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Modern Engineering Thermodynamics

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250 CHAPTER 8: Second Law Closed System Applications<br />

8.1.1 Closed System Indirect Method<br />

1. The entropy produced by a change of state is given by Eq. (7.75) as<br />

Z Z <br />

_q<br />

1ðS P Þ 2<br />

= ms ð 2 − s 1 Þ−<br />

dA dt (8.1)<br />

τ Σ T b act<br />

2. The entropy production rate of a closed system is given by differentiating Eq. (8.1) with respect to time, or<br />

Z <br />

_q<br />

_S P = _S − dA (8.2)<br />

Σ T b act<br />

where T b is the local system boundary temperature evaluated at the point where _q crosses the system boundary Σ.<br />

8.1.2 Closed System Direct Method<br />

1. The entropy produced by a change of state is given by<br />

Z 2<br />

<br />

<br />

Q<br />

1ðS P Þ 2<br />

=<br />

1 T dT + dW <br />

2<br />

act<br />

T<br />

V<br />

irr<br />

<br />

Z Z <br />

=<br />

τ V<br />

<br />

− _q T 2 dT<br />

dx<br />

<br />

+ σ W<br />

<br />

dV dt (8.3)<br />

2. The entropy production rate of a closed system is given by<br />

Z <br />

_S P = − _q <br />

<br />

dT<br />

+ σ<br />

T 2<br />

W dV (8.4)<br />

dx<br />

where σ W =(σ W ) viscouw +(σ W ) electrical +(σ W ) diffusion + ⋯ and T is the local temperature inside the system volume<br />

evaluated at the point where the heat transfer and work irreversibilities occur.<br />

Both the direct method and the indirect method give accurate answers for entropy production if applied correctly.<br />

Which method you choose to solve a particular problem depends entirely on the type of information given to<br />

you in the problem statement (usually only one of the two methods works for a given problem scenario).<br />

In the examples presented in this chapter, we are concerned mainly with determining the amount of entropy<br />

production for a process by calculating it from the closed system entropy balance (i.e., by using the indirect<br />

method).<br />

Last, before we begin the example problems, keep in mind the basic reason why the evaluation of entropy production<br />

is important. The entropy production is a measure of the “losses” within the system, so the larger the<br />

entropy production, the more inefficient the system is at carrying out its function. We continually look for ways<br />

to minimize a system’s entropy production and thus improve its overall operating efficiency.<br />

8.2 SYSTEMS UNDERGOING REVERSIBLE PROCESSES<br />

In Chapter 7, we define a reversible process as any process for which the entropy production or entropy production<br />

rate is zero. Thus, a system is said to be reversible if there are no losses due to friction, viscosity, heat transfer,<br />

diffusion, or the like anywhere within the system. Consequently, the entropy production of a system<br />

undergoing a reversible process is always zero, or<br />

_S P = 1 ðS P Þ 2 = 0 for all reversible processes<br />

and the closed system entropy balance (SB) given in Eq. (7.75) reduces to<br />

Z Z <br />

_q<br />

S 2 − S 1 = ms ð 2 − s 1 Þ =<br />

dA dt (8.5)<br />

τ Σ T b rev<br />

where _q is the heat flux, T b is the system boundary absolute temperature, A is the system boundary area, and τ is<br />

the time required to change from state 1 to state 2. The closed system entropy rate balance (SRB) is obtained<br />

from this equation by differentiating it with respect to time as 1<br />

Z <br />

dS<br />

dt = _S = m_s =<br />

Σ<br />

<br />

_q<br />

T b<br />

dA (8.6)<br />

rev<br />

1 Remember that the system mass m is always constant in a closed system.

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