A New Eulerian Model for Turbulent Evaporating Sprays in Recirculating Flows

A New Eulerian Model for Turbulent Evaporating

Sprays in Recirculating Flows

S. Wittig, M. Hallmann, M. Scheurlen, R. Schmehl

Lehrstuhl und Institut fiir Thermische Stromungsmaschinen

U ni versitc'i.t Karlsruhe (T. H.)

KaiserstraBe 12, D-7500 Karlsruhe, Germany

Summary

A new Eulerian model for the computation of tur-bulent evaporating sprays in recirculating flows is derived. It comprises droplet heating and evapora-tion processes by solving separate transport equa-tions for the droplet's temperature and diameter. Full coupling of the droplet and the gaseous phase is achieved by the exchange of source terms due to momentum, heat and mass transfer. The partial dif-ferential equations describing the droplet's transport and evaporation in the new method can be solved using the same numerical procedure as for the gas phase equations. The validity of the model is estab-lished by comparison with a well known Lagrangian approach and with experimental data. For this pur-pose calculations of a recirculating droplet charged air flow within a model combustor are presented.

Nomenclature

A,B,C C1,C2,C11 CD Cp,p J /(kg K) Cp,v Jf(mol K) Crel m/s Ca kgfkg D m Dx m

ff

Nfm3 Fr h l:!.hv Jjkg

coefficients of the Cox-Antoine vapour pressure equat\on

constants of the k, (-tur-bulence model

drag coefficient

specific heat of droplets molar specific heat of vapour relative velocity vapour concentration diameter characteristic diameter in the Rosin-Rammler drop size distribution external forces Frossling number specific enthalpy of eva-poration k Le M N Nu n

nd

Pr p Re

s

Sc T t ~~ v, w(u;)

v

x,y,z(x;) Vc X m kg/mol K s mfs m3_fs m Greek symbols ( m2js3

r

>.

J.l II w T

m

2

_/s

W/(mK) Nsfm2

m

2

_/s

s-1 s

kinetic energy of turbu-lence

length scale of the k, (-turbulence model Lewis number molecular weight parameter m the Ro-sin- Rammler drop s1ze distribution

Nusselt number molar concentration number of discrete drop-size classes

Prandtl number pressure

Reynolds number source term (equation dependent)

Schmidt number temperature time

velocity components volume flow rate cartesian coordinates volume fraction of the dispersed phase

mol fraction

turbulence energy dissi-pation rate

diffusion coefficient thermal conductivity dynamic viscosity kinematic viscosity particle response fre-quency

density

constants of the k, (-tur-bulence mode\

particle response ( rela-xation) time

Subscripts air eff g l p ref s t v 00 Superscripts air effective gas liquid particle, droplet

reference values (1/3-rule) droplet surface turbulent vapour ambient vector time average turbulent fluctuation

Introduction

Liquid fuel combustion processes in gas turbine com-bustors so far are not well understood and, therefore, object of numerous theoretical and experimental in-vestigations. In particular, the accurate prediction of these processes is extremely difficult due to the combination of complex physical and chemical phe-nomena. Dispersed phase/turbulence interactions and turbulence effects on chemical reactions are only two examples for the complexity. In addition nome-acting two-phase flows include a variety of unsolved problems, e.g. turbulent droplet dispersion, turbu-lence effects on heat and mass transfer between the phases, vapour/air mixing etc.. Besides, nonreac-ting turbulent two phase flows are important by it-self. Premixed-prevaporized gas turbine combustors, Diesel engine sprays and rocket plumes are some ex-amples of this type of flow (Mostafa and Elghobashi (1984)).

Approaches for the prediction of droplet transport and evaporation in combustion systems can be clas-sified by two fundamentally different methods. In the Lagrangian approach, the spray is represented by discrete droplets. Each computed droplet represents a number of physical droplets and is observed on its trajectory until it leaves the calculational domain or it evaporates completely. The equations describing the droplet behaviour can be simplified to ordinary differential equations. In turbulent flows, droplet motion and evaporation is simulated by a stochas-tic or Monte Carlo approach (Gosman and Ioanides (1983), Wittig et al. (1987), Kneer et al. (1990)). In the Eulerian approach, the evaporating spray is treated as an interacting and interpenetrating con-tinuum. The resulting equations are similar to the

equations describing the turbulent gas phase. The application of the Eulerian approach for sprays re-quires the consideration of the continuum assump-tion (Batchelor (1974)). This assumpassump-tion is valid, when each computational element contains a large number of droplets in the way that statistically av-eraged properties can be assigned to the droplets. Crowe (1982) showed that most practical systems satisfy the continuum assumption.

This paper is addressed to the further improvement of a recently developed Eulerian method for the nu-merical simulation of turbulent evaporating sprays (Hallmann et a!. (1993)). Results of the new mo-del are compared with both, experimental data and computations using a Lagrangian approach.

Eulerian methods for the prediction of turbulent two-phase flows have been used by various groups of researchers (e.g. Melville and Bray (1979), El-ghobashi et al. (1984), Chen and Wood (1986), Kramer (1988), Abou-Arab and Rocco (1990), Si-monin (1990)). They all deal with questions con-cerning the dispersion of particles in parabolic flows neglecting heat and mass transfer between the two phases. Mostafa and Elghobashi (1985) as well as Mostafa and Mongia (1987) report on Eulerian methods for the computation of turbulent jets with droplet vaporization. However in these papers rather simplified assumptions were used. The authors as-sumed isothermal flow conditions and a constant droplet (saturation) temperature. In addition, a simplified mass transfer model for the calculation of the droplet diameter was applied. In contrast to these investigations, the present study introduces a new model for the computation of fuel spray cha-racteristics including droplet heating and evapora-tion by solving separate transport equaevapora-tions for the droplet's temperature and diameter. This model is applicable not only to parabolic flows but also to turbulent recirculating non-isothermal evaporating sprays, which are of major interest within the de-sign process of real gas turbine combustors.

Governing equations

Gas phase equations

For the steady mean flow under consideration, the time averaged continuity equ~tion and the stationa-ry Reynolds equations

{)

-(pu;)

a

a:z:/pu,u;) = ~ a ( /JeJJ(-;;-au;

+-;;--

au, 2 -

3

~6i;) au~c )

UZi UZi U:Z:j UZ/c

a

2

a:z:. (p

+

3pk)

+

F;

+

Su;,p (2) 1

in conjunction with the standard k, c:-turbulence mo-del

a

~(pu,k) UZi

a

~(pu,c:) UXi _!_(/Jejj ~)

ax, O"Jc a:z:,

+

G~c- pc+ S~c,,

_!___(

/Jej J !..:..._) aXj (1"( aXj

f

+

k(C1G~c-C2pc:)

+

Sf,p G Jc-/Jt-- au, (au;

- + -

auj)

ax; 8zi az;

(3)

(4)

(5)

are solved numerically by a Finite Volume discretiza-tion method. In the above equadiscretiza-tions

/Jejj =p+iJt (6)

is the frequently used effective viscosity with the laminar viscosity p and the eddy viscosity

(7) External forces acting on the fluid, e.g. gravity are denoted by F; in the momentum equations (2) and are assumed to be negligible in the present investiga-tions. Turbulent transport of enthalpy and vapour concentration is calculated via additional transport equations: with = /Jej I Prell /Jeff = See// (10) (11)

The constants of the standard k, c:-turbulence model used and the turbulent Prandtl and Schmidt num-bers are given in Table 1. Source terms (S~.p) due to gas/droplet interactions are described in combi

-nation with the new droplet model. The wall func-tion method is used to eliminate the large number of grid points needed to resolve the laminar sub layer

Launder and Spalding 19 )).

Ct

c2

c~> 0"1: ul Prz Sc1

1.4 1.92 0.09 0.90 1.30

0

.

9

09

Table 1: Constants of the k, c:-turbulence model

Droplet equations

The equations describing continuity and momentum exchange of the liquid phase for laminar flow condi-tions can be derived from a mass- and momentum balance at an infinitesimal small fluid volume:

with Crel

II

tt-

u-;,

II

(14) Re, Dp Crel v (15) 24 CD 0.36

+

5.48 Re;0 573

+

R

(16) e,

The mass source term due to droplet evaporation is obtained by

(17) The term on the l.h.s. of Eq. (13) represents the inertia force per unit volume due to droplet accele-ration. The first term on the r.h.s. represents the drag force due to the slip between the two phases, th s nd nt "ns tern l f, r lik gravity and the third takes momentum loss due to droplet evapo-ration into account. Pressure gradient terms for the dispersed phase are neglected. An expression for the decreasing diameter of a single droplet in Lagrangian coordinates

dDp

=

-Fr 4M"nreffrefJn

(1- XtJ,oo) (18)

dt p,D, 1-X"·"

with

Fr = 1

+

0.276

.Ji{e;

Set (19) is given by Faeth (1983) and Wittig et al. (1988) assuming a constant liquid density. The calculation of the reference values and the mol fraction of the vapour is shown in Appendix A.

In order to derive a transport equation for the droplet diameter we first write:

=

Including Eqs. (12) and (17) yields:

0

4

OX; (VcPpUi,pDp)

=

3SvcDP (20)

A transport equation for the droplet temperature can be derived via an energy balance in the same manner as the momentum equation:

with Nu Le 6N uFr >.ref (T _ T. ) Vc D2 oo p Cp,p p

+

S (tl.hv T.) IJc

+

p Cp,p 2 Le In ( 1- x • "" ) 1-X.,p (1-Xv,oo)Le _1-X.,p -1

fref Cp,v,ref nref >.ref

(21)

(22)

(23) Fr = 1

+

0.276

-J"Re;

Prk (24) The model implies a uniform temperature distribu-tion in the droplet assuming an infinite thermal con-ductivity. In Eq. (21) the first term of the r.h.s. represents droplet heating due to the temperature gradient between the two phases, the second droplet cooling due to droplet evaporation. The specific heat of the liquid Cp,p is assumed to be constant. An ex-act derivation of the expressions describing droplet heating and cooling in the source term of Eq. (21) is given by Faeth (1983) and Wittig et al. (1988).

Turbulence modelling

Turbulence effects are evaluated by introducing fluc-tuation quantities for the volume fraction and the droplet's velocity, temperature and diameter:

(25) (26)

Inserting Eq. (25) in the l.h.s. of the Eqs. (12), (13), (20) and (21), time averaging and using the follow-ing gradient hypothesis for the second order corre-lations representing the turbulent fluxes of momen-tum, mass and enthalpy

a of for

sprays which can easily be casted in a form identical to commonly used gas phase equations:

0

OX; (vcppUi,pUj,p)

_!_,_ (

l't,p OVc)

+

S (₃₁₎

OXi Sct,p OXi IJc

o (

l't,p OTp) OXi Vc Sct,p OXi

+

Vc 6N

uF;~ref

(T _ Tp) Cp,p p (tl.hv )

+

Svc - - + T p Cp,p (34)

Correlations involving fluctuations in the liquid phase density (ppvc) are taken into account only in the continuity equation.

The closure hypothesis (27) and (28) for the turbu-lent mass flux and momentum transfer have been tested successfully by a lot of researchers (Chen and Wood (1984), Elghobashi et al. (1984), Melville and Bray (1979)) and have been extended by Hallmann et al. (1993) for turbulent heat and mass transfer in evaporating sprays (Eqs. (29), (30)). The turbulent viscosity l't,p of the dispersed phase is modeled using the approach of Melville and Bray (1979):

-

!!£..

kp

with

k -_p-2

~(u'

p 2

+

v'p 2

+

w'p 2) (36) The ratio of the turbulent kinetic energies of the dis-persed and the gas phase is calculated following the approach of Kramer (1988):

kp- 1

k - 1

+w

2_r2 (37)

Since in general the droplets do not follow the motion of the surrounding fluid from one point to another it is expected that the ratio kp/k is different from unity and varies with the particle relaxation time T and lo-cal turbulence quantities (Elghobashi et al. (1984)). Kramer recommends the following equations for the frequency of the particle response

w

~

( jfk

r)o.2s

T Lx (38)

1 Pp

v;

1

18 p 7 1

+

0.133Re~·687

T = (39)

with a characteristic macroscopic length scale of tur-bulence

kl.5

L - C0_·75_{- (40)}

X - 1.1 f

For the turbulent Schmidt number Sct,p of the dis-persed phase Kramer suggests a value of 0.3. In the course of our work we found this value to be particle size dependent. Because more detailed information is not available at the moment we recommend a me-dian value of 2.5 for problems treated in this paper. Coupling with the gas phase equations The gaseous phase is affected by the dispersed phase due to momentum, heat and mass exchange. The easy way of coupling the two phases is an evident advantage of the new model. Due to an identical mathematical formulation the same terms appear for both phases, simply with different sign. A drawing up of the coupling terms for the gas phase equations is given in Table 2.

The expressions are added up for all nd discrete drop-size classes. In addition phase coupling is gua-ranteed by the use of temperature, pressure and va-pour concentration dependent properties in the cal-culations of the gaseous phase.

Source terms (Sk,p, S<,p) for the equations of the k,

!-turbulence model due to the presence of the dis-persed phase were modeled following the approach of Chen and Wood (1984, 1986). Because the calcu-lations presented in this paper do not show any sig-nificant influence of these terms they are not taken

iuLo tJOH/'jitlcnJ-tiuu lwrtl. sc}},p nd p

-

E

(Sv.) k=l nd cc>

-

L:(SuJ

k=l

Uj

- E (

~'Vc:P]f-Cre

l

(u;

- 'Uj,p)

+ Su. Uj,p)

.1:=1 1'

li

_ E ( (

6NuFrJ.,.J (T _ T.)

Cp,p VG "' D'> p

t : l ~· ,

+S.,

.

(~

+Tp)))

Table 2: Coupling terms for the gas phase equations due to droplet transport and evaporation

Solution steps

As mentioned above, the equations of both phases can be solved numerically using the same finite vo-lume discretization method (Noll and Wittig (1991), Noll (1992)). After calculating the droplet field with the preceding values of the gas field properties, the gas field is recalculated with the coupling terms due to droplet transport and evaporation. This proce-dure is repeated until the coupling terms converge, i.e. both phases have statistically constant values.

Results and discussion

For the verification of the new model, measurements of a recirculating droplet charged air flow (Him-melsba h (1987), Wittig et al. (1987), Wittig et al. (1988)) within a model combustor are compared with. The test section used in our laboratory is shown in Fig. 1. It consists of a rectangular flow channel with a cross sectional area of 100*300 mm2_• A prefilming two-dimensional airblast nozzle is in-corporated into the test section. The airflow enters the channel through four slots with 60 m/ s m an air velocity. Two of these slots are charged with a liq-uid film. The experimental studies were performed with ethanol at inlet gas temperatures of 320 K and

520 K.

The symmetric flow field in the model combustor is characterized by a recirculation zone induced by the centerbody of the nozzle (Fig. 2). As reported ear-Hot (WiLLi II l. (19B7)) i1 ~;;~n bs pr11dict~;~d wil.b

sufficient accuracy. In the calculations a 68

*

36 com-putational grid has been used with the MLU-scheme (Noll ( 1992)) for the discretization of the convective terms. The convective terms ofthe droplet equations are discretizised using the well known UPWIND-scheme.

Droplet motion and evaporation were calculated for ten discrete drop-size classes. In Figs. 3 and 4 calcu-lated volume fractions of the liquid phase and cha-racteristic droplet diameters uf the spray are plot-ted against experimental data given by Wittig et al. (1987) for an inlet gas temperature of 320 K. The initial conditions for the droplets were deter-mined from elaborate studies of air-blast atomiz-ers as a function of the operating parametatomiz-ers of the nozzle and the properties of the liquid (Aigner (198()), Sattelmayer (1989)). They can be described with the distribution parameters D63.2 = 63 J.Lm

and N = 2 in the Rosin Rammler distribution (see Appendix B). Both, measurements and predictions show that in the recirculation zone mainly small droplets are found due to turbulent dispersion, while the forward flowing regions are dominated by large droplets. Highest droplet concentrations occur close to the atomizer's edge decreasing rapidly towards the recirculation zone. Turbulent particle dispersion is slightly underpredicted. Nevertheless measurements and predictions are in good agreement.

Fig. 5 shows the predicted spatial distribution of the decreasing droplet diameter in the upper half of the combustor for an inlet gas temperature of 520 K. For comparison results of a Lagrangian approach are shown. Both computational approaches are based on the same evaporation model, the well known 'Uniform Temperature' law (Faeth(1983), Wittig et al. (1988), Hallmann et al. (1993)). The excellent agreement of the two methods in almost all details is proof that the new Eulerian model yields adequate results for the diameter decrease of the evaporating droplets. It should be noted that the shape of the Eulerian solution in Fig. 5 can be influenced by vari-ations of the turbulent Schmidt number Sct,p. As mentioned before, Sct,p is not a constant and fur-ther investigations in the area of turbulent droplet dispersion are necessary to guarantee a general ap-plication of the new model. Fluctuations, which can be seen in the Lagrangian results, are due to the sta-tistical nature of the Monte Carlo sampling method. They could be damped by increasing the number of particles used to evaluate mean droplet diameters. Figs. 6 and 7 show calculated and measured vo-lume fractions (Himmelsbach (1987)) and characte-ristic droplet diameters for T = 520 K (Wittig et al. (1987, 1988)). The larger diameters compared to the cold flow conditions are caused by larger ini-tial diameters {D63 .2

=

78 J.'ffl, N

=

2) but also by

the faster evaporation of small droplets. The compu-tational results are of sufficient agreement with the measurements for both, volume fractions and cha-racteristic diameters of the dispersed phase.

Effects of phase coupling can be seen in Fig. 8 and Fig. 9 for hot flow conditions. The vapour concentra-tion of the gas phase shows a strong increase along the way of the evaporating droplets combined with a decrease of the gas temperature. The recirculating flow transports parts of the cold vapour/air mixture back to the center body of the nozzle resulting in very high vapour concentration and temperature gradi-ents near the atomizing edge. At the channels out-let the vapour concentration reaches approximately 10 % resulting in a temperature decrease of more than 100 degrees.

Conclusions

A new Eulerian model for the computation of tur-bulent evaporating sprays has been developed. In contrast to former Eulerian approaches it comprises transport equations for droplet heating and evapo-ration and is applicable for recirculating flows. The coupling of the gaseous and the droplet phase is gua-ranteed by the exchange of source terms due to mo-mentum, heat and mass transfer and by the calcu-lation of temperature, pressure and vapour concen-tration dependent properties of the gaseous phase. Comparison of the new model with Lagrangian cal-culations and experiments with respect to diameter distributions and concentrations reveal good agree-ment.

However, the physical understanding of the turbu-lent particle dispersion processes need further im-provements for a more general application. Never-theless, the similar structure of the transport equa-tions obtained by the Eulerian approach with the commonly used gas phase equations offers the oppor-tunity for an easy incorporation of our new model in standard CFD-codes. This is a great advantage in contrast to Lagrangian methods which require dif-ferent numerical procedures for the two ph,~<:>es. In addition, the implementation of Lagrangian methods in codes for boundary fitted non-orthogonal coordi-nates or unstructured grids is very complicated, but is easily accomplished with the present formulation.

Acknowledgements

This work was supported by the Arbeitsgemein-schaft Hochtemperatur Gasturbine (Turboflam) and by a grant from the SFB 167 (High Intensity Com-bustors) of the Deutsche Forschungsgemeinschaft.

References

[1] T.W. Abou-Arab, M. C. Roco: Solid Phase Contribution in the Two-Phase Turbulent Ki-netic Energie Equation. ASME-Journal of Flu-ids Engineering, 112, pp. 351-361, 1990. [2] M. Aigner: Charakterisierung der

bestim-menden Einflu8gro6en bei der luftgestiitzten Zerstaubung: Physikalische Grundlagen und meBtechnische Erfassung. Dissertation, Insti-tut fiir Thermische Stromungsmaschinen, Uni-versitat Karlsruhe (T.H.), 1986.

[3] G.K. Batchelor: Transport Properties of Two-Phase Materials with Random Structure. An-nual Review of Fluid Mechanics, 6, pp. 227-255, 1974.

[4] C.P. Chen, P.E. Wood: Turbulence Closure Modeling of Two- Phase Flows. Chemical En-gineering Communications, 29, pp. 291-310, 1984.

[5] C.P. Chen, P.E. Wood: Turbulence Closure Modeling of the Dilute Gas-Particle Axisymet-ric Jet. AIChE-Journal, 32, No. 1, pp. 163-166, 1986.

[6] C.T. Crowe: Review - Numerical Models for Dilute Gas-Particle Flows. ASME-Journal of Fluids Engineering, 104, pp. 297-303, 1982. [7] S.E. Elgobashi, T.W. Abou-Arab, M. Rizk, A.

Mostafa: Prediction of the Particle-Laden Jet with a Two-Equation Turbulence Model. Inter-national Journal of Multiphase Flow, 10, No.6, pp. 697-710, 1984.

[8] G.M. Faeth: Evaporation and Combustion of Sprays. Progress in Energy and Combustion Science, 9, pp. 1-76, 1983.

[9] A.D. Gosman, E. Ioannides: Aspects of Com-puter Simulation of Liquid-Fueled Combustors. Journal of Energy, 7, No. 6, pp. 482-490, 1983. (10] M. Hallmann, M. Scheurlen, S. Wittig:

Com-putation of Turbulent Evaporating Sprays: Eu-lerian Versus Lagrangian Approach. Submitted for presentation: 38th IGTI Conference, Cincin-nati, Ohio, USA, May 24-27, 1993.

[11] J. Himmelsbach: Experimentelle und nu-merische Untersuchungen zur Ausbreitung und Verdampfung fliissiger Brennstofftropfen in einer HeiBgasstromung. Diplomarbeit, lnstitut fiir Thermische Stromungsmaschinen,

Univer-sit""& K rlsrulu: (T.H.), lDB?.

[12] R. Kneer, E. Benz, S. Wittig: Drop Motion Behind a Prefilming Airblast Atomizer: Com-parison of Phase Doppler Measurements with Numerical Predictions. Proceedings of the 5th International Symposium on Applications of Laser Technology to Fluid Mechanics, 1990. [13] M. Kramer: Untersuchungen zum

Bewegungs-verhalten von Tropfen in turbul nL r tromung im Hinblick auf Verbr nnung vorgange. Disser-tation, Institut fUr F u rungst chnik, Univer-sitat Karlsruhe (T.H.), 1988.

[14] B.E. Launder, D.B. Spalding: The Numeri-cal Calculation of Turbulent Flows. Computer Methods in Applied Mechanics and Engineer-ing, 3, pp. 269-289, 1974.

[15] W.K. Melville, K.N.C. Bray: A Model of the Two-Phase Turbulent Jet. International Jour-nal of Heat and Mass Transfer, 22, pp. 647-656, 1979.

[16] A.A. Mostafa, H.C. Mongia: On the Modeling of Turbulent Evaporating Sprays: Eulerian ver-sus Lagrangian Approach. International Jour-nal of Heat and Mass Transfer, 30, No. 12, pp. 2583-2593, 1987.

[17] A.A. Mostafa, S.E. Elghobashi: A Two-Equati-on Turbulence Model for Jet Flows Laden with Vaporizing Droplets. International Journal of Multiphase Flow, 11, No. 4, pp. 515-533, 1985. [18] B. Noll: Evaluation of a Bounded High

Res-olution Scheme for Combustor Flow Compu-tations. AIAA-Journal, 30, No. 1, pp. 64-69, 1992.

[19] B. Noll, S. Wittig: A Generalized Conju-gate Gradient Method for the Efficient Solu-tion of Three Dimensional Fluid Flow

Prob-le s. Nu erical Heat Tran.sfet, Pa:rt B, 20,

No. 2, pp. 207-221, 1991.

[20] T. Sattelmayer, S. Wittig: Performance Cha-racteristics of Prefilming Atomizers in Compar-ison with Other Airblast Nozzles. Encyclopedia of Fluid Mechanics, 8, pp. 1093-1141, 1989. (21] 0. Simonin: Eulerian Formulation for

Partic-le Dispersion in TurbuPartic-lent Two-Phase Flows. Proceedings of the 5th Workshop on Two Phase Flow Predictions, Erlangen, March 19-22, pp. 156-166, 1990.

[22] F.M. Sparrow, J.L. Gregg: The Variable Fluid Property Problem in Free Convection.

[

23)

S. Wi ig, W. Klausmann, B. Noll: Turbulence Effects on the Droplet Distribution behind Air-blast Atomizers. AGARD-CP-422, 1987. [24] S. Wittig, W. Klausmann, B. Noll, J.

Himmels-bach: Evaporation of Fuel Droplets in Tur-bulent Combustor Flow. ASME-Paper-88-GT-107, 1988.

Appendix A

The reference values in the models presented are determined according the 1/3-rule of Sparrow and Gregg (1958) Xv,rej Cp,v,rej >.ref 2 1 3TP

+

3Too 2 1 -Xv p + -Xv oo 3 ' 3 ' n(Trej) f(Tref) Cp,v(Trej) >.(Trej) ( 41) {42) (43) {44) {45) {46)

The mol fraction of vapour at the droplet's surface is given by an exponential law following Cox-Antoine

Figures

Xv,p(p,) =

~

(47)

B

p,(Tp)

=

exp (A- Tp +C) (48) where A, B and C are specific values for the droplet liquid under consideration.

The relation between the vapour concentration c01 and the mol fraction of vapour Xv is

Appendix B

In the R in Rammler dr p-si~e disll·ibuti n

100 1

Dx

=

Ds3.2ln(

100 _ x) liT {50)

x indicates the volume percentage of all droplets with smaller diameters than Dx. Two characteristic

diameters are needed to completely define the dis-tribution. In presenting the results the volumetric mean diameter D50 and a characteristic diameter for

small droplets D1o are used.

I

>. 0.05 0.03 0.00 -0.02 -0.05 0.00 0.100 0.075 >. 0.050 0.025 0.000

-

SOmis

-

--

---

_-

_-

_-

_-

_...

.

.

-; -; -; -; = -;

=

= -

;; ; ; ;

-

-.. .,. -

- -

... ... .. .. .. ~ . ; . . • ~ - - & • • • ' •

lL~l~~~~~~~~~~~~~

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 x [m]

Figure 2: Calculated flow field within the model combustor

Plane 1 (••20 mm) 0.100 ~ 0.075 0.050 0.025 · u 0.000 0 2 J 4 5 Vc

• (v,;v.)

Plane 2 (x•ISO mm)

..._

r---AA

=--~

r

~

~

~

__.-0.0 0.:1 1.0 1.:1 Vc • (VtfVI} Plane l (x•120 mm) 0.000 -t--o-... ...;-. ... --t ... ...,t--...; 2.0 0.0 0.:1 1.0 1.:1 2.0 Vc • (VtfVI} Figure 3: Volume fractions for cold flow conditions; !:::. measurements,-- calculations

Plane 1 (x•20 mm) Plane 2 (x•60 mm) 0.100 0.000 +---... --..;-... +---.-t 0.000 +--... "'+-... ...;,.,.. ... +----t 0 25 50 75 100 0 25 50 7:1 100 0 25 50 7:1 100

o.

[~]

o.

[~]

o.

[~]

.---. ~0 '--' Q. Oil) 0 X

Figure 5: Spatial distribution of the droplet diameter for a starting diameter of 17.67 J-Im ;

upper diagram: Eulerian approach, lower diagram: Lagrangian approach

Plone 1 (••20 mm) Plone 2 (x•60 mm) Plone J (x•120 mm)

0.100 0.100

--

... lf-1 _[:., :::::0 ~ 0.075 ... 0.075

...

>. 0.050 0.050 ~ IIC....A..I: n 0.025 0.025

-L "-0 _.;;;;;;;J 0.000 0.000

,--

0.000 -+----...-f;;,... ... +--.-+-...-i 0 5 0.0 0.5 ~.0 ~.5 2.0 0.0 0.5 ~.0 l.!l 2.0 vc • (V₉/V1) Yc • (V,/VI)

Figure 6: Volume fractions for hot flow conditions ; 6. measurements, - - calculations

Plone 1 (••20 mm) Plone 2 (x•60 mm) PI one J (x•120 mm)

0.100 0.100 0.100 0.075 0.075 0.075 >. 0.050 0.050 0.050 0.025 0.025 0.025 0.000 0.000 0.000 0 25 50 75 100 0 25 50 7:1 100 0 25 50 75 100

o.

[111"11

o.

[111"1]

o.

[111"1)

0.05 0.03

'E

_0.00 ..._. >--0.02 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

x [m]

Figure 8: Calculated fuel vapour distribution for hot flow conditions

0.05 0.03

'E

_0.00 ..._. >--0.02 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

x [m]

Figure 9: Calculated gas temperature distribution for hot flow conditions

Discussion

Question 1. Dr L. Ianovski

Why do you not take account of the thennophoresys force in your model for droplets?

Author's Reply

In the flow considered, these forces are of negligible influence.

Question 2. A. Kleitz

In your experimental device, you use a Malvern which doesn't give a "true" local size measurement. What is the size of the

measuring volume? Author's Reply

Knowing that the Malvern Particle Sizer integrates along the laser beam, we have chosen two-dimensional experimental

conditions which allows us locally resolved measurements.

Question 3. C. Hassa

Have you compared calculations with experimental investigations giving a liquid flux distribution: for instance, Snyder and

Lumley, JFM 1977?

AuthQr's Reply

We compared our predictions with measurements from Hishida et al, which were presented at the "5th Workshop on Two Phase

Flow Modelling" in Erlangen. These measurements include particle velocities and volume fluxes for different flow