Modern Engineering Thermodynamics

9.4 Open System Entropy Balance Equations 281
where ð _S P Þ m
= 0, since we are neglecting diffusion of dissimilar species inside the system. The individual entropy
transport rate terms are given in Eqs. (7.61), (7.63), and (9.1) as
Z
ð_S T Þ Q
=
Σ
½ð_S T Þ m
Š net
= ∑
in
_q
T b
dA (7.61)
act
ð_S T Þ W
= 0 (7.63)
_ms −∑
out
_ms (9.1)
Using the concept of entropy production rate per unit volume, σ, we can express the total entropy production
rate of Eq. (9.3) (neglecting ð _S P Þ m )as
Z
_S P = ð _S P Þ Q + ð _S P Þ W =
V
Z
ðσ Q + σ W ÞdV =
where the total entropy production per unit volume is σ = σ Q + σ W . The relation for σ Q is given by Eq. (7.64) as
V
σdV
σ Q = − _q
dT
T 2 dx
actual
(7.64)
and the equations for σ W for viscous flow and electrical resistance are given by Eqs. (7.70) and (7.73) as
ðσ W Þ vis = μ T
dV 2
and ðσ W Þ
dx
elect = J2 e ρ e
T
Putting Eqs. (7.61) and (9.1) into Eq. (7.3) produces a general open system entropy rate balance (SRB) equation as
General open system entropy rate balance (SRB)
Z
_q
dA +∑ _ms − ∑ _ms + _S P = _S system (9.4)
Σ T b act inlet outlet
and, when the system boundaries are isothermal (i.e., T b is constant), this equation reduces to
Isothermal boundary open system entropy rate balance
_Q
_ms −∑ _ms + _S p = _S system (9.5)
T b act
out
+∑
in
Z
where _Q /T b = ð _q /T b ÞdA: In this equation, _Q is the actual heat transfer rate, and T b is the temperature of the system
Σ
boundary where _Q occurs. The equation for the direct method for determining the entropy production rate is given by
Eq. (9.6) as 1 Z
_S P = Vσ dV > 0 (9.6)
where σ is the total entropy production rate density (EPRD) for the system, given by
σ = σ Q + ðσ W
Þ vis + ðσ W Þ elect +
However, since the use of Eq. (9.6) for the direct determination of _S P is often quite difficult, the example
problems presented in this chapter use the indirect method for determining the entropy production rate ( _S P )in
the entropy rate balance, Eq. (9.5). When this is done, the entropy rate balance can be used to generate
1 In this chapter, we assume that the processes of interest are truly irreversible, so we write _S P > 0, rather than the less restrictive _S P ≥ 0.