modelling of simultaneous mass and heat transfer with chemical reaction using the maxwell-stefan...

8/12/2019 Modelling of Simultaneous Mass and Heat Transfer With Chemical Reaction Using the Maxwell-stefan Theory-II No

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Pergamon Chemical Engineerin0 Science Vol. 50, No. 10, pp. 1661 1671, 1995Copyright 1995 Elsevier Science LtdPrinted in Great Britain. All rights reserved0009-2509/95 9.50 + 0.000 0 0 9 - 2 5 0 9 ( 9 5 ) 0 0 0 1 1 - 9

M O D E L L I N G O F S IM U L T A N E O U S M A S S A N D H E A TT R A N S F E R W I T H C H E M I C A L R E A C T I O N U S I N G T H EM A X W E L L S T E F A N T H E O R Y I I . N O N I S O T H E R M A L

S T U D Y

M . J. W . F R A N K * , J . A . M . K U I P E R S , R . K R I S H N A : a n d W . P . M . V A N S W A A I JDep ar tme nt of Chemical Engineer ing, Twe nte Unive rs i ty of Technology, P.O. Box 217, 7500 AE Enschede ,The N e t he r la nds~/Depar tme nt of Chemical Engineering, Unive rs i ty of Amsterdam, N ieuwe Ach tergracht 166, 1018 WVA m s t e r da m , The N e t he r la nds

Received 28 July 1994; accepted in revised form 14 December 1994)Abstract In Par t I a genera l appl icable model has been developed which ca lcula tes mass and heat t ransferf luxes throu gh a v apo ur /ga s- l iquid in ter face in case a reversib le chemical reac t ion wi th associa ted heatef fec t t akes p lace in the l iquid phase . In th i s mode l the M axwe l l -Stefan theory has been used to d escr ibe themass t ranspor t . Also in Pa r t I the i sotherma l absorpt ion of a pure gas A in a solvent conta ining a reac t ivecom pon ent B has been studied. In this pap er the influence of therm al effects on the mass transfe r rates isinves t iga ted, wi th specia l a t t ent ion to the co ncent ra ted sys tems. T he therm al ef fec ts a r ise as a consequenceof enthalpy changes due to phase t rans i t ions an d ch emical reac tion. Accou nt is t aken of the inf luence oftempe ra ture gradient s o n ( i) the solubi l i ty of the gaseous com pone nt in the l iquid phase , ( i i ) the chemicalreact io n rate an d ( i i i ) the mass tran sfer coefficients in the l iquid phase. N um erica l simulat ion s show tha t ,when com pared to the cor resp onding i sothermal case , the thermal ef fec ts can affect the mass t ransfer ra tesby as mu ch as a fac tor of 30. In case of h igh Lewis numbers the num er ica l ly ca lcula ted mass t ransfer ra tescan v ery well be predic ted f rom an appro xima te analyt ica l expression, which has been presented in th ispaper. In m ost cases this is also a reason able est im ate of the mass transfer rate in case the Lewis num berequals uni ty . In case of a second -order chemical reac t ion i t was shown tha t thermal ef fec t s may change themaximum enhancement fac tor and consequent ly shi f t the absorpt ion f rom the ins tantaneous regime to thepseudo- f i r s t -order regime. Fur ther , i t i s conclud ed tha t there may exis t non- i sothermal g as- l iquid absorp -t ion systems w here min or change s in param eters ap pea ring in the hea t balance, e.g. binary mass transfercoefficients, chemical react ion rate constant , Le number or heat t ransfer coefficients, may result indrast ical ly al tered system behaviour. For si tuat ions in which thermal effects are signif icant , also thevapo r iza t ion o f the l iquid mixture sho uld be t aken in to acco unt , especial ly when the ca lcula ted in ter facetempe ra ture i s near or exceeds the b oi l ing tempera ture of the l iquid.

1. INTRODUCTIONM a n y i n d u s t r i a l p r o c e s s e s in v o l v e m a s s t ra n s f e r p r o -c e s s es b e t w e e n a g a s o r a v a p o u r a n d a l i q u id . O f t e n i ns u c h p r o c e s s e s t h e r m a l e f fe c ts a r e i m p o r t a n t . T h e r m a le f fe c ts a r i se f r o m p h a s e c h a n g e s a c c o m p a n y i n g a b -s o r p t io n , d e s o r p t i o n , c o n d e n s a t i o n o r e v a p o r a t i o n .C h e m i c a l r e a c t i o n s , i f o c c u r r i n g i n t h e l i q u i d p h a s e ,a l s o c o n t r i b u t e t o e n t h a l p y c h a n g e s . R e a c t i v e d i s ti l la -t i o n i s a t y p i c a l e x a m p l e i n w h i c h m a s s a n d h e a tt r a n s f e r p r o c e s s e s b e t w e e n t h e v a p o u r a n d l i q u i dp h a s e s a r e t o b e c o n s i d e r e d s i m u l t a n e o u s l y w i t hl i q u i d - p h a s e c h e m i c a l r e a c t i o n .

I n P a r t I ( F r a n k et al., 1 9 9 5 ) a g e n e r a l a p p l i c a b l em o d e l h a s b e e n d e v e l o p e d w h i c h c a l c u l a t e s m a s s a n dh e a t t r a n s f e r r a t e s t h r o u g h a v a p o u r / g a s - l i q u i d i n t e r -f a c e i n c a s e a r e v e r s i b l e c h e m i c a l r e a c t i o n w i t h a s s o -c i a t e d h e a t e f f e c t t a k e s p l a c e i n t h e l i q u i d p h a s e . I nt h is m o d e l t h e M a x w e l l - S t e f a n t h e o r y h a s b e e n i m p l e -m e n t e d t o d e s c r i b e t h e m a s s t r a n s p o r t . T h e d e s c r i p -t i o n o f t h e t r a n s f e r p r o c e s s e s h a s b e e n b a s e d o n t h e

t Author to whom cor respondence should be addressed.

f il m m o d e l a c c o r d i n g t o w h i c h e a c h p h a s e i s t h o u g h tt o e x is t o f a w e l l - m i x e d b u l k a n d a s t a g n a n t z o n e i nt h e l a tt e r o f w h i c h s i m u l t a n e o u s t r a n s p o r t o f h e a t a n dm a s s o c c u r s . I n P a r t I i s o t h e r m a l s i m u l a t i o n s h a v eb e e n c o n d u c t e d t o s h o w t h e i m p o r t a n t f e a tu r e s o f t h em o d e l f o r m a s s t r a n s f e r w i t h c h e m i c a l r e a c t i o n . W h e nt h e c o n c e n t r a t i o n o f t h e t r a n s f e r r in g s p e c ie s i n t h el i q u i d p h a s e i s h i g h , t h e m a s s t r a n s f e r o f e a c h s p e c i e sw i ll b e i n f l u e n c e d b y t h e m o v e m e n t o f t h e o t h e r t r a n s -f e r ri n g s p e ci e s a n d a p r o p e r f o r m u l a t i o n o f th e d i f fu -s i o n e q u a t i o n s u s i ng t h e M a x w e l l - S t e f a n t h e o r yb e c o m e s i m p o r t a n t . I n t h is p a p e r t h e i n f l u e n c e o f h e a te f f ec t s o n t h e s e ( c h e m i c a l l y e n c h a n c e d ) m a s s t r a n s f e rp r o c e s s e s w i l l b e i n v e s t i g a t e d .

H e a t i s g e n e r a t e d w h e n a g a s a b s o r b s i n t h e l iq u i d ,d u e t o r e l e a s e o f h e a t o f d i s s o l u t i o n . H e a t i s a l s og e n e r a t e d w h e n e x o t h e r m i c c h e m i c a l r e a c t i o n s t a k ep l a c e i n t h e l i q u i d p h a s e . S u c h h e a t e f f e c t s w i l l c a u s ea t e m p e r a t u r e g r a d i e n t t o d e v e l o p n e a r t h e g a s - l i q u i di n te r f ac e . T h e d e v e l o p e d t e m p e r a t u r e p r o f i le w i ll in -f l u e n ce ( i) t h e s o l u b i l i t y o f t h e d i s s o l v i n g c o m p o n e n t ,( ii ) t h e c h e m i c a l r e a c t i o n r a t e c o n s t a n t , a n d ( iii ) t h ed i f f u s iv i t i e s o f t h e t r a n s f e r r i n g s p e c i e s i n t h e l i q u i dp h a s e . A s t h e m a s s t r a n s f e r r a t e d e p e n d s o n a l l t h e s e

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1 6 6 2 M . J . W .p a r a m e t e r s , t h i s r a t e w i l l c h a n g e d u e t o a f o r e m e n -t ioned heat e ffec ts .

I n t h e l i t e r a t u r e s e v e r a l s tu d i e s c o n c e r n i n g s i m u l t a -n e o u s m a s s a n d h e a t t r a n s f e r w i t h c h e m i c a l r e a c t i o nh a v e b e e n r e p o r t e d : S h a h ( 1 9 7 2 ) , M a n n a n d M o y e s(1977) , Al l a n and M an n (1979 , 1982) As a i e t a l . (1985),W h i t e a n d J o h n s ( 19 86 ), B h a t t a c h a r y a e t a l . (1987,1988) , Ch a t t e r j e e and Al tw icke r (1987), E va ns an dSe l im (1990) , AI -Uba id i e t a l . (1990) and A1-Uba id ia n d S e l im (1 99 2) . N o n e o f th e s e s t u d i e s h a s e m p l o y e dt h e M a x w e l l - S t e f a n f o r m u l a t i o n o f d if f us i o n a n dt h e r e f o r e t h e i r v a l i d i t y i s s t r i c t l y l i m i t e d t o c a s e s o fl o w c o n c e n t r a t i o n o f t ra n s f e r r i n g s p e c i e s i n t h e l i q u i dp h a s e . T h e t r a n s f e r m o d e l s u s e d i n t h e l i t e r a t u r e c a nb e d i v i d e d i n t o t h r e e b r o a d c a t e g o r i e s :

(A ) M o d e l s b a s e d o n t h e p e n e t r a t i o n t h e o r y w h i c hi s b a s e d o n n o n - s t a t i o n a r y d e s c r i p t i o n o f t h e t r a n s f e rp r o c e s s e s . F r o m t h e s e m o d e l s i t f o l lo w s t h a t i n l i q u i d sn o r m a l l y th e h e a t p e n e t r a t i o n d e p t h c o n s i d e r a b l y e x -ceeds t he m as s pen e t ra t i on dep th , i .e . ~ >> D~j , andc o n s e q u e n t l y i t i s g e n e r a l l y a s s u m e d t h a t t h e m a s st r a n s f e r p r o c e s s e s t a k e p l a c e a t a c o n s t a n t t e m p e r -a tu re , i . e . t he i n t e r f ace t em pe ra tu re . Due t o t h i s a s -s u m p t i o n t h e h e a t a n d m a s s b a l a n c e s a r e d e c o u p l e da n d c a n b e s o l v e d i n d e p e n d e n t l y .

( B ) M o d e l s b a s e d o n t h e f i l m t h e o r y w i t h t h e a s -s u m p t i o n o f i d e n t i c a l e f f e c t i v e f i l m t h i c k n e s s e s f o rhea t and m ass , i .e . 6h = fro . T he adv an t age o f t h i sm o d e l t y p e i s t h e l e s s i n v o l v e d m a t h e m a t i c s i n c o m -p a r i s o n t o t h e p e n e t r a t i o n - t y p e m o d e l s , b u t t h e f a c tt h a t t h e m a s s t r a n s f e r z o n e a n d t h e h e a t t r a n s f e r z o n ed o n o t n e c e s s a r i l y c o i n c i d e p o s e s a c o n c e p t u a l d i f f i-cu l t y .

( C) M o d e l s b a s e d o n t h e f i lm t h e o r y w h i c h d e f i n ea s t a g n a n t z o n e f o r t h e m a s s t r a n s f e r p r o c e s s e s a n do n e w i t h a m u c h l a r g e r t h i c k n e s s fo r t h e h e a t t r a n s f e rpr oce sse s, i .e . 6h >> 6 m

T h e p r i m a r y g o a l o f t h i s p a p e r i s t o s t u d y s y s t e m -a t i c a l l y t he i n f luence o f hea t e f f ec t s on chem ica l l ye n h a n c e d a b s o r p t i o n . T h e f i lm m o d e l i s a d o p t e d w i t h -o u t m a k i n g a s s u m p t i o n s w i t h r e s p e c t t o

FR NK e t a l .L e = 6 h / J , . . B o t h f i r s t - a n d s e c o n d - o r d e r r e a c t i o n sw i l l b e c o n s i d e r e d , w i t h o u t m a k i n g a s s u m p t i o n so f t h e p r e v a i l i n g r e a c t i o n r e g i m e . I n a d d i t i o n t h eM a x w e i l - S t e f a n t h e o r y w i l l b e u s e d t o d e s c r i b e t h em a s s t r a ns f e r p r o c es s e s . T h e i m p o r t a n c e o f i n c o r p o r -a t i n g t h e r m a l e f f e ct s w i l l b e d e m o n s t r a t e d b y c o m p a r -i n g t h e r e s u lt s o f t h e i s o t h e r m a l m o d e l w i t h m o d e lc a l c u l a t i o n s i n w h i c h n o n - i s o t h e r m a l e f f e c t s a r ep r o p e r l y t a k e n i n t o a c c o u n t .

2 T H E O R YI n P a r t I a n u m e r i c a l s t u d y h a s b e e n c o n d u c t e d o f

t h e i s o t h e r m a l a b s o r p t i o n o f a p u r e g a s A i n a s o lv e n tc o n t a i n i n g a r e a c t i v e c o m p o n e n t B . I n t h i s p a p e r t h es a m e a b s o r p t i o n s y s t e m ( s e e F i g . 1 ) w i l l b e c o n s i d e r e di n d e t ai l u n d e r n o n - i s o t h e r m a l c o n d i t io n s . C o m p o n -e n t A i s a l l o w e d t o r e a c t b y a u n i m o l e c u l a r e x o t h e r -m a l c h e m i c a l r e a c ti o n o r b y a b i m o l e c u l a r e x o t h e r m a lc h e m i c a l r e a c t i o n w i t h c o m p o n e n t B t o p r o d u c e C .T h e c h e m i c a l r e a c t i o n s t a k i n g p l a c e a r e, r e s p e c t i v e ly ,

A - , C (1 )a n d

A + B ~ C (2 )w h e r e t h e r e s p e c t i v e r e a c t i o n k i n e t i c s a r e d e s c r i b e d b y

R kl X A (3)a n d

R = k lX A X s . (4)I t i s a s s u m e d t h a t t h e s o l v e n t , c o m p o n e n t B a n dp r o d u c t C a r e n o t v o l a t i l e a n d t h a t t h e f r a c t io n s o fA a n d C i n t h e l iq u i d b u l k e q u a l z e r o . D u e t o a b s o r p -t i o n a s w e l l a s c h e m i c a l c o n v e r s i o n h e a t w i l l b e p r o -d u c e d i n t h e l i q u i d - p h a s e m a s s t r a n s f e r f i l m , w h i c hw i l l s u b s e q u e n t l y b e t r a n s p o r t e d t o t h e l i q u i d b u l k .F i r s t s o m e c o m m e n t s w i ll b e g i v e n a b o u t t h e i n f lu e n c eof the hea t e ffec ts .

A s m e n t i o n e d b e f o r e t h e r e a r e t w o p o s s i b l e s o u r ce so f h e a t . A t f ir s t, h e a t p r o d u c t i o n d u e t o p h a s e c h a n g e

p u r e g a s s o l v e n tTi

Tlb

X I X b

Z = 0 Z = m Z = g ho r o r o r11=0 11=1 ~ =L e

Fig. l . Schematic representation of the fraction and temperature profiles in the mass and h eat transfer fi lm for the gas-liqu idabsorption system.


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Modell ing of s imultaneous mass and heat t ran sfer - - l lh a s t o b e c o n s i d e r e d . T h i s e f fe c t i s a c c o u n t e d f o r i n t h ep a r t i a l m o l a r e n t h a l p ie s o f c o m p o n e n t A f o r b o t hp h a s e s a n d d e f i n e d b y

H , t = H a , T ,. , + C p , A T - T r e f . ) . 5 )I n e q . (5 ) i t is a s s u m e d t h a t H a i s a l i n e a r f u n c t i o n o ft e m p e r a t u r e T , w h e r e a s HA . r , ,, a n d Cp.A a r e c o n s t a n t sw h i c h w i l l d i f f e r f o r t h e g a s a n d l i q u i d p h a s e . I n o u rs i m u l a t i o n s i t h a s b e e n a s s u m e d t h a t C r , A e q u a l s z e r of o r b o t h p h a s e s a n d i n t h is c a s e t h e h e a t o f s o l u t i o nA H , c a n b e e x p r e s s e d a s

H g a s n l i qA H s = A . T , ,r - - A , T , .t (6)w h i c h i s in d e p e n d e n t o f t e m p e r a t u r e .

T h e s e c o n d s o u r c e o f h e a t i s t h e o n e d u e t o c h e m -i c a l r e a c t i o n . T h i s e f f e c t a l s o c a n b e i n c o r p o r a t e d i nt h e p a r t i a l m o l a r e n t h a l p i e s b y t a k i n g a l l h e a t c a p a c i - N A =t i es e q u a l t o z e r o a n d s e t ti n g H i , r , , o f th e l i q u i d p h a s et o t h e i r a p p r o p r i a t e v a l u es . A s a c o n s e q u e n c e t h e h e a to f r e a c t i o n A H R is g i v e n b y

A H R = - ~ v i l l i r (7)i = A , B ,C

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

t h e r e f o re i n f lu e n c e t h e p h y s i c a l a n d c h e m i c a l p a r a -m e t e r s . It i s g e n e r a ll y a s s u m e d t h a t t h e r e a r e t h r e ei m p o r t a n t p a r a m e t e r s w h i c h a r e s e n s it i v e t o t e m p e r -a t u r e c h a n g e s . T h e s e a r e t h e s o l u b i l i ty o f A i n t h el i q u i d m A , t h e c h e m i c a l r e a c t i o n r a t e c o n s t a n t k~ a n dt h e b i n a r y m a s s t r a n s f e r c o e f f i c i e n t s K i j . T h e s e d e -p e n d e n c i e s h a v e b e e n i n c o r p o r a t e d i n o u r n u m e r i c a lm o d e l b y u s i n g t h e f o l l o w i n g A r r h e n i u s t y p e o fe q u a t i o n s :

m a = m A o e x p F E s ( 1 - - ~ o (8)L R , . , L T

k, = ko , exp ( - E , , ~ ( 9)\ R , . , T /F o t lij = K q .0 exp L ~ - \ ? - . 10 )

T h e t e m p e r a t u r e d e p e n d e n c i e s ar e t h u s d e t e r m i n e d b yse t t in g the v a lues Es , E ,~ an d ED . By u s ing eq . (10) i t i sa s s u m e d t h a t a l l b i n a r y m a s s t r a n s f e r c o ef fi c ie n t s h a v et h e s a m e a c t i v a t i o n e n e r g y o f d i ff u si o n .

D u e t o t h e n o n - l i n e a r i t y o f t h e m o d e l e q u a t i o n st h e i r e x a c t a n a l y t i c a l s o l u t i o n i s n o t p o s s i b l e , w h i c hn e c e s s it a t e s t h e u s e o f a n u m e r i c a l s o l u t i o n p r o c e d u r e .H o w e v e r , in a d d i t i o n t h e d e v e l o p m e n t o f a p p r o x i m -a t e e x p l i c i t a n a l y t i c a l e x p r e s s i o n s , w h i c h c a n b e u s e da s a n a l t e r n a t i v e t o t h e n u m e r i c a l m o d e l , w i l l b ec a r r i e d o u t i n t h e s t u d y . F o r t h i s p u r p o s e i t h a s b e e na s s u m e d t h a t t h e t h i c k n e s s o f t h e h e a t t r a n s fe r f il ms i g n i f i c a n t l y e x c e e d s t h e t h i c k n e s s o f t h e m a s s t r a n s f e rf i lm (L e , > 1). If L e b e c o m e s v e r y la r g e t h e m a s s a n dh e a t b a l a n c e c a n b e d e c o u p l e d a n d t h e h e a t f l u xg e n e r a t e d i n t h e m a s s t r a n s f e r f il m c a n b e i m p o s e d a sa b o u n d a r y f lu x f o r t h e h e a t t r a n s f e r fi lm . I n t h a t c a s et h e m a s s t r a n s f e r e q u a t i o n s s h o u l d b e s o l v e d a t t h e

1 6 6 3i n t e rf a c e t e m p e r a t u r e w h i c h c a n b e p e r f o r m e d a n a -l y t i c a l l y . T h i s w i l l r e s u l t i n e x a c t i m p l i c i t a n a l y t i c a le x p r e s s i o n s f o r t h e a b s o r p t i o n f lu x N A . D u e t o t h e i re a s e o f u s e w e p r e f e r e x p l ic i t a n a l y t i c a l e x p r e s s i o n sa n d t h e r e f o re t h e f o l l o w i n g a p p r o x i m a t e e x p l i c i t e x -p r e s s i o n s f o r th e a b s o r p t i o n f l u x o f A t h r o u g h t h eg a s - l i q u i d i n t e r f a c e N A ( s e e P a r t I ) h a v e b e e n d e -v e l o p e d :- - a b s o r p t i o n w i t h o u t c h e m i c a l r e a ct io n :

- K a , c r l n ( 1 - - X A i)N A = (11)1 + XBb(dAB -- 1)- - a b s o r p t i o n w i t h i n s t a n t a n e o u s b i m o l e c u l a r

c h e m i c a l r e a c t io n :K A s CT XBb

dac - - das)XBb + d ~K A ~ C T ln (1 - - X A i )- 1 2 )

dcs + (dBc -- dcs)XBb(d A c - - 1 )X B b ~ (d sc d a s )X B b + 1- - a b s o r p t i o n w i th c h e m i c a l r ea c t io n :

N A = E A N A ) e q . l l ) 1 3 )w h e r e t h e e n h a n c e m e n t f a c t o r E A i s e x p r e s s e d f o ra f i r s t - o r d e r u n i m o l e c u l a r c h e m i c a l r e a c t i o n a s

H aE a - t a n h ( H a ) (14)w h e r e t h e e x p r e s s i o n f o r t h e H a t t a n u m b e r H a isg i v e n b y

H a = / k16----2~ . (15)~ ] T K A e f f

I n t h e e x p r e s s i o n f o r t h e H a t t a n u m b e r t h e e f f e c t i v em a s s t r a n s f e r c o e f f i c i e n t o f A i n t h e l i q u i d m i x t u r eK A a f i s d e f i n e d f o r t h i s a b s o r p t i o n s y s t e m a s f o l l o w s :

1 x a b 1 x s b4 (16)K A e f f K AB K AsT h e e n h a n c e m e n t f a c t o r E A f o r a s e c o n d - o r d e r b i -m o l e c u l a r c h e m i c a l r e a c t i o n i s ex p r e s s e d a s

H a E /~ _ -_ _ E _aX / E A ~ o - - 1E A = (17)

t a n h ( H a ~ f - ~ E--aw h e r e H a a n d EAoo a r e , r e s p e c t i v e l y , g i v e n b y

H a = ~ (18)~ / C T K A e f fa n d

NA )eq . 1 2 )EAo~ -- (19)N A ) e . i 1 .I t s h o u l d b e k e p t in m i n d t h a t a l l t e m p e r a t u r e - d e p e n -d e n t p a r a m e t e r s i n e q s (1 1 ) -( 1 9 ) m u s t b e e v a l u a t e d a tt h e i n t e r fa c e t e m p e r a t u r e . T h i s i n t e rf a c e t e m p e r a t u r e

C E S 5 0 1 0 g


4/11

1664f o l lo w s f r o m t h e h e a t b a l a n c e o v e r t h e h e a t t r a n s fe rf il m . H e a t g e n e r a t e d d u e t o a b s o r p t i o n a n d c h e m i c a lc o n v e r s i o n i n t h e m a s s t r a n s f e r f i l m h a v e t o b e c o n -d u c t e d t h r o u g h t h e h e a t t r a n s f e r f il m t o t h e l iq u i db u l k ( h e a t l o s s es t o g a s b u l k a r e n e g l e c t e d ) a n d l e a d st o t h e f o l l o w i n g h e a t b a l a n c e :

N A , o A H ~ + ( N a , o - N A , ~ ) A H a = h l ( T i - - T b ) (20)w h e r e N A . o r e p r e s e n t s t h e f l ux t h r o u g h t h e g a s - l i q u i di n t e r f a c e a n d i s e x p r e s s e d b y e q . ( 13 ). N A . ~ r e p r e s e n t st h e m a s s f l u x o f A t h r o u g h t h e m a s s t r a n s f e r f i l m - b u l ki n t e r f a c e , i.e . t h e p a r t o f N A , O w h i c h h a s n o t b e e nc o n v e r t e d i n t h e m a s s t r a n s f e r f il m . N o e x p r e s s i o n i sy e t a v a i l a b l e f o r NA.~ i n c a s e t h e M a x w e l l - S t e f a nt h e o r y i s u s e d t o d e s c r i b e t h e m a s s t r a n s f e r . S i m i l a r t ot h e d e r i v a t i o n o f th e e n h a n c e m e n t f a c t o r in P a r t I , t h ec l a s s i c a l t h e o r y [ s e e e . g . W e s t e r t e r p et a l . ( 1990) ] ha sb e e n a p p l i e d a s a s t a r t i n g p o i n t . I n c a s e F i c k s l a w i su s e d t o d e s c r i b e m a s s t r a n s f e r w i t h f i r s t - o r d e r u n i -m o l e c u l a r c h e m i c a l r e a c t i o n , NA,Z i s g i v e n b y

H aN A 6 - - s i n h ( H a ) (Na)without .. .. tion (21)

w h e r e t h e e x p r e s s i o n f o r H a i s g i v e n b y e q . ( 15 ). N o wi t i s a s s u m e d , s i m i l a r t o t h e e x p r e s s i o n o f t h e e n h a n c e -m e n t f a c t o r , t h a t e q . ( 2 1 ) i s a l s o v a l i d i n c a s e t h eM a x w e l l - S t e f a n t h e o r y i s u s e d t o d e s c r i b e t h e m a s st r an s f e r. T h e m a s s f lu x o f A i n a b s e n c e o f c h e m i c a lr e a c t i o n i s g i v e n b y e q . ( 1 1 ) . A s f o r s e c o n d - o r d e rb i m o l e c u l a r c h e m i c a l re a c t io n s s u c h e q u a t i o n s d o n o te x i s t , i t i s a l s o a s s u m e d t h a t e q . ( 2 1 ) h o l d s f o r s e c -o n d - o r d e r r e a c t i o n s , w h e r e H a n o w i s d e f i n e d b ye q . (1 8). T h i s a s s u m p t i o n w i ll b e c o m e m o r e a c c u r a t ea t h i g h e r v a l u e s o f t h e i n f i n it e e n h a n c e m e n t f a c t o rEAo 0

F r o m e q s (1 1 )- (2 1 ) th e a b s o r p t i o n f lu x o f A t h r o u g ht h e g a s - l i q u i d i n t e r f a c e c a n b e c a l c u l a t e d , t a k i n g i n t oa c c o u n t h e a t e f f ec t s a n d t h e d i f f u s io n a l i n t e r a c t i o n sb e t w e e n t h e t r a n s f e r r in g s p e ci e s. I n a d d i t i o n t h e i n t e r -f a c e t e m p e r a t u r e w i l l b e g iv e n . T h e e q u a t i o n s c a n b e

M. J. W. FRANK et al .s o l v e d i te r a t i v e ly u n t i l t h e a b s o r p t i o n f lu x a n d t h ei n t e r f a c e t e m p e r a t u r e s a t i s f y s o m e c o n v e r g e n c e c r i -t e r i o n . S u m m a r i z i n g i t s h o u l d b e k e p t i n m i n d t h a tt h e s e r e s u lt s a r e b a s e d o n t h e f o l l o w i n g a s s u m p t i o n s :

- - L e >> 1 , i .e . t he t h i c k ne s s o f t he he a t t r a n s f e r f i l mi s m u c h l a r g e r t h a n t h e t h i c k n e s s o f t h e m a s s t r a n s f e rf i lm.- - T h e a p p r o x i m a t e a n a l y ti c a l e x p r es s io n s f or th ea b s o r p t i o n f l u x u n d e r i s o t h e r m a l c o n d i t i o n s a r e v a l i d .T h i s h a s b e e n s h o w n i n P a r t I .

- - T h e a p p r o x i m a t e e x p r e s s i o n f or t h e m a s s fl u x a tt h e m a s s t r a n s f e r f i l m - b u l k i n t e r f a ce , d e d u c e d f r o mc l a s s i c a l t h e o r y f o r m a s s t r a n s f e r w i t h f i r s t - o r d e r r e a c -t i o n , is v a l id f o r b o t h f ir s t- a n d s e c o n d - o r d e r c h e m i c a lr e a c t i o n u s i n g t h e M a x w e i l - S t e f a n t h e o r y . T h i s h a ss t i l l t o b e p r o v e d .

3. RESULTSW i t h t h e n u m e r i c a l m o d e l d e s c r i b e d i n P a r t I s i m u -

l a t i o n s w e r e p e r f o r m e d f o r t h e n o n - i s o t h e r m a l a b -s o r p t i o n o f a p u r e g a s A i n a s o l v e n t c o n t a i n i n ga c o m p o n e n t B (s e e F i g . 1 ). T h e p a r a m e t e r v a l u e s u s e di n t h e s i m u l a t i o n s a r e g i v e n i n T a b l e 1 .

T h e i n f lu e n c e o f t h e h e a t e f fe c ts , d u e t o a b s o r p t i o na n d c h e m i c a l r e a c t i o n , o n t h e m a s s t r a n s f e r r a t e s h a sb e e n s t u d i e d i n d e ta i l b y c o m p a r i n g t h e r e s u l t s o f t h en o n - i s o t h e r m a l a b s o r p t i o n s i m u l a t i o n s ( L e = 10)w i t h t h e r e s u l ts o f i s o t h e r m a l a b s o r p t i o n s i m u l a t io n s .T h i s c o m p a r i s o n h a s b e e n c a r r i e d o u t s y s t e m a t i c a l l yb y g i v i n g o n e o f t h e p a r a m e t e r s E a l , E ~ o r E D a v a lu ed i f fe r i n g f r o m z e r o a n d t a k i n g t h e o t h e r t w o p a r a -m e t e r s o f t h i s se t e q u a l t o z e r o . B y f o l l o w i n g t h i sa p p r o a c h t h e e f fe ct o f e ac h t e m p e r a t u r e - d e p e n d e n tp a r a m e t e r o n t h e m a s s t r a n s f e r r a t e c o u l d b e s t u d i e di n d e p e n d e n t l y . I n a d d i t i o n s i m u l a t i o n s w e r e c a r r i e do u t i n w h i c h a l l t h r e e p a r a m e t e r s ( i.e . E a l , E s a n d E o )d i f f e r e d f r o m z e r o .

T w o s e ts o f b i n a r y m a s s t r a n s fe r c o e f f i c ie n t s h a v eb e e n u s e d t o p e r f o r m t h e s t u d y . I n o n e o f t h e s e t s, a l lb i n a r y m a s s t r a n s f e r c o e ff i c ie n t s w e r e t a k e n e q u a l a n d

Table 1 . Param eter va lues used to ca lcula te the da ta poin t s shown in Figs 2-7 for absorp t ion of a pure gas A in to a solventc on t a i n i ng c om pone n t6m = 1 x 10 -5 mpsHo.i~i. r , . ,. = 50,000 J/m ol

CT = 1 X 104 mol/ m 3H l iq - 0 J/m olO i ~ i T t --

E, = 0 or 25,000 J /molReact ion scheme lA ~ C

Equal binary mass transfer coefficientsK~j = 1 x 10 -4 m/s i = A,B,C,s j = A,B,C,s

Cp.i = 0 J / m o l KH l iq - 0 J / m o lO i~i T~.t -

L e = 1 or 10

Tib = 300 K ht = 500 J /K sn l i q - - 1 0 0 0 0 0 J / m o lo . i ~ i . T . t . - - - -

C om pos i t i on r e g i m e 1XAtO = 0.003 and xnb = 0.006

Eal = 0 o r 25,000 J /m ol E o = 0 o r 10,000 J /m olReact ion scheme 2A + B - - , C

Different binary mass transfer coefficientsK~B = 3 x 1 0-* m/s K ~ c = 3 x 10 -5 m /s

K~, = 1 x 10 -4 m/sKlac = 4 x 10 -4 m /s K~, = 5 x 10 -5 m /sK~s = 2 x 10 -4 m /sC om pos i t i on r e g im e C om pos i t i on r e g im e 3

XAio = 0.003 a nd xsb = 0.6 XAio = 0.3 and Xnb = 0.6


5/11

Modelling of simultaneousin this case the effective mass transfer coefficient ofA in the solvent mixture was composition indepen-dent. The second set has been chosen to provide anextreme test for using the concept of compositionweighted reciprocal mass trans fer coefficients to yieldthe effective t ransfer coefficient KAcff in the expressionfor the H att a nu mber [eq. (15)-I. Besides, three com-position regimes were considered in the simulations.The first one, with low values of Xa~ and XBb, willreduce the Maxwell-Stefan solution to the solutionobtained on basis of Fick s first law. This is also validfor the second composition regime with low XA~ andhigh xab, however in this regime we have to correct forthe effective mass t ransfer coefficient of A. In thesecond composition regime, the bimolecular reactioncan be described as a pseudo-fir st-order reaction withrespect to component A. The third regime with highvalues of both XAi and xnb will lead to interactionsbetween all components present in the liquid phaseand as a consequence Fick s law will not be validanymore which necessitates the use of the Max-well-Stefan theory as already shown in Part I.As L e is given a high value of 10, the approximateanalytical solution derived in the previous section forL e = oo is expected to be valid with a reasonabledegree of accuracy. Whether this is indeed the casewill be studied for all simulations described above. Inadditi on the effect of Le -nu mber on the mass transferrate will be studied by comparing the numerical re-suits of simulations for L e = 10 with numerical re-sults of simula tions for L e = 1. Fro m this compar isonit can also be concluded in which cases the approxim-ate analytic al solution can be used in case L e = 1. Incomparing the results for L e = 10 with the ones forL e = 1, the th ickness of the mass t ransfe r film 6,, iskept constant to maintain the same value for theHatta number. However, the heat transfer coefficientht = 2/6h, which appears in the heat balance [eq. (20)],should also be kept constant to enable proper com-parison. Due to the fact that 6h changes for differentL e values, ). should be adapted to maintain constanth~ values.

Subsequently the results of the simulations will bepresented. Numerically and analytically computed re-sults will be represented in graphical form where ineach figure calculated values of the dimensional ab-sorpti on flux of A, defined as the ratio of the absorp-tion flux of A and the absorption flux of A for thecorrespo nding isothermal case without chemical reac-tion, are shown as a function of the dimensionlessHatta number for the following situations:

1. isothermal absorption,2. non-isothermal absorption with L e = 10,3. non-i sothermal absorption with Le = 1.4. absorption system described by approximate

analytical expressions (11)-(21)C om par i s on o f s imu la t ions w i th equa l and d i f fe r en tb inar y mas s t r ans fe r coe f f i c i en t sFrom the results which will be presented below itbecame evident that the inf luence of heat effects on the

mass and heat transfer--II 1665absorption flux was not biased by the particularchoice of the set of binary mass transfer coefficients.Generally, it could be concluded that the differencesbetween the numerical results for L e = 1 and thosefor L e = 10 were not affected at all and additional lythat the differences between the results obtained fromthe analytical expression and the numerically com-puted results for L e = 10 increase to a small extent ifdifferent instead of equal binar y mass transfer coeffi-cients are used. This is probably due to the appro xim-ate nature of eqs (11)-(19) and (21), which becomesmore pronounced in case of different binary masstransfer coefficients.

Therefore, only the results of the simu lations withdifferent binary mass t ransfer coefficients will be dis-cussed below in detail. In case striking differencesbetween the two sets (i.e. the equal a nd different coeffi-cients) have been found, this will be indicated.

Temper a tur e -dependen t s o lub i l i t yIn this case Ea~ and E o were set equal to zero,

whereas Es was given a value of 25,000 J/mol. Thesolubi lity of the pure gas A is given by the value Of XAi.From eq. (8) it follows that higher interface temper-atures will result in a lower solubility. As XAi de-creases, the driving force for mass transfer decreasesand consequently the absorption flux of A will alsodecrease at higher interface temperatures. The resultsof the simulations are presented in Figs 2(a) and (b)which show the dimensionless absorp tion flux 8 ofcomponent A as a function of the Hatta number.It t urned out that the results for first-order chemicalreaction with low solubility of A and the results forsecond-order chemical reaction with low solubility ofA and an excess of B present in the liquid phase (i.e.pseudo-first-order chemical reaction with respect toA) were very similar. Therefore, only the results of thelatter case are shown [see Fig. 2(a), XBb = 0.6]. Lowsolubilities of A result in low physical absorptionfluxes an d hence in a low heat p roduct ion rate. There-fore no significant change in interface temperatureoccurs and c onsequently heat effects do not playa role. With increasing chemical reaction rate, i.e.increasing Hatta number, enhancement of mass trans-fer occurs resulting in an increased heat productionrate which causes an increasing deviation betweenresults corresponding to isothermal conditions andthe numerical results obtained for L e = 10. Thus, forhigh reaction rates one has to take into account heateffects. Further it can be concluded that in this casethe value of L e has no effect on the absorption flux.This is due to the fact that at high chemical reactionrates the absor ption heat as well as the reaction heat isreleased at the interface for both L e = 10 and Le = 1.Since in this case XA~ is the only temperature-depen-dent quanti ty, not the temp erature profile in the masstransfer film but the interface temperature, which isindependent of Le , becomes important. In case ofa second-order chemical reaction with low solubilityof A and low maximum enhancement factor


6/11

1666$ 0

E

1 0

11 0 " 2 1 0 " 1 10 0 1 0 1 1 0 2

Ha

M. J. W. FRANK t al.as for L e = 1 this is not the case an d accordingly theinterface temperature a nd solubility of A change. Thedifferences are however small (10 ). At both low andhigh H a values heat is released close to the interfaceand will be experienced as a boundary heat flux.

The analyt ical expression produces good results inall three composition regimes and for both reactionschemes: the ma ximum observed deviat ion being 15in the third composition regime of the second-orderchemical reaction. This error is most probab ly causedby the approximate nature of the mass transfer equa-tions.

E

0 .31 0 2

rIsoth .. . I [ " " 'i . . .. . .. . .. . .. . .. . ./ i d i f f c ~ l l t K i jI i / i

. . . c .l . : A o C I [ i/(2):A+B-C[ ~ 2

IIIII:II:::::I:::SISI:~IS:III:I::::::ISI:IIIIS:IS:I:I:II~~:

1 0 " 1 1 0 0 1 0 1 1 0 2H a

Fig. 2. Dimensionlessabsorption flux 8 of component A ob-tained from the numerical model and the analytical expres-sions as a function of the Hatta number in case the solubilityis temperature dependent: (a) second-order chemical reac-tion, different binary mass transfer coefficients and secondcomposition regime; (b) first-order and second-order chem-ical reaction, different binary mass transfer coefficients andthird composition regime. Parameter values are given inTable 1.

(xab = 0.006) heat effects are insign ificant for anychemical reaction rate constant since the absorptionflux is limited due to the low maxim um enhancementfactor.

Figure 2(b) shows the results in case A possessesa high solubil ity. In this case even the physical absorp-tion flux of A is large enough to change the interfacetemperature and thus the solubility, significantly. Ifthe heat balance is not taken into account an error of150 with respect to the abso rpt ion flux will bemade. If a first-order chemical reaction proceeds, thiserror will increase strongly at high reaction rates. Fora second-order chemical reaction it can be seen fromFig. 2(b) that for non -isothermal absorption the masstransfer rate has not reached its maximum attainablevalue. This is due to the fact that with increasing theHatta number also the maxi mum enhancement factorincreases due to the lowered solubility of A. As a con-sequence, the maximum enhancement factor will beattained at higher H a values. For both reactionschemes the influence of L e is most important atintermediate Hatta numbe rs which is expected behav-iour since in this situation the heat of reaction isreleased in the entire mass transfer film. For L e = 10this will be felt as a boundary heat production, where-

T e m p e r a t u r e - d e p e n d e n t b i n a r y m a s s tr a n s fe rc oe f f i c i e n t sIn this case Eal and E~ were set equal to zero,whereas E o was given a value of 10,000 J/mol. Fromeq. (10) it follows that heat produc tion will result inhigher bi nary mass transfer coefficients and hence willincrease the mass transfer rate of component A. Theresults of the simulations are presented in Figs3(a)-(c).

Figure 3(a) shows the results for pseudo-first-orderchemical reaction with low solubility of A. In absenceof chemical reaction heat effects have no influence,which is due to the very low heat production rate.With increasing chemical reaction rate the differencebetween the results for isothermal and non-i sother malconditions becomes more pronounced and eventuallyas high as 70 . Moreover, it can be seen that thevalue of L e has a marked influence on abso rptionrate. The difference between the results for L e = 1and L e = 10 increase with increasing H a , but remainsalways smaller tha n the difference between the resultsfor L e = 10 and the results for isothermal absorpt ion.In case of L e = 10 the mass transfer coefficient willonly be affected by the interface temperature whereasfor L e = 1 the entire temperature profile will affectthe mass transfer process, result ing in lower masstransfer rates. The analytical expressions producemaxim um errors of 30 .

Figures 3(b) and (c) show the results for mass trans-fer with first-order chemical reaction correspondingto the third composition regime where, respectively,different [Fig. 3(b)] and equal [Fig. 3(c)] bina ry masstransfer coefficients have been used. Due to high solu-bility of A large differences between non-is othermalabsorption for L e = 10 and isothermal absorptionalready exist at low H a values. By examining theinfluence of the L e value it follows that for highvalues of the Hatta number, the results for L e = 1and L e = 10 approach each other. This is caused bythe shape of the concentration profile of componentA. The effective film thickness for component A de-creases due to enhancement and consequently thetemp erature which affects the mass trans fer coefficientof A will approach the interface temperature. Theanalytical expression predicts the ab sorpt ion rate verywell for most H a values, but a maxim um error of 50in case of equal mass transfer coefficients and inter-mediate H a values is somewhat high. This is probab ly


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100E

10

10-2 10-1 100

Mo dell ing of s imultaneo us mass and heat t ransfer I I1.2

HWR, tHPR 1.0 F[MJ/m21s] tT o I0.6

0 4 I0 .20101 102Ha 0

................... ................... di fr~-,tK..i s o t h e r m a l t ..................T.................. U

e - - - t ~ : l O l ..................; r - ~ X ~o o . ~100

; : 2 2 i i 2 2 i i il l il l l ii i ii i i i2 2 2110 2 10o 102 Ha

100

10

10-2 10-1 100 101HaFig. 3 . Dim ens ionless absorp t ion f lux 8 of com pone nt A ob-ta ined from the num erica l mod el and the analyt ica l expres-s ions as a funct ion of the Ha t ta n um ber in case the binar ymass transfer coefficients are temperature dependent: (a)second-order chemical reaction, different binary mass trans-fer coefficients and second com pos ition regime; (b) first-orderand second-order chemical reaction, different binary masstransfer coefficients and third composition regime; (c) first-orde r chemical reaction, eq ual bin ary mass transfer coeffi-c ients and th ird composi t ion regime. Parameter va lues aregiven in T able 1 .

d u e t o a s m a l l e r r o r i n c a l c u l a t i n g N ~ f r o m e q . ( 2 0)w h i c h c a u s e s a n e r r o r i n t h e h e a t p r o d u c t i o n r a t e a n dt h e r e b y i n t h e i n t e r f a c e t e m p e r a t u r e . A s a c o n s e -q u e n c e t h e i n t e r f a c e f l u x o f A N o w i l l a l s o c h a n g e .

A n u n e x p e c t e d v e r y l a r g e i n c re a s e i n t h e c o m p u t e da b s o r p t i o n r a t e w a s f o u n d w h e n t h e e f f e c t i v e m a s st r a n s f e r c o e f f i c i e n t o f A w a s i n c r e a s e d b y a f a c t o r o f1 .6 7 [ e q . ( 16 ), b y t a k i n g d i f f e r e n t i n s t e a d o f e q u a lb i n a r y m a s s t r a n s f e r c o e f f i c ie n t s] i n c a s e o f n o n - i s o -t h e r m a l p h y s i c a l a b s o r p t i o n f o r Le = 10 . Th i s s t r ik -

1667

........... l iP equal Kij . ~ .. . . . . . l iP d i fferent Ki j . . /

HW . '"

5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0- - 3 D - T [ KIFig. 4 . Heat w i thdrawal ra te (HWR) an d heat produ ct ionrate (HPR ) as a function of the interface temp eratur e T~calculated from the analytical expressions in case of first-ord er chemical reaction, eq ual or different bin ary masstransfer coeff ic ients and th ird composi t ion regime. Para-meter values are given in Table 1.

i n g p h e n o m e n o n c a n b e e x p l a in e d b y e x a m i n i n ge q . ( 20 ). T h i s e q u a t i o n d e m a n d s a b a l a n c e b e t w e e n t h eh e a t p r o d u c t i o n r a t e p e r u n i t a r e a a n d t h e h e a t f l uxe n t e r i n g t h e l iq u i d b u l k . B o t h q u a n t i t i e s d e p e n d o nt h e v a l u e o f th e i n t e r f ac e t e m p e r a t u r e T i w h i c h h a s t os a t is f y e q . (2 0) . T h i s i n t e r f a c e t e m p e r a t u r e T~ c a n b ed e t e r m i n e d g r a p h i c a l l y b y p l o t t i n g t h e l e f t - a n dr i g h t - h a n d s i d e o f t h i s e q u a t i o n a s f u n c t i o n o f T i . T h i sh a s b e e n d o n e f o r t h e s y s t e m s c o r r e s p o n d i n g t oF i g s 3 ( b ) a n d ( c) , t h e r e s u l t s a r e s h o w n i n F i g . 4 . F r o mt h i s f ig u r e i t f o l l o w s t h a t t h e u s e o f e q u a l b i n a r y m a s st r a n s f e r c o e f f i c i e n t s y i e l d s a l o w - t e m p e r a t u r e s o l u t i o nw h e r e a s t h e u s e o f d i f f e re n t b i n a r y m a s s t r a n s f e r c o e f -f i c ie n t s , r e s u l t i n g i n a s o m e w h a t h i g h e r e f f e c ti v e m a s st r a n s f e r c o e f f i ci e n t o f c o m p o n e n t A , y i e l d s a h i g h -t e m p e r a t u r e s o l u t i o n . D u e t o t h e s h a p e o f th e p r o f i le st h e s e t w o s o l u t i o n s d i ff e r s i g n i f i c a n t l y . T h e h i g h - t e m -p e r a t u r e s o l u t i o n o f F i g . 4 w i l l o f c o u r s e n o t b e e n -c o u n t e r e d i n p r a c ti c e s i n ce v a p o r i z a t i o n o f t h e s o l v e n tw i ll o c cu r . T h e r e s u lt s p r e s e n t e d h e r e a r e j u s t m e a n tt o i l l u s t r a t e t h e p o s s i b l e e ff e ct o f s m a l l c h a n g e s o f t h em a s s t r a n s f e r c o e f f i c i e n t . H o w e v e r , i t s h o u l d b e m e n -t i o n e d t h a t v a p o r i z a t i o n o f t h e li q u i d c a n b e t a k e ni n t o a c c o u n t i n o u r t r a n s f e r m o d e l .

I n c a s e o f t h e s e c o n d - o r d e r c h e m i c a l r e a c t i o n w i t hh i g h s o l ub i l it y o f A a n d l o w m a x i m u m e n h a n c e m e n tf a c t o r h i g h a b s o r p t i o n r a t e s r e s u l t a n d c o n s e q u e n t l yh e a t e f fe c ts p l a y a n i m p o r t a n t r o l e [ s e e F i g . 3 ( b )] . I tc a n b e s e e n t h a t t h e v a l u e o f Le i n f l u en c e s t h e m a x -i m u m a b s o r p t i o n r a t e . I n c a s e o f Le = 1 0 c o m p o -n e n t s A a n d B i n t h e s o l v e n t w i ll e x p e r i en c e t h e s a m et e m p e r a t u r e , w h e r e a s i n c a s e o f Le = 1 A wi l l expe r i -e n c e a h i g h e r t e m p e r a t u r e t h a n B , r e s u l t i n g i n a d if f e r-e n t m a x i m u m e n h a n c e m e n t fa c to r . W h e n u s i n g t h ea n a l y t i c a l e x p r e s s i o n s e r r o r s u p t o 1 5 % c a n b e e x -p e c t e d .Temperature-dependent chemical reaction rateconstant

I n t h i s c a s e Eo a n d Es w e r e s e t e q u a l t o z e r o ,w h e r e a s E a l w a s g i v e n a v a l u e o f 2 5 , 0 0 0 J / t o o l . F r o m


8/11

1668S O

E

1 0

11 0 " 3 1 0 2 1 0 " 1 1 0 0 1 0 1

1 0 0E

1 0

11 0 3 1 0 - 2 1 0 - 1

H a

1 0 0 1 0 1 0 2H a2 A + B = C [ isot I id iffe rentKij [ - -- -- l .~ ' : I0 t ~i

X A i o 0 3 [ . . . . . . . . . . . ~ ' : 1 ia n a l y t i c a l ] i

i ~ 7 i

o.0 i1 0 3 1 0 2 1 0 - I l 0 l 0 1 1 0

H a

5 I [A + B = C [

x . b = 6 1 / / ~ 4 . . . .

I 0 - 2 1 0 - 1 1 0 0 1 0 1 1@H aFig. 5 . Dim ens ionless absorp t ion flux 8 of com pone nt A ob-ta ined from the num erica l model and the analyt ica l expres-s ions as a funct ion of the Hat ta num ber in case the chemicalreac t ion ra te con s tant is tempera ture dependent : (a) second-orde r ebemical reaction , different b ina ry mass transfer coeffi-cients and second composition regime; (b) first-order chem-ical reaction, different binary mass transfer coefficients andthird composition regime; (el second-order chemical reac-tion, different binary mass transfer coefficients and thirdcomposi t ion regime; (d) second-order chemical reac t ion,equal b inary mass t ransfer coeff ic ients and th ird composi-t ion regime. Par ame ter va lues are given in Table 1 .

M. J . W. FRANKet al.e q . ( 9 ) i t c a n b e c o n c l u d e d t h a t t h e c h e m i c a l r e a c t i o nr a t e c o n s t a n t i n c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e .A s a c o n s e q u e n c e , i n c a se o f a n e x o t h e r m a l c h e m i c a lr e a c t i o n , t h e h e a t p r o d u c t i o n r a t e d u e t o c h e m i c a lc o n v e r s i o n e n h a n c e s t h e c h e m i c a l r e a c t i o n r a t e , re -s u i t i n g i n d i f f e r e n t , o f t e n h i g h e r , m a s s t r a n s f e r r a t e s .T h e r e s u lt s o f th e s i m u l a t i o n s a r e p r e s e n t e d i nF i g s 5 ( a ) - ( d ) . I n o r d e r t o e n a b l e p r o p e r c o m p a r i s o n ,t h e a b s o r p t i o n r a t e s h a v e b e e n p l o t t e d a s a f u n c t i o n o fH a e v a l u a t e d a t l i q u i d b u l k t e m p e r a t u r e .

F r o m F i g . 5 (a ), w h i c h s h o w s t h e r e s u l t s o f t h e1 0 2 s e c o n d - o r d e r c h e m i c a l r e a c t i o n i n t h e s e c o n d c o m -p o s i t i o n r e g i m e , i t c a n b e c o n c l u d e d t h a t f o r (p s e u d o )f i r s t- o r d e r c h e m i c a l r e a c t i o n w i t h l o w s o l u b i l i t y o f A ,a l a r g e i n c r e a s e i n a b s o r p t i o n f l u x o c c u r s a t h i g hv a l u e s o f H a d u e t o h e a t g e n e r a t i o n . I f t h e h e a t e f f e ct sa r e n e g l e c t e d l a r g e e r r o r s w i l l r e s u l t ( i. e. 5 0 a n dm o r e ) . T h e v a l u e o f L e h a s n o s i g n i f i c a n t ef f ec t o n t h em a s s t r a n s f e r r a t e s a n d t h e a p p r o x i m a t e a n a l y t i c a ls o l u t i o n p r e d i c t s t h e n u m e r i c a l l y c a l c u l a t e d m o l a rf l u x o f A w i t h i n 1 5 .

I n c a s e o f h i g h s o l u b i l i t y o f A [ s e e F i g . 5 ( b )] t h ea n a l y t i c a l re s u l t s s h o w a s u d d e n i n c r e a s e i n 8 a tH a , ~ 0 .1 . D u e t o n u m e r i c a l p r o b l e m s t h e c o m p u t a -t i o n s c o u l d u n f o r t u n a t e l y n o t b e p e r f o r m e d f o r h i g h e rH a v a l u e s . T h e e x p l a n a t i o n f o r t h i s fa c t f o l l o w s f r o mF i g . 6 w h i c h s h o w s t h e h e a t p r o d u c t i o n r a t e a n d h e a tw i t h d r a w a l r a t e a s a f u n c t i o n o f th e i n t e r f ac e t e m p e r -a t u r e T i f o r d i f f e r e n t ko ~ v a l u e s i n c a se o f e q u a l b i n a r ym a s s t r a n s f e r c o e f f ic i e n ts . F o r l o w k o l v a l u e s , t h e h e a tb a l a n c e p o s s e ss e s a l o w - t e m p e r a t u r e s o l u t i o n , w h e r e-a s f o r h i g h k o l v a l u e s ( > 3 . 1 0 S m o l / m 3 / s ) o n l ya h i g h - t e m p e r a t u r e s o l u t i o n e x i s ts ( th i s l a tt e r s o l u t i o ni s n o t s h o w n ) . T h e c u r v e a l s o p r o v i d e s t h e c l u e t o w h yt h e n u m e r i c a l p r o b l e m s e m e r g e . T h e m o d e l w i l l t r y t og e n e r a t e a s o l u t i o n i n t h e l o w - t e m p e r a t u r e r a n g e ,w h e r e t h e h e a t p r o d u c t i o n c u r v e i s a t m i n i m u m d i s -t a n c e t o t h e h e a t w i t h d r a w a l c u r v e . F u r t h e r i t i s s e e nt h a t f o r ko l = 1 .1 0 s m o l / m 3 / s m u l t i p l e s o l u t i o n s e x i st ,o f w h i c h t h e t w o s o l u t i o n s w i t h t h e h i g h e s t t e m p e r -a t u r e a r e n o n - e x i s t e n t i n p r a c ti c e s i n c e v a p o r i z a t i o n

0 . 5 / AI W R , k o z = 5 . 1 0 t = 3 1 0 sI I P R 0 . 4 - - H P / / k . = I . I 0 sl 0 . 3

0 . 2

0 . 1

0 i I I 1 I2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

~ - T I I K ]Fig. 6 . Heat w i thdrawal ra te (HWR) a nd heat prod uct ionra te (HPR) as a funct ion of the in terface tempera ture T~calculate d from the anal ytical expressions in case of first-order chemical reaction, equal binary mass transfer coeffi-cients and th ird co mp ositio n regime for different kol values.Param eter va lues are given in Table 1 .


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Modelling of simultaneous mass and heat transfer--IIof the solven t will occur, which has not been ac- 100counted for in the present calculations.

EFigures 5(c) and (d) show the results for the sec-ond-order chemical reaction in case of a high solubi-lity of A for, respectively, different and equal binarymass transfer coefficients. Fr om these figures it can be loconcluded that enhancement of mass transfer due toheat production occurs at lower H a values in com-parison to the isothermal case, but the differencesdimini sh at high H a values. In case of different binary lmass transfer coefficients the relative deviation be- 10.2tween the results for L e = 10 and the results forL e = 1 is low, due to low maximum enhancementfactors, whereas in case of equal binary mass transfer 10coefficients, this deviation is high for intermediate H avalues. The latter is due to the steepness of bothcurves.

lOo 1 2H a

1669

A l l t h r e e p a r a m e t e r s a r e t e m p e r a t u r e d e p e n d e n tIn this case all act ivation energies were given non-zero va lues (see Table 1). This will resul t in the follow-

ing three competing effects in case of absorp tion withassociated heat production compared to the isother-mal case: decrease in solubility with increasing tem-perature on one hand and increase in both binarymass transfer coefficients and chemical reaction rateconstant with increasing temperature on the otherhand.

For a second-order chemical reaction with lowsolubility and high maximum enhan cement factor theresults are shown in Fig. 7(a). This figure also repre-sents the results of a first-order chemical reaction withlow solubility. Fro m Fig. 7(a) it can be seen that theabsor ptio n rates for the non- isothe rmal case withL e = 10 coincide with the absorption rates for iso-thermal absorption (differences are smaller than10 ). It can be concluded that the increase in binarymass transfer coefficients a nd chemical reaction rateconstant compensates for the decrease in solubility.The difference between the results for L e = 1 andL e = 10 is smaller than 5 , whereas the approximateanalytical expression predicts the numerically cal-culated absorptio n rates within 10 accuracy forL e = 10. This error may be due to the increase of thefraction of compon ent C in the liquid film or the use ofthe approximate expression for N6.

In case of high solubility of A [see Fig. 7(b)] thethree heat effects do not compensate each other andconsequently the overall thermal effect is more pro-nounced: from Figs 2(b) , 3(b) and 5(c) it can be con-cluded that for L e = 10 and low Hatta number, theheat production changes the absorption rate, respec-tively, with -6 0 (temperature-dependent solubility),+ 3000 (tempera ture-dep endent mass transfer co-

efficients) and 0 (temperature-depende nt chemicalreaction rate constant). Since the overall effect corres-ponds to a decrease in absorption flux of about 50 ,it can be concluded tha t the individual effects cannotsimply be added to obtain the overall effect.

For the second-order chemical reaction it can beseen that the inst anta neou s regime has not been

0.310 2 10 1 100 101 1 2H a

Fig. 7. Dimensionless absorption flux 6~ of component Aobtained from the numerical model and the analytical ex-pressions as a function of the Hatta number in case thesolubility, he binary mass transfer coefficientsand the chem-ical reaction rate constant are temperature dependent: (a)second-order chemical reaction, different binary mass trans-fer coefficientsand second composition regime; b) first-orderand second-order chemical reaction, different binary masstransfer coefficients and third composition regime. Para-meter values are given in Table 1.

reached yet due to heat effects and addition ally thatthe abso rption fluxes almost equal those of the first-order chemical reaction.

For both reaction schemes the infuence of Le' valueon the absor ptio n rates is limited to 20 whereas theanalytical expression predicts the numerically cal-culated results within 10 .

4. DISCUSSIONAND CONCLUSIONSFro m the results presented in this paper, it is clear

that heat effects may play an important role in masstransfer processes with chemical reaction. Heat effectsare especially important in those cases where the masstransfer rates are high, due to a high solubility of theabsorbed gas or due to a high chemical reaction rate.Evidently the value of the mass transfer rate in a situ-ation in which heat effects should be taken into ac-count will depend on the values of the abso rption heatand chemical react ion heat as well as the values of thethree, in this paper mentioned, activation energies.Numerical simulations show that, when compared tothe corresponding isothermal case, the thermal effectscan affect the mass transfer rates by as much asa factor of 30. Due to heat production, three compet-ing effects exist: increase in mass tranfer rate due to


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1670 M .J . W . FRA NKet a l .i n c r e a s e i n b i n a r y m a s s t r a n s f e r c o e f f i ci e n t s a n d i n - gc r e a s e i n c h e m i c a l r e a c t i o n r a t e a n d d e c r e a s e in m a s st r a n s fe r ra t e d u e t o d e c r e a se i n so l u b i l i t y o f t h e a b - E As o r b e d g a s . I t m a y b e p o s s i b l e t h a t t h e s e t h r e e e f f e ct s E a lc o m p e n s a t e e a c h o t h e r r e s u l t i n g i n a z e r o c h a n g e i n E nm a s s t r a n s fe r r a t e i n c o m p a r i s o n t o t h e i s o t h e r m a l E scase. hi

T h e m a s s t r a n s fe r r a t e u n d e r n o n - i s o t h e r m a l c o n d i - H it i o n s w i t h L e = 1 0 c a n b e e s t i m a t e d v e r y w e l l f r o mt h e a p p r o x i m a t e a n a l y t i c a l e x p r e s s io n s , p r e s e n t e d i n A H Rt h i s p a p e r . I n a ll c a s e s , w h i c h h a v e b e e n s t u d i e d , t h e A H sm a x i m u m d i ff er en c e a m o u n t s t o 2 5 , b u t in m o s t H ac a s e s i s s m a l l e r t h a n 1 0 . S o t h e a n a l y t i c a l s o l u t i o n k lp r e s e n t e d i n t h i s p a p e r i s v e r y u s e fu l t o e s t i m a t e t h ei n f l u e n c e o f t h e h e a t e f fe c t s o n t h e m a s s t r a n s f e r r a t e s , k o la s s u m i n g L e = o o . I n m o s t c a s e s t h i s y i e l d s a l s o K i ia r e a s o n a b l e e s t i m a t e o f t h e m a s s t r a n s fe r r a t e i n c a s e L eL e = 1 . m i

I n c a s e o f a s e c o n d - o r d e r c h e m i c a l r e a c t i o n it w a s N ~s h o w n t h a t t h e r m a l e ff ec ts m a y c h a n g e t h e m a x i m u m N i ,oe n h a n c e m e n t f a c t o r a n d c o n s e q u e n t l y s h i f t t h e a b -s o r p t i o n f ro m t h e i n s t a n t a n e o u s r e g i m e t o th e N i . 6p s e u d o - f i r s t - o r d e r r e g i m e .

F r o m F i g s 4 a n d 6 i t f o l lo w e d t h a t m u l t i p l e s o l u - R g a ,t i o n s o f t h e a n a l y t i c a l e x p r e s s i o n s [ e q s ( 1 1 ) -( 2 1 ) ] a r e Rp o s s i b l e . T h e m a x i m u m n u m b e r o f s o l u t i o n s i s t h re e , To f w h i c h t w o w i ll b e s t a b l e a n d t h e in t e r m e d i a t e o n e T iu n s t a b l e . F u r t h e r i t w a s s h o w n t h a t a m i n o r c h a n g e o f x it h e e f f e c ti v e m a s s t r a n s f e r c o e f fi c i e n t o f t h e a b s o r b e ds p e c ie s i n t h e l iq u i d m i x t u r e o r o f t h e c h e m i c a l re a c - x ,t i o n r a t e c o n s t a n t , c a n c a u s e a s i g n i f i c an t c h a n g e i nm a s s t r an s f e r r a t e s. A lo w - t e m p e r a t u r e s o l u t i o n c a n x~ac h a n g e i n t o a h i g h - t e m p e r a t u r e s o l u t i o n , r e s u l ti n g inc o m p l e t e l y d i f f e r e n t m a s s t r a n s f e r r a t e s . A t t h e s e Y ih i g h - t e m p e r a t u re s e v a p o r a t i o n o f th e s o l v en t m i x t u r eb e c o m e s v e ry i m p o r t a n t . A s t h i s h a s n o t b e e n t a k e n zi n t o a c c o u n t i n o u r c a l c u l a t i o n s , t h e p r e s e n t e d h i g h -t e m p e r a t u r e s o l u t i o n d o e s n o t c o r r e s p o n d t o r e a l i t y .W h e n h e a t e f f e c t s a r e t a k e n i n t o a c c o u n t e v a p o r a -t i o n o f t h e l i q u i d m i x t u r e s h o u l d a l s o b e t a k e n i n t oa c c o u n t , e s p e c i a l ly w h e n t h e c a l c u l a t e d i n t e r f a ce t e r n - 6hp e r a t u r e i s n e a r t h e b o i l i n g t e m p e r a t u r e o f t h e l i q u i d . ~F r o m t h e a b o v e i t i s c o n c l u d e d t h a t t h e r e m a y e x i s t qn o n - i s o t h e r m a l g a s - l i q u i d a b s o r p t i o n s y s te m s w h e r e 2m i n o r c h a n g e s i n p a r a m e t e r s a p p e a r i n g i n t h e h e a t v lb a l a n c e , e .g . b i n a r y m a s s t r a n s f e r c o e f f i c ie n t s , c h e m -i c a l r e a c t i o n r a t e c o n s t a n t , L e n u m b e r o r h e a t t r a n s -f e r c o e f f i c i e n t s , m a y c h a n g e t h e s y s t e m b e h a v i o u rd r a s t i c a ll y ; I t w o u l d b e v e r y i n t e r e s t in g t o s h o w t h i sp h e n o m e n o n e x p e r i m e n t a ll y .A c k n o w l e d o e m e n t s - - T h e s e investigations were supp or ted b ythe Foundation for Chemical Research in the Nether lands(S.O.N.). W e also acknowledge P. Schouren for his part inthe theoretical work.

r a t io o f a b s o r p t i o n f lu x a n d c o r r e s p o n d i n gi s o t h e r m a l p h y s i c a l a b s o r p t i o n f l u xe n h a n c e m e n t f a c t o ra c t i v a t i o n e n e r g y f o r w a r d r e a c t i o n , J / m o la c t i v a t i o n e n e r g y o f d i ff u si o n , J /m o la c t i v a t i o n e n e r g y o f d i s s o l u t io n , J / m o lh e a t t r a n s f e r c o e f f i c ie n t , ~ / 6 h , J / K sp a r t i a l m o l a r e n t h a l p y o f c o m p o n e n t i ,J / m o lh e a t o f re a c t i o n , J / m o lh e a t o f d i s s o l u t io n , J / t o o lH a t t a n u m b e r a s d e f i n e d i n e q s ( 1 5 ) o r ( 1 8 )t e m p e r a t u r e - d e p e n d e n t f o r w a r d r e a c t i o nr a t e c o n s t a n t , m o l / m a sf o r w a r d r e a c t i o n r a t e c o n s t a n t , m o l / m 3b i n a r y m a s s t r a n s f e r c o e f f i c i e n t , D i j / ~ m , m / sm o d i f i e d L e w i s n u m b e r , 6h.I/6,, .ts o l u b il i ty o f c o m p o n e n t i , x i / y im o l a r f l u x o f c o m p o n e n t i , m o l / m 2 sm o l a r f lu x o f c o m p o n e n t i th r o u g hg a s - l i q u i d i n t er f ac e , m o l / m 2 sm o l a r f l u x o f c o m p o n e n t i t h r o u g h m a s st r a n s f e r f i l m - b u l k i n t er fa c e , m o l / m 2 sg a s c o n s t a n t , J / m o l Kr e a c t i o n r a t e, m o l / m 3 st e m p e r a t u r e , Ki n t e r f a c e t e m p e r a t u r e , Km o l a r f r a c t i o n ( i n li q u i d p h a s e ) o f c o m p o -n e n t im o l a r f r a c t i o n ( i n l i q u i d p h a s e ) o f c o m p o -n e n t i a t i n t e r f a c em o l a r f r a c t i o n ( in l i q u i d p h a s e ) o f c o m p o -n e n t i i n b u l km o l a r f r a c t io n i n g a s / v a p o u r p h a s e o f c o m -p o n e n t id i s t a n c e i n f i l m , m

G r e e k l e t t e r st h e r m a l d i f f u s i o n c o e f f ic i e n t , 2 / p . C p , m 2 / st h i c k n e s s o f h e a t t r a n s f e r f i l m , mt h i c k n e s s o f m a s s t r a n s f e r f i lm , md i s t a n c e i n f i l m , Z / 6 mc o n d u c t i v i t y , J / m K ss t o i c h i o m e t r i c c o e f f i c i e n t o f c o m p o n e n ti ( r e a c t a n t n e g a t i v e , p r o d u c t p o s i t iv e )

S u b s c r i p t sA c o m p o n e n t Ai c o m p o n e n t ii in te r f acel l i q u i d p h a s e0 r e f e re n c e s t a t e

C TCpid i jD i j

NOTATIONt o t a l c o n c e n t r a ti o n , m o l / m 3h e a t c a p a c i t y o f c o m p o n e n t i , J / t o o l KD A s / D 0b i n a r y d i f f u s i o n c o e f f i c ie n t , m 2 / s

REFERENCESAllan, J. C. and M ann, R., 1979, Reactive exothermic g asabsor p t ion - im pr oved ana lyt i cal predictions f rom a hyp erb o l i c solubil i ty approximation. Chem. Enong Sc i . 34 ,413-415.


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absorpt ion. Can. J. Chem. Engng 65, 454-461.Evans , J .D. and Sel im, M.S., 1990, Pen et ra t ion theory analy-s i s for nonisothermal gas absorpt ion accompanied bya sec ond-ord er che mical reac tion. Chem. Engng Commun.90, 103-124.Frank, M.J.W., Kuipers, J .A.M., Versteeg, G.F. and vanSwaai j , W.P.M. , 1995, Model l ing of s imul taneous massand heat t ransfer wi th chemical reac t ion us ing the Max-wel l -Stefan theory I . Mo del develop men t and i sother -m a l s t udy Chem. Engng Sci. 50 1645-1659.Man n, R . and M oyes , H., 1977, Exo thermic gas absorpt ionwi th chem ical reac t ion. A. I. Ch.E. J. 23, 17-23.Shah, Y.T., 1972, Ga s-l iq uid interface tem pera ture r ise in thecase of t empera ture-d epende nt phys ica l, t ransp or t andreact ion properties. Chem. Engng Sci. 27, 1469-1474.Westerterp, K.R., van Swaaij , W.P.M. and Beenackers,A.A.C.M., 1990, Chemical Reactor Design and Operation.Wiley, Chicester.White, Jr . , D. and John s, L.E., 1986, Th e diffusional l imita-t ion on the t emp era ture r ise in the absorpt ion of chemic-al ly react ive gases. Chem. Engng Commun. 48 177-190.

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