Page 237 - The Mechatronics Handbook

0066_frame_C12 Page 7 Wednesday, January 9, 2002 4:22 PM

Eqs. (12.9b) and (12.9c) give, respectively,

2 2
0 = Q W + m ˙ h i – h e ) +   v i – 2 v e  gz i –( z e ) (12.10a)
(
--------------- +
˙
˙
–
˙
Q
0 = ----- + m ˙ s i –( s e ) + S gen (12.11a)
˙
T b
where for simplicity T b denotes the temperature, or a suitable average temperature, on the boundary
where heat transfer occurs.
When energy and entropy rate balances are applied to particular cases of interest, additional simplifi-
cations are usually made. The heat transfer term Q ˙ is dropped when it is insignificant relative to other
energy transfers across the boundary. This may be the result of one or more of the following: (1) the outer
surface of the control volume is insulated; (2) the outer surface area is too small for there to be effective
heat transfer; (3) the temperature difference between the control volume and its surroundings is small
enough that the heat transfer can be ignored; (4) the gas or liquid passes through the control volume so
˙
quickly that there is not enough time for significant heat transfer to occur. The work term W drops out
of the energy rate balance when there are no rotating shafts, displacements of the boundary, electrical
effects, or other work mechanisms associated with the control volume being considered. The effects of
kinetic and potential energy are frequently negligible relative to other terms of the energy rate balance.
The special forms of Eqs. (12.10a) and (12.11a) listed in Table 12.1 are obtained as follows: When
there is no heat transfer, Eq. (12.11a) gives

˙
S gen
s e – s i = -------- ≥ 0
m ˙ (12.11b)
(no heat transfer)

Accordingly, when irreversibilities are present within the control volume, the specific entropy increases
as mass flows from inlet to outlet. In the ideal case in which no internal irreversibilities are present, mass
passes through the control volume with no change in its entropy—that is, isentropically.
For no heat transfer, Eq. (12.10a) gives

2 2
--------------- +
W = m ˙ h i – h e ) +   v i – v e  gz i –( z e ) (12.10b)
(
˙
2
( no heat transfer)
A special form that is applicable, at least approximately, to compressors, pumps, and turbines results
from dropping the kinetic and potential energy terms of Eq. (12.10b), leaving

˙
W = m ˙ h i – h e )
(
(12.10c)
( compressors, pumps, and turbines)
In throttling devices a significant reduction in pressure is achieved by introducing a restriction into a line
through which a gas or liquid flows. For such devices W = 0 and Eq. (12.10c) reduces further to read
˙

h i ≅ h e
(12.10d)
( throttling process)

That is, upstream and downstream of the throttling device, the specific enthalpies are equal.

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