Schematic of a mixed flow reactor (both advection and dispersion must be accounted for)

Transcription

1 Cas stuy 6.1, R: Chapra an Canal, p Th quation scribin th concntration o any tracr in an lonat ractor is known as th avction-isprsion quation an may b writtn as: Schmatic o a mi low ractor (both avction an isprsion must b account or) c -1 (not within in systm) c c n-1 c n1 ownstram bounary conition c n1 (not within c c i-1 c 1 i c c in systm) i1 n-1 c n c in c in is a assum known L systm bounary NOTES Th systm is in to b coincint with th ractor. Thror concntrtrations at c n-1 an c n1 which aris whn writin mass balancs on ach smnt ar outsi th in systm. Ths ar obtain usin bounary conitions Th concntrations comput in this ocumnt ar at th intracs btwn smnts. Ths locations ar rrr to in this ocumnt as "nos" Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 1 / 8

2 in th "systm" to b coincint with th ractor. Brak th systm into som numbr "n" o smnts or slics, ach with a thicknss. W will writ a mass balanc on a substanc, "s" on ach smnt havin a init lnth,. In wors : accumulation within smnt avction in - avction out - isprsion in isprsion out - cay within smnt W will us Fick's irst Law to scrib isprsiv lu: J isprsiv c. Th nativ is us so that a positiv isprsiv lu will occur in th irction o crasin concntration. V c t Qc( ) Q c( ) A c A c kv c Now, i w combin trms w t : V c t Q ivi thru by th volum V A c * A c c kvc c t Q A c A c A c c kc Now cancl trms an lt an shrink to zro, not that U Q A c t c U c kc Now assum th systm is opratin at stay stat so th lt si o th quation quals zro. This rsults in a n orr, orinary, irntial quation U c kc Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc / 8

3 BOUNARY CONITIONS : Inlt bounary conition: In this problm th contaminant ntrs th tank throuh a pip an w assum isprsion is nliibl. Thus th mass o contaminant ntrin th tank pr tim via avction must qual th mass carri away by avction an isprsion. Qc in Qc o A c c Outlt bounary conition: W spciy that th concntration o th chmical os not chan as it crosss th it bounary at L. In quation orm: c( L, t) NUMERICAL SOLUTION TECHNIQUE - init irnc approach - ivi th ractor into a numbr o smnts, writ a mass balanc on ach smnt. Trat rivativ trms as init irnc approimations. In this ocumnt w will us a cntr irnc approimation or both rivativ trms. Othr possibl approimations ar a orwar irnc an backwar irnc schm. c i 1 c i c i 1 U c i 1 c i 1 kc i W can writ an quation o this orm or ach smnt. Now sparat variabls an rwrit th quation with sparat trms or ach intrnal no c i, c i-1 an c i1 1 c i 1 u U k U c i 1 U c i 1 Notic that ach intrnal no quation contains 3 unknowns, c i 1, c i, c i 1. I w ivi th ractor into n smnts w will hav n unknowns c throuh c n. I w writ an quation or ach intrnal no plus on or ach "bounary no" that will iv n quations. Bcaus ach unknown can appar in as many as 3 quations w will n to solv th ntir systm at onc usin matri albra tchniqus (or o som hllacous albra!) Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 3 / 8

4 Inlt bounary quation : I w writ a stanar no quation or th inlt concntration, c o, w will n up with an quation containin an unknown not in th systm, c 1 1 c 1 u U k U c 1 U c 1 W can liminat this unknown by makin usin o th inlt bounary conition: Qc in Qc o A c c Not : c may b viw as c valuat at c o Now us a cntr irnc approimation o th rivativ trm to writ Qc in Qc o A c c 1 c 1 U solvin or c -1 w t: c 1 c 1 c in U c Substitut th prssion or c 1 into th stanar no quation to obtain an quation at th inlt in trms o unknowns insi th systm an c in which is known. k U c U U U c 1 U c in whr c in, th concntration ntrin th ractor, is assum known Outlt bounary conition I w writ a stanar no quation or th n th no w incur an unknown, c n 1, not within th systm. In a ashion similar to th inlt w will hanl this usin th ownstram bounary conition. 1 c n 1 u U k U c 1 n U c n 1 Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 4 / 8

5 Rcall th outlt bounary conition : c( L, t). In wors this says that th concntration across th ractor it os not chan, thus c n 1 c n 1. I w mak this substitution into our stanar no quation w t: k c n 1 U U U c n Summary so ar: O.K. w brak our ractor (coul b a rivr) into som numbr, n, o smnts. Usin th avction-isprsion quation (rally just a mass balanc) as th ovrnin PE w writ a mass balanc on ach smnt. Bcaus w ar oin a numrical solution w mploy a init irnc approimation o th ovrnin PE. W choos a cntr irnc approimation or th irst an scon rivativs in th PE. W vlop physically ralistic bounary conitions which ar mploy to obtain quations or th concntration at th inlt an outlt o th ractor. At this point w can writ an quation or th inlt, on or ach smnt, an on or th outlt. U k U U c U c U 1 c in...inlt quation, 1 1 c i 1 u U k U c 1 i U c i 1... no quation, 1 or ach smnt k c n 1 U U U c n...outlt quation, 1 An amination o th quations abov inicats that i th systm proprtis,,, U an k ar constant throuhout th ractor thn w hav a st o n linar quations with n unknowns. Th coicints or ach quation or th intrnal quations ar th sam. Ths can b solv usin matri tchniqus. Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 5 / 8

6 An ampl: Q.11 6 al : ay H : 1t B : 1t n : 8 1 k :.1 ay :. t sc U : Q HB Lnth : 1t : Lnth n c in : 1 m litr smnts : Lnth smnts 8 k U U a :, b :, an :...inlt quation coicints U U U 1 k intrnal no quation coicints... :, :, an : U U U U 1 k h :, j :...outlt quation coicints U U U U.967 Comput stability paramtr or chosn paramtrs. I ratr than. systm will bcom unstabl. Now, arran th quations in matri ormat. Not that th coicint matri is triiaonal [coicint matri]*[unknown vctor] [trnal loains vctor] Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 6 / 8

7 a b h j c c 1 c c 3 c 4 c 5 c 6 c 7 c 8 c in Whil thr ar a varity o ways to solv a st o linar quations w will o it by invrtin th coicint matri. c a b h j 1 c in : concntration at th intrac btwn smnts c m litr Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 7 / 8

8 i :.. smnts i : i istanc rom ractor inlt or ach comput concntration Concntration vs istanc alon Ractor Concntration c 31. m 6.16 litr t istanc Cas_stuy_plu_low_ractor.mc last sav 1/5/ / 6:17 PM E:\public_html\Cas_stuy_plu_low_r actor.mc 8 / 8