This is a problem dealing with internal effectiveness factors and diffusion limitations in porous
catalysts.
So here we start with very finely ground catalyst particles, where we expect that there is very
little diffusion limitation, and we observe a certain reaction rate of 4.0 in arbitrary
units, and we use the same mass of particles but in a pelletized form where they're larger
in diameter, reaction rate is 1/4 that of the reaction rate for the small pellets.
So the question then is, for a particle diameter of 1 mm, what is the approximate rate for
the same catalyst mass?
The concept here is that the true rate of reaction, or the intrinsic rate, can be related
to the rate that we actually measure, or that we actually operate at, by an internal effectiveness
factor.
This rate over here on the right hand side is the actual kinetic rate, that is the rate
of reaction at the surface of the catalyst, and this rate on the left hand side that we
would actually observe in our process, in an operating reactor, is the one that can
be lower than the true rate because of slow diffusion, potentially, and that lowering
of the rate is taken into account by the effectiveness factor.
So for the finely ground catalyst, we observe an effectiveness factor of 1, we expect, and
for the 8 mm catalyst, the rate goes down by a factor of 4, and so our effectiveness
factor is 1/4.
Then we're asked about what we expect for a 1 mm catalyst, and here we have to utilize
the fact the effectiveness factor can be plotted against the Thiele modulus, and at low values
of the Thiele modulus, the effectiveness factor is 1, so somewhere over here is where we measure
our effectiveness factor for the finely ground catalyst.
Then, as we increase the Thiele modulus, then the effectiveness factor will start to decrease
pretty rapidly, and it decreases to a value of 0.25, when the Thiele modulus is such that
the catalyst diameter is 8 mm.
Now, the Thiele modulus is proportional, for a spherical catalyst particle, to the diameter
of the particle, and so the Thiele modulus, in going from 8 to 1 mm catalyst particles,
it's going to change by a factor of 1/8, go down by a factor of 8.
Alright, and so that's going to push us in this direction.
And the Thiele modulus here, we can actually read this off of a plot.
So in most textbooks, there is a plot of effectiveness factor versus Thiele modulus, so we can actually
read off that the Thiele modulus for a spherical catalyst particle should be 4.
That means it should be 1/2 when we decrease the catalyst diameter to 1 mm, again we go
down by a factor of 8.
And if you just read off where you are on the plot, you can see on our little schematic
here that it shows up as being close to 1, and that is also observed in the case of actually
reading that off of a real plot, so if you use that effectiveness factor plot, what you
observe is that for that Thiele modulus, the effectiveness factor is very close to 0.9
or maybe even a little bit above that for a Thiele modulus of 1/2.
And so, therefore we expect that the rate we observe is equal to that effectiveness
factor times the intrinsic rate of 4.0, and therefore we expect that the measured rate
is roughly 3.6 in arbitrary units.