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Journal ofEngineeringPhysics and Thermophysics, Vol. 6Z No. 1-2, 1994

M A T H E M A T I C A L M O D E L I N G O F C O N V E C T I V E H E A T

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

P A R T I C L E S I N A B E D

A . I . N ak o r ch ev s k i i , A . N . V y l eg zh an i n , an t i

1. V . G as k ev i chU D C 6 6 4 . 8 . 0 4 7

A c lose d s y s t e m o f e qua t ions i s p ropose d fo r c a l c u la t ing c onv e c t iv e he a t and ma ss t rans f e r i n t he d rying o f

sol id par t ic les by a gaseous heat t rans fer agent in a mov in g bed. A s a n exam ple , the operat ion o f a be l t - type

dryer wi th crossed in terac t ion o f a dry ing agent and a bed o f f ru i t cu t in to c ircular s l ices i s cons idered.

Re su l t s o f a nu me r ic a l so lu t ion o f t he p rob le m are p re se n te d in f i gure s .

T h e d r y i n g o f s o l id p a r ti c le s i n a b e d b y a g a s e o u s h e a t t r a n s f e r a g e n t i s w i d e l y u s e d c o m m e r c i a l ly .

I n t e r ac t i n g f l o w s a r e m o s t o f t en a r r an g ed i n a co u n t e r f l o w ( co l u m n d r y e r s ) o r a c r o s s ed - f l o w ( b e l t co n v ey e r s w i t h

t r a n s v e r s e b l o w i ng ) s c h e m e . T h e b e d t o b e d r ie d i s a tw o - p h a s e s y s t e m o f t h e t y p e o f m o i s t s o l i d - h u m i d g a s , w h e r e

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

s y s t em s . Su b s t an t i a l d i f f i cu lt i e s i n th e m a t h e m a t i ca l d e s c r i p t i o n a r e cau s ed b y t h e g r ea t d i v e r s i t y o f t h e p o s s i b l eg e o m e t r ic a l s t r u c t u r e o f t h e b e d . C o n s e q u e n t l y , s c h e m a t i z a ti o n o f t h e s t r u c t u re o f t h e b e d a n d o r i e n t a t i o n t o s o m e

av e r ag ed ch a r ac t e r i s t i c s a r e i n ev i t ab l e .

T h e ch o i ce o f av e r ag i n g p r i n c i p l e s is an i m p o r t an t s t ep i n th e d ev e l o p m en t o f m e t h o d s o f t h e m ech an i c s o f

h e t e r o g en eo u s m ed i a . C o n c ep t s o f m e t h o d s f o r s p ace an d t i m e av e r ag i n g a r e g i v en i n [ 1 ] an d [ 2 ] , r e s p ec t i v e l y .

T h e l a t t e r ap p r o ach , w h i ch i s m o r e r i g o r o u s m a t h em a t i ca l l y [ 3 ] , w i l l b e u s ed h e r e . I t is a s s u m ed i n t h e d e s c r i p t i o n

t h a t t h e i n t e r a c t in g m e d i a a r e n o n v i s c o u s b u t t h e y e x p e r i e n c e r e s i s ta n c e i n r e la t i ve m o t i o n o f t h e p h a s e s a n d t h e

a c t io n o f A r c h i m e d e a n f o r c es ( t h e h y d r a u l i c a p p r o a c h ) . T r a n s f e r o f m o m e n t u m a n d t h e k i n e ti c c o m p o n e n t o f t h e

e n e r g y a n d d i s s i p a t i o n a r e n e g l e c t e d . O n l y a v e r a g e d p a r a m e t e r s w i t h o u t t h e f l u c t u a t i o n s a r e u s e d . W i t h t h e s e

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

B / + B j = 1 ,

0O--t (Pi Be Wi) = - V '( P i Bi We Ui) - M i j + M j i ,

00"--/ [P i B i (1 - Wi) 1 = - V" [Pi Bi Ui (1 - Wi) l , ( 1 )

00--t (Pi Bi Ui) = - (V 'U i) Pi Bi Ui + Pi Bi Gi - V" (Bi P2) + F j i ,

0O--t (Pi Bi El) = - V '(P i Bi Ui El) - Miy Ei + Myi E] + Qyi - Bi Qi, i , j = 1, 2 .

A cco r d i n g t o t h e s ec o n d an d t h i r d e q u a t i o n s o f s y s t e m ( 1 ) t h e r e l a t i o n

B i ) = - v - ( p i B e u i ) - M q + ,( 2 )

I n s t i t u t e o f E n g i n e e r i n g T h e r m o p h y s i c s , A c a d e m y o f S c i e n c e s o f U k r a i n e , K i e v. T r a n s l a t e d f r o m

I n z h e n e r n o - F i z i c h e s k i i Z h u r n a l , V o l . 6 7 , N o s . 1 - 2, p p . 4 8 - 5 3 , J u l y - A u g u s t , 1 9 9 4. O r ig i n a l a r ti c l e s u b m i t t e d

D ecem b er 1 6 , 1 9 9 2 .

1 0 6 2 - 0 1 2 5 / 9 4 / 6 7 1 2 - 0 7 2 1 $ 1 2 .5 0 9 1 9 9 5 P l en u m Pu b l i s h i n g C o r p o r a t i o n 7 2 1



Y =lq, I I=,l => 0 0

, , 0 = , o O o 0 O oFig. 1. Structural scheme of a bed of a filling.

is obtained, which is used to simplify the equations of motion and energy (the two last equations of system (1)):

O UP i B i O t = - Pi Bi Ui 'VUi + Pi Bi Gi - V (B P2) q- F f i ,

O EP i B i O t = - P i B i ' V E i + M / i ( E l - E i ) + Q f i - B i Q i "

( 3 )

It should be noted that for "adequate" drying M21 = 0. The above equations are expressed in a specific form for

each particular case.

Let us consider the case of stea dy-stat e drying of fruit in dryers with multirow transportation of the material

by belt conveyers. The material is loaded onto the belt in a bed of thickness h, and in the transportation process

it is blown from above by a heat transfer agent, which is conditioned air in this case. The technological scheme of

drying requires that the equations be composed for the plane problem (x, y), where the x axis corresponds to the

direction of movement of the conveyer belt and t he y axis corresponds to that of motion of the heat transf er agent.

For this scheme the system of basic equations (1)-(3) has the form

0 0

B 1 + B2 = 1, ~xx (/91 B1 U l x ) = - M12, 0y (P2 B2 U2y) = M12,

0 0

0x [Pl B1 (1 - W1) U l x ] = 0, ~y [,02 B2 (1 - W2) U 2y ] = 0,

OU2y OP2 0B 2P2B2U2y Oy = - B2 ~- F21'y -p2 0y '

(4 )

O E 1 O E2Pl B1 U l x O x - Q21 , P2 B2 U2y Oy - M12 (E2 - El) - Q21 9

Since the material to be dried is carried by the conveyer with the velocity U l x with the aid of a mechanical drive,

the equation of motion for the first phase is eliminated. Commercial operation of this type of drye r has shown thatthe aer odynamic resistance of the bed of material is low. Therefore, the equation of motion for the second phase

is not important. Nevertheles, the structure of the bed should be well known, since it affects substantially the heat

and mass transfer characteristics.

Before drying, fruits are cut into circular slices. In loading the cut material onto the conveyer belt, the

stable position of an individual element in the filling will most likely be arbit rary. Therefore, a ny spatial o rientation

of this individual element is possible. The structure of the bed must include two limiting cases of orientation -

vertical and horizontal, bo th of which are equiprobable. If the bed is composed of units located only in these limiting

positions, the accounting for their equiprobability will result in the scheme shown in the left-hand side of Fig. 1.

722



I t i s c l e a r f r o m F i g . 1 t h a t t h e s c h e m e i s i n v a r i a n t r e l a t iv e t o t h e x a n d y a x e s a n d r e f l e c ts a z i g z a g fl o w o f t h e h e a t

t r a n s f e r a g e n t p a s t t h e u n i t s . I n t h i s c a s e t h e s t r e a m l i n e s a r e c l o se t o t h o s e o f a fl o w in a s p h e r i c a l f i ll i n g , w h i c h

i s s t u d i e d m o s t t h o r o u g h l y . I f a l l u n i t s o f t h e b e d a r e a s s u m e d t o b e o f t h e s a m e ( r e p r e s e n t a t i v e ) s i z e, c i r c u l a r a n d

s p h e r i c a l f i ll i n g s w i l l b e t h e s a m e , i n t h e s e n s e o f e q u i v a le n c e , c o n c e r n i n g :

a ) a r e a o f c o n t a c t b e t w e e n t h e f i l l i n g a n d t h e h e a t t r a n s f e r a g e n t , i f t h e a r e a s S~ 1) = 4n R s a n d

2S (1) = 4J rRc a r e equ a l , w h ic h i s poss ib l e a t Rs = Re ;

b ) v o l u m e o f f il l in g , i f t h e v o l u m e s ~ 1 ) = 4 / 3 ~ g a s and 2 I / ( c1) = 2 ~ g 2 d c a r e e q u a l , w h i c h i s p o s s i b l e w h e n

c o n d i t i o n ( a) i s s a ti s f i e d a n d t h e t h i c k n e s s o f a c ir c u l a r s li c e 6 c = 2 / 3 R o H e r e R i s t h e r a d i u s o f a u n i t .

E q u a t i o n s ( 4) a r e n o t c l o s ed . T h e r e f o r e , a d d i t i o n a l r e la t i o n s , d e t e r m i n i n g t h e p r o b l e m s t a t e d , a r e r e q u i r e d .

T h e k i n e t i c e q u a t i o n o f d r y i n g t h a t i s u s u a l l y s e t u p o n t h e b a s i s o f e x p e r i m e n t a l d a t a i n t h e f o l l o w i n g f o r m i s t h e

m a i n o n e :

d W 1a t = F 1 ( W I ' T1 . . . . ) ' ( 5)

w h e r e T i s t h e t e m p e r a t u r e . I n o u r c a s e f o r t h e s t e a d y - s t a t e p r o c e s s t h e l e f t - h a n d s i d e o f (5 ) i s U ] x ( O W 1 / O x ) . W i t h

t h e u s e o f t h e s e c o n d a n d f o u r t h e q u a t i o ns o f s y s t e m ( 1 ), t h e r e l at i o n b e t w e e n t h e v o l u m e d e n s i t y o f t h e i n t e r p h a s e

m a s s t r a n s f e r M 1 2 a n d t h e k i n e t i c e q u a t i o n o f d r y i n g is f o u n d a s

M12 - -

P l B 1

f - - W 1 F 1 ( W I ' T1 . . . . )"(6 )

T h e d e n s i t y o f t h e i n t e r p h a s e h e a t f l u x Q 2 1 i s c a l c u l a t e d a s

Q21 = Cr (Z2 - T1) S~ 1) nlk2 1 , (7 )

w h e r e t h e h e a t t r a n s f e r c o e f f i c i en t a 21 f o r a s p h e r i c a l fi l li n g c a n b e d e t e r m i n e d b y D r a k e ' s f o r m u l a a s

. . . . 0 .55 , -, 0 .33, U 2 y 2 R 1 P 2 /~2c22 2 ( 2 + u / c o K e 2 r r 2 ) , R e 2 = p r 2 . ( 8 )

a21 = 2R1 f12 ' = - - ~ -2

H e r e S ~ 1 ) = 4 z~ R 2; n l i s t h e v o l u m e d e n s i t y o f p a r t i c l e s i n t h e s p h e r i c a l f i l li n g :

B1 (9)

n l = 4 / 3 ~ r R ~ ;

k21 is a c o r r e c t i o n f a c t o r f o r d e v i a t i o n o f t h e a c t u a l c o n d i t i o n s o f t h e f l o w f r o m t h o s e a s s u m e d i n s e t t i n g u p f o r m u l a

( 8 ). T o c a l c u l a t e n 1 , t h e c o n d i t i o n o f c o n s e r v a t i o n o f t h e n u m b e r o f p a rt i c le s i n t h e s o l i d p h a s e c a n b e u s e d . T h e

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

v 1 = n l S x U l x = c o n s t , (10 )

w h e r e S x i s t h e c r o s s - s e c t i o n a l a r e a o f t h e b e d p e r p e n d i c u l a r t o t h e x d i r e c t i o n . O n t h e o t h e r h a n d , t h e c o n d i t i o n

4 3B 1 U l x S x = v I -~ ~ R 1 . ( 1 1 )

m u s t b e s a t i s f ie d . C o m b i n i n g E q s . ( 9 ) , ( 1 0 ), a n d ( 1 1 ), w e w i ll f i n d t w o e q u i v a le n t fo r m s o f t h e e q u a t i o n f o r

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

( )U l x S x 1/'3 B11/3 o r R 1 : B ~R 1 = 4n:v1 ~ "

( 12 )

7 2 3



305~

.t07 -

2 9 7 9

29J0

a B1 b

0 .q0

0 . 3 9

0 . 3 8

, ' 0 .37 ,

a03s y o 0 . 0 3 ;

G{ ,

SC

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3t

s1

7

F i g . 2. C h an g es i n t h e t em p er a t u r e T 1 ( a ) an d v o l u m e co n cen t r a t i o n B I ( b )

o f t h e f i l li n g o v e r t h e b e d t h i c k n e s s i n c r o s s s e c t io n s l o c a t e d a t v a r i o u s

d is ta nce s f rom the beg inn in g o f the be l t : x -- 0 (1 ) , x -- 3 .52 m (2 ) ; x ~ 7 .04

(3) ; x = 10.56 m (4) ; x - 14.08 m (5) ; x - - 17.60 m (6 ) . T1, K; y , m.

17

2t

3t

5t{

3"{

G{

!

y

As ap p l i ed to the s y s t e m o f e q u a t i o n s ( 4 ) , t h e p r e s e t functi ons are: P l = P 1 (W1) ,/9 2 = t92 (W2, T2), c i = c i ( V t l i, T i ) ,

H i = H i ( W i , T i ) , P 2 = p2(W 2, T2) , 22 •Az(W2, T2) , Ulx f f i con st , v l ffi con st , S x f fi S x ( x ) , W 1 = W l ( x / U x x , T1) , Q21

= Q21(OC21, T1, T2, BI , R1).

T h e n u m b er o f f u n c t i o n s B 1 , B 2, M 1 2 , U2y, Y / 2, E l , E 2 t o b e f o u n d c o r r e s p o n d s t o t h e n u m b e r o f e q u a t io n s

i n s y s t em ( 4) ( a s w as m e n t i o n ed ea r l i e r , an eq u a t i o n o f m o t i o n h as b een e l i m i n a t ed ) . N a t u r a l l y , i n t h i s ca s e t h e

r e l a t i o n

d E i .= c i d T i , i = 1 , 2 . ( 1 3 )

i s t ak en i n t o co n s i d e r a t i o n .

S y s t em o f eq u a t i o n s ( 4 ) an d ( 5 ) i s s u p p l e m e n t ed b y i n i t i a l - b o u n d a r y co n d i t i o n s ch a r ac t e r i z i n g p h y s i ca l

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

t h e b e d o f m a t e r i a l

I/V (0 , y ) = W10 (y ) , T 1 (0 , y ) = T I0 (y ) , B 1 (0 , y ) = BlO (y ) ,

0 _< y _< h , T 2 (x , 0) = T20 (x ) , W 2 (x , 0) - - 14/20 (x ) ,(14)

U z y ( x , O ) = U 2 o ( X ) , O <_ x <_ l ,

w h er e W l o , T 1 0, B 1 0 , T 20 , W 2 0 , U 2 0 a r e f u n c t i o n s s p ec i f y i n g t h e i n i ti a l d i s t r i b u t i o n o f t h e p h y s i ca l p a r am e t e r s ; h

i s t h e h e i g h t o f t h e b ed o f fi ll in g ; l i s t h e l en g t h o f t h e co n v ey e r b e l t .

E q u a t i o n s ( 4) i s a s y s t em o f p a r t i a l d i f f e r en t i a l eq u a t i o n s o f t h e f i r s t o r d e r , w h i ch can b e ex p r e s s e d a s a

s y s t em t h a t i s s o l v ab l e f o r t h e d e r i v a t i v e s . T o g e t h e r w i t h co n d i t io n s ( 1 4 ) , t h is s y s t e m co n s t i t u t e s a C a u ch y p r o b l em .

D u e t o t h e s p e c if i c p r o p e r t i e s o f th e p h y s i c a l s t a t e m e n t o f t h e p r o b l e m t h e d i r e c t io n s o f m o v e m e n t o f th e c o m p o n e n t s

i n t h e t w o - p h a s e s y s t e m a r e c o n n e c t e d r i g i d l y w i t h t h e c o o r d i n a t e a x e s , a n d b o u n d a r y c o n d i t i o n s ( 1 4) a r e

s i m u l t a n eo u s l y i n it i a l co n d i t i o n s f o r s y s t e m ( 4 ). C o n s e q u en t l y , t h e s p a t i a l v a r i ab l e s x an d y a r e t r an s f o r m ed i n t o

" t i m e l i k e" o n e - s i d ed co o r d i n a t e s .

T h i s d e t e r m i n es t h e s t r u c t u r e o f t h e a l g o r it h m f o r n u m e r i ca l s o l u t i o n o f p r o b l em ( 4 ) , ( 5 ) , an d ( 1 4 ) . T h e

a l g o r i t h m i s b a s e d o n t h e p o s s i b i l i t y of t r e a t i n g t h e t r a n s f o r m e d s y s t e m o f e q u a t i o n s ( 4) a s a s y s t e m t h a t

d e c o m p o s e s l o c a ll y in t o a s y s t e m o f o r d i n a r y d i f fe r e n t ia l e q u a t i o ns i n t h e i n d e p e n d e n t v a r i a b l e s x a n d y t h a t d e p e n d

o n t h e p a r a m e t e r s y a n d x . F o r e a c h o f t h e e q u a t i o n s a C a u c h y p r o b l e m i s fo r m u l a t e d i n t h e c o r r e s p o n d i n g v a r i ab l e .

S o l u t i o n o f t h i s p r o b l em w i t h i n t h e d es c r e t i za t i o n s t ep i s n o t d i f fi cu l t an d can b e p e r f o r m ed w i t h E u l e r ' s , A d am s ' s ,

7 24



7 - z l a 1 W z373 0 . 0 1 I

372 0 . 0 1 0

371 O.O0~

370 O200

5 6 ~ I i 0 .0070 0.035 9 O

b

Y @

. 2

O.O35 g

F i g . 3 . C h a n g es i n t h e t em p er a t u r e T 2 o f t h e h ea t t r an s f e r ag en t ( a ) an d i t s

m o i s t u r e co n t e n t W 2 (b ) o v e r t h e b e d t h i ck n es s i n c r o s s s ec t i o n s l o ca t ed a t

var iou s d i s ta nce s f rom the beg inn ing o f the be l t : x = 0 (1 ) , x - - 3 .52 m (2 ) ; x

= 10 .56 m (3 ) , x = 17 .60 m (4 ) .

o r R u n g e - K u t t a ' s m e t h o d . T h e w h o l e a l g o ri t hm c o n s is t s in s u c c e s s iv e c a l c u la t i on o f u n k n o w n f u n c t i o n s a t n o d e s o f

d r y i n g l o ca t ed a l o n g s t r a i g h t l i n e s p a r a ll e l t o an ax i s co r r e s p o n d i n g t o a p a r t i cu l a r u n k n o w n f u n c t i o n .

A s an ex am p l e , i n F i g s . 2 an d 3 s o m e ca l cu l a t ed r e s u l t s a r e p r e s e n t ed f o r o p e r a t i o n o f a r ea l d r y e r w i t h

t h r ee - r o w t r an s p o r t a t i o n b y a b e l t co n v ey e r o f f r u i ts t o b e d r i ed cu t i n to c i r cu l a r s l ic e s a t t h e f o l l o w i n g p r o ces s

para m eter s : h = 0 .07 m; l = 17 .6 m; R 10 = 0 .04 m; B10 = 0 .406 ; W10 = 0 .87 ; P l 0 = 9 5 0 k g / m 3 ; T 1 0 = 2 9 3 K ; Ulx =

4 .9 .1 0 - 3 m / s ec ; T 20 = 37 3 K ; U 2 0 = 0 .3 4 m / s e c an d w i t h t h e f o ll o w i n g ap p r o x i m a t i o n o f t h e d r y i n g cu r v es :

r( t , T 1 ) = ~ W 1 0 e x p [ - k t / (a - t ) a ] a t t <_ tp,

W1[ x e x p ( - y t ) a t t > t p ,

w h e r e

w.X .

t = xU l - - , -t "= 6 0 [ 2 7 4 - 1 . 1 7 5 ( T 1 - 2 7 3 ) ] " k = - l n, , WI 0

t p '

ka = t p l n - - ~ p l n ~ ; y -

WpWIO ( a - tp) 2 'tc = exp

a ( a - t p ) '

Wp = 0 .002 -~ ; W p/2= 0 .00 13 1 -~0 + 0 .6 2 .

A s f o l lo w s f r o m t h e d a t a s h o w n i n t h e f i g u r e s f o r t h e u p p e r b e l t , a h i g h i n t en s i t y o f b l o w i n g p r o v i d es f o r an a l m o s t

u n i f o r m ( t en d i n g s o m e w h a t t o w ar d a d ec r ea s e ) d i s t r i b u t i o n o f t h e f r u i t t em p er a t u r e T 1 ( F i g. 2 a ) an d t h e v o l u m e

co n cen t r a t i o n B 1 o f t h e f i ll i n g ( F ig . 2 b ) t h r o u g h o u t t h e t h i ck n es s o f t h e b ed ; a s t h e d i s t a n ce f r o m t h e b eg i n n i n g o f

t h e b e l t in c r e a s e s , T i n c r e a s e s a n d B 1 d e c r e a s e s ; t h e p a r a m e t e r s T 2 a n d W 2 o f t h e d r y i n g a g e n t u s e d o n t h e u p p e r

b e l t a r e p r ac t i ca l l y u n c h an g ed a l o n g t h e l en g t h o f th e i n s t a l l a t i o n ( F ig . 3 ) .

N O T A T I O N

x , y , co o r d i n a t e s ; t , t i m e ; p , g a s p r e s s u r e ; E , i n t e r n a l en e r g y ; F , f o r ce o f i n t e r p h as e i n t e r ac t i o n ; G, v o l u m e

f o r ce ; Q , h ea t f l u x ; M , i n t e r p h as e m as s t r an s f e r ; U , Ux, Uy, v e l o c it y a n d c o m p o n e n t s o f t h e v e l o c it y a l o n g x a n d

y ; W , m o i s t u r e co n t en t ; p , d en s i t y ; 2 , t h e r m a l co n d u c t i v i t y ; /~ , v i s co s i ty ; c, h ea t c ap ac i t y ; B , p h a s e i n d i c a t o r f u n c t i o n

7 2 5



( a n a lo g o f t h e v o l u m e c o n c e n t r a t i o n ) . S u b s c r i pt s : 1 , m a t e ri a l ; 2 , d r y i n g a g e n t ; i j , t he t r a ns i t i on i -~ j ; s , sp he r e ; c ,

c i rc le .

R E F E R E N C E S

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2.

3.

P . I . N i g m a t u l i n , F u n d a m e n t a l s o f t h e M e c h a n i c s o f H e t e r o g e n e o u s M e d i a [ in R u s s i a n ] , M o s c o w ( 1 9 7 8 ).

A . I. N a k o r c h e v s k i i , H e t e r o g e n e o u s T u r b u l e n t J e t s [ in R u s s i a n ] , K ie v ( 1 9 8 0 ).

A . I. Na kor c h e vsk i i , i n : P hy s i c s o f Ae r od i sp e r se S ys t e m s , I s sue 33 ( 1990) , pp . 114- 118 .

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