CORRESPONDENCE pubs.acs.org/IECR
Comment on “Molecular Simulation Prediction of Sound Velocity for a Binary Mixture near Miscible Conditions” Luigi Raimondi* Process Simulation Services, 20025 Legnano, Italy
S
ir: The speed of sound is an important property that enters into many areas of single and multiphase fluid flow simulations. In fluid dynamics, its value appears as a critical factor in the Courant Friedrich Levy parameter, which defines the limit of stability for the integration of the partial differential fluid flow equations, as can be found in any textbook on fluid dynamics. Another point where the sound velocity represent an important constraint is the discharge of fluid (vapor and/or liquid) from a pressurized vessel; in this case, its value enters into the calculation of the maximum flow from a pressure safety valve and becomes a critical parameter in process safety analysis. The measurement and evaluation of sound velocity is not straightforward mainly for mixtures near the critical points. For a given mixture, by moving in the temperature pressure (T P) space near the critical point, entering the vapor liquid coexistence region the speed of sound presents a rapid decrease, as linked to the divergence of the specific heat at constant volume to which it is related. A paper recently published in this journal by Fazelabdolabadi and Bahramian1 (FB) presented comparisons between calculated and experimental values of the speed of sound for three binary mixtures of methane and n-butane in the high-pressure region. These binary mixtures were selected by Plantier et al.2 as being representative of volatile oil and gascondensate reservoirs. The FB paper evaluates speed of sound values using a statistical thermodynamic approach (Monte Carlo simulation technique) and compares them with values calculated using the Peng Robinson (PR) cubic equation of state,3 which can be considered to be a standard tool in the simulation of oil and gas production from reservoirs. The reported large differences between calculated values of the sonic speed using the PR equation, with respect to experimental values, have raised this author’s attention as being questionable and probably incorrect. This doubt has been confirmed by recalculating their values with a process simulator developed by the author: the new calculated values are largely different from those presented by FB. This work presents updated values of the speed of sound evaluated using the PR equation of state and compares them with the original experimental values of Plantier et al.2 For validation, calculated values using the PR equation are also compared with those generated by other noncubic and more-complex equations of state.
’ CALCULATION AND SIMULATION METHODS The FB article presents values for the speed of sound of hydrocarbon mixtures calculated by molecular simulations and compares them with experimental and calculated values, using the PR equation of state, which is very common in the field of oil and gas engineering simulation. Latter values are presented without any detail about how they have been calculated or the r 2011 American Chemical Society
Figure 1. Bubble point (BP) and dew-point lines of the three mixtures calculated using the PR equation. The constant temperature value of 311 K is located by one vertical grid line. Cricondenbar (P), cricondentherm (T), and critical (C) points are shown.
simulation tool that was used; what is emphasized is the large difference between experimental and calculated values. The calculation of sonic velocity in multicomponent mixtures has been thoroughly developed by the author some years ago and applied mainly for the simulation of emergency fluid discharge from process equipment. The algorithms for its evaluation apply to both cubic equation of state (Soave Redlich Kwong and Peng Robinson) and noncubic equations. In a paper with extensive applications to the calculation of critical flow conditions resulting in discharging a fluid (gas, liquid, or mixed phase),4 the author has shown how a rigorous calculation of the sonic speed can be used to analyze the discharge of a highpressure fluid through a pressure safety device. The evaluation of sonic speed for both single- and two-phase fluids is presented using both an analytical approach, in the case of the PR equation, or a numerical methodology for more-complex thermodynamic equations, so it is worthwhile to repeat those details here. To verify the goodness of sound velocities values obtained with the PR equation, the same calculations have been repeated using two other more-complex equations: the Lee Kesler (LK) model5 and the GERG-2004 (GERG) model.6 The LK equation is a noncubic equation of state and describes fluids using a corresponding state approach; it predicts the vapor and liquid densities of light hydrocarbons, as well as their thermodynamic properties (enthalpy, entropy, and heat capacity), quite well. Published: September 02, 2011 11455
dx.doi.org/10.1021/ie201386e | Ind. Eng. Chem. Res. 2011, 50, 11455–11458
Industrial & Engineering Chemistry Research
CORRESPONDENCE
Table 1. Sonic Speed for the Methane (x = 0.158) and n-Butane Mixturea Sonic Speed, Calculated Sonic Speed, Exp pressure (MPa)
a
(m/s)
LK (m/s)
GERG rel err (%)
(m/s)
PR
rel err (%)
(m/s)
rel err (%)
13.79
906.3
901.0
0.58
887.6
2.06
730.5
19.40
10.342
866.0
859.7
0.73
848.6
2.01
688.5
20.50
6.895
817.6
813.6
0.49
805.5
1.48
643.4
21.31
6.205
805.6
803.7
0.24
796.3
1.15
633.9
21.31
5.516
793.5
793.5
0.00
786.8
0.84
624.3
21.32
4.826
778.7
782.9
0.54
777.1
0.21
614.4
21.10
4.137
762.3
772.1
1.29
767.1
0.63
604.4
20.71
3.999 3.93
760.7 760.1
769.9 768.8
1.21 1.14
765.1 764.1
0.58 0.53
602.4 601.4
20.81 20.88
Experimental data from Plantier et al.2
The GERG equation is a very complex thermodynamic model, explicit in the Helmholtz free energy, developed recently in a research project by the “Groupe Europeen de Recherches Gazieres”. This equation of state is an accurate and wide range mixture model for natural gases and other mixtures that is based on a multifluid approximation; it uses over 100 parameters to calculate the Helmholtz free energy of each pure compound and a similar number of interaction coefficients for each couple of compounds to describe the mixture free energy residual part. The GERG model is supposed to predict thermodynamic properties of mixtures within experimental error margins. An example, by the author, of the application of the GERG equation can be found in ref 7. All calculated values presented on the following tables and the phase envelopes of Figure 1 have been generated using the XPSIM process simulation tool,8 which contains the PR, LK, and GERG calculation methods.
’ COMPARISON OF CALCULATED AND EXPERIMENTAL DATA Three binary mixture of methane and n-butane, with increasing methane content (molar fractions of 0.158, 0.724, and 0.894), are considered and analyzed at a constant temperature of 311 K but at different pressure ranges. To clarify the PVT behavior of the mixtures and the relative “position” on a pressure temperature diagram, their phase envelopes have been calculated and are shown in Figure 1. Mixture 1. The first mixture considered has a methane molar fraction of 0.158 and has been analyzed in the pressure range between 14 and 4 MPa. In this pressure region, the fluid is liquid and can be representative of a volatile oil (from the point of view of reservoir fluid classification). The critical point temperature is 414.48 K, and the pressure is 5.2422 MPa. Therefore, the mixture is depressurized at a reduced temperature of 0.75. The initial pressure is 2.67, and the final reduced pressure is 0.76. Experimental values of the sound speed vary from ∼900 m/s, at the high-pressure limit, to 760 m/s at low pressure. The PR values of the speed of sound calculated by FB, as shown by their paper,1 are all above the experimental values, with a difference of >150 m/s at the lower pressure side. The values calculated by our simulations are presented in Table 1; PR values are quite distant from experimental values, showing an average error of approximately 20%, but all of the calculated values are
below those of the experimental data. It is interesting to note that values calculated using the LK and GERG equations are similar to the experimental values, showing an average deviation of ∼1%. The large difference with the values calculated using the PR equation of state can be attributed to its insufficient description of the liquid-phase volumetric properties. Mixture 2. This mixture has a methane molar fraction of 0.724, and it is analyzed at a temperature of 311 K in the pressure range between 17.2 and 13.1 MPa. According to the experimental data of Sage9 (also reproduced in a Chemistry Data Series Dechema volume10), it is expected that the mixture critical point is traversed at 13.183 MPa. For this mixture, FB reported a molecular simulation deviation of ∼5% and a minimum deviation of 54.2% for the PR values. The new calculated values are reported in Table 2 and require some discussion that also could include experimental data from Plantier et al.2 In the singlephase region, PR values are quite near the experimental data (2% 5% error) and similar to those calculated with the LK and GERG models. When the two-phase boundary is crossed, the calculated speed of sound decreases almost discontinuously; the PR error increases sharply to an average value of 33%, remaining however consistent with LK and GERG calculated values. Experimental data show a rapid decrease of the sound velocity as the critical point is approached, but there is no evidence of any discontinuity when the two-phase boundary is crossed. This different behavior can be easily explained by noticing that the calculated critical point of the mixture is at some higher pressure and temperature (specifically, at 316.2 K and 13.626 MPa, respectively). Mixture 3. The third mixture has a methane content of 0.894 and is analyzed in a pressure range between 17 and 2 MPa. In this pressure region, the fluid is vapor and can be representative of a gas condensate, from the reservoir fluid classification. The experimental values of the speed of sound change from ∼438 m/s, at high pressure, to ∼381 m/s at low pressure and show a minimum value of ∼365 m/s at ∼7 MPa. An average error of 17.5% is reported by FB for the PR calculated values. Values calculated by our simulations are presented in Table 3. In this case, PR values are very similar to the experimental ones, with an average error of
