Post on 01-Jun-2018
transcript
8/9/2019 Clausius Inequality
1/13
The Clausius Inequality
Expressions of inequality/equality relating toheat ow at a xed temperature.
The expression is required for the derivationof an equation for entropy which is our nextmain topic.
erived from a !thought experiment" using#arnot engines acting in a series.
8/9/2019 Clausius Inequality
2/13
$olume
%ressure
a
b
d
T1
Q1
Carnot Cycle
Q2
V
nRTP
1=
V
constP
.=
T2c
Q=0Q=0
V
nRTP
2=
8/9/2019 Clausius Inequality
3/13
Heat Flows in a Carnot Cycle
&ot 'eservoir( T)
#old 'eservoir( T*
CW
Q1
Q2
2
1
2
1=
Q
Q
T
T
2
1
21= T
T
8/9/2019 Clausius Inequality
4/13
+uch that 02
2
1
1=
T
Q
T
Q
,ne could also consider the small amount of
reversi-leheat owQrevthat ows at atemperature Tat each point in the cycle. The netheat ow is equal to the sum of the dierentialows
02
2
1
1 =
+=
T
Q
T
Q
T
Q
cycle
rev
0or a #arnot cycle( some of the heat into the cycleis converted to wor1 so that Q) 2 Q*. 3e also
1now that2
2
1
1
T
Q
T
Q=
0rom the denition of an integral( we nd for theentire cycle that
0= T
Q rev
4ut is this generally true for anycycle5
8/9/2019 Clausius Inequality
5/13
erivation of the #lausius 6nequality
Principal reservoir T~
C1
T1
Auxiliary
reservoirsW1
Q1
Q1
T~
11
~
T
TQ
1
1
~
T
TQQ
~
Q~
Working
subsance in
a cycle
1 2
8/9/2019 Clausius Inequality
6/13
erivation of the #lausius 6nequality
Principal reservoir T~
C1
T1
W1
Q1
Q1
T~
11
~
T
TQ
11
~
T
TQ
Working
subsance in
a cycle
1 2
W2
Q2
Q2
C2
T2
T~
!
22
~
T
TQ
2
2
~
T
TQ
Add NCarno
engines in oalCo"posi
e
#yse"
8/9/2019 Clausius Inequality
7/13
nalysis of Composite Device
The total heat supplied -y the reservoir to
the composite device in a cycle with Nengines is
6n an entire cycle( Ufor the wor1ingsu-stance and the composite device is 7.Then the 0irst 8aw says
U9 7 9 W: Q
;n important conclusion is thatQ
9
8/9/2019 Clausius Inequality
8/13
erivation of the #lausius 6nequality
%rincipalreservoir
T~
#ompositesystem
i
Q
T
TQ
1
~
= i
$
iWW =i
$
Q= - W
Q = -W2 7
$iolates =elvinstatement ofthe *nd 8aw>
3hat can-eallowed5
8/9/2019 Clausius Inequality
9/13
erivation of the #lausius 6nequality
Principal reservoir T~
#ompositesystem
Q
W
?9 < 3 @7
;llowed
8/9/2019 Clausius Inequality
10/13
erivation of the #lausius 6nequality
Principal reservoir T~
Co"posie
syse"
Q 9 < W9 7
Q
W
;llowed
%#u" o& 've and
(ve Qand W)
8/9/2019 Clausius Inequality
11/13
Results Allowed by SecondLaw
0~
=
1=N
i i
i
T
QTQ
Q@ 0or ? 9 7( thus
0~T4ut * so we can divide -yT~
0
1N
i i
i
T
Q
=
6n the limit of smallQi* we can integrate over the
entire cycle
0
oT
dQ
where Tois the temp. of the reservoir Aexternal heat
sourceB and the circle represents integration over the
entire cycle.
ClausiusInequality
8/9/2019 Clausius Inequality
12/13
Reversible Cycles6f the cycle at the centre of the compositedevice is reversible( then it can -e run in
reverse.The opposite result is then o-tained. 6n thecomposite device( wor1 is done ,C the system( andheat is given ,00 to the reservoir.
T~
Q
W
;llowed
7 Q9 < W
8/9/2019 Clausius Inequality
13/13
Reversible Cycles
Thus( there are restrictions on Qwhen the cycle isoperated in reverse.
4ut the cycle can still also -e operated in theforward direction such that
0~
== 1=
N
i i
i
T
QTQW
,nly one solution satises -oth requirementsfor forward and reverse cycles
0=~
= 1=
N
i i
i
T
QTQ
6ntegrating
6n a reversi-le process( the temperature of thesystem and reservoir are equal T.
0= TdQR
ClausiusEquality