Rachford-Rice Procedure for Isothermal Flash Distillation


In this screen cast we are going to perform
a flash calculation for a multicomponent system using the Rachford Rice solution. Now a flash
distillation is where we send a liquid through some kind of pre-heater, get it nice and hot
and that is going to get sent through to a lower pressure tank. And if we do this we
can create a vapor phase and have vapor liquid equilibrium (VLE). So then we get our liquid
at the bottom of the tank and a vapor forming at the top of the tank. So now we have two
streams. We have a vapor stream and a liquid stream and we will call F our feed. So we
do this to separate multiple components as they will preferentially separate and have
different compositions in both the vapor and the liquid stream. So lets start with all
the variables that we would have to define for this system. So w would have some feed
F, and it is going to have some composition, we are going to say zi, which will be the
mole fraction of component i in the feed. This is going to have some temperature associated
with it and a pressure associated with is which we will label Tf and Pf. We are also
going to have an enthalpy associated with the feed stream. On the outside we are going
to have composition of vapor mole fraction, yi, we will also have a pressure associated
with the vapor stream, a temperature and and enthalpy. We will do the same thing for the
liquid where xi denotes the lurid mole fraction of component i. So you can see we already
have 16 variables here and if we add multiple components we have a variable for one component
i we would add three more variables for each component. But because we have plenty of relationships
between these variables our degrees of freedom isn’t that high. So let’s work through some
of these relationships. First and foremost at equilibrium within our tank the pressure
of the liquid phase has to equal the pressure of the vapor phase. This is also true for
the temperature. These are variables that must be equal between phases at equilibrium.
Now the other thing that is true at phase equilibrium is that the phase equilibrium
ratio, also known as the k value for VLE systems. k of component i is equal to the mole fraction
of i in the vapor phase over the mole fraction of i in the liquid phase. Now fortunately
one thing that is always true is that the sum of the mole fractions in the liquid phase
equals the sum of the mole fractions in the liquid phase which we know is 1. Now if we
tie in our material and energy balances we do an overall balance and we know that the
feed going in has to equal the liquid plus the vapor streams. We could also do a component
balance meaning F times the mole fraction of i in the stream has to be equal to all
of component i leaving. So it is the liquid mole flow, times the mole fraction of i in
the liquid phase plus the vapor mole flow times the mole fraction of i in the vapor
phase. Lastly we have our energy balance, the enthalpy associated with the feed stream
plus any energy we add to the system has to be equal to the enthalpy leaving. Believe
it or not we have everything we need here.Now we can rearrange all this and work our way
to what is known as the Rachford Rice solution. Now typically we are going to have information
associated with our feed stream. So we might know our feed variable F as well as the temperature
the pressure and hopefully our composition of the feed stream. If we know this information
then we typically need two more variables. And those for a flash calculation might be
the pressure in the tank and the temperature in the tank. Maybe it is the pressure of the
vapor stream, maybe it is the pressure of the liquid stream. The nice thing we know
is that they have to be the same. So we could use our first two equations if we are given
this information which basically is our operating conditions. So if given our operating conditions.
Of course, now the question becomes what is our vapor and our liquid flow rates, what
are the compositions associated with them and what is the energy that needs to go into
the system if anything, and what is the heat associated with being added to the feed stream
if necessary. So we are going to take our two material balance streams and rearrange
them. Now the first thing is to get rid of our L variable. So we set L=F-V flow. And
we plug this into our second equation. So we want y on the left side. So if we set y
over here now we can arrange this equation. We get the following. But a more convenient
form is where we have the fraction V/F which is the percent or fraction of our feed stream
that is vaporized. And we typically have seen this also written as psi. So the final form
of the equation for our operating line is going to be the following. The only thing
differently that I am going to do is I am going to plug in psi variable. So I am going
to start with this main equation which I will label 1 and divide everything by xi because
remember yi divided by xi is our ki. So you should get the following equation which I
have arranged on the right. So now we could group terms and set equal to zero, multiply
both sides by psi. We are going to bring our mole fractions to the left side. And change
this to yi by introducing ki because remember ki is yi over xi. So that is one way to get
rid of our xi. And that is going to be equal to the rest of it. We are almost there, one
last thing, we are going to solve for yi. So you can see if we bring yi to the right
side and bring the right side and bring it to the left then we get zi*ki divided by the
rest. And our xi is just equal to yi over ki so the only thing left to do is apply the
equation above that we set for our mole fractions. But rearrange it such that the sum of the
vapor mole fractions minus the sum of the liquid mole fractions has to be equal to 0.
so if we do that it is going to look like the following. And we could combine these
terms into one summation except you might see this numerator written just a little bit
differently. Just distributing a negative sign on both sides. So what does this mean?
Say we are given a flash separation problem, the first thing we do if we are given the
temperature of the tank and the pressure of the tank then we know we could look up the
k values at a given temperature and pressure for our components. Using our k values and
our composition of the feed, we plug that in and we iteratively solve for psi such that
we equate the rest of the components to zero. the rest of it falls into place. Once we have
psi, we know that psi is equal to V/F. So F times psi gives us our vapor stream. F minus
our vapor stream gives us L. We go back to our y and x mole fractions. We solve those
using the information we have already found and the last thing we do is solve our energy
balance. Find out what is our heat that we are adding to our system in the pre-heater.
Since we know the compositions and our mass flow rates we figure out the enthalpies associated
with each stream and calculate q. So right here is the main method for solving a flash
tank separation for multiple components. Now before we get neck deep in this tedious solution
which, highly recommended, you should use some kind of software/solver to do this kind
of work. It is important to check if there is a valid roof for the equation such that
our psi falls in between 0 and 1. Now our first sanity check is to check all of the
k values for the components. So if we have five components in the system and all k values
are greater than 1 then we have a superheated vapor. And it is just the opposite if we have
all k values less than 1, then we have a sub cooled liquid. So to have phase equilibrium
between the vapor and the liquid phase we have to have some k values above 1 and some
k values below 1. Second sanity check is to check our function at the dew point and the
bubble point. We want to make sure we are above the bubble point and below the dew point
to have VLE. At the bubble point, that is where psi is equal to 0 because remember that
is the same thing as V/F. So if we have very little vapor, we are just forming that first
bubble, that is our bubble point. Now our check is, if the function of 0 is greater
than 0 then our mixture is below the bubble point so we are not in VLE. The second check
is at the dew point. This is going to be where we have very little liquid or a lot of vapor.
So the function of 1, if that is less than 0 then we are above our dew point and again
we are not going to have VLE. So we need to have k values that are both above and below
a value of 1 and then we need to make sure our function of psi falls in between those
two conditions. Now to see an example check our other screen cast on using this technique
to solve for multiple component flash tank separation.

18 thoughts on “Rachford-Rice Procedure for Isothermal Flash Distillation

  1. Why does "all k values 1" imply that we have a superheated vapor? All k values 1 makes that implication make sense to me, but if you had a few components at a quality of 75% percent they would each have a k value of 3 and still be within the VLE.

  2. * all k values bigger than one doesnt make sense. all k values much bigger than one does. sorry for the confusion with the comment below. the symbols on my tablet arent working right.

  3. If all K values are larger than 1 for a multicomponent mixture, then the only possible phase is superheated vapor. Not clear on your statement about quality 75% meaning K-values are equal to 3. K-values are dependent on temperature and pressure for the species and in their simplest form give a relationship between vapor and liquid equilibrium (K=y/x). If all k-values are above 1, then the mole fractions cannot sum to 1 and thus there cannot be two phases.

  4. We used them off a DePriester chart. However you can also use vapor equilibria charts like those seen in refining handbooks

  5. are there any books that you would recommend for design and sizing of flash tanks (e.g. residence time, diameter and height, material selection, etc.)

  6. Algebra way to complicated, For crying out loud page 10 guys! http://staff.sut.ac.ir/haghighi/download/documents/Single_Equilibrium_Stages_and_Flash_Calculations.pdf

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