Page 307 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 287
in which ξ is dimensionless packing factor defined by
Velocity of Sound Data
Eq. (5.86) in terms of hard-sphere diameter σ. In general 6.9.2 Equation of State Parameters from
for nonassociating fluids, σ can be taken as a linear func-
tion of 1/T and ρ [50], i.e., σ = d 0 + d 1 ρ + d 2 /T + d 3 ρ/T. From In this section the relations developed in the previous section
Eq. (6.218) we have for the velocity of sound are used to obtain EOS parameters.
These parameters have been compared with those obtained
(6.219) S = S(T, V) from critical constants or other properties in the form of pre-
diction of volumetric and thermodynamic properties. Trun-
Since c is a state function we can write it as a function of only
two independent properties for a pure fluid or fluid mixtures cated virial (Eq. 5.76), Carnahan–Starling–Lennard–Jones
of constant composition in a single phase (see the phase rule (Eq. 5.96), and common cubic equations (Eq. 5.40) have been
in Chapter 5): used for the evaluations and testing of the suggested method.
Although the idea of the proposed method is for heavy hy-
(6.220) c = c(S, V) drocarbon mixtures and reservoir fluids, but because of lack
Differentiating Eq. (6.215) with respect to S at constant V of data on the sonic velocity of such mixtures applicability of
gives the method is demonstrated with use of acoustic data on light
and pure hydrocarbons [8, 44].
V 2 ∂ ∂P !
(6.221) 2cdc =− dS (at constant V)
M ∂S ∂V 6.9.2.1 Virial Coefficients
S V
Applying Eqs. (6.10) to (6.219) gives Since any equation of state can be converted into virial form,
in this stage second and third virial coefficients have been
∂S ∂S

(6.222) dS = dT + dV obtained from sonic velocity for a number of pure substances.
∂T V ∂V T Assuming that the entropy departure for a real fluid is the
which at constant V becomes same as for a hard sphere and by rearranging Eq. (6.218) the
packing fraction of hard sphere can be calculated from real
∂S

(6.223) dS = dT (at constant V) fluid entropy departure:
∂T T
1/2
ig

ig
From mathematical relations we know
2 − S−S − 4 − S−S
π
R
R
3
∂ ∂P ∂ ∂P (6.228) ζ = ρN A σ = S−S ig
(6.224) = 6 3 − R
∂S ∂V S V ∂V ∂S V S
Calculated values of σ from the above equation indicate that
The Maxwell’s relation given by Eq. (6.10) gives (∂P/∂S) V = there is a simple relation between hard-sphere diameter and
−(∂T/∂V) S , where −(∂T/∂V) S can be determined from divid- temperature as in the following form [44]:
ing both sides of Eq. (6.222) by ∂V at constant S as
∂T (6.229) σ = d 0 +
d 1
(6.225) =− (∂S/∂V) T T
∂V
S (∂S/∂T) V Application of the virial equation truncated after the third
Substituting Eqs. (6.223)–(6.224) into Eq. (6.221) and inte- term, Eq. (5.76), to hard sphere fluids gives
grating from T to T →∞, where c → c HS gives the following
relation for c in terms of T and V: HS B HS C HS
(6.230) Z = 1 + V + V 2
2
2
(6.226) c = c HS 2 − V 2 ∞ ∂ T ∂S dT By converting the HS EOS, Eq. (5.93), into the above virial

M ∂V 2 S ∂T V
T form one gets [51]
c HS can be calculated from Eq. (6.216) using the CS EOS, HS 2 3
Eq. (5.93). Derivative (∂S/∂T) V can be determined from B = 3 π N A σ
2 2 (6.231)
Eq. (6.218) or (6.219) as a function of T and V only. (∂ T/∂V ) S HS 5 2 2 6
can be determined from Eq. (6.225) as a function of T and V. C = 18 π N σ
A
Therefore, the RHS of above equation is in terms of only T
and V, which can be written as Substituting Eqs. (5.76) and (6.230) for real and hard-sphere
fluids virial equations into Eq. (6.39) one can calculate en-
(6.227) c = c(T, V) tropy departures for real and hard-sphere fluids as
Equation (6.226) or (6.227) is a cVT relation and is called ve- S − S ig B + TB C + TC
locity of sound based equation of state [44]. One direct appli- (6.232) R =− ln P + V + 2V 2
cation of this equation is that when a set of experimental data ig real fluid HS HS
on cVT or cPT for a fluid or a fluid mixture of constant compo- (6.233) S − S =− ln P + B + C
sition are available they can be used with the above relations R hard sphere V 2V 2
to obtain the PVT relation of the fluid. This is the essence of Since it is assumed that the left sides of the above two equa-
use of velocity of sound in obtaining PVT relations. This is tions are equal, so the RHSs must also be equal, which result
demonstrated in the next section by use of velocity of sound in the following relations:
data to obtain EOS parameters. Once the PVT relation for a

fluid is determined all other thermodynamic properties can (6.234) TB + B = B HS
be calculated from various methods presented in this chapter. TC + C = C HS

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