DIFFUSIVE FLUX, HEAT

1
DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Molecular diffusion is a process by which random
molecular motion moves any quantity down the
concentration gradient, i.e. from high
concentration to low concentration. Diffusion
does not require flow, but it operates in the
presence of flow.
Consider the illustrated container of water. A
dilute concentration of dye (molecules) is placed
in the lower half of the container.
In time, molecular action cause the dye-free
fluid to mix with the dye-laden fluid, so that
the concentration eventually becomes uniform.
2
DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Molecular diffusion is a process by which random
molecular motion moves any quantity down the
concentration gradient, i.e. from high
concentration to low concentration. Diffusion
does not require flow, but it operates in the
presence of flow.
Consider the illustrated container of water. A
dilute concentration of dye (molecules) is placed
in the lower half of the container.
In time, molecular action cause the dye-free
fluid to mix with the dye-laden fluid, so that
the concentration eventually becomes uniform.
3
DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Molecular diffusion is a process by which random
molecular motion moves any quantity down the
concentration gradient, i.e. from high
concentration to low concentration. Diffusion
does not require flow, but it operates in the
presence of flow.
Consider the illustrated container of water. A
dilute concentration of dye (molecules) is placed
in the lower half of the container.
In time, molecular action cause the dye-free
fluid to mix with the dye-laden fluid, so that
the concentration eventually becomes uniform.
4
DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Molecular diffusion is a process by which random
molecular motion moves any quantity down the
concentration gradient, i.e. from high
concentration to low concentration. Diffusion
does not require flow, but it operates in the
presence of flow.
Consider the illustrated container of water. A
dilute concentration of dye (molecules) is placed
in the lower half of the container.
In time, molecular action cause the dye-free
fluid to mix with the dye-laden fluid, so that
the concentration eventually becomes uniform.
5
DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
In the case below the dye is diffusing in the x3
direction. Let c denote the concentration of
dye. Note that c is a decreasing function of x3,
so that
The diffusive flux of dye in the vertical
direction is from high concentration to low
concentration, which happens to be upward in
this case.
c
The simplest assumption we can make for diffusion
is the linear Fickian form where FD,con,3
denotes the diffusive flux of contaminant (in
this case dye) in the x3 direction,
x3
where Dc denotes the kinematic molecular
diffusivity of the contaminant.
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
The units of c are quantity/volume. For example,
in the case of dissolved salt this would be
kg/m3, and in the case of heat it would be
joules/m3.
The units of FD,con,3 should be quantity
(crossing)/face area/time. In the case of
dissolved salt, this would be kg/m2/s, and in the
case of heat it would be joules/m2/s.
The units of Dc are thus
c
These units happen to be the same as those of the
kinematic viscosity of the fluid, i.e. ?.
x3
In the case of heat, Dc is denoted as Dh and
FD,con,3 is denoted as FD,heat, 3.
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
The 3D generalization of the Fickian forms for
diffusivity are
where c is the concentration of the contaminant
(quantity/volume).
The concentration of heat per unit volume
(Joules/m3) is given as ?cp?. Thus
where k ?cpDh denotes the thermal conductivity.
The dimensionless Prandtl number Pr and Schmidt
number Sc are defined as
This comparison is particularly useful because we
will later identify the kinematic viscosity with
the kinematic diffusivity of momentum.
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Some numbers for heat Heat in air
Heat in water
In the above relations ? denotes the dynamic
viscosity of water.
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Some values of Dc and Dh are given as follows.
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Consider a control volume that is fixed in space,
through which fluid can freely flow in and out.
In words, the equation of conservation of
contaminant is
?/?t(quantity of contaminant in control volume)
net inflow rate of contaminant in control volume
Net rate of production of contaminant in
control volume
Contaminant concentration is denoted as c
(quantity/volume). Contaminant can be produced
internally by e.g. a chemical reaction (that
produces heat or some some species of molecule).
Let S denote the rate of production of
contaminant per unit volume per unit time
(quantity/m3/s). Where S is negative it
represents a sink (loss rate) rather then source
(gain rate) of contaminant.
The net inflow rate includes both convective and
diffusive flux terms. Translating words into an
equation,
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
But by the divergence theorem
Thus the conservation equation becomes
or since the volume is arbitrary,
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Now
So the conservation equation reduces to a
convection-diffusion equation with a source term
If the fluid is incompressible, i.e. ?ui/?xi 0,
the relation reduces to
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DIFFUSIVE FLUX, HEAT CONTAMINANT CONSERVATION
Special case of heat, for which c ? ?cp? and Dc ?
Dh, S ? Sh
or thus