A case of particular importance are systems with only one inlet and one exit,
as sketched in Fig. ??, for which the mass balance reduces to
ṁin = ṁout = ṁ .
(1)
There is just one constant mass flow ṁ flowing through each cross section of
the system.
The corresponding forms for energy and entropy balance are1
∙
¸
¢
1¡ 2
2
ṁ h2 − h1 +
(2)
V − V1 + g (z2 − z1 ) = Q̇12 − Ẇ12 ,
2 2
ṁ (s2 − s1 ) −
X Q̇k
Tk
k
= Ṡgen ≥ 0 .
(3)
It is instructive to study the equations for an infinitesimal step within the
system, i.e., for infinitesimal system length dx, where the differences reduce to
differentials,
µ
¶
1 2
ṁ dh + dV + gdz
= δ Q̇ − δ Ẇ ,
(4)
2
ṁds −
δ Q̇
T
= δ Ṡgen .
(5)
Heat and power exchanged, and entropy generated, in an infinitesimal step
along the system are process dependent, and as always we write (δ Q̇, δ Ẇ , δ Ṡ)
to indicate that these quantities are not exact differentials. Use of the Gibbs
equation in the form T ds = dh − vdp allows to eliminate dh and δ Q̇ between
the two equations to give an expression for power,
µ
¶
1 2
δ Ẇ = −ṁ vdp + dV + gdz − T δ Ṡgen .
(6)
2
The total power for the finite system follows from integration over the length of
the system as
¶ Z 2
Z 2µ
1 2
Ẇ12 = −ṁ
T δ Ṡgen
(7)
vdp + dV + gdz −
2
1
1
Z 2
Z
¢
Ẇ12
1¡ 2
1 2
vdp +
T δ Ṡgen
(8)
= −
V1 − V22 + g (z1 − z2 ) −
ṁ
2
ṁ 1
1
The above equation has several implications: First, since T δ Ṡgen ≥ 0, we see–
again–that irreversibilities reduce the power output of a power producing device
(where Ẇ12 > 0), and increase the power demand of a power consuming device
1 The subscripts refer to properties at different locations within the device: “1” denotes the
inlet, “2” denotes the outlet. This must be distinguished from the analysis of closed systems,
where the subscripts normally refer to states assumed at different times t1 , t2 .
1
(where Ẇ12 < 0). Efficient energy conversion requires to reduce irreversibilities
as much as possible.
The corresponding equation for heat is, really, the second law,
δ Q̇
1
= T ds − T δ Ṡgen
ṁ
ṁ
With the Gibbs equation, it reads after integration
Q̇12
= u2 − u1 +
ṁ
δ Q̇
= h2 − h1 −
ṁ
Z
2
1
Z
2
1
1
pdv −
ṁ
1
vdp −
ṁ
Z
2
T δ Ṡgen
1
Z
2
T δ Ṡgen
1
Splitting into heat and work hence depends on compressibility. If incompressible, then dv = 0, and the equations in Herwig paper are ok.
Henning Struchtrup, University of Victoria, Canada, August 2014
2