Cake Filtration Simulation

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Edwin Bishop
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1 ake Filtratio Simulatio Filtratio model a be ued to aalyze ad optimize filtratio operatio. ake filtratio equatio, whih deribe the variatio i filtratio flow rate a a futio of operatig parameter ad phyial propertie, a be derived by applyig Dary law to the filter ake ad medium [1]. The ake filtratio equatio a be writte a follow, d V dt P V 2 R m (1) where t i filtratio time, V i filtrate volume, i liquid vioity, i the peifi reitae of the ake, i the dry ake ma per uit volume of filtrate, i the filtratio area, R i the filter medium reitae, ad m P i the ombied preure drop over the ake ad medium. Several differet mode of operatio are poible. Thee ilude otat rate filtratio, whih orrepod to operatio with a poitive diplaemet pump, ad otat preure filtratio, whih reflet the typial mode of operatio for vauum filtratio. I the latter ae, the filtratio rate dereae with time a the filter ake build up. For ompreible ake, the variatio i ake propertie with time mut alo be oidered i developig a model of the filtratio operatio. The filtratio imulatio tool iorporate model ad olutio method deribed i the literature [1-3]. The priipal feature of thee model are preeted below. otat Rate, Iompreible ake Filtratio For otat rate filtratio, the ake filtratio equatio a be rearraged to obtai a liear relatiohip betwee the filtratio preure ad filtrate volume, q R q m P V 2 (2) where q dv i the otat filtrate flow rate. The peifi ake reitae ad medium dt reitae a thu be determied from the lope ad iterept of a experimetal plot of filtratio preure agait filtrate volume.

2 otat Preure, Iompreible ake Filtratio The performae of a filter beig operated uder otat preure a be deribed by itegratig ad rearragig the ake filtratio equatio to obtai the liearized paraboli rate law equatio, t R m V 2 V 2 P P (3) I thi ae, the peifi ake reitae ad medium reitae a be determied experimetally from a plot of filtrate time over volume agait filtrate volume. otat Rate, ompreible ake Filtratio Whe aalyzig ompreible filter ake it i importat to oider the variatio i ake propertie durig the filtratio proe. Thi a be ahieved by modelig the depoitio of the ake, layer by layer, with the ake propertie beig realulated at eah tep [2]. For a otat rate filtratio, the preure drop required to maitai a otat flow will ireae a the filtratio progree. The ireaig preure will reult i a ompatio of the filter ake, with a oequet ireae i the olid oetratio ad peifi reitae. The variatio i ake propertie with preure a be modeled uig empirial relatiohip uh a thoe give below, (1 ) P, (4) a v o u, (5) a v o (1 u ) P where ad are average value, P i the preure drop aro the ake, ad av av o o, ad u are empirial otat. The imulatio tool model the oditio for the formatio of a ew layer of filter ake. The preure drop aro the ewly formed layer a be alulated from Dary law,, P L q, (6) a v a v

3 where q i the filtrate flow rate ad layer. The thike balae, P i the preure drop aro the ewly formig L of the ewly depoited ake a be determied from a ma L Q t, (7) a v where Q i the feed rate of upeio ad i the feed oetratio expreed a a volume fratio. Thi a be alulated from the ma fratio uig the overio The time t required for the depoitio of the ew ake layer a be alulated by ombiig the preure drop ad ma balae equatio to obtai the followig equatio 2 /. (8) a v t P q Q The filtrate flow rate a be alulated from the feed rate, q Q 1 av. The imulatio alulatio proeed by iremetig the total preure drop aro the ake, omputig the required filtratio time, ad realulatig the ake propertie at eah iremet. otat Preure, ompreible ake Filtratio If medium reitae i egleted, the the etire preure drop applie to the ake. I thi ae a imple olutio i poible, ie the ake oetratio ad peifi reitae will be otat for the duratio of the filtratio tep. For ae where the medium reitae i igifiat, the fratio of the overall preure drop that applie to the ake will ireae with time, reultig i a ireae i ake oetratio ad reitae. iterative alulatio heme for modelig ompreible ake filtratio i propoed by Holdih [3]. The variatio i ake reitae ad oetratio a a futio of preure a be modeled with empirial relatiohip a deribed i the previou etio. The filtrate volume a be alulated by rewritig the rate law a a quadrati equatio,

4 a v 2 Rm V V t 2 2 P P 0. (9) Here, i the average ma of dry ake depoited per uit filtrate volume, whih a be alulated from av a v. (10) The filtrate flow rate a be alulated from the tadard ake filtratio equatio, q dv dt 1 R a v m V 2 P P. (11) The fratio of the overall preure drop that applie to the filter ake a be alulated by ubtratig the preure drop over the medium from the overall preure drop. R m P P q. (12) The iterative olutio proe begi with the aumptio that the etire preure drop applie to the filter ake. Thi lead to a itial etimate of ake propertie whih a the be ued to realulate the preure drop that applie to the filter ake. The ew alulatio iorporate the preure drop aor the medium. Thi lead i tur to a revied etimate of ake propertie, ad the iteratio otiue till the alulatio of the preure drop overge. Give the filtrate flow rate ad ake preure drop value, the height of the filter ake ad the ake oetratio profile a be alulated from the followig equatio L P 1u (1 u ) q o o, (13) u y P y o L u 1 u. (14)

5 Rappture Iterfae The Rappture iterfae ilude a optio meu to hooe from model for ompreible ad iompreible ake, ad operatio mode for otat rate ad otat preure filtratio. The iput form ilude field for the etry of lurry parameter uh a the vioity, filter parameter uh a the urfae area ad medium reitae, ad ake propertie uh a the olid oetratio ad ake reitae (Figure 1). dditioal parameter are required to deribe the variatio i ake propertie with preure for ompreible ake. Operatig oditio uh a the lurry feed rate for otat rate filtratio or the overall preure drop for otat preure filtratio mut alo be provided. Output plot ilude the filtrate flow rate for otat preure filtratio ad the overall preure drop for otat rate filtratio. The filtrate time over volume agait volume for otat preure filtratio i a traight lie illutratig the liearized paraboli rate law. The filtratio imulatio tool a be ued to illutrate ome of the pheomea that our i ompreible filter ake. Figure 2 how the ireae i the filtrate flow rate for a oteibly otat rate filtratio that our a the filter ake i ompreed, forig ome of the retaied liquid out of the filter ake [2]. The olid oetratio veru ake height i plotted at variou time value for otat preure ompreible ake filtratio. The plot i Figure 3 how three profile at proeig time of 1 hour, 2 hour, ad 3 hour repetively. the proeig time ireae the fratio of the overall preure drop that at over the filter ake ireae, reultig i a ireae i the olid oetratio at the urfae of the filter medium [3].

6 NOMENLTURE filtratio area o oeffiiet i ake oetratio orrelatio volume fratio of olid i feed y volume oetratio at height y L dry ake ma per uit volume of filtrate average ma of dry ake depoited per uit filtrate volume height of filter ake L thike of ewly depoited ake expoet i ake reitae orrelatio P ombied preure drop over the ake ad medium P preure drop aro ake P preure drop aro ewly formig layer Q q feed rate of upeio filtrate flow rate R m filter medium reitae t ma fratio of olid i feed filtratio time t time required for depoitio of ew ake layer u V expoet i ake oetratio orrelatio filtrate volume peifi reitae of ake o oeffiiet i ake reitae orrelatio liquid vioity liquid deity olid deity

7 REFERENES 1.. Ruhto,. S. Ward ad R. G. Holdih, Solid-liquid filtratio ad eparatio tehology, VH, Weiheim, Germay, R. Holdih, Software for imulatio ad modelig of eparatio proee, Departmet of hemial Egieerig, Loughborough Uiverity. 3. R. Holdih, Simulatio of ompreible ake filtratio, Filtr Separat 31 (1994),

8 Figure 1 Simulatio of ake filtratio.

9 Figure 2 Ireae i filtrate rate for ompreible ake filtratio.

10 Figure 3 Variatio i olid oetratio with ake height i otat preure ompreible ake filtratio.

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