In this example, we're going to look at how to calculate the effectiveness factor when
we have a catalytic reaction in one catalyst pore. And so we represent this pore as being
very long compared to it's diameter - so this is not a to scale representation, and the
break here in the diagram is meant to indicate that the length is much larger than the diameter.
To make this calculation easier, the catalyst, the green, is on the wall of this pore but
it's not on the end. So what we want to do is determine how effectively we're using the
catalyst. Namely, what is the rate of reaction in this pore divided by the rate if diffusion
was sufficiently fast that the reacting concentration was the same everywhere inside the pore. And
to do this, we have to calculate what is the concentration as a function of distance down
this pore. And we're going to do this for a first order reaction. We're going to do a material balance.
So what I'm going to look at is a small volume, a differential volume. This will be x equals
zero; this will be x; this is x plus dx; and then this is x equals L, the length of the
pore. And I want to do a mass balance for just an A goes to B reaction. I'll do a balance
on A at steady state, so there's no accumulation but I have diffusion of the reactant into
this volume that we're looking at here. I have diffusion out - they are not the same
because I have what I'll call a rate of generation, a rate of reaction, where if I have A being
made in there, would be positive, A's being used up, it's negative. So let's look at the
first term, diffusion in. Well the area for diffusion and then the flux is diffusivity
of A and the concentration gradient. The negative sign is present because diffusion goes from
high concentration to low concentration. This is the area and this is the flux, and this
is evaluated at x. And then I have the same type term evaluated at x plus dx. And then the
last term is the generation term. I have A reacting, so it's negative, it's a rate constant
times the concentration of A - because it's first order. And this rate constant is defined
per area of catalyst, so then I need to multiply by the area of the catalyst in this differential
volume. So that's pi times the diameter times the length of that differential volume. These
are all the terms in the mass balance. Have to have consistent units for K since I'm talking
about the surface available for reaction. And then I need boundary conditions, and the
two boundary conditions are first, that at x equals L, we'll call this concentration
CAs. And at x equals zero, there's no flux through wall, in other words, it's a solid wall. So
A does not diffuse through the wall, which means the concentration gradient at that point
must be zero. I can now simplify the material balance. First thing I'll do is cancel the
pi terms. And I'll cancel one of the diameter terms, and then I'll combine the first two
terms. So I've factored out common terms for the first two terms and reordered - this negative
times this negative means this term is positive. So I can divide both terms by dx, and we recognize
that this term here, then, is a derivative. And so this means we end up with a second derivative.
So I wrote the derivative then I rearranged so the second derivative's on one side. And
then what I've done on the right here is just to define a new variable to make it easier
to manipulate, and I defined this as lambda squared because I already know the answer,
I know that's a more convenient way to define it. So now I can solve this differential equation.
And the solution I'm looking for is, when I differentiate the concentration of A twice,
I get back the concentration of A multiplied by a constant. The solution of this would
be an exponential, so e to the lambda x, or e to the minus lambda x, either one is going
to be a solution. So what we're going to use then is the hyperbolic cosine of lambda x
as our solution, because this is equal to the sum of e to the lambda x, e to the minus
lambda x over 2, and both of those, as we said, were a solution. So what we're going
to say, then, is the concentration of A is come constant times the hyperbolic cosine
of lambda times x. And this has to satisfy our boundary conditions. So when x equals
L, CA is CAs, which means that CAs equals C1 times the hyperbolic cosine of lambda times
L. And so therefore, this is the value of C1, which I can substitute back here and get
the concentration of A, where L, remember, is the length of the pore. And then I can look
at the other boundary condition, that the derivative of the concentration of A with
respect to x equals zero at x equals zero. And so the derivative - CAs is a constant,
this the hyperbolic cosine in the denominator is a constant, and the derivative of the hyperbolic
cosine is going to be lambda times the hyperbolic sine of lambda x, evaluated at x equals zero.
This term then is zero at x equals zero, which means this derivative is zero, so we satisfied our
other boundary condition with this solution. So we've completed one of our objectives,
we've determined the concentration as a function of x. This says, as x increases, the concentration
increases, because remember, x, in our original diagram, x equals zero, we have the lowest
concentration, we have CAs at x equals L. So now what we want to do is calculate the
effectiveness factor, which is the rate of reaction in this pore divided by the rate
of reaction in the pore if the concentration was uniform everywhere and equal to this value
at the inlet of the pore, CAs. We could integrate over the pore, over the changing concentration,
to determine the total rate, or we could just calculate - what's the net flux of A into
to the pore, because everything net that diffuses into the pore must react. And so what we're
going to do then, is calculate this rate of reaction in the pore as being the rate of
diffusion into the pore. And that we can calculate from the diffusivity and the concentration
gradient evaluated at the pore mouth. So this is the flux into the pore. We need to multiply
this by the area to get the total amount diffusing in. We're going to differentiate
our concentration term that we've derived. So I've made this substitution bringing the
constant outside. The derivative of the cosine is lambda times the hyperbolic sine, and this
is evaluated at x equals L. So I combined the terms together and made the substitutions.
This is the hyperbolic tangent of lambda x. Also, substitute back our definition of lambda - we
have this equation. Now, we're interested in the effectiveness factor, which means the
rate of reaction divided by the rate if CA equals CAs everywhere. So I want to divide
this right side, then, by the area for reaction and then rate of reaction per area. And now
I can simplify these terms where this is our effectiveness factor. I'm going to go through
and cancel out terms, try and simplify this. I end up with this expression, which I've
rearranged by putting the constants in the denominator and written them in this form
because notice that this term here is just lambda. So I'm going to end up with that the
effectiveness factor for this pore is the hyperbolic tangent of lambda L over lambda
times L. So what is commonly done is to define a new variable, lambda times L, which is called
the Thiele modulus. And so the effectiveness factor is the hyperbolic tangent of the Thiele
modulus divided by the Thiele modulus. And what I've done is then plot the Thiele
modulus on the x axis and effectiveness factor on the y axis. This is on a linear scale,
showing us the Thiele modulus increases. Thiele modulus increase means our - one way
is the rate constant increases, then the catalyst becomes less effective. Or, as the diffusivity
decreases, then it becomes less effective. Or, as the diameter of the pore gets smaller,
it's more difficult diffusing in, it's less effective. And then it's often plotted on
a log scale. And so this shows that we can determine how effectively to use this catalyst
pore for a reaction if we know the rate constant, the diameter of the pore, and the diffusivity
and the length of the pore.