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The crop water use and nutrient uptake is a central subsystem of aquaponics. The HP part is complex, as pure uptake of water and dissolved nutrients do not simply follow a rather simple linear relationship as, e.g. fish growth. To create a full-functional model, a complete greenhouse simulator is needed. This involves sub-model systems of greenhouse physics including climate controllers and crop biology covering interactive processes with biological and physical stressors.

However, from the HP point of view, greenhouse climate is the main driver for the complete aquaponic system, including, next to the nutrient balances, feedback loops of heat produced by the fish and additional COsub2/sub supplied to the plants as reported by Körner et al. (2017) (Fig. 11.15).

In this model, the fish culture produces heat through metabolic processes. The amount of heat produced by the fish is directly calculated from oxygen consumption that is a function of temperature and a constant for heat production for one unit oxygen consumed (i.e. 13608 J gsup-1/sup fish). Heat from breakdown of organic matter (Qsubbio/sub), e.g. faeces and feed remain, is also contributing to the heat balance. Energy supply to the water system can then be calculated by heat production through the fish calculated from an average oxygen consumption rate (fsubO2,Twb/sub). Additional heat production can then be calculated by biological breakdown of faeces (Fig. 11.16).

COsub2/sub production from the aquatic subsystem (dsubCOsub2/sub/sub, g hsup-1/sup ), i.e. delivery to the aerial environment (d, g hsup-1/sup), can be calculated for the given water temperature (TsubH2O/sub, K) from oxygen delivery to the system (dsubOsub2/sub/sub, g hsup-1/sup) at water base temperature (TsubH2O,b/sub, K) and the Qsub10/sub value of fish respiration (Qsub10,R/sub). The following relationships are used:

Fig. 11.15 Additional symbiotic behaviour of an aquaponic system

$d{O2} = f{fish}\ f{O2} W{O_2}$

$d{CO2} = \frac{[CO2]}{[O2]}\ \ d{O2}\ Q{10,R}(T{H2O,b})/10$ (11.16)

with feed amount for the fish (fsubfish/sub, g hsup-1/sup), oxygen consumption rate at base temperature (fsubO2/sub, kg [Osub2/sub] kgsup-1/sup [feed]), fraction of feed loss wsubO2/sub (sup-/sup) and mass balance of Osub2/sub/COsub2/sub (sup-/sup).

To calculate the basis of aquaponics, i.e. the process flow (indicated with arrows, → ) greenhouse macroclimate → microclimate → evapotranspiration → nutrients uptake, various greenhouse simulators that were developed in the past can be used and combined with aquaculture to an aquaponic system. All greenhouse models include a crop growth model. The model quality, however, can vary a lot from simple empirical regression models, e.g. Boote and Jones (1987), via deterministic models, e.g. Heuvelink (1996), to functional structural plant models (FSPM), e.g. Buck-Sorlin et al. (2011). As current crop growth and development models are inaccurate and have limited predictive power (Poorter et al. 2013), models are occasionally employed in crop management, but then mainly for planning issues in greenhouse simulators, e.g. Vanthoor (2011) and Körner and Hansen (2011). Prediction accuracy is jeopardized by many sources of uncertainty, such as modelling

Fig. 11.16 Aquaculture system implemented in the greenhouse with humidity, temperature and CO2 concentrations of the air (RHsubair/sub Tsubair/sub, COsub2,air/sub), heat from (Q) fish environment (fish), biological breakdown (bio) and heat fluxes ($ɸ$​), taken from Körner et al. (2017)

errors, variability between plants, variability between greenhouses and uncertain external climate conditions. As for predictions, accuracy also varies strongly per situation. However, online feeding of sensor information into the plant model can make plant model predictions considerably more reliable and useful to the grower.

Greenhouse simulators were developed in and for several places, e.g. the Virtual Grower (Frantz et al. 2010), KASPRO (De Zwart 1996), Greenergy Energy Audit Tool (Körner et al. 2008), The Virtual Greenhouse (Körner and Hansen 2011), The Adaptive Greenhouse (Vanthoor 2011), Hortex (Rath 1992, 2011) and the integrated aquaponic greenhouse model (Goddek and Körner 2019). At a research level, some models (i.e. simulation models in combination with certain greenhouse technologies) have been developed that potentially can be used to optimize investments and structural modifications to the production unit and production process. However, most systems entail closed software environments that can only be used by the developers, and many of them only exist in a research mode and lack further development and acceptance from the industry. However, there is yet no common basis for model sharing and collaborative model development. As a result, most modellers and modelling teams work in isolation developing their own models and codes. A shortcoming of that procedure is that greenhouse simulation models are developed in parallel in disparate research environments, which fail in cooperative growth and development.

All HP greenhouse model simulators are a compilation of sub-models that depend on the aim to integrate the interaction of plants and greenhouse equipment. The general two-part differentiation in greenhouse models and also in control and planning is the shoot and the root environment. Rather complicated and differentiated model approaches have been done for the greenhouse climate (Bot 1993; de Zwart 1996), and greenhouse crop growth has been intensively modelled in the 1990s for the main greenhouse crops such as tomatoes (Heuvelink 1996), cucumber (Marcelis 1994) and lettuce (Liebig and Alscher 1993). However, in order to calculate the water and nutrient uptake of crops, the microclimate, i.e. the climate close to and on the plant organs, needs to be known (Challa and Bakker 1999). This is an ongoing issue in greenhouse modelling, as microclimate variables, such as the central leaf temperature, are highly variable and dependent on many parameters and variables. One version of a leaf temperature model used in a crop canopy for crop temperature (Tsubc/sub) integrated over vertical layers (z) by Körner et al. (2007) integrating absorbed irradiative net fluxes (Rsubn,a/sub, Wmsup2/sup), boundary layer and stomata resistances (rsubb/sub and rsubs/sub, respectively, smsup-1/sup) and vapour pressure deficit at the leaf surface (VPDsubs/sub, Pa) in the canopy is shown here, i.e.

$Tc(Z)-Ta = \frac{\frac{1}{\rhoaCP}(rb(Z)+rs(Z))R{n,a}(z)-\frac1{\gamma}VPDs(z)}{1+\frac{\delta}{\gamma}+\frac{rs{z}}{rb(z)}+\frac1{\rhoaCP/4\sigma Ta^3}(rb(z)+r_s(z))}$ (11.17)

with greenhouse air temperature (Tsuba/sub, K), vapour pressure air density (ρsuba/sub, g msup-2/sup), Stefan-Boltzmann constant (σ, Wmsup-2/sup Ksup-4/sup), specific heat capacity of the air (csubp/sub, J gsup-1/sup Ksup-1/sup), the psychrometric constant (γ, Pa Ksup-1/sup) and the slope between saturated vapour pressure and greenhouse air temperature (δ, Pa Ksup-1/sup).

Leaf temperature is the central part of the microclimate model, it has feedback loops to several input variables and especially stomata resistance (often also used as its reciprocal, the conductance), and the calculation needs several simulation steps for equilibrium. For HP, as part of the aquaponic system, however, modelling water and nutrient fluxes is most important. All water and nutrient balances in a closed multi-loop system are controlled based on the evapotranspiration rate of the crop ETsubc/sub (Chap. 8). Commonly ETsubc/sub is calculated as latent heat of evaporation, i.e. in energy terms (λE, Wmsup2/sup), and can be in accordance to leaf temperature expressed in different canopy layer

$\lambda E(z) = (\frac{\frac{\delta}{\gamma}R{n,a}(z)+\frac{\rho _aCp}{\gamma}(\frac1{rb(z)}+\frac1{\rho _aCp/4\sigma Ta^3}VPDs(z))}{1+\frac{\delta}{\gamma}+\frac{rs(z)}{rb(z)}+\frac{1}{\rho _aCp/4\sigma Ta^3}(rs(z)+rb(z))})$ (11.18)

To calculate ETsubc/sub (L msup-2/sup), λE needs to be multiplied with the constant Lsubw/sub (heat of vaporization of water; 2454 ⋅ 103 J kgsup-1/sup) and the specific weight of water (9.789 kN ⋅ msup-3/sup at 20 ˚C).

Fig. 11.17 Input-output system of a greenhouse

Equation (11.18), however, only calculates the water flux through the crop, while the easiest way to estimate nutrient uptake is the assumption that nutrients are taken up/absorbed as dissolved in irrigation water and assuming that no element specific chemical, biological or physical resistances exist. In reality uptake of nutrients is a highly complicated matter. Consequently, to maintain equilibrium, all nutrients taken up by the crop as contained in the nutrient solution need to be added back to the hydroponics system (see Chap. 8). However, Eq. (11.18) only calculates the potential ETc, while too high potential levels can result in a higher transpiration than plants can handle, and then potential water loss may exceed water uptake. For that, the simple assumption of nutrient uptake is not satisfying. As described in Chap. 10, the different nutrients can have different states and change states with, e.g. pH, while the plant availability strongly depends on pH and the relation of nutrients to each other. In addition, the microbiome in the root zone plays an important role, which is not implemented in models yet. Some models, however, differentiate between phloem and xylem pathways. The vast amount of nutrients, however, is not modelled in detail for aquaponics nutrient balancing and sizing of systems, while the easiest way to estimate nutrient uptake is the assumption that nutrients are taken up/absorbed as dissolved in irrigation water and apply the above explained ETc calculation approach.

For control purposes the greenhouse is typically considered as a black box, where outside climate conditions determine the disturbance inputs, COsub2/sub supply, heating and ventilation are the control inputs, and the greenhouse macro- and microclimate define the output of the system (Fig. 11.17).

To control the greenhouse, the actions are directed to minimize the fast impacts of the disturbances, i.e. being ahead of expected changes by smart control. For that, control actions such as feedback and feedforward are used (Chap. 8). The best control, however, can be achieved when using a complete greenhouse model and combine it with weather forecast (Körner and Van Straten, 2008) attaining a modelbased optimal greenhouse climate control, as worked out by Van Ooteghem (2007).

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