Page 103 - Introduction to Computational Fluid Dynamics

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2D BOUNDARY LAYERS
is assumed with equal specific heats then this relationship can be simplified
to [33]

˙ m = (h b − h T ) −1
∂h . (4.58)
b
∂y
y=0
Knowing ˙ m , boundary condition h b can be extracted.

b
Mass Transfer Variables Φ = ω k
The most common boundary condition [33] for these variables at the I boundary,
for example, is

˙ m = r b ˙ m b = (ω k,b − ω k,T ) −1 ∂ω k , (4.59)
b
k
∂y
y=0
where ω k,T refers to the mass fraction deep inside the I boundary. The suffix T, thus,
represents the transferred substance state and ω k,T must be known. Equation 4.59 is
again a flux condition, therefore, it can be treated in the manner of the q b condition
just described. Again, from the converged solution, ω k,b can be extracted.
When heterogeneous chemical reactions occur at the wall, ˙ m b is typically given
by the Arrhenius relationship, which yields
˙ m b = f (ω k,b , T b ). (4.60)

The exact implementation of the boundary condition for a heterogeneous reaction
requires modification of Equation 4.59. This is explained later through an example
of carbon burning (see Equation 4.129).
In problems involving evaporation or condensation, the value of ω k,b itself can
be specified from the equilibrium relation (or saturation condition).
ω k,b = f (T b ). (4.61)

Thus, in mass transfer problems with or without surface chemical reaction, ˙ m b can
be known and this knowledge can be used to evaluate ψ b from Equation 4.54. It is
important to remember, however, that the most general problem of mass transfer is
usually quite complex and, therefore, several manipulations are typically introduced
to simplify the boundary condition treatment [33, 38].

4.6.3 Free Stream

1
The free-stream boundary condition has relevance only when external boundary
layers are considered. The free stream is really a fictitious boundary and is identified

1 In internal flows, only wall or symmetry conditions are relevant because in these flows the flow
width is a priori known. Thus, for developing flow between two parallel plates a distance b apart,
for example, the flow width b remains constant with x. However, in 2D plane diffusers or nozzles,
b may vary with x but still be known a priori.