THE DETERMINATION OF CRITICAL FLOW FACTORS FOR NATURAL GAS MIXTURES

Transcription

1 A REPORT ON THE DETERMINATION OF CRITICAL FLOW FACTORS FOR NATURAL GAS MIXTURES Part 1: The Thermodynamic Properties of the Components of Natural Gas and of Natural Gas Mixtures in the Ideal-Gas State FOR National Measurement System Policy Unit Department of Trade & Industry, London Report No: 72/96 12 December 1996

2 Density Flow Centre Report on THE DETERMINATION OF CRITICAL FLOW FACTORS FOR NATURAL GAS MIXTURES Part 1: The Thermodynamic Properties of the Components of Natural Gas and of Natural Gas Mixtures in the Ideal-Gas State FOR National Measurement System Policy Unit Department of Trade & Industry, London S U M M A R Y This is the first of three reports describing the work on the calculation of C* for natural gas mixtures. The most accurate and wide-ranging equation of state for use with such mixtures can be used only to determine the mechanical properties. A knowledge of the caloric properties is also required in the calculation of the critical flow factor. This report describes the development of the necessary equations and software for determining the properties of natural gas mixtures in the ideal-gas state. Prepared by: Mr J T R Watson... Approved by: Mr J T R Watson... Date: 12 December 1996 for W Paton Director and General Manager Report No: 72/96 Page 1 of 14

3 C O N T E N T S SUMMARY INTRODUCTION THE CALCULATION OF CRITICAL FLOW FACTORS HELMHOLTZ ENERGY 3.1 The Helmholtz Energy of a Real Fluid The Helmholtz Energy in Terms of the Ideal-Gas Heat Capacity The Helmholtz Energy of a Mixture of Real Fluids IDEAL-GAS PROPERTIES 4.1 Review of Available Data for the Isobaric Heat Capacity of Natural Gas Components Functional Form of the Ideal-Gas Heat Capacity Equation Calculation of the Thermodynamic Properties of the Components of a Natural Gas Mixture in the Ideal-Gas State Calculation of the Thermodynamic Properties of a Natural Gas Mixture in the Ideal-Gas State SOFTWARE IMPLEMENTATION REFERENCES LIST OF TABLES Page Report No: 72/96 Page 2 of 14

4 1 INTRODUCTION The objective of this project is to derive reliable values for the critical flow factor, C*, of natural gas mixtures. The programme of work involves the extension of the American Gas Association s equation of state, AGA8, to include the calculation of caloric property data. It is envisaged that at least three reports will be prepared on this topic. These are:- Part 1 Part 2 Part 3 Thermodynamic properties of the components of natural gas in the ideal-gas state; The real gas contribution to the thermodynamic properties of natural gas mixtures; and The calculation of C* for natural gas mixtures. 2 THE CALCULATION OF CRITICAL FLOW FACTORS In recent work at NEL, accurate values of C* have been derived for a number of gases (argon, dry-air, carbon dioxide, methane and nitrogen) using the latest reference-quality formulations for the thermodynamic properties of these fluids. Use of these derived values of C*, which have a significantly lower systematic error than existing literature values, will result in more consistent and reliable meter calibrations. The most accurate and wide-ranging equation of state for use with natural gas mixtures, the AGA8 equation, can be used to determine the mechanical properties (such as density, compressibility etc) but is not of a form suitable for calculating the caloric properties of such mixtures (enthalpy, entropy, speed of sound, isentropic exponent etc). A knowledge of the caloric properties is essential, however, to the calculation of the critical flow factor. Modern formulations for the thermodynamic properties of fluids are expressed as single fundamental equations in terms of the Helmholtz energy of the fluid or fluid mixture. A fundamental equation of this form allows both the mechanical and caloric properties of the fluid to be accurately represented and enables all properties of engineering importance to be derived through differentiation. For consistency with our previous work on critical-flow factors it is necessary to develop a Helmholtz energy formulation for natural gas mixtures which combines the mechanical properties of AGA8 and the caloric properties of the ideal-gas state. This report describes the development of the necessary equations for the properties of natural gas mixtures in the ideal-gas state. Report No: 72/96 Page 3 of 14

5 3 HELMHOLTZ ENERGY 3.1 The Helmholtz Energy of a Real Fluid The Helmholtz energy of a fluid, a function of both density and temperature, is defined as: A( ρ, T) = h( ρ, T) - T.s( ρ, T) - p/ρ, (3.1.1) where ρ T h s p is the molar density, is the absolute temperature, is the molar enthalpy, is the molar entropy, and is the absolute pressure. From thermodynamics we have that p = ρ 2.( A/ ρ) T. On integration of this expression with respect to density at constant temperature we obtain: A( ρ, T) = RT. (p/ ρ 2 ).dρ, where R is the gas constant, and the integration is over the density interval from 0 to ρ. If we assume that the fluid pressure, p, has the functional form: p = p( ρ, T) = ρrt.[1 + φ( ρ, T)], (3.1.2) where φ a function of both density and temperature, is the real fluid contribution to the equation of state, then the Helmholtz energy of a real fluid can be expressed as: A( ρ, T) = RT. (1/ ρ).[1 + φ( ρ, T)].dρ, (3.1.3) where the integration is over the density interval from 0 to ρ. Integration of equation (3.1.3) with these prescribed limits would result in a singularity in the resultant ln( ρ ) term. To avoid this we express the Helmholtz energy of the real fluid relative to that of the fluid in the ideal gas state at the same temperature, T, but at some arbitrary density, ρ ref. The resulting expression is then: A( ρ, T) - A id ( ρ ref, T) = RT. (1/ ρ).[1 + φ( ρ, T)].dρ - RT. (1/ρ).dρ, (3.1.4) where the first integral is for the real fluid in the density interval from 0 to ρ, and the second integral is for the ideal gas in the density interval from 0 to ρ ref. Report No: 72/96 Page 4 of 14

6 Re-arranging equation (3.1.4), we obtain: A( ρ, T) - A id ( ρ ref, T) = RT. (1/ ρ).φ( ρ, T).dρ + RT. (1/ ρ ).dρ, (3.1.5) where the first integral is still from 0 to ρ but the second is now from ρ ref to ρ. Integrating the second term in equation (3.1.5) and rearranging gives: A( ρ, T) = A id ( ρ ref, T) + RT.ln( ρ/ρ ref ) + RT. (1/ ρ).φ( ρ, T).dρ. (3.1.6) To determine the Helmholtz energy of the fluid in the ideal-gas state we can substitute the ideal gas equation, p = ρ.rt, in equation (3.1.1). The Helmholtz energy of the fluid in the ideal-gas state is then: A id ( ρ, T) = h id ( ρ, T) - T.s id ( ρ, T) - RT. (3.1.7) By applying the thermodynamic relation, ( h/ T) p = c p,, where c p is the isobaric heat capacity, and integrating for enthalpy with respect to temperature at some constant pressure, p ref say, in the temperature interval from T ref to T, we obtain: h id ( p ref, T) = h id ( p ref, T ref ) + R. [c p,id (T)/R].dT. Since ( h/ p) T = v - T.( v/ T) p = 0 for an ideal gas, where v is the molar volume, the enthalpy of the ideal gas is independent of both pressure and density. Introducing this simplification, the above expression for the enthalpy of an ideal gas can be written: h id ( T) = h id ( T ref ) + R. [c p,id (T)/R].dT. (3.1.8) Also, by applying the thermodynamic relation, T.( S/ T) v = c v = c p - R, where c v is the isochoric heat capacity, and integrating for entropy with respect to T at some constant arbitrary density, ρ ref say, where the integration is again from T ref to T, we obtain: s id ( ρ ref, T) = s id ( ρ ref, T ref ) + R. [c p,id (T)/RT].dT - R.ln( T ). (3.1.9) Substituting equations (3.1.8) and (3.1.9) into equation (3.1.7), we obtain the following expression for the Helmholtz energy of the fluid in the ideal-gas state at density ρ ref and temperature T: A id ( ρ ref, T) = [ h id ( T ref ) + R. [c p,id (T)/R].dT ] - T.[ s id ( ρ ref, T ref ) + R. [c p,id (T)/RT].dT - R.ln( T ) ] - RT. (3.1.10) The full expression for the Helmholtz energy of a real fluid at density, ρ, and temperature, T, is obtained by combining equations (3.1.6) and (3.1.10), namely: Report No: 72/96 Page 5 of 14

7 A( ρ, T) = [ h id ( T ref ) + R. [c p,id (T)/R].dT ] - T.[ s id ( ρ ref, T ref ) + R. [c p,id (T)/RT].dT - R.ln( T ) ] - RT + RT.ln( ρ/ρ ref ) + RT. (1/ ρ).φ( ρ, T).dρ]. (3.1.11) Expressing the total Helmholtz energy of the fluid in reduced, or dimensionless, terms, we have: A( ρ, T)/RT = A id ( T)/RT + ln( ρ/ρ ref ) + A real ( ρ, T)/RT, (3.1.12) where A id ( T) A real ( ρ, T) a function of temperature only, is the ideal-gas contribution to the Helmholtz energy; a function of both density and temperature, is the real fluid contribution to the Helmholtz energy; A id ( T)/RT = [ h id ( T ref )/RT + (1/T). [c p,id (T)/R].dT ] - [ s id ( ρ ref, T ref )/R + [c p,id (T)/RT].dT - ln( T/T ref ) ] - 1; (3.1.13) A real ( ρ, T)/RT = (1/ ρ).φ( ρ, T).dρ], and (3.1.14) the two temperature integrals in equation (3.1.13) are taken over the interval from T ref to T; and the density integral in equation (3.1.14) is taken over the interval from 0 to ρ. 3.2 The Helmholtz Energy in Terms of the Ideal-Gas Heat Capacity The reduced ideal gas heat capacity, c p,id /R, is normally represented in the form: c p,id /R = α + β.ψ(t), (3.2.1) where the first term, the constant α, relates to the translational contribution to the isobaric heat capacity; and the second, temperature dependent, term relates to the combined rotational, vibrational and electronic contributions to the same. Substituting for the reduced isobaric heat capacity, c p,id /R, in equation (3.1.13) we obtain: A id ( T)/RT = [ h id ( T ref )/RT + α + (β/t). ψ(t).dt ] - [ s id ( ρ ref, T ref )/R + (α - 1).ln( T/T ref ) + β. [ψ(t)/t].dt ] - 1. (3.2.2) The terms in the bold squared brackets in this equation are the enthalpy and entropy contributions, respectively, to the Helmholtz energy of the ideal gas. By convention the enthalpy and the entropy of the ideal gas are both taken to be zero at K and either MPa or, more recently, at 0.1 MPa. Thus: h id ( T) = 0, at T ref = K, and s id ( ρ ref, T) = 0, at T ref = K and ρ ref = p ref /RT ref. Report No: 72/96 Page 6 of 14

8 In practice this is achieved by setting appropriate values for the constants h id ( T ref ) and s id ( ρ ref, T ref ). The final expression for the reduced Helmholtz energy, in terms of the equation of state for the real fluid and the isobaric heat capacity for the fluid in the ideal gas state is of the form: A( ρ, T)/RT = [ h id ( T ref )/RT + α + (β/t). ψ(t).dt ] - [ s id ( ρ ref, T ref )/R + (α - 1).ln( T/T ref ) + β. [ψ(t)/t].dt ] ln( ρ/ρ ref ) + (1/ ρ).φ( ρ, T).dρ], (3.2.3) where the two temperature integrals are taken over the interval from T ref to T; and the density integral is taken over the interval from 0 to ρ. 3.3 The Helmholtz Energy of a Mixture of Real Fluids The Helmholtz energy of a mixture of N fluids in the ideal-gas state differs from the mole fraction average of the individual contributions, A id,i ( T), by an amount which accounts for the entropy of mixing. The reduced Helmholtz energy of a mixture, A id,mix /RT, of N ideal-gas components is thus: A id,mix ( T, x)/rt = x i.{a id,i ( T)/RT + ln( x i ) }, (3.3.1) where x x i is the set of mixture compositions expressed in mole fractions, and is the mole fraction of the i th component of the mixture, and the summation is for i from 1 to N. The total reduced Helmholtz energy of a mixture of N real fluids is given by: A mix ( ρ, T, x)/rt = x i.{a id,i ( T)/RT + ln( x i ) } + ln( ρ/ρ ref ) + A real ( ρ, T, x)/rt, (3.3.2) where A real ( ρ, T, x) is the real fluid contribution to the Helmholtz energy of the mixture at a defined molar density, temperature and composition. 4 IDEAL-GAS PROPERTIES 4.1 Review of Available Data for the Isobaric Heat Capacity of Natural Gas Components A major review of the ideal-gas heat-capacity data for twenty-one components of natural gas mixtures has recently been published by Jaeschke and Schley [1] of Ruhrgas. The data collected by these authors was represented as a function of absolute temperature using an extension of a functional form proposed by Aly and Lee [2]. These high-quality representative equations are based on a critical examination of the available data for each gas. Temperatures are expressed in terms of the ITS-90 temperature scale and the representations are applicable over a wide range of temperatures up to 1000 K. Jaeschke and Schley assigned uncertainty limits on the values of ideal-gas heat capacity derived from their formulations. Report No: 72/96 Page 7 of 14

9 Wagner and co-workers [3, 4, 5, 6] at Bochum University have recently developed high-quality formulations for the thermodynamic properties of argon, carbon dioxide, methane and nitrogen over wide ranges of both pressure and temperature. Accurate formulations were developed for the representation of the ideal-gas heat capacity data also in terms of ITS-90 temperatures. Values of ideal-gas heat capacity derived from the Jaeschke and Schley and the Wagner formulations are in very close agreement and well within the combined uncertainty limits for all four fluids. For convenience Jaeschke and Schley s recommendations for the ideal-gas heat capacities of all twenty-one natural gas components were accepted for use in this work. The lower temperature limits and the assessed uncertainties in the isobaric heat capacities, c p, for each gas are as follows: Gas Lower temperature limit / K Uncertainty in c p / % methane nitrogen carbon dioxide ethane to 0.3 propane to 0.5 water to 0.05 hydrogen sulphide to 0.5 normal hydrogen to 0.1 carbon monoxide oxygen iso-butane to 0.2 n-butane to 1.0 iso-pentane to 1.0 n-pentane to 1.0 n-hexane to 1.0 n-heptane to 1.0 n-octane to 1.0 n-nonane to 1.0 n-decane to 1.0 helium argon Functional Form of the Ideal-Gas Heat Capacity Equation The functional form of equation used for the representation of ideal-gas heat capacity data by Jaeschke and Schley is based on an extended form of equation originally proposed by Aly and Lee [2], namely: Report No: 72/96 Page 8 of 14

10 c p,id (T)/R = b + c.[(d/t)/sinh(d/t)] 2 + e.[(f/t)/cosh(f/t)] g.[(h/t)/sinh(h/t)] 2 + i.[(j/t)/cosh(j/t)] 2, (4.2.1) where c p,id is the ideal-gas heat capacity; T is the temperature on the ITS-90 temperature scale; R is the gas constant; and b, c, d, etc. are the coefficients of the equation for the individual gases as given in Table Calculation of the Thermodynamic Properties of the Components of a Natural Gas Mixture in the Ideal-Gas State From equation (3.1.13) we have that the reduced Helmholtz energy of each fluid in the idealgas state can be expressed in terms of its reduced heat capacity, c p,id (T)/R, by: A id ( T)/RT = [ h id ( T ref )/RT + (1/T). [c p,id (T)/R].dT ] - [ s id ( ρ ref, T ref )/R + [c p,id (T)/RT].dT - ln( T/T ref ) ] - 1; (4.3.1) where both integrations are over the temperature interval T ref to T. The contributions to A id ( T)/RT from enthalpy and entropy, H id /RT and S id /R, respectively, are: H id /RT = [ h id ( T ref )/RT + (1/T). [c p,id (T)/R].dT ], and (4.3.2) S id /R = [ s id ( ρ ref, T ref )/R + [c p,id (T)/RT].dT - ln( T/T ref ) ]. (4.3.3) Substituting equation (4.2.1) in equations (4.3.2) and (4.3.3) we obtain on integration: H id /RT = [h(t ref )/RT] + b + c.(d/t).coth(d/t) - e.(f/t).tanh(f/t) + g.(h/t).coth(h/t) - i.(j/t).tanh(j/t), and (4.3.4) S id /R = s( T ref, ρ ref )/R+ b.ln(t) + c.[(d/t).coth(d/t) - ln(sinh(d/t))] - e.[(f/t).tanh(f/t) - ln(cosh(f/t))] + g.[(h/t).coth(h/t) - ln(sinh(h/t))] - i.[(j/t).tanh(j/t) - ln(cosh(j/t))], (4.3.5) where h( T ref ) and s( T ref, ρ ref ) are the constants of the integration. The calculated values of the integration constants for the twenty-one component gases are given in Table 2. Table 3 contains other physical parameters of the component fluids. Report No: 72/96 Page 9 of 14

11 4.4 Calculation of the Thermodynamic Properties of a Natural Gas Mixture in the Ideal-Gas State The Helmholtz energy of a mixture of N components in the ideal-gas state as given by equations (3.3.1), (4.3.2) and (4.3.3) is: A id,mix ( T, x)/rt = x i.{h id,i /RT - S id,i /R ln( x i ) }, (4.4.1) where x x i H id,i /RT S id,i /R is the set of mixture compositions expressed in mole fractions, is the mole fraction of the i th component of the mixture, the summation is for i from 1 to N, is given by equation (4.3.4) for each of the i components, and is given by equation (4.3.5) for each of the i components. 5 SOFTWARE IMPLEMENTATION Fortran software has been developed to evaluate the Helmholtz energy of mixtures of natural gas components in the ideal-gas state based on the equations given in this report. Ideal-gas values have been derived from the software for four of the components and validated against known values for argon, carbon dioxide, methane and nitrogen. Ideal-gas values have also been derived from the software for a dry-air mixture and validated against known values [7]. REFERENCES 1 JAESCHKE, M. and SCHLEY, P. Ideal Gas Thermodynamic Properties for Natural Gas Applications: Paper presented at the Twelfth Symposium on thermophyscial Properties, June 1994, Boulder, Colorado, USA. 2 ALY, F. A. and LEE, L. L. Self-consistent Equations for Calculating the Ideal Gas Heat Capacity, Enthalpy and Entropy. Fluid Phase Equilibria, 1981, 6, SETZMANN, U. and WAGNER, W. A new equation of state and tables of thermodynamic properties of methane covering the range from the melting line to 625 K at pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 1991, 20(6), SPAN, R., LEMMON, E., JACOBSEN, R. T. and WAGNER, W. Private communication; to be published in J. Phys. Chem. Ref. Datat [Nitrogen]. 5 TEGELER, Ch. and WAGNER, W. Private communication; to be published in J. Phys. Chem. Ref. Data [Argon]. 6 SPAN, R. and WAGNER, W. Private communication; to be published in J. Phys. Chem. Ref. Data [Carbon Dioxide]. 7 JACOBSEN, R. T., PENONCELLO, S. G., BEYERLEIN, S. W., CLARKE, W. P. and LEMMON, E. W. A thermodynamic property forumulation for air. Fluid Phase Equilibria, 1992, 79, Report No: 72/96 Page 10 of 14

12 LIST OF TABLES 1 Coefficients of Jaeschke and Schley s Ideal-Gas Heat Capacity Equation 2 Reference Values of Enthalpy and Entropy for the Ideal-Gas State at K 3 Physical Parameters for the Components of Natural Gas Mixtures. Report No: 72/96 Page 11 of 14

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