Modern Engineering Thermodynamics

284 CHAPTER 9: Second Law Open System Applications
Note that the entropy production rate in Example 9.1 is actually positive if there is no heat transfer into the
water ( _Q = 0). This is because the water temperature could increase from 15ºC to50ºC by internal viscous friction
alone. However, that would require a really long pipe inside the water heater and a pretty big pressure
drop (which we neglected in the solution).
9.5 NOZZLES, DIFFUSERS, AND THROTTLES
In Chapter 6, we see that nozzles, diffusers, and throttling devices are normally steady state, steady flow,
single-inlet, single-outlet open systems with approximately constant surface temperature and they may or may
not be adiabatic. Therefore, Eq. (9.10) can be applied to all three of these types of open systems. Solving
Eq. (9.10) for _S P gives
1. For adiabatic nozzles, diffusers, and throttling devices:
2. For isothermal surface nozzles, diffusers, and throttling devices:
where the second law condition that _S P > 0 has been added.
_S P = _m ðs out − s in Þ> 0 (9.12)
Q
_S P = _m ðs out − s in Þ− _ > 0 (9.13)
T b
If the fluid flowing through these systems is incompressible and has a constant specific heat c, thenfrom
Chapter 7, we have
Equations (9.12) and (9.13) then become
s out − s in = c ln T out
T in
(7.33)
Entropy production rate of an incompressible fluid in an adiabatic nozzle, diffuser, or throttle
_S P
adiabatic = _mc ln T out
> 0 (9.14)
incompressible T
in
fluid
and
Entropy production rate of an incompressible fluid in a nozzle, diffuser, or throttle with heat transfer
_S P
incompressible = _mc ln T out Q
− _ > 0 (9.15)
T in T b
fluid
Equation (9.14) shows us that the outlet temperature must always be greater than the inlet temperature for an
insulated (adiabatic) open system with an incompressible fluid. This is because all the dissipation due to the
irreversibilities within the system simply goes into increasing the temperature of an incompressible fluid.
Nozzles, diffusers, and throttling devices are all physically small (i.e., Z out ≈ Z in ), aergonic systems, so their
modified energy rate balance becomes
_Q = _m h out − h in + V2 out −
V2 in
2g c
and, for constant specific heat incompressible fluids, we have
(9.16)
h out − h in = cT ð out − T in Þ+ vp ð out − p in Þ (6.19)
Combining Eqs. (9.15), (9.16), and (6.19) gives the combined nonadiabatic first and second law relation for an
incompressible fluid with a constant specific heat as
Equation ð9:15Þ with the heat transfer rate evaluated using the ERB
_S P incompressible = _m cln T out
− cT ð out − T in Þ
− vp ð out − p in Þ
− V2 out −
V2 in
(9.17)
fluid
T in T b
T b 2g c T b