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Chapter 14. Topics in Phase EquilibriumChapter 14. Topics in Phase EquilibriumChapter 14. Topics in Phase EquilibriumChapter 14. Topics in Phase Equilibrium
The simplest models for vapor/liquid equilibrium
14.1. The14.1. The14.1. The14.1. The //// formulation of VLE.formulation of VLE.formulation of VLE.formulation of VLE.
Pyf iiv
i =
iii
l
i fxf =
l
i
v
i ff =
iiiii fxPy =
Extension of modified Raoults Law
Overcome the limitation of modified Raoults Law
Introduce vapor-phase fugacity to explain nonideality
From eqn (11.52)
at equilibrium
- Raoults law
- Henrys law
- Modified Raoults law
sat
iii PxPy =
iii HxPy =
sat
iiii PxPy =
=> for a species whose Tc Applied only to low pressure
From eqn (11.90)
=> Vapor phase
=> Liquid phase
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1sat
i
i
i
Usually poynting factor (14.2)
]RT
)P-P(Vexp[Pf
sat
i
l
isat
i
sat
ii =
From eqn (11.44)
satiii
sat
i
l
isat
i
ii PxP]RT
)P-P(V-exp[
y =
1 ii ==
]RT
)P-P(Vexp[PxPy
sat
i
l
isat
i
sat
iiiii =
(14.1) => gamma/phi formulation
If => (14.1) becomes raoults Law
sat
iiii PxPy =
1 i =If => (14.1) becomes modified Raoults Law
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To use eqn(14.1) for the analysis of VLE => need to know the value ofii
sat
i ,,P
i
i
i
sat
iCT
B-APln
+
=
)]-2(yy2
1B[
RT
Pexp jk
j k
jikiii +=
sat
i 0, jkji =
RT
PBexp
sat
iiisat
i
=
2) => need to obtain and
(14.3)
(14.4)
(14.5)
1) => obtained by the Antoine eqnsat
i
ii
sat
i
Values of the pure species vivial coeff. of the species in the mixture
(11.69) ~ (11.74)
: fugacity coeff for pure i as a saturated vapor in eqn(14.4)
RT
)-2(yyP2
1)P-P(B
exp
j k
jkjikj
sat
iii
sat
i
i
i
+
==(14.6)
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3) : can be obtained from the various models ( Van Laar, Wilson, UNIFAC, etc ..)
RT
Py)P-P(Bexp
12
2
2
sat
111
1
+
=
RT
Py)P-P(B
exp 12
2
1
sat
222
2
+
= (14.7b)
(14.7a)
i
For a binary system comprised of species 1 and 2, this becomes
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Dew point and Bubble point Calculation using Gamma/Phi formulatDew point and Bubble point Calculation using Gamma/Phi formulatDew point and Bubble point Calculation using Gamma/Phi formulatDew point and Bubble point Calculation using Gamma/Phi formulations => need iterationions => need iterationions => need iterationions => need iteration
)y,,y,y,P,T( 1-N21i
=
)x,,x,x,T( 1-N21i =
)T(fPsati =
P
Pxy
i
sat
iii
i = sat
ii
ii
iP
Pyx =(14.8) (14.9)
since 1y,1x ii ==
i i
sat
iii
PxP =
i
sat
ii
ii
P
y
1P =(14.10) (14.11)
1) BUBL P calculation
=> Calculation (yi) and P , given xi and T
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Bubble P calculationsBubble P calculationsBubble P calculationsBubble P calculations
Fig. 14.1 Block diagram for the calculation BUBL P
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2) Dew P calculation
=> Calculation xi and P , given yi and T
- BUBL P and Dew P calculation : temperature is given => can calculatesat
iP
i
i
sat
i C-lnP-A
BT =
sat
iP
- BUBL T and Dew T : T is unknown => need initial estimate for T to do iteration
i) (for BUBL T) (for Dew T)i
sat
iiTxT = i
sat
iiTyT =
where (14.12)
ii) Multiply eqn(14.10) (14.11) by (outside summation) and didice by
(inside summation)
sat
iP
i
sat
j
sat
iiii
sat
j)P/P)(/x(
PP = )
P
P(
yPP sat
i
sat
j
i i
iisat
j = (14.14)(14.13)
iii) Calculate corresponding T from eqn(14.3)
jsat
jj
jC-
lnP-A
BT =
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Dew P calculationsDew P calculationsDew P calculationsDew P calculations
Fig. 14.2 Block diagram for the calculation DEW P.
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BUBL T calculationBUBL T calculationBUBL T calculationBUBL T calculation
=> Calculate (yi, T) given xi and P
Fig. 14.3 Block diagram for the calculation BULB T.
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DEW T calculationDEW T calculationDEW T calculationDEW T calculation
=> Calculate (xi , T) given yi and P
Fig. 14.4 Block diagram for the calculation DEW T.
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Flash calculationFlash calculationFlash calculationFlash calculation
)N,,2,1i(1)-K(1
Kzy
i
ii
i =
+
=
In chap 10. Flash calculation is based on Raoults law and K-value correlation
=> Use the gamma/phi formulation
)N,,2,1i(1)-K(1
zx
i
ii
=
+
=
(10.16)
First,yii F01-y,1y ===
i i
ii
y 01-1)-K(1KzF =
+
=
i i
i
x 01-1)-K(1
zF =
+
=
(14.17)
(14.18)
Second,xii F01-x,1x ===
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14.2 VLE from Cubic Equation of State14.2 VLE from Cubic Equation of State14.2 VLE from Cubic Equation of State14.2 VLE from Cubic Equation of State
For VLE, li
v
i ff =
,Px
f
i
i
i = l
ii
v
ii
l
ii
v
ii xyPxPy ==
(11.48)
Since,
Vapor pressure for a pure speciesVapor pressure for a pure speciesVapor pressure for a pure speciesVapor pressure for a pure species
- Usually , can be obtained by experimental measurement
Generic Cubic Equation of state
sat
iP
- at equilibrium can be obtained using EOSsat
iP
b)b)(vv(
)T(a
b-v
RTP
++
=
If P -> , V -> b , if P -> 0, V ->
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Fig. 14-7
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0ln-lnlnln vil
i
l
i
v
i ==
)P,T( ii =
satPP
)T(fPsat
i =
in that case
- For pure species,
and
- For a saturated liquid or vapor )P,T( sat
iii =
P
ZRT
v=
)Z()-Z(
-Zq-1Z
iiii
ii
iiii++
+=
where
- Two widely used cubic eqn of state
Sorve / Redlich / kwung (SPK) eqn
Peng/Robinson (PR) eqn
- Vapor & vapor-like Roots of the Generic Cubic EOS (use )
ii
ii
iiiiii q
Z-1)Z)(Z(Z
+
+++=
(3.52) -> (14.31)
(3.56) -> (14.35)
Liquid & liquid like Roots of the Generic Cubic EOS
Parameter : eqn (3.45), (3.46), (3.50), (3.51)
)q,,b,a( iiii
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Pure number : )Tr(,,,, i
v
i
l
i Zd-Zv
i
l
i ln,ln
iiiiii Iq-)-ln(Z-1-Z=lnFrom (11.37) (11.37)
=> Table 3.1
- Procedure to obtain for pure for a give T, PsatiP
1) Calculate using eqn (14.35)(14.36) before that, calculate parameters
2) Using eqn(11.37), calculate before that, calculate qi, Ii
3) Check if 0=ln-ln v
i
l
i
If yes, P -> Psat
, if no, iterationEqn for calculating vapor pressure -> table 14.3
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14.3 Equilibrium and Stability14.3 Equilibrium and Stability14.3 Equilibrium and Stability14.3 Equilibrium and Stability
If system surrounding => thermal and mechemical equilibrium heat exchange(Q) and expansion work (w) are reversible.
T
dQ-
=T
dQ
=dSsurr
surr
surr
0T
dQ-dS t
0dS+dS surrt
tTdSdQ
tttt PdV+dU=dQPdV-dQ=dW+dQ=dU
0TdS-PdV+dUTdSPdV+dU tttttt
By thermodynamic 2nd law ,
By thermodynamic 1st law
Under these circumstance
dQ : heat transfer for systemT : system =
surrT
(14.65)
(14.66)
Inequality : apply to every incremental change of the system between
nonequilibrium states
equality : holds for changes between equilibrium states (reversible)
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eqn(14.61) : too general to apply to practical problem
=> need more restricted version
0)dS(or0)dU( ttttV,U
t
V,S
t
0)TS-PV+U(d0TdS-PdV+dUP,T
tttttt
tttttt TS-PV+U=TS-H=G
0)G(d P,Tt
At const, T, P
(14.67)
Determination of equilibrium states
1) Express Gt as a function of the numbers of moles of the species in the
several phases
2) Finds the set of values for mole number that minimize Gt subject to the
constraints of mass conversion
At equilibrium 0=)G(d P,Tt (14.68)
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At constant T, P, must be continuous function of x1, and the'''
G,G,G
0>G ''
=> Criterion of stability for single-phase
0>dx
)RT/G(d2
1
2
(14.65)(const. T, P)and
0>dx
Gd 2
1
2
(const. T, P)
From eqn(12.30)
0>dx
)RT/G(d+
xx
1=
dx
)RT/G(d2
1
E2
211
2
iiE xlnxRT-G=G
RTG+xlnx+xlnx=
RTG
E
2211
1
E
21
1 dx)RT/G(d+xln-xln=
dx)RT/G(d
)P,Tconst(xx
1->
dx
)RT/G(d
21
2
1
E2
(14.65)
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2211
E
lnx+lnx=RT
G
1
2
2
1
1
121
1
E
dx
lndx+
dx
lndx+ln-ln=
dx
)RT/G(d
21 ln-ln=
zero Gibbs-Duhem eq.
1
2
1
1
2
1
E2
dx
lnd-
dx
lnd=
dx
)RT/G(d
1
1
2 dx
lnd
x
1=
From eqn(12.6)
In combination with eqn(14.70)
11
1
x
1->
dx
lnd(const T, P)
)P,Tconst(0>dx
d,0>
dx
fd
1
1
1
1
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