The Development of Droplet Spray and Evaporation Models at Coventry University

Transcription

1 The Development of Droplet Spray and Evaporation Models at Coventry University NWABUEZE EMEKWURU, MANSOUR AL QUIBEISSI, ESSAM ABDELFATAH. Coventry University Presentation at the UK Universities Internal Combustion Engines Group (UnICEG) meeting, Coventry University, September 20,

2 Contents Introduction Spray models Particle-source-in-cell method The discrete droplet method The moments spray model Some test cases The future Conclusions Droplet evaporation models Single Component model Multi-Component models Some test cases The future Conclusions References 2

3 Introduction This is an overview of the work being carried out at Coventry University with regards to the development of fuel spray and droplet evaporation models. 3

4 Spray models Various modeling methods exist that describe the behaviour of gas-liquid two-phase flows of the type where discrete droplets in a carrier gas exist. 4

5 Spray models Any numerical spray model needs to account for the mass, momentum, and energy coupling between the gas and liquid phases as sprays are gas-droplet flows. This is not an insignificant undertaking. 5

6 Motivation For many applications, high fidelity simulation of even moderately dense sprays at relatively low computational costs is still a major challenge. 6

7 Current spray models The bulk of the numerical spray models currently available are based on the particlesource-in-cell (Crowe et al., 1977) and discrete droplet model methods (Dukowicz, 1980). 7

8 Particle-source-in-cell method The particle-source-in-cell method presents the droplet phase as a source of mass, momentum, and energy to the gaseous phase. (Image courtesy of Abdelfatah, 2017) 8

9 Particle-source-in-cell method The droplet trajectories are computed from the integration of the droplet motion equation. Simultaneously, the droplet mass, velocity, and temperature are calculated along the trajectories. 9

10 Particle-source-in-cell method Essentially, the trajectories of the droplets are followed in a gaseous field on a fixed mesh. 10

11 Particle-source-in-cell method The trajectories of the droplets are followed in a gaseous field on a fixed mesh. 11

12 Particle-source-in-cell method But we might require to solve for the trajectories of a lot of droplets. (Image courtesy of Abdelfatah, E., 2017) 12

13 The discrete droplet model Droplets are defined into identical packets. Each computational packet contains a large number of droplets of identical physical properties like droplet size, and velocity. 13

14 The discrete droplet model As in the particle-source-in-cell method, the Lagrangian equations of motion for each packet of droplets are solved, the equations of motion for the carrier gas are solved in an Eulerian scheme, and both phases are coupled by regarding the droplets as sources of mass, momentum, and energy in each grid cell. 14

15 The discrete droplet model 15

16 The discrete droplet model But to represent the chaotic turbulent motion of sprays a large number of droplet packets are needed to produce a smooth representation of the spray. 16

17 The discrete droplet model presents higher fidelity simulation outcomes compared to the particle-source-incell but at increased computational costs. 17

18 The moments spray model The moments spray model captures the polydisperse nature of spray flow without using droplet size classes. 18

19 The moments spray model The stochastic nature of sprays has led some researchers to search for a probabilistic formulation to the problem (Williams, 1962; Lundgren, 1967). 19

20 The moments spray model 20

21 The moments spray model The approach presented here is to describe the moments of a droplet number probability distribution n(r), defined as n( r) r In the model, the first four moments, Q 0 are used. Q 0 is the total number of droplets is the total sum of radii of the droplets Q 1 4πQ 2 4πQ3 3 Q i = 0 is the total surface area of the droplets per unit volume is the total volume of the droplets per unit volume i dr to Q 3 21

22 The moments spray model These parameters contain a lot of information about the spray and are used to build up the fully polydisperse spray model. For instance, since mean droplet diameters are often used to characterize the droplet sizes in a spray, these four moments parameters immediately provide all mean droplet diameters from to, since by definition (Sowa, 1992) D 10 D 32 D p q pq = 2 p q Q Q q p 22

23 The moments spray model For example, the liquid volume fraction can easily be calculated from the fourth moment since V V liquid cell = πq

24 The moments spray model In order to construct Eulerian transport equations for the droplet moments, it is necessary to define the speed at which the moments are to be convected. And the mean speed must be different for each moment as the droplets are travelling at different velocities. This is not a trivial problem! 24

25 The moments spray model This is resolved by using the concept of moment-averaging. For example, the net convection of mass should occur at the mass-average velocity, and the net convection of droplet surface area should occur at the surface-area-average velocity. Thus the mass-average velocity is the correct velocity at which to convect Q 3 and the surface-area-average velocity is the correct velocity at which to convect Q 2 25

26 The moments spray model The moment-average liquid velocity vector, over the ith moment Q i is defined as: U 1, averaged U 1i = 0 r i n( r) U dr Q i 1 26

27 The moments spray model The transport equation for the third moment, a liquid phase continuity equation: Q 3, is effectively t ( ρ ( 1 θ )) + ρ ( 1 θ ) l x j ( ) l U l3 j = Sm Q 3 Here,, has been presented in terms of the liquid volume fraction 1-θ θ = 1 V V liquid cell V V liquid cell = 4πQ3 3 27

28 The moments spray model The equations for the remaining moments take a similar form: t x ( Q ) ( ) i + QU i lij = SQ i j Note the use of the appropriate moment-average velocity. The source term is made up of components derived from sub-models of droplet break-up, droplet collisions, droplet evaporation and changes in droplet velocity. 28

29 The moments spray model Then transport equations are solved for the moment-average values of momentum thus: ( ) ( ) Q ( ( )( ) ku lkj + QkU lkiu lkj + Qk U l3i U lki U l3 j U lkj t x x j U + U l3ibq + U i lki ν x j x The third term on the left hand side presents droplet velocities relative to the mass-averaged velocity. The fourth term is due to droplet breakup. The fifth term includes the other source terms such as droplet evaporation and collisions. The source term represents the effect of the droplet drag. j ( ) lki SQ B i Q = Q i kσ v l SU ki j 29

30 The moments spray model Therefore, The droplet number moments are used to provide a representation of the distribution of the droplet sizes at each space and time, and, The moment-average velocities provide the means by which the distribution of the droplet sizes can change in space and time. Both concepts, combined, present the polydisperse nature of a spray. 30

31 The moments spray model Both concepts, combined, present the polydisperse nature of a spray. 31

32 The moments spray model Three schemes involving all or some the four moment transport equations have been implemented in the model (Emekwuru & Watkins, 2010): 1. The 2 moment scheme which uses transport equations for 2 of the moments, 2. The 3 moment scheme which uses transport equations for 3 of the moments, and 3. The 4 moment scheme which uses transport equations for 4 of the moments. 32

33 The 3-moments spray model The last three moments, Q to Q 1 3 are calculated by solving the transport equations for the moments. 33

34 The 3-moments spray model The first moment, Q 0, is then calculated from a Gamma function, which can be described in terms of a shape parameter k, and a scale parameter x, by the integral: Γ x k 1 ( k) = e x dx 0 The moments of this distribution are given by: Q i = Q 0 0 k α Γ( k) r k 32 r k + i 1 e r α ( r 32 ) dr 34

35 The 3-moments spray model The partial integration of the moments of the distribution equation leads to: Q 0 = ( k + ) kr 32 2 Q 1 35

36 The 3-moments spray model Transport equations are written for the last three moments. The first moment is calculated from a gamma distribution function. The velocities for the transport equations are calculated from moment-average velocities. The spray model is closed by equations for the energy of the liquid phase and gas-phase equations, including a k ε turbulence model. All equations are solved in an Eulerian framework, and discretised using the finite volume approach. 36

37 Some recent test cases 37

38 High-pressure narrow-angle sprays Experimental data (Allocca et al., 1992; Dan et al., 1997) in which spray tip penetration and droplet size data from non-evaporating transient high pressure diesel sprays were collected under different ambient conditions using high speed photography and laser light extinction methods, were also used to test the model. 38

39 High-pressure narrow-angle sprays Computational spray structure (contours for SMR) MPa Pinj, at 0.5 ms (left spray), and 1.0 ms (right spray) after the start of injection. (Emekwuru, 2013). 39

40 High-pressure narrow-angle sprays Comparison of predicted and experimental spray tip penetration (left) and droplet SMR (right) MPa Pinj. [11] Allocca et al., 1992, [13] - Beck & Watkins, 2004, [Case A] - Emekwuru,

41 Vaporizing biodiesel fuel sprays A heat and mass transfer model based on the droplet surface-area-averaged temperature is implemented in the model and applied to evaporating biodiesel fuel sprays (Emekwuru, 2016). 41

42 Vaporizing biodiesel fuel sprays The experimental data is from the work of Park et al. (2009) who injected soybean oil methyl ester (SME) fuel into a high pressure chamber filled with Nitrogen at various ambient pressure values and obtained the spray images using high speed cameras. The biodiesel was heated by heat generated from a boiler. 42

43 Vaporizing biodiesel fuel sprays Emekwuru, 2016 Computational grid used and single hole injector Some physical and inlet properties of the nozzle Injector type Single hole Nozzle Diameter (m) 3x10-4 Nozzle orifice depth/hole diameter (L/D) ratio (see the Figure above right)). Fuel quantity injected (mg)

44 Vaporizing biodiesel fuel sprays Computational grid used and single hole injector Fuel spray liquid tip penetration at different ambient gas temperature values. Pamb = 4 MPa, Pinj = 60 MPa, Tfuel = 300 K. (Emekwuru, 2016). Fuel spray liquid tip penetration increases with ambient temperature 44

45 Vaporizing biodiesel fuel sprays Fuel vapor mass fraction at (A) 3 ms and (B) 9 ms after start of injection. Pamb = 4 MPa, Pinj = 60 MPa, Tfuel = 300 K. (Emekwuru, 2016). Fuel vapor mass fraction indicates regions of high concentration in the spray centreline. 45

46 Vaporizing biodiesel fuel sprays Spray liquid tip penetration at a fuel temperature of 300 K. Pamb = 4 MPa, Pinj = 60 MPa, Tfuel = 300 K. (Emekwuru, 2016). Comparison with experimental data and results from a KIVA- 3V code. 46

47 The moments spray model presents less computational costs compared to the discrete droplet models but at some loss of stability due to the necessity of ensuring that all moments are transported to any given control volume. 47

48 Other test cases are available from literature. 48

49 Current work on the moments spray model. Development of other analytically integrable distribution functions, and comparison with distributions from various atomisers. Assessment of the droplet breakup and collision models for medical spray applications. Application of the models to the latest spray test cases, including ECN spray database. Application to liquid film atomisation cases. 49

50 Future cases for the moments spray model Urea injection system (Image courtesy of Abdelfatah, 2017) 50

51 Fuel droplet evaporation modeling 51

52 Fuel droplet evaporation modeling The estimation of the delay in combustion processes due to the heating and evaporation of fuel droplets is essential for the design and performance of internal combustion engines. 52

53 Fuel droplet evaporation modeling There have been compromises between the accuracy of the fuel droplet evaporation models and their computational costs. The models that are presently being developed seek to considerably increase the accuracy of the models without substantial computational expense. 53

54 Simplified droplet evaporation model Takes into account the effects of temperature gradients inside the droplets (Sazhin., et al. 2007). 54

55 Simplified droplet evaporation model Computationally cheap. The multi-component nature of actual fuel droplets are ignored. The species diffusion during evaporation is ignored. The infinite thermal conductivity is ignored. 55

56 Simplified droplet evaporation model Effective Diffusivity/Effective Thermal Conductivity effects - Recirculation (a) (b) (c) The effect of internal recirculation on temperature distribution inside droplets moving with relative velocities (a) 0.2 m s 1, (b) 1 m s 1 and (c) 3 m s 1. (Duret., et al. 2014). 56

57 Multi-component droplet evaporation model - Discrete Component Model (Every) individual fuel component can be analysed. Includes the effects of species diffusivity and recirculation inside droplets (effective diffusivity (ED) and effective thermal conductivity (ETC)). Finite thermal conductivity is accounted for. 57

58 Discrete Component Model Basic equations The time evolution of species mass fractions at any R is derived from: YY llll = DD eff 2 YY llll RR RR YY llll, where D eff is the effective liquid species diffusivity (D eff = D l χ Y ), taking into account the recirculation inside droplets: χχ YY = tanh log 10 Re dd(ll) Sc ll 30, The solution to the first equation above subject the initial and boundary conditions: YY llll = εε ii + 1 RR exp DD eeeeee λλ 0 RR dd exp DD eeeeee nn=1 2 tt qq iii εε ii QQ 0 sinh λλ 0 RR RR dd + λλ nn RR dd 2 tt qq iiii εε ii QQ nn sin λλ nn RR RR dd Effective Diffusivity (ED) model 58

59 Discrete Component Model Basic equations Evaporation rate for an isolated droplet is assumed as the ambient gas density, DD vv is the binary diffusion coefficient of fuel vapour in air, BB MM = YY vvvv YY vv 1 YY vvvv is the Spalding mass transfer number, YY vvvv & YY vv are the vapour mass fractions near and away from the droplet surface, respectively, YY vvvv = ii YY vvvvvv, mm dd = 2ππRR dd DD vv ρρ gg BB MM Sh iso Sh iso is the Sherwood number for an isolated evaporating droplet,

60 Multi-component droplet evaporation model - Discrete Component Model Comparatively high computational costs. Therefore ideally applicable to situations where a small number of components need to be analysed. 60

61 Multi-component droplet evaporation model - Probabilistic models Probabilistic analysis of large numbers of components. It is assumed that the species inside the droplets mix infinitely or do not mix at all. Examples include the Continuous Thermodynamics model, Distillation Curve model. 61

62 Comparison of the Single Component, Probabilistic and Discrete Component Models Droplet surface temperatures and radii. T s (ME) T s (MI) T s (SI) R d (ME) R d (MI) R d (SI) 26.3% The time evolution of Palm Kernel Methyl Ester (PMK) droplet surface temperatures (T s ) and radii (R d ) predicted by DC (ME), PA(MI), SC(SI) models. (Al Qubeissi, 2015).

63 Multi-component droplet evaporation model - Quasi-discrete model To maintain the advantages of the Discrete Component Model but account for large numbers of fuel droplet components. Without huge computational penalties compared to the Discrete Component Model. 63

64 Quasi-discrete model Assume that the thermodynamic properties of the fuel components in a certain range are close and replace the continuous distribution with a discrete one, consisting of quasi-components. Alkanes C n H 2n+2 (Al Qubeissi., et al. 2017). 64

65 Quasi-discrete model Assumes that diesel and gasoline fuels consist only of n-alkanes. However, the total molar fraction of alkanes is only about 40% of the overall composition of diesel fuels! 65

66 Realistic Diesel fuel composition Diesel Fuel (Gun ko., et al. 2013)

67 Therefore yes you guessed it We need another model! 67

68 Multi-dimensional quasi-discrete model This is similar to the quasi-discrete model. But takes into account the realistic composition of fuel droplets. This is achieved by taking into account the relationship between the fuel droplet components and their molar fraction and not just the distribution function of the droplet components like the quasi-discrete model. (Al Qubeissi, 2015). 68

69 Some recent test cases 69

70 FACE gasoline fuel gasoline Fuel (Al Qubeissi, 2015).

71 Surface temperatures and radii gasoline Fuel 67% 6.6% (Al Qubeissi, 2015)

72 CPU efficiency gasoline Fuel The plot of CPU time, required for calculations of stationary droplet heating and evaporation for Intel Xeon (core duo) E8400, 2 GHz and 3 GB RAM for 1 μs time-step. (Al Qubeissi, 2015).

73 diesel-biodiesel fuel blends

74 Diffusion of species 105 components of WCO/Diesel fuels C/QCs CPU time (Sec) Workstation Specs. Z210, Intel core, 64-bit, 3.10 GHz and 8 GB RAM. The time step = 1 µs. (Al Qubeissi., et al. 2017).

75 Other test cases are available from literature. 75

76 The multi-dimensional quasidiscrete component droplet fuel evaporation model presents considerably less (96%) computational costs compared to the Discrete Component model at a loss of negligible predictive capabilities for the estimation of temperature and radii values for fuel droplet types tested thus far. 76

77 Current work on the multi-dimensional quasi-discrete component model Experimental validation of the MDQD model. Application to fuel mixtures. Assessment of the impacts of fuel blends and compositions on their surrogates formulation and ignition. Implementation in ANSYS-Fluent and Converge-CFD commercial codes. 77

78 References Abdelfatah, E., Private Communication (2017). Allocca, L., et al., SAE Technical Paper (1992). Al Qubeissi, M., et al., ILASS Europe (2017). Al Qubeissi, M., WiSA Publishers, Stuttgart (2015). Beck, J.C., and Watkins, A.P., Int. J. of Eng. Research. 15(1): 1-21 (2004). Crowe, C.T., et al., Trans. ASME J. Fluids Eng. 99(2): (1977). Dan, T., et al., SAE Technical Paper (1997). Duret, B., et al., ILASS Europe (2014). Dukowicz, J.K., J. Comp. Phys. 35(2): (1980). Emekwuru, N.G.., SAE Technical Paper (2016). Emekwuru, N.G., SAE Technical Paper (2013). Emekwuru, N.G., and Watkins, A.P., Atomization and Sprays 20(8): (2010). Gun ko, V.M., et al., Fluid Phase Equilibria (356): (2013). Lundgren, T.S., Phys. Fluids 10: (1967). Park, S.H., et al., Int. J. of Heat And Fluid Flow 30(5): (2009). Sazhin, S.S., et al., Proc. Euro. Comb. Meeting. (2007). Sowa, W.A., Atomization and Sprays 2(1):1-15 (1992). Williams, F.A., Eight Symposium (International) on Combustion 8(1): (1961). 78