MARTHA GROVER: We're going to consider
a process called the blending process
and we're going to build a dynamic model
of that process using mass balances.
First let's consider a tank.
And that tank is gonna contain a liquid.
It's gonna be stirred, such that
when the fluid leaves through this exit stream,
it has the same properties in the exit stream
as it does in the tank.
We're gonna consider a mass fraction x
of species A.
And total mass flow rate, w, leaving the tank.
The tank is gonna have a volume, V,
and that volume can change,
but it will also have a density row
that we will assume is constant.
We will assume that we have liquid in the tank,
which is incompressible,
and that the species A of interest is dilute,
such that it's amount doesn't really change
the overall density.
This is the Blending Process
and we are going to blend two streams coming in.
Stream one, that has mass fraction X one
and flow rate W one.
We're gonna blend it with stream two,
that has mass fraction X two
and flow rate W two.
Now for this particular control problem,
we are interested in controlling
the mass fraction X.
That is our control variable, or CV.
And we're not gonna be able
to influence stream one,
that's coming from some upstream process.
So that's our disturbance variable, or DV.
But we will use stream two
to manipulate the process.
W two, in particular, the mass flow rate,
will be our manipulated variable, or MV.
And we are going to build a dynamic model
based on balance equations.
Anytime you do that,
you should really carefully describe
your control volume, so that you know
what is going in
and what is leaving the control volume,
so that you know what to balance.
So here, we'll draw our control volume.
And we're gonna use balance equations
to construct our dynamic model.
Usually we balance either mass or energy.
And here we were not given
any information about energy.
No temperatures or heat flows.
But we were given mass flows.
So let's go ahead and construct a mass balance.
First we will consider the mass
in the entire system, so the total mass here.
So let's look at the accumulation
in the control volume of mass.
So accumulation is equal to what is going in.
W one plus W two
minus what is leaving.
And we can break down that mass variable
into variables that we used in our diagram above.
So, in particular, we can break that down
into d/dt of row
times V, density times volume.
And because density was assumed to be constant
we can pull it out of the time derivative.
And we just have volume inside now.
So, now we can write our total mass equation
as row dv/dt
equals W one
plus W two minus W.
Now this will tell us about how volume change,
which is depending on flow rates in and out,
but it doesn't tell us about
our control variable, our mass fraction, X.
So in order to describe X,
which is the mass fraction of species A,
we should perform a second balance
on the mass of A.
So now if we look at the accumulation of A
in the control volume, that's the total mass,
row V, times the mass fraction, X.
And that will be equal to
the amount of A flowing in,
which is W one X one
plus W two X two
minus WX.
And again, we can break down the mass of A
by pulling out the row,
but now within our d/dt term
we have two time-varying quantities,
volume and mass fraction.
So we'll need to use the product rule
in order to break that down
into our two different variables, X and V.
So be very careful when pulling terms
out of your d/dt.
Make sure they're constant
before you pull 'em out of the d/dt.
So now we can write a balance equation,
a dynamic equation for dx/dt.
So we can row V
dx/dt equals W one X one
plus W two X two
minus WX.
And then we need to move that dv/dt term
to the other side.
So we have minus row
times X times dv/dt.
But we already know what is row dv/dt
from our total mass equation up there at the top.
So we can go ahead and plug that in
and get row V dx/dt
equals W one X one
plus W two X two
minus WX
minus row X
times dv/dt
which is one over row
times W one plus W two
minus W.
This will allow us to get
an equation for dx/dt.
We can cancel those rows.
And we also can cancel this WX
with this W and X out here.
And that will allow us to get our final equation
for dx/dt,
which is one over row V
times W one
times X one minus X
plus W two
times X two minus X.
So this equation for dx/dt,
along with our previous equation
for dv/dt,
which we'll rewrite here,
gives us a set
of two coupled
non-linear ordinary differential equations.
We can't solve one without solving the other
because mass fraction X in volume V
depend upon each other.
But we do here now have a dynamic model
for the Blending Process to describe
the time-varying volume and mass fraction
in the system.