Chapter 14: Phase Equilibria Applications

Part II

If the component is supercritical, then the vapor pressure is not defined

111

101

ˆlim

1f

x

fH x

P

Pxy

P

HxyPyHx

Hxf

sat

l

2

2222

1

11

11

11111

11

11

111

ˆˆ

ˆ

2

example• A binary methane (1) and a light oil (2) at 200K and 30 bar

consists of a vapor phase containing 95% methane and a liquid phase containing oil and dissolved methane. The fugacity of methane is given by Henry’s law, and at 200 K, H1 = 200 bar. Estimate the equilibrium mole fraction of methane in the liquid phase. The second virial coefficient of methane at 200K is -105 cm3/mol

3

solution

:

ˆˆ

ˆ

111

111

sAssumption

Pyf

Hxfv

l

4

Solution (cont)

PyPyf

Hxfv

l

11111

111

ˆˆ

ˆ

Need equation for the fugacity coefficient vapor phase

5

Solution (cont)

827.0)/exp( 111 RTPB

How do we solve for the mole fraction of the solutein the liquid phase?

x1 =0.118

6

Another example

• For chloroform(1)/ethanol(2) at 55oC, the excess Gibbs energy is

2121 )59.042.1( xxxxRT

G E

• The vapor pressures of chloroform and ethanol at 55oC are P1

sat = 82.37 kPa, and P2sat = 37.31 kPa

• Make BUBLP calculations, knowing that B11=-963 cm3/mol, B22 =-1,523 cm3/mol, B12 =52 cm3/mol

7

)1)((2()(exp

)(2()1(exp

12112212

12

11221122

11

xAAAx

xAAAx

satsat2

22

1

11

ˆ;

ˆ

Need equations for the fugacity coefficients

8

2

satsat

and calculate

BBB

RT

PyPB

RT

PB

1

22111212

122211

1

1111

2

expˆ

exp

But we don’t know P

Guess P (avg sat. pressures)

9

12

21222122

11112111

1

)(),,,(

)(),,,(

yy

PxxPyyTPy

PxxPyyTPysat

sat

For example, at x1 =0.25, solve for y1, y2, P

y1 = 0.558y2 = 0.442P = 63.757 kPa

In our web site, there are model spreadsheets that you can download 10

VLE from cubic EOS

PxPy

NDESCRIPTIO EALTERNATIV

...N 2, 1, i ff

lii

vii

li

vi

ˆˆ

,ˆˆ

11

12

Vapor pressure pure species

Liquid branch

Vapor branch

LV transition

li

vi

Using equation (1), the cubic EOS yields Pisat= f(T)

(1)

13

14

Compressibility factors

ii

lii

ilii

lii

li q

ZZZZ

1

))((1

ci

cirii

i

ii

ci

cii

ii

P

TRTTa

RTb

Taq

P

RTb

RT

Pb

22)()(

)(

For the vapor phase there is anotherexpression, (14.36)

15

16

17

How to calculate vapor pressure from a cubic EOS

ili

ilil

i

iiiiii

Z

ZI

where

IqZZ

ln1

)ln(1ln

We solve for the saturation pressure at a given T such that the fugacity coefficients are equal in the two phases: 2 compressibility eqns., two fugacity coefficient eqns., equality of fugacity coefficients, 5 unknowns: Psat, Zl, Zv, l, v

18

Mixture VLE from a cubic EOS

• Equations for Zl and Zv have the same form as for pure components

• However the parameters a & b are functions of composition

• The two phases have different compositions, therefore we could think in terms of two P-V isotherms, one for each composition

19

20

Mixing rules for parameters

2/1jijiij

iji j

ji

ii

i

aaaa

axxa

bxb

21

Also we need “partial” parameters

j

j

j

nTii

nTii

nTii

n

nqq

n

nbb

n

naa

,

,

,

)(

)(

)(

22

The partial parameters are used for the calculation of fugacity

coefficients• Because i is related to the partial molar

property of GR (residual G)

b

b

a

aqq

I

IqZZb

b

iii

ii

i

1

different) are (equations or Z Zis Z

17) (slide before as defined

)ln()1(ˆln

vl

23

example

• Vapor mixture of N2(1) and CH4 (2) at 200K and 30 bar contains 40 mol% N2. Calculate the fugacity coefficients of nitrogen and methane using the RK equation of state.

vvvv

vvvvvv

ZZ

ZqZ

1

For RK, =0 and =1

24

molcmP

Tb

molcm barP

TTa

b

b

a

aayayqq

aaayaya

ayaayyaya

ci

cii

6

ci

cirii

/14.83

08664.0

/)14.83(

42748.0

22

22

2

3

2222/1

1212111

212111

22221211

21

25

Calculate P-x-y diagram at 100 oF for methane(1)-n butane (2) using SRK and mixing rules (14.42) to

(14.44)

Compare with published experimental data (P, x, y)

Initial values for P and yi can be taken from given experimental data

First read critical constants, , from Table B.1 and from Table 3.1

22/12 )1)(176.0574.148.0(1 rSRK T

Calculate b1, b2, a1, a2

In this case T > Tc1 26

27

K value given by

vi

li

iKˆ

ˆ

Step 1

28

The equations for are valid only up to the critical temperature; however is OK toextend the correlation slightly above the critical temperature

Lets calculate the mixture parameters (for step 1). When applied to the liquid phase we use the xi mole fractions

RTb

aq

b

b

a

aaxaxqq

b

b

a

aaxaxqq

bxbxb

axaaxxaxa

l

ll

llll

llll

l

l

2211222

1212111

2211

22221211

21

22

22

2

29

Follow diagram Fig. 14.9

Assume P and yi

Calculate Zl and Zv, and the mixture fugacity coefficients

Calculate K1 and K2 and the Kixi

Calculate normalized yi=Kixi/ Kixi

Reevaluate fugacity coefficients vapor phase, etc

If Kixi > 1, P is too low so increase P; if Kixi < 1, then reduce P

30

Results:Rms % difference between calculated and exp. P is3.9%

Rms deviation between calculated and exp. y1 is0.013

Note that the system consistsof two similar molecules

Where are the largestdiscrepancies with the experimental data?

31