Mass Coupling in the CFD Simulation of Diesel Sprays Combustion Simulation
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Mass Coupling in the CFD Simulation of Diesel Sprays Subtask 1.2H: Fuel Spray Modeling for Diesel Combustion Simulation Krista Stalsberg-Zarling, Kathleen Feigl, Franz X. Tanner Michigan Technological University Martti Larmi Helsinki University of Technology IEA 2005 Acknowledgements: Support provided by the National Technology Agency of Finland, TEKES. Overview Theoretical Considerations: Droplet evaporation model and mass coupling Problems of Nearest Neighbor Method (NNM) of KIVA3 when modeling mass coupling Interpolation of gas properties at droplet locations Distribution of spray source terms Phenomena that occur due to mesh refinements Model description: Lagrange polynomial interpolation Results and discussion Comparison of Lagrange and Kiva’s nearest neighbor method, for mass and momentum coupling Criteria: spray penetrations, velocities, fuel vapor, computational stability Summary and conclusions Future work IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 2 Drop Evaporation Model Rate of change of drop radius: d 2 ( ρD) gas (Tˆ ) r = Bd Shd dt ρd Sherwood number of mass transfer for gas: (1+ Bd ) Shd = (2.0 + 0.6Re Sc )ln Bd 1/ 2 d Mass transfer number: Y1* − Y1 Bd = 1− Y1* IEA 2005 1/ 3 d Droplet Reynold’s number: r 2 ρrv r Red = µgas (Tˆ ) MICHIGAN TECHNOLOGICAL UNIVERSITY Schmidt number: µgas (Tˆ ) Sc d = ( ρD) (Tˆ ) gas 3 Drop Evaporation Model Energy balance equation: 4 πr 3 ρ d c l dTd dr = 4 πr 2Qd + 4 πr 2 L(Td ) dt dt Rate of heat conduction (Ranz-Marshall correlation): (T − Td ) Qd = K gas (Tˆ ) Nud 2r Heat conduction coefficient: K1Tˆ 3 / 2 ˆ K gas (T ) = Tˆ + K 2 Prandtl number: Prd = IEA 2005 µgas (Tˆ )c p (Tˆ ) gas K gas (Tˆ ) Nusselt number for heat transfer: Nud = (2.0 + 0.6Re1/d 2 Prd1/ 3 )ln (1+ Bd ) Bd Latent heat of vaporization: L(Td ) = hl (Td ) − hl ((Td ), pv (Td )) MICHIGAN TECHNOLOGICAL UNIVERSITY 4 Theoretical Considerations: Interpolation of Gas Properties at Drop Location Mass and energy exchange must be modeled Location of gas properties not coincident with drops NNM can’t resolve steep gradients, e.g. velocity and fuel vapor concentration Inaccurately interpolated properties can lead to over- or under-estimated evaporation rates IEA 2005 Node ρ, T, p, etc… v Drop Gas cell Thermodynamic properties stored at cell centers; gas velocities stored at nodes. MICHIGAN TECHNOLOGICAL UNIVERSITY 5 Theoretical Considerations: Distribution of Evaporated Fuel NNM mass exchange: vapor uniformly and instantaneously distributed in computational cell May result in unnatural vapor concentrations and gradients Leads to instabilities of flow solver and inaccurate evaporation rates IEA 2005 droplet s droplets gas cell fuel vapor Effects of uniform vs. non-uniform source/drop distribution on vapor distribution. MICHIGAN TECHNOLOGICAL UNIVERSITY 6 Theoretical Considerations: Mass Coupling Phenomena due to Mesh Refinement Non-uniform drop distribution may lead to Vapor diffusion: coarse meshes may result in low fuel vapor mass fraction and over-estimated evaporation Vapor concentration: cell refinement may result in high fuel vapor mass fraction and underestimated evaporation rate IEA 2005 droplets computational cell fuel vapor Vapor diffusion and concentration for uniform and non-uniform drop/source distribution. MICHIGAN TECHNOLOGICAL UNIVERSITY 7 Theoretical Considerations: Counteracting Momentum Coupling Phenomena Due to Mesh Refinements Non-uniform drop distribution leads to counteracting phenomena Velocity Diffusion: mesh refinements result in greater momentum diffusion Velocity Concentration: same momentum transferred onto smaller gas mass gas cell drops velocity vectors Velocity concentration and velocity diffusion due to mesh refinement IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 8 Theoretical Considerations: Time-Step and Stability Computational instabilities and/or inaccuracies arise due to : Mass coupling: unnatural vapor concentrations lead to under- or over-estimated evaporation rates and steep vapor concentration gradients Momentum coupling: unnatural large velocity concentrations at single nodes and steep velocity gradients Reduce mesh dependence, computational instabilities and increase computational times by Improving accuracy of interpolated gas properties More uniformly distributing source terms IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 9 Lagrange Interpolation Model n V (x, y, z,t) = ∑Vi (t)Gi (x, y, z) i=1 1. 2. 3. 4. 5. (X3,Y3) (X4,Y4) 3 4 A volume or cell is associated with a ‘master’ element. The particle location in this volume is mapped onto the master element. The interpolation polynomials of the master element are evaluated at the particle location and used as weights. These weights are used to interpolate the gas properties at the drop location. The weights are used again to distribute the newly computed droplet source terms. IEA 2005 y (Xp,Yp) 2 1 (X2,Y2) (X1,Y1) x Global coordinates s (-1,1) (1,1) 4 3 r 1 (Rp,Sp) 2 el (1,-1) (-1,-1) v Local coordinates et MICHIGAN TECHNOLOGICAL UNIVERSITY 10 Lagrange Interpolation Model Gas cell Mass coupling cell centered around regular mesh node Gas cell Gas mesh node Gas mesh node Droplet Droplet Mass coupling cell for interpolation of cell-centered gas properties and for mass source distribution. IEA 2005 Momentum coupling cell: cell in which drop is located Gas cell center Momentum coupling cell for interpolation of gas properties stored at nodes and for momentum source distribution. MICHIGAN TECHNOLOGICAL UNIVERSITY 11 Experimental and Computational Details Fluid Density, 15°C Kinematic viscosity, 30°C Fuel pressure Injection velocity Spray angle Gas environment Density Temperature Vessel wall temperature Nozzle diameter Discharge coefficient, Cd Area contraction coefficient, Ca Velocity coefficient, Cv IEA 2005 Philips Research Grade D2 (Cummins diesel #2 for simulations) 843.2 kg/m3 2.83x10-6 mm2/s 142 Mpa 479.4 m/s 14.8° 90.33% N2, 6.11% CO2, 3.56% H2O 13.9 kg/m3 1001 K 452 K 0.257 mm 0.62 0.81 0.76 Number of cells Polar mesh Radial Azi. Axial Total P1 9 30 18 6269 P2 13 30 27 10530 P3 20 30 40 24000 P4 30 30 60 54000 Cylinder: 100 mm dia. x 100 mm length Radius of axial cells: 1.0 mm Injection cell height: 1.0 mm 100,000 parcels, using NTC collision model MICHIGAN TECHNOLOGICAL UNIVERSITY 12 Lagrange Method for Mass & Momentum Coupling vs. NNM: Liquid and Vapor Penetrations Improved mesh independence of liquid spray penetrations using Lagrange method, slight improvement for vapor penetrations IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 13 Lagrange Method for Momentum Coupling vs. Lagrange for Mass/Momentum Coupling More uniform distribution of vapor mass may reduce vapor concentration in areas of dense drops, increasing evaporation Increased evaporation leads to lower spray penetrations IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 14 Lagrange Method for Mass/Momentum Coupling vs. NNM: Gas Velocities and Fuel Vapor Similar mesh independent evaporation rates for Lagrange and NNM Higher spray penetrations for NNM compared with Lagrange method: velocity concentration results in higher gas velocities IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 15 Time-step and Stability Mesh super coarse (P1) CPU Time (hrs.) (max. time-step = 1.0e-6 s) NNM Lagrange Lagrange momentum momentum momentum + + + NNM mass NNM mass Lagrange mass coupling coupling coupling 0.7 1.1 0.3 coarse (P2) 1.9 2.4 1.0 medium (P3) 5.0 5.0 2.9 fine (P4) 11.8 12.1 7.8 More computational effort per parcel required for Lagrange method counteracted by improved stability and smaller time-step Lowest CPU times observed for Lagrange mass and momentum coupling IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 16 Time-step and Stability NNM time-step constrained by evaporation, or strength of fuel vapor source term Reduction in large vapor concentration gradients result in improved computational stability leading to smaller overall timesteps when using Lagrange method IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 17 Summary and Conclusions Method using Lagrange interpolation polynomials was applied to mass and momentum coupling for evaporating sprays and compared with the NNM of KIVA3 Lagrange method lead to improved mesh independence of liquid and vapor spray penetrations Lagrange method lead to improved computational stability resulting in lower overall time-steps and improved simulation times IEA 2005 MICHIGAN TECHNOLOGICAL UNIVERSITY 18
