Modelling of droplet heating and evaporation in computational fluid dynamics codes Sergei SAZHIN*, Irina SHISHKOVA**, Vladimir LEVASHOV **, Morgan HEIKAL* * Sir Harry Ricardo Laboratory, School of Environment and Technology, Faculty of Science and Engineering, University of Brighton, Brighton BN2 4GJ, UK ** Low Temperature Departments, Moscow Power Engineering Institute, Moscow , Russia

2 Presentation overview INTRODUCTION LIQUID PHASE MODELS GAS PHASE MODELS TEST CASES MODELLING VERSUS EXPERIMENTS KINETIC MODELLING RELATED DEVELOPMENTS

3 INTRODUCTION: Processes in Diesel engines Formation of a liquid fuel spray Fuel spray penetration Droplet break-up Heating of fuel droplets Evaporation of fuel droplets Ignition of fuel vapour / air mixture

4 160 MPa injection into 10 MPa gas 100 MPa injection into 10 MPa gas 140 MPa injection into 10 MPa gas Ignition ( experiment )

5 LIQUID PHASE MODELS The Infinite Thermal Conductivity (ITC) models are based on the energy balance equation of the droplet as a whole. The solution to this equation:,

6 Conduction limit. Effective conductivity

7 where θ R is the radiation temperature, R d is the droplet radius, θ R can be assumed equal to the external temperature T ext in the case of an optically thin gas in the whole domain.

8

9 Analytical solution for h=const

10 Numerical algorithms numerical algorithm based on the analytical solution (analytical solution at the end of the time step is considered as the initial condition for the next time step) numerical solution of the discretised heat conduction equation (fully implicit approach) numerical solution based on the assumption of no temperature gradient inside the droplet (conventional approach currently used in CFD codes)

11 Main results (Liquid phase models) The numerical algorithm based on the analytical solution is recommended when a compromise between high accuracy and CPU requirements is essential. The contribution of thermal radiation can be taken into account via the simplified form of the radiation term.

12 GAS PHASE MODELS, Model 4 (Abramzon and Sirignano, 1989):.

13 TEST CASE 1 (Zero-dimensional code),.

14 Plots for Tg0= 880K, pg0=3 MPa, Td0= 300 K, Rd0=10 μm and vd0=1 m/s. The overall volume of injected liquid fuel was taken equal to 1 mm3, and the volume of air, where the fuel was injected, was taken equal to 883 mm3. The results were obtained based on the ETC model and using seven gas phase models. Text=2000K

15 The same as the previous figure but using three liquid phase numerical algorithms: the algorithm based on the analytical solution of the heat conduction equation inside the droplet (1), the algorithm based on the numerical solution of the heat conduction equation inside the droplet (2), the algorithm based on the assumption that the thermal conductivity inside droplets is infinitely large (3).

16 TEST CASE 2 (KIVA 2 CFD code),.

17 The total autoignition delay times observed experimentally and computed using the customised version of the KIVA 2 CFD code at three initial in-cylinder pressures. The initial injected liquid fuel temperature was assumed equal to 375 K. The injection pressure was equal to 160 MPa. The modified WAVE model, two liquid phase models (ETC and ITC) and two gas phase models (model 0 and the AS model) were used for computations.

18 Main results ( Hydrodynamic Heating and Evaporation Models ) The choice of the gas phase model is essential for predicting the evaporation time The choice of the liquid phase model is essential for predicting the initial temperatures and autoignition delay Sazhin, S.S., Kristyadi, T., Abdelghaffar, W.A. and Heikal, M.R. (2006) Models for fuel droplet heating and evaporation: comparative analysis, Fuel, 85 (12-13), Sazhin, S.S., Martynov, S.B., Kristyadi, T., Crua, C., Heikal, M.R. (2008) Diesel fuel spray penetration, heating, evaporation and ignition: modelling versus experimentation, International J of Engineering Systems Modelling and Simulation, 1(1) 1-19.

19 MODELLING VERSUS EXPERIMENTS,

20,

21, Plots of ethanol droplet temperature T d, measured experimentally (solid triangles) and predicted by the model (T ds droplet temperatures at the surface of the droplet, T dav average droplet temperature, and T dc droplet temperature at the centre of the droplet) and gas temperature T g for the initial conditions R do = mm, T do =294 K, C=3.97

22, Plots of ethanol droplet temperature T d and radius R d, measured experimentally (solid triangles and squares) and predicted by the model (T ds droplet temperatures at the surface of the droplet, T dav average droplet temperature, and T dc droplet temperature at the centre of the droplet) for gas average temperature T g equal to 1270 K and for the initial conditions R do = mm, T do =309 K, C=10.5

23 Main results ( MODELLING VERSUS EXPERIMENTS ) For relatively small droplets (initial radii about 65 μm) the experimentally measured droplet temperatures are close to the predicted average droplet temperatures. These temperatures are closer to the temperatures predicted at the centre of the droplets when the droplet diameter is larger than the probe volume size of the two- colour LIF thermometry. C. Maqua, G. Castanet, F. Grish, F. Lemoine, T. Kristyadi, S. S. Sazhin (2008) Monodisperse droplet heating and evaporation: experimental study and modelling, Int J Heat and Mass Transfer, 51(15-16), , 2008

24, KINETIC MODELLING

25 dede 1 2 T s,  s x T Rd,  Kinetic region Hydrodynamic region Rd  j V q

26

27 APPROXIMATION,.

28 Main results (Kinetic Modelling) The kinetic effects predicted by the numerical algorithm turned out to be noticeable and cannot be a priori ignored when modelling droplet evaporation. Shishkova, I.N. and Sazhin, S.S. (2006) A numerical algorithm for kinetic modelling of evaporation processes, J Computational Physics 218 (2), Sazhin, S.S., Shishkova, I.N., Kryukov, A.P., Levashov, V.Yu. and Heikal, M.R. (2007) Evaporation of droplets into a background gas: kinetic modelling, Int J Heat Mass Transfer 50, Sazhin, S.S., Shishkova, I.N. (2008) A kinetic algorithm for modelling the droplet evaporation process in the presence of heat flux and background gas, Atomization and Sprays (in press).

29 Some Related Developments Development of the new model for transient stationary droplet heating [Int J Thermal Science (2007), V. 46 (4), pp ]. Development of the dynamic decomposition method for numerical solution of the system of stiff ODEs [Computers and Fluids (2007), V. 36, pp ]. Development of the new model of particle grouping in oscillating flows [European J of Mechanics B/Fluids (2008), V. 27, pp ; International J Heat and Fluid Flow (2008), V. 29, pp ]. Development of droplet break-up models and their application to modelling transient Diesel fuel sprays [International J of Engineering Systems Modelling and Simulation (2008), V. 1(1), pp Development of vortex ring models and their application to gasoline engines [submitted to J Fluid Mechanics]

30 Unsolved problems Grid dependence of gas/droplets exchange of heat and mass under the Eulerian/Lagrangian approach in CFD codes. Effects of turbulence on droplet heating. Droplet heating during the break-up processes. Heating of non-spherical droplets. Heating of small droplets (geometrical optics approximation is not valid; surface tension is important). Transient heating of moving droplets (boundary layer around the droplet is not fully developed) Evaporation coefficient and inelastic collisions in kinetic models

31 Acknowledgements The original results were obtained in collaboration with our colleagues W. Abdelghaffar, S. Begg, V. Bykov, C. Crua, I. Goldfarb, V. Gol’dshtein, D.Katoshevski, F. Kaplanskii, T. Kristyadi, A. Kryukov, P. Krutitski, E. Sazhina, T.Shakked, V. Sobolev. The authors are grateful to the Royal Society and EPSRC (Project EP/E02243X/1) for financial support.

32 Thank you for your attention Any comments or suggestions would be highly appreciated

33,

Modelling of droplet heating and evaporation in computational fluid dynamics codes Sergei SAZHIN*, Irina SHISHKOVA**, Vladimir LEVASHOV **, Morgan HEIKAL* * Sir Harry Ricardo Laboratory, School of Environment and Technology, Faculty of Science and Engineering, University of Brighton, Brighton BN2 4GJ, UK ** Low Temperature Departments, Moscow Power Engineering Institute, Moscow , Russia

35, where when R RdRd when R d <R<R g COUPLED TRANSIENT SOLUTION

36 and when

37 h=χh 0 Fo= t κ g /R R=R g /R d (no radiation)

38 Main results (Droplet Transient Heating) The correction for convective heat transfer coefficient needs to be taken into account for all Fo. The radiative effects of these corrections are negligibly small for small Fo, but it can become significant for large Fo. Sazhin, S.S., Krutitskii, P.A., Martynov, S.B., Mason, D., Heikal, M.R., Sazhina, E.M. (2006) Transient heating of a semitransparent spherical body, Int J Thermal Science 46 (4),

39 when

40,., A countable set of positive eigenvalues λ n is found from the solution of the equation These are arranged in ascending order 0<λ 1 < λ 2 <.....

41 Generalisation of the expression for h

42 Using Cooper’s (1977) solution (valid for R g —>∞) we obtain:

43, and C >3. where C (ratio of the distance between droplets and their diameters),

44,

45,

46

47 Collision of molecules

48 Boltzmann equations

49 (α=a,v; β=a,v) are collision integrals defined as

50 APPROXIMATION,.

51 APPROXIMATION,