Experimental determination of axial dispersion coefficient in a bubble column

Chmucal Emgmenng P.zr,jamn Ptess Ltd

Scrntce Vol 35 IIP 220?-2205 Pnntcd an Great &dam 1980

EXPERIMENTAL DETERMINATION OF AXIAL DISPERSION COEFFICIENT IN A BUBBLE COLUMN

Chemical

and Nuclear

SOON-JAI KHANG*

and

Engmeenng

Umverslty

Department,

SAMIR

P

KOTHARIt

of Cmcmnati,

Cmcmnatl,

OH 45221, U S A

(Accepted 18 February 1980) Ahstraet-A new graphtcal method of fincbng axial dispersion coefficients m a bubble column IS developed This method requues only the area above an experimental curve, thus IS simpler than the conventional method of curve fittmg The new method IS used to find the dimensIonless chspersion number m the bubble column and shows that the drsperslon number decreases wdb column Reynolds number due to Increasing coalescence of bubbles at higher ps flow-rates

I

INTROLWCTION

The axial dispersion coefficient III hqmd-phase has been used to study the hqmd-phase mlxmg m bubble columns Thus m turn ISrelated to the amal concentration gradient in hquld-phase In spite of the mdustrmlly growmg importance of bubble columns, relatively few stu&es have been reported on this subJect[ 11 The present work developes a new graphical method of findmg the dispersion coefficient usmg a moment method for mean resldencetnne-dlstrrbutlon m a bubble column This method requues only the area above the expenmental tracer curve, thus simpler than the conventlonal method of matching the experlmental and theoretical curves to determine dlsperslon coefficients

0

ACa(x,f)dx=M

for

fz0

where D IS the dlsperslon coefficient In the axial drrectlon For the sake of amphclty, the followmg vanables are introduced to normahze eqn (1))

AXIALDIM’RRSION MODELFORBUBBLECOLUMNS The effect of llquld properties on hqmd-phase axial mlxmg m bubble columns was studied usmg an axml dlsperslon model by Hlluta and Klkukawa[21 The partml dtierentlal equation based on this model has been solved analytically for transient tracer concentrattons by Slems and Werss [3] Figure 1 shows a bubble column where a batch of stagnant hqmd IS contamed and a gas IS bubbled from the bottom of the column A slug of non-volatile hqmd-tracer IS Introduced at the bottom and Its concentration IS measured at the top of the column by a probe A strong inorganic electrolyte solution IS used for the tracer to mmimlze the effects of bubble surface Practically all the tracer maternal IS contamed m hqmd-phase and the bubble surface does not carry any extra amount of tracer A matenal balance of tracer m hqmd-phase gves the followmg differential equation, D a2ca _ aCA =-at

L.

2=X/L

(2)

B = tiT

(3)

t= L/u

(4)

P=uL/D

(9

c = CAlC_y

(6)

where E, U, P and CE are the mean bubble-readencetime, the mean bubble-nse-velocity, the bubble Peclet number and the average concentration of tracer m the column, respectively Substltutmg eqns (2)-(6) to eqn (1)

4J

-

Concentration moomwommnt

llqwd

phase

(1)

CA(X,f)=O~l=OforOO *To whom correspondence should be addressed tPresent address PEDCO Environmentat Inc , Cmcmnatl.

1 Tracer IS rnJected at the bottom and measured of the bubble column

Oluo, U S A

2203

at the top

2204

SOON-JAI KHANGand S~klta P KOTHAR~

becomes 1 a2C ac P a.2 a9

-2=-

@ e=o

C(z,t?)=O $$=O

for

@x=0

B
for

820

for

820

t I cl

C(z,B)dz=l

Ttme

By takmg Laplace transform with respect to 0 and usmg the lmtlal and boundary condltlons, eqn (7) yields the followmg solution m the frequency domain

f? = VW/s) cash (zMPs))lsmh

X/U’S)

For the concentration at the top (z = 0), the above equation further slmpldies to c0 = d(P/s)/smh

t/(Ps)

(8)

Since all the tracer injected at the bottom remams in the column, the normahzed response at the top of the column, Co, IS a cumulative dlstnbutlon The denslt_y function IS a denvatlve of Co whch IS equivalent to SCO m the frequency domam Applymg the moment method, the zeroth and first moments of the density function become

WW = 1 = hm S-DO smh (VU+)) d M, = - tz z (sco) d t/(Ps) = p,fj = - !z z smh (t/P(s)) Therefore the dimensionless tion becomes

( t 1

FIN 2 hsperslon

coefficient IS related to the area between the tracer response curve and the honzontal hne of uruty he&t

Thus, the dlsperslon coefficient D m a given bubble column can be obtamed from the tracer response curve by measunng the area between the response curve and the honzontal lme of umty he&t TRAcEn

-NT

AND RESULTS

Ftgure 3 shows the expenmental setup The glass column of 0 022 m I d and 19 m height contamed distilled water (water level, 1 807 m) and had a very fine fretted glass disc near the bottom of the column An inert gas, hunudlfied mtrogen, generated gas bubbles through the f&ted glass disc Various sizes of bubble nsmg Hrlth various velocltres were observed for a given gas flowrate Due to the expenmental ddlicultles, no attempt was made to measure the various sizes of bubble The mtrogen gas was taken out through a Teflon tubing from the top of the column to a soap bubble flow-meter to measure its outlet flow-rate There was some vapor space and a 90”-bend at the top of the column to disengage hqmd droplets leavmg with gas About 1 cm3 of 4% KC1 solution was introduced wrthm 0 5 second, Just above the fntted glass disc by a synnge pump The changes of KCI concentration were measured at the top of the

(10)

mean of the density func-

pa = M,IM, = PI6 or wth eqns (3) and (4) the mean m the dnnenslon of time becomes (11) The above mean can be convemently the tracer response

curve,

= shaded area avove

measured Co vs t, as follows [4]

the response

curve

from

m FQ

2 (12)

Glass column Platinum prob. To soap-bubble flow motor Canductiwty motor Rocarder

F@

F Frittad glacs dnc G 1 Gas cylinder H * Go8 humidlfter

P ’ Resmtro gouge

1

’ Tkacu syringe Pump

3 Expertmental set-up

Expenmental

Fig 4 Typical

tracer-response

deternunatlon

of axial dlspersron

2205

the column Reynolds number than does the dlsperslon number m packed column This 1s thought to be the results of mcreasmg coalescence of small bubbles as the gas flow-rate increases At the highest gas flow-rate (Re = 33 5), bubbles of large diameter comparable to that of the column were observed It should be noted that the purpose of F@ 5 1s to show the general trends of the drspersron number rather than to gve the exact correlations for various bubbling condttlons For more basic correlation, it should include bubble characterlstlcs such as bubble size dlstnbutlon, bubble frequencies and bubble nse velocities

curve

CONCLUSIONS

00

10 ~.yno,dr

Fig 5 Bsperslon

20 numb..

A new graphical method of finding expenmental dlspersion coefficients for a bubble column was presented This method required only the area above the expenmentat tracer curve, thus simpler than the conventlonal method of curve fittmg The dlmenslonless dlsperslon number was found from the tracer expenments usmg the new graphical method and plotted against the column Reynolds number The results showed that the dlsperslon number decreased with the column Reynolds number due to mcreasmg coalescence of bubbles at higher gas flow-rates

I du.v/r)

number vs Reynolds

number

column Just below the hquld level usmg a specially prepared platmum probe w&h a conductlvlty clrcmt developed by Khang and FltzgeraldP] Figure 4 shows a typical expenmental curve along with theoretical values calculated from the followmg analytlcal solution [3], CO=l+2

5J (-1)“exp n--l

n2v2

[

-7m

1

(13)

The area above the expenmental curve was measured and was used to calculate the dlsperslon coefficient usmg eqn (11) The theoretical values m Fig 4 were then calculated from eqn (13) usmg the experimental dispersion coefficient The experimental curve agrees fairly well mth the theoretical values Thus, the new graphical method mves accurate measurements of dlsperslon coefficient wthout any numerical curve fittmg techmque A dlmenslonless dlsperslon number 1s calculated by dwldmg the measured dlsperslon coefficient by the water depth and the superficial gas-velocity at the ambient condltlon Figure 5 shows the dlsperslon number as a function of column Reynolds number This figure IS somewhat analogous to the dlsperslon number correlations m packed columns(41 In thrs analogy, the bubbles and the bubble rise velocity are considered as solid spheres and interparticle fltud velocity m a packed column An accurate estlmatlon on the dlsperslon coefficient m bubble column cannot be made from the analogy due to their physical differences However, the general trends m dlsperslon number show that both the dlsperslon numbers m bubble column and m packed column decrease with Reynolds number The dispersion number m bubble column decreases more rapidly mth

CES Vat

35 No

I&l

NOTATION

A

CO CA Ci? C D L

M MO MI P t IA UO X Z

liquid-phase cross-sectional area of column, m2 dlmenstonless concentration at the top of aerated water concentration of tracer, kg/m’ average concentration of tracer, kg/m3 = C_.JC,, dlmenslonless concentration axial dispersion coefficient, m2/s height of aerated hquld, m amount of tracer injected, kg the zeroth moment of density function, dimensionless the first moment of density function, dImensionless = uL/D, bubble peclet number, dimensionless time after tracer injection, s bubble rise-velocity, m/s superticlal gas-velocity at the ambient condition, m/s liquid depth, m = x/L, drmenslonless hquld depth

Greek symbols 8 dlmenslonless time J.L the mean of density function, s. or vlscoslty water, kg/m s JL~ the dimensionless mean of density function p density of water, kg/m3

of

[II Elssa S H and Schugerl K , Chem Engng Scr 1975 30 1251 Gl H&~ta H and Klkukawa H , Chcm Engng / 1974 8 191 r31 Slemes W and Weiss W . Chem Ing Tmhnfk 1957 29 727 0 , Chemrcal ReactIon Engmeermg. 2nd Edn, r41 Levensplel bl

Chap 9 Whey, New York 1972 Khang S J and IUzgerald T J Ind Engng Chem Fundls 1975 14 208