Calculation and prediction of fluid phase equilibria from an equation of state

Fluid Phase Equilibria, 10 (1983)173-182 ElsevierScientific PublishingCompany,Amsterdam -Printedin The Netherlands

CALCULATION EQUATION

Ulrich

AND PREDICTION

OF FLUID PHASE EQUILIBRIA

173

FROM AN

OF STATE

K. DEITERS

Department Federal

of Chemistry,

Republic

University

of Bochum,

4630 Bochum,

of Germany

ABSTRACT Liquids or compressed gases consisting of light molecules show deviations from classical mechanics, which are caused by the discontinuity of energy levels. From the assumption that each molecule is confined to a cell with a size depending on the free volume, a quantum correction is derived which extends any van der Waals type equation of state to quantum gases. The correction is applied to a semiempirical equation of state developed by the author. The extended equation yields reasonable critical compressjbility factors and gives a better representation of PVT data than the uncorrected equation. Furthermore high pressure phase equilibria in mixtures containing helium and hydrogen have been calculated. Again the agreement with experimental data is improved; the adjustable binary interaction parameters have values close to the BerthelotLorentz rulesand are less temperature dependent.

INTRODUCTION With the rapid development recent

years

rium data of fluids

and fluid mixtures.

tions of high pressure methods

for phase

even substitution portance.

informations

of experimental

methods

equilib-

investiga-

calculation

the extrapolation

data, are of great

calculation

(e.g. densities,

expensive,

permit

in the

for phase

As experimental

are rather which

technology

demand

also provide

heats of mixing),

which

or

practical

im-

additional

are difficult

experimentally.

Nowadays

calculation

dered reliable "Simple"

systems

equilibria,

Furthermore,

to obtain

of high pressure

there has been an increasing

for mixtures

in this context

electrostatic have always this

are

size

ratios

forces. caused

first

methods

of so-called

means

However,

some trouble

the

so-called

observed

are avai.Lable which may be consi-

small,

in computations.

of

effects, helium

substances. 03783812/83/$03.00

0

and without

strong

the very lsimplest of all molecules

quantum

in mixtures

"simple" molecules.

rigid,

1983Elsevier Science Publishers B.V.

or

and

The reasons second

hydrogen

the with

for large heavier

174 But especially dustrial medium,

such mixtures

applications: and although

helium

compressed

fluid, its extractive cesses;

furthermore

coal gasification QUANTUM

powers

hydrogen

influence

of symmetry

to Fermi-Dirac

natural

contain

effects

properties

or Bose-Einstein

the discontinuity

of energy

here. The discontinuity

pro-

gas from

hydrogen.

can be distinguished,

fluid properties Let us assume by the repulsive

statistics,

molecules particle

leads

and the influence effect

of

is impor-

however,

exists whenever

and therefore

the mo-

contributes

to

at high densities. that a molecule potentials

in a fluid is confined

of its neighbours.

and the collisions

are hard, it is permissible

h*

to a cell

The energy eigen-

are no longer continuous.

is homogeneous

in a cubical =P

the

which

only and shall not be considered

effect,

of such a molecule

ground potential

namely

of the wave functions,

tions of molecules are restricted,

Fig.

gas and product

levels. The symmetry

tant at very low temperatures

'IJK

is not a good extraction

CORRECTIONS

Two kinds of quantum

values

for in-

heat exchanger

are i.mportant for hydrogenation

synthetic

usually

are of great importance

is an interesting

If the backwith neighbour

to use the eiqenvalues

(I~+J~+K~)

(I)

8mkTL2

1. Definition

of a

box of length L:

of the cell size according

to eqn.

(2)

175 The cell length accounts

is related

to the free volume

for cell overlapping,

Vf_ The factor

as shown in fig.

8

1 (Prigogine,

1957): L3

vf

=ay

The partition

(2) function

of a molecule

= Lc exp(- $

qL = CZ exp(- s)13

J

with y = { and A = The partition

JL’2nkT

function

1960) written

Here Q* denotes degrees

partition

=

Qquant

of a classical

the contribution

is constructed

potential

to (4) the quantum

and

mechanical

from the single particle

&C;

qLjNQ*

= Qclass[~

Z exptj

the factor : accounts cells

from

in which

parti-

cally.

It is possible,

expand

its logarithm

in

(5)

may be placed.

Eqn.

(4)

y=O. (5) cannot

to compute

however,

into a power

3N

$ j2y2)l

for the fact that there are

a molecule

(5) by setting

The term in square brackets

series

be evaluated

it numerically

analytiand to

in y:

z exp(- $ j2y2)] = x riyl i

j A table of expansion Eqn.

(e.g.

(3):

distinct

[y

real gas is often

of the attractive

In contrast

as many

In

length).

(4)

In this equation

can be derived

wave

Q”

of freedom.

function

tion function

(thermal de Broglie

as

Q class = j@JN

inner

(3)

j2y2)13

;

j

Hill,

in such a cell is then

ri is available

coefficients

(5) in combination

thermodynamic

properties,

state exists,

which

with

(2) and

provided

(6) permits

from the author. the calculation

that a classical

can be separated

equation

into a repulsion

of

of

and an attrac-

tion part: P=P

rep + 'att

with

P

= NkT(-%& rep

In this case the final expressions sure and molar

Helmholtz

energy

for the quantum

are:

corrected

pres-

176

A

= -kT In Qquant

quant

=A

- 3RT

class

(8)

X r$

i

aA P

PURE

-( quant

P

=

z iriy= rep i

-P

class

(9)

SUBSTANCES

Equation

of state.

The quantum

applied

to any equation

quantum

corrections

derived

to the van der Waals In order

equation

mechanical

of state published region

p[1+cc

P = F

with

the practical

the quantum

the critical

namely

45-

29

Cl-

5)

0

xl

- F

constants

of state has three

of density elsewhere

is not free of quantum

rections

values

(Deiters,

1981, 1982):

from

b, and the number in

(IO) are

and their mean-

1981). argon data, but

the classical

by substracting

gases

are characterized

compressibility

tab. 1, the equation

of state

(9) gives

reasonable

determined

critical

factor.

equation

the quantum

of cor-

agreement

compressibility

compressibilit

by unusually

with

factors.

factors

of several

large

As can be seen from

(IO) in connection

1

Critical

parameters,

(10).

Quantum

of the critical

in

(10)

I] I1

specific

to experimental

effects,

correction

TABLE

(Deiters,

and temperature,

(9) is obtained

for argon

Application.

to work well

c. The other entities

(IO) has been adjusted

by

is known

a

substance

of freedom

ing has been explained

state required

of this approach

[exp(&)-

;I;=cT

1980).

with a semiempirical

well depth a, the covolume

degrees

or functions

As eqn.

which

p2 *

and

have been applied

usefulness

and at high pressures

the attractive

of additional

manner

can be

(7); indeed

eqn.

(Hooper and Nordholm,

corrections

earlier,

p = k Vm '

This equation

argon

of state

deri.ved above

fulfills

in a similar

equation

to demonstrate

we combine

corrections

of state which

with

the quantum

the experimentally

177 The deviation critical neglected

for helium

temperature

is probably

of helium

due to the fact that at the

(5.2 K) symmetry

effects

cannot

be

any longer. 6

3.

. I

-

NAY3

- - -

COlC CbSSICOI

meth

- -.-

coic

mech

quantum

1,

1

).

O-

20

Fig.

2. Density

Furthermore

100

30

of normal

hydrogen

the quantum

of PVT data. Fig.

2 shows

200 300

as a function

correction several

and 1000 K. For both calculated

T/K

improves

isobars

of temperature.

the representation

of hydrogen

lines the parameters

tion of state have been fitted not to the experimental to the critical portance

pressure

of quantum

ties becomes

and temperature.

effects

between

isobars,

From the diagram

at low temperatures

20

of the equabut

the im-

or at high densi-

evident.

MIXTURES Thermodynamics. fluid mixtures, of phase

If we want to calculate

we have to consider

the phase behaviour

the thermodynamic

of

conditions

equilibrium:

p' = P"

(11)

T' = T"

(12)

178

pi;

ll;

=

i = I,2

The chemical potentials Helmholtz

(13) pi can be 'obtained as derivatives

of the

energy:

'i = ($)V,T 1 The Helmholtz

(14)

n ' j+i

energy

is given by the following expression:

A = n,At(Vm+,T)+n2Al( Vf m, T) + RT(n,ln xl + n2ln x2) V - i P(V,T,n;a,b,c,m) V+

dV

(15)

The Af denote the molar Helmholtz

energies

of the pure substances

in the perfect gas state at the temperature T and the very large + volume V The inteqrand is the equation of state of the mixture, m' which depends on a set of variables (volume, temperature, mole number) and a set of parameters, (9) resp.

and for which we shall substitute

(10). The derivation

of (15) has been published

(Deiters 1979, van Konynenburg, Mixing

rules.

As a mixture

state as a pure substance

is described

volume,

as averages

by the same equation

of

(one-fluid theory), it is characterized

by a set of concentration-dependent obtained

elsewhere

11368).

effective

parameters,

of the pure substance parameters.

which are

For the co-

the number of degrees of freedom, and the molar mass, which

depend little on configuration,

the following mixing

rules are pro-

posed: 2 b = x,b,

2 + 2XlX2b12 +- X2b22

(16)

c = xIcl

+ x2c22

(17)

,-0.5

-0.5 -0.5 = Xlmll f X2m22

The average attractive configuration.

(18)

energy, however,

In order to estimate

meter a we assume that each molecule sites. Each nearest S. li

neighbour

account

depends on

of species i has zsii contact

molecule

sites, so that the coordination

different

certainly

the average value of the para-

of the same species occupies z; neighbours

of

kind occupy sij contact sites. This way it is possible

to

for the influence

number remains

of the size of molecules

internal energy. For the contact sites a regular assumed and solved by the quasichemical

method.

on the average

solution model is Details of the de-

179

rivation

have been published

elsewhere

(Deiters,

1982). The result

is: x2az2

a = xlall +

2xlsllq2~a

+

(19)

l+Jl+

Here the qi denote contact exchange

4q,q2rexp(-

+$)-II!

site fractions,

and :\a is the contact

energy: x.

qi =

1

s.

ri

(20)

xlsll +x2=22 2a12 *a=----s12 For mixtures

a11

a22

sll

s22

of spherical

from the covolume of nearest

(21)

molecules

the s. can be calculated rj considerations of the number

ratio. Geometrical

neighbours

(Deiters,

1982) lead to the following

ex-

pressions: (22) 12s s12 = sll

with

+ s22

llS22

+ q

(23)

98s,ls22q

Zij = 2 + lOexpl<

0.828 - $828): (Rij

(24)

(25)

The last step in the thermodynamic ture is the determination state. Usually

description

of the parameters

it is sufficient

to calculate

of a binary mix-

of the equation

of

the pure substance

parameters

a. b. and cii from critical data. The binary interIi' ri' action parameters al2 and b,2 are fitted to binary phase equili-

brium data. The phase equilibrium solving eqn.

The figures

of the helium/nitrogen lated phase boundaries.

3 and 4 show recent experrmental

system

The binary

too. The deviations

of state, but without

interaction

parameters

have been

but apply equally well to the other from the Berthelot-Lorentz

rules is only 1 per cent, whereas equation

data

(Hiibner, 1982) together with calcu-

to the 100 K isotherm,

isotherms,

by numerically

(13).

Application.

adjusted

is calculated

as calculation

quantum

combining

with the same

corrections,

requires

an ad-

180 justment dependent

of 73 per cent in al2 in the classical

this mixing

(in addition,

calculation).

system is that the calculated

phases

agree with the experimental

mental

error.

Fig. 5 shows several system

have been extrapolated. ved, the agreement Omitting shifts

the computed

isotherms

with the experimental

parameters

only. The other

isotherms

In spite of the very high pressure

correction

bubble

about

the experi-

the adjustable

isotherm

is quite good except

the quantum

feature

of coexisting

of the hydrogen/methane

1980). Again

have been fitted to the middle

densities

data almost within

P-x-y isotherms

(Tsang and Streett,

a,2 is temperature

A remarkable

for the critical

in the calculations

invol-

region.

for this system

to the left and spoils the agreement point

line.

P

MPa

Fig. 3. P-x-y diagram of the system helium/nitrogen +, o, 0: exp. data -

02

UL

06 xtie

08

The same holds for the hydrogen/carbon

monoxide

is shown in fig. 6 (exp. data by Tsang and Streett, the adjustable only.

parameters

talc.

system, which 1981). Again

have been fitted to the middle

isotherm

.

He/N,

/’

120K / "

181

Fig. 4. Densities Of coexisting phases in helium/nitrogen mixtures (symbols as in fiig. 3)

I

/' 1OOK

/

9

1

I

0

0.2

0.4

06

08

1

xHe

P

MPc

P

80

dP0 50

60

LO

30

.

20 I.

'O I

0

02

0.L

06 xH..

Fig. 5. P-x-y diagram system H2/CH4

08

of the

+, 0, a: exp. data; -,

1

0

02

0.6

0.L

0.8

xH2

Fig. 6. P-X-Y diagram system H2/CO --- calculated

of the

182

CONCLUSION The inclusion

of quantum

corrections

state not only leads to reasonable sibility

but also markedly

factors,

PVT data over a wide density The calculation tum gases

teraction

firstly

data is achieved,

parameters

come less temperature

obtain

tum corrections.

Therefore isotherms,

of

range. containing

a better

agreement

because

reasonable

quanwith

the binary

values

than in calculations

it is possible

of

compres-

the representation

of mixtures

and secondly

physically

dependent

perimental

P-x-y

improves

because

equation

of the critical

and temperature

of phase equilibria

is facilitated,

experimental

in a classical

values

without

quan-

not only to correlate

but also to extrapolate

in-

and be-

to other

ex-

tem-

peratures.

REFERENCES 1

2

3

4

5 6

7

8

9 IO

11

u.

Deiters, 1979, Entwicklung einer semiempirischen Zustandsgleichung fiir fluide Stoffe und Berechnung von Fluid-Phasengleichgewichten in binaren Mischungen bei hohen Driicken, Bochum. Ph.D. thesis, Ruhr-UniversitZt, U. Deiters, 1981, A new semiempirical equation of state for fluids, part I: Derivation, Chem. Eng. Sci. 36: 1139-1146; part II: Application to pure substances, Chem. Eng. Sci. 36: 1147-1151. U. Deiters, 1982, A new semiempirical equation of state for fluids, part III: Application to phase equilibria in binary mixtures, Chem. Eng. Sci. 37: 855-861. U.K. Deiters, 1982, Coordination numbers for rigid spheres of different size - estimating the number of next-neighbour interactions in a mixture, Fluid F'hase Equil. 8: 123-129. T.L. Hill, 1960, An introduction to statistical thermodynamics. Addison-Wesley Publishing Co., London, p. 287. Generalized van der Waals M.A. Hooper, S. Nordholm, 1980, theory II: Quantum effects on the equation of state, Aust. J. Chem. 33: 2029-2035. H. Hiibner, 1982, Experimentelle Untersuchung des Phasengleichgewichtes fliissig-gasformig fiiirHelium-Stickstoff-Gemische. Ph.D. thesis, Technische Uni\rersit;it Braunschweig. 1968, Critical lines and phase equilibria P.H. van Konynenburg, in binary mixtures. Ph.D. thesis, University of California, Los Angeles. I. Prigogine, 1957, The molecular theory of solutions. North-Holland Publishing Co., Amsterdam, p. 384. C.Y. Tsang, P. Clancy, J.C.G. Calado, W.B. Streett, 1980, Phase equilibria in the hydrogen/methane system at temperatures from 92.3 to 180 K and Pressures to 140 MPa, Chem. Eng. Commun. 6: 365-383. C.Y. Tsang, W.B. Streett, 1981, Phase equilibria in the hydroqen-carbon monoxide system at temperatures from 70 to 125 K and pressures to 53 MPa, Fluid Phase Equil. 6: 261-273.