Page 112 - Mechanical Engineers' Handbook (Volume 4)

3 The Second Law of Thermodynamics for Closed Systems 101

W
1 T /T H
I
II
L
W
maximum (reversible case)
A refrigerating machine or a heat pump operates cyclically between two temperature res-
ervoirs in such a way that during each cycle it receives work and delivers net heat to the
environment,

W Q Q Q 0
H
L
The goodness of such machines can be expressed in terms of a coefficient of performance
(COP)
Q 1
COP refrigerator L T /T 1
W H L
Q H 1
COP heat pump 1 T /T
W L H

Generalizing the second law for closed systems operating cyclically, one can show that
if during each cycle the system experiences any number of heat interactions Q with any
i
number of temperature reservoirs whose respective absolute temperatures are T , then
i
Q i 0
i T i
Note that T is the absolute temperature of the boundary region crossed by Q . Another way
i
i
to write the second law in this case is
Q
T 0
where, again, T is the temperature of the boundary pierced by Q. Of special interest is the
reversible cycle limit, in which the second law states ( Q/T) rev 0. According to the
definition of thermodynamic property, the second law implies that during a reversible process
the quantity Q/T is the infinitesimal change in a property of the system: by definition, that
property is the entropy change
dS or S S
2
Q
Q
T 2 1 1 T
rev rev
Combining this definition with the second law for a cycle, Q/T 0, yields the second
law of thermodynamics for any process executed by a closed system,
S S 2 Q

2
1
1 T 0
entropy
change entropy
(property) transfer
(nonproperty)