Page 479 - Advanced thermodynamics for engineers

20.4 ONSAGER’S RECIPROCAL RELATION 471

Hence
J 1 ¼ L 11 X 1 þ L 12 X 2 (20.9)
where L 12 is the coupling coefficient showing the effect of mass transfer (diffusion) on energy transfer.
In a similar manner, because conduction has an effect on diffusion, the equation for mass transfer
can be written

J 2 ¼ L 21 X 1 þ L 22 X 2 (20.10)
where L 21 is the coupling coefficient for these phenomena.
A general set of coupled linear equations is

X
J i ¼ L ik X k (20.11)
k
The equations are of little use unless more is known about the forces X k and the coefficients L ik .
This information can be obtained from Onsager’s reciprocal relation. There is considerable latitude in
the choice of the forces X, but Onsager’s relation chooses the forces in such a way that when each flow
J i is multiplied by the appropriate force X i the sum of these products is equal to the rate of creation of
entropy per unit volume of the system, q, multiplied by the temperature, T.
Thus
X
Tq ¼ J 1 X 1 þ J 2 X 2 þ . ¼ J i X i (20.12)
i
Equation (20.12) may be rewritten
X
q ¼ J i x i (20.13)
i
where

X i
x i ¼ :
T
Onsager further showed that if the abovementioned condition was obeyed then, in general,
(20.14)
L ik ¼ L ki
This means that the coupling matrix in Eqn (20.8a) is symmetric, i.e. L 12 ¼ L 21 for the particular
case given above. The significance of this is that the effect of parameters on each other is equivalent
irrespective of which is judged to be the most, or least, significant parameter. Consideration will show
that if this was not true then it would be possible to construct a system which disobeyed the laws of
thermodynamics. It is not proposed to derive Onsager’s relation which is obtained from molecular
considerations, it will be assumed to be true.
In summary, the thermodynamic theory of an irreversible process consists of first finding the
conjugated fluxes and forces, J i and x i , from Eqn (20.13) by calculating the entropy production. Then a
study is made of the phenomenological Eqn (20.11) and Onsager’s reciprocal relation (20.14) is used
to solve these. The whole procedure can be performed within the realm of macroscopic theory and is
valid for any process.