Modern Engineering Thermodynamics

282 CHAPTER 9: Second Law Open System Applications
values for only the entropy production rate, _S P , and nothing else. This is the major disadvantage of the
indirect method, since if we could determine _S P directly from Eq. (9.6), we could use the entropy rate balance
to determine other important information about the system, such as _Q , T b , _m , s in ,ands out . Determining the
value of _S P (by either method) provides us with a measure of how well or how poorly a system is operating.
Large _S P values may indicate excessive losses within a particular system or a particularly inefficient mode of
operation. Some of the examples in this chapter explore the possibility of reducing _S P values through alternative
realistic processes that produce the same system operating goals. These analyses lead to methods of
increasing the overall system efficiency (and thus reduce operating costs) by using processes that dissipate less
useful energy as “losses.”
In Chapter 6, we introduce the modified energy balance (MEB) and the modified energy rate balance (MERB),
so that we need not continually deal with the complex general open system first law formulae. These formulae
were designed to work only for steady state, steady flow, single-inlet, single-outlet systems, but the conditions fit
most of the applications we were interested in analyzing. When one or more of these four conditions did not
exist in a particular problem, we carried out the analysis by reverting to the complete, accurate, general energy
rate balance equation.
The general entropy rate balance given in Eq. (9.5) is not as mathematically complex as the general energy rate
balance given in Eq. (6.4), but we still find it convenient to develop similar modified entropy balance and
modified entropy rate balance equations.
The MSB and MSRB formulae require the four conditions used in the MEB and MERB formulae plus one more
condition. In the SRB, the steady state condition requires that
Steady state: _S system =
dS
= 0 (9.7)
dt system
and the steady flow condition requires that
Steady flow: ∑
inlet
_m = ∑
outlet
finally, the steady flow, single-inlet, single-outlet condition requires that
Single-inlet, single-outlet: ∑
inlet
_m (9.8)
_m = ∑
outlet
_m = _m
In addition to these four conditions, we add the fifth condition of isothermal boundaries at all points along the
system boundary where heat transport of energy occurs. Under this condition, the entropy transport term due to
the heat transport of energy becomes
Z
Σ
_q
T b
act
dA = ∑
Σ
_Q
T b
act
= _ Q
T b
(9.9)
In this equation, the simplified notation _Q /T b is used to describe the net (or total) value of the “actual” _Q /T b
summed over the entire system boundary Σ. This simplification is also used in the EB and ERB equations, where
Q, W, _Q ,and _W are used to represent their net (or total) values (e.g., see Eq. (6.5)). The act subscript and the
summation sign has been dropped in the last term of Eq. (9.9) to simplify the notation, but it must always be
considered to be present. Under these five restrictive conditions, the general open system entropy rate balance of
Eq. (9.4) becomes the modified entropy rate balance (MSRB), defined as
Modified entropy rate balance
_Q
+ _m ðs in − s out Þ+ _S P = 0
T b
(9.10)
Multiplying this equation through by dt and integrating over time from system state 1 to state 2 gives the open
system modified entropy balance (MSB) equation as
Modified entropy rate balance
1Q 2
T b
+
Z 2
1
_m ðs in − s out Þdt + 1 S P Þ 2
= 0
(9.11)