Unsteady-State Diffusion

Unsteady-State Diffusion

In this screen cast we are going to look an
herbicide spill over a field. And the fluid remains on the soil for 10 minutes before
being depleted both into the air and into the soil. And we need to find the differential
equations that govern both of these processes as well as stating the initial and boundary
conditions needed to solve. So let’s set our coordinate system where positive z is going
to be in the upward direction. So this is an example of an unsteady state diffusion
problem. How do we know? Because the fluid is being depleted both into the air and into
the soil. It is not an instantaneous process so the concentration of herbicide is going
to change with respect to time. So our plan for both situations is the same. The first
thing we are going to do is simplify the continuity equation. And in this particular case we are
going to start with the equation in cartesian coordinates. The next thing we will do, because
the continuity equation is in terms of the flux or N we are going to simplify Fick’s
Law so that we can find this flux. And we are going to call it Na, where a will be the
herbicide. Next, that expression that we got for Na we are going to substitute it into
the continuity equation and what this allows us to do is state the continuity equation
in terms of the concentration of a instead of in terms of the flux. Then we are going
to find appropriate initial and boundary conditions for both these processes. So let’s start with
the first situation which is the mass transfer into the air. And we are going to assume that
this is stagnant air. Let’s look at the continuity equation. So that is the change in concentration
of a with respect to t plus this divergence of our flux and that is going to equal our
reaction term. So this divergence term looks at the flux in all directions. The x, y and
the z. So the first term stays in, because the concentration of the herbicide does change
with time. Since the mass transfer is only in the z direction that will simplify to the
flux in the z direction with respect to z. And finally there is no reaction in our system,
so that becomes 0. So now what we have to do is we have to find an expression for this
flux in terms of our concentration. So that leads us to Fick’s Law that says that the
flux in the z direction is equal to negative the bulk concentration times the diffusion
coefficient times the change of the mole fraction of the mole fraction of the herbicide with
respect to z, plus, and now this par is what is considered the convective part of Fick’s
Law. So this ya times the Na plus Nb. Na is the flux of a into the air and Nb is the flux
of air into the herbicide. And this is going to equal 0, we have stagnant air. So now we
could write this as minus c*Dab, day/dz plus ya*Na. Now let’s write it in terms only of
this flux. In other words let’s gather our terms for this flux and this is going to equal
minus c*Dab, day/dz. However it asks for this in terms of the concentration of a. So we
take the definition of the mole fraction that says that it is equal to the concentration
of a divided by the total concentration. So now this can be written as minus Dab / 1 over
Ca over c. And this entire thing is multiplied by dCa/dz. Now that we have that let’s go
back here to our continuity equation. And what we are going to do is we are going to
take what we just calculated and substitute it into this. So now dCa/dt plus d/dz and
here comes our substitution, equals 0. So in order to solve this we have one term in
terms of time, but then we have these two terms in terms of z. So we are going to have
to find one initial conditions and 2 boundary conditions. So first of all, remember our
coordinate system says that the positive z is in the upward direction. In addition we
will say that t is going to equal 0 after 10 minutes. the reason for that is that the
fluid stays on the soil for 10 minutes before depletion begins. So now we can put these
together. First is the initial condition. And what it says is at t=0, and so for any
z>0, in other words, above the field, the concentration equals 0. We then have two boundary
conditions, 1 is at any t>0 we look at the surface of the soil and say that the concentration
of a is whatever the surface concentration is. And finally at t>0 as z goes to infinity
our ca is eventually going to become 0. So our diffusion into the soil is similar. We
start with the same continuity equation, however Fick’s Law is going to change because you
can assume that there is no convection in the soil. So we start with dCa/dt +dNaz/dz
equals 0. But now our Na is going to equal minus cDab, day/dz. And the reason for that
is this term ya*Na+Nb is going to equal 0. So again, we are going to write our flux in
terms of our concentration. And that is minus the diffusion coefficient, dCa/dz. And now
when we put it into our continuity equation, what we come out with is the concentration
of the herbicide with respect to t, minus this diffusion coefficient, d^2Ca/dz^2 equals
0. So again we need an initial condition and two boundary conditions. The conditions are
going to be the same as before, only in the opposite direction. The herbicide is going
into the soil, it is going in the downward z direction. So our initial condition is going
to say at t=0, remember that is after 10 minutes, and any z0 at z=0, Ca equals our
surface concentration and that is one of the boundary conditions from our previous situation.
finally at t>0 and as z approaches minus infinity, remember we are going in the downward z direction,
now our Ca equals 0. So between these differential equations and initial conditions and boundary
conditions it is possible to solve for the concentration A. And in this last situation,
this is what is known as a semi-infinite media. In other words there is only 1 definable surface
and it goes to infinity in the rest of the directions. And soil is usually modeled as
a semi-infinite media. And often the solutions include what is known as the error function.

4 thoughts on “Unsteady-State Diffusion

  1. The audio does sound worse than when it was produced. We will redo this one. Any suggestions on other improvements?

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