2001_t.k.sengupta_a taylor vortex photocatalytic reactor for water purification

8/10/2019 2001_T.K.sengupta_A Taylor Vortex Photocatalytic Reactor for Water Purification

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A Taylor Vortex Photocatalytic Reactor for Water Purification

Tapan K. Sengupta,, Mohammad Fazlul Kabir, and Ajay K. Ray*,

Depart m ent of M echani cal E ngi neeri ng and Depart m ent of Chemi cal and E nvi ronment al E ngi neeri ng,

T he N at i onal Un i versi t y of S i ngapore, 10 K ent Ri dge Crescent , S i ngapore 119260

A detailed analysis has been performed for a heterogeneous photocatalytic Taylor vortex reactor

tha t uses flow insta bility to recircula te fluid continua lly from t he vicinity of the rota ting innercyl in d rical su rface to th e stat io n ary o u te r cyl in d rical su rface o f an an n u lu s. I n th e p re se n tresearch, a deta iled t ime-accuratecomputa tion shows t he different sta ges of f low evolution a ndth e e f fects o f th e f in i te len g th of t h e re actor in cre atin g e d d ies, w h ich re sults in a very h ig hovera ll efficiency of photocat a lytic conversion. The physical a rra ngement considered is such t ha tp o llu tan t d e g rad atio n is m axim iz e d b y th e m o tio n o f f lu id p art icle s in a sp e cif ic re g im e o fcentrifugal instability. Also provided are detailed flow structures for the chosen parameterswhen the reactor is started impulsively.

Introduction

Semiconductor photocata lysis is a n adva nced oxida-tion technology for water purification that has recentlybe e n re v ie we d by Ch e n e t a l .1 Heterogeneous photo-

catalytic technology couples low-energy ultraviolet lightwith semiconductors act ing as photocatalysts to over-come many of the drawbacks that exist for tradit ionalwa ter trea tment meth ods. Using t his technology, in situdegradation of toxic organic compounds can be achievedby oxidat ion (using the photogenerated holes 2,3) a n dremoval of toxic metal ions by reduction (using photo-generated electrons4). The appeal of this process tech-nology is the prospect of complete mineralizat ion ofpollutants to environmentally harmless compounds. Inaddit ion, the process uses atmospheric oxygen as theoxidant, and the catalyst (usually TiO2) is cheap, sta blea n d n o n t o x ic a n d c a n be u s e d f o r e x t e n de d p e rio dswit h o ut s u bst a n t ia l los s of a c t ivi t y . M ore ov er , t h e

technique employs very low energy UV-A light ( < 380nm), result ing in energy requirements as low as 1-5W/m 2 of cata lyst surface area , and more importa ntly, i tcan even be act ivated even by sunlight .5

Despite the potential of this promising technology, thedevelopment of a pra ct ical w at er treat ment system ha snot yet been successfully achieved a s a result of var iouslimitat ions.6 There ar e severa l factors tha t impede theefficient design of photocat alyt ic reactors. I n this typeof reactor, in addit ion conventional reactor complica-t ions such as reactant -cata lyst conta ct , flow pa tterns,mixing, mass transfer, reaction kinetics, catalyst instal-lation, temperature control, etc. , an addit iona l engineer-ing factor related to illumination of catalyst becomesrelevant . The illumination factor is of utmost impor-

t a n ce b eca u se t h e a m ou n t of ca t a l y st t h a t ca n b ea c t iva t e d de t ermin es t h e w a t e r t re a t me n t ca p a c it y ofthe reactor. The high degree of interact ion betweentra nsport processes, reaction kinetics, a nd light absorp-t ion lea ds t o a s t ron g cou p lin g of p h y sicoch e mica l

p hen om en a a n d i s o n e o f t h e m a jor h u r dl es i n t h etechnical development of a photocatalytic reactor.

Ph otocata lyt ic react ions ar e promoted by solid pho-t o ca t a ly s t p a rt icles t h a t u s u a l ly con s t i t u t e a disc ret e

phase distributed w ithin a continuous fluid phase in thereactor. Therefore, a t least tw o phases, i .e. , l iquid a ndsolid, a re present in t he rea ctor. The solid phase couldbe dispersed a s a slurry 2 or could be stationary 3 wit h inthe reactor. However, the use of suspensions requiresthe separation and recycling of the ultrafine (submicron-s ize d) c a t a ly s t f ro m t h e t re a t e d l iq u id a n d c a n be a ninconvenient, time-consuming , a nd expensive process.I n a ddit io n , t h e de p t h o f p e n e t ra t io n o f U V l ig h t islimited because of strong absorption by both catalystpar ticles and dissolved pollutan ts. The a bove problemscould be avoided in a fixed-bed photoreactor in whichphotocat alyst part icles are immobilized onto a f ixedtra nsparent surfa ce, such a s the reactor wa ll . However,immobiliza t ion of ca t a ly s t on a s u pp ort g e ne ra t e s a

unique problem. The reaction occurs at the liquid -solidin t e rf a c e , a n d u s u a l ly , o n ly p a rt o f t h e c a t a ly s t is inc o n t a c t wi t h t h e re a c t a n t . H e n c e , t h e o v e ra l l ra t e isl imit ed by t h e ma s s t ra n s port of t h e p ol lu t a n t t o t h eca t a ly s t s u rfa c e.7,8 I n a d d i t ion , t h e ov er a l l r a t e ofre a ct ion is u s u a l ly s low c omp a re d t o con v en t ion a lchemical reaction rates because of the low concentrationlevel of the pollutant.

The s cale-up of fixed-bed photocata lytic r eactors ha sbeen severely limited because reactor configurat ionsha ve not been able to address the issue of mass t ra nsferof pollutants to the catalyst surface. The new reactordesign concepts must deal w ith t his challenge. Ea rlierexperimental studies of our group on catalyst-coated

tube bundles,9

novel immersion-type la mps,10

a n d ro t a t -ing tube bundles revealed that photocatalytic reactionsar e ma inly diffusion-(ma ss-tra nsfer-) controlled. I n ourearlier studies, we enhanced mass t ra nsfer by increas-in g t h e mix in g o f f lu ids t h ro u g h t u rbu le n c e a n d t h eintroduction of baffles.7 I n t h i s w o r k , a n e w r e a c t o rconcept is developed that uses f low instability to in-crease the reaction yield throughout the reactor volume.

Sczechowski et al.11 reported their experimental stud-ies relat ed to enha ncing th e photoefficiency of a Ta ylorvortex reactor (TVR). A schema tic of the reactor iss h own in F ig ure 1a . Th e t o roida l v o rt ice s t h a t a re

* Corresponding author. Tel.: + 65 874 8049. Fax: + 65779 1936. E-ma il: [email protected].

Department of Mechanical Engineering. O n l e ave fr om I nd i an I nst i t ut e of T e c hnol og y , K anpur

208016, I ndia . Department of Chemical and Environmental En gineering.

5268 In d. Eng. Chem. Res. 2001, 40 , 5268-5281

10.1021/ie001120i CCC: $20.00 2001 American C hemical SocietyP ubl ish ed on Web 08/23/2001


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usually formed in th e annula r ga p betw een th e cylindersa r e s h ow n i n F i gu r e 2 a . Th e n on w a v y v or t ex f low appears as a consequence of primary instability givenby appropriate nondimensional numbers such as theTaylor number and Reynolds number. Assuming thea n n u l a r g a p d ) (ro - ri) a n d t h e r o t a t i o n r a t e o fcylinder , the Taylor number is defined a s

When the annula r ga p,d, is small compared to the innerc y l in de r ra diu s wit h v e ry la rg e a s p e c t ra t io s o f t h egeometry (i .e. , very la rge lengths) an d 0 e o/i e 1,Taylor12 showed that the primary instability in the formof the appearance of Taylor vortices occurs at T acrit 1708. When the Reynolds number, defined in terms oft h e g a p w i d t h a s t h e l e n g t h s c a l e a n d t h e s u r f a c evelocity on the inner cylinder surface as the velocityscale, is increased to somewhat higher values, one seeswa v y v o rt e x f lo w, a ls o s h o wn in F ig u re 2a , wh ic h ischaracterized by the appearance of circumferential andaxial wavy motion of the toroidal vortices. Sczechowskiet al .11 observed that, when catalyst particles were usedas a slurry, the useful reaction took place only periodi-cally when the fluid was in contact with the illuminatedinner cylinder surfa ce. They r eported a 3-fold increa seof the photoefficiency when the reactant was illuminatedfor less tha n 150 ms, when it sta yed in the dar k for moret h a n 1 s . Th e ma x imu m p h ot o ef f icien cy t h e y cou lda c h ie ve w a s 30% a t a ra t e of 300 rpm f or t h e in n e rcylinder w hen 10 g/L loading of TiO2 was used. Themajor problem of achieving higher photoefficiencies wasrelated to the transport of purified fluid from the vicinityof the cata lyst . However, use of the cata lyst a s a slurry

raises the problem of separating submicron-sized cata-lyst pa rticles after th e purification sta ge. Moreover, theworking fluid is optically d ense, and t herefore, the lightpenetrat ion depth is restricted to a distan ce tha t is onthe order of the boundary layer thickness of the innercy li n de r. I n v ie w of a l l of t h es e f a c t or s , w e h a v econ s ide red in ou r p res e nt wo rk a TVR of s imila r

g eome t ry , bu t in s t ea d of a s lurry -t y p e re a c t or , t h ephotocata lyst w a s a ssumed t o be immobilized (fixed) onthe outer surfa ce of the inner cylinder a nd a fluorescentla mp wa s a s s u me d t o i l lu min a t e t h e in n e r c y l in de r .Thus, one can use a very low level of catalyst loadingan d, simultan eously, eliminat e the process of separa tingthe catalyst part icles after the purificat ion stage. En-hanced purification can be obtained by utilizing the fluiddy n a mica l in s t a bi li t y a s s ocia t e d wit h t h e ce n t r if u ga lin s t a bi l i t y in t h e c y l in dric a l a n n u la r g e o me t ry . I t isessential to focus on the twin aspects, photocatalysis andcentrifugal insta bility, in a n a nnular cylindrical geom-etry.

Taylor-Couette Flow and Flow I nstabilityThe laminar flow confined within the annulus region

b et w e en t w o coa x i a l cy l in d er s , w i t h t h e i n ner on edifferentially rotating with respect to the outer, sufferscentrifugal instability depending on the geometry (as-pect ra tio) an d rota tion ra tes. The criteria of centr ifugalin s t a bi li t y wa s f i rst e n u ncia t e d by R a y leig h ,13 w h oshowed that an inviscid rotat ing flow is unst able if thee n e rg y o f ro t a t io n a s s o c ia t e d wit h t h e f lu id p a rt ic ledecreases radially outward. Under such unstable con-figura tions, one notices the a ppea ra nce of circumferen-tial toroidal vort ices betw een the t wo cylinders, w hichar e known as Tay lor-Couett e vortices. Ta ylor12 wa s t hefirst t o report experimenta l verifica tion of this phenom-

Figure 1. (a) Schematic diagra m of the Taylor vortex reactor an d (b) locat ion of the observat ion sta t ions in the a nnular region to a nalyzeflow behavior. d*) 0 (rotating inner cylinder), d*) 1 (stat ionary outer w all), y*) 0 (bottom), y*) 1 (top).

T a)rid

3[i2 - o

2]

2

(1)

Ind. Eng . C hem. R es. , Vol. 40, No. 23, 2001 5269


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enon. These vort ices evolve because of t he adversegradient of angula r momentum tha t creat es a potentialu n s t a ble a rra n g e men t of f low (s ee re f 14). Su c h a nunstable condition arises naturally if the outer cylinderis held sta t ionary w hile the inner cylinder is rotated a ta sufficiently high rotat ion ratesan arrangement con-sidered in this paper. In such an unst able condition, thean nulus is filled w ith a pair of counter-rotat ing vortices.This can be viewed in F igure 2b an d c. Figure 2c showsthe formation of a boundary layer on the inner surfa ce.Moreover, i t shows that the boundary layer oscillates

periodically between a lmost zero thickness to a ma xi-mum between the counter-rotat ing vortex pair, wherethe tw o shear layers a pproaching each other spew outa jet of f luid towa rd t he outer w all . The fluid part iclesin t h e ir mot ion a ro u n d t h e t o roida l v ort ice s comeperiodically into conta ct w ith t he inner surfa ce wherethe cat a lyst is immobilized, a nd in t he presence of light,the cata lyst is act ivated, result ing in the occurrence ofthe redox reaction. The residence time of the particlesin t h e i llu min a t e d re gion is t h u s a f u n ct ion of t h eangular velocity of the recirculat ing vortex as well asthe size of the vortex. The la t ter, once a gain, dependson the gap size and the number of vortices formed in agiven length of the rea ctor.

H ow e ver , t h e s it u a t ion ca n b e d if fer en t i f t h eTay lor vort ices a re esta blished in a reactor of shortlen g t h . E x pe rime n t s c a rr ie d o u t by B u rk h a l t e r a n dKoschmieder 15 have revealed that the axial w avelengthof the Taylor vortices for finite cylinders are differentf rom t h os e p redict e d by s t a bi l it y t h e ory f or s in glyperiodic Taylor vort ices in infinitely long cylinders.Furthermore, the effective wavelength also depends onthe end-cap boundar y condit ions. Andereck et al . 16

pointed out that there could be many flow states, whichcould be distinguished by their symmetry under rotation

and reflection, i.e. , their azimuthal and axial wavenum-b er s . Th i s h a s b ee n a t t r i b ut e d t o t h e e xi st e nce ofmu lt ip le s t a ble s olu t ion s of t h e g ov ern in g N a v ier-Stokes equation for f lows fa r from equilibrium.

Wereley and Lueptow17 showed experimentally t ha twavy vortex flow, characterized by both axial and radialazimu tha l deforma tion of the vortices, became str onger,and the outflow between pairs of vortices became jetlike.According to th eir findings, significant tra nsfer of fluidbetw een neighboring vortices occurs in a cyclic fashionat certa in points along an a zimuthal wa ve so tha t , whileon e v ort e x g ro ws in s ize , t h e t wo a dj a c en t v ort ice sbecome sma ller, an d vice versa . This a spect of th e flowis important , because tradit ionally, i t is assumed that

Figure 2. (a) Vort ices without a nd w ith circumferential wa ves, (b) schematic presentat ion of s trea mlines in vort ical cells , an d (c) radialmass transfer in Taylor -Couette f low.

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two Taylor-C ou et t e v or t i ce s m a k in g u p a p a ir a r eindependent of each other a nd of the sa me size, a nd iti s c u st om a r y t o t r e a t t h is a s p lu g f low . I f , i n d ee d,significant mass transfer occurs between adjacent vortexpairs, then it can be used for addit ional benefit in thepresent application for performance enhancement of there a c t o r . I t h a s a ls o be e n re p o rt e d t h a t t h e de g re e o ft ra n s f e r o f t h e f luid is g re a t e r f o r R e) 253 than foreither higher or lower Reynolds numbers. In addit ionto f low in to and out of adjacent vort ices taking placeperiodically, the individual vortex centers also moveboth axially a nd ra dially in a cyclic fash ion. Once aga in,

t h e ma x imu m de p a rt u re in t h e ra dia l dire c t io n is astrong function of the Reynolds number and is found tobe a maximum for R e) 253. It is for th ese reasons tha twe ha ve decided to investigate tw o geometries (aspectratios) that correspond to this optimum Reynolds num-ber. However, to avoid long computat ional t imes, andalso for pract ical ut ility, the establishment of f low iscon s ide red t o occu r in a n impu lsive f a s h ion . I n apractical design, one would like to have th e purifica tionprocess completed in a limited time.

Computational Details

In computing the flow, the three-dimensional Navier-

Stokes equation is solved in a primit ive formulat ionusing t he commercial softw ar e Fluent . This a pproachwa s f o un d t o be a de q u a t e a s t h e f low con s ide red w a slamina r a nd could thus be computed w ithout the needfor resolving large ranges of wavenumbers and circularf req u e n cies . I n s olvin g t h e g ov ern in g e q u a t ion s , n osimplificat ions were made regarding symmetry or re-flection of the solution. In addition, the a priori steady-s t a t e s ol ut i on w a s n ot a s s u m ed ; r a t h e r, t h e t i me - accurate solution was calculated. We considered twodifferent geometric dimensions to stud y t he effect of thean nula r ga p for a consta nt Reynolds number of 253, theoptimum Reynolds number observed experimenta lly byWerely an d Lueptow. 17 Th e dimen s ion s of t h e t woreactors considered in this study are given in Table 1.Reactor configurat ion 1 is identical in its dimensionsto the reactor of Werely and Lueptow17 an d is referredto a s the reference configurat ion. The m ain intentionof this study is to show the efficacy of the immobilizedphotocatalytic reactor for the chosen parameter values.I n r ea c t or con f ig u ra t i on 2, t h e a n n u la r g a p (d) isincreased for a consta nt reactor length, t o increase theam ount of l iquid tha t can be treat ed (i .e. , the capacity)inside the reactor. To be able to compare the results atsame R eynolds number (R e) 253), t he r otat ion speed,, was reduced. When d was increased to 0.0111 m(configuration 2), the rotation speed had to be loweredto 5 rpm to maint a in the Reynolds number a t 253. Thiseffectively increa ses th e ca pa city of the rea ctor by 27.6%,

while r educing the energy cost (lower power requiredto maintain lower rpm, ).

Two different model pollutants were considered int h is w ork , n a me ly , p h en ol a n d S B B dy e . Th e se t w omodel compounds were studied by our group earlier andwill help us to study th e effect of reaction rat e constant son photocat alyt ic degra dat ion. B ecause the init ia l con-centr a tions of pollutan ts a re usua lly very low (100 ppm),i t i s e xp ect e d t h a t t h e ch em i ca l r ea c t ion w i ll n otinfluence the fluid dynamic behavior of the system. Thisw a s ob se rv ed t o b e t h e c a s e, a s s h ow n l a t er . Th ephotocat alyt ic degrada tion rates for phenol and S BB dyewere calculated using the equations given in Table 2.

The density of the system w as assum ed to be constantand equa l to tha t of wa ter, as both of the pollutant s arepresent in t ra ce amount. The va rious locat ions wherethe solutions w ere investigat ed a nd plotted a re shownin Figure 1b. The boundary conditions that are appliedon the inner and outer cylindrical surface correspondto no-slip conditions. The end caps are considered to bea p a rt of t h e o u t er cy l in de r a n d h e n ce a re s t a t io n a ry .

For generating grids within the annulus region of thegeometry, every edge in t hree directions w as defined bycertain nodes. There are high-shear regions near the

inner cylinder wa ll, the outer cylinder w all, a nd t he end-cap regions. Therefore, a xial a nd ra dial derivat ives ofall physical va riables across such layers a re larger tha nthe corresponding derivat ives in the azimuthal direc-t ion, and consequently, more grid points are taken int h e a x ia l a n d ra dia l dire c t io n s c o mp a re d t o t h e a zi-mutha l direct ion. These extragrid points ar e incorpo-ra ted by being more clustered at the tw o ends a nd nextt o t h e in n e r a n d o u t e r wa l ls . T o e n h a n c e t h e dire c tma pping of the grid from the upper wa ll to the bottomwa ll, the total geometry w as separa ted into two volumesby a br ick . Two in t erior p la n e s we re c rea t e d by t h isp roce ss w i t h in t h e a n n u la r s pa c e t h a t w a s u s ed t oanalyze our results at the t ime of postprocessing. Toobt a in t h e in s t a bi li t y p a t t e rn o f flow in a n n u la r a re a ,bot h a f in e r g r id a n d a coa rs e r g r id w e re ch os en t oa n a ly ze t h e g r id de p e n de n c y o f re s u lt s . G rid p o in t salong the t hree direct ions were ta ken as follows: r h) 200 60 150 in the fine-grid case and 100 40 75 in th e coa rse-grid case. The F luent preprocessorG A M B I T 1 . 1 w a s u s e d t o c r e a t e t h e g e o m e t r y a n dgenera te th e grid for both cases. Complete descriptionsof t h e m es h v ol um es a n d n od es g en er a t e d f or t h esimulat ions are as follows:

For the finer grid the mesh volume was 1.8 millionwith a total of 1 821 060 nodes, and the total numberof elements w as once aga in 1.8 million. Crea tion of thisgrid req uired a tota l of 393.66 CP U seconds wit h 431.2M B o f me mo ry . F o r t h e c o a rs e r g r id , t h e re we re 0. 3

Table 1. Geometric Dimensions of the Two ReactorConfigurations Considered

s peci fi ca t i on con fi gu ra t i on 1 con fi gu r a t ion 2

length, L , m 0.4245 0.4245inner radius, r i, m 0.0434 0.0434outer radius, ro, m 0.0523 0.0545a n n u l a r g a p , d, m 0.0089 0.0111aspect rat io, L /d 47.70 38.24rotation speed, , r a d /s (r p m ) 0 .65 5 (6 .2 5) 0 .5 24 (5 .0 )volume of liquid treated, VL, m 3 1.136 10-3 1.449 10-3

Reynolds number, R e 253 253Tay lor number, T a 13 126 16 298

Table 2. Rate Expressions for Phenol and SBB Dye Usedin This Study

phenol3R) -

d Cs

d t ) k0e xp[- ER T]I[

KO2pO2

1 + KO2pO2][ KsCs

1 + KsCs]w h e r e k0 ) 1.19 10-4 mol m -2 s-1, E) 11.80 kJ /mo l

) 0.82, KO2 ) 12.71 at m -1, KS) 4.26 mM-1

S B B d y e5R) -

d Cs

d t ) [

k(I,w)K Cs

1 + K Cs]w h e r e

k(I ,w) )

ksa(w)Ib

1 + a(w)Ib

kS) 0.38 m m ol/m 2/sa ) 3.45, b) 0.85, K) 18.5 m 3/m ol



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million cells, 907 000 total quadrilateral faces (3000in n er wa l l f a c es , 11 000 ou t e r wa l l s pa c es , 15 000interior plane fa ces, a nd 878 000 other interior planefaces). The total number of nodes was 307 040.

The present set of computa tions of the Na vier-Stokesequation for Taylor-Couett e geometr y is expensive. Fort h e f in er g r id , t h e r a n d om a c ce ss m em or y (R AM )requirement is 1.3 GB, which demands a large comput-ing machine. The case with reaction (flow plus reaction)was submitted to a 1.5-GB RAM Windows NTworksta-t ion, and the case without react ion w as submitt ed to ama inframe supercomputer (SG I origin 2000). F or t he

coar se grid, the CPU requirement wa s considerably less,as the num ber of grids wa s 1/6 tha t of the f iner one.

Results and Discussion

Flow Evolution. It has already been mentioned thatthe performa nce of the reactor depends on the photo-efficiency of the photocatalytic process, as well as on thema ss tr a nsfer efficiency of the fluid from th e vicinity oft h e r ot a t i n g i n ne r cy l in d er t o t h e s t a t i on a r y ou t ercy li n de r a n d b a ck w i t h in a s in g le Ta y l or-Couettevortex. The lat ter depends on the primary instability

Figure 3. Grid convergence is shown by the contour of the axial velocity (bottom 11.78%of the reference reactor is shown). The rightb ou n d a r y is t h e r ot a t in g in n e r w a l l (d*) 0 ).



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of t h e f low in e st a bls h in g t h e v ort ica l roles . Th isinstability and th e associat ed mass tra nsfer can be ma demore effective by initiating a secondary motion in whichthere is addit ional mass transfer between neighboringroles through the wavy vortex flow. This has been shownto be a strong function of Reynolds number by Werelyand Lueptow,17 and it was reported that inter-role masstransfer is a maximum for the chosen geometry in theirexperiment for R e) 253. In the present set of computa-t ion s , t h e s a me g eome t ric p a ra me t e rs we re ch os e n,including the a spect r a tio, (AR) L /d) of th e rea ctor forthe reference case. However, in the experiment, thestarting protocol of the inner cylinder rotation was takenas qua si-sta t ic , and the reported r esults w ere recordedafter t he f low wa s a llowed t o develop for an addit ional10 min a f t er t h e a t t a in me n t o f t h e f in a l ro t a t ion ra t e .I n t h e p re se nt com pu t a t i on s , t h e f low w a s s t a r t e dimpulsively for th e practical operat ion of the r eactor in

Figure 4. (a) Velocity vectors shown in the radial plane for y*)0.5 -0 .7 a t v a r ious t i m e s. Th e r ig h t s id e in d ic a t e s t h e r ot a t in ginner wall (d* ) 0). (b). Velocity vectors sh own in the r adia l plan efor y* ) 0.70-0.85 at different times. The right side indicates therotat ing inner wall (d* ) 0). (c). Velocity vectors in the radial planefor y* ) 0.85-1 a t v a r ious t im e s. Th e r ig h t s id e in d ic a t e s t h er ot a t in g in n er w a l l (d*) 0 ).

Figure 5. V e loc i t y v e c t or s s h ow n in t h e r a d ia l p la n e f or t h er a n g e s y*) 0.5 -0.7, y*) 0.7-0.85 and y*) 0.85-1.0 at t) 330s. The right s ide indicates the rotat ing inner wall (d*) 0 ).

Figure 6. In ternal boundary la yer forma tion in Taylor-Couetteflow and the presence of a saddle point . The velocity vectors inthe radial plane for y*) 0.855-0.885 are shown. The right s idein d ic a t e s t h e r ot a t in g in n e r w a l l (d*) 0 ).



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the shortest possible time period. For such an operation,t h e t ra n s ie n t ma s s t ra n s f e r de p e n ds s t ro n g ly o n t h ev o rt ic e s t h a t a re f o rme d a t t h e in i t ia l t ime s n e a r t h efixed end caps of the reactor. Because the flow insidethe reactor is driven by t he rotat ing inner cylinder a ndt h e r ot a t i on r a t es a r e l ow , i t i s q u i t e a d e qu a t e t oconsider t he flow t o be incompressible an d t o solve thegoverning Navier-St okes equation in primitive variableform as

a n d

Furt hermore, isotherm al conditions can be a ssumed forbot h t h e f low e volu t ion a n d t h e ch e mica l re a ct ioncalculat ions a s t he pollutant being oxidized into otherp rodu ct s is p res en t in t ra ce a mou n t s a n d t h e h e a t ofreact ion in photocata lyt ic react ions is usua lly negli-gible.3

The following boundary conditions were used for thenumerical simulations. With reference to Figure 1, on

the inner cylinder surface, r ) ri

wh e re i(t) is t h e in s t a n t a n e o u s ro t a t io n ra t e o f t h einner cylinder. On the outer cylinder (r ) ro) a n d t h eend caps (y) 0 a n d L )

B e c a u s e t h e s imu la t io n is f o r t h e imp u ls iv e s t a r t o frotation of the inner cylinder

The init ial concentra t ions of phenol a nd S BB dye w ereta ken as 100 ppm, while the init ial ma ss fract ion of O2wa s ta ken as tw ice the stoichiometric requirement forthe r eaction. The oxida tion reactions of phenol a nd S B Bdye are given by

respectively. Eq ua tions 2 an d 3 a re solved subject to th eboundary conditions and initial conditions defined in eqs4 a n d 5 wit h in t h e a n n u lus be t we e n t h e c y l in ders , a sshown in F igure 1a a nd Table 1. In computing t he flow,no specific symmetry of the geometry wa s considered,i.e., full th ree-dimensional flow wa s solved even t houghthe geometry is axis-symmetric. I f the f low is indeedaxis-symmetr ic, the solution would reflect t his, an d th isviewpoint w as adopted in solving and int erpreting th es o lu t io n . I n t h e s a me wa y , t h e t ime-accurate solutionwa s s o ug h t , a n d i f in de ed t h e re we re a s t ea dy s t a t e ,the solution would indicate such. In the present case ofc o mp u t a t io n , wh e n t h e f lo w p a t t e rn is s e e n a s s y m-metric, the results a re shown in t he radia l-a x ia l (r-y)plane, as shown by the sha ded region in F igure 1. Forsuch symmetric profiles, the results are shown along

the t hree sets of horizonta l (H1-H 3) and vert ical l ines(V1-V3) indicat ed in Figure 1b.

As a lready m entioned, the commercial softw ar e Flu-ent w as used for t he present simulat ion study. A briefdescription of t he m ethod is given below. The solver is

based on solving eqs 2 a nd 3 in a sequential ma nner bya control-volume-based technique using the followingt h r ee s t ep s: F i rs t , t h e com pu t in g d om a i n w a s d is -cretized into discrete control volume using a computa-t ional grid. Then, the governing equations were t imeadvanced in integral form for each computat ional cellto yield algebraic equations for the discrete dependentvaria bles, such as velocity components, pressure, a ndconserved scala rs. Fina lly, the discrete equations w erelinear ized into a set of algebraic equations, which weresolved to yield upda ted va ria bles. For th e discretiza tionprocess, the familiar QUICK scheme was used for themomentum and species equations, as this is a higher-order accurate scheme with minimum numerical dis-sipat ion t ha t is implicit w ith discret izat ion.

I n t h e s e a rc h f o r t h e a p p ro p ria t e g r id , t h e re s u lt sobta ined by the fine and coarse grids ar e first compared.In Figure 3, the grid convergence is shown with the helpof the plotted axial velocity contours for t he f iner a ndcoa rs e r g r ids a t t wo di ff ere n t t imes . A lt h o ug h t h eplotted contours show some small differences at veryearly t imes (10 s), they are seen to disappear at latert ime s (t ) 30 s) . This shows that the coarser grid isadequate for the present investigat ion. Hence, unlesss t a t e d ot h er w i s e, a l l of t h e r ep or t e d r es u lt s w e r eo bt a in e d u s in g t h e c o a rs e g r id . Simila r ly , u s in g t h iscoar se grid, w e a lso performed calculat ions t o investi-gate the effect of chemical reaction on flow evolution.I n on e ca s e , on ly t h e f lu id f low (n o r ea c t ion ) w a s

VBt

+ (VB)VB ) - 1Fp+ 2VB (2)

VB ) 0 (3)

Vr) Vy) 0 a nd V ) iri (4a)

Vr) Vy) V ) 0 (4b)

te 0: i(t) ) 0 (5a )

t> 0: i(t) ) (5b)

C 6H 5OH + 7O 2 f6C O2 + 3H 2O (6)

C 43H 71O6N 3S 2 + 641/2O2 f

43CO2 + 32H 2O+ 2H 2S O4 + 3HNO3 (7)

Figure7. Flow str ucture of Taylor-Couette flow for an impulsives t a r t a t t) 330 s . The notat ions U, B, and S indicate unidirec-t ion a l , b idir ec t ion a l , a n d s in g ula r f low , r e s pe ct iv ely , a n d E -Jrepresent regions where such f lows exist . The right s ide is ther ot a t in g in n er w a l l (d*) 0 ). b indicates a saddle point .



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computed, whereas in the other case, the computationswere car ried out w ith the react ion proceeding concur-rently. As t he model component, phenol, w as presentat a low concentration (100 ppm), it was expected thatthe chemical react ion w ould not significant ly a lter theflow evolution. This w as indeed found to be t rue, a ndthe results a re not shown here for brevity.

In Figure 4, the deta iled velocity vectors a re shownfor t he top half of the reactor (y*) 0.5-1.0) in a giveny-r plane at different t imes, whereas in Figure 5, thevelocity vector plots ar e shown a t t) 330 s when stea dystate has been reached. The segment between 50 and70% of t h e re a ct o r h e ig h t is s h o wn on t h e lef t , t h esegment between 70 and 85%is shown in the middle,and the rema ining segment of the top half is shown onthe right . One can see weak vort ices forming nea r t he

upper part of the left segment. There are regions alongthe height w here one can see a jetlike flow st ar ting fromt h e in n e r wa l l a n d mo v in g t o wa rd t h e o u t e r wa l l du eto centrifugal action. In the segment for y* ) 0.7-0.85,significant mixing of t he f luid is n oticeable a result ofthe forma tion of coherent vortices in part of the reactor.Once a gain, one can see jet like flow from the inner tothe outer cylinder, alt hough the t ra jectories of the fluidpart icles a re not strict ly stra ight , as ha s been depictedin Figure 2. Moreover, no visible wa ll shear lay er formson t h e in n er w a l l a s h a s be en s h o wn in F ig u re 2c. I nt h e s e gme n t f or y* ) 0.85-1.0, t he velocity vectorsclea rly s h ow re circu la t in g rol ls , a l t h o ug h t h e a x ia llengths va ry significan tly because of end-wa ll effects.Also in this segment , a part from th e jetlike regions, oneca n a l so s ee s m a ll a x ia l r eg ion s w h er e a f low i s

Figure 8. C han ge of axial velocity w ith dimensionless radia l coordinate a t different axial posit ions a nd t imes a t (a) H 1, (b) H 2, and (c)H 3. The figures on the left are for reactor configurat ion 1, and those on the right are for configurat ion 2. The right s ide indicates ther ot a t in g in n e r w a l l .



9/14

established from t he outer t o the inner w all . However,t h is ca n n ot cov er t h e e nt i r e r a d i a l g a p b eca u s e ofcentrifugal force act ing on the fluid par t icles near theinner wall . For the same reasons, one does not see awa l l s h ea r la y e r f ormin g o n t h e in n er cy l in de r a s h a sbe e n s h o wn in t h e s k e t c h o f F ig u re 2c . I n s t e a d, o n ewo u ld n o t ic e t h e f o rma t io n o f a n in t e rn a l la y e r a f t e rsome t ime and the associated full saddle point insidethe flow domain. In retrospect, it appears that, to resolvesuch internal layers, one should have a finer grid in thein t erior t o o. Th is is s h own in F ig ure 6, wh e re t h evelocity vectors a re plotted over a limited segmentbet we e n 85. 5 a n d 88. 5% of t h e re a ct o r len g t h . Aninternal ba nd chara cterizes the forma tion of the internal

la y e r w h e re t h e f low is a lon g t h e a x is o f t h e re a ct o r .This is seen to originate at the saddle point , which ismarked in the figure with a filled circle. The flow is seento be axial in the opposite direct ion across the saddlepoint . Such vert ical l ines are nothing but the internallay er. The velocity vectors inside th e recirculat ing eddiesshow t ime-dependent behavior, although it is not aspronounced near the middle section of the reactor wherethe eddies are either very weak or not formed. All ofthese unsteady events w ould lead to an increase of massexchange between adjacent fluid cells.

The flow structure for the chosen parameters of thiscomputat ion with an impulsive start of the reactor iss h own in F ig u re 7. Th is a s pe ct is imp ort a n t in t h e

Figure 9. C hange of ra dial velocity w ith dimensionless radia l coordinate a t different a xial posit ions and t imes at (a) H 1, (b) H 2, and (c)H 3. The figures on the left a re for configurat ion 1, a nd t hose on t he right are for configurat ion 2. The right s ide indicat es the rota t ingin n er w a l l .



10/14

context of nonunique flow evolution due to differentinit ial condit ions referred to in Andereck et al . 16 Theflow structure computed in this work is significantlydifferent from what has been experimentally visualizedby We rely a n d L u e pt o w,17 a l t h o ug h t h e g eome t ricp a ra me t e rs ( a s pe ct ra t io ) a n d R e y n olds n u mbe r a reiden t ica l . Th is is du e t o t h e impu lsive s t a r t of t h epresent computed case as opposed to the quasi-stat icsta rt in Werely and L ueptow, 17 wh ere the inner cylinderwas accelerated from rest to its f inal value at the veryslow rate of 0.3 R e per second. In Figure 7, the f lowstructure is shown for the segment y*) 0.85-1.0 at t

) 330 s. The time is not important, as it was observedt h a t t h e f low s t ru ct u re re ma in s in v a ria n t bey on d t)180 s. If one looks car efully in F igure 7, th en one noticest h e je t like f low s t ru ct u re bet we e n f lu id ce lls . Th ep o rt io n s ma rk e d by E , G , a n d I a re t h e re g io n wh e reone ca n see unidirectional jets emerging from t he innerw a l l a n d a p pr oa ch in g t h e ou t er w a l l. Th is i s t h eexpected jet str ucture indicat ed in th e sketch of Figure2c. H o we ve r , on e ca n a ls o n ot ice t h e e xis t en ce ofb id ir ect i on a l jet s a s m a r k ed b y F a n d H . A l on g ers eg me n t o r igin a t in g f rom t h e o u t e r w a l l a n d mov in gt o wa rd t h e in n e r wa l l is me t wit h a s ma lle r s e g me n t

Figure 10. Cha nge of axial velocity w ith dimensionless axia l posit ions (y*) 0-1.0) and t ime at locat ion V1. The figures on the left arefor configurat ion 1 w hile f igures on right are for configurat ion 2.



11/14

where the flow is from the inner to outer wall . Then,w e h a v e m a r k ed a l in e J i n t h e f i gu r e, w h i ch i s t h elimit ing case of the bidirect ional jet wh en the width ofthe jet degenerates into a line and the merger line ofthe opposing jet becomes t he sa ddle point. The nota tionsU , B , a n d S i n t h e f i gu r e i n di ca t e t h e l oca t i on s ofunidirectional, bidirectional, and singular zones, respec-tively.

In F igures 8 and 9, respectively, the axial an d ra dialv eloci t ies a s f u n ct ion s of t h e dimen s ion les s ra dia ldist a n c e ( a n n u la r g a p ) a re p lot t e d f or v a r iou s t imein s t a n t s a t t h e o bs erv a t ion s t a t ion s H 1, H 2, a n d H 3 ofFigure 1b for t wo different a spect ra tios, configurat ions1 and 2 of Table 1. Because the middle section (H 2) isat the center (y* ) 0.5), the a xial velocity is sma llest inma gnitude there as compared to the other tw o stat ions

wh ere the ma ximum velocity component is on the order5-6%of the velocity sca le. The results a lso indica te th a tt h e f low is n ot f u lly de ve lop ed e ve n a t t ) 330 s,a l t h ou g h t h e r a t e of ch a n g e of f low h a s d ecr ea s e dsignificant ly. For the m iddle plan e (H2), t h e a x ia l a n dra dial velocit ies demonstra te th at the Taylor-Couettevort ices move faster toward the middle part when theaspect ratio is lower (configuration 2). Figures 8 and 9also indicate tha t there is an inflow of f luid towa rd theinner cylinder for all of the indicat ed times for th e rad ialvelocity w herea s for the a xial velocity component , inflowoccurs at certain instants while outflow occurs at otheri ns t a n t s w i t hi n t h e a n n ul a r g a p. I n a d d it ion , t h ema gnitud e of the ad ial velocity component is 1 order ofma gnitude lower tha n t hat of the axial velocity compo-nent .

Figure 11. C hange of ra dial velocity w ith dimensionless axial posit ions (y*) 0 to 1.0) and t ime at locat ion V1. Figures on the left arefor configurat ion 1, and those on the right are for configurat ion 2.



12/14

I n F ig u re s 10 a n d 11, re s p e c t iv e ly , t h e a x ia l a n dra dial velocity components a re plotted along t he dimen-sionless axial direct ion at observat ion stat ions V1 a tthree different t imes. At shorter t imes, the Taylor-Co u e t t e v o rt ic e s a re f o rme d a t t h e e n ds , a n d a s t imeprogresses, more and more of them are formed, coveringthe middle par t of the a nnulus. I f , indeed, the vort icesformed were like a plug flow, then the velocity compo-nent a long t his line would ha ve been zero. Once a gain,we observe tha t vort ices move faster towa rd t he centerwhen the aspect rat io is lower. The very fact that thevelocity component alternates in sign indicates that thevortex centers not only execute axial waviness but alsoshow significant radial motion. This was also observed

experimentally by Werely an d Lueptow,17 where theyshowed such motions for a ll Reynolds numbers betw een131 and 1221. However, i t has to be noted th at , in thepresent investigat ion, the f low is started impulsivelyand not accelerated qua si-sta t ically and the flow is notcompletely fully developed. This a spect is clearly dem-

Figure 12. F low development (axial velocity contour) for tw oreactor configurat ions. The t op pictures are for configurat ion 1,and the bottom pictures are for configurat ion 2. Only t he bottomhalf of the reactor (y*) 0-0.5) is shown . The right side indicat est h e r ot a t in g in n e r w a l l .

Table 3. Comparison of Performance of a Taylor VortexReactor (TVR ) with that of a Classical Annular Reactor(CAR) for Photocatalytic Decomposition of Phenol

C AR18 TVR 1 TVR 2

volume of rea ctor, m 3 3.48 10-3 3.65 10-3 3.96 10-3

c atal y st surfac e are a, m 2 0.18 0.116 0.116I C D ,a , m 2/m 3 69 102 80volume of liquid trea ted, m 3 2.6 10-3 1.136 10-3 1.449 10-3

elect rica l en er gy input , W 400 100 100efficiency,b s-1 m -3 W-1 1.0 10-3 2.67 10-2 1.44 10-2

%in cr ea se in efficiency 0 2570 1340

a I lluminated catalyst density is defined as illuminated catalystsurface area (m 2) per unit volume of liquid treated (m 3) in t h ereactor (see ref 10). b Ef f icien c y is d e fin ed a s 5 0% p ollut a n tconverted per unit time per unit volume of liquid treated per unitelectrical energy input.

Figure 13. Contour of photocata lyt ic degradat ion of phenol an dS B B d y e a t t) 330 s. The entire r eactor length is divided into th etop 30%, m iddle 40% an d bottom 30%. The scale of polluta ntconcentrat ion is shown on the left (1 1 0-4 t 100 ppm).

Figure 14. P hotocat alyt ic degradat ion of phenol and SB B dye inreactor configurat ions 1 a nd 2.



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on s t ra t e d in t h e con t o ur p lot s of t h e a x ia l v eloci t ycomponent a long th e entire length of the reactor shownin thr ee segments in Figure 12. The figure shows th atflow develops slowly for configuration 2 (lower aspectra tio), but vortices move fa ster t owa rd t he center of thereactor.

To analyze the performance of the TVR, phenol andSBB dye were considered as pollutants present in waterwith initial concentrations of 100 ppm. The photocata-lytic degrada tion rates for phenol and SB B dye are given

in Table 2. The degrada tion rat e of the SB B dye is about400 times slower than that of phenol. In Figure 13, thepollutant mass fract ion contours are shown at t) 240s for the photocata lytic degrada tion of phenol in reactorconfigurat ions 1 and 2 and of SBB dye in the referencerea ctor. A close scrutin y of Figur e 12, w here t he velocityvectors a re shown, explains t he ma ss fra ct ion contourplots of the pollutant s in F igure 13. I t reveals tha t t hereaction is faster at either end where well-developedvort ices ar e present , whereas the degrada tion rates a reslower where Taylor-Co ue t t e v ort ice s a re n ot we ll-formed. This type of flow evolution d irectly influencesthe photocatalytic oxidation of pollutant and can be seenin Figure 13. The rea ction ta kes place only in t he shea rlayer of the inner cylinder where no Taylor -Couettevort ices are present . Pollutant degradation integratedover the full reactor volume is shown as a function oftime in Figure 14. This plot clea rly ind icates t he role ofTaylor -Couette vort ices in enhancing the ra te of pol-lutant degradation. The figure clearly illustrates thatthe photocata lytic reactions a re diffusion-contr olled, ast h e ra t e o f de g ra da t io n o f SB B dy e is s a me a s t h a t o fp h e n o l e v e n t h o u g h t h e ra t e c o n s t a n t f o r t h e dy e isabout 400 times lower t ha n th a t for phenol. The overa llreact ion is primarily controlled by mass transfer andnot by intr insic kinetics. The figure also illustr at es tha tthe degra da tion ra te for phenol in TVR 2 is slower t ha nthat in the reference-configuration reactor (TVR 1). Thisis du e t o t wo f a c t ors . F irs t , e ve n t h ou g h , f or re a ct o rcon f ig u ra t ion 2, t h e v ort ice s mo v e f a s t e r t o wa rd t h ecenter from t he end caps, the ma gnitud e of the vorticesis s ma lle r . Se con d, t h e a mo u n t of ca t a ly s t p er u n itvolume of liquid t reated (the pa ra meter, ) is lower inreactor 2. The volume of liquid treated in TVR 2 is 27.5%more tha n th a t of TVR 1. Figures 13 and 14 also indicatethe possibility of operat ing the rea ctor in the t ra nsientmode by periodically switching the reactor on and offbeca u s e , du rin g t h e t ra n s ie n t p h a s es , t h e re a ct ionproceeds at a ra pid rat e because of the vortices near th ee n d w a l ls . Ta ble 3 c omp a re s t h e p erf orma n c e o f aclassical annular reactor (CAR)18 with those of the tw oTVR configurat ions considered in this work for thep h ot o ca t a ly t ic de g ra da t io n of p h en ol , wh ile Ta ble 4compares the performa nce of a TVR w ith those of a

CAR,18 a slurry reactor (SR),18 a mult iple tube rea ctor(MTR), 9 and a tube light reactor (TLR)10 for the photo-ca t a ly t ic de g ra da t io n o f SB B dy e. Th e t a ble clea rlyillustra tes the efficiency of the TVR over the otherreactors. One can observe a n increase in efficiency ofthe TVR of 60%over the tube light rea ctor10 and 125%over t he mult iple tube r eactor.9

Conclusions

Heterogeneous photoca ta lysis on semiconductor pa r-t i cl es h a s b ee n s h ow n t o b e a n e ff ect i ve m ea n s ofremoving toxic organic pollutants from water. Earlierexperimental studies of our group on catalyst-coatedtube bundles, novel immersion-type lam ps an d rota tingtube bundles revealed that photocatalyt ic react ion isma inly diffusion-contr olled. The rea ction occurs a t thefluid-cata lyst interface, and in m ost cases, the overa llra te of react ion is l imited by the ma ss tra nsport of thepollutant to the cata lyst surfa ce. In our earlier studies,we enha nced ma ss tra nsfer by increasing the mixing offluids through turbulence and introduction of baffles.I n t h is w ork , a n e w p h ot o ca t a ly t ic rea c t or de sig n isconsidered where an increase in conversion for the insitu degra da tion of toxic polluta nts is a chieved through

introduction of f luid f low instability. The new idea ofusing fluid flow insta bility to increase the rea ction yieldt h ro ug h ou t t h e re a ct o r v olu me is s h own t o p rodu ceeffect ive pollutant reduction. In this work, w e consid-ered unsteady Taylor-Couette flow between tw o coaxialcylinders, wh ere the inner cylinder (coat ed w ith TiO2catalyst) is rotated to achieve the desired instability.The ma in a dvant age of considering t his part icular typeof flow pattern as our design consideration is its creationof wavy vortex flow in the laminar-flow regime. Signifi-ca n t t ra n s f e r of f lu id bet we e n n e ig h borin g v ort ice soccurs in a cyclic fashion along a certain wave, and netaxial f low also occurs in which fluids wind a round thevort ices. Two different reactor configurat ions wereconsidered that depict two different aspect ratios of thereactor. The flow fields were calculated for a constantReynolds number, and it wa s observed that , in the loweraspect ra tio case, vortices move faster t owar d the centera l t h o u g h t h e ma g n it u de s a re s ma lle r . T wo di f f e re n tpollutants were considered, namely, phenol and a textiledy e, t h a t h a v e t wo v e ry di ff ere n t ra t e c on s t a n t s . Th era te of photocata lyt ic oxidat ion increased dra mat ically.It was observed that the reaction is completely diffusion-controlled, as both phenol and SBB dye degraded at thesame rate. The Taylor vortex reactor was found to bean excellent r eactor, a s t he increase in rea ctor perfor-ma nce efficiency wa s found to be 60 and 125%over th oseof a tube light reactor 10 and a mult iple tube reactor,9

respectively.

Table 4. Comparison of Performance of a TVR with that of a Classical Annular Reactor (CAR), a Slurry Reactor (SR), aMultiple Tube Reactor (MTR), and a Tube Light R eactor (TLR) for Photocatalytic Decomposition of SBB Dye

C AR 18 S R 18 MTR 9 TLR 10 TVR

volume of reactor, m 3 3.48 10-3 3.4 10-3 1.23 10-3 5.36 10-3 3.65 10-3

catalyst surface area, m 2 0.18 3.7 0.51 0.15 0.116I C D ,a , m 2/m 3 69 6139 1087 618 102volumetric f low rat e, m3/s 8.42 10-5 B a t ch oper a tion 3.0 10-5 1.67 10-5 Batch operat ionvolume of liquid tr eated, m 3 2.6 10-3 6.0 10-4 4.69 10-4 2.43 10-4 1.136 10-3

elect r ica l en er gy in put , W 400 960 40 126 100efficiency, b s -1 m -3 W1- 9.43 10-4 3.1 10-3 1.18 10-2 1.67 10-2 2.67 10-2

% in cr ea se in efficien cy 0 229 1157 1668 2730a I lluminated cata lyst density is defined as illuminated cata lyst surface area (m2) per unit volume of liquid treated (m 3) in the r eactor

(see ref 10). b Efficiency is defined a s 50%pollutant converted per unit t ime per unit volume of liquid tr eated per unit electrical energyinput .



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Notation

AR) a s pect r a t iod) a n n u l a r g a p , mL ) lengt h, mr) ra dius , m; ra dia l direct ion, mR e ) Reynolds numbert) t ime, sT a) Tay lor numberV) velocit y, m/s

Symbols

) i llumina t ed ca t a lys t dens it y, m 2/m 3 ) rotat ion ra te, ra d/sF ) densi ty , kg /m 3) kinemat ic viscosity, m 2/s

Subscripts/ Superscri pts

i) innero) outer

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(17) Werely, S. T.; Lueptow, R. M. Spatio-temporal Characterof Non-Wavy and Wavy Taylor C ouette Flow. J . F l u i d M ech .1998,364, 59.

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Received for review December 20, 2000Revised ma nu scri pt received J une 6, 2001

Accepted J une 6, 2001

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2001_t.k.sengupta_a taylor vortex photocatalytic reactor for water purification

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