Application of simple dynamic response analysis to a recirculating aquaculture system — A preview

AquaculturalEngineertng 1 (1982) 93-113

APPLICATION OF SIMPLE DYNAMIC RESPONSE ANALYSIS TO A RECIRCULATING AQUACULTURE SYSTEM - A PREVIEW

L. R. WEATHERLEY

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

ABSTRA CT

This paper describes the application of dynamic response analysis to a simple first order recireulating aquaculture system and demonstrates how the distribution of ammonia throughout the system may be predicted following dynamic changes. The approach is validated by experimental data obtained for a reeireulating system under con trolled conditions of mixing and disturbance. It is concluded that unsteady state predictive methods have significant potential application in the context of aquaculture, subject to further refinement in the modelling of biochemical la'neties and liquid mixing.

NOMENCLATURE

a

F/VF defined in eqn (5)

A A6(t) b

Magnitude of impulse function Impulse function Step change in ammonia production rate

h -1 mg litre-a m g h -1 litre -1 mg h -~ litre -1

ci

Ammonia concentration in effluent

mglitre -~

Unsteady state deviation in ammonia concentration in effluent from fish tank

mg litre -1

Ammonia concentration in influent to fish tank

mg litre -l

Unsteady state deviation in ammonia concentration to influent to fish tank CNH 3 Unsteady state deviation in ammonia concentration d F/VB defined in eqn (5) 93

mg litre -1 mg litre -1 h -1

t

Ciss ! Ci c°

Coss

t

Co

Aquacultural Engineering 0144-8609/82/0001-0093/$02.75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain

94

L.R. WEATHERLEY

f F I k m n p q~

Rate constant defined in eqn (5) Recirculation rate Total impulse magnitude First order rate constant for ammonia oxidation Parameter defined in eqn (9) Parameter defined in eqn (9) a + f + d defined in eqn (7) Ammonia production rate disturbance function q af defined in eqn (7) rv Ammonia production rate rvs s Steady state value of ammonia production rate Rv Step disturbance in ammonia production rate s Laplace transform variable $1, $2 Quadratic roots defined in eqn (11) t Time VB Filter volume Vr Fish tank volume

h -1 litre h -1 mg h -1

mg h -1 litre -1 mg h -1 litre -1 mg h-1 litre -~ mg h -a litre -a

litre litre

Subscripts i o ss B F

Refers to effluent stream from fish tank Refers to inlet stream to fish tank Steady state value Refers to filter Refers to fish tank

INTRODUCTION

The rapidly accelerating evolution of novel recirculating systems in aquaculture has given rise to increasingly complex flow sheets involving a number o f separate processes linked in an attempt to optimise water use and quality. For example, Rosenthal (1980) describes the growing use of multiple recirculating cycles for each separate type of water treatment including denitrification, oxygenation, ozonation and metabolite removal, but concludes that in spite of these developments, the design and operation of recirculating systems are still largely an art rather than a science. The current work seeks to demonstrate in a relatively simple way one method of breaking down an aquaculture system into individual modules and analysing theoreticaUy the dynamic behaviour by classical process control techniques, and to identify likely sources of disturbance o f interest to both designer and operator. The paper is introductory in its approach, and the detailed problem discussed is conf'med to removal and distribution of ammonia in a recirculating system, although it is specifically intended to expand this work in the future to analyse distribution of

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

95

nitrite, oxygen, heat and other key operating quantities likely to be sensitive to operational and environmental disturbances. Other work in this area, Mantle (1980), has adopted a pseudo steady state approach to modelling of aquaculture systems, calculating the steady state response to imposed changes. While this approach may be useful to the designer, it neglects consideration of the important transient behaviour, especially relevant to the consecutive reactions for the oxidation of ammonia to nitrate ion and the intermediate formation of nitrite. Other attempts at design mode calculations based entirely on the correlation of collected data for a particular system have been made. Hirayama (1974) presents correlations to calculate the relative sizing of fish tanks and biofilters, but the data upon which the correlation coefficients are calculated are from an aquarium with a small experimental filter and are not considered to be useful for the general case of a commercial aquaculture system. Speece (1973), Liao and Mayo (1972) also present steady state correlations for design mode calculations based upon existing data.

SYSTEM ANALYSIS

Unless attempts at mathematical description of a system are to be little more than statistical correlation of observed data, the behavioural characteristics of each discrete unit comprising the whole system require analysis. A simple recirculating aquaculture system may comprise fish tank, effluent pipe, biological filter and return pipe; see Fig. 1. Consideration of the fish tank alone requires accurate knowledge of the volume, the degree of mixing (i.e. the residence time distribution) and the rate of metabolite production for a successful response analysis. In this preliminary investigation the contents of the tank are assumed to be well mixed. Similarly for the biological filter unit, the degree of mixing exerts an important influence on the unsteady state response behaviour. In purely flow terms the degree of mixing is measured experimentally and is a function of the type of filter, i.e. ring packing/gravel/sand, trickle downflow, upflow, etc., and the specific liquid flow-rate. In the analysis presented here the filter is assumed to be well mixed and the experimental data examined later on (Donaldson, 1979) bear out this assumption. In considering the distribution of ammonia in the system the rate of removal in the filter is the second important aspect of triter behaviour to be considered. Kinetics of biochemical ammonia oxidation do not conform to simple kinetic analysis in a general sense due to the role of bacterial reproduction and death rates in determining the overall rate of reaction (Atkinson, 1974). However, deviations in ammonia removal rates due to changes in bacterial population density are likely to be very much slower compared to deviations resulting from operational disturbances. Srna and Baggaley (1975) successfully correlated kinetic data for both ammonia and nitrite oxidation in a

96

L.R. WEATHERLEY CI

co

l r=kCo VB

VF

Air lift system

Fish tank

I

Biological filter

Fig. 1. Simplerecirculating aquaculture system. marine system using first order equations. More recent work by Donaldson (1979) confirmed first order behaviour under constant population conditions. While the fish tank and biological filter represent the most important modules of the system under consideration, the interconnecting pipework and air lift section (Fig. 1) can potentially influence the response of the system to dynamic changes, and pipework can constitute a significant time lag between fish tanks and filter. The effect of transportation lag is ignored in this preliminary analysis. It is now necessary to examine some of the dynamic disturbances to which a recirculating system may be subject. Major disturbances following start-up are most commonly induced by the operator and include actions such as feeding, introduction or removal of fish population, seeding and nurturing of the biological filter unit. Other disturbances caused by changes in temperature, and changes in ammonia production due to stress changes in the population are likely to be less rapid. In order to predict the unsteady state response to any of these disturbances, the time dependent nature of each must be known. For example, spike mode nurturing of a biological filter can be approximated by an impulse disturbance in respect of ammonia to the fish tank. Rapid introduction of stock to a fish tank may be approximated by a step disturbance in ammonia production rate. The time function representation of the other disturbances must be interpreted from experimental observations in a known fish tank, and while the relationship between feeding cycle and ammonia output may potentially be numerically quantified, the representation of stress induced disturbances in ammonia output provide an altogether more difficult problem.

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

97

The following analysis is conf'lned to impulse and step disturbances in ammonia output in the fish tank.

IMPULSE DISTURBANCE

Referring to the Appendix, unsteady state material balance for ammonium ion across the fish tank yields

dCi

VF "-~-t ~ F(Co -- Ci) +A6(t) Vv

(1)

where A 6 ( t ) is the impulse of ammonia added to the well mixed fish tank at t = 0. Co and Ci are unsteady state deviation variables referring to change in concentration of ammonium ion in the streams entering and leaving the fish tank, respectively. By definition, therefore, Ci(t = O) = O, Co(t = O) = O. A similar mass balance for ammonium ions may be conducted across the biological filter. Assuming first order kinetics and that the fluid in the filter is well mixed, unsteady state mass balance yields: VB ~dt°= F(Ci -- Co) -- kCoVB

(2)

where k is the specific first order rate constant for the oxidation of ammonia to nitrite ion. Taking Laplace transforms of eqns (1) and (2) and rearranging, yields: F

sc~s = ~ ( C o s - c l )

+A

(3)

F SC°s = VB (Cis -- C°s) -- kC°s

(4)

Letting F

F

a=--" d=--" f=k Vv' VB'

(5)

substitution and rearrangement yield: Ad

Cos - sZ + sp + q

(6)

where p~-a+f+d;

q=af

(7)

98

L.R. WEATHERLEY

Equation (6) is general and the specific nature of the system response is dictated by the values o f p and q. If p 2 < 4q, the roots of the denominator of eqn (6) are complex and Co as a function of time has an undamped response and is oscillatory. Partial fraction reduction and inversion of eqn (6) yields in this case: Co =

--Ad

sin (nt) exp (rot)

(8)

n

where (p2 __ 4q)1/2

n=

2

, m=--p/2

(9)

Thus, the consequent changes in ammonia concentration under these conditions would fluctuate in accordance with eqn (8). If pZ > 4q the roots of the denominator of eqn (6) are real and the response Co is a smooth exponential decay indicated in the solution to eqn (6) under these conditions; thus: Co -

Ad

[exp ($1 t) -- exp ($2 t)]

(10)

$1 - $2 where Sl =

2

; $2 =

(11)

2

The corresponding expression for Ci is found by substitution and inversion of eqn (4), and thus:

Ci-sl_s2[Slexp(Slt)--S2exp(S2t)]+

+1

Co

(12)

The type of response to the disturbances considered here, depends entirely on the values of p and q, which are functions of the sizes, flow-rates and biochemical ldnetics of the system. In the examples considered in this paper p 2 > 4q in all cases and damped responses are thus obtained.

STEP D I S T U R B A N C E

This case is a consideration of the response of the simple recirculating system shown in Fig. 1, to a step change in the rate of ammonia production in the fish tank. As in the previous case, first order reaction kinetics and ideal mixing are assumed.

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

99

Referring to the Appendix, unsteady state material balance for ammonium ions across fish tank and filter yield eqns (13) and (14), respectively, for a step change in ammonia production rate OfRF mg h -1 litre -1. dCi

VF ~t = F(C° -- Ci) + VvRF

(13)

dCo V~ ~-t = F(G - Co) -/cCo

(14)

Taking Laplace transforms as before yields: b

sG~ = a(Cos -

qs) + -

S

(15)

sCos = d ( C i s - Cos) -fCos

(16)

b =R F

(17)

where

As before, the case of over-damped response is of primary interest and partial fraction reduction and inversion yields:

Co

] bd [ 1 -exp 1 S~$2L~-~2 (1 S2t) --SI (1 -- exp Slt)]

(18)

and Ci =

Co + ~

,$1--$2

(exp Sat - exp $2 t)

(19)

OTHER DISTURBANCES

Disturbances in ammonia production rate of known time form can be readily handled in a modified form of eqn (13) and if the deviation in ammonia output from steady state is expressed as an explicit function of t, then eqn (13) may be rewritten: dCi

VF -~t -- F(C° -- Ci) + VF~b(t)

(20)

where ~(t) is the known function describing the disturbance. Depending on the nature of q~, numerical methods for the solution of eqn (20) may be necessary. A fully detailed derivation of the material balance equations (1)-(19) is presented in the Appendix, together with a description of the disturbances considered.

100

L.R. WEATHERLEY

0.12

0.11

0.10

0.09

T

o

C=- 0.08

E 0.07 .C

0.06

+ o

E

/

I.)

'o

005

E

+

'< 0.04

1

0.03 n + 0.02

-

0.01

I

4

8

12

16 TIME, h

1

I

I

20

24

28

I 32

Fig. 2. Response to impulse disturbance according to computed and Donaldson's results. ( - - ) , Computed response (eqn (12)). Donaldson's results: +, run 2; A , run 3; o, run 4.

Fig. 3.

i

0

.

2

~ I 6

I 8

F iO

I 12 TIME, h

I -14

I 16

~ ~

I 2

I 4

I 18

I 20

I 22

I 24

E f f e c t o f fish t a n k size o n i m p u l s e d i s t u r b a n c e . V B = 29.45, F = 84, k = 0.35. (a) V F = 5 0 0 , I = 8 3 3 . 3 ; (b) V F = 100, I = 1 6 6 . 6 7 ; (c) V F = 50, I = 8.33, (d) V F = 10, I = 1 6 . 6 7 ; (e) V F = 3, I = 5.

o4

~3

°

z=

u~

~

•o0.6 -

Z

"~

10-

)

8_

E

1.4

0 " 6 ~ ( 0

102

L.R. WEATHERLEY CALCULATED RESULTS

The 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 in a m m o n i a o u t p u t is s h o w n in Figs 2 a n d 3. Figure 2 shows t h e r e s p o n s e c a l c u l a t e d for 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 Donaldson ( 1 9 7 9 ) . This s y s t e m c o m p r i s e d a trickling biological filter a n d an air lift s y s t e m simply recirculating w a t e r f r o m t h e base o f the filter to t h e top. T h e filter was

TABLE 1 Input parameters of systems calculated. (a) Response to impulse d i s t u r b a n c e - influence of fish tank volume Filter volume (VB) = 29.45 litre Recirculation rate ( F ) = 84 litre h -1 Rate constant (k) = 0-35 h -t Fish tank volume VF (litres)

1repulse (rag NH3)

3 10 50 100 500

5 16.67 83.33 166.67 833.33

(b) Response to impulse disturbance - influence of recirculation rate (for results see Fig. 4) Parameter

Value

Fish tank volume (VF) Filter volume (VB) Rate constant (k) Impulse (/)

500 litre 29-45 litre 0-35 h -1 833.3 mg NH 3

(i) Recirculation rate (F) (ii) Recirculation rate (F) (iii) Recirculation rate (F)

84 litre h -~ 150 litre h -~ 1000 litre h -~

Parameter

Value

Fish tank volume (VE) Filter volume (VB) Rate constant (k) Impulse (/)

50 litre 29.45 litre 0.35 h -~ 83.33 mg NH 3

(i) Recirculation rate (F) (ii) Reeirculation rate (F) (iii) Recirculation rate (F) (iv) Recirculation rate (F) (v) Recirculation rate (F)

5 20 50 150 1000

litre h- l litre hlitre h -~ litre hlitre h- 1

0

0.2

._ 0.6' i

; 0.8

.E

i

"=

1.0

1.2:

3

1.4,!

1.6

~

I

2

~

I

4

F

6

I

8

I

=

10

I

F--E)

'12 TIME,h

I,

F:84

F=5

84

~

I

14

F--150

F=150

I

16

18

I

~

F=IO00

Effect of recireulation rate on impulse response. VB -- 29.45; k = 0.35.

F =1000

Fig. 4.

~

20

22

~

I

24

=50

VF=500

~o

~-.

>

"~ 0Z

.~

0

1.2

0.2

0.4

io.°

E 0.8

c

E 1.0

T

1.4

1.6

Fig. 5.

002

I

I 004

-

I 006

I 0.08

Filter I 0.10

TIME,h

I 0.12

I 0.14

I 0.16

i 0.18

I 0.20

1 0.22

E f f e c t o f initial m i x i n g o n a m m o n i a d i s t r i b u t i o n f o l l o w i n g i m p u l s e . V F = 3; V B = 2 9 - 4 5 ; F = 8 4 ; I = 5.

~

k

I 0.24

0

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

105

sustained by periodic impulse injection o f ammonium solution into the line at the top of the filter. The operating parameters o f the system are shown in Table 1. Sampling o f liquor along the length of the filter demonstrated a flat concentration profile with respect to ammonia, thus indicating a high level of liquid mixing. The experimental response in the ammonia concentration entering the filter is shown as experimental points in Fig. 2, and the continuous line indicates the response calculated using eqn (12) and a first order rate constant o f 0.33 h -1. To demonstrate the more general application o f eqns (10)-(19), a series o f results were calculated for a range of systems listed in Table 1, and the effect of varying fish tank size and flow-rate upon the response to an impulse disturbance studied. The magnitude of the disturbance per unit volume of fish tank was held constant. The results are shown in Fig. 3 and serve to illustrate the effect of fish tank size upon the ability of the fixed size biological filter unit to respond to an impulse disturbance in ammonia output, while the biochemical rate constant is held constant. Table l(b) indicates the parameters used to study the effect of recirculation rate upon response characteristic. Tank sizes of 500 litres and 50 litres were chosen with a fixed filter size to examine the relative importance o f recirculation rate upon response. The results are shown in Fig. 4. Figure 5 shows the pattern of ammonia distribution immediately following injection using the system parameters of Donaldson. The time axis is essentially an expansion o f that in Fig. 3, and the response curve corresponds to that for VF = 3 and I = 5 in Fig. 3. The initial distribution and dilution o f ammonia throughout the system is very rapid and a pseudo steady state value o f ammonia concentration is apparent after approximately 0.2 h. The responses of well mixed recirculating systems to a step disturbance (see Table 2) in ammonia production rate are plotted in Fig. 6, which illustrates the effect of increasing fish tank size relative to filter size upon the system response. The size of

TABLE 2 Response to step disturbance in ammonia production rate-influence of fish tank volume Filter volume (VB) = 29.45 litre Recirculation rate (F) = 84 litre h -1 Rate constant (k) = 0.35 h -~ Fish tank volume V F (litres)

Step change R F (rag N H 3 h- ~ litre-')

10 50 100 500

1.67 8.33 16.67 83.33

8

Fig. 6.

2

l 4

I 6

I 8

I 10

I 12

I i4

I 16

I 18

I 20

TIME,h (c) V F = 50, R F = 8.33; (d) V F = 100, R F = 1.667.

Resp•nset•stepchangeinamm•niafeedrate•VB=29•45;F=-84•k=••35•(a)VF=5•••RF=83•33;C•)VF=••••RF=•6•67;

0

0.2

0.4

E 0.6

o.8-

I 22

I 24

(a

r-

t'-'

N

(c)

(bl

g

/

.c

1.0-

~- 1 . 2 E

"7

1.6-

(a)

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

107

the input disturbance was scaled in accordance with the tank size to maintain the change per unit volume at a constant value.

DISCUSSION OF RESULTS

Experimental points measured by Donaldson ( 1 9 7 9 ) - see Fig. 2 - show excellent agreement with the curve calculated using eqn (12) for response to impulse disturbance. Assumption of ideal mixing in both fish tank and filter, inherent in the model, are valid which is not surprising as Donaldson's 'fish tank' was an air lift system which is well mixed. Measurement of ammonia concentrations along the axis of the filter indicated a flat profile also indicating excellent axial mixing within the bed. Donaldson's system was small compared to a full fish rearing system, and the effects of dynamic lag in the connecting pipe-work are very small compared to the time constants of filter and 'fish tank'. A second major difference is in relative size of fish tank and filter. A realistically sized fish tank employed in conjunction with the 30 litre filter considered here may not have such favourable mixing characteristics as assumed in the derivation of eqns (10) and (12). It is thus argued that experimental measurement of fish tank mixing characteristics must be an essential ingredient of model validation and refinement. On this assumption the ability of the biological filter to respond to changes in a much larger fish tank is shown in Fig. 3, where a family of curves is plotted showing ammonia concentration response to impulse disturbance in fish tanks of increasing size. The concentration response over a period of 24 h is shown, although clearly it is the length of the period for which the ammonia level remains high which is critical in deciding the acceptability of a particular response characteristic. It is important to "notice that the fall in concentration following the initial disturbance is due to initial mixing and distribution of ammonia throughout the whole system, and due to the action of the biological t~ilter. The distinction between these two effects is demonstrated by closer examination of the responses at small values of Vv. The response shown in Fig. 3 for VF = 3 shows a very rapid reduction in the value of CNH3 from 1.67 mg litre -1 to approximately 0.15 mg litre -1. This is demonstrated more clearly in Fig. 5 where the same response is plotted as the top curve with a greatly expanded time axis. The lower curve shows the increase in concentration experienced in the filter due to initial mixing, this value converging with the value for the fish tank after a time of approximately 0.2 h. In this case the dilution of ammonia injected is very rapid compared to the rate of removal by filter action shown by the relatively slow decay in concentration in Fig. 3. As the relative size of fish tank to filter increases, the dilution effect becomes much less marked and the reduction in concentration is primarily controlled by filter action, denoted by smoother characteristics; see Fig. 3.

108

L.R. WEATHERLEY

The responses to impulse disturbance at various recirculation rates are shown in the family of curves plotted in Fig. 4. Two systems are studied, each having a 30 litre filter and fish tank volumes of 50 litres and 500 litres, respectively. Curves for the 50 litre system show that ammonia removal is enhanced as the recirculation rate is increased, but the degree of enhancement falls off asymptotically as higher flow rates are reached. The predictions, as before, do not account for nonidealities in mixing behaviour and the data shown represent limiting cases, and indeed the occurrence of non-ideal mixing is much more likely at the low flow-rates indicated. Thus, the enhancement experienced in practice at increased recirculation rates is likely to be greater than predicted by this method. Responses to step-wise disturbances in ammonia addition rates to the fish tank are shown in Fig. 6, and show the level to which the concentration of ammonia is increased as a consequence of changes in production rate. The values of major interest are the steady state values and these may be readily calculated from eqns (18) and (19) as t ~ oo. The family of curves show directly the ability of a fixed size filter to respond to a step change in ammonia input to the system, and clearly, as the gross amount of ammonia being fed increases, the steady state values of ammonia concentration rise in accordance with the ability of the filter to remove the ammonia. The accurate use of eqns (10)-(19) to predict the response in real systems depends upon the validity of the assumption of ideal mixing in the two major vessels considered. While ideal mixing may be readily approached in a coarse gravel filter such as used by Donaldson (1979), that of the fish tank is much less likely to be so. The mixing r6gime within a fish tank, as in any other vessel, may be qualitatively described by the residence time distribution (Danckwerts, 1958; Bischoff and Levenspiel, 1963) and measured using tracer response techniques. Detailed information of this nature is an essential prerequisite to model refinement. A further refinement would be the inclusion of transportation lags allowing for the finite time required for flow of liquor between vessels which may be quite significant in long pipe runs and at reduced flow-rates. The crudest way of achieving this adjustment is described by Coughanowr and Koppel (1965), and requires rewriting eqns (1) and (2) on different time bases. It is clear that Donaldson's experiment involved insignificant time lag between different parts of the system, and the match of experimental points and predicted results bears this out.

CONCLUSIONS

In principle the flow configurations of recirculating aquaculture systems are well suited to the application of response analysis techniques. Accurate predictions are subject to the availability of kinetic data, tank and filter mixing data and precise knowledge of the types of disturbance encountered in real systems.

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

109

The work presented here shows that given idealised disturbance conditions and ideal mixing, accurate predictions of the changes in ammonia concentration throughout the system can be made. The predictive techniques discussed here illustrate the potential of response analysis, subject to further ref'mement, as a tool in the design and operation of recirculating aquaculture systems. Obvious refinements would include the incorporation of qualitative non-ideal mixing models into the analysis, an updating of kinetic data linked to known changes in the fdter bacterial population and consideration of transportation lag between major items.

REFERENCES Atkinson, B. (1974). Biochemical Reactors, Pion Press, London. Bischoff, K. B. & Levenspiel, O. (1963). Patterns of flow in chemical process vessels. Adv. Chem. Eng., 4, 95-198. Coughanowr, D. R. & Koppel, L. B. (1965). Process Systems Analysis and Control, McGraw-Hill, New York. Danckwerts, P. V. (1958). Continuous flow systems. Distribution of residence times. Chem. Eng. Sci., 2. 1-13. Donaldson, D. (1979). Construction o f a Marine Experimental Biological Filter and its Nitrification Kinetics, Heriot-Watt University Honours Research Report, Edinburgh, UK. Hirayama, K. (1974). Water control by filtration in closed water systems. Aquaculture, 4, 369-85. Liao, P. B. & Mayo, R. D. (1972), Salmonid hatchery water reuse systems. Aquaculture, 1, 31735. Mantle, G. (1981). Simulation Model o f Closed Water Circulation System for Fish Culture, Applied Biology Research Unit Research Report, The Open University, Milton Keynes, UK. Rosenthal, H. (1980). Recirculation systems in western Europe. Proc. World Symp. on Aquaculture in Heated Effluents and Reeirculation Systems, Vol. II, Berlin, pp. 305-15. Speece, R. E. (1973). Trout metabolism characteristics and the rational design of denitrification facilities for water reuse in hatcheries. Trans. Am. Fish. Soe., 102 (2), 223-34. Srna, R. F. & Baggaley, A. (1975). Kinetic response of perturbed marine nitrification systems. J. Water Pollut. Control Fed., 47,472-86.

APPENDIX

Mass balances on the fish tank

Figure A1 shows the nature of the disturbances in ammonia production rate studied in this paper and with reference to the simple flow diagrams the unsteady state mass balance equations for ammonia may be derived. Impulse disturbance. In time element dr.

Ammonia in = inflow from filter + impulse addition = Fe o dt + A 6 ( t ) dtVF

1l0

L. R. WEATHERLEY

Ammonia production rate (A ~ (t ])

Impulse Disturbance

To filter Ammoniaout ( F c i d!)

Ammonia in (AS(t)dtVF)

]Fish tank [ e (V F)

.~lvolum

t

TAmmonia in (FCodl) -4 From filter Step Disturbance

Ammonia production rate rFss

To filter ~" Ammonia out (Fc, d; )

fF --

RF

Ammoni p r oVdr duatc t i o n r F :~ Lv°Fishumetank( VF)

rFss [Ammonioin (Fc o d/ ) /

1

~ "

Fromfilter

Other Disturbances Ammonia production rate

To filter

~ FF =

Ammonia out ( F c i d! )

(r F ]

Ammoniaproduction [,, r (r F VFdI ) Fish tank ~ [volume ( VF )

rFss t

lAmmonio ~ in (FCodt) From filter

Fig. A1.

Mass balance on fish tank.

Ammonia out = outflow to filter

= Fci dt Therefore, nett accumulation of ammonia = F ( c o - - ci) dt + A ~ ( t ) dtVF. The change in ammonia concentration in the tank, dci, thus caused is given by VF dc i =

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

111

F(co --ci) dt + A6(t) dtVv, i.e. dc i

Vv ~-t = F(c° -- el) + Af(t) Vv

(21)

Since we are primarily concerned with the deviations from some initial steady state, the concentrations of interest are conveniently expressed as deviation variables. Co -- Co - Coss

(22)

Ci = cl -- Ciss

(23)

where Coss and Ciss are the initial steady state values of influent and effluent concentrations, respectively. By definition, Coss =Cis s

dCoss

and

dt

-

dCiss dt

- 0

(24)

and dCo

-

dt

dco

(25)

dt

dCi

dei -

dt

(26)

dt

Substitution of eqns (22)-(26) into eqn (21) then yields:

dC~ Vv dt = F ( C o - - C i ) + A6(t) Vv

Fromfishtank

r / Ammoniain(F cidt)

JBiologicalfilter ] volume V'B)

~ Ammoniaoxidation ~

( V8 k Co dt )

Ammonia out ( F ¢o dt } To fishtank

Fig. A2.

Mass balance on biological filter.

(1)

112

L.R. WEATHERLEY

Step disturbance. In time element dt.

Ammonia in = inflow from filter + ammonia produced = Fe o dt + VFr F dt

Ammonia out = outflow to filter = Fe i dt

Therefore, nett accumulation of ammonia = F(eo -- el) dt + Vvrv dt. As before, the accumulation of ammonia is manifest as a change in concentration in the tank VF dei. Thus, dei VF ~ t = F(ei -- e ° ) + VFrF

(27)

The change in ammonia production rate is defined by the deviation variable RF = r F - - rFs s-

For the steady state values ofei, %, rF becomes: F(eis s --Coss) + Vvrvs s = 0

(28)

Combination of eqns (27) and (28) yields: dcl VF ~ t = F [(e i -- eiss) - - (Co - - eoss)] + V F ( r F - - r F s s )

i.e. in terms of deviation variables: dCi Vv ~-t = F ( G - Co) + VFRF

(13)

Other disturbanees in ammonia produetion rate. The treatment for a step disturbance in ammonia production rate is general for other time varying disturbances in ammonia production rate, hence eqn (20):

dCi Vv ~ t = F(Co -- G ) + VF ~9(t)

(20)

Mass balances on the biological filter With reference to Fig. A2 the unsteady state mass balance for ammonia across the filter is of the same form irrespective of the nature of the disturbance in the fish tank. Thus, in time element dt:

Ammonia in = inflow from fish tank = Fcl dt

APPLICATION OF DYNAMIC RESPONSE ANALYSIS

1 13

Ammonia out = outflow to fish tank + ammonia oxidised = Fc o dt + kc o V B dt

Accumulation = VB dci Therefore, dci

VB ~

= F ( c i - - Co) - kco VB

(29)

In terms of deviation variables Ci, Co, see eqns (22) and (23), (29) may be rewritten:

dC~ VB --~t = F ( Ci -- C°) -- kC° VB

(2), (14)