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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