chaoter 14 Phase Equilibrium

8/12/2019 chaoter 14 Phase Equilibrium

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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

chaoter 14 Phase Equilibrium

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