Advertisement

Probability Density Function Modeling of Turbulent Spray Combustion

  • Rana M. Humza
  • Yong Hu
  • Eva GutheilEmail author
Conference paper
Part of the ERCOFTAC Series book series (ERCO, volume 19)

Abstract

Spray processes play a crucial role in liquid fueled combustion devices such as Diesel or fueled rocket engines and industrial furnaces. The combustion occurs under turbulent conditions, and a wide dynamic range of length and time scales characterize these processes, where the scales of the flow field and chemical reactions typically differ considerably. Moreover, a strong interdependence of liquid breakup and atomization, turbulent dispersion, droplet evaporation, and fuel-air mixing makes the spray modeling a challenging task. In the present chapter, a one-point one-time Eulerian statistical description of a joint mixture fraction—enthalpy probability density function (pdf) model for the gas phase is derived and modeled. A Lagrangian Monte Carlo method is used to solve the high-dimensional joint pdf transport equation. Two different mixing models, the interaction-by-exchange-with-the-mean and an extended modified Curl model, are employed in order to evaluate molecular mixing in the context of two-phase reacting flows. Moreover, a modified β function for application in turbulent spray flames, which has been proposed in an earlier study of non-reacting spray flows, is discussed in comparison with the standard β function and the transported pdf method. The modified β function is defined through two additional parameters compared to the standard form, and the choice of these parameters is discussed in the present study. A steady, two-dimensional, axisymmetric, turbulent liquid fuel/air spray flame is investigated, where both methanol and ethanol are studied. The numerical results are compared and discussed in context with the experimental data.

Keywords

Nozzle Exit Mixture Fraction Spray Flame Scalar Dissipation Rate Conditional Moment Closure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

The authors gratefully acknowledge the financial support of Heidelberg School of Mathematical and Computational Sciences for their financial support. YH acknowledges funding through the China Scholarship Council.

References

  1. 1.
    Abramzon, B., Sirignano, W. A.: Droplet vaporization model for spray combustion calculation. Int. J. Heat Mass Transfer. 32, 1605–1618 (1989).CrossRefGoogle Scholar
  2. 2.
    Anand, G., Jenny, P.: Stochastic modeling of evaporating sprays within a consistent hybrid joint PDF framework. J. Comput. Phys. 228, 2063–2081 (2009).CrossRefzbMATHGoogle Scholar
  3. 3.
    Atkins, P., Paula, J. D.: Atkins Physical Chemistry, Oxford Higher Education, 7th Edition, (2001).Google Scholar
  4. 4.
    Batchelor, G. K.: An Introduction to Fluid Dynamics. Cambridge University Press, London, (1967).zbMATHGoogle Scholar
  5. 5.
    Cao, R. R., Wang, H., Pope, S. B.: The effect of mixing models in PDF calculations of piloted jet flames. Proc. Combust. Inst. 31, 1543–1550 (2007).CrossRefGoogle Scholar
  6. 6.
    Crowe, C. T., Sharma, M. P., Stock, D. E.: The particle-source-in cell (PSI-Cell) model for gas-droplet flows. J. Fluids Eng. 99, 325–332 (1977).CrossRefGoogle Scholar
  7. 7.
    Curl, R. L.: Dispersed phase mixing: 1. Theory and effects in simple reactor. AIChE J. 9(2), 175–181 (1963).CrossRefGoogle Scholar
  8. 8.
    De, S., Lakshmisha, K. N., Bilger, R. W.: Modeling of nonreacting and reacting turbulent spray jets using a fully stochastic separated flow approach. Combust. Flame. 158, 1992–2008 (2011).CrossRefGoogle Scholar
  9. 9.
    Demoulin, F. X., Borghi, R.: Assumed PDF modeling of turbulent spray combustion. Combust. Sci. Technol. 158, 249–271 (2000).CrossRefGoogle Scholar
  10. 10.
    Dopazo, O., O’Brien, E. E.: An approach to the auto-ignition of a turbulent mixture. Acta Astronaut. 1, 1239–1266 (1974).CrossRefzbMATHGoogle Scholar
  11. 11.
    Durand, P., Gorokhovski, M., Borghi, R.: An application of the probability density function model to diesel engine combustion. Combust. Sci. Technol. 144, 47–78 (1999).CrossRefGoogle Scholar
  12. 12.
    Düwel, I., Ge, H.-W., Kronemayer, H., Dibbel, R., Gutheil, E., Schulz, C.: Experimental and numerical characterization of a turbulent spray flame. Proc. Combust. Inst. 31, 2247–2255 (2007).CrossRefGoogle Scholar
  13. 13.
    Eckstein, J., Chen, J. Y., Chou, C. P., Janicka, J.: Modeling of turbulent mixing in opposed jet configuration: one-dimensional Monte Carlo probability density function simulation. Proc. Combust. Inst. 28, 141–148 (2000).CrossRefGoogle Scholar
  14. 14.
    Garg, R., Narayanan, C., Subramaniam, S.: A numerically convergent Lagrangian-Eulerian simulation method for dispersed two-phase flows. Int. J. Multiphase Flow. 35, 376–388 (2009).CrossRefGoogle Scholar
  15. 15.
    Ge, H.-W., Gutheil, E.: PDF simulation of turbulent spray flows. Atomization Sprays. 16, 531–542 (2006).CrossRefGoogle Scholar
  16. 16.
    Ge, H.-W., Gutheil, E.: Simulation of a turbulent spray flame using coupled PDF gas phase and spray flamelet modeling. Combust. Flame. 153, 173–185 (2008).CrossRefGoogle Scholar
  17. 17.
    Ge, H.-W., Düwel, I., Kronemayer, H., Dibble, R. W., Gutheil, E., Schulz, C., Wolfrum, J.: Laser based experimental and Monte Carlo PDF numerical investigation of an ethanol/air spray flame. Combust. Sci. Technol. 180, 1529–1547 (2008).CrossRefGoogle Scholar
  18. 18.
    Ge, H.-W., Hu, Y., Gutheil, E.: Joint gas-phase velocity-scalar PDF modeling for turbulent evaporating spray flows. Combust. Sci. Technol. 184, 1664–1679 (2012).CrossRefGoogle Scholar
  19. 19.
    Gordon, R. L., Masri, A. R., Pope, S. B., Goldin, G. M.: A numerical study of auto-ignition in turbulent lifted flames issuing into vitiated co-flow. Combust. Theor. Model. 11, 351–376 (2007).CrossRefzbMATHGoogle Scholar
  20. 20.
    Gutheil, E.: Structure and extinction of laminar ethanol-air spray flames. Combust. Theor. Model. 5(2), 131–145 (2001).CrossRefzbMATHGoogle Scholar
  21. 21.
    Gutheil, E., Sirignano, W. A.: Counterflow spray combustion modeling with detailed transport and detailed chemistry. Combust. Flame. 113, 92–105 (1998).CrossRefGoogle Scholar
  22. 22.
    Heye C. R., Raman, V., Masri, A. R.: LES/probability density function approach for the simulation of an ethanol spray flame. Proc. Combust. Inst. 34, 1633–1641 (2013).CrossRefGoogle Scholar
  23. 23.
    Hollmann, C., Gutheil, E.: Modeling of turbulent spray diffusion flames including detailed chemistry. Proc. Combust. Inst. 26, 1731–1738 (1996).CrossRefGoogle Scholar
  24. 24.
    Hollmann, C., Gutheil, E.: Flamelet-modeling of turbulent spray diffusion flames based on a laminar spray flame library. Combust. Sci. Technol. 135, 175–192 (1998).CrossRefGoogle Scholar
  25. 25.
    Hubbard, G. L., Denny, V. E., Mills, A. F.: Droplet evaporation: effects of transient and variable properties. Int. J. Heat Mass Transfer. 18, 1003–1008 (1975).CrossRefGoogle Scholar
  26. 26.
    Kung, E. H., Haworth, D. C.: Transported probability density function (tPDF) modeling for direct-injection internal combustion engines. SAE Paper 2008-01-0969; SAE Int. J. Engines. 1, 591-606 (2009).Google Scholar
  27. 27.
    Liu, Z. H., Zheng, C. G., Zhou, L. X.: A joint PDF model for turbulent spray evaporation/combustion. Proc. Combust. Inst. 29, 561–568 (2002).CrossRefGoogle Scholar
  28. 28.
    Lundgren, T. S.: Model equation for non-homogeneous turbulence. Phys. Fluids. 12, 485–497 (1969).CrossRefzbMATHGoogle Scholar
  29. 29.
    Luo, K., Pitsch, H., Pai, M. G., Desjardins, O.: Direct numerical simulations and analysis of three dimensional n-heptane spray flames in a model swirl combustor. Proc. Combust. Inst. 33, 2143–2152 (2011).CrossRefGoogle Scholar
  30. 30.
    Masri, A., Gounder, J.: Details and Complexities of Boundary Conditions in Turbulent Piloted Dilute Spray Jets and Flames. In: Bart, Merci, Dirk, Roekaerts and Amsini, Sadiki (Eds.), Experiments and Numerical Simulations of Diluted Spray Turbulent Combustion, 41–68, New York, Springer, (2011).Google Scholar
  31. 31.
    McDonell, V. G., Samuelsen, G. S.: An experimental data base for computational fluid dynamics of reacting and nonreacting methanol sprays. J. Fluids Eng. 117, 145–153 (1995).CrossRefGoogle Scholar
  32. 32.
    Mehta R. S.: Detailed Modeling of soot formation and turbulence-radiation interactions in turbulent jet flames. Ph.D. thesis, The Pennsylvania State University, (2008).Google Scholar
  33. 33.
    Menon, S., Fureby, C.: Computational Combustion. In: Encyclopedia of Aerospace Engineering, Wiley, (2010).Google Scholar
  34. 34.
    Miller, R. S., Bellan, J.: On the validity of the assumed probability density function method for modeling binary mixing/reaction of evaporated vapor in gas-liquid turbulent shear flow. Proc. Combust. Inst. 27, 1065–1072 (1998).CrossRefGoogle Scholar
  35. 35.
    Mortensen M., Bilger R.: Derivation of the conditional moment closure equation for spray combustion. Combust. Flame. 156, 62–72 (2009).CrossRefGoogle Scholar
  36. 36.
    Naud, B.: PDF modeling of turbulent sprays and flames using a particle stochastic approach. Ph. D. Thesis, TU Delft, (2003).Google Scholar
  37. 37.
    Olguin, H., Gutheil, E.: Influence of evaporation on spray flamelet structures, Combust. Flame (2013), http://dx.doi.org/10.1016/j.combustflame.2013.10.010.
  38. 38.
    Pai, G. M., Subramaniam, S.: A comprehensive probability density function formalism for multiphase flows. J. Fluid Mech. 628, 181–228 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Peters, N.: Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10, 319–339 (1984).CrossRefGoogle Scholar
  40. 40.
    Pope, S. B.: The relationship between the probability approach and particle models for reaction in homogeneous turbulence. Combust. Flame. 35, 41–45 (1979).CrossRefGoogle Scholar
  41. 41.
    Pope, S. B.: Transport equation for the joint probability density function of velocity and scalars in turbulent flow. Phys. Fluids. 24, 588–596 (1981).CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Pope, S. B.: PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192 (1985).CrossRefMathSciNetGoogle Scholar
  43. 43.
    Raju, M. S.: Application of scalar Monte Carlo probability density function method turbulent spray flames. Numer. Heat Transfer A. 30, 753–777 (1996).CrossRefGoogle Scholar
  44. 44.
    Richardson, J. M., Howard, H. C., Smith, R. W.:The relation between sampling-tube measurements and concentration fluctuations in a turbulent gas jet. Proc. Combust. Inst. 4 814–817, (1953).CrossRefGoogle Scholar
  45. 45.
    Rumberg, O., Rogg, B.: Full PDF modeling of reactive sprays via an evaporation-progress variable. Combust. Sci. Technol. 158, 211–247 (2000).CrossRefGoogle Scholar
  46. 46.
    Schiller, L., Neumann, A. Z.: A drag coefficient correlation. VDI Zeitschrift 77, 318–320 (1933).Google Scholar
  47. 47.
    Wang, H., Pope, S. B.: Lagrangian investigation of local extinction, re-ignition and auto-ignition in turbulent flames. Combust. Theory Modeling. 12, 857–882 (2008).CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Zhu, M., Bray, K. N. C., Rumberg, O., Rogg, B.: PDF transport equations for two-phase reactive flows and sprays. Combust. Flame. 122, 327–338 (2000).CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany

Personalised recommendations