Modeling Local and Advective

Diffusion of Fuel Vapors:

Progress Report

Author:

Andrew Brandonasbrando@math.umd.edu

Advisor:

Dr. Ramagopal Ananth

Naval Research Laboratory, Washington D.C.ramagopal.ananth@nrl.navy.mil

Outline

• Background

• Equations and Domain

• Current Model

– Species Fraction Algorithm• Results

• Testing

– Vorticity and Stream Function Algorithm

• Timeline

• Future Work

Background

• Fuel Pool Fires

• Filming Foam

– Issues with current

surfactants

• Foam Layer and Ghosting

– Diffusion of Fuel Vapors

– Degradation of Film Layer

• Project Goal

– Effective Diffusion Constant and Flux

Earlier Timeline

• October – November– Code Species Fraction and Stream Function/ Vorticity Algorithms

– Stagnation flow solution from Leonard

• December – February– Verify code against Fuel Vapor in Air data

• March – April– Apply code to Film and Foam data

• May– Prepare report and final presentation

Governing Equations

∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

∂∂

+∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

2

2

2

2

2

2

2

2

2

2

2

2

1:

11:

11:

z

Y

r

Y

rr

YD

z

Yw

r

Yu

t

YY

z

w

r

w

rr

w

z

P

z

ww

r

wu

t

ww

z

u

r

u

rr

u

r

P

z

uw

r

uu

t

uu

ρµ

ρ

ρµ

ρ(1)

(2)

(3)

Transformed Governing Equations

∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

∂Ω∂

+∂Ω∂

+Ω

−∂Ω∂

+Ω

=∂Ω∂

+∂Ω∂

+∂Ω∂

Ω

∂∂

+∂∂

+∂∂−

=Ω−

∂∂

=∂∂−

=

2

2

2

2

2

2

2

2

2

2

2

2

2

2

1:

1:

111:

1,

1

z

Y

r

Y

rr

YD

z

Yw

r

Yu

t

YY

zrrrrr

u

zw

ru

t

zrrrrr

rrw

zru

η

ψψψψ

ψψ

(4)

(5)

(3)

The Experimental Domain

Fuel Pool

Aqueous

Film or

Foam Layer

(Domain 1)

Domain 2

Height = 20cm

Radius = 5cm

The Experimental Domain

Fuel Pool

Aqueous

Film or

Foam Layer

(Domain 1)

Domain 2

Height = 20cm

Radius = 5cm

Current Simplified Domain

Axis-Symmetric

Cylinder

Radius = 5cm

Height = 20cm

Species Fraction Algorithm• Upwind Differencing Scheme– Backward Differencing on the convective terms

– Centered Differencing on the diffusive terms

•

• Constant flow – slip conditions

• Diffusion of into

s

cmcwu == ,0

2N2O

s

cmD

2

25.0=

∆−

∆−+

∆+−

=∆−

+∆−

⇒

∂∂

+∂∂

=∂∂

+∂∂

−+−+−+

r

YY

rir

YYYD

r

YYU

t

YY

r

Y

rr

YD

r

YU

t

Y

n

i

n

i

n

i

n

i

n

i

n

i

n

i

n

i

n

i

2)1(

12

1

11

2

111

1

2

2

Species Fraction Model

• Boundary

conditions

• Initial

conditions

0=∂∂z

Y

1=Y

0=∂∂r

Y0=∂∂r

Y00 =Y

Vorticity Algorithm

• Solve

– interior and exterior points

• Upwind Differencing Scheme

• With , solve

r

w

z

u nnn

∂∂

−∂∂

=Ω

1+Ωn

∂Ω∂

+∂Ω∂

+Ω

−∂Ω∂

+Ω

=∂Ω∂

+∂Ω∂

+∂Ω∂

2

2

2

2

2

1

zrrrrr

u

zw

ru

tη

2

12

2

121

2

1 111

zrrrrr

nnnn

∂∂

+∂

∂+

∂∂−

=Ω−+++

+ ψψψ(4)

(5)

Stream Function Algorithm

•

• Successive Over Relaxation (SOR)

resijdijcijb

ijaijhije

rrrrrr

==−+++−

+++Ω+⇒∂∂

+∂∂

+∂∂−

=Ω−

0),1(),1()1,(

...)1,(),(),(

1112

2

2

2

2

ψψψψψ

ψψψ

e

resijij ωψψ −= ),(),(

(4)

Stream Function Algorithm

• SOR with Chebyshev Acceleration

– Determining the relaxation parameter

where

ω

12

0

25.01

1

1

−−=

=

k

k

q ωω

ω

2

2

1

coscos

∆∆

+

∆∆

+

=

z

r

nz

r

mq

ππ

2

1

5.01

1

q−=ω

Vorticity and Stream Function

Boundary Conditions

0=∂∂

=∂∂

z

w

z

u

s

cmcwu == ,0

s

cmwu 0==0=

∂∂

=∂∂

r

w

r

u

Vorticity and Stream Function

Boundary Conditions

0=∂Ω∂

=∂∂

zz

ψ

0),(2

22 =Ω−= Rrc

ψ

0

0

=∂Ω∂

=

r

ψ0=

∂Ω∂

=∂∂

rr

ψ

Timeline

• October – November– Code Species Fraction Algorithm

– Code Stream Function/ Vorticity Algorithms

– Testing Species Fraction Code

• December – February– Validation

• Stagnation flow solution from Leonard

• Species Fraction Test Cases

• Diffusion of Fuel Vapors in air

• March – April– Apply code to Film and Foam data

• May– Prepare report and final presentation

Future Work

• Debug Stream Function/Vorticity Code

– Test Cases

• Validation

– Stagnation flow Solution

– Species Fraction Test Cases

– Diffusion of Fuel Vapors in air

• Introduce Pipe and Fritted Ceramic Disk

• Create the Film/Foam Layer Domain

References

1. Ananth, R, and Farley, J.P. Suppression Dynamics of a Co-Flow Diffusion Flame with High Expansion Aqueous Foam. Journal of Fire Sciences. 2010

2. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. Transport Phenomena. 1960

3. Leonard, J.T., and Burnett, J.C. Suppression of Fuel Evaporation by Aqueous Films of Flourochemical Surfactant Solutions. NRL Report 7247. 1974

References

4. Panton, R.L. Incompressible Flow. 1984

5. Pozrikidis, C. Introduction to Theoretical and

Computational Fluid Dynamics. 1997

6. Press, W.H., Teukolsky, S.A.,Vetterling, W.T.,

Flannery, B.P. Numerical Recipes in Fortran.

1992

7. Williams, B.A, Sheinson, R.S., and Taylor, J.C.

Regimes of Fire Spread Across an AFFF –

Covered Liquid Pool. NRL Report. 2010

Questions?