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17 min readSizing an aquaponics system requires balancing the nutrient input and -output. Here, we basically apply the same principle as sizing a one-loop system. Yet, this approach is a bit more complicated, but will be fully illustrated with the aid of an example.

Fig. 8.5 Scheme that shows the mass balance within a four-loop aquaponics system; where msubfeed/sub are the dissolved nutrients added to the system via feed. Add labels: QsubDIS/sub - QsubX/sub to distillate returned to HP; 'sludge' for nutrients entering reactor

Figure 8.5 illustrates the mass balance diagram for our system approach. In the optimal situation, the system has only one input and output. However, in practice, one will have to add additional nutrients to the hydroponics part to optimize plant growth. This model can be used to size the system, e.g. based on phosphorus, which is a non-renewable resource (Chap. 2). The input to the system (msubfeed/sub) is the fraction of a nutrient that the fish excrete in a dissolved form. The remainder accumulates in the fish as biomass or ends up as sludge (see previous section). The output is the plant nutrient uptake. Determining nutrient uptake of plants depends on many factors and is very complex; the easiest way to give a rough estimate is to consider plant respiration as the main driver of nutrient uptake (Goddek and Körner 2019).

Evapotranspiration rate is highly climate dependent and is either directly or indirectly influenced by absorbed shortwave radiation, relative humidity, temperature, and COsub2/sub concentration. Due to the high complexity of a multi-loop system, we assume that the plants are located in a climate-controlled greenhouse, and therefore we only need to consider global radiation as the dynamic variable determining how much shortwave radiation is absorbed. In other words, we first need to determine how much of the added nutrients become available for the plants, and then determine how much the plants actually take up.

The fish feed rate depends on the total biomass in the system and the feed conversion ratio (FCR). Timmons and Ebeling (2013) provide a simple approach for determining fish growth rates for different fish species. However, we recommend taking industrial data to determine the biomass more precisely. Lupatsch and Kissil (1998) (Eq. 8.10) provide a general growth formula, for which Goddek and Körner (2019) determined the growth coefficients by curve fitting using the mathematical software environment MATLAB (internal function 'fitnlm') with empirical data for Nile __Tilapia__ (__Oreochromis niloticus__). Additional initial and final weights, water temperature of the system, and the output for the species-specific growth coefficients can be found in Table 8.2. Inserting these parameters into Eq. 8.10 gives us the weight at a specific day for this fish species.

$W__{t} = [W__0^{1-\beta w}+(1-\beta __w)\alpha__wexp{\gamma_wT}T]^{\frac{1}{1-\beta w}}$ (8.10)

where Wsubt/sub (g) is the fish weight at a specific time (days), Wsub0/sub (g) is the initial fish weight, T is the water temperature (in ˚C), αw βw and γw are species-specific growth coefficients (no units), and t is the time in days.

**Table 8.2** Fish growth parameters for Eq. 8.10 for a given water temperature (T). Wsub0/sub and Wsubf/sub can be adjusted to one's own needs

table
thead
tr class="header"
thFunction/th
thParameters/th
thDescription/th
thValue/th
thSource/th
/tr
/thead
tbody
tr class="odd"
td rowspan="4"Fish growth/td
tdWsub0/sub/td
tdInitial weight of the *Tilapia* fingerlings (in g)/td
tdFor example, 55/td
tdGoddek and Körner (2019)/td
/tr
tr class="even"
tdWsubf/sub/td
tdTarget harvest weight of the fish (in g)/td
tdFor example, 600/td
tdGoddek and Körner (2019)/td
/tr
tr class="odd"
tdT/td
tdWater temperature of the RAS (in ˚C)/td
td30/td
tdTimmons and Ebeling (2013)/td
/tr
tr class="even"
tdαw; βw; γw/td
tdSpecies-specific growth coefficients/td
td0.0261; 0.4071; 0.0827/td
tdGoddek and Körner (2019)/td
/tr
/tbody
/table

Based on the output of the equation above, we were able to determine how much feed the fish will require per growth stage. Most of the times, the feed rate (X% of the body weight) or FCR is mentioned by the species-specific feed manufacturer. However, Timmons and Ebeling (2013) provide a rough guideline for FCR for tilapia: 0.7—0.9 for __Tilapia__ that weigh less than 100 g and 1.2—1.3 for tilapia that weigh more than 100 g. This is done via the following equation.

$Feed\ rate\ (g)=FCR\times WG__t\times m__{fish}$ (8.11)

where FCR is the feed conversion ratio, WGsubt/sub is the weight gain (per day), and msubfish/sub is the amount of fish in the fish tank.

The weight gain (WG) per day can be determined with Eq. 8.10 by subtracting the weight of, e.g. day 10 from the weight of day 11. This can be done for each tank. Figure 8.6 shows the fish feed input to the system for __Tilapia__ using the equations above. The average feed input per day after the system is totally cycled is 165 kg.

**Fig. 8.6** Example of biomass balance for __Tilapia__ reared in 13 tanks in cohorts with a total volume (including biofilter and sump) of 482.000 L at a max. Total biomass of 80 t for a period of 2 years including start-up phase with average fish weight (a) (each line represents one tank/cohort) and the daily total feed rate (b) (data taken from Goddek and Körner 2019)

Neto and Ostrensky (2013) report a soluble N excretion of 33% and a soluble P excretion of 17% of feed input when rearing Nile __Tilapia__ (__Oreochromis niloticus__, L.). These are the nutrients that finally accumulate in the RAS system and can be taken up by the plants.

Table 8.3 gives an overview of the crop-specific evapotranspiration (ETsubc/sub) rates that are linked to global radiation. One mm of ET per square meter equals 1 L. For simple sizing, one should take the annual daily average (see next section).

**Table 8.3** Overview of outside global radiation levels subarctic, temperate maritime, and arid conditions (based on Goddek and Körner 2019) and their respective crop evapotranspiration __(ETc, mm day-1)__ rates for __lettuce__ and __tomato__ grown in a controlled greenhouse environment of 20˚C and 80% relative humidity. Lettuce was cultivated with continuous planting year round; tomato was planted in January and removed in December (Faroe Islands and the Netherlands) or July and June (Namibia)

table thead tr th rowspan="3"Month/th th colspan="3"Faroe Islands/th th colspan="3"The Netherlands/th th colspan="3"Namibia/th /tr tr class="header" thGlobal radiation/th thETsubc/sub/lettuce/th thETsubc/sub/tomato/th thGlobal radiation/th thETsubc/sub/lettuce/th thETsubc/sub/tomato/th thGlobal radiation/th thETsubc/sub/lettuce/th thETsubc/sub/tomato/th /tr tr class="header" thmol msup-2/sup daysup-1/sup/th th colspan="2"kg msup-2/sup daysup-1/sup/th thmol msup-2/sup daysup-1/sup/th th colspan="2"kg msup-2/sup daysup-1/sup/th thmol msup-2/sup daysup-1/sup/th th colspan="2"kg msup-2/sup daysup-1/sup/th /tr /thead tbody tr class="odd" tdbJanuary/b/td td1.4/td td0.78/td td0.52/td td4.5/td td0.78/td td0.53/td td54.2/td td2.74/td td4.55/td /tr tr class="even" tdbFebruary/b/td td5.2/td td0.85/td td1.38/td td9.1/td td0.93/td td1.40/td td53.7/td td2.70/td td4.47/td /tr tr class="odd" tdbMarch/b/td td13.7/td td1.20/td td2.12/td td17.0/td td1.28/td td2.14/td td51.2/td td2.42/td td3.96/td /tr tr class="even" tdbApril/b/td td30.6/td td1.90/td td3.05/td td27.9/td td1.82/td td2.90/td td40.2/td td3.05/td td5.38/td /tr tr class="odd" tdbMay/b/td td39.2/td td2.29/td td3.57/td td32.2/td td2.40/td td3.74/td td30.0/td td2.70/td td4.59/td /tr tr class="even" tdbJune/b/td td39.6/td td2.33/td td3.60/td td26.6/td td2.52/td td3.91/td td30.5/td td2.28/td td3.80/td /tr tr class="odd" tdbJuly/b/td td34.5/td td2.17/td td3.37/td td36.4/td td2.54/td td3.51/td td37.5/td td2.61/td td3.92/td /tr tr class="even" tdbAugust/b/td td21.3/td td1.67/td td2.73/td td31.7/td td2.28/td td3.51/td td37.5/td td2.61/td td3.92/td /tr tr class="odd" tdbSeptember/b/td td13.2/td td1.20/td td2.04/td td23.1/td td1.75/td td2.77/td td43.2/td td2.00/td td3.02/td /tr tr class="even" tdbOctober/b/td td6.0/td td0.91/td td1.77/td td13.3/td td1.17/td td1.94/td td51.6/td td2.07/td td3.09/td /tr tr class="odd" tdbNovember/b/td td2.1/td td0.78/td td1.60/td td6.2/td td0.87/td td1.62/td td57.9/td td2.30/td td3.58/td /tr tr class="even" tdbDecember/b/td td0.4/td td0.79/td td1.66/td td3.5/td td0.77/td td1.52/td td59.8/td td2.44/td td3.95/td /tr tr class="odd" td buAverage/u/b /td tdu17.3/u/td tdu1.41/u/td tdu2.28/u/td tdu20.6/u/td tdu1.59/u/td tdu2.50/u/td td3.83/td tdu2.47/u/td tdu3.83/u/td /tr /tbody /table

Balancing the loops is necessary for sizing the system. The input should be equal to the output (Fig. 8.5). In a decoupled aquaponics system incorporating a bioreactor unit, we have two nutrient inflow streams: (1) the fraction of the feed that is excreted to the RAS system in a soluble form and (2) the fraction of the nutrients in the fish sludge that the bioreactor(s) manage to mineralize and mobilize. The major outflow stream (apart from the periodic removal of demineralized sludge) of nutrients is the nutrient uptake of the plants. The differential Eq. 8.12. expresses this balance:

$Mineralization\ (Eq.8.2)+m__{feed}=\frac{\underline{Q__{HP}}\times \rho_{HP}}{1000}$ (8.12)

$(\underline{\eta__{feed}}\times 10000\times \pi__{feed}\times \pi__{sludhe}\times \eta__{min})+\underline{m__{feed}}=\frac{\underline{Q__{HP}}\times \rho _{HP}}{1000}$ (8.13)

wheren $\underline{\eta_{feed}}$ is the average feed (in kg) entering the RAS system, πsubfeed/subis the proportion of the nutrient in the feed formulation, πsubsludge/sub is the proportion of a specific feedderived element ending up in the sludge, and ηsubmin/sub is the mineralization and mobilization efficiency of the reactor system, umsubfeed/sub/u is the average amount of a nutrient that the fish defecate in a dissolved form,uQsubHP/sub/u is the average total evapotranspiration, and ρsubHP/subis the target (i.e. optimal) nutrient concentration for a specific nutrient in the hydroponic subsystem.

However, to be able to determine the required area, there are two variables that need to be redefined in order to solve this equation. Equation 8.14 shows how to calculate the soluble nutrient excretion. In Eq. 8.15, we show that the average total evapotranspiration is a product of the area and the plant-specific evapotranspiration rate (here shown as an average) per msup2/sup.

$\underline{m__{feed}} = \underline{n__{feed}}\times \pi__{feed}\times \eta__{excr}$ (8.14)

where ηsubexcr/sub represents the fraction of the nutrient excreted by the fish in a soluble form.

$\underline{Q__{HP}}=A\times \underline{ET__c}$ (8.15)

where $\underline{Q__{HP}}$ represents the average total evapotranspiration (in L), A the area, and $\underline{ET__{c}}$ the average crop-specific evapotranspiration in mm/msup2/sup (i.e. L/msup2/sup).

Solving Eq. 8.13 by incorporating Eqs. 8.14 and 8.15 to find A, we are able to calculate the required plant area with respect to the average feed input (Eq. 8.15).

$A=\frac{(\underline{\eta__{feed}\times 1000\times \pi__{feed}\times \eta__{excr}\times 1000}) + (\underline{\eta__{feed}}\times 1000\times \pi__{feed}\times \pi__{sludge}\times \eta__{min}\times 1000)}{\underline{ET__c}\times \rho_{HP}}$ (8.16)

Example 8.2For this example, we want to size (i.e. balance) the system with respect to P. We assume that the RAS component of our system requires an average daily feed input of 150 kg. The manufacturer reports the P content of the fish feed to be 1%. We estimate the P ending up in the sludge to be 55% and the P that fish excrete in a soluble form to be 17%. The bioreactors perform quite well and mineralize around 85% of the P.

On the output side, we calculated the average crop-specific evapotranspiration rate for lettuce (by, e.g. using the FAO Penman-Monteith equation). At our location, it is around 1.3 mm/day (i.e. 1.3 L/day). The optimal P composition of the nutrient solution is reported to be 50 mg/L (Resh 2013). Finding the area of plant cultivation needed to uptake the P produced by the system is then solved by:

$A=\frac{(\underline{\eta

{feed}\times 1000\times \pi{feed}\times \eta{excr}\times 1000}) + (\underline{\eta{feed}}\times 1000\times \pi{feed}\times \pi{sludge}\times \eta{min}\times 1000)}{\underline{ETc}\times \rho_{HP}}$$A=\frac{(150000\times 0.01\times 0.17\times 1000)+(150000\times 0.01\times 0.55\times 0.85\times 1000)}{1.3\times 50}$

$\ \ \ =\frac{255000+701250}{65}$

$\ \ \ =14711m^2 = 1.47ha$

The example above shows that the majority of the P in the hydroponics unit originates from the bioreactors. Thus, implementation of a bioreactor within a decoupled system has a very high impact on P sustainability. By contrast, in order to size simple one-loop aquaponics systems, a rule of thumb is usually applied. For leafy plants approx. 40—50 g and for fruity plants approx. 50—80 g of feed is required per msup2/sup cultivation area (FAO 2014). When looking at the feed input in the given example above = 150 kg, and dividing it by 45 (the average of the leafy plant approximation), the proposed cultivation area is around 3750 msup2/sup. Leaving out the sludge mineralization, our example would suggest a cultivation area of 3333 msup2/sup when sizing the system on P.

The role of the distillation unit is to keep the nutrient concentration of the RAS system and the hydroponics system at their respective desired levels. Since nutrient accumulation and the corresponding specific nutrient density are dynamic in RAS systems (i.e. depending on the ETsubc/sub rates) that depend on the QsubHP/sub and QsubX/sub flow (Fig. 8.5), the size of the distillation unit cannot be determined using a differential equation. Thus, a time series model is required to determine the nutrient concentration in the RAS over time. The nutrient concentration at a specific time is necessary to be able to execute mass balance equations within the system (Sect. 8.3).

**Fig. 8.7** Simulations comparing NO3-N concentration in the RAS water system on the impact of distillation flows (no, solid line; 5000 L h-1, dashed line) on hydroponics (yellow, —-) and RAS (blue, —-) nutrient solution concentrations in **(a)** Namibia and **(b)** the Netherlands, i.e. in low and high latitudes (Namibia 22.6˚S and the Netherlands, 52.1˚N, respectively) within a 36-month period (including the system run-up phase) using local climate data and climate-adjusted greenhouses as model input

For the system to be balanced (i.e. input = output), we can give a general guideline on the required capacity of such a distillation unit. The objective is to avoid nutrient accumulation in the RAS system. Figure 8.7a, b shows the impact of distillation flows on the hydroponics and RAS nutrient solution __without a mineralization loop in two different latitudes__. Both systems have the same feed input (in average 158.6 kg daysup-1/sup; see Fig. 8.6). However, by taking the environmental conditions and climate-adjusted greenhouses into account, the necessary and optimal hydroponic area differs between geographical locations (see Chap. 11). Hydroponic systems with low potential evaporation rates, as are common in locations at high latitudes (i.e. far from the equator) would need larger cultivation areas than places closer to the equator. At the same time, a higher annual variation in irradiation and thus transpiration is common in these regions, thus a higher demand on seasonal variability on water and nutrients is present (see Fig. 8.7). In greenhouse cultivation, however, supplementary lighting may be necessary, and in countries such as Norway, vegetable cultivation without supplementary lighting hardly takes place. In addition, the total crop leaf surface makes a difference; crops with a high leaf area per unit ground area (i.e. leaf area index) transpire more than crops with smaller leaf areas, and a distinct difference can be seen between tomato and lettuce crops. All of these factors need to be considered when planning and sizing the aquaponic system.

In the following we provide an overview of the optimized hydroponic area size for the above described aquaponics systems: The cultivation area for monocultures simulated with scenarios in steps of 250 msup2/sup to find the fitting area of either lettuce or tomatoes in order to balance the system appropriately was without supplementary lighting (for lettuce or tomato, respectively):

17.000 msup2/sup or 11.750 msup2/sup for Faroe Islands

15.500 msup2/sup or 11.000 msup2/sup for the Netherlands

8750 msup2/sup or 6500 msup2/sup for Namibia

Even though the size of the systems differs, the average annual nutrient uptake is similar. However, when integrating a digester system, we have to take the additional nutrient source into consideration (Fig. 8.1c). Changing one component inevitably leads to imbalances of the system, yet the system must aim to provide optimal nutrients to both RAS and HP. For example, NO3-N in RAS must be below a certain threshold \200 mg L<sup-1/sup for, e.g. tilapia, while PO4-P in HP should be as close as possible to the recommended concentration of 50 mg Lsup-1/sup for good-quality plant cultivation. Thus, simulation studies help determine sizing of components in a decoupled closed multi-loop aquaponics system in order to achieve optimal nutrient supplies for both fish and plants. For that purpose, Goddek and Körner (2019) created a numeric aquaponics simulator.

However, planning an aquaponic system involves some basic system understanding in order to reach a balance that minimizes the unwanted peaks in nutrient demand and supply. Since the driving force for nutrient dynamics is the evapotranspiration of the crop (ETsubc/sub in the HP system), that is largely driven by microclimate and absorbed light. In a perfectly balanced system, this would be a fully automated and controlled (see Sect. 8.5 Monitoring and Control) environment with 24-h lighting. Plants need a certain dark period of about 4—6 h, so the best-balanced system is realistically to carry out aquaponics in closed plant factories solely with artificial light sources. This, however, demands high electrical input and investment costs and is only feasible with very high product prices. Therefore, we recommend greenhouse production with supplementary lighting (if necessary and if it pays off) as a practical and economically feasible way of building an aquaponic unit. Placing both plants and fish in the same physical construction results in additional synergies including reduced heating and increased plant growth through elevated COsub2/sub (Körner et al. 2017).

In addition to these technical issues, plant cultivation procedures (the practical horticultural part of the system) have to be adjusted to the needs of aquaponics such that there is a constant crop nutrient demand (assuming same climate and light) as shown in Table 8.3. Cultivation of lettuce and other leafy greens are carried out continuously (Körner et al. 2018), while larger crops, like fruit vegetables such as tomato, cucumber, or sweet pepper, are usually sown in winter, and the first harvest is often in late winter/early spring followed by removal of plants and another crop sown for harvest in winter again. Without interplanting, i.e. either various crop types in the same system or batches of fruit vegetables planted throughout the year in order to sustain nutrient demand, periods of low nutrient demand and high nutrient levels will occur. Based on Goddek and Körner (2019), we show the variation of NOsub3/sub-N in RAS for tomato (often not adjusted in aquaponics) and lettuce when no supplementary lighting is used for three climate zones (Faroe Islands, the Netherlands, and Namibia) (Fig. 8.8). System balance is achievable by increasing the daily light integral (i.e. sum of mol light received during a 24-h period) with dynamic supplementary lighting control (Körner et al. 2006).

**Fig. 8.8** NO3-N in RAS combined with HP growing tomato or lettuce in three climate zones and decreasing latitudes (Faroe Islands 62.0N, the Netherlands 52.1˚N, Namibia 22.6˚S) with optimized area for hydroponics (see above) in a 36-month simulation using local climate data and climate-adjusted greenhouses as model input

Applying distillation/desalination technologies can contribute to significant reductions in nutrient levels in the RAS while adjusting levels in the HP system closer to optima, i.e. the unit concentrates nutrients to levels required by plants. Figure 8.9 illustrates the effect of a desalination unit on RAS NOsub3/sub-N concentration when applying between 0 and 5000 L hsup-1/sup and systemsup-1/sup. It is obvious that with increasing desalination flux, the NOsub3/sub-N concentration in the RAS system is decreasing. The unit, however, is controlled by the demand of POsub4/sub in the HP system. Peaks need to be avoided and, as stated above, this can be achieved by creating a stable climatic environment with dynamic light controls. It is obvious that in climate regions with fewer annual differences in solar radiation, there is less variation in ETsubc/sub and the complete system is more stable. Installing lamps and keeping a daily light integral of at least 10 mol msup-2/sup can compensate for seasonal variations. Interplanting and mixed crop production help level the peak resulting from the traditional tomato cultivation protocol with young plants in winter when both

**Fig. 8.9** NO3-N in RAS combined with HP with tomato (right) or lettuce (left) with desalination between 0 and 5000 L h sup-1/sup supply in three climate zones and decreasing latitudes (Faroe Islands 62.0˚N, the Netherlands 52.1˚N, Namibia 22.6˚S) with adjusted area for HP (see above) in a 36month simulation using local climate data and climate-adjusted greenhouses as model input

climate (low radiation) and cultivation (small plants, low potential ETsubc/sub) contribute to nutrient accumulation.