PNG 520
Phase Relations in Reservoir Engineering

Expressions for Fugacity Calculation

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It is clear that, if we want to take advantage of the fugacity criteria to perform equilibrium calculations, we need to have a means of calculating it. Let us develop a general expression for fugacity calculations. Let us begin with the definition of fugacity in terms of chemical potential for a pure component shown in (16.21a):

dμ=RTd In f @ const T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.26)
 

The Maxwell’s Relationships presented in equation (15.27c) is written for a pure component system as:

( μ P ) r = V ¯ = v ˜ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.27)
 

Consequently,

dμ= v ˜ dP @ const T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.28)
 

Substituting (16.28) into (16.26),

RTd In f= v ˜ dP @ const T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.29)
 

Introducing the concept of fugacity coefficient given in equation (16.23a),

ϕ= f P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.23a)
 
ln ϕ=ln f- ln P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.30)
 

We end up with:

RTd ln ϕ= v ˜ dPRTd ln P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.31a)
 

or equivalently,

RTd ln ϕ= v ˜ dPRT dP P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.31b)
 

Integrating expression (16.31b),

ln ϕ m lnϕ d ln ϕ= P m P { v ˜ RT 1 P } dP This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.32)
 

It is convenient to define the lower limit of integration as the ideal state, for which the values of fugacity coefficient, volume, and compressibility factor are known.

At the ideal state, in the limit P>0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ,

ϕ * >1ln ϕ * >0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.33)
 

Substituting into (16.32),

E ln ϕ= 0 P { v ˜ RT 1 P }dP This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.34)
 

Equation (16.34) is the expression of fugacity coefficient as a function of pressure, temperature, and volume. Notice that this expression can be readily rewritten in terms of compressibility factor:

ln ϕ= 0 P ( P v ˜ RT 1 P )dP= 0 P { Z1 P } dP This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.35)
 

Let us also derive the expression for the fugacity coefficient for a component in a multicomponent mixture. Following a pattern similar to that which we have presented, beginning with the definition of fugacity for a component in terms of chemical potential:

d μ i =RTd ln  f i @const T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.36)
 

This time, it is more convenient to use the Maxwell’s Relationships presented in equation (15.27d):

( μ i V ) T,n = ( P n i ) T,V, n i1 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.37)
 

After you introduce the definitions of fugacity coefficient and compressibility factor:

ϕ i = f i y i P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.38a)
 
P= ZnRT V This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.38b)
 

and recalling that our lower limit of integration is the ideal state, for which, at the limit P>0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. :

V * > This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.39c)
 
ϕ i * >1 and hence lnϕ i * >0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.39a)
 
z * >1 and hence ln Z * >0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.39b)
 

it can be proven that:

ln  ϕ i = 1 RT v { RT V ( P n i ) T,V, n i1 }dVlnZ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.40)
 

The multi-component mixture counterpart of equation (16.35) becomes:

ln  ϕ i = 0 P { Z ¯ i 1 } dP P This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.41a)
 

where:

Z ¯ i = ( Z n i ) P,T, n i1 = P RT ( V n i ) P,T, n i1 = P V i ¯ RT This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(16.41b)
 

Equations (16.34), (16.35), (16.40), and (16.41) are very important for us. Basically, they show that fugacity, or the fugacity coefficient, is a function of pressure, temperature and volume:

f=f(P,V,T) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
 

This tells us that if we are able to come up with a PVT relationship for the volumetric behavior of a substance, we can calculate its fugacity by solving such expressions. It is becoming clear why we have studied equations of state — they are just what we need right now: PVT relationships for various substances. Once we have chosen the equation of state that we want to work with, we can calculate the fugacity of each component in the mixture by applying the above expression. Now that we know how to calculate fugacity, we are ready to apply the criteria for equilibrium that we just studied! That is the goal of the next module.