CHEE 470 Course Notes
Thermodynamics Notes
1. SINGLE STAGE CONCEPTS
1
1.1 K Value- the vapour liquid equilibrium ratio
1
1.2 Relative Volatility
1
1.3 Bubble Point
1
1.4 Dew Point
2
2. FLASH CALCULATIONS
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3. NON IDEALITY
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4. THERMODYNAMIC DATA
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4.1 K Values
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4.2 Gamma Values
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4.3 Equations of State
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SUMMARY
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PAST EXAM QUESTIONS
14
Lecture Notes: Selecting Thermodynamic Methods
CHEE 470 Course Notes
VAPOUR / LIQUID EQUILIBRIA
Review: Pages 497 to 502 of Turton, Bailie, Whiting, Shaeiwitz
An excellent review can be found in Section 7.2 - 7.6 , Beigler, Grossmann and Westerberg
Systematic Methods of Process Design (‘97 text)
Today's Objectives: A Review of Thermodynamic methods
1. SINGLE STAGE CONCEPTS
1.1 K Value- the vapour liquid equilibrium ratio
The vapour liquid equilibrium ratio (K value) is defined as:
Ki = yi /xI
where y and x are the vapour and liquid mole fractions of component i.
For a system where Raoult’s Law applies (an ideal system):
Ki = pi /
Where pi is the vapour pressure of component i and  is the system pressure.
1.2 Relative Volatility
The relative volatility of components i and j is defined as:
 ij = Pi/Pj (1) = Ki/Kj
This only applies for ideal systems.
1.3 Bubble Point
This is the temperature at which a liquid first begins to boil.
 yi =  Ki xi = 1.0
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
1.4 Dew Point
This is the temperature at which a vapour begins to condense.
 xi =  (yi/Ki) = 1.0
2. Flash Calculations
Refer to Figure1, a typical flash. We can assume that we have a liquid feed of known
composition at a temperature and pressure such that the mixture is below its bubble point. If we
hold the pressure constant and gradually increase the temperature, we will move in a horizontal
fashion across the diagram. At first, the  Kixi will be less than 1.0. The mixture is subcooled.
As we continue to increase the temperature this summation becomes precisely unity and the
mixture will be at its bubble point. As we continue to further increase the temperature, the
mixture will vaporize and we will be in the two-phase region. In this zone, both  Kixi and 
(yi/Ki) are greater than 1.0, but the former will be increasing while the latter will be decreasing.
When the temperature reaches the point where all the mixture has vaporized we have now
reached the dew point. At this point  (yi/Ki) = 1.0, and any further increase in temperature will
cause this summation to become less than 1.0. The system is said to be superheated.
Figure 1. A TYPICAL FLASH DIAGRAM
In a flash calculation, the composition of the feed mixture is known (this could be a two-phase
mixture) however the composition of the product is not known. There are basically two types of
flash calculations, isothermal and adiabatic. In an isothermal flash, the final temperature and
pressure are known. An example is passing a feed of known composition through a heat
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
exchanger to a new temperature and pressure. For this situation, as for any single or multistage
operation at steady state, we can write a molal balance.
i = Lxi + Vyi
where:
i
Lxi
Vyi
is the feed rate in moles per unit time of component i
is the Liquid rate in moles per unit time times vapour mole fraction of component i, and
is the vapour rate in moles per unit time times vapour mole fraction of component i.
From this relationship, and from the definition of a K value, we can arrive at the following
general equation for an isothermal flash:
L =  Lxi =  fi/ (1 + V/L Ki) =  fi/(1+fi)
The second type of flash is an adiabatic flash, in which there is no change in total enthalpy.
These are significantly more difficult to solve than the isothermal. For this type of flash, the final
pressure and initial heat content are known, the final temperature and phase compositions are
not known. The usual method used is to assume a final temperature and calculate an
isothermal flash, then compare the heat content of the feed and products. An iterative
calculation is required to converge on a heat balance. In general, when solving either type of
flash a numerical method (such as Newton’s method) is used to achieve more rapid
convergence than a simple substitution procedure.
3. NON IDEALITY
One of the most basic process engineering issues is the question of non-ideality. In the
Chemical Process Industries and related fields such as Oil Refining and Waste treatment, the
Process Engineer deals mainly with liquid solutions and mixtures. Crude oil is a mixture of an
enormous number of compounds. Crude Ethanol from a fermentation process contains a
surprising number of compounds in addition to ethanol and water. Separation and purification of
these streams is of paramount importance. It really doesn’t matter whether one is dealing with a
classic distillation or extraction processes, or one of the more exotic processes such as
biochemical separations, supercritical extraction, reactive distillation and so on, the
understanding of liquid/liquid and liquid/vapour behaviour is most important. Vapour liquid
equilibria considerations often enter into the design of thermosiphon reboilers, a further
example of the need for the understanding of these concepts in a related field.
Prior to the universal availability of computers, many systems were assumed to be ideal since
the mathematics of non-ideal systems were often just too complex to handle by less
sophisticated methods. (Unfortunately, the ideal state only seems to exist in the minds of some
politicians.) In the field of chemical engineering separations, there are few if any cases where a
truly ideal system can be assumed. The degree of non- ideality will vary from a very modest
(and in some cases hardly observable) situation when the components are very similar, such as
a mixture of normal paraffins of differing molecular weight, to systems containing polar and
non-polar species which are extremely non-ideal. If one cannot deal with this latter situation in a
satisfactory manner, it is simply impossible to ‘design’ any separation equipment to handle
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
these systems. Azeotropic and extractive distillations are examples of where we have taken
advantage of this non-ideality to either design systems to carry out separations which otherwise
would be impossible, or to improve the economics of some separations by a great degree. For
example, it would be essentially impossible to produce pure ethanol by fractionation without the
use of an azeotropic dehydration step, which takes advantage of a ternary heterogeneous
azeotrope.
These notes will recap some of the thermodynamics relating to non- ideal systems that have
been introduced in previous courses. In plant operation, design or rating of systems where nonideality is a significant consideration, it will become obvious that this is not some abstract idea
to confuse students but it is very much a part of the real world.
4. THERMODYNAMIC DATA
Thermodynamic data can be defined as the methods used to calculate vapour/liquid
equilibrium constants (K values), liquid/liquid equilibrium constants, enthalpies (H values),
entropies (S values), and the transport properties. The Gibb’s Free Energy can be a most
important number as well.
K values and Enthalpy values are required for practically any problem where mass and energy
balances are required. Entropy is generally not required unless compressor or expander
calculations are carried out, or for certain reaction calculations.
4.1 K Values
There are generally three approaches to the calculation of K values. The first is the many
general-purpose data generators, which are usually restricted to mixtures of non-polar
hydrocarbons. They fall into several distinct categories - Equations of State, methods based on
Corresponding states and Empirical methods.
The second procedure is the assumption of ideal behaviour. This is a reasonable assumption
for mixtures of a limited number of similar compounds at low pressure. For example, a mixture
of higher olefins oligomers could be assumed an ideal mixture, particularly if being processed at
low pressure. Another example would be fractionation under vacuum, provided no water was
present. For this assumption, accurate vapour pressure is all that is required for a reasonable
estimate of the K values.
PRO/II™ or any of the comparable programs have the option of assuming Ideal behaviour. As
an alternative, the user may provide specific K value, Enthalpy and Entropy data, provided it is
available in either tabular form or can be correlated by one of the many methods available.
Probably one of the most significant areas where the separation technology applicable to the
manufacture of chemicals deviates from the separation of hydrocarbons in the oil refining
processes is in the area of highly non-ideal polar liquid mixtures. When a system contains a
mixture of oxygenated hydrocarbons such as ketones, alcohols and water, none of the
generalized methods such as equations of state or assumption of ideal behaviour are usually
suitable for estimating K values.
To handle these systems a third approach, the concepts of fugacity and liquid phase activity
coefficients, is introduced. As a rule, the use of an Equation of State or Equivalent method is
recommended, rather than activity coefficients, when the following statement applies. The
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
Equation of State attempts to estimate K values at high pressure for a limited number of
compounds. The Activity Coefficient method estimates equilibria for a wider spectrum of
compounds at relatively low pressures.
In order to re-examine the concepts used in the Activity Coefficient approach, Chapters 11, 12
and 13 in Smith and Van Ness Introduction to Chemical Engineering Thermodynamics cover
this subject in considerable detail, and accordingly only the basic rules will be highlighted.
The general equation for Equilibrium K values is as follows.
Ki =  i i0i/ i
Where
 I = component i liquid phase activity coefficient
i0i = component i standard state fugacity
 = system pressure
i= component i vapour fugacity coefficient
For the purpose of this particular treatment it is assumed that i0i is equal to the component i
vapour pressure. In PRO/II the more rigorous approach, where the standard state fugacity is
calculated, is applied.
The liquid phase activity coefficient or gamma value is a function of temperature, pressure and
composition of the mixture. In practical terms, it is a multiplier that is applied to the fugacity (or
in a simplified case vapour pressure) to represent the ‘activity’ of the component in the mixture.
Sometimes the gamma value has been described as the ‘Escaping Tendency’ or a measure of
just how desperately a particular molecular species wants to get out of the liquid solution. The
determination of the appropriate activity coefficients to correctly represent the degree of nonideality of mixtures is regarded by many as arcane art.
Let’s take a look at the various elements of the above equation. Fugacity is a concept that was
developed to represent chemical potential or free energy in equilibrium calculations. It is a term
with units of pressure. By definition for two phases which are at equilibrium with respect to one
another:
Tv = TI
Pv = PI
jv = il
In other words, at equilibrium the temperature and pressure of the two phases must be equal,
and the fugacities of each component must be the same in both phases. In addition by
definition fi  p0i =  yi as   0 that is , the fugacity of a component approaches the partial
pressure of that component as the pressure of the system approaches zero. There are several
methods to calculate fugacities, Reid and Sherwood cover quite a few. PRO/II uses an
Equation of State to calculate fugacities .
The standard state fugacity for component i is calculated using the following formula.

01
i
  pi Vi l

 p i exp 
RT 

PoyntingCorrection
0
i
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
where
Pi = component vapour pressure or Henry’s Constant
 I = pure component fugacity coefficient at Pi and the system temperature
 = system pressure
Vil = liquid molar volume of component i
R = Gas constant
T = system temperature
It is important that the pure component fugacity coefficient  i and the component vapour
fugacity coefficient i be calculated using the same equation of state. For lower pressures it is
often appropriate to assume the component vapour pressure will adequately represent the
standard state fugacity.
From the above, it can be seen that a simple relationship such as Raoult’s Law can get
complicated when we try to “massage” the concept to represent the real world. All sorts of weird
and wonderful terms emerge, standard state fugacities, gamma values and so on. In the
relationship immediately above, it can be seen that rather than use a simple vapour pressure in
the already complicated equation, a better representation can be attained by expanding it to
represent a fugacity. This is where computers have been a mixed blessing. Since they are
"immune" to number crunching fatigue, these more sophisticated relationships can be applied.
All that is necessary is the appropriate coding and data, the algorithm does the rest. However,
this is where the ‘Black Box’ syndrome can take over. As these systems get more involved and
incorporate more and more sophisticated procedures, it becomes more and more difficult for
the average process engineer to understand what the algorithms are about all the time and to
be able to interpret the data and the solutions correctly.
A second issue is a question of simplifying assumptions and ‘short- cut’ back of the envelope
calculations. There will be many occasions where a quick and dirty estimate is all that is
justified, and there are many occasions where one can simplify an otherwise very complex
system by making the appropriate assumptions, as has been done here where it is agreed to
assume that vapour pressures are adequate rather than calculating the fugacities by the
preceding equation.
There are really only two realistic choices, the simplistic back of the envelope approaches, and
the full blown rigorous simulation, modified where experience has indicated that simplifying
assumptions can be made. In either case, you MUST understand what you are doing. ‘Black
Boxes’ are not the answer and could represent a real trap if the user is not careful. Even the
‘lowly slide rule’ was not much use to someone who did not know how to use it.
Returning to the question of Activity Coefficients, since the component vapour fugacity
coefficient  i in the first expression is calculated by an Equation of State, it can be assumed
that in the lower temperature and pressure ranges that this fugacity can be assumed to be
unity. Referring to the Lee and Kesler charts (on pages 336 through 339 in Smith and Van
Ness), it can be seen that this assumption is reasonable.
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
4.2 Gamma Values
Gamma values, or liquid phase activity coefficients - the correct thermodynamic relationship is
given on page 345 of Smith and Van Ness. Most of the activity coefficient data is available from
regression of experimental VLE data.
Unfortunately, there isn’t a great deal of data available. The number of possible binary
combinations of compounds is, in effect, infinite. Very often in industrial practice one finds that
the appropriate data for the specific systems of interest may be available in-house, and if not
available, one must determine it by regression of experimental data. There is a reference given
in Smith and Van Ness (J. Gmehling, U. Onken and W. Arlt ‘Vapour-Liquid Equilibrium Data
Collection Chemistry Data Series, vol. 1, Parts 1-8, DECHEMA, Frankfurt/Main 1977-1984)
which provides many published data. This reference is available in the library.
Gamma values are a function of composition and much effort has been expended in attempts
to represent this dependence on composition by an appropriate mathematical expression. The
earlier expressions were in essence empirical, however some of the more recent procedures do
have some theoretical basis. One of the more exciting developments to come along is the
development of ‘Group Counting’ methods, which are remarkably good estimates for systems
where actual binary data is not available. We will spend some time on these.
Several methods that are most satisfactorily used to estimate the activity coefficients for the
various components at. various compositions, temperatures and pressures are given in
PRO/II These are as follows:
Wilson Equation
Margules Equation
Non-random Two Liquid (Renon-NRTL)
Regular Solution Equation (Scatchard-Hildebrand)
UNIQUAC Equation
van Laar
Modified van Laar
UNIFAC
With the exception of the Regular Solution Equation, it is necessary in all cases above to
provide some form of binary interaction coefficients, for all the possible binaries in the mixtures.
UNIFAC is a Group Contribution Method, which was mentioned briefly above. All of these
equations and a detailed description of the UNIFAC system are given in the PRO/II manual.
Note that the form given in PRO/II is the multicomponent form rather than the binary. The
binary equation is slightly different.
As a general rule, when dealing with chemical mixtures, particularly where it is known that polar
compounds are present, the more rigorous methods requiring binary interaction coefficients are
to be preferred. However, if these coefficients simply are not available or if there is a time
constraint, the Group Contribution method UNIFAC most certainly should be used rather than
trying to rely on an Equation of State. In no circumstances should one assume an ideal system
unless there is most valid reason for doing so. As pointed out earlier, mixtures of very similar
compounds at low pressure could be assumed to be ideal, such as a mixture of n-paraffins;
provided the system was anhydrous.
Unfortunately there are no foolproof "EXPERT SYSTEMS’ yet available for the estimation of
activity coefficients and a good deal of judgement is often required. This is most important when
dealing with a system for which there may be no previous studies which established some form
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
of confidence level in the approach taken. An examination of the system by someone
experienced in this field may very likely lead to a judgement call which could eliminate many of
the binaries from consideration as not being significant for the separation desired. If that is the
case, these binaries could be assumed to be ideal should there be no data available. The fill
option in PRO/II™ allows estimation of data by the UNIFAQ method.
When one considers the enormous amount of calculation involved in solving a multistage multicomponent non-ideal system (It has been estimated that some 50 to 90 % of the computation
time in a rigorous process simulation is taken up in computing the phase relationships) it can be
appreciated that before the advent of the computer these rigorous solutions were more or less
impossible by straight-forward desk calculations. Although most of the concepts and in many
instances the algorithms have been available for some time, before the availability of machines
the short-cut methods had to be relied upon. Even today with the enormous amount of machine
power available, considering the fact that the bulk of the number-crunching is simply
recalculating K values for each stage for all components every time there is a temperature or
composition change, there is considerable interest in methods that will approach the solution
using simplified estimation of the K values. The rigorous methods are often not invoked until the
problem has approached a solution to within a certain level of tolerance. Even though there is a
great deal of effort required to put together the data for a rigorous calculation, in most instances
it nearly always proves to be worthwhile as compared to a short-cut method solution for final
design purposes’ The capital cost involved in plant construction and the long term grief that a
poorly designed separation unit can cause, generally can justify the best solution possible.
Among the many forms of liquid activity procedures, the following three are most often applied.
The Wilson Equation
The NRTL Equation
The UNIQUAC Equation
Please refer to the literature for the form of these equations (The Properties of Gases and
Liquids by Reid, Prausnitz and Sherwood). We will examine more closely one of the equations,
the Wilson Equation, which is as follows.
 12

 21
ln  1   ln  x1  x2 12   x2 


 x1  x2 12 x2  x1 21 
 12

 21
ln  2   ln x2  x1 21   x1 


 x1  x2 12 x2  x1 21 
where
 ij 
j
  
exp  12 
i
 RT 
The constants i and  j are the molar volumes of the pure liquids i and j and the constants  ij
are independent of composition and temperature and are specific to the binary in question.
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
The gamma values for Isopropanol and water using the coefficients given in Example 12.1 on
page 388 of Smith and Van Ness. The limiting values or the gammas at infinite dilution are very
important numbers and this data is often available where the full range VLE data is not. These
values can be used to determine the values for the lamdas from the following equations.
ln  1   ln 12  1   21
ln  2   ln  21  1  12
The UNIFAC group contribution procedure is available in PRO/II . As has been pointed out
before there are many systems where the binary data simply is not available in order to use one
of the liquid phase activity coefficient (gamma value) methods. The UNIFAC procedure (there
are other similar methods) will give a quite reasonable estimate for the gamma value provided
the system is non-electrolytic, temperatures are between 300 and 450 Kelvin, and pressure is
up to a few atmospheres. The basic method cannot be used for polymers (there is a specialised
package for these systems) and is restricted to compounds with less than ten structural groups.
The database in PRO/II™ contains sufficient data on the compounds in the mixture to use the
UNIFAC method without the user being required to supply additional information. This data is
as follows:



Component Structural Descriptions
van der Waals area and volume parameters
Group Interaction Parameters
4.3 Equations of State
The concept of non-ideality and the use of liquid-phase activity coefficients for well-defined
systems at low to moderate pressure levels have been introduced. Systems where we have
assumed that Raoult's Law applied (ideal solutions) have also been discussed.
The third approach is the use of more general methods such as the concept of Corresponding
States, completely empirical methods or an Equation of State. A review of chapter # 3 and 14 in
Smith and Van Ness will be helpful. Equations of state are useful, but mainly we will
concentrate on their use for the generation of K values. The Equations of State, methods of
Corresponding States and some empirical methods, which are available in PRO/II, are used
mainly when one is dealing with mixtures of hydrocarbons, particularly at elevated pressures.
They are used most extensively in the Gas Processing and Petroleum Refining industries. As
mentioned previously, they very often fall apart when one has to deal with polar compounds. A
great deal of work in recent years has resulted in a convergence of Activity Coefficient methods
and Equations of State. Many equations of state can be modified by applying binary coefficients
in order to enhance their ability to estimate K values, for broader range of components, and for
broader regions of temperature and pressure.
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
An Equation of State is an analytical formulation of the relationships between P, V, and T. The
simplest Equation of State is the Ideal Gas Law (PV = nRT). There are very few instances
where the Ideal Gas Law is applicable, and consequently the many expansions to this equation
were formulated in order to better represent real behaviour. Figure 2 is a familiar one, being a
typical textbook example showing the liquid/vapour dome and the subcritical isotherm for a pure
fluid.
Before further discussing the Equations of State, the method of corresponding states will be
reviewed. The basis of the corresponding state correlations is the observation of graphs of P
versus V, with isotherms that are qualitatively similar for all gases. The graphs for many
substances can be made quantitatively similar by superimposing the critical points and properly
modifying the P and V scales of ordinate and abscissa. Most of the graphical relationships allow
one to estimate the compressibility factor Z as a function of reduced temperature and pressure.
There are many of these generalised methods available in PRO/II. An extensive list of the
various correlations available in this program is given in the keyword input manual. Again we
must emphasizes that although we refer to PRO/II, these comments apply to most if not all
Flowsheet Simulators. There are eleven Generalised Correlation methods in the PRO/II
Reference Manual, and at least as many Equations of State.
A set of guidelines is provided in the PRO/II manual as to the limitations for each of the above
methods. In general the three commonly used equations of state, B-W-R, Peng Robinson, and
the Soave Redlich-Kwong may be applied over the full temperature and pressure range,
however the other methods have temperature and pressure ranges beyond which they are not
specifically recommended. Once an Equation of State has adequately described the PvT
behaviour of a system, additional physical and thermodynamic properties of the system can be
calculated for the apparent phases, such as:
1. Enthalpy
2. Entropy
3. Isochoric Heat Capacity
4. Isobaric Heat Capacity
5. Isothermal Coefficient of Bulk Compressibility
6. Adiabatic Coefficient of Bulk Compressibility
7. Thermal Pressure Coefficient
8. Coefficient of Thermal Expansion
9. Equilibrium Sound Velocity
10. Joule Thomson Coefficient
For an EOS to have significant engineering utility it should adequately represent three features
of the isotherm. (See Figure 2 below.)
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
Figure 2 Isotherms as given by a cubic equation of state. (from Smith and Van Ness, p. 81)
1. It must adequately represent the "Steepness" of the isotherm as v approaches some
small non-zero value.
2. It must adequately represent multiple values at the saturation pressure.
3. It must approach the ideal gas law as v approaches infinity.
For the understanding of how an Equation of State is used to calculate K values, let’s select the
first one in the list above, the Redlich-Kwong method, which has been extensively used in
industry with generally good results. To be precise, when the Redlich-Kwong equation is used it
is most likely used in one of its modified forms, very often the Soave modification. Experience
has proven this one of the better relationships. The Peng-Robinson method, a more recent
development, has also performed very well.
With the RK EOS, the general expression for the K value is:
Ki 
 li
 iv
The Vapour Liquid Equilibrium K value for component i is the ratio of the fugacity coefficient of
component i in the liquid to the fugacity coefficient of component i in the vapour. For the
definition of the fugacity coefficient of species i in solution refer to page 333 of Smith and Van
Ness. We are going to calculate the fugacity coefficient using an Equation of State, the RedlichKwong Soave.
The following is the generic form of the Redlich Kwong Equation,
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CHEE 470 Course Notes
a 0.5
RT
T
p

V  b V V  b) 
For the Soave modification, the term a/T0.5 is replaced by a temperature dependent term srk
(Page 491 Smith and Van Ness). When we are dealing with VLE considerations, in essentially
all cases we are dealing with mixtures.
There are generally two problems associated with the use of an Equation of State. The first
problem is related to mixing rules. As pointed out earlier, an Equation of State can be used to
calculate a considerable number of derived thermodynamic properties. When one uses an
Equation of State to calculate Vapour Liquid Equilibria (K values) one of course is dealing with
more than a single component. It is necessary to apply mixing rules to arrive at values for the
various parameters that can be used for mixtures. The mixing rules recommended in the
original Soave paper are relatively simple, as follows.
a = ( xiai0.5)2
b =  Xebio
On the other hand, the Peng Robinson mixing rules are somewhat more complex and an
empirically determined binary interaction coefficient, which characterizes the ij binary, is taken
into consideration.
a =  i j xixjaij
b =  ixibi
aij = 1 -  ij ai0.5aj0.5
The reference manual has PRO/II™ for introduction of Binary Interaction Data for both the
Peng Robinson and Soave Redlich-Kwong equations. We will not pursue the issue any further
at this time. To summarize, there are occasions when one is preparing a final design when it is
appropriate to introduce these binary coefficients to improve the accuracy of the calculation.
This situation arises most often when non-hydrocarbons such as N2, C02, and or H2S are
present in a hydrocarbon mixture. High-pressure gas processing units will often have this
situation.
The second major problem with the use of an Equation of State is the determination of the
correct values for the compressibility factor for the liquid and vapour phases. If you refer to
page 80 of Smith and Van Ness, there is a discussion of Cubic Equations of State. The
significance of this form of equation is that there can be three roots. If the temperature and
pressure being considered, results in the two phase region, the only region where VLE has any
meaning, one root should represent the saturated liquid, another the saturated vapour and the
third root is imaginary and has no real meaning. It is feasible to solve for these roots analytically
although in most instances numerical methods are used to converge on the roots. The cubic
form of the SRK Equation expressed in compressibility Z is as follows.
Z3 - Z2 + Z(A - B - B2 ) - AB = 0
The equivalent form of the Peng-Robinson Equation is as follows.
Lecture Notes: Selecting Thermodynamic Methods
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CHEE 470 Course Notes
Z3 - (1-B)Z2 + (A-3B2 - 2B)Z - ( AB-B2-B3 ) = 0
To review; before we calculate the fugacity coefficients for the liquid and vapour in equilibrium,
required to evaluate the K value, it is necessary to solve the above equations for the liquid and
vapour compressibility’s Z.
Analytical solutions for cubic equations can be found in the literature. For example, Perry's
Chemical Engineering Handbook discusses solutions to equations of this type in the section on
Algebra. Example 3.3 in "Applied Numerical Methods” by Carnahan, Luther and Wilkes
describes a solution of the Beattie-Bridgeman EOS using Newton’s method.
We have not discussed the other common Equation of State that is available, the BenedictWebb-Rubin. This is probably the Cadillac of Equations of State, however it is significantly more
complex than either of the two other, the SRK and PR. The equation expressed in terms of
molal density is as follows.
P = RTp + (B0RT - A0 - C0/T2 ) 2 + (bRT-a)  3 + a  6 + c 3/T2 (1+  2) e(-  2)
This is an extremely useful equation for correlating data for light hydrocarbons and their
mixtures. The major problem of course is the complexity of the relationship and the necessity of
the appropriate constants. It is in no way as suitable for general application as the SRK and PR,
however should the data be available, by all means use this equation provided it is supported
by the program being used such as PRO/II™ . The results are in most instances superior to any
other procedure when the data is available.
There is a section in the Keyword Input Manual on Applications Guidelines. This reference
presents some simple heuristic rules for selecting for selecting an appropriate thermodynamic
method for calculating the required properties as a function of various systems and
environments.
SUMMARY
Although this lecture is intended to be a simple review of material covered previously, there is
one particularly good reference that one could review in order to better understand some of the
issues and techniques of EOS construction. The reference is an article that appeared in
Chemical Engineering Progress in February 1989 entitled "Thirteen Ways of Looking at the van
der Waals Equation". This article begins on page 25. Other articles are referenced on the
course description.
Lecture Notes: Selecting Thermodynamic Methods
13
CHEE 470 Course Notes
Past Exam Questions
1996
An understanding of vapour liquid equilibria is essential for any process engineer
working in the design or analysis of chemical process units. The following systems are
among the more common methods of estimating vapour/liquid equilibria. BenedictWebb-Rubin, Peng-Robinson, and Ideal. Briefly describe these methods and give an
indication of which you would choose for the system that you have been working with
this past term. UNIFAC is a special case of one of the above. Describe this method
and when it would be most appropriate.
1997
One of the most critical issues when attempting to simulate a separation, distillation in
particular, is phase equilibria. Phase equilibrium is determined when the Gibb’s Free
Energy of the overall system is at a minimum.
a) Describe the issues involved in the above two statements (phase equilibrium and
Gibb’s Free Energy) and give an example of the three categories of thermodynamic
methods for determining phase equilibria. (12 pts.)
b) Why are there three approaches? (8 pts.)
1997 (supplemental)
We have the three following situations:
1. Separation of an alcohol, a ketone and a paraffin from an aqueous mixture
2. Separation of a mixture of C8 to C12 linear alpha olefins
3. Separation of a natural gas mixture at high pressure but below the pseudo-critical
point of the mixture.
What would be an appropriate method of estimating the K values for each of the three
scenarios, and give us your reasons for selecting the method chosen. (20 pts)
1998
Describe equations of state, liquid phase activity methods and generalized correlation
methods. Give an example for each and describe the appropriate conditions for their
application. (15 marks)
Lecture Notes: Selecting Thermodynamic Methods
14
CHEE 470 Course Notes
Today’s Question
“What thermodynamic system (or systems) should we use to model the methanol process”
In addressing this issue keep in mind that the main components are:
 Methane
 Water
 Hydrogen
 Carbon monoxide
 Carbon Dioxide
 Methanol and higher alcohols
 Acetone
Also, there are two other facts to consider
 The process operates at moderate temperature and pressure
 Chemists found the SRK method worked best in lab reaction studies
Lecture Notes: Selecting Thermodynamic Methods
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