MARTHA: Here we'll consider the blending process
and describe deviation variables.
So it's often desired to operate
a process at a constant steady state,
but sometimes there are deviations that happen.
So here we're gonna consider the blending process
with change in volume, the function of time,
described as one over the density
times the flow rate in through stream one,
the flow rate in through stream two,
and the flow rate leaving.
And then our second dynamic equation
is for the mass fraction of A in the exit stream.
We have one over rho V in the denominator,
then we have w1 times x1 minus x
plus w2, x2 minus x.
So the first equation is linear
and the second equation is nonlinear
since we have terms like w1 times x1.
So here, we're gonna introduce deviation variables
for any variable that could change in time.
And so to simplify our analysis here,
we're gonna assume that x1 and x2 are constant,
so those mass fractions in the inlet streams
are not going to change in time.
But we will introduce deviation variables for volume,
so we'll have the constant steady state,
volume V bar plus a deviation, V prime.
Same for our mass fraction x, that's x bar plus x prime,
and then we'll also introduce
deviation variables for our flow rates.
W is w bar plus w prime.
And we can introduce those similarly for w1 and w2.
So now we can plug into our original
equations our deviation variables.
Instead of V, we'll have V bar plus V prime,
and then that's equal to one over rho times w1 bar
plus w1 prime plus w2 bar plus w2 prime
minus w bar minus w prime.
And we can simplify this somewhat.
The derivative operator is a linear operator,
so we can distribute it within our two terms here.
And V bar is a constant, so that term is zero.
And then we also know from our steady state calculations
that w bar is equal to w1 bar plus w2 bar.
So at steady state, the outlet flow
is equal to the inlet flow.
And that allows us to then cancel out these bar terms here.
Then we are left with the equation,
so this happens to look just like the original equation
except we have primes on all of our terms,
on V and on the w's.
The reason this looks mathematically similar
to the original equation and deviation variables
is because the original equation was linear.
But when we have a nonlinear equation,
then it's not going to look similar.
So if we move on to our equation for x,
we write that in deviation variables instead of x.
we can still write x bar plus x prime,
and then we can still distribute the ddt
to give us the x prime ddt, since x bar is a constant.
And now we have one over rho V,
but we'll replace that with V bar plus V prime
times w1, which is now w1 bar
plus w1 prime quantity times x1 bar.
Now we assumed x1 and x2 are both constants,
so we do not have a deviation variable for x1,
but we do for x, so the outlet mass fraction
can still change over time.
Then we have an additional term, w2 bar plus w2 prime
times x2 bar minus x bar minus x prime.
So this equation does not simplify in the way
that the first linear equation did.
We have V bar plus V prime in the denominator,
then we also have products of terms
like w1 bar times x prime or x1 prime times w1 prime.
Now if these deviations from steady state are small,
which from a practical processing point of view,
then we might assume that the product
of two small terms is small.
And that's what we will do going forward,
so we will use deviation variables to perform
a linearization using a Taylor Series expansion
in the next video to formalize this idea
of deviation variables and to compute
linear approximate models that will be accurate
if we stay very near steady state with small deviations.