Page 101 - gas transport in porous media

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We now define the intrinsic interfacial area average according to
Whitaker
1
ψ γ γκ = ψ γ dA (6.128)
A γκ
A γκ
and use this definition with c As and R As so that Eq. (6.127) takes the convenient form
given by
γ
∂ c Aγ γ ∂ c As γκ
ε γ +∇ · c Aγ v Aγ = R Aγ − a v (6.129)
∂t ∂t

transport homogeneous
accumulation adsorption
reaction
+ a v R As γκ

heterogeneous
reaction
One must keep in mind that this is a general result based on Eqs. (6.120) and (6.121) ;
however, only the first term in Eq. (6.129) is in a form that is ready for applications. On
the right hand side of Eq. (6.129) we are confronted with the necessity of chemical
kinetic constitutive equations for the reaction rate terms and a suitable connection
between the surface concentration, c As , and the bulk concentration, c Aγ . In this
development, we will avoid these problems and concentration on a single aspect
of Eq. (6.129), the determination of the species velocity, v Aγ .
The manner in which v Aγ is determined depends on the particular characteristics
of the species momentum equation given by Eq. (6.84). When Eqs. (6.49) and (6.87)
are a valid representation of the N species momentum equations, we make use of
the total momentum equation to determine the mass average velocity, v γ , and the
Stefan-Maxwell equations to determine the N − 1 independent species velocities.
Under these circumstances, Eq. (6.129) takes the form
γ
∂ c Aγ γ
ε γ +∇ · c Aγ v γ = ∇ · J Aγ + R Aγ (6.130)
∂t
∂ c As γκ
− a v + a v R As γκ
∂t
and we are confronted with the average of a product, c Aγ v γ , whereas it is the product
of the averages that we desire. Here we follow the approach used in turbulence
modeling and decompose the point values of the concentration and velocity into the
intrinsic average values and the spatial deviation values. These decompositions are
given by
γ
γ
c Aγ = c Aγ +˜c Aγ , v γ = v γ +˜v γ (6.131)
so that the volume averaged convective transport is described as
+ ,
γ γ γ γ
v γ c Aγ = v γ c Aγ +˜v γ c Aγ + v γ ˜c Aγ +˜v γ ˜c Aγ (6.132)