Activity Coefficient Calculation for Binary Systems Using UNIQUAC

Project Work in the Course

Advanced Thermodynamics: With Application to Phase and Reaction Equilibria

KP8108

Department of Chemical Engineering, NTNU

Ugochukwu E. Aronu

July, 2009

Table of Contents 2 1.0 Introduction 3 2.0 Thermodynamic Framework 4 2.1 Thermodynaic Equilibrium Condition 4 2.2 Partial Molar Gibbs Free Energy 5 2.3 Gibbs-Duhem Equation 6 2.4 Equilibrium in Heterogenous Closed System 7 2.5 Chemical Potential 8 2.6 Fugacity and Fugacity Coefficient 9 2.7 Activity and Activity Coefficient 10 2.7.1 Normalization of Activity Coefficient 11 2.8 Excess Functions 12 3.0 Vapour Liquid Equilibrium Calculation 13 3.1 Chemical Equilibria 14 3.1.1 Chemical Equilibrium Constant 15 3.2 Phase Equilibria 15 3.3 Model for Activity Coefficient 17 EG 3.3.1 Local Composition Models 18 4.0 UNIQUAC 18 4.1 UNIQUAC Implementation 20 4.1.1 MEA-H20 System 21 4.1.2 Acetic Acid-H20 System 21 4.1.3 Other Systems 21 5.0 Results 25 Summary 25 Future Work 25 References 25 Derivation of Activity Coefficient Expression Based on UNIQUAC 27

2

1.0 Introduction Design of industrial chemical process separation equipment such as absorption and distillation

columns as well as simulation of chemical plants require reasonably accurate correlation and

prediction of phase equilibria because it is an integral part of vapour liquid equilibrium (VLE)

modeling. It is thus necessary to develope thermodynamic models suitable for practical phase

equilibrium calculations.

In general, VLE models are based on fundamental equations for phase and chemical

equilibria. The basic quantities required for VLE calculations are; chemical reaction

equilibrium constants, Henry’s law constant, fugacity coefficients and activity coefficients

(Poplsteinova, 2004), phase equilibria calculation requires Henry’s law constant, fugacity

coefficients and activity coefficients. The differences in VLE models are mainly in the way

the phase non-idealities are treated. In the vapor phase non-ideality is either neglected or

represented by a fugacity coefficient calculated from one of the well-established equations of

state such as Soave-Redlich-Kwong equation of state, the models will then differ on the type

of equation of state that were applied. These differences, however are of minor importance

since the calculation of vapor phase fugacities are not crucial for the model performance. The

treatment of the liquid phase, on the other hand, is very important. Accurate prediction of the

liquid phase composition plays a key role in the VLE modeling. Since the evaluation of

equilibrium constants and fugacity coefficients are reasonably well established, the activity

coefficients were identified as the most important variables of the VLE model. Thus proper

representation of activity coefficients is desirable (Poplsteinova, 2004).

The system of interest in my research group is mainly CO2-H20-Alkanolamine. An activity

coefficient model for such system must be able to represent the non-ideality of an electrolyte

solution. Some activity coefficient models for non-electrolyte systems include, Wilson,

NRTL, UNIFAC, UNIQUAC. Some have been modified for electrolyte systems. Review of

electrolyte activity coefficient models were present by Maurer, 1983; Renon, 1986 and

Anderko et al., 2002. For the purpose of this course the universal quasi-chemical

(UNIQUAC) equations proposed by Abrams and Prausnitz, 1975 is used in calculating

activity coefficient for non-electrolyte systems. Further work will involve extending the

model for calculation activity coefficient for electrolyte solutions.

3

2.0 Thermodynamic Frame Work

Vapour liquid equilibria calculation requires simultaneous solution of phase and chemical

equilibria for reactive systems. Ths section shows briefly the mathematical framework for the

VLE calculations which is built around basic concepts of thermodynamic. Most of the

discussions here were taken from Elliot and Lira, 1999; Prausnitz et al., 1999; Kim, 2009;

Hartono, 2009 and Poplsteinova, 2004.

2.1 Thermodynamic Equilibrium Condition

Homogenous Closed system

A homogeneous is a system with a uniform properties ie. properties such as density is same

from point to point, in a macroscopic sense example is a phase. A closed system do not

exchange matter with the surrounding eventhough it may exchange energy (Prausnitz et al.,

1999). An equilibrium state is one with no tendency to depart spontenously (having in mind

certain permissible changes or process such as heat transfer, work of volume displacement

and for open systems mass transfer across phase boundry). It’s properties are independent of

time. A change in equilibrium state of a system is called a process and a reversible process is

one that maintains a state of virtual equilibrium throughout the process, it is often referred to

as one connecting a series of equilibrium states.

For a reversible process, the general condition for thermodynamic equilibrium is derived by

combination of first and second law of thermodynamics as:

dU TdS PdV 1

This condition could be expressed in terms of all extensive thermodynamic functions; internal

energy (U), enthalpy (H), Helmholtz energy (A) and Gibbs free energy (G): The condition for

equilibrium is usually maintained by keeping two of the thermodynamic variables constant.

Conditions for equilibrium is often expressed in terms of Gibbs free energy because the two

constant variables are often the temperature and pressure. Using the fundamental

thermodynamic relation for Gibbs free energy

G U TS PV 2

4

Eq. 1 could be written as

dG SdT VdP 3

Thus at constant temperature and pressure

0dG 4

which implies that for thermodynamic equilibrium at constant T and P, the Gibbs free energy

of a closed system reaches its minimum.

2.2 Partial Molar Gibbs Free Energy

Homogenous open System

For a closed system, U is considered a function of S and V only; that is U = U(S, V) but for

open system, there are additional indepedent variables. For these variables we use the mole

numbers of the various compnents present, thus we consider U as the function

1 2( , , , ..., )mU U S V n n n , where m is the number of components present. Total differential is

then

, ,i i

iiV n S n S V n

U U UdU dS dV dn

S V S

, , j

5

where subsript ni refers to all mole numbers and nj to all mole numbers other than ith (the

term under differenciation). Writing eq. 5 in the form

dU TdS PdV + 6 i ii

dn

where , , j

ii S V n

U

n

Eq. 6 is the fundamental equation for an open system corresponding to eq. 1 for a closed

system. The function μi is an intensive property which depends on temperature, pressure and

composition from its position in the equation as the coefficient of dni it can be refered to as

mass or chemical potential just as T (the coefficient of dS) is a thermal potential and P (the

5

coefficient of dV) is a mechanical potential. Similar expression can be derived in terms of

Gibbs free energy G (and other extentive thermodynamic properties).

dG SdT VdP + i ii

dn 7

where

, , , , , , , ,j j j

ii i i iS V n S P n T V n P T n

U H A G

n n n n

j

8

The quantity i is the partial molar Gibbs free energy, but not partial molar internal energy,

enthalpy, or Helmholtz energy, because the independent variables T and P chosen for defining

of partial molar quantities are also fundamental independent variables for the Gibbs free

energy G (Prausnitz et al., 1999). The quantity, , , j

ii P T n

G

n

, the partial molar Gibbs free

energy is also called the chemical potential.

2.3 Gibbs-Duhem Equation

At constant temperature and pressure eq. 7 reduces to

= ,

i ii P T

dn dG 9

For equilibrium, at constant P and T, the Gibbs free energy is minimized (i.e dG = 0) (Elliot

and Lira, 1999). Also for a closed system dni = 0 thus equation 7 equal to zero at equilibrium.

From equation 9, equilibrium condition in terms of chemical potential could then be deduced

,

i ii P T

dG dn = 0 10

This is called Gibbs-Duhem Equation. It is a thermodynamic consistency relation for a

heterogenous system that is used for experimental data evalution and theory development. It

expresses the fact that among + 2 variables consisting of temperature, pressure and

6

chemical potentials of each component present in the system. Only +1 are independent

variables and the last variable is a dependent variable calculated in such a way that Gibbs-

Duhem equation is satisfied.

In terms of activity coefficient Gibbs-Duhem equation is expressed as

ln 0i ii

x d 11

Equation 11 is a differential relation between the activity coeffiecients of all the components

in solution (Prausnitz et al., 1999). For a binary solution, the Gibbs-Duhem equation may be

written as:

11 2

1 2

ln lnd dx x

dx dx2

12

2.4 Equilibrium in a Heterogeneous Closed System

A heterogeneous, closed system is made up of two or more phases with each phase considered

an open system within the overall closed system. If the system is in internal equilibrium with

respect to the three processes of heat transfer, boundry displacement and mass transfer,

neglecting special effects such as surface forces; semipermeable membranes; and electric,

magnetic or gravitational forces (Prausnitz et al., 1999), the general result for a closed system

consisting of phases (where two of the phases are liquid (L) and vapour (V)) can be written

as

...

...

...

v L

v L

v L

T T T

P P P

13

This is the basic criterion for phase equilibrium, which states that at equilibrium, the

temperature, pressure and chemical potentials of all species are uniform over the whole

system.

7

2.5 Chemical Potential

The task of phase-equilibrium thermodynamics is to describe quantitatively the distribution at

equilibrium of every component among all the phases present. Gibbs obtained the

thermodynamic solution to the phase-eqiulibrium problem by introducing the abstract concept

of chemical potential. The task is then to relate the abstract chemical potential of a substance

to physically measurable quantity such as temperature, pressure and composition.

For a pure substance i, the chemical potential is related to the temperature and pressure by the

differential equation

i i id s dT v dP 14

where is the molar entropy and the molar volume. Integrating and solving for is iv i at some

temperature T and pressure P, we have

( , ) ( , )r r

T Pr r

i i i

T P

T P T P s dT v dP i 15

where superscript r is an arbitrary reference state. The integrals can be solved from thermal

and volumetric data over temperature range to T and pressure range to P but the

chemical potential is unknown. Chemical potential at T and P can thus only be

evaluated relative to an arbitrary reference states give as and these refence states are

often known as standard states.

rT rP

( , )r ri T P

rT rP

Chemical potential does not have an immediate equivalent in the physical world and it is

desirable to express the chemical potential in terms of some auxilaiary function that might be

more easily identified with physical reality. The term fugacity (f) was introduced by G.N.

Lewis in trying to simplify the equation of chemical equilibrium by first considering the

chemical potential for a pure, ideal gas and then generalized the result to all systems

(Prausnitz et al., 1999). Auxiliary thermodynamic functions such as fugacities and activities

are often used in thermodynamic treatment of phase equilibria.

8

2.6 Fugacity and Fugacity Coefficient

The fugacity of component i in a mixture is defined as (Elliot and Lira, 1999)

In i iRTd f d at constant T 16

where fi is the fugacity of component i in a mixture and i is the chemical potential of the

component. For a pure ideal gas, the fugacity is equal to the pressure, and for a component i in

a mixture of ideal gases, it is equal to its partial pressure yiP. The definition of fugacity is

completed by the limit:

1i

i

fas P

y P 0 17

By integrating eq. 16 at constant T for any component in any system, solid, liquid or gas or

pure mixed, ideal or non-ideal. For vapour phase we have

lnoo

fRT

f 18

While either o and fo is arbitrary, both may not be chosen independently; when one is

chosen, the other is fixed. Writing an analogous expression for the liquid and vapour phase

and equating the chemical potentials using equation 13 obtain:

ln 0v

v LL

fRT

f 19

This transformation consequently leads to additional criteria for equilibrium called

isofugacity:

...v Lf f f 20

This tell us that the equilibrium condition in terms of chemical potential can be replaced

without loss of generality by equation in terms of fugacity.

9

Fugacity coefficient, is the ratio of fugacity to real gas pressure. It is a measure of non-

ideality.

ii

i

f

y P 21

It is a way of characterizing the Gibbs excess function at fixed T, P. For a mixture of ideal

gases i = 1.

2.7 Activity and Activity Coefficient

Activity concept is an alternative approach to express the chemical potential in a real solution.

The activity of component i at given temperature, pressure and composition is defined as the

ratio of the fugacity of i at these conditions to the fugacity i at standard state. Activity of a

substance gives an indication of how active a substance is relative to its standard state, it is

expressed as:

0i

ii

fa

f 22

Substituting equation 22 into 18 gives relationship between chemical potential and activity.

0 lni i iRT a 23

A general expression for the chemical potential in an ideal solution in terms of ideal mixing

could be written as

0 lnidi i iRT x 24

Activity coefficient i gives a measure of non-ideality of solution, it is the ratio of activity of

component i to its concentration, usually the mole fraction

ii

i

a

x 25

10

In an ideal solution the activity is equal to the mole fraction and the activity coefficient is

equal to unity. Introducing equation (25) into (23)

0 ln lni i iRT x RT i

i

26

Activity coefficient relates chemical potential in an ideal solution to the chemical potential in

a real solution, thus representing a measure of non-ideality as illustrated by combining

equations (26) and (24)

lnidi i RT

1i

27

2.7.1 Normalization of Activity Coefficient

It is convenient to define activity in such a way that for an ideal solution activity is qual to the

mole fraction or equivalently, that the activity coeffiicent is equal to unity. Because we have

two types of ideality (one leading to Raoult’s law and the other leading to Henry’s law), it

implies there will be two ways of normalizing activity coefficient.

Symmetric Convention

This convention applies when all the components both solutes and solvent at the system

temperature and pressure are liquids in their pure state (reference state). The activity

coefficient of each component i then approaches unity as its mole fraction approaches unity.

This convention leads to an ideal solution in the Raoult’s law sense. It follows that:

1i as x 28

Unsymmetric Convention

This convention applies when pure component cannot be used as a reference state for instance

when some component are solid or gaseous at the system temperature and pressure. In this

case it is convenient to define the reference state as the infinite dilute state of the component

at system temperature and reference pressure. This convention therefore leads to an ideal

dilute solution in the sense of Henry’s law.

11

for solvent 1 1s sas x 29

for ionic and molecular solutes 30 1 0i ias x

Subscripts s and i refer to solvent and solute respectively while asterisk(*) shows that the

activity coefficient of the solute approaches unity as mole fraction approaches zero. This

convention is said to be unsymmetric because solvent and solute are not normalized in same

way.

2.8 Excess Functions

Excess functions are thermodynamic properties of a solution that are in excess of an ideal (or

ideal dilute) solution at the same conditions of temperature, pressure and composition. For an

ideal solution all excess properties are zero (Prausnitz, 1999). A general excess function is

defined as

E real ideale e e 31

One particularly important excess function is the excess Gibbs energy defined by ( )EG

( , ) ( ,E

actual solution atT Pand x ideal solutionat the )sameT Pand xG G G 32

Similar definitions hold for excess volume , excess entropy , excess enthalpy ,

excess internal energy , and excess Helmholtz energy . Relations between these excess

functions are exactly the same as those between the total functions, for example:

EV ES EH

EU EA

E EG H TS E 33

Also, partial derivative of extensive excess function are analogous to those of the total

functions. For example:

,

EE

P x

GS

T

34

12

Partial molar excess functions are defined in a manner similar to partial molar thermodynamic

properties. If M is an extensive thermodynamic property, then im , the partial molar M of

component i, is defined by

, ,

i

ji P T n

Mm

n

similarly , ,

i

j

EE

i P T n

Mm

n

35

From Euler’s theorem, we have that

iii

M n m similary i

EEi

i

M n m 36

From excess function definition, it can be seen from eq. 27 that ln iRT is equal to excess

chemical potential Ei . Since chemical potential at constant T and P is equal to the partial

molar Gibbs free energy as shown in section 2.2, we obtain a very important relationship

between the activity coefficient and the partial molar excess Gibbs free energy:

lnE

iig RT 37

Using equation (37) in (36) gives an equally important relation

lnEi

i

g RT x i 38

Eq. 38 forms the basis for calculation of activity coefficients from models as will

described in section 3.3.

EG

3.0 Vapour Liquid Equilibrium Calculation

Vapour liquid equilibrium calculation requires simultanoeaus solution of chemical and phase

equilibria, activity and fugacity coefficients is required in chemical equilibria calculations

while fugacity cofficient is required in phase equilibria. Both are used to express chemical

potentials in liquid and vapour phase respectively

13

3.1 Chemical Equilibria

Molecular electrolytes dissociates or react in the liquid phase to produce ionic species to an

extent governed by the chemical equilibrium. Chemical reaction in the liquid phase enhances

the mass transfer rate and the solubility of CO2 and thus affects the phase equilibrium and

vice versa, the distribution of species between the two affects the chemical equilibrium in the

liquid phase. A system is in equilibrium when there is no driving force for a change of

intensive variables within the system. Chemical reaction moves towards a dynamic

equilibrium in which both reactants and products are present but have no further tendency to

undergo net change.

The generalized chemical reaction:

1 1 2 2 1 1... ...m m m mA A A A 39

could be written as (Ott and Goates, 2000):

0i ii

A 40

where the coefficients i are positive for the products of the reaction and negative for the

reactants. The condition for equilibrium in a chemical reaction is given by:

0i ii

41

The Gibbs free energy change in the chemical reaction is given by:

ln ivor r

i

G G RT a i 42

where is the Gibbs free energy change with reactants in their standard states. orG

14

3.1.1 Chemical Equilibrium Constant

The chemical equilibrium is traditionally defined by a chemical equilibrium constant. At

equilibrium, = 0 so that equation eq. 42 becomes, (Ott and Goates, 2000): rG

lnorG RT K 43

where K is the equilibrium constant and is given by:

ivi

i

K a 44

The activities in eq. (44) are now the equilibrium activities. An equilibrium constant K

expressed in terms of activities (or fugacities) is called a thermodynamic equilibrium constant.

Equilibrium constants are required for each of the reactions occuring in solutions. They are

related to the activities of each species as:

bB cC dD eE 45

d eD Eb cB C

a aK

a a 46

Here the lower case letters are the stoichiometric coefficients, i , and the capital letters are

labels for the chemical species.

3.2 Phase Equilibria

The solubility of gas in a liquid is often proportional to its partial pressure in the gas phase,

provided that the partial pressure is not large. The equation that describes this is known as

Henry’s law:

i iP y P Hx i 47

where Henry’s constant (H) is the constant of proportionality for any given solute and solvent,

depending only on temperature (Prausnitz et al., 1999). At high partial pressures, Henry’s

constant must be multiplied by activity coefficient and pressure by fugacity cofficient.

15

A measure of how chemical species distributes itself between liquid and vapour phase, is the

ratio (Perry, 1997):

i i

i i

iy fK

x P

48

The condition for phase equilibrium in a closed heterogeneous system at constant temperature

and pressure is given by eq. 20.

vi

Lif f 49

Where vi

Lif and f are fugacities of component i in the vapor and liquid phase respectrively.

From eq. 48 or by inserting eqs. 21, 22, and 25 into eq. 49 we obtain:

oLi i i i iy P x f 50

where i and i are fugacity and activity coefficient respectively. oLif is reference state

fugacity coefficient defined either by symmertic conventions Raoult’s law or unsymmetric

convention Henry’s law.

A correction term used to relate fugacities at the different pressures is called Poynting factor,

written as and at constant temperature and pressure change from P1 to P2 can further be

shown to be:

2

1

2 2

1 1

( , )exp

( , )

p

i

P

v dPf T P

f T P RT

51

The complete phase equilibria for vapour – liquid equilibrium of solute and solvent at systems

temperature and pressure can thus be respectively written as (Austgen 1989):

For solute: 2(

expo

i H Oi i i i i

v P Py P x H

RT

) 52

16

For solvent: (

expo

o o o i ss s s s s s

v P Py P x P

RT )

53

where Hi and iv represent Henry’s law constant, partial molar volume of molecular solute i

at infinite dilution in pure water at the system temperature and at saturation water vapour

pressure, 2

oH OP while sv is the molar volume of pure solvent at system temperature and

saturation pressure. The exponential correction (Poyning factor) here takes into account the

fact that liquid is at a pressure P different from the saturation pressure . For molecular

solutes such as carbon dioxide, Henry’s constant represents reference-state fugacities.

oP

To solve the phase equilibrium equation, we need to evaluate the fugacity and activity

coefficients. Gas phase fugacity coefficient can be calculated from equation of state such as

Soave-Redlich-Kwong equation of state. The real modeling task lies in the calculation of

activity coefficient. In this work activity coefficient calculation using, UNIQUAC is

presented.

3.3 GE Model for Activity Coefficient

Non-ideality in liquid phase is represented by activity coefficient as described in section 2.8.

They are usually obtained from excess Gibbs free energy models using eq.38. An apprioprate

excess Gibbs energy function most take into consideration the molecular interactions between

all species in the system. For electrolyte solutions, diverse species are usually present and

interactions among them must be represented. At high concentrations, interactions between

neutral molecules or between ions and neutral molecules are very short-range in character and

dominates while at low concentrations it is interactions between ions which are very strong

long-range electrostatic interactions that dominates. The usually practice is to assume that the

contributions of the various types of interactions are independent and additive (Poplsteinova,

2004). The excess Gibbs energy is then calculated as the sum of short-range and long-range

contributions:

E ESR LRG G G E 54

17

Most modeling applications combine the Debye-Hückel electrostatic theory for the long-range

term with modifications of well known non-electrolytes models for the short-range term. In

this work the local composition model for non-electrolytes, UNIQUAC will be presented in

more detail since they were applied in this work.

3.3.1 Local Composition Models

Regular solution theory assumes that the mixture of interactions were independent of each

other such that quadratic mixing rules provide reasonable approximations. However in some

cases, the mixture interaction can be strongly coupled to mixture composition. One way of

treating this is to recognize the possibility that the local compositions in the mixture might

deviate strongly from the bulk composition (Elliot and Lira, 1999). Some of the well-known

local composition models for non-electrolytes are Wilson, NRTL, UNIFAC and UNIQUAC

for these models to be used in electrolyte solutions, several assumptions have to be made

regarding the local composition in the presence of ions.

4.0 UNIQUAC

For this work the UNIQUAC equation described by Abrams and Prausnitz, 1975 was derived,

implemented and used for calculation of activity coefficients for non-electrolyte systems.

Main advantages of UNIQUAC is that it uses only two adjustable parameters per binary to

obtain reliable estimates for both vapor-liquid and liquid-liquid equilibria for a large variety

of multicomponent systems using the same equation for the excess Gibbs energy (Abrams and

Prausnitz, 1975).

Experimental data for typical binary mixtures are usually not sufficiently plentiful or precise

to yield three meaningful binary parameters, various attempts were made (Abrams and

Prausnitz, 1975; Maurer and Prausnitz, 1978; Anderson, 1978; Kemeny and Rasmussen,

1981) to derive a two-parameter equation for that retains at least some advantages of

Wilson equation without restriction to completely miscible mixtures. Abrams derived an

equation that in a sense, extends the quasichemical theory of Guggenheim for

nonrandommixtures to solutions containing molecules of different size. This is called

universal quasi-chemical (UNIQUAC) theory.

Eg

18

UNIQUAC equation for consists of two parts combinatorial part that describe the

dominant entropic contribution, and residual part that accounts for intermolecular forces

which are responsible for the enthalpy of mixing. The combinatorial part is determined only

by composition and by the sizes and shapes of the molecules, it requires only pure-component

data. The residual part depends on intermolecular forces, thus the two adjustable binary

parameters appear only in the residual part. The UNIQUAC equation is

Eg

E E E

combinatorial residual

g g g

RT RT RT

55

For a binary mixture

1 2 11 2 1 1 2 2

1 2 1

ln ln ln ln2

E

combinatorial

g zx x x q x q

RT x x2

2

56

1 1 1 2 21 2 2 2 1 12ln( ) ln( )E

residual

gx q x q

RT

57

where coordination number z is set equal to 10. Segment fraction, , and the area fractions,

, are given by

1 11

1 1 2 2

x r

x r x r

2 2

21 1 2 2

x r

x r x r

58

1 11

1 1 2 2

x q

x q x q

2 2

21 1 2 2

x q

x q x q

59

The parameters r and q are the two pure-component structural parameter per component

representing volume and area respectively. They are dimensionless and are evaluated from

bond angles and bond distances. For a binary mixture, there are two adjustable parameters,

12 21and . They are given in terms of charateristic energies 12 12 22u u u ; . 21 21 11u u u

1212 exp

u

RT

21

21 expu

RT

60

19

For many cases eq. 60 gives the primary effect of temperature on 12 21and . are

often weakly dependent on temperature.

12 21u and u

Writing eq. 38 in terms of mole number ni:

lnET

i

n g RT ni i 61

where is the total number of moles. The final expression for activity coefficient is obtained

by taking the partial derivative of excess Gibbs energy with respect to mole number.

Tn

Eg

, , ( )

lnj

ET

ii T P n j ì

n gRT

n

62

For a multicomponent system, activity coefficient expression based on UNIQUAC is then

derived to be:

ln ln 1 ln 12

1 ln

k k k kk k

k k k k

i kik j jk

j i j jij

zq

x x

q

63

Detailed derivation of this UNIQUAC expression for activity coefficient is shown in

Appendix

4.1 UNIQUAC Implementation

The expression for activity coefficient was added to the thermodynamic function library

developed by Tore Haug-Warberg and Bjørn Tore Løvfall at Department of Chemical

Engineering, NTNU. The model was coded in an in-house language called RGrad supporting

automatic gradient calculations, and from there exported to C-code which is compiled into a

set of DLL’s accessible from Matlab, Octave and Ruby (Haug-Warberg, 2008).

20

It has a folder <MODEL>/src/mex that contains two makefiles which compiles and set up the

model called <model>. The model is then simply run as <model>_mexmake or

<model>_octmake depending on whether you want a matlab/MEX interface or an Octave

interface.

4.1.1 MEA-H2O System

Monoethanolamine(MEA) is an important solvent for CO2 absorption, so for this work,

UNIQUAC equation was applied in calculation of activity coefficient in binary system of

MEA and water. In the model, the parameter adjustable energy interaction parameters of

the UNIQUAC enthalpy term are assumed to be temperature dependent and are fitted to the

following temperature function

( )iju

0 ( 298.15)Tij ij iju u u T 64

The r and q parameters for MEA and water as well as and were taken from Faramarzi

et al. 2009. Other parameter values were set to large values while parameters were set to

zero. Result from the calculation was compared with literauture data of Tochigi, K et al.,

1999, Belabbaci, A et al. 2009 and Kim, I et al., 2008 in figure 1. Molar Excess Gibbs energy

function was also calculated and presented in figure 5.

oiju T

iju

oiju T

iju

4.1.2 Acetic Acid-H20 System

The r and q parameters as well as energy interaction parameter for acetic acid-water

system were taken from Prausnitz, et al. 1999. The activity coefficient plot for acetic acid-

water system is shown in figure 2. The molar excess Gibbs energy function was also

calculated and presented in figure 6.

ija

4.1.3 Other Systems

The predicted model for formic acid – acetic acid sysems as well as acetone – chloroform

syetems are presented in figure 3 and 4 respectively. All parameters were taken from Prautnitz

et al. 1999. It has not been easy to get experimental data to use to compare the model result

from this work, however the shapes of the plots agrees very well with plots in Prausnitz, et al.

1999.

21

5.0 Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.4

0.5

0.6

0.7

0.8

0.9

1

xMEA

Tochigi K. et al. 1999

Belabbaci A. et al. 2009

Kim I. et al. 2008H20 This work

MEA This work

Figure 1: Activity coefficient plot for MEA-H20 system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

xH20

log

Ellis & Bahari 1956Sebastiani & Lacquaniti 1967

Hansen et al. 1955

Arich & Tagliavini 1958

Marek & Standart 1954

H20 This workACETIC This work

22

Figure 2: Activity coefficient plot for Acetic Acid – H20 system.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

xFORMIC

ACETIC

FORMIC

Figure 3: Activity Coefficient plot for Acetic Acid – Formic Acid System

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.4

0.5

0.6

0.7

0.8

0.9

1

xACETONE

CHLOROFORM

ACETONE

Figure 4: Activity Coefficient plot for Acetone -Chloroform System

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-700

-600

-500

-400

-300

-200

-100

0

xMEA

Exc

ess

Gib

bs E

nerg

y (g

E)

Figure 5: Excess molar Gibbs free energy plot for MEA-H20 System

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

xACETIC

Exc

ess

Gib

bs E

nerg

y gE

Figure 6: Excess molar Gibbs free energy plot for Acetic Acid -H20 System

24

Summary

UNIQUAC activity coefficient model was derived and successfully implemented for

calculating activity coefficient for different non-electrolyte systems. Literature activity

coefficient data for MEA-H20 system appear to vary among the different authors presented,

the model appear to fit much better to the data from Tochigi K et al 1999. Activity coefficient

data for acetic acid – H20 system however is more consistent. Model prediction agrees well

with the literature values.

Future Work

Further work will involve the calculation of activity coefficient for electrolyte systems by

extending the UNIQUAC model for electrolyte system through addition of the long-range

term.

References

Abrams, Denis S.; Prausnitz, John M., 1975; Statistical thermodynamics of liquid mixtures. New expression for the excess Gibbs energy of partly or completely miscible systems. AIChE Journal, 21(1), 116-28. Anderko, Andrzej; Wang, Peiming; Rafal, Marshall., 2002; Electrolyte solutions : from thermodynamic and transport property models to the simulation of industrial processes. Fluid Phase Equilibria, 194-197 123-142. Anderson, T. F.; Prausnitz, J. M., 1978; Application of the UNIQUAC equation to calculation of multicomponent phase equilibriums. 2. Liquid-liquid equilibriums. Industrial & Engineering Chemistry Process Design and Development , 17(4), 561-7. Arich, Guido; Tagliavini, Giuseppe., 1958; Liquid-vapor equilibrium isotherms for the water-acetic acid system. Ricerca sci., 28 2493-500. Belabbaci, Aouicha; Razzouk, Antonio; Mokbel, Ilham; Jose, Jacques; Negadi, Latifa., 2009; Isothermal Vapor - Liquid Equilibria of ( Monoethanolamine + Water ) and (4-Methylmorpholine + Water ) Binary Systems at Several Temperatures. Journal of Chemical & Engineering Data ACS ASAP. Elliot, J.R.; Lira, C.T., 1999; Introductory Chemical Engineering Thermodynamics; Prentice Hall PTR: New Jersey. Ellis, S. R. M.; Bahari, E. P., 1956 Vapor-liquid equilibrium at low concentrations; acetic acid-water; nitric acid-water. British Chemical Engineering , 1 210-11.

25

26

Faramarzi, Leila; Kontogeorgis, Georgios M.; Thomsen, Kaj; Stenby, Erling H., 2009; Extended UNIQUAC model for thermodynamic modeling of CO2 absorption in aqueous alkanolamine solutions. Fluid Phase Equilibria (2009), 282(2), 121-132. Hansen, Robert S.; Miller, Frederick A.; Christian, Sherril D., 1955; Activity coefficients of components in the systems water-acetic acid, water-propionic acid, and water-butyric acid at 25� Journal of Physical Chemistry, 59 391-5. Hartono, Ardi; 2009, Characterization of diethylenetriamine(DETA) as absorbent for carbon dioxide. Ph.D thesis., Chemical Engineering Department, Norwegian University of Science and Technology, Trondheim, Norway. Kemeny, Sandor; Rasmussen, Peter., 1981; A derivation of local composition expressions from partition functions. Fluid Phase Equilibria , 7(2), 197-203. Kim, Inna; 2009, Heat of reaction and VLE of post combustion CO2 absorbents. Ph.D thesis., Chemical Engineering Department, Norwegian University of Science and Technology, Trondheim, Norway. Marek, Jan; Standart, George., 1954; Effect of association on liquid-vapor equilibria. I. Equilibrium relations for systems involving an associating component. Collection of Czechoslovak Chemical Communications, 19 1074-84. Maurer, G., 1983; Electrolyte solutions. Fluid Phase Equilibria (1983), 13 269-96. Maurer, G.; Prausnitz, J. M., 1978; On the derivation and extension of the UNIQUAC equation. Fluid Phase Equilibria, 2(2), 91-9. Ott, J. B., Boerio-Goates J., 2000: Chemical Thermodynamics: Advanced Applications. Academic Press Perry, R.H., Green, D.W.,1997; Perry’s Chemical Engineering Handbook (7th edition), McGraw-Hill. Poplsteinova, Jana; 2004, Absorption of carbon dioxide – modelling and experimental characterization. Ph.D thesis., Chemical Engineering Department, Norwegian University of Science and Technology, Trondheim, Norway. Prausnitz, J. M., Lichtenthaler, R.N., and de Azevedo, E. G., 1999; Molecular Thermodynamics of Fluid-Phase Equilibria (3rd ed.), Prentice Hall PTR; New Jersey. Renon, Henri., 1986; Electrolyte solutions. Fluid Phase Equilibria , 30 181-95. Sebastiani, Enzo; Lacquaniti, L. 1967; Acetic acid - water system thermodynamic correlation of vapor-liquid equilibrium data. Chemical Engineering Science, 22(9), 1155-62. Thomsen, Kaj; Rasmussen, Peter., 1999; Modeling of vapor-liquid-solid equilibrium in gas-aqueous electrolyte systems. Chemical Engineering Science , 54(12), 1787-1802. Tochigi, Katsumi; Akimoto, Kentarou; Ochi, Kenji; Liu, Fangyhi; Kawase, Yasuhito., 1999; Isothermal Vapor - Liquid Equilibria for Water + 2 - Aminoethanol + Dimethyl Sulfoxide and Its Constituent Three Binary Systems. Journal of Chemical and Engineering Data, 44(3), 588-590. Tore Haug-Warberg, 2008; KP8108 Advanced Thermodynamics: With applications to Phase and Reaction Equilibria, Lecture Handout, Department of Chemical Engineering, NTNU.

Derivation of Activity Coe�cient Expression

Based on UNIQUAC

Ugochukwu Aronu

July 13, 2009

UNIQUAC Expression for Excess Gibbs Energy

gE

RT=

Combinatorialz }| {Xi

xi ln�i

xi+

Z

2

Xi

qixi ln�i

�i

Residualz }| {�

Xi

qixi ln(Xj

�j�ji) (1)

where �ji = exp�huji�uiiRT

i

�i =qixiPj qjxj

=qiniPj qjnj

(2)

�i =rixiPj rjxj

=riniPj rjnj

(3)

writing eq.1 in terms of the number of moles 'n' and factoring out 'n'.

ngE

RT=

Az }| {Xi

ni ln

��in

ni

�+

Bz }| {Z

2

Xi

qini ln�i

�i�

Cz }| {Xi

qini ln(Xj

�j�ji) (4)

ln k =

@n( g

E

RT)

@nk

!T;ni 6=k

=

�@A

@nk

�T;ni 6=k

+

�@B

@nk

�T;ni 6=k

+

�@C

@nk

�T;ni6=k

(5)

1

Simplify Eq. 4A and di�erentiate with respect to mole number.

A =Xi

ni ln

��in

ni

�

=Xi

ni (ln�in� lnni)

=Xi

ni (ln�i + lnn� lnni)

=

A1z }| {Xi

ni (ln�i � lnni)+

A2z }| {n lnn

@A

@nk=

@A1

@nk+

@A2

@nk(6)

@A1

@nk=

@ [P

i ni(ln�i � lnni)]

@nk

Using kronecker-delta to di�erenciate

@A1

@nk=Xi

�(ln�i � lnni)�ik + ni(

1

�i

@�i

@nk�

1

ni�ik)

�

@A1

@nk= ln

��k

nk

�+Xi

ni

�i

@�i

@nk� 1 (7)

@A2

@nk=

@(n lnn)

@nk

= lnn@n

@nk+ n

@ lnn

@nk

= lnn+1

nn

@A2

@nk= lnn+ 1 (8)

Substituting eq. 7 and eq. 8 back in eq. 6

@A

@nk= ln

��k

nk

�+ lnn+

Xi

ni

�i

@�i

@nk

@A

@nk= ln

��k

xk

�+Xi

ni

�i

@�i

@nk(9)

2

From 3, �i =riniPj rjnj

; thus

@�i

@nk=

�Pj rjnj

�ri�ik � (rini)

Pj rj�jk�P

j rjnj

�2@�i

@nk=

ri�ikPj rjnj

�rinirk�Pj rjnj

�2 (10)

Put eq. 10 into eq. 9

@A

@nk= ln

��k

xk

�+Xi

ni

�i

264 ri�ikP

j rjnj�

rinirk�Pj rjnj

�2375

= ln�k

xk+

Pi niri

�iP

j rjnj�

Pi rin

2i rk

�i(P

j rjnj)2

= ln�k

xk+

nkrk

�kP

j rjnj�

Pi rin

2i rk

�i(P

j rjnj)2

Factoring out �k and �i and simplifying

= ln�k

xk+ 1�

Pi nirkPj rjnj

= ln�k

xk+ 1�

nrkPj rjnj

Multiplying numerator and denominator by nk and factor out �k

= ln�k

xk+ 1�

nrkPj rjnj

:nk

nk�@A

@nk

�T;ni 6=k

= ln�k

xk+ 1�

�k

xk(11)

3

Simplify Eq. 4B and di�erentiate with respect to mole number.

B =z

2

Xi

qini ln�i

�i=

z

2

Xi

qini(ln �i � ln�i)

@B

@nk=

z

2

Xi

qi

�(ln �i � ln�i)�ik + ni(

1

�i

@�i

@nk�

1

�i

@�i

@nk)

�

@B

@nk=

Az }| {z

2qk ln

�k

�i+

Bz }| {z

2

Xi

qi

�ni

�i

@�i

@nk�

ni

�i

@�i

@nk

�(12)

From 10 @�i@nk

= ri�ikPj rjnj

�rinirk

(P

j rjnj)2 ; therefore

ni

�i

@�i

@nk=

niri�ik

�iP

j rjnj�

rinirk

�i(P

j rjnj)2

= �ik �nirk

(P

j rjnj)(13)

Similarly;

@�i

@nk=

(P

j qjnj)qi�ik � qiniqk

(P

j qjnj)2

=qi�ikPj qjnj

�qiniqk

(P

j qjnj)2

ni

�i

@�i

@nk=

niqi�ik

�iP

j qjnj�

niqiniqk

(�iP

j qjnj)2

= �ik �niqk

(P

j qjnj)(14)

Substitute eq. 13 and eq. 14 into eq. 12B

z

2

Xi

qi

" �ik �

niqkPj qjnj

!�

�ik �

nirkPj rjnj

!#

=z

2

Xi

qi

nirkPj rjnj

�niqkPj qjnj

!

=z

2

Pi qinirkPj rjnj

�

Pi qiniqkPj qjnj

!

4

Multiply numerator and denominator of by nk then factor out �k and �i.

=z

2

Pi qinirkPj rjnj

nk

nk�

Pi qiniqkPj qjnj

!

Factor out �k and �i.

=z

2

Pi qini�k

nk�

Xi

�iqk

!

= qkz

2

Pi qini�k

qknk�

Xi

�i

!

Factoring out �k and settingP

i �i = 1.

=z

2qk

��k

�k� 1

�Therefore

@B

@nk=

z

2qk ln

�k

�k+

z

2qk(

�k

�k� 1)

�@B

@nk

�T;ni 6=k

=z

2qk

�ln

�k

�k+

�k

�k� 1

�(15)

Di�erenciate Eq. 4C with respect to mole number.

C = �Xi

qini ln(Xj

�j�ji)

@C

@nk= �

Xi

qi

24ln

0@X

j

�j�ji

1A @ni

@nk+ ni

1Pj �j�ji

Xj

�ji@�j

@nk

35

= �

2666664

Az }| {qk ln

0@X

j

�j�jk

1A+

Bz }| {Xi

qi

�niP

j �ji@�j@nk

�P

j �j�ji

3777775 (16)

But; �j =qjnjPl qlnl

. Note that subscript 'l ' is used in place of subscript 'i ' for

clearity.@�j

@nk=

(P

l qlnl)qj�jk � qjnjqk

(P

l qlnl)2

5

@�j

@nk=

qj�jkPl qlnl

�qjnjqk

(P

l qlnl)2

(17)

Substituting eq. 17 in eq. 16B

Xi

qini

Pj �ji

hqj�jkPl qlnl

�qjnjqk

(P

l qlnl)2

iP

j �j�ji

)

Xi

hqini�kiqkP

l qlnl�

qiniP

j qjnjqk�ji

(P

l qlnl)2

iP

j �j�ji

Factor out �i then substitute and simplify further;

) qkXi

�i�kiPj �j�ji

� qkXi

�iP

j �ji�jPj �j�ji

Simplify further and substitute back into Eq.16

@C

@nk= �

24qk ln(X

j

�j�jk) + qkXi

�i�kiPj �j�ji

� qkXi

�i

35

ButP

i �i = 1

�@C

@nk

�T;ni 6=k

= qk

241� ln(

Xj

�j�jk)�Xi

�i�kiPj �j�ji

35 (18)

Substituting eq. 11, eq. 15 and eq. 18 into eq. 5 and obtain the �nal

expression for activity coe�cient based on UNIQUAC

ln k = ln�k

xk+ 1�

�k

xk

+z

2qk

�ln

�k

�k+

�k

�k� 1

�

+ qk

241� ln(

Xj

�j�jk)�Xi

�i�kiPj �j�ji

35

6