7. excess gibbs energy models

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.


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


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


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


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


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


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


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


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


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


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.


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


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


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


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


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


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


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


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


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


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


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


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



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



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


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


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



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



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


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


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

7. excess gibbs energy models

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excess gibbs energy

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