In this example I'm going to describe the steps we would use to calculate the heat removal
to condense a mixture of vapors to a liquid mixture at constant pressure, so I've shown
a diagram of the type of calculation we'd like to do where we have a vapor coming in,
in this case it's at its dew point, we want a liquid leaving at its bubble point and what
we're going to use is the idea of state function to do this calculation, the overall calculation
is just delta H is equal to Q and this is, for example, per mole of feed and the question
is how do we account for the fact that there's interactions in the liquid phase that the
liquid's not ideal whereas the vapor, the molecules are not interacting, and so what
I'll do is I'll go through the steps of taking advantage of the idea of state function to
do this calculation.
First let's look at what we're doing on a T, XY diagram, so we're starting here, the
inlet is at the bubble point, this is a constant pressure plot, pure methanol here has a lower
boiling point than pure water over here, and we're going from this inlet to this outlet
and the question is what is the delta H so therefore calculate Q. So we're going to be
systematic about this and break it down using state function to a number of steps, the first
step is say we will separate the gasses into the pure gases, well, reasonable to assume
these are ideal gases so delta H for this process would be zero so delta H of mixing
for ideal gases is zero, and then we're going to condense each of the species, we know the
heat of vaporization for the pure components, so I've written down to condense water delta
H, we're going to pick as our basis one mole, so we have 0.4 of a mole of water times minus
delta H of vaporization for water, that will be delta H2, and this is at 80 degrees C so
we could look this up in the steam tables and then the same thing for methanol, we have
six tenths of a mole of methanol, minus delta H of vaporization of methanol, again at 80
degrees C and then we need a source for this so we would have to look up somewhere.
Our next step is to cool liquid water from 80 degrees to 70 degrees because our outlet
temperature is 70 degrees, delta H4 then we have four tenths of a mole times the heat
capacity of liquid water, 70 minus 80, and then we could look this up in steam tables
or use a constant heat capacity which are a pretty good approximation for liquids, small
temperature range, and then we have the same thing for methanol, so I've written down the
same calculation for methanol and again we have to find the heat capacity for methanol,
now we have the two pure liquids at 70 degrees, what we're going to do is mix them to get
to our final condition at 70 degrees C, well there would be an enthalpy change if this
is not an ideal solution so delta H6, delta H of mixing, which we would need data that
we would have to look up somewhere to do this part of the calculation, and then our overall
Q is the overall delta H which is just the sum of the delta H's and then we can substitute
in for each of the values that we calculated and then we can group together the methanol
and the water terms, and so we have our final answer, the heat removal to condense a mole
of 60 percent methanol and 40 percent water and we have to then look up heats of vaporization
heat capacities and then delta H of mixing allows us to account for the fact that the
liquid's not going to be ideal, and so we can take advantage of state function to simplify
the calculation significantly.