either, because the thickness of liquid film on theplain resin disks was not measured.

Conclusions

The rotating-disk gas-liquid contactor equipped withdisks madeof appropriate wire meshis suitable forexperimental studies of liquid-phase mass transfer ingas absorption, mainly because the penetration modelcan be applied almost perfectly to this type of equip-ment for the range of gas-liquid contact time from

0.2 to 1.5 seconds, which can bevaried by the rota-tional speed of the disks. The kLa values for this typeof contactor are comparable with those for gas bubblecolumns.

Nomenclatureerfc(x) = complementary error function of x=l-erf(aj)

'e-v*dy

>o

ierfc(#) = integral complementary error function of xSoo

erfc (y)dy X= -Le-x2-x erfc {x)

in

A = total gas-liquid contact area [cm2]a == specific gas-liquid contact area [cm2/cm3]C = concentration of solute in liquid [g-mole/cm3]D = diffusivity of solute in liquid [cm2/sec]E = meter reading [-]

H = radial width ofwettedsurface of rotating disks [cm]I, I0, Ih I2, h == integral values defined by Eqs.(18),(19),

(20), and (21), respectively [-]JcLa = volumetric liquid-phase mass transfer coefficient

[sec"1]

kL = liquid-phase mass transfer coefficient [cm/sec]kL = mean of kL for a ring element of contact surface

[cm/sec]kL = mean ofkL for entire contact surface [cm/sec]

= depth in liquid film [cm]M = amount of gas absorbed [g-mole]Nt = mass flux at time t [g-mole/cm2à"sec]Nt = mean of Nt for a ring element of contact surface

[g-mole/cm2 à"sec]

Nt = meanof Nt for entire contact surface[g-mole/cm2-sec]

n = rotational speed of disks [r-p.s.]R = radiusofdisk [cm]r = radius [cm]t = gas-liquid contact time [sec]0 = cos-1-^^- [radian]<Subscripts>1 = initial value

L = liquid phase<Superscript)

* = equilibrium value

Literature Cited

1) Crank, J: "The Mathematics of Diffusion", p.45,Oxford at the Clarendon Press (1955)

2) Higbie, R: Trans. A.I.Ch.E., 31, 365 (1935)

3) Ohshima, N: Private Communication, Department ofChemical Engineering, Kyoto University, Kyoto (1969)

EFFECT OF LIGHT-SOURCE CHARACTERISTICS ON THEPERFORMANCE OF CIRCULAR ANNULARPHOTOCHEMICAL REACTOR*

TAKASHI AKEHATA AND TAKASHI SHIRAI

Research Laboratory of Resources Utilization, TokyoInstitute ofTechnology, Tokyo, Japan

Three light source models, i.e., radial, specular, and diffuse line source models, arecompared for light intensity profile, overall rate of reaction, and radial scale-up ratioof annular photochemical reactors, with the assumptions of constant absorption coefficientand constant quantum efficiency. The necessary condition for the radial light model to

hold is that, when the ratio of inner radius to height of the reactor m, is 0.1, that of outerradius to inner radius p should be less than 3, and that when m is larger p should beclose to 1. Published reaction data are analyzed according to this condition.

Introduction

There are two fundamental types of photochem-ical reactors in terms of reactor-lamp configuration,

Received on February 25, 1972Presented at the 36th Annual Meeting of the Soc.

of Chem. Engrs., Japan, April 4, 1971

i.e. cylindrical reactor and circular annular reactor.

The cylindrical reactor has lamps appropriately posi-tioned outside the reactor to provide uniform radia-tion to the reactor. The circular annular photoreactorreceives radiation from a. cylindrical lamp inside theannulus, and this configuration is of interest for bothresearch and industrial purposes.

VOL. 5 NO. 4 1972 (59) 385

G

(a )

>I Jo

( b ) /< /

(c )

Fig. 1 Light emanating from three types of lamps:specular(a)/ diffuse(b), and radial(c) light sources

A few investigations of the performance of photo-chemical reactors have been reported. To predictreactor performance, the light intensity profilethroughout the reactor should be specified as wellas mass balance considerations regarding the react-ing species. Most previous investigations employan assumption of uniform radial incident light (radiallight model).1.2.MXs.io

As for annular photochemical reactors, DranofT andhis co-workers7-8-9) investigated radial scale-up of aperfectly-mixed reactor. They carried out the pho-tohydrolysis of chloroplatinic acid in dilute aqueoussolution in annular reactors of several sizes, using a

nominal 8-watt black light bulb, a General ElectricModel F8T5/BLB. They first7 8) assumed the radiallight model, defined a radial scale-up ratio Rji{c),

Rn(c)=P,(c) _2J?U-e-"*'(r'- å .)}

Pi(c)SI\{l-e-ft..(n-n)}

(1)

and analyzed the experimental results in terms ofRji(c). Recently Jacob and Dranoff9) reported afurther detailed study to determine the quantumefficiency of the photohydrolysis as a function of lightintensity and reactant concentration, and analyzedthe reaction data on the basis that the lamp is a"normal" line source. The term "normal" corre-sponds to "specular" in the present paper.Since the radial light model is simple in every re-

spect, it is worthwhile to know under what conditionsit may be applied. It is quite clear that a largeammountof non-radial radiation emanates from the

lamp and passes through the reactor. Matsuura andSmith12) proposed a diffuse light model to representthe light intensity profile within an elliptical reactor.Harada et al.5) also pointed out in their study of anelliptical reactor that the radial light model shouldbe carefully employed in the calculation of the light

intensity profile, and they presented a two-dimensionallight distribution model. Very recently, Zolmer andWilliams15) presented a three-dimensional light dis-

<k

B

c

T

X

x i -蝣u t ~

A

L

'< p ->i-

> J -H

l " - r o -! y

in e s o u r c e is A B

Fig. 2 Reactor geometry

tribution model for the same problem. Since the

lamp employed by Dranoff and his co-workers was afluorescent lamp, the calculation should be made on

the basis that the light source is diffuse.The purpose of the present paper is to predict the

light intensity profile in a circular annular reactor bylight sources of three kinds, i.e., specular, diffuse, andradial sources, to find the necessary condition under

which the radial light model may be employed, andto analyze published reaction data in trems of thepresent theory.

Light Intensity Profile in Annular Reactor

Characteristics of line light sources

Light emitted from mercury and neon lamps isusually considered specular, and that from fluorescentlamps, diffuse3). Radiation from three types of lampsare shown schematically in Fig. 1. The differential

length dx of the specular line source radiates uniformlyin all directions in the same way as a uniform pointsource, and that of the perfect diffuse line source em-anates in the a-direction to the normal of the sourceaxis,

/-/o cos a(2)

where /0 is the normal intesnity of radiation flux.Since the projected length of dx on the normal tothe ^-direction is dx'cos a, brightness of radiationbecomes independent of a, i.e., B=J0cosaldx cos a=Joldx.

When the source is one to which the radial lightmodel can be applied, the intensity shows no varia-

tion with height.Light intensity profile by specular line sourceConsider the annular reactor shown in Fig. 2. A-long the centerline of the reactor whose height is Land inner radius r0, a specular line source of lengthL is placed. A reacting solution of absorption coef-ficient fjt fills the reactor volume from r0 to r. It isassumed that the absorption coefficient is constant inthe course of reaction and that light absorption andscattering by reactor inner wall and ' lamp-coolingjacket are negligible.The light intensity at a point P(r,z) due to radia-tion from the dx section is given by13)

386 ;eo) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

Tnti-nQi'tv-/Diminution due toVAttenuation due \ (oIntensity-^the inverse law ^to Lambert's lawy1 (da

The diminution term is calculated as

Diminutions ^^J of radiation) =jAfx (3b)(Distance)2 r2-\- X2

The intensity of radiation /J is related to the sourcestrength SL of the lamp per unit lamp length. Thetotal flux of radiation, SLdx, from the dx-section ofthe specular lamp is equal to the sumof the radiationcrossing the sphere of unit-length radius, whoseoriginis at the J#-section, that is,

SLdx=2\n/2{Jsod%)2.x cos ada =AnJldxJo

HenceJi=SL[4z(3c)

The attenuation of light occurs only in the absorbingmediumwhose optical path length is (r-r0)lcosa.The attenuation factor then becomes

Attenuation=exp {-v(r-rQ)i/r2+x2lr } (3d)

From Eqs. (3a) through (3d), Eq.(3a) takes the form^Intensity=^|^* exp{-^(r-r.J^!±»!) (3)

and, therefore, the intensity at P due to the radia-tion from the entire line source is found by integrat-ing Eq.(3) over the source length AB,

or2+x2L~z 1

o r2+x2exp(-/^(r-ro)1 V2+a2

rfOJexp (-//(r-ro)^2+^2 )^] (4)

Whendimensionless variablesm=r0IL9 T=pr0, £=zlL9 7]=x/L, and p=rlr0 (5)

are introduced, Eq.(4) becomes

r(p, Z)=AnLm2p'[\(/(y, p)dv+\l~:gs(y, p)dv](6)

where

«"(*' P]= 1+vimY) exp H^V1*Jp.(7)

The mean light intensity Is(p) at p may be definedas

^=tJ>' z)^(8)

and is expressed asI'(p)=-

4tt LmV-G°(P)

(9)

where

G»= £[JV(% P)dv+\l'igs(v, p)dy]d£(10)

And the volume-mean light intensity over the reac-

tor volumewhose radius is p becomes

SLAn Lm2(p2-1)

f'å 2-l)JiGS(P)dp

(ll)(12)

Light intensity profile by diffuse line sourceWhen the source is perfectly diffuse, the diminutionterm should be calculated by the equation

Diminution =,(Normal intensity) (cos a)(Distance) 2_Jjdx r

(r2+x*) i/rz+x*(13a)

The normal intensity /jf is related to SL and is ob-tained by a treatment similar to Eq.(3c), that is,

S7T/2Utdx coso)2tz cos ada=7T2J%dx

0

and thusJt=SJz'(13b)

The intensity at P due to the Jx-section therefore be-comes

Intensity =

SLdx7T2 (r2+^2)3/2exp-tHr-ro&± *

(13)

and the intensity at P due to the entire source is givenby

I d(p, n=- Lnf-p'{JV (V, p)dv+\1o'Cga(v, p)dy\

(14)

where

{f(V, p)=-

(l+vV(mV))snexp-r(p-l)^/

1+my.

(15)

The mean light intensity Id{p) at p isI d(p) = ^t L nfip'-Gd{p)(16)

where

Ga{p) =\I\y {v' p^+H'^^v, P)dPy?(17)

The volume-mean light intensity isldo,M =-

n*Lm\(?-\)C'-1)JiGd(P)dp (18)

Light intensity profile by radial light modelIn this case the intensity does not vary in the z-

direction and Tr(p) is identical to Ir(p, f),

r(p, Q=ir(P)_sL_ sL

1

2n Lmpexp {-Mr-ro)}exp l-r^-l)}

(19)

And the volume-mean intensity becomes

I raM= 2jt Lm%p2(20)

where

Gr(p)=mp exp {-T(p-1)} (21)Calculated resultsCalculated profiles for the case of m=0. 1 and z=1.0 are shown in Fig. 3. Solid lines are for the spec-ular source and are obtained by Eq.(6)? and brokenlines for the perfect diffuse source by Eq.(14). Tocalculate the intensity at £*(-z/L), the lamp length

L was divided into 10 sections and the contributionfrom each section was calculated by the five-point

numerical integration methodof Gauss. These con-tributions from 10 sections were finally summedup togive the intensity due to the entire lamp length. The

VOL. 5 NO. 4 1972 (61) 367

0.5

0.4

0.3

N0.2

0.1

0

F ull lin es ar

3

I

����������������������������������������������������������������w

�" ' I

I I I I I / /i / / / 蝣蝣'/

0

d d ash ed

0.5 I/In

e b y sp ecu lar sou rce anlin es b y d iffu se so u rce.

Fig. 3 Light intensity profiles (m=0.1 and r=1.0)

1. 0

: o .5

T Jo

0

m = i. O

T

^ - 0 / 1

- 1 0 m = 0 .5

T0)

A O m = 0. 1

,- ' " .. .. 蝣 T^ __^- 0

_ - 1- 1 0

1 8

2 3 4 5 6

pFig . 4b Gd(p) lp2 vs . ' p

2

a _T J

- S P E C U L A R

0

- D IF F U S E C ^ > v '

w A i P ^

% K

. 1

. 0 ^ ^ T - 0

<i * ,/ ' y / *o _

" b |a

CL_

0

,/ , / y - ^ 0 .1

' A ^ r - - " ^ 0 .5

< J & & ^ r " ~ " ^ Ti rj 5* 3 2 ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ S S S I

?

Fig. 5a [P(Gs/p)dp and ^{Gdjp)6p vs. p

0 -JfFC ^ j <r ¥ m = 0 .1

5- S P E C U L A R - D I F F USE

o r

1

X - R A D IA L

I I I I I I I I I I I I I I I I I

1 0H T

Fig. 6a Radial scale-up ratio vs. v (m-0.1)

1. 0

CMa _

蝣0 .5mo

0

m = 1. 0

U

. 0- -- '., 1

1 0 m = 0 .5

/ / T/ ^ - o

/ ,.-'- ""' ./ 1- 1 0 m = 0.1

z ^0

- 1 0

? 3 A 5

pFig . 4a Gs{p) jp2 vs . p

1. 0

<u. 0.5

m = i,0r

. 0

/ - - " > 1 m r O .5

. - '" r

y ¥

" , 1

1 0 m = 0 .1

o

0

/ " X " - ' " -c

/ 一蝣 ¥ / - o

- 1 0

¥ ^^ X - '

2 3 4 5 6

pFig . 4c Gr(p) /p2 vs . p

2

O -._ p

o

0

" r-W o

.// too o

F

｣ 1k .

' rtj

CU E

0

W

0 .2

0 .5

1

vm *1 5 ... p - 1

Spme~T^"^dpvs.p

1 0

5

m = 0 . 5

f= 6 ^ - S P E C U L A R

- D I F F U S E

c c

1

_ ^ 3 - R A D I A L

3 . - _ _ J T ¥

- : ^ - - - $ ^

= ^ = ^ ^ ｧ = 1 ! ^ [ I I I I

1 0 ' T

Fig. 6b Radial scale-up ratio vs. t (m=0.5)

388 (62) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

calculation was done by FACOM 270-20 at thisResearch Laboratory. The intensity is shown in a

relative quantity to the maximumintensity in eachcase. The maximumintensities are observed at thepoint ofz/L=0.5 and rjro=1.0, and their values are

Im.J(SLl47:L)=27. 5

by the specular source and

/m«/(Si/VZ,)= 19. 6

by the diffuse source.

It is seen that there is a portion uniformly radiatedas long as about 40%of the reactor length around themid-height, and the diffuse light model gives a muchflatter ^-distribution.

Gs{p)lp2 and Gd{p)jp2 are representatives of themeanlight intensity at p, since Ii(p)=(SLlni7rLm2){Gl(p)lp2), where i=s and d. The curves inFigs.4aand 4b show that there is not much difference betweenthose of the specular and of the diffuse sources. Thecurves in Fig. 4c are of the uniform radial model. Itcan be seen that the radial light model gives consider-ably different values than do the other two cases.

The quantity corresponding to the volume mean

Sp{Gijp)dp, is

plotted against p in Fig. 5. Whenboth the absorp-tion coefficient and the quantum efficiency are con-stant irrespective of reactant concentration, overallrate of reaction occurring within the reactor is givenby

Pt= <P^(r2-rl)LPav

2SLL* f G\p)=<PP-dp (22)

(23)

where ni^=2i ir, and 4 for i=r, d, and 5, respectively.The value ofPl may be calculated from Fig. 5 withtheknowledgeof<py ft, and SL. For the caseofm=0.land r=0.1 at ^=5, the values of (Pil(p[Ji2SLL2') forthe three kinds of sources are close and become

r:d:5=^:A^: 0^=0.l65:0.153:0.1482 7T 4

At large m, however, these values considerably differ.For the case ofm=0.5 and t=0.1 at p=5,

r:d:5=

1.65 .1.13 .1.20=0.83 : 0.36 : 0.30,

being very different values.It maybe concluded from similar comparisons thatthe radial light model is valid when ^<3 for #2=0.1,within 10%accuracy, from the viewpoint of the totalamount of reaction per unit time. Whenm is larger,p should be close to 1.

Radial Scale-Up Ratio

As stated in the introduction, Dranoff and his co-workers proposed the radial scale-up ratio Rji(c) as anindex relating size of the annular reactor to total

amount of reaction. Whenthe reactor volume from^=1.0 to pi is taken as a reference volume, the scale-up ratio is calculated from Eq.(23) and equations for/*w, and expressed as

Rin{c)=^{Gtlp)dpl\f\Gilp)dP i=s, d, and r (24)

As a matter of convenience, px maybe taken to be1.5 after the experiments by DranofTet al. For theradial light model the ratio becomes

i?51(c)= (e-'t"-»-l)/(e-°-5r-l)(25)

The relationship between Rn and t for m=0.l andm=0.5 is shown in Figs. 6a and 6b, respectively. Itis seen that when m=0Athe three kinds of sourcesgive indistinguishable values of Rn for p<3, whereasthey differ greatly even at p=3 when m=0.5. It isconcluded that from the standpoint of the RjX value,the radial light model may be employed when p<3 form-0. 1 for both specular and diffuse sources. Whenm is larger, p should be close to pi.

Discussion

Experimental results by previous investigatorsHarris and Dranoff7) investigated the effect of reac-tor size on the total reaction amount using the de-composition of chloroplatinic acid. The quantum ef-ficiency of the reaction is a complex function of variousfactors. Harris and Dranoff first assumed it con-stant, and later Jacob and Dranoff9) proposed anempirical relationship between <p and light intensity

and reactant concentration.First, the assumption of constant quantum efficiencywill be employed for simplicity and the variable quan-tum efficiency will be taken into account in later dis-cussions.

Since the lamp they used was a fluorescent one,data analysis is made in a more rigorous way on thebasis of the diffuse light source. Their experimentalconditions were

m=r0IL=1.25 in./10.5 in.=O. 119

pi=r1lr0=S. 125 in./l.25 in.=2.5

^2=4.875 in./l.25 in.=3.9 and

r=r0i"w=(1. 25 in.)(O. 224 in.)-1=0. 28

where pm is an average absorption coefficient of the

reaction mixture over the wave length of interest andwas given7) as 0.224 in."1 From these values of pa-rameters, particularly from m, it maybe said thattheir assumption of the uniform radial incidence oflight was permissible.

Jacob and Dranoff8) carried out similar experi-ments with m=0.394, z=0.4, and ^=1.58, p2=2.12,^3=2.84, and ^4=3.40. In Table 1, example dataofi?31, where p!=l. 58 and ^=2.84, are shown. Whenthe reaction proceeds Rohs decreases. Rr31 calculatedby Jacob and Dranoff8) agrees relatively well withRobs and decreases in the same way as i?Obs. Ac-cording to Jacob and Dranoff9), the quantum effi-

ciency of the photohydrolysis increases as the light in-tensity decreases. This particular situation wouldmake the apparent applicability of the radial lightmodel wider. However, the radial light model isvalid in the light of the conclusion obtained in thepreceding section.

It is then necessary that the data be analyzed fromVOL. 5 NO. 4 1972 (63) 389

Table 1 Data by Jacob and Dranoff8) and calculatedratios

C o n d . o fs o l n . 9 . 5 1 0 . 5 1 1 . 5 1 2 . 5 1 3 . 5 R e f .

[ x 1 0 5 , m h o s]

R * 2 .8 6 2 .7 0 2 .6 0 2 .5 2 2 .4 0 (8 )

* s i 2 .6 3 2 .5 8 2 .5 2 2 .4 8 2 .4 3 (8 )

2 .4 0 2 .3 6 2 .3 2 2 .2 8 2 .2 5

R U 2 .3 4 2 .3 0 2 .2 6 2 .2 3 2 .1 9

R ¥ 1, co rr 2 .8 2 2 .6 7 2 .5 6 2 .5 0 2 .4 5 (9 )

2 .9 4 2 .7 4 2 .6 2 2 .5 4 2 .4 8

/ fc rn - 1] 0 .0 7 6 0 0 .0 8 5 3 0 .0 9 6 0 0 .1 0 5 0 .1 1 6

0 .3 2 7 9 0 .2 5 1 3 0 .2 1 2 3 0 .1 9 3 1 0 .18 9 0

the diffuse source standpoint. R31 was calculatedand Rs31 was also calculated for comparison. Bothare shownin Table 1. For calculating Rfx and Rszl,the values of f*m shown in the table were used, whichvalues were determined from R31 and the reactorconfiguration. Either R31 or Rs3l does not agreewith Rohs. Rfi, however, is closer to Rohs than Rs3l.Variable quantum efficiency and effect of light re-flection and refraction Jacob and Dranoff9) reexamined the decomposi-tion of the acid and determined an empirical relation-ship between <p and the light intensity. With thisknowledge of <p, they reanalyzed the reaction datareported in 1966 from the viewpoint of a specularlight source. In predicting the scale-up ratio, theyalso considered the effect of light reflection andrefraction at the outer and inner surfaces of the

inner wall of the reactor. The empirical equation for <p, which is not givenin a closed form, may be approximated by the equation

p = (///» (26)

where /3 becomesa function of the composition of thereaction mixture. The values of /3 can be determinedfrom Fig. 7 of reference (9), and are shown in Table 1.

To take the effect of light reflection and refraction

into account, Jacob and Dranoff10) measured the lightintensity profile in a reactor filled with water anddefined a correction function. The function was

simply determined by dividing the observed intensityby the intensity predicted by Eq.(4). Using anempirical equation for (p and a correction function,they predicted corrected values of the radial scale-upratio, i?3lfeorrjwhich are shown inTable 1. The pre-diction was said to be satisfactory.

Strictly speaking, however, since the lamp theyemployed was a diffuse one, the correction functionshould be obtained by dividing the observed intensityby that predicted by Eq.(14). The analysis accord-ing to this consideration gives a correction function

Cd{r, z)=1. O+(r-l.615)(0. 06+0. 035z)

+(0. ll-0. 055*)(27)

where r and z are in inches.With this Cd{r,z) and <p, corrected values of R31,

Rii,corrj were calculated and are shown in Table 1. Itis seen that i?3ifCOrr> which becomes essentially the

same as RlltCoiT in this case, agrees well with Rohs.

Conclusion

The light intensity profile, the overall rate of reac-tion, and the radial scale-up ratio of the annular pho-tochemical reactor were predicted for the cases wherethe specular and the diffuse line sources were placed

along the centerline of the reactor.

It was found that, from the standpoint of the totalamount of reaction and of the radial scale-up ratio,

the radial light model may be employed within 10%accuracy whenthe ratio of the reactor inner radius tothe lamp lengthm is 0.1 and the ratio of the reactorouter radius to the inner radius pis less than 3. Whenm is larger, p should be close to 1. For reactor con-figurations which do not satisfy these conditions, thespecular or the diffuse line source model should be

used according to the lamp characteristics.

Nomenclature

C(r,z) = correction function defined by Eq.(27)c = reactant concentration [moles/cm3]G - function defined by Eq.(7) or (15)g = function defined by Eqs.(10), (17), or (21)/ = light intensity [Einsteins/sec à"cm2]/ = intensity of radiation [W or W/sterad]L = height of reactor [cm or in]m = rJL ^ [-]

fti = 2 for i=r, nfor i=d, and4for i=s, respectivelyPj[c) = overall rate of reaction [moles/sec]Rji(c) = radial scale-up ratio defined by Eq.(l) [-]r = radial distance as shown in Fig. 2 [cm or in]r0 = inner radius of reactor [cm or in]SL = source strength [Einsteins/cm- sec]x . = vertical distance as shown in Fig. 2 [cm]z = vertical distance as shown in Fig. 2 [cm or in]a = angle as shown in Fig. 1 [rad]j8 = power index in Eq.(26) [-]C = z/Lv = xjLp = absorption coefficient [cm"1]P = r/r0 [_]

t = /*r0 [-]

<P = quantum efficiency [moles/Einstein]<Superscript)

i = r for radial, d for diffuse, and 5 for specularmodel

Literature Cited

1) Cassano, A. E., P. L. Silveston and J. M. Smith: Ind.Eng. Chem., 59(1), 18 (1967)

2) Cassano, A. E. and J. M. Smith: A.I.Ch.E.Journal,12, 1124 (1966)

3) Denki Gakkai (The Institute of Electrical Enginneersof Japan) : "Shomei Kogaku" (Illumination Engineering)

(1963)

4) Gaertner, R. F. and J.A. Kent: Ind. Eng. Chem., 50,1223 (1958)

5) Harada, J., T. Akehata and T. Shirai: Kagaku Kogaku,35, 233 (1971)

6) Harano, Y. and J. M. Smith: A.I.Ch.E.Journal, 14,584 (1968)

7) Harris, P. R. and J. S. Dranoff: A.I.Ch.E. Journal,ll, 497 (1965)

8) Jacob, S. M. andJ. S. Dranoff: Chem. Eng. Prog., Sympo-sium Ser., No.68, 62, 17 (1966)

9) Jacob, S. M. andJ. M. Dranoff: ibid., No.89, 64, 54(1968)

390 ;64) JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

10) Jacob, S. M. and J. S. Dranoff: A.LCh.E. Journal,16, 359 (1970)

ll) Japan Exlan Co., Ltd.: Japanese Patent 1971-3584512) Matsuura, T. and J. M. Smith: A.LCh.E.Journal, 16,

359 (1970)

13) Rockwell, T., Ill: "Reactor Shielding Design Manu-

al", van Nostrand Co., Inc., Princeton, N.J. (1956)14) Schechter, R. S. and E. H. Wissler: Appl. Sci. Res., A9,

334 (1960)15) Zolmer, W.J., Ill and J. A. Williams: A.LCh.E.

Journal, 17, 502 (1971)

AN ANALYSIS OF THE LIGAMENT-TYPE DISINTEGRATION OFTHIN LIQUID FILM AT THE EDGE OF A ROTATING DISK'

TAMOTSU KAMIYA

Department of Chemical Engineering, Kogakuin University, Tokyo, Japan

An analysis of ligament-type disintegration was made by equating the rate of dis-sipation of kinetic energy, surface energy and viscous deformation energy of disturbanceof the liquid film to the motive power by centrifugal force at the edge of a rotatingdisk.

The number of ligaments generated in disintegration of liquid film is given as a func-tion of the Weber number and the stability number.

This analysis gives an interpretation on the basis of the energy balance for the ex-perimental results by Hinze and his co-worker, who used a rotating cup and representedthe same expression as the present paper.

Introduction

Theoretical studies of the disintegration of thinliquid film and liquid column in a gas-liquid systemhave been made by Squire8), Hagerty3), Dombrowski1 2), and Webern) on the basis of the wave theory.When the liquid is fed at a moderate rate througha nozzle to the center of a rotating disk, the liquidspreads out on the disk in a thin circular film whichbreaks up into ligaments at the edge of the disk.This is a form of disintegration of liquid film dif-ferent from that studied by Squire and Dombrowski.The stream of ligaments further breaks up into drops.In the event that the final step of breakup into dropsis due to the disintegration of liquid column, the pri-mary disintegration of liquid from film to ligamentsfollows an important rule in the sense that it forms

ligamentary bridges to the final step of breakup intodrops.

Experimental results on the number of ligamentsgenerated in disintegration of liquid film at the edgeof a rotating disk have been reported4-5), but no ana-lytical solution of how many ligaments should be gen-erated at the edge of a disk has been published.In the present study, themotion of liquid was treatedtwo-dimensionally, and analytical results on the num-ber of ligaments were obtained and compared withthe experimental ones. To test the analytical results,the author further expanded them to the estimationof the growth of ligaments at the time of disintegration,

Received on March 7, 1972

and compared the results with those reported byWebern) in the case of liquid column disintegration.

1. Theory

The liquid flowing down at the center of a rotat-ing disk spreads out on the surface in the form of cir-cular film and reaches the edge of the disk.Surface tension works on the film edge under thestate of a gas-liquid-gas system. The liquid film be-comes unstable departing from the edge, and changesinto several ligamentary streams due to the growthof disturbance. The stream of ligaments actuallyobserved is close to the involute of a circle for thereason that the radial velocity of the film is negligiblysmall in comparison with the tangential velocity ofthe disk. (cf. Appendix)

This property, based on experimental knowledge,

gives an advantage in analysis in that only the growthrate of disturbance may be considered in the radialdirection.

Here, the disturbance potential is taken into ac-count at the edge of the liquid film and an analysisis then made with the aid of the small-disturbancemethod3-7), on the assumption that the number ofligaments is decided by the wave number which givesthe maximumgrowth rate of disturbance. Assumingthat the motion along the z-axis is uniform in thecircular cylindrical coordinates (r, 6, z), the analysisis treated as a two-dimensional problem.

The disturbance potential can readily be derivedfrom the Laplace equation as in Eq.(1).

VOL. 5 NO. 4 1972 [65] 391