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TN#: 238582
Undergraduate
Chemical and Biological Engineering
cmdundas

Christopher Dundas
cmdundas@buffalo.edu
7327572137

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Journal Title: Fluid Phase Equilibria
Volume: 78 Issue:
MonthNear: October 1992 Pages: 157-190
Article Author:
Article Title: New applications of Kahl's VLE analysis to engineering phase behavior
calculations

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Fluid Phase Equilibria, 78 (1992) 157-190

157

Elsevier Science Publishers B.V., Amsterdam

New applications of Kahl's VLE analysis to engineering
phase behavior calculations
Johannes M. Nitsche
Department of Chemical Engineering,
New York 14260 !USA)

State University of New York at Buffalo, Buffalo,

(Received June 10, 1991; accepted in final form April 6, 1992)

ABSTRACT
Nitsche, J .M., 1992, New applications of Kahls' VLE analysis to engineering phase behavior
calculations. Fluid Phase Equilibria, 78: 157-190.
In a little known but profound paper, G.D. Kahl (1967) {Phys. Rev., 155; 78-80)
developed a compelling argument questioning the validity of isothermal integration of the
van der Waals equation between the spinodal points in accordance with Maxwell's equal-area
construction. This work generalizes Kahl's approach and shows conclusively that it is
entirely possible to describe vapor-liquid equilibria without lending credence to the
thermodynamic properties of unstable states. The analysis is structured around a new free
energy parameter .6.(T) that allows adjustment of calculated vapor pressures without
alteration of the equation of state itself, as is demonstrated using a novel modification of
the Redlich-Kwong equation. The analysis opens the door to greater flexibility in using
excess free energy models to devise mixing rules for equations of state. An extension of the
theoretical development leads to a new type of VLE procedure whereby unrelated equations for liquid and vapor phases can be combined for essentially exact volumetric description of the pure fluids comprising a mixture. Introduction of a parameter .6.(T) also makes it
possible to formulate a conjecture explaining the apparent nonexistence of liquid-solid
critical points.

PROLOGUE

It was the first warm, sunny day of the spring -

the kind that cannot be
spent indoors, or so thought the wiry little man known to all as Elroy. Thus,
the morning saw him walking to The Meadow instead of staying inside his
Correspondence to: J.M. Nitsche, Department of Chemical Engineering, State University of
New York at Buffalo, Buffalo, New York 14260, USA.

0378-3812/92/$05.00

© 1992 Elsevier Science Publishers BV All rights reserved

J.M. Nitsche / Fluid Phase Equilibria 78 (J992) 157-/90

158

cottage. His left hand held a writing tablet and a fountain pen. Clutched
under his right arm was a large bound volume of thermodynamic tables, so
heavy that an observer would have noticed him listing markedly to starboard as he trod down the Sandy Path. Twenty minutes later Elroy reached
his destination, a shady spot under a willow at the bank of the Bubbling
Brook. There he stopped, dropped his things and settled back against the
thick tree trunk. He sat pensively for a few moments, then picked up pen
and paper and began to scribble, stopping every so often to flip through the
tome on the grass beside him. After three hours he paused, looked rather
pleased with himself. For he had much to be pleased about.
On this fine morning, Elroy had decided to busy himself with a routine
but interesting calculation. He would estimate the vapor pressure of water
at temperature To = 373.15 K using the van der Waals equation
RT
P= V-b

a

-

V'

(1)

with a = 27R2T/ / 64Pe = 5.525 X 106 bar ern" mol "? and b = RTj8Pe =
30.42 cm' mol-I. He started by guessing a value for the vapor pressure Po
and then computing the corresponding saturated liquid and vapor molar
volumes VJ and Vo' as the smallest and largest roots of the equation P( V,
To) = Po. In order to ascertain whether or not his guess had been correct,
he then had to calculate the difference in chemical potential between the
states (Vu" To) and (VJ, To). Here Elroy was stumped for a while, for he
had to find a physically accessible path from (V,i, To) to (VO To), and this
could obviously not be the isotherm at temperature To, for this is interrupted by the inaccessible interval between the spinodal points. But eventually a solution presented itself in the form of a simple three-step path: (i)
heating at constant volume from (VJ, To) to (VJ, Tel; (ii) isothermal expansion from (VJ, Tel to (Va" Tel; and finally (iii) cooling at constant volume
from (Vo', Tel to (J.~lv, To). Elroy's hasty sketch is reproduced here as Fig. 1.
For this path, application of standard thermodynamic identities together
with integration by parts (see the Appendix) led him to the explicit formula
V

,

f-'(Vo" To) - f-'(~:, To)

=

fVO[(Te
vri

+

-

To)(

ap)
aT

((1- ;)[Cv(VJ,

+ Po(Vo' - VJ)

(V,
v

TJ -

P(V, Te)] dV

T)-Cv(Vo"

T)] dT
(2)

The volume integral was readily computed using the equation of state,
eqn. (1). Evaluation of the temperature integral required heat capacity

J.M. Nitsche / Fluid Phase Equilibria 78 (992) 157-190

159

p

-+-+-\-+----+----7

v;

Fig. 1. Path for integration
(VlJ', TlJ)·

v

of the Helmholtz

energy from the state (Vr~, To) to the state

data. For C v( Vd, T), Elroy thought it an eminently reasonable approximation to assume ideal-gas behavior, so he used the correlation (Smith and
Van Ness, 1987, p. 109)
C;i'(T) ~ R[ 2.470 + (1.450

X

10-3 K-I)T

+ (0.121

X

105 K')r']

(3)

Values of C v for the liquid side were a bit more complicated to come by,
requiring numerical differentiation of data taken from a steam table
(Sandler, 1989, pp. 596-597) and application of the identity C v = Cp +
T(aV/aT)~/(aV/ap)T'
Elroy found Cv to be rather insensitive to pressure
(or molar volume) at all temperatures except values near critical, so he
ignored the pressure dependence and fitted his calculated values with the
empirical expression
(4)
this being satisfactory for a rough calculation in nice weather. With these
approximations, the temperature integral in eqn. (2) was simply a constant,
independent of V,: and Vo' and therefore of the trial value of Po' its
explicit value being 18887 bar crn' mol "'. After some trial-and-error
calculations, Elroy came up with the value Po ~ 7.90 bar as the common
pressure at which the states (Vo" To) and (Vd, To) have equal chemical
potentials. This was considerably higher than the experimental value,

160

1.M. Nitsche! Fluid Phase Equilibria 78 (1992) 157~190

1.01325 bar ~ 1 atm, but it was near to the proper order of magnitude, and
this is why Elroy was pleased, especially because it is well known that the
van der Waals equation performs poorly in quantitative predictions of the
volumetric properties of fluids.
It was only now that he looked up, noticed the position of the sun, and
realized that it was time for lunch. So he got up, collected his things, and
began to step over the soft grass back to the trail. But after ten paces he
stopped suddenly in his tracks, for a disturbing thought had crept into his
mind and started alarm bells ringing. He stood frozen for several minutes,
frowning and puzzling.
Eventually the resolution came to him. He brightened visibly, and
continued to the trail and happily followed it home. So ends the tale.
And of Elroy's unsettling question? It was just this: What about Maxwell's
equal-area construction?

INTRODUCTION

The purpose of this communication is to explore several novel applications of a profound but little known analysis by Kahl (1967), which has
important implications for the modelling of phase equilibria using
pressure-explicit equations of state.' The overall approach is best addressed by first recalling two well-known graphical constructions expressing
the conditions for equilibrium (equality of pressure and chemical potential)
of a liquid phase with molar volume Vi and a vapor phase with molar
volume V', both at temperature T. The first is the Maxwell construction,
according to which a horizontal (constant pressure) line segment connecting the points at V I and V' on the isotherm "cuts off equal areas from the
curve above and below" (Clerk-Maxwell, 1875, p. 359) (Fig. 2(a)). The
second requires that, in a plot of the molar Helmholtz energy A as a
function of molar volume V, the points at Vi and VV share a common
tangent line (Fig. 2(b)) whose slope is minus the vapor pressure. Both
constructions make either direct or indirect use of the equation of state in
the interval between the spinodal points where 3P/3V> 0, and where it is
devoid of physical significance. Many have wondered about the reliance on
the unphysical portion of the isotherm, i.e. the unstable region, but in
practice no difficulties arise. The usual procedure for fitting equations of

A volume-by-volume examination of the Science Citation Index indicates that Kahrs paper
has been cited only three times since its publication, once by Kahl himself, and only in
contexts largely removed from VLE calculations.
a

J.M. Nitsche / Fluid Phase Equilibria 78 (J9921157-190

161

p

(al

--t-+-B'f------'-----'»

v

-++------1------,>

v

(hi

v'

V'

Fig. 2. Graphical constructions

for vapor-liquid

equilibrium: (a) Maxwell; (b) tangent line.

state to vapor pressure data is to adjust the values of parameters appearing
therein so that equal areas are achieved with P close to the experimental
vapor pressure. Skogestad (1983, p. 180) assesses the procedure succinctly:
" ... the two-phase region is 'free to use', and what is in fact done when
applying equations of state to VLE is to use this region to 'store' the vapor
pressures. This is completely empirical and is obtained by forcing the
volumetric behavior inside the two-phase region to obey Maxwells equalarea rule ... " In the case of van-der-Waals-like equations of state, the
attraction energy parameter a is usually made temperature-dependent to
achieve this end (see, for example, Soave, 1972; Peng and Robinson, 1976;
Schmidt and Wenzel, 1980; Soave, 1980; Marchand et al., 1982; Patel and
Teja, 1982; Adachi and Lu, 1984; Soave, 1984; Stryjek and Vera, 1986a,c;
Watson et al., 1986; Heidemann and Kokal, 1990; Soave, 1990; Twu et al.,
1991), and the same basic procedure is invariably followed with other
equations of state having different origins (see, for example, Sanchez and
Lacombe, 1976; Kleintjens and Koningsveld, 1980; Lee and Chao, 1988).
But nagging questions remain. Consider, for example, the following
hypothetical situation: Suppose, after considerable labor, we succeed in
developing an expression that gives the pressure as a function of volume

162

J.M. Nitsche / Fluid Phase Equilibria 78 (992) 157-190

and temperature very accurately (in the experimentally accessible portions
of the phase space) for a given substance. Suppose further that we carry
out the Maxwell construction and find that the calculated vapor pressure
differs considerably from the experimental value. Should we dismiss our
equation of state as a failure? Hardly, for it is an outstanding success
insofar as PVJ behavior is concerned. Should we adjust the values of some
of the parameters in order that equal areas are achieved at a pressure
closer to the true vapor pressure? Certainly not, for we have just expended
great effort in finding parameter values that reproduce the experimental
isotherms (outside the unstable region). What, then, should we do? Later
we shall find ourselves in precisely this situation, so coming up with a
resolution to this quandary represents a real priority.

KAHL'S GENERALIZATION

OF THE MAXWELL

CONSTRUCf10N

In truth, the sole purpose of an equation of state is to reproduce the
measured PVJ behavior. Requiring that it perform in phase behavior
calculations is asking too much. Certainly, the integration of - PdV over
the unstable interval at any prescribed temperature T gives the difference
in Helmholtz energy A( V, T) between the vapor and liquid at the spinodal
points, but the value so obtained is predicated upon information removed
from reality. With sufficient heat capacity data, this free energy difference
can be calculated exactly and unambiguously without involving the unstable
region, by following a path around the two-phase coexistence dome as was
done in the Prologue. It will generally not be true that the integral of
- PdV at the original temperature T happens to equal the exact value of
the free energy difference. Utilization of Maxwell's construction amounts
to attempting phase behavior calculations without the benefit of any heat
capacity data. This can be done (in fact, this is what is always done (Soave,
1972; Peng and Robinson, 1976; Sanchez and Lacombe, 1976; Kleintjens
and Koningsveld, 1980; Schmidt and Wenzel, 1980; Soave, 1980; Kubic,
1982; Marchand et aI., 1982; Patel and Teja, 1982; Adachi et aI., 1983;
Adachi and Lu, 1984; Soave, 1984; Stryjek and Vera, 1986a,c; Watson et
aI., 1986; Dohrn and Prausnitz, 1990; Heidemann and Kokal, 1990; Soave,
1990; Twu et aI., 1991)), but fitting parameters via a scheme that involves
phase equilibrium (as opposed to purely PVJ fitting) violates the basic
premise of the equation of state. Skogestad (1983) could have been more
severe in his criticism: not only is the volumetric behavior inside the
two-phase region forced to obey the Maxwell construction; the volumetric
behavior everywhere is compromised, since adjusting parameter values
modifies the shape of the entire isotherm.

J.M. Nitsche / Fluid Phase Equilibria

163

157-190

78 (1992)

Kahl's (1967) approach can be expressed as follows (with some changes
in formalism and in greater generality): Since the pressure is minus the
partial derivative of the molar Helmholtz energy A with respect to molar
volume V, the Helmholtz energy can be obtained from the equation of
state by integration, i.e.

A(V, T) ~AW,

f vP(V',

T) -

(5)

T) dV'

V"

where V" is an arbitrarily selected reference volume. For definiteness, this
can be regarded as being a large value V v.o characteristic
of the vapor
phase, this choice being suggested by the' superscript. Thus, for a vapor
phase at molar volume
the Helmholtz energy is given by

V"

A(VV, T) =A(V'O,

T) - f;',p(V,

T) dV

(6)

For a liquid phase with molar volume VI, standard procedure would decree
that the same formula be used with Vi substituted for V'. However, the
equation of state is devoid of significance between the spinodal points. It
follows that the difference in Helmholtz energy between liquid and vapor
phases calculated by integrating - PdV over an interval that includes the
unstable region is pure fiction. This value must, in general, be corrected, so
it is more proper to write

A(VI,

T) =A(Vv,o, T) - r'oP(V,

T) dV + A(T)

(7)

where A( T) serves to bring the liquid-phase Helmholtz energy to a realistic
value irrespective of the peculiarities
of the equation of state in the
unstable region, Utilization of Maxwell's construction amounts to assuming
that A(T) es 0 (Kahl, 1967). In general, the Helmholtz energy correction
A(T) must, however, be determined
from the equation of state together
with heat capacity data, and Kahl's key observation is that A(T) is generally nonzero, With the benefit of the calculations in the Prologue, one can
give the explicit formula

-A(T)

=

fV"'[(T, -

T)( ap) (V,
aT v

+ f;'(1-

;,

Vi,

T

e)

-

P(V, TJ + P(V, T)] dV

)[CV(VI,O, T')-Cv(Vv,o,

T')] dT'

(8)

with V 1,0 an arbitrary liquid-phase reference volume, The best way to
conclude that this can be non-zero is by example, Owing to the simple way
in which temperature enters the van der Waals equation, it is easily seen
that the first integrand is identically zero, Thus, for the case considered in
the Prologue, A(373.15K) ~ -18887 bar cm ' mol-I,

164

J.M. Nitsche / Fluid Phase Equilibria 78 (J992) /57-190

A
Ll(T) = 0 (Maxwell)

--t-----~r";~
\

"
\

....
---__':>

' ",

.................

'-

Fig. 3. Tangent-line

construction

V

----

for different values of aCT).

Although little can be said a priori about t. in the absence of heat
capacity data, it is clear that this function is defined only for temperatures
below the critical temperature T" and that it must tend to zero as T -> T,
(see Remarks, below). The important practical implication of Kahl's type of
approach is that one can also view t. as an adjustable parameter. At any
given temperature T, changing t.(T) amounts to shifting vertically the
Helmholtz energy curve for the liquid, which alters the tangent-line construction (Fig. 3) and thereby the calculated vapor pressure. We are thus
led to the following appealing prospect: There is no need to adjust
constants in the equation of state to fit experimental vapor pressures.
Rather, the equation of state can be optimized solely to PVT data. Later,
values of t. as a function of temperature can always be found to bring the
calculated vapor pressures into agreement with the experimental values.
To summarize: phase behavior calculations generally require two inputs:
(i) an equation of state, of which the sole purpose is the representation of
PVT data; and OJ) a Helmholtz energy correction t.(T) which indicates
how far the liquid Helmholtz energy curve should be shifted vertically at
each temperature in order to reproduce experimental vapor pressures (Fig.
3).
It will be observed that if one calculates heat capacities C v( V, T) from
ideal-gas correlations C~(T) following the usual procedure (Modell and

Reid, 1983, p. 158; Sandler, 1989, p. 151)
Cv(V,

T)

=

C~(T)

+

r
00

T

a2p)

( aT
-2

v

(V',

T) dV'

(9)


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