Liquid-Liquid Equilibrium Digram for Aniline - Methyl-cyclopentane - Hexane at 45 °C and 1 atm
​Authors: Housam Binous and Ahmed Bellagi

We compute the LLE diagram and tie-lines for Aniline - Methyl-cyclopentane - Hexane at 45 °C and 1 atm . Calculations use NRTL activity coefficients and are based on a new approach, which is thoroughly described in Ref . 1-2.
​
​aniline (component 1)
n-hexane (component 2)
methyl-cyclopentane (component 3)​
​
​References:​
[1] H. Binous and A. Bellagi, Calculation of ternary liquid-liquid equilibrium data using arc-length continuation. Engineering Reports. 2021; 3:e12296. https://doi.org/10.1002/eng2.12296​
[2] H. Binous , K. Mejbri , and A. Bellagi , Application of graduate-level numerical tools to teach phase equilibria of liquid ternary systems. Comput Appl Eng Educ. 2021; 29: 1589–1601. https://doi.org/10.1002/cae.22411
NRTL model
In[]:=
TK=273.15+45;​​​​GAMMA1=Exp[(t21G21x2+t31G31x3)/(x1+G21x2+G31x3)+x1/(x1+G21x2+G31x3)(-(x2t21G21+x3t31G31)/(x1+G21x2+G31x3))+x2G12/(G12x1+x2+G32x3)(t12-(x1t12G12+x3t32G32)/(G12x1+x2+G32x3))+​​x3G13/(G13x1+G23x2+x3)(t13-(x1t13G13+x2t23G23)/(G13x1+G23x2+x3))]/.x31-x1-x2;​​GAMMA1p=Exp[(t21G21xp2+t31G31xp3)/(xp1+G21xp2+G31xp3)+xp1/(xp1+G21xp2+G31xp3)(-(xp2t21G21+xp3t31G31)/(xp1+G21xp2+G31xp3))+xp2G12/(G12xp1+xp2+G32xp3)(t12-(xp1t12G12+xp3t32G32)/(G12xp1+xp2+G32xp3))+​​xp3G13/(G13xp1+G23xp2+xp3)(t13-(xp1t13G13+xp2t23G23)/(G13xp1+G23xp2+xp3))]/.xp31-xp1-xp2;​​GAMMA2=Exp[(t12G12x1+t32G32x3)/(G12x1+x2+G32x3)+x1G21/(x1+G21x2+G31x3)(t21-(x2t21G21+x3t31G31)/(x1+G21x2+G31x3))+x2/(G12x1+x2+G32x3)(-(x1t12G12+x3t32G32)/(G12x1+x2+G32x3))+x3G23/(G13x1+G23x2+x3)(t23-(x1t13G13+x2t23G23)/(G13x1+G23x2+x3))]/.x31-x1-x2;​​GAMMA2p=Exp[(t12G12xp1+t32G32xp3)/(G12xp1+xp2+G32xp3)+xp1G21/(xp1+G21xp2+G31xp3)(t21-(xp2t21G21+xp3t31G31)/(xp1+G21xp2+G31xp3))+xp2/(G12xp1+xp2+G32xp3)(-(xp1t12G12+xp3t32G32)/(G12xp1+xp2+G32xp3))+xp3G23/(G13xp1+G23xp2+xp3)(t23-(xp1t13G13+xp2t23G23)/(G13xp1+G23xp2+xp3))]/.xp31-xp1-xp2;​​GAMMA3=Exp[(t13G13x1+t23G23x2)/(G13x1+G23x2+x3)+x1G31/(x1+G21x2+G31x3)(t31-(x2t21G21+x3t31G31)/(x1+G21x2+G31x3))+x2G32/(G12x1+x2+G32x3)(t32-(x1t12G12+x3t32G32)/(G12x1+x2+G32x3))+x3/(G13x1+G23x2+x3)(-(x1t13G13+x2t23G23)/(G13x1+G23x2+x3))]/.x31-x1-x2;​​GAMMA3p=Exp[(t13G13xp1+t23G23xp2)/(G13xp1+G23xp2+xp3)+xp1G31/(xp1+G21xp2+G31xp3)(t31-(xp2t21G21+xp3t31G31)/(xp1+G21xp2+G31xp3))+xp2G32/(G12xp1+xp2+G32xp3)(t32-(xp1t12G12+xp3t32G32)/(G12xp1+xp2+G32xp3))+xp3/(G13xp1+G23xp2+xp3)(-(xp1t13G13+xp2t23G23)/(G13xp1+G23xp2+xp3))]/.xp31-xp1-xp2;​​G12=Exp[-a12t12];G21=Exp[-a12t21];G13=Exp[-a13t13];G31=Exp[-a13t31];G32=Exp[-a32t32];G23=Exp[-a32t23];​​​​t13=A13+B13/TK;t23=A23+B23/TK;t12=A12+B12/TK;​​t32=A32+B32/TK;t31=A31+B31/TK;t21=A21+B21/TK;​​
Parameters for the NRTL model obtained from Aspen-Properties
In[]:=
param={a120.30,a130.20,a320.30,​​​​B212291.8179,B121410.9323,A12-2.8027,A21-5.2486,​​​​B23-153.9573,B32190.1605,A320.0,A230.0,​​​​B13857.3718,B312650.6843,A13-1.9693,A31-7.1321};​​
In[]:=
K={GAMMA1/GAMMA1p,GAMMA2/GAMMA2p,GAMMA3/GAMMA3p}/.param;
In[]:=
z={0.4,0.5,0.1};
In[]:=
sol=FindRoot
3
∑
i=1
(K[[i]]-1)z[[i]](1+(K[[i]]-1)ϕ)0,x1z[[1]](1+(K[[1]]-1)ϕ),x2z[[2]](1+(K[[2]]-1)ϕ),xp1K[[1]]z[[1]](1+(K[[1]]-1)ϕ),xp2K[[2]]z[[2]](1+(K[[2]]-1)ϕ),{ϕ,0.4},{x1,0.15},{x2,0.87},{xp1,0.87},{xp2,0.15},MaxIterations->1000
Out[]=
{ϕ0.391481,x10.12261,x20.741532,xp10.831177,xp20.124561}
In[]:=
z={0.4,a,1-0.4-a}/.aa[s];
In[]:=
eq1=
3
∑
i=1
(K[[i]]-1)z[[i]]/(1+(K[[i]]-1)ϕ)0/.x1x1[s]/.x2x2[s]/.xp1xp1[s]/.xp2xp2[s]/.ϕϕ[s];
In[]:=
eq2=x1z[[1]]/(1+(K[[1]]-1)ϕ)/.x1x1[s]/.x2x2[s]/.xp1xp1[s]/.xp2xp2[s]/.ϕϕ[s];​​eq3=x2z[[2]]/(1+(K[[2]]-1)ϕ)/.x1x1[s]/.x2x2[s]/.xp1xp1[s]/.xp2xp2[s]/.ϕϕ[s];​​eq4=xp1K[[1]]z[[1]]/(1+(K[[1]]-1)ϕ)/.x1x1[s]/.x2x2[s]/.xp1xp1[s]/.xp2xp2[s]/.ϕϕ[s];​​eq5=xp2K[[2]]z[[2]]/(1+(K[[2]]-1)ϕ)/.x1x1[s]/.x2x2[s]/.xp1xp1[s]/.xp2xp2[s]/.ϕϕ[s];
In[]:=
eq6=x1'[s]^2+x2'[s]^2+xp1'[s]^2+xp2'[s]^2+ϕ'[s]^2+a'[s]^21;
In[]:=
derivinit={x1'[0]0,x2'[0]0,xp1'[0]0,xp2'[0]0,ϕ'[0]1,a'[0]-0.4};
In[]:=
init={x1[0](x1/.sol),x2[0](x2/.sol),xp1[0](xp1/.sol),xp2[0](xp2/.sol),ϕ[0](ϕ/.sol),a[0]0.3};
In[]:=
system=Flatten[Join[{{eq1,eq2,eq3,eq4,eq5,eq6},derivinit,init}]];
In[]:=
SOL=Quiet@NDSolve[system,{x1,x2,xp1,xp2,ϕ,a},{s,-100,1000},AccuracyGoal->5]
Out[]=
x1InterpolatingFunction
Domain: {{-3.33,19.1}}
Output: scalar
,x2InterpolatingFunction
Domain: {{-3.33,19.1}}
Output: scalar
,xp1InterpolatingFunction
Domain: {{-3.33,19.1}}
Output: scalar
,xp2InterpolatingFunction
Domain: {{-3.33,19.1}}
Output: scalar
,ϕInterpolatingFunction
Domain: {{-3.33,19.1}}
Output: scalar
,aInterpolatingFunction
Domain: {{-3.33,19.1}}
Output: scalar

In[]:=
tbl1=Flatten[Table[{x1[s],1-x1[s]-x2[s],x2[s]}/.SOL,{s,-3.3,19,0.02}],1];
In[]:=
tbl2=Flatten[Table[{xp1[s],1-xp1[s]-xp2[s],xp2[s]}/.SOL,{s,-3.3,19,0.02}],1];
In[]:=
p1=TernaryListPlot[{tbl1,tbl2},PlotRangeAll,GridLines->Automatic,PlotStyleThickness[0.0075],Joined->True,FrameLabel->{Style["Aniline",16],Style["C5H9CH3",16],Style["n-hexane",16]},PlotRange->All,Method->{"Backdrop"->Lighter[Yellow,.85]}];
25 Tie lines computed with Aspen-Properties
In[]:=
tbltieaspen25={{"tie line",Style["aniline",Darker@Red],Style["C5H9CH3",Darker@Red],Style["n-hexane",Darker@Red],Style["aniline",Darker@Blue],Style["C5H9CH3",Darker@Blue],Style["n-hexane",Darker@Blue]},{1,0.0948185,0,0.9051814,0.8728427,0,0.1271573},{2,0.0993864,0.025371,0.8752425,0.8657988,0.00750088,0.1267004},{3,0.1041562,0.050487,0.8453568,0.85853,0.0152341,0.1262358},{4,0.1091434,0.0753336,0.8155229,0.8510207,0.0232155,0.1257638},{5,0.1143646,0.0998951,0.7857402,0.8432524,0.0314625,0.125285},{6,0.1198413,0.1241533,0.7560054,0.8352058,0.0399947,0.1247995},{7,0.1255959,0.1480875,0.7263166,0.8268578,0.0488341,0.1243081},{8,0.1316552,0.1716741,0.6966707,0.8181827,0.0580057,0.1238116},{9,0.1380482,0.1948863,0.6670655,0.8091511,0.0675379,0.1233109},{10,0.1448099,0.2176924,0.6374976,0.7997282,0.0774637,0.1228081},{11,0.1519829,0.2400557,0.6079614,0.789874,0.0878207,0.1223052},{12,0.1596153,0.2619328,0.5784518,0.779541,0.0986534,0.1218055},{13,0.1677657,0.2832721,0.5489621,0.7686728,0.1100139,0.1213133},{14,0.1765045,0.3040117,0.5194838,0.757201,0.1219646,0.1208344},{15,0.1859182,0.3240759,0.4900059,0.7450418,0.134581,0.1203772},{16,0.1961144,0.3433716,0.460514,0.7320908,0.1479559,0.1199534},{17,0.2072304,0.3617812,0.4309884,0.7182146,0.1622056,0.1195799},{18,0.2194449,0.3791536,0.4014014,0.703239,0.1774794,0.1192816},{19,0.2329989,0.3952888,0.3717122,0.6869287,0.1939747,0.1190967},{20,0.2482303,0.4099119,0.3418578,0.6689521,0.2119625,0.1190855},{21,0.2656403,0.4226252,0.3117345,0.6488157,0.2318351,0.1193492},{22,0.2860336,0.43281,0.2811564,0.6257243,0.2542043,0.1200714},{23,0.3108708,0.4393845,0.2497446,0.5982297,0.2801419,0.1216285},{24,0.3434493,0.4400144,0.2165363,0.5630529,0.311967,0.1249801},{25,0.3941572,0.4275496,0.1782932,0.5038525,0.3604729,0.1356746}};
In[]:=
TableForm[tbltieaspen25]
Out[]//TableForm=
tie line
aniline
C5H9CH3
n-hexane
aniline
C5H9CH3
n-hexane
1
0.0948185
0
0.905181
0.872843
0
0.127157
2
0.0993864
0.025371
0.875243
0.865799
0.00750088
0.1267
3
0.104156
0.050487
0.845357
0.85853
0.0152341
0.126236
4
0.109143
0.0753336
0.815523
0.851021
0.0232155
0.125764
5
0.114365
0.0998951
0.78574
0.843252
0.0314625
0.125285
6
0.119841
0.124153
0.756005
0.835206
0.0399947
0.1248
7
0.125596
0.148088
0.726317
0.826858
0.0488341
0.124308
8
0.131655
0.171674
0.696671
0.818183
0.0580057
0.123812
9
0.138048
0.194886
0.667066
0.809151
0.0675379
0.123311
10
0.14481
0.217692
0.637498
0.799728
0.0774637
0.122808
11
0.151983
0.240056
0.607961
0.789874
0.0878207
0.122305
12
0.159615
0.261933
0.578452
0.779541
0.0986534
0.121806
13
0.167766
0.283272
0.548962
0.768673
0.110014
0.121313
14
0.176505
0.304012
0.519484
0.757201
0.121965
0.120834
15
0.185918
0.324076
0.490006
0.745042
0.134581
0.120377
16
0.196114
0.343372
0.460514
0.732091
0.147956
0.119953
17
0.20723
0.361781
0.430988
0.718215
0.162206
0.11958
18
0.219445
0.379154
0.401401
0.703239
0.177479
0.119282
19
0.232999
0.395289
0.371712
0.686929
0.193975
0.119097
20
0.24823
0.409912
0.341858
0.668952
0.211963
0.119086
21
0.26564
0.422625
0.311735
0.648816
0.231835
0.119349
22
0.286034
0.43281
0.281156
0.625724
0.254204
0.120071
23
0.310871
0.439385
0.249745
0.59823
0.280142
0.121629
24
0.343449
0.440014
0.216536
0.563053
0.311967
0.12498
25
0.394157
0.42755
0.178293
0.503853
0.360473
0.135675
Our Tie lines computed with Mathematica
In[]:=
i=1;While[i<=24,{solt[i]=FindRoot[(x2[s]/.SOL)==tbltieaspen25[[i+1]][[4]],{s,-0.4},MaxIterations->1000];Print[i," ",solt[i]];i++}]
1 {s-0.299792}
2 {s-0.26368}
3 {s-0.227572}
4 {s-0.191459}
5 {s-0.155332}
6 {s-0.119175}
7 {s-0.0829729}
8 {s-0.046709}
9 {s-0.0103619}
10 {s0.0261069}
11 {s0.0627031}
12 {s0.0994795}
13 {s0.136484}
14 {s0.173778}
15 {s0.211443}
16 {s0.249584}
17 {s0.288353}
18 {s0.327959}
19 {s0.368712}
20 {s0.411108}
21 {s0.456012}
22 {s0.505135}
23 {s0.562663}
24 {s0.643533}
In[]:=
mytie=Flatten[Table[{i,x1[s],1-x1[s]-x2[s],x2[s],xp1[s],1-xp1[s]-xp2[s],xp2[s]}/.SOL/.solt[i],{i,1,24}],1]
Out[]=
{{1,0.0948197,-1.05312×
-6
10
,0.905181,0.872841,5.70778×
-7
10
,0.127158},{2,0.099388,0.0253695,0.875243,0.865799,0.00750035,0.1267},{3,0.104157,0.0504861,0.845357,0.85853,0.0152341,0.126236},{4,0.109144,0.0753328,0.815523,0.85102,0.0232158,0.125764},{5,0.114365,0.0998943,0.78574,0.843252,0.0314623,0.125285},{6,0.119842,0.124153,0.756005,0.835206,0.0399944,0.1248},{7,0.125597,0.148087,0.726317,0.826857,0.0488342,0.124308},{8,0.131655,0.171674,0.696671,0.818183,0.0580058,0.123812},{9,0.138048,0.194887,0.667066,0.809151,0.0675381,0.123311},{10,0.14481,0.217692,0.637498,0.799728,0.0774637,0.122808},{11,0.151983,0.240056,0.607961,0.789874,0.0878208,0.122305},{12,0.159616,0.261933,0.578452,0.779541,0.0986536,0.121806},{13,0.167765,0.283272,0.548962,0.768672,0.110015,0.121314},{14,0.176501,0.304016,0.519484,0.757198,0.121966,0.120835},{15,0.185908,0.324086,0.490006,0.745033,0.134587,0.12038},{16,0.196118,0.343368,0.460514,0.732095,0.147953,0.119952},{17,0.207231,0.36178,0.430988,0.71822,0.162202,0.119578},{18,0.219425,0.379174,0.401401,0.703215,0.177497,0.119289},{19,0.232998,0.395289,0.371712,0.686929,0.193974,0.119097},{20,0.248229,0.409913,0.341858,0.668951,0.211963,0.119086},{21,0.265634,0.422631,0.311735,0.648804,0.231844,0.119353},{22,0.286034,0.43281,0.281156,0.625724,0.254204,0.120071},{23,0.310872,0.439383,0.249745,0.598235,0.280138,0.121627},{24,0.343448,0.440016,0.216536,0.563046,0.311972,0.124982}}
In[]:=
TableForm[Join[{{"tie line",Style["aniline",Darker@Red],Style["C5H9CH3",Darker@Red],Style["n-hexane",Darker@Red],Style["aniline",Darker@Blue],Style["C5H9CH3",Darker@Blue],Style["n-hexane",Darker@Blue]}},mytie]]
Out[]//TableForm=
tie line
aniline
C5H9CH3
n-hexane
aniline
C5H9CH3
n-hexane
1
0.0948197
-1.05312×
-6
10
0.905181
0.872841
5.70778×
-7
10
0.127158
2
0.099388
0.0253695
0.875243
0.865799
0.00750035
0.1267
3
0.104157
0.0504861
0.845357
0.85853
0.0152341
0.126236
4
0.109144
0.0753328
0.815523
0.85102
0.0232158
0.125764
5
0.114365
0.0998943
0.78574
0.843252
0.0314623
0.125285
6
0.119842
0.124153
0.756005
0.835206
0.0399944
0.1248
7
0.125597
0.148087
0.726317
0.826857
0.0488342
0.124308
8
0.131655
0.171674
0.696671
0.818183
0.0580058
0.123812
9
0.138048
0.194887
0.667066
0.809151
0.0675381
0.123311
10
0.14481
0.217692
0.637498
0.799728
0.0774637
0.122808
11
0.151983
0.240056
0.607961
0.789874
0.0878208
0.122305
12
0.159616
0.261933
0.578452
0.779541
0.0986536
0.121806
13
0.167765
0.283272
0.548962
0.768672
0.110015
0.121314
14
0.176501
0.304016
0.519484
0.757198
0.121966
0.120835
15
0.185908
0.324086
0.490006
0.745033
0.134587
0.12038
16
0.196118
0.343368
0.460514
0.732095
0.147953
0.119952
17
0.207231
0.36178
0.430988
0.71822
0.162202
0.119578
18
0.219425
0.379174
0.401401
0.703215
0.177497
0.119289
19
0.232998
0.395289
0.371712
0.686929
0.193974
0.119097
20
0.248229
0.409913
0.341858
0.668951
0.211963
0.119086
21
0.265634
0.422631
0.311735
0.648804
0.231844
0.119353
22
0.286034
0.43281
0.281156
0.625724
0.254204
0.120071
23
0.310872
0.439383
0.249745
0.598235
0.280138
0.121627
24
0.343448
0.440016
0.216536
0.563046
0.311972
0.124982
Ternary diagram for ANILINE/METHY-CYCLOPENTANE/HEXANE mixture (obtained with Mathematica).
Tie lines with Aspen-Properties in blue; our tie lines using Mathematica are in red.
Partition of the solute (METHYL-CYCLOPENTANE) between ANILINE-rich and n-HEXANE-rich phases. Blue Dots : Aspen-Properties; Red Curve : our calculated distribution line using method described in Ref . 1 - 2.