Effect of Radiative Heat Transfer on Thermal Ignition

7/24/2019 Effect of Radiative Heat Transfer on Thermal Ignition

1/6

260

J.THERMOPHYSICS

VOL.3, NO. 3

Effect

of

Radiative Heat Transfer

on

Thermal Ignition

W .W.Yuen*and S. H.Zhu|

University of

California,

Santa Barbara, Santa Barbara,

California

The

effect

of

radiative heat transfer

on

thermal ignition

is

analyzed numerically.

Tw o

specific problems

are

considered. They are 1) the semi-in finite problem in which a large mass of hot burnt gas is suddenly brou ght

into contact with

a

large mass

of

cold unburnt air/fuel mixture,

and 2) the finite-hot-slab

problem

in

which

a finite slab of hot burnt gas is suddenly immersed in a sea of cold unburnt air/fuel mixture. Numerical results

show that

in

both problems

theradiative

emission

and

absorption

by

physical boundaries play essential roles

in

determining the effect of radiation on thermal ignition. In general, radiation is shown to have important effects

on the

ignition delay.

For the finite-hot-slab

problem,

the

minimum slab thickness required

for

ignition

is

also

shown to depend strongly on the radiation effect. Both the optical thickness of the burnt mixture and the

unburnt

mixture

are

important

in

determining

the

actual effect

of

radiation.

Nomenclature

x

b

(t )

a = absorption coefficient,1/m *s(0

B

=

dimensionless parameter,

Eq .

(11

a) #

C

= dimensionless parameter, Eq.(lib)

c

=

specific heat

of

gas, KJ/(Kg-K) *

E

a

=

activation energy,K J / K m o l

E

n

(x) = exponential integral function ^ o

f ( r f ) =

dimensionless temperatureprofile, Eq. ^

(2 1 ) 6/,m

F n,\

c

)

=

function utilizedin Eq. (12)

e

G/(i

=

1,2,3,4,5)

=

parameters defined

by

Eqs. (A3-A7) f

/ / ( /=1,2,3,4)

=

param eters utilized

in

Eqs. (22)

an d

(23)

ft

k - thermal conductivityofgas, W/(m -K)

K

u

= initial optical thick ness of the unbur n t

f t ,

mixture

K

b

= initial

optical

thickness of the burnt

ft>(

r

)

mixture

l

c

=characteristic length defined

by Eq.

&

(3b), m

L =total length of the one-dimensional

* ?

system defined

in

Fig.

1, m

m

=

mass loading

of the

mixture, Kg/m

3

0

M = dimensionless mass loading definedby c

Eq. (10)

M

bu

= mass loading ratio,m

b

/m

u

K

n

= complex index of refractionof particles

K

b - i

N(n)

=function defined

by Eq.

(A10)

*o

q

r

=radiative heat flux, W /m

2

* 2 5

q

r

=

dimensionless rad iative heat

flux

K

L

Q

=

volumetric heat generation rate,

W/m

3

V

Go

= characteristic heat of reaction,

W/m

3

P

R =dimensionless heat generation rate

defined

by Eq. (4) r

R =total dimensionless heat generation

defined by Eq. (5)

T

R

c

=universal gas constant, K J/(Kmo l-K)

T

m,o

S = dimensionless temperature gradient, 0(

T

)

/'(O)

t =t ime,

s

t

c

=characteristic time d efined by Eq. (3a), s Subscripts

T = temperature,

K

u

x =space co ordinate, m b

coordinate

of the

foot

of the

f lame,

m

half-thickness of the f lame, m

dimensionless temperature ratio defined

by

Eq. (9)

dimensionless half-thickness

of

f lame,

=

value

of 6 at

which

dr

m

/d0

= 0

= minimum value of 6 ,-

= activation number,E

a

/R

c

T

b

= dimensionless length defined by Eq. (2)

= dimensionless initial position of the foot

of the

f lame

=

minimum initial slab thickness

fo r

ignition

= d imensionless position of the foot of the

flame

= dimensionless length of the one-dimen-

sional system

= dimensionless variable defined by

Eq. (20)

=

dimensionless time defined

by Eq. (2)

= dimensionless time at which the two

combustion zones are in contact

= optical thickness defined by Eq. (9)

=

optical thickness defined

by Eq.

(A8a)

= optical thickness defined by Eq. (A8b)

= optical thickness defined by Eq. (A8c)

=

optical thickn ess defined by Eq.

(A8d)

= characteristic wavelength, m

= gas

density,

Kg/m

3

= Stefan-Boltzmann constant, W/(m

2

-K

4

)

=

dimensionless temperature defined

by

Eq. (2)

=r(0)

= value of

T

m

at whichdr

m

/d0= 0

=

dimensionless heat generation rate used

in

Eq. (1)

:

properties of theun burnt mixture

properties

of the

burnt mixture

Received

Feb.

9,

1988;

revision received Ju ne

23,

1988.

C opyright

American

Institute of

Aeronautics

and

Astronautics, In c., 1988.

All

rights reserved.

*Professor,

Department

of M echanical and Environmental Engi-

neering. M ember AIAA.

fGraduate Student,

D epartment

of

M echanical

and

Environmental

Engineering.

I.

Introduction

T

HE present work is the second in a series of two papers

that deals with

th e

general

effect of

radiation

on

f lame

propagation and ignition. In the first paper

1

the effect of radi-

ation

on

f lame speed, f lame thickness,

a nd fuel

consumption


2/6

JULY

1989

RADIATIVE

THERMAL IGNITION

261

rate

(flame

propagation parameters that are important after

ignition has

occurred)

w as

demonstrated.

The radiative prop-

erties

of the combustion products and the reacting mixture,

together with

the

radiation

from th e

physical boundaries,

were

demon strated to have

significant

effectson theflamepropaga-

tion behavior. The present

work

considers the

effect

of radi-

ative heat transfer on thermal ignition. Tw o specific

one-di-

mensional problems

will

be

analyzed. First,

a large mass of

hot burnt gas is

suddenly

brought into

contact

with

a

large

mass of

coldunbu rnt ai r / fuel mixture.

The

conditions

under

which

ignition occurs

and the

m agnitude

of the

ignitiondelay

will

be determined.Second,aslabof hot bu rnt gas is

suddenly

immersed in a sea of cold unburnt air/fuel

mixture.

Thecrit-

icalconditionsrequired

fo r

ignition will

be

developed. Based

on numerical results

generated

for the two

cases,

th e fund-

amental

effect ofradiation (both fromthe medium and

physi-

ca l

boundaries)

on general

ignition

phenomena

will

be estab-

lished.

In recent years, a number ofworks

2

'

6

have been

reported

dealing

with

the general effect of radiation on thermal igni-

tion. For aone-dimensionalplanar absorbing-emittinghomo-

geneous medium with volumetric Arrhenius

heat

generation,

the critical ignition parameters

(similar

to that derived by

Frank-Kamenetskii

7

in the

no-radiation

case)

f or

thermal

igni-

tiona re

calculated

b y

various

nume rical techniques or approx-

imation procedures. In these analyses, the radiative heating

(or cooling) of the medium b y a hot (cold)wallwas considered

as the major

physical

mechanism leading to

ignition

(extinc-

tion). In manypractical situations,

however,

the combustion

in a

premixed

air/fuel

mixture

is promoted by the use of

sparks or pilot flames.

Ignition

also can be induced by the

simultaneous conductive and radiative heating of the pre-

mixed air/fuel mixture with

a hot

wall

and hot

combustion

products .

Even in an initially

hom ogeneous air/fuel

mixture,

th e heat generation is a function of the

concentration

of the

reacting mixture and temperature. Results from the existing

works

2

'

6

are clearly not

applicable

fo r

such

situations. The

objective

to thepresent

work

is to illustrate the

effect

ofradia-

tion on theseimportant practicalignition problem s.

II . Semi-Infinite

Problem

The

physical

modelisillustratedby Fig. la. A large slab

of

hot combustion products at temperature T

b

is brought into

contact

with a

large slab

of

cold

unburn t

air/fuel mixture

at

temperature

T

u

.

T heobjectiveof the analysis is to determine

conditions under which

the energy

generated

by chemical

reactioni s

sufficient

toovercome thermaldiffusion and ignite

the unburnt

m ixture.

As

discussed

in the previous

work,

1

th e

assumption

of a

semi-infinite

absorbing medium is

accurate

only in the optically thick limit. In order to illustrate the

general

effect of

radiation

for systems

with arbitrary optical

thickness, theanalysismu st include the existence of two black

walls

of temperature

T

u

an d

T

b

in the considered

one-dimen-

sional system as

shown

inFig.

la.

As in the previous

wor k ,

1

the energy an d

species

conserva-

tion

equationcan besimplified

based

on the

Shvab-Zeldovich

transformation and a procedure developed originally by

Spalding

8

to become

2

r

dq

r

(1)

where r, 0, and

f

are dimensionless temperature, time, an d

distance defined

by

T T

(2a)

T

b

-T

u

P ( x ) d x

Pule

(2b)

(2c)

The subscripts

u

an d b stand for propertiesevaluated for the

upstream

unburned

reacting mixtures (T=0) and the down-

stream

burned products ( j =

1), respectively. Note that the

region

0


3/6

262 W.

W. Y U E NAND S. H. ZHU

J. THERMOPHYSICS

particles

9

),q

r

can be generatedreadily fromthe solution to the

radiative transfer equation as

u

= C f

L

- fc )

B

r(=2-

K

+

0]

4

E

2

(K - K' ) d K

f

- [T(K') +(3 ]

4

E

2

(

K

f

- K)

d*'

(8)

with

T

b

-T

u

9

d / c

=CM(f) df

an d5, Caredimensionless parameters

given

by

C =a

u

l

c

a

u

=

F n,\

c

)p

u

m

u

(9)

(10)

(lla)

(lib)

(12)

Physically,

small particles in the

reacting

mixture and prod-

ucts are

assumed

to be the prim ary species responsible for the

radiativeabsorption and emission in thedevelopmentofEqs.

(10-12);

m( ) represents the mass loading of the small par-

ticles. The dimensionless parameter

B

represents th eratio of

radiative

flux

to reactive

heat

generation,

while

C stands for

the optical thickness of the system based on the unburned

absorption

coefficient a

u

.

The two

ends

of the

one-dimen-

sional systemare assumed to be bounded byblack boundary

at

temperature

T

u

an d

T

b

,

respectively. The

parameter

F n,

A

c

)

is a function of theoptical constant of the

solid

particles

evaluated at a

characteristic

wavelength X

c

. Th e definition of

various dimensional

parametersa re

given

in the

Nom enclature

section. A more detailed

discussion

of the physical assump-

tions and the

mathematical

development of the

preceding

equationswerepresented in the earlier work.

1

The initialcon-

dition, as represented byFig. la, can bewritten as


4/6

JULY 1989

R A D I A T I V E

THERMAL

IGNITION

263

III. Finite-Hot-Slab Problem

The physical model is illustrated by Fig. 3. A slab of hot

combustion products at temperature T

b

and thickness 2x/ is

initially

surrounded by a cold

unburn t

air/fuel mixture at

temperature T

u

o n

both

sides.Becauseof thesymmetryof the

problem,x =0 ischosento be thecenterof theinitialhot slab,

and the

totalthickness

of the medium is taken to be2 L.

Using

the same physical assumptions and definition of

dimensionless

variables as the

semi-infinite

problem,

the dimensionless

ener-

gy

equation isagaingivenbyEq.(1).Because of the change in

coordinate

system

(f is measured

from

the midpoint

between

the two

walls,

an d K

L

no w represents the optical

thickness

from theoriginf= 0 to thewallat

f

=

f

L

) ,

thedimensionless

radiative

heat flux

becomes

K

L

)- E

3

(K

L

- i c ) ]+

[T(K')

+

0]*E

2

(

K

- K ' ) d/c '- [r(/O+

/3]

4

2

(

K

'

- K )

The initial

condition,

as represented by

Fig.

3,

is

0

(17)

(18)

Forcasesinwhichignitionoccurs,the expected temperature

profile

after ignitioni sillustrated

qua litatively

b yFig.4a. The

transient

flame

propagating behavior

is similar to that of the

semi-infinite

problem

as

presented

in the previous

work.

1

For

the extinction

and almost-ignition

case,

the expected

tempera-

ture

profile

is shown

qualitatively

byFig. 4b. The two com-

bustion zones are in contact, and the temperature at

f

= 0 ,

T

m

is

lessthan 1.0.

In the

extinctioncase,r

m

decreases mono-

tonically

toward

zero

after th e

initial

contact. If ignition

occurs, on the other hand, r

m

will

decrease to a minimum

valuegreater

than zero

and,

subsequently,

increase. The

med-

iu m

will eventually

achieve a temperature

profile similar

to

that inFig. 4a, and the flame willpropagate. Sincet he

objec-

tive of thepresent analysis is to determine the minimum slab

thickness required for ignition, the present work willconsider

only th e

flame behavior after

the initial

contact

of the two

combustion

zones.

The integral

method

will

be

used

to

deter-

mine the behavior of r

m

an d

d.

Specifically, assuming

that

the temperature

profile

has the

following

generalized form,

0 < f < 2 6

where

r j

is

given

by

r - s

(19)

(20)

As

demonstrated in Spalding

8

and solutions to the semi-in-

finite

problem,

numerical

predictions

for the

ignition

an d

flame propagat ion

behavior are

insensitive

to the

assumed

dimensionless temperature profile

/(r/).

The same expression

used

in the

previous work

1

is ,

thus, assumed

to be

applicable

in the present

case.

It is

1

(21)

M athematically, Eq.

(21)

is

generated

by the

combination

of

tw o

pieces

of error functions.

Numerical

experiments

show

that

it is an

dequ te

approximation of the temperature pro-

file within

the

combustion zone. Notethat

6, in

addition

to

being the dimensionless thickness of the

combustion

zone, can

also be interpreted as the dimensionless

flame

center for the

present problem. Substituting

Eq.

(21)into

Eq.

(1),

and

inte-

grating

from

0 to

L

an d

from

6 to

& ,

,

a set of two

nonlinear

ordinary

differential

equations can begenerated fo r

6(0)

an d

T

m

(0).

They

are

y

+

7- *IA

-

/

2

(/3

d(r

m

5)

d9

(22)

(23)

with

S

=/'(0) being thedimensionless temperature gradient

atf = 0. Theparameters

/ ,

7

2

,

/

3

,

and7

4

,whichare

related

to

the integral of the dimensionless temperature

profile

and heat

generation

across

the combustion zone, have

been

evaluated in

the previous

wor k .

1

Expressionsforq

r

(d) and

q

r

( L)

are

given

indetailin Appen dix A. TheinitialconditionforEqs.(22)and

(23)

are

r

m

(0

c

)=1.0, 6(0

C

)

=

S

7

(24)

where

0

C

stands fo r the dimensionless time required by the

initial

step-function temperature

profile (a s

shown

in

Fig.

3) to

develop into a

profile

in

which

the two combustion

zones

ar e

in

contact

(as

shown inFig.4b); B

c

an d

< 5 /(notethat

in

general,

6

f

. ft) can be determined from an early time

solution

to

Eq. (1), which is equivalent to the

semi-infinite

case.

As in the semi-infinite

problem,

the important dimension-

less

parameters for thepresent problem ar e B,

C ,

e,

f t ,

f

L

, |8,

and themass loadingratioM

bu

. Physically,theparameter

B

illustrates the

importance

of the radiativeeffect relative to the

chemical

heat

generation

effect. The parameter C is a char-

acteristic

optical thicknessthatalso

represents

the importance

of

th e radiative

effect

compared to the conduction effect.

Again, in

contrast

to

previous

ignition analyses

that

do not

include

th e effect ofradiation, th e size and theoptical

thick-

ness

of the

system(and

the

radiative

emission and

absorption

by

the two cold walls at ft, an d f t , ) play important roles in

determining

th e

effect

of radiation on ignition.

To illustrate the

mathematical

behaviorof T

m

,it isuseful to

expand Eq. (23) and combine with Eq . (22) to yield

(25)

I-28

Fig .

4a Exp ected temperature profile for the finite-slab problem in

which ignition

occurs.

Fig.

4 b

Expected

temperature

profile

for the

finite-slab

problem in

which

possible extinction occurs.


5/6

264

1.0

0.8

0.6

0 4

0 2

W. W. YUEN AN D S . H . ZHU J . THERMOPHYSICS

100 r

1rnr

n

r

~

r

n

r

~

r

~ n

r

~

r r

i

r

8

F ig. 5 Schematic representation of the numerical

procedure

used to

determine the minimu m ignition thick ness. (Curve A represents the set

of

points at

which

dr

m

/d 5

=

0.

Curves

i 2 and 3

represent

the

variation

of

r

m

with

6 for

various starting values

of 5. The starting

value

of 5 for curve 1 corresponds approximately to the minimum

ignition thickness.)

Since dd/dd

is

always

positive [as it can be

observed readily

from

Eq. (22), and is also consistent

with

physical expecta-

tion]

, a nd

dT

m

/dO

always

starts

out to be negative at

0

=

d

c

[a s

it

can be

observed

readily from Eq.

(23)],

dr

m

/d6 is

also

negative at 0 =

6

C

.

If ignition

occurs,

T

m

must eventually in-

crease

back

to

1.0.

This

requires

a positive

dr

m

/dd.

A

neces-

sary condition

fo r

ignition

is ,

thus, dr

m

/d d

= 0 .

M athemati-

cally,

the

smallest f /

for which

such condition

can be satisfied

will

have the physical interpretation of a minimum ignition

thickness.

Specifically,

the

following

mathematical

procedure

is used

to determine the minimum thickness

r

quired for

ignition.

For

a

given

set of specified dimensionless

parameters,

a set of

points T

W > O

^o)atwhich

dr

m

/dS

= 0 are

first

determined from

Eq .

(25).

A

typical

set of solutions for

(r

w> 0

,

S o )

are

illustrated

inFig.

5. For

each

set of

r

m

6

0

),Eqs. (22)

a nd

(23)

a re

then

solved

( in

backward

time

step)

to determine the initial value of

6 / [which is identical to 5(0

C

)]at whichr

m

= 1.0.Some typical

solutions

for the (j

mt

d)

history generated

from

Eqs.

(22) and

(23)arealsoshown inFig.5 asillustrations.It can be observed

readily [andalso

can be shown rigorously

from Eqs. (22)

an d

(23)] that as 6

0

increases, the corresponding 6 / approaches

asymptotically to a

constant

6

/> m

.

M athematically,

no

solu-

tions with

6 / m

isdetermined, th e early-time solution can beused

to relate 6 /

> /n

to theminimum

initial

slabthickness f /

> / w

. Numer-

ical results,

however,

showthat in all cases considered in the

present work ,

6 /

> w

isessentiallyidentical to

f/,

m

(i.e., the flame

does

not move very much during the initial diffusion

stage);

6 /

> m

thus can be interpreted as the

minimumhot-slabthickness

required for ignition.

As in the previous work,

1

a

survey

of the physicochemical

dataand typicalflame

temperatures

f or

common hydrocarbon

fuels

10

suggests values of 5.0 and

10.72

foreandB/C,respec-

tively.Num ericaldatafo r

other

values of e (which are of

some

practical

interest in terms of analyzing

different

fuels an d

air/fuel ratios) ar e

presented

elsewhere.

11

Assuming that f

L

=

1500,

th e

minimum ignition

thicknesses for

different

ab -

sorption

coefficients

for the burnt mixture are calculated and

presented in

Fig.

6.The

no-radiation result,

which is

identical

to that generated bySpalding,

8

is presented in the same

figure

for

comparison. The absorption

coefficient

of the

unburnt

mixture

does not appear as a parameter in the

figure

because

it

has

practically

n o

effect

on the minimum

ignition

thickness.

1.0

1.0

0 1

1 0

10

10

I0

3

K

K

Fig .6 Effect of the absorp tion coefficient of the burnt

mixture

on

the minimum

ignition thickness.

The broken line represents the no-

radiation

result.

1 0

0 5

K

U

=O.OI

0 01

0 1

1 0

Fig.

7 Effects of

optical

thicknesses of the

burnt

and unburnt

mixtures on the

ignition

delay for the

finite-slab

problem (the broken

line represents the

no-radiation

results).

Physically,

th e

major

effect of

radiation

on the

minimum

ignition

thickness is to

increase

th e

heat

loss from the initial

burnt mixture to the unburnt mixture an d

cold

wall. As the

optical

thickness of the

burn t

m ixture increases, the radiative

heat loss increases,

and a

larger slab

of

burnt mixture

is re-

quired fo r ignition. The lack of dependence on the optical

property of the unburnt mixture

suggests

that th e

radiative

heattransfer from the hotburn tmixture to thecoldwallis the

dominating cooling

effect controlling ignition.

Another important physical resultassociated wtih th e igni-

tion

process is the

ignition

delay.

Assuming f /

= 50 (which is

greater than the minimum ignition thickness calculated an d

presented in

Fig.

6

and, therefore,

th e

two-combustion zone

does no t meet , an d ignition always occurs), results for the

ignition delay

ar e

plotted

fo r different initial optical

thick-

nesses

K

b

an d

K

u

inFig.7 .Notethat

K

b

an dK

u

ar e

still defined

byEqs. (15)

and (16). Although the

optical

properties of the

unburn t

mixture have

only a small effect on the mini-

mum

ignition thickness,

results in

Fig.

7

show

that

i ts

effect

on

the

ignition

delay is

quite significant. Because

of the increased

preheating

of the

unburn t

mixture by

radiative

absorption,the

ignition delay decreases with

increasing

optical thickness

of

th eunburnt mixtureK

u

.Increasingthe opticalthicknessof the

burn t

mixture,

on the

other

hand,

first increases

an d

then


6/6

JULY

1989

RADIATIVE

THERMAL IGNITION

265

decreases th e

ignition

delay. Physically, the increase in the

optical thickness of theburntmixtureleads to two

com peting

effects

on ignition delay.First, it increases the radiative heat

lossfrom the burnt mixtureand,second, it leads to additional

preheatingof the unburnt mixture. For a burnt

mixture

with

small optical

thickness, the

first effect

dominates, and the

ignition

delay thus increases with increasing

optical

thickness

of th e

burnt mixture.

In the limit of an

optically

thick burnt

mixture, the

increase

in

heat

loss caused

by an

increase

in

opti-

cal thickness

is

small compared

to the

total energy radiated

into the unburnt mixture by the burnt

mixture.

The second

effect dominates and leads, therefore, to a reduction in

ignition delay.

IV . Conclusions

An

approximate

analysis is

performed

to

determine

th e

igni-

t ion behavior

of two simple problems: 1) a semi-infinite

one-

dimensional problem in which a slab of hot

burnt

mixture is

brought

into contact with a slab of cold

unburn t

air/fuel

mixture, and 2) a finite-hot-slab problemin which afinite slab

of

hot burntmixtureis surrounded by a cold unburntair /fuel

mixture. Inboth cases, the present work can be considered as

an

extension

of a

previous

work by

Spalding

8

with the inclu-

sion

of the

radiative

heat transfer effect. Th e following

con-

clusions

ar e

generated:

1) In general, the radiation

effect

increases

ignition delay

fo r both

problems,

and for the finite-hot-slab problem it in-

creases

the minimum

thickness required

fo r

ignition. Physi-

cally, this is because of the heat-loss

enhancement

effect of

radiation.

2)

Radiative emission and absorption by physical bound-

ariesplayessential roles indetermining the

effect

ofradiation

on

ignition. Analysesthat assume

semi-infinite medium

with-

out boundary correspond only to cases with

infinite optical

thickness.

Such

analyses

ar e

thus

not useful in illustrating the

general effect ofradiation.

3) For the semi-infinite problem, the ignition delay in-

creases

with

th einitial

optical

thickness of the

burnt

mixture.

The

optical thickness

of the

unburn t

mixture has

only

minor

effect

on ignition delay, except in the limit of an

optically

thick burnt mixture. In the limit of an

optical

thick burnt

mixture

and an optically

thin

unburn t

mixture,

ignition

fails.

4) For the finite-hot-slab problem, the minimum thickness

required for ignition increases with

increasing

absorption

coefficient for the hotburn tgas.T heabsorptioncoefficient of

the unburnt m ixture has

practically

no effect on the

minimum

ignition thickness.

5)

When ignition

occurs in the finite-hot-slab

problem,

th e

ignition delay

depends on

both

the

opticalthickness

of the hot

burnt

gas and of the cold unburnt

mixture.

The ignition delay

decreases withincreasingoptical

thickness

of the unb urnt mix-

ture. For a

fixed

opticalthicknessof the cold

unbu rnt mixture,

increasing the optical thickness of the hot burnt ga s first in -

creases, and then decreases the

ignition

delay.

Appendix

Substituting

Eq.

(19)

intoEq.

(17)

andevaluatingat f

L

and

< ,

one

obtains

K2 8

)- E

3

(

KL

-

K 2 d

) ]

2Bd(G

l

+ G

2

)

=2

(Al)

2Bd(G

3

+G

4

-

G

5

) (A2)

=

\i[l-fW]T

m

+0}

4

E

2

[K

L

+K

b

-

r

, -ri)]N ri)dr

)

(A3)

-i

i

dr ,

-/di)]T

m

1

0

i = \[ T

m

f(ri)

J

0

with

=

C6[l+(M

bu

-

X L

=C[ f

L

+ (M

bu

-

(A4)

(A5)

(A6)

(A7)

(A8a)

(A8b)

(A8c)

(A8d)

where

Physically, K

O

is the optical

thickness

form f= 0 to the

flame

center

f

= 6; K

L

is the optical

thickness

of the half-domain

ranging

from f

= 0 to

f

=

f

L

; K

2d

is the

opticalthickness

of the

flame,

an d

/ ^ - ^ ( r? )stands

for the

opticalthickness from f

= 0

to an

arbitrary

f.

Th e

functions A/0?)

an d

7 V r ? )

ar e

given

by

(A9)

(A10)

References

Yuen,

W. W. and Zhu , S .

H.,

TheEffect of

Thermal R adiation

on the Propagation of Laminar Flames,

Proceedings

of the 8th

International

Heat Transfer Conference,

Vol.

2, 1986, pp.

835-842.

2

Khalil,H., Shultis,J. K .,and Lester,T. W., StationaryThermal

Ignitiono f ParticleSuspensions, ASME Transactions of

th e

Journal

of

Heat Transfer, Vol. 105, M ay 1983, pp.

288-294.

3

Arpaci, V.S., RadiationAffected ThermalIgnition, American

Society of M echanical Engineers Paper 83-WA/HT-61,

1983.

4

Arpaci,

V. S. and

Tabaczynski,

R . J.,

"Radiat ion-Affected

Laminar FlameQuenching, Combustionand Flames,

Vol.

57 , 1984,

pp.

169-178.

5

Arici, O .,

Radiation

Affect

Thermal Ignition Near th e

Wall,

Fundamental an d

Applications

of

Radiation Heat Transfer, H TD-

Vol.

72 ,

edited

b y A . M .

Smith

and T. F.

Smith, American Society

of

M echanicalEngineers, New York, 1987, pp.53-57.

6

Crosbie, A. L. and Pattabongse, M .,

Radiative Ignition

in a

Planar

Medium,

Journal

of Quantitative Spectroscopy and Radia-

tive

Transfer, Vol.

37, No. 2, 1987, pp . 192-203.

7

Frank-Kamenetski i , D. A., Diffusion

and

Heat Transfer

in

Chemical

Kinetics,

Plenum

Press ,

1969.

8

Spalding, D. B ., Approximate

Solutions

of Transient and

Two-Dimensional

F lame Phenomena:

Constant-Enthalpy Flames,

Philosophical Transactions of the Royal Society of London, Vol.

245A,

1957,

pp. 352-372.

9

Yuen,

W. W . and

Tien,

C .L., "A SimpleCalculation Scheme fo r

Luminous-Flame Emissivity," T he

Sixteenth

International

Sympo-

sium

on

Combustion, The Combustion Inst . ,

Pit tsburgh,

P A ,

1976,

pp. 1481-1487.

10

Kanury, A .

M ., Introduction

to

Combust ion Phenomena, Gor-

don and Breach Sciences Publisher, 1977.

H

Zhu, S . H., "A Study of the Effect of Thermal Radiation on

Laminar

Flame Propagation and Ignition,

Ph.D. Thesis,

Dept. of

M echanical and Environm ental Engineering,

Univ.

of Cal ifornia,

Santa Barbara,

CA,

1986.

Documents

Effect of Radiative Heat Transfer on Thermal Ignition