Modifications of Cubic Equations of State for Polar Compounds
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Transcript of Modifications of Cubic Equations of State for Polar Compounds
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Modification of Cubic Equations of State for Polar Compounds
Seminar by:Hitanshu Sachania12BCH025
Guided by:Dr. Milind JoshipuraAssociate Professor
DEPARTMENT OF CHEMICAL ENGINEERING INSTITUTE OF TECHNOLOGY
NIRMA UNIVERSITY
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Equations of State
Equations of the form f(P,V,T) = 0 are known as
equations of state.
Such equations yield the volumetric properties of
the system and hence, describe its P-V-T behaviour.
“Volumetric Properties” are essential parts of
calculations used to compute thermodynamic
properties like internal energy (U), enthalpy (H),
entropy (S), etc.
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During process plant design these aid the engineer in
sizing of pipelines and process equipment like: [5]
Distillation Columns
Reaction Tanks
Heat Exchangers, etc.
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Cubic Equations of State
Cubic equations of state are a genus of equations
of state. They are in a cubic power of molar
volume (V). CEOS in the form of compressibility
factor Z, can be written as:
Z = Zrepulsive + Zattractive
Cubic equations of state are valid only for a
hypothesized gas known as an “Ideal Gas”.
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Pragmatic gases manifest a non-ideal behaviour
and to study their P-V-T behaviour i.e. to determine
their volumetric properties, cubic equations of
state need to be modified.
An ideal gas is such a gas which has no
intermolecular interaction since all of its
constituent molecules are separated infinitely from
each other.
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Van der Waals Cubic Equation of State
227
64C
C
RTa
P
1
8C
C
RTb
PWhere and
The concept of separation of repulsive forces due to
size of molecules and the cohesive forces due to
molecular attraction proposed by van der Waals, even
today remains as the basis of theories about the
prediction of fluid properties.
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Real GasesPolarity in a molecule arises
due to the combined effect of
electronegativity difference of
the constituent atoms and the
spatial arrangement of the
molecule.
H2O is polar since it has a
net dipole moment of the
two Oδ- - Hδ+ dipoles.Non
-Ioni
c Co
mpo
unds
Polar
Non-Polar
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CH4 is non-polar due to a null net dipole moment
since all four Cδ- - Hδ+ dipoles are oriented towards
the vertices of a tetrahedron.
There are three different types of intermolecular
interactions:
Dipole – Dipole: When a polar compound interacts
closely with another polar compound, the electric
fields of the two dipoles overlap and hence, a
net
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mutual attractive force is experienced by the
two molecules.
Dipole – Induced Dipole: When a polar compound
comes in a close proximity to a non-polar
compound, the permanent dipole of the polar
molecule induces an instantaneous dipole in the
non-polar molecule resulting in the two molecules
experiencing a net attractive force.
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Induced Dipole – Induced Dipole: Even when two
non-polar molecules come close enough to each
other, they induce evanescent dipoles into each
other which too results in a net attractive force
between the two molecules.
However, intermolecular interactions are not always
attractive; they may be repulsive as well. This can
easily be understood with the help of Lennard-Jones
Potential.
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Lennard – Jones Potential[1]
12 6
( ) 4U LJr r
Where r = intermolecular separation distance, ε =
maximum negative potential’s magnitude or the
depth of potential well and σ = separation distance
at which the intermolecular interaction between the
two molecules is zero.
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Schematic of Intermolecular Potential Energy for a pair of Non-ionic Molecules[1]
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The first term in the LJ potential equation i.e. the r-
12 term represents repulsive interactions while the
second term i.e. the r-6 term represents attractive
interactions.
For practical purposes intermolecular interactions
possess considerable magnitudes at separations up
to ten times the molecular diameter. ε and σ are
characteristic parameters corresponding to the
substance.
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General Methodology for Modification of Cubic Equations of State [3]
Cubic equations of state are generally modified in
three different ways:
Modification of the temperature dependent
function α(TR) in the attractive term of the EOS.
Volume Translation: Modification of the volume
dependence of the attractive pressure term.
Use of a 3rd substance dependent parameter.
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Three Parameter Cubic Equations of State
(C3EoS) A major limitation of the two parameter cubic
equations of state (C2EOS) is the prediction of the
same critical compressibility factor zc for all
substances.
Clausius’ Modified CEoS:
This equation is based on the notion that at lower
temperatures form clusters with a strong mutual
attraction in lieu of moving freely. [2,3]
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The a/V2 term in VdW CEoS is too small at low
temperatures and hence, doesn’t represent the
intermolecular attraction accurately.
To let zc be substance dependent, Clausius in his
EoS included a 3rd putative substance dependent
parameter (c) in the attractive volume term.
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Clausius’ three parameter cubic equation of state
laid the radix for the genesis of a new class of
CEoS, the C3EoS.
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Facets of Parameters in C3EoS
It’s easier to understand C3EoS parameters with
reference to the equation proposed by Schmidt and
Wenzel:
Where a(T) is the cohesive energy parameter and b
is the volumetric parameter.
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For constant u and w, this equation becomes a
C2EoS.
When one of u or w is fixed or if there exists a
relationship between the two, the equation
becomes a C3EoS.
The parameters of Schmidt and Wenzel equation
can be obtained by:
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1. Imposing the thermodynamic condition:
at Tc, Pc, Vc
2. Imposing the mathematical constraint:
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Relationship between u and w CEOS
u = 0, w = 0 Van der Waals
u = 1, w = 0 Redlich/Kwong , Soave/Redlich/Kwong
u = 2, w = -1 Peng/Robinson
w = 0 Fuller, Usdin and McAuliffe
w = u2/4 Clausius (VT - VDW)
u + w = 1 Heyen , Schmidt and Wenzel, Harmens and Knapp , Patel and Teja
u - w = 3 Yu et al. , Yu and Lu
u - w = 4 Twu et al.
w = 2(u+2)2/(9 - u -1) VT – SRK
w = (u-2)2/(8-1) VT –PR
Table 3.1 - Systematics of C2EOS and C3EOS [2]
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The C3EOS pertaining to the relation u + w = 0
stand out to best predict the PVT behaviour of both
non-polar and polar substances within a range of
critical compressibility factor (zc) values. [2]
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Modification of
Various modifications of function of the reduced
temperature for the SRK and PR EOS have been
proposed for better predictions of vapour pressure of
polar compounds.
Expression for Model
Redlich and Kwong
Wilson
Selected models of in Cubic Equations of State [3,4]
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Expression for Model
Heyen
Boston and Mathias
Usdin and McAuliffe
Joshipura
Twu
Soave
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The variation of some of these with disparity in temperature[3]:
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Cubic Plus Association Equation of State[13]
xAi is obtained from:
Where ΔAiBj, the association solidity between the site A of molecule and site B of molecule is given by:
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CPA EoS surmounts other equations in determining values of properties like density, heat capacity, enthalpy, Joule-Thomson coefficient and velocity of sound for reservoir fluids which contain polar substances like water and methanol. [13]
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SummaryThere are various methods for the modification of
CEoS to transpire.Though varied in manner, all such methods have a
ubiquitous idea as their base principle.The principle is the inclusion of additional
parameters and/or manipulation of the equation’s mathematical construct, so as to render the equations substance dependent.
No single equation has proved to be superior since, all of them have their own domain of supreme performance.
Moreover, molecular equations of state developed recently have superseded CEoS.
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References1. "Lennard-Jones Potential," [Online]. Available:
http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mecha
nics/Atomic_Theory/Intermolecular_Forces/Lennard-
Jones_Potential. [Accessed August 2014].
2. W.-R. Ji and D. Lempe, "A systematic study of cubic three-
parameter equations of state for deriving a structurally optimized
PVT relation," Fluid Phase Equilibria, pp. 85-103, 1998.
3. J. O. Valderrama, "The State of the Cubic Equations of State,"
Industrial & Engineering Research, pp. 1603 - 1618, 20 March
2003.
4. M. H. JOSHIPURA, "PREDICTIONS OF VAPOR PRESSURES OF TEN
IONIC LIQUIDS USING PATEL TEJA EQUATIONS OF STATE,"
International Journal of Research in Engineering & Technology
(IJRET), pp. 49-54, July 2013.
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5. "Chapter 2: Volumetric Properties of Real Fluids," [Online].
Available: http://nptel.ac.in/courses/103101004/downloads/chapter-
2.pdf. [Accessed 23 August 2014].
6. Heyen G. A Cubic Equation of State with Extended Range of
Application. In Chemical Engineering Thermodynamics by
Newman,175 1983.
7. Boston, J. F.; Mathias, P. M. Phase Equilibria in a Third- Generation
Process Simulator. Presented at the 2nd International Conference on
Phase Equilibria and Fluid Properties in the Chemical Industry,
Berlin, Germany, Mar 17-21, 1980.
8. Usdin, E.; McAuliffe, J. C. A One-Parameter Family of Equations of
State. Chem. Eng. Sci. 1976, 31, 1077.
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9. Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V.
An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev.
1949, 44, 233.
10. Wilson, G. M. A New Expression for the Excess Free Energy of
Mixing. J. Am. Chem. Soc. 1964, 86, 127.
11. Soave, G. Application of a Cubic Equation of State to Vapor-Liquid
Equilibria of Systems Containing Polar Compounds. Inst. Chem. Eng.
Symp. Ser. 1979, 56, 1.2/1.
12. Twu, C. H. A Modified Redlich-Kwong Equation of State for Highly
Polar, Supercritical Systems. In Proceedings of the International
Symposium on Thermodynamics in Chemical Engineering and
Industry; Academic Periodical Press: Beijing, China, 1988; p 148.
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13. C. Lundstrøm, M. L. Michelsen, G. M. Kontogeorgis, K. S. Pedersen and
H. Sørensen, "Comparison of the SRK and CPA equations of state for
physical properties of water and methanol," FLUID PHASE EQUILIBRIA,
pp. 149-157, 2006.
14. W. Yan, G. M. Kontogeorgis and E. H. Stenby, "Application of the CPA
equation of state to reservoir fluids in presence of water and polar
chemicals," FLUID PHASE EQUILIBRIA, pp. 75-85, 2009.