Page 283 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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AT029-Manual
AT029-06
6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 263
E
E
where f is the fugacity of pure i at T and P of the system.
i S AT029-Manual-v7.cls June 22, 2007 20:46 Eq. (6.138). Similarly V and H can be calculated from γ i
In wax precipitation usually the solid solution is considered and H and V of the solution may be calculated from the fol-
S
S
ideal and γ is assumed as unity [17]. x is the mole fraction lowing relations:
i
i
of solid i in the solid phase solution. Here the term solution
∂ ln γ i
id
means homogeneous mixture of solid phase. As it will be seen H = H − RT 2 x i

in the next chapter these relations can also be used to deter- (6.166) ∂T P,x i
mine the conditions at which hydrates are formed. id ∂ ln γ i
Calculation of fugacity of pure solids through Eq. (6.155) V = V + RT x i ∂P
T,x i
is useful for SLE calculations where the temperature is above
the triple-point temperature (T tp ). When temperature is less Once G, H, and V are known, all other properties can be cal-
than T tp we have solid–vapor equilibrium as shown in Fig. culated from appropriate thermodynamic relations discussed
5.2a. For such cases the relation for calculation of fugacity of in Section 6.1.
pure solids can be derived from fugacity of pure vapor and Another common way of determining thermophysical pro-
effect of pressure on vapor phase fugacity similar to deriva- perties is through thermodynamic diagrams. In these dia-
L
L
tion of Eq. 6.136, where f , P i sat , and V should be replaced grams various properties such as H, S, V, T, and P for both
i
i
S
S
by f , P sub , and V , respectively. However at T < T tp , P sub or liquid and vapor phases of a pure substance are graphically
i i i i
solid–vapor pressure is very low and φ i sat is unity. Furthermore shown. One type of these diagrams is the P–H diagram that
S
molar volume of solid, V is constant with respect to pressure is shown in Fig. 6.12 for methane as given by the GPA [28].
i
(see Problem 6.15). Such diagrams are available for many industrially important
pure compounds [28]. Most of these thermodynamic charts
and computer programs were developed by NIST [29]. Val-
6.7 GENERAL METHOD FOR CALCULATION ues used to construct such diagrams are calculated through
OF PROPERTIES OF REAL MIXTURES thermodynamic models discussed in this chapter. While these
diagrams are easy to use, but it is hard to determine an accu-
Two parameters have been defined to express nonideality of rate value from the graph because of difficulty in reading the
a system, fugacity coefficient and activity coefficient. Fugac- values. In addition they are not suitable for computer appli-
ity coefficient indicates deviation from ideal gas behavior and cations. However, these figures are useful for the purpose of
activity coefficient indicates deviation from ideal solution be- evaluation of an estimated property from a thermodynamic
havior for liquid solutions. Once residual properties (devia- model. Other types of these diagrams are also available. The
tion from ideal gas behavior) and excess properties (deviation H–S diagram known as Mollier diagram is usually used to
from ideal solution behavior) are known, properties of real graphically correlate properties of refrigerant fluids.
mixtures can be calculated from properties of ideal gases or
real solutions. Properties of real gas mixtures can be calcu-
lated through residual properties. For example, applying the 6.8 FORMULATION OF PHASE EQUILIBRIA
definition of residual property to G we get PROBLEMS FOR MIXTURES
ig
(6.162) G = G + G R In this section equations needed for various phase equi-
ig
R
where G is the residual Gibbs energy (defined as G − G ). librium calculations for mixtures are presented. Two cases
R
G is related to ˆ φ i by Eq. (6.128), which when combined with of vapor–liquid equilibria (VLE) and liquid–solid equilibria
the above equation gives (LSE) are considered due to their wide application in the
petroleum industry, as will be seen in Chapter 9.

ig
(6.163) G = G + RT y i ln ˆ φ i
Furthermore from thermodynamic relations one can show 6.8.1 Criteria for Mixture Phase Equilibria
that [1]
The criteria for phase equilibrium is set by minimum Gibbs

ig 2 ∂ ln ˆ φ i free energy, which requires derivative of G to be zero at the
H = y i H − RT y i
i
∂T conditions where the system is in thermodynamic equilib-
(6.164) P,y i rium as shown by Eq. (6.108). Gibbs energy varies with T, P,
ig ∂ ln ˆ φ i
V = y i V + RT y i and x i . At fixed T and P, one can determine x i that is when G is
i
∂P
T,y i minimized or at a fixed T (or P) and x i , equilibrium pressure
(or temperature) can be found by minimizing G. At different
Calculation of properties of ideal gases have been discussed
in Section 6.3, therefore, from the knowledge of fugacity co- pressures functionality of G with x i at a fixed temperature
efficients one can calculate properties of real gases. varies. Baker et al. [29] have discussed variation of Gibbs en-
Similarly for real liquid solutions a property can be cal- ergy with composition. A typical curve is shown in Fig. 6.13.
culated from the knowledge of excess property. Properties of To avoid a false solution to find equilibrium conditions, there
ideal solutions are given by Eqs. (6.89)–(6.92). Property of is a second constraint set by the second derivative of G as
a real solution can be calculated from knowledge of excess [17, 20, 30]
property and ideal solution property using Eq. (6.83): (∂G) T,P = 0
(6.167) 2
ig
(6.165) M = M + M E ∂ G T,P > 0
E
where M is the excess property and can be calculated from This discussion is known as stability criteria and it has re-
E
activity coefficients. For example, G can be calculated from ceived significant attention by reservoir engineers in analysis
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