Page 236 - The Mechatronics Handbook

0066_frame_C12 Page 6 Wednesday, January 9, 2002 4:22 PM

within systems. A control volume form of the extensive property balance for entropy is

˙
dS Q j m ˙ is i ∑
˙
------ = ∑ ----- + ∑ – m ˙ es e + S gen (12.8)
dt T j
j i e
------------------------------ ----
rates of entropy rate of entropy
transfer generation
where dS/dt represents the time rate of change of entropy within the control volume. The terms m ˙ is i
account, respectively, for rates of entropy transfer into and out of the control volume accom-
and m ˙ es e
˙
panying mass flow. Q j represents the time rate of heat transfer at the location on the boundary where
˙
the instantaneous temperature is T j , and Q j/T j accounts for the accompanying rate of entropy transfer.
˙
S gen denotes the time rate of entropy generation due to irreversibilities within the control volume. An
entropy rate balance for closed systems is obtained by dropping the terms of Eq. (12.8) involving mass
flow rates.
When applying the entropy balance in any of its forms, the objective is often to evaluate the entropy
generation term. However, the value of the entropy generation for a given process of a system usually
does not have much significance by itself. The significance normally is determined through comparison:
the entropy generation within a given component would be compared with the entropy generation values
of the other components included in an overall system formed by these components. This allows the
principal contributors to the irreversibility of the overall system to be pinpointed.

Control Volumes at Steady State
Engineering systems are often idealized as being at steady state, meaning that all properties are unchanging
in time. For a control volume at steady state, the identity of the matter within the control volume changes
continuously, but the total amount of mass remains constant. At steady state, the mass rate balance
Eq. (12.5) reduces to

∑ m ˙ i = ∑ m ˙ e (12.9a)
i e
At steady state, the energy rate balance Eq. (12.7a) becomes

2 2



0 = Q W + ∑ m ˙ i h i + ---- + gz i ∑ m ˙ e h e + ---- + gz e   (12.9b)
v e
˙
v i
˙
–
–



i 2 e 2
At steady state, the entropy rate balance Eq. (12.8) reads
˙
˙
0 = ∑ ----- + ∑ m ˙ is i ∑ m ˙ es e + S gen (12.9c)
Q j
–
T j
j i e
Mass and energy are conserved quantities, but entropy is not generally conserved. Equation (12.9a)
indicates that the total rate of mass flow into the control volume equals the total rate of mass flow out
of the control volume. Similarly, Eq. (12.9b) states that the total rate of energy transfer into the control
volume equals the total rate of energy transfer out of the control volume. However, Eq. (12.9c) shows
that the rate at which entropy is transferred out exceeds the rate at which entropy enters, the difference
being the rate of entropy generation within the control volume owing to irreversibilities.
Many applications involve control volumes having a single inlet and a single exit. For such cases
the mass rate balance, Eq. (12.9a), reduces to m ˙ i = m ˙ e . Denoting the common mass flow rate by , m ˙
©2002 CRC Press LLC