Vapor-Liquid Equilibrium Data Using Arc Length Continuation

​
ethanol/water
ethanol/ethyl-acetate
isobaric VLE diagram
equilibrium curve
azeotrope locus
pressure in mmHg
760
Consider two binary mixtures: (1) ethanol and water and (2) ethanol and ethyl acetate. This Demonstration computes the isobaric vapor-liquid diagram as well as the equilibrium curve at user-set values of the total pressure
P
(expressed in
mmHg
). The modified Raoult's law is used along with the van Laar model and Antoine equation. Both systems present a positive pressure-sensitive azeotrope. When present, this azeotrope is indicated on the equilibrium curve by a red dot. The loci of the azeotrope versus pressure
P
is given in a separate plot. In both cases, the azeotrope disappears at a low enough total pressure. One particular feature of the present calculation is that it uses the arc length continuation method (see the Details section) to find the bubble/dew point temperatures versus liquid/vapor phase compositions. This takes advantage of a new function as of Mathematica 9.0, WhenEvent, which determines the loci of the azeotropes; indeed they verify
T'(s)=0
, where
s
is the arc length parameter.

Details

For vapor-liquid equilibrium data computations, the nonlinear equation
f(T,x)=0
, where
x
is the liquid mole fraction and
T
is the bubble temperature, is the bubble point equation derived from Dalton's law and the modified Raoult's law. Introduce an arc length parameter
s
. The nonlinear algebraic equation becomes
f(T(s),x(s))=0
. We use the built-in Mathematica function NDSolve to solve this equation together with the differential equation (called the arc length constraint)
T'
2
(s)
+x'
2
(s)
=1
in order to find
x(s)
and
T(s)
. A simple initial condition is found by taking
x(0)=1
and
T(0)
equal to the boiling temperature of pure ethanol at
P
.

Permanent Citation

Housam Binous, Ahmed Bellagi, Brian G. Higgins
​
​"Vapor-Liquid Equilibrium Data Using Arc Length Continuation"​
​http://demonstrations.wolfram.com/VaporLiquidEquilibriumDataUsingArcLengthContinuation/​
​Wolfram Demonstrations Project​
​Published: December 2, 2013