Page 257 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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AT029-Manual
20:46
6. THERMODYNAMIC RELATIONS FOR PROPERTY ESTIMATIONS 237
For practical applications the above equation is converted
into dimensionless form in terms of parameters Z defined by Solution—By substituting the Maxwell’s relation of Eq. (6.15)
into Eq. (6.30) we get
Eq. (5.15). Differentiating Z with respect to T at constant P, ∂ H ∂V
from Eq. (5.15) we get (6.40) = V − T
∂P T ∂T P
∂ Z P ∂V PV 1

(6.34) = + − differentiating this equation with respect to T at constant P
∂T RT ∂T R T 2 gives
P P
2
Dividing both sides of Eq. (6.33) by RT and combining with ∂ ∂ H ∂V ∂V ∂ V
= − + T
Eq. (6.34) gives ∂T ∂P ∂T ∂T ∂T 2
T P P P P
∂ V
2
H − H ig P ∂ Z dP (6.41) =−T ∂T 2
(6.35) =−T (at constant T ) P
RT ∂T P P
0 From mathematical identity we have
∂ ∂ H ∂ ∂ H
It can be easily seen that for an ideal gas where Z = 1, (6.42) =
ig
Eq. (6.35) gives the expected result of H − H = 0. Similarly ∂T ∂P T P ∂P ∂T P T
for any equation of state the residual enthalpy can be calcu- Using definition of C P through Eq. (6.20) and combining the
lated. Using definitions of T r and P r by Eq. (5.100), the above above two equations we get
equation may be written as
2
∂ V
(6.43) ∂C P =−T 2
H − H ig P r ∂ Z dP r ∂P T ∂T P
(6.36) =−T r 2 (at constant T )
RT c ∂T r P r P r Upon integration from P = 0 to the desired pressure of P at
0 constant T we get
where the term in the left-hand side and all parameters in the P 2
ig
RHS of the above equation are in dimensionless forms. Once (6.44) C P − C = −T ∂ V 2 dP
P
the residual enthalpy is calculated, real gas enthalpy can be P=0 ∂T P T
determined as follows:
Once C ig is known, C P can be determined at T and P of in-
H − H terest from an EOS, PVT data, or generalized corresponding
ig P
ig
(6.37) H = H + RT c
RT c states correlations.
In general, absolute values of enthalpy are of little interest
and normally the difference between enthalpies in two differ- 6.1.4 Fugacity and Fugacity Coefficient
ent conditions is useful. Absolute enthalpy has meaning only for Pure Components
with respect to a reference state when the value of enthalpy Another important auxiliary function that is defined for cal-
is assigned as zero. For example, tabulated values of enthalpy culation of thermodynamic properties, especially Gibbs free
in steam tables are with respect to the reference state of satu- energy, is called fugacity and it is shown by f . This parameter
rated liquid water at 0 C [1]. As the choice of reference state is particularly useful in calculation of mixture properties and
◦
changes so do the values of absolute enthalpy; however, this formulation of phase equilibrium problems. Fugacity is a pa-
change in the reference state does not affect change in en- rameter similar to pressure, which indicates deviation from
thalpy of systems from one state to another. ideal gas behavior. It is defined to calculate properties of real
A relation similar to Eq. (6.33) can be derived in terms of gases and it may be defined in the following form:
volume where the gas behavior becomes as an ideal gas as

V →∞: (6.45) lim f = 1
P→0 P
∂P With this definition fugacity of an ideal gas is the same as
V

ig
(6.38) (H − H ) T,V = T − P dV + PV − RT
∂T V its pressure. One main application of fugacity is to calculate
Gibbs free energy. Application of Eq. (6.8) at constant T to an
V→∞
Similar relation for the entropy departure is ideal gas gives
ig
(6.46) dG = RTdln P
∂P R
V
ig
(6.39) (S − S ) T,V = − dV For a real fluid a similar relation can be written in terms of
∂T V
V fugacity
V→∞
(6.47) dG = RTdln f
Once H is known, U can be calculated from Eq. (6.1). Sim-
ilarly all other thermodynamic properties can be calculated where for an ideal gas f ig = P. Subtracting Eq. (6.46) from
from basic relations and definitions. (6.47), the residual Gibbs energy, G , can be determined
R
through fugacity:
Example 6.1—Derive a relation for calculation of C P from G R G − G ig f
PVT relation of a real fluid at T and P. (6.48) = = ln = ln φ
RT RT P
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