Modern Engineering Thermodynamics

16.8 Linear Momentum Rate Balance 673
c. For entropy, X = S and x = S/m = s. Entropy is not conserved, so σ s ≠ 0: The non-mass flow entropy flux is identified in
Chapter 7 as J s = _q /T b : Then, for an isothermal (i.e., one-dimensional) boundary, Eq. (16.31) reduces to
where
is the net non-mass flow entropy transport rate, and
Q
ðdS/dtÞ sys = _ +∑ _ms −∑ _ms + _S P
T b in out
_Q /T b = ∑
in
_q/T b −∑ _q/T b
out
Z
_S p = σ s dV
V
is the entropy production rate. These results are identical to those developed in Chapter 9 in for the one-dimensional
entropy rate balance originally presented in Eq. (9.6).
Exercises
22. Use Eq. (16.31) to produce a general closed system rate balance equation for an arbitrary property X. Answer:
∑
in
Z
J x A −∑ J x A +
out
V
σ x dV =
23. Apply the general closed system rate balance equation developed in Exercise 22 to produce the steady state closed
system entropy rate balance equation. Answer:
or
∑
in
_q
T b
−∑
out
Z
_q
+ σ s dV =
T b
V
_Q
T b
+ _S P = 0
dS
dt
dX
dt
system
system
= 0
where
_Q
= ∑
T b in
_q
T b
−∑
out
Z
_q
and _S P = σ s dV
T b
V
24. Use Eq. (16.31) to produce an availability rate balance for a steady state closed system. Answer:
_Q − _W +∑ _ma f −∑ _ma f + _I =
dA
= 0
in out
dt system
16.8 LINEAR MOMENTUM RATE BALANCE
The linear momentum rate balance (LMRB) for a one-dimensional open system can be easily developed from
Eq. (16.31) by letting X = mV and x = mV/m = V. Since momentum is conserved, σ x = σ V = 0: External forces are
the source of the non-mass flow momentum transport rate across the system boundary, so the one-dimensional
momentum flux, J x = J V , is the external force per unit area, or
Then, Eq. (16.31) gives the LMRB as 3
d
dt ðmVÞ open
system
J V = ðF ext /AÞg c
= ∑ðF ext Þg c +∑ _mV −∑ _mV (16.32)
net
in out
3 The vector nature of momentum necessarily causes Eq. (16.32) to be three dimensional. The one-dimensional restriction on this
equation implies only that V and F ext are area averaged quantities over the surface of the system. However, components of V and F ext
may be in each of the three coordinate directions.