fluid-phase-equilibria-74-1992-l-15

1. Fluid Phase Equilibria, 74 (1992) 1-15
Elsevier Science Publishers B.V., Amsterdam
Calculation of Thermodynamic Equilibrium Properties.
Jorgen M. Mollerup and Michael L. Michelsen.
Department of Chemical Engineering.
Building 229, DTH, 2800 Lyngby, Denmark.
(Received October 4, 1991; accepted in final form January 27 1991)
Keywords: Calculation procedure, Helmholtz surface, derived properties.
ABSTRACT
A formalism for calculation of equilibrium properties from a model of the Helmholtz function
is presented. A modular approach which enables modification of single features of the model
without rewriting the entire computer code is emphasized. The formalism ensures a fully
thermodynamic consistent set of relations and leads to efficient code. Identity tests for checking
calculated fugacities and their partial derivatives are presented.
INTRODUCTION
Calculation of thermodynamic properties from an equation of state may appear a trivial
problem which only requires adherence to basic definitions. The increasing complexity of
thermodynamic models, however, requires a systematic approach in order to avoid inefficient
or even incorrect code, and we therefore find it appropriate to present some ideas which we
have found valuable for generating modular and efficient code for calculation of thermodyna-
mic properties.
Our main aim is to describe a procedure which ensures that the resulting thermodynamic
model is fully consistent. An additional objective is to present a formalism based on a modular
approach, which enables us to modify a single feature of the model, e.g. a mixing rule for one
of the model parameters, without rewriting the entire code. Finally we wish to demonstrate
that the suggested approach leads to a computationally efficient code, in particular when
derived properties, such as composition derivatives of fugacity coefficients, are desired.
0378-3812/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved

2. 2 J.M. Mollerup and M.L. Michelsen /Fluid Phase Equilibria 74 (1992) l-15
THE HELMHOLTZ SURFACE AND ITS DERIVATIVES
Given an equation of state
P - WPJ) (1)
where x is the vector of mixture mole fractions, the textbook approach to calculate mixture
fugacity coefficients is by means of an integral, i.e.
(2)
where Z is the compressibility factor, Z=Pv/RT. Interchange of the order of integration and
differentiation in eqn. (2) leads to the equivalent, more convenient expression
V
RTLn9,--$/(P-F)dV-RTenz
1*
- aA ‘(T,V,n)/%, - RT &Jz
where
A’(T,V,@ - -
(3)
(4)
A’ is the residual Helmholtz function, i.e. the Helmholtz function of the mixture, given as a
function of temperature T, total volume V, and the vector of mixture mole numbers II, minus
that of the equivalent ideal gas mixture at (T,V,n). R is a homogeneous function of degree 1
in the extensive variables (V,n) and, given an expression for A’, all other properties can be
derived solely by differentiation. The pressure equation itself, normally used to define the
‘equation of state’, is actually just one of these derivatives given by
pm_- a,,+?av
(5)
The expression for the residual Helmholtz energy is thus the key equation in equilibrium
thermodynamics (Michelsen 1981), where all other residual properties are calculated as partial
derivatives in the independent variables T, V, and n, as shown in Table I in the Appendix. In

3. J.M. Mollerup and M.L. Michelsen /Fluid Phase Equilibrfa 74 (1992) 1-15 3
particular, it is important to recognize that mole numbers rather than mole fractions are the
independent variables. Derivatives with respect to mole fractions are best avoided, as they re-
quire a definition of the ‘dependent’ mole fraction and in addition lead to more complex ex-
pressions missing many important symmetry properties.
Should we wish to modify our thermodynamic model, it is far most convenient to introduce
such modifications directly in the expression for A’. Modified mixture parameters should
depend only on T, V, and n and preserve the homogeneity of A’. This approach reduces the
risk of errors and inconsistencies or misconceptions such as ‘pressure dependent interaction
coefficients’.
In addition many model concepts relate directly to the Hehnholtx function rather than to a
pressure equation. This applies for example to a corresponding states model, to a ‘chemical
model, and to a model based on statistical mechanics, where the Helmholtx function is given
directly in terms of the canonical partition function, A = - kT In Q.
Finally since the Helmholtx energy is a first order homogeneous function in the extensive
variables V and n we can from Euler’s theorem of a homogeneous function write the following
mathematical constraint
A,dT - WA, - Crt&“, - 0
i
In general we use the notation F,, for the partial derivative of the function or parameter F with
respect to the variable y keeping all other variables constant. That is
A,- a4
( 1aT “y-
s
A,- M
( 1av rr - -
P
(7)
(9)
Equation (6) is known as the Gibbs-Duhem equation.

4. 4 J.M. Mollerup and M.L. Michelsen /Fluid Phase Equilibria 74 (19%) l-15
CALKULATION OF THE DERIVATIVES OF THE HELMHOLTZ SURFACE.
When a particular mathematical model is chosen for the Hehnholtz energy derived properties
such as fugacity coefficients, enthalpy, heat capacity etc. are obtained as partial derivatives with
respect to the independent variables T, V, and n, cf. Table I in the Appendix. Although not
necessary it is somewhat more convenient to work with the reduced residual Helmholtz energy,
defined by
F-A’
RT
(10)
Normally our model provides an expression for F in terms of T, V, total moles n, and one or
more ‘mixture parameters’, a, b, .... e.g.,
where the mixture parameters are prescribed functions of T, V, and n. It is important to realize
that making these parameters functions of other thermodynamic properties, such as e.g. pres-
sure, converts the algebraic model into a probably intractable differential equation. In the
following we shall assume that mixture parameters are specified explicit functions of tempera-
ture, volume, and composition.
We recommend that the calculation of the derivatives of F is performed as a two-step proce-
dure. In the first step; F is differentiated with respect to its primary variables, i.e. T, V, n, and
the mixture parameters, and in the second step the derivatives of the mixture parameters are
evaluated. This results in a set of ‘model derivatives’ independent of the chosen mixing rules
and a set of ‘parameter derivatives’ which are not directly affected by the form of the chosen
model.
The partial derivatives of F needed for calculation of thermodynamic properties are:
(aF/m)v,,- FT + Fsbr+ F8, (12)
(13)
(14)

5. .I.M. Mollerupand M.L. Michelsen/Fluid Phase Equilibria 74 (I 992) I-15 5
where we use bi and a, as abbreviated notations for the composition derivatives of b and a.
The ‘model derivatives’ are thus the terms F,,,F,, Fv, Fb,and F,, and the ‘parameter derivati-
ves’ are b, b,., bv and the corresponding terms for the a-parameter.
Second order derivatives are found by repeated differentiation and straightforward application
of the chain rule. As an example the second composition derivatives are given by
(=)
an,anj IT
- F, + F&b, + F,bu + F,,,,ai + &a,
(15)
where we have utilized the symmetry in the second order model derivatives, i.e. Fsb=F,,, etc.
Adaption of the procedures described here does not only lead to an easier and better struc-
tured approach for deriving thermodynamic properties but is also very likely to provide an
efficient code, in particular when derivatives of fugacity coefficients are required. An illustra-
tion is given in Table II in the Appendix for the Density Dependent Local Composition Model
developed by Mollerup (1983, 1985)and Math& and Copeman (1983). Additional examples
can be found in Michelsen and Mollerup (1986).
An example: The Kedlich-Kwong Equation of State (Mollerup 1986)
The pressure equation is:
p_RT_- o(T)
v-b v(v+b)
(16)
were v is the molar volume, and mixingrules for mixture parameters b and a are conventional-
ly given by:
(17)
(18)

6. 6 J.M. Mollerup and M.L. Michelsen / Fluid Phase Equilibria 74 (1992) I-15
where xi and xj are mixture mole fractions.
First we substitute mole numbers for mole fractions and define the two new ‘molar’ mixture
parameters B and D, where
B - nb - c n,b,
I
(19)
D - n*a(T) - cc nj~p~(T)
i I
and substitute the total volume V for the molar volume v, yielding
P
nRT D----
V-B v(V+B)
We integrate according to equation (4) to obtain the residual Hehnholtz energy
A’(T,V,n) - nRTllL -
and F, the reduced residual Helmholtz energy
F(n,T,VWJ) - n ln v -
V-B
(20)
(21)
(22)
(W
The ‘model’ derivatives needed for calculation of fugacity coefficients are thus F,, FP F,, F,
and Fn, given by
F,-lay
V-B
(24)
(25)
F” - n(lV
_‘)_D(‘_1)
V-B BRT V+B V
(26)
F,, _ n _
V-B
(27)

7. J.M. Moilerup and M.L. Micheisen /Fluid Phase Equilibria 74 (1992) 1-15
Derivatives of the fugacity coefficients with respect to temperature, pressure, and composition
require the second order model derivatives F,,, F,,r, F,,,, F F F F F FnB, nD, TP TV, TB, TIM F,, Fva,
htDt Fee, FBD, and FDD- These are readily found by repeated differentiation, and it is imme-
diately evident that the terms F,,, FnT, F,,u, and Fun are identically zero.
These model derivatives of F are combined with the derivatives of the mixture parameters to
yield the derivatives required according to Table I in the Appendix, i.e.
- F,, + F,B, + F,D,
8F
l-1a@” y
- F,, + FybB1 + F,D,
- (Fm + Fm Da B, + FmD, + FDDrr
V
- F,(B,+B$ +F,B,B,+F,B,,+F,,(B,D,+B,D,)+F,D,,
The derivatives of the mixture parameters are
(33)
Dii - 2au (35)
(29)
(30)
(31)
(32)
(36)

8. 8 J.M.Mollerupand M.L. Michelsen/Fluid PhaseEquilibria 74 (1992) I-15
D, - 7 ni 7 n,@*a~l~*) (37)
and the derivatives of B,
B, - b, BU - 0 (38)
In order to improve model performance the covolume parameter b is often calculated using
a quadratic rather than a linear mixing rule, i.e.
b - CCx*Jr& (39)
i i
B is in this case given by
nB - n*b (40)
and its derivatives are found from
B + nBi - 2 cnjbU
I
(41)
B, + Bj + nB,, - 2bu (42)
The expressions derived above can be simplified further. The explicit temperature dependence
in the model can be removed by absorbing the RT-term in the D-parameter, defining a new
mixture parameter D’ = D/(RT). This reduces F to a function of only four independent vari-
ables V, n, B and D’.
Introduction of new mixing rules.
An alternative possibility is to introduce a new mixture parameter E, defined by
E-D_
qvj~
BBT Cnjbj
j
(43)
This substitution results in simpler ‘model derivatives’ at the expense of more complicated
expressions for the derivatives of E.

9. J.M. Mollerupand M.L. Michelsen/Fluid Phase Equilibria 74 (1992) 1-15 9
As an additional example we shall consider the use of a Huron-Vidal type mixing rule for the
a-parameter (Huron and Vidal, 1979):
where g’ is a local composition based expression for the excess Gibbs energy at infinite pres-
sure. The only modifications required to utilize the derivatives derived earlier is the expressions
for the D-parameter and its composition and temperature derivatives:
It is important to realize that D is still treated as an independent parameter in the evaluation
of the model derivatives, although it formally depends on both n and B.
If desired, D is readily replaced by D’, but in the present case a formulation in terms of the
parameter E is more convenient, as it leads to simpler expressions for both the model deriva-
tives and the parameter derivatives. The mixing rule for E becomes:
E-D-C 411 1 ?lg’
BRT i
n,-- - GRT
RTb,
TEST OF CALCULATED FUGACITY COEFFICIENTS AND PARTIAL DERIVATIVES.
Unfortunately, even a systematic approach does not prevent coding errors, and it is important
that computer codes for calculation of fugacity coefficients and their partial derivatives are
tested for internal consistency. We have found the identities listed in Table III in the Appendix
very useful.
The analytical derivatives can always be tested by numerical evaluation of the derivatives,
preferably carried out by means of central differences, i.e.
q fin,* ,...,n, + c,..n> - finI,%,...p, - e,..nJ
i)n,- 2s
(47)
For composition derivatives e should be chosen as about l@ times the sum of moles, which
should yield results accurate to 8-10 digits. Note that the composition derivatives are homoge-
neous functions of degree -1 in the composition.
Algorithms for calculation of mgacity coefficients which, given temperature, pressure, and
composition, return only fugacity coefficients and the compressibility factor Z can be subjected

10. 10 J. M. Mollerup and M.L. Michelsen /Fluid Phase Equilibria 74 (1992) l-15
to the test relations
and
NOTATION
A=
a
b
B
G&V
C
D
E
F
G’
g’
H’
MW
“iaj
n
n
P
R
S’
V
V
XI4
x
Z
Total residual Hehnholtz energy
Mixture parameter in Cubic Equation of State
Mixture parameter (covolume) in Cubic Equation of State
Scaled b-parameter B = nb
Heat capacities for n moles
Number of components in mixture
Scaled a-parameter D = n2a
Modified mixture parameter E = D/(BRT)
Reduced residual Hehnholtz energy AT/(RT)
Total residual Gibbs Energy
Excess molar Gibbs energy expression
Total residual Enthalpy
Mixture molar weight
Molar amounts of components i and j
Total moles in mixture
Vector of total moles
Pressure
The Gas constant
Total residual Entropy
Total volume
Molar volume
Mole fractions of components i and j
Vector of mixture mole fractions
Mixture compressibility factor Z = PV/(nRT)
(48)
(49)

11. J.M. Mollerup and M.L. Michelsen /Fluid Phase Equilibria 74 (1992) l-15
SUperscriptS
11
r Residual property = real property minus ideal gas property at the same state
variables.
Greek
Joule-Thompson coefficient
Component fugacity coefficient
REFERENCES
Huron, M.J. and Vidal,J., 1979. “New mixing rules in simple equations of state for representing
vapour-liquid equilibria of strongly non-ideal mixtures”. Fluid Phase Equilibria, 3: 255-271.
Michelsen, M.L., 1981. “Partial derivatives of thermodynamic properties from equations of
state”. Department of Chemical Engineering, DTH.
Mollerup, J.M., 1983. “Correlation of thermodynamic properties using a random-mixture
reference state.” Fluid Phase Equilibria, 15: 139-154.
Math& P.M. and Copeman,T.W., 1983. “Extension of the Peng-Robinson equation of state
to complex mixtures: Evaluation of various forms of the local composition concept.” Fhrid
Phase Equilibria, 13: 91-108.
Mollerup, J.M., 1985. “Correlation of gas solubilities in water and methanol at high pressure.”
Fluid Phase Equilibria, 22: 139-154.
Michelsen, M.L. and Mollerup, J.M., 1986. “Derivatives of thermodynamic properties”. AJChE
Journal, 32: no. 8, 1389-1392.
Mollerup, J.M., 1986. “Thermodynamic properties from a cubic equation of state”. Department
of Chemical Engineering, DTH.

12. 12
APPENDIX
J.M. Molierupand M.L. Michelsen/Fluid Phase Equilibtia 74 (1992) I-IS
TABLE I
Thermodynamic properties calculated as derivatives of the residual Helmholtz surface
A’(T,V,n)--
Pressure and derivatives
2 - PV7nRT
dP
( 1 P_JQ A??
dT,=-? ( 1avar "
0)
(2)
(3)
(4)
(9
(7)

13. J.M. Mollerup and M.L. Michelsen /Fluid Phase Equilibria 74 (1992) I-15
The fugacity coefficient and derivatives
Other thermodynamic properties ’
13
(8)
(9)
(10)
(11)
(12)
(13)
04)

14. 14 J.M. Molierup and ML Michelsen /Fluid Phase Equilibria 74 (1992) 1-IS
S’(Tg,n) - S’fzV+) + nR ln2
H’(T,P,n) ” A ‘(T,V$l) + zs’(T,Vp] + PV - MT
G ‘(TT,n) - A ‘(T,V,nj + PV - nZtT - nRT faZ
(16)
07)
(18)
(19)
TqT$,n) - RThlcp, (21)
*Since the ideal gas energy, enthalpy, and heat capacity do not depend on pressure or
volume, their residual properties are independent of whether they are calculated at
specified (n,T,P) or (n,T,V). This is not the case with the entropy, the Gibbs energy and
the Helmholtz energy since g = - 4 and -$ - $ for an ideal gas.

15. J.M. Mollerup and M.L. Michelsen /Fluid Phase Equilibria 74 (1992) 1-15 15
TABLE II
Relative computer time consumption of the DDLC model
Number of components 2 14
Calculated properties Time consumed
Mathias and Copeman, (1983); ZandPnQi 1 75
Mollerup, (1985); ZandFn$, 1 16
Z, On+,, and derivatives 1.4 32
TABLE III
Identity tests of calculated fugacity coefficients and partial derivatives
i- 1,2,...,c
(2)
(3)
(4)
(5)