REACTOR. Source: Fundamentals of Water Treatment: Unit Processes Physical, Chemical, and Biological 2

Transcription

1 Reacor Models 1

2 REATOR A reacor is he place where he desired reacion akes place ha is, he removal of a seleced conaminan. onainers, vessels or anks in which chemical or biological reacions are carried ou. Source: Fundamenals of Waer Treamen: Uni Processes Physical, hemical, and Biological 2

3 MATHEMATIS OF REATORS Reacor mahemaics is based upon wo principles: (1) Maerials balance and (2) reacion kineics. Maerials Balance: oncep The basic idea of a reacor model is quie simple and is embodied in he following equaion. 3

4 Maerials Balance (mass balance ) The firs sep involved in preparing a mass balance is o define he sysem boundary so ha all he flows of mass ino and ou of he sysem boundary can be idenified. Source: Fundamenals of Waer Treamen: Uni Processes Physical, hemical, and Biological 4

5 hp://engineering.darmouh.edu/~d3345d/courses/engs37/massbalance.pdf 5

6 Maerials Balance Maerials Balance (mass balance ) quaniaive descripion of all maerials ha ener, leave and accumulae in a sysem wih defined boundaries. based on he law of conservaion of mass (mass is neiher creaed nor desroyed) is developed on a chosen conrol volume. 6

7 There may be mass generaion in he conrol volume also. hps:// _sudy_guide.pdf hp:// hp://ocw.mi.edu/courses/civil-and-environmenal-engineering/1-77-waer-qualiyconrol-spring-26/lecure-noes/chaper5lecure.pdf Please find hese pdf documens and read!!! 7

8 hp://ocw.mi.edu/courses/civil-and-environmenal-engineering/1-77-waer-qualiyconrol-spring-26/lecure-noes/chaper5lecure.pdf 8

9 Rae of Rae of flow Rae of flow Rae of mass accumulaion of mass ino of mass ou generaion/eliminaion of mass wihin = he sysem of he sysem + wihin he sysem boundary boundary he sysem boundary Accumulaion = Inflow - Ouflow + Generaion/Eliminaion - 9

10 Accumulaion = Inflow - Ouflow + Generaion/Eliminaion Generaion/Eliminaion erm. can be + or - [Mos of he maerials of ineres disappear and herefore generaion erm is - in mos cases.] Symbolic Represenaion: Accumulaion = Inflow - Ouflow + Generaion/Eliminaion d d ( Q o ) - Q r volume of he reacor, m 3 d rae of change of reacan concenraion wihin he reacor (g / m 3 sec) d Q = flow ino and ou of he reacor ( m 3 /sec) =concenraion of reacan in he reacor and effluen (g/ m 3 ) r = rae of generaion (g/ m 3 sec) 1

11 Operaional saes ha mus be considered in he applicaion of maerials balances: Seady sae: There is no accumulaion in he sysem. Raes and concenraion do no vary wih ime. A Example: Pump discharging consan volume of waer wihin ime. Unseady (ransien) Sae: Rae of accumulaion is changing wih ime A Example: Filling a reservoir pumping of he conens of a ank. 11

12 Types of reacors 12

13 REATOR MODELS ommon reacor configuraions include (a) compleely mixed bach reacors, (b) compleely mixed flow reacors (MFRs), and (3) plug flow reacors (PFRs). Source: MWH s Principles of Waer Treamen, Third Ediion, h4 Kerry J. Howe, David W. Hand, John. rienden, R. Rhodes Trussel and George Tchobanoglous, 212 John Wiley & Sons, Inc. 13

14 REATOR MODELS 5 principal reacor models used in Environmenal Engineering operaions: 1. Bach reacor 2. omplee-mix reacor(coninuous flow sirred ank reacor),(fstr) 3. Plug-flow reacor (PFR) (ubular-flow reacor) 4. ascade of complee mix reacor (complee mix reacors in series) 5. Packed- bed reacor 14

15 Types of Reacors Used in Waer Treamen Source: MWH s Waer Treamen: Principles and Design, Third Ediion, h6 John. rienden, R. Rhodes Trussell, David W. Hand, Kerry J. Howe and George Tchobanoglous, 212 John Wiley & Sons, Inc. 15

16 Source: MWH s Waer Treamen: Principles and Design, Third Ediion, h6 John. rienden, R. Rhodes Trussell, David W. Hand, Kerry J. Howe and George Tchobanoglous, 212 John Wiley & Sons, Inc. 16

17 BATH REATORS The simples reacor ype Flow is neiher enering nor leaving he reacor The liquid conens are mixed compleely and uniformly Ref: hp:// Applicaions: Is a non-coninuous and perfecly mixed closed vessel where a reacion akes place. A common use of bach reacors in laboraories is o deermine he reacion equaion and rae consan for a chemical reacion. The kineic informaion deermined in a bach reacor can be used o design oher ypes of reacors and full-scale reamen faciliies. 17

18 Bach reacors have no inpus or oupus. 18

19 The reacion rae equaion can be subsiued for r and Eq can be inegraed o yield an equaion for as a funcion of. For a firs-order reacion, r = - k 19

20 For a second-order reacion, r = - k 2 2

21 OMPLETE-MIX REATORS (FSTR=oninuous-Flow Sirred Tank Reacor) Fluid paricles ha ener he reacor are insananeously dispersed hroughou he reacor volume Q, o. Q, Fluid paricles leave he reacor in proporion o heir saisical populaion conc of any maerial leaving = conc. a any poin in he reacor d Q d No concenraion gradien wihin he sysem. o QrV Maerial enering is uniformly dispersed hroughou he reacor. 21

22 ONTINOUS FLOW STIRRED TANK (FSTR) REATOR MODELS Q, o Q, Maerial inpu may be conservaive (non-reacive) non-conservaive (reacive), NOTE For conservaive (non-reacive) maerial inpu having o conc., eff. conc. is iniially (no o ) due o unseady sae condiion. When seady-sae is reached effluen conc. () = o 22

23 onservaive (non-reacive) = Tracer ess Some reacor analyses are conduced wih conservaive (or nonreacive) consiuens. I may seem ha conservaive chemicals would be of lile ineres in reacor analysis. However, hey provide a mechanism for undersanding he hydraulic characerisics of a reacor. Since conservaive consiuens do no reac, hey flow wih he waer and say in a reacor as long as he waer says in he reacor. Thus, a curve of effluen concenraion of a conservaive consiuen reveals he residence ime disribuion of he waer in he reacor. onservaive consiuens are commonly called racers, and ess o deermine he residence ime disribuion of a reacor are called racer ess. Source: MWH s Principles of Waer Treamen, Third Ediion, h4 Kerry J. Howe, David W. Hand, John. rienden, R. Rhodes Trussel and George Tchobanoglous, 212 John Wiley & Sons, Inc. 23

24 Tracers Tracers (dyes, elecrolyes, radioacive isoopes) are used o characerize he degree of mixing. mus be conservaive does no paricipae in any reacion i is no adsorbed or absorbed by reacor or is conens are assumed o be moved abou in he same manner as he waer molecules heir flow paern will mimic liquid flow paern. 24

25 Tracer ess Two ypes of echniques are used for racer ess: Pulse inpu: A he beginning of he esing period (i.e., ime = ), a known mass of racer is added o he reacor influen insananeously (i.e., added as a pulse or slug) and hen flows hrough he reacor. Measuremen of he effluen concenraion coninues unil he pulse has compleely passed hrough he reacor. Sep inpu: A ime =, a feed pump is urned on and feeds a racer ino he reacor influen. The concenraion of he racer in he influen says consan over he duraion of he es. Measuremen of he effluen concenraion coninues unil i is he same as he new influen concenraion. Source: MWH s Principles of Waer Treamen, Third Ediion, h4 Kerry J. Howe, David W. Hand, John. rienden, R. Rhodes Trussel and George Tchobanoglous, 212 John Wiley & Sons, Inc. 25

26 Tracer ess Source: MWH s Principles of Waer Treamen, Third Ediion, h4 Kerry J. Howe, David W. Hand, John. rienden, R. Rhodes Trussel and George Tchobanoglous, 212 John Wiley & Sons, Inc. 26

27 Response of FSTR o Pulse Tracer Inpu When a pulse inpu is inroduced ino a FSTR, he effluen racer concenraion insanly reaches a maximum as he racer is uniformly disribued hroughou he reacor. As clean waer (conaining no racer) coninues o ener he reacor afer ime =, he racer gradually dissipaes in an exponenial manner as he racer maerial leaves he effluen. The exponenial shape of he racer curve can be demonsraed using a mass balance analysis of a MFR.

28 Response of FSTR o Pulse Tracer Inpu Divide by V d d -Q d - Q d d Q V r d since he racer is non-reacive Source: MWH s Principles of Waer Treamen, Third Ediion, h4 Kerry J. Howe, David W. Hand, John. rienden, R. Rhodes Trussel and George Tchobanoglous, 212 John Wiley & Sons, Inc.

29 Response of FSTR o Pulse Tracer Inpu The hypoheical ime,, ha waer says in a reacor is: V/Q is defined as he hydraulic residence ime (HRT). I is he ime ha he influen feed spends inside he reacor. Every molecule enering he reacor will have he exac same amoun of ime in he reacor V/Q may be denoed wih he symbols, R, D, HRT, τ. HRT is an imporan design parameer. Process efficiency is dependen on hydraulic residence (deenion) ime. HRT affecs he operaional and invesmen coss and energy requiremens, and in general, higher HRTs will lead greaer invesmen coss.

30 d Q V d A =+ (ime immediaely afer racer is added), he racer is uniformly dispersed wihin he FSTR. Thus, inegrae for FSTR wih beginning a =, and a = d Q V d o Ln Ln Ln o Q V Q V Ln o Q V 3

31 Response of FSTR o Pulse Tracer Inpu e Q Q V 1 e / 31

32 e Q e / This equaion demonsraes ha he effluen concenraion from a FSTR will be = a =, = a infinie ime, and decay exponenially beween hose exremes. Source: MWH s Principles of Waer Treamen, Third Ediion, h4 Kerry J. Howe, David W. Hand, John. rienden, R. Rhodes Trussel and George Tchobanoglous, 212 John Wiley & Sons, Inc. 32

33 Response of FSTR o SepTracer Inpu d d ( Q )-Q i r since he racer is non-reacive i is he influen racer concenraion. Divide by Q d ( Q ) d d Q d i - i - Q

34 Response of FSTR o SepTracer Inpu Q d d Inegrae for FSTR wih = beginning a =, and a = i - d i Q d 1 1 dx lnaxb axb a 34

35 1 Lni 1 Q Q Ln Ln i i Ln Q Ln i i Ln Q Ln i i Ln i i Q. i - e 1- i i i e - Q - Q 1 (Divide boh sides by i) e Q i 1 e / 35

36 Response of FSTR o SepTracer Inpu hp://ceae.colorado.edu/~silvers/cven5534/ideal%2reators.pdf 36

37 Response of FSTR o SepTracer Inpu 3 Q m /sec 3 m 1 sec 1 R Hydraulic reenion ime (HRT) 1 e / R 3737 / R,5 1e,5 =, e 1 =, e2 =, e3 =, e4 =, e5 =,99 =, = Afer his ime, / does no change. 37

38 Response of FSTR o SepTracer Inpu 1,2 1 Afer his ime, / does no change /o,8,6,4, / R

39 1 e / R hp://ceae.colorado.edu/~silvers/cven5534/ideal%2reators.pdf 39

40 hp://ceae.colorado.edu/~silvers/cven5534/ideal%2reators.pdf 4

41 Response of FSTR o Non-onservaive (Reacan) Inpu Unseady Sae Unseady Sae Analysis Reacors of concern in waer reamen engineering ypically operae a seady-sae condiions. Accumulaion = Inflow - Ouflow + Generaion/Eliminaion d d ( Q o ) - Q r A seady sae he accumulaion erm is zero. Therefore, here is no change in concenraion wihin he reacor wih ime. 41

42 Response of FSTR o Non-onservaive (Reacan) Inpu Unseady Sae Unseady Sae Analysis However, someimes he reacors operae a unseady-sae condiions When a reacor is firs brough online, Afer mainanance or inoperaion, When he inle concenraion, o, changes. Unseady sae: concenraions vary wih ime & accumulaion is non-zero The goal of unseady sae analysis is o deermine he ime necessary o reach seadysae operaion. 42

43 Oule oncenraion, o, mg/l ime Response of a FSTR o an increase in he inle concenraion from 5 mg/l o 1 mg/l (1) and from 5 mg/l o 75 mg/l (2) 43

44 Response of FSTR o Non-onservaive (Reacan) Inpu Unseady Sae Q, o Q, A reacion A, B, known o be firs order, is o be carried ou in a FSTR. Waer is run hrough he reacor a a flow rae Q (m 3 /sec ) and a = he reacan A is added o he inpu sream on a coninuous basis. Deermine he oupu concenraion of A as a funcion of ime and plo reacoroupu response curves for reacan A. 44

45 Response of FSTR o Non-onservaive (Reacan) Inpu Unseady Sae Q, o Q, The unseady-sae analysis begins wih a mass balance equaion, which is wrien for a firs-order reacion assuming consan volume and using deenion ime, τ = V /Q, as follows:, Maerials balance for he sysem: d Q Qeliminaion d d Q Qk.. d Firs-order rxn r=-k eliminaion erm =r = -k (Divide boh sides by ) 45

46 dc d Q. k. Q 1 ( = ) R dc d 1 R k.. dc d k.. R R dc d dc d dc d dc d +c R R (1 k. ( R R R (1 k. - 1 R. 1 R +k) ) R k = ) R dy dx P(x)y Q(x) Inegraion facor= P(x). dx e Muliply boh sides w/inegraion facor P(x(/dx d[ye ] Lef hand side = 46 Q(x)

47 1 R k e. d e. dc d +c ( 1 R +k) = R Muliply boh sides wih inegraion facor. e dc 1.. e.... e. o d R d.e 1. d d R. o.e 1.e..e. d R.e. 1 R 1..e. 1 e ax dx e ax a 47

48 ( 1 1k R e / R ) FSTR, UNSTEADY-STATE FOR NON-ONSERVATIVE REATANT HAVING 1ST ORDER REATION RATE 48

49 /o Sar-up of an ideal FSTR. o =1 mg/l R = 1 min ( 1 e 1k R / R ),6,5,4,3,2 k=.1 k=.2 k=.3 k=.4,1, ime (min) 49

50 Effluen oncenraion,, mg/l Sar-up of an ideal FSTR. o =1 mg/l R = 1 min ( 1 e 1k R / R ) k=.1 k=.2 k=.3 k= ime (min) 5

51 Response of FSTR o Non-onservaive (Reacan) Inpu Seady Sae Q, o Q, As approaches infiniy ( ) seady-sae soluion is approached ( 1 e 1k R 1k. / R R ), e / R FSTR, seady-sae, nonconservaive (reacive) reacan having 1 s order reacion rae. 51

52 For seady-sae condiion (1 s order reacion): dc =Q d -Q-k (Divide boh sides by ) dc Q = d Q - - k dc d A seady-sae 1 k= R ( -) ( ) k = R - k += R 1k. R FSTR, seady-sae, non-conservaive (reacive) reacan having 1 s order reacion rae. 52

53 ASADE OF OMPLETE MIX REATORS (FSTR in series) Oupu of he firs reacor is he inpu of he second reacor. In environmenal engineering, i is common o employ a series of MFRs o improve he hydraulic performance of a reacor. The firs FSTR operaes a a higher concenraion and, herefore, a higher reacion rae is possible. 53

54 ASADE OF OMPLETE MIX REATORS (FSTR in series) n A seady-sae: 1 s reacor dc d =Q -Q + r 1 1 dc Q = d - Q r 1 1 = - +r 1 R1 R1 1 For 1 s order k reacion: = - ( +k) 1 R1 R1 R1 R1 = 1 R1-1 1+k R1 R1 1 1k R1

55 dc d 2nd reacor =Q -Q + r dc Q = d - Q r = - +r 1 2 R2 R2 1 2 n For 1 s order 1 1 reacion: - k = - ( +k) R2 1 2 R2 R2 R2 = 1 R k R2 R k R2 1 = 1+k R 1 2 1k. 1k R1 R2 55

56 dc d 3 rd reacor =Q -Q + r dc Q = d - Q r n 1 1 = - +r 2 3 R3 R3 For 1 s order 1 1 reacion: 2-3 k3 = = 1 R3 R3 1 R R3 ( 3 1 R3 +k) 1+k R3 R k R3 3 2 = ( 1+k )( 1+k ) R1 R2 1k 1 k 1 k R1 R2 R3 56

57 1 1k R1 2 1k. 1k R1 R2 3 1k 1 k 1 k R1 R2 R3 n h reacor n = ( 1+k )( 1+k )...1+k ( ) R1 R2 Rn FSTR in series under seady saes and for 1 s order rxn. 57

58 ASADE of OMPLETE MIX REATORS (omplee Mix Reacor in Series) n-1 n n+1 is used o model he flow regime ha exiss beween he hydraulic flow paerns corresponding o he complee and plug flow reacors. If he series is composed of one reacor If he series consiss of an infinie number of reacors in series complee mix regime prevails plug-low regime prevails Applicaion: In modeling rivers wihin small incremens (segmens) 58

59 V1=8.68 x 1 ^5 m3 V2=25.9x 1 ^5 m3 V3=17.28 x 1 ^5 m3 v4=8.64 x 1 ^5 m3 V5=25.92x 1 ^5 m3 Ref: Tchobanoglous and Scroeder, 1985, Addison-Wesley Publishing ompany EXAMPLE 1: The river reach shown has been divided ino 5 segmens based on measured velociies and dephs. An indusrial faciliy is planned jus upsream of he 1 s segmen and i is necessary o esimae effec of ww discharge. A series of dye experimens have been run and each of he segmens was found o behave as an approximae FSTR. The polluan is expeced o disappear according o 1s order reacion. For he daa given deermine he seady-sae polluan conenraion in each segmen. Q 5m 3 /sec river k,2day 3g/m

60 PLUG FLOW REATOR-(PFR) No mixing in he axial direcion. Fluid paricles pass hrough he reacor and are discharged in he same sequence in which hey enered he reacor. Each fluid paricle remains in he reacor for a ime period equal o he heoreical deenion ime. This ype of flow is approximaed in long anks wih a high lengh/widh raio in which longiudinal dispersion is minimal or absen. Applicaion: Used o sudy river sysems 6

61 61 hp://ceae.colorado.edu/~silvers/cven5534/ideal%2plug%2flow%2reator.pdf

62 PLUG FLOW REATOR-(PFR) An effluen racer (conservaive) signal is exacly he same as he inpu, excep ha is ransposed in ime by R. 62

63 PLUG FLOW REATORS (PFR) Are ideally mixed in laeral direcion and unmixed longiudinally Unrealisic assumpion for mos real-world sysems bu can be approximaed closely The mean HRT ime = rue HRT ime 63

64 PF condiions are achieved by designing long and narrow reacors or placing baffles in a reacor. Ref: Tchobanoglous and Scroeder, 1985, Addison-Wesley Publishing ompany 64

65 In a PF siuaion he mass balance mus be aken over an incremenal volume because a longiudinal concenraion gradien exiss (since here is no longiudinal mixing. Maerials Balance: Accumulaion = Inflow - Ouflow + Generaion c - r Q x (Divide boh sides o ) Q x x c = Q Δ ( ) + r x- x+δx Ref: Tchobanoglous and Scroeder, 1985, Addison-Wesley Publishing ompany c = Q AΔx ( ) + r x- x+δx 65

66 66 r x c A Q c r c Q c PFR Unseady-sae condiions r x A Q c x x x

67 c seady-sae condiions r c c r Qc c R PFR seady-sae condiions 67

68 EXAMPLE 2: A plug flow reacor (PFR) is o be used o carry ou he reacion A B The reacion is firs order and he rae is characerized as r a =-k A Deermine he seady-sae effluen concenraion as a funcion of R. 68

69 Soluion for Example 2 (Derivaion of Effluen ocnenraion Equaion) hp://ceae.colorado.edu/~silvers/cven5534/ideal%2plug%2flow%2reator.pdf 69

70 hp://ceae.colorado.edu/~silvers/cven5534/ideal%2plug%2flow%2reator.pdf 7

71 EXAMPLE 3: Deermine he volume of a FSTR required o give a reamen efficiency of 95% for a subsance ha decay according o half order kineics wih a rae consan of.5 (mg/l) 1/2. The flow rae is seady a 3L/hr and he influen concenraion is 15mg/L. EXAMPLE 4: Deermine he volumes of wo idenical FSTR reacors in series o provide he same degree of reamen for he condiions given in Example 1. EXAMPLE 5: Deermine he volume of a PFR o provide he same degree of reamen for he condiions given Example 1. 71

72 Volume omparison For Examples 3-5 Example 1 Example 2 Example 3 FSTR 2 FSTR in series PFR ( m 3) When he same reacion model (excep for zero-order rxns) applies, regardless of he mixing regime a PF sysem is always he mos efficien (less volume requiremen) 72

73 PAKED BED REATORS These reacors are filled wih some ype of packing medium ( e.g.rock, slag, ceramic or plasic) Wih respec o flow, compleely filled (anaerobic filer) inermienly dosed (rickling filer) Ref: hp:// When he pore volume of he medium is filled wih a liquid flow is said o be SATURATED When he pore volume is parially filled flow is said o be UNSATURATED 73

74 PAKED BED REATORS (coninue) Applicaion: Used o sudy he movemen of waer and conaminans in groundwaer sysems. FLUIDIZED-BED reacors Packed bed reacors in which he packing medium is expanded by he upward movemen of fluid (air or waer) hrough he bed. Example: Filer backwashing Ref: hp:// 74