INSTRUCTOR: In this video,
we're going to consider the blending process,
which is a non-linear process.
And we're gonna calculate an approximate linear model
for that process.
We're going to consider the dynamic equation
for mass fraction X leaving the tank.
We derive this dynamic model previously
using mass balances.
This is a non-linear model.
And we're gonna refer to this right-hand side
as our function F.
It depends on time varying quantities.
Volume V,
mass fraction X.
Now, this is the mass fraction in the tank
and leaving the tank.
Because the tank is well-mixed.
We also have inlet mass flow rates W one and W two.
And in this case we're going to assume
that our density row is constant.
We're also going to assume that the inlet mass fractions,
X one and X two,
are constant for this example.
And now we're going to calculate the linearization
of this non-linear dynamic equation.
For the left-hand side,
we'll substitute in deviation variables,
so instead of X,
we'll have X bar, the steady state value,
plus X prime, the deviation from steady state.
And because X bar is a constant,
we can then simplify that to the DX prime DT.
So the deviation is time varying.
But the steady state value is a constant.
And then we're gonna approximate the right-hand side
by a first order Taylor series approximation.
So we start off by evaluating F at the steady state.
Then we add on linear terms associated
with each of the independent variables
in our function F.
So we have DF,
DV evaluated at steady state,
we'll use SS for steady state
times V prime
plus DF,
DX
evaluated at steady state
times X prime
plus DF, DW one
evaluated at steady state
Times W one prime
plus the FW two
evaluated at steady state
times W two prime.
So one thing to notice here is that
when we evaluate the derivative of F
with respect to each of these independent variables,
we are evaluating at the steady state value
and those are gonna be constant values.
So the only time varying quantities
in this new model
are going to be associated with the deviation variables.
And those all enter linearly.
So, the reason that this model is gonna be linear
is because we're using only
a first order Taylor series approximation.
If we want it to be more accurate,
we can include higher order terms,
but then we would be back to having a non-linear model.
Now, let's go ahead and plug in
our function F and its derivatives
into this approximate linear model.
First, we need to evaluate this function
at a steady state value.
But the function is just the time derivative of X,
and at steady state we can plug in zero
DX, DT equals zero,
and the bar variables on the right-hand side
and so,
this term is always going to be zero
for any linearization around steady state.
So, that's very convenient.
We can always cross off that term
as long as we're linearizing around steady state.
But we do need to go ahead and evaluate these derivatives.
Now, the first derivative is going to be the messiest
because we have one over V
in our differential equation.
We'll go ahead and evaluate the derivative of F
with respect to V at steady state.
So, we're basically just gonna rewrite all these other terms
that we're going to hold at their constant value
for this derivative.
And then we're gonna need to take the derivative
of one over V.
So, we're gonna get
minus one V to the minus two
and we're gonna evaluate that again.
The bar value.
And then we're gonna multiply this entire derivative,
DF, DV by V prime.
So that is our first term.
We need to do that three more times.
But fortunately,
the function is not as complicated in order to do that.
So, that is the linearization for any
arbitrary steady state of the system.
But let's plug in some numbers.
So, we'll use the same values
that we had used previously.
And from these steady state inlet conditions,
we can then calculate the steady state outlet conditions.
So W bar is just the sum of the two inlets streams.
So that's four kilograms per second.
And X bar is 0.025,
which is an intermediate value
between X one and X two,
which we are blending.
Now that we have numerical values
for all of our variables,
let's go ahead and plug them in.
So now we can go ahead and simplify that.
It actually turns out
that the V prime term cancels.
We have a 0.075 term here
and another here.
So those two cancel out
and we actually don't have a V prime equation
in our final equation.
And then we can solve this for X prime,
and then that's in deviation variables,
but then we can go back and solve for X
is approximately equal to X bar plus X prime of T.
But it is important to remember
that this model in a linear approximate model
and the approximation should only be good
for small deviations from steady state.
Small values of our deviation variables.
That's because the first order Taylor series approximation
is only going to be accurate
for small deviation terms.