The derivatives of the general NRTL equations are very useful for VLE calculations, but are incredibly cumbersome to perform by hand in the general form.
General NRTL equations
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The general equation for
ln
(
γ
i
)
{\displaystyle \ln(\gamma _{i})}
for species
i
{\displaystyle i}
in a mixture of
n
{\displaystyle n}
components is[1] :
ln
(
γ
i
)
=
∑
j
=
1
n
x
j
τ
j
i
G
j
i
∑
k
=
1
n
x
k
G
k
i
+
∑
j
=
1
n
x
j
G
i
j
∑
k
=
1
n
x
k
G
k
j
(
τ
i
j
−
∑
m
=
1
n
x
m
τ
m
j
G
m
j
∑
k
=
1
n
x
k
G
k
j
)
{\displaystyle \ln(\gamma _{i})={\frac {\displaystyle \sum _{j=1}^{n}{x_{j}\tau _{ji}G_{ji}}}{\displaystyle \sum _{k=1}^{n}{x_{k}G_{ki}}}}+\sum _{j=1}^{n}{\frac {x_{j}G_{ij}}{\displaystyle \sum _{k=1}^{n}{x_{k}G_{kj}}}}{\left({\tau _{ij}-{\frac {\displaystyle \sum _{m=1}^{n}{x_{m}\tau _{mj}G_{mj}}}{\displaystyle \sum _{k=1}^{n}{x_{k}G_{kj}}}}}\right)}}
(1.1 )
with
G
i
j
=
exp
(
−
α
i
j
τ
i
j
)
{\displaystyle G_{ij}={\text{exp}}\left({-\alpha _{ij}\tau _{ij}}\right)}
(1.2 )
α
i
j
=
α
i
j
0
+
α
i
j
1
T
{\displaystyle \alpha _{ij}=\alpha _{ij_{0}}+\alpha _{ij_{1}}T}
(1.3 )
τ
i
j
=
A
i
j
+
B
i
j
T
+
C
i
j
T
2
+
D
i
j
ln
(
T
)
+
E
i
j
T
F
i
j
{\displaystyle \tau _{ij}=A_{ij}+{\frac {B_{ij}}{T}}+{\frac {C_{ij}}{T^{2}}}+D_{ij}\ln {\left({T}\right)}+E_{ij}T^{F_{ij}}}
(1.4 )
For derivative calculations, it becomes convenient to further compartmentalize the general NRTL equation by abstracting the summation terms. While this does introduce an additional substitution for chain rule differentiation, it does make the final equation more readable.
ln
(
γ
i
)
=
S
1
i
j
S
2
i
k
+
∑
j
=
1
n
x
j
G
i
j
S
2
j
k
(
τ
i
j
−
S
1
j
m
S
2
j
k
)
{\displaystyle \ln(\gamma _{i})={\frac {S_{1_{ij}}}{S_{2_{ik}}}}+\sum _{j=1}^{n}{\frac {x_{j}G_{ij}}{S_{2_{jk}}}}{\left({\tau _{ij}-{\frac {S_{1_{jm}}}{S_{2_{jk}}}}}\right)}}
(1.5 )
with
Sum type 1:
S
1
i
j
=
∑
j
=
1
n
x
j
τ
j
i
G
j
i
{\displaystyle S_{1_{ij}}=\displaystyle \sum _{j=1}^{n}{x_{j}\tau _{ji}G_{ji}}}
(1.6 )
Sum type 2:
S
2
i
j
=
∑
j
=
1
n
x
j
G
j
i
{\displaystyle S_{2_{ij}}=\displaystyle \sum _{j=1}^{n}{x_{j}G_{ji}}}
(1.7 )
Derivatives of system variables
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The system variables are intrinsic to a system being observed or predicted. These include several directly measurable variables and many indirectly measurable variables. Only the directly measurable system variables need to be calculated for the NRTL model.
Directly measurable system variables:
Temperature
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These, along with the composition derivatives, are the primary derivatives of interest, the reason for which will become obvious when reading the pressure derivatives section.
Adjustable parameters
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Since the adjustable parameters are treated as constants during typical VLE calculations, their derivatives are straight forward degenerate solutions:
Non-randomness
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(
∂
α
i
j
0
∂
T
)
P
,
N
→
=
0
(
∂
α
i
j
1
∂
T
)
P
,
N
→
=
0
{\displaystyle {\begin{matrix}\displaystyle \left({\frac {\partial \alpha _{ij_{0}}}{\partial T}}\right)_{P,{\vec {N}}}=0&\displaystyle \left({\frac {\partial \alpha _{ij_{1}}}{\partial T}}\right)_{P,{\vec {N}}}=0\end{matrix}}}
(2.1-1 )
Interaction
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(
∂
A
i
j
∂
T
)
P
,
N
→
=
0
(
∂
B
i
j
∂
T
)
P
,
N
→
=
0
(
∂
C
i
j
∂
T
)
P
,
N
→
=
0
(
∂
D
i
j
∂
T
)
P
,
N
→
=
0
(
∂
E
i
j
∂
T
)
P
,
N
→
=
0
(
∂
F
i
j
∂
T
)
P
,
N
→
=
0
{\displaystyle {\begin{matrix}\displaystyle \left({\frac {\partial A_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=0&\displaystyle \left({\frac {\partial B_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=0&\displaystyle \left({\frac {\partial C_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=0&\displaystyle \left({\frac {\partial D_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=0&\displaystyle \left({\frac {\partial E_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=0&\displaystyle \left({\frac {\partial F_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=0\end{matrix}}}
(2.1.1-1 )
Non-randomness term
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The non randomness term has a similarly simple series of temperature derivatives, being linear with respect to temperature.
First order:
(
∂
α
i
j
∂
T
)
P
,
N
→
=
α
i
j
1
{\displaystyle \left({\frac {\partial \alpha _{ij}}{\partial T}}\right)_{P,{\vec {N}}}=\alpha _{ij_{1}}}
(2.1.2-1 )
Second and higher order:
(
∂
n
α
i
j
∂
T
n
)
P
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}\alpha _{ij}}{\partial T^{n}}}\right)_{P,{\vec {N}}}=0}
(2.1.2-2 )
Interaction term
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The interaction term can be conveniently expressed in terms of the derivative order, which becomes more or less obvious after a couple of iterations:
First order:
(
∂
τ
i
j
∂
T
)
P
,
N
→
=
−
B
i
j
T
2
−
2
C
i
j
T
3
+
D
i
j
T
+
E
i
j
F
i
j
T
F
i
j
−
1
{\displaystyle \left({\frac {\partial \tau _{ij}}{\partial T}}\right)_{P,{\vec {N}}}=-{\frac {B_{ij}}{T^{2}}}-{\frac {2C_{ij}}{T^{3}}}+{\frac {D_{ij}}{T}}+E_{ij}F_{ij}T^{F_{ij}-1}}
(2.1.3-1 )
Second order:
(
∂
2
τ
i
j
∂
T
2
)
P
,
N
→
=
2
B
i
j
T
3
+
6
C
i
j
T
4
−
D
i
j
T
2
+
E
i
j
F
i
j
(
F
i
j
−
1
)
T
F
i
j
−
2
{\displaystyle \left({\frac {\partial ^{2}\tau _{ij}}{\partial T^{2}}}\right)_{P,{\vec {N}}}={\frac {2B_{ij}}{T^{3}}}+{\frac {6C_{ij}}{T^{4}}}-{\frac {D_{ij}}{T^{2}}}+E_{ij}F_{ij}\left(F_{ij}-1\right)T^{F_{ij}-2}}
(2.1.3-2 )
Higher order: {{NumBlk||
(
∂
n
τ
i
j
∂
T
n
)
P
,
N
→
=
(
−
1
)
n
(
n
!
)
B
i
j
T
n
+
1
+
(
−
1
)
n
(
(
n
+
1
)
!
)
C
i
j
T
n
+
2
+
(
−
1
)
n
+
1
(
(
n
−
1
)
!
)
D
i
j
T
n
+
E
i
j
T
F
i
j
−
n
∏
k
=
0
n
−
1
(
F
i
j
−
k
)
{\displaystyle \left({\frac {\partial ^{n}\tau _{ij}}{\partial T^{n}}}\right)_{P,{\vec {N}}}=\left(-1\right)^{n}\left(n!\right){\frac {B_{ij}}{T^{n+1}}}+\left(-1\right)^{n}\left((n+1)!\right){\frac {C_{ij}}{T^{n+2}}}+\left(-1\right)^{n+1}\left((n-1)!\right){\frac {D_{ij}}{T^{n}}}+E_{ij}T^{F_{ij}-n}\prod _{k=0}^{n-1}{\left(F_{ij}-k\right)}}
|2.1.3.-3 }
First order
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Interaction energy term
edit
(
∂
G
i
j
∂
T
)
P
,
N
→
=
−
(
α
i
j
(
∂
τ
i
j
∂
T
)
P
,
N
→
+
τ
i
j
(
∂
α
i
j
∂
T
)
P
,
N
→
)
exp
(
−
α
i
j
τ
i
j
)
{\displaystyle \left({\frac {\partial G_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=-\left(\alpha _{ij}\left({\frac {\partial \tau _{ij}}{\partial T}}\right)_{P,{\vec {N}}}+\tau _{ij}\left({\frac {\partial \alpha _{ij}}{\partial T}}\right)_{P,{\vec {N}}}\right){\text{exp}}\left({-\alpha _{ij}\tau _{ij}}\right)}
(2.1.4.1-1 )
(
∂
G
i
j
∂
T
)
P
,
N
→
=
−
(
α
i
j
(
∂
τ
i
j
∂
T
)
P
,
N
→
+
τ
i
j
(
∂
α
i
j
∂
T
)
P
,
N
→
)
G
i
j
{\displaystyle \left({\frac {\partial G_{ij}}{\partial T}}\right)_{P,{\vec {N}}}=-\left(\alpha _{ij}\left({\frac {\partial \tau _{ij}}{\partial T}}\right)_{P,{\vec {N}}}+\tau _{ij}\left({\frac {\partial \alpha _{ij}}{\partial T}}\right)_{P,{\vec {N}}}\right)G_{ij}}
(2.1.4.1-2 )
Sum type 1
edit
(
∂
S
1
i
j
∂
T
)
P
,
N
→
=
∑
j
=
1
n
x
j
(
τ
j
i
(
∂
G
j
i
∂
T
)
P
,
N
→
+
G
j
i
(
∂
τ
j
i
∂
T
)
P
,
N
→
)
{\displaystyle \left({\frac {\partial S_{1_{ij}}}{\partial T}}\right)_{P,{\vec {N}}}=\displaystyle \sum _{j=1}^{n}{x_{j}\left(\tau _{ji}\left({\frac {\partial G_{ji}}{\partial T}}\right)_{P,{\vec {N}}}+G_{ji}\left({\frac {\partial \tau _{ji}}{\partial T}}\right)_{P,{\vec {N}}}\right)}}
(2.1.4.2-1 )
Sum type 2
edit
(
∂
S
1
i
j
∂
T
)
P
,
N
→
=
∑
j
=
1
n
x
j
(
∂
G
j
i
∂
T
)
P
,
N
→
{\displaystyle \left({\frac {\partial S_{1_{ij}}}{\partial T}}\right)_{P,{\vec {N}}}=\displaystyle \sum _{j=1}^{n}{x_{j}\left({\frac {\partial G_{ji}}{\partial T}}\right)_{P,{\vec {N}}}}}
(2.1.4.3-1 )
Main equation
edit
(
∂
ln
γ
i
∂
T
)
P
,
N
→
=
S
2
i
k
(
∂
S
1
i
j
∂
T
)
T
,
N
→
−
S
1
i
j
(
∂
S
2
i
k
∂
T
)
T
,
N
→
S
2
i
k
2
+
∑
j
=
1
n
x
j
S
2
j
k
(
G
i
j
(
(
∂
τ
i
j
∂
T
)
P
,
N
→
−
S
2
j
k
(
∂
S
1
j
m
∂
T
)
P
,
N
→
−
S
1
j
m
(
∂
S
2
j
k
∂
T
)
P
,
N
→
S
2
j
k
2
)
+
(
τ
i
j
−
S
1
j
m
S
2
j
k
)
(
∂
G
i
j
∂
T
)
P
,
N
→
)
+
G
i
j
(
τ
i
j
−
S
1
j
m
S
2
j
k
)
(
∂
S
2
j
k
∂
T
)
P
,
N
→
S
2
j
k
2
{\displaystyle \left({\frac {\partial \ln {\gamma _{i}}}{\partial T}}\right)_{P,{\vec {N}}}={\frac {S_{2_{ik}}\left({\frac {\partial S_{1_{ij}}}{\partial T}}\right)_{T,{\vec {N}}}-S_{1_{ij}}\left({\frac {\partial S_{2_{ik}}}{\partial T}}\right)_{T,{\vec {N}}}}{{S_{2_{ik}}}^{2}}}+\sum _{j=1}^{n}{x_{j}{\frac {S_{2_{jk}}\left(G_{ij}\left(\left({\frac {\partial \tau _{ij}}{\partial T}}\right)_{P,{\vec {N}}}-{\frac {S_{2_{jk}}\left({\frac {\partial S_{1_{jm}}}{\partial T}}\right)_{P,{\vec {N}}}-S_{1_{jm}}\left({\frac {\partial S_{2_{jk}}}{\partial T}}\right)_{P,{\vec {N}}}}{{S_{2_{jk}}}^{2}}}\right)+\left(\tau _{ij}-{\frac {S_{1_{jm}}}{S_{2_{jk}}}}\right)\left({\frac {\partial G_{ij}}{\partial T}}\right)_{P,{\vec {N}}}\right)+G_{ij}\left(\tau _{ij}-{\frac {S_{1_{jm}}}{S_{2_{jk}}}}\right)\left({\frac {\partial S_{2_{jk}}}{\partial T}}\right)_{P,{\vec {N}}}}{{S_{2_{jk}}}^{2}}}}}
(2.1.4.4-1 )
Second order
edit
Third order
edit
Pressure
edit
These are the least interesting, though still very important and useful, derivatives. Since the pressure derivatives are evaluated at constant temperature and composition and none of the terms involved are explicit functions of pressure, all of the derivative expressions are essentially derivatives of a constant scalar term, which is zero. All of the partial derivatives are listed below for sake of complete coverage of the topic.
Adjustable parameters
edit
The pressure derivatives for the adjustable parameters are analogous to their respective temperature derivatives.
(
∂
n
ϕ
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}\phi }{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.1-1 )
where
ϕ
{\displaystyle \phi }
is an adjustable parameter.
Non-randomness term
edit
Unlike the temperature derivatives, the pressure derivatives of the non-randomness term are all zero.
(
∂
n
α
i
j
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}\alpha _{ij}}{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.2-1 )
Interaction term
edit
(
∂
n
τ
i
j
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}\tau _{ij}}{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.3-1 )
Interaction energy term
edit
(
∂
n
G
i
j
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}G_{ij}}{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.4-1 )
Sum type 1
edit
(
∂
n
S
1
i
j
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}S_{1_{ij}}}{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.5-1 )
Sum type 2
edit
(
∂
n
S
2
i
j
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}S_{2_{ij}}}{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.1-6 )
Main equation
edit
This is the most important derivative resulting from the pressure derivatives due to its use in simplifying calculations.
(
∂
n
ln
γ
i
∂
P
n
)
T
,
N
→
=
0
{\displaystyle \left({\frac {\partial ^{n}\ln {\gamma _{i}}}{\partial P^{n}}}\right)_{T,{\vec {N}}}=0}
(2.2.7-1 )
Composition
edit
Adjustable parameters
edit
(
∂
n
N
ϕ
∂
N
i
n
)
T
,
P
,
N
j
≠
i
=
0
{\displaystyle \left({\frac {\partial ^{n}N\phi }{\partial {N_{i}}^{n}}}\right)_{T,P,N_{j\neq i}}=0}
(2.3.1-1 )
where
ϕ
{\displaystyle \phi }
is an adjustable parameter.
Non-randomness term
edit
(
∂
n
N
α
i
j
∂
N
k
n
)
T
,
P
,
N
m
≠
k
=
0
{\displaystyle \left({\frac {\partial ^{n}N\alpha _{ij}}{\partial {N_{k}}^{n}}}\right)_{T,P,N_{m\neq k}}=0}
(2.3.2-1 )
Interaction term
edit
(
∂
n
N
τ
i
j
∂
N
k
n
)
T
,
P
,
N
m
≠
k
=
0
{\displaystyle \left({\frac {\partial ^{n}N\tau _{ij}}{\partial {N_{k}}^{n}}}\right)_{T,P,N_{m\neq k}}=0}
(2.3.3-1 )
Interaction energy term
edit
(
∂
n
N
G
i
j
∂
N
k
n
)
T
,
P
,
N
m
≠
k
=
0
{\displaystyle \left({\frac {\partial ^{n}NG_{ij}}{\partial {N_{k}}^{n}}}\right)_{T,P,N_{m\neq k}}=0}
(2.3.4-1 )
Sum type 1
edit
(
∂
N
S
1
i
j
∂
N
k
)
T
,
P
,
N
m
≠
k
=
τ
k
i
G
k
i
{\displaystyle \left({\frac {\partial NS_{1_{ij}}}{\partial N_{k}}}\right)_{T,P,N_{m\neq k}}=\tau _{ki}G_{ki}}
(2.3.5-1 )
Sum type 2
edit
(
∂
N
S
1
i
j
∂
N
k
)
T
,
P
,
N
m
≠
k
=
G
k
i
{\displaystyle \left({\frac {\partial NS_{1_{ij}}}{\partial N_{k}}}\right)_{T,P,N_{m\neq k}}=G_{ki}}
(2.3.6-1 )
Main equation
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First order
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Second order
edit
Third order
edit
Derivatives of adjustable parameters
edit
References
edit