First Order Reaction in a Spherical Pellet
The second pdf file for this Learning Module titled Collocation Primer describes in considerable detail how collocation is used to solve boundary value problems, such as reaction with diffusion in a spherical pellet. Here we will explore the effect of changing the value of the Thiele modulus on the concentration profile in a spherical pellet for a first-order reaction.
As detailed in the text, increasing the value of the Thiele modulus leads to a steeper gradient of the concentration within the pellet. We will see that for even modest values of the Thiele modulus, the concentration within the pellet quickly becomes vanishingly small. Then by exploring this dependence you should get a sense of the importance of this parameter in deciding what is important or controlling in a given situation.
Since there is an analytical solution to this problem, viz.,
we can compare the numerical solution against the analytical solution. This will allow you to examine how the number of collocation points needs to increase as the concentration profile becomes steeper near the pellet surface.
Note that the curves are plotted in a semi-log format so you can better see the magnitude of the concentrations within the pellet. In the handout Collocation Primer, the curves appear to approach zero, but they never do.
THE QUESTIONS
As you run the simulation you will be able to separately set the value of the Thiele modulus, Φ, and the number of collocation points.
- Change the value of Φ from the default. Select values that change by an order of magnitude to see how the concentration varies with Φ. This should give you a sense of when you could reasonably assume the reaction is kinetically controlled versus diffusion controlled.
- Make a point of examining a value of Φ = 10 and note how small the dimensionless concentration becomes within the pellet. What is the concentration of a gas at 1 atm and 200 °C? What is the absolute concentration with in the pellet?
- Holding Φ at the same value, increase the number of collocation points to see how more points are needed to approximate the analytical solution. Notice that with enough points, the fit is very good. You should also notice that as you select more and more collocation points, the computations take longer and longer so there is a price to pay for selecting too many points.
