Idea Transcript
PETE 310 Lectures # 36 to 37 Cubic Equations of State
…Last Lectures
Instructional Objectives Know the data needed in the EOS to evaluate fluid properties Know how to use the EOS for single and for multicomponent systems Evaluate the volume (density, or z-factor) roots from a cubic equation of state for Gas phase (when two phases exist) Liquid Phase (when two phases exist) Single phase when only one phase exists
Equations of State (EOS) Single Component Systems Equations of State (EOS) are mathematical relations between pressure (P) temperature (T), and molar volume (V).
Multicomponent Systems For multicomponent mixtures in addition to (P, T & V) , the overall molar composition and a set of mixing rules are needed.
Uses of Equations of State (EOS) Evaluation of gas injection processes (miscible and immiscible) Evaluation of properties of a reservoir oil (liquid) coexisting with a gas cap (gas) Simulation of volatile and gas condensate production through constant volume depletion evaluations Recombination tests using separator oil and gas streams Many more…
Equations of State (EOS) One of the most used EOS’ is the PengRobinson EOS (1975). This is a threeparameter corresponding states model.
P
P
RT V b V (V
Prep
Pattr
a b) b(V
b)
Equations of State (EOS) Peng-Robinson EOS is a three-parameter corresponding states model. Critical Temperature Tc Critical Pressure Pc Acentric factor
PV Phase Behavior Pressurevolume behavior indicating isotherms for a pure component system
Tc
Pressure
CP
T2
v P1 T1 L 2 - Phases V
L
V
Molar Volume
Equations of State (EOS) The critical point conditions are used to determine the EOS parameters
P V
0 Tc
2
P 2 V
0 Tc
Equations of State (EOS) Solving these two equations simultaneously for the Peng-Robinson EOS provides 2
a
a
2 c
RT Pc
and
b
b
RTc Pc
Equations of State (EOS) Where and with
a
0.45724
b
0.07780 1 m1
Tr
m 0.37464 1.54226
2
0.2699
2
EOS for a Pure Component
Pres sur e
CP
T2 4 v
P 1
3
1 L
A1
2
10 0
5
A2 P ~ V
0
V
T
2 - P has hases
L
7
1 2
V
Mo la r V o lum e
T1
6
EOS for a Pure Component
Maxwell equal area rule (Van der Waals loops) For a fixed Temperature lower than Tc the vapor pressure is found when A1 = A2 Equations of State cannot be quadratic polynomials Lowest root is liquid molar volume, largest root is gas molar volume Middle root has no physical significance
CP Pressure
T2
v 3 P1 1 L A1 1 10 0 2
4 A2 P V~
5 0
V
T
2 - Phas es
L 7
1 2
V
Molar Volum Volume e
T1
6
Equations of State (EOS) Phase equilibrium for a single component at a given temperature can be graphically determined by selecting the saturation pressure such that the areas above and below the loop are equal, these are known as the van der Waals loops.
Equations of State (EOS) PR equation can be expressed as a cubic polynomial in V, density, or Z.
Z
3
( B 1) Z
( A 3B
2
2
2 B) Z
( AB B
2
3
B) 0
A
with B
a P 2 RT bP RT
Equations of State (EOS) When working with mixtures (a ) and (b) are evaluated using a set of mixing rules The most common mixing rules are: Quadratic for a Linear for b
Quadratic MR for a 0.5
Nc Nc
a
xi x j ai a j
m
i
j
1 ki j
i 1 j 1
where kij’s are the binary interaction parameters and by definition
kij k ji kii 0
Linear MR for b
Nc
bm
xibi i 1
Example For a three-component mixture (Nc = 3) the attraction (a) and the repulsion constant (b) are given by a
m
2 x1 x2 a1a2
0.5 1
2 x1 x3 a1a3 x32 a3
bm
x1b1
2
0.5 1 3
3
x2b2
(1 k12 ) 2 x2 x3 a2a3
x3b3
(1 k13 )
x12 a1
1
0.5 2
3
x22 a2
(1 k23 ) 2
Equations of State (EOS) The constants a and b are evaluated using Overall compositions zi with i = 1, 2…Nc Liquid compositions xi with i = 1, 2…Nc Vapor compositions yi with i = 1, 2…Nc
Equations of State (EOS) The cubic expression for a mixture is then evaluated using
Am
a
P m
RT
2
Bm
bm P RT
Analytical Solution of Cubic Equations The cubic EOS can be arranged into a polynomial and be solved analytically as follows.
Z
3
( B 1) Z
( A 3B
2
2
2 B) Z
( AB B
2
3
B) 0
Analytical Solution of Cubic Equations Let’s write the polynomial in the following way
x
3
a1 x
2
a2 x a3
0
Note: “x” could be either the molar volume, or the density, or the z-factor
Analytical Solution of Cubic Equations When the equation is expressed in terms of the z factor, the coefficients a1 to a3 are:
a1 a2 a3
( B 1) ( A 3B
2
( AB B
2 B) 2
3
B )
Procedure to Evaluate the Roots of a Cubic Equation Analytically Let
Q R
3a2 9 9a1a2
S
3
T
3
R R
2 1
a
3 1
27 a3 54 Q
3
Q
3
2a R
2
R
2
Procedure to Evaluate the Roots of a Cubic Equation Analytically The solutions are,
x1 x2 x3
S T
1 a1 3
1 S T 2 1 S T 2
1 1 a1 i 3 S T 3 2 1 1 a1 i 3 S T 3 2
Procedure to Evaluate the Roots of a Cubic Equation Analytically If a1, a2 and a3 are real (always here) The discriminant is D = Q 3 + R2 Then One root is real and two complex conjugate if D > 0; All roots are real and at least two are equal if D = 0; All roots are real and unequal if D < 0.
Procedure to Evaluate the Roots of a Cubic Equation Analytically x1 If D 0
x2
where
cos
R Q3
x3
2
1 Q cos 3
2
1 Q cos 3
2
1 Q cos 3
1 a1 3 120
1 a1 3
240
1 a1 3
Procedure to Evaluate the Roots of a Cubic Equation Analytically
x1
x2
x1 x2 x1 x2 x3
x3
a1
x2 x3
x3 x1
a2
a3
where x1, x2 and x3 are the three roots.
Procedure to Evaluate the Roots of a Cubic Equation Analytically The range of solutions useful for engineers are those for positive volumes and pressures, we are not concerned about imaginary numbers.
Solutions of a Cubic Polynomial
We are only interested in the first quadrant.
Solutions of a Cubic Polynomial http://van-der-waals.pc.unikoeln.de/quartic/quartic.html contains Fortran codes to solve the roots of polynomials up to fifth degree.
Web site to download Fortran source codes to solve polynomials up to fifth degree
EOS for a Pure Component
Pres sur e
CP
T2 4 v
P 1
3
1 L
A1
2
10 0
5
A2 P ~ V
0
V
T
2 - P has hases
L
7
1 2
V
Mo la r V o lum e
T1
6
Parameters needed to solve EOS Tc, Pc, (acentric factor for some equations i.e. Peng Robinson) Compositions (when dealing with mixtures) For a single component Specify P and T determine Vm Specify P and Vm determine T Specify T and Vm determine P
Tartaglia: the solver of cubic equations
http://es.rice.edu/ES/humsoc/Galileo/Catalog/Files/tartalia.html
Cubic Equation Solver
http://www.1728.com/cubic.htm
WWW Cubic Equation Solver Only to check your results You will not be able to use it in the exam if needed Special bonus HW will be invalid if using this code, you MUST provide evidence of work Write your own code (Excel is OK)
Two-phase VLE The phase equilibria equations are expressed in terms of the equilibrium ratios, the “K-values”.
Ki
yi xi
ˆl i ˆv i
Dew Point Calculations Equilibrium is always stated as: l ˆ xi i P
v ˆ yi i P
(i = 1, 2, 3 ,…Nc)
with the following material balance constraints Nc
Nc
xi 1, i 1
Nc
yi 1, i 1
zi 1 i 1
Dew Point Calculations At the dew-point l ˆ xi i
xi Ki
v ˆ zi i
zi
(i = 1, 2, 3 ,…Nc)
Dew Point Calculations Rearranging, we obtain the Dew-Point objective function Nc i 1
zi Ki
1 0
Bubble Point Equilibrium Calculations For a Bubble-point
Nc
zi Ki 1 0 i 1
Flash Equilibrium Calculations Flash calculations are the work-horse of any compositional reservoir simulation package. The objective is to find the fv in a VL mixture at a specified T and P such that Nc i
zi ( K i 1) f v ( K i 1) 11
0
Evaluation of Fugacity Coefficients and K-values from an EOS The general expression to evaluate the fugacity coefficient for component “i” is
RT ln ˆ
P
v i
Vi 0
RT dP P
T fixed
Evaluation of Fugacity Coefficients and K-values from an EOS The final expression to evaluate the fugacity coefficient of component ‘i’ in the vapor phase using an EOS is.
RT ln ˆiv
Vtv
P v ni
T , n vj i
RT v dV t v Vt
RT ln Z v
A similar expression replacing v by l is used for the liquid
Equations of State are not perfect… EOS provide self consistent fluid properties Density (o & g) trends are correctly predicted with pressure, temperature, and compositions (and all derived properties…) Same phase equilibrium model for gas and liquid phases (material balance consistency)
Equations of State are not perfect… However… predicted fluid property values may differ substantially from data EOS are routinely “calibrated” to selected & limited experimental data After “calibration” EOS predictions beyond range of data can be used with confidence EOS are extensively used in reservoir simulation
What is EOS calibration? Minimization of squared differences between experimental and predicted fluid properties Ndata
gi
predicted
gi
exp erimental
2
min
i 1
These Properties (gi) include: Densities, saturation pressures Relative amounts of gas and liquid phases Compositions, etc.
What is EOS calibration? Accomplished by changing within certain limits selected EOS parameters Minor adjustments (1 to 2%) of binary interaction parameters (kij) can change saturation pressures by 20 to 30% Different properties of the C7+ fraction affect liquid dropout and densities. These properties include Molecular weight (uncertainty is +/- 10%) Specific gravity Critical properties and acentric factors which are highly dependent on correlations – Cannot be easily measured and not usually done.
Pre and post calibration predictions from an EOS
Pre and post calibration predictions from an EOS
Pre and post calibration predictions from an EOS
Pre and post calibration predictions from an EOS
Problems to Think About… Determine the equilibrium ratio of C1 from multiple flash calculations using SOPE. Select a mixture and a suitable pressure temperature range Discuss the trends, how does kC1change with T at a fixed P? Discuss the trends, how does kC1change with P at a fixed T? Provide well documented graphs
Problems to Think About… Compare the equilibrium ratio of C1 at 4000 psia and at 200 oF with that of the convergence pressure chart using.
A mixture of C1 and C2 A mixture of C1 and C4 A mixture of C1 and C8 Discuss the results obtained and provide overlapped plots
Calibrate one of EOS’s in SOPE to the bubble point data reported by Standings in the following table
Problems to Think About…
Problems to Think About… Mole fraction of C1 Dew point pressure Bubblepoint pressure Z-factors of mixture (gas and liquid) Molar volumes of mixture gas & liquid All at T = 160oF (not shown here)
Problems to Think About… Select one EOS Select the best kij (Vdw, RK, SRK, PR, that matches the or Cubic-4G) bubble point pressure Select one bubble point pressure for Compare the one composition of values of methane experimental vs. predicted molar Plot pb predicted vs volumes binary interaction parameter selected
You should be obtaining a plot like this one… Bubble Point Pressure C 1-C 4 Mixture (10% C1) at T = o
160 F 2500.0
Experimental pb is 339 psia
Pressure, psia
2000.0
1500.0
You CANNOT use this same composition in Your homework
1000.0
500.0
0.0 -0.5
-0.3
Cubic-4G
-0.1
P-R
kij
0.1
S-R-K
0.3
R-K
0.5
VDW
This is the end, we survived!!!