Dissolved oxygen model
The 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).
Dissolved oxygen model.
The 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, h
After 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 model
Fish 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 < 1, dimensionless). к is the photoperiod factor (0 > к < 1, dimensionless). δ is the dissolved oxygen factor (0 > δ < 1, dimensionless). φ is the unionized ammonia factor (0 > φ < 1, dimensionless). h is the coefficient of food consumption (g1-m day−1). ƒ is the relative feeding level (0 > ƒ < 1, dimensionless). K is the coefficient of catabolism.h, m, n are constants.
Water temperature affects the food intake14. Caulton15 described the relationship between temperature and feed intake for tilapias. Food intake rate reaches the maximum value when the temperature is in an optimal range. If the temperature is outside the optimal range, the food intake rate decreases. Food intake stops when the temperature is the limit range. The temperature factor (from 0 to 1) can be described as16,17.
$$ tau = EXPleft{ { – 4.6left[ {frac{{T_{opti} – T}}{{T_{opti} – T_{max } }}} right]^{4} } right}quad {text{if}};;{text{T}} prec {text{T}}_{{{text{opti}}}} , $$
(13)
$$ tau = EXPleft{ { – 4.6left[ {frac{{T – T_{opti} }}{{T_{max } – T_{opti} }}} right]^{4} } right}quad {text{if}};;{text{T}} ge {text{T}}_{{{text{opti}}}} $$
(14)
where Tmin is the below this temperature fish stop eating, °C. Tmax is the above this temperature fish stop eating, °C. Topti is the optimum temperature for fish taking food, °C.
The catabolism term is also affected by temperature. The effect is described as18:
$$ K = K_{min } {text{ exp}}left[ {{text{s}}left( {{text{T}} – {text{T}}_{{{text{min}}}} } right)} right] $$
(15)
where Kmin is the coefficient of fasting catabolism at Tmin, g1−n h−1. s is a constant.
The effect of DO on fish growth is described in three stages. When DO is below the minimum limits level, DOmin fish feeding stops. When DO is above a critical level, DOcrit, DO has no effect on feeding. When DO is between DOmin and DOcrit feeding is affected by DO18.
$$ delta = 1.0quad {text{if}};;{text{DO}} succ {text{DO}}_{{{text{crit}}}} $$
(16)
$$ delta = frac{{DO – DO_{min } }}{{DO_{crit} – DO_{min } }}quad {text{if}};;{text{DO}}_{{{text{min}}}} le DO le DO_{crit} $$
(17)
$$ delta = 0.0quad {text{if}};;{text{DO}} prec {text{DO}}_{{{text{crit}}}} $$
(18)
Unionized ammonia, NH3, is toxic to fish19. The effects of unionized ammonia can be simulated using an equation similar to that for DO18. When NH3 is higher than NH3max, then the fish stop feeding. When NH3 is lower than the critical value, NH3crit, then there is no effect on feeding. When the concentration of NH3 is higher than the critical value, NH3crit and lower than a maximum value, NH3max, then food intake will decrease as the concentration of NH3 increases. The function can be decreased as18.
$$ varphi = 1.0quad {text{if}};;{text{NH}}_{{3}} prec NH_{{{text{3crit}}}} $$
(19)
$$ varphi = frac{{NH_{3max } – NH_{3} }}{{NH_{3max } – NH_{3crit} }}quad {text{if}};;{text{NH}}_{{{text{3crit}}}} le NH_{3} le NH_{3max } $$
(20)
$$ varphi = 0.0quad {text{if}};;{text{NH}}_{{3}} succ NH_{{{text{3crit}}}} $$
(21)
Caulton20 indicates that many cultured fish species including tilapias tended to feed only during daylight hours. Photoperiod factor (к), based on 12:12 h of light–dark cycle and used for adjusting daily food consumption, is expressed as follow:
$$ kappa = {text{photoperiod/12}} $$
(22)
where, photoperiod is the day time between sunrise and sunset (h), which can be estimated from sunrise and sunset hour angle calculations21. The constant of 12 is the photoperiod in the 12:12 h of light dark cycle.
The fish growth rate is dependent on the amount of food and the quality of available. To determine the value of the relative feeding level “ƒ” to be used in our case, we used the model at progressive values of “ƒ” starting from zero, step 0.01 up to 1.0 and compare the results with those obtained by22.
Equation is used to calculate the accumulate growth starting by one gram of individual fish to the marketable weight of 250 g.
$$ {text{W}}_{{text{n}}} = W_{n – 1} + FGR $$
(23)
$$ {text{Amount}};{text{of}};{text{feeding}};{text{(kg/day)}} = F_{r} times W_{n} times {text{No}}.;{text{of}};{text{fish/100,000}} $$
(24)
where n is the number of day from the start
All computational procedures of the model were carried out using Excel spreadsheet. The computer program was devoted to mass balance for predicting the dissolved oxygen consumed through aquacultural recirculating system. Figure 2 shows the flowchart of the model. The parameters used in the model that were obtained from the literature are listed in Table 1.
Flowchart of the model.
Source: Ecology - nature.com
