Page 270 - Numerical Methods for Chemical Engineering

BVPs from conservation principles 259

∂Ω ndar

d
Ω n

v
Figure 6.1 Fixed control volume (CV) in space.

The left-hand side is the rate of change of the total amount of within . The first term
on the right-hand side is the net convective transport of across ∂ into by the velocity
field of the medium v(r, t), n being the outward normal vector at the boundary. The second
term is the net diffusive transport of across ∂ into , with the flux vector J D often being
related to the local field gradient by a constitutive equation of the form of Fick’s law,
e
anisotropic diffusion J D =− · ∇ϕ = m n mn e [m] [n]
(6.3)
isotropic diffusion J D =− ∇ϕ ∈
The third term on the right-hand side is a source term, s(r, t, ϕ) being the amount of
generated per unit volume per unit time at (r, t). Equation (6.2) is known as a macroscopic
balance, as it applies to a control volume of finite size. A corresponding microscopic balance
is obtained through use of the divergence theorem,
' '
n · wdS = ∇ · wdV (6.4)
∂
to convert the surface integrals into volume integrals,
' ' ' '
∂ϕ
dr =− [∇ · (ϕv)]dr − [∇ · J D ]dr + s(r, t,ϕ)dr (6.5)
∂t

As all integrals are over the same domain, we can combine them,
∂ϕ
'
+ ∇ · (ϕv) + ∇ · J D − s(r, t,ϕ) dr = 0 (6.6)
∂t

This balance must hold for any arbitrary fixed control volume, requiring the field to satisfy
everywhere the partial differential equation
∂ϕ
=−∇ · (ϕv) − ∇ · J D + s(r, t,ϕ) (6.7)
∂t
Substituting (6.3) for the diffusive flux yields
∂ϕ
=−∇ · (ϕv) + ∇ · [ ∇ϕ] + s(r, t,ϕ) (6.8)
∂t
For isotropic diffusion with a constant , this takes the form of a classic convection/diffusion
equation with a source term,
∂ϕ 2
=−∇ · (ϕv) + ∇ ϕ + s(r, t,ϕ) (6.9)
∂t