Wikipedia [1] describes dry ice as “the solid form of carbon dioxide (CO₂). It is colorless, odorless, non-flammable, and slightly acidic [2]. At temperatures below −69.9°F (−56.6°C) and pressures below 75.2 psia (518 kPa), the triple point, CO₂ changes from a solid to a gas with no intervening liquid form, through a process called sublimation. The opposite process is called deposition, where CO₂ changes from the gas to solid  phase (dry ice). At atmospheric pressure, sublimation/ deposition occurs at  −109.3°F (−78.6°C). The density of dry ice varies, but usually ranges between about 87 and 100 lb_m/ft³ (1400–1600 kg/m³) [3]. The low temperature and direct sublimation to a gas makes dry ice an effective coolant, since it is colder than water ice and leaves no residue as it changes state [4]. Its enthalpy of sublimation is 245.5 Btu/lb_m (571 kJ/kg).”

While dry ice has many good features and applications, its formation can plug up equipment and cause severe operational problems in gas processing plants. Therefore, accurate predictions of conditions for dry ice formation are required. In order to prevent dry ice formation, a good knowledge and understanding of phase behavior of systems containing carbon dioxide are essential in cryogenic gas processing as in turboexpander plants for deep natural gas liquid (NGL) recovery. Thermodynamic modeling based on the equality of chemical potentials for each component in all phases and application of an equation of state with tuned parameters is normally used for accurate prediction of dry ice formation conditions.

In this tip of the month (TOTM), we will study the phase behavior of gas mixtures containing carbon dioxide. A description of phase behavior at different conditions of pressure and temperature is presented.

The Peng-Robinson (PR) [5] equation of state (EOS) option of ProMax [6] was used to perform all of the calculations in this study. In dealing with dry ice, reference [7] discusses the importance of using the right tools in process simulation software. The same reference also demonstrates the accuracy of ProMax against experimental data, including GPA RR 10 experimental data [8], for prediction of dry ice formation at different conditions.

Case Studies:

The composition of the two mixtures containing CO₂ considered in this study is shown in Table 1. Figure 1 also presents a simplified process flow diagram that was used to study dry ice formation in this study.  The feed gas (stream 1) enters Sep-100 from which the vapor stream (stream 2) is cooled in HEX-100. The stream leaving this cooler is passed through Sep-101 for separation of gas and liquid.

Figure 2 presents a complete phase envelope for mixture A (see Table 1) in which the state of each region has been identified.

The feed gas (stream 1) enters Sep-100 at -96˚F and 300 psia (-71.1˚F and 2069 kPa) which is point “A” on Figure 2. At this condition, it is all vapor and all of the feed leaves the separator as vapor. In the HEX-100, the vapor stream (stream 2) is cooled at constant pressure to -160˚F (-106.7˚C), which is represented by point “E” (stream 4). The horizontal dotted straight line identifies the cooling path. During the cooling process when point “B”, the dew point, on Figure 2 is reached, the first drop of liquid is formed. Between points “B and C”, mixture of liquid + vapor coexist at equilibrium. At point C, the incipient point of dry ice, solid phase will also form. Between points “C and D”, three phases of solid + liquid + vapor will coexist at equilibrium. Further cooling to point “E” results in a mixture of solid + liquid at equilibrium. Finally, the stream leaving this cooler is passed through Sep-101 for separation of any gas from and liquid.

Table 1. The composition of the two mixture studies

Figure 1. A simplified process diagram for the case study

 If mixture A enters the cooler at a pressure less than 167 psia (1152 kPa) and cools down, it will form dry ice without forming any liquid. As an example, let’s  assume the mixture is at     -100˚F and 100 psia (-73˚C and 690 kPa), point “x” on Figure 2. If this gas is cooled at constant pressure of 100 psia (690 kPa), it forms dry ice at a temperature of about -133˚F (-92˚C). Further cooling below about -137˚F (-94˚C) will form solid + liquid + vapor at equilibrium. Finally, cooling below -200 ˚F (-129˚F) results in a mixture of solid + liquid in equilibrium.

Figure 2. Complete phase envelope for mixture A.

At a pressure of 300 psia (2069 kPa), starting at -90°F (-68°C) (Point “A”), the fluid is 100% vapor.  Cooling at constant pressure results in liquid formation when the temperature reaches about -113°F (-81°C) at Point “B”.  Further cooling results in dry ice formation at Point “C” and the temperature is approximately -119°F (-84°).  The last vapor bubble would disappear at Point “D” (about -156°F, -104°C).  Below this point, the fluid exists as dry ice and liquid.

For the cooling process described above for a constant pressure of 300 psia,  the cooling temperature and vapor fraction of mixture as a function of heat removed from the process fluid (mixture A) in HEX-100 are shown in Figures 3A (Field Units) and 3B (SI Units).

Figure 3A. Temperature and vapor fraction of mixture A as it passes through HEX-100 (Field Units).

Figure 3B. Temperature and vapor fraction of mixture A as it passes through HEX-100 (SI Units).

Each mixture has a unique phase envelope and dry ice formation curve. As the mixture composition changes, the shape of the phase envelope and the dry ice curve will change. Similarly, a complete phase envelope for mixture B with the cooling path is shown in Figures 4, 5A, and 5B.

Figure 4. Complete phase envelope for mixture B.

Conclusions:

In cryogenic processes such as turboexpander plants for deep NGL recovery, accurate prediction of dry ice formation conditions is important. A good knowledge of phase behavior and thorough understanding of dry ice formation can prevent severe operational problems. On the phase envelope, any operating condition that lies on, to the left or below the dry ice curve (the dotted black curves on Figures 2 and 4) will form a solid phase and may cause severe operational problems, damage the equipment and lead to human casualty.

It is important to use the right tools and an accurate equation of state within simulation software to generate the correct phase envelope and dry ice curve. It is recommended to check the accuracy of the thermodynamic models against experimental data before generating any phase envelope or performing process simulation.

Figure 5A. Temperature and vapor fraction of mixture B as it passes through HEX-100 (Field Units).

To learn more about similar cases and how to minimize operational problems, we suggest attending our G40 (Process/Facility Fundamentals), G4 (Gas Conditioning and Processing), PF81 (CO₂ Surface Facilities), and PF4 (Oil Production and Processing Facilities) courses.

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

By: Dr. Mahmood Moshfeghian

Figure 5B. Temperature and vapor fraction of mixture B as it passes through HEX-100 (SI Units).

Reference:

1. http://en.wikipedia.org/wiki/Dry_ice
2. Yaws, C. Matheson gas data book (7th ed.). McGraw-Hill Professional. p. 982, 2001
3. Häring, H-W. Industrial Gases Processing. Christine Ahner. Wiley-VCH, 2008
4. Treloar, R., Plumbing Encyclopedia (3rd ed.). Wiley-Blackwell, 2003.
5. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.
6. ProMax 3.2, Bryan Research and Engineering, Inc, Bryan, Texas, 2011.
7. Hlavinka, M. W., Hernandez, V. N., and McCartney, D., “Proper Interpretation of  Freezing and  Hydrate  Prediction Results From Process Simulation,” Proceedings of the Eighty-Fifth GPA Annual Convention. Grapevine, TX: Gas  Processors Association, 1999:121-127 GPA 2006.
8. Kurata, F., “Solubility of Solid Carbon Dioxide in Pure Light Hydrocarbons and Mixtures of Light Hydrocarbons,” GPA Research Report RR-10, Gas Processors Association, 1974

In the January and February 2012 tips of the month (TOTM) we discussed the isothermal and non-isothermal transportation of pure carbon dioxide (CO₂) in the dense phase region. We illustrated how thermophysical properties changed in the dense phase and studied their impacts on pressure drop calculations. The pressure drop calculation results utilizing the liquid phase and vapor phase equations were exactly the same. We showed that the effect of the overall heat transfer coefficient on the pipeline temperature is significant. In this TOTM, we will study the same case study in the presence of nitrogen impurities under non-isothermal conditions. The Joule-Thompson expansion effect and the heat transfer between pipeline and surroundings have been considered. Specifically, we will report the effect of nitrogen impurities on the pressure and temperature profiles. The Peng-Robinson equation of state (PR EOS) was utilized in this study.

For a pure compound above critical pressure and critical temperature, the system is often referred to as a “dense fluid” or “super critical fluid” to distinguish it from normal vapor and liquid (see Figure 1 for carbon dioxide in December 2009 TOTM [1]).

Calculation Procedure:

The same step-by-step calculation procedure described in the February 2012 TOTM [2] was used to determine the pressure and temperature profiles in a pipeline considering the Joule-Thompson expansion effect and heat transfer between the pipeline and surroundings.

In the following section we will illustrate the pressure drop calculations for transporting CO₂ in dense phase using the gas phase pressure drop equations. For details of pressure drop equations in the gas and liquid phases refer to the January 2012 TOTM [3].

Case Study:

For the purpose of illustration, we considered a case study [also described in reference 2] for transporting 160 MMSCFD (4.519×10⁶ Sm³/d) CO₂ using a 100 miles (160.9 km) long pipeline with an inside diameter of 15.551 in (395 mm). The inlet conditions were 2030 psia (14 MPa) and 104˚F (40˚C). The following assumptions were made:

1. CO₂, with nitrogen impurities of 0, 1, 5, 10, and 15 mole %.
2. Horizontal pipeline, no elevation change.
3. Inside surface relative roughness’s (roughness factor), ε/D, of 0.00013.
4. The ambient/surrounding temperature,Ts, is 55 ˚F and (12.8 ˚C)
5. Overall heat transfer coefficients of 0.5 Btu/hr-ft²-˚F (2839 W/m²-˚C).

Properties: The dense phase behavior and properties were calculated using the Peng-Robinson equation of state (PR EOS) [4] in ProMax [5] software. ProMax was also used to determine pressure and temperature profiles along the pipeline.

Results and Discussions:

Figures 1 through 4 present the phase envelope, dry ice (CO₂ freeze out) curve, and pipeline pressure and temperature profile for 1, 5, 10, and 15 mole % N₂ impurities, respectively, the relative roughness (ε/D) of 0.00013, and the overall heat transfer coefficient (U) of 0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C).

Figure 1. Phase envelop and dense phase pipeline pressure-temperature profile for 99 mole % CO₂ + 1 mole % N₂, ε/D=0.00013, and U=0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C).

Figure 2. Phase envelop and dense phase pipeline pressure-temperature profile for 95 mole % CO₂ + 5 mole % N₂, ε/D=0.00013, and U=0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C).

Figure 3. Phase envelop and dense phase pipeline pressure-temperature profile for 90 mole % CO₂ + 10 mole % N₂, ε/D=0.00013, and U=0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C).

Figure 4. Phase envelop and dense phase pipeline pressure-temperature profile for 85 mole % CO₂ + 15 mole % N₂, ε/D=0.00013, and U=0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C).

The effect of N₂ impurities on the line temperature profile is shown in Figure 5. This figure indicates that N₂ impurities have negligible effect on the pipeline temperature profile.

Figure 6 presents the effect of N₂ impurities on the pipeline pressure profile. This figure indicates that as the N₂ impurities increases the pressure drop increases. This can be explained by the fact as the N₂ impurities increase, the mixture density decreases, consequently the velocity increases. Note the pressure drop is proportional to square of velocity and inverse of density. While viscosity decreases with increase in N₂ impurities, its effect is not as large as the density effect. Table 1 presents variation of the mixture density and viscosity as a function of N₂ mole %.

Table 1. Effect of N₂ impurities on density (ρ) and viscosity (µ) of mixture at the inlet condition of 2030 psia (14 MPa) and 104˚F (40˚C)

Conclusions: 

Analyzing Table 1 and Figures 1 through 6, the following conclusions can be made:

1. For the range 0 to 15 mole % N₂, the effect of the N₂ impurities on the pipeline temperature profile is negligible.
2. As the N₂ impurities increase, the pipeline pressure drop increases due to the change in thermophysical properties of mixture.
3. Care should be taken to use accurate thermophysical properties and the phase envelope should be plotted to avoid any operating problem.

Figure 5. Variation of the pipeline temperature profile with the N₂ impurities and U=0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C)

Figure 6. Variation of the pipeline pressure profile with the N₂ impurities and U=0.5 Btu/hr-˚F-ft² (2.839 W/m²-˚C)

To learn more about similar cases and how to minimize operational problems, we suggest attending our G40 (Process/Facility Fundamentals), G4 (Gas Conditioning and Processing), P81 (CO₂ Surface Facilities), and PF4 (Oil Production and Processing Facilities) courses.

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

By: Dr. Mahmood Moshfeghian

Reference:

1. Bothamley, M.E. and Moshfeghian, M., “Variation of properties in the dese phase region; Part 1 – Pure compounds,” TOTM, http://www.jmcampbell.com/tip-of-the-month/2009/12/variation-of-properties-in-the-dense-phase-region-part-1-pure-compounds/, Dec 2009.
2. Moshfeghian, M., ”Transportation of CO₂ in the Dense Phase,” TOTM, http://www.jmcampbell.com/tip-of-the-month/2012/02/ , Feb 2012
3. Moshfeghian, M., ”Transportation of CO₂ in the Dense Phase,” TOTM, http://www.jmcampbell.com/tip-of-the-month/2012/01/, Jan 2012
4. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.

ProMax 3.2, Bryan Research and Engineering

 In the January 2012 tip of the month (TOTM) we discussed the isothermal transportation of carbon dioxide (CO₂) in the dense phase. We illustrated how thermophysical properties changed in the dense phase and studied their impacts on pressure drop calculations. The pressure drop calculation results utilizing the liquid phase and vapor phase equations were exactly the same. In this TOTM, we will study the same case study under non-isothermal conditions. The Joule-Thompson expansion effect and the heat transfer between pipeline and surroundings have been considered. Specifically, we will report the effects of the overall heat transfer coefficient and the relative roughness on the pressure and temperature profiles. The Span and Wagner CO₂ equation of state (EOS) was utilized in this study.

For a pure compound, above critical pressure and critical temperature, the system is oftentimes referred to as a “dense fluid” or “super critical fluid” to distinguish it from normal vapor and liquid (see Figure 1 for carbon dioxide in December 2009 TOTM [1]).

Calculation Procedure:

The following step-by-step calculation procedure may be used to determine the pressure and temperature profiles in a pipeline considering the Joule-Thompson expansion effect and heat transfer between the pipeline and surroundings.

1. Divide the pipeline into n segments. The segments may be different lengths, but should be carefully chosen to provide the information sought through the calculations to be made. The more segments, the longer the calculation time. Time, however, is a relatively small investment compared to the importance of adequate representation of the line profile.
2. Assume an outlet pressure for each segment by use of a linear interpolation along the length of the line. Note that the outlet pressure of the first segment automatically becomes the inlet pressure to the second segment.
3. For the first iteration calculation, assume the flow in the segment is isothermal.
4. Calculate the average temperature, T_avg= (T_out + T_in)/2, and pressure, P_avg= (P_out + P_in)/2, for the first segment in the line. For the first iteration the inlet and outlet temperatures for the segment will be the same since isothermal flow is assumed.
5. Using the EOS, determine the state of the flow at T_avg and P_avg to establish whether or not the flow is in the dense phase.
6. Using the gas phase or liquid phase equations, calculate the segment pressure drop.
7. Compare the calculated pressure at the end of a segment with the pressure that was assumed in step 2. If the difference between these pressures is sufficiently small, proceed to step 8. If the difference between the assumed and calculated pressure at the end of segment is too large (1 psi or 6.9 kPa), replace the assumed outlet pressure with the calculated value of the outlet pressure and repeat steps 4 through 7 as many times as necessary to calculate s suitable segment outlet pressure.
8. Calculate the enthalpy at the end of the segment by writing an energy balance around the segment using the following procedure:
   H_out = H_in+ Q                        (1)
   Where:
   Q = UA (T_avg-T_s)                     (2)
   H_out= Enthalpy of the fluid at the outlet of the segment
   H_in= Enthalpy of the fluid at the inlet of the segmentQ = The overall heat transfer to or from the segment
   U = The overall heat transfer coefficient between the external surface of the pipe and its surroundings
   A = The external surface area of the segment of pipe
   T_avg = The average temperature of the segment
   T_s = The temperature of material surrounding the pipe
9. Using the EOS, determine the segment outlet temperature based on the calculated H_out and P_out.
10. If the temperature calculated in step 9 is the same as the assumed value in step 3, the calculations proceed to the next segment of the line. If the temperature is different from that assumed in step 3, calculations revert to step 4 using the newly calculated value for segment outlet temperature.

When conditions at the outlet end of the last segment of the line have been calculated to a satisfactory small difference (less than 1 psi or 6.9 kPa for pressure and less than 0.1 ˚F or 0.05 ˚C for temperature), calculations for flow conditions in the pipeline are complete.

In the following section we will illustrate the pressure drop calculations for transporting CO₂ in dense phase using the gas phase pressure drop equations. For details of pressure drop equations in the gas and liquid phases refer to the January 2012 TOTM [2].

Case Study:

For the purpose of illustration, we considered a case study [also described in reference 2] for transporting 160 MMSCFD (4.519×10⁶ Sm³/d) CO₂ using a 100 miles (160.9 km) long pipeline with an inside diameter of 15.551 in (395 mm). The corresponding mass flow rate is 214.7 lb_m/sec (97.39 kg/s). The inlet conditions were 2030 psia (14 MPa) and 104˚F (40˚C). The following assumptions were made:

1. Pure CO₂, ignored any impurities such as N₂.
2. Horizontal pipeline, no elevation change.
3. Five different inside surface relative roughness’s (roughness factor), ε/D, were studied (0.00004, 0.00013, 0.0002, 0.0004, and 0.001).
4. The ambient/surrounding temperature,Ts, is 55 ˚F and (12.8 ˚C)
5. Six different overall heat transfer coefficients ranging from 0 to 1 Btu/hr-ft²-˚F (0 to 5.678 W/m²-˚C) were studied.

Properties: The dense phase behavior and properties were calculated using the Span and Wagner CO₂  EOS [3] in ProMax [4] software. ProMax was also used to determine pressure and temperature profiles along the pipeline.

Results and Discussions:

Figure 1 presents the pressure drop per unit length as a function of relative roughness (ε/D) and the overall heat transfer coefficient (U). In this figure, the values of U₁ through U₆ are 0, 0.125, 0.25, 0.5, 0.75, and 1.0 Btu/hr-˚F-ft² (0, 0.71, 1.42, 2.839, 4.259, and 5.678 W/m²-˚C), respectively.

Figure 1. Variation of pressure drop with the relative roughness and the overall heat transfer coefficient.

            Figure 1 indicates that as the overall heat transfer coefficient increases, pressure drop decreases. This is because the line temperature drops more quickly at higher overall heat transfer coefficients. Note that as the U approaches 1.0 Btu/hr-˚F-ft² (5.678 W/m²-˚C) its effect vanishes.

As an example, Tables 1 and 2 present the impact of relative roughness on the pressure drop for an overall heat transfer coefficient of 0 and 0.50 Btu/hr-˚F-ft² (0 and 2.839 W/m²-˚C), respectively. These tables also present the line outlet temperatures.

Table 1. Impact of relative roughness on pressure drop (Number of segments=10).

Table 2. Impact of relative roughness on pressure drop (Number of segments=10).

These two tables and Figure 2 indicate that while the relative roughness has great impact on the pressure drop, its effect on temperature is small. On the other hand, the effect of overall heat transfer coefficient on the outlet temperature is more significant. The impact of U on the line temperature profile is shown in Figure 3. This figure also indicates that U has great impact on the line temperature profile. Figure 4 also indicates that the effect of relative roughness on the line temperature is negligible. Figure 5 presents the effect of the overall heat transfer confident on the line pressure profile. As can be seen in this figure, the increase in the overall heat transfer coefficient results in lower pressure drop. This is because the line temperature drops more quickly at the higher values of overall heat transfer coefficients.

Figure 2. Variation of the outlet temperature with the relative roughness and the overall heat transfer coefficient.

Figure 3. Variation of temperature profile with the overall heat transfer coefficient

Figure 4. Variation of the temperature profile with the pipe relative roughness

Figure 5. Variation of the line pressure profile with the overall heat transfer coefficient

Conclusions:

Analyzing Tables 1 and 2 and Figures 1 through 5, the following conclusions can be made:

1. The effect of the overall heat transfer coefficient on the pipeline temperature is significant.
2. As the overall heat transfer coefficient increases, the outlet temperature decreases.
3. As the overall heat transfer coefficient increases, the outlet pressure increase (line pressure drop decreases).
4. As the value of the heat transfer coefficient approaches 1.0 Btu/hr-˚F-ft² (5.678 W/m²-˚C) its effect on the pipeline pressure drop vanishes.
5. While pipeline roughness factor has great impact on the pressure drop, it has little effect on the pipeline temperature profile.

To learn more about similar cases and how to minimize operational problems, we suggest attending our G40 (Process/Facility Fundamentals), G4 (Gas Conditioning and Processing), P81 (CO₂ Surface Facilities), and PF4 (Oil Production and Processing Facilities) courses.

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

By: Dr. Mahmood Moshfeghian

Reference:

1. Bothamley, M.E. and Moshfeghian, M., “Variation of properties in the dese phase region; Part 1 – Pure compounds,” http://www.jmcampbell.com/tip-of-the-month/2009/12/variation-of-properties-in-the-dense-phase-region-part-1-pure-compounds/, December 2009.
2. Moshfeghian, M., ”Transportation of CO₂ in the Dense Phase,” http://www.jmcampbell.com/tip-of-the-month/
3. Span, R.; Wagner, W. – Equations of State for Technical Applications. I. Simultaneously Optimized Functional Forms for Nonpolar and Polar Fluids. Int. J. Thermophys. 2003,24(1), 1-39
4. ProMax 3.2, Bryan Research and Engineering, Inc, Bryan, Texas, 2011.

 In this tip of the month (TOTM) we will discuss transportation of carbon dioxide (CO₂) in the dense phase. We will illustrate how thermophysical properties change in the dense phase and their impacts on pressure drop calculations. The pressure drop calculations results utilizing the liquid phase and vapor phase equations will be compared. The application of dense phase in the oil and gas industry will be discussed briefly. In a future TOTM, we will discuss the dense phase transportation of natural gas.

When a pure compound, in gaseous or liquid state, is heated and compressed above the critical temperature and pressure, it becomes a dense, highly compressible fluid that demonstrates properties of both liquid and gas. For a pure compound, above critical pressure and critical temperature, the system is oftentimes referred to as a “dense fluid” or “super critical fluid” to distinguish it from normal vapor and liquid (see Figure 1 for carbon dioxide in December 2009 TOTM [1]). Dense phase is a fourth (Solid, Liquid, Gas, Dense) phase that cannot be described by the senses. The word “fluid” refers to anything that will flow and applies equally well to gas and liquid. Pure compounds in the dense phase or supercritical fluid state normally have better dissolving ability than do the same substances in the liquid state. The dense phase has a viscosity similar to that of a gas, but a density closer to that of a liquid. Because of its unique properties, dense phase has become attractive for transportation of CO₂ and natural gas, enhanced oil recovery, food processing and pharmaceutical processing products.

The low viscosity of dense phase, super critical carbon dioxide (compared with familiar liquid solvents), makes it attractive for enhanced oil recovery (EOR) since it can penetrate through porous media (reservoir formation). As carbon dioxide dissolves in oil, it reduces viscosity and oil-water interfacial tension, swells the oil and can provide highly efficient displacement if miscibility is achieved. Additionally, substances disperse throughout the dense phase rapidly, due to high diffusion coefficients. Carbon dioxide is of particular interest in dense-fluid technology because it is inexpensive, non-flammable, non-toxic, and odorless. Pipelines have been built to transport CO₂ and natural gas in the dense phase region due to its higher density, and this also provides the added benefit of no liquids formation in the pipeline.

In the following section we will illustrate the pressure drop calculations for transporting CO₂ in dense phase using liquid phase and vapor phase pressure drop equations.

Case Study:

For the purpose of illustration, we considered a case study for transporting 160 MMSCFD (4.519×10⁶ Sm³/d) CO₂ using a 100 miles (160.9 km) long pipeline with an inside diameter of 15.551 in (395 mm). The corresponding mass flow rate is 214.7 lb_m/sec (97.39 kg/s). The inlet conditions were 2030 psia (14 MPa) and 104˚F (40˚C). The following assumptions were made:

1. Pure CO₂, ignored any impurities such as N₂.
2. Horizontal pipeline, no elevation change.
3. Inside surface relative roughness (roughness factor), ε/D, is 0.00004.
4. Isothermal transportation of CO₂.

Properties: Dense phase behavior is unique and has special features. The thermophysical properties in this phase may vary abnormally. Care should be taken when equations of state are used to predict thermophysical properties in dense phase. Evaluation of equations of state should be performed in advance to assure their accuracy in this region. Many simulators offer the option to use liquid-based algorithms (e.g. COSTALD [2]) for this region. Dense phase is a highly compressible fluid that demonstrates properties of both liquid and gas. The dense phase has a viscosity similar to that of a gas, but a density closer to that of a liquid. This is a favorable condition for transporting CO₂ and natural gas in dense phase as well as carbon dioxide injection into crude oil reservoir for enhanced oil recovery.

Figures 1 and 2 present variation of density and viscosity of CO₂ with pressure at constant temperature of 104 ˚F (40 ˚C) calculated by the SRK EOS and COSTALD liquid density option in ProMax [3] and the Span and Wagner CO₂  EOS in REFPROP [4] software. Note, ProMax also has the Span and Wagner CO₂  EOS option which produced practically the same results as the REFPPROP.

Figure 1. Density-Pressure diagram for CO₂ at 104˚F (40˚C) by the SRK EOS and COSTALD liquid in ProMax and Span and Wagner CO₂  EOS in REFPROP

Figure 2. Viscosity-Pressure diagram for CO₂ at 104˚F (40˚C) by the SRK EOS and COSTALD liquid in ProMax and Span and Wagner CO₂  EOS in REFPROP

For the sake of easier calculation steps, these diagrams were fitted to the following 3^rd degree polynomials for density and viscosity, respectively:

In these equations, ρ is density (kg/m³), µ is viscosity (cP) and P_avg is the average pipeline segment pressure calculated by:

The fitted coefficients for equations 1 and 2 are presented in Table 1.

Table 1. The fitted coefficients for CO₂ density and viscosity (Equations 1 & 2) at 104˚F (40˚C)

Figures 1 and 2 clearly indicate that there are large differences between predicted properties using two different sources. In the following section, we will illustrate the impact of these differences on pressure drop calculations.

Liquid Phase Pressure Drop Equations: The pressure drop for a liquid phase is calculated as follows.

Where:

Vapor Phase Pressure Drop Equations: In addition to Equations 5 through 8, which are also valid and used for the gas pipeline, the following equations are also used.

Where:

Results and Discussions:

The pressure drop calculations were performed using the liquid phase and vapor phase equations. First, the pipeline cross sectional area was calculated with Equation 8 and the gas density at the standard condition was calculated with equation 10.  In each case the calculation was trial and error and the following step-by-step procedure was followed:

1. The line was divided into n segments (e.g. n = 1, 10, 20, or 100).
2. For segment 1, an outlet pressure was guessed.
3. Segment average pressure was calculated with Equation 3.
4. CO₂ density and viscosity were calculated using Equations 1 and 2, respectively.
5. CO₂ velocity was calculated with Equation 7.
6. Reynolds number was calculated with Equation 6.
7. Friction factor was calculated with Equation 5 (this is also trial and error).
8. Liquid phase pressure drop was calculated with equation 4.
9. Calculate average gas compressibility factor with equation 11.
10. Calculate segment gas outlet pressure by Equation 9 and segment pressure drop with Equation 12.
11. If the calculated outlet pressure is not the same as the guessed outlet pressure in step 2, replace the guessed outlet pressure with the calculated outlet pressure and repeat steps 3 through 10 until the calculated outlet pressure becomes equal to the guessed value.
12. Use the calculated outlet pressure of segment “1” for the inlet of segment “2” and repeat the above steps for each segment till the end of line is reached.

Table 2 summarizes the pressure drop calculation results for four cases in which the pipeline was divided into 1, 10, 20, and 100 segments. Table 2 indicates that for the cases of 10 segments and higher no change in pressure drop is observed.

Table 2. Summary of pressure drop calculation results for different number of segments and different sources of properties.

For all cases tested, both the liquid phase and the vapor phase pressure drop equations gave exactly the same pressure drop. Note that there is at least 100 psi (690 kPa) difference in pressure drops calculation using REFPROP (Span and Wagner CO₂ EOS) or ProMax (SRK EOS and COSTALD liquid density) because the EOS options were different. However, the Span and Wagner CO₂ EOS in both software would result in the same pressure drop. A sample calculation in MathCad format is attached: Dense Phase CO2 Pipeline 1 Segment ProMax.

Table 3 presents the impact of relative roughness on pressure drop. Typical / generally accepted numbers for relative roughness are (and these are regarded as conservative) for steel pipes are:  new or clean service  =  0.00004, mildly corroded  =   0.0002, corroded / dirty service =  0.0004.

Table 3. Impact of relative roughness on pressure drop (Number of segments=10).

Conclusions:

As discussed in December 2009, dense phase behavior is unique and has special features. The thermophysical properties in this phase may vary abnormally. Care should be taken when equations of state are used to predict thermophysical properties in dense phase. Evaluation of equations of state should be performed in advance to assure their accuracy in this region. Many simulators offer the option to use liquid-based algorithms (e.g. COSTALD) for this region. It is very important to use the most appropriate option.

Dense phase is a highly compressible fluid that demonstrates properties of both liquid and gas. The dense phase has a viscosity similar to that of a gas, but a density closer to that of a liquid. This is a favorable condition for transporting CO₂ and natural gas in dense phase. It was also found that either the liquid phase or vapor phase pressure drop equations can be used to calculate CO₂ pressure drop in the dense phase. Both set of equations gave exactly the same pressure drop. Due to high density of CO₂ in the dense phase, pressure drop due to elevation change should not be ignored.

To learn more about similar cases and how to minimize operational problems, we suggest attending our G40 (Process/Facility Fundamentals), G4 (Gas Conditioning and Processing), P81 (CO₂ Surface Facilities), and PF4 (Oil Production and Processing Facilities) courses.

By: Dr. Mahmood Moshfeghian

Reference:

1. Bothamley, M.E. and Moshfeghian, M., “Variation of properties in the dese phase region; Part 1 – Pure compounds,” http://www.jmcampbell.com/tip-of-the-month/2009/12/variation-of-properties-in-the-dense-phase-region-part-1-pure-compounds/, December 2009.
2. Hankinson, R. W., Thomson, G. H., AIChE J., Vol. 25, no. 4, pp. 653-663, 1979.
3. ProMax 3.2, Bryan Research and Engineering, Inc, Bryan, Texas, 2011.
4. NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP): Version 9.0, 2011.

In the November tip of the month (TOTM) we presented a single-stage compressor calculation result of a case study. We compared the rigorous method with the values from the short cut methods. The rigorous method was based on the Soave-Redlich-Kwong (SRK) for calculating the required enthalpies and entropies.

In this TOTM, we will present a case study of multistage stage compression with interstage cooling using the rigorous method. The rigorous method will be based on the Soave-Redlich-Kwong (SRK), Peng-Robinson (PR), Lee Kesler (LK) and Benedict-Webb-Rubin-Starling (BWRS) equations of state. The K-values, enthalpies, and entropies are calculated by these EOSs to perform vapor-liquid-equilibrium (VLE) and the energy balance calculations to determine the power requirement, the discharge temperatures and the cooling load requirements. We will compare the compressor power and cooling load requirements based on the rigorous equations of state.

Power Calculations

The theoretical power requirements are independent of compressor type; the actual power requirements vary with the compressor efficiency. In general the power is calculated by:

From a calculation viewpoint alone, the power calculation is particularly sensitive to the specification of flow rate, inlet temperature and pressure, and outlet pressure. Gas composition is important but a small error here is less important providing it does not involve the erroneous exclusion of corrosive components. A compressor is going to operate under different values of the variables affecting its performance. Therefore the most difficult part of a compressor calculation is specification of a reasonable range for each variable and not the calculation itself. Reference [1] emphasizes that using a single value for each variable is not the correct way to evaluate a compression system.

Normally, the thermodynamic calculations are performed for an ideal (reversible) process. The results of a reversible process are then adapted to the real world through the use of efficiency. In the compression process there are three ideal processes that can be visualized: 1) an isothermal process, 2) an isentropic process and 3) a polytropic process. Any one of these processes can be used suitably as a basis for evaluating compression power requirements by either hand or computer calculation. The isothermal process, however, is seldom used as a basis because the normal industrial compression process is not even approximately carried out at constant temperature.

Step-by-Step Computer Solution

For known gas rate, pressure (P₁), temperature (T₁), and composition at the inlet condition and discharge pressure (P_2­) or compression ratio, computation of compressor power requirement is based on an EOS using a computer and involves two steps:

1. Determination of the ideal or isentropic (reversible and adiabatic) enthalpy change of the compression process. The ideal work requirement is obtained by multiplying mass rate by the isentropic enthalpy change.
2. Adjustment of the ideal work requirement for compressor efficiency.
   The step-by-step calculation based on an EOS is outlined below.
   1. Assume steady state, i.e.   and the feed composition remain unchanged.
   2. Assume isentropic process, i.e. adiabatic and reversible
   3. Calculate specific enthalpy h₁=f(P₁, T₁, and composition) and suction specific entropy s₁=f(P₁, T₁, and composition) at the suction condition by EOS
   4. For the isentropic process . Note the * represents ideal value.
   5. Calculate the ideal specific enthalpy  at outlet condition for known composition, P₂ and .
   6. The ideal work is 
   7. The actual work is the ideal work divided by efficiency or 
   8. The actual enthalpy at the outlet condition is calculated by 
   9. The actual outlet temperature is calculated by EOS for known h₂, P₂, and composition.

The efficiency of the compressor, and hence, the compression process obviously depends on the method used to evaluate the work requirement. The isentropic efficiency is in the range of 0.70 to 0.90.

If the manufacturer provides the compressor head curve and efficiency curve, the head is determined from the actual gas volume rate at the inlet condition. Second, from the head, the actual work, discharge pressure and finally the discharge temperature are calculated.

Case Study

The gas mixture with the composition shown in Table1 at 105 °F (40.6 °C) and 115 Psia (793 kPa) is compressed to 1015 psia (7000 kPaa) using a multistage centrifugal compressor. The total feed gas volumetric flow rate was 101 MMSCFD (2.86×10⁶ Sm³/d). This is the same feed used in the November TOTM.

Table 1. Feed gas analysis

A simplified process flow diagram is shown in Figure 1. The dry feed gas is saturated with water, passed through a scrubber (knockout drum) before entering the first stage of compressor. Each compression stage is followed by cooling and subsequent knockout drum before entering the next stage. An equal compression ratio of 3 was used for each stage. The polytropic efficiency of 86, 80, and 79 % based on the actual inlet volumetric rate (from Figure 13.23 of GPA Data book [2]) was specified for stages 1, 2, and 3 respectively. After each compression stage, the gas was cooled to the feed temperature of 105 °F (40.6 °C).

Figure 1. Three-stage compression with interstage cooling

Results and Discussions

The feed composition, suction temperature and pressure, volumetric flow rate at standard condition along with the compressor polytropic efficiency for each stage, and pressure drop for each cooler were specified. For this study, the above PFD was simulated using the Soave-Redlich-Kwong (SRK) [3], Peng-Robinson (PR) [4], Lee Kesler (LK) [5] and Benedict-Webb-Rubin-Starling (BWRS) [6] equations of states. These data were entered into the ProMax software [7] to perform the rigorous calculations based on the EOS. The program calculated discharge temperature, power for each stage, and cooling loads for each cooler. For the actual gas flow rate at the inlet condition, the polytropic efficiency was specified from the GPA data book. The calculated results for the four EOSs are presented in Table 2 (bold numbers with white background).

Table 2 (FPS units). Summary of the rigorous results for four EOSs using ProMax

The bold numbers with white background are the calculated values

 

Table 2 (SI Units). Summary of the rigorous results for four EOSs using ProMax

The bold numbers with white background are the calculated values

For the case of LK EOS, the wet feed volume flow rate at standard condition to the first stage of compressor is lower than the other cases because this EOS has not been revised to handle water content.

For the case studied, Table 2 indicates that there is 0.8 to 1.4 percent deviation in total compression power requirements among these 4 EOSs. The deviation in total heat removal using different EOSs is 1.7 to 2.2 percent. For facility type calculations and planning purposes, these differences are negligible. However, for cases with large power requirement, these small differences in terms of total HP or kW could be significant; therefore, care should be taken to choose an appropriate EOS for handling VLE calculations and accurate predictions of enthalpy and entropies for the system under consideration. The deviation range could be different for other cases depending on the flow rates, condition composition and compression ratio.

To learn more about similar cases and how to minimize operational problems, we suggest attending the John M. Campbell courses; G4 (Gas Conditioning and Processing), G5 (Gas Conditioning and Processing-Special) and ME 44 (Fundamentals of Pump and compressors).

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

By Dr. Mahmood Moshfeghian

References:

1. Maddox, R. N. and L. L. Lilly, “Gas conditioning and processing, Volume 3: Advanced Techniques and Applications,” John M. Campbell and Company, 2^nd Ed., Norman, Oklahoma, USA, 1990.
2. Gas Processors Association Data Book, 12^th Edition, GPA, Tulsa, Oklahoma.
3. Soave, G., Chem. Eng. Sci., Vol. 27, pp. 1197-1203, 1972.
4. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.
5.  Lee B.I., Kesler M.G., “A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States”, AIChE J., 21(3), 510-527, 1975
6. Starling, K. E., Fluid Thermodynamic Properties for Light Petroleum Systems,  Gulf Publishing Co., Houston, 1973.
7. ProMax 3.2, Bryan Research and Engineering, Inc, Bryan, Texas, 2011.

In this tip of the month (TOTM) we will present the compressor calculations of a case study. We will compare the rigorous method results with the values from the short cut methods. The rigorous method is based on an equation of state like the Soave-Redlich-Kwong (SRK) for calculating the required enthalpies and entropies. The enthalpies and entropies are used to determine the power requirement and the discharge temperatures.  The results indicate that the accuracy of the shortcut method is sensitive to the value of heat capacity ratio, k.

Power Calculations

The theoretical power requirements are independent of compressor type; the actual power requirements vary with the compressor efficiency. In general the power is calculated by:

where  mass flow rate and h is specific enthalpy.

From a calculation viewpoint alone, the power calculation is particularly sensitive to the specification of flow rate, inlet temperature and pressure, and outlet pressure. Gas composition is important but a small error here is less important providing it does not involve the erroneous exclusion of corrosive components. A compressor is going to operate under varying values of the variables affecting its performance. Thus the most difficult part of a compressor calculation is specification of a reasonable range for each variable and not the calculation itself. Reference [1] emphasizes that using a single value for each variable is not the correct way to evaluate a compression system.

Normally, the thermodynamic calculations are performed for an ideal (reversible process). The results of a reversible process are then adapted to the real world through the use of an efficiency. In the compression process there are three ideal processes that can be visualized: 1) an isothermal process, 2) an isentropic process and 3) a polytropic process. Any one of these processes can be used suitably as a basis for evaluating compression power requirements by either hand or computer calculation. The isothermal process, however, is seldom used as a basis because the normal industrial compression process is not even approximately carried out at constant temperature.

For an isentropic (reversible and adiabatic) process, equation 1 can be written as:

and based on the polytropic process:

The isentropic head is calculated by equation 3A:

Similarly, the polytropic head is calculated by equation 3B:

The actual discharge temperature based on the isentropic path is calculated by equation 4A.

The actual discharge temperature based on the polytropic is calculated by equation 4B.

where η and η_P are the isentropic (or adiabatic) and polytropic efficiency, respectively, P₁ suction pressure, P₂ discharge pressure, T₁ and T₂ arethe suction and discharge temperatures, respectively, q is gas volume flow rate at standard condition of P_S and T_S, Z_a average gas compressibility factor, k heat capacity ratio, R the gas constant, and n is the polytropic path exponent. Equations 1 and 2 are equally correct theoretically. The practical choice depends on the available data, although it is somewhat arbitrary. The power calculation should be made per stage of compression and then summed for all stages connected to a single driver. For general planning purposes the graphical solutions shown in reference [2] produce results comparable to these equations.

Equation of State (EOS)

The heart of any commercial process flow simulation software is an equation of state. Due to their simplicity and relative accuracy, normally a cubic EOS such as Soave Redlich-Kwong (SRK) [3] or Peng-Robinson [4] is used. These equations are used to calculate phase behavior, enthalpy, and entropy. With proper binary interaction coefficients, the process simulation results of these two equations are practically the same. Therefore, only the SRK was used in this work.

Step-by-Step Computer Solution

For known gas rate, pressure (P₁), temperature (T₁), and composition at the inlet condition and discharge pressure (P_2­), computation of compressor power requirement is based on an EOS using a computer and involves two steps:

1. Determination of the ideal or isentropic (reversible and adiabatic) enthalpy change of the compression process. The ideal work requirement is obtained by multiplying mass rate by the isentropic enthalpy change.
2. Adjustment of the ideal work requirement for compressor efficiency.
   The step-by-step calculation based an EOS is outlined below.
   1. Assume steady state, i.e.   and the feed composition remain unchanged.
   2. Assume isentropic process, i.e. adiabatic and reversible
   3. Calculate enthalpy h₁=f(P₁, T₁, and composition) and suction entropy s₁=f(P₁, T₁, and composition) at the suction condition by EOS
   4. For the isentropic process . Note the * represents ideal value.
   5. Calculate the ideal enthalpy () at outlet condition for known composition, P₂ and .
   6. The ideal work is 
   7. The actual work is the ideal work divided by efficiency or 
   8. The actual enthalpy at the outlet condition is calculated by 
   9. The actual outlet temperature is calculated by EOS for known h₂, P₂, and composition.

The efficiency of the compressor, and hence, the compression process obviously depends on the method used to evaluate the work requirement. The isentropic efficiency is in the range of 0.70 to 0.90.

If the compressor head curve and efficiency curve are provided by the manufacturer,  the head is determined from the actual gas volume rate at the inlet condition. Second, from the head, the actual work, discharge pressure and finally the discharge temperature are calculated.

Case Study

The gas mixture with the composition shown in Table1 at 105 °F (40.6 °C) and 115 Psia (793 kPa) is compressed using a single-stage centrifugal compressor with the polytropic head and efficiency curves shown in Figures 1 and 2 at a speed of 7992 rpm. The total feed gas volumetric flow rate was 101 MMSCFD (2.86×10⁶ Sm³/d).

Table 1. Feed gas analysis

Figure 1. Compressor polytropic head and best efficiency point

Figure 2. Compressor polytropic efficiency

Results and Discussions

SRK (Rigorous Method): The feed composition, temperature, pressure, volumetric flow rate at standard condition along with the compressor polytropic head and efficiency curves data were entered into the ProMax  software [5] to perform the rigorous calculations based on the SRK EOS. The program calculated polytropic and isentropic efficiencies, heads, compression ratio (discharge pressure), discharge temperature and power. For the actual gas flow rate at the inlet condition, the polytropic efficiency is close to the compressor best efficiency point (BEP). The program also calculated the gas relative density, heat capacity ratio (k), and polytropic exponent (n). These calculated results are presented in the SRK columns of Table 2 (bold numbers with white background).

Table 2. Summary of the rigorous and shortcut calculated results

The bold numbers with white background are the calculated values

Short-1 (Shortcut Method): In this method, we used equations 2 through 4 to calculate the polytropic and isentropic heads, the discharge temperature and power. We used the ProMax calculated polytropic and isentropic efficiencies, compression ratio (P₂/P₁), heat capacity ratio (k) and polytropic exponent (n) to calculate head, power, and the discharge temperature. The results are presented in the short-1 columns of Table 2. Note the short-1 results (discharge temperature, adiabatic and polytropic heads and power) are very close to the SRK values. The calculated actual discharge temperature by equation 4A (isentropic path: 265.3˚F=129.6˚C) was slightly lower than by equation 4B (polytropic path: 265.9 ˚F=129.9 ˚C).

Short-2 (Shortcut Method): Similar to short-1 method, we used equations 2 through 4 to calculate the polytropic and isentropic heads, the actual discharge temperature and power. We used only the ProMax calculated values of polytropic efficiency (n_P), compression ratio (P₂/P₁), and relative density (y). The heat capacity ratio (k) was estimated by equation 5:

The polytropic exponent (n) was estimated by equation 6.

The isentropic (adiabatic) efficiency () was estimated by equation 7.

The results for this method are presented in the short-2 columns of Table 2. The calculated discharge temperature by equation 4A (isentropic path) was exactly the same as by equation 4B (polytropic). Note the short-2 results (discharge temperature, adiabatic and polytropic heads and power) are deviated from the SRK values.

The results in Table 2 indicate that an increase of 2.2% in k (from 1.224 to 1.251) results in power increase of 1.42%. The polytropic exponent (n) increased by 3% and isentropic efficiency () decreased by 0.5 %. The difference in the actual discharge temperatures of the SRK and short-2 values is 17.5 ˚F (9.7˚C).

With the exception of actual discharge temperature, these differences between the SRK and short-2 methods results for facilities calculations and planning purposes are negligible. Note that the accuracy of the shortcut method is dependent on the values of k and n. In Short-1 method in which we used the k and n values from the SRK method the results were identical to those of SRK method.

To learn more about similar cases and how to minimize operational problems, we suggest attending the John M. Campbell courses; G4 (Gas Conditioning and Processing), and G5 (Gas Conditioning and Processing-Special).

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

By Dr. Mahmood Moshfeghian

Reference:

1. Maddox, R. N. and L. L. Lilly, “Gas conditioning and processing, Volume 3: Advanced Techniques and Applications,” John M. Campbell and Company, 2^nd Ed., Norman, Oklahoma, USA, 1990.
2. Campbell, J. M., “Gas Conditioning and Processing, Vol. 2, the Equipment Modules, 8^th Ed., Campbell Petroleum Series, Norman, Oklahoma, 2001
3. Soave, G., Chem. Eng. Sci., Vol. 27, pp. 1197-1203, 1972.
4. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.
5. ProMax 3.2, Bryan Research and Engineering, Inc, Bryan, Texas, 2011.

In the March 2011 tip of the month (TOTM) we studied a constant volume translation of liquid density method presented by Peneloux et al. [3] and demonstrated its application for pure components. Considerable improvements were obtained, specifically for the low temperature range (reduced temperature < 0.8), of saturated specific volume (or liquid density) predicted by Soave-Redlich-Kwong (SRK) [1] and Peng-Robinson (PR) [2], but, the constant volume shift fails near the critical temperature. In the July 2011 TOTM, we reviewed several temperature dependent volume correction methods [3-11]. They improved the accuracy of liquid density prediction considerably near the critical point region.

As the extension of the March and July 2011 Tips of the Month in this TOTM we will demonstrate application and accuracy of some of these methods to predict liquid density of light hydrocarbon mixtures encountered in gas processing. We will compare their accuracy against both experimental data and a few correlations.

Equations of state (EoS) are used in commercial simulation software for predicting phase behavior and thermodynamic properties. The cubic equations of state give relatively accurate results for predicting vapor-liquid equilibria, especially for non-polar or slightly polar systems. These equations can be used to accurately predict vapor densities, enthalpy and entropy. These advantages encourage the researchers to augment EoS ability more than before, especially liquid density, although their accuracy for liquid density prediction is generally poor. The popular EoSs such as SRK [1] and PR [2] predict liquid density with an average absolute error of about 8%, much higher than several good density correlations. This large magnitude of error is not acceptable by industry; therefore, they are not used for this purpose. In order to overcome this deficiency, volume translated methods have been developed.

The correlations and volume translated methods [3-11] used in this study are the same as those presented in the July 2011 TOTM. Only those methods, which gave the best results for the cases studied in this TOTM, are described briefly here.  More details about these methods can be found in the corresponding references.

1. Correlations
   The following correlations were used in this study. 
   1. COSTALD, 1979: The COSTALD correlation by Hankinson and Thomson [12] requires two parameters: w_SRK, the optimized value of the acentric factor based on the SRK equation of state (EoS) and; V*, the pure component characteristic volume.
   2. RSD, 1972: Spencer and Danner [13] improved the liquid density correlation of Rackett [14]. The improved correlation for saturated liquid density calculation requires only Z_RA, the improved compressibility factor.
   3. NM, 1998: Nasrifar and Moshfeghian [15, 16] presented an equation and a set of mixing rules for predicting the liquid density of pure refrigerants and liquefied natural gas.
2. Volume Translated EoS Methods
   Equation 1 is the basic volume translated (shift) method proposed by Peneloux et al. [3] and used in this study. Equations 2 and 3 utilize the Kay’s rule to estimate mixture molecular weight (MW) and volume shift correction parameter (c).

In the above equations, is the corrected liquid specific volume, is the liquid specific volume calculated by the EoS, MW is the molecular weight, ρ^L is the liquid density, and the correction term or volume shift factor “c” is determined from experimentally measured liquid density. The volume shift factor is normally regressed against several experimental data points. The following methods were used to determine c for the mixtures.

1. Temperature Independent, PRF, 1982: Peneloux et al. [3] proposed the following expression to estimate the constant temperature volume shift correction for each component “i” in the mixture. In the absence of experimentally regressed value, it can be estimated as follows:
    

   where Z_RA, is the Rackett [15] parameter, R is the gas constant, and T_C and P_C are the critical temperature and pressure, respectively.

2. Temperature Dependent, AG, 2001: Ahlers and Gmehling [6] temperature dependent correction factor, c, is calculated as follows:(5)(6)

In the above equations, Tr is the reduced temperature, ω is acentric factor, T_C, P_C and Z_C are component i critical temperature, pressure and compressibility factor, respectively. The correction term, c, from the above methods is substituted into equation 1 to calculate the corrected density.

Results and Discussion:

We applied the preceding methods to several natural gas mixtures shown in Table 1. The experimentally measured temperature, pressure, composition and relative liquid density for each mixture are shown in this table [17]. These mixtures and corresponding conditions represent those encountered in the cryogenic processes. As an example, the condition and phase envelope for mixture number 1 of Table 1 are presented in Figure 1.

Table 1. Experimentally measured composition, temperature, pressure and relative density for the mixtures studied [17]

Table 2 presents the summary of the error analysis for the liquid density prediction by different methods for the natural gas mixtures shown in Table 1. As can be seen in Table 2, both SRK and PR EoSs give poor results; however, considerable improvements are observed by applying temperature-independent volume translated SRK (SRK-PRF) and temperature-dependent volume translated PR (PR-AG). The volume translated results for these mixtures are much closer to the results obtained by the three correlations of RSD, NM, and COSTALD.

To learn more about similar cases and how to minimize operational problems, we suggest attending the John M. Campbell courses; G4 (Gas Conditioning and Processing), and G5 (Gas Conditioning and Processing-Special).

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

By Dr. Mahmood Moshfeghian

Figure 1. Phase envelope and liquid density condition for mixture 1

Table 2: Comparison of acuracy of EoS, volume translated EoS, and correlations for predicting liquid density of mixtures prsented in Table 1

Reference:

1. Soave, G., Chem. Eng. Sci., Vol. 27, pp. 1197-1203, 1972.
2. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.
3. Peneloux, A. E., Rauzy, E., and Freze, R., Fluid Phase Equilib., Vol. 8, pp. 7-23, 1982.
4. Tsai, J. and Y.P. Chen, J. of Fluid Phase Equilibria, Vol. 145, pp. 193-215, 1998.
5. Ahlers, J. and J. Gmehling, J. of Fluid Phase Equilibria, Vol. 191, pp. 177-188, 2001.
6. Lin, H. and Y.Y. Duan, J. of Fluid Phase Equilibria, Vol. 233, pp. 194-203, 2005.
7. Ji, W.R. and D.A. Lempe, J. of Fluid Phase Equilibria, Vol. 130, pp. 49-63, 1997.
8. Pfohl, O., J. of Fluid Phase Equilibria, Vol. 163, pp. 157-159, 1999.
9. Magoulas, K. and D. Tassios, J. of Fluid Phase Equilibria, Vol. 56, pp. 119-140-445, 1990.
10. Frey, F., Augustine, C., Ciccolini, R.P., Paap, S., Modell, M., and J. Tester, , J. of Fluid Phase Equilibria, Vol. 260, pp. 316-325, 2007.
11. Frey, F., Modell, M., and J. Tester, J. of Fluid Phase Equilibria, Vol. 279, pp. 56-63, 2009.
12. Hankinson, R. W., Thomson, G. H., AIChE J., Vol. 25, no. 4, pp. 653-663, 1979.
13. Spancer, C. F., and Danner, R. P., J. Chem. Eng. Data, vol. 17, no. 2, pp. 236-241, 1972.
14. Rackett, H. G., J. Chem. Eng. Data, vol. 15, No. 4, pp. 514-517, 1970.
15. Nasrifar, Kh. and Moshfeghian, M., Fluid Phase equilibria Vol. 153, 231-242, 1998.
16. Nasrifar, Kh. and M. Moshfeghian, J. of Fluid Phase Equilibria, Vol. 158-160, pp. 437-445, 1998.
17. Haynes, W.M.J., J. Chem. Thermodyn., Vol 14, pp. 603–612, 1982.

In the April, 2011, Tip of the Month (TOTM), we looked at a simple, graphical representation of process safety competency.  This TOTM will follow up on that by asking a simple question:

“When examining catastrophic incidents, what are the typical mistakes that engineers make?”

This question was asked of me at a lunch and learn session I conducted for a client where I had described the competency pyramid introduced in the April TOTM.  At the time, I thought, what a good question.  I replied, I have to think about that.  After reflecting for a bit, I went to the pyramid.

Looking at the pyramid, it seems to me that a lot of an engineer’s duties revolve around the “Equipment” and “Mitigation” levels.  It is here that separators are sized; pumps are chosen; inherently safer design is incorporated into a process; safety instrumented systems (SIS) are designed; pressure safety valve (PSV) sizing is calculated.  Refer to the April TOTM again.  “To become a well rounded professional in the oil and gas industry, no matter what specialty position a person works in, requires varying degrees of competency in many different areas of expertise.  Obtaining higher level competencies is a continuous process of training and performing tasks, sometimes under the direction of a mentor or coach.”

It could be that the typical mistakes made by engineers are a result of competency with equipment and mitigation measures.  Consider some information from JMC’s Process Safety Engineering course, PS 4.  This table lists the area of responsibility for incidents [1].

Notes:

1. The item “inspection during operation” includes some items that are not always the responsibility of the inspection department:
   1. Vibration monitoring for rotating equipment.
   2. Corrosion probes
2. The item “inspection of process fluids” includes:
   1. Flammable-gas detection in shutdown, and in tanks during operation
   2. Inspection of purchased and process fluids to determine whether they are the ones specified.
3. The average incident has 1.56 responsibilities

How many of the items on the list are engineering functions and which of them could be related to competency?  I suppose it could be argued that almost all are related to engineering functions if it is accepted that engineers provide the design, recommend inspections and maintenance tasks, provide significant information needed for development of operating procedures and have historically been assigned management responsibilities in the oil and gas industry.  Which are related to competency?  That is much more difficult to answer and could only be answered by individual organizations based on performance reviews and competency mapping.

The April, 2010, TOTM discussed the need to perform a good job task analysis to identify personal and process safety hazards.  While the checklist presented there can be modified to allow engineering personnel to analyze engineering tasks, there is a simpler way to insure that engineers reduce the likelihood of mistakes in their work.  Ask the following six questions about any project or job that is being done.  If each question is answered fully, the job should be performed mistake free.

• What are we doing? — A very simple description of what is required to perform the task.
• What is the most dangerous part? — To find the most dangerous part, all hazards will have to be identified.
• What will we do to protect ourselves? — Plan for the worst.
• How will we know that we are changing what we are doing?—Insures that scope creep doesn’t happen.
• What will we do about it? — Contingency planning prior to problems being encountered.
• How will we know we are done? — Should be able to identify everything that needs to be in place when finished.

It is difficult to determine the typical mistakes, related to technical competency, made by engineers that cause or could reasonably cause a catastrophic incident.  Root cause analysis can usually discover breakdowns in an organization’s process safety management system.  Using the six questions to analyze work prior to starting and periodically throughout the life of a project may help to keep personnel focused on the consequences of failure and reduce the likelihood of failure.  Thus, reducing risk.

Several JMC courses develop competencies associated with the equipment and mitigation levels of the competency pyramid.  To learn more about process safety for engineers, consider attending a  session of our PS 4, Process Safety Engineering course.

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. With the most sought after consultants in the oil and gas industry, JMCC provides first choice consulting services to select clients. For more information about services offered by Campbell, visit our website at www.jmcampbellconsulting.com.

By:  Clyde Young

Reference:

1. Ian Duguid, “Take This Safety Database to Heart”, Chemical Engineering Magazine, July 2001.

As discussed in the May 2011 Tip Of The Month (TOTM), for hydrate inhibition, the most commonly used equilibrium inhibitors used in the upstream and midstream sectors of the oil and gas business are:  monoethylene or diethylene glycol (MEG or DEG) and methanol.  In general, glycols are more commonly used in systems requiring continuous inhibition. The glycol is typically recovered, regenerated and recirculated.  Methanol is more commonly used in systems that do not require continuous inhibition, i.e. systems only requiring inhibition during cold weather or upset conditions. Methanol is not commonly recovered and reused because of the difficulty of separation of the methanol from water. There are obviously exceptions to this. For example, methanol is used as a continuous inhibitor in a few offshore installations and in a handful of gas processing facilities. Another significant disadvantage of methanol relative to glycol is the high methanol losses to both the liquid hydrocarbon and vapor phase.

In the May TOTM, we reviewed experimental VLE data for methanol-hydrocarbon systems. We also revisited Figure 6.20 of reference [1] for methanol loss to the vapor phase using the experimental vapor liquid equilibrium data reported in the Gas Processors Association Research Report 117 (GPA RR 117) [2].

In this Tip of the Month, we will investigate further the methanol loss to the vapor phase and present a simple correlation to estimate methanol K-values. The K-value is defined as the mole fraction of methanol in vapor phase/mole fraction of methanol in aqueous liquid phase. Since the effect of gas composition is small, the K-values will be expressed only in terms of pressure and temperature. The estimated K-value can be used to calculate the rate of methanol loss to the vapor phase.  The objective of this TOTM is to develop a simple and generalized model for estimation of methanol losses in terms of K-values and present a generalized chart which is less dependent on the weight percent of methanol in the liquid aqueous phase.  A step-by-step calculation procedure using K-values will be presented as well.

Figure 1 (FPS). Effect of methanol concentration on methanol loss at -10 °F.

Figure 1 (SI). Effect of methanol concentration on methanol loss at -23.3 °C.

Methanol Loss in Terms of K-Value:

The polar version of Peng-Robinson [3] equation of state (PR EOS) in ProMax [4] was used to generate the required data in the following sections.

Figure 1 indicates that presenting charts for ratio of vapor to liquid composition in terms lb_m of methanol per MMSCF/(weight % methanol in aqueous phase) or kg of methanol per 10⁶ Sm³/(weight % methanol in aqueous phase) is very sensitive to the methanol concentration in liquid phase. Similar methanol concentration dependencies, as shown in Figure 1, were also observed for other isotherms. An alternative is to use the K-Values for the y-axis. However, we have found that the range of ratio of K-Values at 15 weight % to 25 weight % MeOH is from 0.95 to 1.0. Similarly, the range of ratio of K-Values at 50 weight% to 25 weight % MeOH is from 1.0 to 1.03. These ranges are much smaller than the corresponding ratios of methanol losses. Therefore, in the subsequent charts as well as for modeling we will use K-values as the dependent variables.

The K-Values over 25 weight % methanol solution are presented in Figure 2 as a function of pressure and temperature. It should be noted that Figure 2 can be used for methanol concentration up to 70 weight % in aqueous phase.

As shown in Figure 2, at lower temperatures considerable curvature is observed, which makes modeling rather difficult. However, if the same chart is replotted in terms of pressure times K-value, (P)(K) on the y-axis, as shown in Figure 3, much less curvature is observed.

In order to use Figure 2 to calculate the rate of methanol loss to vapor phase, the following steps should be followed:

1. At specified pressure (P) and temperature (T), read methanol K-Value from Figure 2.
2. Convert weight % of methanol (wt%) in aqueous phase to mole fraction (x) by: (1A)
3. Calculate methanol mole fraction in the gas phase (y) by y = (K)(x) (1B)
4. Calculate mass of methanol in vapor phase

FPS: (MeOH lb_m/MMSCF) = [(y lbmole MeOH)/(Total lbmole of gas)](32 lb_m/lbmole

MeOH) (lbmole of gas/379.5SCF) (10⁶ SCF/MMSCF) = 84 321.5 y ≈ 84 322 y    (1C)

SI: (MeOH kg/10⁶ Sm³) = [(y kmole MeOH)/(Total kmole of gas)](32 kg/kmole MeOH) (kmole of gas/23.64 Sm³) (10⁶) = 1 353 638 y ≈ 1 353 640 y     (1D)

A worked example is shown in Appendix A.

Figure 2 (FPS). Variation of methanol K-Values as a function of pressure and temperature.

Figure 2 (SI). Variation of methanol K-Values as a function of pressure and temperature.

Development of Mathematical Model

An alternative to Figure 2 is a simple and generalized correlation which can estimate the K-values easily. This will be handy when one intends to use spreadsheet calculations to estimate methanol vapor losses. A simple model presented by Wilson [5] for light hydrocarbon mixtures is shown in Equation 2.

(2)

where:

Tc_i critical temperature, °R or K

Pc_i critical pressure, psi, kPa or bar

ω_i acentric factor

P system pressure, psia, kPa or bar

T system temperature, °R or K

This correlation is applicable to low and moderate pressure, up to 500 Psia (3.5 MPa), and the K-values are assumed to be independent of composition.

We propose to use a similar equation in the form of equation 3 to estimate methanol K-value at specified pressure and temperature.

(3)

In the above equation, P^*, T^* and ω^* are the normalized pressure, temperature, and acentric factor; respectively. The same data that were used to generate Figure 2 were also used regress the above equation parameters using a non-linear program and the following expressions were developed.

P^* =  P/35  with P in Psia  (4)

T^* = T/615 with T in °R (5)

ω^* = 2.95-(0.02607)P^*+(8.92828×10^-5)(P^*)²-0.851257/ T^* (6)

We will refer to the above model as the “K-Value Express”. Spreadsheet estimation of methanol vapor losses can be developed by using equation 3 to determine methanol K values, and then applying the calculation method as discussed for the application of Figure 2.

Overall, 156 data points covering temperature range of -10 to 100 °F, and pressure range of 100 to 5000 psia for  25 weight % methanol solution were used. The overall Average Absolute Percent Error (AAPE) for the K-Values was 3.6% with a Maximum Absolute Percent Error (MAPE) for K was 19.7%. The MAPE occurred at -10 °F and 2000 psia where ProMax K was 0.001 and K-Value Express K was 0.0008.

Figure 3 presents the comparison between the methanol K-Values calculated by ProMax (solid lines in Figure 3) and those estimated by K-Value Express (identified by dashed line in Figure 3).

Results and Discussion:

The K-Value Express model with the parameters shown in Equations 4 to 6 were used without any further fitting to predict K-values over 15 and 50 weight % methanol for wide ranges of pressures and temperature. For the case of 15 weight % methanol with 149 data points, the AAPE for K was 5.2% with a MAPE of 14.7%. For the case 50 weight % methanol with 155 data points, the AAPE for K was 3.6% with a MAPE for K was 22.1%.

The maximum average absolute % error occurred at -10 °F and 2000 psia where ProMax K was 0.00098 and K-Value Express K was 0.00077 for the case of 50 weight % methanol.

Figure 4 presents the K-Value Express K-Values vs ProMax K-Values for more than 500 data points over 15, 25, and 50 weight % methanol solution. This figure indicates relatively good agreement between the two methods.

Figure 5 is a revised and extended version of Figure 6.20 in reference [1]. Notice y-axis and x-axis variables are switched.  In this way the dependent variable is on the y-axis and independent variable is on x-axis.

Conclusion:

ProMax was used to reproduce Figure 6.20 in reference 1 and presented here in this work as Figure 5.  This figure covers wider ranges of pressure, temperature, and methanol weight percent (up to 70 weight %). However, we suggest using Figure 2 as a better chart since it is less sensitive to methanol weight % in aqueous phase.  In addition, we developed a simple and generalized K-Value Express model that can be used to estimate methanol K-values for wide ranges of pressure, temperature, and methanol weight %. As shown in Figures 3 and 4, the proposed model is in good agreement with the results obtained from ProMax. The sample calculations in Appendix indicate good agreement between the methanol losses to vapor phase obtained from Figures 2, 3, 5, and the K-Values Express model.

Figure 3 (FPS). Comparison of predicted methanol K-Values by ProMax and the proposed Express K-Value model.

Figure 3 (SI). Comparison of predicted methanol K-Values by ProMax and the proposed Express K-Value model.

Figure 4. Accuracy of the proposed Express K-Value model against ProMax

Figure 5. Variation of methanol loss to vapor phase with pressure and temperature for methanol concentration of 25 weight % in the aqueous phase

Figure 5 (SI). Variation of methanol loss to vapor phase with pressure and temperature for methanol concentration of 25 weight % in the aqueous phase

To learn more about similar cases and how to minimize operational problems, we suggest attending the John M. Campbell courses; G4 (Gas Conditioning and Processing) and G5 (Gas Conditioning and Processing-Special).

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. With the most sought after consultants in the oil and gas industry, JMCC provides first choice consulting services to select clients. For more information about services offered by Campbell, visit our website at www.jmcampbellconsulting.com.

By Dr. Mahmood Moshfeghian

Reference:

1. Campbell, J. M. “Gas conditioning and processing, Volume 1: Fundamentals,” John M. Campbell and Company, Norman, Oklahoma, USA, 2001.
2. Ng, H. J., Chen, C. J., and D. B. Robinson, D.B.; RR-117, “The Solubility of Methanol or Glycol in Water – Hydrocarbon Systems,” Gas Processors Association (Mar. 1988).
3. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.
4. ProMax 3.1, Bryan Research and Engineering, Inc, Bryan, Texas, 2010.
5. Wilson, G., “A modified Redlich-Kwong equation of state applicable to general physical data calculations,” Paper No15C, 65^th AIChE National meeting, May, (1968).

Appendix A – Sample Calculations

Determine mass of methanol in vapor per MMSCF (10⁶ Sm³) at 1000 psia (6 897 kPa) and 10 °F (-12.2 °C) over a rich solution containing 25 weight % methanol.

Solution: Method 1 (Figure 5)

At 1000 psia (6 897 kPa) and 10 °F (-12.2 °C) using Figure 5;

FPS:

lbm MeOH/MMSCF/Wt%=0.5  or  lb_m MeOH/MMSCF = (0.5) (25) = 12.5

SI:

kg MeOH/10⁶ Sm³/Wt% = 8.2  or  MeOH/10⁶ Sm³ = (8.2) (25) = 205

Solution: Method 2 (Figure 2 or 3)

At 1000 psia (6 897 kPa) and 10 °F (-12.2 °C) using Figure 2; K=0.00095 or

Figure 3; PK=0.95 psia which gives K=0.00095

Convert 25 wt% to mole fraction by Eq 1A:

Calculate y by Eq 1B: y = (K)(x) = (0.00095)(0.1579) = 0.00015

FPS: Using Eq 1C: (MeOH lb_m/MMSCF) = 84 322 y = 84322(0.00015)  =  12.6

SI: Using Eq 1D: (MeOH kg/10⁶ Sm³) = 1 353 640 y = 1 353 640(0.00015) = 203

Solution: Method 3 (Express K-Value Model)

At 1000 psia (6 897 kPa) and 10 °F (-12.2 °C) using Eqs 3 through 6 calculate K.

Using Eq 4: P^* =  P/35 = 1000/35 = 28.57

Using Eq 5: T^* = T/615 = (460+10)/615 = 0.762

Using Equation 6:

ω^* = 2.95-(0.02607)P^*+(8.92828×10^-5)(P^*)²-0.851257/ T^*

ω^* = 2.95-(0.02607)( 28.57)+(8.92828×10^-5)( 28.57)²-0.851257/ 0.762

ω^* = 1.161

Using Eq 3:

Calculate methanol mole fraction in gas phase ( y) by Eq 1B:

y = (K)(x) = (0.00093)(0.1579) = 0.000147

FPS: Using Eq 1C: (MeOH lb_m/MMSCF) = 84 322 y = 84322(0.000147)  =  12.4

SI: Using Eq 1D: (MeOH kg/10⁶ Sm³) = 1 353 640 y = 1 353 640(0.000147)  = 199

Equations of state (EoS) are used in commercial simulation software for predicting phase behavior and thermodynamic properties. The cubic equations of state (EoS) give relatively accurate results for predicting vapor-liquid equilibria, especially for non-polar or slightly polar systems. Furthermore, these equations can be used to accurately predict vapor densities, enthalpy and entropy. These advantages encourage the researchers to augment EoS ability more than before, especially liquid density, although their accuracy for liquid density prediction is generally poor. The popular EoSs such as Soave-Redlich-Kwong (SRK) [1] and Peng-Robinson (PR) [2] predict liquid density with an average absolute error of about 8%, much higher than several good density correlations. This large magnitude of error is not acceptable by industry; therefore, they are not used for this purpose. In order to overcome this deficiency, volume translated methods have been developed.

In the March 2011 tip of the month (TOTM) we studied a constant volume translation of liquid density Method presented by Peneloux et al. [3] and demonstrated its application for hydrocarbons such as pure methane, n-pentane, decane, pentadecane and carbon dioxide. Considerable improvements, specifically for the low temperature range (T r < 0.8), of saturated specific volume (or liquid density) predicted by PR and SRK were obtained. On the other hand, the constant volume shift fails near the critical temperature, because the slope of volume with respect to temperature greatly increases in this region.

Since Peneloux et al. presented their constant volume translation (shift) method in 1982, several temperature dependent volume correction methods [4-11] have been proposed. In this TOTM we will demonstrate application and accuracy of some of these methods to predict liquid density of common hydrocarbons and non-hydrocarbons in gas processing. We will compare their accuracy against both experimental data and a few correlations.

1. CorrelationsThe following correlations were used in this study.
   1. COSTALD, 1979: The COSTALD correlation by Hankinson and Thomson [12] requires two parameters: SRK, the optimized value of the acentric factor based on the SRK equation of state (EoS) and; V*, the pure component characteristic volume.
   2. RSD, 1972: Spencer and Danner [13] improved the liquid density correlation of Rackett [14]. The improved correlation for saturated liquid density calculation requires only ZRA, the improved compressibility factor.
   3. NM, 1998: Nasrifar and Moshfeghian [15, 16] presented an equation and a set of mixing rules for predicting the liquid density of pure refrigerants and liquefied natural gas.
2. Volume Translated EoS Methods The following volume translated (shift) methods were used in this study.
   1. PRF, 1982: Peneloux et al. [3] proposed the following constant volume shift correction. (1)In the above equation, V^L is the corrected liquid specific volume, V^EoS is the liquid specific volume calculated by the EoS, MW is the molecular weight, p^L is the liquid density, and the correction term or volume shift factor “c” is determined from experimentally measured liquid density. The volume shift factor is normally regressed against several data points. In the absence of experimentally regressed value, it can be estimated as follows: (2)where Z_RA, is the Rackett [15] parameter, R is the gas constant, and T_C and P_C are the critical temperature and pressure, respectively.We determined the optimum value of “c” for each compound by the procedure described in
      the March 2011TOTM.
   2. MT, 1990: Magoulas and Tassios [4] temperature dependent correction factor is calculated as follows: (3) (4) (5) (6)We will refer to this method as MT-VTPR.
   3. TC, 1998: Tsai and Chen [5] temperature dependent correction factor is calculated as follows: (7) (8) (9) (10)
   4. AG, 2001: Ahlers and Gmehling [6] temperature dependent correction factor, c, is calculated as follows:
      (12) 

      (13)

      (14)

      (15)

      (16)

      (17)

   5. LD, 2005: Lin and Duan [7] presented a temperature dependent factor, c, as follows: (18) (19) (20) (21)

In the above equations, Tr is the reduced temperature, ω is acentric factor, T_C, P_C and Z_C are critical temperature, pressure and compressibility factor, respectively. The correction term, c, from the above methods is substituted in equation 1 to calculate the corrected density.

Results and Discussion:

A simple MathCad program was written to perform all of the calculations based on the above methods. We applied the preceding methods to several pure compounds shown in Table 1. The reduced temperature (Tr) and number of points (N) for each compound are also shown in Table 1. This Table presents the summary of the error analysis for different methods for the pure compounds. As can be seen in Table 1, these generalized temperature dependent volume shift methods improve the accuracy but yet not as good as the generalized correlation methods shown in the last three column of Table 1.

Table 1. Summary of error analysis for different methods studied

Figures 1 through 4 present graphical comparisons between the predicted and experimental [17] liquid density values of methane, n-pentane, decane and pentadecane; respectively. Similar trends were observed for the other compounds shown in Table. For clarity, only the results for PR EoS, PRF-VTPR (constant volume shift) and LD-VTPR (temperature dependent volume shift) are presented in these figures. A much better accuracy is obtained near the critical region by applying the temperature dependent volume shift.

Figure 1. Comparison of predicted liquid density of CH₄ by PR EoS, volume translated PRF-VTPR and LD-VTPR against experimental data [17]

Figure 2. Comparison of predicted liquid density of C₅H₁₂ by PR EoS, volume translated PRF-VTPR and LD-VTPR against experimental data [17]

Figure 3. Comparison of predicted liquid density of C₁₀H₂₂ by PR EoS, volume translated PRF-VTPR and LD-VTPR against experimental data [17]

Figure 4. Comparison of predicted liquid density of C₁₅H₃₂ by PR EoS, volume translated PRF-VTPR and LD-VTPR against experimental data [17]

In order to show the sensitivity of the LD-VTPR method and the applicability of tuning its parameters, the Z_C value for pentadecane was changed from 0.243, represented by the solid black curve in Figure 4, to 0.231 which is represented by the dashed black curve in Figure 4. The curve for Z_C=0.231 is labeled as LD-VTPR*. This sensitivity is used for practical applications to tune the volume translated model parameters (e.g. Z_C) to match the predicted liquid density with the experimentally measured data.

Table 1 indicates that considerable improvements are obtained by applying temperature dependent volume shift corrections to the liquid specific volume (or liquid density) near the critical point region. However, the accuracy of the COSTALD, RSD and NM correlations are still by far much better than the volume translation applied to these two EoSs. As shown in Figure 4, further improvement of volume shift methods are obtained by tuning the parameters of volume shift methods with experimental measurement.

To learn more about similar cases and how to minimize operational problems, we suggest attending the John M. Campbell courses G4 (Gas Conditioning and Processing) and G5 (Gas Conditioning and Processing-Special).

John M. Campbell Consulting (JMCC) can provide thermodynamic expertise for gas processing projects to ensure that the developed process model is as accurate as possible. For more information about services offered by JMCC, visit our website at www.jmcampbellconsulting.com.

A free copy of the MathCad Version 14 file showing the calculations steps for methane is available upon request.

By Dr. Mahmood Moshfeghian

Reference:

1. Soave, G., Chem. Eng. Sci., Vol. 27, pp. 1197-1203, 1972.
2. Peng, D. Y., and Robinson, D. B., Ind. Eng. Chem. Fundam., Vol. 15, p. 59, 1976.
3. Peneloux, A. E., Rauzy, E., and Freze, R., Fluid Phase Equilib., Vol. 8, pp. 7-23, 1982.
4. Magoulas, K. and D. Tassios, J. of Fluid Phase Equilibria, Vol. 56, pp. 119-140-445, 1990.
5. Tsai, J. and Y.P. Chen, J. of Fluid Phase Equilibria, Vol. 145, pp. 193-215, 1998.
6. Ahlers, J. and J. Gmehling, J. of Fluid Phase Equilibria, Vol. 191, pp. 177-188, 2001.
7. Lin, H. and Y.Y. Duan, J. of Fluid Phase Equilibria, Vol. 233, pp. 194-203, 2005.
8. Ji, W.R. and D.A. Lempe, J. of Fluid Phase Equilibria, Vol. 130, pp. 49-63, 1997.
9. Pfohl, O., J. of Fluid Phase Equilibria, Vol. 163, pp. 157-159, 1999.
10. Frey, F., Augustine, C., Ciccolini, R.P., Paap, S., Modell, M., and J. Tester, , J. of Fluid Phase Equilibria, Vol. 260, pp. 316-325, 2007.
11. Frey, F., Modell, M., and J. Tester, J. of Fluid Phase Equilibria, Vol. 279, pp. 56-63, 2009.
12. Hankinson, R. W., Thomson, G. H., AIChE J., Vol. 25, no. 4, pp. 653-663, 1979.
13. Spancer, C. F., and Danner, R. P., J. Chem. Eng. Data, vol. 17, no. 2, pp. 236-241, 1972.
14. Rackett, H. G., J. Chem. Eng. Data, vol. 15, No. 4, pp. 514-517, 1970.
15. Nasrifar, Kh. and Moshfeghian, M., Fluid Phase equilibria Vol. 153, 231-242, 1998.
16. Nasrifar, Kh. and M. Moshfeghian, J. of Fluid Phase Equilibria, Vol. 158-160, pp. 437-445, 1998.
17. Vargaftik, N.B., Handbook of Physical Properties of Liquids and Gases (Pure Substances and Mixtures), 2^nd ed., English Translation, Hemisphere Publication, 1975.