Page 238 - The Mechatronics Handbook

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TABLE 12.1 Energy and Entropy Balances for One-Inlet, One-Outlet Control
Volumes at Steady State and No Heat Transfer
Energy balance
2
2 v e
v i –
˙
W = m ˙ ( h i – h e ) +   --------------- + gz i –( z e ) (12.10b)

2
a
Compressors, pumps, and turbines
W = m ˙ h i – h e ) (12.10c)
(
˙
Throttling
h e ≅ h i (12.10d)
Nozzles, diffusers b
(
v e = v i + 2 h i – h e ) (12.10e)
2
Entropy balance
˙
S gen
s e – s i = -------- ≥ 0 (12.11b)
m ˙
a
For an ideal gas with constant c p , Eq. (1′) of Table 12.4 allows Eq. (12.10c) to
be written as
˙
W = m ˙ c p T i –( T e ) (12.10c′)
The power developed in an isentropic process is obtained with Eq. (5′) of Table 12.4 as
˙
W = m ˙ c p T i 1 –[ ( p e /p i ) ( k−1)/k ] ( s = c) (12.10c′′)
where c p = kR/(k−1).
b
For an ideal gas with constant c p , Eq. (1′) of Table 12.4 allows Eq. (12.10e) to be
written as
(
2
v e = v i + 2c p T i – T e ) (12.10e′)
The exit velocity for an isentropic process is obtained with Eq. (5′) of Table 12.4 as
–
[
2
v e = v i + 2c p T i 1 – ( p e /p i ) ( k 1)/k ] ( s = c)
(12.10e′′)
where c p = kR/(k − 1).
A nozzle is a flow passage of varying cross-sectional area in which the velocity of a gas or liquid increas
es in the direction of flow. In a diffuser, the gas or liquid decelerates in the direction of flow. For such
devices, W = 0 . The heat transfer and potential energy change are generally negligible. Then Eq.
˙
(12.10b) reduces to
2 2
v i – v e
0 = h i – h e + ---------------
2
Solving for the exit velocity

v e = v i + 2 h i – h e )
(
2
(12.10e)
( nozzle, diffuser)
The steady-state forms of the mass, energy, and entropy rate balances can be applied to control volumes
with multiple inlets and/or exits, for example, cases involving heat-recovery steam generators, feedwater
heaters, and counterflow and crossflow heat exchangers. Transient (or unsteady) analyses can be con-
ducted with Eqs. (12.5), (12.7a), and (12.8). Illustrations of all such applications are provided by Moran
and Shapiro (2000).

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