02 - pr 01 - thermo forces and efficiency: Rewrite the linear response equations in terms of thermodynamic
forces,
x1
/ T and x 2 T / T 2 . Solution: Recognizing that (1 / T ) T / T 2 , we have, J N L11 J Q L21
L12 T 1
L11 L22 (1/ T ) L21
L12 x1
(1.1)
;
L22 x2
Maximize the efficiency with respect to x1 , such that x | x2 0 , showing, then, that max C 1
JQ
eT ; e
Introduce a new thermodynamic potential, u
L11 L22 L12 L21
/
Je
L11T JQ
J N 0
2
Z
. Solution:
;
(1.2)
J e / ( e T ) , and use this to define another
thermodynamic potential, u 1 TS e . We want to show that JQ
2 S e T
ZT 11 ZT 11
JQ
Using Fourier’s law (1.2) for J e T J S . Using
, we have, u
Je JQ
E0
J e ( J Q )E0
Je J e J Q TS e J e u 1J e J e TJ S ; TS e u 1 ; eT e T
(1.3)
Now, using (1.3) and (1.14), we can re-write r in the following form, which is handy later, E Je
r
JS
T
J
(J Q / T ) T
(J Q / ) () ( J Q ) (T / T )
J Q / J Q T / T
/ ; T / T
We want to re-write the relative efficiency (1.14) in terms of the thermodynamic potential
u
(1.4)
. Recalling that
(u) , we solve the definition of u, (1.3), for by inserting J e e ( JQ J E ) (see conservation law (1.11), r
Appendix) and Fourier’s law (1.2), in which e u
e ( J Q J E )
e
e T
S e2 T Z
e
(see (1.2)), yielding,
Z
Z r Z r 1 T S u 1 1 1 ; e Se T Se Se SeT Se SeT
(1.5)
Solving (1.5) for r (which follows Goupil’s work), we get, Z
2
S T S SeT Z u 1 Z u u TSe u r TS u 1 1 1 u 1 ; 1 Se SeT e S T
r
e
1
e
(1.6)
e
Extremizing this efficiency (1.6) with respect to u, 1 1 Z S1T (1 (1 ZT ) S e r ST ZT 0 1 u Z Se u ( S T u )2 e
e
e
Putting this critical u into (1.6), we compute an efficiency of,
1 ZT ZT
)
1 S eT
ZT 1 1 S eT
;
(1.7)
1 Ze (
ZT 1 1 SeT
1 S1eT (
SeT
S
max r
(u )
)
( ZT 1 1) 1
( ZT 1 1) 1
) ZT 1 1
( y 1)( y 1) ZT
ZT 11 ZT
(
ZT 1 1) (
2
ZT 1) ZT 1 ZT
ZT 1
2
( y 1)( y 2 y ) ZT y
(1.8)
2
1 1 1 y 1 ZT 1 1 3 1 1 O (a ) ZT a ZT ZT ZT 2
Show that ZT corresponds to max C , the Carnot efficiency. Solution: it is trivial to write, ZT 1 1
lim max C lim
ZT
ZT
ZT 1 1
ZT
C
ZT
C lim max C ;
(1.9)
ZT
Appendix I – the relative efficiency from first principles
we need the following statements of mass and energy conservation,
J N N 0;
Je
eJ N ;
(1.10)
J E E 0 (JQ e J N ) JQ e J N ;
(1.11)
Poynting’s theorem (another statement of conservation) for a non -radiating electromagnetic system says P
Je
E,
in which, for a chemical voltage
E e
/ e , the conservation law (1.11) implies,
J E (J Q e J N ) 0 J Q e J N eE J N E ( eJ N ) E J e ;
(1.12)
1
the relative efficiency is computed from conservation-principles, and is defined as r Pe / J Q . Here, P e E J e (the electric power), and J Q J s T . Using (1.13) in (1.11), and using E Je
J Q J , and r
Ee QC
Ee QC
J
JS
J Q / T , and
J Q / , we have,
E e
(QH Ee )
JQ (T (S ) J Q )
Appendix II – extremizing efficiency with respect to
Using E e E Je JQ (from (1.12)),
JS
x1
JQ JQ TS
E Je
(J S T ) T J S
E Je
JS
T
; (1.14)
T 1
J Q / T , Q H TS , and considering a quasi-
static/reversible/linear thermoelectric device so that J S S , the efficiency (1.14) becomes, r
E Je
T J S
(e / e) ( eJ N )
T (J Q / T )
( e / T ) J N (T / T 2 ) J Q
x1 ( L11x1
L12 x 2 )
x 2 ( L12x1
L22x 2 )
r (x1 , x 2 );
1 Extremizing the efficiency (1.15) with respect to the ith component of x1 T , denoted
use the handy abbreviation 1
1
L11L22 L122
1 ZT 1 ,
This is the efficiency relative to the Carnot efficiency: r
/ C , in which C 1 TC / T H .
i
x1 ,
(1.15)
in which we
L L L 1 L L12 L11L22 r 1 L x x1 1 A x i x2 i L i 2 | 0 i 1 2 i i x L x 1 L22 L12 x2 x1 x1 1 L ( x1 ) xi 1 B x ( x2 L x1 ) 11
11 22 2 12
12 2
j 2
L12
22 2
12
12
22
1
L L 1 ; i L i 2 i x2i 2 1 ( x2i L x1i ) 2 x1i 22 x2i L22 ( x2 L x1 ) x2 L12 1 x2i
(1.16)
12
22
12
22
Putting this critical
i
x1
into the expression for efficiency, and series-expanding in the last step in order to verify
that lim ZT r ( ZT ) 1 , we have, 1 L 11x
L22 L12
1 L22 x2
L12 L22
L
max r
12 2
L x
12 2
x2 (
(
(1 1 ZT ) 2 ZT
1
1
1)
1) 1
1 ( 1)( 1 (
1
1
1)
1) 1
1 ZT ZT 1 1 (1 1 ZT ) ZT
1 2 ZT 1/ 2 O1 (ZT ) lim r 1 ; ZT
(1.17)
