x1
127
Figure 6-35: Comparison between the experimentally determined liquid-phase activity coefficients and those calculated from the NRTL-HOC model combination for 2- pentanone (1) + 2-methylpropan-1-ol (2) system at 343.15 K, 358.15 K and 363.15 K.
(―) 343.15 K, (…) 358.15 K and (- - -) 363.15 K for NRTL-HOC model combination;
(○) 343.15 K, (□) 358.15 K and (Δ) 363.15 K for experimental γ1; (●) 343.15 K, (■) and (▲) 363.15 K for experimental γ2
Figure 6-36 shows the illustrative plot of the comparison between the experimentally calculated activity coefficients and those derived from the NRTL-HOC model combination for the three isotherms. It can also be seen from the plot that the NRTL-HOC model combination used for the regression of the isothermal VLE data sets provided good fit.
-0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6
0,0 0,2 0,4 0,6 0,8 1,0
ln(γ1,γ2)
x1
128
Figure 6-36: Temperature dependence of the NRTL-HOC model combination parameters for 2-pentanone (1) + 2-methylpropan-1-ol (2) system. (g12 – g22) (♦), (g21 – g11) (■)
Figure 6-37: Temperature dependence of the WILSON-HOC model combination parameters for 2-pentanone (1) + 2-methylpropan-1-ol (2) system. (λ12 – λ22) (♦), (λ21 – λ11) (■)
∆g12= -6.1761x2+ 4384.7x - 776899
∆g21= 5.8341x2- 4164.4x + 742992 -400
-200 0 200 400 600 800 1000 1200 1400 1600
340 345 350 355 360 365
(g12-g22), (g21-g11)/Jmol-1
Temperature/K
∆g21= 5.8303x2- 4161.3x + 742302
∆g12= -6.2058x2+ 4405.1x - 780328
-400 -200 0 200 400 600 800 1000 1200 1400 1600
340 345 350 355 360 365
(λ12 -λ22), (λ21 -λ11)/Jmol-1
Temperature/K
129