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A rearing model for salmonids McLean, William Eric 1979

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A REARING MODEL FOR SALMONIDS by WILLIAM ERIC McLEAN B. Sc., University of British Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Interdisciplinary Program) The Departments of Bio-Resource Engineering and Civil Engineering. We accept this thesis as* conforming to the required standard ( P a c i f i c B i o l o g i c a l Station, Nanaimo) ( C i v i l Engineering) (Bio-Resource Engineering) THE UNIVERSITY OF BRITISH COLUMBIA August, 1979 (c) William Eric McLean, 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e DE-6 BP 75-51 1 E i i ABSTRACT The rearing of salmon fry for release to the ocean stands out as the most complex, expensive, and critical feature of hatchery production. This complex process not only involves decisions concerning space, food, and water flow requirements, but also involves choosing optimum release times and sizes and management strategies. For a rearing program to be consistently successful, decision making must be based on some sort of rational understanding of how these require-ments are influenced by the rearing environment. At present, this under-standing is often based on intuition and site specific experience. A formalized model developed from fundamental knowledge and pooled experience can augment the present knowledge base. This thesis presents a framework within which the rearing process can be understood more clearly. Where reliable data or basic physiological understanding exists, simple deterministic models have been developed. These models quantify the relationships between the most important rearing requirements and the web of environmental factors which affect them. The key environmental factors have been identified as water temperature, ration level, time, and the degree of hatchery disturbance. Under normal hatchery conditions these factors have been used to predict fish growth or weight at a particular time by means of a generalized growth model. i i i This information has been combined with environmental factors to determine oxygen consumption and metabolite production rates. The meta-bolites considered are ammonia-N, un-ionized ammonia-N, carbon dioxide, and suspended solids. Translating these consumption and production rates into predictions about pond environmental conditions is an important element of the model,.,because i t is this information which can be used directly by decision makers (design and operations personnel). This has been achieved by developing a simplified picture of flow through a rearing pond; ponds have been assumed to approximate either ideal "plug flow" or ideal "mixed flow" type reactors. Using this approximation plus background water quality information, a model of the distribution and concentration of metabolites and oxygen within the rear-ing pond has been developed. The combined equations can be used to predict fish weight, pond density and oxygen and metabolite concentrations over the rearing period. Conversely, i f constraints are placed on these factors, space, flow, and ration requirements can be similarly predicted. Realistic, constraints for salmon culture have been discussed briefly. These reflect some of the most recent information on the effects of time and size at release and pond environmental conditions on smolt viability. It is emphasized that the primary function of the "Rearing Model" is to provide a framework of predictive relationships; i t is not designed to set guidelines for rearing. iv TABLE OF CONTENTS Page ABSTRACT 11 TABLE OF CONTENTS iv LIST OF TABLES v i LIST OF FIGURES v i i ACKNCWLEDGEMENTS x i CHAPTER I INTRODUCTION .. 1 II THE REARING PROCESS 5 III A REARING MODEL 8 A. Introduction 8 B. Some Guidelines for Rearing 8 C. Model Formulation 16 D. Individual Components 24 1. Background and Initial Conditions . . . . 24 a. Water quality 25 b. Fish culture and rearing pond information . 27 c. Assumptions 27 2. Pond Flow Characteristics 28 3. Fish Growth 35 a. Development of a growth model . . . . 35 b. Solutions of the growth equations . . 40 c. Comparisons with measured values. . . 43 d. Application of the growth model . . . 46 4. Oxygen Concentration 47 a. Development of an oxygen model. . . . 47 b. Comparisons with other models . . . . 53 c. Comparisons with measured average daily oxygen consumption rates. . . . 57 d. Development of a safety factor. . . . 59 e. Application 62 5. Carbon Dioxide and pH 65 a. Carbon dioxide 65 b. pH changes 65 V Page 6. Ammonia-N 66 a. Introduction 66 b. Ammonia-N model components 68 c. Comparisons with other models . . . . 71 d. Comparisons with measured values. . . 73 7. Un-ionized Ammonia-N. . 78 8. Suspended Solids 79 a. Review 79 b. Theoretical solids model 80 c. Solids discharge rates and comparisons with other models . . . . 82 9. Density 86 IV SYNTHESIS OF MODEL COMPONENTS 92 a. Illustrative example 92 b. Application 96 V DISCUSSION AND CONCLUSIONS 108 BIBLIOGRAPHY 116 APPENDIX I 120 APPENDIX II 121 APPENDIX III 122 APPENDIX IV . 124 APPENDIX V 127 APPENDIX VI 129 APPENDIX VII 133 v i LIST OF TABLES TABLE Page 1 General release weights and dates and rearing periods for each of the four species of salmonids cultured in Federal and Provincial hatcheries in British Columbia 1 2 Model relating factors identified in the "conceptual framework" 21 3 A comparison of predicted and actual weights for hatchery coho on a reduced ration 45 4 Mean percentage differences between predicted oxygen consumption rates and measured average daily values. . . 58 5 Abrupt dissolved oxygen drops resulting from typical hatchery disturbances to crowded rearing ponds . . . . . 6l 6 Ammonia-N excretion rates for starved fish; model projections vs. actual measurements. The proposed ammonia-N model is represented by equation 54 77 7 Model predictions of suspended solids output rates at a feed rate of one percent per day and various temperatures and fish weights. Equations 67 and 72 are sensitive to fish weight and temperature whereas equations 63 and 66 depend only on the feeding level . . 84 8 Typical fish densities at a number of British Columbia hatcheries. These values have been plotted on the density vs. weight curves shown in Figure 26 91 9 Basic parameters (independent variables) and Factors (dependent variables) required to characterize the rearing process at a given time 98 10 A comparison of results as a function of the method used to solve the growth equation; % error=P-E x100%, c = correction factor E 126 v i i LIST OF FIGURES FIGURE Page 1 A schematic representation of the rearing process. Blocks represent model components; l i n e s represent information flow. 6 2 Dissolved oxygen c r i t e r i a f o r freshwater rearing developed by Davis (1975). Minimum oxygen con-centrations to meet protection levels A, B, and C have been plotted against water temperature. 10 3 Effect of migrant weight on s u r v i v a l rates f o r four species of salmon reared at Big Qualicum Sainon Development Project (F. J. Fraser, pers. comm.) 13 4 Y i e l d of adult biomass from different sized smolts (gm) released i n A p r i l , May.j and June (B i l t o n and Jenkinson, 1978) 14 5 Percent survival vs. smolt weight at release f o r hatchery reared steelhead i n C a l i f o r n i a , Oregon, and Washington (Royal, 1972) 15 6 The c y c l i c chain of events required to develop a model - t h i s i s referred to as a "Policy Validation Cycle" by Van Gigch (1974) I 8 7 Contrasting temperature regimes of two hatchery water supplies: A, Chilliwack River; B, Cold Creek. . . 26 8 The oxygen consumption rate as a function of the environmental oxygen l e v e l . Above the "in c i p i e n t l i m i t i n g tension" the oxygen con-sumption rate i s independent of the environ-mental oxygen l e v e l (Davis, 1975) 31 9 Metabolite concentrations within i d e a l plug flow and i d e a l mixed flow type ponds. The idealized concentration " c 0 " has been plotted against the pond length (measured from inflow to outflow); a background metabolite concentration of "c^" has been assumed 33 v i i i FIGURE Page 10 The concentration profile at the pond outflow after injection of a "slug" of tracer. "A" represents an ideal plug flow reactor and "B" represents an ideal mixed flow reactor 34 11 Concentration profile for non-ideal tanks. "A" rep-resents extreme short circuiting and "B" represents parallel paths 34 12 Fraction of the maximum specific growth rate as a function of the ration level: values shown have been calculated for 10 gm fish at a water temperature of 10°C. The maintenance ration "Fma" gives zero growth while the maximum ration "Fmax" yields max-imum growth (l.O). The relationship has been described by a sine curve 37 13 Specific growth rate x 100 for 1 gm fish at the maximum ration as a function of water temperature. The relationship is described by a polynomial 39 14 Percent error between the weights predicted by the incremental method (P) and the exact method (E) as a function of the exact weight (% error =(P-E)/E X 100%). Predictions were made using one or five day time in-crements; weight projections were made assuming an in i t i a l weight of one gram, a maximum ration and. a constant temperature of 10°C 42 15 Comparison of predicted and measured growth at reduced ration levels. Ninety-five % confidence intervals of selected measured values are shown 44 16 Weight projections over a 170 day rearing period. Predictions have been made at three different ration levels and a hypothetical temperature regime 48 17 Oxygen consumption rate as a function of the feeding level. In this case, the feeding level is expressed as the fraction of the maximum ration. The linear relationship is shown for two different fish weights . . . 52 ix FIGURE Page 18 Average daily oxygen consumption projections at 10°C vs. fish weight. A number of different models have been used. The present model (equation 33) has been calculated at f u l l ration (f = l) and at half of the maximum ration (f - 0.5) 56 19 Oxygen consumption rate pattern for a typical undisturbed hatchery rearing pond over a 24-hour period. This curve was measured for coho smolts in April; the average daily temperature was 9.5°C . . . 60 20 Water flow requirements per 1,000,000 ten gram fish as a function of temperature. Maximum ration and 60% ration projections have been made; an average daily oxygen level of 8 mg/l has been maintained in the pond effluent. 64 21 Effect of load rate on pond outflow pH levels. Projections were made at a temperature of 11.7 °C, a fish weight of 9.1 gm, and feeding levels of 60% of the maximum ration. A background carbon dioxide concentration of 1.0 mg/l and an alkalinity of 40 mg/l as CaCO^  were assumed. , . 67 22 A simplified schematic representation of the flow of nitrogen in the rearing process , 69 23 Ammonia-N excretion rates vs. feeding levels for 15 gm fish at a water temperature of 15°C The proposed model (equation 54) is shown as a dashed line. Other models are labelled by their equation (eq.) number 74 24 Average daily ammonia-N excretion rates vs. pre-dicted values over a period of major pond dis-turbances (April 14, 15) and over a period of stability (April 20 - May 5) 7 5 25 Suspended solids output rates as a function of the feeding level. Predictions have been made for 10 gm fish at a water temperature of 10°C 83 26 Density as a function of fish weight as predicted by various theoretical models. Actual densities from Table 6 have been displayed and labelled by number. . . 90 X FIGURE Page 27 Growth projection assuming an i n i t i a l weight of 0.3 gm. The temperature regime and ration level are also indicated: the notation "f = 0.9" indicates that 90% of the maximum daily allowance is being fed. A marking program at day 230 is assumed to c suppress growth for a short time period 99 28 Maximum load rate and minimum water requirements -per 1,000,000 fish to meet "Level A" oxygen criteria developed by Davis (1975) 100 29 Maximum load rate and minimum water requirement per 1,000,000 fish to meet "Level B" oxygen criteria developed by Davis (1975) . . 101 30 Un-ionized ammonia-N concentrations in the pond outflows; ponds are loaded to meet either Level A or B oxygen criteria 102 31 Total ammonia-N concentration in the pond outflows; ponds are loaded to meet either Level A or B oxygen criteria 103 32 Carbon dioxide concentrations in the pond outflows; ponds are loaded to meet either Level A or B oxygen criteria 104 33 Suspended solids concentrations in the pond outflows; ponds are loaded to meet either Level A or B oxygen criteria 105 34 pH of pond inflow and outflows; ponds are loaded to meet either Level A or B oxygen criteria 106 35 Volume requirements per 1,000,000 fish over the rearing period: conservative density criteria based on experience have been assumed 107 36 Oxygen concentration in the pond outflow as a function of the load rate. As the pond dissolved oxygen level falls below "Cr", the oxygen uptake rate is suppressed by the reduced environmental oxygen level. The "solid" line incorporates this effect while the "dashed" line assumes that the uptake rate is independent of the oxygen concentration . 112 x i ACKNOWLEDGEMENTS The author i s grateful to Dr. J. R. Brett of the Fisheries and Marine Service, Professor S. 0. Russell of the Department of C i v i l Engineering, and Professor J. W. Zahradnik of the Department of Bio-Resource Engineering for their guidance and encouragement. Comments and criticisms from their diverse areas of expertise have been invaluable in making this interdisciplinary study possible. Many of the physiological relationships which this work has been based on have been investigated by Dr. J. R. Brett and his colleagues at the Pacific Biological Station. Special thanks are due to this group for their generosity in providing information and support. The f i e l d measurements and observations cited i n this thesis were made at a number of production f a c i l i t i e s ; however, intensive monitoring programs were carried out at the Quinsam and Big Qualicum hatchefies. The author would like to extend his gratitude to managers Jim Van Tine and Dick Harvey for their cooperation, during the pre-liminary stages of this work. Operations biologists Don Sinclair, Dave Wilson, and Ted Perry of Fisheries and Marine Service have also been very helpful. Their input has been extremely valuable i n defining the problems related to the design and operation of production rearing f a c i l i t i e s . Special acknowledgement i s made to Miss Crystal Spicer for her care in preparing the f i n a l draft of the thesis. 1 I INTRODUCTION The production of salmonids for ocean release frcm British Col-umbia hatcheries has reached 17*000,000 fish per year (Fisheries and Marine Service, 1978), With the implementation of the Salmonid En-hancement Program this figure should increase to 68,000,000 fish per year by 1985. The most expensive and critical feature of hatchery production is the "Rearing Process". Simply stated, this process in-volves feeding fry to the release or smolt stage. Table 1 summarizes major aspects of the rearing process for various species. TABLE 1. General release weights and dates and rearing periods for each of the four species of salmonids cultured in Federal and Provincial hatcheries in British Columbia. Species Initial Weight (gm) Final Weight at Release (m) Release Date Approximate Rearing Period (days) Coho 0.30 15 - 30 May-June 400 Steelhead 0.25 40 - 60 May-June 400 Chinook 0.50 4-10 June 100 Chum 0.25 0.8 - 1.5 April-May 60 Hatchery rearing is carried out in order to increase the ocean survival of the released smolts. It is now generally felt that ocean survival is to a large extent, dependent on the rearing strategy; op-timum survivals occur i f healthy smolts of the "right size" are released at the "correct time". 2 A r t i f i c i a l rearing activities take place under a tremendous variety of background conditions. Temperatures vary from 0.5°C to 25°C. Some systems are characterized by nearly year round constant temperature re-gimes while others have extreme temperature fluctuations. Water quality is also variable between systems. The alkalinity, pH, solids, ammonia-N, carbon dioxide, hardness, metal concentration and disease incidence are often site specific. Many rearing pond types are possible. Some of the pond parameters which can be varied are dimensions, shape, flow, outflow and internal baffle structures and construction material. Concrete ponds usually rely on a standard hatchery diet as the major food input while some gravel bottom ponds may have significant inputs of natural food. At the onset of a new rearing program, hatchery designers, opera-tions biologists, and fish Culturists usually have only background in-formation such as temperature, water quality and disease monitoring data available. With this information, optimum pond designs and rearing strategies must be developed. Decisions faced by designers involve specification of: a) water flow requirements; given water quality, disease back-ground and production targets. b) space (volume) requirements. c) construction material, inflow and outflow structures, pond shape, flow measurement and alarm systems. d) inflow water pretreatment requirements - temperature or pH may have to be manipulated for example, before production targets can be met. 3 e) flexibility in pond operation - for example, recycling pond effluent may be feasible. f) effluent treatment requirements. The operation of a rearing facility and achievement of production goals involves daily decisions concerning: a) choice of ration level to achieve a smolt of a given weight by a given release date. b) allocation of limited space and water flow resources among different groups of fish - for example, a multi-species hatchery may have to deal simultaneously with coho fry (0.4 gm), coho smolts (20 gm), Chinook fry (0.6 gm), steelhead fry (0.3 gm), and steelhead smolts (40 gm). c) effects of fish culture practices and the hatchery environment on disease susceptibility and the condition of the fish. d) effects of background conditions on general health and growth rates. Designers and fish culturists inevitably face these concerns during the implementation and operation of a large scale fish culture program. Often, intuition and experience must be the sole basis of decision making. This is especially true where fundamental understanding or critical data are lacking. A rearing model attempts to identify and integrate what is fundament-ally known about the growth and survival of fish in the fish culture envir-onment; i t must then express these relationships quantitatively. This ex-ercise attempts to augment rather than replace the intuition and experience of the hatchery manager or design biologist. A model is based on prin-ciples common to a l l rearing processes. It should interpret the perfor-mance of large scale production units in terms of fundamental knowledge. 4 Because many data voids inevitably exist in complex real l i f e situations, intuition and experience are critical in final decision making. However, with the research and development of the past twenty years enough biological knowledge .exists to develop a simplified rearing model. This model wil l provide, at least, a framework around which designers and fish culturists can make quantitative projections. It should also help identify data voids to research personnel. 5 II THE REARING PROCESS The primary objective of any hatchery rearing program involves the release of an optimum number of healthy fish at a certain size and time. The factors affecting this process are numerous and have been thoroughly reviewed by Klontz (1978) • A few of the main factors are water temp-erature regime, water quality, pond flow, pond volume, pond dimensions, ration level, and ration quality. Fish culturists have through exper-ience, learned how these factors interrelate qualitatively. The process of modelling wil l attempt to analyze Rearing as a system of components; relationships between components w i l l then be expressed quantitatively. A simplified view of the rearing process is presented in Figure 1. On a given day against specified environmental conditions, a ration in-put results in metabolic activity. A weight increase may result, fish density (biomass/vol.) may increase, oxygen w i l l be consumed and waste products wil l be produced. These changes vary with time as background conditions and fish weight vary. In Figure 1, each box represents a set of relationships whereas lines represent the information flow re-quired to interrelate each component. The pond condition at a particular time represents the overall state of the rearing system. Fish culturists must interpret these conditions in light of the primary objectives of the rearing operation. For example, i f a weight release target of 20 grams at 400 days has been set and the actual weight at 200 days is 7 grams, a judgement must be made in light of anticipated background conditions. Ration levels or water temperatures may have to be adjusted. Background and Initial Conditions temp, regime species number in i t i a l weight water quality disease back-ground Growth Ration pensitf tkyged A U ^ I Uh-ioniped Suspended KN Bolids arbon bioxid Pond Condition Ammonial-N f > t \t \ f Pond Flow Characteristics > f FIGURE 1. A schematic representation of the rearing process. Blocks represent model components; lines represent information flow,. 7 If tin-ionized ammonia levels are judged too high, a fish culturist may consider one of the following actions: reducing ration level, changing ration composition, increasing flow, decreasing numbers of fish, decreas-ing the water temperatures, decreasing pH. Here again, judgements must be made in light of the primary rearing objectives. Primary objectives can be restated as follows. The goal is to release fish: 1. at a particular weight 2. at a particular time 3. in such numbers (n) that the returns are maximized. The number returning can be expressed as the product of n times the survival rate. Therefore, the objective is to maximize returns "R": R = n x s(n, wr, tr) (l) where: n = number of smolts released wr = weight at release tr = time of release s(n,wr,tr) = the fraction surviving; which, from a facility with limited water and space, is itself a function of the number of smolts released. It should be noted that a model must be able to predict pond conditions given certain background information and ration levels. However, i f constraints on the pond environment are made, then the model should be able to specify limits on background conditions. Because of this, background and i n i t i a l conditions should not be thought of as fixed. Often, water temperatures can be manipulated to some degree. Also, the number of fish, water flow, and water quality can sometimes be altered in attempting to maintain a satisfactory pond environment. 8 ' III A REARING MODEL A. Introduction The rearing process will be considered as a system of interacting components. The choice of components is somewhat arbitrary; however, they are meant to reflect key factors which are known to dramatically affect the success of rearing programs. Model components and their interactions are schematically presented in Figure 1. The main focus of the thesis is to determine quantitatively how the various compon-ents interact in large-scale rearing operations. For instance, the model should be able to predict whether an increase in temperature would cause a decrease in pond oxygen levels, and i f so, by how much. Of course, the model cannot judge whether this is undesirable; i t is only designed to be a predictive tool. The judgement as to whether specific environmental conditions are desirable or not depends on the perception and experience of the fish culturist or operations biologist. In order to put specific oxygen levels, ammonia concentrations, growth rates, etc., in perspective, a brief review of some useful rearing criteria will be made. B. Some Guidelines for Rearing In order to meet rearing program objectives (page 7), a salmon culture operation must pay attention to environmental conditions, growth rates and the timing of smolt releases. The oxygen concentration has long been recognized as a key envir-onmental factor in determining satisfactory pond conditions. It is sug-gested that oxygen criteria developed by Davis (1975) be adopted for hatchery use. '9 Davis ( 1 9 7 5 ) has thoroughly reviewed the literature and derived realistic minimum oxygen requirements based on sublethal responses. Briefly, these criteria are based on establishing both a sufficient oxygen tension (partial pressure in units of mm of Hg) and a sufficient oxygen content (mg/l). The oxygen tension establishes a pressure gradient for driving oxygen across the g i l l s , whereas the oxygen con-tent f u l f i l l s the needs of fish metabolism. Cxygen criteria for fresh water salmon rearing are shown in Figure 2 as a function of temperature. The three protection levels reflect the fraction of the population exhibiting effects of reduced dissolved oxygen. If levels A, B, or C are met, then 1%, 50%, and 8 5 % respectively, of the individuals in a population would be affected. It should be noted that a level of safety is exceeded i f the oxygen concentration drops below the established c r i -teria for more than a few hours a day. Ammonia-N is the major end product of protein catabolism in fish and so is always present during rearing activity. The un-ionized form is recognized as a powerful cellular poison even at low concentrations (Fromm and Gillette, 1968). Burrows (1964),using Chinook fingerlings, demonstrated sublethal effects after prolonged exposure to un-ionized ammonia-N concentrations of 3 ug/l. The Environmental Protection Agency (1972) recommends an upper limit of 20 ug/l while Westers (1976) suggests an upper limit of 12 ug/l for salmon rearing. Recent work has indicated that 2 ug/l un-ionized ammonia caused reductions in the growth rates of pink salmon alevins (D. F. Alderdice, pers. comm.) Establishing a safe level is difficult; however, a limit of 2 ug/l is tentatively suggested (Sigma Resource Consultants Ltd, 1979). 12 3-2 i i • I i > i i i i i i 1 1 1 1 — — i 1 1 1 0 1 2 3 k 5 6 7 8 9 10 11 12 13 Ik 15 16 17 18 19 20 Tempterature (°C) FIGURE 2. Dissolved oxygen criteria for fresh water rearing developed by Davis (1975). Minimum oxygen ^ concentrations to meet protection levels A, B, and C have been plotted against water temperature. o 11 The fraction of the total ammonia-N in the un-ionized form depends on the pH and temperature. Pond pH levels, however, are directly affected by the carbon dioxide production of the fish. It is this interaction between ammonia-N and pH that gives the carbon dioxide component of the model its importance. Levels of carbon dioxide or pH shifts caused by the carbon dioxide are generally not of sufficient magnitude to affect rearing operations. However, carbon dioxide levels should be kept below 25 mg/l and pH levels should be maintained between 6.5 and 8.5 (Sigma Resource Consultants Ltd., 1979). There is seme evidence that carbon dioxide and pH may have subtle effects ©n the toxicity of the un-ionized ammonia itsel f . Lloyd and Herbet (i960) found that un-ionized ammonia toxicity increased when the carbon dioxide concentration increased from 3.2 mg/l to 48.0 mg/l (pH dropped from 8.2 to 7.0). This suggests that when comparing rearing environments, un-ionized ammonia, carbon dioxide, and pH information are required. Waste food and feces cause increases in the suspended solids con-centration of the rearing environment. Meaningful criteria are difficult to establish because the effects depend on the physical and chemical nature of the solids. The Environmental Protection Agency (1972) recom-mends a limit of 25 mg/l. However, personal observation and communication with fish culturists suggests that "fine" waste food particles cause sig-nificant g i l l irritation at concentrations far below 25 mg/l. 12 These factors along with the density (number or weight of fish per unit volume) and water velocity determine the condition of the rearing environment. Fish culture experience has taught that high environmental quality increases the probability of a successful rearing operation while a poor environment can be lethal. It should be pointed out, however, that during normal operation, environmental quality rarely degenerates to the point of causing outright mortalities. For example, only in the case of an accident (pump failure) would oxygen ever drop to a lethal level*. More often, the rearing environment is looked on as exerting some degree of sublethal "stress". These stresses, depending on their sev-erity, can precipitate disease outbreaks (Wedemeyer, 1974). As would be expected, this indirect mechanism is impossible to quantify at this time; as experience with ocean released fish is accumulated, a probabil-ist i c relationship between the quality of the rearing environment and the smolt viability will eventually be established. Insight into optimum growth rates and release times is slowly being gained. It i s now generally felt that ocean survival i s , to a large extent, dependent on the rearing strategy; optimum survivals occur i f smolts of the "right size" are released at the "correct time". Obser-vations made at the Big Qualicum River Salmon Development Project (F. J. Fraser, pers. comm.) over a number of years illustrates in a general way, the relationship between survival and release size (Figure 3). Bilton (1978) has performed extensive, controlled exper-iments with coho salmon and has demonstrated a clear relationship between survival and release sizes and times (Figure 4). FIGURE 3. Effect of migrant weight on survival rates for four species of salmon reared at Big Qualicum Salmon Development Project (F. J. Fraser, pers. comm.). 14: FIGURE 4. Yield of adult biomass from different sized smolts (gm) released in April, May, and June (Bilton and Jenkinson, 1978) Steelhead Smolt Weight (gm) at Release Percent survival vs. smolt weight at release for hatchery reared steelhead in California, Oregon, and Washington (Royal, 1972). 16 Royal (1972) has correlated adult returns for Washington State steelhead with smolt weight at release (Figure 5 )» General release sizes and times cannot be stated; however, at the present time, British Columbia hatcheries are attempting to release salmon smolts near weights and times shown in Table 1. In the next decade, rearing strategy wil l undoubtedly be refined as new information becomes available. This introductory discussion was intended to present the main factors involved in the rearing model and emphasize their significance in the production of young salmon. C. Model Formulation The procedures used to develop the model have been formalized by Van Gigch (1974). These techniques provide a systematic way of trans-forming qualitative statements or observations into reliable quantita-tive projections. In essence, this scheme is an extension of the "scientific method". This traditional sequence of events (problem or event defined; observa-tions made; hypothesis postulated; experiment designed; measurements made; hypothesis accepted or rejected) is viewed by Van Gigch (1974) as a "hierarchy of models"! The The Policy The > The The Numerical Decision Process Theory Experiment v Function ^nd Action Within this new terminology, "the process" involves developing a conceptual framework to describe observations concerning the event or problem being studied. This step must not only define relevant factors but also set boundaries to the problem being considered. 17 The next step in the process involves posing a cogent explanation or theory of the preliminary observations. Building a theory w i l l necessarily involve a number of simplifications and assumptions, but in essence w i l l attempt to describe in a qualitative way at least, how the various factors interact. The derivation of a "numerical function" or mathematical model requires testing of the theory against measurement data or ideally, against the results of controlled experiments. The "experiment model" requires postulating a hypothesis concerning the form of the relation-ship between model factors and a r i g i d statement of the conditions under which the hypothesis i s to be tested. If the hypothesis i s accepted, a mathematical function or model of the data can be developed. This allows the modeller to make inferences and predictions about new situations. Because the consequences of alternative combinations of factors can be projected, Van Gigch (1974) terms this stage of the model "the policy decision and action component". A detailed view of this concept of model development has been presented schematically in Figure 6. Before considering the mathematical model in de t a i l , i t w i l l be useful to take an overview of the rearing model i n light of the cycle presented i n Figure 6. The development of the conceptual framework i s a major task because the problem or phenomenon being studied must be stated i n a way that allows analysis and quantification. This process draws extensively on the fi s h culture literature and on the experience of operations and research personnel. Conceptual Framework Phenomenon or Process Observation Concept Definition Concept > Def initio] Concept Definitioi Theory or Model[ -S± owing Among Factors And Variables Relationship -^Hypothesis ^ Measurement — * — ^ — Yes {Accept Theory as Explanation of] Causal Relationship No Results Testing the Hypothesis Mapping of Property into Numbers No reneralization — r -Prediction I 1 <olicy Decision and Actiony | FIGURE 6. The cyclic chain of events required to develop a model; this is referred to as a "Policy Validation Cycle'! by-Van Gigch (1974). 19 The present framework is built around the concept of smolt survival (s(wr,tr,n)). As stated previously, the objective of a rearing program is to maximize adult returns (n x s(wr, tr, n), see equation l ) . The development of the conceptual framework is summarized below: Premise - the survival rate (s(wr,tr,n)) is influenced by the rearing history of the smolts. Assumption 1 - the most significant aspects of this rearing history-involve: a) pond water quality b) space availability c) time and size at release. This is an important assumption; i t could be argued that exposure to natural food or manipulation of the photoperiod also have important effects on survival. If future work demonstrates such effects, then the present framework would have to be expanded. It should be noted that the mechanisms whereby survival rates are affected by smolt history are not understood at this times. Assumption 2 - rearing pond water quality can normally be charac-terized by concentrations of the following parameters: a) dissolved oxygen b) ammonia-N i c) un-ionized ammonia-N d) pH e) carbon dioxide f) suspended solids g) background water quality. 20 Assumption 3 - space availability is simply related to the volume of the rearing pond. Assumption 4 - the weight of a smolt at a particular release time can be achieved by either temperature and/or ration manipulation. These assumptions and simplifications, along with the previous review of the rearing process, lead to a simple "conceptual framework" involving the following components: (see Figure l) a) background and i n i t i a l conditions b) pond flow characteristics c) fish growth d) oxygen concentration e) carbon dioxide and pH f) ammonia-N g) un-ionized ammonia-N h) suspended solids i) density. If the conceptual framework is accepted, the next step involves postulating relationships between the factors and variables identified in the framework. Table 2 shows model components and postulated re-lationships (theories or models). It should be noted that these theor-etical relationships are stated in terms of undefined functions f^, fg, f^... . Determining the form of these functions and the numerical values of associated constants is an essential step in developing the mathemat-ical model. TABLE 2. Model relating factors identified in the "conceptual framework" Components Postulated Relationships 1) background and i n i t i a l conditions T WQ = f i ( t ) 4 ( t ) 2) pond flow characteristics c = f 3(V, Q, n, w, T, WQ) 3) fish growth w = f 4(w 0, T, P, H) 4) oxygen concentration xo = f 5 ( x i , T, F, w, H, f 3 ) 5) carbon dioxide Co = f6(xo> c i * f3) 6) pH pHo f 7(Co, WQ, T) 7) ammonia-N No f 8(Ni, F, T, w, H, f 3) 8) un-ionized ammonia-N (No)u f9(No, pHo, T, f 3 ) 9) suspended solids So = fl0(^» F» T> w> f3) 10) density D = f l » f^, f3».. = functions T - temperature t = time WQ background water quality co = pond outflow metabolite concentration w = fish weight w o i n i t i a l fish weight <? Q = water flow V = pond volume n = number of fish F = ration level xo outflow oxygen concentration x i = inflow oxygen concentration 22 TABLE 2. (cont'd) Co = outflow carbon dioxide concentration Ci = inflow carbon dioxide concentration H = hatchery disturbance No outflow ammonia-N concentration Ni = inflow ammonia-N concentration (No)u = outflow un-ionized ammonia-N concentration pHo = outflow pH level So = outflow suspended solids concentration Si inflow suspended solids concentration 23 The problem of determining these relationships w i l l be attacked in a variety of ways. The main feature the various approaches have in com-mon is that they involve reliable, numerical information. Such infor-mation has been derived from: a) primary scientific and fish culture literature - much of this information has been collected under controlled conditions (e.g. Brett, 1974) and has often been related to basic physiology, etc. b) technical literature relating to fish culture - this information often reflects previous modelling efforts and usually involves empirical relationships based on data collected at production, hatcheries. c) measured values - this refers to measurements made at a number of production facilities by the author. d) basic theory - this type of information is derived from basic physical, chemical or biological relationships. For example, the relation-ship between the un-ionized ammonia-N fraction and the pH and temperature can be derived frcm basic chemical theory. Often, information from a number of sources w i l l be used to develop a relationship. Assumptions wil l be required] these will be emphasized where they occur in the text. Attempts wil l be made to generalize the relationships; the model w i l l be extended .. beyond the test conditions. Predictions made under new con-ditions w i l l be tested i f possible, by comparisons with measured values. This procedure w i l l be limited by the range and quality of the data available. 24 It must be emphasized that the process of model building is cyclic (see Figure 6). Numerical laws are only accepted until new conditions or new interactions give results that cannot be correlated with measured values; at this stage^, a-, new hypothesis must be proposed and the cycle repeated. The set of fully developed relationships (postulated in Table 2) together with a set of imposed constraints allows a wide range of infer-ences to be made. Seme of the possibilities are explored as the individ-ual components are being developed. In summary, the objectives of the present study are to: a) develop a conceptual framework within which the rearing process can be understood. b) develop quantitative relationships between these factors and related variables. D* Individual Components In this section, model components are described individually. Each set of relationships are developed quantitatively; whenever possible components are justified by comparison with recorded data. Some appli-cations of the isolated components w i l l be demonstrated by reference to realistic fish culture problems. 1. Background and Initial Conditions This complex component includes a variety of factors and assumptions which are prerequisite to any attempt at making quantitative or even qualitative projections. 25 a. Water quality Obviously, water quality must be .satisfactory before a rearing program can be successful. Water quality standards for salmon rearing have been defined in "Summary of Water Quality Criteria For Salmonid Hatcheries" (Sigma Resource Consultants Ltd., 1 9 7 9 ) . Most water supplies have a unique pattern of water quality par-ameters; some of these parameters vary throughout the year while others are stable over long periods. If a supply is generally acceptable, the most important "variable" water parameter is usually temperature. Among existing water supplies, temperature regimes vary widely. The temperature characteristics of a typical groundwater source (Cold Creek) have been contrasted with a surface water supply (Chilliwack River) in Figure 7 » Although both systems w i l l successfully produce hatchery fish, the rearing strategies and pond designs wil l be dramatically affected by the temperature differences. Temperature often appears as an independent variable in subsequent model components and so for convenience, has been described as a function of time by means of a Fourier Series. These concise mathematical state-ments adequately describe the complex yearly temperature fluctuations characteristic of most natural water supplies. The technique is described in Appendix I. As an illustration of this method, Cold Creek and C h i l l i -wack River temperatures have been described by equations 2 and 3» 27 T = 9.53 + 0.2023cos2trt + 0.4419cos4flt - 0.9543sin2irt - 0.8575sin4trt (2) 547 547 547 " 547 (t » 0,Dec 15) T = 7.7056 + 3.9067cos2trt + 0.2278cos4fft + 1.6891sin2trt + 0.9911sin^b (3) 365 365 365 365 (t . 0,Jan 15) where: T = temperature (°C) t = time (days) Disease incidence i s often dependent on the water supply or hatchery site selection. Information about the incidence of disease i n brood stock and i n wild stocks inhabiting the hatchery water supply- often indicates the potential for disease problems during hatchery rearing operations. I t i s well known that hatchery rearing stress usually magnifies disease prob-lems. Background disease information should play an important part in determining a r e a l i s t i c rearing strategy; production targets, temperature regimes, etc., may have to be modified to prevent outbreaks. b. Fish culture and rearing pond information Details about the i n i t i a l weight of the f i s h , species, target weight at release, release time, release number, pond volume, pond flow capacity, and pond shape are often necessary before projections about the rearing environment can; be made. Information about major fish disturb-ances (marking programs, etc.) and pond cleaning routines may also be required. c. Assumptions Model projections usually assume relatively stable conditions: prediction about normal oxygen consumption w i l l have l i t t l e meaning during 28 large scale fish handling operations or during massive disease outbreaks. Also, normal diet quality has been assumed. Modelling efforts can take into account the effects of diet quantity or ration level but can not pre-dict the effects of rancidity or degenerated vitamin levels. A l l project-ions are based on the high quality diets normally used in production hatch-eries. Proximate analysis of some typical production diets are shown in Appendix II. 2. Pond Flow Characterisitcs When considering pond flow characteristics, i t is useful to think in terms of the concepts and terminology of chemical reaction engineering. Within this framework, rearing ponds approximate one of two types of ideal reactors. - Plug flow reactor:: Raceway type ponds approximate this class of reactors. Water is introduced at one end at a steady flow rate. Flow through the reactor is orderly with l i t t l e mixing along the flow path. The residence time in the reactor is the same for a l l fluid elements (Levenspiel, 1972). The mean residence time t is given by: I - V (4) Q> where: t = mean residence time (min) V = volume of reactor (l) Q = flow (l/min). A slug of tracer in the inflow of a pond approximating plug flow con-ditions would also appear at the outflow as a slug (Figure 10). 29 - Mixed flow reactor: Circulating type ponds often approximate this type of reactor. Water is introduced and is uniformly mixed within a short period of time. For a mixed flow reactor, the mean residence time is also given by V/Q. A slug of tracer introduced into the inflow would be quickly mixed throughout the entire volume to give a peak concentration "co". The con-centration of the tracer in the outflow would then begin to decrease expo-nentially (Figure 10). If we consider a rearing pond approximating one of these ideal reactors, then the concentration of a metabolite at the outflow can be predicted from: dc = RD or c 0 = c i + RDt (5) dt Zo HO where: c Q = concentration at outflow (mg/l) C j [ = inflow concentration (mg/l) R = rate of metabolite production or oxygen consumption / mg \ kg of fish*hr D = uniform density of fish over pond volume or total biomass ~ volume (B/v) (kg/l) t = mean residence time (min). If we assume steady flow and no stagnant areas, then t = V/Q, so: c = C ; + RDV 0 1 Q60 substituting D = B gives: V c Q = c ± + RL (6) 6-0 where: L = B/Q, load rate or biomass per flow (kg per l/min). Equation 6 is the fundamental relationship between the effluent metabolite concentration (c Q) and the fish load rate (L). 30 Equation 5 involves a serious limitation; metabolite production or oxygen consumption rates have been assumed to be independent of the meta-bolite concentrations. In the case of oxygen consumption, i t is well known that below a certain concentration (referred to as the incipient limiting tension) the rate of oxygen uptake is dependent on the environmental oxygen level. This relationship has been described graphically in Figure 8. It should be noted that equation 5 is not valid below the incipient limiting tension. In normal chemical terminology, the crit i c a l oxygen level (denoted as "Cr" in Figure 8) represents the transition from zero order kinetics to first order kinetics as the oxygen concentration drops below "Cr". The simple zero order reaction represented by equation 5 becomes: dc = -D(Kc+b) (7) dt oO where: K = slope of the line describing the relationship between "R" and "c" at c<Cr b = intercept D o density (kg/l). Solving equation 7 gives: ln(Kc0+b) - ln(Kc.+b) = -KDl = -KL (8) 1 "ST5" oO (c Q, c^, R. D, t, and L are as previously defined.) Neglecting this shift in reaction order leads to over estimation of the oxygen consumption rates at low dissolved oxygen levels. Because hatchery rearing is often carried out at levels above or near the incipient limiting tension and because this error wi l l tend to make load projections conservative, a l l model predictions have been based on the simplest equation (equation 5). FIGURE 8. The oxygen consumption rate as a function of the environmental oxygen level. Above the "incipient limiting tension", the oxygen consumption rate is independent of the environmental oxygen level (Davis, 1975). 32 In other words, metabolite production rates have been assumed to be inde-pendent of the metabolite concentrations. Accepting the limitations of equation 5, the theoretical metabolite concentration over the length of a plug flow (raceway) type pond is given by: c = C - : + RLx (9) 0 1 "50 where: x = fraction of the pond length. This assumes uniform fish density. The concentration at the outflow (x=l) of a plug flow type pond is simply given by: C Q = c i + RL/60. For a mixed flow type pond with the same flow rate, volume and load rate, the metabolite concentration wil l be uniform over the entire pond and wil l also be given by: co = c i + These relationships for "plug flow" and "mixed flow" type ponds are contrasted in Figure 9• It should be emphasized that equation 6 predicts only the outflow concentration. Actual concentrations within the pond (assuming a constant density) depend on the flow distribution within the tank. Consider a pond with a flow distribution approximating two parallel paths. A slug of tracer at the inflow would appear as two peaks in the outflow (Figure 11). This is an extreme case, but i t could result i f the majority of the flow was directed across the top half of a tank. In this case the average effluent concentration is given by equation 6 but the con-centration at the bottom of the pond would be greater than that at the top. This must be stressed because i t is a fundamental limitation of the model. This model predicts environmental conditions only from the metabolite con-centrations of the pond outflow. 33 FIGURE 9. Metabolite concentrations within ideal plug flow and ideal mixed flow type ponds. The idealized concentration "Co" has been plotted against the pond length (measured from inflow to outflow); a background metabolite concen-tration of " c i " has been assumed. 34 E=V/Q t=V/Q t i m e FIGURE 10. The concentration profile at the pond outflow after injection of a "slug" of tracer. "A" represents an ideal plug flow reactor and "B" represents an ideal mixed flow reactor (Levenspiel, 1972). A B FIGURE 11. Concentration profile for non-ideal tanks. "A" represents extreme short-circuiting; "B" represents parallel paths (Levenspiel, 1972). 35 If stagnant areas within the pond occur, then the mean residence time t is not given by V/Q and equation 6 is not valid. In this case, a tracer slug at the inflow would appear in the outflow earlier than expected (Fig-ure 11). Hence, effluent concentrations could not be predicted by the model. Although such deviations would not be expected in actual rearing ponds, additional tracer information would be useful. 3. Fish Growth a. Development of a growth model The specific or instantaneous growth rate "g" is defined as: g = 1 dw (lb) w dt solving gives: w = wQec ^ (11) where: wQ = i n i t i a l weight (gm) w = final weight (gm) t = time (days). Equation 11 predicts fish growth accurately only under conditions where "g" is constant. Since "g" varies with the weight of the fish itself and the pond environmental conditions, equation 11 is valid only over short time intervals. In this model, time intervals of 1 or 5 days have been used. Stauffer (1973) has provided a method for relating "g" to "w" and certain environ-mental conditions. Hence "g" can be recalculated on a daily basis and a new weight can be .predicted using equation 11. 36 Stauffer (1973) has provided an excellent analysis of coho and Chinook growth derived in part from studies carried out at Washington State hatch-eries. Of the many factors considered; diets, disease, maturity, photoperiod, ration and feeding frequency, social hierarchy, species and race, swimming activity and exercise, size and age, and water temperature, i t was concluded that ration, fish weight and water temperatures have the most important predictable influence on fish growth. Standard hatchery diets such as O.M.P. (Oregon Moist Pellet) were assumed. The relationship between specific growth rate "g" and ration level "F" has been investigated by Brett et. al. ( 1 9 6 9 ) . Figure 12 shows the general shape of the curve relating "g" and "F". It can be seen that the maintenance ration corresponds to zero growth, while the maximum ration by definition is required to achieve the maximum growth rate. If efficiency is defined as g/F for a given ration input "F" (the slope of a line passing from the origin to the curve at point F), then the optimum ration occurs at F opt. (Figure 1 2 ) . Stauffer has mathematically described this curve by: g = G s i n ^ f (F - Fma ) (12) 2(Fmax-Fma) where g = specific growth rate G = maximum specific growth rate F = ration level in units of gm dry food per 100 gm fish flesh per day - fFmax (f = fraction of maximum ration) Fma = maintenance ration (gm dry food per 100 gm fish per day) Fmax = maximum ration (gm dry food per 100 gm fish per day or percent biomass per day). 3 7 -0.254 FIGURE 12. Fraction of the maximum specific growth rate as a function of the.ration level; values shown have been calculated for 10 gm fish at a water temperature of 10°C. The maintenance ration "Fma" gives zero growth while the maximum ration "Fmax" yields maximum growth (1.0). The relationship has been described by a sine curve. 38 A 75 % moisture content has been assumed in the conversion from oven dried fish to fish flesh. The maximum growth rate "G" is a function of temperature and is apparently best described by a polynomial (Stauffer, 1973): G = [a1+a2(l.8T+32)+a3(l.8T+32)2+a4(l.8T+32)3+a5(l.8T+32)4] • [0.0304w"°*3333] (13) wherev a^ = -15.949 a 2 = 1.3849 a3 = -O.O46018 a4 = 6.9698 x 10~4 a5 0 -3.8991 x 10""6 T = temperature (daily average)°C w = weight (gm) The relationship between "G" and temperature for 1 gm fish is shown graphically in Figure 13. Stauffer has thoroughly reviewed the literature to find expressions for Fma and Fmax. These expressions w i l l merely be stated: Fma - 0.1844(10 )°-°^V 0 - 2 0 0 (14) Fmax = w"°*333[-37.71+10.731n(l.8T+32)]. (15) It should be noted that equations 14 and 15 are not valid at temperature extremes. At temperatures within 2 or 3°C of the upper lethal limit, loss of appetite occurs and equation 15 is not valid. Also, at temperatures near 0°C equation 15 predicts negative results. Although exact limits cannot be specified, these equations have been used only between 3°C and 20°C in the present study. 39 FIGURE 13. Specific growth rate x 100 for 1 gm fish at the maximum ration as a function of water temperature. The relation-ship is described by a polynomial. 40 b. Solutions of the growth equations At maximum ration and a constant temperature, the growth equations can be solved explicitly. Equation 12 reduces to: g = G at F = Fmax and equation 13 reduces to: G * Aw"B (16) where: A = a function of temperature (0.0304)(a1+a2(l.8T+32)+...see equation 13) B = 0.3333. Since the expression for "g" is known, equation 10 becomes: 1 dw = k\fB. (17) w dt Solving for "w" as a function of time "t" gives: w = (W q B + ABt) 1 / / B (18) where: wQ = i n i t i a l fish weight (gm) w = final fish weight (gm) t = time (days). At 10°C, the relationship between "w" and "t" is: w - (wo°-3333+,o.010174lt)3-OG°3. ( 1 9 ) Equation 19 provides an exact solution of the growth equations at 10°C and maximum ration. It is interesting to note that equation 13 (Figure 13) is almost linear between 2°C and 12°C. Because of the linearity, a constant temperature within this range is not required for an explicit solution of equation 10; a good approximation can be obtained by using the average temperature. In other words, a predicted final weight based on a constant temperature of 10°C is the same as the predicted weight based on an average temperature of 10°C (as long as the temperatures range between 2°C and 12°C). 41 If the ration is not maximum and temperatures are not constant, equation 10 cannot be solved explicitly (Appendix III). Solutions have been obtained using an incremental method involving 1 or 5-day time intervals. In this method, conditions at the beginning of a time interval (i.e. i n i t i a l weight, temperature, and ration level) are used to calculate "g" from equations 12, 13, 14* and 15. This value is substituted into equation 11 and the final weight is predicted. The process is repeated until the weight at the desired time is found. These operations can be quickly and simply performed with a small programmable calculator/printer. Because "g" decreases continuously with V , the incremental approach will introduce positive systematic errors. It is revealing to see how these errors propagate through the calculations. This can be accomplished by comparing incremental growth projections at a constant temperature of 10°C and maximum ration with the weight predictions offered by the explicit (exact) solution (equation 19). Figure 14 shows plots of the percentage error incurred by using the incremental method as a function of the exact fish weight. Percentage error has been calculated from: % error = P - E x 100% (20) E where: P = predicted weights by the incremental method E = exact weight. Differences between the "incremental" and the "exact" methods are less than 3% i f a five day time interval is used and less than 1% for a one day interval. For most applications, either method will give acceptable results. On the other hand, a correction factor may be included to eliminate this error (Appendix TV). 5-day increments 1-day increments 20 30 40 50 60 70 80 90 100 Exact Weight (gm) 110 120 130 1/+0 Percent grror between the weights predicted by the incremental method (P) and the exact method (E) as a function of the exact weight (% error = P-E x 100%). Predictions were made using 1 or 5-^ day time increments; weight projections were made assuming an in i t i a l weight of one gram, a maximum ration, and a constant temperature of 10°C. 43 c. Comparisons with measured values Model predictions must be compared with the actual growth records of production hatcheries in order to verify and define the limitations of the growth equations. Stauffer has compared growth model predictions with the growth data from many U. S. coho and chinook hatcheries. Given the uncertainty of hatchery data, i t is remarkable that growth was accurately predicted in nine out of fifteen chinook runs. Predicting coho growth was less success-ful; however, by making one reset during the run, growth was accurately predicted in eight out of fifteen trials. The growth of coho salmon in a 75-foot Burrows pond located at the Quinsam Hatchery was also compared to model predictions. Pond 5 (1976 brood) was monitored intensively from June 17, 1977 to May 3, 1978. Oregon Moist Pellet (O.M.P.) ration levels were approximately 80% of those specified by the O.M.P. Feeding Schedule. Inventory values were based on an.initial weight count on June 17, 1977, followed by a statistical estimation of pond numbers on November 30, 1977. The estimate was based on the marked to unmarked ratio of a sample of 1169 fish from a population containing a known number of marks. The pond population estimate on November 30, 1977 was 48400 while the 95% confidence interval ranged between 45817 and 51291 fish. Figure 15 shows actual growth and predicted growth from June 1977 to May 1978. Predicted growth was calculated by updating the specific growth rate "g" daily using Stauffer's growth equations, the average daily temp-erature and the daily ration level. -p x; to • H 0> Mean measured values and confidence i n t e r v a l s P r e d i c t e d values June 17 1977 160 180 200 Time (days) 240 300 320 340 360 May 3 1978 FIGURE 15. Comparison of predicted and measured growth at reduced r a t i o n l e v e l s . N i n e t y - f i v e percent ; confidence i n t e r v a l s of selected measured values are shown. 45 It can be seen frcm Figure 15 that growth is accurately predicted up until early October. At this time some effect other than fish size, temp-erature, or ration level suppressed growth and the model began to predict higher than actual values. However, by restarting the model at a mean weight of 13.2 gm on day 154, accurate projections were made to the end of the rearing period. Actual and predicted weights on days 94 and 321 are compared in Table 3. Both days are near the end of computing periods when differences should be greatest. In both cases differences between predicted and actual values were not significant at the.0.05 level. Actual mean weights were estimated frcm a random sample of 150 fish. The 95% confidence intervals were calculated in the usual manner (mean + 1.96S/^i50'). Because of uncertainties in the pond inventory and hence in the feeding levels, confidence limits were also calculated for predicted values. TABLE 3. A comparison of predicted and actual weights for hatchery coho on a reduced ration. Day Predicted 95% Confidence Measured 95% Confidence Weight (gm) Interval Mean Weight Interval (gen) 94 10.87 10.73 to 11.01 11.28 10.62 to 11.93 321 25.06 24.05 to 26.06 24.83 23.68 to 25.98 46 The period between the middle of October and the end of November i s important because i t i d e n t i f i e s a l i m i t a t i o n of the model. I t i s i n t e r -esting t o note that t h i s i s a period of diminishing l i g h t i n t e n s i t y and shortening photoperiod. I t should also be noted that an intensive marking program and pond transfer were begun on October 2 3 r d . During the next f i v e days, the ration was d r a s t i c a l l y reduced and intensive f i s h handling occurred. To thoroughly define the li m i t a t i o n s of the growth model, many com-parisons should be made over a wide range of temperatures and rati o n l e v e l s . This process has begun but i s d i f f i c u l t due to the uncertainty associated with hatchery data. At present, i t w i l l be hypothesized that the growth equations provide a r e a l i s t i c means f o r specifying the pot e n t i a l for growth under a given temperature regime and ra t i o n l e v e l . d. Application of the growth model Stauffer's equations can be used to predict the expected growth at new hatchery s i t e s . I f the monthly mean temperatures, the ponding date, and the i n i t i a l f i s h size are known, then the potential growth can be eas i l y predicted f o r various ration l e v e l s . Maximum growth i s usually desirable f o r Chinook and steelhead so rations are often near the th e o r e t i c a l maximum rat i o n . This i s often required i f the target weight i s not to be exceeded at release time. I f the temperature regime of a system i s described by a continuous function, then the growth equations become simple and convenient to use. In t h i s presentation, growth projections f o r new systems have been made using a Fourier series approximation of the yearly temperature regime. (Appendix I) 47 These techniques have been used to predict the weight vs time curves shown in Figure 16; projections have been made at various ration levels on a hypothetical temperature regime. If the ration provided is expressed as a fraction of the maximum ration and the temperature regime is described as a continuous function, growth projections can be simply obtained in the field using a minimum of computing equipment (hand-held programmable calculator). However, the model must be used with an appreciation for its limitations. Because of the simplicity of obtaining model projections, many simulation trials at various ration levels can be run. If the target weights and release times are known, the model will indicate the required ration level. This can immediately be translated into a daily feeding chart. 4. Oxygen Concentration a. Development of an oxygen model The oxygen concentration component of the rearing model is based on a relationship between the oxygen consumption rate "R0" and the ration level "F", fish weight "w", and the water temperature "T". If "Ro" is known, then the oxygen concentration at the rearing pond outflow can be predicted: xft = x. - RoL (21) ° 1 "So where: x^ = inflow dissolved oxygen (mg/l) L = load rate (kg per l/min) Ro = oxygen consumption ( mg ) . kghr If the fraction saturation of the inflow water is "S", then the oxygen concentration can be related to temperature by: x-L = S 475 (Weber 1970) (22) T + 32.035 49 So i f reaeration is insignificant, the oxygen at the pond outflow "x0" is given by: x n = S 475 - RoL (23) T + 32.035 60 where: x Q = oxygen concentration (mg/l) S = fraction of saturation T = temperature °C Ro = oxygen consumption(_mg_) kghr L = load rate (kg per l/min) . However, i f aeration within the pond is significant, then the Streeter Phelps equation must be used (Nemerow 1974)* In this case, oxygen at the outflow is given by: x Q = x g - RoM(l-e- k V/ Q) - (x - X i ) e - k V / Q (24) V60k s 1 where: k = aeration constant (min""^ ") Q = flow (l/min) V = volume (1) x a « saturated oxygen concentration (mg/l) (Appendix V). In this presentation, equation 23 w i l l be used to predict oxygen concentration. In other words, reaeration will not be considered. Enough basic experimental work has been performed to relate oxygen con-sumption to ration level, fish weight and water temperature. Brett (1974) has measured "feeding metabolic rates" of young sockeye at various tempera-tures. Oxygen consumption "Ro" increased linearly with ration level: (Ro) 2 Q = B + AF (25) where: F = ration level % dry food/day A, B = constants (Ro)2Q = average daily oxygen consumption ( mg )for 20 gm fish. kghr If equation 25 is expressed in terms of the fraction of the maximum ration fed "f" (rather than F), then: (Ro) 2 0 = B + [A(Fmax)20]f (26) where: f = F/(Fmax)2o (Fmax)2Q - maximum ration for 20 gm fish (equation 15). Brett's (1974) measurements were carried out at temperatures of 5, 10, 15, 20, and 23°C over ration levels ranging from 0 to excess. The average fish weight was 20 grams t 10 grams. The constants A and B have been modelled after Brett's (1974) data, (a fish moisture content of 75% has been assumed). Slopes and intercepts are closely approximated by: B = exp(0.6491nT+2.760) = 15.7998T0*649 and, (27) A = exp(0.328lnT+4.097) - 60.1595T0*328 (28) where: T = temperature (°C), 5°C<T^23°C. Brett (1965) has explored the relationship between metabolic rate and fish weight. The basic expression is given by: Y = kwb, or (29) log Y = log k + b log w where: Y = mg oxygen consumed per hour w = fish weight (gm) k, b = constants In terms of oxygen consumption rate "Ro" ( mg ), equation 29 becomes: b-1 ^ Ro = Y = kw 1 (30) 51 The value of "b" depends on the activity level of the fish. Brett (1965) found that "b" varied from 0.78 to 0.97 at maximum swimming speed. Under hatchery conditions, a "b" value of 0.8 has been assumed (J. R. Brett, pers. comm.). To obtain a general feeding metabolic rate expression, oxygen consumption must be related to fish weight "w" and the ration fraction " f " . Consider the experimentally derived relationship between "Ro" and "f" for 20 gram fish at a constant temperature; the relationship is presented graphically in Figure 17. At a different fish weight "w", the feeding metabolic rate relationship would be given by the parallel line: (Ro)w = B' + [A(Fmax)20]f (31) It should be noted: that the slope of the line is unchanged (see equation 26). Also, "f" is the fraction of the maximum ration at tempera-ture "T" and fish weight "w". The intercept B' can be found by noting that at f = 0: -0 2 (Ro)w = B* = kw * and (Ro^0= B = k(20)~ 0* 2 so k = (Ro)20 = 1.82(Ro)oo* substituting this value for "k" into 20-0.2 the expression for (Ro)w gives: (Ro)w = B' = 1.82w~0,2B (32) The general expression for feeding oxygen consumption levels "Ro" is given by substituting expression 32 into equation 31: Ro = (Ro)w = 1.82w"0,2B + [A(Fmax)20]f (33) where: A and B are given by equations 27 and 28 respectively. Ro = oxygen consumption ( mg ) kghr w = fish weight (gm) f = the fraction of the maximum ration fed for fish at weight "w" and water temperature "T"; or F/Fmax, 0£f «1. 52 fraction of maximum ration FIGURE 17. Oxygen consumption rate as a function of the feeding level. In this case, the feeding level is expressed as the fraction of the maximum ration. The linear relationship is shown for two different fish weights. 53 (Fmax)2Q = maximum ration for 20 gm fish at water temperature 11T" in units of gm dry food per 100 gm fish flesh per day. Fmax = maximum ration from equation 15. The average daily oxygen consumption value "Ro" given by equation 33 can be substituted into equation 21 to give a general expression for the oxygen concentration of the pond outflow. The temperature range over which equation 33 is valid is determined by the ranges over which equations 15 and 25 have been tested; in this presentation, equation 33 (Ro) will be used only between 5°C and 20°C. In summary, the average daily oxygen concentration at the pond outflow, "x0" can be predicted by substituting the expression for "Ro" (equation 33) into equation 21 i f reaeration is insignificant, or into equation 24 i f reaeration is considered. "x0" w i l l depend on the: - average daily water temperature (°C) - fish weight (gm) - feed rate (as a percentage of dry food to wet body weight per day) - fraction saturation of inflow water (usually near 1.0) - load rate or kg of fish per unit of flow (l/min) - reaeration constant (if equation 24 is used). b. Comparisons with other models Westers and Pratt (1977) base their projections of oxygen uptake solely on the feed rate "F". The basis for their model is the quantitative relationship between food metabolized and oxygen consumed; this is given by: 0.25 lb. of oxygen is required to metabolize 1 lb. of trout pellets. (34) If a 10 % moisture content is assumed for the food and units are trans-formed to mg , then equation 34 becomes: kghr oxygen consumption "Ro" = 115.7F (35) 54 where: Ro = oxygen consumption ( mg ) kghr F = feed rate (percentage of dry food to wet body weight per day). The two expression for oxygen uptake, equation 33 and equation 35* can now be compared directly. Figure 18 shows plots as a function of fish weight at a constant temperature of 10°C and a maximum ration. Significant differences occur for fish below 2 gm and above 15 gm. Japanese experience with chum salmon rearing has led to the following empirical formula for oxygen consumption: (C. West, pers. comm.) Ro = 25.564T + 32.224 (36) where: Ro = oxygen consumption ( mg ) kghr T = water temperature (°C) . This equation is apparently valid for 1 gm fish at hatchery feeding levels. Assuming a typical feeding rated of 3% per day (Moberly and Lium, 1977), a water temperature of 10°C and also assuming that equation 36 represents the average daily uptake value, this relationship can be compared with equation 33: F = 3% per day w = 1 gm T = 10°C Ro from equation 33 = 271 mg kghr Rb from equation 36 = 288 mg . kghr Under these conditions, similar results are obtained. Comparison over a broad range of conditions is difficult because equation 36 is not sensitive to either the fish weight or ration level. 55 Liao (1971) has related oxygen consumption "r^" to fish weight and water temperature: Ro = k(l.8T + 32) nw m (37) where: Ro = oxygen consumption in mg kghr w = fish weight (gm) T - water temperature (°C) and the constants k, n, and m, are given below: k m n T $ 10 9.829 x 10~4 -0.194 3.20 T> 10 6.689 x 10"2 -0.194 2.12 Ro (equation 37) has been plotted as a function of fish weight "w" at 10°C in Figure 18. It should be noted that the feed rate has not been considered in this model. Elliott (1969) also considered oxygen consumption independent of feed-ing level. Oxygen consumption levels "Ro" have been related empirically to water temperature at various fish weights by the following set of equations: weight (gm) equations 1.85$w^ 5.90 Ro = (21.807T+1.28) - (1.1191T-2.8526) (w-1.85) (38) 5.90<w^ .7.5 Ro = (17.275T+12.90)- (0.3312T+0.0805)(w-5.90) (39) where: Ro = oxygen consumption ( mg ) kghr T = temperature (°C) w -= fish weight (gm) . Although Elliott used the values predicted by "Ro" to calculate hatchery carrying capacity at "normal activity levels", the feed level is not clearly specified. "Ro" values have also been plotted in Figure 18 as a function of fish weight (up to 17.5 gm)• 56 600 , • ' ' » ' 0 5 10 15 20 25 Weight (gm) FIGURE 18. Average daily oxygen consumption projections at 10 °C vs. fish weight. A number of different models have been used. The present model (equation 33) has been calculated at f u l l ration (f = l) and at half of the maximum ration (f = 0.5). 57 It is interesting to compare Liao's (1971) and Elliott's (1969) results with "Ro" values calculated for a 50% ration level (f = 0.5, equation 33). These models do not take feed rates into account but give similar results to "Ro" calculated from equation 33 at half the maximum ration (Figure 18). Figure 18 shows obvious and fundamental differences between various oxygen prediction models. It is clear that a general oxygen consumption model must not only be sensitive to fish weight and water temperature, but also to ration level. c. Comparisons with measured average daily oxygen consumption rates. Continuous dissolved oxygen monitoring was carried out on 75-foot concrete Burrows Ponds loaded with coho salmon fingerlings. Accurate records of fish weights, water temperatures, and feed rates were kept. Pond inventories were estimated by the mark recapture method. Briefly, a known number of marked fish were placed in a pond; a random sample was then drawn and the ratio of marked fish in the sample was calculated. From this ratio, and the known number of marks originally added, the pond inventory was estimated. The daily ration was dispensed by two automatic feeders per pond. These operated every 15 minutes from dawn until dusk. The pond effluents were monitored continuously over a total of 24 days. Measurements were made with a Y.S.I, dissolved oxygen meter and recorded on a Houston Instruments strip chart recorder. The performance of the probe was checked frequently against Winkler titrations. The oxygen level of the ponds inflows were estimated by Winkler titrations on grab samples. 58 It should be emphasized that the 24 sample days represent typical hatchery conditions. The consumption rates obtained reflect the kind of disturbances fish experience around a production facility (drains cleaned, dead picked, feeders f i l l e d , etc.). The percentage differences between predicted oxygen consumption (Ro, etc.) and measured values (M) have been calculated according to: Ro - M x 1Q0 % Ro The mean percentage differences and their 95% Confidence Intervals (using "t" values) are shown in Table 4. TABLE 4« Mean percentage differences between predicted oxygen consumption rates and measured average daily values. Prediction Model Mean Percentage Difference 95 % C I. n "Ro", (equation 33) - 4.7 -9.3 to 0.0 24 "Ro", (equation 35) -79.7 -96.4 to -63.0 24 "Ro", (equation 37) -62.5 -71.2 to -53.8 24 "Ro", (equations 38 and 39) -50.4 -29.5 to -71.3 5 A l l models gave, on the average, low results. However, "Ro" values based on equation 33 were closest to measured values. The reasonable agreement between "Ro" values and the 24 measured values does not constitute verification of the model. The 24 sample days represent only a very narrow range of temperatures (8.3 to 10.8°C) and feed levels (0.8 to 1.7 %/day). Also, fish weights varied only between 12 and 33 grams. However, the results are at least encouraging; further monitoring is required to firmly establish equation 33 as a useful relationship; valid for production rear-ing f a c i l i t i e s . 59 d. Development of a safety factor A simple deterministic oxygen model has been developed based on the feeding level, water temperature, and f i s h weight. The model predicts the average daily oxygen consumption or the average daily oxygen concentration in the pond effluent. One of the most important and d i f f i c u l t factors to quantify affecting the oxygen levels of hatchery rearing systems are disturbances caused by essential routine hatchery a c t i v i t i e s . These include disturbances caused by pond transfers, dead picking, and f i s h sampling. Although the effects of such activities are random, they must be taken into account i f safe projections are to be made. Table 5 shows the effects of some typical hatchery disturbances on pond effluent oxygen levels. It should be noted that even walking along the length of a crowded pond has a measurable affect on the oxygen level. It must be emphasized that peak daily oxygen consumption levels (cor-responding to minimum daily oxygen concentrations) may be as important as ' average daily values. In setting oxygen c r i t e r i a , Davis (1975) has ident-i f i e d risks when f i s h are continuously exposed to low oxygen levels for more than a few hours. The diurnal variation i n oxygen consumption which accompanies daily temperature, light, and feeding changes also must be taken into account. Figure 19 shows a typical daily oxygen consumption pattern. In this case, no major hatchery disturbances occurred and the mean value (199.8 mg/kghr) was accurately predicted by the oxygen consumption model. However, the peak value, which occurred at 1500, was much higher at 286 mg/kghr. 400 FIGURE 19. Oxygen consumption rate pattern for a typical undisturbed hatchery rearing pond over a 24-hour period. This curve was measured for coho smolts in Aprili the average daily temperature was 9.5 °C. 61 TABLE 5 . Abrupt dissolved oxygen drops resulting from typical hatchery disturbances to crowded rearing ponds. "Date Disturbance and resulting oxygen drop (mg/l)  Walk length Hand Pond Dead Fish of Pond Feed Sample Picked Transfer 1 3 / 0 9 / 7 5 ° » 5 2 3 / 0 9 / 7 5 0 . 7 2 3 / 0 9 / 7 5 1.3 17/09/75 0 . 4 0 . 6 0 . 5 0 9 / 0 2 / 7 6 1 . 4 1 3 / 0 2 / 7 6 2 . 2 1 4 / 0 2 / 7 6 1 . 2 2 . 1 0 4 / 0 3 / 7 6 0 9 / 0 3 / 7 6 1 . 7 1 1 / 0 3 / 7 6 2 . 1 If a safety factor of 1 . 3 5 is used, we can be reasonably confident that the oxygen consumption rates w i l l be below the corrected values most of the time. This factor was derived by comparing 1 0 2 predicted and measured oxygen con-sumption values. In this case, the measured values represented both the results of afternoon grab sampling and continuous daily measurements. These observations were taken over a wide range of hatchery conditions. "Safe" predictions can be made i f the "corrected" oxygen consumption rate rather than the mean daily rate is used: Corrected Predicted Value = 1 . 3 5 x mean daily predicted value ( 4 0 ) 62 In summary, measured oxygen consumption rates, which are affected by diurnal variations and hatchery disturbances, wi l l be less than the cor-rected predicted values ninety percent of the time. In other words, use of the correction factor wi l l insure that on a daily basis,oxygen levels are at least equal to specified values ninety percent of the time. e. Application Knowledge of the mean daily -oxygen consumption rate (equation 33) allows prediction of the average daily pond effluent oxygen concentration. Also, predictions of the maximum pond load rate "L" (kg per l/min or number per l/min) can be made i f the minimum desired effluent oxygen level is specified. Solving equation 23 for "L" gives: L = wn = 285Q0S - 6Qxn (T + 32.035) (41) 1000 Ro (T + 32.035) where: L = kg per l/min S = fraction of oxygen saturation x D = desired oxygen level (mg/l) T = temperature (°C) Ro = oxygen consumption rate ( mg ) kghr w = average fish weight (gm/fish) n = number of fish. In this way, maximum safe loadings can be predicted. For example, at T = 10°C, S = 1, and with the average daily effluent oxygen concentration "x0" specified at 8 mg/l, the maximum load rate "L" can be found by substituting these values into equation 40 to get: L = 198 (42) Ro 63 where "Ro" is the average daily oxygen consumption rate which depends on temperature, feeding level, and fish weight (equation 33). The water flow requirement (l/min) per 1,000,000 fish can also be predicted from equation A l t flow per 1,000,000 fish = wlOOO . (43) L It should be noted that applying the "safety factor" to the average daily oxygen consumption rate implies that the oxygen level of the pond effluent would be at least 6.9 nig/l ninety percent of the time. To summarize, equation 41 and 42 allow the rapid prediction of a number of operation and design parameters, namely: - oxygen consumption rate - loading rates - oxygen concentration - water flow requirements. Such predictions may be made with complete flexibility over any combination of water temperatures, fish weights, feeding levels, dissolved oxygen levels or load rates. As an illustration of model output, consider the water flow required per 1,000,000 fish as a function of temperature (Figure 20). An average daily dissolved oxygen level in.the pond .effluent of 8mg/l, 10 gm fish, a feeding level of 100 % of maximum, and an oxygen saturation level of 1.0 have been assumed in the calculations. As would be expected, the water required to maintain 8 mg/l increases with temperature and ration level. In fact, an extra 2000 l/min is required i f the ration is increased from 60 percent to 100 % (naximum ration) at 15°C 64 Temperature °C FIGURE 20. Water flow requirement per 1,000&000 ten gram fish as a function of temperature. Maximum ration and 60% ration projections have been made; an average daily oxygen level of 8 mg/l has been maintained in the pond effluent. 65 5. Carbon Dioxide Concentration and pH a. Carbon dioxide The rate of carbon dioxide production i s related to the oxygen consumption rate by the Respiratory Quotient (R. Q.). I f a normal R. Q; of 0.9 i s assumed (Brett and Groves 1978), then the expression for the carbon dioxide production rate "Ro" (by weight) becomes: Rc = (0.9 x 4Jt)Ro = 1.238Ro (44) 32 where: Ro = oxygen consumption rate ( mg ) given by equation 33 kghr Rc - carbon dioxide consumption rate ( mg ) kghr 1.238 = factor found by converting the volumetric R. Q. to a weight based R. Q. The carbon dioxide concentration in the outflow can be found by substituting the expression for Rc into equation 6 to giver Co = C i + RcL (45) "60" where: Co = carbon dioxide outflow concentration (mg/l) C i = carbon dioxide inflow concentration (mg/l) Rc = carbon dioxide production rate ( mg ) kghr L = load rate kg per l/min. b. pH changes In the pH range of most natural waters, the carbonate-bicarbonate system has a major influence on pH regulation (gtumm and Morgan 1970). If this simplification i s accepted, a general expression relating carbon dioxide and pH can be derived. A complete derivation i s included, i n Appendix I I I . 66 In summary: (pH)i = -log 1.1364^ (46) A (pH)o = -log 1.13641^ (47) A where: (pH)i = pH at pond inflow (pH)o = pH at pond outflow Ci = inflow carbon dioxide (mg/l) C 0 = outflow carbon dioxide (mg/l) A = bicarbonate alkalinity (mg/l as CaCO^) a f i r s t ionization constant of ^CO^ at 10°C » 3.436 x 10 r 7 (Stumm and Morgan 1970). The importance of being able to predict pH changes l i e s i n the fact that the toxicity of ammonia, the major nitrogen excretory product of metabolism i n f i s h , i s dramatically affected by pH. Figure 21 shows the predicted pH drop as a function of pond loading (kg per l/min); this projection has been made for 9.1 gram f i s h fed at 60 percent of the max-imum ration. 7. Ammonia-Nitrogen a. Introduction Ammonia-N i s the primary end product of protein metabolism in f i s h . It i s also a cellular poison at low concentrations and so is an important factor i n the rearing process. To predict the rate of ammonia-N production, many empirical hatchery models have been developed. These models were often developed over a narrow range of conditions and for specific purposes. The present theoretical model i s based on a consid-eration of the nitrogen balance that must exist i n any rearing process. 7.0« • • • -•— « • « 1 — 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Load Rate (kg per l/min) F I G U R E 21. Effect of load rate on pond outflow pH levels. Projections were made at a temperature of 11.7°C, a fish weight of 9.1 gm, and feeding levels of 60% of the maximum ration. A background carbon dioxide concentration of 1.0 mg/l and an alkalinity of 40 mg/l as CaCO-3 were assumed. 68 A simplified picture of nitrogen flow is presented in Figure 22. The model is based on the assumption that the nitrogen excretion rate is a function of the difference between the nitrogen content of the ration and the nitrogen assimilated as growth. To quantify this relationship, the model w i l l be developed along theoretical lines and then compared with measured results at specific conditions. From these comparisons, a scaling factor w i l l be derived and a general model developed. b. Ammonia-N model components (i) nitrogen input If the protein fraction of a typical hatchery ration (on a dry weight basis) is denoted by "p", then the daily nitrogen input per gram of fish is given by: Nj. = pF(l.6 x 10-3) ( 4 8 ) where:: p = protein fraction of diet F = feeding rate (gm dry food per 100 gm fish per day) Nj = nitrogen input (gm-N per gm fish per day) assuming 16 % of dietary protein is nitrogen. (ii) nitrogen assimilated as growth The fish weight after n+1 days of rearing is given by: w n + 1 = w ne§ (49) where: w n + i = weight at the end of the (n+1 )th day wn = weight at the beginning of the day g = specific growth rate. 69 N assimilated as growth Protein-N in ration N waste (immediate pond waste) Protein-N processed & excreted Y N waste (undigested) FIGURE 22. A simplified schematic representation of the flow of nitrogen in the rearing process. 7 0 The protein content of fish has been related to fork length by Groves (1970). If a general relationship is assumed between length and weight, the protein content of fingerlings can be related to fish weight by: p = 0.1309w L 0 9 2 (50) where: p = protein content (gm) w = fish weight (gm). The increase in protein "AP" over the (h+l)th day is given by substituting wn+l a n c* w into equation 50: AP - - P, - C U O ^ V 1 - 0 9 2 - ! ] (51) The daily increase in nitrogen per gram of fish is approximated by 0.l6Ap/wn. (i i i ) nitrogen excretion The theoretical nitrogen excretion rate "N^ " is given by: NR = DNj - 0.16&P (52) w n where:: D = digestibility of the protein. If the nitrogen excretion rate is converted to units of mgN , and the kghr expressions for Nj and p are substituted into equation 52, the expression for the nitrogen excretion rate "NR" becomes: % = D35.3F - 872.7w n 0- 0 9 2[eS 1- 0 9 2-l] (53) where: NJJ = average daily nitrogen excretion rate on the nth day (mg-N) kghr D = digestibility of dietary protein (D20.90(Windell et al. , 1978)) F = feed rate gm dry food per 100 gm fish per day wn = fish weight on nth day g = specific growth rate (equation 10) p = 0.53 (protein fraction of diet). Ammonia-N accounts for between 55% and 60% of the total nitrogen excreted (Fromm, 1963;). If an average value of 0.575 is used, the rate of ammonia-N excretion "RNH3" becomes: 71 RNH3 = 0 ' 5 7 5 N R ^ ( 5 4 ) where: oc = factor NJJ = nitrogen excretion rate given by equation Rj^o = nitrogen excreted as ammonia (mg-N). kghr Calculation of the factor . " , < X . " was not possible because of the insufficient average daily ammonia-N excretion data. However, comparison of predicted results with 47 grab samples taken at the Quinsam Hatchery showed that i f c* = 0.828, predicted results were on the average 19% lower than mea-sured values. The grab samples were a l l taken between 1300 and 1500 hours and so represent values somewhat higher than the daily average. It should also be noted that the measured values represent only a narrow range of temp-eratures (8 - 11°C?). Because of inadequecies of the data, a factor of 0.828 was tentatively accepted. Equation 54 (witho<,= 0.828) will be compared with a number of other hatchery ammonia excretion models. Following this, model projections w i l l be compared with same measured values. c. Comparisons with other models A number of empirical hatchery models will be described. In a l l cases: F = feed rate (gm dry food per 100 gm fish per day) T = average daily water temperature (°C) w = fish weight (gm) Rj^ 3 = ammonia-N production rate ( mg ) kghr r = multiple regression correlation coefficient. 72 Empirical Hatchery models: (i) Equation 55 was developed at the Quinsam Hatchery for coho salmon (unpublished data). Twenty-seven pairs of measurements on pond inflow and outflow taken between 1300 and 1500 hours were used to cal-culate the following regression line: RNH3 = 2 , 0 3 2 °'^94w + I.468T + 4.109F (55) r = 0.951 • Temperatures varied between 8 and 11 °C, feeding levels were normally above 50% of the maximum ration,.' and fish weights varied between 1 and 25 grams. The food involved in this study was a "dry" formulation with about a 7% moisture content. (ii) R M 3 = 14.48 - 13.37F - 0.17w + 1.23FT(r = 0.81) . (56) Equation 56 was developed by B. C. Research Council (1976) for rainbow trout. This model was developed from average daily measurement over a temperature range of 13 to 15QC and weight range of 1 to 11 grams. ( i i i ) RJJJJ = 12.03F (57) Equation 56 was presented by Liao and Mayo (1974). This model was developed from a large number of measurements over many U. S. salmon hatcheries. It is assumed that R j ^ represents average daily excretion rate and that "F" represents feed rate on a dry weight basis. (iv) = 6.67F (58) This equation was developed from data presented by Hartman(l976). Hartman found that for brown trout, 0.0144 gm of ammonia nitrogen was excreted per gm of food consumed. This factor is equivalent to equation 57 i f unit conversions are made and a standard "dry" diet formulation with 73 about 10% moisture is assumed. Equation 5 7 predicts the average daily ammonia excretion rate. It was developed over a temperature range of 8°C to 11°C and a weight range of 5 gm to 25 gm. (v) Speece ( 1 9 7 3 ) expressed the ratio of ammonia produced to food consumed as a function of temperature: gm NHo - N = 0.22 + 8 . 1 x 1 0" 4T . gm food Converting to common units and assuming a standard dry diet gives: HJJJJ = F ( 1 0 . 1 9 + 0 . 3 7 5 T ) . ( 5 9 ) This model was developed for rainbow trout. These models have been used to predict the ammonia-N excretion rate as a function of feeding level (Figure 2 3 ) . Projections have been made for 1 5 gm fish at 9°C and 1 5°C d. Comparisons with measured values It is difficult to find accurate hatchery data including average daily ammonia excretion rates, water temperatures, fish weights, and feeding levels. Often, only afternoon grab samples are taken for ammonia determination and usually the stated feeding levels are unreliable because of uncertainties in the pond inventories. A limited continuous ammonia monitoring program was carried out at the Big Qualicum River Experimental Hatchery during the Spring of 1 9 7 2 (McLean and Fraser, 1 9 7 4 )• Although these data suffer from the inventory uncertainties common to a l l production facilities, i t is interesting to compare measured and predicted values using equation 5 4 (Figure 24). The predictions are accurate only between April 2 0 and May 5 . Ammonia-N excretion projections for April 1 7 were badly underestimated. 74 50 y 0 0.5 1.0 1.5 2.0 2.5 Feed Rate (gm dry food per 100 gm fish per day) FIGURE 23. Ammonia-N excretion rates vs. feeding levels for 15 gm fish at a water temperature of 15°C The proposed model (equation 54) is shown as a dashed line. Other models are labelled by their equation (eq.) number. 75 Major Pond Disturbance (37,000 fish removed) FIGURE 24. Average daily ammonia-N excretion rates vs. predicted values over a period of major pond disturbances (April 14, 15) and over a period of stability (April 20 - May 5). r 76 It should be noted that on April 14 and 15 the ration level was sharply reduced and 37*000 fish were removed (over two days). This major disturbance inflated the ammonia-N excretion rate of both the April 15 and April 17 samples. The ammonia-N model can not take such random disturbances into account; only the effects of temperature, fish weight, and ration level have been considered. Average daily ammonia-N excretion rates have been accurately measured for starved fish by Brett and Zala (1975) and Fromm (1963). These values along with model predictions are shown in Table 6. It should be noted that the values predicted by equation 54 are closest to the measured values. An ammonia model based on a simple N-balance (equation 54,0.828) provides reasonable estimates of ammonia-N excretion rates over the range of conditions tested. This model succeeds in predicting ammonia excretion rates of hatchery coho except during periods of disturbance (Figure 24). It also provides reasonable estimates of the endogenous (starvation) N-excretion rate (Table 6). Although comparisons have been made against a very limited amount of measurement data, this equation has been accepted as the ammonia-N component of the present Rearing Model. Equation 54 can be substituted into equation 6 to predict the total ammonia-N concentration in the pond effluent: (N 0) T (60) where: total ammonia-N concentration in the pond outflow (mg/l) total ammonia-N concentration in the pond inflow (mg/l) Ri •NH3 rate of ammonia-N excretion, equation 54 ( mg ) kghr L load rate (kg of fish per l/min)« 77 TABLE 6. Ammonia-N excretion rates for starved fish; model projections vs actual measurements. The proposed ammonia-N model is rep>-resented by equation 54* Source Temp. Weight Measured (°C) (gm) NH4-N excretion rate (_mgj. kghr Projected NH,-N excretion rate (_mg_) kghr Brett and Zala 15 • 29 7.27 * 0.20 eq. 54 5.2 (1975) eq. 55 9.7 (sockeye) eq. 56 9.6 eq. 57 0 eq. 58 0 eq. 59 0 Fromm (1963) 13 129 3.4 eq. 54 4.1 (•rainbow trout) (12-14) - eq. 55 -42 eq. 56 - 7 eq. 57 0 eq. 58 0 eq. 59 0 78 7. Un-ionized Ammonia-N Aqueous ammonia exists in the following equilibrium (Thurston et. a l . , 1974). NH3(g) + nH20(l) ^=t NH3 • nH20(aq) ^ NH^ + OH" + (n-l) H20 (6l) An ammonia solution therefore contains both ionized (NH4+) and un-ionized (NH^(g)) ammonia. This distinction is important because the toxicity of ammonia solutions has been associated with the un-ionized form (Fromm and Gillette, 1968). Thurston et. a l . (1974) has derived equations relating the un-ionized ammonia-N concentration (Nu) to the total ammonia concen-tration (N T), pH, and temperature (T): Nu * Hp (1 + lO^'^yho3 (62) where: Nu = un-ionized ammonia-N concentration in ug/l (ppb). N^, = total ammonia-N concentration in mg/l (ppm). pka = the negative log of the ionization constant: pka = 0.0901821 + 2729.92 . T+273.16 In summary, the rate of ammonia-N excretion can be related to the fish culture environment (temperature, fish weight, etc.) by equation 54. Knowledge of the load rate (L) and background ammonia-N levels (N^) allows calculation of the total ammonia-N at the pond outflow (equation 60). Equations 47 and 62 can then be used to predict the un-ionized ammonia-N levels (Nu). 79 8. Suspended Solids Suspended solids generated from rearing facilities consists mainly of feces and waste food. Because fine particles can irritate gi l l s ; suspended solids plays an important part in determining the overall environmental condition of the pond. a. Review Liao and Mayo (1974) have related the rate of suspended solids generation "Rs" to the feeding level: Rs - 216.67F (63) where: Rs = mg kghr F = feed rate % body weight per day. Willoughby et. a l . (1972) has also developed an empirical relationship between the increase in the suspended solids concentration through a rear-ing unit and the feed rate: Concentration increase in mg/l (C) = food fed (lb) per USGPM x 25 (6k) Converting this expression to metric units yields: C = 2.086FB (65) Q where: C = concentration increase (mg/l) F = feed rate (% body wt. per day) B = biomass (kg) Q = flow (l/min) . This expression can be rearranged to give: Rs = CQ60 = 125.16F (66) B 80 Klontz et a l . (1978) has related suspended solids production rates to both feeding level and dietary efficiency: Rs = 4l6.67[0.95(lbs. fed - (lbs. fed x efficiency))] (67) where: Rs = mg kghr efficiency = lbs, growth . lbs. fed The factor "o.95" apparently indicates that 95% of the metabolic wastes will be solids. Equations 63, 66, and 67 represent empirical models frequently used to predict the solids output of rearing facilities. b. Theoretical solids model The previous attempts to relate growth, oxygen consumption, ammonia production and carbon dioxide generation to environmental conditions, naturally leads to questions about the solids balance in the rearing process. Construction of a simple theoretical balance using previously developed model components provides some insight into the generation of solids during rearing. Because of the complex behavior of solids in production ponds, this effort can not be expected to yield an accurate predictive model. Unlike dissolved substances, solids tend to settle (depending on the water velocity and pond depth) and are therefore flushed out of the ponds dis-continuous ly (during cleanings, etc.). Also, settled material may dissolve over time. Calculated outputs may only be useful as indicators of potential long term average discharge rates. 81 The solids generated during rearing depends on the difference between the ration input and the food either assimilated as growth or metabolized: Rs = Food - Growth - M (68) where: Rs = solids production Food = feed rate Growth = beiomass increase M = material metabolized. If these quantities are expressed in grams of dry material per grams of fish per day, the difference "Rs" should represent the theoretical solids production. The three terms in the equation have been quantified previously: Food = _F_ 100 (69) where: Food = gm dry food per gm fish per day F = gm dry food per 100 gm fish per day. The "Growth" term in equation 68 represents the gain of dry material per day per gram of fish (a fish moisture content of 75% has been assumed): Growth = 0.25(wn+1-wn) = 0.25(wneg~wn) = 0.25(eg-l) (70) wn wn where:- Growth = gm dry growth per gm per day g = specific growth rate (equation 10) wn = fish weight on the nth day wn-^  = fish weight one day later. The weight of material metabolized "M" can be directly related to the oxygen consumption and ammonia-N and carbon dioxide production (see Appendix VII). "M" is given by: M = 1.957% + 0.691 Ro (71) where: M = material metabolized (gm dry material per gm of fish per day) Ro = oxygen consumption in units of gm of oxygen per gm fish per day 82 RN = nitrogen per gm of fish per day. Substituting expressions for "Food" (equation 69), "Growth" (equation 70) and "M" (equation 71) into equation 68 gives the suspended solids output in units of grams per grams of fish per day. Converting the terms in equation 71 to more common units of mg/kghr and expressing the nitrogen excretion rate in terms of ammonia nitrogen production gives: Rs = 416.667F - 10,4l6.668(eg-l) - 3.403RNH3 - 0.691Ro (72) where: Rs = suspended solids output rate (• mg ) kghr F = feed rate (gm per 100 gm fish per day) Fmax g = specific growth rate RNH^ = ammonia-N production rate ( mg ) kghr Ro = oxygen consumption rate ( mg ). kghr Food in excess of the maximum ration is assumed to contribute directly to the suspended solids output. c. Solids discharge rates and comparisons with other models Solids production rates are plotted against feed rates for 10 gram fish at 10°C in Figure 25. The circled points are predicted from the theoretical model (equation 72). Above the maximum ration of 2% per day the solids output increases rapidly. Equation 67 (Klontz et a l . 1978) predicts a minimum output at intermediate feeding levels; this reflects increasing growth efficiency as the "optimum" ration is approached (Figure 25). 1 Feed Rate (gm food per 100 gm fish per day) FIGURE 25. Suspended solids output rates as a function of the feeding level. Predictions have been made for 10 gm fish at a water temperature of 10°C using the equations discussed in the text. 00 84 It should be noted that equation 63 (Liao and May, 1974) and equation 65 (Willoughby et a l . , 1972) depend only on the feeding level. Because equations 67 and 72 involve the effects of growth and metabolism on food utilization, they are also sensitive to fish weight and temperature. Solids production rates at a feed rate of about 1% per day at various fish weights and temperatures are shown in Table 7. It should be emphasized that model predictions may only represent the potential long term average output. The day to day solids discharge rate would depend on the water velocity, depth, dimensions, and on the construc-tion material of the rearing pond; the rate would also depend on pond cleaning practices. TABLE 7. Model predictions of suspended solids output rates at a feed rate of 1% per day and various temperatures and fish weights. Equations 67 and 72 are sensitive to fish weight and tempera-ture, whereas equations 63 and 66 depend only on the feeding level. Weight Temp. Feed Rate Suspended Solids Output Rate (kgnr) (gm) (°c) (%/day) eq. 63 eq. 66 eq. 67 eq. 72 5 10 1 215 124 82 175 2 10 1 215 124 132 186 30 5 1 215 124 267 271 1 15 1 215 124 427 217 85 In summary, the suspended solids predictions "Rs" given by equation 72 are based simply on the solids balance dictated by the previously developed "Growth" and "Metabolite models. For example, 10 gm fish at 10°C and a ration level of 0.991 assimilated 0.212 gm of dry material as growth and metabolized 0.386 gm of food per 100 gm of fish per day. The difference of 0.393 gm represents the potential solids discharge rate. These values are summarized below: W = 10 gm T = 10°C - Food:, 0,991 gm dry matter per 100 gm fish per day. - Metabolized: 0,386 gm dry matter per 100 gm fish per day. - Growth (dry): 0.212 gm dry matter per 100 gm fish per day. - Solid Waste: 0.393 gm dry matter per 100 gm fish per day = (0.991 - 0.386 - 0.212). If the fish size and/or water temperatures are altered, the solids balance is shifted. If the temperature were increased to 15°C and the fish weight reduced to 1 gm in the previous example, maintenance requirements would increase and a loss in weight would occur; the resulting distribution is shown below: W = 1 gm T = 15°C - Food: 0.991 gm dry matter per 100 gm fish per day. - Metabolized: 0.492 gm dry matter per 100 gm fish per day. - Growth (dry): -0.022 gm dry matter per 100 gm fish per day. - Solids Waste: 0.521 gm dry matter per 100 gm fish per day. 86 Solids projections based on equation 72 represent theoretical discharge rates consistent with the other model components. Although these rates are untested by comparison with actual data, predictions are comparable to those made using established empirical models. As with previous rate expressions, pond outflow concentration (mg/l) can be calculated by substituting "Rs" into equation 6. 9. Density Density is usually expressed in terms of the number of fish per cubic foot of rearing space or in terms of the pounds of fish per cubic foot. In metric units this becomes the number of fish per liter or the grams of fish per l i t e r (or kg per m ). In symbols: Dn « n (73) V where: Dn = number density (fish/liter) n = number of fish V = volume (l) Dw = nw (74) V where: Dw = weight density (gm/l) w = fish weight (gm) Most density or space requirement criteria are based solely on experience. Density guidelines are often rationalized by reference to such crowding effects as stress (which acts as a disease trigger), disease transmission, and reduced growth rates. 87 Three frequently used methods f o r calculating maximum allowable densities w i l l be reviewed. These projections w i l l be contrasted with some load densities currently used i n B r i t i s h Columbia hatcheries. Be-cause the physiological basis of density effects i s not w e l l understood, a general density model w i l l not be developed. Westers and Pratt (1976) have based density c r i t e r i a on "R", the pond exchange rate. "R" i s defined as the number of pond volume changes per hour: R = 60Q = 60 (75) V t where: R = exchanges per hour Q = flow (l/min) V = pond volume ( l ) t = mean residence time (min). I f both sides of equation 75 are mul t i p l i e d by the biomass B(kg) and the expression i s rearranged, we get: Dw = 16.67LR (76) where: Dw = density (gm/l) = B1000 V L = load rate (kg per l/min) = B . Q The load rate "L" i s determined so that the effluent oxygen levels are maintained above 5 mg/l and the un-ionized ammonia concentrations are kept below 12.5 u g / l . The space required (V, l i t e r s ) f o r "n" f i s h i s given by equation 77* V = nw (77) 16.67LR 88 Westers suggests that for salmon culture, an exchange rate of "4" be maintained. Burrows (1968) has expressed maximum allowable densities as a function of fish weight "w" . Dw = 11.54 + 0.914w ; w<5 gm. (78) Dw = 14.43 + 0.353w ; w>5 gm. (79) These criteria were developed for salmon. Klontz (1978) has derived density criteria as a function of fish length and species. The density index "a" or maximum density per unit of fish length for various species is shown below: Species Density Index "a" ( gm ) lite r cm Coho 2.52 Chinook 1.89 Rainbow 3*16 Cutthroat 1.89 The density in "gm/liter"' is given by: Dw = a X 3 If a typical relationship between length and weight of w = (0.01l)j£ is assumed, then density can be expressed in terms of weight "w" by: Dw = 11.33W1/3 (coho) (80) o l/3 Dw = 8.50w ' (chinook) (8l) Dw = 14.21W1/3 (rainbow) (82) l/3 Dw = 8.50w ' (cutthroat) (83) 89 These different methods of calculating the maximum allowable density (or minimum space requirement) are contrasted i n Figure 26 where density pro-jections have been made as a function of f i s h weight. The load rate "L" for use i n equation 76 has been calculated assuming a temperature of 10°C, a maximum ration and a minimum dissolved oxygen of 8 mg/l. Densities found i n some t y p i c a l B r i t i s h Columbia rearing operations are summarized i n Table 8 (D. "Wilson, pers. comm.). A l l of these f a c i l i t i e s have experienced some success; these densities have been displayed i n Figure 26 to emphasize the tremendous range of densities i n common use. General density c r i t e r i a are impossible to establish at t h i s time. Maximum densities can only be established by experience; c r i t e r i a w i l l be affected by the pond design, species, disease backgound of the stock, background water q u a l i t y , load rate and rearing program objectives. FIGURE 26 Density aa a function of fish weight as predicted by various theoretical models. Actual densities from Table 6 have been displayed and labelled by number. TABLE 8 . Typical fish densities at a number of British Columbia hatcheries. These values have been plotted on the density V3 weight curves shewn in Figure 2 6 . Location Pond Type Volume (1) Species II J N Load Rate (kg per l/min) "R" Exchange Rate (hr - 1) Fish Weight (gm) Dw Density (gm/l) Dn Density (no./l) 1) Abbotsford Provincial Hatchery Circular 0.99 m deep 6.13 m dia. 30,000 Steelhead 1.20 2.0 55 40 0.727 2) Robertson Creek Hatchery 3) Robertson Creek Hatchery Circular 1.52 m deep 3.05 m dia. Earthen Channel 11,129 293,000 Chinook Coho 0.29 0.20 1.2 5.2 4.5 2 5 . 2 4.0 17.1 0.90 0.68 4) Robertson Creek Hatchery Concrete Raceway 424,700 Chinook 0.26 3.6 5.3 15.7 2.94 5) Big Qualicum Hatchery Earthen Channel 1 , 5 6 1 , 0 0 0 Coho 0.28 2.1 18.2 9.8 0.537 6) Big Qualicum Hatchery Concrete Raceway 339,800 Chinook 0.20 6.0 5.9 19.7 3.86 7) Quinsam Hatchery Burrows Circulating 102,000 Coho 0.92 1 . 6 1 2 5 24.5 0..98 8) Quinsam Hatchery Burrows Circulating 102,000 Coho 0.46 1 . 6 1 2 5 1 2 . 2 5 0.49 9) Quinsam Hatchery Burrows Circulating 102,000 Chinook 0.769 1 . 6 1 7 20.6 2.94 10) Quinsam Hatchery Trough 1,584-8 Chinook 0.396 8.6 1.2 5 6 . 8 46.2 11) Quinsam Hatchery Trough 1,584.8 Chinook 0.499 8.6 1.7 71.1 4 2 . 1 92 IV SYNTHESIS OF MODEL COMPONENTS Individual components can easily be combined to give additional insight into the rearing process. This can be illustrated by means of a hypothetical example. a. Illustrative example Consider the typical situation where a hatchery designer has infor-mation about the background water quality and temperature regime of a new water supply and is faced with the problems of determining: - water flow requirements - space requirements - feeding and rearing schedules to meet target release weights. The first step in solving these problems is to compile required background fish culture and water quality information. Although this is a hypothetical example, numerical values are typical of hatchery water supplies, (i) background fish culture information: In this hypothetical case, coho are ready for ponding around March 15; this date therefore, is the beginning of the rearing period (day 0). The average weight of newly ponded fry is 0.3 gm. The incidence of bacterial diseases in the system is high enough that a substantial disease risk to hatchery stocks exists. As part of a routine assessment procedure, 15% of the coho fingerlings w i l l be marked (handled) sometime during the rearing period. 93 ( i i ) background water qu a l i t y information: I t w i l l be assumed that the inflow water i s of high q u a l i t y . Some background levels are shown below: Parameter Background Concentrations Dissolved Oxygen 100 % saturated (s = l ) Ammonia-N 10 ug/l Carbon Dioxide 1.5 mg/l Bicarbonate A l k a l i n i t y 50 mg/l as CaCO^ Suspended Solids 2 mg/l These parameters are assumed to be constant except for a b r i e f period during the spring. Suppose the carbon dioxide concentration drops to 0.75 nig/l and the ammonia-N value increases to 30 ug/l f o r a brie f period around the 100th day of rearing. The temperature regime i s shown i n Figure 27 and i s described by the Fourier series: T = 7.7056 + 3.9067cos2Kt + 0.2278cos4trt + 2.0sin2ltt + 0.991sin4TTt (84) 365 363 363 365 ( i i i ) rearing strategy: Previous experience of the designer suggests that optimum su r v i v a l w i l l r e s u l t from a release of 25 gm smolts i n mid May the following year (after 420 days of rearing). I t i s also f e l t that because of the disease r i s k , oxygen levels should not drop below protection l e v e l A (see c r i t e r i a developed by Davis (1975), Figure 2 ) . With background information and constraints established, a rearing and feeding schedule can be developed using the growth model (equation 11). 94 Figure 27 shows one possible feeding regime to achieve a 25 gm smolt at 420 days. Ninety percent of the maximum daily ration is fed for the fi r s t 100 days; 60% is fed over the next 100 days, while 45% of the maximum daily allowance is fed over the remainder of the rearing period. Marking is performed during a low temperature period (day 230); l i t t l e growth takes place during this disturbance. This is only one possibility for achieving release objectives; often temperatures can be manipulated to achieve similar results. The maximum possible load rates (and therefore minimum water requirements) to meet level A protection criteria for oxygen were then calculated using equations 40, 41, and 43« Results have been plotted in Figure 28. Both the load rates (kg of fish per l/min) and water requirements per 1,000,000 fish (l/min per 1,000,000) have been shown. It can be seen that water requirement maximums occur at days 150 and 420. To achieve "level A protection" for a population of 1,000,000 fish, peak flows of 19,000 l/min and 31,000 l/min are required on the 150th and 420th day. Considerably less water is required i f a level B protection criterion is judged acceptable. Water requirements to meet level B are shown in Figure 29. It can be seen that flows of 10,000 l/min on the 150th day and 18,000 l/min on the 420th day are required. A water requirement curve based on level A (Figure 28) will be an acceptable statement of water flow needs as long as the un-ionized ammonia-N, carbon dioxide, suspended solids, ammonia-N, and pH values are within limits. Although setting limits on these parameters is complex and somewhat subject-ive, the following standards are tentatively proposed: (see page 8) 95 - suspended solids - un-ionized ammonia-N - carbon dioxide * 2 ug/1 - 25 mg/l - 25 mg/l - pH 6.5 - pH < 8.5 It should be emphasized that the function of the model is to predict the values of these parameters over the rearing period; i t is not designed to set water quality standards. As noted previously, the setting of standards will be influenced by the disease history of the fish, water temperatures, pond design, and the experience and perceptions of the operator. Un-ionized ammonia-N, total ammonia-N, carbon dioxide, suspended solids, and pH levels have been predicted from equations 6 2 , 6 0 , 45, 72, 6 , and 47j these parameters are plotted against time in Figures 30, 31, 32, and 33. Because concentrations were within acceptable limits over the entire rearing period, no further modifications to the water use curves shown in Figure 28 and Figure 29 are required. In this example, oxygen concentration is the first limiting factor; water flow requirements have been predicted from the oxygen consumption rate which in turn has been influenced by the water temperature, ration level, degree of hatchery disturbance, and the average weight of the fish. As was noted previously, rearing space requirements (at the present time) are based more on experience than on basic principles. If a conser-vative density "Dw" vs weight "w" relationship of "Dw = 11 + 0.25w" was assumed (see broken line in Figure 2 6 ) , then space requirements could be 3 predicted from equation 73. Volume requirements (m ) per 1,000,000 fish are plotted against time in Figure 35. 96 This example illustrates a rational approach to determining the require-ments of a hypothetical rearing program; projections are based on the "Rearing Model", the best available water quality criteria, and the exper-ience of the designer. b. Application The general rearing model can be used to solve a variety of problems. Two basic approaches can be taken. Firstly, the model can be used to predict pond conditions (e.g. oxygen concentration, etc.) i f basic fish culture objectives and water quality background information are available. Secondly, , i f constraints are imposed on environmental conditions (e.g. un-ionized ammonia-N - 2ug/l, etc.), then the model gives predictions as to the min-imum water flow and space requirements (see Illustrative Example). In either case, a certain minimum amount of information is required. Examination of equations 2 through 73 shows that twelve parameters must be specified in order to characterize the rearing process at a particular time. This is illustrated in Table 9. Because the relationships between "basic parameters" and "factors" shown in Table 9 are expressed quantitatively, a great deal of flexibility exists in making model projections. In the previous "Illustrative Example", c* 3 T, wo, wf, Ci, A, (Ni)-p, arid Si were specified; constraints were placed on x<>*Nu, Rs, Dw, Co, t, and pHo and allowable R, Q, V, and n values were predicted (ration, flow, and space requirements per 1,000,000 fish). We could have just as easily specified crt., F, wo, wf, Ci, A, (Ni)^. and Si; placed constraints on t, xo, Nu, Rs, Dw, Co, and pHo and predicted T, Q, V, and n (temperature regime plus space and flow requirements). The manipulation of parameters simply depends on the requirements of the model user. 97 Model equations are simple enough that useful predictions can be made with a minimum of computing equipment. A l l projections in this report were made with a small programmable calculator/printer. Programs were stored on magnetic cards. Obviously, more sophisticated equipment would allow greater speed, convenience, and flexibility. 98 TABLE 9. Basic parameters (independent variables) and Factors (dependent variables) required to characterize the rearing process at a given time. BASIC PARAMETERS t % oxygen saturation of inflow Q * flow V, volume T » temperature t time wo > i n i t i a l weight wf i f i n a l weight C i > inflow CO2 A > background a l k a l i n i t y ( N i ) T * background t o t a l NH^ -N S i background solids n t numbers of f i s h FACTORS . , inflow oxygen concentration x Q , outflow oxygen concentration Ro , oxygen consumption rate L , load rate F , ration l e v e l Rc , CO2 production rate Co , C0 2 outflow concentration pHi, pH inflow pHo, pH outflow , t o t a l ammonia production rate (No) T, t o t a l outflow ammonia concentration Nu , outflow un-ionized ammonia cone. Rs , solids production rate So , outflow solids concentration Dw , weight density Dn , number density RNH3 25 r FIGURE 27. Growth projection assuming an i n i t i a l weight of 0.3 gm. The temperature regime ? and ration level are also indicated: the notation "f = 0.9" indicates that 90% of the maximum daily allowance is being fed. A marking program at day 230 is assumed to suppress growth. ^ S Q Water Requirements (l/min per 1,000,000 fish) ^ ^ water requirement^per 1 ^ * 0 0 0 0 50 100 150 200 D a y s « . U k 250 300 350 FIGURE 29. Maximum load rate and minimum water requirements per 1,000,000 f i s h to meet Level B oxygen c r i t e r i a developed by Davis (1975). Loaded to meet Level B oxygen criteria Loaded to meet Level A oxygen criteria 150 250 300 350 200 Days FIGURE 30. Un-ionized ammonia-N concentrations in the pond outflows; ponds are loaded to meet exther Level A or B oxygen criteria. 400 800 r D a y s FIGURE 3 1 . Total ammonia-N concentration i n the pond outflows; ponds are loaded to meet either Level A or B oxygen c r i t e r i a . o VjJ FIGURE 32. Carbon dioxide concentrations in the pond outflows; ponds are loaded to meet either Level A or B oxygen criteria. 20r Pond loaded to meet Level B Pond loaded to meet Level A 0 50 100 150 200 250 300 350 400 420 Days FIGURE 33. Suspended solids concentrations in the pond outflows; ponds are loaded to meet either Level A or B oxygen criteria. o 0 50 100 150 200 250 300 350 400 420 Days FIGURE 3U» pH of pond infow and outflow; ponds are loaded to meet either Level A or B oxygen criteria. 2000 108 V DISCUSSION AND CONCLUSIONS The approach adopted to modelling the rearing process concentrates on establishing simple deterministic relationships between the most significant factors known to affect rearing success. This has produced a set of rela-tionships that can be useful in facility design and operation but which are s t i l l very dependent on the judgement and experience of the user. For example, oxygen constraints, although reflecting sound criteria established by Davis (1975), s t i l l require a decision whether to accept level "A", "B", or "C" criteria. If level "CV rather than "A" oxygen criteria are used, the number of smolts released may be doubled or tripled. Although this trade-off seems profitable the implications to adult production are not clear. As noted previously, reduced environmental quality may have subtle effects on the ability of the smolts to survive. At present, specification of environmental constraints, release sizes, etc., often reflect the hatchery operators intuitive feelings about how to optimize adult production. The problem of objectively setting rearing goals and constraints stands as an important limitation of the present "Rearing Model". This limitation should not be interpreted as a criticism; i t is a natural consequence of the simple deterministic approach taken to predict such factors as fish weight and oxygen concentration. It is unlikely that any factor as complex as ocean survival wi l l ever be related deterministically to the rearing environment. As more survival information becomes available both through assessment of production facilities and controlled experiments, the possibility of developing probabilistic relationships between survival and rearing conditions wi l l develop. 109 This is a necessary step in the establishment of optimum rearing criteria. Until such a probabilistic model is developed, rearing objectives must s t i l l be determined through intuition and experience on a site specific basis. The relationships presented in this thesis represent the first stage in the development of a general rearing model. Basically, a conceptual framework has been proposed and quantitative relationships have been pre-sented. Although these relationships have been built frcm a detailed analysis of fundamental information, they have only been tested against a limited amount of data. In effect, one loop of Van Gigch1s (1974) "model validation cycle" has been made (see Figure 6). To accomplish this, a minimum of computing equipment has been used. Individual equations are simple enough that they can be conveniently ex-plored with a hand-held programmable calculator. The next step in the evolution of the model should involve integrating the equations into a single flexible program. This would allow rapid comparison of model predictions with new data; deviations would eventually lead to the modification and refinement of the present equations. Development of an integrated, flexible program would also allow the model to be used in solving hatchery operations and design problems. At present, using equations one at a time on a programmable calculator is awkward; requir-ing an intimate knowledge of the model. Within the bounds of the present deterministic model, same serious weaknesses exist. Pond environmental conditions are based on the average effluent concentrations. If pond hydraulic characteristics deviate sub-stantially from those of ideal containers, metabolite concentrations could 110 be much higher in specific areas of the pond than predicted by equations 6 and 9. Additional "tracer" information is necessary to identify significant deviations from ideal flow conditions. Water velocity considerations have also been omitted from the structure of the model. Although rearing pond velocities were not considered an important factor to fish growth (Stauffer, 1973)* i t is felt that velocity may have important effects on fish "condition". Velocity may also affect the solids concentrations in ponds (Westers, 1977). Westers maintains that low water velocities (0.25 ft/sec) allow solids to be removed (by settling on bottom), thus leading to improved environmental conditions. Burrows (1970) on the other hand, recommends relatively high water velocities in circulat-ing ponds in order to achieve a self cleaning action. Although there is no agreement on the lower water velocity limits, upper limits impose obvious constraints. Flow requirements must not only satisfy water quality criteria, they must also establish velocity patterns which allow the fish to hold and maneuver. It is these upper velocity limits that often prevent fish culturists from forcing a high flow through a limited rearing volume. The present model could be refined to predict oxygen concentrations below the critical oxygen level "Cr" (see Figure 8) . As noted previously, oxygen predictions neglect the effects of oxygen concentration on the rate of oxygen consumption. This leads to inflated uptake values at oxygen concentrations below the incipient limiting-tension. I l l This problem can be i l l u s t r a t e d by an example. I f pond outflow oxygen lev e l s are above "Cr", then the outflow concentration "x 0" i s a simple l i n e a r function of the load rate "L" (see equation 6, page 23);, x 0 = X j _ - ROL/60. At outflow concentrations below "Cr" however, "x 0" values are predicted using the " f i r s t order" equation 7 (see page 30). Separating variables and solving t h i s equation over the appropriate l i m i t s of integration gives: (85) since L = Dt, ln(Kx n+b) - ln(KCr+b) = - K [L-L P J . (86) 60 L r where: Cr = c r i t i c a l oxygen concentration L C r = "I"oad r a t e r e c l u i r e c * t o S i v e t n e c r i t i c a l oxygen concentration at the pond outflow x Q = outflow oxygen concentration at load rate "L". I f t y p i c a l R , x^, Cr, K, and b values of 280 mg , 11.33 mg/l, 6.5 mg/l, kghr 43 l i t e r s / k g h r , and 0 mg/kghr are assumed respectively, then outflow oxygen concentrations as a function of load rate "L" can be predicted: x 0 * 11.33 - 4.67L , i f x Q > C r (or L ^ L ^ ) x Q = 0.0233 exp(6.375 - 0.717L) , i f x Q < C r (or L > L C r ) . These equations are plotted i n Figure 36. The dashed l i n e represents the "low" estimates of pond outflow oxygen concentrations based on the assumption that uptake rates are independent of the oxygen concentration. This omission does not severely affect the accuracy of model predictions unless rearing i s being carried out at oxygen levels below 5 mg/l (Figure 36), x D = 11.33 - A.67L •Cr" 0 0.5 1.0 1.5 Load Rate (kg per l/min) 2.0 2.5 FIGURE 36. Oxygen concentration in the pond outflow as a function of the load rate. As the pond dissolved oxygen level falls below "Cr", the oxygen uptake rate is suppressed by the reduced environmental oxygen level. The "solid" line incorporates this effect, while the "dashed" line assumes that the uptake rate is independent of the oxygen concentration. 113 Incorporating this refinement would require detailed uptake vs concentration relationships at oxygen levels below Crj such relationships, being complex, would probably be affected by water temperature and fish weight. With pond oxygen levels most often being the first limiting factor, maximum load rate projections will usually be based on equation 21; x Q = x i - RoL/60 , or: L = (x. - x )60 ( 87) Ro where: x^ = inflow oxygen concentration (mg/l) x 0 = outflow oxygen concentration (mg/l) Ro = average daily oxygen consumption rate ( mg ) kghr L = load rate (kg per l/min). By setting the outflow oxygen concentration "x0" at the desired level, the allowable load rate can be calculated. It should be noted however, that the consumption rates "Ro" and therefore the pond outflow concentrations, are average daily values. This implies that for half of the day, oxygen levels may be below the levels specified in equation 87. This problem was approached by applying a safety factor to the average daily consumption rate "Ro". The safety factor was developed from oxygen data taken at the Quinsam Hatchery and guarantees that oxygen concentrations wil l exceed the specified levels for no more than 2.4 hours per day (10%). This factor reflects daily oxygen consumption peaks which in turn reflect both fish cultural activities around the ponds and the daily temperature pulse. Because daily temperature regimes vary widely between water supplies, this safety factor should be considered site specific. 114 There are several obvious limitations associated with the proposed model. The pH equation involves a simplification in that only the bicar-bonate/ carbonate buffer system has been considered. Although this approx-imation is adequate for many natural water supplies, i t may be suspect in the case of ground water. Also, in modelling pond suspended solids and ammonia-N output, bacterial action on the pond bottom and walls has not been considered. This is probably an insignificant factor in concrete ponds but in the case of large semi-natural channels bacterial or algal action would probably have important effects on output rates. Further-more, none of the equations have been refined to the point where they are sensitive to species differences. The ammonia-N excretion model is based on the difference between the nitrogen in the ration and the nitrogen assimilated as growth. It was found that measured ammonia values only accounted for about 40% of this theoretical nitrogen waste. Therefore, a scaling factor was derived to relate ammonia-N excretion to the theoretical waste nitrogen. This implies that 62% of the nitrogen input could not be accounted for as growth or as ammonia-N. This discrepency was assumed to represent other nitrogenous waste products (e.g. urea), fecal losses and direct food wastage. The scaling factor unfortunately, has only been tested over a limited range of conditions; additional data is required for small fish at high ration levels. The simple deterministic approach used to develop this model necessarily concentrates on parameters that are easily quantified. Consequently, chemical and physical needs have been stressed. Such factors as the degree of bank cover, diet composition, construction material, access to a range of water velocities and fish densities and access to natural food have not been discussed. 1 1 5 These factors may exert significant indirect effects. To date, their role is not clear as successful rearing programs have been carried out in both highly a r t i f i c i a l concrete ponds and in semi-natural channels. The present "Rearing Model" has attempted to relate the prime fish culture parameters (fish weight, water temperature, ration level, and hatchery disturbance) to factors which are vital to rearing success. It stresses the effects of ration level because this factor has often been omitted in previous modelling efforts and because i t is easily manipulated by fish culturists. The model also emphasizes the dynamic nature of the rearing process; variations in oxygen level, fish weight, etc., can easily be projected over the entire rearing period. Hopefully, this undertaking has at least established a framework around which this complex process can be understood more clearly. This framework should remain intact although the details w i l l undoubtedly change as new information becomes available. BIBLIOGRAPHY ALDERDICE, D. F. Resources Service Branch, Fisheries and Marine Service, Dept. of Fisheries and Oceans Canada, Pacific Bio-logical Station, Nanaimo, B. C, personal communication. BILTON, H. T. 1978. Returns of adult coho salmon in relation to mean size and time at release of juveniles. Fish Res. Board Can. Tech. Rep. 8 3 2 : 73 p. BRETT, J. R., J. E. SHELBOURN AND C. T. SHOOP. 1 9 6 9 . Growth rate and body composition of fingerling sockeye salmon, Oncorhyn-chus nerka, in relation to temperature and ration size. J. Fish. Res. Board Can. 26: 2363-2394. BRETT, J. R. 1 9 6 5 . The relation of size to rate of oxygen consump-tion and sustained swimming speed of sockeye salmon (Oncorhyn-chus nerka). J. Fish. Res. Board Can., 2 2 : 1 4 9 1 - 1 5 0 1 . BRETT, J. R. 1 9 7 6 . Feeding metabolic rates of sockeye salmon, Oncorhynchus nerka, in relation to ration level and temperature, Envir. Can. Fish. Mar. Serv. Tech. Rep. No. 675* 18 p. BRETT, J. R. AND T. D. D. GROVES. 1 9 7 9 . Physiological energetics, p. 2 7 9 - 3 5 2 . In W. S. Hoar and D. J. Randall [ed.] Fish physiol-ogy. Vol. VIII. Academic Press Inc., New York, N. Y. BRETT, J. R. AND C. A. ZALA. 1975. Daily pattern of nitrogen excre-tion and oxygen consumption of sockeye salmon (Oncorhynchus nerka) under controlled conditions. J . Fish. Res. Board Can. 3 2 : 2 4 7 9 - 2 4 8 6 . BRITISH COLUMBIA RESEARCH COUNCIL. 1 9 7 6 . Enhancement of the Water Reclamation System for Abbotsford Trout Hatchery. Report pre-pared by: Division of Applied Biology, B. C. Research, Vancouver, B. C. for British Columbia Dept. of Public Works. BROWN, W. C. AND G. BRENGEIMANN. 1 9 6 5 . Energy metabolism, p. 1 0 3 0 - 1 0 4 9 . In T. C. Ruch and H. D. Patton [ed.] Physiology and Biophysics. W. B. Saunders Co., Philadelphia and London. BURROWS, R. E. 1 9 7 0 . The rectangular circulating rearing pond. Prog. Fish-Cult., 3 2 : 6 7 - 8 O . BURROWS, R. E. 1964. Effects of accumulated excretory products on hatchery reared salmonids. U. S. Fish and Wildlife Service, Re-search Report No. 6 6 . 1 2 p. BURROWS. R. E. AND B. D. COMBS. 1968. Controlled environment for sal-mon propagation. Prog. Fish-Cult., July: 1 2 3 - 1 3 6 . 117 CLARK, d. W., W. VIE3SMAN (Jr.) AND M. J. HAMMER. 1971. Water Supply and Pollution Control. International Textbook Company, Scranton, Pennsylvania, 66l p. DAVIS, J. C. 1975. Minimal dissolved oxygen requirements of aquatic l i f e with emphasis on Canadian species: a review. J. Fish. Res. Board Can. 32: 2295-2332. ECKENFELDER, W. W. (Jr.). 1970. Water Quality Engineering for Prac-ticing Engineers. Barnes and Noble, Inc., New York, N. V/. 328 p. ELLIOTT, J. W. 1969. The oxygen requirements of chinook salmon. Prog. Fish-Cult., 31: 67-73. ENVIRONMENTAL PROTECTION AGENCY. 1972. Water Quality Criteria, 1972. U. S. Environmental Protection Agency, Washington, D. C. 594 p. FISHERIES AND MARINE SERVICE. 1978. A Report on the Proceedings of the Salmonid Culture Managers Conference. Dept. of Fisheries and Environment, Vancouver, B. C. FRASER, F. J. Resource Service Branch, Fisheries and Marine Service, Vancouver, personal communication. FRQMM, P. 0. 1963. Studies on renal and extra-renal excretion in a freshwater teleost, Salmo Gairdneri. Comp. Biochem. Physiol. 10: 121-128. FROMM, P. 0. AND J. R. GILLETTE. 1968. Effect of ambient ammonia on blood ammonia and nitrogen excretion of rainbow trout (Salmo Gairdneri). Comp. Biochem. Physiol. 26: 887-896. GROVES, T. D. D. 1970. Body composition changes during growth in young sockeye (Oncorhynchus nerka) in fresh water. J. Fish. Res. Board Can. 27: 929-942. HARTMAN, J. 1976. Ammonia production and oxygen consumption of Brown Trout (Salmo trutta fario) in three pass water reuse system. Michigan Department of Natural Resources, Lansing, Michigan 48909. KLONTZ, G. W., I. R. BROCK AND J. A. McNAIR. 1978. Aquaculture Tech-niques: Water Use and Discharge Quality. Research technical com-pletion report, project A-054-IDA, Idaho Water Resource Research Institute, University of Idaho, Moscow, Idaho. 114 p. LEVENSPIEL, 0. 1972. Chemical Reaction Engineering, 2nd ed. John Wiley and Sons, Inc., New York, N. Y. 578 p. LIAO, P. D. 1971. Water requirements of salmonids. Prog. Fish-Cult., 33: 210-220. 118 LIAO, P. B. AND R. D. MAYO. 1974. Intensified fish culture combining water reconditioning with pollution abatement. Aquaculture, 3: 61-85. LLOYD, R. AND D. W. M. HERBERT. I960. The influence of carbon dioxide on the toxicity of urn-ionized ammonia to Rainbow Trout (Salmo Gairdnerii Richardson). Ann. appl. Biol. 48, (2): 399-404i McLEAN, W. E. AND F. J. FRASER. 1974. Ammonia and urea production of coho salmon under hatchery conditions. Dept. of Environment, Fisheries and Marine Service, Vancouver, B. C. Report No. EPS 5-PR-74-5. MOBERLY, S. A. AND L. ROBERT. 1977. Japan Salmon Hatchery Review. Alaska Department of Fish and Game, Division of Fisheries Rehab-ilitation, Enhancement and Development. 124 p. NEMEROW, N. L. 1974. Scientific Stream Pollution Analysis. McGraw-Hill Inc. New York, N. Y. ROYAL, L. A. 1972. An examination of the anadromous trout program of the Washington State Game Department. Department of Game, Olympia Washington. 176 p. SIGMA RESOURCE CONSULTANTS LTD. 1979. Summary of water quality criteria for salmonid hatcheries. Report to the Department of Fisheries and Oceans, Vancouver, B. C. SPEECE, R. E. 1973. Trout metabolism characteristics and the rational design of nitrification facilities for water reuse in hatcheries. Trans. Amer. Fish. Soc. No.2r 323-334. STAUFFER, G. D. 1973. A growth model for salmonids reared in hatchery environments. PhD. Thesis. University of Washington, Seattle, Washington, 213 P» STUMM, W. AND J. J. MORGAN. 1970. Aquatic Chemistry, Wiley-Inter science, New York, N. Y. 583 p. THURSTON, R. V., R. C. RUSSO AND K. EMERSON. 1974. Aqueous Ammonia Equilibrium Calculations. Fisheries Bioassay Laboratory, Montana State University, Bozeman, Montana, Tech, Rept. No. 74-1. (republished in J. Fish. Res. Board Can., 32:: 2379-2383, 1975). VAN GIGCH, J. P. 1974. Applied General Systems Theory. Harper and Row, Publishers, New York, N. Y. 439 p. WAYLAND, H. 1975. Differential Equations Applied in Science and Eng-ineering. D. Van Nostrand Company Inc. New York, N. Y. 353 P« 119 WEBER, W. J. (JR.). 1970. Physicoehemical Processes for Water Quality Control. Wiley-Interscience, New York, N. Y. 640 p. WEDEMEYER, G. A. 1974. Stress as a predisposing factor in fish diseases. U. S. Fish and Wildlife Service, Division of Cooperative Research, Washington, D. C. 8 p. WEST, C. Enhancement Services Branch, Fisheries and Marine Service, Dept. of Fisheries and Oceans Canada, Vancouver, B. C, personal communication. WESTERS, H. AND K. H. PRATT. 1976. The rational design of fish hatch-eries based on characteristics,of fish metabolism. Michigan Depart-ment of Natural Resources, Fisheries Division. 10 p. WILLOUGHBY, H., N. LARSEN AND J. T. BROWN. 1972. The pollutional effects of fish hatcheries. American Fishes and U. S. Trout News, 17: 6-20. WILSON, D. Enhancement Services Branch, Fisheries and Marine Service, Dept. of Fisheries and Oceans Canada, Vancouver, B. C, personal c cmmunication. WINDELL, J. T., J. W. FOLTZ AND J. A. SAROKON. 1978. Effect o f f i s h size, temperature and amount fed on nutrient digestibility of a pelleted diet by rainbow trout, Salmo gairdneri. Trans. Amer. Fish. Soc. 107: 6l3-6l6. 120 Appendix I Fourier analysis can be used to obtain a mathematical -approximation of a periodic empirical curve (Wayland, 1957). If the temperature curve is described by mean weekly or monthly temperatures and the curve is assumed to be periodic with period "P", then the continuous representation of the temperature curve "f( t ) " is given by: An adequate approximation of the curve can often be obtained i f only five coefficients are evaluated by numerical integration. This operation can be easily performed on most programmable calculators using Simpson's or the Trapazoidal rule. The continuous approximation to the temperature regime becomes: where the fourier coefficients a, o* an* n* can be calculated from: This series can be expanded i f a closer approximation is desired. 121 Appendix II Proximate analysis of some typical hatchery diets as stated by the manufacturer; Oregon Moist Swedish Ewos Pellet (Astra Chemicals) Abernathy Protein (%) > 35.0 52.0 > 45 Moisture (%) 4 35.0 7.0 4 10 Carbohydrate (%) 17.5 Fat (%) £.5.0 10.0 > 8 Ash (%) Cellulose (%) ^ 4.0 3.5 122 Appendix III The growth equation could be solved explicitly (weight vs time) i f the ration level was expressed in terms of the fraction of the difference between the maximum and maintenance levels, rather than as a fraction of the maximum ration itself. At present, a feeding level of "0.6" means "0.6" of the maximum ration (0.6Fmax). In this notation, an unknown fraction (which depends on temperature and fish weight) is involved in maintenance while the remainder represents the potential for growth. Another approach might involve "Fmax - Fma". In this case, a feed-ing level of "0.6" would mean "0.6" of the difference between maximum and maintenance rations (0.6(Fmax - Fma)); the actual daily feed rate "F" would be: F = Fma + 0.6(Fmax - Fma). If this approach to feeding was adopted, explicit solutions to the growth equations could be simply obtained. At feeding level the daily feed rate "F" would be: F = Fma + ot(Fmax - Fma) (l) Substituting this expression into equation 12_(see page 36) gives: g = GsinTf_oc (2) 2 If temperatures are within the range of linearity of the growth polynomial (equation 13), then the average daily temperature "Ta" over a time period may be considered. Therefore: G = Aw~B (3) where:: A * 0.0304(aj+a2(l.8Ta+32) + ....(see equation 13) w = fish weight (gm) B = 0.3333 Ta = average daily temperature (°C) t = time (days) 123 Substituting aquation 3 into equation 2 gives: g = Aw""BsinK tx = 1 dw (4) 2 wdt Solving for "w" yields a simple expression for weight against time: w = (w_B + ABsinT£* t ) X / B (5) 2 If the average daily temperature over a time period is 10°C, then the expression for "w" vs "t" becomes: w = ( w ° , 3 3 3 3 + 0.0101741sinTXoct)3*0003 (6) 2 At a feeding level of c<= 0, only the maintenance ration would be fed (F = Fma) and growth would be G. At ©<. = 0.5* the daily feed rate would be "F = Fma + 0.5(Fmax - Fma)" and the prediction of weight vs time would become: w - ( w 0 0 s 3 3 3 3 + 0.0071942t) 3 # 0 0 0 3 (7) Appendix IV The small error that results when the "incremental method" is used to solve the growth equation can be eliminated by means of a correction factor. The problem arises because the specific growth rate decreases continuously as the fish weight increases. Consider growth over a 5-day time increment. The final weight "w^" is predicted from equation 11:: wx = w Qe 5 g (1) where: wQ = i n i t i a l weight g = specific growth rate However, "g" is not only based on the ration level and average daily temperature over the 5-day interval, but also on the i n i t i a l weight "w0" (see equations 12 and 13). If the average daily ration level and temperatures are known, then "g" is given by: g - Aw'0*3333 (2) where: A = constant derived frcm equations 12 and 13. It can be seen that the growth rate decreases from g Q = Aw0""°*3333 at the beginning of the time interval to g^ = Aw^ ""°*3333 f i v e days later. Use of "g 0" over the entire interval introduces a small positive error; this error is compounded over each increment. The error can be eliminated by using the average growth rate "g" rather than "g0" in equation 1. "I" is given by: g = cg D (3) where: c = correction factor g 0 = i n i t i a l growth rate g = average growth rate over the 5-day increment, g is also given by: A \ w-°' 3 3 3 3dw , A L ° » 6 6 6 7 . w 0.66671 ( 4 ) (w, - wj \ -0.6667K-™ ) L 0 J 1 o' Jw 0 1 O' 125 If "w " is approximated by "UQe^g°" and "g 0 = Aw0~0*333", then: c - g - e 3' 3 3 3 5 go - 1 i (5) go O.6667(e5go-i) The corrected weight at the end of a time increment is given by: W l = w o e 5 C S ° <6) where: c= correction factor calculated from equation 5. If the exact expression for w^ , -(w^  = w e 0 ^ 0 ) , is substituted into equation 4, the expression involving the correction factor "c" becomes: c = e3.3335g0c . , ^ ( ? ) O.6667(e5cgo-1) "c" can be solved to any degree of accuracy by expanding the exponentials in a MacLaurin series. For example, i f the exponentials are expanded to the third term: e3.3335gQc s x + 3 #333 5 g o C + (3^3335^2 ( g ) 2 e^goc * ! + 5 g o c + (5g o C)2 (9) 2 and substituted into equation 7, "c" can be solved for: c = 5.556lgQ - 3.3335 + (11.1122 + 74.0797go + 30.8702g/)2 . (10) l6.6675g0 Different methods of making growth projections are compared in Table 10. In order to make exact predictions using equation 19, an in i t i a l weight of 1 gram, a maximum ration and a constant temperature of 10°C have been assumed. Use of correction factors given by equations 5 or 8 gives results that are within 0.1 % of the exact values. It should be noted that the corrected incremental method allows accurate, and general solutions of the growth equations to be made. TABLE 10. A ccmparison of results as a function of the method used to solve the growth equation; % error = P - E x 100%, c = correction factor. E Exact (gm) Time • "E" Incremental Method ( T=5 days) (days) (Equation 19) c •= 1 % error c=equation 5 % error c=equation 8 % error 100 8.2125 8.4381 2.75 8.2055 -0.085 8.2226 0.123 200 27.9604 28.7635 2.87 27.9397 -0.074 27.9906 0.108 400 130.3596 133.6157 2.50 130.2908 -0.053 130.4572 0.075 127 Appendix V The oxygen deficit incurred as water moves through a rearing pond ideally depends on the oxygen consumption of the fish and the reaeration effects of the pond. The rate at which the deficit is generated can be expressed by the Streeter Phelps equation: dD = A - kD (1) dt where A = rate of oxygen concentration decrease ( mg ) Imin k = aeration constant (min" ) D * oxygen deficit (mg/l) or the difference between saturated value and actual concentration. The rate of decrease in the oxygen concentration "A" can be related to fish culture parameters by: A » RoQL (2) V60 where: Q = flow (l/min) V = volume (l) L = load rate (kgitin or kg per l/min) 1 Ro = oxygen consumption ( mg ) kghr Substituting equation 2 into equation 1 and solving for the oxygen deficit at the pond outflow "Do" yields: Do = Ro^(l-e- k V/ Q) +Die" k V/Q (3) VoOk where: Do = deficit at outflow = x a - x Q Di = deficit at inflow = x s - x^ x g = saturation values Substituting expressions for "Do" and "Di" into equation 3 and solving for "xo" gives: x 0 = x s - RoLQ (1 - e-^/Q) - (x s - x J e ^ / Q (4) VoOk If the inflow water is 100 % saturated (XJ_ = x s ) , equation 4 reduces to: x o - x - RoLQ(l - e" k V/ Q) (5) ° 3 V60k Reaeration effects can be estimated i f "k" is approximated by: k = CUn (Weber, 1970) (6) where: k = aeration constant (day"1) U = average stream velocity (ft/sec) H = average stream depth (ft) n, m, and C are constants. In this example, "n", "m", and "C" are assumed to be 0.5, 1.5, and 12.9 respectively (Eckenfelder, 1970; Clark et al, 1971). Consider a typical pond with a mean residence time of 30 minutes, a mean water velocity of 1 ft/sec and an average depth"of 3 feet. Assume the pond is loaded at 1.2 kg per l/min and the fish are consuming oxygen at a rate of 300 mg/kghr. The combined effects of oxygen consumption and reaeration are predicted by substituting appropriate values into equations5 and 6: x Q = x. - 5.85 If the inflow oxygen concentration is 11 mg/l, the outflow level, in this case, would be 11 - 5.85 or 5.15 mg/l. If reaeration is not considered, the outflow concentration would be simply given by equation 21: x Q = Xj_ - RoL = 11 - 6 = 5.0 mg/l. o0~ In this example, reaeration effects (0.15 mg/l) are small compared to the 6.0 mg/l deficit caused by the metabolism of the fish. Neglecting reaeration effects in simple flow-through rearing ponds would not introduce serious errors in predicting outflow oxygen concentrations. 129 Appendix VI Carbon dioxide production tends to lower the pH of natural waters. The effects can be quantified i f the carbonate-bicarbonate system i s assumed to be predominant. This i s usually a valid assumption for natural waters i f a conservative pH range (6.5 to 8.5) i s considered. In aqueous carbonate systems, carbon can be i n the ILjCO^, CC^', or HCOj form depending on pH and temperature. The notation B^CO^ denotes the t o t a l analytical concentration of dissolved carbon dioxide. H2C03" = [CG 2(aq) + H 2C0 3] (l) It should be noted that less than 0.3 % of the carbon dioxide i s hydrated (H 2C0 3) at 25°C (Stumm and Morgan, 1970). If an amount of carbon dioxide "C" i s added to a solution, the pH drops such that:: C = ( C T ) p H l - ( CT)pH 2 ( 2) where: (CT)pH^ = t o t a l concentration at pH-j_ (C-r.)pjj2 = t o t a l concentration at pH2. Note: C T = [H2C03*] + [HC03"] + [C0 3 =] (3) Let od = [HC03"] (4) Cip <*2 " Cg0 3=] (5) but, \ = [H +I][HC0 3~]/[H 2C0 3 ] ,1st'ionization constant (6) and, K 2 = [H +][C0 3 =] /[HCC>3~ ] , 2nd ionization constant (7) so, [HC03~] = K ^ C C y ] (8) [H +] and, [C0 3 =] = K2[HC03"]. (9) [H +] 130 Substituting equations 8 and 9 into 4 and 5 gives: « 1 = ( c ! ? + 1 \ k f ( 1 0 ) c t 2 = ( m 2 • [ H i j + i y - 1 ( I D V Ki K 2 K 2 y Neglecting the small amount of ammonia present, the alkalinity can be expressed as: [Alk] = [HCO3] + 2[G03=] = [OH"] - [H+] (12) or [Alk] = C T(«1 + 2<*2) + [ 0 H r ] - [H+] (13) solving for Cp gives: C T = [Alk] - [OH"] + [H+] ( 1 4 ) + 2<y2 This expression for Cp can be substituted into equation 2 to obtain a relationship between carbon dioxide addition and pH drop. Several approximations can be made to greatly simplify this relationship. Approximation 1 In typical natural waters, the numerator in equation 14 can be approximated by: [Alk] = [Alk] - [OH"] + [H+] (15) —Zi. _ o [Alk] is typically about 10 to 10 moles/liteij whereas at pH's in the 6.5 to 8.6 range, the [H+] and [OH ] concentrations would be much smaller. 131 Approximation 2 0^2 c a n D e considered negligible in comparison toCXp even at pH = 8.0: ° ^ = 158 . (16) <*2 This approximation wil l introduce an unacceptable error at high pH's. Also, i f [C0-j~] is neglected, equation 4 becomes: ^ |H2C0f j + LHCO3-J substituting [H2CQ3"] = [H+] into equation 17 [H2C03~] Kx gives j OCi » L _ • (18) % + [H+] These approximations lead to a greatly simplified expression for dp (equation 14). GT = [ A l J c ] = [Alk](K1 + [H+]) ( 1 9 ) (Xl h The relationship between the amount of carbon dioxide produced "C" and the pH change becomes: C = [AlkKK! + fHiJ) - f A l k K ^ + (20) % % Rearranging gives: C = [Alk]£Hjl - [AlkirHol = Cx - C 2 (21) Ki Ki where: C 2 = i n i t i a l carbon dioxide concentration. = final carbon dioxide concentration 132 C l = [Alkl [K*] and (22) Kl C 2 - [Alk] [fl*] , so (23) Kl [Hj] = ; [Hj] - a n d [Alk] [Alk] pHx = -logK1G1 ; (24) [Alk] pH2 = -log . (25) [Alk] [Alk] can be approximated by [HCO^ ] at intermediate pH's. Equations 24 and 25 allow prediction of the effects of carbon dioxide addition on pH levels. 133 Appendix VII Knowledge of the oxygen consumption, nitrogen excretion, and carbon dioxide production rates allow calculation of the rates of oxidation of protein, lipid, and carbohydrate (Brown and Brengelmann, 1965). Constants used throughout the derivation have been taken from Brown and Brengelmann, 1965 and are summarized in the table below. Food Gxygen Carbon Dioxide  liters 0? gm 0? liters CO? gm CO? gm gm gm gm Carbohydrate 0.81 1.157 0.81 1.591 Protein 0.94 1.343 0.75 1.473 Lipid 1.96 2.800 1.39 2.730 Let "Ro", "Rc", and "RNH3" be the oxygen, carbon dioxide, and ammonia-N production rates in units of gram per gram of fish per day and let "C", "L", and "P" denote the weight (gms) of carbohydrate, lipid, and protein oxidized per gram of fish per day. The nitrogen excretion rate "R^ " can be estimated from the ammonia-N production rate "NH-^ " and the knowledge that about 57.5 % of the nitrogen wastes are excreted as ammonia-N (Fromm, 1963). % " 1 - 7 3 9 R N H 3 (1) If protein is assumed to be 16 % nitrogen by weight, then the weight of protein metabolized is given by: Rw = 0 . 1 6 P (2) 134 The oxygen consumption rate "Ro" results from the oxidation of carbohydrate, lipid, and protein. Using the constants previously defined, "Ro" can be expressed in terms of P, C, and L. RQ = 1.157C + 2.8L + 1.343P (3) The carbon dioxide production rate can also be expressed in terms of P, C, and L . RQ + 1.591C + 2.73L + 1.473P (4) Equations 2, 3, and 4 can be solved simultaneously for the weights of protein "P", carbohydrate "C", and lipid " L " oxidized for a given nitrogen excretion, oxygen consumption, and carbon dioxide production rate. P = 6.25RN (5) C = 2.l6lRc - 2.107Ro - 2.207RJJ (6) L = 1.228RQ - 0.893RQ - 2.086RN (7) The total weight (gm) of material oxidized per gram of fish per day is given by "M": M = P + C + L = 1.957RN + 1.268RC - 0.879RQ (8) If a standard R.Q. is assumed and RQ = 1.238RQ (equation 8), "M" reduces to: M = 1.957% + 0.691R0. 

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