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7. Excess Gibbs Energy Models. Practicing engineers find most of the liquid-phase information needed for equilibrium calculations in the form of excess Gibbs Energy models. These models: reduce vast quantities of experimental data into a few empirical parameters, - PowerPoint PPT Presentation
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CHEE 311 J.S. Parent 1
7. Excess Gibbs Energy Models
Practicing engineers find most of the liquid-phase information needed for equilibrium calculations in the form of excess Gibbs Energy models. These models:
reduce vast quantities of experimental data into a few empirical parameters,
provide information an equation format that can be used in thermodynamic simulation packages (Provision)
“Simple” empirical models Symmetric, Margule’s, vanLaar No fundamental basis but easy to use Parameters apply to a given temperature, and the models
usually cannot be extended beyond binary systems.
Local composition models Wilsons, NRTL, Uniquac Some fundamental basis Parameters are temperature dependent, and multi-
component behaviour can be predicted from binary data.
CHEE 311 J.S. Parent 2
Excess Gibbs Energy Models
Our objectives are to learn how to fit Excess Gibbs Energy models to experimental data, and to learn how to use these models to calculate activity coefficients.
sat11
11 Px
Pylnln
sat22
22 Px
Pylnln
2211E lnxlnxRT/G
CHEE 311 J.S. Parent 3
Margule’s Equations
While the simplest Redlich/Kister-type expansion is the Symmetric Equation, a more accurate model is the Margule’s expression:
(11.7a)
Note that as x1 goes to zero,
and from L’hopital’s rule we know:
therefore,
and similarly
21212121
E
xAxAxRTx
G
1
210xln
xRTxG
lim
E
1
12
0x21
E
AxRTx
G
1
112 lnA 221 lnA
CHEE 311 J.S. Parent 4
Margule’s Equations
If you have Margule’s parameters, the activity coefficients are easily derived from the excess Gibbs energy expression:
(11.7a)
to yield:
(11.8ab)
These empirical equations are widely used to describe binary solutions. A knowledge of A12 and A21 at the given T is all we require to calculate activity coefficients for a given solution composition.
21212121
E
xAxAxRTx
G
]x)AA(2A[xln 1122112221
]x)AA(2A[xln 2211221212
CHEE 311 J.S. Parent 5
van Laar Equations
Another two-parameter excess Gibbs energy model is developed from an expansion of (RTx1x2)/GE instead of GE/RTx1x2. The end results are:
(11.13)for the excess Gibbs energy and:
(11.14)
(11.15)
for the activity coefficients.
Note that: as x10, ln1 A’12
and as x2 0, ln2 A’21
2/121
/21
/21
/12
21
E
xAxA
AAxRTx
G
2
2/21
1/12/
121xA
xA1Aln
2
1/12
2/21/
212xA
xA1Aln
CHEE 311 J.S. Parent 6
Local Composition Models
Unfortunately, the previous approach cannot be extended to systems of 3 or more components. For these cases, local composition models are used to represent multi-component systems.
Wilson’s Theory Non-Random-Two-Liquid Theory (NRTL) Universal Quasichemical Theory (Uniquac)
While more complex, these models have two advantages: the model parameters are temperature dependent the activity coefficients of species in multi-component liquids
can be calculated from binary data.
A,B,C A,B A,C B,C
tertiary mixture binary binary binary
CHEE 311 J.S. Parent 7
Wilson’s Equations for Binary Solution Activity
A versatile and reasonably accurate model of excess Gibbs Energy was developed by Wilson in 1964. For a binary system, GE is provided by:
(11.16)
where(11.24)
Vi is the molar volume at T of the pure component i.aij is determined from experimental data.
The notation varies greatly between publications. This includes, a12 = (12 - 11), a12 = (21 - 22) that you will encounter in
Holmes, M.J. and M.V. Winkle (1970) Ind. Eng. Chem. 62, 21-21.
)xxln(x)xxln(xRTG
2112212211
E
RTa
expVV
RTa
expVV 21
2
121
12
1
212
CHEE 311 J.S. Parent 8
Wilson’s Equations for Binary Solution Activity
Activity coefficients are derived from the excess Gibbs energy using the definition of a partial molar property:
When applied to equation 11.16, we obtain:
(11.17)
(11.18)
2112
21
1221
12212211 xxxx
x)xxln(ln
2112
21
1221
12121122 xxxx
x)xxln(ln
jn,P,Ti
EEii n
nGGlnRT
CHEE 311 J.S. Parent 9
Wilson’s Equations for Multi-Component Mixtures
The strength of Wilson’s approach resides in its ability to describe multi-component (3+) mixtures using binary data.
Experimental data of the mixture of interest (ie. acetone, ethanol, benzene) is not required
We only need data (or parameters) for acetone-ethanol, acetone-benzene and ethanol-benzene mixtures
The excess Gibbs energy is written:
(11.22)
and the activity coefficients become:
(11.23)
where ij = 1 for i=j. Summations are over all species.
i j
ijji
E
xlnxRTG
k
jkjj
kik
iijji x
xxln1ln
CHEE 311 J.S. Parent 10
Wilson’s Equations for 3-Component Mixtures
For three component systems, activity coefficients can be calculated from the following relationship:
Model coefficients are defined as (ij = 1 for i=j):
3322311
i33
2332211
i22
1331221
i113i32i21i1i
xxx
x
xxx
x
xxx
x)xxxln(1ln
RT
aexp
V
V ij
i
jij
CHEE 311 J.S. Parent 11
Comparison of Liquid Solution Models
Activity coefficients of 2-methyl-2-butene + n-methylpyrollidone.
Comparison of experimental values with those obtained from several equations whose parameters are found from the infinite-dilution activity coefficients. (1) Experimental data. (2) Margules equation. (3) van Laar equation. (4) Scatchard-Hamer equation. (5) Wilson equation.
CHEE 311 J.S. Parent 12
8. Non-Ideal VLE to Moderate Pressure 12.4 text
We now have the tools required to describe and calculate vapour-liquid equilibrium conditions for even the most non-ideal systems.
1. Equilibrium Criteria: In terms of chemical potential
In terms of mixture fugacity
2. Fugacity of a component in a non-ideal gas mixture:
3. Fugacity of a component in a non-ideal liquid mixture:
li
vi
li
vi f̂f̂
P)y,...,y,y,P,T(ˆy)y,...,y,y,P,T(f̂ n21viin21
vi
RT
)PP(VexpP)x,...,x,x,P,T(x
f)x,...,x,x,P,T(x)x,...,x,x,P,T(f̂satiisat
isatin21ii
lin21iin21
li
CHEE 311 J.S. Parent 13
Formulation of VLE Problems
To this point, Raoult’s Law was only description we had for VLE behaviour:
We have repeatedly observed that calculations based on Raoult’s Law do not predict actual phase behaviour due to over-simplifying assumptions.
Accurate treatment of non-ideal phase equilibrium requires the use of mixture fugacities. At equilibrium, the fugacity of each component is the same in all phases. Therefore,
or,
determines the VLE behaviour of non-ideal systems where Raoult’s Law fails.
satiii PxPy
RT
)PP(VexpPxPˆy
f̂f̂satiisat
isatiii
vii
li
vi
CHEE 311 J.S. Parent 14
Non-Ideal VLE to Moderate Pressures
A simpler expression for non-ideal VLE is created upon defining a lumped parameter, .
12.2
The final expression becomes,
(i = 1,2,3,…,N) 12.1
To perform VLE calculations we therefore require vapour pressure data (Pi
sat), vapour mixture and pure component fugacity correlations (i) and liquid phase activity coefficients (i).
satiiiii PxPy
sati
vi
sati
li
sati
vi
i
ˆ
RT
)PP(Vexp
ˆ
CHEE 311 J.S. Parent 15
Non-Ideal VLE to Moderate Pressures
Sources of Data:1. Vapour pressure: Antoine’s Equation (or similar correlations)
12.3
2. Vapour Fugacity Coefficients: Viral EOS (or others)
12.6
3. Liquid Activity Coefficients Binary Systems - Margule, van Laar, Wilson, NRTL, Uniquac Ternary (or higher) Systems - Wilson, NRTL, Uniquac
i
ii
sati CT
BAPln
RT
)2(yyP5.0)PP(Bexp
jkj k
jikjsatiii
i
CHEE 311 J.S. Parent 16
Non-Ideal VLE Calculations
The Pxy diagram to the rightis for the non-ideal system ofchloroform-dioxane.
Note the P-x1 line representsa saturated liquid, and is commonly BUBL LINEreferred to as the bubble-line.
P-y1 represents a saturatedvapour, and is referred to as thedew line (the point where a liquid DEW LINEphase is incipient).
i i
satiii
BUBL
PxP
isatii
iiDEW
Py
1P
CHEE 311 J.S. Parent 17
Non-Ideal BUBL P Calculations
The simplest VLE calculation of the five is the bubble-point pressure calculation.
Given: T, x1, x2,…, xn Calculate P, y1, y2,…, yn
To find P, we start with a material balance on the vapour phase:
Our equilibrium relationship provides:
12.9 from 12.1
which yields the Bubble Line equation when substituted into the material balance:
or12.11
ni
1ii 1y
PPx
yi
satiii
i
i i
satiii Px
P
i i
satiii
ii P
Px1y
CHEE 311 J.S. Parent 18
Non-Ideal BUBL P Calculations
Non-ideal BUBL P calculations are complicated by the dependence of our coefficients on pressure and composition.
Given: T, x1, x2,…, xn Calculate P, y1, y2,…, yn
To apply the Bubble Line Equation:
requires:
?
Therefore, the procedure is: calculate Pi
sat, and i from the information provided assume i=1, calculate an approximate PBUBL use this estimate to calculate an approximate i repeat PBUBL and i calculations until solution converges.
)T(PP
)x,...,x,x,T(
)y,...,y,y,P,T(
sati
sati
n21i
n21i
i i
satiii Px
P
CHEE 311 J.S. Parent 19
Non-Ideal Dew P Calculations
The dew point pressure of a vapour is that pressure which the mixture generates an infinitesimal amount of liquid. The basic calculation is:
Given: T, y1, y2,…, yn Calculate P, x1, x2,…, xn
To solve for P, we use a material balance on the liquid phase:
Our equilibrium relationship provides:
12.10 from 12.1
From which the Dew Line expression needed to calculate P is generated:
12.12
ni
1ii 1x
satii
iii
P
Pyx
isatii
ii
Py
1P
CHEE 311 J.S. Parent 20
Non-Ideal Dew P Calculations
In trying to solve this equation, we encounter difficulties in estimating thermodynamic parameters.
Given: T, y1, y2,…, yn Calculate P, x1, x2,…, xn
?
?
While the vapour pressures can be calculated, the unknown pressure is required to calculate i, and the liquid composition is needed to determine i
Assume both parameters equal one as a first estimate, calculate P and xi
Using these estimates, calculate i
Refine the estimate of xi (12.10) and estimate i Refine the estimate of P Iterate until pressure and composition converges.
)T(PP
)x,...,x,x,T(
)y,...,y,y,P,T(
sati
sati
n21i
n21i
isatii
ii
Py
1P
CHEE 311 J.S. Parent 21
8. Non-Ideal Bubble and Dew T Calculations
The Txy diagram to the rightis for the non-ideal system ofethanol(1)/toluene(2) at P =1atm.
Note the T-x1 line representsa saturated liquid, and is commonly DEW LINEreferred to as the bubble-line.
T-y1 represents a saturatedvapour, and is referred to as thedew line (the point where a liquidphase is incipient).
BUBL LINE
i i
satiii
BUBL
PxP
isatii
iiDEW
Py
1P
CHEE 311 J.S. Parent 22
Non-Ideal BUBL T Calculations
Bubble point temperature calculations are among the more complicated VLE problems:
Given: P, x1, x2,…, xn Calculate T, y1, y2,…, yn
To solve problems of this sort, we use the Bubble Line equation:
12.11
The difficulty in determining non-ideal bubble temperatures is in calculating the thermodynamic properties Pi
sat, i, and i.
Since we have no knowledge of the temperature, none of these properties can be determined before seeking an iterative solution.
i i
satiii Px
P
)T(PP
)x,...,x,x,T(
)y,...,y,y,P,T(
sati
sati
n21i
n21i
CHEE 311 J.S. Parent 23
Non-Ideal BUBL T Calculations: Procedure
1. Estimate the BUBL T Use Antoine’s equation to calculate the saturation
temperature (Tisat) for each component at the given pressure:
Use TBUBL = xi Tisat as a starting point
2. Using this estimated temperature and xi’s calculate
Pisat from Antoine’s equation
Activity coefficients from an Excess Gibbs Energy Model (Margule’s, Wilson’s, NRTL)
Note that these values are approximate, as we are using a crude temperature estimate.
lnln
sat sati ii i i i
i i
B BP A T C
T C A P
CHEE 311 J.S. Parent 24
Non-Ideal BUBL T Calculations: Procedure
3. Estimate i for each component.We now have estimates of T, Pi
sat and i, but no knowledge of i. Assume that i=1 and calculate yi
’s using:
12.9
Plug P, T, and the estimates of yi’s into your fugacity
coefficient expression to estimate i.
Substitute thesei estimates into 12.9 to recalculate yi and continue this procedure until the problem converges.
Step 3 provides an estimate of i that is based on the best T, Pisat, i,
and xi data that is available at this stage of the calculation. If you assume that the vapour phase is a perfect gas mixture,
all i =1.
P
Pxy
i
satiii
i
CHEE 311 J.S. Parent 25
Non-Ideal BUBL T Calculations: Procedure
4. Our goal is to find the temperature that satisfies our bubble point equation:
(12.11)
Our estimates of T, Pisat, i and i, are approximate since they are
based on a crude temperature estimate (T = xi Tisat)
Calculate P using the Bubble Line equation (12.11)» If Pcalc < Pgiven then increase T» If Pcalc > Pgiven then decrease T» If Pcalc = Pgiven then T = TBUBL
The simplest method of finding TBUBL is a trial and error method using a spreadsheet.
Follow steps 1 to 4 to find Pcalc. Change T and repeat steps 2, 3, and 4 until Pcalc = Pgiven
i i
satiii Px
P
CHEE 311 J.S. Parent 26
Non-Ideal DEW T Calculations
The dew point temperature of a vapour is that which generates an infinitesimal amount of liquid.
Given: P, y1, y2,…, yn Calculate T, x1, x2,…, xn
To solve these problems, use the Dew Line equation:
12.12
Once again, we haven’t sufficient information to calculate the required thermodynamic parameters.
Without T and xi’s, we cannot determine i, i or Pi
sat.
isatii
ii
Py
1P
)T(PP
)x,...,x,x,T(
)y,...,y,y,P,T(
sati
sati
n21i
n21i
CHEE 311 J.S. Parent 27
Non-Ideal DEW T Calculations: Procedure
1. Estimate the DEW T Using P, calculate Ti
sat from Antoine’s equation
Calculate T = yi Tisat as a starting point
2. Using this temperature estimate and yi’s, calculate
Pisat from Antoine’s equation
i using the virial equation of state
Note that these values are approximate, as we are using a crude temperature estimate.
ii
isati C
PlnA
BT
CHEE 311 J.S. Parent 28
Non-Ideal DEW T Calculations: Procedure
3. Estimate i, for each component Without liquid composition data, you cannot calculate activity coefficients using excess Gibbs energy models.A. Set i=1
B. Calculate the Dew Pressure:
C. Calculate xi estimates from the equilibrium relationship:
D. Plug P,T, and these xi’s into your activity coefficient model to
estimate i for each component.
E. Substitute these i estimates back into 12.12 and repeat B through D until the problem converges.
satii
iii P
Pyx
isatii
ii
Py
1P
CHEE 311 J.S. Parent 29
Non-Ideal DEW T Calculations: Procedure
4. Our goal is to find the temperature that satisfies our Dew Line equation:
(12.12)
Our estimates of T, Pisat, i and i, are based on an approximate
temperature (T = xi Tisat) we know is incorrect.
Calculate P using the Bubble Line equation (12.11)» If Pcalc < Pgiven then increase T» If Pcalc > Pgiven then decrease T» If Pcalc = Pgiven then T = TDew
The simplest method of finding TDew is a trial and error method using a spreadsheet.
Follow steps 1 to 4 to find Pcalc. Change T and repeat steps 2, 3, and 4 until Pcalc = Pgiven
isatii
ii
Py
1P
CHEE 311 J.S. Parent 30
9.3 Modified Raoult’s Law
At low to moderate pressures, the vapour-liquid equilibrium equation can be simplified considerably.
Consider the vapour phase coefficient, i:
Taking the Poynting factor as one, this quantity is the ratio of two vapour phase properties:
Fugacity coefficient of species i in the mixture at T, P Fugacity coefficient of pure species i at T, Pi
sat
If we assume the vapour phase is a perfect gas mixture, this ratio reduces to one, and our equilibrium expression becomes,
or12.20
RT)PP(V
expˆ sat
ili
sati
vi
i
satiiii
satiiiii
PxPy
PxPy
1
CHEE 311 J.S. Parent 31
Modified Raoult’s Law
Using this approximation of the non-ideal VLE equation simplifies phase equilibrium calculations significantly.
Bubble Points:
Setting i =1makes BUBL P calculations very straightforward.
Dew Points:
isatii
i
Py
1P
i i
satiii Px
P i
satiii PxP
isatii
ii
Py
1P
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