Modeling real gas equations of state in high density combustion

Transcription

1 26 th ICDERS July 3 th August 4 th, 217 Boston, MA, USA Modeling real gas equations of state in high density ombustion Chenwei Zheng, Deshawn Coombs, Ben Akih-Kumgeh Department of Mehanial and Aerospae Engineering Syrause University Syrause, New York, USA 1 Motivation and Objetives Combustion systems inreasingly operate at high pressure and low temperature onditions. These extreme operating onditions lead to high densities of the fuel and air mixtures whih invalidate the ideal gas modeling often used in analysis. The ideal gas model an indue errors in predited ombustion properties. For more physially realisti reating flow simulations, real gas effets must therefore be taken into aount. It is preferable that any method used to address this problem be omputationally onvenient. There are many models of real gas equations of state (EOS), starting from the more theoretially grounded Van der Waals equation to the very omplex empirial EOS, suh as the Benedit-Webb-Rubin (BWR) equation. It an be observed that most pratial real gas EOS are modifiations of the Van der Waals equation of state. Currently, in order to simplify omputational tasks assoiated with EOS evaluations, some researh projets use ubi equation of state suh as the Redlih-Kwong (RK) model [1], Soave-Redlih-Kwong (SRK) [2] and Peng-Robinson (PR) model [3]. Aording to Reitz [4], the general form of ubi equation of states an be written as : P = RT V b a V 2 + qbv + wb 2 (1) where V is the volume of the system and R is the universal gas onstant, q and w are onstants depending on the type of ubi EOS. For the PR-EOS their values are: q = 2, w = 1, and q = 1, w = for the RK and the SRK-EOS. For pure speies, a and b are funtions of temperature. The expressions for EOS parameters are presented in Table 1, where T and P are ritial temperature and ritial pressure, ω is the aentri fator, κ is a onstant that depends on the aentri fator and T r is defined as redued temperature, T r = T/T. Correspondene to: bakihkum@syr.edu 1

2 Table 1: Parameters for ubi EOS Equation b a q w.867rt.4278r 2 T 2.5 Redlih Kwong (RK) 1 P T P.8664RT.42747R2 T 2.5 [1 + κ(1 T Soave Redlih Kwong (SRK) r.5 )] 2 P P 1 κ = ( ω.176ω 2 ).778RT.45724R2 T 2 [1 + κ(1 T Peng Robinson (PR) r.5 )] 2 P P 2-1 κ = ω.26992ω 2 The binary mixing rule with regards to a and b suggested by Peng and Robinson [3]: a = x i x j a ij i j b = x i b i i (2) a ij = (1 δ ij )a 1/2 i a 1/2 j where δ ij is an empirially determined binary interation oeffiient haraterizing the binary formed by omponent i and omponent j. δ ij is proposed at the range between.2 to.2 by Abudour et al [5]. Another method to deal with the mixing rule is the pseudoritial method based on orresponding state priniple (CSP). To obtain equation of state, we first alulate the pseudoritial properties for the mixture. Aording to Polling et al. [6] the pseudoritial temperature T m an be estimated by the simplest mole fration average method, Kay s rule [7]: T m = n i=1 y it i. As for the pseudoritial pressure, P m, a mole-fration average of pure-omponent ritial pressures is normally unsatisfatory. This is beause the ritial pressure for most systems goes through a maximum or minimum with omposition. The only exeptions are if all omponents of the mixture have quite similar ritial pressures or ritial volumes. The simplest rule whih an give aeptable P m values for two-parameter or three-parameter CSP is the modified rule of Prausnitz and Gunn [8]: P m = Z mrt m V m = ( n i=1 y iz i ) R ( n i=1 y it i ) ( n i=1 y iv i ) where all of the mixture pseudoritials Z m, T m, and V m are given by mole-fration averages (Kay s rule) and R is the universal gas onstant. The mixture pseudo aentri fator is ommonly given by a mole fration average Joffe [9]: (3) ω m = n y i ω i (4) i=1 The need to aount for real gas effets in Computational Fluid Dynamis (CFD) simulations have been realized, with the result that inreasingly more studies onsider this problem. The text by Sirignano [1] 26 th ICDERS July 3 th August 4 th, 217 Boston, MA 2

3 desribes the use of ubi equations of state by the spray ombustion ommunity. Reent omprehensive reviews of work on real gas behavior in multiphase reating flows have been presented by Bellan [11], Yang [12] and Sazhin [13]. Also, vapor-liquid equilibrium (VLE) proesses of n alkanes from C2 to C2 were desribed by Zhu and Reitz [4]. Both PR- and SRK-EOS were found to be superior to RK-EOS. Oefelein and Yang [14] employed Large Eddy Simulation (LES) to investigate two dimensional mixing and ombustion of hydrogen and oxygen. Thermodynami non-idealities and transport anomalies was evaluated diretly from fundamental theories over the fluid regime. A 32-term BWR EOS proposed by Jaobsen and Stewart [15] was used in the viinity of the ritial point while elsewhere the SRK EOS from Poling et al. [6] was used. With the need to inlude more hemial kineti effets in ombustion simulations, one enounters a high number of speies whose ritial properties are not well known or would be too um for manual assignment. This paper develops a framework to generate parameters of real gas EOS from transport and thermodynamis data already used in ombustion hemial kineti models. We first demonstrate through simulations that using real gas EOS leads to differenes in the flow field at ombustion relevant onditions, ompared to simulation based on ideal gas EOS. We then desribe the method to obtain required real gas EOS parameters from hemial kineti model and their assoiated data. We further highlight an existing problem in the way urrent transport data are estimated, sine signifiant differenes are observed. 2 Researh Approah The researh onsists of two parts. The first part aimed at justifying the motivation above. We show using LES of propane jet that resulting flow fields an be dependent on the type of EOS used. The seond part disusses how the properties of a real gas EOS an be estimated from available ombustion hemistry model resoures. Speifi Volume [m 3 /kg] NIST Data Peng-Robinson Redlih-Kwong Soave-Redlih-Kwong Idea Gas Speifi Volume [m 3 /kg] NIST Data Peng-Robinson Redlih-Kwong Soave-Redlih-Kwong Idea Gas Temperature [K] (a) T-V variation at P = 1 atm Temperature [K] (b) T-V variation at P = 5 atm. Figure 1: Variation of propane speifi volume with temperature, highlighting importane of real gas EOS. 2.1 Propane EOS Estimation Propane temperature and speifi volume relationship is shown in Fig.1. Real gas EOS and ideal gas EOS results are ompared with NIST experimental data. The temperature range is hosen based on the availability of experimental data. 26 th ICDERS July 3 th August 4 th, 217 Boston, MA 3

4 Fig.1a shows that at 1 atm, the speifi volume predited by real gas EOS is very lose to experimental data, while the ideal gas EOS shows some deviation (up to 2%). When the pressure is inreased to 5 atm as shown in Fig.1b, the error in speifi volume indued by the ideal gas assumption is about 7%, while the real gas EOS is lose to the measured data. The impat of this on flow field preditions an be assessed by arrying out omparative flow simulations. 2.2 Simulation of a Propane Jet The example problem we onsider here onsists of a propane jet introdued into a hot hamber of air. The onditions are suh that the hamber is at T = 9 K, and the subsoni propane jet is at T = 4 K. Simulations are arried out at 1 atm and at 5 atm, using both ideal gas EOS and real gas (PR) EOS for air and propane. The real gas EOS require ritial state parameters whih for propane are T = K, P = bar, ω =.153; and for air are T = K, P = bar, ω =, here ω is the aentri fator. In order to properly resolve the flow field, the turbulene model hosen here is LES with the subgrid-sale (SGS) model, Wall-Adapting Loal Eddy (WALE). The mesh onsists of 2, 393, 42 ells in total with a minimum ell size x min =.1 mm. The injetion veloity are the same between ideal gas EOS and real gas EOS, the speed of jet is m/s with 1 ms injetion duration. Simulation is arried out using the segregated solver of Star-CCM+. The time disretization is impliit, and to obtain good onvergene of the solver at eah time step the following unrelaxation fators were used,.8 for veloity,.2 for pressure,.9 for both the speies and energy equations. A maximum of 5 iterations for the iterative matrix solvers was used to ensure the solution reahed a onverged state at eah time step. The time step was set to whih lead to a maximum loal CFL number of approximately 2.4. The resulting flow field an be ompared using salars suh as the propane mass fration and temperature. Figure 2a and 2b show the mass fration field at 1 atm, whih is to be ompared with the fields in Fig.2 and 2d, obtained at a pressure of 5 atm. We see that in both ases, the flow fields are different, sine the spread and penetration depth in the ase of PR-EOS are greater than those of ideal gas EOS. Further at high pressure, hene higher density, the differene is more pronouned as would be expeted. At 1 atm, PR EOS is prediting slightly lesser density than ideal gas, the differene is within 1%. In Figure 2, the penetration depth of 1 atm is about.6 for either real gas eos and ideal gas EOS. However, the spread angle is different. The differene in the flow pattern an be introdued by the departure funtion in energy: du = C v dt + [ T [ dh = C p dt + v T ( ) P T P v ( ) v T P ] dv (5) ] dp (6) where the seond terms are the departure terms. While ideal gas EOS will result in zero for the departure term, Real Gas EOS an ontribute a value in the funtion. Therefore, the energy funtion would be slightly different for the two EOS, and eventually ausing the variation in the flow field. The temperature fields also reflet differenes that align with the mixing field. These fields are affeted through the use of the EOS to relate the pressure, density, and temperature, whih appear in the onservation equations. 26 th ICDERS July 3 th August 4 th, 217 Boston, MA 4

5 Y_C3H8 1. Y_C3H (a) P = 1 atm, ideal gas EOS. () P = 5 atm, ideal gas EOS Y_C3H8 1. Y_C3H (b) P = 1 atm, real gas (PR) EOS. (d) P = 5 atm, real gas (PR) EOS. Figure 2: Propane mass fration field at 1 ms after injetion onset. 3 Method of EOS Parameter Estimation Our goal is to determine for eah speies in a given mehanism its real gas EOS parameters, i.e. T, P, ω. Using an appropriate mixing rule, the EOS parameters for a given mixture ombustion an be determined from those of individual speies. There is a onnetion between the ritial state properties and the parameters of intermoleular potentials (e.g. Lennard-Jones 12-6 potential). The latter are used in ombustion flows to generate required transport properties (visosity, ondutivity and thermal diffusivity). One an reverse-engineer the transport data to obtain the ritial state parameters needed for the EOS. Many empirial relations between parameters of intermoleular potentials and the ritial state properties have been proposed [16]. Some require knowledge of the aentri fator while others do not. The most widely used of these equations are in the form: σ ( P T ) 1/3 = a σ b σ ω; and ε k B T = a ε b ε ω (7) where ε is the depth of the potential well and σ is the intermoleular separation at whih the potential energy is zero. In hemial kineti models, the estimation rules proposed in Kee et al. [17] are often used. It appears that the Tee et al. [16] performs better than the Kee [17] when predited transport variables are ompared 26 th ICDERS July 3 th August 4 th, 217 Boston, MA 5

6 Table 2: Parameters for estimation Referene Tee [16] Kee [17] Holley [18] a σ b σ a ε b ε with observations. We note that T, P, ω are observable quantities whereas the auray of σ, ε an only be inferred from their predition of transport properties. This irularity an lead to a verifiation problem. Holley et al. [18] have suggested updated oeffiients for the Tee orrelation. What this means is that with an appropriate orrelation, one an obtain neessary T and P from transport data. The aentri fator, ω, poses a hallenge that an only be resolved through struture-ativity orrelations. For instane, for alkanes, it an established the ω depends on the number of arbon (N ) [19]: ω =.4423[ln( N )] The neessary arbon number an be obtained from the model thermodynami data file. Taken together, the above offers ways to generate ritial state properties, hene EOS parameters from urrently available resoures T [K] (a) T based on Kee method. T [K] () T based on Tee method. T [K] (e) T based on Holley method P [atm] 4 3 P [atm] 4 3 P [atm] (b) P based on Kee method (d) P based on Tee method (f) P based on Holley method. Figure 3: Critial properties estimation based on intermoleular onstants. 26 th ICDERS July 3 th August 4 th, 217 Boston, MA 6

7 4 Critial Properties Estimation One might hek the effetiveness of the proposed approah by omparing predited ritial state parameters with measured data for some speies of relevane to ombustion. An example is shown in Fig.3 for C1-C12 n-alkanes, with ritial properties from the literature [6]. Intermoleular potential parameters σ and ε are hosen from the JetSurf 2. model [2] and n-heptane mehanism. The observed differenes in T C and P trae bak to differenes in the transport data σ and ε, whih are often not given ritial attention in model evaluations. As shown in the Figure 3, Tee s orrelation with hemial kineti database provides aurate ritial properties. While for large arbon alkanes, Holley s orrelation with JetSurf 2. shows good areement with experimental data, for small arbon alkanes, the orrelation results in inonsistany. Kee s orrelation does not require aentri fator but errors our as arbon number inreases due to deviation from spherial shape. Therefore, we reommend using Tee s orrelation with database. 5 Conlusion In this paper, the problem of real gas equations of state for high density flow simulations is onsidered. The importane of suh equations of state is underlined by omparing the large eddy simulation results of a propane jet into an air vessel realized using large eddy simulations with ideal and real gas equations of state. It is then suggested that parameters of a ubi equation of state, namely, the ritial state variables, an be determined from the transport data of hemial kineti models. It is further observed that the transport data of various hemial kineti models are not neessarily onsistent, suggesting further attention to this modeling problem. Aknowledgement The first author is supported through a Syrause University initiative funded by the National Siene Foundation under Grant No. DGE Referenes [1] Redlih, O., Kwong, J. N. (1949). On the thermodynamis of solutions. V. An equation of state. Fugaities of gaseous solutions. Chemial reviews, 44(1), [2] Soave, G. (1972). Equilibrium onstants from a modified Redlih-Kwong equation of state. Chemial Engineering Siene, 27(6), [3] Peng, D. Y., Robinson, D. B. (1976). A new two-onstant equation of state. Industrial Engineering Chemistry Fundamentals, 15(1), [4] Zhu, G. S., Reitz, R. D. (22). A model for high-pressure vaporization of droplets of omplex liquid mixtures using ontinuous thermodynamis. International Journal of Heat and Mass Transfer, 45(3), th ICDERS July 3 th August 4 th, 217 Boston, MA 7

8 [5] Abudour, A. M., Mohammad, S. A., Robinson Jr, R. L., Gasem, K. A. (214). Generalized binary interation parameters for the PengRobinson equation of state. Fluid Phase Equilibria, 383, [6] Poling, B. E., Prausnitz, J. M., & O onnell, J. P. (21). The properties of gases and liquids (Vol. 5). New York: Mgraw-hill. [7] Kay, W. B. (1936). Density of hydroarbon gases and vapors at high temperature and pressure. Ind. Eng. Chem, 28(9), [8] Prausnitz, J. M., & Gunn, R. D. (1958). Volumetri properties of nonpolar gaseous mixtures. AIChE Journal, 4(4), [9] Joffe, J. (1971). Combining rules for the third parameter in the pseudoritial method for mixtures. Industrial & Engineering Chemistry Fundamentals, 1(3), [1] Sirignano, W. A. (21). Fluid dynamis and transport of droplets and sprays. Cambridge University Press. [11] Bellan, J. (2). Superritial (and subritial) fluid behavior and modeling: drops, streams, shear and mixing layers, jets and sprays. Progress in energy and ombustion siene, 26(4), [12] Yang, V. (2). Modeling of superritial vaporization, mixing, and ombustion proesses in liquidfueled propulsion systems. Proeedings of the Combustion Institute, 28(1), [13] Sazhin, S. S. (26). Advaned models of fuel droplet heating and evaporation. Progress in energy and ombustion siene, 32(2), [14] Oefelein, J. C., Yang, V. (1998). Modeling high-pressure mixing and ombustion proesses in liquid roket engines. Journal of Propulsion and Power, 14(5), [15] Jaobsen, R. T., Stewart, R. B. (1973). Thermodynami properties of nitrogen inluding liquid and vapor phases from 63K to 2K with pressures to 1, bar. Journal of Physial and Chemial Referene Data, 2(4), [16] Tee, L. S., Gotoh, S., Stewart, W. E. (1966). Moleular parameters for normal fluids. Lennard-Jones 12-6 Potential. Industrial Engineering Chemistry Fundamentals, 5(3), [17] Kee, R. J., Coltrin, M. E., Glarborg, P. (25). Chemially reating flow: theory and pratie. John Wiley Sons. [18] Holley, A. T., You, X. Q., Dames, E., Wang, H., Egolfopoulos, F. N. (29). Sensitivity of propagation and extintion of large hydroarbon flames to fuel diffusion. Proeedings of the Combustion Institute, 32(1), [19] Han, B., Peng, D. Y. (1993). A groupontribution orrelation for prediting the aentri fators of organi ompounds. The Canadian Journal of Chemial Engineering, 71(2), [2] H. Wang, E. Dames, B. Sirjean, D. A. Sheen, R. Tango, A. Violi, J. Y. W. Lai, F. N. Egolfopoulos, D. F. Davidson, R. K. Hanson, C. T. Bowman, C. K. Law, W. Tsang, N. P. Cernansky, D. L. Miller, R. P. Lindstedt, A high-temperature hemial kineti model of n-alkane (up to n-dodeane), ylohexane, and methyl-, ethyl-, n-propyl and n-butyl-ylohexane oxidation at high temperatures, version 2., September 19, 21 ( 26 th ICDERS July 3 th August 4 th, 217 Boston, MA 8