Modern Engineering Thermodynamics

19.3 Linear Phenomenological Equations 765
Table 19.3 Entropy Production Rate Density One-Dimensional and Three-Dimensional
Formulae
Entropy Production Rate Density Due To
Heat transport of energy
Viscous dissipation
Electrical energy dissipation
Diffusion of n dissimilar chemical species
EPRD Formula
σ Q = − _q
dT
= ! q .∇ ! 1
T 2 dx T
ðσ W Þ vis
= μ
dV 2
= τ: ! ∇ ! V
T dx e
ðσ W Þ elect
= ρ e J 2 E /T = − E! . ! J E /T
ðσ m Þ diff
= ∑ n
i =1
J_
ix
. d dx
^μi
= ∑ n
T
i =1
!
Ji .∇ ! μ
i
T
Table 19.4 Generalized EPRD One-Dimensional Flux and Force Formulae
Entropy Production
Rate Density (σ)
Generalized Energy
Transport Flux (J)
Generalized Energy
Transport Force (X)
σ Q _q = ! q /A = heat flux vector − 1
dT
T 2 dx
ðσ W Þ vis μ dV
1
dx
T dV
dx
ðσ W Þ J E = I/A = electron flux ρ e J e
elect = −
T T
1
dϕ
dx
ðσ m Þ _
diff
J ix = mass flux of chemical
d ^μi
species i in the x direction
dx T
In Table 19.3 J E = I/A = the current (electron) flux. Note that each σ listed in Table 19.3 has the form
σ = JX = J ! . X ! =
e
J :
e
X (19.1)
where J, ! J , J = generalized (scalar, vector, or tensor) energy transport flux, and X, ! X , X = generalized (scalar, vector,
or tensor) e energy transport driving force producing the flux.
e
Table 19.4 identifies the various flux and force terms associated with the entropy production rate density formulae
given in Table 19.3.
For any system containing a number of entropy production rate densities resulting from the simultaneous operation
of a number of irreversible processes, the total EPRD is simply the sum of all the individual EPRDs present
within the system. Also, the second law of thermodynamics requires that the total EPRD be positive, so
σ total = σ Q + ðσ W Þ vis + ðσ W Þ elect + ðσ m Þ diff + … > 0 (19.2)
Using the generalized flux-force formulation, this can be written for m such densities as
σ total = J Q X Q + ðJ W X W Þ vis + … = ∑ m
J i X i > 0 (19.3)
19.3 LINEAR PHENOMENOLOGICAL EQUATIONS
We begin by postulating that all simultaneously occurring physical phenomena that can be coupled are coupled
to each other in some way. This is succinctly stated in the “coupling postulate.”
For example, consider a system that has two energy transport fluxes, J 1 and J 2 , which result from two generalized
forces, X 1 and X 2 . The coupling postulate states that
and
J 1 = J 1 ðX 1 , X 2 Þ
J 2 = J 2 ðX 1 , X 2 Þ
Expanding J 1 about the “equilibrium” state X 1 = X 2 = 0 using a Taylor’s series gives
J 1 ðX 1 , X 2 Þ≃J 1 ð0, 0Þ + ∂J 1
∂X 1
dX 1 + ∂J 2
∂X 2
dX 2 + higher order terms
i=1