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 111250, 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:

dllgsgs Rc

htTTTTT

3

exp)( 0,

6

Conduction limit. Effective conductivity

7

3 3 20 1 R 2 R= a + a /10 a ( /10 ) ,a

6 43 10d

bd R l lR R

P R a R c

where θR is the radiation temperature, Rd is the droplet radius,

3 3 20 1 R 2 R= b + b /10 b ( /10 ) ,b

θR can be assumed equal to the external temperature Text in the case of an optically thin gas in the whole domain.

0dR R

P R

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

,

3/1)PrRe1[(1()1ln(

2 ddT

T

B

BNu

)])(2/[]1]Re,1[max 077.0Td BF

3/1)Re1[(1()1ln(

2 ddM

M ScB

BSh

)])(2/[]1]Re,1[max 077.0Bd BF

Model 4 (Abramzon and Sirignano, 1989):

.

MT

MTMTMT B

BBBF

,

,7.0,,

)1ln()1()(

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), 1613-1630.

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

,

Membrane and orifice

Piezoceramic

Manometer

Ethanol tank

Filter

Monodisperse droplet stream

Wave generator

Temperatureregulation by

water circulation

Pressurized air tank

Heated coil (combustion ignition)

21

,

Plots of ethanol droplet temperature Td, measured experimentally (solid triangles) and predicted by the model (Tds droplet temperatures at the surface of the droplet, Tdav average droplet temperature, and Tdc droplet temperature at the centre of the droplet) and gas temperature Tg for the initial conditions Rdo= 118.65 mm, Tdo=294 K, C=3.97

22

,

Plots of ethanol droplet temperature Td and radius Rd, measured experimentally (solid triangles and squares) and predicted by the model (Tds droplet temperatures at the surface of the droplet, Tdav average droplet temperature, and Tdc droplet temperature at the centre of the droplet) for gas average temperature Tg equal to 1270 K and for the initial conditions Rdo= 52.25 mm, Tdo=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), 3932-3945, 2008

24

,

KINETIC MODELLING

25 de

1 2

T s, s

x

T Rd , Rd

Kinetic region

Hydrodynamic region

Rd

jV

q

26 )/()(~

;/~

;1500m;5 0000 scrssddgd TTTTTRRRKTR

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

R~

T~

R~

T~

time (ms)

Kinetic: j, q

Kinetic: j Hydrodinamic

27

APPROXIMATION

,

.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

R0=5 mkm, T

g=1000 K

calculation approximaton (I)

(Rkt-R

h)/R

0

Rh/R

0

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

calculation approximaton (II)

R0=5 mkm, T

g=1000 K

(Rkt-R

h)/R

0

Rh/R

0

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), 635-653.

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, 2675-2691.

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. 444-457].

Development of the dynamic decomposition method for numerical solution of the system of stiff ODEs [Computers and Fluids (2007), V. 36, pp. 601-610].

Development of the new model of particle grouping in oscillating flows [European J of Mechanics B/Fluids (2008), V. 27, pp. 131-149; International J Heat and Fluid Flow (2008), V. 29, pp. 415-426].

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. 1-19

Development of vortex ring models and their application to gasoline engines [submitted to J Fluid Mechanics]

30

Unsolved problemsGrid 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

,

PhD studentships, £12,940 tax free stipend per annum Applications are sought for two PhD Studentships within the Centre of Automotive Engineering (CAE) at the University of Brighton, in collaboration with CORIA (France). You will join a well-established, internationally recognised and dynamic research group with top-class facilities. You will be part of a specialist research team working on a collaborative European project between the Universities of Brighton (UK) and Rouen (France) entitled: ‘Cross-Channel Centre for low Carbon Combustion’.

Optical investigation of automotive fuel atomisation and combustion Your work will focus on the application of laser diagnostic techniques to characterise Diesel sprays, combustion and emissions. The research will be conducted on optical engine test-cells using high-speed cameras and state-of-the-art laser-based equipment. You should have a first or upper second class honours degree or equivalent in Engineering, Physics or Chemistry. Good analytical and computing skills are essential as is practical aptitude and the ability to work well within a team. Knowledge of internal combustion engine theory and/or fuel chemistry would be an advantage. Informal enquiries should be addressed to Dr. C Crua: C.Crua@brighton.ac.uk

Modelling heat and mass transfer processes in Diesel fuel droplets Your work will focus on the development of analytical and numerical models for fluid dynamics and heat and mass transfer processes in droplets and sprays with particular emphasis on multi-component droplets. The implementation of the models into the computational fluid dynamics code KIVA and the comparison of the predictions of the models with experimental data provided by the partners will form a key part of the work. You should have a first or upper second class honours degree or equivalent in Mathematics, Engineering or Physics. Competence in analytical and numerical modelling of fluid and thermal dynamics and related processes, together with experience in developing CFD codes are essential. Informal enquiries should be addressed to Professor S Sazhin: S.Sazhin@brighton.ac.uk

These studentships are for 3 years and the proposed start date is 1 January 2009. Closing date: 17 November 2008.

Enquiries and applications, including an up-to-date CV, full contact details (including e-mail addresses), details of 2 referees (at least one of whom must be from your last academic institution) and a one page personal statement should be sent to: Sharon Gunde Senior Research Administrator University of Brighton Centre for Automotive Engineering Cockcroft Building – C215 Lewes Road Brighton BN2 4GJ Tel: +44 (0) 1273 642443 Fax: +44 (0) 1273 642285/642330 Email: S.Gunde@brighton.ac.uk Web: www.brighton.ac.uk/iceg

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 111250, Russia

35

,

22

( , )T T

R P t Rt R R R

where /l l l lk c

/g g g gk c

when RRd

when Rd<R<Rg

COUPLED TRANSIENT SOLUTION

36

0 0 ( )t dT T R R dR

and

00 gt TT when RRd gR

d dR R R RT T

dd RRgRRl R

Tk

R

Tk

0gRRTT

g

when

37

h=χh0 Fo= t κg /R R=Rg/Rd (no radiation)

2d

0( ( , ))g dq h T T R t

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), 444-457.

39

dR

gnn

ng Tv

tR

TtRT0

021

20 ((

||||

1)[exp(

1),(

),(])())(exp()())(0

20 RvdptdRcRRvRT nn

t

nllnd

)sin(

)sin()(

ddn

dnn

Ra

RaRv

))(sin(

))(sin()(

gdgn

ggnn RRa

RRaRv

when dR R

when

d gR R R

40

,

.

,

))((sin

)(

)(sin2||)(||

22

2

gdgn

dggpg

ddn

dlln

RRa

RRc

Ra

RcRv

22 nd

gl

R

kk

.)(),(||||

)(0

2cdRRvRtPR

v

ctp n

R

n

lln

d

A countable set of positive eigenvalues λn is found from the solution

of the equation

nd

glgdggpgg

R

kkRRack

))((cot)(cot ddlll Rack

These are arranged in ascending order 0<λ1 < λ2 < .....

l

lld k

ca

g

gpgg k

ca

41

Generalisation of the expression for h

0( ( , ))g dq h T T R t

g

d

kh

R

0

1( , )

c r

g d

T T

T T R t

20 0

1

( ) exp( ),c g d l d g n nn

T T T k a a A t

)],exp(1[103

2

12

416

tA

a

aRaT n

n n

n

d

gRb

mdr

))]1

)(cot[||||

12

dnddnd

nn a

RaRv

A

))((cot gdgn RRa

42

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

2

0

2

0

( ) exp( Fo ) ( cos sin )

( ) exp( Fo ) sin

d l

g g

l

g

kdu u u u u u

k

du u u u

222 ]sin)sin)sincos([sin

sincos)(

uuuuuk

kuu

uuuu

g

l

l

g

43

,

iso iso

Nu ShC

Nu Sh

tanh 0.36 0.82C C and C >3.

where

C (ratio of the distance between droplets and their diameters),

44

,

a) b) c)

r (mm)

z (mm)

1002003004005006000

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5-2 -1 0 1 2- 20

10

20

30

40

50

60

70

80

z (mm)

T (°C) V (m/s)

T (°C)100 200 300 400 500

a) b) c)

r (mm)

z (mm)

1002003004005006000

10

20

30

40

50

60

70

80

0 0.5 1 1.5 2 2.5-2 -1 0 1 2- 20

10

20

30

40

50

60

70

80

z (mm)

T (°C) V (m/s)

T (°C)100 200 300 400 500

45

,

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 20 50

T(K)

r/D

Streaming axis

Flamefront

46

fuel

air

47

Collision of molecules

1 v

1 v

v v

48

,

,

b

a a

bb ba

ab

b

a a

b

af f

J

fJ

x

x

J

J

tf

t

ξ

ξ

Boltzmann equations

aaJ abJ

bbJ

baJ

, ,a a af f t x

, ,b b bf f t x 2

,

1

3

i i

ii i x i x

ii x i x i

i

n f d

n u f d i a b

T u f dRn

49

|,|)(ddsind2 11

'1

'2

00

1

2

vvv

ffffJ

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

50

APPROXIMATION

,

.

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

R0=5 mkm, T

g=1000 K

calculation approximaton (I)

(Rkt-R

h)/R

0

Rh/R

0

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

calculation approximaton (II)

R0=5 mkm, T

g=1000 K

(Rkt-R

h)/R

0

Rh/R

0

51

APPROXIMATION

,

.

KTKTmR cr 659,300;5 00

;,;,00

mRRR

RRY

R

RX kth

hkth

)(;exp1

1 It

XaY

)(;expexp

22

11 II

t

Xa

t

XaY

2a 2t

750 0.02096 0.08142 0.14800 0.02802

1000 0.13070 0.29023 0.02404 0.08400

1500 0.54256 0.50583 -0.13500 1.72373

1aKTg , 1t