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BASIC CONCEPTS IN THE ANALYSIS OFDIFFUSION CONTROLLED

COMBUSTION

Amable Liñán

Escuela de Ingenieros AeronáuticosUniversidad Politécnica de Madrid

6thInternational Congress on Fire SafetyEngineering

Madrid, 23-25 February, 2011

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FIRE

Rapid burning of a combustiblematerial with the evolution of heat,and normally accompanied by flame.

FLAME

Reacting body of gas that gives offenergy in the form of heat and light.

COMBUSTION

Chemical reaction between a fuel(solid, liquid or gas) and the oxygenof the air. It generates heat and light.

LAVOISIER 1775

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COMBUSTION

Chemical reaction between a fuel (solid, liquid orgas) and the oxygen of the air. It generates light and heat

LAVOISIER 1775

Example: HEPTANE/AIR

7 16 2 2 211 7 8C H O CO H O

The detailed mechanism includes hundreds ofintermediate species (like CH3, CH. H2, O, OH, H, HO2,CO …) in addition to the fuel

A minimal reduced mechanism would include

7 16 2 2

2 2

74 7 8

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7 72

C H O CO H O

CO O CO

complemented with

2 2 2N O NO

resulting from the NO thermal production mechanism ofZeldovich

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2

( )( )

N O NO NN O NO

SFO

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PRODUCTION AND DESTRUCTION OF O3

IN THE UPPER ATMOSPHERE

2

2 3

O 240Prod

h O O nm

O O M O M

3 2 240 315O h O O nm

3 2

2

( )

( )

Cl O ClO O

ClO O

S

lF C O

3 2 2

2 2

( )

( )

S

F

NO O NO O

NO O NO O

Destr, 3 22O O O

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CO2 emissions (in 2000)

28200 Mton

Annual absorption of CO2

17000 Mton

_________________

Pollutants in the troposphere

380 ppm, CO2

1,75 ppm, CH4

0,3 ppm, SO2

0,05 ppm, O3 (4 ppm at 30 Km)

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DESCRIPTION OF THE COMBUSTIONPHENOMENA WITHIN THE FRAMEWORK

OF FLUID MECHANICS

710 ;m L 9010 ;t s t

410ct s

(Local thermodynamic near-equilibrium as a chemicallynon reacting mixture)

________________________

VARIABLES CHARACTERIZING THEDISTRIBUTION OF MASS, MOMENTUM AND

ENERGY

( , ) ,t Y x ( , ),tv x ( , )e tx

________________________

OTHER THERMODYNAMIC VARIABLES

, , /p T h e p p

Equations of state , , , , ,p T Y h h p T Y

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CONSERVATION EQUATIONS

MASS

( ) 0divt

v

MASS OF SPECIES / Y

( )dY div Y wt

v v

Local acum. Transp. by convect and diff Chem. React.

MOMENTUM CONSERVATION

-div grad pt

v v v g

Local acum. Convect. and viscous trans. grav. and press. forc.

ENERGY CONSERVATION

2 2/2 /2e v div e vt v

Local accumulation convect transport

( ) ( )raddiv p div div g v- v)+ ( v q qWork of gravity, pressure and viscous forces. Heat trans.

by conduction and radiation

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EQUATIONS OF STATE FOR GASEOUSREACTIVE MIXTURES

1

/ ; 1/ /N

p TR M M Y M

0

0

1 1

/ ( )N N T

pTe p h h Y h Y Y c T dT

Chemical and thermal enthalpy

0 Th h h

0

( )TT

pTh c T dT

1

N

p pc c Y

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NON-EQUILIBRIUM TRANSPORT PHENOMENA(Simplified description)

NAVIER-POISSON LAW

23ij ij ij i j j i v ijp v v

v

________________________

FICK’S LAW for the diffusion velocities d v v v

, 1 , 1dY D NY j v (approximate form for quasi-binary mixtures; neglecting Soret effects)

___________________

FOURIER’S LAW FOR HEAT TRANSFER

1

N

k T h

q j________________________

( ), ( ), ( ), ( )v vT T k k T D f T for a quasi-binary mixture

r/ , / / 1 /p p Tc k P D c k D D L L(Lewis number of the species )

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CONSERVATION EQUATIONS FOR THEMASS OF THE SPECIES

( ) ( )dY div Y wt

v v

________________________

d v v v

v /d dY D gradY D L v

( , , ) / cw w p T Y t

0, ,x L t t Uv ________________________

CHARACTER. VELOCITYREYNOLDS NUMBER

DIFFUSION VELOCITYULD

RESIDENCE TIMEDAMKÖHLER NUMBER

REACTION TIMEc

LUt

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METHANE COMBUSTION IN AIR

Global reaction

light

4 2 2 22 2CH O CO H O heat

Reduced kinetic scheme

4 2 2

2 2

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212

CH O CO H O

CO O CO

NO production (Zeldovich)

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2

N O NO N S

N O NO O F

2 2 2N O NO

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JET DIFFUSION FLAMES

20 AY

20Y

FY0

0F

T

Y

2

0

0 A

T

Y

FU

2

0

0 A

T

Y

AU

AU

T

Overall reaction

2 2 2

2 1 2 2 2

1 2

( )

/ / / /TO CO H O h F

F sO sCO s H O q

w s w s w s w q w

Thermochemical parameters

2

/ , /FO O A S O OS sY Y T T T

with / (1 )S O FO pT T qY c S

2/F OsY Y Equivalence ratio

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Overall Arrhenius reaction rate

2

/ OF nnE RTF F Ow Be Y Y

Reaction time 1 /E RTcht Be

Residence time /m Ft a U , a inject. radius

Damköhler number// E RT

m chF

aDa t t Be

U

Combustion Reactions / 1, / 1S O O ST T T E RT

Zeldovich number

2

( ) ( )/

S O S O

S S S

T T T TERT E RT T

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Conservation Equations 2 1M

0

ˆˆ ˆ

v

F T FF F

F FO

t

pt

Y D wY Y

t L Y

v

vvv g

v

ˆ

ˆ ˆO T FO F

O FO

Y D wY Y S

t L Y

v

pp T p F

c Tc T D c T qw

t

v

/ (constant), T

T SS

p MD T

RD T

T

/ˆ ˆ ˆ/ OF nnE RTF FO F Ow Y Be Y Y ,

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OF

A

nnF OB BY Y

2 2

ˆ ˆ/ , /F F FO O O O AY Y Y Y Y Y

0 0( ) / ( )sT T T T , 0 0( ) / (1 )s F pT T qY c S

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/F C F OU U U t t

, ,C Oa U t

2 r(1 P )) (F sO s q

/ ˆ ˆ/ OF nnE RTF F Ow Be Y Y (Arrhenius reaction)

Chemical time1 /E RT

cht Be

´

2

/´

11 /

ˆ 1ˆ ˆ ˆ OF

C F

nnE R TFF T F

c

F

hO

OY Y D Y

U Dt

t a

B e Y Y

a

t

v

2

2 2

1

CC e CO

O OF

v

O

UU p Ug

t

pt

a aD

a

v

v v g

MAIN PARAMETERS

Strouhal nº / ;C Oa U t Froude nº /CU gaReynolds nº /C OU a

Damköhler nº / C ch aa U t D

If 1:aD Frozen chemical reaction

If 1 :aD Chemical equilibrium:2

0F OY Y

FUEL

AIR

FU2a

AU

AIR

AU

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Characteristic timesResidence / Fa UDiffusion 2 / Fa UCharacteristic time Ot

12/

/1

1

/

1/

2

/

/

ˆ 1ˆ ˆ ˆ ˆ( )

min , / , /

chFFO

O

m

F

tU aD at

t

nnE RTFF F F F F F O

tm

m O F F

YL Y Y D Y Be Y Yt

t t a U a D

v

Damköhler number /E RTa mD t Be

¨ 01: F FaD L Y (chem frozen flow)

ˆ ˆ0 0ˆ ˆ(chem equil flow) 0 or

1:

0

OF nnF F O

c O

a W Y Y

Y Y

D

/ 1C

u C

T TEE RT RT T

m at Be D e

1aD if , 1c aT T D if cT T

FUEL

0FY

0 0Y

0FY

Diffusion flame

Premixed flame sheet0 0FY Y

F

u

U

FUELT

uT

AIR

AIR2a

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Planar premixed flame front

2O uY

FuY

uT

bT

0FbY

2O bY

LU L u bU

2bRT

E

rL

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0 u Lu

u Ux

/ ˆ OF nnE RTF T Fu L F O

F

Y D YU Be Y Y

x x L x

Equivalence ratio2

/Fy O usY Y

Lean flames ( 1): 0, /Fb b u Fu pY T T qY c

b sT T , L LU S for 1Rich flames

2 2( 1): 0, /O b b u O b pY T T qY sc

Flame velocity

( 1)

1/ 2/F O

b

n n

E RTL Tb

b

EU D Be

RT

Flame thickness /L Tb LD U

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IN THE BURKE-SCHUMANN LIMIT, /E RT

F

aBe

U

2

2

/ˆ 1ˆ ˆ ˆˆ

ˆˆ 0

OF

F O

nnE R

Da

TF TF F

F F F

F OY Y

a Y D aY Y Be Y Y

U t L U

v

20 0Y

FU

T0

0

FY

0Y

0Y

FY

T

fT

F

2a

Flamesheet

1

1

The fuel and oxygen do not coexist the production term is

a Dirac delta source along the flame surfacef with a

strength to be determined, together withf . The

reactants reach the flame in stoichiometric proportions

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Burke-Schumann analysis for 1F OL L

ˆˆ

ˆˆ

F FT F

FO

O FT O

FO

DY wD Y

Dt Y

DY wD Y

Dt Y

ˆ ˆ ˆ ˆ 0F O T F O

DSY Y D SY Y

Dt

(Schwab-Zeldovich coupling funtion)

Mixture fractionˆ ˆ 1

1F OSY Y

ZS

(Z=1 in the fuel stream, Z=0 in the air stream)

0TDZ

D ZDt

Burke-Schumann limit

/ ˆˆ: 0E RTF O

F

aDa Be Y Y

U

Flame sheet at 1/( 1)stZ Z S

ˆ ˆ/ 1O S O F OH T T T T Y Y

0 0T

DHD H H

Dt ,

if the injector wall is adiabatic; then / 1 atS O CO p stT T T qY c S Z Z

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Burke-Shumann analysis for 1F OL L ˆ 1 ˆ

ˆ 1 ˆ

F FT F

F FO

O FT O

O FO

DY wD YD t L Y

DY wD Y S

D t L Y

ˆˆ

ˆ ˆ ( ) 0OFF O T

F O

YYDSY Y D S

Dt L L

ˆ ˆ ˆ ˆ1 1

,1 1

F O F OSY Y SY YZ Z

S S

ˆ ˆ0; 0TF O

M

DDZZ Y Y

Dt L

( )Z Z Z

11 1,

1O

M O F

S LS S

L L S L

ˆ ˆ0 for ; 0 forF S O SY Z Z Y Z Z

Flame sheet at1/( 1)sZ Z S or

at 1/ 1sZ Z S

1

1

Z

SZ

SZ Z0

Z

FY

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COMBUSTION INSTABILITIES(HYDRODYNAMIC)

RAYLEIGH

ATOMIZATION OFLIQUIDJETS DUE TO SURFACETENSION INSTABILITIES.

HELMHOLTZ-KELVIN 0/ 1e fR U a

2F AU U

FU

AU

2F AU U

2F AU U

FU

AU

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0

0

1NU l

,0

1L LS

1/ 2m N Nl x l

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