Equations of State
-
Upload
sunil-saini -
Category
Documents
-
view
214 -
download
0
description
Transcript of Equations of State
PETE 310PETE 310Lectures # 32 to 34Lectures # 32 to 34
Cubic Equations of StateCubic Equations of State
……Last LecturesLast Lectures
Instructional ObjectivesInstructional Objectives Know the data needed in the EOS to Know the data needed in the EOS to
evaluate fluid propertiesevaluate fluid properties Know how to use the EOS for single and Know how to use the EOS for single and
multicomponent systemsmulticomponent systems Evaluate the volume (density, or z-factor) Evaluate the volume (density, or z-factor)
roots from a cubic equation of state forroots from a cubic equation of state for
Gas phase (when two phases exist)Gas phase (when two phases exist) Liquid Phase (when two phases exist)Liquid Phase (when two phases exist) Single phase when only one phase existsSingle phase when only one phase exists
Equations of State (EOS)Equations of State (EOS)
Single Component SystemsSingle Component Systems
Equations of State (EOS) are Equations of State (EOS) are mathematical relations between mathematical relations between pressure (P) temperature (T), and molar pressure (P) temperature (T), and molar volume (V). volume (V).
Multicomponent SystemsMulticomponent Systems
For multicomponent mixtures in For multicomponent mixtures in addition to (P, T & V) , the overall molar addition to (P, T & V) , the overall molar composition and a set of mixing rules composition and a set of mixing rules are needed.are needed.
Uses of Equations of State Uses of Equations of State (EOS)(EOS)
Evaluation of gas injection processes Evaluation of gas injection processes (miscible and immiscible)(miscible and immiscible)
Evaluation of properties of a reservoir Evaluation of properties of a reservoir oil (liquid) coexisting with a gas cap oil (liquid) coexisting with a gas cap (gas)(gas)
Simulation of volatile and gas Simulation of volatile and gas condensate production through condensate production through constant volume depletion evaluations constant volume depletion evaluations
Recombination tests using separator Recombination tests using separator oil and gas streamsoil and gas streams
Equations of State (EOS)Equations of State (EOS)
One of the most used EOS’ is the One of the most used EOS’ is the Peng-Robinson EOS (1975). This is a Peng-Robinson EOS (1975). This is a three-parameter corresponding three-parameter corresponding states model.states model.
)()( bVbbVV
a
bV
RTP
attrrep PPP
PV Phase BehaviorPV Phase Behavior
Pressure-Pressure-volume volume behavior behavior indicating indicating isotherms for isotherms for a pure a pure component component systemsystem
Pre
ssu
r e
Mo lar V olume
Tc
T2
T1
P1v
L
2 - P has es
CP
V
L
V
Pre
ssu
r e
Mo lar V olume
Tc
Equations of State (EOS)Equations of State (EOS)
The critical point conditions are The critical point conditions are used to determine the EOS used to determine the EOS parametersparameters
0
0
2
2
c
c
T
T
V
P
V
P
Equations of State (EOS)Equations of State (EOS)
Solving these two equations Solving these two equations simultaneously for the Peng-simultaneously for the Peng-Robinson EOS providesRobinson EOS provides
c
ca P
TRa
22
c
cb P
RTb andand
Equations of State (EOS)Equations of State (EOS)
Where
and
with
07780.0
45724.0
b
a
211 rTm
22699.054226.137464.0 m
EOS for a Pure ComponentEOS for a Pure Component
-10
0
12
A1
A2
Pre
ssu
re
Mo la r V olum e
T2
T1P1
v
L
2 - P hases
CP
V
L
V
1
2
3
4
7 6
5
0
TV~P
-10
12
A1
A2
Pre
ssu
re
112
A1
A2
Pre
ssu
re
Mo la r V olum e
T2
T1P1
v
L
2 - P has
CP
V
L
V
1
2
3
4
7 6
5
0
TV~P
Equations of State (EOS)Equations of State (EOS)
PR equation can be expressed PR equation can be expressed as a cubic polynomial in V, as a cubic polynomial in V, density, or Z. density, or Z.
RT
bPB
RT
PaA
2
3 2
2
2 3
( 1)
( 3 2 )
( ) 0
Z B Z
A B B Z
AB B B
withwith
Equations of State (EOS)Equations of State (EOS)
When working with mixtures When working with mixtures (a(a) and (b) are evaluated ) and (b) are evaluated using a set of mixing rulesusing a set of mixing rules
The most common mixing The most common mixing rules are:rules are: Quadratic for a Quadratic for a
Linear for bLinear for b
Quadratic MR for aQuadratic MR for a
where the kij’s are called interaction where the kij’s are called interaction parameters and by definitionparameters and by definition
0.5
1 1
1Nc Nc
i j i j i j i jmi j
a x x a a k
0
ij ji
ii
k k
k
ExampleExample
For a three-component For a three-component mixture (Nc = 3) the attraction mixture (Nc = 3) the attraction (a) and the repulsion constant (a) and the repulsion constant (b) are given by(b) are given by
1
0.5 0.5
1 2 1 2 1 2 12 2 3 2 3 2 3 23
0.5 2 21 3 1 3 1 3 13 1 1 2 2 2
23 3 3
2 (1 ) 2 (1 )
2 (1 )
ma x x a a k x x a a k
x x a a k x a x a
x a
1 1 2 2 3 3
mb x b x b x b
Equations of State (EOS)Equations of State (EOS)
The constants a and b can be The constants a and b can be evaluated usingevaluated using
Overall compositions Overall compositions zzii with with ii = = 1, 2…Nc1, 2…Nc
Liquid compositions Liquid compositions xxii with with ii = = 1, 2…Nc1, 2…Nc
Vapor compositions Vapor compositions yyii with with ii = = 1, 2…Nc1, 2…Nc
Equations of State (EOS)Equations of State (EOS)
The cubic expression for a mixture The cubic expression for a mixture is then evaluated usingis then evaluated using
2 mm
m m
a P b PA B
RTRT
Analytical Solution of Analytical Solution of Cubic EquationsCubic Equations
The cubic EOS can be arranged The cubic EOS can be arranged into a polynomial and be into a polynomial and be solved analytically as follows. solved analytically as follows.
3 2
2
2 3
( 1)
( 3 2 )
( ) 0
Z B Z
A B B Z
AB B B
Analytical Solution of Analytical Solution of Cubic EquationsCubic Equations
Let’s write the polynomial in Let’s write the polynomial in the following waythe following way
Note:Note: “x” could be either the “x” could be either the molar volume, or the density, or molar volume, or the density, or the z-factorthe z-factor
3 2 31 2 0x a x a x a
Analytical Solution of Analytical Solution of Cubic EquationsCubic Equations
When the equation is When the equation is expressed in terms of the z expressed in terms of the z factor, the coefficients factor, the coefficients aa11 to to aa33 are:are:
1
22
2 33
( 1)
( 3 2 )
( )
a B
a A B B
a AB B B
Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation
AnalyticallyAnalytically2
2 1
31 2 3 1
3 23
3 23
3
9
9 27 2
54
a aQ
a a a aR
S R Q R
T R Q R
LetLet
Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation
AnalyticallyAnalytically
1 1
2 1
3 1
1
31 1 1
32 3 21 1 1
32 3 2
x S T a
x S T a i S T
x S T a i S T
The solutions are,The solutions are,
Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation
AnalyticallyAnalytically
If If aa11, , aa22 and and aa33 are real and if are real and if D = D = QQ33 + R + R22 is the discriminant, then is the discriminant, then
One root is real and two complex One root is real and two complex conjugate if conjugate if D > 0D > 0;;
All roots are real and at least two All roots are real and at least two are equal if are equal if D = 0D = 0; ;
All roots are real and unequal if All roots are real and unequal if D D < 0< 0..
Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation
AnalyticallyAnalytically
wherewhere
1 1
2 1
3 1
1 12 cos
3 3
1 1If 0 2 cos 120
3 3
1 12 cos 240
3 3
x Q a
D x Q a
x Q a
3cos
R
Q
Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation
AnalyticallyAnalytically
where where xx11, , xx22 and and xx33 are the three are the three roots.roots.
1 2 3 1
1 2 2 3 3 1 2
1 2 3 3
x x x a
x x x x x x a
x x x a
Procedure to Evaluate the Procedure to Evaluate the Roots of a Cubic Equation Roots of a Cubic Equation
AnalyticallyAnalytically
The range of solutions that are The range of solutions that are used for the engineer are used for the engineer are those for positive volumes and those for positive volumes and pressures, we are not pressures, we are not concerned about imaginary concerned about imaginary numbers.numbers.
Solutions of a Cubic Solutions of a Cubic PolynomialPolynomial
From the shape of the polynomial we are only
interested in the first
quadrant.
Solutions of a Cubic Solutions of a Cubic PolynomialPolynomial
http://www.uni-koeln.de/math-http://www.uni-koeln.de/math-nat-fak/phchem/deiters/nat-fak/phchem/deiters/quartic/quartic.htmlquartic/quartic.html contains contains Fortran codes to solve the Fortran codes to solve the roots of polynomials up to fifth roots of polynomials up to fifth degree.degree.
Web site to download Fortran Web site to download Fortran source codes to solve polynomials source codes to solve polynomials
up to fifth degreeup to fifth degree
EOS for a Pure ComponentEOS for a Pure Component
-10
0
12
A1
A2
Pre
ssu
re
Mo la r V olum e
T2
T1P1
v
L
2 - P hases
CP
V
L
V
1
2
3
4
7 6
5
0
TV~P
-10
12
A1
A2
Pre
ssu
re
112
A1
A2
Pre
ssu
re
Mo la r V olum e
T2
T1P1
v
L
2 - P has
CP
V
L
V
1
2
3
4
7 6
5
0
TV~P
Parameters needed to Parameters needed to solve EOSsolve EOS
Tc, Pc, (acentric factor for some Tc, Pc, (acentric factor for some equations I.e Peng Robinson)equations I.e Peng Robinson)
Compositions (when dealing Compositions (when dealing with mixtures)with mixtures) Specify P and T Specify P and T determine Vm determine Vm
Specify P and Vm Specify P and Vm determine T determine T
Specify T and Vm Specify T and Vm determine P determine P
Tartaglia: the solver of cubic equations
http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/tartalia.html
Cubic Equation Solver Cubic Equation Solver
http://www.1728.com/cubic.htmhttp://www.1728.com/cubic.htm
Equations of State (EOS)Equations of State (EOS)
Phase equilibrium for a single Phase equilibrium for a single component at a given component at a given temperature can be graphically temperature can be graphically determined by selecting the determined by selecting the saturation pressure such that saturation pressure such that the areas above and below the the areas above and below the loop are equal, these are known loop are equal, these are known as the van der Waals loops.as the van der Waals loops.
Two-phase VLETwo-phase VLE
The phase equilibria equations The phase equilibria equations are expressed in terms of the are expressed in terms of the equilibrium ratios, the “K-equilibrium ratios, the “K-values”. values”.
ˆ
ˆ
li i
i vi i
yK
x
Dew Point CalculationsDew Point Calculations
Equilibrium is always stated Equilibrium is always stated as:as:
(i = 1, 2, 3 ,…Nc) (i = 1, 2, 3 ,…Nc)
with the following material with the following material balance constrainsbalance constrains
ˆ ˆl vi i i ix P y P
1 1 1
1, 1, 1Nc Nc Nc
i i ii i i
x y z
Dew Point CalculationsDew Point Calculations
At the dew-pointAt the dew-point
ˆ ˆl vi i i i
i i i
x z
x K z
(i = 1, 2, 3 ,…Nc)(i = 1, 2, 3 ,…Nc)
Dew Point CalculationsDew Point Calculations
Rearranging, we obtain the Rearranging, we obtain the Dew-Point objective functionDew-Point objective function
1
1 0Nc
i
i i
z
K
Bubble Point Equilibrium Bubble Point Equilibrium CalculationsCalculations
For a Bubble-pointFor a Bubble-point
1
1 0Nc
i ii
z K
Flash Equilibrium Flash Equilibrium CalculationsCalculations
Flash calculations are the work-Flash calculations are the work-horse of any compositional horse of any compositional reservoir simulation package. reservoir simulation package.
The objective is to find the The objective is to find the ffvv in in a VL mixture at a specified T a VL mixture at a specified T and P such thatand P such that
1
( 1)0
1 ( 1)
cNi i
i v i
z K
f K
Evaluation of Fugacity Evaluation of Fugacity Coefficients and K-values from Coefficients and K-values from
an EOSan EOSThe general expression to The general expression to
evaluate the fugacity evaluate the fugacity coefficient for component “coefficient for component “ii” ” isis
fixedT
P
ivi dP
P
RTVRT
0
ˆln
The final expression to The final expression to evaluate the fugacity evaluate the fugacity coefficient using an EOS is.coefficient using an EOS is.
vvtv
inT
v
V
vi ZRTdV
V
RT
n
PRT
tvj
i
vt
lnˆln,
Evaluation of Fugacity Evaluation of Fugacity Coefficients and K-values from Coefficients and K-values from
an EOSan EOS