78
Heat and Mass Transport
Darcy 's law for fluid (water) flow in a porous medium (soil):
In Eqs. (6.1) through (6.4) t is the shear stress ( ~ l m ~ )
between layers of
a moving fluid with dynamic viscosity ,u and velocity gradient duldz; Fi
is the flux density (kg m-2s-') of a diffusing substance with molecular
diffusivity D j (m 2 /s) and concentration or density (kg/m
3 ) gradient of
dpj/dz; H is the heat flux density (w/m2) in a substance with thermal
conductivity k ( ~ m - '
K-') and temperature gradient d Tldz (Clm); and
J, is the water flux density (kg m-2 s-') in soil with hydraulic conductivity K(@) (kg s m-3) and water potential gradient d@/dz (J kg-' m-'
or m/s2). We use Eq. (6.2) mainly for describing diffusion of gases in air.
The subscript j represents the different substances that diffuse through
air. Here we are concerned mainly with water vapor, C02, and oxygen for
which we use the subscripts v, c and o. The negative signs in Eqs. (6.2)
through (6.4) indicate that the flux is in the positive direction when the
gradient is negative.
The strong dependence of hydraulic conductivity on water potential in
unsaturated soil is indicated by K (@) in Eq. (6.4). The other coefficients
are almost constant. Equations (6.1) through (6.3) therefore express a
nearly linear relationship between a flux density and a driving force (a
"concentration" gradient).
6.1 Molar Fluxes
In order for us to use the mass transport equation, it needs to be converted
to the form for molar fluxes. Substituting Eq. (3.2) into 6.2 gives
where Fj is in mol m-2 s-'. This is the form we use throughout this
book. Note, however, that mole fluxes are easily converted to mass fluxes
through multiplication by the molecular mass of the diffusing gas.
There are several advantages to expressing the heat equation in the
same form as Eq. (6.5). The mathematical manipulations are then the
same for both transport processes, and the units (m
2
/s) are the same for
both diffusivities. The diffusivities are roughly the same size and they
have similar temperature and pressure dependence (which can be derived
from kinetic theory). For many conditions of interest in environmental
biophysics, the diffusivities are constant multiples of each other so if one
is known, the other is easily found. Equation (6.3) can be converted to a
form similar to Eq. (6.5) by multiplying and dividing by c,, where c, is
the molar specific heat of air (29.3 J mol-' C-I). The quantity klc, is
the thermal diffusivity DH, SO Eq. (6.3) becomes
Heat and Mass Transport
Darcy 's law for fluid (water) flow in a porous medium (soil):
In Eqs. (6.1) through (6.4) t is the shear stress ( ~ l m ~ )
between layers of
a moving fluid with dynamic viscosity ,u and velocity gradient duldz; Fi
is the flux density (kg m-2s-') of a diffusing substance with molecular
diffusivity D j (m 2 /s) and concentration or density (kg/m
3 ) gradient of
dpj/dz; H is the heat flux density (w/m2) in a substance with thermal
conductivity k ( ~ m - '
K-') and temperature gradient d Tldz (Clm); and
J, is the water flux density (kg m-2 s-') in soil with hydraulic conductivity K(@) (kg s m-3) and water potential gradient d@/dz (J kg-' m-'
or m/s2). We use Eq. (6.2) mainly for describing diffusion of gases in air.
The subscript j represents the different substances that diffuse through
air. Here we are concerned mainly with water vapor, C02, and oxygen for
which we use the subscripts v, c and o. The negative signs in Eqs. (6.2)
through (6.4) indicate that the flux is in the positive direction when the
gradient is negative.
The strong dependence of hydraulic conductivity on water potential in
unsaturated soil is indicated by K (@) in Eq. (6.4). The other coefficients
are almost constant. Equations (6.1) through (6.3) therefore express a
nearly linear relationship between a flux density and a driving force (a
"concentration" gradient).
6.1 Molar Fluxes
In order for us to use the mass transport equation, it needs to be converted
to the form for molar fluxes. Substituting Eq. (3.2) into 6.2 gives
where Fj is in mol m-2 s-'. This is the form we use throughout this
book. Note, however, that mole fluxes are easily converted to mass fluxes
through multiplication by the molecular mass of the diffusing gas.
There are several advantages to expressing the heat equation in the
same form as Eq. (6.5). The mathematical manipulations are then the
same for both transport processes, and the units (m
2
/s) are the same for
both diffusivities. The diffusivities are roughly the same size and they
have similar temperature and pressure dependence (which can be derived
from kinetic theory). For many conditions of interest in environmental
biophysics, the diffusivities are constant multiples of each other so if one
is known, the other is easily found. Equation (6.3) can be converted to a
form similar to Eq. (6.5) by multiplying and dividing by c,, where c, is
the molar specific heat of air (29.3 J mol-' C-I). The quantity klc, is
the thermal diffusivity DH, SO Eq. (6.3) becomes