Modelling of straitened sedimentation process in bidisperse suspension with inter-fractional coagulation

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1 American Journal of Chemical Engineering 04; (): 4-0 Publihed online June 0 04 (htt:// doi: 0.648/j.ajche Modelling of traitened edimentation roce in bidiere uenion with inter-fractional coagulation Tulegen Amanbaev South Kazakh State Univerity Shymkent 5 Tauke Khantr. 600 Intitute of Mathematic and Mathematical Modeling Almaty 5 Puhkintr Kazakhtan addre: tulegen_amanbaev@mail.ru To cite thi article: Tulegen Amanbaev. Modelling of Straitened Sedimentation Proce in Bidiere Suenion with Inter-Fractional Coagulation. American Journal of Chemical Engineering. Vol. No doi: 0.648/j.ajche Abtract: In thi aer we tudy the general characteritic of deoition of large article (or aggregate) that reult from the mutual aggregation of mall and large fraction in bidiere (or double fractional) uenion. We give equation of motion and change in ma of the large article in the reence of inter-fractional coagulation roce and effect of traitened edimentation. In the limiting Stoke and Newton mode (relevant for mall and large Reynold number) we have movement formula for eed of edimentation and their analyi. Dicu ome of the reult of calculation obtained by numerical integration of the equation of motion of a large article in uenion. Keyword: Suenion Brownian Motion Sedimentation Coagulation. Introduction In variou area of chemical technology (roduction of catalyt hotograhic material coagulative urification of liquid etc.) are wideread rocee of edimentation of a olid hae. Thu quite often there are uenion with trongly different ize of article (o-called bidiere or two-fractional uenion). For examle in [] it i noted that under certain condition in diere ytem there can be a litting of a range of article of a diere hae on two different cale fraction. Beide there i a wide cla of bidiere uenion in which article of fine (mall) fraction can be taken for a homogeneou medium in which the article ha a coare(large) fraction (e.g. two-tage grinding coal-water lurrie which are ued in ieline tranort [] two-fractional olymeric uenion [3] etc). Earlier uenion with a twofractional diere hae wa oberved in [4] to reearch the degree of dierion of eat. Nowaday with the advent of increaed interet in nanouenion (liquid with nano-cale article) becaue of their unique roertie tudying of the edimentation of the olid hae i quite imortant. Thi i due to the fact that nano-uenion i quite untable i.e. for variou reaon the article can coagulate and when it reache a certain critical ize begin to reciitate. The roce of edimentation in the uenion of heterogeneou article with very different ize i alo ued to roduce nanocomoite material. Some aect of two-fractional model of the coagulating diere ytem are baed on Smolukhovky kinetic equation are dicued in [5 6] and in [7] the kinetic of edimentation of a low-concentrated coagulating uenion i invetigated. The aim of thi work i tudying of influence of coagulation of article of fine (mall) and coare (large) fraction in bidiere uenion on dynamic of gravitational edimentation of aggregate (aearing in uenion by the addition of mall article) for the effect of contraint. Further along with the term "large article" will alo ue the term "aggregate.. Baic Aumtion and Equation.. The Key Aumtion Let uenion be Newtonian (untructured) i.e. it rheological roertie are determined by only hydrodynamic factor. Condition for the exitence of uch uenion are analyzed in detail in []. In relation to the large article (aggregate) environment diere ytem "liquid-fine article" i viewed a a continuum with an effective vicoity and adduced denity. In thi cae the influence of Brownian motion of mall article on the

2 American Journal of Chemical Engineering 04; (): effective vicoity of the uenion i neglected. Thi aroach i often ued for the roce of edimentation lurry [ 8 9]. Let' conider coagulating dierion ytem which wa already lit into two fraction. Following [5] we make the following aumtion: the article of different fraction vary coniderably in ize; aggregate grow by the addition of fine (mall) article; the roce of coagulation growth of fine article to aggregate ize ha a negligible rate comared to the direct exchange of material between the two fraction. We aume that the roce of interfractional coagulation i due to two main mechanim: Brownian motion of fine (mall) article and their engagement by large article becaue of the eed difference of fine and coare fraction of the diered hae of the uenion. Aggregate conit of fine article (interval are filled with liquid) have a herical form and aren't ubject to Brownian motion. Fragmentation of aggregate i not conidered in thi connection be taken into account the effect of traitened edimentation of large fraction a by cature of mall article the volume art of the aggregate can be ubtantial. For imlicity we aume that the number denity of large article i contant and uniform throughout the ace filled with a tationary uenion of fine article (atially homogeneou uenion). Under the latter aumtion to decribe the motion of the entire coare fraction i ufficient to conider the equation of motion of only one large article (taking into account the effect of other article). Further the bottom indexe l and are noted arameter of uenion it liquid and fine comonent. Parameter of coare fraction (aggregate) and the mall article attached to it are noted by the bottom indexe and reectively... The Equation Thu within the acceted aumtion the equation decribing change of ma and velocity of aggregate in the field of gravity can be written down in a look (the coordinate axi i directed againt gravity acceleration) f dm dt j du m f + f + f + f ju () dt g A m C πr uu f g V g j nm( KB + Kc) V 4/ 3) 3 du V dt f m V g + f A l l + ( ) l + l ( πr m V + l + Vn n cont where t i time m V r u are ma volume radiu and eed of the large article j i the intenity of the deoition of the fine fraction on a urface of large article f f g f A f are reectively force of vicou m reitance weight Archimede and the force of attached ma C i coefficient of reitance of a large article n m are numerical concentration of the fine fraction of the uenion and the ma of mall article l l are volume fraction and the true denity of the liquid and the fine fraction in the uenion reectively are the volume concentration and the true denity of the uenion of fine article (a mixture of liquid and fine article without aggregate) n are the volume and numerical concentration of large article in the lurry and their true denity i the volume fraction of olid (fine article) in the aggregate ( remained hare i the liquid hae) which can uually be aumed contant the denity of large aggregate will alo be contant (in articular the random acking of herical article 0.6 [0]). Rate of change of volume concentration of the fine fraction i decribed by equation d dt n ( K B + K ). () The frequencie of colliion of fine article with a large article in Brownian motion of mall article and due to the difference in the eed of article of different fraction are given by [6 ] K B 4πkBT( r + r) B( r) K ( r r) u c πη + B ( r ) (6π lr) where k i the Boltzmann contant T i temerature B r i radiu of mall article. Value B r ) characterize the mobility of fine article and i determined by Stoke law. Efficiency of catching of fine article by aggregate due to the difference between their velocity η can be calculated by the following formula [ 3] (taking into account that the ratio of article ize of fine and coare fraction δ i ufficiently mall). 3δ / Re ( B A)Re + (A B) <Re η η( r u) + Re δ < Re 80 (3) 5 E D Re + (5D 4E) 80 < Re δ Re > 00 A 3δ / B.94δ D 7.75δ E 3δ δ r /r ( c

3 6 Tulegen Amanbaev: Modelling of Straitened Sedimentation Proce in Bidiere Suenion with Inter-Fractional Coagulation where Re i Reynold number of relative movement of aggregate. In the formula (3) the exreion at the to and bottom line meet the limit vicou and otential mode of motion of the aggregate and the exreion in the middle row - tranient. Mechanim of catching occur mainly becaue the fine article have a finite ize and at the ditance order of the radiu of fine article from the urface of a large article the normal comonent of velocity of liquid i not zero [6]. Effect of contraint of motion of aggregate in cae of the large volumetric concentration uually i accounted by correction ψ in the reitance coefficient [0 4 5] C C ψ( ) ( ) ( ) m m 5 ψ ru C Re. Re Re To calculate the effective vicoity of the uenion with the fine fraction of article it i adviable to ue a Mooney formula which decribe quite well the numerou exerimental data in a wide range of volume content of the diered hae and ha the form [8 6 7].5 ex / l where i the volume art of the fine fraction of the uenion at the mot dene acking of the article (henceforth referred 0.74). Coefficient.5 rovide a tranition Mooney formula to Eintein' famou formula for mall volume content of the fine fraction [0]. We define the following initial condition for the ytem () (): t 0 m m0 u 0 n. 0 0V0 3. Reult and Dicuion 3.. A Fit Integral It i eay to ee that the ytem of equation () () ha the firt integral which reflect the law of conervation of ma θ l + cont (4) θ + ( ) where the contant value on the right ide i determined for examle by the initial data. Equation (4) can be ued in articular to comute the volume content of the coare fraction and the radiu of the aggregate l + θ ) 0 ( / 3 / 3 r θ( ) r +. r0 0 0 Hence in articular derive an exreion for the limiting or teady-tate (when the fine fraction i fully joined to the large aggregate i.e. when 0) and the radiu of the aroriate limit the volume content of aggregate in uenion / 3 / 3 + r θ 0 + θ. 0 0 With the growth value be / arameter θ tend to it imum. The correonding imum value + 0. will Thi formula reflect the fact that the limit of volume fraction of large article (aggregate) i alway greater than the um of the initial volume content of fine and coare fraction of the diered hae (a in the aggregate there i alo a liquid hae which fill the ace between the article of aggregate). Becaue of the aumtion made in the framework the volume fraction of aggregate in uenion may not exceed the limit value correonding to the mot dene acking o that mut be done the relation <0.74 or 0 + < When we give the value and 0 in mind the obtained ratio. 3.. A Cae of Non-Sedimentation (5) we hould kee In the abence of the edimentation ( u 0 ) conolidation of aggregate occur due to Brownian coagulation. Such a ituation can occur in thoe cae where for examle aggregate ize are ufficiently mall or the article ubtance denity i not much different from the denity of the liquid. In thi cae the ytem ()() ha an analytical olution that can be rereented in an imlicit form ϕ( z) t ln 3[ ω( z) ω( z0) ] ϕ( z0) 3 3 [ ( z) ] / / 3 ( z) + ϕ ( z) ω( z ) arctg z 3

4 American Journal of Chemical Engineering 04; (): θ r kt r z t 0 t r π r r l 0 ( z <z < 0 ). The grah of deendence z(t) at different z0 i hown in Fig.. Fig.. Grah z (t) at different z 0: -z Further it i ueful to analyze the limiting Stoke and Newton regime of the teady motion of the aggregate Stoke Movement Mode of Aggregate In the Stoke regime when the Reynold number of the relative motion of large article Re are mall enough for the drag coefficient of the aggregate i allowed the following aroximate exreion C 4/ Re. In thi cae edimentation rate can be exreed in the formula v u ( ) r ψ( ) u l U gr U 9 u 0 ( 0 < 0 v > l u ). (6) Analyi howed that the rate v defined by the formula (6) with increaing monotonically decreae 0 wherea with a change behave not monotonically and ha imum at the following value of (7) θ 7 Limit volume concentration of the coare fraction correonding to thi value 0 i equal to θ. The correonding imal rate of edimentation can be calculated by the formula / 3 0 v For examle for we get 0.07 v 0. A 0 a imum of deendence v ) /7 0 ( exit when the condition. It i obviou that the value 0 can only be in the range from 0 to / 7. When >/ 0 7 the velocity v with increae monotonically decreae. The reult of calculation according to formula (6) below (ee Fig. 5). The non-monotonic behavior of the limit eed i exlained by the fact that with increaing increae limit volume fraction of large article conequently the limit ize of aggregate r. Growth r lead to an increaed rate of edimentation of aggregate and magnification lead to increaed influence of traitened edimentation. For mall value of arameter q θ 0/ 0 << for the deoition rate of the aggregate v there are following aymtotic rereentation 4 7 ( )( 0) 0 + q. (8) 3 7 v 0 Thi how that when </ 0 7 the eed v with increaing q i alo growing wherea when >/ 0 7 in the contrary it decreae. If / 0 7 the edimentation rate v doe not deend on a mall arameter q it i 5 exreed in a more comact form v ( )(. 0) 3.4. Newtonian Motion Mode of the Aggregate In other extreme condition correonding to high Reynold number of the relative motion of the aggregate (when you can take C 0. 5 ) for a teady rate of edimentation we have v ( ) r γψ( ) / 3U γ. (9) 8gr Similarly to Stoke mode in thi cae rate v during increae monotonically decreae and with a change 0 behave non-monotonically and ha imum at the following value of (0) θ 6 Limiting volume concentration of the coare fraction correonding to thi value 0 i equal to 0

5 8 Tulegen Amanbaev: Modelling of Straitened Sedimentation Proce in Bidiere Suenion with Inter-Fractional Coagulation θ. The correonding imum rate i calculated by the formula / v / 6 γ 0 T93 K. The initial volume concentration of fine and coare fraction and were choen taking into account the 0 contraint (5) and ranged from 0 to 0.35 and from 0-4 to 0. reectively. Note that the imum rate in the Newton mode i achieved at a lower value than in the Stoke regime (comare formula (7) and (0)). For mall value of q<< for the edimentation rate of the aggregate in the Newtonian mode imilar to a Stoke regime the aymtotic rereentation i 8 β( 3/ 0) 0 + q 3 6 v 0 ( β γ / ) () from which in articular follow that with </ 0 6 the value v with growth of q i increaed wherea when >/6 converely decreae. When / 0 0 6the rate of deoition of aggregate doe not deend on a mall 5/ arameter q and ha a imler form v β( 0). Comaring formula (6) and (9) it hould be noted that the teady rate (dimenionle) of aggregate edimentation in the Newtonian regime i roortional to r / wherea it i roortional to r in the Stoke mode. From thi we can conclude that in any mode of movement of the aggregate (when the Reynold number take not very mall and not very large value) v r where 0.5 ω. In thi 0 ω cae the value correonding to the imum of tationary eed of aggregate atifie condition θ θ 7 or for limit volume art of coare article - the condition 6. 7 Note that the exreion (6) (9) can be ued in articular for an aroximate calculation of the teady rate of edimentation reectively for mall and large cale aggregate and the aymtotic formula (8)()- at low volume concentration of the fine fraction (when << / θ ) Numerical Reult After thi there are dicued ome reult obtained by numerical integration of the ytem of equation () and () under the following fixed arameter: 0 Fig.. Deendence of deoition rate of the aggregate at time when 0.00: Fig. how time variation of the rate of deoition aggregate v u > 0 for the initial concentration of the coare fraction Curve -3 correond to different initial volume content of the fine fraction (curve ) 0. () 0.3 (3). For comarion a dotdahed line how the teady-tate eed of a large article in the abence of inter-fractional coagulation (for 0 0.) and the dahed line - without the fine fraction ( 0 0). It i evident that deending on the volumetric concentration of the fine fraction the deoition rate of the aggregate behave differently. In non-highly concentrated uenion aggregate eed gradually increae reaching it teady-tate value for a long time (curve ). Wherea in the roce of edimentation lurry concentrated coare fraction can be divided into two tage (curve 3). In the firt tage rate increae becaue larger article to increae it ma by addition of fine article and the econd tage where it eventually reache a certain ize the edimentation rate decreae due to the effect of contraint. In thi range of the concentration of the initial volume fraction of the edimentation rate of mall aggregate in the reence of the coagulation roce i ubtantially higher than in it abence. However we note that an increae rate of deoition of the aggregate due to the influence of traitened edimentation effect may be le than in the ure liquid. The effect of the teady rate on a large article in a uenion i dicued below (ee Fig. 5) r 0 50 m r 0.05 m

6 American Journal of Chemical Engineering 04; (): larger a in the vicou regime it i roortional to δ and in the otential regime to δ (δ<<). Fig. 3. Deendence of the volume concentration of the fine fraction of the uenion from time when 0.00: Fig. 3 how the variation in time of the volume concentration of fine in it variou initial value (curve ) 0.05 () 0. (3) 0. (4) 0.3 (5). Value i the ame a in Fig.. It i een that at time 0 5 ec 0 deite the different eed of aggregate at different (ee Fig. ) the volume concentration of the fine fraction decreae ignificantly. Fig. 4. The ame a in Fig. 3 but with 0.000: Fig. 4 illutrate the behavior at lower value (comared with the reviou cae) the initial value of the volume concentration of the coare fraction Curve correond to 0.0 (curve ) 0.05 () 0. (3) 0.35 (4). It can be een that not very concentrated lurry volume fraction of fine due to coagulation i reduced a in the reviou cae lowly enough (curve ). While more concentrated lurry coagulation of fine and coare fraction occur lowly at firt then unlike the reviou cae in a relatively hort time the fine fraction i almot comletely converted into aggregate (curve -4). The latter i due to the fact that reducing the volume content of coare article (fixed ize) of the number denity of the lurry i reduced. Conequently by joining aggregate of fine article comared with the reviou cae reach the larger ize to which the flow regime change from a vicou to otential. Thu in accordance with formula (3) the intenity of tranfer of fine fraction to the coare fraction ignificantly Fig. 5. Deendence of tationary velocity of aggregate on volume concentration of fine article fraction for different : / Fig. 5 how the edimentation rate of limit aggregate v from an initial concentration of fine in uenion at different initial concentration of a large volume fraction 0.0 (curve ) 0.05 () /7 (3) 0. (4). The olid 0 curve correond to the numerical integration of the ytem ()() and dahed curve are baed on the formula (6) and the correonding Stoke regime of movement of the aggregate. Calculation how that in general imilar to Stoke and Newtonian regime the deendence v ) ( from the value behave qualitatively different. For 0 examle at about 0 < ~/7 conidering deendence varie non-monotonically (curve ) and ae through a imum wherea with 0 > ~/7 it monotonically decreae with increaing (curve 3 and 4). Moreover with a decreae the imum oint hift to higher 0 (a evidenced by the behavior of the dot-dahed line). Note that with increaing initial volume content of the coare fraction difference in behavior 0 v( ) according to the Stoke and general cae decreae. Thi i undertandable becaue with the increae due to the 0 effect of contraint deoition rate of the aggregate fall and the condition for movement cloe to Stoke regime Concluion Thu in the model there are analyzed ome feature of the dynamic of aggregate in a atially homogeneou coagulating uenion with the traitened edimentation effect. Note the given work i theoretical. The reult of the analyi and the reulting aroximate formula can be ued for etimation and calculation the general arameter of ediment uenion in the reence of the coagulation roce.

7 0 Tulegen Amanbaev: Modelling of Straitened Sedimentation Proce in Bidiere Suenion with Inter-Fractional Coagulation Acknowledgment Thi work wa uorted by a grant ( 0750/GF and 7/GF) of the Minitry of Education and Science of the Reublic of Kazakhtan. Reference [] T.R. Cam Floc volume concentration J. Amer. Water Work Aoc.60 (96) 6. P [] N.B. Uriev and A.A. Potanin Fluidity of uenion and owder Mocou: Chemitry 99. (in Ruian) [3] L.V. Ravitchev A.V. Bealov and V.J. Loginov Simulation of vicou roertie of concentrated uenion Theoretical Found. of Chemical Engineering 008. V P. 36. [4] M.P. Volarovich and N.V. Churaev Invetigation of the degree of dierion of eat Proceeding of the Mocow Intitute of eat Mocow 955. Iue 3. P. 33. (in Ruian) [5] B.M. Dolgonoov The kinetic of coagulationfragmentation and the equilibrium ectrum of aggregate in moving uenion Theoretical Found. of Chemical Engineering 00. V P [6] V.A. Galkin I.V. Galkin D.Y. Oetky D.A. Rizhikov and A.V. Galkin Mathematical modeling of intering of owder material and the growth of Aggregate Proceeding of the regional cometition of reearch roject in the field of natural cience Kaluga 007. Iue. P. 4. (in Ruian) edimentation uenion Theoretical Found. of Chemical Engineering 005. V P [8] B.I. Brounhteyn and V.V. Schogolev Hydrodynamic ma and heat tranfer in the column aaratue Mocou: Chemitry 980. (in Ruian) [9] J. Hael and H. Brenner Hydrodynamic at low Reynold number Sringer-Verlag 976. [0] R.I. Nigmatulin Fundamental of mechanic of heterogeneou media Mocow: Nauka 978. (in Ruian) [] V.M. Volohchuk and Y.S. Sedunov Coagulation rocee in diere Sytem Leningrad: Gidrometeoizdat 975. (in Ruian) [] Theory and technology of flotation Ed. Bogdanov O.S. Mocow: Nedra 990. (in Ruian) [3] R.H. Yoon and G.H. Luttrell The effect of bubble ize on fine article flotation Miner. Proce. Extr. Metal. Rev P. 0. [4] G.K. Batchelor Sedimentation in a dilute dierion of here J. Fluid Mech. 97. V. 5. P. 45. [5] A.M. Golovin and V. Chizhov Calculation of the deoition rate of a homogeneou uenion J. Al. Math. and Mechanic 978. V. 4.. P.05. [6] M. Mooney The vicoity of concentrated uenion of herical article J. Coll. Sci. 95. V. 6.. P. 6. [7] T.L. Smith and C.A. Bruce The vicoity of concentrated uenion J. Coll. Interface Sci V. 7.. P. 3. [7] B.M. Dolgonoov The kinetic of the coagulating