## Binary Vapor Liquid Equilibrium (VLE)

This article shows how to prepare Pxy and Txy diagram for binary mixtures in excel spreadsheet based on Wilson, NRTL and UNIQUAC activity coefficient model.

For low to moderate pressure vapor liquid equilibrium (VLE) is described by modified Raoult's Law -

`y`

= x_{i}P_{i}γ_{i}P_{i}^{sat}

where, y_{i} is vapor mol fraction, P is system
pressure, x_{i} is liquid mol fraction, γ_{i} is
activity coefficient and P_{i}^{sat} is vapor
pressure for a pure component i. Vapor pressure is calculated
based on Antoine equation.

`ln P`_{i}^{sat} = A_{i} - B_{i} /( T + C_{i} )

A_{i} , B_{i} and C_{i} are Antoine
equation constants and T is temperature at which vapor pressure
is to be calculated.

Txy Diagram

Txy diagram plots bubble and dew point curves at constant
pressure P. Put down the liquid mol fraction x_{1} from
0.0 to 1.0 with increment of 0.01 in spreadsheet. Iteration is
done for each liquid mol fraction to estimate equilibrium
temperature T and activity coefficient γ_{i}.

For first iteration, T_{1}^{sat} and T_{2}^{sat} are calculated from Antoine equation.

`T`_{i}^{sat} = B_{i}/ (A_{i} - ln P_{i}^{sat}) - C_{i}

Equilibrium temperature is estimated as following -

`T = x`_{1} T_{1}^{sat} + (1 - x_{1})T_{2}^{sat}

Based on temperature T, activity coefficient γ_{1} and
γ_{2} are calculated from activity coefficient model
selected e.g. Wilson, NRTL and UNIQUAC. For ideal mixture γ_{1 }and γ_{2} are 1.

Saturation pressure for a component is calculated using following equation -

`P`_{1}^{sat} = P/(x_{1}γ_{1} +(1-x_{1})γ_{2} P_{2}^{sat}/P_{1}^{sat})

Temperature corresponding to the vapor pressure P_{1}^{sat} is calculated from Antoine equation.

`T = B`_{1}/ (A_{1} - ln P_{1}^{sat}) - C_{1}

Temperatue thus calculated is used for next iteration and
activity coefficients γ_{1} and γ_{2} are
calculated. Iterations are repeated till there is no change in
subsequent temperature estimations. Typically temperature
difference becomes negligible within 10 iterations.

Above steps are repeated for all liquid mol fractions, thereby
giving a table of x_{1} and corresponding temperature T.
Vapor mol fraction y_{1} is calculated as following -

`y`_{1} = x_{1} γ_{1} P_{1}^{sat}/ P

Plot of T, x_{1} & y_{1} gives Txy Diagram -

Pxy Diagram

Pxy diagram plots bubble and dew point curves at constant
temperature T. Put down the liquid mol fraction x_{1}
from 0.0 to 1.0 with increment of 0.01 in spreadsheet. Calculate
activity coefficients γ_{1} and γ_{2} based on
activity coefficient model selected from Wilson, NRTL and
UNIQUAC.

Calculate partial pressure of each component P_{1} and
P_{2} as following -

`P`

_{1}= x_{1}γ_{1}P_{1}^{sat}`P`

_{2}= (1 - x_{1}) γ_{2}P_{2}^{sat}

Equilibrium pressure is obtained as following -

`P = P`_{1} + P_{2}

Vapor mol fraction is calculated as per below equation.

`y`_{1} = P_{1} / P

Plot of P, x_{1} & y_{1} gives Pxy Diagram -

Wilson Model

Activity coefficient for binary system are defined as -

Wilson parameter is provided by following equation -

where, λ_{12} - λ_{11} and λ_{21} -
λ_{22} are binary interaction parameters available from
literature for a binary pair.

Modified Rackett equation is used to estimate liquid molar
volume V_{1} & V_{2}.

`V = (RT`_{c}/P_{c})Z_{RA} ^{[1 + (1-Tr)^(2/7)]}

where, T_{c} and P_{c} are critical temperature
and pressure. T_{r} is the reduced temperature. Z_{RA}
is Rackett equation parameter, if it is not available, it can be
estimated from accentric factor ω as following.

`Z`_{RA} = 0.29056 - 0.08775ω

NRTL Model

Activity coefficient for binary system are defined as -

Parameter g_{12} - g_{22} and g_{21} -
g_{11} are binary parameters available from literature.
α_{12} is related to non-randomness in mixture and is
available from literature for binary pairs.

UNIQUAC Model

Activity coefficient for binary system are defined as -

Parameter u_{12} - u_{22} and u_{21} -
u_{11} are binary parameters available from literature.
Remaining parameters are calculated as following -

where z is set equal to 10 and r, q & q' are pure component UNIQUAC parameters.

Resources

- Spreadsheet for Binary Vapor Liquid Equilibrium