Page 276 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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256 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
If truncated virial equation (Eq. 5.75) is used, ˆ φ i is calcu-
gas phase is nearly ideal.
lated from the following relation as derived from Eq. (6.124): —Good approximation for gases at low pressure where the
—Good approximation at any pressure whenever i is present

P
in large excess (say, y i > 0.9). The Lewis rule becomes exact
(6.127) ln ˆ φ i = 2 y j B ij − B
j RT in the limit of y i → 1.
—Good approximation over all range of pressure and com-
where B (for whole mixture) and B ij (interaction coefficients) position whenever physical properties of all components
should be calculated from Eqs. (5.70) and (5.74), respectively. present in the mixture are the same as (i.e., benzene and
As discussed earlier Eq. (5.70) is useful for gases at moderate toluene mixture).
pressures. Equation (6.127) is not valid for liquids. —Good approximation for liquid mixtures whose behavior is
like an ideal solution.
Example 6.9—Suppose that fugacity coefficient of the whole —A poor approximation at moderate and high pressures
mixture, φ mix , is defined similar to that of pure components. whenever the molecular properties of components in the
Through mixture Gibbs energy, derive a relation between f mix mixture are significantly different from each other (i.e., a
and f i for mixtures. mixture of methane and a heavy hydrocarbon).
Solution—Applying Eq. (6.80) to residual molar Gibbs free Lewis rule is attractive because of its simplicity and is usu-
R ig R ally used when the limiting conditions are applied in certain
¯ R
energy (G = G − G ) gives G = y i G and since μ i = ¯ G i
i
ˆ
from Eq. (6.119) d ¯ G i = RTdln f i and for ideal gases from situations. Therefore when the Lewis rule is used, fugacity of
ig
Eq. (6.117) we have dG = RTdln y i P. Subtracting these two i in a mixture is calculated directly from its fugacity as pure
i
relations from each other gives d ¯ G = RTdln ˆ φ i , which after component. When Lewis rule is applied to liquid solutions,
R
i
R
integration gives ¯ G = RT ln ˆ φ i . Therefore for the whole mix- Eq. (6.114) can be combined with Eq. (6.131) to get γ i = 1
i
ture we have (for all components).
R
(6.128) G = RT x i ln ˆ φ i
6.6.4 Calculation of Fugacity of Pure Gases
where after comparing with Eq. (6.48) for the whole mixture and Liquids
we get
Calculation of fugacity of pure components using equations

(6.129) ln φ mix = x i ln ˆ φ i of state was discussed in Section 6.5. Generally fugacity of
pure gases and liquids at moderate and high pressures may
or in terms of fugacity for the whole mixture, f mix , it can be be estimated from equations given in Table 6.1 or through
written as
generalized correlations of LK as given by Eq. (6.59). For pure
ˆ
f i gases at moderate and low pressures Eq. (6.62) derived from
(6.130) ln f mix = x i ln virial equation can be used.
x i
To calculate fugacity of i in a liquid mixture through
This relation can be applied to both liquid and gases. f mix is Eq. (6.114) one needs fugacity of pure liquid i in addition to
useful for calculation of properties of only real mixtures but the activity coefficient. To calculate fugacity of a pure liquid i
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
is not useful for phase equilibrium calculation of mixtures at T and P, first its fugacity is calculated at T and correspond-
except under certain conditions (see Problem 6.19). ing saturation P sat . Under the conditions of T and P sat both
vapor and liquid phases of pure i are in equilibrium and thus
6.6.3 Calculation of Fugacity from Lewis Rule L sat V sat sat sat
(6.133) f (T, P ) = f (T, P ) = φ i P
i
i
Lewis rule is a simple method of calculation of fugacity of a sat is the fugacity coefficient of pure vapor at T and
component in mixtures and it can be used if the assumptions where φ i
sat
made are valid for the system of interest. The main assump- P . Effect of pressure on liquid fugacity should be consid-
L
sat
L
i
i
tion in deriving the Lewis fugacity rule is that the molar vol- ered to calculate f (T, P) from f (T, P ). This is obtained
ume of the mixture at constant temperature and pressure is by combining Eq. (6.8) (at constant T) and Eq. (6.47):
a linear function of the mole fraction (this means ¯ V i = V i = (6.134) dG i = RTdln f i = V i dP
constant). This assumption leads to the following simple rule
ˆ
for f i known as Lewis/Randall or simply Lewis rule [21, 22]: Integration of this equation from P sat to desired pressure of
P for the liquid phase gives
ˆ
(6.131) f i (T, P) = y i f i (T, P)
L
f (T, P) P V L
where f i (T, P) is the fugacity of pure i at T and P of mixture. (6.135) ln i = i dP
L
Lewis rule simply says that in a mixture ˆ φ i is only a function of f (T, P sat ) sat RT
i
T and P and not a function of composition. Direct conclusion P i
of Lewis rule is Combining Eqs. (6.133) and (6.135) leads to the following
relation for fugacity of pure i in liquid phase.
(6.132) ˆ φ i (T, P) = φ i (T, P)
⎛ ⎞
which can be obtained by dividing both sides of Eq. (6.131) by L sat sat P V i L
y i P. The Lewis rule may be applied to both gases and liquids (6.136) f (T, P) = P i φ i exp ⎝ RT dP⎠
⎜
⎟
i
with the following considerations [21]: P i sat
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