In order to solve equilibrium stage problems, you must have a model or correlation for the vapor-liquid equilibrium (VLE) and physical properties of the system. When two phases are in equilibrium, VLE data enables us to relate the composition of a liquid phase to the composition of the vapor phase.
Equilibrium data may be obtained by experiment, by thermodynamic calculation, or in published sources. It is typically presented either in tabular form or as an equilbrium diagram. Diagrams may take several forms:
The figure (also see Fig 21.3 in MSH6) shows one common way of plotting equilibrium data -- the Txy diagram. It represents a binary mixture, and all compositions are expressed as mole fractions of the more volatile component; x in the liquid phase or y in the vapor phase. (The less volatile component is thus 1-x and 1- y.)
The lower curve plots the bubble point of the binary mixture as a function of composition. The upper curve is the dew point. For a given temperature and composition, this diagram tells us the nature and composition of each phase of the mixture that is present.
Similar drawings can be constructed at constant temperature while allowing pressure to vary. You should have seen some of these diagrams in your material balance and stoichiometry class (for instance, see Felder & Rousseau (3rd ed.), Section 6.4d).
In this class, we will often work with plots of vapor phase composition vs. liquid phase composition, y vs. x (also see Figures 21.2 and 21.20 in MSH6).
An xy diagram like this may be constructed from a Txy diagram by picking a temperature, reading the corresponding y and x and plotting them against each other.
Tip: When the envelope enclosed by the equilibrium curve and the 45 degree line is "fat", distillation will probably be an easy way to make separations of the mixture.
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Problems can be greatly simplified in cases where the VLE behavior of the system is ideal or can be represented in a simple form. This usually means we can work with one of three main alternatives:
Raoult's Law (review your notes from material balances and from thermodynamics) relates the vapor and liquid equilibrium compositions of a mixture to the pure component vapor pressure and system pressure:
There are few theoretical tools for predicting the equilibria of gases dissolved in liquid solvents. One way of correlating the data is to use Henry's Law, which takes the same form as Raoult's Law but replaces the vapor pressure with a Henry's Law coefficient (also a function of temperature). Henry's Law applies to low solubility gases dissolved in liquids (it would be a bad choice for ammonia, say ...).
A "K-value" or equilibrium coefficient is the ratio of the vapor composition to the liquid composition. These are often correlated as functions of temperature and can be found in published data, as nomographs, or as equations, such as:
A related concept, the "distribution coefficient", can be used to relate the equilibrium compositions of two liquid phases.
The "relative volatility" is the ratio of the K values for two components:
Relative volatility is generally a much less strong function of temperature than the component vapor pressures; in many systems, it is acceptable to assume that the relative volatility is constant over a range of temperatures and compositions.
High relative volatilities produce xy diagrams with a great deal of separation between the equilibrium curve and the 45 degree line. A relative volatility less than one is probably "upside down" -- the more volatile component is in the denominator.
Computer calculation aids immensely in construction of equilibrium diagrams. I've prepared examples of two approaches on MATHCAD. You may down load them by following the links:
In cases where the VLE cannot be treated as ideal, more rigorous models based on multicomponent thermodynamics are required. We won't worry about these for awhile.
Most of the time, we deal with systems where the equilibrium curves have no inflection points; that is, as the concentration of the less volatile component increases, so do the dew point and bubble point.
If, however, there are strong physical or chemical interactions between components, diagrams may look different.
In some such systems, there is a critical composition where the liquid and vapor compositions are identical. Once this composition is reached, separation cannot continue without changing pressure. These mixtures are called azeotropes -- they will have minimum or maximum boiling points. Minimum boilers are more common. Composition of the vapor produced from an azeotrope is the same as the liquid, so an azeotrope can be boiled at constant pressure without changing composition.
For a binary mixture, cp, , , etc. are functions of composition. All of this information can be used to construct an enthalpy concentration diagram (see MSH6 Fig 21.24 for another example):
Developing Hxy diagrams isn't particularly difficult, but it does require attention to detail. To begin with, reference temperatures must be selected and applied with consistency. It is also important to remember that liquid solution enthalpies must include both sensible heat and heats of mixing:
Remember that enthalpy is a state function, so you may calculate it along any "path" that you choose; and usually want to choose the one that makes calculation easiest. Vapor enthalpies are typically calculated using heat capacities and the latent heat of vaporization:
Mixture values are usually hard to find, so as an approximation, assume the unmixed liquids are heated separately to the dew point, vaporized at that temperature, and then mixed:
There is a direct relationship between Hxy and xy diagrams. If you have tie lines connecting equilibrium liquid and vapor compositions on an Hx diagram, you can use it to construct an xy diagram. The endpoints of the tie lines are at the equilibrium compositions needed/used on the xy diagram.
Constructing an equilibrium curve from an Hxy diagram (or adding tie- lines to an Hxy diagram from data on an Hxy diagram) is a nice construction. Place the equilibrium diagram below the Hxy and line up the scales. Starting from a tie line liquid endpoint on the Hxy, drop a vertical line onto the equilibrium plot. This will be an x-coordinate. Next drop a line from a vertical endpoint on the Hxy. When you hit the 45 degree line, turn at a right angle. The intersection between this line and the previous line will give you one (x,y) point on the equilibrium curve.
You can reverse the construction and use an xy diagram to get the tie lines on your Hx plot. Take x coordinates straight up and mark the intersection with the bubble point curve. Take y-coordinates over to the 45-degree line, turn and carry them up and mark the intersection with the dew point curve. Connect the two intersections to get a tie line.
References:
R.M. Price
Original: 3/6/97, 1/5/98; 8/13/2002
Modified: 1/8/98, 1/8/99; 2/10/2003
Copyright 1998, 1999, 2002, 2003 by R.M. Price -- All Rights Reserved