Modern Engineering Thermodynamics

11.11 Is Steam Ever an Ideal Gas? 397
WHERE DID THE STEAM TABLES COME FROM?
There is no record of who first took interest in measuring the p-v-T properties of steam, but the development of the steam
engine and the associated Industrial Revolution it produced created a strong practical need for such information. By 1683,
Samuel Morland (1625–1695) is said to have acquired data on the pressure and temperature of saturated steam near atmospheric
pressure. In 1662, Robert Boyle (1627–1691) developed the equation pV = constant for isothermal “elastic fluids”
(i.e., compressible gases), and this equation was used over 100 years later for steam by James Watt (1736–1819) in his
improved steam engine patent of 1782. In about 1787, Jacques Charles (1746–1823) developed the equation for isobaric
gas behavior, V/T = constant, and soon thereafter the laws of Boyle and Charles were combined into the ideal gas
equation 5 that we have today, pV = mRT:
The combined Boyle-Charles ideal gas equation continued to be used for steam engine design until the end of the 19th century,
when engines were operating at sufficiently high pressures and temperatures to render the equation noticeably
inaccurate.
In the 1840s, the French scientist Henri Victor Regnault (1810–1878) was sponsored by his government to carry out a series
of precise measurements of the saturation properties of various substances, including water. He found that the Boyle-
Charles ideal gas equation was only approximately true for real substances. By 1847, he had correlated his experimental
results for the saturation pressure and temperature of steam with the formula given in the problem set at the end of this
chapter. Regnault’s data was considered to be an accurate authoritative source for over 60 years, and many others made
mathematical correlations from it. By the end of the 19th century, many steam tables based on various correlations of
increasing complexity of Regnault’s data had become available for engineering use.
5 Boyle’s law was independently discovered in 1676 by Edme Mariotte (1620–1684) and is sometimes known as Mariotte’s law. Also, Charles’s law was
independently discovered in 1802 by Joseph Louis Gay-Lussac (1778–1850) and is sometimes known as Gay-Lussac’s law. Carnot, Clapeyron, and many
others were using the ideal gas equation in the form pv = R(T + A), where T was in °C and A = 273 °C was then an empirical constant. Later, in
1848, William Thomson (Lord Kelvin) recognized that –273 °C corresponded to absolute zero temperature.
any state near the vapor dome and below the critical point. Under this definition, superheated steam was now a
vapor, not a gas, and it would be unforgivable for a student to apply ideal gas equations to a vapor.
For many years, this subterfuge was successful. But growing student computer literacy and the availability of complex
software containing all the equations necessary to generate accurate steam properties will eventually make the use of
printed tables obsolete. Yet, there remains the nagging question of whether or not the ideal gas equations can be used
to describe the thermodynamic properties of steam with reasonable accuracy in
low-pressure or high-temperature situations. If so, then engineering students could
write relatively simple computer programs to solve challenging thermodynamic
steam (or any other “vapor”) problems without using elaborate software for generating
property values.
It was the pragmatic Scottish engineer William John Macquorn Rankine
(1820–1872), in his Manual of the Steam Engine and Other Prime Movers, 6 who
noted that
Steam attains a condition which is sensibly that of a perfect gas, by means
of a very moderate extent of superheating; and it may be inferred that the
formulae for the relations between heat and work which are accurate for
steam-gas are not materially erroneous for actual superheated steam; while
they possess the practical advantage of great simplicity.
The concept of a region of steam ideal gas behavior is illustrated in Figure
11.12. This figure is a Mollier diagram that shows the regions in which the
equation’s v g = RT sat /p sat , v = RT/p, h = h g + C p (T – T sat )ands = s g + c p ln(T/
T sat ) – R ln(p/p sat ) are accurate to within about 1% or less of the actual steam
table values (where h g and s g are evaluated at T sat ). The use of these ideal gas
equations for steam in the regions shown produces errors of only a few percent
in an analysis, which is often quite acceptable for many engineering thermodynamic
applications. The reader can easily define the regions shown in
Figure 11.12 by using the preceding equations and a steam table.
h, Btu/lbm
1650
1550
1450
1350
1250
1150
1050
950
850
Error in v
less than 1%
Critical
point
Saturated
vapor line
Error in s
less than 1%
Error in h
less than 1%
750 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
s, Btu/lbm . R
FIGURE 11.12
The ideal gas equations are accurate to 1% or less in
the regions to the right of the dashed and dotted lines
shown on this Mollier diagram.
6 This book has the honor of being the first comprehensive engineering thermodynamics textbook. It was first printed in 1859 and
went through 17 editions.