This reaction deals with the sizing of a PFR. We have an elementary irreversible reaction,
A plus B goes to C so 1 to 1 stoichiometry. We're given conditions of the pressure of
10 bars and temperature of 650K. We're given that we have an equimolar feed of A and
B and that they in sum represent 60% of the feed, we'll assume on a molar basis, and 40%
of the feed is inert. The conversion of A in an existing PFR is measure to be 50%
but how much we would need to increase that volume of the PFR to get the conversion up to 75%?
To start out with this problem we need to first have a material balance
for the PFR, and here's a material balance equation. The molar flow rate of
A, the limiting reactant, I guess both reactants are equal limiting, times the change in conversion
with respect to volume is equal to the rate of reaction and since this is an elementary
reaction it's equal to a rate constant times the concentration of A times the concentration
of B. Our next step is to express these concentrations in terms of conversion
and we'll do that in the next step here. It's useful to write this out, that C_A is
equal to C_A0 times 1 - X and if this were a reaction in which we could have no
change in the volumetric flow rate with reaction then we could stop there, but in fact we can
get a change in the volumetric flow rate with reaction, so I'll use the nomenclature
of Fogler in solving this problem and this has been described in previous screencasts,
that the change in volumetric flow rate can be accounted for by this term 1 plus Y_A0
which is the initial mole fraction of A in the feed and that's equal to 30%, 0.3, times
the change in total number of moles in the system when 1 mole of A is reacted and so
when 1 mole of A reacts then you lose a total of 1 moles in the system because for each mole
of A that reacts you consume a mole of B but you only produce a mole of C and so the total
change in the system is negative 1 and so we see that expressed here and so now, because
C_B is also equal to C_A since we have an equimolar feed of 30% and 30%, and they're
consumed at the same rate in the reaction, then we just square that term and we end up with
a k times C_A0 squared times 1 - X squared and then our 1 - 0.3X quantity squared
here. Now we have this differential equation, we just need to set it up for a solution and
so we'll pull all the X terms to the left hand side of the equation and pull all the volume
terms, of which there is only one, over to the right hand side of the equation and then
this is what we end up wit, though we can collect a number of constants here. As we'll see in
a minute we can refer to this collection of constants as a Damkohler number. It's
a dimensionless quantity. The X containing terms look like this once we assemble them
and the back of the Fogler textbook or a standard integration handbook will show you that you
can express that in the terms shown over here for an arbitrary X. We're initially
interested in an X at value of 50% conversion, because that will describe the current volume
of our system. So at an X of 0.5 we can evaluate the integral just by plugging in
here and then we can see that this number is equal to 0.83, this dimensionless number.
We could try to take some further steps. we're asked to determine by what factor the
volume needs to change. Presumably if this was a real reactor we would know the volume from
the manufacturer or we could measure it. We're not given any information on the rate constant,
we're not really told what the inlet flow rates are. We could solve for the concentration
using the ideal gas law but we'll see that's not really necessary. The reason
for that is that nothing here, we don't expect anything to change, we're just going to operate
the reactor exactly the same way and the only thing that's going to change under these quantities
is going to be the volume. So we want to know how much to increase the volume with everything
else held equal in order to get a conversion of 75%. So the k won't change because
this is an isothermal reaction and so that's going to remain constant, since k depends only on
temperature, we're probably not going to operate at a more concentrated feed conditions since
we're not told that we're going to, for example, increase the pressure or increase the amount
of inert. Then the molar flow rate of component A we presumably won't change, we're
just going to change the volume. So if we know by how much the Damkohler number changes,
because nothing else is changing, the volume is going to have to be changed by the same
amount. So if we have a conversion of 75%, then we can just use this same expression
and plug in a different value of X to solve for a Damkohler number and that turns out
to be 2.12. So we can set this equal to a new volume times this same quantity.
Therefore, the new volume that we're going to need to get a conversion of 75% over
the old volume is equal to the new Damkohler number over the old one and that's equal to
2.6. So to get from a conversion of 50% to 75% in this system we're going to have
to more than double the volume. So what we see in this problem is that sometimes
it's not necessary, even though we could solve for things like the concentration, it's not
really necessary to do so if we're just looking at a process change. If lots of variables
are remaining constant during that change and in this case that only things that are
changing are the volume and the conversion then we can usually just keep the other constants
along for the ride and they'll ratio out later when we look at how the volume or some other
parameter changes.