CHEE 311 Lecture 2 1 VLE Calculations Purpose of this lecture : To demonstrate how Raoult’s 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. Raoult’s 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 (7 th edition), or Section 10.4, pp. 338- 348 (6 th edition)
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
CHEE 311 Lecture 2 1
VLE Calculations
Purpose of this lecture:
To demonstrate how Raoult’s 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.
Raoult’s 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)
CHEE 311 Lecture 2 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
CHEE 311 Lecture 2 3
Phase Rule in VLE: Single Component Systems
For a two phase (=2) system of a single component (N=1):
F = 2- + N
F = 2- 2 + 1 = 1
Therefore, for the single component system, specifying either T or P fixes all intensive variables.
VLE for Pure Components
0
200
400
600
800
270 320 370 420Temperature: K
Pre
ssu
re: k
Pa
Acetonitrile Nitromethane
CHEE 311 Lecture 2 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
CTB
APln sati
224C/T
47.29452724.14kPa/Pln sat
1
209C/T
64.29722043.14kPa/Pln sat
2
CHEE 311 Lecture 2 5
(General Case)
For a two phase (=2), binary system (N=2):
F = 2- 2 + 2 = 2
Therefore, for the binary case, two intensive variables must be specified to fix the state of the system.
Phase Rule in VLE: Ideal Binary Mixtures
CHEE 311 Lecture 2 6
Phase Rule in VLE: Binary Systems (Pxy diagrams)
Example: Acetonitrile (1) / Nitromethane (2) system
Acetonitrile(1) - Nitromethane(2) @ 75C
40
50
60
70
80
90
0.0 0.2 0.4 0.6 0.8 1.0x1,y1
Pre
ssu
re,
kPa
y1 x1
CHEE 311 Lecture 2 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.
Acetonitrile(1) Nitromethane(2) @ 70kPa
65.0
70.0
75.0
80.0
85.0
90.0
0.00 0.20 0.40 0.60 0.80 1.00x1,y1
Tem
p,
deg
C
y1 x1
CHEE 311 Lecture 2 8
VLE Calculations using Raoult’s Law
Raoult’s Law for ideal phase behaviour relates the composition of liquid and vapour phases at equilibrium through the component vapour pressure, Pi
sat.
Given the appropriate information, we can apply Raoult’s 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)
PP
xy sat
i
i
i
CHEE 311 Lecture 2 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
CHEE 311 Lecture 2 10
• 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) Raoult’s law for ideal phase behaviour:
2) Antoine’s Equation
satiiii PxPyP **
i
ii
sati CT
BAP
)ln(
(1)
(2)
VLE Calculations - Introduction
CHEE 311 Lecture 2 11
- Calculate and from Antoine’s Equation
- For the vapour-phase composition (bubble) we can write:
y1+y2=1 (3)
- Substitute y1 and y2 in Eqn (3) by using Raoult’s law:
(4)
- Re-arrange and solve Eqn. (4) for P
- Now you can obtain y1 from Eqn (1)
- Finally, y2 = 1-y1
BUBL P Calculation (T, x1 known)
sat1P sat
2P
1P
Px1
P
Px
P
Px
P
Px sat21
sat11
sat22
sat11
*)(***
CHEE 311 Lecture 2 12
- Calculate and from Antoine’s Equation
- For the liquid-phase composition (dew) we can write:
x1+x2=1 (5)
- Substitute x1 and x2 in Eqn (5) by using Raoult’s law:
(6)
- Re-arrange and solve Eqn. (6) for P
- Now you can obtain x1 from Eqn (1)
- Finally, x2 = 1-x1
DEW P Calculation (T, y1 known)
sat1P sat
2P
1P
Py1
P
Py
P
Py
P
Pysat2
1sat
1
1sat2
2sat
1
1
*)(***
CHEE 311 Lecture 2 13
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 Antoine’s 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 P’=P; If not, try another T’-value- Calculate y1 from Raoult’s law
BUBL T Calculation (P, x1 known)
ii
isati C
PA
BT
)ln(
)'()'( TPandTP sat2
sat1
'PP
sat2
sat1 TTT '
CHEE 311 Lecture 2 14
Same as before, calculation of T, x1 requires an iterative approach:
- Re-arrange Antoine’s 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 Antoine’s Eqn.
- Solve Eqn. (6) for pressure P’
- If , then P’=P; If not, try another T’-value
- Calculate x1 from Raoult’s law
DEW T Calculation (P, y1 known)
sat2
sat1 TTT '
)'()'( TPandTP sat2
sat1
'PP
CHEE 311 Lecture 2 15
P, T Flash Calculation
- Calculate and from Antoine’s Equation
- Use Raoult’s law in the following form:
(8)
- Re-arrange and solve Eqn. (8) for x1
- Now you can obtain y1 from Eqn (1), i.e.,
sat1P sat
2P
1P
Px1
P
Pxy
sat21
sat11
i
*)(*
P
Pxy
sat11
1
*
CHEE 311 Lecture 2 16
Example
Assuming Raoult’s Law to be valid, prepare
(a) a Pxy diagram for T=90oC, and
(b) a Txy diagram for P=90 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
CHEE 311 Lecture 2 17
P (kPa)
0.0 0.0
… … …
1.0 1.0
• The construction of Pxy diagram requires multiple P, T Flash calculations, where T is held constant and P is varied from P2
sat to P1sat.
• The results can be tabulated as shown below:
Construction of Pxy diagrams
sat2
sat1
sat2
1 PP
PPx
P
Pxy
sat11
1
*
sat2P
sat1P
This type of calculations can also be performed by keeping T constant and varying x1 or y1 from 0.0 to 1.0
CHEE 311 Lecture 2 18
Example* – (a) Generation of Pxy Data
CHEE 311 Lecture 2 19
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
0.00 0.20 0.40 0.60 0.80 1.00
P (
kPa)
x1
y1
liquid
VLE
vapor
Example – (a) Construction of a Pxy Plot
CHEE 311 Lecture 2 20
T (oC)
0 0
… … …
1.0 1.0
• 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:
Construction of Txy diagrams
sat2
sat1
sat2
1 PP
PPx
P
Pxy
sat11
1
*
sat2T
sat1T
This type of calculations can also be performed by keeping P constant and varying x1 or y1 from 0.0 to 1.0
Txy diagram for 1-chlorobutane (1) and chlorobenzene (2) at P = 90 kPa (assuming validity of Raoult's law)
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
0.00 0.20 0.40 0.60 0.80 1.00
x1,y1
T (
deg
C)
x1
y1liquid
VLE
vapor
Example – (b) Construction of a Txy Plot
CHEE 311 Lecture 2 23
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: