Page 86 - gas transport in porous media

Chapter 6: Conservation Equations
which is identical to Cauchy’s second equation. For a Stokesian fluid (Serrin, 1959)
the stress tensor can be decomposed according to 79
T =−p I + τ (6.48)

in which τ is the viscous stress tensor. Use of Eq. (6.48) in Eq. (6.46) leads to the
viscous stress equations of motion.

∂
(ρv) +∇ · (ρvv) = ρb −∇p +∇ · τ (6.49)
∂t

To complete the analysis one needs a constitutive equation for the viscous stress tensor
so that Eq. (6.49) can be used to determine the mass average velocity and this is an
extremely important result for many mass transfer and fluid mechanics problems. For
Newtonian fluids and incompressible flows, the viscous stress tensor takes the form

T
τ = µ(∇v +∇v ) (6.50)
and one obtains the Navier-Stokes equations

∂v 2
ρ + ρv∇· v = ρg −∇p + µ∇ v (6.51)
∂t
Here body forces have been limited to the gravitational force in which the gravita-
tional acceleration is represented by g. In general, an equation of state is required to
determine the pressure in Eqs. (6.49) and (6.51), and for an N-component system we
suggest a generic form given by

p = p(T, c A , c B , ... , c N ) (6.52)

When the flow can be treated as incompressible, this equation of state can be discarded
in the classical manner.

6.1.6 Energy
The energy for a multicomponent system represents a complex matter for two reasons.
First, the axiomatic statement for the thermal and kinetic energy of a species body is
complex in its own right, and second, one must subtract the mechanical energy from
the total energy in order to obtain the desired thermal energy equation. Because of the
complexity of the subject, this presentation will only provide an outline of the steps
involved. The axioms for the thermal and mechanical energy of a species body can