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5 2D Convection – Cartesian Grids May 20, 2005 12:28

5.1 Introduction

5.1.1 Main Task
In the previous chapter, we considered convective–diffusive transport in long
(x direction) and thin (y direction) flows. This implied that although convective
fluxes were significant in both x and y directions, significant diffusion fluxes oc-
curred only in the y direction; diffusion fluxes in the x direction are negligible. We
now turn our attention to flows in which diffusive fluxes are comparable in both x
1
and y directions. Thus, the general transport Equation (1.25) may be written as
∂(ρ ) 1 ∂(rq j )
+ = S, j = 1, 2, (5.1)
∂t r ∂x j
where
∂
q j = ρ u f j −
eff . (5.2)
∂x j
In Equation 5.2, the first term on the right-hand side represents the convective
flux whereas the second term represents the diffusive flux. Note that suffix f is
attached to the velocity appearing in the convective flux; the significance of this
suffix will become clear in a later section. In Equation 5.1, r stands for radius.
This makes the equation applicable to axisymmetric flows governed by equations
written in cylindrical polar coordinates. When plane flows are considered,r = 1 and
Equation 1.25 is readily recovered. By way of reminder, we note that may stand
for 1, u i (i = 1, 2), u 3 (velocity in the x 3 direction), ω k , T or h, and e and , and

eff is the effective exchange coefficient (see Equation 4.89).
Flows with comparable convective–diffusive fluxes in each direction occur rou-
tinely in most practical equipment although they are usually three dimensional.
Here, only 2D situations are considered for convenience and because the primary

1 Note that ρ m signifying mixture density is now written as ρ for convenience.

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