In this video we are going to look at interpolating tie lines on a ternary diagram. Here we have
a ternary diagram for three liquids, IPA, water and ether at a specific temperature and pressure.
We have our phase equilibrium data here. We may want to use something like this for liquid-liquid
extraction systems in terms of solving for maybe the number of stages or what our compositions
might be after mixing two different solutions together. The tie lines we might get from
our equilibrium data but we may need to draw other tie lines and it is probably not the
best thing to do to try to look at the line and gauge on how it should connect between two equilibrium
points. In this video we are going to show two different ways to use this diagram to
create more tie lines. We are going to start with method 1. Method 1 we are going to draw
a parallel lines to the lines AB and BC. I am going to highlight line AB and draw a green
line parallel to that to demonstrate this. I will use purple for the line BC. We are
going to take any of our tie lines and a good number of them so that we can draw what is
known as a conjugate curve. Let's start with this first tie line shown here. What we are
going to do is draw a green parallel line to AB and then we will draw a line parallel
to BC from the other point. Where these intersect we will draw a point. We are going to repeat
this process for each of our tie lines. Here I show our second tie line. I will highlight
that yellow. We have drawn two parallel line and we will mark that point. We will do this
again for the next one. Draw a point where those intersect. We could continue with more
tie lines that we have if we wanted to. I am more interested in drawing an interpolated
line between the three I have just chosen. The point that we are missing here is what
is known as the plait point where the compositions of the raffinate and the extract are equal.
We are not told what point that is but we could estimate that it is going to be somewhere
around here. As you can see this is going to be a curve. We have to do the best we can
at matching these points up along that curve. Now we want to draw more tie lines. We choose
anywhere one that curve and then just reverse the process. For instance, I could pick a point
right between 2 of our points. I will draw a parallel line to our equilibrium curve.
I will draw another parallel line to our other equilibrium curve. You can see where these
two lines meet on the equilibrium curve so we will connect those two points. Now you
see we have another tie line. We could do this again. Where those two lines meet we
will draw another tie line. You can see that based on this conjugate curve that we have
made, which I will now delete, we have interpolated the tie lines we might have had from data
and created more that we could use to read from our plot. Again this is what we are going
to say is method 1 which is using the lengths AB and the other side BC. Let me show you
method 2. In method two we are going to use 2 different sides. We are going to use AB
again but this time we are going to use AC. Our intersection is actually occur within
our triangle. This method is used less just because it can get a little cluttered. This
time I am going to highlight side AC blue. The first thing we will do is choose our plait
point again. We will choose our tie lines. Here I am going to choose our first tie line
and we are going to draw parallel to AC. Now we are going to take our other tie line point
and draw it parallel to AB. Where these intersect we are going to put a point. We repeat this
for each of the tie lines just like we did in method 1. Here I have chosen three tie
lines from our plot and draw the intersection of these three lines. You can see our curve
is going to be within the diagram and again here we use our conjugate curve and reverse
the process. I could choose any where along that curve and draw lines parallel to the
lengths we started with. For instant here is 1 point line along the curve that we have
drawn two parallel lines and where those intersect on our equilibrium graph is where we are going
to draw our new tie line. We could repeat this process for as many tie lines as we want
and we could create that. Again as you see since it is inside of our triangle this can
get cluttered. Hopefully these two methods showed you how we can interpolate tie lines
in our phase equilibrium diagram and use this maybe later in a liquid - liquid extraction
problem where we need more tie lines to solve the problem or maybe we are looking for composition
from our mixture and it helps to be able to read them right off of tie lines.