The continuous phase heat and mass-transfer properties of dispersions

Chemcal

En@neenng Snence,

1981,

1’01

19, pp

The continuous

39 to 54

Press Ltd.

Per@mmn

Pnntrd III Great Brltnln

London

phase heat and mass-transfer dispersions

P Department

H.

of

and M. B MOO-YOUNG

CALDERBANK

of Chemical Technology,

(Receaoed 2 September

properties

Umrerslty

of Edmburgh,

1960, an rewed form 22 December

Scotland 1960)

Abstract-Techmques have been developed for measunng the mterfaclal area m gas-hquld dlsperslons It has thus been possible to measure the hqmd phase mass transfer coefficients m gas-hqmd tisperslons such as are produced m aerated nuxmg vessels, and Eneve and smtered plate columns The results have been combmed with other pubhshed data for heat and mass-transfer m hquld-hqmd and sohd-gas hperslons m which the dispersed phases are free to move under the actlon of gravity, and also with data on transfer by free convectlon from spheres These data can all be correlated

by

(1) For large gas bubbles which do not behave hke rIgId spheres as the smaller ones do k, (N&l2

= 0 42

(2)

If the particles of the dlsperscd phase are not free to move under gravity due to turbulence m the surroundmg fluld, k,

(~~~)3/3

=

5 CP

p5p

=

0 18

PC

(PhJ) PC 11/d 2

[

and transfer IS

(3)

PC

where P/v IS the power dlsslpatlon per unit volume Tius correlation applies to heat and mass transfer m nuxmg vessels where the sohd phase 1s m the form of a single fixed submerged body and also prebcts the small increase m mass transfer coefficients as the power Qssipatlon level IS increased beyond that needed for lust completely suspendmg hspersed solid particles m mixing vessels The correlation also apphes to heat and mass transfer in turbulent fluids flowmg through fixed beds of particles and through pipes RBsumB-Des mCthodes ont CtCd&eloppCes pour mesurer la surface d’dchangedans Lesdlsperslons gaz-hqulde C’est amsl qu’d a CtC possible de mesurer les coefficients de transfert de masse en phase hqmde dens des dlsperslons gaz-hqmde telles que celles qu& l’on rencontre dans des mClangeurs B air et dans des colonnes B grilles, et S plateaux fr&t& Les r&ultats ont CtC compar& avec d’autres don&es pubhCes pour les transferts de masse et de chaleur dans les dlsperslons hqulde-hqmde et sohde-gaz oh les phases dlsper&es sont hbres de se deplacer sous l’actlon de la pcsanteur et Cgalemrnt avec les donnCes qur le transfert par convection hbre de spheres Ces don&es peuvent toutcq &tre mlqes sous Is formr k,, (A~+)~/~ = $-

(~$4~1~

= 0,31

-

l/9

(-3-JPC

P pc

Pour de grosses bulles gazeuses cpu ne se cornportent trairement aux petltes bulles, on a h, (Nsc)‘/2

API*,&!

0,42

39

pas comme des sphPres rIgIdes con-

P H CAI.DERBANKand M. B Moo-You~a Sr les partmules de la phase drsper&e ne sont pas hbres de se d&placer sous l’actlon pesanteur et $1 le transfert est du a la turbulence dans le flulde envuonnant, on a kL (~7~~1313=

2

h

CP

(h$,)2/3

= 0~3

de la

PC 1

(?f?!$

PC

[

ou P/v cst 1~ pmssance dtssrpee par umte de volume Cette relatron s’apphque au transfert dc chaleur et de masse dans les melangeurs ou la phase sohde se presente sous forme dun corps umque fixe et submerge, et pr6vort aussr le petrt accroissement des coefficrents de transfert de masse quand la quantrte d’energre drssrpee est augmentee au-dela de ce qur est necessarre pour disperser completement les partrcules sohdes dans les melangeurs La relation s’applique aussi au\ transferts de chaleur et de masse dans les fhudrs turbulents s’ecoulant a travers les hts fixes et a travers les tubes Zusammenfassung-Es wurden Verfahren entwrckelt zur Messung von Phasengrenzflachen m Gas-Flussrgkeits-Dispersionen Hierdurch war es moglmh, die Stofftransportkoefhrrenten fur dir flussrge Phase be1 Gas-Flussrgkerts-Dispersronen zu messen, wre sre m begasten Ruhrkesseln sower m Kolonnen mrt Sreb- und Smterplatten vorkommen Die Ergebmsse wurden kombmrert mit anderen veroffenthchten Datm uber den Warmcund Stoffaustausch m Flusslg-Flusslg- und Gas-Fcststoff-Drspersronen, bei denen die 1 ertellten Phasen such frei unter dem Emfluss drr Schwerkraft bewegen, und solchen Daten, be1 denrn der Transport durch frele Konvektron von Kugeln erfolgt Alle Daten konnrn nle folgt knrrrhrrt wrrden l/3

I,,

(~~~1313 = 2

(spry p PC

= 0,51

%;

g

(

(1)

1

Fur grosse Blasen, dre such mcht wre kleme starre Kugeln verhalten l/3

k, (Nsc)‘13 = 0,42

Wenn die Partrkel der vertedten Phase such mcht trer unter dem Emlluss der Schwerkraft beaegen

und der Stofftransport von der Turbulenz des umgebenden Fluldums abhangt, gilt

wober P/v der Kraftverbrauch pro Volumenemhert 1st Drese Korrelatlon kann angewandt werden auf den Warme- und Stofftransport m Ruhrkesseln, m denen drefeste Phase m Form emesemzelnen festen Korpers vorhegt, und fur die Voraussage des lerchten Anstelgens der Stofftransportkoeffizrenten berm Anstergen des Kraftverbrauchs brs unterhalb des Wertes, der benotrgt wud, urn dre Feststofftedchen m volhger Suspension zu halten Die Korrelatron kann aurh berm Warmr- und Stofftrdnsport turbulenter Medren m Srhuttschrchten und Rohren

INTRODUCTION THE

work separately

present

evaluate

TO PREVIOUS

WORK

was undertaken m order to the mass transfer coefficients

and interfacial areas m gas-hqmd dispersions such as obtain m fermentation vessels and dlstdlatlon plate columns Hltherto it has not been possible to arrive at these values because of a lack of means of measuring mterfaclal areas. The present authors have described a hght transmission method for measuring mterfaclal areas m not too optically dense dispersions [I]

and

also

a light-reflection

technique

dlsperslons of optically transparent In addition, a y-ray transmission

for dense

phases [2] method was

described for measuring the gas hold-up m these systems 121, from which the Sauter mean bubble size could be deduced The latter value was checked m many cases by a statIstica analysis of high-speed flash photographs Details of the photographic techniques and statlstlcal analysis wdl be described m a subsequent pubhcatlon Experiments were performed m which sparingly

The contmuous

soluble volatile

the gaseous

for extreme variations of dlffuslon coeficlents and Although the authors’ results hqmd vlscosltles

m the hqmd

as a bubble

supported

solutes were desorbed

m liquid

phase being dispersed cloud facial

equipment

were

determmed

vessels and smtered

m aerated

mutually

of

aqueous

glycol

smaller varlatlons Dlffuslon

and

glycerol

as

solutions,

and

solutes

the

mass

transfer

the

had no effect on

coefficient,

but

that

lated

whose mass transfer

well

with

those

coefficients

observed

for

large

flow m the boundary

layer

coefficient,

[P] and others. For large bubbles drag

predommated

corre-

the

diameter)

hydrogen

increased

coeficlent

reactions

coefficients

as solute

measuring

tion

to

wlthout

for

carbon

with the Stokes to mclude

obtained

using

This has been achieved

the rates

of catalytic

by

hydrogenation

in various solvents contammg

an excess

was determined

reaction

rate was fast

and the progress of the by the rate of solution

and followed by the mfra-red absorp-

characterlstlcs

of the product

of reactlon.

Fmally, published data on contmuous phase heat and mass transfer to dispersions of sohds and

to

single

combmed

with

bubbles

and

the present

more comprehensive

form of

dlffus~on

dlffuslon

of hydrogen,

sense, and

condltlons

high

reaction

sohd-hqmd

as found by FROESSLING

( > 2 5 mm and

same

ensured that the chemical

that under these circumstances the mass transfer coefficient was proportional to the 2/3 power of the dlffuslon

the

and not rate determining,

dispersions It was concluded that small “ rigidsphere ” bubbles experienced frlctlon drag causing hindered

and largely

agent

the vlscoslty

of catalyst and through which hydrogen bubble swarms were passed The excess of catalyst

bubbles ( > 2 5 mm diameter) gave greater mass transfer coefficients than small bubbles (< 2 5 mm diameter)

thickening

which enabled

diffusion cell) The data have also been extended

hquld phase vlscoslty was not, however, revealed It was clear that agltatlon intensity, bubble size free rlsmg velocity

(Polyacrylamlde)

dioxide as pure water (determmed

Influenced the value of the mass The independent effect of coefficient.

and bubble

WoicK

firstly by the use of an aqueous

have

cell.

which

transfer

parameters

Solutions havmg vlscosltles up to 87 cP were used and found to be Newtonian m rheologlcal behavlour, and to

The above work showed that the value of the liquid phase dlffuslon coefhclent was the major factor

these variables

property

affectmg the dlffuslon coeficlent

of carbon dloxlde m aqueous

using the Stokes dlffuslon

PKEYENT

be independently

glycol and glycerol were measured by the hqmd Jet technique [3], and these results have now been confirmed

so that

by physlcal

In the present work the previous mass transfer data for bubble swarms have been extended,

by the use

by the use of various

coefficients

the effect

on the mass transfer

mixing

and sieve plate columns,

m the latter being achieved

compensatmg,

were replaceable

has been previously described [2, 31 This work covered a wide range of bubble sizes and liquid phase diffusion coefficients, extreme variations

as regards

coefficient

coefficient, they also showed that the effects of bubble size and shp velocity were such as to be

being such as to ensure

that the hquld phase was perfectly mixed In this way values of the liquid phase mass transfer coefficients

these equations

of the dlffuslon

The rate of material transfer and the mterarea were simultaneously observed, the

experimental

propertles of dlsperslons

solvents,

or gaseous

from or dissolved

phase heat and mass-transfer

drops work

and convmcmg

have

been

to achieve

a

correlation

RESULTS

unFig

hindered flow envisaged by HIGBIE [5] obtamed, and as postulated by HIGBIE the mass transfer coefficient was proportional to the 4 power of the dlffuslon coefficient It IS important to appreciate that the FROESSLING and HIGBIE equations have never hitherto been tested with systems of gas bubbles

I shows two correlations

for hquld phase

mass transfer coefficients for bubble swarms passing through hqmds m sieve and smtered plate columns and m aerated mlxmg vessels The upper correlation line refers to bubble swarms of average bubble diameter greater than about 23 mm such as are produced when pure liquids are aerated m 41

P. H. CALDERBANK

mlxmg vessels and sieve plate columns, to the result NSh = 0 42 k,(Ns,)1'2 =

OF

and leads

NSolla NLr113

The data for ” small ” bubbles are m agreement with those reported by previous workers for both heat and mass transfer

(2)

The lower correlation lme refers to bubble swarms of average bubble diameter less than about 2; mm

such

solutions mlxmg

as

are

produced

of hydrophyhc vessels,

or

when

aerated with smtered

hqulds

m

generally

are

m density

between

the fluid at the interface, as compared

and

with the

difference m density between the dispersed and contmuous phases which 1s employed when the former 1s free to move under gravity. This result 1s obvrously due to the fact that m both cases the

l/8

= o 31 (y)

with those on heat and mass transfer

the bulk m free convection

This leads to the result

(N#3

solid m liquid and

m free turbulent convection from stationary spheres, if the term Ap 1s taken as the difference

when aqueous are aerated

with

hquld m liquid dlsperslons as shown m Fig 1 Fig 2 also shows that the present data are m agreement

plates or plates containing

very small perforations kL (Ns,)2’3 = &C

solutes

MOO-YOUNG

” Small ” bubble correlataon

(2)

API-G "a a ( PC )

0 42

andM.B

(1)

relative rate of flow between the phases 1s induced, It will be seen that both correlations

show that

the liquid phase mass transfer coefficients independent of bubble size and slip velocity depend

only

system

In

coefficients

on the physical mixing

vessels

are independent

properties the

mass

Fig. 2 demonstrates

of the

dlfferelc Sherwood

transfer

of the power

dlssl-

pated by the agitator,

and m sieve-plate

they

of the flmd flow rates

are independent

and its magnitude determined, by the appropriate value of the density difference.

are and

value

columns

of

the fact that as the density

between the and Nusselt 2

appropriate

phases dlmmlshes, numbers approach to

Allowing for this, correlation form becomes

natural (1)

_.A FIG 1

convection

in more

precise

1

Correlation of heat and mass transfer coefficients for dlsperslons of small bubbles and ngd spheres (lower hne) and large bubbles (upper line) See “ Source of data ” and “Range of variables ” 12

the the

,

,

Water - 0,

Water - 0,

Bnne - 0,

Polyacrylamlde

Wax - H, Aq ethanol solution ~ CsHs Water - Resm beads

Aq glyderol solutions - Hg Water - OS

+

l

+

d

C-a

:

[ii]

(small bubbles and n@d spheres) This work (small bubbles and rl@d spheres) This work (small bubbles and n@d spheres) TIM work (small babbles and n@d spheres) [81

This work

T~u work (large bubbles)

Rkference

45 “C

Water - CO,

X

D

15 “C

Glycol - CO,

#

T *

ao: a

i

Legend

Absorption

-

0 07

-

-

00127 -00215 0 00435 00042 -000627 0 00291

-00138

-

0 0394

-

00059

I

1

-0048

00187

00253 0 011

0 00122-o

0 122

-0 010

0 0024

0007

and desorptlon

459

-

93

2 8 4 5 115 62

106

x10-s

x 10-s

1 84 x lo-! 43 x 10-s -175 x 10-s

-

4 75 - 14 25 x lo-! 135 x 10-5

65

x105

9 5 x 10-l

17 8 x 10-4

1 x10-6

1 13 x 10-s

2 6 -

3 4 -

1 s-48

DL (cma/sec)

I

/

(Cal/cm set) ‘C

A

Range of Vanables (see Fig 1)

H, - water CO, - water

100-l

Af

0 772

-198 1165 0 588 0 266-O 36 0 69

130

12 27-12 55 0 867 0 89

0 174-O 177

100

0 698-

1160

178

k/cm3)

111-

_-

Sand - air Carborundum - air Alovlte - air Air - water

This work

Water - An

Water - an

This work This work This work

F.B.

System Water - 0. Water - d, , CO, Toluene - H, Bnne - NaCl Water - KC1 Water - Urea Brme - NaCl Water - sucrose Solvents - benzolc acid Water - sodmm thlosulphate

StIrred tank

This work

This work

This work

This work

This work

This work

Reference Legend This work A

ST

Souree of data (see Fig 1) Free fall

Process Absorption and desorptlon (FR.ST) Absorption aid de&ptlon (FR,ST)

F F.

(FR,ST) Absorption and desorptlon (FR.,ST) Absorption a&d des&ptlon IF R . S T.) Absorption aid de&rptlon (FR,ST) Absorption and desorptlon (FR,ST) Absorption and des&ptlon (FR,ST) Absorption (F R ) Absorption (F R ) Ion exchange (S T ) Heat transfer (F F ) Absorption (F R )

0 007-O 080

@m/s&

leL

solutions - CO,

Aq glycol solutions - CO,

%

syste?n glycerol solutions - CO,

Free nse

A

Legend 0

FR

1 11-l

126 1

10

j 0791;;(:11

I

I

1 0 -1 249 0 867 1 178

10

100

772

160

1 OOC-1 178

-105

0 89-

1 52

100

199-89

113

0 6 -87

-

d (cm)

11 415 65 47

-12 0 0 33-O ,0 45-O 0 l&O

009

0 225 0 14 0 014 065

0 09

0 073

020-080

Wk3) I (c”p) _-

I 0 698-o

I

Reference Process Absorption (F R ) WI Absorption (F R ) K! Absorption (F R ) Duzsolutlon (S T ) Dlssolutlon (F B , S T ) Dlssolutlon (F B , S T ) [i4j Crystalhzatlon (F B ) Dlssolutlon (S T ) f::j Dlssolutlon (S T ) Dlssolutlon (S T ) [I71 Large ram drops evaporatmg [I81 (F F ) Small ram drops evaporatmg WI (F F ) Particles falhng through hot air undergomg heat transfer (FF) Dlssolutlon m centnfugal apparatus Absorption (F R.) Absorption (F R )

Fhudmed bed

1' H iv,,

=

N,,

=

2 0 +

as rigid

0 31 N&l/s

It may be shown that this relatlonshlp ally

identical

1s form-

to FROESSLING’S equation

Figs

and

1

2 show

if the

been employed Figs

a comprehensive

and for suspended

or rlsmg particles, This correlation single

bubbles

particles

or free falling

and drops 1s obtained

holds equally and

for dispersions

shows

that

and

“ small ”

bubbles move m liquids as rlgld spheres under condltlons of hindered flow m the boundary layer It may

be noted

that

the particle

heat and mass transfer coefficients uous phase may properties only The correlation independent

size and

m the correlation,

be

from

[6] has proposed

for the dlssolutlon

of solid suspen-

sions m mlxmg vessels which 1s of the same form as correlation (l), while SCHMIDT [7] has proposed for turbulent

free convective

heat transfer

so that ” Large ” bubble correlatzon

m the contm-

predicted

may be noted that ZDANOVSKII a correlation

a similar equation

sense velocity are not involved

m this connexlon

and 2 show the extent to which a wide

1

variety of previously published data agree with the present observations and the range of variables In addition it which the correlation includes

for heat and mass transfer m turbulent

free convection

for

that

bodies, and where relative movement the phases 1s retarded as m boundary In the absence of adequate experllayer theory mental proof FROESSLING'S equation has previously between

drag coefficient for spheres 1s inversely proportional to the square root of the Reynolds number correlation

hfOO-YOUNG

C'ALDhRBANK aIldi!lB

physical

To produce

“ large ” bubble

clouds

m aqueous

glycerol solutions and thereby effect major changes 1s seen to hold accurately

for

and extreme changes m the variables,

m the diffusion

coefficients

of solutes undergoing

and has been tested for heat and mass-transfer

mass transFer it was necessary to employ the pulsed sieve-plate column described m a previous

m turbulent

pubhcatlon

free convection,

mlxmg

vessels

where

bubbles

are

undergomg

transfer sohds,

m hqmd

for mass transfer m

suspended

and gas

dlssolutlon,

flmdlsed

for heat transfer

solids

for

mass

beds of crystalhzmg

when

hqmd

drops

fall

through other liquids, and for several other similar processes The

correlation

where the particles

FIG

2

(1)

apphes

to those

of the dispersed

Correlation

of heat

and

systems

phase behave

mass transfer

ngld

spheres wth

[3]

Without

means of producing difficult aqueous impossible

the use of a posltlve

large bubbles

it 1s extremely

to prevent small bubbles forming m solutions of hydrophlhc solvents and to do so m aerated

The independent

mlxmg

effect of hqmd

vessels.

vlscoslty

was

evaluated agent

using polyacrylamlde as a thickening it was established that the for water,

solutions employed

coefficients natural

for

dwperwons

convection

were Newtonian

of small

bubbles

m rheologlcal

and

for

The contmuous

phase heat and mass-transfer

propertles of dlsperslons

behavlour and were without Influence on the dlffuslon coefficient of the solute bemg transferred Mass transfer coefficients for bubbles greater m diameter than about 29 mm are notably larger The than for small “ rigid-sphere ” bubbles mass transfer coefficient 1s proportional to the square root of the hqmd phase dlffuslvlty, lmplymg that the HIGBIE model [5] of unsteady state transfer applies, and that the unhindered flow sltuatlon envisaged m this model IS responsible for the high transfer coefficients obtained. In this region of bubble size the bubbles are deformed from spherical to ehpsoldal shapes

that surface active agents and trace lmpurltles mhlblt the transition from the hindered to the unhmdered flow condltlon The present observations have not included the effect of surface active agents In aerated mixing vessels as the agitator power IS increased the gas-hquld mterfaclal area increases and the bubble size dlmmlshes, while the mass transfer coefficient remains constant until a crltlcal point IS reached, at which the transfer coefficient begms to fall rapidly as the bubbles pass through the transltlon size reg?on Ultimately the transfer coefficient again becomes constant while the mterfaclal area continues to Transztzon regaon between LLsmall ” and “ large ” increase with increasing power dissipation The bubbles same behavlour ts observed with sieve-plate columns as the gas.+flow rate IS increased Fig. 3 shows some of the mass transfer data Fig 4 shows the results of many new observafor bubbles of a size intermediate between those tions carried out on sieve trays operating as previously classified as “ large ” and “ small,” previously described [2] under simulated dlstlllaexhibiting unhindered and hmdered flow respectlon condltlons, and for all of which the data have tlvely Slmdar results have been observed by GARNER [23] for gas bubbles, and by GRIFFITH been found to he m the transltlon region Fig 5 shows that m this region mean values of lCLa [24] for hquld drops. are almost independent of all variables, except It has also been observed by the above workers Absorption

of

Water contamng 69 5 % by wt

CO,

at 25%

tn

glycerol

watw contamng 63 0 % by wt glycerol water contolnlng 52 2 % by vd glycerol water contammg 38 0 % by wt glycerol water contamng 26 0 % by wt glycerol water contomng 14 0 % by wt glycerol

FIG 8

Typical data for transltlon from “ large ” bubble mass transfer coefficients to “ small ” bubble values Chm

45

Engng Sea

Vol

10, Nos

1 and 2

December,

lOG1

P H CA~JIERBANK and M B MOO-YOUNG

10-l -

Fro 4

+

Wax-H,

“ Transmon” repon mass transfercoefflclentsobservedwith froths in sieve-plateeolunms.

the Schmidt group. It was found that under dlstlllatlon condltlons producing a large vapour hold-up m the foam the transltlon from unhindered to hindered flow occurred with bubbles of greater size than m aerated mlxmg vessels, where the gas hold-up was comparatively low It 1s evident that further work remams to be done m order to throw light on this interesting region of bubble behavlour, but some relevant observations are worth making at this stage (a) The high mass transfer coefficients observed with large bubbles are only obtamed in shallow hqmd pools With deep pools a progressive decrease m the pomt values of the mass transfer coefficient with height or residence time 1s observed and these values finally decline to those obtained with small ngld-sphere bubbles These observations have been made by BAIRD [22], GRIFFITHS[24] and DOWNING[25] and the present authors. (b) Surface active agents cause the high mass transfer coefhclents for large bubbles to be reduced

to those obtained with rigid spheres [22, 24, 251 (c) As reported by CALDERBANK121 high rates of gas flow in neve-plate columns cause large gas bubbles to rise at rates considerably m excess of their free rlsmg velocltles when their mass transfer coefficients approach the rigid-sphere values The unhindered flow sltuatlon appears to be most readily reahsed with large bubbles at low rates of hqmd flow around them (free rise m viscous hqmds or flow round trapped bubbles as studied by HIGBIE [5]). The higher the Reynolds number of the bubble the more rapidly does the mass transfer coefficient decrease with residence time These phenomena are probably associated with the properties of the wake behind the bubble as discussed by GARNER and TAYEBAN [26] who observed a slmllar behavlour m hquld drops. As shown m Fig 1, the mass transfer coefficients for large and small bubbles approach each other at low values of the Schmidt group. This fact agrees with the conclusions of TOOR and 46

The contmuous

phase heat and mass-transfer

propertles of dlsperslons

cpr vclZ-6f p s

FIG

5

Average kLa values observed with froths m

n

Wafer-CO,bf 41)

meve

plate-columns

density difference between phases 1s sufficiently large to cause the force of gravity to determine Such exclusively the rate of mass transfer dlsperslons Include those m which solid particles are Just completely suspended m murmg vessels. It 1s known that an increase m agitation intensity above the level needed for complete suspension of particles results m a small increase m the mass transfer rate [lil, 161 When the particles m a mixing vessel are Just completely suspended, turbulence forces JUSt balance those due to gravity and the mass transfer rates are the same as for part&es movmg freely under gravity. At higher agltatlon levels turbulence forces are greater than those due to gravity and can no longer be equated with them so that a separate treatment, described hereunder, has to be applied. This approach which treats tur-

MARXHELLO 1271 who predict that the boundary layer and penetration mechamsms of mass transfer lead to ldentlcal results at low values of the Schmidt group. An important practical conclusion from these observations 1s that the high mass transfer coefficlents for large bubbles m unhindered flow can only be realized m special circumstances and that m mdustrlal sltuatlons of high gas flow rates surface active lmpurltles and deep hqmd pools the mass transfer coefficients wdl approach the lower values obtammg for rigid spheres

~~PERSEDPHASEUNDERTHEINELUENC~

of TURBULENCE The subject matter dealt with m previous parts of this paper refers to dlsperslons m which the

Chmi

47

Engng

SC*

Vol 16,Nos land2

December,1961

P. H. CALDERBANK and M B. MOO-YOUNG bulent

forces

as those

which

determine

mass

transfer rates has also been applied to fixed bodies submerged m mixing vessels In the case of gas-hqmd practical to exceed mechamcally induced operate Local

poorly

dlsperslons

1t 1s 1m-

gravltatlonal forces by turbulence since agitators

m gas-hqmd

dlsperslons

flow produces

primary

of the mam flow stream

by viscous

flow

When

the

Reynolds number of the main flow 1s high, most of the kmetlc energy 1s contained m the large eddIes, but nearly all of the d1sslpatlon occurs m the smallest eddies 1s large compared

If the scale of the mam flow

with that of the energy dlswpat-

energy. In the process the directional the primary eddies 1s gradually lost

of them,

operatmg

at

fluid flow from

1t 1s approx-

of a mass

transfer rates between the particle and the fluid, and d the particle IS itself fluid may lead to 1ts break-up The theory

of local isotropy

on the turbulent

mtenslty

around the particle. over

gives mformatlon

m the small volume

Thus, 2/[ucdj3] 1s a statlst~cal

descrlbmg

mg velocity

the variation

a distance

m the fluctuat-

comparable

to the

particle diameter (d) and may be used m place of the velocity of the particle m correlations of

nature of

KOLMOGOROFF [28] concludes that all edd1es which are much smaller than the primary eddies independent

vessel,

Turbulence m the immediate v1cm1ty particle of a dispersion affects heat and

parameter

mg eddies, a wide spectrum of mtermedlate eddies exist which contam and d1sslpate little of the total

are statlstlcally

m a flmd mixing

an energy input of 10 h.p /lOOO gall. of water, I IS 25 TV If the agitator has blades 5 cm wide the Thus, L/Z = 2000 which 1s large enough to allow local isotropy to be assumed.

These large primary eddies are unstable and dlsmtegrate mto smaller eddies until all their energy 1s dissipated

example,

1mately 5 cm

eddles which

have a wave length or scale of similar magmtude to the dimensions

(:)3”

The condltlon for local isotropy where the above equation applies 1s frequently encountered For

scale of the mam

zsotropac turbulence

Turbulent

= const (F).,.

3

heat and mass transfer

rates.

Thus, the Reynolds

number

1s given

for local isotropy

by

and that

the properties of these small eddies are determined solely by the local energy dlss1patlon rate per umt mass of flmd Thus, 1f a volume of flmd 1s considered

whose dlmenslons

or, alternatively

are small compared

N,; =

with the scale of the mam flow the fluctuatmg components of the velocity are equal, and this so-called local isotropic turbulence exists even though may

the turbulent

be far from

motion

as expressed hcation [2]

of the larger edd1es

lsotroplc

In

local

l=S

0

If we now assume the usual functional relatlonship between the Sherwood, Schmidt and Reynolds

-l/4

-

PC

pub-

mass transfer rates m turbulent fluids by OYAMA [15] and subsequently by KOLAR [SO] and CALDERBANK [2]

KOLMOGOROFFas p

by CALDERBANK m a previous

The turbulence Reynolds number above was apparently first employed for the correlation of

lsotroplc

turbulence the smallest eddies are responsible for most of the energy dlss1patlon. The scale of these smallest eddies 1s given by

3/4

pC1f3( P/v)~‘~ d2j3 1/2 CLe

numbers for mass transfer we arrive at the relation

V

Moreover, the mean square fluctuating velocity over a distance d m a turbulent fluid field where L > cl > 1 (L being the scale of the primary eddies and 1 that of the smallest eddies) 1s grven by BATCHELOR [29] as

k,d DL

tll

(-I[ PC

PC DL

1

pc1/3(~/~)1/3 d213 Y PC

112

Experiments on mass transfer m m1xmg vessels [12, 16, 311 have led to the conclusion that 48

The contmuous

phase heat and mass-transfer propertres of dlspersrons

kL a DL213

the analogous

9

expression

for heat transfer

being

and kL does not depend on d. In order

sary

that

to satisfy these condltlons x = + and y = Q, whence

k,d CTc p,l’3 (W;!‘” DL [ or

k, cc {E;$;

data

$iy2,,‘4 LT C c

n A v ri @

6

I

Contmuous

Heat transfer Water Water Glycerol LM 011 A-12 011 Hot water I Nltratmg hquor

[32]

0 0 0

Ref [33]

Ref

z ti f c

[34]

Mass transfer Sohd drssolutron a Sucrose - water * Sodmm thiosulphate

- water

I

IO

I 0

8

Ref Ref

[15] [17]

agltatlon

Its speed-power

I

IO-'

the

phase heat and mass transfer coefficrents Ref

(3) heat and mass transfer

m these forms It was sometlmes

measure

/

10-Z

IO-’

equation

to reconstruct

@a)

I

IO-

0 0

Followmg

data observed by many workers m mlxmg vessels are plotted m Fig 6 In order to present published

deis-jai2 (+J”’

lo-”

FIG

WI it IS neces-

Benzorc Benzorc Benzorc Benzoic Succmrc Succmrc Salycrhc Salycrhc Benzom

characterlstlcs

49

and

to

with

a

1

IO*

IO’

111mlsmg

vessels

Mass transfer acrd - sperm 011 acrd - cotton seed 01 acrd - rape seed orl acrd - glycol acrd - acetone acrd - n-butanol acrd - benzene acrd - water acrd - water

Lrqmd-hqmd extractron Et Benzorc acrd - kerosene - water Chm

necessary

eqmpment

En@y+' Scz Vol

16, Nos

I and

2

Ref

[16]

Ref

[35]

Ref

[36]

December,

1961

P. H. CALDERBANX and M B MOO-YOUNG dynamometer, as only infrequently mformatlon been reported Fig

6 reveals

considerable 107-fold merits

a correlation

scatter

change

which

m the data

this

despite

extends

m the variables

serious consideration

has

and

It may

eddies the mass transfer

a

for a

complete

therefore

observed

be argued

that some of the scatter of the experimental

through

the data

kL (Nsc)2’3 =

m Fig

with a standard

regression

number

hne

drawn

power PC

of 66 per cent.

of

vessels.

number correlation.

(4) does not

the

impeller

heat

and mass

However,

tur-

with geometrically

the

the

(4)

however,

until

is given by equation

lust

(1).

are

of the

Schmidt

group

IN

PACKED

BEDS

through is

them m turbulent

reached

from

the

flow. mass

recently of some

determmations. may

unit volume work.

on the

compute

the

power

dissipated

of flmd m a packed

P _=_- AP V

power is

become

the mass transfer

system

correlation for packed beds by FALLAT 1371 at a result

which is not revealed m

the particles

suspended,

careful

that the type of

when the agitator

for a given

exponent

conclusion

transfer proposed

is

agitation

used may have a mmor mfluence

mcreased

This

We

mass transfer coefficients

pletely

of

factor

small, so that the figure confirms

to fluids flowing

similar

equation

geometry

equipment It appears probable,

Thus,

m

It is perhaps surprlsmg to find that equation (4) also holds for mass transfer from packed beds

such a correlation

whereas

equipment

(4)

due to variations

m the several experiments

MASS TRANSFER

transfer

For fully developed

agitation

equation

of the results shown

Reynolds

may only be employed

agitator

has been

As a result of these

but does not provide precise evidence for the form of the power factor used.

bulence, where P a N3, equation (4) may be transformed to give a conventional Reynolds

of

dissipation

the two-thuds

for correlatmg

independent

some scatter

6 is to be expected

comparatively

use

rates m mlxmg

behavlour

ables m Fig 6 is largely made up of variations m the Schmidt group and that variations m the

gives

deviation

the

will remam

It should be made clear that the range of varithe

It should be noted that equation invalidate

This

agitator characteristics reported.

$P

suspension.

by PIRET et al [12].

considerations

data

is due to uncertamtles m the values of the mterfacial areas for suspended particles as obtained by screen analysis. The equation of

coefficients

independent of power dissipation until the latter is raised considerably above that needed for

L

per

bed as the flow

G

(5)

PCZ

The pressure drop m packed beds has been car-

com-

related by CHU [38] as

coefficient

As the power IS further

increased equation (4) IS followed, but with the practical hmltatlons of allowable power dlsslpation, the Increase m mass transfer coefficient not

be easily

observed

However,

whence,

may

it 1s almost

from equation

P

certam that agitators producmg high ratios of flow to shear for a given power are most effective

-

=354

(5)

,+O 44 (1 _ a144

V

Inserting

for suspendmg particles, so that particles suspended with these agitators come under the m-

this equation

z)l 44 6'2.56 PC

2z4

for P/v

mto equation

(4) and using 0 13 as the value of the dimensionless constant we find

fluence of power at a lower value of power dlsslpation than with agitators producmg high shear and low flow rates This question IS complicated by the fact that small particles are most easily suspended, and if

= 0 313 (sL)‘i”

their size is of the order of the scale of the smallest 50

(9”

(9”’

(6)

The contmuous

phase heat and mass-transfer

a range

turbulence

with

a conventional

Z = 0 41,

correlation

and

by

Smce the work of FALLAT [87] covered of values of Z of 0 40 to 0 42, taking

findings with mwng

we find Ns, = 069 N,, 2/3 Nsl/S ja

Ol-

stants

0 69 NRe-1’3

=

This result may be compared

The agreement

as well represented

between

turbulence

POWER

REQUIRED

(8)

equations

(7) and (8)

flow

m

As previously

not be supposed

TO SUSPEND

SOLIDS IN

VESSELS

discussed

the

pipes

the

agitator

dissipation

Rlasms

power,

the rate of solution

of

o 0396

small increase

m solution

correlations

rate may be realized

of compete

the mass transfer

(N&-l/*

although as the power increased a comparatively

is further

At the pomt

-

the conhas been

particles lust completely suspended m a mixing vessel IS dependent on factors which do not include

equation

LGa

could

MIXING

Heal and muss transfer zn pope $0~

AP.dp.Pc_

The result

by

complete

turbulent

vessels to evaluate

ates to interrelate energy dissipation and the specific rates of turbulent transport phenomena

data reported

jd = 0.67 Nne-l’3

is substantially

transfer

experimental

to apply. It would appear that a more general law oper-

jd = o 626 Nne-‘J 322

by him can be almost

mass

some

m this correlation

local isotropic

with that given

which, for the range of experimental

usmg

found to apply reasonably well to turbulent flow m packed beds and pipes where the theory of

(7)

by FALLAT [37] as

For

propertIes of dispersions

suspension

of sohds

is given

by either

coefficient

or (P), so that on equatmg

(1)

may be used to predict the pressure drop and hence the power

coefficient, we arrive at an expression for the power required to Just suspend the particles, which

dissipation

per umt volume

If this result IS then combined m the same fashion

of fluid

with equation

as m the previous

(4)

we find

which may be compared

P _ = V

(9)

with the correlation

CHILTON and COLBURN [39] which

of

Reynolds

number

range

2000-20,000,

with that proposed

P _ = const (g APY’ V

which

finds

is

data have while

Hl”

D1ja

by ZWIETERING

p _ = const* (g AP)’ 35 P2 3 c2 V

been reported The sigmficance of the present findings is the reahzation that power dissipation

a2

p

and with that proposed

12 per cent over the

the range over which most experimental

(10)

PC

by KNEIJLE [13] who finds

This equation agrees with equation (9) at a value of Nne = 10,000 and results m a maximum from it of only f

(gAP)~~~~‘3

const*

This result may be compared

gives

Jo, ,, = 0 023 Nne-‘J 2o

deviation

the mass transfer

takes the form

Section

j4h = 0 058 Nne-O 31

so as to ehmmate

these

relationships

PC

OYAMA

and ENDOH

P _ = V

per umt volume of fluid is the factor which allows a comprehensive correlation of heat and mass

const

[PO] who

Ho 3g

0 35

[41] report

(gAp);y2 PC

The most obvious discrepancy between present result and the formulae previously

transfer coefficients for turbulent fluids to include fluids where the turbulence is induced by mechanical agitation

the pro-

posed is that equation (10) does not contam the particle size as a variable, and it would seem reasonable to suppose that it ought

It should be noted that equation (4) was derived by combmmg the theory of local isotropic

Chem

51

Engng

Scz

Vol

16, Nos

1 and 2

December,

1961

P H

CALDERBANK~~M

However, If the square root of the particle diameter IS mcluded m equation (10)as suggested by the experimental correlatxons above, this might imply that correlatxon (4)should m&de a factor of dl’* It IS unhkely, with the present preclslon of experimental mass transfer data obtainable with sohd dlsperslons, that this factor could be identified with certamty The same argument suggests that It 1s only shghtly more hkely that an effect of sohds hold-up on the mass transfer coefficient could be easily identified, although m this connexlon both PIRET et al [12] and WILHEI~~ et al [42] report a small influence CONCLUSIONS

0F THE

PRACTICAL

INTERxwr

FROM

FOREGOING

OIle clear conclusion IS that when a dispersed phase ISsuspended, or m free rise or fall and under the influence of gravity, the heat and mass transfer coefficients are almost completely unaffected by mechamcal power dissipated m the system Thus, for example, m dlssolvmg sohd particles m hqmds, little advantage can be gamed by mcreasmg the agl~tlon intensity above the pomt at which the particles are freely suspended Clearly, mlxmg impellers should be deagned to secure suspension of particles with a minimum power consumption which should not greatly exceed this level For gas-hqmd dlsperslon power dlsslpatlon increases the gas-hqmd mterfaclal area and mcreases the mass transfer rate for this reason although, as before, httle effect on the mass transfer coefliclent can be reahzed Correlation (1) shows the important influence of the difference m density between the suspended dispersed and the contmuous phases on the transThis means that where this fer coefficients difference m den&y 1s small transfer rates are extremely low This situation IS most likely to be realized m some hqmd-hqmd extractlon processes and can be countered by mcreasmg the gravltatlonal field strength as m centnfugal extractors The problem IS also met with m dlssolvmg polymers m solvents, where the density difference between polymer solution and sohd polymer can be quite small, and the rate of solu-

B MOO-YOUNG

further impeded by the high vlscoslty of the solution as would be expected from correlation

tion 1s

(1) An Important further example IS found m aerobic fermentation processes, where the over-all rate process can be determmed by the rate of supply of oxygen to the growmg orgamsm, whose density may not differ greatly from that of the surrounding nutrient fluid Thus, although the rate of supply of oxygen from aerating gas bubbles to hqmd may be increased by agltatron (by mcreasmg the gas-hqmd mterfaclal area), the rate of assimllatlon of dissolved oxygen by the organism 1s unaffected by agltatlon, and 1s seriously restricted by the small solid-hqmd density dlfference, and frequently also by the high vlscoslty of the hqmd medium In this case the very high mterfaclal area between the orgamsms and the oxygenated fhud sur~undlng them assists the rate of oxygen consumption It may be found, however, that the product of the high orgamsmhqmd mterfaclal area and low mass transfer coefficient may be less than the product of the comparatively low gas-hquld mterfaclal area and high mass transfer coe&clent, m whteh case the rate of oxygen supply IS no longer under the control of the plant operator or designer, but 1s determined by the physlcal propertles of the system When dlffuaon-controlled crystalhzatlon processes are earrled out m suspended beds of crystals the maximum rate of erystalhzatlon 1s aehleved, and where the density difference between crystal and liquor 1ssmall, as m some organic preparations, rather low rates will have to he tolerated because the mass transfer coefficients are almost unaffected by agltatlon. It may be shown that Note added zrz proof correlation (1) can be derived from the Froesshng equation when this IS combmed with the equation proposed by ALLEN [4$] for the free rtsmg or falling velocltles of small partlcles

A constant of o 28 instead of 0 31,as proposed, IS thereby obtamed Again correlation (I ) may also be derived by 52

The contmuous phase heat and mass-transfer propertles of dlsperslons combmmg

with

mass transfer

Stokes’

Law

the

equation

for

Jd = -kL (Nsc)2/s J, v, k, = contmuous-phase

from spheres moving m the Stokes’

region, as deduced theoretically by FRTEDLANDER [50], [NsL = 0 89 (dV/D,)‘/‘] when a constant of 0 34 IS obtamed Recent correspondence

has

revealed

diameters

theoretical

equation

WP,)~‘~

mass transfer cocfficrent

1 = scale of smallest eddies m turbulent

fluid

L = scale of prrmary eddies m turbulent

fluid

dunenslonless

‘VR,’ =

Reynolds

&/3

Number

up to 5 cm as well as the proposed

dvnensronless

by this worker N Nu = Nusselt

K, = 0 975

1.e.

p PCVs

that

correlation (2) represents the data of BAIRD [22] obtamed with spherlcal cap bubbles of equivalent

h

= &

0‘I4

IlL1j2 f

Number

NSh = Sherwood

dunenslonless

Number

lmensionless

Acknowle@nents-It has recently eome to the authors’ notlce that the general pnnclple employed m this and preVIOUSwork of measurmg gas-hqmd mterfaclal areas by hght transnusslon 1s sumlar to that described by SAUTER[43] for measurmg these properties m sprays Dr R EDGE and Mr F EVANS were responsrble for the mass transfer

data

obtamed

m sieve-plate

dlmenslonless

dunensionless

drmenslonless

columns

This paper 1s published by pernusslon of the DIrector, Warren Sprmg Laboratory, Stevenage, Herts and the work was carned out whde the authors were employed by the Department of Scientific and Industnal Research

dunensronless (P/u)

of contmuous

Cp = Specific heat at constant

\/[a-2(d)]

pressure of contm-

= root mean

square

fluctuatmg

velocity

over

dispersed

and

drstance d

uous phase flmd

pi = dispersed phase densrty pe = continuous phase density

d = particle drameter of drspersed phase dfi = pipe diameter DL = drffuslon coefficient m contmuous phase g = acceleratron

phase

Ap/L = pressure drop per unit length of pope V, = supefficral flmd velocity m packed bed V = velocrty of particles of disperse phase

NOTATIONS

G = mass super&al

= power drssrpated by agrtator per unit volume

pc = contmuous Ap = hfference

flmd flow rate

contmuous

due to gravity

he = heat transfer coefficient m contmuous

phase vlscosrty m density

X = volume fraction

phase

voids m packed bed

X = thermal conductlvlty

H = mass fractron of sohds m dupersion

x, y = dlmensronless

Jd, ,, = mass or heat transfer factor drmensronless

between

phases of contmuous phase fluid

constants

REFERENCES

PI PI PI 141 PI PI 171 PI PI WI Pll WI WV P41

CALDERBANK P H

Trans

Inst

Chem

EngTs

1958 36 448

Symposaum on Dzstzllatwn, European

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, MATTERN R

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England,

1960

Trans

Theses, Umverslty

Chena -Ing

U

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and BAIN J

of Washmgton

of Edinburgh

and BILOUS 0

1953

1956

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Trans

Inst

Chem

Engrs

1966 38 11 Chem En@%?SC% Vol 16, Nos

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December,

1991

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54