Weatherley 1982 Aquacultural-Engineering

7/28/2019 Weatherley 1982 Aquacultural-Engineering

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AquaculturalEngineertng 1 (1982) 93-113

A P P L I C A T I O N O F S I M P L E D Y N A M I C R E S P O N S E A N A L Y S I S T OA R E C I R C U L A T I N G A Q U A C U L T U R E S Y S T E M - A P R E V I E W

L. R. WEATHERLEYAquaculture Engineering Research Group, Department of Chemical and Process Engineering,Heriot-Watt University, Chambers Street, Edinburgh EH1 1HX, UK

ABSTRA CTThis paper describes the application o f dyna m ic response analysis to a s imple f irs torder recireula ting aquaculture sys tem and dem ons trates ho w the d is tr ibut ion o famm onia th roughou t the s y s t em ma y be pred ic ted fo l low ing dynamic changes. Theapproach is val idated by exper im ental data obtain ed for a ree ireula ting sys tem undercon tro l led co ndi t ions o f m ix ing and d is turbance.

I t i s conc luded that uns teady s ta te predic t ive methods have s igni f icant potent ia lapplicat ion in the con tex t o f aquaculture , subjec t to fur th er re f in em ent in themod el l ing o f b iochem ical la 'ne ties and l iquid m ix ing.

NOMENCLATUREa F / V F d e f i n e d i n e q n ( 5 ) h - 1A M a g n i t u d e o f i m p u l s e f u n c t i o n m g l it re - aA 6 ( t ) I m p u l s e f u n c t i o n m g h - 1 l it re -1b S t e p c h a n g e i n a m m o n i a p r o d u c t i o n r a te m g h -~ l it re - 1c i t A m m o n i a c o n c e n t r a t i o n i n e f f lu e n t m g l i t r e -~C i s s !C i U n s t e a d y s t at e d e v i at i on i n a m m o n i a c o n c e n t r a t i o n i n

e f f l u e n t f r o m f is h t a n k m g l it re - 1c t A m m o n i a c o n c e n t r a t i o n i n i n f lu e n t t o f is h t a n k m g l it re - lC o s sC o U n s t e a d y s t at e d e v i a ti o n i n a m m o n i a c o n c e n t r a t i o n t o

in f l u e n t t o f i sh t a n k m g l i tr e - 1C N H 3 U n s t e a d y s ta t e d e v i a ti o n i n a m m o n i a c o n c e n t r a t i o n m g l it re -1d F / V B d e f i n e d i n e q n ( 5 ) h - 19 3Aquacultural Engineering 0144-8609/82/0001-0093/$02.75 Applied Science Publishers Ltd,England, 1982Printed in Great Britain


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9 4 L .R . W E AT HE RLE Yf R a t e c o n s t a n t d e f i n e d i n e q n ( 5 ) h -1F Re c i r c u l a t i o n r a t e l i tr e h - 1I T o t a l i m p u l s e m a g n i t u d e m gk F i rs t o r d e r r a t e c o n s t a n t f o r a m m o n i a o x i d a t i o n h -1m P a r a m e t e r d e f i n e d i n e q n ( 9 )n P a r a m e t e r d e f i n e d i n e q n ( 9 )p a + f + d d e f i n e d i n e q n ( 7 )q~ A m m o n i a p r o d u c t i o n r a t e d i s t u r b a n c e f u n c t i o n m g h -1 l it re -1q a f d e f i n e d i n e q n ( 7 )r v A m m o n i a p r o d u c t i o n r a t e m g h -1 l it re - 1rvs s S t e a d y s t a te v a lu e o f a m m o n i a p r o d u c t i o n r a t e m g h - 1 l it r e -~R v S t e p d i s t u r b a n c e i n a m m o n i a p r o d u c t i o n r a t e m g h - a l it re - as L a p l a c e t r a n s f o r m v a r ia b le$ 1 , $ 2 Q u a d r a t i c r o o t s d e f i n e d in e q n ( 1 1 )t T i m eVB F i l t e r v o lu m e l i tr eV F i s h t a n k v o l u m e l it r eSubscr ip t si R e f e r s t o e f f l u e n t s t r e a m f r o m f is h t a n ko Re f e r s t o i n l e t s t r e a m to f i s h t a n ks s S t e a d y s t a t e v a lu eB Re f e r s t o f i l te rF Re f e r s t o f i s h t a n k

INTRODUCTION

T h e r a p i d l y a c c e l e ra t i n g e v o l u t i o n o f n o v e l r e c i r c u la t i n g s y s t e m s i n a q u a c u l t u r e h a sg i ve n r is e t o i n c r e as i n g l y c o m p l e x f l o w s h e e t s i n v o l v in g a n u m b e r o f s e p a r a t e p r o c e s s e sl i n k e d in a n a t t e m p t t o o p t i m i s e w a t e r u s e a n d q u a l i t y . F o r e x a m p l e , R o s e n t h a l( 1 9 8 0 ) d e s c r ib e s t h e g r o w i n g u se o f m u l t i p l e r e c i r c u la t i n g c y c l e s fo r e a c h s e p a r at et y p e o f w a t e r t r e a t m e n t i n c l ud i n g d e n i tr i fi c a ti o n , o x y g e n a t i o n , o z o n a t i o n a n dm e t a b o l i t e r e m o v a l , b u t c o n c l u d e s t h a t i n s p it e o f t h e s e d e v e l o p m e n t s , t h e d e si g n a n do p e r a t i o n o f r e c i r c u l a t i n g s y s t e m s a r e s t il l l a r g e ly a n a r t r a t h e r t h a n a s c i e n c e .

T h e c u r r e n t w o r k s ee ks t o d e m o n s t r a t e i n a r e la t iv e ly s im p l e w a y o n e m e t h o d o fb r e a k i n g d o w n a n a q u a c u l t u r e s y s t e m i n t o i n d i v i d u a l m o d u l e s a n d a n a l y s i n g t h e o r e t i-c a U y t h e d y n a m i c b e h a v i o u r b y c l a s si c al p r o c e s s c o n t r o l te c h n i q u e s , a n d t o i d e n t i f yl i k e ly s o u r c e s o f d i s t u r b a n c e o f in t e r e st t o b o t h d e s i g n e r a n d o p e r a t o r . T h e p a p e r i si n t r o d u c t o r y i n it s a p p r o a c h , a n d t h e d e t a i l e d p r o b l e m d i s cu s s ed is c o n f ' m e d t or e m o v a l a n d d i s t r i b u t i o n o f a m m o n i a i n a r e c i r c u la t i n g s y s t e m , a l t h o u g h i t iss p e c i fi c a ll y i n t e n d e d t o e x p a n d t h is w o r k i n t h e f u t u r e t o a n a ly s e d i s t ri b u t i o n o f


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A P P L I C A T I O N O F D Y N A M I C R E S P O N S E A N A L Y S I S 95n i t r i t e , oxygen , hea t and o the r key ope ra t i ng quan t i t i e s l i ke ly t o be s ens i t i ve t oo p e r a t i o n a l a n d e n v i r o n m e n t a l d i s t u r b a n ce s .

O t h e r w o r k i n th i s a r ea , M a n t l e (1 9 8 0 ) , h a s a d o p t e d a p s e u d o s t e a d y s t a t e a p p r o a c ht o m o d e l l i n g o f a q u a c u l t u r e s y s t e m s , c a lc u l a ti n g t h e s t e a d y s ta t e r e sp o n s e t o im p o s e dchanges . Whi le t h i s app roac h ma y be u se fu l t o t he de s igne r , i t neg l ec ts cons ide ra t i on o fthe impor t an t t r ans i en t behav iou r , e spec i a l l y r e l evan t t o t he consecu t ive r eac t i ons fo rt h e o x i d a t i o n o f a m m o n i a t o n i t ra t e i o n a n d t h e i n te r m e d i a t e f o r m a t i o n o f n i tr i te .

O t h e r a t t e m p t s a t d e s ig n m o d e c a l cu l a ti o n s b a s e d e n t ir e l y o n t h e c o r r e l a ti o n o fc o l l e c t e d d a t a f o r a p a r t ic u l a r s y s t e m h a v e b e e n m a d e . H i r a y a m a (1 9 7 4 ) p r e s en t scor r e la t ion s to ca lcu la te th e re la t ive s iz ing o f f ish tanks and b iof i l te rs , bu t the da taup on wh ich t he co r r e l a t i on coe f f i c i en t s a r e c a lcu l a t ed a r e f rom an aqua r ium w i th asma l l expe r im en ta l f i l te r and a r e no t co ns ide red t o be u se fu l f o r t he gene ra l c a se o f ac o m m e r c i a l a q u a c u l t u re s y s t e m .Speece (1973) , L i ao and Mayo (1972) a l so p r e sen t s t e ady s t a t e co r r e l a t i ons fo rdes ign mode ca l cu l a t i ons ba sed upon ex i s t i ng da t a .

S Y S T E M A N A L Y S I S

Unless a t t em p t s a t ma th em a t i ca l d e sc r ip t i on o f a sy s t em a re to be l i tt l e m ore t hans t a t is t i c a l co r r e l a t i on o f o bse rved da t a , t he behav iou ra l cha rac t e r i st i c s o f e ach d i s c r e teun i t com pr i s ing the who le sy s t em r equ i r e ana ly s is . A s imp le r ec i r cu la t i ng aquacu l tu r esys t em may compr i s e f i sh t ank , e f f l uen t p ipe , b io log i ca l f i l t e r and r e tu rn p ipe ; s eeFig. 1.

C o n s i d e r a t io n o f t h e f is h t a n k a l o n e r e q u i re s a c c u r a t e k n o w l e d g e o f t h e v o l u m e ,the deg ree o f mix ing ( i. e . t he r e s idence t ime d i s t r i bu t i on ) and t he r a t e o f m e tabo l i t ep roduc t ion fo r a succes s fu l r e sponse ana ly s i s . I n t h i s p r e l imina ry i nves t i ga t i on t hec o n t e n t s o f th e t a n k a r e a s s u m e d t o b e w e l l m i x e d .

S imi l a r l y fo r t he b io log i ca l f i lt e r un i t , t he deg ree o f mix in g exe r t s an im po r t an ti n f luence on t he uns t e ady s t a t e r e sponse behav iou r . I n pu re ly f l ow t e rm s t he deg reeo f mix ing is me asu red e xpe r im en ta l l y and i s a f unc t i on o f t he t ype o f f i l te r , i .e . r ingpack ing /g rave l / s and , t r i ck l e dow nf low , up f low , e t c ., and t he spec i f i c l i qu id f l ow- ra t e .In t he ana ly si s p r e sen t ed he re t he f i l t e r i s a s sumed to b e we l l m ixed and t he expe r i -m e n t a l d a t a e x a m i n e d l a t e r o n ( D o n a l d s o n , 1 9 7 9 ) b e a r o u t t h i s a ss u m p t i o n .

I n c o n s i d e r in g t h e d i s t ri b u t io n o f a m m o n i a in t h e s y s t e m t h e r a t e o f re m o v a l i n t h ef i l t e r i s t he s econd impor t an t a spec t o f t r i t e r behav iou r t o be cons ide red . K ine t i c s o fb i o c h e m i c a l a m m o n i a o x i d a t i o n d o n o t c o n f o r m t o s i m p l e k i n e t ic a n a l y si s i n a g e n e ra ls ense due t o t he ro l e o f bac t e r i a l r ep rodu c t ion and dea th r a t e s i n de t e rmin ing t heo v e ra ll r a te o f r e a c t i o n ( A t k i n s o n , 1 9 7 4 ). H o w e v e r , d e v i a t io n s i n a m m o n i a re m o v a lr a te s d u e t o c h a n g e s i n b a c t e ri a l p o p u l a t i o n d e n s i t y a re l i k e ly to b e v e r y m u c h s l o w e rcompared t o dev i a t i ons r e su l t i ng f rom ope ra t i ona l d i s t u rbances . S rna and Bagga l ey( 1 9 7 5 ) s u c c es s fu l ly c o r r e la t e d k i n e t ic d a t a f o r b o t h a m m o n i a a n d n i t ri t e o x i d a t i o n i n a


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96 L.R . WEATHERLEYCI

co

lr=kCo VBVF

A i r l i f t F i s h Isys tem tan k B io log ica lf i l t e rFig. 1. Sim ple ecirculating aquaculture system.

m a r i n e s y s t e m u s i n g f i r s t o r d e r e q u a t i o n s . M o r e r e c e n t w o r k b y D o n a l d s o n ( 1 9 7 9 )c o n f i r m e d f ir s t o r d e r b e h a v i o u r u n d e r c o n s t a n t p o p u l a t i o n c o n d i t io n s .

Whi l e t he f i sh t ank and b io log i ca l f i l t e r r ep r e sen t t he mos t impor t an t modu le s o ft h e s y s t e m u n d e r c o n s i d e r a ti o n , t h e i n t e r c o n n e c t in g p i p e w o r k a n d a ir l if t s e c t io n( F ig . 1 ) c a n p o t e n t i a l l y i n f lu e n c e t h e r e s p o n s e o f t h e s y s t e m t o d y n a m i c c h an g e s , a n dp ip ew ork can con s t i t u t e a s i gn if i c an t t ime l ag be tw een f i sh t anks and f i l te r . The e f f ec to f t r ans po r t a t i o n l ag i s i gno red i n t h is p r e l im ina ry ana ly s i s.

I t is n o w n e c e s s a ry t o e x a m i n e s o m e o f t h e d y n a m i c d i s t u r b a n c e s t o w h i c h ar ec i r cu l a ti ng sys t em m ay b e sub j ec t . M a jo r d is t u rbances fo l l owing s t a r t - up a r e m os tc o m m o n l y i n d u c e d b y t h e o p e r a t o r a n d i n c lu d e a c t io n s s u c h a s f e e d in g , i n t r o d u c t i o no r r em ova l o f f ish pop u l a t i on , s eed ing and nu r tu r i ng o f t he b io log i ca l f i l te r un i t .O t h e r d i s tu r b a n c e s c a u s e d b y c h a n g es i n t e m p e r a t u r e , a n d c h an g e s in a m m o n i a p ro -duc t i on due t o s t re s s changes i n t he pop u l a t i on a r e l i ke ly to be le ss r ap id .In o rde r t o p r ed i c t t he uns t eady s t a t e r e sponse t o any o f t he se d i s t u rbances , t het i m e d e p e n d e n t n a t u r e o f e a c h m u s t b e k n o w n . F o r e x a m p l e , s p ik e m o d e n u r t u r i n go f a b io log i ca l f i l t e r c an be app rox ima ted by an impu l se d i s t u rbance i n r e spec t o fa m m o n i a t o t h e f is h t a n k . R a p i d i n t r o d u c t i o n o f s t o c k t o a f is h t a n k m a y b e a p p r o x i-m a t e d b y a st e p d i s t u rb a n c e i n a m m o n i a p r o d u c t i o n r a te .

T h e t i m e f u n c t i o n r e p r e s e n t a t i o n o f t h e o t h e r d i s t u r b a n c e s m u s t b e in t e r p r e t e df rom expe r imen ta l obse rva t i ons i n a known f i sh t ank , and wh i l e t he r e l a t i onsh ipb e t w e e n f e e d in g c y c le a n d a m m o n i a o u t p u t m a y p o t e n t i a l ly b e n u m e r i c a ll y q u a n t i fi e d ,t h e r e p r e s e n t a t i o n o f s tr e ss i n d u c e d d i s t u rb a n c e s in a m m o n i a o u t p u t p r o v i d e a na l t o g e t h e r m o r e d i f fi c u l t p r o b l e m .


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A P P L I C A T I O N O F D Y N A M I C R E S P O N S E A N A L Y S I S 9 7

The fo l lowing ana lys i s i s conf ' lned to impulse and s tep d i s turbances in ammoniaou t pu t i n t he f ish t ank .

I M P U L S E D I S T U R B A N C E

Ref e r r i ng to t he A p pend i x , u ns t ead y s t a t e m a t e r i a l ba lance f o r am m oni um i on acr os st he f ish t ank y i e ld s

dC iV F " - ~ - t ~ F ( C o - - C i) + A 6 ( t ) V v ( 1 )w h e r e A 6 ( t ) is t he i m pu l se o f am m oni a add ed t o t he w e ll m i xed f ish tank a t t = 0 .

Co and Ci a r e uns t ea dy s t a te dev i a t ion va r iab le s r e f e r ri ng to change i n con cen t r a t i ono f am m oniu m ion in the s t reams enter ing and leaving the f i sh t ank , respec t ive ly .

By de f i n i t i on , t he r e f o r e , C i ( t = O ) = O , C o ( t = O ) = O .A s i m i la r m ass ba l ance f o r am m o ni um i ons m ay be cond uc t ed acr os s t he b i o log i ca l

f i lt e r . Assuming f i r st o rder k ine t ics and tha t the f lu id in the f i l te r i s wel l m ixed ,uns teady s ta te mass ba lance y ie lds :

VB ~dt = F ( C i - - C o) - - k C o V B (2 )w he r e k is t he spec i f i c f i rs t o r de r r a te cons t an t f o r t he ox i da t i on o f am m o ni a t o n i t ri t eion .

Taking Laplace t ransfo rm s of eqns (1) and (2) and rear ranging , y ie lds :Fs c ~ s = ~ ( C o s - c l ) + A ( 3 )F

S C s = VB (Cis - - C s) - - k C s (4 )Le t t i ng

F Fa = - - " d = - - " f = k ( 5 )V v ' V B 'subs t i t u t i on and r ea r r angem en t y i e l d :

A dCos - s Z + s p + q (6 )w h e r e

p ~ - a + f + d ; q = a f (7 )


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98 L . R . W E A T H E R L E YEqu a t ion (6 ) i s gene ra l and t he spec i f i c na tu r e o f t he sys t em r e sponse i s d i c t a t ed byt h e v al u es o f p a n d q .

I f p 2 < 4 q , t h e r o o t s o f t h e d e n o m i n a t o r o f e q n ( 6 ) ar e c o m p l e x a n d C o a s afunc t i on o f t ime has an un dam ped r e sponse and i s o sc i l l a t o ry . Pa r t ia l f r a c t i on r educ -t i on and i nve r s ion o f eqn (6 ) y i e ld s i n t h is c a se :

- - A dCo = sin ( n t ) e x p ( ro t ) (8 )n

w h e r e(p 2 __ 4q)1/2n = , m = - - p / 2 (9 )2

T h u s , t h e c o n s e q u e n t c h a n g e s i n a m m o n i a c o n c e n t r a t i o n u n d e r t h e s e c o n d i t i o n swou ld f l uc tua t e i n a cco rdance w i th eqn (8 ) .

I f pZ > 4q t he roo t s o f t he d eno m ina to r o f eqn (6 ) a r e r eal and t he r e sponse Co i s as m o o t h e x p o n e n t i a l d e c a y i n d i c a t e d i n t h e s o lu t i o n t o e q n (6 ) u n d e r t h e s e c o n d i t i o n s ;t hus :

A dCo - [exp ($1 t ) - - exp ($2 t ) ] (10)$1 - $2w h e r e

Sl = ; $2 = (11)2 2The co r r e spo nd ing exp re s s ion fo r C i is f oun d by sub s t i t u t i on an d i nve r s ion o f eqn (4 ) ,and t hus :

C i - s l _ s 2 [ S l e x p ( S l t ) - - S 2 e x p ( S 2 t ) ] + + 1 C o (12 )T h e t y p e o f r e s p o n se t o t h e d i s t u r b a n c e s c o n s i d e r e d h e r e, d e p e n d s e n t i re l y o n th e

va lue s o f p and q , wh ich a r e func t i ons o f t he s iz es , f l ow- ra t e s and b ioch em ica l l dne ti c so f t he sys t em. In t he exam ple s cons ide r ed i n t h i s p ape r p 2 > 4q in all c a ses andd a m p e d r e s po n s e s ar e th u s o b t a i n e d .

ST E P D I ST U RBA N CE

Thi s ca se i s a cons ide r a t i on o f t he r e sponse o f the s im p le rec i r cu l a ti ng sys t e m show n inF ig . 1 , t o a s tep change i n the r a t e o f am m on ia p ro duc t i on i n t he f i sh t ank . As i n t hep rev ious ca se , f i rs t o rde r r e ac t i on k ine t i c s and i dea l m ix ing a r e a s sumed .


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A P P L I C A T I O N O F D Y N A M I C R E S P O N S E A N A L Y S I S 9 9Ref e r r i ng t o t he Append i x , uns t eady s t a t e m a t e r i a l ba l ance f o r am m oni um i ons

across f ish t ank and f i l t e r y ie ld eqns (13 ) and (14) , r espec t ive ly , for a s tep chan ge inam m oni a p r o du c t i on r a t e O f R F m g h -1 l i tr e - 1.

dCiV F ~ t = F ( C - - Ci) + V v R F ( 1 3 )dC oV ~ ~ - t = F ( G - C o ) - / c C o ( 1 4)

Tak i ng Lap l ace t r ans f o r m s a s be f o r e y i e l d s:b

s G ~ = a ( C o s - q s ) + - (1 5)S

s C o s = d ( C i s - C os ) - f C o s ( 1 6 )w h e r e

b = R F ( 17 )As b e f o r e , t he case o f o ve r - dam ped r e sponse is o f p r i m ar y i n t e r e s t and pa r t ia l

f r ac t i on r educ t i o n and i nve rs i on y i e l d s :b d [ 1 - e x p 1 ]

Co S ~ $ 2 L ~ - ~ 2 (1 S 2 t ) - - S I (1 - - exp S l t ) ] ( 18 )and

Ci = Co + ~ (exp S a t - exp $2 t ) (19), $ 1 - - $ 2

O T H E R D I S T U R B A N C E S

Di s t u r bances in am m on i a p r o duc t i on r a t e o f kn ow n t im e f o r m can be r ead i l y hand l edi n a m o d i f i e d f o r m o f e q n ( 1 3 ) a n d i f th e d e v i a t io n i n a m m o n i a o u t p u t f r o m s t e a d ys t a te i s exp r e ssed as an exp l i c i t f unc t i on o f t , t hen e qn ( 13 ) m ay be r ew r i t ten :

dCiV F - ~ t - - F ( C - - Ci) + VF~b(t) (2 0)whe r e ~ ( t ) i s t he known f unc t i on de sc r i b i ng t he d i s t u r bance . Depend i ng on t he na t u r eo f q~, num er i ca l m e t ho ds f o r t he so l u t i on o f eqn ( 20 ) m ay be neces sa r y .

A f u l l y de t a i l ed de r i va t i on o f t he m a t e r i a l ba l ance equa t i ons ( 1 ) - ( 19 ) i s p r e sen t edi n t he Ap pend i x , t og e t he r wi t h a de sc r i p t ion o f t he d i s t u r bances cons i de red .


8/21

1 0 0 L . R . W E A T H E R L E Y

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Co m puted response (eqn (12) ) . D onaldson 's result s : + , ru n 2 ; A , run 3 ; o , run 4 .


9/21

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C A L C U L A T E D R E S U L T ST h e r e s p o n s e t o i m p u l s e d i s t u r b a n c e i n a m m o n i a o u t p u t is s h o w n i n F ig s 2 a n d 3 .F i g u re 2 s h o w s t h e r e s p o n s e c a l c u l a t ed f o r t h e e x p e r i m e n t a l s y s t e m u s e d b y D o n a l d -s o n ( 1 9 7 9 ) . T h i s s y s t e m c o m p r i s e d a t r i c k l i n g b i o l o g i c a l f i l t e r a n d a n a i r l i ft s y s t e ms i m p l y r e c i r c u l a t i n g w a t e r f r o m t h e b a s e o f t h e f i l t e r t o t h e t o p . T h e f i l t e r w a s

T A B L E 1I n p u t p a r a m e t e r s o f s y s t e m s c a l c u l a t e d . ( a )R e s p o n s e t o i m p u l s e d i s t u r b a n c e - i n f lu e n c e o ff i sh t a n k v o l u m eF i l t e r vo lum e ( V B) = 29 . 45 l i t r eR e c i r c u l a t i o n r a te ( F ) = 8 4 l i t r e h -1R a te c on s t a n t ( k ) = 0 - 35 h - tFish tankvolume 1repulseVF (litres) (rag NH3)

3 510 16 . 675 0 8 3 . 3 3100 166 . 675 0 0 8 3 3 . 3 3

( b ) R e s p o n s e t o i m p u l s e d i s tu r b a n c e - i n f l u e n c e o fr e c i r c u l a t i on r a t e ( f o r r e su l t s s e e Fig . 4 )Parameter Value

F i s h t a n k v o l u m e ( V F ) 500 l i t r eF i l t e r vo lu m e ( V B ) 29 - 45 l i t r eR a te c on s t a n t ( k ) 0 - 35 h -1I m p u l s e ( / ) 8 3 3 . 3 m g N H 3( i) R e c i r c u l a t i on r a t e ( F ) 84 l i t r e h -~( ii ) R e c i r c u l a t i on r a t e ( F ) 150 l i t r e h -~( i ii ) R e c i r c u l a t i on r a t e ( F ) 1000 l i t r e h -~

Parameter ValueF i s h t a n k v o l u m e ( V E ) 5 0 l i tr eF i l t e r v o l u m e ( VB) 29 .45 l i t r eR a te c on s t a n t ( k ) 0 . 35 h -~I m p u l s e ( / ) 8 3 . 3 3 m g N H 3

( i ) R e c i r c u l a t i on r a t e ( F ) 5 l i t r e h - l( ii ) R e e i r c u l a t i on r a t e ( F ) 20 l i t r e h -( i ii ) R e c i r c u l a t i on r a t e ( F ) 50 l i t r e h -~( iv ) R e c i r c u l a t i on r a t e ( F ) 150 l i t r e h -( v ) R e c i r c u l a t i o n r a t e ( F ) 1 0 0 0 l i t r e h - 1


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12/21

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13/21

APPLICATION OF DYNAM IC RESPONSE ANALYSIS 105s u s ta i n e d b y p e r io d i c i m p u ls e i n j e c t io n o f a m m o n i u m s o l u t i o n i n t o t h e l in e a t t h et o p o f t h e f i l te r . T h e o p e r a t i n g p a r a m e t e r s o f t h e s y s t e m a r e s h o w n i n T a b l e 1 .

S a m p l i n g o f li q u o r a l o n g t h e l e n g t h o f t h e f i l t e r d e m o n s t r a t e d a f l a t c o n c e n t r a t i o np r o f i le w i t h r e s p e c t t o a m m o n i a , t h u s i n d i c a t i n g a h i g h le v el o f l i q u i d m i x i n g .

T h e e x p e r i m e n t a l r e s p o n s e in th e a m m o n i a c o n c e n t r a t i o n e n t e r i n g t h e fi l te r iss h o w n a s e x p e r i m e n t a l p o i n t s i n F i g . 2 , a n d t h e c o n t i n u o u s li n e i n d i c a t e s t h e r e s p o n s ec a l c u l a t e d u s i n g e q n ( 1 2 ) a n d a f i rs t o r d e r r a t e c o n s t a n t o f 0 . 3 3 h - 1.

T o d e m o n s t r a t e t h e m o r e g e n e r a l a p p l ic a t i o n o f e q n s ( 1 0 ) - ( 1 9 ) , a s e ri es o f r e s u lt sw e r e c a l c u l a t e d f o r a r a n g e o f s y s t e m s l is t e d i n T a b l e 1 , a n d t h e e f f e c t o f v a r y i n gf is h t a n k s iz e a n d f l o w - r a te u p o n t h e r e s p o n s e t o a n i m p u l s e d i s t u r b a n c e s t u d i e d . T h em a g n i t u d e o f t h e d i s t u r b a n c e p e r u n i t v o l u m e o f fi sh t a n k w a s h e ld c o n s t a n t . T h er e s u l ts a r e s h o w n in F ig . 3 a n d s e r v e t o i l l u s t r a t e t h e e f f e c t o f f i s h t a n k s i z e u p o n t h ea b i l i t y o f t h e f i x e d s i ze b io lo g i c a l f i l te r u n i t t o r e s p o n d t o a n im p u l s e d i s t u r b a n c e i na m m o n i a o u t p u t , w h i le t h e b i o c h e m i c a l r a t e c o n s t a n t i s h e l d c o n s t a n t .

T a b l e l ( b ) i n d i c a te s t h e p a r a m e t e r s u s e d t o s t u d y t h e e f f e c t o f r e c i r c u l a ti o n r a teu p o n r e s p o n s e c h a r a c t e r i s ti c . T a n k s iz e s o f 5 0 0 l it re s a n d 5 0 l it r es w e r e c h o s e n w i t h af i x e d f i lt e r si ze t o e x a m i n e t h e r e la ti v e i m p o r t a n c e o f r e c i r c u l a ti o n r a t e u p o n r e s p o n se .T h e r e s u l t s a r e s h o w n in F ig . 4 .

F i gu r e 5 s h o w s t h e p a t t e r n o f a m m o n i a d i s tr i b u t io n i m m e d i a t e l y f ol l o w i n gi n j e c t i o n u s i n g t h e s y s t e m p a r a m e t e r s o f D o n a l d s o n . T h e t i m e a x is is e s s e nt i al l y a ne x p a n s i o n o f t h a t i n F i g . 3 , a n d t h e r e s p o n s e c u r v e c o r r e s p o n d s t o t h a t f o r V F = 3a n d I = 5 i n F i g . 3 . T h e in i ti a l d i s tr i b u t i o n a n d d i l u t i o n o f a m m o n i a t h r o u g h o u t t h es y s t e m is v e r y r a p id a n d a p s e u d o s t e a d y s t a t e v a l ue o f a m m o n i a c o n c e n t r a t i o n i sa p p a r e n t a f t e r a p p r o x i m a t e l y 0 . 2 h .

T h e r e s p o n s e s o f w e ll m i x e d r e c i r c u la t i n g s y s t e m s t o a s t e p d i s t u r b a n c e ( s eeT a b l e 2 ) i n a m m o n i a p r o d u c t i o n r a t e a r e p l o t t e d i n F i g. 6 , w h i c h i l lu s t ra t e s t h e e f f e c to f i n c r e a s in g f i sh t a n k s i z e re l a t iv e t o f i l t e r s i ze u p o n t h e s y s t e m r e s p o n s e . T h e s iz e o f

TABLE 2Response to step disturbance in ammoniap r o du c ti o n r a t e - i n f l u e n c e o f f i s h t a nkvolume

Filter volum e (VB) = 29.45 litreRecirculation rate (F) = 8 4 litre h -1Rate constant (k) = 0.35 h -~Fish tank volume Step change

V F (litres) R F (rag NH 3 h- ~ litr e-')10 1.6750 8.33100 16.67500 83.33


14/21

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15/21

APPLICATION OF DYNAMIC RESPONSE ANALYSIS 107t h e i n p u t d i s t u r b a n c e w a s s c a l e d i n a c c o r d a n c e w i t h t h e t a n k s i z e t o m a i n t a i n t h echange pe r un i t vo lume a t a cons t an t va lue .

DISCUSSION OF RESULTS

E x p e r i m e n t a l p o in t s m e a s u r e d b y D o n a l d s on ( 1 9 7 9 ) - s ee F ig . 2 - s h o w e x c e ll e n ta g r e e m e n t w i t h t h e c u r v e c a lc u l a t ed u s i n g e q n ( 1 2 ) f o r r es p o n s e t o i m p u l s ed i s t u rbance . As sum pt ion o f idea l m ix ing i n bo th f i sh t an k and f i l t e r, i nhe r e n t i n t hem ode l , a r e va l id wh ich i s no t su rp r is i ng as Dona ld son ' s ' f i sh t ank ' was an a i r l i f t sy s t emw h i c h is w e l l m i x e d . M e a s u r e m e n t o f a m m o n i a c o n c e n t r a t i o n s a l o n g t h e a x i s o f t h ef i l te r i nd i ca t ed a f l a t p ro f i le a l so ind i ca t i ng exc e l l en t ax i a l m ix ing w i th in t he bed .

D o n a l d s o n ' s s y s t e m w a s s m a l l c o m p a r e d t o a f u l l fi sh r e a ri n g s y s t e m , a n d t h ee f f e c ts o f d y n a m i c l a g in t h e c o n n e c t i n g p i p e - w o r k a re v e r y s m a ll c o m p a r e d t o t h et ime cons t an t s o f f i l te r and ' f i sh t ank ' . A s eco nd ma jo r d i f f e r ence i s i n r e l a ti ve s iz e o ff i sh t ank and f i l t e r . A r ea l i s t i c a l l y s i z ed f i sh t ank emp loyed i n con junc t i on w i th t he30 l i tr e f i l te r cons ide r ed he r e m ay n o t h ave such f avou rab l e mix ing cha rac t e r is t i c s a sa s sumed i n t he de r i va t i on o f eqns (10 ) and (12 ) . I t i s t hus a rgued t ha t expe r ime n ta lm eas u re m en t o f f ish t ank m ix ing cha rac t e r i s t i c s m us t be an e s sen t ia l i ng red i en t o fm o d e l v a l i d a ti o n a n d r e f i n e m e n t .

On t h is a s sum pt ion t he ab i l i t y o f t he b io log i ca l f i l te r t o r e spo nd t o changes i n am uc h l a rge r f ish t ank i s show n in F ig . 3 , whe re a f am i ly o f cu rves is p lo t t ed show inga m m o n i a c o n c e n t r a t i o n r e s p o n se t o i m p u l s e d i st u r b a n c e i n f i sh t a n k s o f i n cr e as in gsize.

The con cen t r a t i on r e sponse ove r a pe r i od o f 24 h i s show n , a l t ho ugh c l ea r l y i t is t hel eng th o f t he pe r i o d fo r wh ich t he am m on ia l evel r ema ins h igh wh ich i s c r i ti c a l i ndec id ing t he ac cep t ab i l i t y o f a pa r t i cu l a r r e spo nse cha r ac t e r i s ti c .

I t i s im po r t an t t o "no t ic e t ha t t he f all in con cen t r a t i on fo l l owing t he i n i ti a ld i s t u rb a n c e is d u e t o i n i ti al m i x i n g a n d d i st r ib u t i o n o f a m m o n i a t h r o u g h o u t t h e w h o l esys t em , a nd due t o t he ac t i on o f t he b io log i ca l t~ ilt er . The d i s t i nc t i on be tw een t he set w o e f f e c t s is d e m o n s t r a t e d b y c l o s er e x a m i n a t i o n o f t h e r e s po n s e s a t sm a l l v a lu e s o fVv. The r e sponse shown in F ig . 3 f o r VF = 3 shows a ve ry r ap id r edu c t i on i n t he va lueo f CNH 3 f rom 1 .67 m g li t re - 1 t o app rox im a te ly 0 .15 m g li tr e - 1. Th i s is d em ons t r a t edmore c l ea r l y i n F ig . 5 whe re t he s ame r e sponse i s p lo t t ed a s t he t op cu rve w i th ag r ea t l y expa nde d t ime axis . The l owe r cu rve shows t he i nc r ease i n con cen t r a t i onexpe r i enced i n t he f i lt e r due t o i n i ti a l m ix ing , t h i s va lue conve rg ing w i th t he va lue fo rt h e f i s h t a n k a f t e r a t i m e o f a p p r o x i m a t e l y 0 .2 h .

I n t h i s c a s e t h e d i l u ti o n o f a m m o n i a i n j e c te d is v e r y r a p id c o m p a r e d t o t h e r a t e o fr em ova l by f i l t e r a c t i on show n by t he r e l a ti ve ly s l ow decay i n con cen t r a t i on i n F ig . 3 .

As t he r e l a ti ve s iz e o f f i sh t an k t o f i lt e r i nc r ea se s , t he d i l u t i on e f f e c t becom es mu chle ss m arke d and t he r educ t i on i n con cen t r a t i on is p r im ar i l y con t ro l l ed by f i l t e r a c t i on ,deno t ed by smoo the r cha r ac t e r i s t i c s ; s ee F ig . 3 .


16/21

108 L.R. WEATHERLEY

The r e sponses t o impu l se d i s t u rbance a t va r i ous r ec i r cu l a t i on r a t e s a r e shown in t hef am i ly o f cu rves p lo t t ed i n F ig . 4 . T wo sys t em s a re s t ud i ed , e ach hav ing a 30 l i tr ef i l te r and f i sh t an k vo lum es o f 50 l i t r e s and 500 l i tr e s , r e spec t ive ly .

Curves fo r t he 50 l i tr e sy s t em show tha t am m on ia r emo va l i s enhan ced a s t her ec i r cu l a t i on r a t e i s i nc r ea sed , bu t t he deg ree o f enhan cem en t f a ll s o f f a sym pto t i c a l l ya s h ighe r f l ow r a t e s a r e r eached . The p r ed i c t i ons , a s be fo r e , do no t a ccoun t f o r non -idea l it i es i n m ix ing beha v iou r and t he da t a sh ow n r ep re sen t l im i t i ng ca se s, and i ndeedthe occu r r ence o f non - idea l mix ing is m uc h m ore l i ke ly a t t he l ow f l ow - ra t e s i nd i ca t ed .Thus , t he enh anc em en t ex pe r i ence d i n p r ac t i c e a t i nc r ea sed r ec ir cu l a t i on r a t e s i s l i ke lyt o b e g r e a t e r t h a n p r e d i c t e d b y t h i s m e t h o d .

Respon ses t o s t ep -wi se d i s tu rbances i n am m on ia ad d i t i on r a t e s t o t he f ish t ank a r es h o w n i n F ig . 6 , a n d s h o w t h e l e ve l t o w h i c h t h e c o n c e n t r a t i o n o f a m m o n i a isi nc r ea sed a s a consequen ce o f changes in p rod uc t i o n r a t e . The va lue s o f ma jo r in t e r e s ta r e t he s t e ad y s t a t e va lue s and t he se m ay be r ead i l y ca l cu l a t ed f rom eqns (18 ) and (19 )as t ~ oo. The fam i ly of curves show di rec t ly the ab i l i ty of a f ixed s ize f i l t e r to re spo ndto a s t ep change i n am m on ia i npu t t o t he sy s t em , and c l ea r ly , a s t he g ross am oun t o fam m onia be ing f ed i nc r ea se s, t he s t e ady s t a t e va lues o f amm on ia con cen t r a t i on r is e i na c c o r d a n c e w i t h t h e a b i l it y o f t h e f i l te r to r e m o v e t h e a m m o n i a .

T h e a c c u r a te u s e o f e q n s ( 1 0 ) - ( 1 9 ) t o p r e d i c t th e r e s p o n s e i n r ea l s y s t e m s d e p e n d sup on t he va l i d i t y o f t he a s s um pt ion o f i deal mix ing i n t he two m a jo r ves se lscons ide r ed . Whi le i dea l m ix ing ma y be r ead i ly app roach ed i n a coa r se g rave l f i l te r suchas u sed by Dona ldson (1979) , t ha t o f t he f i sh t ank i s much l e s s l i ke ly t o be so . Themix ing r6g ime wi th in a fi sh t ank , a s i n any o the r ve s se l , m ay be qua l i t a t i ve ly de sc r i bedby t he r e s idence t ime d i s t r i bu t i on (Dan ckw er t s , 1958 ; B i scho f f and Levensp i e l , 1963 )and measu red u s ing t r a ce r r e sponse t e chn iques . De t a i l ed i n fo rma t ion o f t h i s na tu r e i san e s sen t i a l p r e r equ i s i t e t o mode l r e f i nemen t .

A f u r t h e r r e f i n e m e n t w o u l d b e t h e i n cl u s io n o f t r a n s p o r t a t io n l ag s a l lo w i n g f o r t h ef i n i te t im e r equ i r ed fo r f l ow o f l i quo r b e twe en ves se ls wh ich m ay be qu i t e s i gn i fi c an ti n long p ipe runs and a t r edu ced f l ow- ra t e s . The c rudes t wa y o f a ch i ev ing t h i s ad ju s t-m e n t i s d e s c r i b e d b y C o u g h a n o w r a n d K o p p e l ( 1 9 6 5 ) , a n d r e q u i r e s r e w r i t i n g e q n s(1 ) and (2 ) on d i f f e r en t t ime ba se s . I t i s c l e a r t ha t Dona ldson ' s expe r imen t i nvo lvedins ign i f ic an t t ime l ag be twee n d i f f e r en t pa r t s o f t he sy s t em , and t he m a tch o f expe r i -men ta l po in t s and p r ed i c t ed r e su l t s bea r s t h i s ou t .

CONCLUSIONS

I n p r i nc ip l e t he f l ow con f igu ra t i ons o f re c i r cu la t i ng aquac u l t u r e sy s t em s ar e we l lsu i t ed t o t he app l i ca t i on o f r e sponse ana ly s is t e chn iques . Accu ra t e p r ed i c t i ons a resub j ec t t o t he ava i l ab i li t y o f k ine t i c da t a , t ank and f i lt e r mix ing da t a and p r eci s ek n o w l e d g e o f t h e t y p e s o f d i st u r b a n c e e n c o u n t e r e d i n r ea l s y s t e m s .


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A P P L I C A T I O N O F D Y N A M I C R E S P O N S E A N A L Y S I S 1 0 9T h e w o r k p r e s e n t e d h e r e s h o w s t h at g i ve n id e a li se d d i s t u r b a n c e c o n d i t i o n s a n d

i d ea l m i x in g , a c c u r a te p r e d i c t io n s o f t h e c h a n g e s i n a m m o n i a c o n c e n t r a t i o n t h r o u g h -o u t t h e s y s t e m c a n b e m a d e .T h e p r e d i c t i v e t e c h n i q u e s d i s c u s s e d h e r e il lu s t r a te th e p o t e n t i a l o f r e s p o n s ea n a l y s is , s u b j e c t t o f u r t h e r r e f 'm e m e n t , a s a t o o l i n t h e d e s ig n a n d o p e r a t i o n o fr e c i rc u l a t in g a q u a c u l t u r e s y s t e m s . O b v i o u s r e f i n e m e n t s w o u l d in c l u d e t h e i n c o r p o r a -t i o n o f q u a l i t a ti v e n o n - i d e a l m i x i n g m o d e l s i n t o t h e a n a l y si s , a n u p d a t i n g o f k i n e t i cd a ta l i n k e d t o k n o w n c h a n g e s in t h e f d t e r b a c te r ia l p o p u l a t i o n a n d c o n s i d e r a t i o n o ft r a n s p o r t a ti o n l a g b e t w e e n m a j o r i te m s .

R E F E R E N C E SA t k i n s o n , B . ( 1 9 7 4 ). B i o c h e m i c a l R e a c t o r s , P ion P re ss , Lo ndo n .B i sc ho f f , K . B . & Le ve nsp ie l , O . ( 1963 ) . P a t t e r n s o f f l ow in c he m ic a l p r oc e s s ves se ls . Adv . C he m .Eng . , 4 , 9 5 - 1 9 8 .C ougha now r , D . R . & K oppe l , L . B . ( 1965 ) . P r o ce s s S y s t e m s A n a l y s i s a n d C o n t r o l, McGraw-Hil l ,N e w Y o r k .D a n c k w e r t s , P . V . ( 1 9 5 8 ) . C o n t i n u o u s f l o w s y s t em s . D i s t r i b u t i o n o f r e s i d en c e t im e s . Chem. Eng.Sc i . , 2 . 1-13 .D ona ldson , D . ( 1979 ) . C o n s t r u c t i o n o f a M a r i ne E x p e r i m e n t a l B i o l o g i c a l F i l t e r a n d i t s N i t ri f ic a -t i o n K i n e t i c s , H e r i o t - W a t t U n i v e rs i t y H o n o u r s R e s e a rc h R e p o r t , E d i n b u r g h , U K .H i r a y a m a , K . ( 1 9 7 4 ) . W a t e r c o n t r o l b y f i l t r a t i o n i n c l o s ed w a t e r s y s t e m s . A q u a c u l t u r e , 4 ,3 6 9 - 8 5 .L ia o , P . B . & M a yo , R . D . ( 1972 ) , S a lm on id ha t c he r y w a te r r e use sy s t e m s . A q u a c u l t u r e , 1 , 3 1 7 -35.M a n t l e , G . ( 1981 ) . S i m u l a t i o n M o d e l o f C l o s e d W a t e r C i r c u l a t i o n S y s t e m f o r F i s h C u l t u r e ,A p p l i e d B i o lo g y R e s e a r c h U n i t R e s e a rc h R e p o r t , T h e O p e n U n i v e r s i ty , M i l t o n K e y n e s , U K .R o s e n t h a l , H . ( 1 9 8 0 ) . R e c i r c u l a t i o n s y s t em s i n w e s t e r n E u r o p e . Proc . W or ld Sy m p . on Aqua-c u l t u r e in H e a t e d E f f l u e n t s a n d R e e i r c u l a t io n S y s t e m s , V ol . I I , B e r li n , pp . 3 05 - 1 5 .S pe e c e , R . E . ( 1973 ) . T r o u t m e ta bo l i sm c ha r a c t e r i s t i c s a n d t h e r a t i o n a l d e s i gn o f d e n i t r i f i c a t i o nf a c i l it i e s f o r w a te r r e use i n ha t c he r i e s . Trans. Am. Fish. Soe. , 102 ( 2 ) , 223 - 34 .S r na , R . F . & B a ggale y , A . ( 1975 ) . K ine t i c r e sponse o f pe r tu r b e d m a r ine n i t r i f i c a t i on sy s t e m s .J . W a te r Po l lu t . C o n t ro l Fe d . , 4 7 , 4 7 2 - 8 6 .

A P P E N D I X

M a ss b a l a n c e s o n t h e i s h t a n kF i g u r e A 1 s h o w s t h e n a t u r e o f t h e d i s tu r b a n c e s i n a m m o n i a p r o d u c t i o n r a te s tu d i e d

i n th i s p a p e r a n d w i t h r e f e r e n c e t o t h e s i m p l e f l o w d i a g r a m s t h e u n s t e a d y s t a t e m a s sb a l an c e e q u a t i o n s f o r a m m o n i a m a y b e d e r iv e d .

I m p u l s e d i st u r b a n c e . I n t im e e l e m e n t d r.A m m o n i a in = i n f l o w f r o m fi lt er + i m p u l s e a d d i ti o n

= F e o d t + A 6 ( t ) d t V F


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1 l 0 L. R. WEATH ERLEY

A m m o n ia I m p u ls e D i s tu r b a n c ep r o d u c t io n T o f il te rr a t e(A ~ ( t ] ) A m m o n ia o u t ( F c i d ! )

A m m o n i a in ] F is h t a n k [( A S ( t ) d t V F ) . ~ l v o l u m e ( V F )

t T A m m o n ia in ( F C o d l )- 4 F r o m f i l te r

S t e p D is t u r b a n c e T o f il t e rA m m o n i a ~ "p r o d u c t i o nr a te A m m o n ia o u t ( F c , d ; )r F s s f F

- - R F A m m o n iar o d u c t i o n r Fr d t : ~ L Fishum etank(F )

r F s s[ A m m o n io i n ( F c o d / )/

1 ~ " F r o m i l t e rO t h e r D i s tu r b a n c e s

T o f i l t e rA m m o n i a ~p r o d u c t io n F F =r a te A m m o n ia o u t ( F c i d ! )( r F ]

A m m o n i a p r o d u c t io n [ , , r( r F V F d I ) F i s h t a n kr F ss ~ [ v o lu m e ( V )

t lA m m o n i o in ( F C o d t )F r o m f i l t e r

Fig. A1. Mass balance on fish tank.

Amm oni a out = outflow to filter= F c i d t

Therefore, nett accumulation of ammonia = F ( c o - - ci) dt +A ~( t) d t V F . Thechange in am moni a conc ent rat ion in the ta nk, dci, thus caused is given by VF dc =


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APPLICATION OF DYNAMIC RESPONSE ANALYSIS 1 1 1F( co - - c i ) d t + A 6 ( t ) d t Vv , i .e .

dc iVv ~-t = F(c -- e l ) + A f ( t ) V v ( 2 1 )

S i n c e w e a r e p r i m a r i l y c o n c e r n e d w i t h t h e d e v i a t i o n s f r o m s o m e i n i ti a l s t e a d y s ta t e ,t h e c o n c e n t r a t i o n s o f i n t e re s t a r e c o n v e n i e n t l y e x p r e s s e d a s d e v i a ti o n v a r ia b le s .

C o - - C o - C o s s ( 2 2 )Ci = cl - - Ciss (2 3)

w he re Cos s a nd Ciss a r e t h e i n i t ia l s t e a d y s t a t e v a lu e s o f i n f l u e n t a n d e f f l u e n t c o n c e n -t r a t i o n s , r e s p e c t i v e ly . By d e f in i t i o n ,

d Co s s d Ci s sCoss = C i s s a n d - - 0 ( 2 4 )d t d ta n d

d C o d c o- ( 2 5 )d t d tdCi de i

- ( 2 6 )d t d tS u b s t i t u ti o n o f e q ns ( 2 2 ) - ( 2 6 ) i n t o e q n ( 2 1 ) t h e n y i e l d s:

dC~V v d t = F ( C o - - C i ) + A 6 ( t ) V v ( 1 )

F ro m is h t a n k r / A m m o n ia in F c i d t )

J B io l o g i c a l i lt e r ] ~ A m m o n ia x id a t io nv o lu m e V ' ) ~ ( V 8 k C o d t )

A m m o n i a o ut ( o d t }

To fish ank

Fig. A2. Mass balance on biological filter.


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1 1 2 L . R . W E A T H E R L E Y

S t e p d i s t u r b a n c e . I n t i m e e l e m e n t d t .A m m o n i a i n = i n f lo w fr o m fi lt er + a m m o n i a p r o d u c e d

= F e o d t + V F r F d tA m m o n i a o u t = o u t f l o w t o f i lt er

= F e i d tT h e r e f o r e , n e t t a c c u m u l a t i o n o f a m m o n i a = F ( e o - - e l ) d t + V v r v d t . A s b e f o r e ,

t h e a c c u m u l a t i o n o f a m m o n i a is m a n i fe s t a s a ch a n g e in c o n c e n t r a t i o n i n t h e t a n kVF de i . Thus ,

deiV F ~ t = F ( e i - - e ) + V F r F ( 2 7 )

T h e c h a n g e i n a m m o n i a p r o d u c t i o n r a t e is d e f i n e d b y t h e d e v i a t io n v a r ia b le R F =r F - - r F s s -F o r t h e s t e a d y s t a t e v a l u e s o f e i , % , r F b e c o m e s :

F ( e i s s - - Co s s ) + V v r v s s = 0 ( 2 8 )C o m b i n a t i o n o f e q ns ( 2 7 ) a n d ( 2 8 ) y i e ld s :

d c lV F ~ t = F [ ( e i - - e i s s ) - - ( C o - - e o s s ) ] + V F ( r F - - r F s s )

i .e . in t e r m s o f d e v i a t i o n v a r i a b l e s :dC iV v ~ - t = F ( G - C o ) + V F R F ( 1 3 )

O t h e r d i s tu r b a n e e s i n a m m o n i a p r o d u e t i o n r ate . T h e t r e a t m e n t f o r a s t e pd i s t u r b a n c e i n a m m o n i a p r o d u c t i o n r a t e is g e n e ra l f o r o t h e r t i m e v a r y i n g d i s t u r b a n c e si n a m m o n i a p r o d u c t i o n r a t e , h en c e e q n ( 2 0 ) :

dC iV v ~ t = F ( C o - - G ) + V F ~ 9(t) ( 2 0 )

M a s s b a l a n c e s o n t h e b i o l o g i c al f i l t e rW i t h r e f e r e n c e t o F i g. A 2 t h e u n s t e a d y s t a te m a s s b a l a n c e f o r a m m o n i a a c r os s t h e

f i lt e r is o f t h e s a m e f o r m i r r es p e c t iv e o f t h e n a t u r e o f t h e d i s t u r b a n c e i n t h e f i sh t a n k .T h u s , i n t i m e e l e m e n t d t :

A m m o n i a i n = i n f lo w f ro m f is h t a n k= F c l d t


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A P P L I C A T I O N O F D Y N A M I C R E S P O N S E A N A L Y S I S 1 1 3

A m m o n i a o u t = o u t f lo w t o f is h t a n k + a m m o n i a o x i di se d= F c o d t + k c o V B d t

A c c u m u l a t i o n = VB d c iT h e r e f o r e ,

d c iVB ~ = F ( c i - - C o ) - k c o V B ( 29 )I n t e r m s o f d e v i a t i o n v a r i a b le s C i , C o , s e e e q n s ( 2 2 ) a n d ( 2 3 ) , ( 2 9 ) m a y b e r e w r i t t e n :

dC~VB - -~ t = F ( C i - - C ) - - kC VB ( 2 ) , ( 1 4 )

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Weatherley 1982 Aquacultural-Engineering