APPLICATION OF THE WILSON ACTIVITY COEFFICIENT MODEL IN PHASE EQUILIBRIA CALCULATIONS
EMEMERURAI [email protected]
Paper Presented at the NSChE Edo/Delta Chapter Meeting for January 2012
Friday, April 7, 2023
INTRODUCTIONActivity coefficients of either the liquid, or solid is
usually introduced into thermodynamic calculations to correct for the non-ideality of a mixture.
The use of activity coefficients is akin to the use of compressibility factor to correct for the non-ideality of a compressible fluid in PVT calculations.
An activity coefficient of 1 (unity) shows that the interactions between dissimilar or same molecules are always identical and the mixture is in the ideal state
Activity coefficient models are derived based on excess Gibbs free energy
i
nPTi
E
RTnnG ln
,,
INTRODUCTIONThe relationship between activity coefficient
and excess Gibbs free energy is
n = Composition of componentsGE= Excess Gibbs free energyThe following are the thermodynamic activity
coefficient models some of which are available in most modular simulators such as HYSYS, Chemcad and PRO II:
a) Regular solution modelb) Flory-Huggins modelc) Margules equation
INTRODUCTIOND) Van Laar equationE) Wilson equationF) Nonrandom Two Liquid (NTRL) modelG) Uniquac (Universal Quasi-chemical) model andH) Unifac (UNIQUAC Functional- group ) modelModels A and B work well for non-polar systems
such as HC mixtures, isomers and homologuesModels C-H are used non-polar and /or polar
systemsActivity coefficient models are used for the
determination of equilibrium ratio in both VLE and SLE (Prausnitz et al,2005)
INTRODUCTIONIn special distillation processes the K value is
used to calculate the relative volatility (Lei, et al;2005)
In wax formation modeling, activity coefficient of both the liquid and solid are needed to calculate the equilibrium ratio KSL after which flash calculations are done with the Rachford-Rice equation
The purpose of this paper is to demonstrate the application of the Wilson model in the calculation of activity coefficients using Excel spreadsheet and Matlab.
The Wilson model is given in equation 2:
WILSON EQUATION (EQUATION 2)
nc
inc
jjij
ikinc
jkjjk
x
xx1
1
1
ln0.1ln
INTRODUCTIONγK = Activity coefficient for component kΛij, Λji =Wilson coefficients for the binary
pair , i, and jAccording to Sinnot (2005) the use of
Wilson model presents an advantage in phase equilibria since it can be used to calculate the equilibrium compositions for multicomponent mixtures using only the binary coefficients obtained for the binary pairs that comprise the mixture.
APPLICATION OF THE WILSON MODELComputer programmes are most suitable for
solving the Wilson model and Sinnot (2005) presented a GWBASIC programme and an Excel spreadsheet for this purpose, although the later was not complete.
A complete solution of the Excel spreadsheet will be presented in this paper as well as the Matlab solution of the Wilson model
The components involved in the problem given by Sinnot (2005) include methanol (MeOH), ethanol (EtOH), isopropyl alcohol (IPA) and water with the liquid molar composition 0.05, 0.05, 0.18, and 0.72, respectively.
APPLICATION OF THE WILSON MODEL
The Wilson coefficients for the binary pairs are given in the Table below:Λ=
The spreadsheet for the complete solution is given below:
1 2.3357 2.7385 0.4180
0.1924 1 1.65 0.1108
0.2419 0.5343 1 0.0465
0.9699 0.9560 0.7793 1
APPLICATION OF THE WILSON MODEL-SPREADSHEET SOLUTIONQ11 Q12 Q13 Q14 X
1 0.192 0.242 0.97 0.05 0.01 0.012 0.048 0.05 Methanol2.336 1 0.534 0.956 0.117 0.05 0.027 0.048 0.05 Ethanol2.739 1.65 1 0.78 0.493 0.297 0.18 0.14 0.18 Isopropyl acohol0.418 0.111 0.047 1 0.301 0.08 0.033 0.72 0.72 Water
Q 0.961 0.436 0.252 0.957
MeOH EtOH IPA WATERQ21 0.052 0.122 0.143 0.022Q22 0.022 0.115 0.189 0.013Q23 0.173 0.381 0.713 0.033Q24 0.73 0.72 0.587 0.753Q2 0.977 1.337 1.632 0.82
ACTIVITY COEFFICIENT→γ 1.065 1.636 2.107 1.251
nc
inc
jjij
ikinc
jkjjk
x
xx1
1
1
ln0.1ln
APPLICATION OF WILSON MODEL-MATLAB SOLUTION The Matlab code for the solution
of the Wilson liquid activity coefficient is shown below:
THIS EXAMPLE IS TAKEN FROM RICHARDSON AND COULSON VOL.6 PAGE 344-348
n=4; for i=1:n A=[1 0.1924 0.2419 0.9699; 2.3357 1 0.5343 0.956; 2.7385 1.65 1 0.7795; 0.418 0.1108 0.0465 1 ]; X=[ 0.05 0.05 0.18 0.72]; end for i=1:n A2(:,i)=X(i).*A(i,:); end for i=1:n A3(i,:)=X(i).*A(:,i)./sum(A2(i,:)); end
for i=1:n G(i)=exp(1-
log(sum(A2(i,:)))-sum(A3(:,i))); end disp('THE ACTIVITY
COEFFICIENT OF METHANOL, ETHANOL, ISOPROPYL ALCOHOL')
disp(',AND WATER ARE GIVEN AS')
disp(' GMeOH GEtOH GIPA GH20')
disp(G)
OUTPUT OF MATLAB PROGRAMME
THE ACTIVITY COEFFICIENT OF METHANOL, ETHANOL, ISOPROPYL ALCOHOL, AND WATER ARE GIVEN AS
GMeOH GEtOH GIPA GH20 1.0655 1.6361 2.1074 1.2512After determination of the activity
coefficient, the equilibrium ratio in VLE can then be calculated using the equation:
PP
xyK
si
Li
i
ii
APPLICATION OF THE WILSON MODEL
Where γi = Liquid activity coefficientPs= Vapour pressure of liquidP = Pressure of system In SLE, e.g. ; in wax formation modelling the
liquid activity coefficient may be calculated with the Unifac model while the solid activity coefficient is found from the Wilson model and then K is calculated as:
Si
Li
Si
LiSL
i ffK
APPLICATION OF WILSON MODELWhere f L = fugacity of liquidf S = fugacity of solid which is a function
of the melting point, heat of fusion and solid-solid transition temperature.
CONCLUSION
The utility of the Wilson activity coefficient in phase equilibria calculations has been demonstrated in this presentation.
The calculation of equilibrium ratio is done using activity coefficient models such as the Wilson’s before flash calculations using the Rachford-Rice equation are carried out to determine the liquid and gas composition in VLE and the liquid and solid composition in SLE.
Hand calculations of the Wilson model is tedious, and thus computer programmes and spreadsheets are used to ease the calculations as demonstrated in this presentation
REFERENCESLei, Z; Chen, B; and Ding, Z (2005).Special
Distillation Processes, Elsevier, Amsterdam, pp 1-36
Prausnitz, J.M; Lichtenthaler, R, N; and de Azevedo, G (2005). Molecular Thermodynamics of Fluid-Phase Equilibria, Third Edition, Prentice Hall International Series, Chapter 6.
Sinnot, R.K (2005). Richardson and Coulson’s Chemical Engineering Design, Vol.6, Butterworth-Heinemann, Oxford, PP342-348.
