Mass Transfer in Bubble Columns a Comparison of Correlations

8/12/2019 Mass Transfer in Bubble Columns a Comparison of Correlations

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P , gamo. 0043-1354(94)00253-3Wat. Res.Vol. 29, No. 4, pp. 1129-1138, 1995Copyright 1995 Elsevier ScienceLtdPrinted in Great Britain. All rights reserved0043-1354/95 $9.50 + 0.00

M A S S T R A N S F E R I N B U B B L E C O L U M N S AC O M P A R I S O N O F C O R R E L A T I O N S

J . DUDLEY@WRc, Frankland Road, Swindon SN5 8YF, England

First received December 1993; accepted in revised orm September 1994)Abstract--Several correlations were evaluated for predicting the gas hold-up and oxygen mass transfercoefficient in bubble columns. A recent hold-up correlation was greatly superior to the others used in thisstudy. Only one of the mass transfer coefficient correlations included the effect of the sparger area inrelationship to the total surface area, yet it was found that this relationship did have a significant effecton the measured mass trans fer coefficients. Most of the mass -transfer correlations therefore required acorrection factor. As with gas hold-up it was a recent correlation, based in part on theoretical principles,that gave the best results. This study also developed a new correlation for the depression to be expectedfrom detergents on the mass transfer coefficient.Key words--bubble column, hold-up, mass-transfer, alpha-factor, Kea

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

a = gas-liquid interracial area (m2/m3)B = bubble column diameter (m)B0 = Bond number = pLgd~/~D L = diffusivity of gas in liquid (m2/s)d b = Sauter mean bubble size diamete r (m)F = diffuser diameter (m)Fr = Froude number = V6/ ~b)g = acceleration due to gravity (m/s2)H = liquid depth (m)KL = specific mass transfer coefficient (m/s)KLa = mass transfer coefficient (l/s)P~ = pressure at diffuser orifice (Pa)Pc = gassed power input (W)P~ = pressure at liquid surface (Pa)QG = gas flowrate at standard temperature and pressure(m3/s)SAA = surface active agent concentration as Manoxol OTequivalent (kg/m 3)Sc = Schmidt number = #L/PLDL)T = temperature (C)Vb = bubble free rise velocity (m/s)V6 = superficial gas velocity (m/s)lie = [in equation (1)] liquid volume (m3)V = superficial liquid velocity (m/s)

Greek symbols= ratio of mass transfer coefficient in liquid to that intap waterEG = gas hold-up, dimensionless fractioni' = surface active agent concentratio n on bubble surface

/~L = l iquid viscosity (Pa s)PG = gas densi ty (kg/m 3)PL = liquid density (kg /m 3)a = gas-liquid surface tension (N/m)

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

When designing aeration systems it is frequentlynecessary to be able to calculate the mass transfer

coefficient, KLa. Many correlations are available forthese calculations, but with no guidance as to whichcorrelation sho uld be used. The purpose o f this studywas to evaluate a range of such correlations todetermine the preferred one. Many of these corre-lations required the calculation of gas bubble diam-eters and gas hold-up. These were again calculatedusing correlations, and the study was thereforewidened to include an evaluation of gas hold-upcorrelations. All these correlations assume tap water(or comparabl e clean fluids ), yet oxygen masstransfer is rarely required under these conditions.As an example, in the activated sludge processthere are frequently traces of detergents. While it isknown that these detergents will depress the masstransfer coefficient, there has been no met hod torelate detergent concentration to the expected de-pression. This study theref ore developed such a corre-lation.

Many activated sludge models require the use ofKLa values. Fo r man y designers converting the aera-tion requirements predicted by the model, as a rangeof Kta values, is problematic. The correlations in thiswork allow this problem to be tackled. They alsoallow the effect of detergents in sewage on KLa to beestimated. The results of this study are inte nded to beincluded in the activated sludge model in WRc'ssewage works model STOAT.

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

The correlations used in this evaluation for gashold-up are given in Table l, and for mass transferin Table 2. These correlations requir ed the bubble size

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KL corre lations 1131

O.

~Dr 40t

O

XIO-10 . 5 0

0 . 40 . ~ j ~ . ~ . ~/X A

0 . 3 0 ~ ~ ~4a ~ /

0 . 2 0 .

0 . I 0 .

0 . 0 0 ,0 00 0 . 1 0 0 40. 2 0 0 . 3 0 X I 0 - IE x p e n i m e n t a l h o l d u p

0 , 5 0

Fig. 2. Hold-u p: experimental da ta vs the K oide correlation.d i a m e t e r , w h i c h w a s c a l c u l a t e d u s in g t h e C a i d e r b a n kc o r r e l a t i o n (J o s h i a n d S h a r m a , 1 97 9)

0.0 .6a ~ = 4 .1 5 - ~ + 9 . l O - . (1 )i , , , IT h i s c o r r e l a t i o n w a s d e v e l o p e d f o r a 6 - b l a d e d d i s k

t u r b i n e i n a b a f fl e d t a n k , b u t h a s b e e n r e c o m m e n d e d

b y J o s h i a n d S h a r m a ( 1 97 9 ) f o r d if f u s e d a i r s y s t e m s .O n e o t h e r b u b b l e s i z e c o r r e l a t i o n w a s s t u d i e d , d e v e l -o p e d b y K e i l a n d R u s s e l l (1 9 8 7) f o r a c t i v a t e d s l u d g e -t y p e s y s te m s . K e i l ' s c o r r e l a ti o n w a s f o u n d t o p r e d i c tm u c h l a r g e r b u b b l e s iz e s t h a n w e r e o b s e r v e d i n t h i sw o r k . T h e b u b b l e s iz e s p r e d i c t e d b y C a l d e r b a n k ' sc o r r e l a t i o n d i d m a t c h v i s u al e s t im a t e s o f t h e b u b b l es i zes i n t h i s wo rk .

X10-10 . 6 0

0 . 5 0 .

0 . 40 .C ~m

o 0 . 3 04-, I4.a

2 oJ

0 . 1 0 J

0 . 0 0, 0 0

z x A Z~/x tx &

z z

0 . I 0 0 . 20 0 . 30 0 . 4 0 0 . 5 0 0 6 0 1 0 - IExperimenta holdupFig. 3. Ho ld-up : experimental da ta vs the Lockett correlation .


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1132 J. Du dley

n

OC-

C-CJ

cUO3

X10-10 . 5 0

0 ,~0.

0 . 3 0 .

0 , 2 0 .

0 . 1 0 .

0 , 0 00 . 0 0

,x A

0 , 10 0 .20 0 .30 0 .40 0 .5 0X10-1E x p e p i m e n t a ] h o l d u pF i g 4 Hold-up: experimental data vs the Schuger I correlation.

T h e g a s p o w e r i n p u t i s g i v e n b yP c = Q c ' P , I n P .P

T he gas - l iqu id in te r fac ia l a rea by6 . E 6O= du

a n d t h e f r e e -r i se v e l o c i t y o f a b u b b l e b y ( W i n k l e r,1981)

( 2 ) [ 2 . t rVb = 4 P - - ~ b + 0 . 5 g db . (4 )M a n y o f t h e se c o r r e l a t i o n s w e r e t a k e n f r o m s ec -( 3) o n d a r y s o u r c e s r e c o m m e n d i n g t h e i r g e n e r a l a p p l i -

c a t i o n , p r i n c i p a l ly b e c a u s e o f t h e t h e o r e ti c a l s u p p o r t

cx13Oc-

Oc_02L

3

X I 0 - 10 . 5 0

Z ~ ~ z~A0 40

0 .3 0 _ J ~

0 , 2 0 ~

0 . 10_A A

0 . 0 00 O0 0 .10 0 .20 0 .30 0 .40 0X10-1E x p e r i m e n t a l h o l d u p

F i g 5 Hold-up: experimental data vs the Kulkarni correlation.

5 0


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Lacorrelations 1133

to

oC'13r

1 2 0

1 0 0 .

8 0

6 0

4 0

2 0 .

Z~

o ~ o 4 0 ~ o ~ 0 0E x p e r i m e n t a l K l a

Fig. 6. La cor re la t ion : K hud enk o vs exper imenta l.

2 0

b e h i n d t h e c o r r e l a t i o n s . K h u d e n k o a n d S h p i r t ( 1 9 8 6 )d e v e l o p e d t h e i r c o r r e l a t i o n p u r e l y f r o m n o n - d i m e n -s i o n a l a n a l y s i s , a n d f i t t e d p a r a m e t e r s t o t h e i r m o d e lf r o m s m a l l - s c a l e e x p e r i m e n t s ; a s w i t h t h e o t h e r c o r r e -l a t i o n s t h e r e w e r e n o e f f e c t i v e l i m i t s p l a c e d o n t h er e s u l t i n g a p p l i c a t i o n . B e c a u s e o f t h i s a l l t h ec o r r e l a t i o n s h a v e b e e n t e s te d a s i f t h e y w o u l d b ea p p l i c a b l e o v e r t h e r a n g e o f c o n d i t i o n s d e v e l o p e dw i t h i n t h e t e s t e q u i p m e n t . A l l t h e K L a n d K L a c o r r e -

l a t i o n s w e re d e v e l o p e d a r o u n d o x y g e n i n c l e a n w a t e r ,w i t h n o m e n t i o n o f i o n ic effe ct s. K h u d e n k o a n dS h p i r t ' s c o r r e l a t i o n i s t h e e x c e p t i o n h e r e , w h e r e t h ea l p h a - f a c t o r i s e x p l i c i t l y i n c l u d e d i n t h e c o r r e l a t i o n .

E X P E R IM E N T L P R O C E D U R E

A colum n 200 mm i .d . by 4 m dep th was used to r epresen ta ver t ica l s l ice of an ac t iva ted s ludge tank . Th e co lu mn was

120

I00

8 0

tox 6 0

4-~

2 0 .

00

t,

/ -~o o so ~o ioo

E x p e r i m e n t a l K l aFig. 7. La cor re la t ion : Oz turk vs exper imenta l da ta .

2 0


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1 1 34 J . D u d l e y

ror - Iv

-,,-,I

gO807060.5040302010_

0 0

A

i o ~ o ~ o 4 0 ~ o ~ o o ~ o 9 0Exper imen ta l K la

Fig . 8 . K L a c o r r e l a t io n s : H i g b i e v s e x p e r i m e n t a l .

c h o s e n t o h a v e a d e p t h c o m p a r a b l e t o a c t i v a t e d s l u d g et a n k s a n d t o r e p r e s e n t t h e p i t c h a r o u n d a s i n g l ed i f f u s e r . T h i s c o l u m n s i z e h a s b e e n u s e d p r e v i o u s l y( D o y l e et aL 1 98 3) t o m i n i m i s e s c a l e e f f e ct s o n t h e m e a s u r e dh o l d - u p . T w o d i f fu s e r s w e r e u s e d , o n e I 0 0 m m d i a m e t e r ,t h e o t h e r 5 0 m m . L i q u i d l e v e ls in t h e c o l u m n c o u l d b ev a r i e d a n d l i q u i d d e p t h s o f 1 .5 , 2 .5 a n d 3 .5 m w e r e u s e df o r t h e t ri a ls . F o r t h e h o l d - u p e x p e r i m e n t s w o r k i n gf l u id s w e r e t a p w a t e r , s e t tl e d s e w a g e a n d m i x e d l i q u o rf r o m a b a t c h a c t i v a t e d s l u d g e t a n k . T h e m a s s t r a n s f e re x p e r i m e n t s u s e d t a p w a t e r a n d t a p w a t e r s p i k e d w i t hd e t e r g e n t .

F o r t h e h o l d - u p t e s t s t h e c o l u m n w a s f i l l e d t o 1 . 5 , 2 . 5 o r3 .5 m a n d a e r a t i o n s t a r t e d , r e c o r d i n g t h e g a s f lo w . A t a p w a so p e n e d a t t h e l i q u i d l ev e l, a n d a e r a t i o n f o r c e d e x c e s s l i q u ido u t o f t h e c o l u m n . T h i s l i q u i d w a s c o ll e c t e d a n d t h e v o l u m em e a s u r e d , f r o m w h i c h t h e a i r v o l u m e a n d t h e h o l d - u p c o u l db e c a l c u l a te d . T h e g a s f l o w r a t e s u s e d w e r e c h o s e n t o m a i n -t a i n t h e s a m e d i f f u s e r l o a d i n g s a s w o u l d b e f o u n d w i t h t h e s ed i f f u s e r s i n a n a c t i v a t e d s l u d g e p l a n t . T h e a i r f l o w s u s e dw e r e t h e r e f o r e 2 - 5 l / m i n w i t h t h e 5 0 m m d i f fu s e r a n d5 - 2 0 1 / m i n f o r t h e 1 0 0 m m d i f fu s e r. F l o w r a t e w a s m e a s u r e di n a r o t a m e t e r , a n d c o r r e c t i o n s t o t h e r o t a m e t e r r e a d i n gw e r e m a d e f o r t h e g a s p r e s s u r e .

1

80_

m 60_r - ivvt-,rD

40_O313I 'D

2 0 .

0

A A&

5o o 6o @oExper imen ta l K la

Fig . 9 . La c o r r e l a t io n : C a l d e r b a n k v s e x p e r i m e n t a l .

0 0


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L acorrelations 1135The mass t r ans fe r exper iments used the same appara tus .Water was de-oxygena ted by s t r ipp ing with n i t rogen , witha rap id switchover a f te r de -oxy gena t ion f rom the n i t rogens t r ea m to a n a i r s t re a m. T h i s i s th e p r o c e d u r e r e c o mme n d e dby Spr ie t and Botte rman (1984) .De-ox ygena t ion cont inue d unt i l the d isso lved oxygenleve l was in the range 0 .05~) .5 mg/1 . Then a s tep changein th e g a s o x y g e n c o n c e n t r a t io n wa s m a d e a s a i r wa sinjected. T he dissolved oxy gen levels were me asured a tin te rva ls o f 30 s in i t ia l ly , inc reas ing as the ra te of inc reaseof d is so lved oxygen conc entra t io n decreased . T his isth e p r o c e d u r e r e c o mme n d e d b y Br o wn ( 1 9 7 9 ) . W h e nsubs tan t ia l ly cons tan t d is so lved oxygen leve ls werereached the co lumn was aga in s t r ipped of oxygen us ingni t rogen , and another exper iment begun a t a d if fe ren tflowrate.Cor rec t ion s to the equ i l ib r ium d issolved oxygen concen-tra t ions for dep th were made , us ing the approach of Lis te rand Boon (1973) . The dynamics of the d is so lved-oxygenprobe , a pHOX probe , was mode l led as a f i r s t -order lag(D un n and Einse le, 1975) . Other approaches a re to use asecond-order lag (Shioya and D un n, 1979) or a de tai ledrepresen ta t ion of the chemical r eac t ions tak ing p lace with inthe probe 1 (Linek et al. 1987). The mea sured mass t r ans fe rcoeff icients were corrected to 20C usin g the wa ter- ind ustrys t a n d a r d a p p r o a c h ( Br o wn , 1 9 79 ) o f

KLaT = KLa20C' 1.024r-m- (5)The ca lcu la t ion procedure used with the cor re la t ions isdescr ibed in A ppend ix A .

COMPARISONS OF CORRELATIONST h e c o r r e l a t i o n s w er e c o m p a r e d b y p l o t t i n g

t h e m e a s u r e d a n d p r e d i c t e d g a s h o l d - u p a n do x y g e n m a s s t r a n s f e r c o e f f i c i e n t s a g a i n s t e a c ho t h e r . T h e s e g r a p h s a r e g i v e n a s F i g s 1 - 1 0 . T h ef o l l o w i n g d i s c u s s i o n o f c o r r e l a t io n s i s b a s e d o n t h es y s t e m :

C o r r e l a t i o n f o r d pC o r r e l a t i o n f o r V b

C o r r e l a t i o n f o r EGC o r r e l a t i o n f o r K t a .

T h e r e f o r e t h e d i s c u s s i o n i s n o t o f t h e a d e q u a c y o f t h ec o r r e la t i o n s w h e n p r o v i d e d w i t h d a t a o n g a s h o l d u po r b u b b l e d i a m e t e r, b u t w i t h t h e e n s e m b l e o fc o r r e l a t i o n s f o r d e s i g n u s e .G a s h o l d - u p

V i s u a l e x a m i n a t i o n re a d i l y s h o w s t h a t t h eK u i k a r n i c o r r e l a t i o n w a s a b e t t e r f it t o t h e d a t a t h a nt h e c o m p e t i n g h o l d - u p c o r r e l a t i o n s . T h e s i m p l e J o s h ic o r r e l a t i o n p r o v i d e s a g o o d f i r s t a p p r o x i m a t i o n , a n dd o e s n o t n e e d t h e i t e r a t i v e s o l u t i o n m e t h o d s r e q u i r e db y K u l k a r n i ' s c o r r e la t i o n .

T h e d i f fe r e n t f l u id s a n d d i f f u s e r d i a m e t e r s d i d n o ta ff e ct t h e g a s h o l d - u p . K u l k a r n i ' s c o r r e l a t i o n p r e d i c t st h a t t h e r e w i l l b e a d i f f e r e n c e b e t w e e n d i f f e r e n t f l u i d s ,d e p e n d e n t o n t h e v a l u e o f t h e s u r f a c e - a c t i v e a g e n tp a r a m e t e r , 7 - B u t t h i s d i f f e r e n c e o n l y r e a c h e s t h em e a s u r e m e n t a c c u r a c y l e v e ls in t h i s s t u d y f o r g a sh o l d - u p s a b o v e 8 % ; b e l o w t h i s , ~, c a n b e s e t to z e r oa n d t h e h o l d - u p s u s e d f o r a l l w a t e r m i x t u r e s ; t a p ,s e w a g e o r a c t i v a t e d s l u d g e .O xygen m ass - t rans fer coe f f i c ien t

T h e m a s s t r a n s f e r c o e f f i c i e n t s w e r e f i r s t e x a m i n e d( B o x et al. 1 9 7 8 ) f o r t h e e f f e c t o f d i f f u s e r d i a m e t e r ,w a t e r d e p t h , p r o b e l o c a t i o n a n d g a s fl o w r a t e. F r o mt h i s a n a l y s i s t h e p r o b e l o c a t i o n w a s n o t s i g n i f i c a n t a tt h e 5 % s i g n i f i c a n c e l e v e l, w h i l e t h e o t h e r f a c t o r s w e r ea l l s i g n i f ic a n t b e y o n d t h e 0 . 5 % l e ve l . A v e r a g e v a l u e sf o r m a s s - t r a n s f e r c o e ff i c i e n ts w e r e t h e r e f o r e t a k e nf o r e a c h s e t o f f l o w r a t e , w a t e r d e p t h a n d d i f f u s e rc o m b i n a t i o n .

1 2 0

100 .

8 0 .ro. - i 60

4 0

20 .

00 2 0 4 0 6 0 8 0 1 O 0 1 2 0

E x p e n i m e n t a l K l aFig. 10. K t a correlation: Kawase vs experimental.


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1136 J. DudleyT h e c o m p a r i s o n b e t w e e n m e a s u r e d a n d p r e d i c t e d

m a s s - t r a n s f e r c o e f f i c ie n t s i s g i v e n i n F i g s 6 - 1 0 .M o s t o f t h e c o r r e l a t i o n s h a d t o b e c o r r e c t e d f o rt h e e f f e c t o f t h e d i f f u s e r d i a m e t e r . T h a t d i f f u s e rd e n s i t y a f f e c t s t h e m e a s u r e d KLa v a l u e h a s b e e nr e c o g n i s e d b y K h u d e n k o a n d S h p i r t (1 98 7) a n db y o n e o f t h e r e v i e w e r s o f t h i s p a p e r . T h e e f f e c to f d i f f u s e r d e n s i t y i s t h e r e s u l t i n g p l u m e w i d t h o ft h e a i r : t h e g r e a t e r t h e p l u m e w i d t h , t h e g r e a t e rt h e m i x i n g w i th i n t h e a e r a t e d v o l u m e a n d t h eg r e a t e r t h e t u r b u l e n c e , f a c t o r s t h a t l e a d t o h i g h e rm a s s t r a n s f e r c o e f f i c i e n t . R a t i o s o f 8 F / B f o rK h u d e n k o a n d O z t u rk , a n d 4 F / B f o r H i g b ie a n dC a l d e r b a n k w e r e u s e d . T h u s t h e p r e d i c t e d KLaf r o m t h e s e c o r r e l a t i o n s w a s m u l t i p l i e d b y 8 F / Bo r 4 F / B t o o b t a i n r e v i s e d KLa p r e d i c t i o n s . F o rt h e 5 0 m m d i ff u s e r th i s m e a n t t h a t t h e c o r r e c t i o nf a c t o r w a s 1 .0 f or H i g b i e ' s a n d C a l d e r b a n k ' sc o r r e l a t i o n s . K a w a s e ' s c o r r e l a t i o n w a s n o t a d j u s t e d .W i t h t h i s c o r r e c t i o n C a l d e r b a n k ' s c o r r e l a t i o n w a st h e b e s t ( m o s t o f t h e p r e d i c t e d v a l u e s l y in g w i t h int h e _+ 1 0 % l i n e s m a r k e d o n F i g . 9 ) , f o l l o w e d b yH i g b i e . T h e K a w a s e c o r r e l a t i o n d i d n o t r e q u i r e ac o r r e c t i o n f a c t o r b u t s t i ll p r o d u c e d a r e a s o n a b l ea g r e e m e n t w i t h t h e e x p e r i m e n t a l d a t a . T h e r e i sn o t h i n g a p p a r e n t i n K a w a s e ' s c o r r e l a ti o n toe x p l a i n w h y i t d i d n o t r e q u i r e a c o r r e c t i o n f a c t o r f o rd i f f u s e r d e n s i t y , o t h e r t h a n t h a t a t l o w KLa v a l u e s( l es s t h a n 1 0 / h) i t h a s a g r e a t e r s p r e a d t h a n t h e( c o r r e c te d ) p r e d i c t i o n s f r o m t h e o t h e r c o r r e l a t io n s .B e c a u s e i t h a s b e e n d e v e l o p e d f r o m a t h e o r e t i c a la n a l y s i s , a n d b e c a u s e i t d i d n o t r e q u i r e a c o r r e c t i o nf a c t o r i t i s th e c o r r e l a t i o n r e c o m m e n d e d f o r g e n e r a lu se .

A LP H A -F A CTO R C O R R E L A T I O N

A l p h a - v a l u es t h e r a t i o o f t h e o x y g e n t r a n s f e rc o ef f ic i e nt i n a t e s t l i q u i d t o t h a t i n t a p w a t e r w e r ea l s o m e a s u r e d i n o u r a p p a r a t u s . F o r t h e s e t e s t s w eu s ed a c o n s ta n t l i qu i d d e p t h - - 3 . 5 m - - a n d f ix e dd i ff u s e r d i a m e t e r - - 5 0 m m . K L a v a l u e s f o r w a t e r w e r ec a l c u l a t e d u s i n g t h e c o r r e c t e d C a l d e r b a n kc o r r e l a t i o n .

T h e e f f e c t o f s u r f a c t a n t s o n KLa i s t h r e e - f o l d . T h ef i rs t e ff e c t i s t o r e d u c e t h e g a s - l i q u i d s u r f a c e t e n s i o n ,s o ' t h a t t h e b u b b l e s i z es a r e s m a l l e r f o r a g i v e na e r a t i o n p o w e r . T h i s r e s u l t s i n a n i n c r e a s e d s p e c i f i ca r e a f o r m a s s t r a n s f e r , a . T h e s e c o n d e f f e c t i s t h a tg e n e r a l l y t h e d i f f u s i v i ty o f o x y g e n t h r o u g h t h e s u r f a c -t a n t l a y e r a r o u n d t h e a i r b u b b l e s i s l e s s t h a n t h a t i nw a t e r , s o t h a t t h e m a s s t r a n s f e r c o e f fi c i e n t g L i s l o w e rt h a n i n c l e a n w a t e r . T h e t h i r d e f f e c t i s t h a t t h es u r f a c t a n t c a n m a k e t h e b u b b l e s u r f a c e m o r e r i g i d ,w h i c h f u r t h e r r e d u c e s t h e K L v a l u e s . T h e r e i s th e n a ni n t e r p l a y b e t w e e n b u b b l e s i z e , m a s s t r a n s f e r a n dc h a n g e s i n f l o w r e g i m e s c a u s e d b y t h e s m a l l e r a n dm o r e r i g i d b u b b l e s ( A n d r e w s et al., 1 9 8 8 ) . Th e n e te f f ec t o n KLa i s u s u a l ly , f o r d i f f u s e d a e r a t i o n s y s t e m s ,t o r e d u c e t h e K t a v a l u e .

T h e a l p h a - f a c t o r w a s c o r r e l a t e d u s i n g th e e q u a t i o not = bo+ b2 e x p - b 3 ' S A A . EG h\ v o /

( S A A . % - h ) (6 )+ b 3 e x p - b 4 a Vow i t h t h e c o n s t r a i n t b 0 + b I + b 3 = 1. T h i s c o n s t r a i n te n s u r e s t h a t i n t h e a b s e n c e o f a n y s u r f a c e - a c t i v e a g e n t( S A A = 0 .0 ) t h e a l p h a - f a c t o r i s 1 .0 .

0 . 8 0

t Of -C ~u

4 Jr -I DEL.c'3Xhi

0 . 7 0

0 6 0

0 5 0

0 . 4 0

0 3 00 3 0

AAA

0 .~rO

a

Cornelation alphaFig. 11. A lpha correlation: experimental da ta com pare d with the correlation.

. 5 0 0 . 8 0 0 . 7 0 0 . 8 0

AA


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Parameterrb0blb2b3b4

KL a cor re la t ionsTable 3. A lpha-factor correlation parameter values

Change in sum -of-squares for change in param eter (%)Value -2 0 - 10 + 10 +200.111 +30 +8 +10 +350 .362 +2 -0 .3 +3 +86.54 + 17 +4 +2 +80.527 +544 +133 +143 +5630.0726 + 32 + 8 + 6 + 24

1137

T h e c o r r e l a t i o n i s b a s e d o n t h e e x p e c t a t i o n t h a t t h ea l p h a f a c t o r i s a f f e c te d m a i n l y b y t h e s u r f a c t a n tc o n c e n t r a t i o n o n t h e s u rf a ce , S A A / a , a n d t h e b u b b l er e s i d e n c e t i m e , E c h / V c . W i t h t h i s f o r m t h e a l p h af a c t o r d e c r e a s e s w i t h i n c r e a s i n g s u r f a c t a n t c o n c e n -t r a t i o n , a n d w i t h d e c r e a s i n g f l o w r a t e : t h e o b s e r v e db e h a v i o u r .

T h i s e q u a t i o n w a s f i t t e d t o o u r d a t a u s i n g an o n l i n e a r l e a s t s q u a r e s a l g o r i t h m . T h e r e s u l t i n g c o r -r e l a t i o n i s s h o w n i n F i g . 1 1, a n d t h e p a r a m e t e r v a l u e sa r e g i v e n in T a b l e 3 . F r o m t h i s a n a l y s i s t h e m o d e l isi n s e n s i t i v e t o t h e v a l u e f o r b j a n d f a i r l y i n s e n s i t i v e t ot h e v a l u e f o r b 2 , i m p l y i n g t h a t t h e s e d o n o t a d d m u c ht o t h e p e r f o r m a n c e o f t h e c o r re l a t io n . T h e c a l c u la t e d

v a l u e s a r e s i g n i fi c a n t ly a f f e c te d b y t h e v a l u e s f o r t h eo t h e r p a r a m e t e r s . T h e c o r r e l a t i o n w a s e v a l u a t e d w i tho n l y t h re e p a r a m e t e r s b u t t h e p e r f o r m a n c e w a s s i g-n i f i c a n tl y w o r s e ( s e t ti n g b t t o 0 i n c r e a s e d t h e s u m o fs q u a r e s b y 1 2 0 % ) . A d d i n g a n a d d i t i o n a l e x p o n e n t i a lt e r m t o t h e c o r r e l a t i o n d i d n o t s t a t i s t i c a l l y i m p r o v et h e p r e d i c t i o n s .

C O N C L U S I O N S

1 . G a s h o l d - u p i s a f u n c t i o n o f su p e r f i ci a l v e l o c it yw i t h a w e a k e f f ec t o f d e p t h . T h e r e a r e n oe f f e c t s f r o m t h e d i f f u s e r s i z e o r a d d i t i v e s t ot h e l i q u i d , a t le a s t u p t o g a s h o l d - u p s o f a r o u n d8 % .

2 . H o l d - u p c a n b e a d e q u a t e l y c o r r e l a te d b yt h e K u l k a r n i c o r r e l a t io n i n c o n j u n c t i o n w i t hc o r r e l a t i o n s f o r t h e b u b b l e d i a m e t e r a n d r i s ev e l o c i t y .

3 . T h e m a s s t r a n s f e r c o e f f i c i e n t , K L a i s a f u n c t i o no f d e p t h , s u p e r fi c i a l g a s v e l o c i t y a n d a l s o t h ee f f e c t i v e g a s s e d a r e a .

4 . C a l d e r b a n k ' s c o r r e l a t i o n w i t h a c o r r e c t i o n f o rt h e g a s s e d a r e a c a n a d e q u a t e l y c o r r e l a t e 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 .

5 . K a w a s e ' s m a s s t r a n s f e r c o e f f i c ie n t c o r r e l a t i o ns u p p l i e s a r e a s o n a b l e c o r r e l a t i o n o f t h eo b s e r v e d c o e f f ic i e n t w i t h o u t n e e d i n g a c o r r e c -t i o n f o r t h e g a s s e d a r e a .

6 . C o r r e l a t i o n s b a s e d o n th e o r e t i c a l s u p p o r t h a v ep e r f o r m e d b e t t er i n t h is c o m p a r i s o n t h a n t h o s ed e v e l o p e d b y c u r v e - f i t t i n g .

7 . T h e a l p h a - f a c t o r i s a f u n c t i o n o f g a s v e l o c i t ya n d s u r f a c t a n t c o n c e n t r a t i o n .8 . E q u a t i o n ( 6 ) c a n c o r r e l a t e t h e a l p h a f a c t o r , w i t h

t h e c o r r e c t b e h a v i o u r f o r z e r o s u r f a c t a n t c o n -

c e n t r a t i o n . T h i s c o r r e l a t i o n h a s b e e n d e v e l o p e df o r a s pe c i fi c s y s t e m , a n d t h e p a r a m e t e r v a l u e ss h o u l d w h e r e p o s s i b le b e c o n f i r m e d f o r o t h e rs y s t e m s .

A c k n o w l e d g e m e n t s - - T h e a u t h o r w o u l d l i k e t o t h a n k W R cplc fo r permiss ion to publ i sh th i s paper and the rev iewers fo rthe i r he lpfu l comments .

REFERENCESA n d r e w G . F . , F i k e R . F . a n d W o n g S . ( 1 9 8 8 ) B u b b l eh y d r o d y n a m i c s a n d m a s s t r a n s f e r a t h i g h R e y n o l d s n u m -ber and sur fac tan t concent ra t ion . Chem. Engng Sci . 47,1467-1477.Box G. E . P . , Hunte r W. G. and Hunte r J . S . (1978)S t a t i s ti c s f o r E x p e r i m e n t e r s pp . 503-504 . Wi ley , NewYo r k .Brown L . B . (1979) Oxygen t ransfe r paramete r es t imat ion .I n P r o c . W o r k s h o p t o w a r d a n o x y g e n t r a n s fe r s ta n d a r dEPA /600/9-78 /021 , pp . 27-40. A me r ican Soc ie ty o f Civ i lEngineers.D o y l e M . I . , B o y le W . C . , R o o n e y T . a n d H u i b r e g a t e G . L .(1983) P i lo t p lan t de te rm ina t ion of oxygen t ransfe r in f ine

bubble ae ra t ion . J . W P C F 55, 1435-1440.Du nn I . J . and Einse le A . (1975) Oxygen t ransfe r coef f ic ien tsb y t h e d y n a m i c m e t h o d . J . AppL Chem. Bio t ech . 2 57 0 7 - 7 2 0 .J o s h i J . B . a n d S h a h Y . T . ( 1 98 1 ) H y d r o d y n a m i c a n d m i x i n gm o d e l s f o r b u b b l e c o l u m n r e a c t o r s Chem. Engng. Com-m u n . 11, 165-199.Josh i J . B . and Sharma M. M. (1979) A c i rcu la t ion modelfor bubble co lumns . Trans. I ns t. Chem Engrs 57,244-251.K a w a s e Y . , H a l a r d B . an d M o o - Yo u n g M . ( 19 8 7 ) T h e o r e t i -ca l p red ic t ion of vo lum et r ic mass t ransfe r coeff icien ts inb u b b l e c o l u m n s f o r N e w t o n i a n a n d n o n - N e w t o n i a nfluids. Chem. Engng Sci . 42, 1609-1617.Kei l Z . O. and Russe l l T . W. F . (1987) Des ign of comm erc ia lsca le gas - l iqu id con tac tors ' . A. I. Ch~ E. Jl 33,488~496.K h u d e n k o B , M . a n d S h p i r t E . ( 1 9 8 6 ) H y d r o d y n a m i cpara me ters o f diffused air systems. War. Res . 20905-915.K u l k a r n i A . , S h a h Y . T . a n d K e l k a r B . G . ( 1 9 8 7 ) G a shold-up in bubb le co lum n wi th sur face-ac t ive agen ts : Atheore t ica l mode l . A . I . C h . E . J l 33, 690-693.Linek V. , Vacek V. and Benes P . (1987) A c r i t i ca l rev iewand exper imenta l ve r i f i ca t ion of the cor rec t use o ft h e d y n a m i c m e t h o d f o r t h e d e t e rm i n a t i o n o f o x y g e nt ransfe r in ae ra ted ag i ta ted vesse l s to wate r , e lec t ro ly teso lu t ions and v i scous l iqu ids . Chem. E ngng J . 341 1 - 3 4 .Lis te r A . R . and Boo n A . G. (1973) Aera t ion in deep tanks :A n evalu ation of a f ine-bubble diffused-air system. Wat .Pol lut . Control pp. 59ff4i05.Lock e t t M. J . and K i rkpa t r ick R~ D. (1975 ) Idea l bubblyf low and ac tua l f low on bubble co lumns . Trans. Inst.Chem. Engrs 53, 267-273.


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1138 J . Du dleyM ot a r j e m i M . J . a nd J a m e s on G . J . ( 1978) M a s st r a n s fe r f r o m v e r y s m a ll b u b b l e s - - t h e o p t i m u mbubb l e s i z e f o r a e r a t i on . Chem. Engng. Sci. 33,1415-1423.O z t u r k S . S . , Sc hum pe A . a nd D e c kw e r A . - D .( 1987) O r ga n i c l i q u i ds i n a bubb l e c o l um n: ho l d - up sa n d m a s s t r a n s f e r c o e f f i c i e n t s . . 4 . L Ch.E. Jl 33,1473-1489.Sh i oya S . a nd D unn I . J . ( 1979) M ode l c om pa r i s ons f o rd y n a m i c KLa m e a s u r e m e n t s w i t h i nc om pl e t e l y m i xe dphases . Chem. Engng Commun. 3 , 41 - 52 .Sp r i e t J . A . a n d B o t t e r m a n J . H . ( 1984) C or r e c t i on f a c t o r sf o r t h e d y n a m i c m e a s u r e m e n t o f t h e v o l u m e t r i c m a s s -t ransfer coeff ic ient . J. Chem. Tech. Biotechnol. 34A137-153.W i nk l e r M . ( 1981) Biological Treatment of Waste-Water I s t edn , pp. 55, 56, 66. El l i s s Ho rwo od,C h i c he s t e r .

P P END I XT he c a l c u l a t i on p r oc e du r e a dop t e d w a s, g i ve n t he a i rf lowra te :

I . Ca lcula t e V~ .2 . U s i ng e q ua t i on ( 2) , c a l c u l a t e PG3 . C a l c u l a t e t he ga s ho l d - up u s i ng J os h i ' s c o r r e l a t i on i nT a b l e 1 ( o r K o i de - - t he o t he r c o r r e l a t i ons w i ll bea dd r e s s e d a t po i n t 7 ) .4 . C a l c u l a t e t he bu bb l e d i a m e t e r f r om e q ua t i on ( 1 ) .5 . Ca lcula t e the spec i f i c sur f ace a rea f rom equa t ion (3) .6 . Ca lcula t e the r i s e ve loc i ty f rom equa t ion (4) .7 . Fo r ga s ho l d - up e q ua t i ons o t he r t ha n J os h i ' s o r K o i de ' s ,c a l c u la t e t h e g a s h o l d - u p a n d c o m p a r e w i t h t h e c u r r e n tva lue . Repea t s t eps 4 , 5 , 6 , 7 unt i l t he gas hold-up va lueha s c onve r ge d .8 . C a l c u l a t e t he KLa va l ue f r om t he r e l e va n t e q ua t i on i nT a b l e 2 .

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