Environmental Engineering Reference
In-Depth Information
velocity of the species. The diffusion term describes the motion of the species relative
to the mean mixture. In the literature, the advection term often is called the convection
term. In the context of heat and mass transfer problems, the notion
refers
to combined effects of advection and diffusion, as is discussed in Section 4.4 on con-
vective heat and mass transfer.
The diffusion velocities in general depend on the set of binary diffusion coeffi-
cients associated with each pair of components of the mixture, but are also influenced
by pressure gradients, differences in body forces between components and tempera-
ture gradients; this is the so-called Soret effect (Poinsot and Veynante, 2011). Keeping
only the effect of the binary diffusion coefficients, the Stefan
convective
Maxwell equations are
obtained as a set of equations determining the diffusion velocities:
-
! X i = X
N
X i X j
D ij
! j
! i
r
ð
Eq
:
4
:
2
Þ
j =1
Here, N is the number of species, D ij is the binary mass diffusion coefficient of species
i into species j, and X i denotes the mole fraction of species i. The mole fractions can be
calculated from the mass fractions and vice versa using the molecular weights (MW)
of the species. The Stefan
Maxwell equations form a linear system of size N 2 to be
solved in each of the three spatial directions at each point and at each instant for
unsteady flow. If the mixture contains only two species (N = 2), the system can be
solved exactly and the well-known Fick
-
'
s law is obtained (Kuo, 2005):
! 1 Y 1 =
! Y 1
D 12 r
ð
Eq
:
4
:
3
Þ
Another case where Fick
s law is exact is multispecies diffusion (N > 2) with all
binary diffusion coefficients equal. In a situation with unequal binary diffusion coef-
ficients in principle, one should solve the Stefan
'
Maxwell equations, but to avoid
the complexity and cost for doing so, usually an approximation in the form of a gen-
eralized Fick
-
s law is used. In this approach, for each species, a diffusion coefficient
relative to the mixture D i , m is considered, and one uses, e.g., ! i Y i =
'
! Y i .An
expression for the coefficients D i , m has to be provided to complete the model. The best
first-order approximation to the solution of the Stefan
D i , m r
Maxwell equations is the
Hirschfelder and Curtis approximation (Poinsot and Veynante, 2011):
-
! i X i =
! X i
D HC
i , m r
ð
Eq
:
4
:
4
Þ
In the absence of detailed information on the composition of the mixture as is often
the case in engineering applications, an estimated effective diffusion coefficient D ,
often assumed equal for all species, is used.
In the species mass fraction Equation (4.1), the transient term and the advection
term on the left-hand side are balanced by the diffusive term and the source term
on the right-hand side. The complexity of the mass transfer problem decreases when
one or more of these terms vanish. This leads to a classification of mass transfer
problems. Some examples of simplification are:
Search WWH ::




Custom Search