In this video I will demonstrate how to use nonlinear regression to determine kinetic
parameters from kinetic data. In the actual calculations I am going to use POLYMATH because
it is a very flexible program that allows us to do more complicated nonlinear regression
problems. The situation we are going to look at is we have kinetic data that we obtain
using a continuous stirred tank reactor. To accomplish this what we do is we change the inlet
volumetric flow rate, v_0, and we keep the concentration of that stream constant.
We measure the concentration at the outlet and with a mass balance on the CSTR we can
then directly obtain the rate of reaction in this CSTR. And the mass balance, we are at steady state so we
will pick a number of different steady state conditions, so steady state means the volumetric
flow rate in times the concentration of our reactant in, that's a molar flow rate in, the volumetric flow rate out
is going to be the same for this liquid phase system. The concentration is going to be different, and it's going to be different because we have reaction.
We can rearrange this equation to say the rate of reaction, I'm going to write this with a minus sign as we'll see why in a few seconds here,
this is the equation. I'm going to call this the rate of reaction, so this is positive. C_A0 is greater than C_A.
What we are going to obtain is data at a number of conditions. What I have listed here is
the concentration. This is the concentration of reactant A, so some reaction of A goes to
B. This is our experimental rate and we do not have to worry about units. We got this
rate by measuring the concentration for different volumetric flow rates. What we want to do
is determine how does the rate depend on the concentration of A in some power log expression.
We want to determine the value of "k" (the rate constant). so for a given temperature,
and everything we are doing here is isothermal, it's a constant. This is
the order of reaction, first order, second order, et cetera. The way we are going to do this is we're going to guess values. We will guess
"k" and "n" and then we will calculate from this rate expression what rate we get. For
example I am showing the same data again. Then I guess k as 0.020. I guess first order,
n=1. Then I calculated "rcalc". You can see it varies and as we get to the higher
concentration vary significantly, so this is certainly not the right guess. What we
will do is use an iteration. An iteration says let's take the difference of r_experimental
minus r_calculated, square it for each of these concentration values and we'll sum all these, and
this is what we want to minimize. We square it because the sign should not be important as
to whether the calculated rate is higher or lower than the experimental rate. We want
to have the minimum by having all of the calculated values close to the experimental value. We
are going to use a regression technique to do this. This is a nonlinear regression and
the program that we use is POLYMAYTH, a very robust program that allows us to do more complicated problems.
We put values into the program and we will use a
separate video to demonstrate how to use the program. Here is the output from the program.
It shows our initial guess. The answer says that our rate is 0.0859 times the concentration
of A to the 1.48 power. 1.5 order is certainly a reasonable result for these data. It gives us a confidence limits and a number of other properties.
It also shows, here now is our measured value, our experimental value, here is our calculated value. Now you notice the values
are much closer than showed earlier. This is the difference between the two. This then allows us to take
kinetic data we started out with and determine order of reaction and rate constant from kinetic
data.