Modern Engineering Thermodynamics

302 CHAPTER 9: Second Law Open System Applications
CASE STUDIES OF ENTROPY PRODUCTION IN OPEN SYSTEMS
The following examples are typical case studies in open system
applied engineering thermodynamics. They are included here to
provide the student with an exposure to a second law analysis of
complex systems typical of 21st century engineering technology.
Energy and entropy rate balance equations are used as tools to
understand these technologies.
Case study 9.1. Temperature magic
A vortex tube is a seemingly magical device that separates the temperature
of its inlet flow stream into hot and cold outlet flow
streams. Remarkably, it contains no moving parts other than the
flowing gas itself.
The inlet flow enters the vortex tube chamber tangentially, and the
resulting swirling motion causes the gaseous core along the chamber
centerline to become extremely cold while the gas near the
chamber wall becomes very hot. Internal baffles allow the core gas
to exit through one tube (cold outlet) while the wall gas exits
through the other (hot outlet), thus producing flow stream temperature
separation. The illustration in Figure 9.18 shows the operation
of this simple device.
3
Hot outlet
Cold side outlet
1
Inlet
(a)
Cold outlet
Compressed air inlet
Flow control valve
(b)
FIGURE 9.18
(a) A typical vortex tube. (b) A schematic showing how the off-center inlet
causes the flow to form a swirl (vortex) inside the tube. The center of the
vortex becomes cold and exits one end of the tube, and the outer part of
the vortex becomes hot and exits the other end on the tube.
2
Hot side outlet
This remarkable device was discovered by Georges Joseph Ranque and
was first described in a French patent in 1931. 6 In 1933, Ranque presented
a paper to the Societe FrancaisedePhysiqueonthisdevice,
and nothing more was heard about it until 1945, when a vortex tube
was found by an American and British investigation team at the end
of World War II in the laboratory of Rudolph Hilsch at the University
of Erlangen, Germany. Hilsch had begun research on the vortex tube
in 1944, after reading Ranque’s paper, and he published his results in
Germany in 1946 and in the United States in 1947. Since then, interest
in the vortex tube has remained high, and it is now frequently
used in industry for inexpensive localized cooling applications.
Applying the energy rate balance to the adiabatic and aergonic vortex
tube shown in Figure 9.18 and assuming ideal gas behavior
with constant specific heats yields
yT ð 1 − T 2 Þ+ T 2 − T 3 = 0
where y = _m 3 / _m 1 = _m hot / _m cold is the hot-side mass fraction defined
by Eq. (9.27). Note that this result is exactly the same as Eq.
(9.30), which was developed for the mixing operation. Thus, the
first law of thermodynamics is insensitive to whether the fluids are
being mixed or separated. It yields the same result in either case.
However, application of the entropy rate balance to the same system
produces
_S P
vortex tube = _m 3½ys ð 1 − s 2 Þ+ s 2 − s 3 Š> 0 (9.49)
and comparing this with Eq. (9.29) yields the remarkable result that
_S P
vortex tube = − _S P
mixing
(9.50)
yet both entropy production rates must be positive. This means that
these two processes cannot simply be the reverse of each other. They
cannot both follow the same thermodynamic path. The vortex tube
separation phenomena must occur by a process that is unavailable
to the simple mixing process and vice versa; otherwise, one of these
devices violates the second law of thermodynamics.
To produce temperature separation, the vortex tube must have a significant
pressure drop between the inlet and the outlet flow streams.
It does not work isobarically. This pressure drop is not necessary in
the mixing operation. Mixing is usually nearly isobaric, and emulation
of the vortex tube separation operation requires a higher mixer outlet
pressure than inlet pressure. This cannot be done without introducing
heat or work energy transport into the system, which would alter the
basic nature of the simple mixing device. Therefore, it is clear that the
vortex tube inlet pressure is the source of the energy needed to produce
the observed temperature separation. It is also the source of the
entropy generation needed to allow Eq. (9.50) to be valid, as shown
in Example 9.9, which follows.
Though the vortex tube is not an isobaric device, its two exit pressures
are essentially equal (i.e., p 1 ≈ p 2 ). Combining this condition
with Eq. (7.37) for ideal gases and substituting the result into Eq.
(9.49) gives
_S P
vortex tube = _m ðT 1 /T 2 Þ y
3 c p ln
1 + yT ð 1 /T 2 − 1Þ + R ln p
3
6 In 1932, he applied for a U.S. patent, which was awarded March 27, 1934
(U.S. patent number 1,952,281).
p 2
(9.51)