Microsoft Word - Cengel and Boles TOC _2-03-05_.doc

7–1 ■ ENTROPY

The second law of thermodynamics often leads to expressions that involve

inequalities. An irreversible (i.e., actual) heat engine, for example, is less

efficient than a reversible one operating between the same two thermal

energy reservoirs. Likewise, an irreversible refrigerator or a heat pump has a

lower coefficient of performance (COP) than a reversible one operating

between the same temperature limits. Another important inequality that has

major consequences in thermodynamics is the Clausius inequality.It was

first stated by the German physicist R. J. E. Clausius (1822–1888), one of

the founders of thermodynamics, and is expressed as

That is,the cyclic integral of dQ/T is always less than or equal to zero. This

inequality is valid for all cycles, reversible or irreversible. The symbol 
(inte-

gral symbol with a circle in the middle) is used to indicate that the integration

is to be performed over the entire cycle. Any heat transfer to or from a system

can be considered to consist of differential amounts of heat transfer. Then the

cyclic integral of dQ/Tcan be viewed as the sum of all these differential

amounts of heat transfer divided by the temperature at the boundary.

To demonstrate the validity of the Clausius inequality, consider a system

connected to a thermal energy reservoir at a constant thermodynamic (i.e.,

absolute) temperature of TRthrough a reversiblecyclic device (Fig. 7–1).

The cyclic device receives heat dQRfrom the reservoir and supplies heat dQ

to the system whose temperature at that part of the boundary is T(a vari-

able) while producing work dWrev. The system produces work dWsysas a

result of this heat transfer. Applying the energy balance to the combined

system identified by dashed lines yields

where dWCis the total work of the combined system (dWrev
dWsys) and

dECis the change in the total energy of the combined system. Considering

that the cyclic device is a reversibleone, we have

where the sign of dQis determined with respect to the system (positive if to

the system and negative if fromthe system) and the sign of dQRis deter-

mined with respect to the reversible cyclic device. Eliminating dQRfrom the

two relations above yields

We now let the system undergo a cycle while the cyclic device undergoes an

integral number of cycles. Then the preceding relation becomes

since the cyclic integral of energy (the net change in the energy, which is a

property, during a cycle) is zero. Here WCis the cyclic integral of dWC, and

it represents the net work for the combined cycle.

WCTR (^) 
dQ
T
dWCTR¬
dQ
T
dEC
dQR
TR

dQ
T
dWCdQRdEC
¬
dQ
T
 0
332 | Thermodynamics
Combined system
(system and cyclic device)
Reversible
cyclic
device
δQR
Thermal reservoir
TR
δWrev
δWsys
T
δQ
System
FIGURE 7–1
The system considered in the
development of the Clausius
inequality.
SEE TUTORIAL CH. 7, SEC. 1 ON THE DVD.
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