Page 403 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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9. APPLICATIONS: PHASE EQUILIBRIUM CALCULATIONS 383
(i.e., the last pseudocomponent of a C 7+ fraction). If Eq. (9.24)
is applied to all N components in the mixture the number in the initial fluid. The ratio of S/F is the same as S F used in
Eq. (9.17). The ratio of V/F in Eqs. (9.26) and (9.27) is the
of components that satisfy this equation is designated as N S same as V F in Eqs. (9.17) and (9.18).
(<N). If N S = N it means that the mixture at T and P is initially The above set of equations can be solved by converting them
in a solid phase (100% solid). All precipitating components into equations similar to Eqs. (9.17)–(9.19). For precipitating --`,```,`,``````,`,````,```,,-`-`,,`,,`,`,,`---
L
must satisfy the following isofugacity equations: components, x can be calculated directly from Eq. (9.26),
i
while for nonprecipitating components they must be calcu-
ˆ L
S
L
(9.25) f (T, P, x ) = f (T, P) i = (N − N S + 1), ... , N
i i i lated from Eq. (9.27) after finding V/F and S/F. Moles of
S
The material balance equation for the nonprecipitating com- solid formed for each component, n , must be calculated from
j
ponents is Eq. (9.27). Values of V/F and S/F must be found by trial-and-
error procedure so that Eq. (9.29) is satisfied.

N n S V V
x
z i − x L 1 − j − − K VL L = 0 The CPT of a crude can be calculated directly from
i F F i i F
j=(N−N S +1) Eq. (9.24) using trial-and-error procedure as follows:
(9.26) i = 1, ... ,(N − N S )
a. Define the mixture and break C 7+ into appropriate number
S
where n is the moles of solid phase j and F is the number of pseudocomponents as discussed in Chapter 4.
j
of moles of feed (initial fluid mixture). For the precipitating b. P and z i are known for all component/pseudocomponents.
components where all solid phases are pure
c. Guess a temperature that is higher than melting point of the
N n V n V heaviest components in the mixture so that no component
S
S

z i − x i L 1 − j − − j − K VL L i = 0 in the mixture satisfies Eq. (9.24).
x
i
j=(N−N S +1) F F F F
d. Reduce the temperature stepwise until at least one com-
(9.27) [i = (N − N S + 1), ... , N − 1], (N S > 1) ponent (it must be the heaviest component) satisfies the
equality in Eq. (9.24).
In addition, all components must satisfy the following VLE e. Record the temperature as calculated CPT of the crude oil.
isofugacity:
ˆ L
ˆ V
L
(9.28) f (T, P, y i ) = f (T, P, x ) i = 1, ... , N A schematic of CPT and wax precipitation calculation using
i
i
i
There are two constraint equations for component i in the this model is illustrated in Fig. 9.17. To simplify and reduce
liquid and vapor phases: the size of the calculations, Lira-Galeana et al. [43] suggest
that solid phases can be combined into three or four groups
N N
L where each group can be considered as one pesudocompo-
(9.29) x = y i = 1
i
i=1 i=1 nent. As the temperature decreases, the amount of precip-
Equation (9.28) is equivalent to Eq. (6.201) in terms of VLE itation increases. Compositions of six crude oils as well as
ratios (K VL ). There are N S equations through Eq. (9.24), their experimental and calculated values of CPT according
i
(N − N S ) equations through Eq. (9.26), (N S − 1) equations to this model are given in Table 9.10. Calculated values of
through Eq. (9.27), N equations through Eq. (9.28), and two CPT very much depend on the properties (especially molecu-
equations through Eq. (9.29). Thus the total number of equa- lar weight) of the heaviest component in the mixture. For oils
L
tions are 2N + N S + 1. The unknowns are x (N unknowns), the C 7+ fractions should be divided into several pseudocom-
i
S
y i (N unknowns), n (N S unknowns), and V/F (one unknown), ponents according to the methods discussed in Chapter 4. In
j
with the sum of unknown same as the number of equations such cases, the heaviest component in the mixture is the last
(2N + N S + 1). Usually for crude oils and heavy residues, pseudocomponent of the C 7+ and the value of its molecular
where under the conditions at which solid is formed, the weight significantly affects the calculated CPT. In such cases,
amount of vapor is small and V/F can be ignored in the above the molecular weight of last pseudocomponent C 7+ may be
equations. For such cases Eq. (9.28) and y i = 0 in Eq. (9.29) used as one of the adjustable parameters to match calculated
can be removed from the set of equations. On this assumption, amount of wax precipitation with the experimental values.
the number of equations and unknowns reduces by N + 1 and Prediction of the amount of wax precipitation for oils 1 and
y i and V/F are omitted from the list of unknowns. Total num- 6 in Table 9.10 are shown in Fig. 9.18 as generated from the
ber of moles of solid formed (S) is calculated as data provided in Ref. [43].
In the calculation of solid fugacity through Eq. (6.155), C P
N
S is required. In many calculations it is usually considered as
(9.30) S = n j
j=(N−N S +1) zero; however, Lira et al. [43] show that without this term,
considerable error may arise in calculation of solute compo-
The amount of wax precipitated in terms of percent of oil is sition in liquid phase for some oils as shown in Fig. 9.19. Ef-
calculated as fects of temperature and pressure according to the multisolid-
N M i n S phase model are clearly discussed by Pan et al. [17] and for
i=1
(9.31) wax wt% in oil = 100 × N j several oils they have compared predicted CPT with exper-
F z i M i
i=1 imental data at various pressures. They conclude that for
where F is the total number of moles of initial oil and n S j heavy oils at low pressure or live oils (where light gases are
is the moles of component i precipitated as solid. M i is the dissolved in oil) the increase in pressure will decrease CPT as
molecular weight of component i and z i is its mole fraction shown in Fig. 9.20. However, for heavy liquid oils (dead oils)
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