K. KAUPERT, OC Turboexpanders, Irvine, California
E
very year turboexpanders generate millions of Euros in rev-enue for hydrocarbon processing plants by removing heat from gas streams, also known as the “turboexpander refrig-eration benefit.” To maximize this financial benefit, accurate gas dynamic performance predictions for turboexpanders are a neces-sity. This requires an accurate thermodynamic equation of state that uses a real-gas model. But, which real-gas model is best? For example, an ideal gas assumption can cause horribly wrong perfor-mance predictions due to gas compressibility at high pressures and low temperatures.1 As a result, all turboexpander manufacturers use real-gas models in their simple gas dynamic sizing predictions.However, beyond the simple sizing predictions, advanced tur-boexpander manufacturers continue to apply real-gas modeling in the detailed computational fluid dynamics (CFD) design of its turboexpanders. This permits turboexpanders, such as shown in Fig. 1 to attain high efficiency levels. Historically, the application of real-gas models in commercially available CFD packages has been problematic or even non-existent. This is due to:
1. Most turbomachinery manufacturers (e.g., those producing gas turbines or turbochargers) are content with an ideal gas law or a simple real-gas model since their compressibility effect is modest and the flow is single phase.
2. Increased computational time is required when a real-gas model is applied in CFD (slower code).
3. Numerical robustness is decreased when a real-gas model is applied in CFD (code can more easily diverge).
4. Difficulty modeling two-phase flow in a wet gas expansion region.
The turboexpander manufacturer is, therefore, confronted with somewhat unique challenges, as the expander inlet gas can exhibit substantial compressibility while the expander outlet gas can exhibit two-phase wet gas flow.2 For example, Fig. 2 shows an expander impeller connected to a rotating assembly. This particular impeller is subjected to both compressible gas and wet gas, which requires accu-rate real-gas modeling in CFD to maximize expander efficiency. But there are many real gas models available in the open literature. So, the question again arises, which model is best? In this article, we will assess several real-gas models used in CFD for turboexpanders.
CONSIDERING THE EQUATION OF STATE
An equation of state relates a fluid’s state variables.3 For turboex-panders, the equation of state is a thermodynamic equation, which mathematically describes the interaction of the macroscopically measurable properties of the process fluid (e.g., the thermodynamic variables of pressure, temperature, density and composition).
Ideal gases and liquids. The simplest, most popular equa-tion of state for gases is the Ideal gas law; it states:
P = RT (1)
For liquids, the bulk modulus and coefficient of thermal expan-sion combine for the equation of state given as:
Turboexpander designed with real-gas modeling in CFD.
FIG. 1
Expander side of a turboexpander rotating assembly.
FIG. 2
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NOVEMBER 2010 HYDROCARBON PROCESSINGKl = dP/(d/) and ␣l =(1/V)dV/dT (2) Over limited ranges of pressure and temperature and without phase changes, these two equations ( Eqs. 1 and 2) give reasonable property predictions (e.g., the ideal gas law for ethane at less than 5 bar pressure and higher than 100°C temperature gives less than 2%
error on the density). But turboexpanders routinely handle gases outside the limited range of the Ideal gas law and with two-phase flow. In essence, real gas and liquid modeling is obligatory.
Real gases and liquids. The need for accurate equations of state has resulted in an abundance of real-gas models in the open literature. So numerous are the real-gas models that they could not possibly all be evaluated in this short article. However, these five real gas equations of state do find widespread application:
1. Peng-Robinson (PR) 2. Soave-Redlich-Kwong (SRK)
3. Benedict-Webb-Ruben-Starling (BWRS) 4. Lee-Kesler-Plöcker (LKP)
5. GERG.
These five equations of state are compared here with CFD results through the expander side of a turboexpander. The
math-ematics of these equations of state are presented elsewhere; they are large in size and require detailed explanation, among them Refs. 4 and 5. The American Gas Association (AGA-8) is not evaluated here as it poorly predicts the weight fraction of liquid in the wet-gas region at the expander outlet.
Another very common and simple real gas equation of state utilizes the compressibility factor, Z and is called the universal gas law, given as:
P = ZRT (3)
Unfortunately, the determination of Z for gas mixtures in turboexpanders is very difficult; this equation (containing Z ) is not evaluated here.
CFD WITH REAL-GAS EQUATIONS OF STATE
CFD using a real-gas equation of state is a time-consuming task. In the past, this author has pursued three methods to incor-porate real-gas equations in CFD. The first method is to program the real-gas equation of state and patch it into the CFD code.
While this method is the most straightforward, it frequently leads to numerical convergence difficulties in the transition region from single-phase gas to two-phase wet gas. This concern can be addressed with numerical damping routines that are ingeniously applied by some CFD vendors. The second method is to use the real-gas equation of state to prepare a set of “look-up” tables in the desired pressure and temperature range of the gas. This method is more tedious for pre-processing the CFD but tends to avoid convergence problems while running the code. A third method to follow uses the tables from the second method to create a set of polynomials to approximate the equation of state. But the third method can lead to large errors outside a prescribed temperature and pressure range and convergence difficulties for the CFD.
Density as a function of pressure and temperature from a real-gas model and used in a “look-up” table for CFD.6 FIG. 3
Enthalpy vs. entropy diagram for a hydrocarbon gas example.
FIG. 4
TABLE 1. Gas conditions for the expander side of the NGL hydrocarbon example
Gas dynamics
Job: Example NGL fractionation
Component MW Expander, mol%
Methane 16.0430 81.000%
Ethane 30.0700 11.000%
Propane 44.0970 5.000%
i-Butane 58.1230 0.800%
n-Butane 58.1230 0.900%
i-Pentane 72.1500 0.344%
n-Pentane 72.1500 0.100%
n-Hexane 86.1770 0.050%
C7+ 110.00 0.002%
Nitrogen 28.0134 0.800%
Carbon dioxide 44.0100 0.003%
Water 18.0153 0.001%
Total 100%
Given process conditions Rated case expander
Molecular weight 20.087
Inlet pressure, P1, MPa a 6.91
Inlet temperature, T1, °C 0.2
Outlet pressure, P2, MPa a 3.48
Mass flow, kg/sec 13
Volume flow, Nm3/h 52036
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71Most CFD vendors have followed the direction of the second method or at least opened up their code for input from such “look-up” tables. It is the second method that the author has applied here. Essentially, the real-gas equation of state is used to generate the thermodynamic “look up” tables for P=P (u,), T =T(u,), P
= P (h,s), = (h,s), s=s(h,P), h=h(s,P), u=u (P,T) and = (P,T) where the variables are:
P Pressure T Temperature
Density u Internal energy h Enthalpy s Entropy.
An example of such a “look-up” table is plotted in Fig. 3, taken from Ref. 6.
Example: CFD for real gases in natural gas liquids.
To protect client confidentiality, a generic gas composition and condition were selected for this example. However, both gas composition and condition are representative of a natural gas liquids feedstock to the expander side of a turboexpander. The gas composition and condition are seen in Table 1. The enthalpy vs. entropy diagram is seen in Fig. 4 along three lines for the gas expansion. The red line “a-b” represents an isenthalpic gas expan-sion as would be experienced through a Joule-Thompson valve.
The blue line “a-c” represents an isentropic, or perfect, expan-sion of the gas. The green line “a-d” represents the gas expanexpan-sion through the turboexpander and indicates a lower outlet tempera-ture than the isenthalpic red line expansion.
This highlights the benefit of using a turboexpander vs. a Joule-Thompson valve for the gas expansion—a colder gas outlet tem-perature with the green line and heat removal from the expander gas stream. Table 2 shows the outlet gas conditions at the point “d”
compared between the ideal gas and five real gas models. It is seen immediately that the ideal gas model gives the lowest expander outlet temperature along with the largest
expander wheel (impeller) output power.
However, the compressibility Z is seen as 1.0, which is not realistic for the expander inlet or outlet gas conditions. Accordingly, applying an ideal gas model would lead to a substantial modeling error for this gas.
Among the five real-gas models, the PR model gives the lowest expander outlet tem-perature, lowest output power, and lowest efficiency. The GERG, LKP and BWRS models all give similar outlet temperature, output power and efficiency, as well as, simi-lar weight liquid percentages. Based on
mea-surements for similar gas compositions and conditions, the GERG model was found to be most accurate for this gas case. Although, the LKP and BWRS models would be acceptable as well.
Results from applying the GERG model through the use of generated property tables in the CFD are seen in the values given in Figs. 5 and 6 at the expander design point. Fig. 5 shows the distribution of temperature as predicted with the GERG real-gas equation of state in the natural gas mixture. A rapid temperature change is seen through the expander nozzles; it is typical for turbo expanders, as the temperature decreases due to the acceleration of the flow. In Fig. 6, the relative velocity vectors in the expander impeller are seen, again as computed with the GERG real-gas model. The overall image is shown on the left side of the figure and a zoom at the expander impeller trailing edge is seen at the
CFD results for the temperature reduction in the expander per GERG real-gas model.
FIG. 5
Left: CFD results using the GERG real-gas model for the relative velocity vectors in the expander impeller (wheel). Three blades are shown along the shroud showing the parallel flow to the blade at the outlet. A zoom is shown to the right at the impeller outlet.
FIG. 6
TABLE 2. Comparison table of CFD results from five real gas models at the expander outlet
Ideal-gas model Real-gas model Real-gas model Real-gas model Real-gas model Real-gas model
Computed outlet conditions Ideal GERG PR LKP SRK BWRS
Outlet temperature, T2, °C –33.7 –28.8 –29.4 –28.9 –28.3 –28.7
Zin/Zout 1.0/1.0 0.72/0.78 0.69/0.76 0.71/0.79 0.70/0.77 0.71/0.78
Specific enthalpy ⌬Hs, kJ/kg 86.84 54.03 52.12 54.37 52.52 53.69
Expander isentiopic efficiency, % 82.7 85.3 84.6 85.4 84.8 85.2
Expander impeller power, kW 933.2 599.1 573.2 603.6 579.0 594.7
Outlet weight liquid, % unable 13.60 12.74 13.52 13.32 13.64
PLANT SAFETY AND ENVIRONMENT SPECIALREPORT
right, showing that the relative velocity vectors closely follow the blade surface without any recirculation zones. Such visualizations are the key to optimizing the gas path and efficiency of expanders by reducing any unwanted entropy generation.
Real-world view. Real-gas modeling is needed for accurate gas dynamic performance predictions in turboexpander CFD to optimize expander efficiency. As the five models have shown, dif-ferent results are attained by applying difdif-ferent real-gas models. It is important for turboexpander manufacturers to use test results and to also obtain detailed field feedback. Together, this will allow selecting the real-gas model best suited to a particular application.
In this article, only one example was presented. It is not wise to generalize on the basis of just this one example. The scope is widened with other published or in-house data. Table 3 offers a number of generalized recommendations as to which equation of state should be considered for CFD in modern turboexpanders.
Finally, interested readers are always encouraged to review comprehensive texts on turbomachines, (such as Ref. 7) or more elementary books on turboexpanders (Ref. 8).
As energy conservation has become one of the world’s foremost priorities, the importance of efficient turboexpanders continues to increase. HP
LITERATURE CITED
1 Beinecke, D. and K. Lüdtke, Die Auslegung von Turboverdichtern unter Berücksichtigung des realen Gasverhaltens, VDI-Berichte 487, VDI-Verlag Düsseldorf, pp. 271–279, (in German), 1983.
2 Kaupert, K. A., “Design of Two-Phase Flow Air Separation Turboexpanders,”
Cryogenic Technology Journal China, Vol. 1, pp. 47–52 (in Chinese), 2010.
3 Zemansky, M. W., Heat and Thermodynamics: An Intermediate Textbook, McGraw-Hill Book Co., 6th edition, 1981.
4 Modisette, J. L., “Equation of State Tutorial,” Pipeline Simulation Group (PSIG), Paper 0008-2000, 2000.
5 Kunz, O., R. Klimeck, W. Wagner and M. Jaeschke, The GERG-2004 Range Equation of State for Natural Gases and Other Mixtures, Berichte 557 Reihe 6, VDI-Verlag Düsseldorf, 2007.
6 Numeca, Numeca Fine Users Manual Version 6.1-1, Numeca International, February 2003.
7 Lüdtke, K., Process Centrifugal Compressors, Springer Verlag, Heidelberg, 2004.
8 Bloch, H. P. and C. Soares, Turboexpanders and Process Applications, Gulf Publishing Co., Houston, ISBN 0-88415-509-9, 2001.
TABLE 3. Generalized recommendations regarding applicable equations of state
Gas type GERG PR LKP SRK BWRS
Hydrocarbon +/– 2% +/– 5% +/– 2% +/– 3% +/– 2%
Air +/– 2% +/– 5% +/– 3% +/– 3% +/– 2%
CO2 +/– 3% +/– 6% +/– 4% +/– 4% +/– 1%
H2 +/– 4% +/– 7% +/– 4% +/– 2% +/– 5%
NH3 not good not good not good not good not good
Note: the PR equation of state was the least accurate for all gas types but it is also the sim-plest to apply. For ammonia, none of the five real-gas models performed satisfactorily and in-house models are still relied upon.
Dr. Kevin Kaupert is the director of technology at OC Tur-boexpanders. He holds a doctorate in turbomachinery engineering from the ETH Zurich Swiss Federal Institute of Technology. He has over 25 years of experience in turbomachinery for cryogenics, power generation and aerospace applications.
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