Page 281 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 261
S
L
alcohol mixtures [21]. For hydrocarbon systems, UNIQUAC
(universal quasi chemical) model that is based on a group and 6.11 is shown by δ , then δ may be calculated from the
following relation [24]:
contribution model is often used for calculation of activity f
2 2 H
coefficient of compounds with known structure. More details (6.154) δ S i = δ L i + i
on activity coefficient models and their applications are dis- V i
3 1/2
cussed in available references [4, 21]. The major application in which δ is in (J/cm ) , H is in J/mol, and V i is in
f
i
3
of activity coefficient models is in liquid–liquid and solid– cm /mol.
liquid equilibria as well as low pressure VLE calculations Calculation of fugacity of solids through Eq. (6.151) re-
when cubic equations of state do not accurately estimate liq- quires calculation of f . For convenience the standard state
◦
i
uid fugacity coefficients. for calculation of f is considered subcooled liquid at temper-
◦
1
L
ature T and for this reason we show it by f . In the following
1
discussion solute component 1 is replaced by component i
6.6.6 Calculation of Fugacity of Solids to generalize the equation for any component. Based on the
In the petroleum industry solid fugacity is used for SLE cal- SLE for pure i at temperature T it can be shown that [21, 25]
culations. Solids are generally heavy organics such as waxes f (T, P) = f (T, P)
S
L
and asphaltenes that are formed under certain conditions. i i f
Solid–liquid equilibria follows the same principles as VLE. × exp H i 1 − T Mi − C Pi 1 − T Mi − C Pi ln T Mi
Generally fugacity of solids are calculated similar to the meth- RT Mi T R T R T
ods that fugacity of liquids are calculated. In the study of (6.155)
solubility of solids in liquid solvents usually solute (solid) is where f (T, P) is the fugacity of pure solid at T and P, H is
S
f
i
i
shown by component 1 and solvent (liquid) is shown by com- the molar heat of fusion of solute, T Mi is the melting or freezing
ponent 2. Mole fraction of solute in the solution is x 1 , which is point temperature, and C Pi = C L − C , which is the differ-
S
Pi
Pi
the main parameter that must be estimated in calculation of ence between heat capacity of liquid and solid solute at av-
solubility of solids in liquids. We assume that the solid phase erage temperature of (T + T Mi )/2. Derivation of Eq. (6.155)
is pure component 1. In such a case fugacity of solid in the is similar to the derivation of Eq. (6.136) for calculation of
ˆS
solution is shown by f , which is given by fugacity of pure liquids but in this case equilibrium between
1
solid and liquid is used to develop the above relation. Firooz-
S
ˆ S
(6.151) f (solid in liquid solution) = x 1 γ f ◦
1 1 1 abadi [17] clearly describes calculation of fugacity of solids.
Since methods of calculation of f were discussed in the pre-
L
where f is the fugacity of solute at a standard state but tem- i
◦
1 vious section, f can be calculated from the above equation.
S
S
perature T of solution. γ is the activity coefficient of solid i
1 Values of T M and H for some selected compounds are given
f
component in the solution. Obviously for ideal solutions γ 1 S i
S
is unity. Model to calculate γ is similar to liquid activity co- in Table 6.10 along with liquid molar volume and solubility
1 parameter. Estimation of freezing point T M for pure hydro-
efficients, such as two-suffix Margules equation:
carbons was discussed in Section 2.6.4. From Eq. (2.42) and
A 2 coefficients given in Table 2.6 we have
S
(6.152) ln γ = (1 − x 1)
1 RT T M = 397 − exp (6.5096 − 0.1487M 0.47 )
A more accurate activity coefficient model for nonpolar so- n-alkanes (C 5 − C 40 )
lutes and solvents is given by the Scatchard–Hildebrand rela- T M = 370 − exp (6.52504 − 0.04945M 2/3 )
tion (Eq. 6.145): (6.156)
n-alkylcyclopentanes (C 7 − C 40 )
2
L
V (δ 1 − δ 2 ) 2 T M = 375 − exp (6.53599 − 0.04912M 2/3 )
S
(6.153) ln γ = 1 2
1 RT n-alkanes (C 9 − C 42 )
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
L
where V is the liquid molar volume of pure component 1 at where T M is in kelvin. Average deviation for these equations
1
T and P, δ 2 is solubility of solvent, δ 1 is the solubility parame- are 1.5, 1.2, and 0.9%, for n-alkanes, n-alkylcyclopentanes,
ter of subcooled component 1, and 2 is the volume fraction and n-alkylbenzenes, respectively. Similarly based on the data
f
of solvent and is given by Eq. (6.146). Methods of calculation given for H in Table 6.10 the following relations are devel-
i
of δ 1 and 1 have been discussed in Section 6.6.5. δ 1 can be oped for estimation of heat of fusion of pure hydrocarbons
calculated from Eq. (6.147) from the knowledge of heat of for the PNA homologous families.
vaporization of solute, H 1 vap . Values of the solubility param-
eter for heavy single carbon number components are given in ln H f =−71.9215 + 70.7847M 0.01
Table 4.6. When the liquid solvent is a mixture δ 2 is replaced RT M for n-alkanes (C 2 − C 36 )
S
by δ mix and γ is calculated through Eq. (6.150). It should be
1
noted that when Eq. (6.153) is used to calculate fugacity of a ln H f = 0.8325 + 0.009M
solid in a liquid solution value of δ can be obtained from Table RT M for n-alkylcyclohexanes (C 7 − C 16 )
6.10 from liquid solubility data. However, when this equation
is applied for calculation of fugacity of a solid component i H f = 1.1556 + 0.009M + 0.000396M − 6.544 × 10 −7 M 3
2
in a homogeneous solid phase mixture (i.e., wax) then solid RT M
S
solubility, δ , should be used for value of δ as recommended for n-alkylbenzenes(C 6 − C 24 )
by Won [24]. If a value of liquid solubility given in Tables 6.10 (6.157)
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