A Universal Model of A Universal Model of Droplet Vaporization Droplet Vaporization Applicable to Applicable to Supercritical ConditionSupercritical Condition

November 19, 1999

Zhou JiAdvisor: Dr.Jiada Mo

Sections:Sections:

• 1 Introduction1 Introduction

• 2 Model Establishment2 Model Establishment

• 3 Implementation & Tests3 Implementation & Tests

• 4 Conclusions4 Conclusions

IntroductionIntroduction

The phenomenon of droplet vaporization and combustion has many different applications. Behavior of liquid fuel in combustor is the one that motivates most researches including this work.

There are two aspects of this problem: Droplet vaporization & supercritical phenomenon.

Droplet VaporizationDroplet Vaporization

Basic physical process

•No convection, no combustion

Basic physical processBasic physical process

• with combustionwith combustion

Basic physical processBasic physical process

• with convectionwith convection

Basic physical processBasic physical process

• higher rate of blowinghigher rate of blowing

Droplet VaporizationDroplet Vaporization

Classic Model: d 2-law

co n stan t d

)(d 2

t

dK s

To predict:• evaporation rate K

• flame temperature Tf

• flame front standoff ratio rf/rs

Simplifying Assumptions of d 2-law

• Constant temperature of droplet• Simultaneous gasification and combustion• Constant transport properties• Gas-phase quasi-steadiness• No supercritical condition

Supercritical PhenomenonSupercritical Phenomenon

Equation of state p=p(v,T)

• p-v diagram

Equation of state p=p(v,T)

• p-T diagram

Equation of state p=p(v,T)

• p-v-T diagram

Conceived image of “blurred” droplet

Supercritical PhenomenonSupercritical Phenomenon

It is an open question in physics.

• universality• long range correlation

Model EstablishmentModel Establishment

• Conservation Equations

• Transport relations

• Equations of State

• Other properties (coefficients)

• Continuity equation:

• Momentum equation: u=1, 2, 3

• Species equation: i, i=1, 2, …, k-1

• Energy equation: h

•ConservationConservation EquationsEquations

General form of the relation between fluxes and driven force.

• Driven force: gradient of chemical potential and temperature

• Fluxes of mass and heat

•Transport RelationsTransport Relations

• Thermal Equation of State:

p=p(v,T, X1)

• Caloric Equation of State:

h=h(v,T, X1)

•Equation of statesEquation of states

Some can be derived from thermal or caloric EOS, like coefficient D, heat capacity; others need to be modeled individually.

• Diffusivity, Thermal-to-Mutual Diffusion Ratio (transport matrix);

• Viscosity; conductivity; expansion rate;

• Mixing rule of critical properties.

•Other coefficientsOther coefficients

Reorganized modelReorganized modelFor spherically symmetric case with two species

)(1

)( 22 rrr

rrr

p

r

uu

t

u

)()(1 2

2 r

mu

t

m

m

nunr

rrt

n

)(1

12

22

11 Jrrrnm

m

r

Xu

t

X

)(1

)(

)(1

)()(

12

22

2

1

11

22

Jrrrm

h

m

hm

qrrrr

pu

t

pT

r

Tu

t

TC

v

rVp

Reorganized model (continued)Reorganized model (continued)

where)(

3

4

r

u

r

urr

2)(3

4

r

u

r

uv

]1

)1([ 112

1 r

T

TnDkXXJ

m

mJ mTb

r

TkJRTkq bTr

)(

]}1

)()[()1(

{1

1

2

2

2

2

1

111211

r

T

Tm

h

m

h

r

p

m

V

m

V

m

XXmm

r

XnDJ Dmb

Implementation and TestsImplementation and Tests

• Two species: droplet O2, environment H2

• Discontinuity at the initial instant

• In some cases, the spherical region occupied by O2 is actually in gas phase or supercritical condition at the specific temperature and pressure.

Equation of State p=p(T, v, XEquation of State p=p(T, v, X11))

Peng-Robinson Equation of State

Hydrogen

•plus Evaporation Part.

)()( bVbbVV

a

bV

RTP

Species concentrationSpecies concentration

Initial temperature 200K

DensityDensity

Initial temperature 200K

TemperatureTemperature

Initial temperature 200K

VelocityVelocity

Initial temperature 200K

Simulated ImagesSimulated Images

• Species concentrationSpecies concentration

• TemperatureTemperature

DensityDensity

Initially two phases

TemperatureTemperature

Initially two phases

Species concentrationSpecies concentration

Initially two phases

ConclusionsConclusions• A universal model is established.A universal model is established.

• Numerical test indicates that such a Numerical test indicates that such a model can be implemented.model can be implemented.

• Temperature change shows different Temperature change shows different pattern for liquid droplet and gas pattern for liquid droplet and gas sphere. sphere.

• Equations of state and formulations of Equations of state and formulations of other coefficients are critical in the other coefficients are critical in the model.model.

Thank you for your Thank you for your attention.attention.Any questions or Any questions or comments?comments?