Modern Engineering Thermodynamics

766 CHAPTER 19: Introduction to Coupled Phenomena
THE COUPLING POSTULATE
All the generalized flows (J i ) are dependent on (or coupled to) all the generalized forces (X i ) present within the system. 1
That is, a functional “coupling” relationship exists between all generalized fluxes and all generalized forces within a system
of the form
J i = J i ðX 1 , X 2 , X 3 , X 4 , …Þ (19.4)
1 This is limited by the Curie principle, which states that, in an isotropic system, the forces and fluxes must be of the same tensor rank before they can be
coupled.
Now J 1 (0, 0) = 0 (if there are no forces, there are no flows, or fluxes) and very near the equilibrium state of the
system, we can write
and
dX 1 ≈ ΔX 1 = X 1 − 0 = X 1
dX 2 ≈ ΔX 2 = X 2 − 0 = X 2
so that near equilibrium, we have
and, similarly,
J 1 = ∂J
1
∂X 1
J 2 = ∂J
2
∂X 1
X 1 + ∂J 1
∂X 2
X 1 + ∂J 2
∂X 2
X 2 (19.5)
X 2 (19.6)
Now, we define the “primary” coefficients, L 11 and L 22 ,as
L 11 = ∂J 1
∂X 1
L 22 = ∂J 2
∂X 2
and the “secondary” or “coupling” coefficients, L 12 and L 21 ,as
L 12 = ∂J 1
∂X 2
Then, Eqs. (19.5) and (19.6) become
or, in general,
L 21 = ∂J 2
∂X 1
J 1 = L 11 X 1 + L 12 X 2 (19.7)
J 2 = L 21 X 1 + L 22 X 2 (19.8)
J i = ∑ m
j=1
L ij X j (19.9)
where the summation notation implies that the summation is to take place over all m fluxes and forces present
in the system.