Page 478 - Advanced thermodynamics for engineers

470 CHAPTER 20 IRREVERSIBLE THERMODYNAMICS

where n i ¼ amount of substance, i, C i ¼ concentration of component, i,and k ¼ diffusion
coefficient. It will be shown later that this equation is not as accurate as one proposed by
Hartley, in which the gradient of the ratio of chemical potential to temperature is used as the driving
potential.
Other similar relationships occur in physics and chemistry but will not be given here. The three
equations given above relate the flow of one quantity to a difference in potential: hence, there is a flow
term and a force term as suggested by Eqn (20.4). It will be shown that although Eqns (20.5)–(20.7)
appear to have the correct form, they are not the most appropriate relationships for some problems.
Equations (20.5)–(20.7) also define the relationship between individual fluxes and potentials, whereas
in many situations the effects can be coupled.

20.4 ONSAGER’S RECIPROCAL RELATION
If two transport processes are such that one has an effect on the other, e.g. heat conduction and
electricity in thermoelectricity; heat conduction and diffusion of gases; etc., then the two processes are
said to be coupled. The equations of coupled processes may be written as

J 1 ¼ L 11 X 1 þ L 12 X 2
(20.8)
J 2 ¼ L 21 X 1 þ L 22 X 2
Equation (20.8) may also be written in matrix form as

#" #
X 1
J 1 L 11 L 12
¼ (20.8a)
J 2 L 21 L 22 X 2
It is obvious that in this equation the basic processes are defined by the diagonal coefficients in the
matrix, while the other processes are defined by the off-diagonal terms.
Consider the situation, where diffusion of matter is occurring with a simultaneous conduction of
heat. Each of these processes is capable of transferring energy through a system. The diffusion
processachievesthisbymasstransfer, i.e. each molecule of matter carries some energy with it.
The thermal conductivity process achieves the transfer of heat by the molecular vibration of the
matter transmitting energy through the system. Both achieve a similar result of redistributing
energy but by different methods. The diffusion process also has the effect of redistributing
the matter throughout the system, in an attempt to achieve the equilibrium state in which the
matter is evenly distributed with the minimum of order (i.e. the maximum entropy or minimum
chemical potential). It can be shown, by a more complex argument that thermal conduction will
also have an effect on diffusion. First, if each process is considered in isolation the equations can
be written

J 1 ¼ L 11 X 1 the equation of conduction without any effect due to diffusion;
J 2 ¼ L 22 X 2 the equation of diffusion without any effect due to conduction:
Now, the diffusion of matter has an effect on the flow of energy because the individual, diffusing,
molecules carry energy with them, and hence an effect for diffusion must be included in the term for
thermal flux, J 1 .