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Modern Engineering Thermodynamics

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284 CHAPTER 9: Second Law Open System Applications<br />

Note that the entropy production rate in Example 9.1 is actually positive if there is no heat transfer into the<br />

water ( _Q = 0). This is because the water temperature could increase from 15ºC to50ºC by internal viscous friction<br />

alone. However, that would require a really long pipe inside the water heater and a pretty big pressure<br />

drop (which we neglected in the solution).<br />

9.5 NOZZLES, DIFFUSERS, AND THROTTLES<br />

In Chapter 6, we see that nozzles, diffusers, and throttling devices are normally steady state, steady flow,<br />

single-inlet, single-outlet open systems with approximately constant surface temperature and they may or may<br />

not be adiabatic. Therefore, Eq. (9.10) can be applied to all three of these types of open systems. Solving<br />

Eq. (9.10) for _S P gives<br />

1. For adiabatic nozzles, diffusers, and throttling devices:<br />

2. For isothermal surface nozzles, diffusers, and throttling devices:<br />

where the second law condition that _S P > 0 has been added.<br />

_S P = _m ðs out − s in Þ> 0 (9.12)<br />

Q<br />

_S P = _m ðs out − s in Þ− _ > 0 (9.13)<br />

T b<br />

If the fluid flowing through these systems is incompressible and has a constant specific heat c, thenfrom<br />

Chapter 7, we have<br />

Equations (9.12) and (9.13) then become<br />

s out − s in = c ln T out<br />

T in<br />

(7.33)<br />

Entropy production rate of an incompressible fluid in an adiabatic nozzle, diffuser, or throttle<br />

_S P<br />

adiabatic = _mc ln T out<br />

> 0 (9.14)<br />

incompressible T<br />

<br />

in<br />

fluid<br />

and<br />

Entropy production rate of an incompressible fluid in a nozzle, diffuser, or throttle with heat transfer<br />

_S P<br />

incompressible = _mc ln T out Q<br />

− _ > 0 (9.15)<br />

T in T b<br />

fluid<br />

Equation (9.14) shows us that the outlet temperature must always be greater than the inlet temperature for an<br />

insulated (adiabatic) open system with an incompressible fluid. This is because all the dissipation due to the<br />

irreversibilities within the system simply goes into increasing the temperature of an incompressible fluid.<br />

Nozzles, diffusers, and throttling devices are all physically small (i.e., Z out ≈ Z in ), aergonic systems, so their<br />

modified energy rate balance becomes<br />

<br />

_Q = _m h out − h in + V2 out − <br />

V2 in<br />

2g c<br />

and, for constant specific heat incompressible fluids, we have<br />

(9.16)<br />

h out − h in = cT ð out − T in Þ+ vp ð out − p in Þ (6.19)<br />

Combining Eqs. (9.15), (9.16), and (6.19) gives the combined nonadiabatic first and second law relation for an<br />

incompressible fluid with a constant specific heat as<br />

Equation ð9:15Þ with the heat transfer rate evaluated using the ERB<br />

<br />

_S P incompressible = _m cln T out<br />

− cT ð out − T in Þ<br />

− vp ð out − p in Þ<br />

− V2 out − <br />

V2 in<br />

(9.17)<br />

fluid<br />

T in T b<br />

T b 2g c T b

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