New correlation for hydrogen-natural gas mixture compressibility factor

Transcription

1 New correlation for hydrogen-natural gas mixture compressibility factor Zahreddine Hafsi 1, Sami Elaoud 2, Mohsen Akrout, Ezzeddine Hadj Taïeb Laboratory of Applied Fluids Mechanics Process and Environment Engineering, ENIS P.O. Box, W, Sfax, 3038, Tunisia 1 hafsi.zahreddine@gmail.com 2 elaoudsa@yahoo.fr Abstract. In this paper, new correlation of hydrogen-natural gas mixture compressibility factor will be built. Based on Soave Redlich-Kwong (SRK) real gases equation of state (EOS) and under an isothermal condition, the evolution of gas compressibility factor will be followed up and compared to experimental data. The established correlation will be used to solve equation of motions in steady state and to evaluate the pressure drop through hydrogen natural-gas mixture pipeline. Keywords: SRK equation of state, compressibility factor, hydrogen-natural gas mixture, linear regression. 1 Introduction The pressure-volume-temperature (PVT) behavior of gases was the subject of numerous researches along the four previous centuries. In 1845, Victor Regnault and by applying Avogadro s hypothesis on the volume occupied by one mole of an ideal gas, introduced the general equation of state on ideal gases, where is the number of moles of gas and is the universal gas constant. This relationship is the general equation that governs the PVT behavior of all substances that behave as ideal gases. In 1873, Johannes Diderik Van der Waals proposed correction terms to the Regnault ideal gas equation ( ). He added additional terms to both the (pressure) and

2 2 Z. Hafsi et al. (volume) variables in the ideal gas equation. these new correction terms introduced the effect of molecular attraction forces on the pressure term and took into account the size of real molecules on the volume term, in that way, Van der Waals replaced the pressure in ideal gas equation with ( and the volume with ( ), so that his equation of state was written under the form: (1) where the terms and described, respectively, the molecular interaction forces and the actual volume occupied by the gas molecule. Inspired from Van der Waals efforts, a large number of equations of state was constructed and several modifications were made to the original VDW equation, one can cite Pitzer EOS (1955), Beattie-Bridgeman EOS (1927), Benedict-Webb- Rubin (BWR) EOS (1940) and Redlich-Kwong EOS (1949). In 1955, Pitzer introduced the concept of acentric factor. This presents a measure of the configuration and sphericity of the molecule. In 1972, Giorgio Soave modified the Redlich-Kwong equation and introduced what known as SRK EOS. In this paper, the SRK EOS will be the basis to establish a linear correlation of the compressibility factor evolution in binary gas mixture. The established factor expression will be used to integrate motions equation and to develop an expression of the pressure drop through pipeline of gas mixture. 2 SRK equation of state For computing purposes, a correction factor is introduced into the ideal gas EOS of state so, for real gases, the equation of state in its general form is written: (2) Where is the pressure, is the molar volume of the gas, is the compressibility factor is the universal gas constant and is the temperature The major limitation of the above equation is that the gas compressibility factor is not a constant but it s a function of the pressure and the temperature of the considered gas, it was usually determined experimentally. However many correlations were established to evaluate Z with direct relations (Papp Correlation,

3 New correlation for hydrogen-natural gas mixture compressibility factor ; Beggs and Brill correlation, 2005) or iterative relations (Hall-Yarborough Correlation, 1973; Dranchuk-Abu-Kassem Correlation, 1975) or even by considering a pseudo-constant Z factor in isothermal condition (Tabkhi, 2011). Added to that, the SRK equation of state is written as follows (Tester and Modell, 1996; Prausnitz et al., 1998; Sandler, 1999): (3) where (4) (5) and are respectively the critical temperature and the critical pressure of the considered gas, is the reduced temperature defined by: (6) and where is Pitzer s acentric factor defined as follows (7) (8) is the saturated vapor pressure of the gas at Equation (2) can be written as follows (9) Equation (9) combined with equation (3) (the SRK EOS) gives: (10) Equation (10) is a 3 rd order equation in term of, its development leads to: (11) The cubic equation (11) was numerically solved and the evolution of the compressibility factor versus pressure for different temperature values was compared to curves issued from experimental data. Figure 1 shows a good concordance between numerical and experimental results for methane compressibility factors mainly for medium and high pressure range.

4 4 Z. Hafsi et al. Figure 1. Methane compressibility factor evolution (Experimental data source: Jeffrey L. Savidge, 2000) In the following section, and using Matlab software for solving equation (11), a new correlation of factor expression for binary gas mixture will be established. 3 Compressibility factor of hydrogen gas mixture in isothermal condition For pure substances, SRK EOS parameters and are determined by using the critical properties and the acentric factor. For mixtures, the parameters are expressed through defined mixing rules. Most of the applications of EOS to mixtures used the classical Van der Waals mixing rules which give accurate results [Adachi and Sugie, 1985; Trebble, 1988; Shibata and Sandler, 1989]. The classical VDW mixing rules are given by the following expressions: and (12) (13) where is the mole fraction of the component, and are parameters corresponding to pure component while and are called the unlike-interaction parameters. Customarily, the geometric mean is used for the force parameter arithmetic mean is used for the volume parameter : while the

5 New correlation for hydrogen-natural gas mixture compressibility factor 5 Taking into consideration equation (15), the expression of following one-summation simple form: For a binary mixture, using equations (12) and (16), one obtains: (14) (15) will be reduced to the (16) (17) (18) Considering a binary mixture of hydrogen and natural gas and, Table 1 shows the physical properties and the SRK EOS parameters and of hydrogen and natural gas calculated using equations (4) and (5) in a temperature. Let Table 1. Physical properties and SRK EOS parameters of hydrogen and natural gas Hydrogen (H2) Methane(CH4) Molecular weight ( Critical temperature ( Critical pressure ( Acentric factor ( be the mole fraction of hydrogen in the mixture, applying the simple mixing rules given by equations (17) and (18) for different mole fractions, the values of the SRK EOS parameters and of the binary mixture are reported in table 2. Unit Table 2. SRK EOS coefficients values for the mixture (T=288K) Considering the calculated SRK EOS parameters of the mixture in an isothermal condition ( ) and in a range of pressure, equation (11) was numerically solved and the evolution of the compressibility factor versus pressure for different values is shown in figure 2.

6 Compressibility factor Compressibility factor Compressibility factor Compressibility factor 6 Z. Hafsi et al Actual variation Linear approximation 0.05 Linear: norm of residuals = T=288K, y= Pressure(bar) Actual variation Linear approximation 0.05 Linear: norm of residuals = T=288K, y= Pressure(bar) Linear: norm of residuals = Actual variation 0.96 T=288K, y= Linear approximation Pressure(bar) Linear: norm of residuals = Actual variation T=288K, y=1 Linear approximation Pressure(bar) Figure 2: Linear regression of the compressibility factor evolution 0 The above curves show that, in the considered range of pressure, a linear approximation of factor s isothermal evolution is acceptable. Actually, a linear regression is less accurate than a quadratic or a cubic one but it will be very useful to analytically integrate motions equation in steady state. Then is written: (19) where and depend on the temperature and the mole proportions of the gas mixture components. Table 3 shows the calculated coefficients of equation (19) at and for different mole fractions of hydrogen in the mixture.

7 New correlation for hydrogen-natural gas mixture compressibility factor 7 Table 3. Coefficients of the linear approximation Steady state analysis The mass conservation and momentum laws, applied to an element of fluid between two sections of abscissa and of an horizontal pipe, lead to the following equations that describe one-dimensional adiabatic compressible gas flow (Wylie et al, 1993): (20) (21) where is the coefficient of friction, is the velocity of the gas, is its density, is its pressure and Mass flow rate is the pipe internal diameter., also called pipe throughput, is expressed as function of the density, the gas velocity and the cross section area of the pipe : (22) Using equation (9), the gas density can be expressed, by introducing the compressibility factor and the average molecular mass of the gas, as follows: (23) For gas mixture, can be calculated using the simple mixing rule (Kay s rule): (24) Where s are the molecular masses of species. Considering equation (23), equations (21) and (22) can be, rearranged in the basis of mass flow rate and pressure, expressed as follows: (25) In the case of the steady state ( ), equations (25) and (26) become: (26)

8 8 Z. Hafsi et al. (27) (28) Considering equation (27), equation (28) is written: (29) Taking into account equation (19), in isothermal condition it comes: (30) Which leads to: (31) Denoting, (32) It can be written then: (33) Integrating equation (33) between ( ) and ( ), one obtains (34) The friction factor is given by Colebrook-White equation (Giles 1962; Fox 1992; Shames 1989; White 1994): (35) where is the internal roughness of the pipe and is the Reynolds number. For fully turbulent flow or a rough turbulent flow (, the second term of equation (35) can be neglected and it can be, simply, written: (36)

9 Pressure (Pa) pressure (Pa) New correlation for hydrogen-natural gas mixture compressibility factor 9 5 Case study The studied case is an installation composed of a compressor pumping the hydrogen-natural gas mixture through a steel Sch80 pipeline ( ).as shown in figure 3. The pipe characteristics and the friction factor calculated using equation (36) are reported in table 4. Table 4. Tubular pipes material properties: steel Sch 80 pipe (ANSI standards) Figure 3: Gas mixture installation An entry pressure Designation Value Unit Internal roughness ( ) Nominal Pipe Size ( ) 500 Internal diameter ( ) Friction factor ( ) is applied in the upstream side with a mass flow rate of the gas mixture. Figure 4 shows a comparison of the pressure drop along the pipeline between the developed model and the model that considers the compressibility factor as a constant (Tabkhi, 2011). The comparison between these two models is considered for natural gas ( ). In the model of (Tabkhi, 2011), the compressibility factor is a function of the critical properties of the gas mixture, average pressure of the pipe segment and the temperature that has been considered as constant (Mohring et al., 2004): (37) (38) x 106 Our model Traditional model x x T=288 K y=0 y=0 y=0.5 y= x(m) x(m) Figure 4. Comparison between models Figure 5: Pressure drop through the duct

10 10 Z. Hafsi et al. According to figure 4, it can be noted that assuming a constant compressibility factor may be sufficient in short pipes but in long and relatively long ducts, the factor variations have to be taken into account. In fact, the latter influence the pressure drop value in steady state which will affect the transient regime behavior. Figure 5 illustrates the evolution of pressure through the pipe for different values of hydrogen mole fractions. It shows that the injection of hydrogen in the natural gas duct increases the pressure drop value. 6 Conclusion In this study, a linear approximation of hydrogen-natural gas mixture compressibility factor evolution was carried out and validated. Thus, a new expression of pressure drop through gas duct was established and used to evaluate the pressure evolution in a gas mixture installation for different mole fractions. The compressibility factor approximation was based on resolution of cubic equation derived from SRK EOS in isothermal condition using Matlab software. A linear regression using Matlab fitting curves was adopted and validated. References Guo B, Ghalambor A (2005). Natural Gas Engineering Handbook, Gulf Publishing Company. Yarborough L, Hall K R. (1973). Oil and Gas Dranchuk P M, Abou-Kassem (1975) J H.J Can Petrol Tech. Sandler, S.I. (1999). Chemical and Engineering Thermodynamics, Wiley USA. Jeffrey L. Savidge (2000).The first requirement of gas measurement is accuracy. D.A.Tefankjian ISHM. Lawrence Reid Award Recipient. Tester, J.W and M. Modell (1996). Thermodynamics and Its Applications, 3rd edition. Prentice Hall. Englewood Cliffs, USA. Fox RW, McDonald AT. (1992) Introduction to Fluid Mechanics, Wiley,USA Shames IH. Mechanics of Fluids, Singapore, McGraw-Hill Book Co, 1989:692.