Page 274 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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254 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
where T M is the melting point temperature at pressure P.If
o
case every phase has different composition but all have the
atmospheric pressure is shown by P (1.01325 bar) and the equilibrium similar criteria must apply to every phase. In this
o
melting point at P is shown by T Mo (normal melting point), same T and P. We know for mechanical equilibrium, total
o
integration of the above equation from P to pressure P gives energy (i.e., kinetic and potential) of the system must be min-
V × (P − P ◦)
M imum. The best example is oscillation of hanging object that
(6.107) T M = T Mo exp it comes to rest when its potential and kinetic energies are
H M
minimum at the lowest level. For thermodynamic equilib-
where in deriving this equation it is assumed that both V M rium the criterion is minimum Gibbs free energy. As shown
M
and H are constants with respect to temperature (melting by Eq. (6.73) a mixture molar property such as G varies with
point). This is a reasonable assumption since variation of T M T and P and composition. A mathematical function is mini-
with pressure is small (see Fig. 5.2a). Since this equation is mum when its total derivative is zero:
derived for pure substances, T M is the same as freezing point (6.108) dG(T, P, x i ) = 0
--`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
M
(T f ) and H is the same as heat of fusion ( H f ).
(a) To calculate melting point of n-C 18 at 300 bar, we have Schematic and criteria for VLE of multicomponent systems
P = 300 bar, P o = 1.01325 bar, T Mo = 301.4K,and V M = are shown in Fig. 6.7b. Phase equilibria calculations lead to
3
M
L
S
1/ρ − 1/ρ = 0.1313 cm /g. H = 242.4597 J/g, 1/J = 10 determination of the conditions of T, P, and composition at
3
bar · cm , thus from Eq. (6.107) we have T M = 301.4 × exp which the above criteria are satisfied. In this section general
[0.1313 × (300 − 1.013)/10 × 242459.7] = 301.4 × 1.0163 = formulas for phase equilibria calculations of mixtures are pre-
301.4 × 1.0163 = 306.3 K or T M = 33.2 C. This indicates sented. These are required to define new parameters such as
◦
that when pressure increases to 300 bar, the melting point activity, activity coefficient, and fugacity coefficient of a com-
of n-C 18 increases only by 5 C. In this temperature range as- ponent in a mixture. Two main references for thermodynam-
◦
sumption of constant V M and H M is quite reasonable. ics of mixtures in relation with equilibrium are Denbigh [19]
(b) To calculate the triple point temperature, Eq. (6.107) and Prausnitz et al. [21].
must be applied at P = P tp = 3.39 × 10 −5 kPa = 3.39 ×
10 −7 bar. This is a very low number in comparison
6.6.1 Definition of Fugacity, Fugacity
with P o = 1 bar, thus T M = 301.4 × exp(−0.1313 × 1.013/10 × Coefficient, Activity, Activity Coefficient,
242459.7) = 301.4 × 0.99995 = 301.4 K. Thus, we get triple and Chemical Potential
∼
point temperature same as melting point. This is true for most
of pure substances as P tp is very small. It should be noted that In this section important properties of fugacity, activity, and
Eq. (6.107) is not reliable to calculate pressure at which melt- chemical potential needed for formulation of solution ther-
ing point is known because a small change in temperature modynamics are defined and methods of their calculation are
causes significant change in pressure. This example explains presented. Consider a mixture of N components at T and P
why melting point of water decreases while for n-octadecane and composition y i . Fugacity of component i in the mixture
ˆ
it increases with increase in pressure. As it is shown in Sec- is shown by f i and defined as
tion 7.2 density of ice is less than water, thus V M for water ˆ
is negative and from Eq. (6.107), T M is less than T Mo at high (6.109) lim f i → 1
pressures. y i P P→0
where sign ∧ indicates the fact that component i is in a mix-
ˆ
ture. When y i → 1 we have f i → f i , where f i is fugacity of
6.6 PHASE EQUILIBRIA OF
MIXTURES—CALCULATION pure i as defined in Eq. (6.45). The fugacity coefficient of i in
OF BASIC PROPERTIES a mixture is defined as
ˆ
(6.110) ˆ φ i = f i
Perhaps one of the biggest applications of equations of state y i P
and thermodynamics of mixtures in the petroleum science ˆ
is formulation of phase equilibrium problems. In petroleum where for an ideal gas, ˆ φ i = 1or f i = y i P. In a gas mixture
production phase equilibria calculations lead to the determi- y i P is the same as partial pressure of component i. Activity of
nation of the composition and amount of oil and gas produced component i,ˆa i , is defined as
at the surface facilities in the production sites, PT diagrams to f i ˆ
determine type of hydrocarbon phases in the reservoirs, solu- (6.111) ˆ a i = f ◦
bility of oil in water and water in oils, compositions of oil and i
◦
gas where they are in equilibrium, solubility of solids in oils, where f is fugacity of i at a standard state. One common
i
and solid deposition (wax and asphaltene) or hydrate forma- standard state for fugacity is pure component i at the same
◦
tion due to change in composition or T and P. In petroleum T and P of mixture, that is to assume f = f i , where f i is the
i
processing phase equilibria calculations lead to the determi- fugacity of pure i at T and P of mixture. This is usually known
nation of vapor pressure and equilibrium curves needed for as standard state base on Lewis rule. Choice of standard state
design and operation of distillation, absorption, and stripping for fugacity and chemical potential is best discussed by Den-
columns. bigh [19]. Activity is a parameter that indicates the degree of
A system is at equilibrium when there is no tendency to nonideality in the system. The activity coefficient of compo-
change. In fact for a multicomponent system of single phase nent i in a mixture is shown by γ i and is defined as
to be in equilibrium, there must be no change in T, P, and ˆ a i
x 1 , x 2 ,..., x N−1 . When several phases exist together while at (6.112) γ i =
x i
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