PETE 310 Lectures # 36 to 37

Transcription

1 PETE 310 Lectures # 36 to 37 Cubic Equations of State Last Lectures

2 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

3 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.

4 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

5 Equations of State (EOS) One of the most used EOS is the Peng- Robinson EOS (1975). This is a threeparameter corresponding states model. P V RT b a V ( V b) b( V b) P P rep P attr

6 Equations of State (EOS) Peng-Robinson EOS is a three-parameter corresponding states model. Critical Temperature T c Critical Pressure P c Acentric factor

7 PV Phase Behavior Pressure P 1 v L T c CP 2 - Phases T 2 T 1 V Pressurevolume behavior indicating isotherms for a pure component system L V Molar Volume

8 Equations of State (EOS) The critical point conditions are used to determine the EOS parameters P V 2 V P 2 T c T c 0 0

9 Equations of State (EOS) Solving these two equations simultaneously for the Peng-Robinson EOS provides a a R 2 P T c 2 c and b b RT P c c

10 Equations of State (EOS) Where and a b with 1 m1 Tr 2 m

11 Pres sur e EOS for a Pure Component CP T 2 v P 1 1 L A A2 V ~P T T 1 V P hases L V 7 Mo la r V olum e 6

12 Pre sure EOS for a Pure Component Maxwell equal area rule (Van der Waals loops) For a fixed Temperature lower than T c the vapor pressure is found when A 1 = A 2 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 v P 1 1 L CP T A2 5 - A V ~P T Phases L V 7 Molar Volume 6 T1 V

13 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.

14 Equations of State (EOS) PR equation can be expressed as a cubic polynomial in V, density, or Z. Z ( B 1) Z ( 3 2 ) A B B Z 2 3 ( AB B B ) 0 with B a RT bp RT P A 2

15 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

16 Quadratic MR for a Nc Nc 1 j a x x a a 1 k m i i j i j i j i j where k ij s are the binary interaction parameters and by definition k k ij ii k 0 ji

17 Linear MR for b b Nc x b m i i i 1

18 Example For a three-component mixture (Nc = 3) the attraction (a) and the repulsion constant (b) are given by a 2 x x a a (1 k ) 2 x x a a (1 k ) m x x a a (1 k ) x a x a x a b x b x b x b m

19 Equations of State (EOS) The constants a and b are evaluated using Overall compositions z i with i = 1, 2 Nc Liquid compositions x i with i = 1, 2 Nc Vapor compositions y i with i = 1, 2 Nc

20 Equations of State (EOS) The cubic expression for a mixture is then evaluated using A m a P bp RT m B m 2 m RT

21 Analytical Solution of Cubic Equations The cubic EOS can be arranged into a polynomial and be solved analytically as follows. 3 2 Z ( B 1) Z 2 ( A 3B 2 B) Z 2 3 ( AB B B ) 0

22 Analytical Solution of Cubic Equations Let s write the polynomial in the following way 3 2 x a x a x a Note: x could be either the molar volume, or the density, or the z-factor

23 Analytical Solution of Cubic Equations When the equation is expressed in terms of the z factor, the coefficients a 1 to a 3 are: a ( B 1) a ( A 3B 2 B) a ( AB B B )

24 Procedure to Evaluate the Roots of a Cubic Equation Analytically Let Q R 2 3a2 a1 9 9a a 27a 2a S 3 R 3 Q 2 R T 3 R 3 Q 2 R

25 Procedure to Evaluate the Roots of a Cubic Equation Analytically The solutions are, 1 x S T a x S T a i 3 S T x S T a i 3 S T

26 Procedure to Evaluate the Roots of a Cubic Equation Analytically If a 1, a 2 and a 3 are real (always here) The discriminant is Then D = Q 3 + R 2 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.

27 Procedure to Evaluate the Roots of a Cubic Equation Analytically where 1 1 x 2 Q cos a If D 0 x 2 Q cos 120 a 3 3 cos R Q x 2 Q cos 240 a

28 Procedure to Evaluate the Roots of a Cubic Equation Analytically x x x a x x x x x x a x x x a where x 1, x 2 and x 3 are the three roots.

29 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.

30 Solutions of a Cubic Polynomial We are only interested in the first quadrant.

31 Solutions of a Cubic Polynomial contains Fortran codes to solve the roots of polynomials up to fifth degree.

32 Web site to download Fortran source codes to solve polynomials up to fifth degree

33 Pres sur e EOS for a Pure Component CP T 2 v P 1 1 L A A2 V ~P T T 1 V P hases L V 7 Mo la r V olum e 6

34 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

35 Tartaglia: the solver of cubic equations

36 Cubic Equation Solver

37 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)

38 Two-phase VLE The phase equilibria equations are expressed in terms of the equilibrium ratios, the K-values. K i y x i i ˆ ˆ l i v i

39 Dew Point Calculations Equilibrium is always stated as: x ˆl P y ˆv P i i i i (i = 1, 2, 3, Nc) with the following material balance constraints Nc Nc Nc x 1, y 1, z 1 i i i i 1 i 1 i 1

40 Dew Point Calculations At the dew-point x ˆ ˆ l z v i i i i x K z (i = 1, 2, 3, Nc) i i i

41 Dew Point Calculations Rearranging, we obtain the Dew-Point objective function Nc i 1 z i K i 1 0

42 Bubble Point Equilibrium Calculations For a Bubble-point Nc i 1 zk i i 1 0

43 Flash Equilibrium Calculations Flash calculations are the work-horse of any compositional reservoir simulation package. The objective is to find the f v in a VL mixture at a specified T and P such that N c z ( K 1) i 1 f ( K 1) i 1 v i i 0

44 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 v i P 0 V i RT P dp T fixed

45 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 ˆ v i V v t P v n i T, n v j i RT v V t dv v t RT ln Z v A similar expression replacing v by l is used for the liquid

46 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)

47 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

48 What is EOS calibration? Minimization of squared differences between experimental and predicted fluid properties Ndata i 1 g i predicted g exp erimental i 2 min These Properties (g i ) include: Densities, saturation pressures Relative amounts of gas and liquid phases Compositions, etc.

49 What is EOS calibration? Accomplished by changing within certain limits selected EOS parameters Minor adjustments (1 to 2%) of binary interaction parameters (k ij ) can change saturation pressures by 20 to 30% Different properties of the C 7+ 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.

50 Pre and post calibration predictions from an EOS

51 Pre and post calibration predictions from an EOS

52 Pre and post calibration predictions from an EOS

53 Pre and post calibration predictions from an EOS

54 Problems to Think About Determine the equilibrium ratio of C 1 from multiple flash calculations using SOPE. Select a mixture and a suitable pressure temperature range Discuss the trends, how does k C1 change with T at a fixed P? Discuss the trends, how does k C1 change with P at a fixed T? Provide well documented graphs

55 Problems to Think About Compare the equilibrium ratio of C 1 at 4000 psia and at 200 o F with that of the convergence pressure chart using. A mixture of C 1 and C 2 A mixture of C 1 and C 4 A mixture of C 1 and C 8 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

56 Problems to Think About

57 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 = 160 o F (not shown here)

58 Problems to Think About Select one EOS (Vdw, RK, SRK, PR, or Cubic-4G) Select one bubble point pressure for one composition of methane Plot p b predicted vs binary interaction parameter selected Select the best k ij that matches the bubble point pressure Compare the values of experimental vs. predicted molar volumes

59 Pressure, psia You should be obtaining a plot like this one Bubble Point Pressure C 1 -C 4 Mixture (10% C1) at T = o F Experimental p b is 339 psia You CANNOT use this same composition in Your homework k ij Cubic-4G P-R S-R-K R-K VDW

60 This is the end, we survived!!!