a simplified model for the pyrolysis of charring materials

8/10/2019 A Simplified Model for the Pyrolysis of Charring Materials

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COMBUSTION ND FL ME 6 8 : 2 3 1 - 2 4 7 1 9 8 7 )

231

Simpl i fi ed M odel for the Pyrolys i s o f harr ing M ater ia l s

I N D R E K S W I C H M A N a n d A R V I N D A T R E Y A

Department o f Mechanical Engineering Michigan State University East Lansing MI 48824-1226

A s i m p l i f ie d m o d e l o f t h e p y r o l y s i s o f c h a r r i n g m a t e r i a l s is a n a l y z e d . T h e e f f e c t s o f m o i s tu r e a r e n e g l e c t e d , a n d t h e

hea t o f py ro lys i s i s a s s um ed equa l to ze ro . F ou r s t ages o f py ro lys i s a r e ob ta ined : ( i ) ine r t hea t ing , ( i i ) in it i a l py ro ly s i s ,

( i i i) th in cha r , and ( iv ) th ick char . F o rm ulas fo r the vo la t i l e m as s e f f lux , m , a r e ob ta ine d in s t ages (i i ) , ( i i i) , and ( iv ) ;

m = 0 in the f i r s t s t age . Th e ca lcu la t ions ind ica te tha t the s u r f ace t em p era tu re con t ro l s the vo la t i l e p roduc t ion r a te in

the in i t i a l py ro lys i s s t ages ( the k ine t i ca l ly con t ro l l ed r eg im e) , w h i le the t em pera tu re g r ad ien t con t ro l s the vo la t i l e

p r o d u c t i o n r a t e i n t h e t h i c k c h a r s t a g e ( t h e d i f f u si o n - c o n t r o l l e d r e g i m e ) . C o m p a r i s o n s o f t h e c a l c u l a t e d r e s u l t s w i t h

n u m e r i c a l c o m p u t a t i o n s a r e m a d e f o r t h e v o l a t i l e m a s s e f f l u x , t h e s u r f a c e t e m p e r a t u r e , a n d t h e d e n s i t y .

I . I N T R O D U C T I O N

W o o d a n d o t h e r c e l l u l o s i c m a t e r i a l s c o n s t i t u t e a

subs tan t i a l f rac t ion of the fue l l oad in many

bui ld ing f i re s , and a re inc reas ing ly be ing u t i l i zed

a s r e n e w a b l e s o u r c e s o f e n e r g y . T h u s , i t i s

i m p o r t a n t t o u n d e r s t a n d t h e i r b e h a v i o r u n d e r h i g h -

t e m p e r a t u r e b u r n i n g c o n d i t i o n s . W h e n h e a t e d

s u f f i c i e n t l y t h e s e m a t e r i a l s e v e n t u a l l y u n d e r g o

t h e r m a l d e c o m p o s i t i o n , p r o d u c i n g c o m b u s t i b l e

g a s e s v o l a t il e s ), w a t e r v a p o r , a n d c h a r . Q u a n t i -

t i e s th a t a r e o f p a r t i c u l a r im p o r t a n c e i n t h e f i e l d o f

f i r e r e s e a r c h a r e t h e e v o l u t i o n r a t e o f t h e v o l a t i l e

g a s e s a n d th e s u r f a c e t e m p e r a t u r e o f th e p y r o l y z -

i n g s a m p l e . K n o w l e d g e o f th e s e q u a n t i ti e s i s

r e q u i r e d i n both t h e i g n i t io n a n d g r o w t h s t ag e s o f

the f i re .

A s u b s t a n t ia l a m o u n t o f e x p e r i m e n t a l a n d t h e o -

r e t ic a l w o r k o n t h e p y r o l y s i s o f t h e r m a l l y t h i c k

s a m p l e s o f w o o d a n d c e l l u l o s i c m a t e r ia l s h a s

a l re a d y b e e n d o n e . T h e r e c e n t ex p e r i m e n t a l w o r k

i s r e v i e w e d i n [ 1 ] , a n d w i l l n o t b e c o n s i d e r e d

f u r t h e r h e r e . T h e p r e v i o u s t h e o r e t i c a l s tu d i e s c a n

g e n e r a l l y b e d i v i d e d i n to t w o g r o u p s : 1 ) d e ta i l ed

n u m e r i c a l s t u d i e s t h a t a t t e m p t t o p r o v i d e a c o m -

p r e h e n s i v e d e s c r i p t i o n o f w o o d p y r o l y s i s b y i n -

e l u d i n g a s m a n y p h y s i c a l p r o c e s s e s a s p o s s i b le a n d

C o p y r i g h t 1 9 87 b y T h e C o m b u s t i o n In s t i tu t e

P u b l i s h e d b y E l s e v i e r S c i e n c e P u b l i s h i n g C o . , I n c .

5 2 V a n d e r b i l t A v e n u e , N e w Y o r k , N Y 1 0 01 7

2) s impl i f i ed ana ly t i ca l s tud ies tha t a t t empt to

d e v e l o p p r a c t i c a l a n d u s e f u l f o r m u l a s f o r q u a n t i -

t i e s such a s the vo la t i l e mass e f f lux . The in i t i a l

numer ica l s tud ies [2] inc luded vo la t i l e hea t con-

v e c t i o n , A r r h e n i u s d e c o m p o s i t i o n k i n e t i c s , n e t

d e c o m p o s i t i o n e x o t h e r m i c i t y , a n d v a r i a b l e t h e r -

m o p h y s i c a l p r o p e r t i e s . S u b s e q u e n t s t u d i e s h a v e

a d d e d t h e e f f e c t s o f v o l a t i l e g a s m o m e n t u m [ 3 ] ,

m o i s t u r e , a n d a s i m u l a t io n o f c h a r c r a c k i n g [ 4 ] .

T h e n u m e r i c a l s t u d y o f R e f . [ 5] d e v e l o p s a n

o v e r a l l e n e r g y b a l a n c e b y c o m p u t i n g i n d i v i d u a l

e n e r g y b a l a n c e s a c r o s s N t h i n s l ic e s o f t h e

pyr o lyz ing ma te r i a l . The ind iv idua l s l i ces re t a in

the i r i den t i t i e s a s the ma te r i a l shr inks and c racks

d u r i n g p y r o l y s i s a n d c h a r f o r m a t i o n . T h e p r o b -

l e m w i t h t h e n u m e r i c a l m o d e l s i s t h a t t h e y a r e

n e i t h e r s i m p l e t o u s e n o r c o m p r e h e n s i v e e n o u g h t o

i n c lu d e a l l o f t h e r e l e v a n t p h y s i c s a n d c h e m i s t r y .

N o t a b l e o m i s s i o n s f r o m t h e m o d e l s a r e t h e e f f e c ts

o f a m b i e n t o x y g e n c o n c e n t r a t i o n , v a r i a b le c h a r

y ie ld , and mois ture conten t . In addi t ion , i t i s

d i f f i cu l t t o desc r ibe the re l a t ionsh ips be tween the

v a r i o u s p a r a m e t e r s o f t h e p r o b l e m . O n t h e o t h e r

hand , the ex i s t ing ana ly t i ca l s tud ies [6] ignore a l l

p h y s i c a l a n d c h e m i c a l p r o c e s s e s e x c e p t t h e t r a n -

s i en t p r o p a g a t i o n o f a c o n s t a n t t e m p e r a t u r e p y r o -

lys i s f ron t in to the so l id , which advances in to the

0 0 1 0 - 2 1 8 0 / 8 7 / 0 3 . 5 0


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I N D R E K S . W I C H M A N a nd A R V I N D A T R E Y A

v i r g i n so l i d w h i l e l e a v i n g b e h i n d a c h a r l a y e r .

T h u s , t h e a n a l y t i c a l m o d e l s i g n o r e t h e t r a n s i e n t

c h e m i c a l d e c o m p o s i t i o n p r o c e s s , w h i c h l i m i t s

t h e i r r a n g e o f a p p l i c a b i l it y a n d m a k e s t h e m u n su i t -

a b l e f o r a n a l y z i n g c e r t a i n p h y s i c a l p r o c e s se s , su c h

as ign i t i on .

T h e p u r p o se o f t h i s s t u d y i s t o a n a l y z e a

s i m p l i f i e d m o d e l o f w o o d p y r o l y s i s t h a t i n c l u d e s

t h e e f f e c t s o f g l o b a l d e c o m p o s i t i o n c h e m i s t r y a n d

v a r i a b l e t h e r m o p h y s i c a l p r o p e r t i e s . T h e a n a l y s i s

p r o d u c e s f o r m u l a s f o r t h e v o l a t i l e m a ss f l u x t h a t

a re appl i cab le dur ing a l l s t ages of the pyro ly s i s

p r o c e s s , a n d e x h i b i t i t s d e p e n d e n c e o n t h e m o s t

i m p o r t a n t t h e r m a l , c h e m i c a l , a n d t h e r m o p h y s i c a l

p a r a m e t e r s .

The phys ica l conf igura t ion s tud ied he re i s i l l us-

t ra t ed in F ig . 1 . Th e the rma l ly th i ck ( semi inf in i t e )

sample , i n i t i a l ly a t equ i l ib r ium wi th the ambien t

sur rou nding s , is sub jec t ed a t / = 0 to a cons t an t

inc iden t rad ia t ive hea t f l ux F . I t is a ssum ed tha t

the gas phase i s i ne r t (no ox ida t ive e f fec t s ) and tha t

t h e i n f l u e n c e s o f so l i d - p h a se c r a c k i n g , sh r i n k a g e ,

su r f a c e r e g r e s s i o n , a n d g r a i n d i r e c t io n a r e n e g l ig i -

b l e . T h e i n t e r a c t i o n b e t w e e n / a n d t h e g a se o u s

p r o d u c t s o f p y r o l y s i s i s a lso n e g l e c t e d ; t h u s , ~ i s

a s su m e d t o b e t h e t i m e - a v e r a g e r a d i a t i v e h e a t f l u x

re ching

t h e su r f a c e . T h e s a m p l e i s a s su m e d t o b e

n o n d i a t h e r m i c a n d o p a q u e t o t h e i n c i d e n t r a d ia t iv e

f lux .

The in i t i a l hea t ing phase , he re in ca l l ed the ine r t

hea t ing s t age , i s cha rac t e r i zed by the cons t an t

dens i ty hea t u p o f t he su r face to 7~p , t he py ro lys i s

t e m p e r a t u r e , d e f i n e d a s t h e v a l u e o f 7 a t t h e

i n c e p ti o n o f p y r o l y s i s . N o c h e m i c a l r e a c t io n s

occur in th i s s t age , and s ince the ma te r i a l i s

c o n s i d e r e d d r y ( i . e . , o n l y w a t e r o f c o n s ti t u ti o n ) ,

t h e r e a r e n o e f f e c t s o f m o i s t u r e . I n t h e n e x t h e a t in g

phase ( the in it i a l pyro lys i s s t age ) t he in i t ia l r e l ease

of vo la t i l e s f rom the sur face co inc ides w i th the

f o r m a t i o n o f a p y r o l y s i s f r o n t . I n t h i s p h a se a r a p i d

r i se in the vo la t i l e mass f lux i s obse rved , and hea t

l o s ses f r o m t h e su r f a c e b y c o n v e c t i o n a n d r e r a d i a -

t i o n b e c o m e i m p o r t a n t . T h e su r f a c e t e m p e r a t u r e

a n d t h e v o l a t i l e m a ss e f f l u x c o n t i n u e t o i n c r e a se

unt i l a t h in cha r l aye r fo rms the re . In th i s hea t ing

s t age ( th in cha r ) t he mass f lux a t t a ins i t s maxim um

va lue . The f ina l hea t ing s t age ( th i ck cha r ) i s

c h a r a c t e r i z e d b y a g r a d u a l d e c l i n e i n t h a n d a

c o r r e sp o n d i n g l y g r a d u a l i n c r e a se i n t h e c h a r l a y e r

t h i ck n e s s . T h e e x p e c t e d t i m e sp a n o f e a c h o f t h ese

four s t ages i s i l l us t ra t ed qua l i t a t ive ly in the

ve rsus f p lo t o f F ig . 2 .

A . 4 ~4. A * A

~ - o ~ Ts - T = . H ( T s - T = )

o / /

/ / -xa _ F-H(T-T. )

I I

.

~ d T

-E /RT

4 .A

p- -p . T= T.

F i g 1 T h e p h y s i c a l c o n f i g u r a t i o n T h e d i m e n s i o n a l g o v e r n i n g e q u a t io n s a n d b o u n d a r y

c o n d i t i o n s a r e a l s o s h o w n


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P Y R O L Y S I S M O D E L O F C H A R R I N G M A T E R I A L S

2 3 3

o

0 _ I n e r t _ l T r a n s i t i o n _ _ i

hin h ick

?

r h e a t i n g l - r e g i m e - I - - ]

char char

F i g . 2 . A q u a l i t a t i v e p l o t o f t h e v o l a t i l e ma s s f l u x , r h , v e r s u s t i me , / , i n d i c a t i n g t h e s e p a r a t e

pyrolysis regimes to be examined.

I I. F O R M U L A T I O N

B e f o r e s t a t i n g t h e n o n d i m e n s i o n a l g o v e r n i n g

e q u a t i o n s a n d b o u n d a r y c o n d i t i o n s , t h r e e i m p o r -

t an t a s sumpt ions made in th i s s tudy mus t be

d i scus sed . The f i r s t a s sumpt ion i s t ha t t he in t e rac -

t ions be tween the vo la t i l e gases and the reac t ing

mate r i a l can be neg lec ted . I t i s known tha t t he

g e n e r a t i o n o f v o l a t i l e g a s e s i n s i d e t h e s o l i d p r o -

d u c e s h i g h p r e s s u r e s ( u p t o 0 . 3 a t m [ 7 ] , d e p e n d i n g

o n t h e w o o d p o r o s i t y ) , w h i c h f o r c e t h e v o l a t i l e s

t o w a r d b o t h t h e h o t c h a r l a y e r a n d t h e i n t e r i o r o f

the so l id . The gases t rave l ing in to the in t e r io r o f

t h e s o l id c o n d e n s e , o n l y t o b e s u b s e q u e n t l y r e g a s i -

f l ed [8] . The ne t hea t t rans fe r be tween the vo la t i l e s

a n d t h e h o t c h a r , w h o s e m a g n i t u d e i s m e a s u r e d b y

the quant i ty

C p g ( T s - T c ) ,

i s i g n o r e d f o r t w o

reasons . F i r s t , fo r sma l l cha r l aye r th i cknes s ,

(~pgT c)(7~s/T'c - 1) is ne glig ibl e beca us e ( T s / 7 c

- 1 ) , ~ O ( 1 ) . S e c o n d , w h e n t h e c h a r - l a y e r

th i cknes s i s l a rge , and Ts /Tc >> O( 1) , mo s t o f the

v o l a t il e s a r e i s s u e d t h r o u g h c r a c k s t h a t f o r m i n t h e

c h a r l a y e r , w h i c h m a k e s t h e n e t h e a t t r a n s f e r

b e t w e e n v o l a t i l e s a n d c h a r n e g l i g i b l e . T h u s , t h e

hea t t rans fe r be tween the vo la t i l e s and the so l id

d e p e n d s o n t h e e x t e n t o f c h a r c r a c k i n g , w h i c h h a s

b e e n d i s r e g a r d e d f r o m t h e o u t s e t i n t h i s m o d e l .

T h e e f f e c t s o f c o n d e n s a t i o n a n d r e g a s i fi c a t io n a r e

i g n o r e d b e c a u s e t h e v o l a t i l e g a s p h y s i c a l p r o p e r -

t i e s , s u c h a s c o n d e n s a t i o n t e m p e r a t u r e s a s f u n c -

t io n s o f p r e s s u r e , a r e u n k n o w n ; t h e r e f o r e , t h e

ca lcu la t ion of the i r i n f luence i s imposs ib le .

T h e s e c o n d a s s u m p t i o n i s t h a t t h e c h e m i c a l

p r o c e s s e s o c c u r r i n g i n t h e d e c o m p o s i t i o n o f

d r y w o o d to c h a r c a n b e m o d e l e d by a

s i n g l e ,

o n e - s t e p r a t e e q u a t i o n c o n t a i n i n g t h r e e f i x e d p a -

r a m e t e r s , t h e c h a r d e n s i t y , t h e p r e e x p o n e n t i a l

f a c t o r , a n d t h e a c t i v a ti o n e n e r g y . T h i s a s s u m p t i o n

i s an idea l i za tion of the ac tua l p roces s s ince wo od

d e c o m p o s e s i n a c o m p l e x m a n n e r , p r o d u c i n g

h u n d r e d s o f c o m p o u n d s . H o w e v e r , t h e r e i s p r e s -

e n t l y m u c h c o n t r o v e r s y c o n c e r n i n g i t s p r e c i s e

na ture (even for pur e ce l lu lose , s ee , e .g . , [9 , 10] ) ,

a n d n o g e n e r a l l y a c c e p t e d r e a c ti o n p a t h w a y , a n a l-

o g o u s t o t h o s e u s e d t o m o d e l g a s - p h a s e r e a c t i o n s ,

h a s y e t b e e n d e v e l o p e d . T h u s , s i n c e t h e p h y s i c a l

p a r a m e t e r s a p p e a r i n g i n t h e o n e - s t e p r a te e q u a t i o n

h a v e d i r e c t p h y s i c a l m e a n i n g , a n d s i n c e m u c h o f

t h e r e c e n t e x p e r i m e n t a l w o r k h a s b e e n d e v o t e d t o

the i r me asure me nt [11] , i t is fe l t t ha t use of the

o n e - s t e p s c h e m e i s a p p r o p r i a t e .

T h e t h i r d a s s u m p t i o n i s t h a t t h e h e a t o f p y r o l y -

s i s t e rm, Q , which appea rs in the equa t ion for

c o n s e r v a t i o n o f e n e r g y , i s n e g li g i b le . A s d i s c u s s e d

i n R e f . [ 4 ] , t h e r e i s m u c h c o n f u s i o n c o n c e r n i n g Q .

R e p o r t e d v a l u e s v a r y b e t w e e n 1 8 0 c a l / g ( e n d o -

t h e rm i c ) a n d - 4 5 0 0 c a l /g ( e x o th e r m i c ). T h e

e x o t h e r m i c i t y a p p a r e n t l y a r i s e s f r o m s e c o n d a r y


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r e a c t i o n s b e t w e e n t h e o x y g e n a n d t h e h o t c h a r , o r

t h e p y r o l y s i s g a s e s . H o w e v e r , t h e n e t e x o t h e r m i c -

i t y o f th e p r o c e s s m u s t b e r e l a t i v e l y s m a l l, s i n ce

t h e r m a l r u n a w a y ( i . e . , e x p l o s i o n ) h a s n e v e r o c -

c u r r e d d u r i n g w o o d p y r o l y s i s . A n e s t i m a t e f o r th e

u p p e r b o u n d o f Q

i s c p T p - T a , ) ~

O ( 1 0 0 c a l/

g ) , which i s t he s ens ib le hea t requi red to ra i s e lg

o f w o o d t o i ts p y r o l y s i s t e m p e r a t u r e . T h u s , Q _>

1 0 0 c a l / g c o u l d l e a d t o t h e r m a l r u n a w a y . T h e

d e t a i le d n u m e r i c a l s t u d y o f R e f . [ 1 2 ] , u s i n g

p r e v i o u s e x p e r i m e n t a l d a t a , g i v e s o n l y s l i g h t l y

e n d o t h e r m i c Q . I n t h e a b s e n c e o f a n o x i d i zi n g

a t m o s p h e r e , t h i s s e e m s t o b e t h e m o r e r e a s o n a b l e

c o n c l u s i o n , s i n c e w o o d p y r o l y s i s i s a d e g r a d a t i v e

c h e m i c a l p r o c e s s . T h e r e f o r e , s i n c e [ Q [ - O ( 1 0

c a l / g ) , w h e t h e r e x o t h e r m i c o r e n d o t h e r m i c , i s

g e n e r a l l y m u c h s m a l l e r t h a n t h e t h e r m a l d i f f u si o n

t e r m i n t h e e q u a t i o n f o r c o n s e r v a t i o n o f e n e r g y [ ~

O E T / O . ~ 2 - C p ( 7 p - T e a ) - O(1 00 cal /g)] , Q = 0 is

a s s u m e d h e r e . T h i s a s s u m p t i o n h a s a l s o b e e n

e m p l o y e d b y p r e v i o u s a u t h o r s [ 4 , 5 ] .

W i t h t h e s e a s s u m p t i o n s t h e m o d e l e q u a t i o n s a n d

b o u n d a r y c o n d i t i o n s d e s c r i b i n g t h e p y r o l y s i s o f

w o o d r e d u c e t o th o s e s h o w n i n F i g . 1 . I f i t i s

a d d i t i o n a ll y a s s u m e d t h a t t h e t h e r m a l c o n d u c t i v i ty

i s p r o p o r t i o n a l to th e d e n s i ty , i . e . , ~ = ~ , w h e r e

~ = cons tan t = ~oo/~= , then the nond imen s iona l

e q u a t i o n s b e c o m e

E n e r g y : P at cgx P

Op

D e c o m p o s i t i o n : - - = - A ( p - 6 ) e - E / r ,

O t

a m a p

Mass :

a x a t

Boundary and in i t i a l :

T ( x ,

0 ) = p ( x , 0 ) = T ( ~ , t ) = 1

a T

- P ~xx ( 0 , t ) = 1 - r ( T s - 1 ).

(1)

(2)

3 )

(4)

H e r e, T = T / ~ * * , p = ~ / ~ , x = / [ / S =

05=/~)7~**/P] , and t = / ' / ? [? = 2 / =] w ere

u s e d f o r n o n d i m e n s i o n a l i z a t i o n . T h e f o u r n o n d i -

m e n s i o n a l p a r a m e t e r s a p p e a r i n g i n E q s . ( 1 ) - ( 4 )

a r e E = / ~ / / ~ 7 ~ a n d A = A ? , th e n o n d i m e n -

s io n a l a c t iv a t i o n e n e r g y a n d p r e e x p o n e n t i a l f a c t o r

f o r th e o v e r a l l d e c o m p o s i t i o n r e a c t io n , t5 = ~ c / ~ . ,

t he c h a r y i e l d , a n d F = / - I T ~ / P + ( ~ C 4 / / F ) ( f s

+ 1)(Ts2 + 1) , the l inea rized heat loss term ,

c o n t a i n in g t h e e f f e c t s o f c o n v e c t i v e a n d l i n e a r iz e d

rad ia t ive hea t los ses f rom the sur face . He re , ~ i s

a n a v e r a g e n o n d i m e n s i o n a l s u r f a c e t e m p e r a t u r e ,

chosen to ma tch the hea t l os ses to the i r ac tua l

v a l u e s o v e r t h e a p p r o p r i a t e t e m p e r a t u r e r a n g e .

I I I . A N A L Y S I S

T h e r e a r e f o u r d i s t i n c t s ta g e s o f p y r o l y s i s th a t

wi l l be cons ide red in the l imi t o f h igh ac t iva t ion

e n e r g y , E ~ ~ . T h e i m p o r t a n t f e a t u r e s o f e a c h

a re i l l us t ra t ed qua l i t a t ive ly in F igs . 3a -3d . In the

ine r t hea t ing s t age the dens i ty i s cons tan t , because

Ts < Tp is a lwa ys sat is f ied (Fig. 3a) . Th e ini t ia l

p y r o l y s i s s t a ge c o n t a i n s t h e p r e l i m i n a r y e f f e c t s o f

c h e m i c a l r e a c t i o n . A n a r r o w r e a c t i o n z o n e d e -

ve lops nea r the sur face , i n which 6 < p < 1 and

Tp < Ts < Tc (F ig . 3b) . W hen Ts pas ses th roug h

To, a th in cha r l aye r fo rm s a t the sur fac e (F ig . 3c ) .

In th i s s t age the condi t ions s a t i s f ied by p and T ar e

p = ~, Ts -> Tc in the char layer , and t5 < p < 1,

Tp < T < Tc in the pyro lys i s zon e preced ing it .

T h e t e m p o r a l a n d s p a ti a l m a t c h i n g o f t h es e t h r e e

reg imes i s eas i ly ach ieved because they fa l l i n the

in i t i a l , o r k ine t i ca l ly cont ro l l ed , pyro lys i s reg ime .

T h e t h i c k c h a r s t a g e, h o w e v e r , r e p r e s e n t s t h e lo n g

t i m e b e h a v i o r o f th e s y s t e m ( F i g . 3 d ) . H e r e , t h e

s u r f a c e t e m p e r a t u r e a p p r o a c h e s i t s m a x i m u m

v a l u e , T m ax = 1 + 1 / r , o b t a i n e d f r o m th e s e c o n d

o f E q s . ( 4 ) b y p u t t in g a T s / S x = 0 . T e m p o r a l

ma tch ing of the so lu t ion in th i s s t age to tha t in the

k ine t i ca l ly cont ro l l ed reg ime i s ve ry d i f f i cu l t ,

b e c a u s e t h e t im e t a p p e a r s o n l y a s a p a r a m e t e r i n

the equa t ions .

III 1 The Inert Heating Stage

H e r e , o n e d e f i n e s D = A e x p ( - E / T p ) as the

D a m k 6 h l e r n u m b e r , t h e r e b y a l l o w i n g E q . ( 2 ) t o

b e r e w r i t t e n a s O p / a t = - D ( p - ~ ) e x p [ ( - E /

T p ) ( T p / T - 1 )] . T h u s , f o r l a r g e v a l u e s o f E , a p /

a t w i l l be ve ry sma l l un t i l T i s ve ry c lose to the

pyro lys i s t em pera tu re , Tp , g iv ing p = 1 a s the

a p p r o x i m a t e s o l u t i o n f o r t h e d e n s i t y . T h e r e f o r e ,


5/17


235

Ts< TO

pT

{a

Reac t ion

z o l ~ m

p= T=

b )

Is ,.Tp

klye ---x

p s = p f - - . \ T s> T c T s'T m a x

' t ~ ~ 1 ~ ~-Tc ~ T c

layerzone

Tc 4._.------eaction

z ' & I

p= T p=,T=

(c) (d)

Fig. 3. The four pyrolysis regimes: a) the inert-heating stage, in which no reaction occurs

and the density is constant, b) the transition from inert heating to charring conditions, c) the

thin-char regime, in which Ts > To, and d) the thick-char regime.

the so lu t ion for the tempera tu re , f rom Eqs . (1) and

(4), is

[

I = 1 + erfc - e rx+r2t

X e r fc I v~ + . (5)

Consequent ly , the sur face tempera ture , TI (0 , t ) ,

r eaches Tp wh en t = tp, the pyro lys is t ime . The

def in i t ion of Tp (or tp) i s p resent ly a rb i t r a ry , bu t

f rom Eq. (5) the spec i f ica tion of one immed ia te ly

de te rmines the o the r . The procedure for ca lcu la t -

ing Tp i s now desc r ibed .

C o n s id e r t p a s a k n o w n p a r a me te r . T h e n t h e

expans ion of Eq . (5) for x < O(1) and t - tp 0 since -

O T / O x = O 0 / O S > O .

N o t e a l so f r o m E q . ( 4 1 ) th a t t h e h e a t f l u x a c ro s s

t h e r e a c t i o n l a y e r , O

O 0 / O S ,

is

c o n s t a n t ,

s ince Q

= 0 ( see the d i scuss ion of Sec t ion I I ) . Equa t ion

(45) sa t i s f i e s t he condi t ions

l i m O ( S , t ) =

[ g ( t ) / 6 ]

l i m S = oo

a n d

l im O ( S , t ) =

[ g ( t ) / 6 ]

l im S = - oo.

T h u s , t h e j u m p c o n d i t i o n f o r th e t e m p e r a t u r e

g r a d i e n t a c r o s s t h e r e a c t i o n z o n e i s g i v e n b y

= ( ~ 6 6 ) g ( t ) , (46)

w h i c h w i ll b e u se d i n t h e f o l l o w i n g su b se c t i o n t o

m a t c h w i t h th e so l u t i o n s i n t h e t w o o u t e r z o n e s .

Th e so lu t ion for p i s ob ta ined by subs t i t u t ing

E q . ( 4 5 ) i n to E q . ( 4 2 ) , w h i c h c a n t h e n b e w r i tt e n

a s t h e f o l l o w i n g f i r s t - o r d e r sy s t e m o f e q u a t i o n s:

o

( i) ~ -~ =q , p (S - - , - oo) = 1 ,

(ii) d q q Z q

- - - ~ g ( t ) - ,

d S 0 - 6 o

q ( S ~ - oo ) -

(1 -

6 ) K ( t ) e g in s.

(47)

T h i s sy s t e m i s i d e n t i c a l t o t h e o n e d e r i v e d i n

S e c t i o n I I I . 2 , e x c e p t f o r t h e f a c t o r s

g ( t )

a n d

K ( t ) .

T h e a n a l y s i s o f t h e f o l l o w i n g su b se c t i o n sh o w s

that

g ( t ) - t - 1 /2

a n d

K ( t ) ~ t l / 2 ;

t hus , Eqs . (47)

w e r e i n t e g r a t e d n u m e r i c a l l y w i t h a f o u r t h - o r d e r

R u n g e - K u t t a s c h e m e , s t a r t i n g w i t h

K ( t ) = g ( t )

= 1 , and then inc reas ing t . F igu re 7 show s a p lo t

o f A S , t h e r e a c t i o n - z o n e th i c k n e s s , a n d

K ( t )

a n d

g ( t ) ,

v e r su s

t x/z.

A n e a r l y l i n e a r i n c re a se o f A S

wi th

t 1/2

i s o b se r v e d , i n d i c at i n g t h a t th e a s su m p -

t ion of an in f in i t e s ima l ly th in reac t ion l aye r

b e c o m e s p r o g r e s s i v e l y w o r se a s t i n c r e a se s .

1 1 1. 4. 2. T h e O u t e r H o t C h a r a n d U n b u r n t

W o o d Z o n e s

I n t h e t w o o u t e r z o n e s t h e d e n s i t y i s c o n s t a n t ( s e e

F ig . 6 ) ; t hus , t he t empera ture f i e lds obey Eq . (1 ) ,

w i th p = 6 in the ho t -ch a r l aye r and p = 1 in the

u n b u r n t - w o o d l a y e r . T h e b o u n d a r y c o n d i t i o n s a r e

T H ( 0 , t ) = T ma x = 1 + 1 - I ,

T n ( x c ( t ) , t ) = T c

( h o t - c h a r l a y e r ) a n d

T w ( x c ( t ) , t ) = T o

Tw( ao, t )

= 1 ( u n b u r n t - w o o d l a y e r ) . I n t e g r a t io n o f t h e tw o

e n e r g y e q u a ti o n s g i v e s T H = A + B e r f ( x / 2 x / ~ ) ,

T w = C + D e r f ( x / 2 x / ~ , w h e r e A , B , C , a nd D

a r e c o n s t a n t s . A p p l i c a t i o n o f t h e b o u n d a r y c o n d i -

t i ons and the jum p con di t ion [Eq . (46) ] requi re s

x ~ ( t ) = 2 F x / t ( 4 8 )

a n d

g ( t ) = J / ' f t t , (49)

wh ere F an d J a re cons t an t s . S ince the re a re s ix

c o n s t a n ts ( A , B , C , D , F , J ) a n d o n l y f i v e

b o u n d a r y c o n d i t i o n s ( i n c l u d in g t h e ju m p c o n d i -

t i on) , an addi t iona l condi t ion i s necessa ry . Thi s i s

o b t a i n e d f r o m E q . ( 4 1 ) b y n o t i n g t h a t t h e h e a t f l u x

a c r o s s t h e p y r o l y s i s z o n e i s c o n s t a n t , w h e r e b y

- - 6 0 T H / O X ) x R t ) = - -

a T w / a X ) x R ( t ) ,

which , w i th

the above so lu t ions for TH and Tw l eads to the

req uire m ent , , ,/ '6-B ex p( - F 2 / 6 ) = D e x p ( - F 2 ) .

T h u s ,

f s - _

T m a x - Z c ~ / ~

e - F2 /~ ,

(50)

J = e r f (F /x / -6)

a n d F i s d e t e r m i n e d f r o m t h e t r a n sc e n d e n t a l

e q u a t i o n

e_F2 [( l /6)_ l I e r f c F 1 T o - 1

e r f ( F / x / 6 ) - x / 6 T m~ x- T c ( 51 )

F o r sm a l l F , E q . ( 5 1 ) r e d u c e s t o

F = 2

T - I /

(52)

T h e f o r m u l a f o r m i n t h i s h e a t i n g r e g i m e i s

ob ta ined by subs t i t u t ing Eq . (48) i n to Eq . (38) ,


12/17

2 4 2


8

A

5 5

K t )

~ AS t)

I

1.5

Fig. 7. The reaction-zone thickness, AS, as a function t wz, indicating that AS -- t v2. The

functions K t ) and g t) , used in the calculation of AS, a re also plotted against t j/2.

t.O

K t),

q t)

v i z . ,

1 - t S ) F

m = 53)

I V . N U M E R I C A L C O M P U T A T I O N S

C O M P A R I S O N A N D D I S C U S S IO N

E q u a t i o n s 1 ) - 4 ) w e r e a l s o i n t e g r a t e d n u m e r i c a l l y

w i t h a n i t e r a t e d C r a n k - N i c o l s o n s c h e m e [ 2 ] .

S h o w n i n F i g . 8 , f o r t h e c a s e , 4 , E , ~ , r ) =

1 0 n , 4 0 , 0 . 3 , 0 . 4 ) i s t h e r e s u lt i n g m v e r s u s t p l o t

a n d i ts c o m p a r i s o n t o t h e p r e d i c t i o n s o f E q s . 2 0 ) ,

22) , and 23) . The m a ss f lux r i s e s un t i l p = ~ a t

t h e s u r f a c e , a n d d e c r e a s e s m o n o t o n i c a l l y t h e r e a f -

t e r . B y u s i n g p p = 0 . 9 9 t h e p a r a m e t e r s s h o w n i n

T a b l e 1 c a n b e c a l c u l a te d . T h e m v e r s u s t p r o f i l e s

o f E q s . 2 2 ) a n d 2 3 ) a p p r o x i m a t e t h e e x p o n e n t i a l

r i s e o f m w e l l f o r s m a l l t - t p. H o w e v e r , t h e y r i s e

m u c h f a s t e r a s t - t p i n c r e a s e s , s i n c e r e a c t a n t

d e p l e t i o n h a s b e e n n e g l e c t e d . T h e e s t i m a t e f o r m

g i v e n b y E q . 2 0 ) i s also a n o v e r p r e d i c t i o n ;

a l t h o u g h r e a c t a n t d e p l e t i o n h a s b e e n i n c l u d e d , t h e

a s y m p t o t i c f o r m u l a t i o n u s e s t h e l inearized A r r h e -

n i u s e x p o n e n t , - E / T p ) + E / T p 2 ) T - Tp ,

w h i c h f o r T > T p i s a l w a y s less than i t s ac tua l

v a l u e , - E / T ) . T h u s , t h e r e a c t i o n r a t e i n t h e

a s y m p t o t i c f o r m u l a i s a lw a y s h i g h e r a n d t h e m a s s

f l u x p e a k m u s t t h e r e f o r e o c c u r s o o n e r . T h e

a s y m p t o t i c a l l y p r e d i c t e d v a l u e s o f m r ~ x [ E q . 2 5 )]

a n d t c t h e r e f o r e u n d e r e s t i m a t e t h e a c t u a l v a l u e s

[ see Tab le 1 ] .

A p l o t o f th e s u r f a c e t e m p e r a t u r e a n d d e n s i t y

p r o f i l e s f o r t h e s a m e c a s e i s s h o w n i n F i g . 9 , a l o n g

w i t h t h e m v e r s u s t p r o f i l e o f F i g . 8 . T h e

a s y m p t o t i c e s t i m a t e o f p s t ) i s o b t a i n e d f r o m t h e

in tegr a t ion of Eqs . 19) , wi th ps tp ) = 1 . Th e r a te

o f d e c r e a s e o f t h e a s y m p t o t i c e s t i m a t e i s f a s t e r

t h a n t h e n u m e r i c a l p r e d i c t i o n s o f E q s . 1 ) - 4 ) , a s

p r e v i o u s l y d i s c u s s e d f o r m . F o r Ts t), the ine r t

p r o f i l e o f E q . 5 ) fo l l o w s t h e n u m e r i c a l s o l u ti o n

unt i l t = tp , wh en Ts r i s e s r ap id ly . F o r t _> tp ,

Tis t) i s a p o o r e s t i m a t e o f T s . T h e a s y m p t o t i c

e s t i m a t e o f T s f o r t > t p i s o b t a i n e d b y u s i n g E q s .

5 ) , 1 5 ) , a n d 1 8 ) i n T = T I + O . T h e a g r e e m e n t

w i t h n u m e r i c a l p r e d i c t i o n s i s g o o d u n t i l t = 0 . 7 .

T h e c h a r f o r m a t i o n t e m p e r a t u r e , f r o m F i g . 9 , i s

Tc = 1.97 i .e . , 7~c = 600 K ) , a t tc -- -- ~ .7. Th en ,

f r o m Eqs . 35) and 36) , kc = 0 . 74 and m m ax ---

0 . 73 . Th is po in t i s a l so p lo t t ed in F ig . 8 .

F o r t h e t h i c k c h a r s t a g e , t h e s u b s t i t u t i o n o f T

= 1 . 97 , {5 = 0 . 3 , and Tm ax = 1 + 1 / r = 3 . 5


13/17


0 .5

t

m A /-,~ x E bE

~ m = ~ [ /_ , ,~ o , r

LO L5

/.6

m

.5.

F i g . 8 . T h e m a s s f l u x m v e r s u s t f o r t h e c a s e A = 1 0 n , E = 4 0 , 6 = 0 . 3 , a n d F = 0 . 4 .

S h o w n a r e t h e n u m e r i c a l e v a l u a t i o n o f E q s . ( 1 ) - ( 4 ) , t h e a s y m p t o t i c p r e d i c t io n s o f E q s . ( 2 2 )

a n d ( 2 3 ) , a n d t h e n u m e r i c a l l y e v a l u a t e d m a s s - f l u x p r o f il e o f E q . ( 2 0 ) . T h e n u m e r i c a l

c o m p u t a t i o n s g i v e t p = 0 . 2 7 , t c = 0 . 6 9 , a n d r nm a x = 0 . 8 2 5 . T h e b e s t - f i t c u r v e , m = 0 . 5 2 / ~ ,

i s p l o t t e d i n t h e t h i c k - c h a r r e g i m e . T h e p o i n t r n ~ x , e v a l u a t e d f r o m E q . ( 3 6 ) , i s al s o s h o w n .

43

T A B L E 1

C a l c u l a t i o n o f P a r a m e t e r V a l u e s f o r E a c h H e a t i n g S ta g e , fo r t h e C a s e A = 1 0 u , E = 4 0 , 6 = 0 . 3 , l = 0 . 4 ( W h e r e P o s s i b l e ,

C o m p a r i s o n s A r e M a d e b e t w e e n C a l c u l a t e d a n d C o m p u t e d V a l u e s , a n d P e r c e n t D i f f e r e n c e I s S h o w n )

H e a t i n g C a l c u l a t e d N u m e r i c a l P e r c e n t D i f f e r e n c e

S t a g e Q u a n t i t y V a l u e S o u r c e V a l u e [ ~NUM -- t~CALC)/~NUM)] X 1 0 0

Ine r t Tp 1 .5 F__ .qs. (8 ) - (1 l ) - - - -

t p 0 . 2 7 E q s . ( 8 ) - ( 1 1 ) - - - -

a 0 . 8 0 F .q . 8 ) - - - -

b 0 . 7 6 E q . ( 9 ) - - - -

I n i t ia l p y r o l y s i s s * 0 . 6 F i g . 4 - - - -

t c 0 . 6 1 E q . ( 2 4 ) 0 . 7 1 3

r n , ~ , 0 . 6 6 E q . ( 2 5 ) 0 . 8 2 5 2 0

A x 0 . 2 2 E q . ( 2 6) - - - -

T h i n c h a r a ~ 2 . 5 8 E q s . ( 8 ) , ( 2 9 i ) ; - - - -

u s e d tc = 0 . 7

b ~ 2 . 1 6 E q s . ( 9 ) , ( 2 9 i i ) ; - - - -

u s e d tc = 0 . 7

T 1 . 7 6 E q s . ( 5 ) , ( 1 5 ) , 1 . 9 7 i 0

1 8 ) , a t t = 0 . 6 1

k c 0 . 7 4 E q . ( 3 5 ); u s e d - - - -

T = 1 .97

m m ax 0 . 7 3 E q . ( 3 6 ) ; u s e d 0 . 8 2 5 1 2

Tc = 1 .97

T h i c k c h a r F 0 . 2 7 E q . ( 5 1 ) - - - -

m 0 . 2 / x / t E q . ( 53 ) 0 . 5 2 / ~ - 6 2


14/17


15/17


t = . 2

t . O , [

2 C

1 .5 . 0 1 . 0 2 x . 0 3 . 0 4 . 0 5

F i g . l O a

0 . . . . . . o .b 5 0 ' / 0 ' 2 0 3

~

F i g . 1 0 b

2 4 5

L 3 .

2 .

I

0

n - . 5 4

7 / . 8 3

] ~ m L = A n - 8 ) e - E / T

o . 1 o . 5 , . b -

, , ~

F i g . 1 0 c

F i g . 1 0 . N u m e r i c a l l y c a l c u l a t e d [ f r o m E q s . ( 1 ) - ( 4 ) ] p r o f i l e s o f p , T , a n d m L = ( p - - ~i) e x p ( - E / T ) a s

f u n c t i o n s o f x a n d t , f o r t h e c a s e A = l 0 n , E = 4 0 , ~ = 0 . 3 , 1 = 0 . 5 . N o t e t h a t t h e m a x i m u m o f r n

o c c u r s w h e n t = 0 . 8 ( i . e . , w h e n p ~ = ~5 = 0 . 3 i s f i rs t a t t a i n e d ) . T h e w a v e l i k e c h a r a c t e r o f p y r o l y s i s i n t h e

t r a n s i t i o n s t a g e i s e v i d e n t f r o m ( a ) . T h e m a x i m a o f m L f o r t = 0 . 8 a n d t = 1 . 0 o c c u r w h e n p = 0 . 5 5 .

T h e r e f o r e , m L a n d p a r e no t f u n c t i o n s o f T a l o n e .


16/17

2 46 I N D R E K S. W l C H M A N a nd A R V I N D A T R E Y A

r e g i m e s . F o r m u l a s f o r t h e v o l a t i l e m a s s e f f l u x ,

re t ) ,

w e r e d e r i v e d i n t h e k i n e t i c a l l y c o n t r o l l e d

s tage [ Eqs . ( 20) , ( 22) , ( 23) ] and in the d i f f us ion

c o n t r o l l e d s t a g e [ E q . ( 5 3 ) ]. A n e s t i m a t e f o r t h e

c h a r f o r m a t i o n t i m e i s g i v e n b y E q . ( 2 4 ) . T h e

i n f l u e n c e s o f t h e a c t i v a t i o n e n e r g y , p r e e x p o n e n t i a l

f a c t o r , c h a r y i e l d , a n d c h a r r i n g t e m p e r a t u r e o n

m m a x a r e g i v e n b y E q . ( 3 6 ) , w h i c h h a s b e e n

d e r i v e d i n t h e th i n c h a r s t a g e . T h e w a v e l i k e n a t u r e

o f t h e i n it ia l p y r o l y s i s p r o c e s s i s e v i d e n t f r o m t h e

a s y m p t o t i c a n a l y s i s , w h e r e t h e s i m i l a r it y v a r i a b l e

s = r - ~ e d u c e s t h e m a t h e m a t i c a l d e s c r i p t i o n o f

p y r o l y s i s t o o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . T h e

t - 1/2 d e p e n d e n c e o f t h e m a s s f l u x i n t h e t h i c k c h a r

s t a g e h a s b e e n d i s c u s s e d b y p r e v i o u s a u t h o r s [ 6 ] .

T h e a n a l y s i s o f S e c t i o n 1 1 1 . 4 s h o w s t h a t t h e

a s s u m p t i o n o f a n i n f i n i t e s i m a l l y th i n r e a c t i o n z o n e

b e c o m e s p o o r e r a s t i n c r e a s e s , s i n c e t h e r e a c t i o n

z o n e t h i c k n e s s i n c r e a s e s a s t ~/2.

T h e p r e s e n t m o d e l c a n b e m a d e m o r e r e a l i s t i c

b y i n c l u d i n g t h e f u l l n o n l i n e a r r e r a d i a t i o n t e r m i n

t h e b o u n d a r y c o n d i t i o n s ( r e c a l l th a t t h e r e r a d i a t i o n

t e r m w a s l i n e a r i z e d ) , a n d b y a d d i n g t h e e f f e c t s o f

m o i s t u r e a n d c h a r o x i d a t i o n . T h e l i n e a r i z e d r e r a -

d i a t i o n t e r m a n d t h e a s s u m p t i o n )x = p k a r e

b e l i e v e d r e s p o n s i b l e f o r t h e a n o m a l o u s t e m p e r a -

tu r e r i s e in the in i t i a l pyr o lys i s s t age ( s ee F ig . 9 ,

f o r 0 . 4 < t < 0 . 7 ) , w h i c h is n o t o b s e r v e d

e x p e r i m e n t a l l y [ 1 4 ] . T h e a v a i l a b l e s u r f a c e t e m p e r -

a t u re p r o f i l es s h o w a m o n o t o n i c a n d m u c h s t e e p e r

r i s e i n T s t o i ts m a x i m u m v a l u e . T h e i n f l u e n c e s o f

a t m o s p h e r i c o x y g e n a n d t h e o x y g e n o f m o l e c u l a r

d e c o m p o s i t io n o n t h e r a t e o f p y r o l y s i s a re b e l i e v e d

t o b e s i g n i f ic a n t [ 1 4 ] . T h e i n f l u e n c e s o f m o i s t u r e ,

w h i c h i s d r i v e n f r o m t h e w o o d i n th e i n i t ia l h e a t i n g

p h a s e , c a n b e d e s c r i b e d b y a k i n e t i c e q u a t i o n f o r

t h e m o i s t u r e d e n s i t y a n a l o g o u s t o E q . ( 2 ) [ 4 ] . A

t h e o r y b a s e d o n t h e a s s u m p t i o n o f l o c a l th e r m o d y -

n a m i c e q u i l i b r i u m b e t w e e n t h e r a t e s o f a d s o r p t i o n

a n d d e s o r p t io n o f m o i s t u r e in t h e p o r o u s w o o d

s a m p l e c a n a l s o b e d e v e l o p e d .

N O M E N C L A T U R E

c

A

n o n d i m e n s i o n a l s u r f ac e t e m p e r a t u r e g r a d i -

en t , Eq . ( 8 )

see Eq . ( 29 i )

n o n d i m e n s i o n a l p r e e x p o n e n t i a l f a c t o r , A

b n o n d i m e n s i o n a l t i m e r a t e o f c h a n g e o f T xs,

Eq . ( 9 )

bc s ee Eq . ( 29 i i )

Cp spec i f i c hea t

D D a m k 6 h l e r n u m b e r , D = A e x p ( - E ~ Tp)

D o D a m k 6 h l e r n u m b e r , D o = D / b E / T p 2)

E n o n d i m e n s i o n a l a c t i v a t i o n e n e r g y ,

inc iden t hea t f lux

g t ) see Eqs . ( 46) and ( 49)

c o n v e c t i v e h e a t - l o s s c o e f f i c i e n t

K t ) see Eq . ( 43)

kc n u m e r i c a l c o n s t a n t i n t h e t h i n - c h a r s t a g e

L c h a r a c t e r i s t i c l e n g t h , / S = X f ' o o / F

m n o n d i m e n s i o n a l m a s s e f f l u x

r~ r~ = m / b / a )

q q = d a / d s , see Eqs . ( 19)

h e a t s o u r c e t e r m ; Q = 0 is u s e d h e r e

Q s n o n d i m e n s i o n a l n e t h e a t f l u x i n t o s u r f a c e

/ ~ i d e a l g a s c o n s t a n t

s s im i la r i ty va r ia b le , s = r - ~

so s ev alu ate d a t t = tp, So = In D o - ~

S s t r e t c h e d s p a t i a l c o o r d i n a t e i n t h i c k - c h a r

r e g i m e , s e e E q . ( 4 0 )

A S r e a c t i o n - z o n e t h i c k n e s s

s * v a l u e o f s f o r w h i c h m = m m a x ;S * d e p e n d s

on 8 [ s ee F ig . 5 ]

t n o n d i m e n s i o n a l t i m e , t =

f /

T n o n d i m e n s i o n a l t e m p e r a t u r e , T = 7 ~ / f '~

T Cn AR n o n d i m e n s i o n a l s o l u t i o n f o r T u p t o c h a r -

r i n g t i m e

OR n o n d i m e n s i o n a l r e a c t i o n - z o n e p r o p a g a t i o n

ve loc i ty , OR = d X R /d t

X R n o n d i m e n s i o n a l r e a c t i o n - z o n e l o c a t i o n

r e e k

&

r

8

~ x

0

the r m a l d i f f us iv i ty , & = ~ ,/~dp

n o n d i m e n s i o n a l l i n e a r i z e d h e a t - l o s s p a r a m -

e t e r

n o n d i m e n s i o n a l c h a r y i e l d , 8 = p c / ~

p y r o l y s i s z o n e t h i c k n e s s

n o n d i m e n s i o n a l t e m p e r a t u r e c o r r e c t i o n i n

t h e i n i t i a l p y r o l y s i s s t a g e

t h e r m a l c o n d u c t i v i t y

s t r e t c h e d n o n d i m e n s i o n a l s p a t i a l c o o r d i -

na te


17/17


2 4 7

P

?

xI,

n o n d i m e n s i o n a l d e n s i t y , p = ~ / ~

S t e f a n - B o l t z m a n n c o n s t a n t

c h a r a c t e r i s t i c t i m e , ? = / S 2 / ~ ,

n o n d i m e n s i o n a l t e m p e r a t u r e c o r r e c t i o n i n

t h e t h i n - c h a r s t a g e

Partial support for the second author was

provided by the National Science Foundation

under Grant Number CBT-8415423.

R E F E R E N E S

Subscripts

c c h a r

H h o t - c h a r

I i n e r t

m a x m a x i m u m

o l o w e s t - o r d e r a p p r o x i m a t i o n

p p y r o l y s i s

s s u r f a c e

w u n b u r n t - w o o d

o o a m b i e n t

Superscripts

d i m e n s i o n a l q u a n t i ty

a v e r a g e

This research was per formed while the fir st

author was an NR C postdoctoral research sta ff

member at the National Bureau of Standards

(Fire Research program). The opportunity pro-

vided by this program to study problems o f

one s own choosing is greatly appreciated.

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Flame 1986).

2 . Kung, H . , Combust. Flame 18:185 1972).

3. Kansa , E. J . , Per lee , H. E. , and Chaik en, R. , Combust.

Flame 29:311 1977) .

4 . A t re ya , A . , NBS-GCR -83-449, 1984.

5. Parker , W. J . , NBSIR 85-316 3, 1985.

6. Delichats ios , M. A. , and deRis , J . , Factory Mutual

Re se a rc h , OKOJ1.BU, 1984.

7. Lee , C. , Chaik en, R. , and Singer , J . M. , Sixteenth

Symposium International) on Combustion, T h e

Com bu s t ion Ins t i tu te , 1976, pp . 1459-1470.

8 . Min , K . , a nd Em m ons , H . , Proc . He a t Tra ns fe r a nd

Fluid Mech. Ins t . , 1972.

9. Shafizadeh, F. , in The Chemistry of Solid Wood R .

Rowe l l , Ed . ) , ACS, W a shington D .C . , 1984, pp . 489-

529.

10. Hira ta , T. , NBSIR 85-3218, 1985.

11. Roberts , A. F. , Combust. Flame 14:261 1970).

12. Kun g, H. , and Kalekar , A. S. , Combust. Flame 20:91

1973).

13. Lif if in,A . , a nd W i l l ia m s , F . A . , Comb. Sci. Tech. 3:91

1971).

14. Kashiwag i, T. , Ohlem il ler , T. J . , and Atreya , A. , 8th

U.S . - Ja pa n Na t iona l Re sourc e s Me e t ing on Pa ne l of F i r e

Research, 1985.

Received 14 April 1986; revised 22 January 1987

a simplified model for the pyrolysis of charring materials

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