In this interactive simulation we do a degree
of freedom analysis for a distillation process, and the simulation is set up where we can
do analysis around just one part, like the condenser I’ve shown here, and that’s selected
with this button, and the blue values correspond to unknowns, the mass flow rate and the mass
fraction, and the mass fraction here of the liquid. And so the number of unknowns is listed,
the balances, two species balances, and then the relation from phase equilibrium between
this mole fraction of vapor and this liquid mole fraction that’s assumed to be in equilibrium,
and for these conditions we’ve selected the number of unknowns that says the system can
be solved. We have the option of a partial condenser or a total condenser so that we
condense all liquid and means this mass fraction here would be the same as this mass fraction
here if we had a total condenser, and then the simulation allows us to click new problem
and we have different unknowns, so we’re solving a different system. And we can do a balance
now around the distillation column, the re-boiler, or around the whole system, and we’ll see
how we do that when we look at the interactive simulation. So here we’re looking at the interactive
simulation, and we selected the condenser, we have the option of the column or the re-boiler
also, or an overall balance. The blue are the unknowns, so say mass fraction, vapor
coming in, mass flow rate leaving, and then this mass fraction of the liquid that’s condensed.
So we have three unknowns. We have two species balances because we have two components, and
we have additional information, vapor/liquid equilibrium that relates the mass fraction
in the vapor to the mass fraction in the liquid, so this says that we have no degrees of freedom,
or in other words this problem can be solved, and this is what the degree of freedom analysis
does is tell us whether we can now sit down and calculate the unknowns, the blue values.
Now if I select a new problem what this does is create different unknowns. Now the unknowns
are the inlet flow rate and the outlet flow rate, these mole fraction are known. Again
this happens to be a solvable case. Now another problem, a new problem, it’s over specified.
We have three equations and only two unknowns, so it’s over specified. We can do likewise,
we can look at the re-boiler, for these particular conditions it’s over specified. If I go to
a new problem now it’s solvable. Over specified again, and let me decrease the number of unknowns
and look at a new problem. Solvable for these conditions. And then over specified for other
conditions. Let’s look at the, finally an overall balance, and now this tell’s for this
problem the unknowns again are blue, change conditions, let’s increase the number of unknowns.
Well if I have seven unknowns now our system is under specified pretty much for all the
conditions we could pick, we don’t have enough mass balances to be able to solve for this
system. The other option here is to consider both A and B mass fractions as unknowns, and
have additional equations that say the mass fractions have to add to one, and this is
the case where different, basically some people like to do it one way where they have both
mass fractions unknowns, others like to assume we’ve already taken into account that the mass
fractions add to one, and therefore only have, for example here one unknown, not two unknowns.
And so this degree of freedom analysis is valuable for understanding systems and how
we can do mass balances on them, and depending on which part we select we can have something
that’s under specified, or over specified, or solvable.