VLE Calculations

1
VLE Calculations

• Purpose of this lecture
• To demonstrate how Raoults law can be used in
  the prediction of the VLE behaviour of ideal
  mixtures
• Highlights
• Phase rules gives the number of variables we need
  in order to determine the intensive state of a
  system at equilibrium
• Saturation pressures can be calculated by means
  of the Antoine Eqn.
• Raoults law can be used for constructing Pxy,
  Txy diagrams and performing dew point and bubble
  point calculations
• Reading assignment Section 10.4, pp. 347-357
  (7th edition), or
• Section 10.4,
  pp. 338-348 (6th edition)

2
Phase Rule for Intensive Variables
SVNA-10.2

• For a system of ? phases and N species, the
  degree of freedom is
• F 2 - ? N
• variables that must be specified to fix the
  intensive state of the system at equilibrium
• Phase Rule Variables
• The system is characterized by T, P and (N-1)
  mole fractions for each phase
• Requires knowledge of 2 (N-1)? variables
• Phase Rule Equations
• At equilibrium ?i? ?i ? ?i ? for all
  N species
• These relations provide (?-1)N equations
• The difference is F 2 (N-1)? - (?-1)N
• 2- ? N

3
Phase Rule in VLE Single Component Systems

• For a two phase (p2) system of a single
  component (N1)
• F 2- ? N
• F 2- 2 1 1
• Therefore, for the single component system,
  specifying either T or P fixes all intensive
  variables.

4
Correlation of Vapour Pressure Data

• Pisat, or the vapour pressure of component i, is
  commonly represented by Antoine Equation
  (Appendix B, Table B.2, SVNA 7th ed.)
• For acetonitrile (Component 1)
• For nitromethane (Component 2)
• These functions are the only component properties
  needed to characterize ideal VLE behaviour

5
Phase Rule in VLE Ideal Binary Mixtures

• (General Case)
• For a two phase (?2), binary system (N2)
• F 2- 2 2 2
• Therefore, for the binary case, two intensive
  variables must be specified to fix the state of
  the system.

6
Phase Rule in VLE Binary Systems (Pxy diagrams)

• Example Acetonitrile (1) / Nitromethane (2)
  system

7
Phase Rule in VLE Binary Systems (Txy diagrams)

• Alternatively, we can specify a system pressure
  and examine the VLE behaviour as a function of
  temperature and composition.

8
VLE Calculations using Raoults Law

• Raoults Law for ideal phase behaviour relates
  the composition of liquid and vapour phases at
  equilibrium through the component vapour
  pressure, Pisat.
• Given the appropriate information, we can apply
  Raoults law to the solution of 5 types of
  problems
• Dew Point Pressure or Temperature
• Bubble Point Pressure or Temperature
• P,T Flash calculation of equilibrium composition
  (P, T, zi given)

9
Dew and Bubble Point Calculations

• Dew Point Pressure
• Given a vapour composition at a specified
  temperature, find the composition of the liquid
  in equilibrium
• Given T, y1, y2,... yn find P, x1, x2, ... xn
  
• Dew Point Temperature
• Given a vapour composition at a specified
  pressure, find the composition of the liquid in
  equilibrium
• Given P, y1, y2,... yn find T, x1, x2, ... xn
  
• Bubble Point Pressure
• Given a liquid composition at a specified
  temperature, find the composition of the vapour
  in equilibrium
• Given T, x1, x2, ... xn find P, y1, y2,... yn
  
• Bubble Point Temperature
• Given a vapour composition at a specified
  pressure, find the composition of the liquid in
  equilibrium
• Given P, x1, x2, ... xn find T, y1, y2,... yn

10
VLE Calculations - Introduction

• For now, we are going to employ these
  calculations only for
• identifying the state and composition of binary
  and ideal mixtures
• As we are going to see later in the course, the
  aforementioned
• VLE calculations are also applicable to
  non-ideal or/and
• multi-component mixtures
• The calculations revolve around the use of 2 key
  equations
• 1) Raoults law for ideal phase behaviour
• 2) Antoines Equation

(1)
(2)
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BUBL P Calculation (T, x1 known)

• - Calculate and from Antoines
  Equation
• For the vapour-phase composition (bubble) we can
  write
• y1y21
  (3)
• Substitute y1 and y2 in Eqn (3) by using
  Raoults law
• 
  
  (4)
• - Re-arrange and solve Eqn. (4) for P
• Now you can obtain y1 from Eqn (1)
• Finally, y2 1-y1

12
DEW P Calculation (T, y1 known)

• - Calculate and from Antoines
  Equation
• For the liquid-phase composition (dew) we can
  write
• x1x21
  (5)
• Substitute x1 and x2 in Eqn (5) by using
  Raoults law
• 
  
  (6)
• - Re-arrange and solve Eqn. (6) for P
• Now you can obtain x1 from Eqn (1)
• Finally, x2 1-x1

13
BUBL T Calculation (P, x1 known)

• Since T is an unknown, the saturation pressures
  for the
• mixture components cannot be calculated
  directly. Therefore,
• calculation of T, y1 requires an iterative
  approach, as follows
• Re-arrange Antoines equation so that the
  saturation temperatures
• of the components at pressure P can be
  calculated
• 
  
  (7)
• Select a temperature T so that
• Calculate
• Solve Eqn. (4) for pressure P
• If , then PP If not, try
  another T-value
• Calculate y1 from Raoults law

14
DEW T Calculation (P, y1 known)

• Same as before, calculation of T, x1 requires an
  iterative approach
• Re-arrange Antoines equation so that the
  saturation temperatures
• of the components at pressure P can be
  calculated from Eqn. (7)
• - Select a temperature T so that
• Calculate
  from Antoines Eqn.
• Solve Eqn. (6) for pressure P
• If , then PP If not, try
  another T-value
• Calculate x1 from Raoults law

15
P, T Flash Calculation

• - Calculate and from Antoines
  Equation
• Use Raoults law in the following form
  
• 
  
  (8)
• - Re-arrange and solve Eqn. (8) for x1
• Now you can obtain y1 from Eqn (1), i.e.,

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Example

• Assuming Raoults Law to be valid, prepare
• a Pxy diagram for T90oC, and
• a Txy diagram for P90 kPa
• for a mixture of 1-chlorobutane (1)
  /chlorobenzene (2)
• Antoine Coefficients

A B C
1-chlorobutane (1) 13.9600 2826.26 224.10
Chlorobenzene (2) 13.9926 3295.12 217.55
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Construction of Pxy diagrams

• The construction of Pxy diagram requires
  multiple P, T Flash
• calculations, where T is held constant and P is
  varied from P2sat to P1sat.
• The results can be tabulated as shown below

P (kPa)
0.0 0.0

1.0 1.0
This type of calculations can also be performed
by keeping T constant and varying x1 or y1 from
0.0 to 1.0
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Example (a) Generation of Pxy Data
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Example (a) Construction of a Pxy Plot
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Construction of Txy diagrams

• The construction of Txy, diagram requires
  multiple P, T, Flash
• calculations, each one of which provides a set
  of equilibrium y1, x1
• values for a given value of temperature (at
  fixed P)
• The results can be tabulated as shown below

T (oC)
0 0

1.0 1.0
This type of calculations can also be performed
by keeping P constant and varying x1 or y1 from
0.0 to 1.0
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Example (b) Generation of Txy Data
22
Example (b) Construction of a Txy Plot
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VLE Calculations - Summary

• Why? To completely identify the thermodynamic
  state
• of a mixture at equilibrium (single phase, 2
  phases..?)
• How? Through the calculation of its P, T, and
  composition
• - The type of calculation that we need to
  perform is subject
• to the variables we are looking to evaluate
• - These calculations are classified as follows

Specified/Known Variables Unknown Variables Calculation
T, x P, y BUBL P
T, y P, x DEW P
P, x T, y BUBL T
P, y T, x DEW T
P, T x, y P, T Flash