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8. VLE Flash Calculations for Non-Ideal Systems
We introduced flash calculations for ideal systems (Raoult’s Law) in
lecture 10. With our revised models of chemical potential, we are now
able to handle non-ideal systems quite accurately.
The basic P,T-flash problem:
Given: P,T, z1,z2,…, zn
Feed
z1
z2
z3=1-z1-z2
Tf, Pf
Find: V,L, x1, x2,…xn, y1, y2,…, yn
P,T
Vapour
y1
y2
y3=1-y1-y2
Liquid
x1
x2
x3=1-x1-x2
 Use a flash calculation whenever the overall composition of the
system is known, but the composition of each phase is not.
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VLE Flash Calculations from a Phase Diagram
To the right is the Txy diagram for the
highly non-ideal system of Ethanol(1)Toluene(2) at P=1 atm.
1. Given a feed stream containing 25%
ethanol, between what temperatures
do we have two phases?
2. At 90°C, what are the compositions
of the liquid and vapour streams?
3. Under these conditions, what fraction
of the system exists as a vapour?
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VLE Flash Equation
When a phase diagram is not at hand, a flash calculation using a
model for the phase behaviour is required.
Whenever confronted with a flash problem, apply one of the
general flash equations:
ziK i
1

1

V
(
K

1
)
i
i
12.27
or
zi
1

1

V
(
K

1
)
i
i
12.28
This is the most versatile approach to solving flash problems
 In all but simple P,T flashes for binary systems, the general
flash equation will produce the quickest answer.
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Partition Coefficient, Ki
The partition coefficient, Ki = yi / xi, is used to simplify the general
flash equation.
 It reflects the tendency of a component to vapourize. Those
components with a large partition coefficient (Ki >1)
concentrate in the vapour, while those with Ki <1 concentrate
in the liquid phase.
The partition coefficient in a VLE system is provided by our phase
equilibrium expression (derived from equivalence of chemical
potential). Recall,
yiiP  xi  iPisat
Therefore,
y i  iPisat
Ki  
xi
 iP
Note that for a non-ideal system, Ki is a function of P,T and the
compositions of both the liquid and vapour phase.
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Solving Non-Ideal Flash Problems
The “Classic” P,T-flash problems involves:
Given: P,T, z1,z2,…, zn
Find: V,L, x1, x2,…xn, y1, y2,…, yn
For a three component system, the VLE flash equation is:
z3
z1
z2


1
1  V(K1  1) 1  V(K 2  1) 1  V(K 3  1)
12.28
Or, substituting for the partition coefficients:
z3
z1
z2


1
sat
sat
sat
P
 P
 P
1  V( 1 1  1) 1  V( 2 2  1) 1  V( 3 3  1)
 1P
 2P
 3P
The general solution involves:
 Find the vapour phase fraction (V0) that satisfies 12.28.
 Substitute V into:
xi 
CHEE 311
zi
and solve for yi using: y i  K i x i
1  V(K i  1)
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Solving Non-Ideal Flash Problems
The non-ideal flash equation requires knowledge of the vapour and
liquid compositions to evaluate i and i, respectively.
 These are the unknowns that we are attempting to calculate
 Therefore, flash calculations always require iteration.
Suppose you are given P,T, z1, z2, z3 and you are asked to find V,
and the phase compositions.
1. Calculate the DEWP and BUBLP of the feed at the given
temperature to ensure that two phases exist.
2. Use Raoult’s law to simplify the flash problem to the degree that
it can (easily?) be solved.
 This involves setting i = 1 and i =1 for all components
 Calculate Pisat for each component at the given T
 Solve for V using the flash equation, 12.28
 Solve for xi, and yi
These are ESTIMATES that serve only to get us started!
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Solving Non-Ideal Flash Problems
3. Using the latest estimate of xi and yi, along with the P,T given,
calculate:
 i for each component using an activity coefficient model
 i for each component using an equation of state for the
vapour.
 Calculate Ki = iPisat / iP for each component
4. Using the revised partition coefficients, calculate V through a trial
and error procedure on the general flash calculation.
zi
1

1

V
(
K

1
)
i
i
5. Recalculate xi,and yi for each component.
xi 
zi
1  V(K i  1)
yi  K i xi
6. Repeat steps 3 through 5 until the solution converges.
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8. Azeotropic Mixtures
SVNA 12.3,12.5
Large deviations from ideal liquid
solution behaviour relative to the
difference between the pure component
vapour pressures result in azeotrope
formation.
In CHEE 311, we are interested in:
1. Describing azeotropic mixtures
both physically and in
thermodynamic terms.
2. Detecting azeotropic conditions
and calculating their composition.
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Azeotropic Mixtures
Water / Hydrazine, P=1atm
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Water / Pyridine, P=1atm
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Azeotropes - Impact on Separation Processes
Separation processes that exploit
VLE behaviour (flash operations,
distillation) are influenced greatly
by azeotropic behaviour.
An azeotropic mixture boils
to evolve a vapour of the
same composition and,
conversely,condenses to
generate a liquid of the
same composition.
Ethanol(1)/Toluene(2) at P=1 atm
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Predicting Whether an Azeotrope Exists
To determine whether an azeotrope will be encountered at a given
pressure and temperature, we define the relative volatility. For a
binary system, a12 is
a12 
y1 x1
y2 x2
12.21
where xi and yi are the mole fractions of component i in the liquid
and vapour fractions, respectively.
At an azeotrope, the composition of the vapour and liquid are
identical. Since, y1=x1 and y2=x2 at this condition,
a12  1
To determine whether an azeotropic mixture exists, we need to
determine whether at some composition, a12 can equal 1.
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Predicting Whether an Azeotrope Exists
We can derive an expression for a12 using modified Raoult’s Law
as our phase equilibrium relationship,
y iP  x i  iPisat
which when substituted into the relative volatility, yields
a12
 1P1sat

 2P2sat
12.22
a12 is therefore a function of T (Pisat, i) and the composition of the
liquid phase. Calculation of a12 therefore requires:
 Antoine’s equation
 an activity coefficient model (Margule’s, Wilsons, …)
 a liquid composition
Our goal is to determine whether an azeotrope exists.
 At some composition, can a12=1?
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Predicting Whether an Azeotrope Exists
One means of determining
whether a12=1 is possible
is to evaluate the function
(Eq’n12.22) over the entire
composition range.
This is plotted for the
ethanol(1)/toluene(2)
system using Wilson’s
equation to describe liquid
phase non-ideality.
According to this plot,
a12=1 at x1 = 0.82,
meaning that an azeotrope
exists at this composition.
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Predicting Whether an Azeotrope Exists
Because equation 12.22 is continuous and monotonic, we do not
need to evaluate a12 over the whole range of x1.
 It is sufficient to calculate a12 at the endpoints, x1=0 and x1=1
At x1 = 0, we have
a12
x1  0
 1P1sat

P2sat
and at x1 = 1, we have
a 12
x1 1
P1sat
  sat
 2 P2
If one of these limits has a value greater than one, and the other
less than one, at some intermediate composition we know a12 =1.
 This is a simple means of determining whether an azeotrope
exists.
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Determining the Composition of an Azeotrope
For an azeotropic mixture, the relative volatility equals one:
a12
 1P1sat

1
sat
 2P2
at an azeotrope.
To find the azeotropic composition, two methods are available:
 trial and error (spreadsheet)
 analytical solution
Rearranging 12.22 as above yields:
 1 P2sat
 sat
 2 P1
The azeotropic composition is that which satisfies this equation.
 Substitute an activity coefficient model for 1, and 2.
 Solve for x1.
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8. Dilute Liquid Solution Approximations
There are many non-ideal systems in which one component is
significantly more volatile than others. In these cases, the liquid
phase is very rich in the heavy components, and lean in the light
component.
 The most common system encountered in environmental and
biochemical engineering is H2O (1) / O2 (2).
The very low O2 content found in the liquid phase under practical
engineering conditions allows us to simplify phase equilibrium
calculations:
for H2O
fˆHv2O  fˆHl 2O
fˆ
v
O2
 fˆ
l
O2
for O2
Since the origin of most non-ideal behaviour is found in the liquid,
we will focus on simplifications to liquid phase properties.
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Rigorous Treatment of Liquid Mixture Fugacity
Shown is a plot of the mixture fugacities
of MEK (1) and Toluene (2) at 50°C as a
function of liquid composition
For component 1, MEK:
l
sat

V
(
P

P
)
1
1
ˆf1l   1x11satP1sat exp 

RT


For component 2, Toluene:
l
sat

V
(
P

P
2
2 )
l
sat
sat
fˆ2   2 x 2  2 P2 exp 

RT


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Dilute Liquid Solution Approximations
Figure 11.3 of SVNA illustrates the
fugacity of a component in a liquid
mixture.
Note that the Lewis-Randall rule
applies for the predominate
component of a liquid solution:
 x11: 1 1, f1l  f1l x1
If the component is present in very
small amounts (x1 < 0.02), its mixture
fugacity can be approximated by a
linear relationship, such that:
 x10: f1l  k1x1
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Dilute Solution Simplifications: Heavy Component
For the heavy component (1) of a dilute solution, we can apply the
Lewis-Randall rule:
fˆ1l   1x1f1l
 x1f1l
as x1  1.0
The equilibrium relationship for the heavy component (such as H2O
in the water-oxygen system) becomes:
or,
or,
fˆ1v  fˆ1l
y ˆ P  x  P
v
1 1
sat sat
1 1
1
 V1(P  P1sat ) 
exp 

RT


y1component
1P  x1Pin1 solution (x1>0.98)
for the predominate
sat
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Dilute Solution Simplifications: Light Component
For the light component (2) of a dilute liquid solution, we create a
new construct, the Henry’s constant, k2
fˆ2l  dfˆ2l 

 
 k2
lim
x 2 0 x 2
 dx 2  x 2 0
(11.2)
This is the slope of the f2l vs x2 curve as x2  0.
 The Henry’s constant is tabulated for a specific system at a
given temperature.
The equilibrium relationship for the light component (such as O2 in
the water-oxygen system) becomes:
fˆ2v  fˆ2l
y 2 ˆ 2P kx2 is
where the Henry’s constant,
of oxygen in water at the
2kthat
2
temperature of interest.
v
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Dilute Liquid Solution Approximations: Example 1
Suppose we are designing a bioreactor in which pure O2 is bubbled
through an aqueous medium to replenish the oxygen consumed by
the cell culture.
 The process is operated at atmospheric pressure and a
temperature of 25°C.
 How do we define a thermodynamic system that provides
relevant phase equilibrium data?
 What is the concentration of oxygen in the liquid phase, given
a Henry’s Coefficient, kO2 = 4400 MPa at 25°C?
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Dilute Liquid Solution Approximations: Example 2
As a research engineer in a hydrogenation plant, you are asked to
compile VLE data on the H2 / chlorobenzene system in a way that
plant engineers can readily apply.
 The objective is to conduct VLE experiments on the system,
treat the data according to thermodynamic theory, and
summarize the results in a simple manner.
 How will you start?
 What simplifications can be made?
 In what form will you summarize the data?
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