Plotting P-x-y diagram for binary system Acetone/water at temperatures 25,100,and 200
C using UNIFAC method and comparing it with experimental results.
Unifac Method:
The UNIFAC method is based on the UNIQUAC equation, for which the
activity coefficients are given by the formula
lnγ (i) ＝ lnγc (i) + lnγr (i)
(1)
where c(i) is combinatorial term to account for molecular size and shape differences, and r(i)
is a residual term to account for molecular interactions.
Function c(i) contains pure species parameters only, whereas function r(i) incorporates two
binary parameters for each pair of molecules.For a multicomponent system,
⎛
⎛ J (i) ⎞⎞
J (i)
+ ln ⎜――
lnγc (i) ＝ 1 − J (i) + ln (J (i)) − 5 ⋅ q (i) ⋅ ⎜1 − ――
⎟⎟
(
)
L i
⎝
⎝ L (i) ⎠⎠
⎛
⎛
⎛ β (ik) ⎞⎞⎞
β (ik)
lnγr (i) ＝ q (i) ⋅ ⎜1 − ∑ ⎜θ (k) ⋅ ――− e (ki) ⋅ ln ⎜――⎟⎟⎟
s (k)
⎝ s (k) ⎠⎠⎠
⎝
k ⎝
(2)
(3)
where
r (i)
J (i) ＝ ―――――
∑ (r (j) ⋅ x (j))
(4)
j
q (i)
L (i) ＝ ―――――
∑ (q (j) ⋅ x (j))
(5)
j
where
r (i) ＝ ∑ ⎛v ⋅ (i) ⋅ R (k)⎞
k
⎠
k ⎝
Non-Commercial Use Only
(6)
q (i) ＝ ∑ ⎛v ⋅ (i) ⋅ Q (k)⎞
k
⎠
k ⎝
(7)
v ⋅ (i) ⋅ Q (k)
k
e (ki) ＝ ――――
q (i)
(8)
∑ (x (i) ⋅ q (i) ⋅ e (ki))
i
θ (k) ＝ ―――――――
∑ (x (j) ⋅ q (j))
(9)
j
s (k) ＝ ∑ (θ (m) ⋅ τ (mk))
(10)
m
⎛ −a (mk) ⎞
τ (mk) ＝ exp ⎜―――
⎟
T
⎝
⎠
(11)
T --- Temperature
β (ik) ＝ ∑ (e (mi) ⋅ τ (mk))
(12)
m
Subscript i identifies species, and j is a dummy index running over all species. Subscript k
identifies subgroups, and m is a dummy index running over all subgroups.The quantity v ⋅ (i)
k
is the number of subgroup parameters of type k in a molecule of species i. Values of the
subgroup parameters R(k) and Q(k) and of the group interaction parameters a(mk) come from
tabulations in the literature.
From tables of Appendix G in Smith and Van Ness
Subgroups for acetone are 1
CH
3
(k=1) and
1 COCH
3
Subgroups for water is the molecule itself (k=17).
In our problem
Non-Commercial Use Only
(k=19)
In our problem
k
R(k)
v ⋅ (1)
Q(k)
v ⋅ (2)
k
k
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
CH
1
3
0.9011
0.848
1
0
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
COCH
3
19
1.6724
1.488
1
0
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
H ⋅O
2
17
0.9200
1.400
0
1
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Calculations
From equations 6 and 7
r1 ≔ 1 ⋅ 0.9011 + 1 ⋅ 1.6724
r2 ≔ 1 ⋅ 0.9200
r1 = 2.574
r2 = 0.92
q1 ≔ 1 ⋅ 0.848 + 1 ⋅ 1.488
q2 ≔ 1 ⋅ 1.400
q1 = 2.336
q2 = 1.4
From the above values using equation 8 e(ki) table is prepared
−
−−−−−−−−−−−−−−−−−−−−−−−−−−−
e (ki)
−
sample calculation
for e(19,1)
−−−−−−−−−−−−−−−−−−−−−−−−−−−
k
i=1
i=2
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
1
0.363
0
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
19
0.637
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
0
Non-Commercial Use Only
1 ⋅ 1.488
―――
= 0.637
2.336
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
17
0
1
−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Interaction parameters from tables
a (1 , 1) ＝ a (19 , 19) ＝ a (17 , 17) ＝ 0
a (1 , 19) ＝ 476.4
a (19 , 1) ＝ 26.76
a (1 , 17) ＝ 1318.00
a (17 , 1) ＝ 300.00
a (19 , 17) ＝ 472.50
a (17 , 19) ＝ −195.40
bar ≡ 10 5 ⋅ Pa
joule
Rgas ≔ 8.314 ⋅ ―――
mole ⋅ K
acetone ---- 1
water
---- 2
Experimental values of liquid phase composition (x1exp25), vapor phase composition
(y1exp25) of acetone and total pressure of the system are given as shown below.
At 25C
i ≔ 0 ‥ 16
Non-Commercial Use Only
⎡ 0.0000 ⎤
⎢ 0.0001 ⎥
⎢
⎥
0.0194
⎢
⎥
⎢ 0.0289 ⎥
⎢ 0.0449 ⎥
⎢ 0.0556 ⎥
⎢ 0.0939 ⎥
⎢ 0.0951 ⎥
⎢
⎥
x1exp25 ≔ ⎢ 0.131 ⎥
i
⎢ 0.147 ⎥
⎢ 0.1791 ⎥
⎢ 0.2654 ⎥
⎢ 0.3538 ⎥
⎢
⎥
⎢ 0.5808 ⎥
⎢ 0.7852 ⎥
⎢ 0.9999 ⎥
⎢⎣ 1.0000 ⎥⎦
⎡ 0.0000 ⎤
⎢ 0.0001 ⎥
⎢
⎥
⎢ 0.5234 ⎥
⎢ 0.6212 ⎥
⎢ 0.7168 ⎥
⎢ 0.7591 ⎥
⎢ 0.8351 ⎥
⎢ 0.8416 ⎥
⎢
⎥
y1exp25 ≔ ⎢ 0.8618 ⎥
i
⎢ 0.8768 ⎥
⎢ 0.8782 ⎥
⎢ 0.8856 ⎥
⎢ 0.8954 ⎥
⎢
⎥
⎢ 0.9158 ⎥
⎢ 0.9421 ⎥
⎢ 0.9999 ⎥
⎢⎣ 1.0000 ⎥⎦
⎡ 0.0314 ⋅ bar ⎤
⎢ 0.031597 ⋅ bar ⎥
⎢
⎥
⎢ 0.066795 ⋅ bar ⎥
⎢ 0.082393 ⋅ bar ⎥
⎢ 0.108391 ⋅ bar ⎥
⎢ 0.121457 ⋅ bar ⎥
⎢ 0.168120 ⋅ bar ⎥
⎢ 0.168786 ⋅ bar ⎥
⎢
⎥
Pexp25 ≔ ⎢ 0.192384 ⋅ bar ⎥
i
⎢ 0.200784 ⋅ bar ⎥
⎢ 0.213049 ⋅ bar ⎥
⎢ 0.234781 ⋅ bar ⎥
⎢ 0.245847 ⋅ bar ⎥
⎢
⎥
⎢ 0.265445 ⋅ bar ⎥
⎢ 0.284643 ⋅ bar ⎥
⎢ 0.307856 ⋅ bar ⎥
⎢⎣ 0.308000 ⋅ bar ⎥⎦
x2exp25 ≔ 1 − x1exp25
Calculation of activity coefficients by UNIFAC-method
r1 ≔ 2.574
r2 ≔ 0.9200
q1 ≔ 2.336
q2 ≔ 1.400
r1
J1 ≔ ――――――――
i
r1 ⋅ x1exp25 + r2 ⋅ x2exp25
i
J1 =
i
⎡ 2.798 ⎤
⎢ 2.797 ⎥
⎢
⎥
2.704
⎢
⎥
⎢ 2.66 ⎥
⎢ 2.589 ⎥
⎢ 2.544 ⎥
⎢ 2.394 ⎥
⎢ 2.389 ⎥
⎢
⎥
⎢ 2.265 ⎥
⎢ 2.213 ⎥
⎢ 2.116 ⎥
⎢ 1.894 ⎥
⎢⎣ ⋮
⎥⎦
i
i
i
From equations 6 and 7
r2
J2 ≔ ――――――――
i
r1 ⋅ x1exp25 + r2 ⋅ x2exp25
i
J2 =
i
⎤
⎡1
⎢1
⎥
⎢
⎥
0.966
⎢
⎥
⎢ 0.951 ⎥
⎢ 0.925 ⎥
⎢ 0.909 ⎥
⎢ 0.856 ⎥
⎢ 0.854 ⎥
⎢
⎥
⎢ 0.809 ⎥
⎢ 0.791 ⎥
⎢ 0.756 ⎥
⎢ 0.677 ⎥
⎢⎣ ⋮
⎥⎦
Non-Commercial Use Only
i
From equation 4
q1
L1 ≔ ――――――――
i
q1 ⋅ x1exp25 + q2 ⋅ x2exp25
i
i
q2
L2 ≔ ――――――――
i
q1 ⋅ x1exp25 + q2 ⋅ x2exp25
i
L1 =
i
⎡ 1.669 ⎤
⎢ 1.668 ⎥
⎢
⎥
1.647
⎢
⎥
⎢ 1.637 ⎥
⎢ 1.62 ⎥
⎢ 1.609 ⎥
⎢ 1.57 ⎥
⎢ 1.569 ⎥
⎢
⎥
⎢ 1.534 ⎥
⎢ 1.519 ⎥
⎢ 1.49 ⎥
⎢ 1.417 ⎥
⎢⎣ ⋮
⎥⎦
L2 =
i
⎤
⎡1
⎢1
⎥
⎢
⎥
0.987
⎢
⎥
⎢ 0.981 ⎥
⎢ 0.971 ⎥
⎢ 0.964 ⎥
⎢ 0.941 ⎥
⎢ 0.94 ⎥
⎢
⎥
⎢ 0.919 ⎥
⎢ 0.911 ⎥
⎢ 0.893 ⎥
⎢ 0.849 ⎥
⎢⎣ ⋮
⎥⎦
x1exp25 ⋅ q1 ⋅ 0.363
Non-Commercial Use Only
i
From equation 5
x1exp25 ⋅ q1 ⋅ 0.363
i
θ1 ≔ ――――――――
i
x1exp25 ⋅ q1 + x2exp25 ⋅ q2
i
x1exp25 ⋅ q1 ⋅ 0.637
i
θ19 ≔ ――――――――
i
x1exp25 ⋅ q1 + x2exp25 ⋅ q2
i
i
x2exp25 ⋅ q2 ⋅ 1
i
θ17 ≔ ――――――――
i
x1exp25 ⋅ q1 + x2exp25 ⋅ q2
i
θ1 =
i
⎤
⎡0
⎢ 6.057 ⋅ 10 −5 ⎥
⎢
⎥
⎢ 0.012
⎥
⎢ 0.017
⎥
⎢ 0.026
⎥
⎢ 0.032
⎥
⎢ 0.054
⎥
⎢
⎥
⎢ 0.054
⎥
⎢ 0.073
⎥
⎢ 0.081
⎥
⎢ 0.097
⎥
⎢ 0.137
⎥
⎢
⎥
⎣⋮
⎦
i
From equation 9
i
⎤
⎡0
⎢ 1.063 ⋅ 10 −4 ⎥
⎢
⎥
⎢ 0.02
⎥
⎢ 0.03
⎥
⎢ 0.046
⎥
⎢ 0.057
⎥
⎢ 0.094
⎥
⎢
⎥
⎢ 0.095
⎥
⎢ 0.128
⎥
⎢ 0.142
⎥
⎢ 0.17
⎥
⎢ 0.24
⎥
⎢
⎥
⎣⋮
⎦
θ19 =
i
θ17 =
i
⎤
⎡1
⎢1
⎥
⎢
⎥
0.968
⎢
⎥
⎢ 0.953 ⎥
⎢ 0.927 ⎥
⎢ 0.911 ⎥
⎢ 0.853 ⎥
⎢ 0.851 ⎥
⎢
⎥
⎢ 0.799 ⎥
⎢ 0.777 ⎥
⎢ 0.733 ⎥
⎢ 0.624 ⎥
⎢⎣ ⋮
⎥⎦
From equation 10
τ (mk)
s1 ≔ θ1 ⋅ 1 + θ19 ⋅ 0.914 + θ17 ⋅ 0.365
i
i
i
's in equation 10 are
calculated from equation 11
i
Sample calculation
s19 ≔ θ1 ⋅ 0.202 + θ19 ⋅ 1 + θ17 ⋅ 1.926
i
i
i
i
τ (1 , 19) ＝
s17 ≔ θ1 ⋅ 0.012 + θ19 ⋅ 0.205 + θ17 ⋅ 1
i
i
i
i
Non-Commercial Use Only
⎛ −476.4 ⎞
exp ⎜―――
⎟ = 0.202
⎝ 298 ⎠
s1 =
i
⎡ 0.365 ⎤
⎢ 0.365 ⎥
⎢
⎥
0.384
⎢
⎥
⎢ 0.392 ⎥
⎢ 0.407 ⎥
⎢ 0.417 ⎥
⎢ 0.451 ⎥
⎢ 0.452 ⎥
⎢
⎥
⎢ 0.482 ⎥
⎢ 0.495 ⎥
⎢ 0.52 ⎥
⎢ 0.583 ⎥
⎢⎣ ⋮
⎥⎦
s19 =
i
⎡ 1.926 ⎤
⎢ 1.926 ⎥
⎢
⎥
1.887
⎢
⎥
⎢ 1.868 ⎥
⎢ 1.838 ⎥
⎢ 1.817 ⎥
⎢ 1.747 ⎥
⎢ 1.745 ⎥
⎢
⎥
⎢ 1.682 ⎥
⎢ 1.655 ⎥
⎢ 1.602 ⎥
⎢ 1.469 ⎥
⎢⎣ ⋮
⎥⎦
s17 =
i
⎤
⎡1
⎢1
⎥
⎢
⎥
0.972
⎢
⎥
⎢ 0.959 ⎥
⎢ 0.937 ⎥
⎢ 0.923 ⎥
⎢ 0.872 ⎥
⎢ 0.871 ⎥
⎢
⎥
⎢ 0.826 ⎥
⎢ 0.807 ⎥
⎢ 0.769 ⎥
⎢ 0.675 ⎥
⎢⎣ ⋮
⎥⎦
⎛
⎛ J1 ⎞⎞
J1
i
i ⎟⎟
⎜
⎜
lnγc1 ≔ 1 − J1 + ln ⎛J1 ⎞ − 5 ⋅ q1 ⋅ ⎜1 − ――
+ ln ⎜――
i
i
L1
L1 ⎟⎟
⎝ i⎠
i
⎜⎝
⎜⎝ i ⎟⎠⎟⎠
From equation 2
⎛
⎛ J2 ⎞⎞
J2
i
i ⎟⎟
⎜
⎜
lnγc2 ≔ 1 − J2 + ln ⎛J2 ⎞ − 5 ⋅ q2 ⋅ ⎜1 − ――
+ ln ⎜――
i
i
L2
L2 ⎟⎟
⎝ i⎠
i
⎜⎝
⎜⎝ i ⎟⎠⎟⎠
⎛
⎛⎛
0.945
⎛ 0.945 ⎞⎞
⎛
0.71
Non-Commercial Use Only
⎛ 0.71 ⎞⎞
⎛
0.135 ⎞⎞⎞
⎛
⎛⎛
⎛ 0.945 ⎞⎞ ⎛
⎛ 0.71 ⎞⎞ ⎛
0.945
0.71
0.135 ⎞⎞⎞
lnγr1 ≔ q1 ⋅ ⎜1 − ⎜⎜θ1 ⋅ ――− 0.363 ⋅ ln ⎜――⎟⎟ + ⎜θ19 ⋅ ――
− 0.637 ⋅ ln ⎜――
+ ⎜θ17 ⋅ ――⎟⎟⎟
⎟
⎟
i
i
i s19
i
s1
s17 ⎟⎟⎟
⎜⎝
⎜⎝⎜⎝
⎜⎝ s1i ⎟⎠⎟⎠ ⎜⎝
⎜⎝ s19i ⎟⎠⎟⎠ ⎜⎝
i
i
i ⎠⎠⎠
From equation 3
⎛
⎛ 1 ⎞⎞⎞⎞
⎛⎛
0.365 ⎞ ⎛
1.926 ⎞ ⎛
1
− ln ⎜――
lnγr2 ≔ q2 ⋅ ⎜1 − ⎜⎜θ1 ⋅ ――⎟ + ⎜θ19 ⋅ ――⎟ + ⎜θ17 ⋅ ――
⎟⎟⎟⎟
i
i
i
i
s1 ⎟ ⎜
s19 ⎟ ⎜
s17
s17 ⎟⎟⎟⎟
⎜⎝
⎜⎝⎜⎝
⎜
i ⎠
i
i
i
⎝
⎠ ⎝
⎝
⎠⎠⎠⎠
lnγr1 =
i
⎡ 1.342 ⎤
⎢ 1.342 ⎥
⎢
⎥
1.247
⎢
⎥
⎢ 1.205 ⎥
⎢ 1.138 ⎥
⎢ 1.096 ⎥
⎢ 0.965 ⎥
⎢ 0.961 ⎥
⎢
⎥
⎢ 0.859 ⎥
⎢ 0.818 ⎥
⎢ 0.744 ⎥
⎢ 0.585 ⎥
⎢⎣ ⋮
⎥⎦
lnγr2 =
γ1 ≔ exp ⎛lnγc1 + lnγr1 ⎞
i
i
i⎠
⎝
⎡ 11.485 ⎤
⎢ 11.473 ⎥
⎢
⎥
9.407
⎢
⎥
⎢ 8.597 ⎥
⎢ 7.466 ⎥
⎢ 6.838 ⎥
γ1 = ⎢ 5.185 ⎥
i
⎢ 5 145 ⎥
i
sample calculation
β (ik) 's are
calculated by
(
)
β 1 , 19 ＝
equation 12.
0.363 ⋅ 0.202 + 0.637 ⋅ 1 = 0.71
⎤
⎡0
⎡ 1.099 ⎤
⎢ 2.594 ⋅ 10 −8 ⎥
⎢ 1.098 ⎥
⎤
⎡0
⎢
⎥
⎢
⎥
−4
⎢
−8 ⎥
9.15
⋅
10
0.994
2.846 ⋅ 10
⎢
⎥
⎢
⎥
⎢
⎥
⎢ 0.002
⎥
0.001
⎢ 0.947 ⎥
⎢
⎥
⎢ 0.005
⎥
⎢ 0.872 ⎥
0.002
⎢
⎥
⎢ 0.007
⎥
⎢ 0.826 ⎥
⎢ 0.005
⎥
⎢
⎥
⎢ 0.681 ⎥
⎢
⎥
lnγc1
=
0.017
0.007
⎢
⎥
i
⎢ 0.677 ⎥
⎢
⎥
0.018
lnγc2
=
0.019
⎢
⎥
⎢
⎥
i
⎢
⎥
⎢ 0.031
⎥
0.02
⎢ 0.565 ⎥
⎢
⎥
⎢ 0.037
⎥
⎢ 0.522 ⎥
0.034
⎢
⎥
⎢ 0.052
⎥
⎢ 0.444 ⎥
0.041
⎢
⎥
⎢ 0.097
⎥
⎢ 0.287 ⎥
⎢ 0.056
⎥
⎢
⎥
⎢⎣ ⋮
⎥⎦
⎢ 0.1
⎥
⎣⋮
⎦
⎢
⎥
⎣⋮
⎦
γ2 ≔ exp ⎛lnγc2 + lnγr2 ⎞
i
i
i⎠
⎝
⎡1
⎤
⎢1
⎥
⎢
⎥
1.002
⎢
⎥
⎢ 1.004 ⎥
⎢ 1.01 ⎥
⎢ 1.014 ⎥
γ2 = ⎢ 1.037 ⎥
i
⎢ 1 038 ⎥
Non-Commercial Use Only
γ1
i
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
γ2
5.185
5.145 ⎥
⎥
4.153 ⎥
3.818 ⎥
3.282 ⎥
2.392 ⎥
⎥⎦
⋮
Psat1 ≔ 0.308 ⋅ bar
i
Psat2 ≔ 0.0314 ⋅ bar
Pcal25 ≔ x1exp25 ⋅ γ1 ⋅ Psat1 + x2exp25 ⋅ γ2 ⋅ Psat2
i
i
i
i
i
x1exp25 ⋅ γ1 ⋅ Psat1
i
i
――――――
y1cal25 ≔
i
Pcal25
i
y1cal25 =
i
⎤
⎡0
⎢ 0.011 ⎥
⎢
⎥
0.646
⎢
⎥
⎢ 0.714 ⎥
⎢ 0.773 ⎥
⎢ 0.796 ⎥
⎢ 0.836 ⎥
⎢ 0.836 ⎥
⎢
⎥
⎢ 0.852 ⎥
⎢ 0.857 ⎥
⎢ 0.863 ⎥
⎢ 0.874 ⎥
⎢⎣ ⋮
⎥⎦
Pcal25 = ? bar
i
Non-Commercial Use Only
1.037
⎢ 1.038 ⎥
⎢
⎥
⎢ 1.066 ⎥
⎢ 1.081 ⎥
⎢ 1.113 ⎥
⎢ 1.218 ⎥
⎢⎣ ⋮
⎥⎦
from literature at 25 C
rms ≔
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
2
∑ ⎛Pexp25 − Pcal25 ⎞
i
i⎠
i ⎝
Root mean square error calculation.
rms = ? bar
10
8
6
4
Pexp25
i
―――
bar
Pexp25
i
―――
bar
2
0
-2
Non-Commercial Use Only
Pcal25
i
―――
bar
Pcal25
i
―――
bar
-4
-6
-8
-10
-10
-8
-6
-4
-2
0
x1exp25
y1exp25
x1exp25
2
i
i
i
y1cal25
i
Similar calculations were carried on at 100 and 200 C
At 100C
⎡ 0.00000 ⎤
⎢ 0.00330 ⎥
⎢
⎥
0.00400
⎥
⎢
⎢ 0.00450 ⎥
⎢ 0.00800 ⎥
⎢ 0.04800 ⎥
⎢ 0.08200 ⎥
⎢ 0.10800 ⎥
⎥
⎢
x1exp100 ≔ ⎢ 0.22000 ⎥
i
⎢ 0.30800 ⎥
⎢ 0 31600 ⎥
Non-Commercial Use Only
4
6
8
10
⎢ 0.48000 ⎥
⎢ 0.69500 ⎥
⎢
⎥
⎢ 0.74200 ⎥
⎢ 0.85400 ⎥
⎢ 0.97100 ⎥
⎢⎣ 1.00000 ⎥⎦
⎡ 0.00000 ⎤
⎢ 0.09020 ⎥
⎢
⎥
0.10900
⎢
⎥
⎢ 0.11800 ⎥
⎢ 0.20700 ⎥
⎢ 0.54500 ⎥
⎢ 0.61300 ⎥
⎢ 0.63200 ⎥
⎢
⎥
y1exp100 ≔ ⎢ 0.70500 ⎥
i
⎢ 0.71500 ⎥
⎢ 0.71900 ⎥
⎢ 0.74700 ⎥
⎢ 0.80100 ⎥
⎢
⎥
⎢ 0.82300 ⎥
⎢ 0.87800 ⎥
⎢ 0.97200 ⎥
⎢⎣ 1.00000 ⎥⎦
⎡ 1.013000 ⋅ bar ⎤
⎢ 1.110056 ⋅ bar ⎥
⎢
⎥
1.130734 ⋅ bar
⎢
⎥
⎢ 1.145439 ⋅ bar ⎥
⎢ 1.303107 ⋅ bar ⎥
⎢ 2.240790 ⋅ bar ⎥
⎢ 2.447639 ⋅ bar ⎥
⎢ 2.785478 ⋅ bar ⎥
⎢
⎥
Pexp100 ≔ ⎢ 3.068162 ⋅ bar ⎥
i
⎢ 3.199164 ⋅ bar ⎥
⎢ 3.206057 ⋅ bar ⎥
⎢ 3.474955 ⋅ bar ⎥
⎢ 3.571481 ⋅ bar ⎥
⎢
⎥
⎢ 3.599065 ⋅ bar ⎥
⎢ 3.674899 ⋅ bar ⎥
⎢ 3.681805 ⋅ bar ⎥
⎢⎣ 3.722000 ⋅ bar ⎥⎦
x2exp100 ≔ 1 − x1exp100
r1 ≔ 2.574
r2 ≔ 0.9200
q1 ≔ 2.336
q2 ≔ 1.400
i
r1
J1 ≔ ―――――――――
i
r1 ⋅ x1exp100 + r2 ⋅ x2exp100
i
J1 =
i
⎡ 2.798 ⎤
⎢ 2.781 ⎥
⎢
⎥
2.778
⎢
⎥
⎢ 2.775 ⎥
⎢ 2.758 ⎥
⎢ 2.576 ⎥
⎢ 2.438 ⎥
⎢ 2.343 ⎥
⎢
⎥
⎢ 2.005 ⎥
⎢ 1.801 ⎥
⎢ 1.784 ⎥
⎢ 1.502 ⎥
⎢⎣ ⋮
⎥⎦
r2
J2 ≔ ―――――――――
i
r1 ⋅ x1exp100 + r2 ⋅ x2exp100
i
i
J2 =
i
⎤
⎡1
⎢ 0.994 ⎥
⎢
⎥
0.993
⎢
⎥
⎢ 0.992 ⎥
⎢ 0.986 ⎥
⎢ 0.921 ⎥
⎢ 0.872 ⎥
⎢ 0.837 ⎥
⎢
⎥
⎢ 0.717 ⎥
⎢ 0.644 ⎥
⎢ 0.638 ⎥
⎢ 0.537 ⎥
⎢⎣ ⋮
⎥⎦
Non-Commercial Use Only
i
i
q1
L1 ≔ ―――――――――
i
q1 ⋅ x1exp100 + q2 ⋅ x2exp100
i
L1 =
i
⎡ 1.669 ⎤
⎢ 1.665 ⎥
⎢
⎥
1.664
⎢
⎥
⎢ 1.664 ⎥
⎢ 1.66 ⎥
⎢ 1.617 ⎥
⎢ 1.582 ⎥
⎢ 1.556 ⎥
⎢
⎥
⎢ 1.455 ⎥
⎢ 1.384 ⎥
⎢ 1.378 ⎥
⎢ 1.263 ⎥
⎢⎣ ⋮
⎥⎦
i
q2
L2 ≔ ―――――――――
i
q1 ⋅ x1exp100 + q2 ⋅ x2exp100
i
L2 =
i
Non-Commercial Use Only
⎤
⎡1
⎢ 0.998 ⎥
⎢
⎥
0.997
⎢
⎥
⎢ 0.997 ⎥
⎢ 0.995 ⎥
⎢ 0.969 ⎥
⎢ 0.948 ⎥
⎢ 0.933 ⎥
⎢
⎥
⎢ 0.872 ⎥
⎢ 0.829 ⎥
⎢ 0.826 ⎥
⎢ 0.757 ⎥
⎢⎣ ⋮
⎥⎦
i
x1exp100 ⋅ q1 ⋅ 0.363
i
θ1 ≔ ―――――――――
i
x1exp100 ⋅ q1 + x2exp100 ⋅ q2
i
i
i
i
x2exp100 ⋅ q2
i
―――――――――
θ17 ≔
i
x1exp100 ⋅ q1 + x2exp100 ⋅ q2
⎤
⎡0
⎢ 0.002 ⎥
⎢
⎥
0.002
⎢
⎥
⎢ 0.003 ⎥
⎢ 0.005 ⎥
⎢ 0.028 ⎥
⎢ 0.047 ⎥
⎢ 0.061 ⎥
⎢
⎥
⎢ 0.116 ⎥
⎢ 0.155 ⎥
⎢ 0.158 ⎥
⎢ 0.22 ⎥
⎢⎣ ⋮
⎥⎦
θ1 =
x1exp100 ⋅ q1 ⋅ 0.637
i
θ19 ≔ ―――――――――
i
x1exp100 ⋅ q1 + x2exp100 ⋅ q2
i
⎤
⎡0
⎢ 0.003 ⎥
⎢
⎥
0.004
⎢
⎥
⎢ 0.005 ⎥
⎢ 0.008 ⎥
⎢ 0.049 ⎥
⎢ 0.083 ⎥
⎢ 0.107 ⎥
⎢
⎥
⎢ 0.204 ⎥
⎢ 0.271 ⎥
⎢ 0.277 ⎥
⎢ 0.386 ⎥
⎢⎣ ⋮
⎥⎦
θ19 =
i
i
θ17 =
i
⎤
⎡1
⎢ 0.995 ⎥
⎢
⎥
0.993
⎢
⎥
⎢ 0.993 ⎥
⎢ 0.987 ⎥
⎢ 0.922 ⎥
⎢ 0.87 ⎥
⎢ 0.832 ⎥
⎢
⎥
⎢ 0.68 ⎥
⎢ 0.574 ⎥
⎢ 0.565 ⎥
⎢ 0.394 ⎥
⎢⎣ ⋮
⎥⎦
s1 ≔ θ1 ⋅ 1 + θ19 ⋅ 0.9308 + θ17 ⋅ 0.4474
i
i
i
i
s19 ≔ θ1 ⋅ 0.2788 + θ19 ⋅ 1 + θ17 ⋅ 1.6885
i
i
i
i
Non-Commercial Use Only
i
s17 ≔ θ1 ⋅ 0.0292 + θ19 ⋅ 0.2818 + θ17 ⋅ 1
i
s1 =
i
i
⎡ 0.447 ⎤
⎢ 0.45 ⎥
⎢
⎥
0.451
⎢
⎥
⎢ 0.451 ⎥
⎢ 0.454 ⎥
⎢ 0.487 ⎥
⎢ 0.513 ⎥
⎢ 0.533 ⎥
⎢
⎥
⎢ 0.61 ⎥
⎢ 0.664 ⎥
⎢ 0.669 ⎥
⎢ 0.756 ⎥
⎢⎣ ⋮
⎥⎦
i
i
s19 =
i
⎡ 1.689 ⎤
⎢ 1.683 ⎥
⎢
⎥
1.682
⎢
⎥
⎢ 1.681 ⎥
⎢ 1.676 ⎥
⎢ 1.615 ⎥
⎢ 1.565 ⎥
⎢ 1.529 ⎥
⎢
⎥
⎢ 1.384 ⎥
⎢ 1.284 ⎥
⎢ 1.275 ⎥
⎢ 1.112 ⎥
⎢⎣ ⋮
⎥⎦
s17 =
⎛
⎛ J1 ⎞⎞
J1
i
⎜
⎜ i ⎟⎟
lnγc1 ≔ 1 − J1 + ln ⎛J1 ⎞ − 5 ⋅ 2.336 ⋅ ⎜1 − ――
+ ln ⎜――
i
i
i
L1
L1 ⎟⎟
⎝ ⎠
i
⎜⎝
⎜⎝ i ⎟⎠⎟⎠
⎛
⎛ J2 ⎞⎞
J2
i
⎜
⎜ i ⎟⎟
lnγc2 ≔ 1 − J2 + ln ⎛J2 ⎞ − 5 ⋅ 1.400 ⋅ ⎜1 − ――
+ ln ⎜――
i
i
L2
L2 ⎟⎟
⎝ i⎠
i
⎜⎝
⎜⎝ i ⎟⎠⎟⎠
Non-Commercial Use Only
i
⎤
⎡1
⎢ 0.996 ⎥
⎢
⎥
0.995
⎢
⎥
⎢ 0.994 ⎥
⎢ 0.989 ⎥
⎢ 0.937 ⎥
⎢ 0.895 ⎥
⎢ 0.864 ⎥
⎢
⎥
⎢ 0.741 ⎥
⎢ 0.655 ⎥
⎢ 0.647 ⎥
⎢ 0.509 ⎥
⎢⎣ ⋮
⎥⎦
⎛
⎛ 0.956 ⎞⎞ ⎛
⎛ 0.738 ⎞⎞ ⎛
⎛⎛
0.956
0.738
0.190 ⎞⎞⎞
lnγr1 ≔ q1 ⋅ ⎜1 − ⎜⎜θ1 ⋅ ――− 0.363 ⋅ ln ⎜――⎟⎟ + ⎜θ19 ⋅ ――− 0.637 ⋅ ln ⎜――⎟⎟ + ⎜θ17 ⋅ ――⎟⎟⎟
i
i
i
i
s1
s19
s17 ⎟⎟⎟
⎜⎝
⎜⎝⎜⎝
⎜⎝ s1i ⎟⎠⎟⎠ ⎜⎝
⎜⎝ s19i ⎟⎠⎟⎠ ⎜⎝
i
i
i ⎠⎠⎠
⎛
⎛ 1 ⎞⎞⎞⎞
⎛⎛
0.4474 ⎞ ⎛
1.6885 ⎞ ⎛
1
lnγr2 ≔ q2 ⋅ ⎜1 − ⎜⎜θ1 ⋅ ―――
+ ⎜θ19 ⋅ ―――
+ ⎜θ17 ⋅ ――
− ln ⎜――
⎟
⎟
⎟⎟⎟⎟
i
i
s1 ⎟ ⎜ i s19 ⎟ ⎜ i s17
⎜⎝
⎜⎝⎜⎝
⎜⎝ s17i ⎟⎠⎟⎠⎟⎠⎟⎠
i
i ⎠
i
⎠ ⎝
⎝
lnγc1 =
i
⎡ 1.099 ⎤
⎢ 1.08 ⎥
⎢
⎥
1.076
⎢
⎥
⎢ 1.073 ⎥
⎢ 1.054 ⎥
⎢ 0.859 ⎥
⎢ 0.723 ⎥
⎢ 0.634 ⎥
⎢
⎥
⎢ 0.362 ⎥
⎢ 0.231 ⎥
⎢ 0.222 ⎥
⎢ 0.09 ⎥
⎢⎣ ⋮
⎥⎦
lnγc2 =
i
⎤
⎡0
⎢ 3.069 ⋅ 10 −5 ⎥
⎢
⎥
−5
⎢ 4.5 ⋅ 10
⎥
⎢ 5.686 ⋅ 10 −5 ⎥
⎢ 1.778 ⋅ 10 −4 ⎥
⎢
⎥
⎢ 0.006
⎥
⎢ 0.015
⎥
⎢ 0.024
⎥
⎢ 0.076
⎥
⎢ 0.123
⎥
⎢
⎥
⎢ 0.127
⎥
⎢ 0.211
⎥
⎣⋮
⎦
lnγr1 =
i
γ1 ≔ exp ⎛lnγc1 + lnγr1 ⎞
i
i
i⎠
⎝
⎡ 11.058 ⎤
⎢ 10.707 ⎥
⎢
⎥
10.634
⎢
⎥
⎢ 10.583 ⎥
⎢ 10.235 ⎥
⎢ 7.265 ⎥
γ1 = ⎢ 5.697 ⎥
γ2 =
⎢
⎥
⎡ 1.304 ⎤
⎢ 1.291 ⎥
⎢
⎥
1.288
⎢
⎥
⎢ 1.286 ⎥
⎢ 1.272 ⎥
⎢ 1.124 ⎥
⎢ 1.017 ⎥
⎢ 0.944 ⎥
⎢
⎥
⎢ 0.694 ⎥
⎢ 0.548 ⎥
⎢ 0.536 ⎥
⎢ 0.333 ⎥
⎢⎣ ⋮
⎥⎦
γ2 ≔ exp ⎛lnγc2 + lnγr2 ⎞
i
i
i⎠
⎝
⎡1
⎤
⎢1
⎥
⎢
⎥
1
⎢
⎥
⎢1
⎥
⎢1
⎥
⎢ 1.01 ⎥
⎢ 1.027 ⎥
⎢
⎥
Non-Commercial Use Only
lnγr2 =
i
⎤
⎡0
⎢ 2.257 ⋅ 10 −5 ⎥
⎢
⎥
−5
⎢ 3.31 ⋅ 10
⎥
⎢ 4.184 ⋅ 10 −5 ⎥
⎢ 1.311 ⋅ 10 −4 ⎥
⎢
⎥
⎢ 0.004
⎥
⎢ 0.012
⎥
⎢ 0.019
⎥
⎢ 0.068
⎥
⎢ 0.12
⎥
⎢
⎥
⎢ 0.125
⎥
⎢ 0.259
⎥
⎣⋮
⎦
γ1 =
i
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
5.697 ⎥
4.847 ⎥
⎥
2.875 ⎥
2.178 ⎥
2.133 ⎥
1.526 ⎥
⎥⎦
⋮
γ2 =
i
⎢ 1.027 ⎥
⎢ 1.045 ⎥
⎢
⎥
⎢ 1.155 ⎥
⎢ 1.275 ⎥
⎢ 1.287 ⎥
⎢ 1.601 ⎥
⎢⎣ ⋮
⎥⎦
Psat1 ≔ 3.722 ⋅ bar
Psat2 ≔ 1.013 ⋅ bar
Pcal100 ≔ x1exp100 ⋅ γ1 ⋅ Psat1 + x2exp100 ⋅ γ2 ⋅ Psat2
i
i
i
i
i
x1exp100 ⋅ γ1 ⋅ Psat1
i
i
y1cal100 ≔ ――――――
i
Pcal100
i
y1cal100 =
i
⎤
⎡0
⎢ 0.115 ⎥
⎢
⎥
0.136
⎢
⎥
⎢ 0.149 ⎥
⎢ 0.233 ⎥
⎢ 0.571 ⎥
⎢ 0.645 ⎥
⎢ 0.674 ⎥
⎢
⎥
⎢ 0.721 ⎥
⎢ 0.736 ⎥
⎢ 0.738 ⎥
⎢ 0.764 ⎥
⎢⎣ ⋮
⎥⎦
Pcal100 = ? bar
i
Non-Commercial Use Only
rms ≔
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
2
∑ ⎛Pexp100 − Pcal100 ⎞
i
i⎠
i ⎝
rms = ? bar
10
8
6
4
Non-Commercial Use Only
Pexp100
i
―――
bar
Pexp100
i
―――
bar
Pcal100
i
―――
bar
Pcal100
i
―――
bar
2
0
-2
-4
-6
-8
-10
-10
-8
-6
-4
-2
0
x1exp100
2
4
6
i
y1exp100
i
x1exp100
i
y1cal100
i
At 200C
⎡ 0.0000 ⎤
⎢ 0.0013 ⎥
⎢
⎥
⎢ 0.0182 ⎥
⎢ 0.0450 ⎥
⎢ 0.0920 ⎥
⎢ 0.2260 ⎥
⎢ 0.3620 ⎥
⎢ 0.4460 ⎥
⎥
⎢
x1exp200 ≔ ⎢ 0.5510 ⎥
i
⎢ 0.6590 ⎥
⎢ 0 7580 ⎥
⎡ 0.0000 ⎤
⎢ 0.0160 ⎥
⎢
⎥
⎢ 0.1360 ⎥
⎢ 0.2680 ⎥
⎢ 0.3540 ⎥
⎢ 0.4550 ⎥
⎢ 0.5020 ⎥
⎢ 0.5400 ⎥
⎥
⎢
y1exp200 ≔ ⎢ 0.5950 ⎥
i
⎢ 0.6650 ⎥
⎢ 0 7430 ⎥
⎡ 15.552476 ⋅ bar ⎤
⎢ 15.995808 ⋅ bar ⎥
⎢
⎥
⎢ 18.202126 ⋅ bar ⎥
⎢ 21.511604 ⋅ bar ⎥
⎢ 24.200554 ⋅ bar ⎥
⎢ 27.303190 ⋅ bar ⎥
⎢ 28.751086 ⋅ bar ⎥
⎢ 29.440560 ⋅ bar ⎥
⎥
⎢
Pexp200 ≔ ⎢ 30.267930 ⋅ bar ⎥
i
⎢ 30.543720 ⋅ bar ⎥
⎢ 30 336877 ⋅ bar ⎥
Non-Commercial Use Only
8
10
⎢ 0.8160 ⎥
⎢ 0.9300 ⎥
⎢
⎥
⎢ 0.9400 ⎥
⎢ 0.9780 ⎥
⎢ 0.9790 ⎥
⎢⎣ 1.0000 ⎥⎦
⎢ 0.7940 ⎥
⎢ 0.9190 ⎥
⎢
⎥
⎢ 0.9310 ⎥
⎢ 0.9760 ⎥
⎢ 0.9790 ⎥
⎢⎣ 1.0000 ⎥⎦
⎢ 29.992140 ⋅ bar ⎥
⎢ 28.475296 ⋅ bar ⎥
⎢
⎥
⎢ 28.406349 ⋅ bar ⎥
⎢ 27.923717 ⋅ bar ⎥
⎢ 27.923717 ⋅ bar ⎥
⎢⎣ 26.699899 ⋅ bar ⎥⎦
x2exp200 ≔ 1 − x1exp200
i
r1 ≔ 2.574
i
r2 ≔ 0.9200
q1 ≔ 2.336
q2 ≔ 1.400
r1
J1 ≔ ―――――――――
i
r1 ⋅ x1exp200 + r2 ⋅ x2exp200
i
i
r2
J2 ≔ ―――――――――
i
r1 ⋅ x1exp200 + r2 ⋅ x2exp200
i
J1 =
i
⎡ 2.798 ⎤
⎢ 2.791 ⎥
⎢
⎥
2.709
⎢
⎥
⎢ 2.588 ⎥
⎢ 2.401 ⎥
⎢ 1.989 ⎥
⎢ 1.695 ⎥
⎢ 1.553 ⎥
⎢
⎥
⎢ 1.406 ⎥
⎢ 1.281 ⎥
⎢ 1.184 ⎥
⎢ 1 134 ⎥
i
J2 =
i
⎡1
⎤
⎢ 0.998 ⎥
⎢
⎥
0.968
⎢
⎥
⎢ 0.925 ⎥
⎢ 0.858 ⎥
⎢ 0.711 ⎥
⎢ 0.606 ⎥
⎢ 0.555 ⎥
⎢
⎥
⎢ 0.502 ⎥
⎢ 0.458 ⎥
⎢ 0.423 ⎥
⎢ 0 405 ⎥
Non-Commercial Use Only
⎢⎣ ⋮
⎥⎦
⎢⎣ ⋮
q1
L1 ≔ ―――――――――
i
q1 ⋅ x1exp200 + q2 ⋅ x2exp200
i
L1 =
i
⎡ 1.669 ⎤
⎢ 1.667 ⎥
⎢
⎥
1.649
⎢
⎥
⎢ 1.62 ⎥
⎢ 1.572 ⎥
⎢ 1.45 ⎥
⎢ 1.343 ⎥
⎢ 1.285 ⎥
⎢
⎥
⎢ 1.219 ⎥
⎢ 1.158 ⎥
⎢ 1.107 ⎥
⎢ 1.08 ⎥
⎢⎣ ⋮
⎥⎦
i
⎥⎦
q2
L2 ≔ ―――――――――
i
q1 ⋅ x1exp200 + q2 ⋅ x2exp200
i
L2 =
i
Non-Commercial Use Only
⎤
⎡1
⎢ 0.999 ⎥
⎢
⎥
0.988
⎢
⎥
⎢ 0.971 ⎥
⎢ 0.942 ⎥
⎢ 0.869 ⎥
⎢ 0.805 ⎥
⎢ 0.77 ⎥
⎢
⎥
⎢ 0.731 ⎥
⎢ 0.694 ⎥
⎢ 0.664 ⎥
⎢ 0.647 ⎥
⎢⎣ ⋮
⎥⎦
i
x1exp200 ⋅ q1 ⋅ 0.363
i
θ1 ≔ ―――――――――
i
x1exp200 ⋅ q1 + x2exp200 ⋅ q2
i
i
⎤
⎡0
⎢ 7.867 ⋅ 10 −4 ⎥
⎢
⎥
⎢ 0.011
⎥
⎢ 0.026
⎥
⎢ 0.052
⎥
⎢ 0.119
⎥
⎢ 0.177
⎥
⎢
⎥
⎢ 0.208
⎥
⎢ 0.244
⎥
⎢ 0.277
⎥
⎢ 0.305
⎥
⎢ 0.32
⎥
⎢
⎥
⎣⋮
⎦
θ1 =
i
x1exp200 ⋅ q1 ⋅ 0.637
i
θ19 ≔ ―――――――――
i
x1exp200 ⋅ q1 + x2exp200 ⋅ q2
i
x2exp200 ⋅ q2
i
θ17 ≔ ―――――――――
i
x1exp200 ⋅ q1 + x2exp200 ⋅ q2
i
i
θ19 =
i
⎡0
⎤
⎢ 0.001 ⎥
⎢
⎥
0.019
⎢
⎥
⎢ 0.046 ⎥
⎢ 0.092 ⎥
⎢ 0.209 ⎥
⎢ 0.31 ⎥
⎢ 0.365 ⎥
⎢
⎥
⎢ 0.428 ⎥
⎢ 0.486 ⎥
⎢ 0.535 ⎥
⎢ 0.561 ⎥
⎢⎣ ⋮
⎥⎦
s1 ≔ θ1 ⋅ 1 + θ19 ⋅ 0.945 + θ17 ⋅ 0.53
i
i
i
i
s19 ≔ θ1 ⋅ 0.365 + θ19 ⋅ 1 + θ17 ⋅ 1.512
i
i
i
i
i
Non-Commercial Use Only
θ17 =
i
⎡1
⎤
⎢ 0.998 ⎥
⎢
⎥
0.97
⎢
⎥
⎢ 0.927 ⎥
⎢ 0.855 ⎥
⎢ 0.672 ⎥
⎢ 0.514 ⎥
⎢ 0.427 ⎥
⎢
⎥
⎢ 0.328 ⎥
⎢ 0.237 ⎥
⎢ 0.161 ⎥
⎢ 0.119 ⎥
⎢⎣ ⋮
⎥⎦
s17 ≔ θ1 ⋅ 0.062 + θ19 ⋅ 0.368 + θ17 ⋅ 1
i
s1 =
i
i
⎡ 0.53 ⎤
⎢ 0.531 ⎥
⎢
⎥
0.543
⎢
⎥
⎢ 0.562 ⎥
⎢ 0.593 ⎥
⎢ 0.672 ⎥
⎢ 0.742 ⎥
⎢ 0.779 ⎥
⎢
⎥
⎢ 0.822 ⎥
⎢ 0.862 ⎥
⎢ 0.895 ⎥
⎢ 0.913 ⎥
⎢⎣ ⋮
⎥⎦
i
i
s19 =
i
⎡ 1.512 ⎤
⎢ 1.51 ⎥
⎢
⎥
1.49
⎢
⎥
⎢ 1.458 ⎥
⎢ 1.405 ⎥
⎢ 1.269 ⎥
⎢ 1.151 ⎥
⎢ 1.086 ⎥
⎢
⎥
⎢ 1.013 ⎥
⎢ 0.945 ⎥
⎢ 0.889 ⎥
⎢ 0.858 ⎥
⎢⎣ ⋮
⎥⎦
⎛
⎛ J1 ⎞⎞
J1
i
⎜
⎜ i ⎟⎟
lnγc1 ≔ 1 − J1 + ln ⎛J1 ⎞ − 5 ⋅ 2.336 ⋅ ⎜1 − ――
+ ln ⎜――
i
i
L1
L1 ⎟⎟
⎝ i⎠
i
⎜⎝
⎜⎝ i ⎟⎠⎟⎠
⎛
⎛ J2 ⎞⎞
J2
i
⎜
⎜ i ⎟⎟
⎛
⎞
――
lnγc2 ≔ 1 − J2 + ln J2 − 5 ⋅ 1.400 ⋅ ⎜1 −
+ ln ⎜――
i
i
L2
L2 ⎟⎟
⎝ i⎠
i
⎜⎝
⎜⎝ i ⎟⎠⎟⎠
Non-Commercial Use Only
s17 =
i
⎤
⎡1
⎢ 0.998 ⎥
⎢
⎥
0.978
⎢
⎥
⎢ 0.946 ⎥
⎢ 0.893 ⎥
⎢ 0.757 ⎥
⎢ 0.639 ⎥
⎢ 0.574 ⎥
⎢
⎥
⎢ 0.501 ⎥
⎢ 0.433 ⎥
⎢ 0.376 ⎥
⎢ 0.345 ⎥
⎢⎣ ⋮
⎥⎦
⎛
⎛ 0.965 ⎞⎞ ⎛
⎛ 0.769 ⎞⎞ ⎛
⎛⎛
0.965
0.769
0.257 ⎞⎞⎞
lnγr1 ≔ q1 ⋅ ⎜1 − ⎜⎜θ1 ⋅ ――− 0.363 ⋅ ln ⎜――⎟⎟ + ⎜θ19 ⋅ ――− 0.637 ⋅ ln ⎜――⎟⎟ + ⎜θ17 ⋅ ――⎟⎟⎟
i
i
i
i
s1
s19
s17 ⎟⎟⎟
⎜⎝
⎜⎝⎜⎝
⎜⎝ s1i ⎟⎠⎟⎠ ⎜⎝
⎜⎝ s19i ⎟⎠⎟⎠ ⎜⎝
i
i
i ⎠⎠⎠
⎛
⎛ 1 ⎞⎞⎞⎞
⎛⎛
0.53 ⎞ ⎛
1.512 ⎞ ⎛
1
lnγr2 ≔ q2 ⋅ ⎜1 − ⎜⎜θ1 ⋅ ――
+ ⎜θ19 ⋅ ――⎟ + ⎜θ17 ⋅ ――
− ln ⎜――
⎟
⎟⎟⎟⎟
i
i
s1 ⎟ ⎜ i s19 ⎟ ⎜ i s17
⎜⎝
⎜⎝⎜⎝
⎜⎝ s17i ⎟⎠⎟⎠⎟⎠⎟⎠
i ⎠
i ⎠
i
⎝
⎝
lnγc1 =
i
⎡ 1.099 ⎤
⎢ 1.091 ⎥
⎢
⎥
1
⎢
⎥
⎢ 0.872 ⎥
⎢ 0.687 ⎥
⎢ 0.351 ⎥
⎢ 0.174 ⎥
⎢ 0.11 ⎥
⎢
⎥
⎢ 0.059 ⎥
⎢ 0.028 ⎥
⎢ 0.012 ⎥
⎢ 0.006 ⎥
⎢⎣ ⋮
⎥⎦
lnγc2 =
i
⎤
⎡0
⎢ 4.792 ⋅ 10 −6 ⎥
⎢
⎥
−4
⎢ 8.921 ⋅ 10 ⎥
⎢ 0.005
⎥
⎢ 0.018
⎥
⎢ 0.079
⎥
⎢
⎥
⎢ 0.151
⎥
⎢ 0.194
⎥
⎢ 0.245
⎥
⎢ 0.292
⎥
⎢ 0.33
⎥
⎢ 0.35
⎥
⎢
⎥
⎣⋮
⎦
lnγr1 =
i
γ1 ≔ exp ⎛lnγc1 + lnγr1 ⎞
i
i
i⎠
⎝
⎡ 10.344 ⎤
⎢ 10.222 ⎥
⎢
⎥
8.81
⎢
⎥
⎢ 7.122 ⎥
⎡ 1.238 ⎤
⎢ 1.233 ⎥
⎢
⎥
1.176
⎢
⎥
⎢ 1.091 ⎥
⎢ 0.96 ⎥
⎢ 0.67 ⎥
⎢ 0.456 ⎥
⎢ 0.351 ⎥
⎢
⎥
⎢ 0.241 ⎥
⎢ 0.148 ⎥
⎢ 0.08 ⎥
⎢ 0.049 ⎥
⎢⎣ ⋮
⎥⎦
γ2 ≔ exp ⎛lnγc2 + lnγr2 ⎞
i
i
i⎠
⎝
⎤
⎡1
⎢1
⎥
⎢
⎥
1.001
⎢
⎥
⎢ 1.008 ⎥
Non-Commercial Use Only
lnγr2 =
i
⎤
⎡0
⎢ 2.971 ⋅ 10 −6 ⎥
⎢
⎥
−4
⎢ 5.656 ⋅ 10 ⎥
⎢ 0.003
⎥
⎢ 0.013
⎥
⎢ 0.067
⎥
⎢
⎥
⎢ 0.155
⎥
⎢ 0.227
⎥
⎢ 0.337
⎥
⎢ 0.479
⎥
⎢ 0.645
⎥
⎢ 0.761
⎥
⎢
⎥
⎣⋮
⎦
γ1 =
i
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢⎣
⎥
7.122 ⎥
5.193 ⎥
2.774 ⎥
1.878 ⎥
1.585 ⎥
⎥
1.349 ⎥
1.192 ⎥
1.096 ⎥
1.056 ⎥
⎥⎦
⋮
γ2 =
i
Psat1 ≔ 26.699899 ⋅ bar
⎢
⎥
⎢ 1.008 ⎥
⎢ 1.032 ⎥
⎢ 1.158 ⎥
⎢ 1.359 ⎥
⎢ 1.524 ⎥
⎢
⎥
⎢ 1.789 ⎥
⎢ 2.162 ⎥
⎢ 2.65 ⎥
⎢ 3.04 ⎥
⎢⎣ ⋮
⎥⎦
Psat2 ≔ 15.552476 ⋅ bar
Pcal200 ≔ x1exp200 ⋅ γ1 ⋅ Psat1 + x2exp200 ⋅ γ2 ⋅ Psat2
i
i
i
i
i
x1exp200 ⋅ γ1 ⋅ Psat1
i
i
y1cal200 ≔ ――――――
i
Pcal200
i
y1cal200 =
i
⎤
⎡0
⎢ 0.022 ⎥
⎢
⎥
0.219
⎢
⎥
⎢ 0.364 ⎥
⎢ 0.467 ⎥
⎢ 0.546 ⎥
⎢ 0.574 ⎥
⎢ 0.59 ⎥
⎢
⎥
⎢ 0.614 ⎥
⎢ 0.646 ⎥
⎢ 0.69 ⎥
⎢ 0.726 ⎥
⎢⎣ ⋮
⎥⎦
Pcal200 = ? bar
i
Non-Commercial Use Only
rms ≔
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
2
∑ ⎛Pexp200 − Pcal200 ⎞
i
i⎠
i ⎝
rms = ? bar
10
8
6
Non-Commercial Use Only
4
2
Pexp200
i
―――
bar
0
Pexp200
i
―――
bar
-2
-4
Pcal200
i
―――
bar
-6
Pcal200
i
―――
bar
-8
-10
-10
-8
-6
-4
-2
0
x1exp200
i
y1exp200
i
x1exp200
i
y1cal200
i
x1exp25 =
i
⎤
⎡0
⎢ 1 ⋅ 10 −4 ⎥
⎢
⎥
⎢ 0.019 ⎥
⎢ 0.029 ⎥
⎢ 0.045 ⎥
⎢ 0.056 ⎥
⎢ 0.094 ⎥
⎢
⎥
⎢ 0.095 ⎥
⎢ 0.131 ⎥
⎢ 0.147 ⎥
⎢ 0.179 ⎥
⎢ 0.265 ⎥
⎢
⎥
⎣⋮
⎦
⎡0
⎤
⎢ 0.011 ⎥
⎢
⎥
0.646
⎢
⎥
0 714
Non-Commercial Use Only
2
4
6
8
10
y1cal25 =
i
x1exp100 =
i
i
i
Pcal25 = ? bar
i
⎤
⎡0
⎢ 0.003 ⎥
⎢
⎥
0.004
⎢
⎥
⎢ 0.005 ⎥
⎢ 0.008 ⎥
⎢ 0.048 ⎥
⎢ 0.082 ⎥
⎢ 0.108 ⎥
⎢
⎥
⎢ 0.22 ⎥
⎢ 0.308 ⎥
⎢ 0.316 ⎥
⎢ 0.48 ⎥
⎢⎣ ⋮
⎥⎦
y1cal100 =
x1exp200 =
0.646
⎢
⎥
⎢ 0.714 ⎥
⎢ 0.773 ⎥
⎢ 0.796 ⎥
⎢ 0.836 ⎥
⎢ 0.836 ⎥
⎢
⎥
⎢ 0.852 ⎥
⎢ 0.857 ⎥
⎢ 0.863 ⎥
⎢ 0.874 ⎥
⎢⎣ ⋮
⎥⎦
⎤
⎡0
⎢ 0.115 ⎥
⎢
⎥
0.136
⎢
⎥
⎢ 0.149 ⎥
⎢ 0.233 ⎥
⎢ 0.571 ⎥
⎢ 0.645 ⎥
⎢ 0.674 ⎥
⎢
⎥
⎢ 0.721 ⎥
⎢ 0.736 ⎥
⎢ 0.738 ⎥
⎢ 0.764 ⎥
⎢⎣ ⋮
⎥⎦
Pcal100 = ? bar
i
⎡0
⎤
⎢ 0.001 ⎥
⎢
⎥
0.018
⎢
⎥
⎢ 0.045 ⎥
⎢ 0.092 ⎥
⎢ 0.226 ⎥
⎢ 0.362 ⎥
⎢ 0.446 ⎥
⎢
⎥
Non-Commercial Use Only
⎢ 0.551 ⎥
⎢ 0.659 ⎥
⎢ 0.758 ⎥
⎢ 0.816 ⎥
⎢⎣ ⋮
⎥⎦
y1cal200 =
i
⎤
⎡0
⎢ 0.022 ⎥
⎢
⎥
0.219
⎢
⎥
⎢ 0.364 ⎥
⎢ 0.467 ⎥
⎢ 0.546 ⎥
⎢ 0.574 ⎥
⎢ 0.59 ⎥
⎢
⎥
⎢ 0.614 ⎥
⎢ 0.646 ⎥
⎢ 0.69 ⎥
⎢ 0.726 ⎥
⎢⎣ ⋮
⎥⎦
Pcal200 = ? bar
Pcal ≔ augment (Pcal25 , Pcal100 , Pcal200)
Pcal = ? bar
Non-Commercial Use Only
i
x ≔ augment (x1exp25 , x1exp100 , x1exp200)
y ≔ augment (y1cal25 , y1cal100 , y1cal200)
i ≔ 0 ‥ 16
j≔0‥2
⎡ 25 + 273 ⎤
z ≔ ⎢ 100 + 273 ⎥
⎢
⎥
⎣ 200 + 273 ⎦
T ≔z
i,j
j
⎡ Pcal ⎤
Non-Commercial Use Only
⎢
⎥
⎢ bar ⎥
⎢ x ⎥
⎣ T ⎦
⎡ Pcal ⎤
⎢ ――
⎥
⎢ bar ⎥
⎢ y ⎥
⎣ T ⎦
Conclusions
1) This metod is more accurate when compared to the other methods available.This can be
seen
by comparing the "root mean square" values of different methods with this method at
same
temperature.
2)
This method is most accurate at low temperatures.
3) This method doesn't require the knowledge of experimental values of x (liquid phase
composition) and y (gas phase composition) to plot phase equilibrium plot. We can select our
own x values and find corresponding y values to construct phase equilibrium plot. The only
values needed are the saturation pressures of the components which can be estimated
from Reidel corresponding states method.
4) This method doesn't require the knowledge of critical temperature and critical pressure
values.
5) This method just requires the knowledge of structure of the compound to plot the phase
diagrams of the compounds.Based on the structure, the compound is divided into subgroups
and calculations are performed.
Non-Commercial Use Only