[Law_C.K.] Combustion physics

6.3. Condensed Fuel Vaporization and the Stefan Flow 209
Y , T 1,
x =
Y 1,o, To
x = 0
Component 1
in liquid phase
Figure 6.3.1. Schematic of vaporization from a one-dimensional chamber.
is another kind of convection, called the Stefan flow, which is internally generated
and can be present even in the absence of any externally imposed flow. Such a flow
frequently arises as a consequence of the gasification of a condensed fuel. Here a
fuel vapor source is present at the surface of the condensed fuel, and the ambience,
or the flame, which is located away from the surface, is typically low in the fuel vapor
concentration and thus represents a sink for the fuel vapor. The fuel vapor then
continuously diffuses from the surface to either the ambience or the flame. Since the
consequence of this diffusion is a net transport of mass, a convective motion, that is,
the Stefan flow, is induced. This motion can be quite significant and has to be accounted
for, especially for rapid rates of vaporization when the condensed fuel is
volatile, or when it is placed in a hot environment or undergoes combustion. This
phenomenon is also particularly relevant for nonpremixed combustion because practical
fuels are frequently present in the condensed phase, for example fuel droplets
and coal particles.
To demonstrate this phenomenon we consider the following chemically nonreactive
example. Here a pool of pure liquid, say water, designated by i = 1, undergoes
vaporization in an open container (Figure 6.3.1), with its height l fixed. Its vapor
concentration at the surface is Y 1,o .There is a constant breeze over the container
such that at its edge the gas composition is the same as that of the environment,
consisting of Y 1,l of species 1 and Y 2,l = 1 − Y 1,l of a noncondensable species, say air.
We aim to determine the mass vaporization flux, f = (ρu) o .
From continuity, d(ρu)/dx = 0, we have
ρu = f = constant, (6.3.1)
which shows that this internally generated convection is a constant, and as such
directly yields the mass vaporization flux.
From species conservation, Eq. (5.3.14), we have
f dY i
dx − d
dx
(
ρ D dY i
dx
)
= 0. (6.3.2)