Here we have a problem where we're going to feed a liquid mixture - ethanol and methanol.
We're going to lower the pressure and carry out this flash adiabatically so we end up
with a vapor - liquid mixture at 2 bar pressure. We want to know then, what's the temperature
where this happens? The effluent from this flash drum. What fraction of the liquid that
we feed is turned into vapor? And then, what are the compositions of the liquid and the
vapor? We're going to need additional information, and so you should think about what that information is.
Here are the values that I'm going to use. We need heat capacities for both liquid
and vapor. We need heats of vaporization. And we need Antoine equations and a couple
things to point out in this solution that we need to keep in mind, we're going to work
in pressure bar, that's been given in the problem statement. The Antoine equations I've
found, however, are in mmHg so I'm going to have to correct for that. Also going to work
in Kelvin but the Antoine equation is in Centigrade so I have to correct for that also in my final
equations. I ended up solving this numerically, I'm going to set up the equations then in
a separate screencast show numerical solutions to the final values. But let's look at what a
diagram represents for feeding methanol and ethanol, ideal solution, these are very similar
molecules. We're going to assume constant heat capacities, the values I've given. And
then we'll assume these heats of vaporization are at 300 Kelvin and we're going to use 300
Kelvin as our reference temperature. That's an arbitrary, convenient value. One mole of
feed, so V is the moles of vapor and L is the moles of liquid. We have to have these
two phase liquid and vapor in equilibrium. So we're going to need an energy balance.
Now this Q is zero, no heat removed from the system. We're going to need an energy balance
which is just total energy in equals total energy out. We need mass balances, and the
mass balances are that we're feeding one mole and it leaves either as vapor or liquid, so
they have to add to one. I'll use Z to represent the mole fraction of methanol, so Z of methanol
is 0.30 times 1, that's the number of moles of methanol coming into the system. The number
of moles of methanol leaving are the mole fraction in the vapor phase times the number
of moles in the vapor phase. Mole fraction of methanol in the liquid phase and then the
number of moles in the liquid phase. We could write another equation for ethanol but it
would be redundant with the total mass balance. And then the third thing we're going to need
is vapor-liquid equilibrium because we have this mixture, our mole fractions are going
to in this case be modeled by Raoult's Law because ideal solution. So we're going to
say the mole fraction of methanol in the liquid phase, saturation pressure of methanol at
the temperature, the T-out temperature. Mole fraction of the vapor phase, the pressure
out, which is 2 bar. And then we have this same type of equation for ethanol. A couple
more things we need for this vapor-liquid equilibrium, the mole fractions in the liquid
phase must add to one and the mole fractions in the vapor phase must add to one. And then
we need the saturation pressures as a function of temperature. So the log base 10, since
we're given methanol saturation is in the form A minus B over T plus a constant. So
A, B and C are constants and are different for ethanol. So we have a corresponding equation
for ethanol, don't need to write that down. Keep in mind temperature here is in degrees
Centigrade, pressure mmHg, we're going to have to correct for that. The big thing that
we need more information is this energy balance. Let's look at the terms. So we have energy
coming in for methanol and ethanol. Energy in for methanol, I'm going to write that down.
This is the total energy so I have Z moles, Z is the mole fraction, number of moles is
1, so Z times one is Zm, that's the mole fraction and that's the number of moles of methanol.
And they're coming in with some enthalpy. What I'm going to do is pick my reference
temperature and say enthalpy of methanol at 300 plus heat capacity of methanol in the
liquid phase, the actual inlet temperature minus 300, working in Kelvin. But enthalpies
are arbitrary, only differences are important. I'm going to set that equal to zero and I'm
going to do the same thing for the total ethanol in. Of course that's Z of ethanol and the corresponding
terms so I won't write all of those down. And then we have the enthalpy of methanol
in the liquid phase leaving the system. So that's the number of moles, so that's a mole
fraction times the number of moles of liquid, that's the moles of methanol. And the same
idea, enthalpy of 300 is my reference, so heat capacity, the temperature out now minus
300. And likewise I won't write that down, we have the corresponding term for ethanol.
And then the final thing we need is the enthalpy out in the vapor phase. Mole fraction of ethanol
in the vapor phase times the moles of vapor. Now I need something different. I'm to, again
the same reference, but I'm going to now vaporize that ethanol, the heat of vaporization per
mole. It vaporizes at 300 degrees Kelvin. I'm going to heat up the vapor, so this is
for methanol. It's going to heat up, use the heat capacity of the vapor and T-out minus
300. And then likewise I have corresponding term for ethanol where heats of vaporization
and heat capacities are for ethanol. Now I have all the terms for the energy balance,
mass balance, vapor-liquid equilibrium non-linear equations with this saturation pressure temperature
dependence. I'm going to solve these numerically, I'll show a couple different approaches in
other screencasts. One is to use POLYMATH, the other is to use a spreadsheet. What I'm
going to show here are the final results I got from that solution. So here are the values,
I've only reported them here to 2 significant figures and in the other screencasts you can
see the details of these calculations and see more significant figures.