Okay, so in this screencast we are going to try to do an example related to crystallization, and
try to read this problem. Work through it on your own before watching this screencast.
So in this example we have a cooling crystallizer. I will just draw like this and it’s operating
at 30 degrees C. We are feeding the fluid that contains 100 kg total. It is given in
the problem statement and that contains some mass of water and a mass of salt. In
the crystallizer we are told that we produce 25 kg of the salt, which is KNO3. We still
have a liquid solution that also exits, and all the water exits there. So we have the
same mass of water. Then we have some smaller mass of salt. Basically “m_s” minus 25
kilograms in the aqueous solution. I will call that “m_saq”. We know that this should
be a saturated amount of salt inside our liquid water. Otherwise we wouldn’t have equilibrium
between a solid phase and this liquid phase. Saturation implies equilibrium and we are
assuming that we go all the way to equilibrium at 30 degrees C. So we produce both solid
and a liquid state in this case. What we are asked is how many degrees C above saturation
we have in the feed. So how many degrees of superheat do we have. Alright we will get
to that in a minute. First we basically want to solve for all of these flow rates. That
is going to help us answer that question. Alright, so if we do in overall balance, which
is just in equals out because there is no reaction. We have 100 kg total coming in.
That is equal to the 25 kg of the salt coming out plus the mass of the water plus the mass
of the salt in the aqueous phase. Alright, at saturation at 30 degrees C we can look
on our table. We are about right here. Of course there is some error reading off this
plot, but I am going to assume that the solubility is 48 g per 100 g of water or another words
0.48 g/g of water. Alright so 0.48 is equal to the mass of the salt in the aqueous solution
over the mass of water. We can then solve for the mass of the salt in the aqueous solution
here. It will be 75 minus “m_w” and we can plug that into this equation. So now we
have one equation and one unknown, and the mass of water turns out to be 51 kg, and therefore
the mass of salt in the aqueous solution turns out to be 75 minus 51. If we use this top
equation here, which is equal to 24 kg. Finally the mass of salt that we must have had in
the original solution is therefore 24 plus 25. So we have 24 kg of salt in this stream
and 25 grams in this stream. So that means that we must have had 49 kg to start off with.
To go with our 51 kg of water. Now what we are asked is the amount of superheat in the
entering solution. You are not given any information that it is in equilibrium with any solid phase.
So it maybe a superheated system that is well above in temperature were you would expect
the salting out to occur, and the degrees of superheat is basically the degrees above
the saturation condition at the inlet concentration. The inlet concentration at the inlet that
would be equal to the 49 kg that we have of salt over the 51 kg of water. That we determined
from our balance. If we take that ratio that turns out to be 0.96. And we can look up 0.96
on this plot about 96 grams of KNO3 over 100 grams of water, and we see that it approximately
again it’s hard to read this with a high degree of accuracy. We are approximately at
55 degrees C. Alright so at 55 degrees C the maximum amount of salt we be able to dissolve
in our water would be 49 kg per 51 kg of water but our inlet solution actually enters at
100 degrees C. So that means that we are superheated. We could accommodate a lot more
salt in the entering solution, but we don't have a saturated system in that case. So we are less than half
the actual amount of salt that we could accommodate at 100 degrees C. Alright at 55 degrees C
we would be saturated. So to determine the degrees of superheat we simply; I am going to call this delta T of superheat; and then we simply
look at out 100 degrees C, which is our inlet temperature minus the temperature that corresponds
to saturation. At the equilibrium conditions and that is 55 degrees C. So we have 45 degrees
of superheat in this crystallization process when this enters at 100 degrees C. The temperature
is going to have to fall all the way to 55 degrees C before you start to form any crystals because above
55 degrees C we don't expect any kind of equilibrium between the solid and liquid state.