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Dissolved oxygen modelThe dissolved oxygen in this model had a number of interactions to consider. Oxygen consumption through the processes of both respiration and nitrification. On the other hand, the water receives oxygen through water agitation as it is pumped through the system and from the oxygen generator. Oxygen is added to the water by oxygen generator and flow aeration (Fig. 1).Figure 1Dissolved oxygen model.Full size imageThe required oxygen supplementation is a sum of the pervious components as follows:$$ DO_{FR} + DO_{B} + DO_{N} = DO_{sup } + DO_{PF} $$
(1)
where DOFR is the dissolved oxygen consumption through fish respiration, g O2 m−3 h−1. DOB is the dissolved oxygen consumption through the biofilter, g O2 m−3 h−1. DON is the dissolved oxygen consumption through nitrification, g O2 m−3 h−1. DOPF is the dissolved oxygen addition through pipe flow, g O2 m−3 h−1. DOsup is the required oxygen supplementation (oxygen generator), g O2 m−3 h−1.The rate of change in DO concentration in fish tank:$$ frac{dDO}{{dt}} = DO_{FR} + DO_{B} + DO_{N} – DO_{PF} $$
(2)
where (frac{dDO}{{dt}}) is the rate of change in DO concentration during the time interval, g O2 m−3 h−1. dt is the rate of change in the time interval, hAfter calculating oxygen concentration for each element at each time step, the net oxygen change is then added to or subtracted from the previous time step`s oxygen concentration. DO concentrations can be calculated at any time (t) as:$$ DO_{t} = DO_{t – 1} + left( {frac{dDO}{{dt}} cdot dt} right) $$
(3)
where DOt is the DO concentration (g m−3) at time t. DOt−1 is the DO concentration (g m−3) at time t−1.The rate of oxygen consumption through fish respiration can be calculated on water temperature and average fish weight. This calculation is shown in the following equation10:$$ FR = 2014.45 + 2.75W – 165.2T + 0.007W^{2} + 3.93T^{2} – 0.21WT $$
(4)
$$ DO_{FR} = frac{FR times SD}{{1000}} $$
(5)
where FR is rate of oxygen consumption through fish respiration, mg O2 kg−1 fish. h−1. W is average of individual fish mass, g. T is water temperature, °C. SD is the stocking density of fish, kg m−3.The correlation coefficient for the equation was 0.99. Data used in preparing the equation ranged from 20 to 200 g for fish weight and from 24 to 32 °C.The rate of oxygen consumption through nitrification is calculated in terms of Total Ammonia Nitrogen (TAN) that is converted from ammonia to nitrate. The rate found in the literature is 4.57 g O2 g−1 TAN6.The oxygen consumption in nitrification process can be calculated as11:$$ DO_{N} = 4.57 times K_{NR} times {{{text{Nr}}} mathord{left/ {vphantom {{{text{Nr}}} {text{V}}}} right. kern-nulldelimiterspace} {text{V}}} $$
(6)
$$ K_{NR} = 0.1left( {1.08} right)^{{left( {T – 20} right)}} $$
(7)
$$ Nr = frac{{0.03 times F_{r} times W times N_{F} }}{24 times 1000} $$
(8)
where KNR is the coefficient of nitrification. Nr is the nitrification rate, g TAN h−1. Fr is the feeding ratio, % of body fish day−1. NF is the number of fish. V is the water volume, m3.The feeding ratio can be calculated as the following equation:$$ F_{r} = 17.02 times e^{{left[ {{raise0.7exhbox{${left( {ln W + 1.14} right)^{2} }$} !mathord{left/ {vphantom {{left( {ln W + 1.14} right)^{2} } { – 19.52}}}right.kern-nulldelimiterspace} !lower0.7exhbox{${ – 19.52}$}}} right]}} $$
(9)
The bacteria in the biofilter are a second source of oxygen consumption. Lawson explains that the biofilter oxygen demand is approximated 2.3 times the BOD5 production rate of fish6. The oxygen consumption of the biofilter is calculated using following equation:$$ DO_{B} = frac{{(2.3)left( {BOD_{5} } right)left( {W_{n} } right)}}{{left( V right)left( {24} right)left( {1000} right)}} $$
(10)
where BOD5 is average unfiltered BOD5 excretion rate, 2160 mg O2 kg−1 fish day−1. Wn is biomass, kg fish.The water pumping cycle was a source of oxygen addition to the system. The amount of oxygen addition through the water pumping cycle was calculated on an hourly basis. The method of calculating aeration from a pipe is detailed by12:$$ DO_{PF} = frac{PC times f times E times OTR}{V} $$
(11)
where PC is pump cycle length, h. f is pumping frequency, h−1. E is efficiency, %. OTR is oxygen transfer rate, g O2 h−1.This model sums the DOFR, DOB, DON, and DOPF to determine the supplemental DO demand in kg h−1. This number can be used to estimate the oxygen consumption if pure oxygen transfers system is used.Fish growth modelFish growth is affected by environmental and physical factors, such as water temperature, dissolved oxygen, unionized ammonia, photoperiod, fish stocking density, food availability, and food quality.In order to calculate the fish growth rate (g day−1) for individual fish, the following model was used13 as it includes the main environmental factors influencing fish growth. These factors are temperature, dissolved oxygen and unionized ammonia.$$ FGR = left( {0.2919 , tau , kappa , delta , varphi , h , f , W^{m} } right) – K.W^{n} $$
(12)
Where FGR is the fish growth rate, g day−1. τ is the temperature factor (0  > τ  к  δ  φ  ƒ  More

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