Page 231 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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TABLE 5.4—Second virial coefficients for several gases [41].
Another form of Eq. (5.59) for calculation of B mix can be writ-
ten as following:
Temperature, K 17:42 5. PVT RELATIONS AND EQUATIONS OF STATE 211
Compound 200 300 400 500
N N
N
N 2 −35.2 −4.2 9 16.9 B mix = y i B ii + 1 y i y j δ ij where δ ij = 2B ij − B ii − B jj
CO 2 ... −122.7 60.5 −29.8 i=1 2 i=1 j=1
CH 4 −105 −42 −15 −0.5
C 2 H 6 −410 −182 −96 −52 (5.70)
C 3 H 8 ... −382 −208 −124 There are several correlations developed based on the the-
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Note: Values of B are given in cm /mol.
ory of corresponding state principles to estimate the second
virial coefficients in terms of temperature. Some of these rela-
where B, C, D, . . . are called second, third, and fourth virial co- tions correlate B/V c to T r and ω. Prausnitz et al. [6] reviewed
efficients and they are all temperature-dependent. The above some of these relations for estimation of the second virial
two forms of virial equation are the same and the second coefficients. The relation developed by Tsonopoulos [42] is
equation can be derived from the first equation (see Problem useful to estimate B from T c , P c , and ω.
5.7). The second form is more practical to use since usually T
and P are available and V should be estimated. The number BP c = B (0) + ωB (1)
of terms in a virial EOS can be extended to infinite terms RT c
but contribution of higher terms reduces with increase in (0) 0.330 0.1385 0.0121 0.000607
power of P. Virial equation is perhaps the most accurate PVT B = 0.1445 − T r − T r 2 − T r 3 − T r 8
relation for gases. However, the difficulty with use of virial 0.331 0.423 0.008
equation is availability of its coefficients especially for higher B (1) = 0.0637 + T 2 − T 3 − T 8
terms. A large number of data are available for the second r r r
virial coefficient B, but less data are available for coefficient (5.71)
C and very few data are reported for the fourth coefficient where T r = T/T c . There are simpler relations that can be used
D. Data on values of virial coefficients for several compounds for normal fluids [1].
are given in Tables 5.4 and 5.5. The virial coefficient has firm BP c
basis in theory and the methods of statistical mechanics allow = B (0) + ωB (1)
derivation of its coefficients. RT c
B represents two-body interactions and C represented (5.72) B (0) = 0.083 − 0.422
three-body interactions. Since the chance of three-body in- T r 1.6
teraction is less than two-body interaction, therefore, the im- (1) 0.172
portance and contribution of B is much greater than C. From B = 0.139 − T r 4.2
quantum mechanics it can be shown that the second virial
coefficient can be calculated from the knowledge of potential Another relation for prediction of second virial coefficients of --`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
function ( ) for intermolecular forces [6]: simple fluids is given by McGlashan [43]:
BP c
(5.73) = 0.597 − 0.462e
∞ 0.7002/T r

2
(5.67) B = 2π N A 1 − e − (r)/kT r dr RT c
A graphical comparison of Eqs. (5.71)–(5.73) for prediction
0
of second virial coefficient of ethane is shown in Fig. 5.11.
−1
where N A is the Avogadro’s number (6.022 × 10 23 mol ) and Coefficient B at low and moderate temperatures is negative
k is the Boltzman’s constant (k = R/N A ). Once the relation for and increases with increase in temperature; however, as is
is known, B can be estimated. For example, if the fluid seen from the above correlations as T →∞, B approaches a
follows hard sphere potential function, one by substituting positive number.
Eq. (5.13) for into the above equation gives B = (2/3)π N A σ . To predict B mix for a mixture of known composition, the in-
3
Vice versa, constants in a potential relation (ε and σ) may teraction coefficient B ij is needed. This coefficient can be cal-
be estimated from the knowledge of virial coefficients. For culated from B ii and B jj using the following relations [1, 15]:
mixtures, B mix can be calculated from Eq. (5.59) with a being
replaced by B. For a ternary system, B can be calculated from RT cij (0) (1)
B ij = B + ω ij B
Eq. (5.64). B ij is calculated from Eq. (5.67) using ij with σ ij P cij
and ε ij given as [6] B (0) and B (1) are calculated through T rij = T/T cij
1
(5.68) σ ij = (σ i + σ j ) ω i + ω j
2 ω ij = 2
(5.69) ε ij = (ε i ε j ) 1/2 1/2
T cij = (T ci T c j ) (1 − k ij )
(5.74)
TABLE 5.5—Sample values of different virial coefficients for Z ij RT cij
several compounds [1]. P ij =
V cij
3
6
Compound T, C B,cm /mol C,cm /mol 2 Z ci + Z c j
◦
Methane a 0 −53.4 2620 Z cij =
Ethane 50 −156.7 9650 2
Steam (H 2 O) 250 −152.5 −5800 V 1/3 + V 1/3 3
Sulfur dioxide (SO 2 ) 157.5 −159 9000 V cij = ci ci
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a For methane at 0 C the fourth virial coefficient D is 5000 cm /mol . 2
3
◦

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