Page 256 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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236 CHARACTERIZATION AND PROPERTIES OF PETROLEUM FRACTIONS
There are two other measurable properties: coefficient of
P
thermal expansion, β, and the bulk isothermal compressibility, Values of C ig are known for many compounds and they are
given in terms of temperature in various industrial handbooks
κ. These are defined as [5]. Once C ig is known, C , U , H , and S can also be de-
ig
ig
ig
ig
P V
1 ∂V termined from thermodynamic relations discussed above. To
(6.24) β =
V ∂T P calculate properties of a real gas an auxiliary function called
residual property is defined as the difference between prop-
1 ∂V ig
(6.25) κ =− erty of real gas and its ideal gas property (i.e., H − H ). The
V ∂P T difference between property of a real fluid and ideal gas is
since ∂V/∂P is negative, the minus sign in the definition of also called departure from ideal gas. All fundamental relations
κ is used to make it a positive number. The units of β and also apply to residual properties. By applying basic thermo-
−1
κ in SI system are K −1 and Pa , respectively. Values of β dynamic and mathematical relations, a residual property can
and κ can be calculated from these equations with use of an be calculated through a PVT relation of an equation of state.
equation of state. For example, with use of Lee–Kesler EOS If only two properties such as H and G or H and S are known
(Eq. 5.104), the value of κ is 0.84 × 10 −9 Pa −1 for liquid ben- in addition to values of V at a given T and P, all other prop-
zene at temperature of 17 C and pressure of 6 bar, while the erties can be easily determined from basic relations given in
◦
actual measured value is 0.89 × 10 −9 Pa −1 [6]. Once β and this section. For example from H and G, entropy can be calcu-
κ are known for a fluid, the PVT relation can be established lated from Eq. (6.6). Development of relations for calculation
for that fluid (see Problem 6.1). Through the above thermo- of enthalpy departure is shown here. Other properties may be
dynamic relations and definitions one can show that calculated through a similar approach.
Assume that we are interested to relate residual enthalpy
TVβ 2 (H − H ) into PVT at a given T and P. For a homogenous
ig
(6.26) C P − C V =
κ fluid of constant composition (or pure substance), H can be
Applying Eqs. (6.24) and (6.25) for ideal gases (Eq. 5.14) considered as a function of T and P:
ig
ig
gives β = 1/T and κ = 1/P. Substituting β ig and κ ig into (6.28) H = H(T, P)
Eq. (6.26) gives Eq. (6.23). From Eq. (6.26) it is clear that
C P > C V ; however, for liquids the difference between C P and Applying Eq. (6.10) gives
C V is quite small and most thermodynamic texts neglect this ∂ H ∂ H
difference and assume C P = C V . Most recently Garvin [6] has (6.29) dH = dT + dP
∼
reviewed values of constant volume specific heats for liquids ∂T P ∂P T
and concludes that in some cases C P − C V for liquids is signif- Dividing both sides of Eq. (6.5) to ∂P at constant T gives
icant and must not be neglected. For example, for saturated
liquid benzene when temperature varies from 300 to 450 K, (6.30) ∂ H = V + T ∂S
the calculated heat capacity ratio, C P /C V , varies from 1.58 ∂P T ∂P T
to 1.41 [6]. Although these values are not yet confirmed as Substituting for (∂S/∂P) T from Eq. (6.15) into Eq. (6.30)
they have been calculated from Lee–Kesler equation of state, and substitute resulting (∂ H/∂P) T into Eq. (6.29) with use of
but one should be careful that assumption of C P = C V for liq- Eq. (6.20) for (∂ H/∂T) P , Eq. (6.29) becomes
∼
uids in general may not be true in all cases. In fact for ideal

incompressible liquids, β → 0 and κ → 0 and according to (6.31) dH = C P dT + V − T ∂V dP
Eq. (6.26), (C P − C V ) → 0, which leads to γ = C P /C V → 1. ∂T P
There is an EOS with high accuracy for benzene [7]. It gives where the right-hand side (RHS) of this equation involves
C p /C v for saturated liquids having a calculated heat capacity measurable quantities of C P and PVT, which can be deter-
ratio of 1.43–1.38 over a temperature range of 300–450 K. mined from an equation of state. Similarly it can be shown
Another useful property is Joule–Thomson coefficient that that
is defined as
dT ∂V
∂T (6.32) dS = C P − dP

(6.27) η = T ∂T P
∂P
H
Equations (6.31) and (6.32) are the basis of calculation of en-
This property is useful in throttling processes where a fluid thalpy and entropy and all other thermodynamic properties
passes through an expansion valve at which enthalpy is nearly of a fluid from its PVT relation and knowledge of C P .Asanex-
constant. Such devices are useful in reducing the fluid pres-
sure, such as gas flow in a pipeline. η expresses the change of ample, integration of Eq. (6.31) from (T 1 , P 1 )to(T 2 , P 2 ) gives
change of enthalpy ( H) for the process. The same equation
temperature with pressure in a throttling process and can be can be used to calculate departure functions or residual prop-
related to C P and may be calculated from an equation of state erties from PVT data or an equation of state at a given T and
(see Problem 6.10).
P. For an ideal gas the second term in the RHS of Eq. (6.32)
is zero. Since any gas as P → 0 behaves like an ideal gas, at a
6.1.3 Residual Properties and fixed temperature of T, integration of Eq. (6.31) from P → 0
Departure Functions to a desired pressure of P gives
Properties of ideal gases can be determined accurately P ∂V
ig
through kinetic theory. In fact all properties of ideal gases are (6.33) (H − H ) T = V − T dP (at constant T )
known or they can be estimated through the ideal gas law. ∂T P
0
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