Page 83 - gas transport in porous media

Whitaker
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The first of these axioms represents the linear momentum equation for species A, and
the point equation associated with Eq. (6.23) is the governing differential equation
for v A . Axiom I simply indicates that the time rate of change of the linear momentum
of a species A body is equal to the force acting on the body plus the rate of production
of momentum owing to chemical reaction. In Eq. (6.23), the term P AB represents the
force per unit volume exerted by species B on species A, and the sum from B = 1to
B = N represents the force exerted by all the other molecular species on species A.
The angular momentum equation given by Eq. (6.24) will lead to the symmetry
of the species stress tensor, and should be thought of as simply a constraint on the
contact force. The axiom given by Eq. (6.25) indicates that the total sum of the
intermolecular forces is zero and it is satisfied by the condition P AB =−P BA that
appears in the Stefan-Maxwell equations. Axiom IV requires that linear momentum
is neither created nor destroyed by chemical reactions.
In order to derive the governing differential equation associated with Eq. (6.23), one
follows the classical developments concerning stress (Cauchy’s lemma and Cauchy’s
fundamental theorem) to obtain
(6.27)
t A(n) =−t A(−n)
t A(n) = n ·T A (6.28)
One then makes use of the appropriate form of the general transport theorem so that
the following differential equation can be extracted from Eq. (6.23):
∂
(ρ A v A ) +∇ · (ρ A v A v A ) = ρ A b A +∇ ·T A (6.29)
∂t

convective body surface
local
acceleration force force
acceleration
B=N

+ P AB + r A v A A = 1, 2, ... , N

B=1
source of momentum

diffusive owing to reaction
force
Here we have identified the forces associated with P AB as the diffusive force since
this force is so closely related to the process of diffusion.
The analysis of the second axiom is algebraically complex but the final result
simply indicates that the species stress tensor is symmetric
T
T A = T , A = 1, 2, ... , N (6.30)
A
The third and fourth axioms remain unchanged and are ready for use.

6.1.4 Total Momentum Equation
In order to compare the results given by Eqs (6.29) and (6.30) with those for single
component systems, we first sum Eq. (6.29) over all N species in order to develop