Equilibrium

Report
Ref 1: Physical Property Methods and Models, Aspen Technology, Inc., 2006
Ref 2: Smith et. Al, Introduction to Chemical Engineering Thermodynamics,
McGraw-Hill, 1996
1
VLE (EOS Method)
fi  fi
v
 At Equilibrium:
 Where
 Therefore
f i   y i Pt
v
v
i
k
vl
i

,
yi
xi

l
f i   x i Pt
l


l
i
l
i
v
i
2
VLE (EOS Method)

ln  i 

P
0

 ( nZ


  n i

Z  compressib

)
 dP 


1 


 P 
 T , P ,n j

ility factor
Cubic Equations of State in the Aspen Physical Property System
Redlich-Kwong(-Soave) based
Peng-Robinson based
Redlich-Kwong (RK)
Standard Peng-Robinson(PENG-ROB)
Standard Redlich-Kwong-Soave(RK-SOAVE ) Peng-Robinson(PR-BM)
Redlich-Kwong-Soave (RKS-BM)
Peng-Robinson-MHV2
Redlich-Kwong-ASPEN(RK-ASPEN)
Peng-Robinson-WS
Schwartzentruber-Renon
Redlich-Kwong-Soave-MHV2
Predictive SRK (PSRK)
Redlich-Kwong-Soave-WS
3
VLE (EOS Method)
Standard RK-SOAVE
PV
Z 

RT
V
V b

aV
RTV (V  b )
 Where
a

i
xi x j ( a i a j )
0 .5
(1  k ij ) ,
b
 xb
j
i
i
i
2
a i   i 0 . 42747
Pci
 i (T )  [1  m i (1  T ri )] ,
0 .5
2
R T ci
2
,
bi  0 . 08664
RT ci
Pci
m i  0 . 48  1 . 57  i  0 . 176  i
2
4
VLE (EOS Method)
Standard PENG-ROB
Z 
V
V b

aV
RT [V (V  b )  b (V  b )]
 Where
a

i
xi x j ( a i a j )
0 .5
(1  k ij ) ,
b
 xb
j
i
i
2
a i   i 0 . 45724
 i (T )  [1  m i (1  T ri )] ,
0 .5
i
2
2
R Tci
Pci
,
bi  0 . 07780
RT ci
Pci
m i  0 . 37464  1 . 54226  i  0 . 26992  i
2
5
VLE (EOS Method)
Advantages and Disadvantages
 Equations of state can be used over wide ranges of
temperature and pressure, including subcritical and
supercritical regions.
 Thermodynamic properties for both the vapor and liquid
phases can be computed with a minimum amount of
component data.
 For the best representation of non-ideal systems, you must
obtain binary interaction parameters from regression of
experimental VLE data. Binary parameters for many
component pairs are available in the Aspen databanks.
6
VLE (EOS Method)
Advantages and Disadvantages…
 Equations
of state are suitable for modeling
hydrocarbon systems with light gases such as CO2 , N2
and H2 S .
 The assumptions in the simpler equations of state
(SRK, PR, Lee-Kesler , … ) are not capable of
representing highly non-ideal chemical systems, such
as alcohol-water systems. Use the activity-coefficient
options sets for these systems at low pressures. At high
pressures, use the predictive equations of state.
7
VLE (Activity Coefficient Method)
fi  fi
v
 At Equilibrium:
f i   i y i Pt
v
 Where
 Therefore
v
k
vl
i

yi
f i   i xi f i
l
,

l
 i fi
*, l
*, l
 Pt
v
i
xi
 F0r ideal gas and liquid
 i  1,   1  k
v
i
vl
i

yi
xi

Pi
*
Pt
Raoult ' s Law
8
VLE (Activity Coefficient Method)
Liquid Phase Reference Fugacity
 For solvents: The reference state for a solvent is defined as
pure component in the liquid state, at the temperature and
pressure of the system.
fi
*, l

*, v
i
(T , Pi ) Pi q
*, l
*, l
*, l
i
,
(  i  1 as x i  1)
 i*,v = Fugacity coefficient of pure component i at the system
temperature and vapor pressures, as calculated from the vapor phase
equation of state
 qi*,l = Poynting factor
q
*, l
i
 1
 exp 
 RT

*, l
V
dP

Pi*,l i

P
9
VLE (Activity Coefficient Method)
Liquid Phase Reference Fugacity
 For dissolved gases: Light gases (such as O2 and N2 ) are
usually supercritical at the temperature and pressure of the
solution. In that case pure component vapor pressure is
meaningless and therefore it cannot serve as the reference
fugacity.
f i  xi  i H i
l
*
and
 i  1 as x i  0
*
 Using an Empirical Correlation: The reference state fugacity
is calculated using an empirical correlation. Examples are the
Chao-Seader or the Grayson-Streed model.
10
VLE (Activity Coefficient Method)
Multi-component Mixtures
 Multicomponent vapor-liquid equilibria are calculated from
binary parameters. These parameters are usually fitted to binary
phase equilibrium data (and not multicomponent data) and
represent therefore binary information. The prediction of
multicomponent phase behavior from binary information is
generally good.
 Multi-component liquid-liquid equilibria cannot be reliably
predicted from binary interaction parameters fitted to binary
data only. In general, regression of binary parameters from
multi-component data will be necessary.
11
VLE (Activity Coefficient Method)
NRTL (Non-Random Two-Liquid)
 The NRTL model calculates liquid activity coefficients for the
following property methods: NRTL, NRTL-2, NRTL-HOC,
NRTL-NTH, and NRTL-RK. It is recommended for highly
nonideal chemical systems, and can be used for VLE, LLE and
VLLE applications.
x
j
ln  i 
ji
G ji

j
x
k
k
G ki

j





x j G ij
x
k
k
G kj


  ij 


x
m
 mj G mj  
m
x
k
k
G kj




12
VLE (Activity Coefficient Method)
NRTL (Non-Random Two-Liquid)
x
j
ln  i 
G ji

j
x
k
 Where
ji
k
G ki

j





G ij  exp(   ij ij )
x j G ij
x
k
,
k
G kj


  ij 


x
m
 mj G mj  
m
x
k
G kj
k




G ii  1
 ij  a ij  bij T  e ij ln T  f ij T
,
 ii  0
 ij  c ij  d ij (T  273 . 15 )
 The binary parameters aij, bij, cij, dij, eij and fij can be determined
from VLE and/or LLE data regression. The Aspen Physical Property
System has a large number of built-in binary parameters for the
NRTL model.
13
VLE (Activity Coefficient Method)
Advantages and Disadvantages
 The activity coefficient method is the best way to represent
highly non-ideal liquid mixtures at low pressures.
 You must estimate or obtain binary parameters from
experimental data, such as phase equilibrium data.
 Binary parameters are valid only over the temperature and
pressure ranges of the data.
 The activity coefficient approach should be used only at low
pressures (below 10 atm).
 The Wilson model cannot describe liquid-liquid separation at
all; UNIQUAC, UNIFAC and NRTL are suitable.
14
Principle Steps in Selecting the Appropriate
Method for VLE
1. Choosing the most suitable method based on the
nature of chemical species.
2. Comparing the obtained predictions with data from
the literature.
3. Estimate or obtain binary
experimental data if necessary.
parameters
from
4. Generation of lab data if necessary to check the
property model.
15
VLE
Bubble Point and Dew Point
PTxy diagram for VLE for a two
component mixture is as shown.
If one starts with a liquid at F
(subcooled liquid) and reduces the
pressure at constant temperature
and composition along vertical line
FG, the first bubble of vapor
appears at point L, which lies on
the upper surface. Thus, L is a
bubblepoint (saturated liquid),
and the upper surface is the
bubblepoint surface.
16
VLE
Bubble Point and Dew Point
 The state of the vapor bubble in
equilibrium with the liquid at L
must be represented by a point on
the under surface at the
temperature and pressure of L. This
point is indicated by V. Line VL is
an example of a tie line, which
connects points representing phases
in equilibrium (saturated liquid to
saturated vapor).
17
VLE
Bubble Point and Dew Point
 As the pressure is further
reduced along line FG, more and
more liquid vaporizes until at W
the process is complete. Since W is
the point at which the last drops of
liquid (dew) disappear, it is a
dewpoint (saturated vapor), and
the lower surface is the dewpoint
surface. Continued reduction of
pressure merely leads into the
superheated vapor region.
18
VLE
Bubble Point and Dew Point
Pxy diagram
Txy diagram
19
VLE
Ideal Model
At equilibrium:
_ V
f
j
_ L
 f
j
y P  xf
V
j
j
j
j
yP  xP
1
1
P  xP  xP
s
1
1
2
2
s
s
2
s
2
s
1
s
1
2
 x P  (1  x ) P
 P  (P  P )x
s
j
1
yP  xP
2
L
1
1
s
2
s
2
1
At fixed temperature
20
VLE
Ideal Model
Example – Phase diagrams
for benzene-toluene
mixture at 90 oC
21
VLE
Homogeneous Azeotropes
For non-ideal mixtures, the
activity coefficients are
different from unity:
yP  xP
1
1
1
1
yP  x P
2
2
2
s
1
1
1
2
S
2
P  x  P  (1  x )  P
1
S
s
2
If   1 the mixture may have a minimum-boiling azeotrope
i
Example – Phase diagrams for Isopropyl ether-Isopropyl alcohol
22
VLE
Homogeneous Azeotropes
For non-ideal mixtures, the
activity coefficients are
different from unity:
yP  xP
1
1
1
yP  x P
2
2
2
s
1
1
1
2
S
2
P  x  P  (1  x )  P
1
S
1
s
2
If   1 the mixture may have a maximum-boiling azeotrope
i
Example – Phase diagrams for Acetone-Chloroform
23
VLE
Heterogeneous Azeotropes
For a minimum-boiling azeotrope with large deviation from
Raoult’s law ( i  1, usually  7 ), phase splitting may occur
and a minimum-boiling heterogeneous azeotrope forms,
having a vapor phase in equilibrium with two liquid phases.
Homogeneous Azeotrope
Heterogeneous Azeotrope
24
VLE
Dew and Bubble Point Calculations
At Equilibrium:
Where:
yi 
x i  i Pi
sat
v
(I)
 P
v
i
y i i P
or
xi 
 i Pi
sat
( II )
 i   (T , P , y 1 , y 2 ,  , y N 1 )
v
 i   (T , x 1 , x 2 ,  x N 1 )
Pi
Since 
yi  1
sat

Bi 


 exp  A i 
T  C i 

and  x i
i
1
Antoine
Eq.
, we also have:
i
P 

i
x i  i Pi

v
i
sat
(III)
and
1
P 

i
y i
i
Pi
v
i
(IV)
sat
25
VLE
BUBL P: Calculate {yi} and P, given {xi} and T
26
VLE
DEW P: Calculate {xi} and P, given {yi} and T
xi 
xi
x
i
i
27
VLE
BUBL T: Calculate {yi} and T, given {xi} and P
P 

x i  i Pi

i
v
i
sat
 Pj
sat
P


i

i
P
sat
j
Saturation
 Pj
sat
y i
i
i Pj
v
sat
y i i P j
v
1
P 
x i  i Pi
Pi
v
i
 P
i
( V)
sat
 i Pi
sat
( VI)
sat
sat

Bj 
Bj


 exp A j 
 T 
C
sat


T Cj 
A j  ln P j

tempertau re of pure species : T i
sat

Bi
A i  ln P
j
( VII )
 Ci
( VIII )
28
VLE
BUBL T: Calculate {yi} and T, given {xi} and P
29
VLE
DEW T: Calculate {xi} and T, given {yi} and P
30
VLE
Flash Calculation (for known T and P)
Consider an equilibrium system containing one mole of nonreacting chemical
species with overall composition of {zi}. Let L and V be the moles of liquid and
vapor with compositions {xi} and {yi}, respectively. The material-balance are:
L V 1
eliminate
L
from VLE : K i 

i
and
yi
xi
z i  x i L  y i V ( i  1, 2 ,  , N )
z i  x i (1  V )  y i V ( i  1, 2 ,  , N )

zi K i

 y i  1  V ( K  1)

i

zi
x 
 i 1  V ( K i  1)
( yi  xi )  0  F 
( i  1, 2 ,  , N )
z i ( K i  1)
 1  V (K
i
i
 1)
 0
( IX )
(X )
31
VLE
Flash Calculation (for known T and P)
Initial guess:
V 
Pbubble  P
Pbubble  Pdew
 i   i , dew
 i , bubble   i , dew
 i   i , dew
 i , bubble   i , dew


P  Pdew
Pbubble  Pdew
P  Pdew
Pbubble  Pdew
32

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