Physical Property Modeling from Equations of State

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Physical Property Modeling from Equations of
State
Evaluation of Series Coefficients for the
Peng-Robinson Equation

• David Schaich
• Hope College REU 2003

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Summary of the Project

• The goal of this project was to determine series
  coefficients for an analytic power series
  expansion of the Peng-Robinson equation of state.
• This series can be used to give general
  expressions for accurate estimation of
  thermodynamical properties, as explicit functions
  of temperature, in the vapor-liquid coexistence
  region away from the critical point.
• Results for terms up to 13th order in temperature
  were determined and tested for accuracy.
• Only pure substances were considered in the
  course of this project.

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Equations of State

• An equation of state is a functional relationship
  among the variables P, V, T
• Example Ideal Gas Law
• Example The Generalized Compressibility Factor
  (Z1 for ideal gases)

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Vapor-Liquid (Phase) Equilibrium

• Conditions under which liquid and vapor can
  coexist.

The line from the triple point to the critical
point in the phase diagram to the right is the
vapor-liquid equilibrium line.
Diagram courtesy of the Swedish National Testing
and Research Institute
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Using Equations of State

• All equations of state can predict one of the
  variables P, V, or T, given the other two.
• However, in order to predict phase equilibrium,
  one needs an equation of state capable of
  describing substances in both liquid and vapor
  phases.
• To do this, we can use cubic EoS, which take the
  form of cubic polynomials in molar volume.

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A Few Notes Regarding Cubic EoS

• The three major cubic EoS have the same general
  form a repulsive term derived from the ideal gas
  law, and an attractive term modeling van der
  Waals forces.
• All have a substance-specific constant bgt0 that
  corrects for the volume occupied by the molecules
  themselves.
• All also have a term agt0 that influences the
  attractive van der Waals force. For more complex
  cubic EoS, a is a function of temperature and
  acentric factor.
• The Pitzer acentric factor (w) is a
  substance-specific constant that reflects the
  geometry and polarity of a molecule.

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The Major Cubic EoS

• The van der Waals Equation
• The Soave-Redlich-Kwong (SRK) Equation
• The Peng-Robinson Equation

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Advantages of Cubic EoS

• They are applicable over a wide range of
  pressures and temperatures.
• They are capable of describing substances in both
  liquid and vapor phases
• They can therefore be used to predict phase
  equilibrium properties, such as vapor pressure,
  heat of vaporization, enthalpy departure and
  various other results.

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Also...

• Although cubic equations of state can do all of
  this, they are still relatively simple, unlike,
  for example, the Benedict-Webb-Rubin equation

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The Series Approach

• Cubic equations of state can be represented as
  Taylor series around the critical point

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Simplifying the Series

• First we used thermodynamical relationships to
  eliminate one set of partial derivatives
• Then we used the following series relationship to
  eliminate one independent variable
• Note All terms have a positive sign for liquid
  densities for vapor, the terms alternate in sign.

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The Series Simplified

• Eliminating the variables and partial derivatives
  from the
• pressure and chemical potential series gives the
  equations
• where the coefficients Ci and Gi contain only the
• unknown coefficients Bj, jlti, and the known
• partial derivatives Pm,n, m2ni.
• The goal of my research was to determine the
• coefficients Ci for the series corresponding to
  the Peng-
• Robinson equation.

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Solution Method
It has been shown that the coefficients of the
odd half powers of (-?T) are zero. This
presented a convenient method for calculating the
density coefficients Bi they could be determined
by setting the odd-subscripted C and G
coefficients to zero and solving for the unknown
B coefficients. Since all the odd terms are
zero, the series can be written more
compactly At this point, all of the elements of
the pressure series coefficients are known and Aj
can be evaluated. All of these calculations
however, are very complex, and were performed
with programs written with the Maple mathematical
software package.
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Results
In this project, I determined series coefficients
for density up through the 12th order in
temperature (B24) and for vapor pressure up
through the 13th order in temperature (A13).
Listed below are decimal values precise to 10-5
for the polynomial coefficients of f(?) appearing
in the density and vapor pressure coefficients.
For odd subscripted density coefficients, the
ratios Bi/(1f(?))1/2 are given in the polynomial
form. That is, B1 2.50615(1f(?))1/2 B2
1.35419 1.35419(f(?)) Results are given below
for density coefficients up to B5 and vapor
pressure coefficients up to A5. Complete
results can be found at http//www.amherst.edu/d
aschaich/reu2003/results.htm
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Decimal coefficients of polynomials in acentric
factor function f(?)
Coefficients of f(w) polynomial Coefficients of f(w) polynomial Coefficients of f(w) polynomial Coefficients of f(w) polynomial Coefficients of f(w) polynomial Coefficients of f(w) polynomial
Constant f(w) f(w)2 f(w)3 f(w)4 f(w)5
B1/(1f(w))1/2 2.50615
B2 1.35419 1.35419
B3/(1f(w))1/2 -0.65944 -1.59925
B4 -0.62593 -2.26750 -1.64157
B5/(1f(w))1/2 0.08584 0.75691 1.27803

A0 1.00000
A1 4.35530 3.35530
A2 6.16148 11.54414 5.35266
A3 2.77903 11.85224 14.94798 5.87477
A4 0.42392 4.90733 14.63759 15.98672 5.83255
A5 1.00060 5.49346 15.06487 24.14467 19.88211 6.30944
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Checking the Results
We checked our results by comparing the series'
predictions for the vapor pressure of water over
a range of temperatures with those of the
equation itself (generated using an iterative
solution method). If all coefficients in the
series were correct, the difference between these
two values would fit a curve one degree greater
than that of the last term included in the
series. That is, if the series were truncated
after the quadratic term in temperature, this
difference should fit a cubic polynomial with
respect to temperature deviation. If the series
included terms up to the 13th power of
temperature, the difference should be 14th
power, as illustrated below.
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Sample Error Check, for Series Through A13
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Future Plans
Currently, we are assessing the accuracy of the
Peng-Robinson series over the whole range of
temperatures and acentric factors. The chart
below shows that the accuracy of the series
decreases as temperature decreases and acentric
factor increases. In the future, the series can
be used to generate predictions of equilibrium
properties such as heat of vaporization and
enthalpy departure. At some point we will also
attempt to move beyond pure solutions and adapt
the series solutions for the Peng-Robinson and
SRK equations to mixtures.
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Relative error in vapor pressure predictions for
PR equation - series truncated after A5 term
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Acknowledgments

• I would like to thank
• Dr. Misovich, for his expert guidance and
  assistance
• The National Science Foundation, for its generous
  funding
• The Physics and Engineering Department at Hope
  College
• Scott Gangloff, John Alford and others who have
  worked on this topic