Page 245 - The Mechatronics Handbook

0066_frame_C12 Page 15 Wednesday, January 9, 2002 4:22 PM

The subscript f denotes the saturated liquid state at the temperature T, and p sat is the corresponding
saturation pressure. The underlined term of Eq. (12.17c) is usually negligible, giving h(T, p) ≈ h f (T).
Graphical representations of property data also are commonly used. These include the p-T and p-v
diagrams of Fig. 12.1, the T-s diagram of Fig. 12.2, the h-s (Mollier) diagram of Fig. 12.3, and the p-h
diagram of Fig. 12.4. The compressibility charts considered next use the compressibility factor as one of
the coordinates.

Compressibility Charts
The p-v-T relation for a wide range of common gases is illustrated by the generalized compressibility
chart of Fig. 12.5. In this chart, the compressibility factor, Z, is plotted vs. the reduced pressure, p R , reduced
where
temperature, T R , and pseudoreduced specific volume, v′ R

pv
Z = ------- (12.18)
RT
3
R
v
In this expression is the specific volume on a molar basis (m /kmol, for example) and is the universal
⋅
gas constant (8314 N m⋅ /kmol K, for example ). The reduced properties are
T
p
v
p R = ----, T R = -----, v ′ R = -------------------- (12.19)
p c T c ( RT c p c )
where p c and T c denote the critical pressure and temperature, respectively. Values of p c and T c are obtainable
from the literature—see, for example, Moran and Shapiro (2000). The reduced isotherms of Fig. 12.5
represent the best curves fitted to the data of several gases. For the 30 gases used in developing the chart,
the deviation of observed values from those of the chart is at most on the order of 5% and for most
ranges is much less.

Analytical Equations of State
Considering the isotherms of Fig. 12.5, it is plausible that the variation of the compressibility factor might
be expressed as an equation, at least for certain intervals of p and T. Two expressions can be written that
enjoy a theoretical basis. One gives the compressibility factor as an infinite series expansion in pressure,

Z = 1 + B T()p + C T()p + D T()p + … (12.20a)
ˆ
3
2
ˆ
ˆ
and the other is a series in 1/ , v
BT() CT() DT()
Z = 1 + ------------ + ------------ + ------------- + … (12.20b)
v v 2 v 3
ˆ
ˆ
ˆ
Such equations of state are known as virial expansions, and the coefficients , C D … and B, C, D…
B
,
are called virial coefficients. In principle, the virial coefficients can be calculated using expressions from
statistical mechanics derived from consideration of the force fields around the molecules. Thus far the
first few coefficients have been calculated for gases consisting of relatively simple molecules. The coeffi-
cients also can be found, in principle, by fitting p-v-T data in particular realms of interest. Only the first
few coefficients can be found accurately this way, however, and the result is a truncated equation valid
only at certain states.
Over 100 equations of state have been developed in an attempt to portray accurately the p-v-T behavior
of substances and yet avoid the complexities inherent in a full virial series. In general, these equations
exhibit little in the way of fundamental physical significance and are mainly empirical in character. Most
are developed for gases, but some describe the p-v-T behavior of the liquid phase, at least qualitatively.

©2002 CRC Press LLC