Modern Engineering Thermodynamics

294 CHAPTER 9: Second Law Open System Applications
Similarly, the appropriate entropy rate balance equation for an isothermal system boundary is
_S P
mixing = _m Q
3s 3 − _m 1 s 1 − _m 2 s 2 − _ T b
Q
= _m 3 ½ys ð 2 − s 1 Þ+ ðs 3 − s 2 ÞŠ− _
Many mixing systems use the inlet flow streams to induce mixing and thus tend to be isobaric (i.e., p 1 = p 2 = p 3 ). 4 To
simplify the preceding results somewhat, consider the adiabatic, aergonic, isobaric mixing of two flow streams of the
same material but at different temperatures. Then, these formulae reduce to
and
T b
yh ð 1 − h 2 Þ+ ðh 2 − h 3 Þ = 0 (9.28)
Entropy production rate in mixing two nonreacting flow streams
_S P
mixing = _m 3½ys ð 2 − s 1 Þ+ ðs 3 − s 2 ÞŠ> 0 (9.29)
If these materials are identical incompressible liquids with negligible mixer pressure loss or identical ideal gases
with constant specific heats, then Eqs. (6.19) and (7.33) or Eqs. (6.22) and (7.37) can be used to give
yT ð 1 − T 2 Þ+ T 2 − T 3 = 0 (9.30)
and
_S P
mixing = _m 3c yln T 2
+ ln T
3
T 1 T 2
(9.31)
where c = c p in the case of ideal gases. Combining these two equations by eliminating T 3 gives
Entropy production rate in mixing two flow streams of the same
incompressible liquid or the same ideal gas in this case, c = c p
_S P
mixing = _m 3c ln 1 + y T
1
− 1
T 2
−y
T1
T 2
(9.32)
For a given T 1 /T 2 ratio, there is a critical mass fraction (y c )thatproducesamaximum rate of entropy production. 5 The
critical mass fraction can be found from Eq. (9.32) by setting d _S P /dy = 0andsolvingfory = y c . The result is
which gives
ð
y c = 1 − T 1/T 2 Þ+ lnðT 1 /T 2 Þ
ð1 − T 1 /T 2 ÞlnðT 1 /T 2 Þ
T _S P mixing = _m 3 c ln 1 + y 1
c − 1
ðmaxÞ
T 2
− yc
T1
T 2
(9.33)
(9.34)
Figure 9.11 shows the variation in y c with the absolute temperature ratio T 1 /T 2 , and Figure 9.12 shows the
resulting _S P
mixing ðmaxÞ vs. T 1/T 2 relation.
Note that y c is limited by its definition to be less than or equal to 1 in Figure 9.11 and that, in Figure 9.12, it is
impossible to have a mixer whose entropy production rate falls in the region above the curve shown.
Finally, the analysis of the adiabatic, aergonic, isobaric (i.e., viscousless) mixing of identical ideal gases with
constant specific heats produces formulae identical to Eqs. (9.32), (9.33), and (9.34) except c is replaced with
the constant pressure specific heat c p .
4 Isobaric mixing also requires negligible viscous friction losses.
5 It is a maximum because d 2 _ S P /dy 2 < 0 when evaluated at y = y c . Also, the minimum value of _S P is always zero, which occurs here
when T 1 = T 2 .