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Correlation of Liquid Phase Data

SVNA 12.1

Purpose of this lecture: To show how activity coefficients can be calculated by means of (published) tabulated data of GE vs. mixture composition for binary mixtures Highlights • In binary systems, excess Gibbs energy data, plotted in the form GE/RTx1x2, often follows a linear relationship with respect to mixture composition for a wide range of mole fraction values •Simple models for GE , such as the Redlich/Kister and Symmetric Equation, can be used satisfactorily for binary mixtures Reading assignment: Section 12.1 and 11.9 (refresher)

Fig 12.3

Fig 12.2

fˆi  dfˆi  lim   Hi  dx  xi 0 x i  i  xi  0 fˆi  vs  xi

Henry’s constant, the limiting slope of the curve at xi = 0. ˆ Henry’s law expresses: f i  xi H i  yi P , it is approximate valid for small values of xi

Henry’s law

Lewis/Randall rule Gibbs/Duhem equation

x1 → 0

x2 → 1

Gibbs/Duhem equation for binary mixture at const. T and P: x1dM 1  x2 dM 2  0

 dfˆ1     dx   f1 The Lewis/Randall rule,  1  x1 1 fˆi  fˆi id  xi f i 1  dfˆ1  1 f1  dx1  x 1 1



1  dfˆ1 f1  dx1

    x1 1





x2  0



2

2

xi 1

di  RT ln fˆi x1d ln f 1  x 2 d ln f 2  0

x1

d ln fˆ1 d ln fˆ2  x2 0 dx1 dx1

dx1  dx 2  0

fˆ1  f1

2

x1d 1  x2 d 2  0

Division by dx1



dfˆ1 / dx1 dfˆ2 / dx 2 lim  lim x1 1 ˆ x2  0 f /x fˆ / x 1



dfˆ2 / dx 2 x 2 0  lim fˆ / x

when x1 = 1,

1

M i  Gi   i

2

limit

dfˆ1 / dx1 dfˆ2 / dx2  fˆ2 / x2 fˆ1 / x1

d ln fˆ1 d ln ˆf2 x1  x2 dx1 dx2

7. Correlation of Liquid Phase Data

SVNA 12.1

The complexity of molecular interactions in non-ideal systems makes prediction of liquid phase properties difficult.  Experimentation on the system of interest at the conditions (P,T,composition) of interest is needed.  Previously, we discussed the use of low-pressure VLE data for the calculation of liquid phase activity coefficients. As practicing engineers, you will rarely have the time to conduct your own experiments.  You must rely on correlations of data developed by other researchers.  These correlations are empirical models (with limited fundamental basis) that reduce experimental data to mathematical equations.

Correlation of Liquid Phase Data Recall our development of activity coefficients on the basis of the partial excess Gibbs energy : E

id

Gi  G i  Gi

where the partial molar Gibbs energy of the non-ideal model is provided by equation 10.42:

Gi  li  i (T )  RT ln ˆf li and the ideal solution chemical potential is: id

l Gi   id i  i ( T )  RT ln x i fi

Leaving us with the partial excess Gibbs energy: E Gi  RT ln ˆfil  RT ln xi fi l ˆfil  RT ln l xi fi

 RT ln  i

(11.91)

Correlation of Liquid Phase Data The partial excess Gibbs energy is defined by: E Gi

(nGE )  ni T,P,nj

In terms of the activity coefficient,

 ( nG E / RT ) ln  i   ni T ,P ,nj

(11.96)

ln(i) is a partial molar property GE/RT and activity coefficients are related using the summability relationship for partial properties.

GE   x i ln  i RT i

(11.99)

This information leads to useful correlations for activity coefficients.

Correlation of Liquid Phase Data We can now process our MEK/toluene data one step further to give the excess Gibbs energy,

GE/RT = x1ln1 + x2ln2

Correlation of Liquid Phase Data Note that GE/(RTx1x2) is reasonably represented by a linear function of x1 for this system. This is the foundation for the simplest correlations for experimental activity coefficient data

 yP  ln  1  ln 1 sat   x1P1   yP  ln  2  ln 2 sat   x 2P2 

GE / RT  x1 ln 1  x 2 ln  2

Positive deviation from Raoult’s law behavior: The dimensionless excess Gibbs energy: The value of GE/RT is zero at both x1= 0 and x1 =1

d ln 1 d ln  2 x1  x2  0 (const . T , P ) dx1 dx1

 i  1 ln  i  0 GE  (0) ln  1  (1)( 0)  0 lim x1 0 RT

GE lim  ln  1 x1 0 x x RT 1 2 GE lim  ln  2 x1 1 x x RT 1 2

Models for the Excess Gibbs Energy Models that represent the excess Gibbs energy have several purposes:  they reduce experimental data down to a few parameters  they facilitate computerized calculation of liquid phase properties by providing equations from tabulated data  In some cases, we can use binary data (A-B, A-C, B-C) to calculate the properties of multi-component mixtures (A,B,C)

A series of GE equations for activity coefficients are derived from the Redlich/Kister expansion at const. T:

Equations of this form “fit” excess Gibbs energy data quite well. However, they are empirical and cannot be generalized for multicomponent (3+) mixtures or multiple temperatures.

Symmetric Equation for Binary Mixtures

To calculate activity coefficients, we express GE in terms of moles: n1 and n2. An n nG E

RT



1 2

(n1  n2 )

And through differentiation,

(nGE / RT ) ln 1  n1 T,P,n 2 we find:

Excess Gibbs Energy Models Practicing engineers usually get information about activity coefficients from correlations obtained by making assumptions about excess Gibbs Energy. These correlations:  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, Unisym, Aspen) Simple empirical correlations  Symmetric, Margules, van Laar  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  Wilson, NRTL, Uniquac  Some fundamental basis  Parameters are temperature dependent, and multicomponent 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.

 yP  ln  1  ln 1 sat   x1P1   yP  ln  2  ln 2 sat   x 2P2 

GE / RT  x1 ln 1  x 2 ln  2

Margules’ Equations While the simplest Redlich/Kister-type correlation is the Symmetric Equation, but a more accurate equation is the Margules correlation: (12.9a)

GE  A 21x1  A12 x 2 RTx1x 2

Note that as x1 goes to zero,

GE RTx1x 2

Also,

 A 12 x1 0

E

so that

G  ln 1 x 1 0 RTx1x 2 lim

and similarly

A 12  ln 1

A 21  ln  2

Margules’ Equations If you have Margules parameters, the activity coefficients can be derived from the excess Gibbs energy expression: GE  A 21x1  A12 x 2 RTx 1x 2

(12.9a)

to yield:

ln 1  x22 [ A12  2( A 21  A12 )x1 ]

(12.10ab)

ln  2  x12 [ A 21  2( A 12  A 21 )x 2 ] 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.

van Laar Correlation Another two-parameter excess Gibbs energy model was developed from an expansion of (RTx1x2)/GE instead of GE/RTx1x2. The end results are: GE A12/ A21/

RTx1 x2



A12/ x1  A21/ x2

for the excess Gibbs energy and: / A 12 x  /  ln 1  A12 1  / 1   A 21x2 

(12.16)

2

(12.17a)

2

A /21x 2  /   ln  2  A 21 1  / A x  12 1  for the activity coefficients. Note that:

as x10, ln1  A’12

and

as x2  0, ln2  A’21

(12.17b)

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 using information from binary data.

•

A,B,C

• tertiary mixture CHEE 311

A,B binary Lecture 18

A,C

B,C

binary

binary 33

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: GE  x1 ln( x1  x2 12 )  x 2 ln( x 2  x1 21) RT • (12.18) • where V  a  • 12  2 exp  12  V1  RT  • •

21 

V1  a  exp  21  V2  RT 

(12.24)

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), a21 = (12 - 22) that you will encounter in Holmes, M.J. and M.V. Winkle (1970) Ind. Eng. Chem. 62, 21-21. CHEE 311

Lecture 18

34

Wilson’s Equations for Binary Solution Activity • Activity coefficients are derived from the excess Gibbs energy using the definition of a partial molar property:

RT ln  i 

E Gi

nGE   ni

T,P,n j

• When applied to equation 11.16, we obtain:

•

•

CHEE 311

12   21    ln 1   ln( x 1  x 212 )  x 2  x x x x      1 2 12 2 1 21   12   21    ln  2   ln( x 2  x1 21)  x1  x x x x      1 2 12 2 1 21 

Lecture 18

(12.19a)

(12.19b)

35

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 for multicomponent mixtures is GE written:   x ln( x  ) RT

 i

i



j

ij

j

•

(12.22)

• and the activity coefficients become: x  ln  i  1  ln  x j  ij   k ki i k  x j kj •

(12.23)

j

• where ij = 1 for i=j. Summations are over all species. CHEE 311

Lecture 18

36

Wilson’s Equations for 3-Component Mixtures • For three component systems, activity coefficients can be calculated from the following relationship:

ln  i  1  ln( x1  i1  x 2  i2  x 3  i3 ) 

x 1 1i x1  x2 12  x3 13



x 2 2 i x1 21  x 2  x 3  23



x3  3 i x1 31  x2  32  x 3

• Model coefficients are defined as (ij = 1 for i=j):

 ij 

CHEE 311

  a ij  exp  Vi  RT  Vj

Lecture 18

37

NRTL(Non- Random Two-Liquid)

Comparison of Liquid Solution Models • Activity coefficients of 2-methyl2-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

Lecture 18

42

Non-Ideal VLE to Moderate Pressures: Overview

SVNA 14.1

• 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

iv   li

 In terms of mixture fugacity

ˆf v  ˆf l i i

• 2. Fugacity of a component in a non-ideal gas mixture:

fˆiv (T,P, y 1, y 2 ,..., y n )  y i ˆ vi ( T,P, y 1, y 2 ,..., y n ) P • 3. Fugacity of a component in a non-ideal liquid mixture:

ˆfil (T,P, x1 , x2 ,..., xn )  x i  i (T,P, x 1, x 2 ,..., x n ) fil CHEE 311

P, x118, x 2 ,..., x n )  P  x i  i (T,Lecture

sat sat i i

 Vi (P  Pisat )  exp43  RT  

, Formulation of VLE Problems • To this point, Raoult’s Law was only description we had for VLE behaviour: sat

y iP  x iPi

• We know 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,

ˆf v  ˆf l i i y ˆ P  x   P v i i

sat sat i i i i

 Vi (P  Pi sat ) exp  RT  

• determines the VLE behaviour of non-ideal systems where Raoult’s Law fails. CHEE 311

Lecture 18

44

Non-Ideal VLE to Moderate Pressures • A simpler expression for non-ideal VLE is created upon defining a lumped parameter, F.

  Vil(P  Pisat )  ˆiv F i  sat exp   RT i  

•

ˆiv  sat i

• The final expression becomes, •

yi FiP  x i iPisat

(i = 1,2,3,…,N)

14.1

• To perform VLE calculations we therefore require vapour pressure data (Pisat), vapour mixture and pure component fugacity correlations (Fi) and liquid phase activity coefficients (i).

CHEE 311

Lecture 18

45

Non-Ideal VLE to Moderate Pressures • Sources of Data: • 1. Vapour pressure: Antoine’s Equation (or similar correlations) •

ln Pisat  A i 

Bi T  Ci

14.3

• 2. Vapour Fugacity Coefficients: Viral EOS (or others) •

 Bii (P  Pisat )  0. 5 P  y j yk (2 ji jk ) j k  Fi  exp    RT  

14.6

• 3. Liquid Activity Coefficients  Binary Systems - Margules,van Laar, Wilson, NRTL, Uniquac  Ternary (or higher) Systems - Wilson, NRTL, Uniquac CHEE 311

Lecture 18

46...