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 2.2 2.3 2.4 2.5 2.6 2.7
Thermodynaic Equilibrium Condition Partial Molar Gibbs Free Energy Gibbs-Duhem Equation Equilibrium in Heterogenous Closed System Chemical Potential Fugacity and Fugacity Coefficient Activity and Activity Coefficient
4 5 6 7 8 9 10
2.7.1
11
2.8 3.0
5.0
Excess Functions
12
Vapour Liquid Equilibrium Calculation
13
3.1
Chemical Equilibria
14
3.1.1
15
3.2 3.3
4.0
Normalization of Activity Coefficient
Chemical Equilibrium Constant
Phase Equilibria G E Model for Activity Coefficient
15 17
3.3.1
18
Local Composition Models
UNIQUAC
18
4.1
UNIQUAC Implementation
20
4.1.1 4.1.2 4.1.3
21 21 21
MEA-H20 System Acetic Acid-H20 System Other Systems
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 separati on 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 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 CO 2-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
dG 0
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
U U ( S , V , n1 , n2 ..., nm ) , where m is the number of components present. Total differential is then
U U dU dS dV S V V ,ni S ,ni
U
S i
dni
5
S ,V ,n j
where subsript ni refers to all mole numbers and n j to all mole numbers other than ith (the term under differenciation). Writing eq. 5 in the form
dU TdS PdV +
dn i
i
6
i
U ni S ,V ,n
where i
j
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 +
dn i
7
i
i
where
U H A G ni S ,V ,n ni S ,P ,n ni T ,V ,n ni P ,T ,n
i
j
j
j
8
j
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
G , the partial molar Gibbs free n i P ,T ,n
energy G (Prausnitz et al., 1999). The quantity, i
j
energy is also called the chemical potential.
2.3 Gibbs-Duhem Equation
At constant temperature and pressure eq. 7 reduces to
dn
dG =
i
9
i
i
P ,T
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
dn
dG
i
i
i
= 0
10
P ,T
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 GibbsDuhem equation is satisfied. In terms of activity coefficient Gibbs-Duhem equation is expressed as
x d ln i
i
0
11
i
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:
x1
d ln 1 dx1
x2
d ln 2
12
dx2
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
T v T L ... T
P v P L ... P
13
v L ...
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
d i si dT vi dP
14
where si is the molar entropy and vi the molar volume. Integrating and solving for i at some temperature T and pressure P , we have
T
i (T , P ) i (T , P ) r
r
s dT i
T
r
P
v dP i
15
r
P
where superscript r is an arbitrary reference state. The integrals can be solved from thermal and volumetric data over temperature range T r to T and pressure range P r to P but the chemical potential i (T r , P r ) is unknown. Chemical potential at T and P can thus only be
evaluated relative to an arbitrary reference states give as T r and P r these refence states are often known as standard states.
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) RTd In f i d i
at constant T
16
where f i 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 y P i . The definition of fugacity is completed by the limit:
f i yi P
1
P 0
as
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
o RT ln
f
18
f o
While either o and f o 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:
RT ln v
L
f v f L
0
19
This transformation consequently leads to additional criteria for equilibrium called
isofugacity:
f v f L ... 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 nonideality.
f i yi P
i
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:
ai
f i f i 0
22
Substituting equation 22 into 18 gives relationship between chemical potential and activity.
0 i i RT ln ai
23
A general expression for the chemical potential in an ideal solution in terms of ideal mixing could be written as
iid i0 RT ln xi
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
i
ai xi
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)
i i0 RT ln xi RT ln 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)
i iid RT ln i
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:
i 1
as
xi 1
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
s
1 as xs 1
29
for ionic and molecular solutes i
1 as xi 0
30
for solvent
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 E e real eideal
31
One particularly important excess function is the excess Gibbs energy (G E ) defined by
G E G( actual solution at T ,P and x ) G(ideal solution at t he sameT , Pand x )
32
Similar definitions hold for excess volume V E , excess entropy S E , excess enthalpy H E , excess internal energy U E , and excess Helmholtz energy A E . Relations between these excess functions are exactly the same as those between the total functions, for example:
G E H E TS E
33
Also, partial derivative of extensive excess function are analogous to those of the total functions. For example:
G E E S T P , x
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 mi , the partial molar M of component i, is defined by
M m ni P ,T ,n
similarly
i
j
M E m ni P ,T ,n E i
35
j
From Euler’s theorem, we have that
M
n m i
similary
i
i
M E
n m
E
i
i
36
i
From excess function definition, it can be seen from eq. 27 that RT ln i is equal to excess chemical potential E i . 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:
E
g i RT ln i
37
Using equation (37) in (36) gives an equally important relation
g E RT
x ln i
i
38
i
Eq. 38 forms the basis for calculation of activity coefficients from G E models as will described in section 3.3.
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 A1 2 A2 ... m Am m 1 A m 1 ...
39
could be written as (Ott and Goates, 2000):
A i
i
0
40
i
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:
0 i
41
i
i
The Gibbs free energy change in the chemical reaction is given by:
Gr Gr o RT ln aiv
i
42
i
where Gr o is the Gibbs free energy change with reactants in their standard states.
14
3.1.1 Chemical Equilibrium Constant
The chemical equilibrium is traditionally defined by a chemical equilibrium constant. At equilibrium, Gr = 0 so that equation eq. 42 becomes, (Ott and Goates, 2000):
Gr o RT ln K
43
where K is the equilibrium constant and is given by:
K
a
vi i
44
i
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 K
e a Dd aE c a Bb aC
45 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 :
Pi yi P Hxi
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):
K
yi
i
i f i
48
i P
The condition for phase equilibrium in a closed heterogeneous system at constant temperature and pressure is given by eq. 20.
f i v f i L
49
Where f i v and f i L 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:
i i P i xi fi oL
50
where i and i are fugacity and activity coefficient respectively. f i oL 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 P 1 to P 2 can further be shown to be:
p vi dP exp P RT f1 (T , P1 ) f 2 (T , P2 )
2
51
1
The com plete phase equilibria for vapour – liquid equilibrium of solute and solvent at systems temperature and pressure can thus be respectively written as (Austgen 1989):
vi ( P P Ho O ) For solute: i yi P i xi H i exp RT 2
52
16
vi ( P P so ) For solvent: s ys P s x P exp RT o s
o s
o s
53
where H i and vi 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 o pressure, P while v s is the molar volume of pure solvent at system temperature and H 2O
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 P o . For molecular solutes such as carbon dioxide, Henry’s constant represents reference-state fugacities.
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.
E
3.3 G 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 G E G E GSR LR
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 g E 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.
18
UNIQUAC equation for g E 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
gE gE RT RT combinatorial RT residual gE
55
For a binary mixture
g E 1 2 1 2 z ln ln ln ln x x x q x q 1 2 2 2 1 1 RT 2 2 x1 x2 combinatorial 1 g E RT residual
x1q1 ln(1 2 21 ) x2q 2 ln( 2 1 12 )
56
57
where coordination number z is set equal to 10. Segment fraction, , and the area fractions, , are given by
1
1
x1r 1 x1r1 x2 r2
x1q1 1q1 x2 q2
2
2
x2 r 2 1r1 x2 r 2
x2 q2 x1q1 x2 q2
58
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 and 21 . They are given in terms of charateristic energies u12 u12 u22 ; u21 u21 u11 .
u 12 exp 12 RT
u 21 exp 21 RT
60
19
For many cases eq. 60 gives the primary effect of temperature on 12 and 21 . u12 and u21 are often weakly dependent on temperature.
Writing eq. 38 in terms of mole number ni:
nT g E RT
n ln i
61
i
i
where nT is the total number of moles. The final expression for activity coefficient is obtained by taking the partial derivative of excess Gibbs energy g E with respect to mole number.
nT g E RT ln i ni T , P ,n
j
62 ( j ì )
For a multicomponent system, activity coefficient expression based on UNIQUAC is then derived to be:
1 2 k k xk xk i ki qk 1 ln j jk j i j ji j
ln k ln
k
1
k
z
k
qk ln
k
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 /src/mex that contains two makefiles which compiles and set up the model called . The model is then simply run as _mexmake or _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 CO 2 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 (uij ) of the UNIQUAC enthalpy term are assumed to be temperature dependent and are fitted to the following temperature function
uij uij0 uijT (T 298.15)
64
The r and q parameters for MEA and water as well as uijo and uijT were taken from Faramarzi et al. 2009. Other uijo parameter values were set to large values while uijT 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.
4.1.2 Acetic Acid-H20 System
The r and q parameters as well as energy interaction parameter aij for acetic acid-water system were taken from Prausnitz, et al. 1999. The activity coefficient plot for acetic acidwater system is shown in figure 2. The molar excess Gibbs energy function was also calculated and presented in figure 6.
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 1
0.9
0.8
0.7
0.6 Tochigi K. et al. 1999
0.5
Belabbaci A. et al. 2009 Kim I. et al. 2008 H20 This work
0.4
MEA This work 0
0.1
0.2
0.3
0.4
0.5 xMEA
0.6
0.7
0.8
0.9
1
0.8
0.9
1
Figure 1: Activity coefficient plot for MEA-H20 system 0.6 Ellis & Bahari 1956 Sebastiani & Lacquaniti 1967
0.5
Hansen et al. 1955 Arich & Tagliavini 1958
0.4
Marek & Standart 1954 H20 This work ACETIC This work
0.3
g o l
0.2
0.1
0
-0.1
0
0.1
0.2
0.3
0.4
0.5 xH20
0.6
0.7
22
Figure 2: Activity coefficient plot for Acetic Acid – H20 system. 1.35 ACETIC FORMIC
1.3
1.25
1.2
1.15
1.1
1.05
1
0
0.1
0.2
0.3
0.4
0.5 0.6 xFORMIC
0.7
0.8
0.9
1
Figure 3: Activity Coefficient plot for Acetic Acid – Formic Acid System 1
0.9 CHLOROFORM ACETONE 0.8
0.7
0.6
0.5
0.4
0
0.1
0.2
0.3
0.4 0.5 0.6 xACETONE
0.7
0.8
0.9
1
Figure 4: Activity Coefficient plot for Acetone -Chloroform System
23
0
-100
) -200 E g ( y g r e -300 n E s b b i G -400 s s e c x E -500
-600
-700
0
0.1
0.2
0.3
0.4
0.5 xMEA
0.6
0.7
0.8
0.9
1
0.9
1
Figure 5: Excess molar Gibbs free energy plot for MEA-H20 System
250
200 E g y g r 150 e n E s b b i G s 100 s e c x E
50
0
0
0.1
0.2
0.3
0.4
0.5 0.6 xACETIC
0.7
0.8
Figure 6: Excess molar Gibbs free energy plot for Acetic Acid -H20 System
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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
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