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10 min readGlobal fish aquaculture reached 50 million tons in 2014 (FAO 2016). Given the growing human population, there is a growing demand for fish proteins. Sustainable growth of aquaculture requires novel (bio)technologies such as recirculating aquaculture systems (RAS). RAS have a low water consumption (Orellana 2014) and allow for a recycling of excretory products (Waller et al. 2015). RAS provide suitable living conditions for fish, as a result of a multistep water treatment, such as particle separation, nitrification (biofiltration), gas exchange and temperature control. Dissolved and particulate excretory products can be transferred to secondary treatment such as plant (Waller et al. 2015) or algae production in integrated aquaagriculture (IAAC) systems. IAAC systems are sustainable alternatives to conventional aquaculture systems and in particular are a promising expansion to RAS. In RAS it would be necessary to circulate the process water which has special implications for the process technology in both, the RAS and the algae/plant system. To combine RAS and algae/plant system, a deep understanding of the interaction between fish and water treatment is prerequisite and can be derived from dynamic modelling. The metabolism in fish follows a daily pattern which is well represented by the gastric evacuation rate (Richie et al. 2004). Particle separation, biofiltration and gas exchange are subjected to the same pattern. For design purposes the characterization of the basic components of a RAS treatment system should be investigated through simulation models. These simulation models are highly complex. Available numerical models for RAS capture only a small part of the complexity and consider only a part of the components with corresponding mechanisms. Hence, in this chapter, only a small part of a dynamic RAS model will be presented, i.e. nitrification-based biofiltration. The conversion of toxic ammonia into nitrate is a central process in the water treatment process in RAS. In the following, the dynamic modelling of the mass balance of ammonia excretion of fish and the conversion of ammonia into nitrate will be demonstrated as well as the transfer of the nutrient into an aquaponic system. With this it is possible not only to engineer a RAS but also to integrate fish production into an IAAC system based on valid parameters.

The model is subdivided into a fish model for European seabass, Dicentrarchus labrax, a model describing the time- dependenting excretion of ammonia, and a nitrification model (Fig. 11.8). The fish excretion pattern is introduced into the model through the input vector u (Eq. 11.15), similar to the approach used by Wik et al. (2009). The complexity of the fish model is kept low to be able to explain its method of implementation. Nonetheless, a short introduction into modelling fish is presented in Sect. 11.3.2. Four basic aspects important to describe the nutrient flow in RAS (Badiola et al. 2012) are:

- The flow Q, which is the total process water flow per unit time through the RAS, determines the mass transfer of all dissolved and particulate matter, including ammonia and nitrate.
- The excretion of the fish input ammonia to the RAS process water and is depicted by the product of matrix B and vector u (Eq. 11.15).
- The ammonia conversion into nitrate, taking place in the nitrification, is depicted in the nitrification vector n (Eq. 11.15).
- The nutrient transfer from the RAS to a connected HP system is depicted in vectoru (Eq. 11.15). Other important aspects of the RAS process chain such as solid removal, dissolved oxygen concentration and carbon dioxide concentration are not considered here. Hints for modelling these can be found in Sects. 3.1.1 and 3.2.2 of this book.

**Fig. 11.8** RAS setup with fish tank, pump, nitrification reactor and water transfer to hydroponic system

A variety of models in the scientific literature predict growth and feed intake of different aquatic species. The models describe growth as weight gain per day, as percentage growth increment or as specific growth rate based on an exponential growth model. Models are often valid for specific life stages. Feed consumption, biomass and gender are influencing the model output as well as the environmental conditions such as temperature, oxygen level and nutrient concentration (Lugert et al. 2014). Careful research is needed to identify the correct model used for the specific application. Commercial RAS that consists of several cohorts of fish in different life stages require the modelling to incorporate cohorts into the model (Fig. 8.6) (Halamachi and Simon 2005). The excretory mass flow for the European seabass (Dicentrarchus labrax) can be estimated with algorithms published by Lupatsch and Kissil (1998).

Here the net nitrogen mass flow into the process water is estimated from the feed composition (protein content), the amount of given feed and the nitrogen retained in body tissues through the growth (weight increment) of fish. The faecal nitrogen losses are not included in the model, but excretion rate is corrected assuming a share of 0.25 and 0.75 of nitrogen excretion for faecal loss and ammonia excretion, respectively. The nitrogen input through the feeding of fish is estimated from the protein content and the average relative nitrogen content of proteins which is assumed to be 0.16. The protein content of seabass tissue is reported at around 0.17 g proteins gsup-1/sup seabass (Lupatsch et al. 2003). For a fish gaining body weight by consuming a given amount of feed, the nitrogen excretion (XsubN,excreted/sub, g) can be calculated from Eq. (11.9). It is assumed that the feed (Xsubfeed/sub) contains 0.5 g protein gsup-1/sup fish. It is further assumed that the feed conversion rate equals 1, i.e. 1 g of feed consumption is resulting in 1 g of body weight increase (Fig. 11.9):

$X__{N,excreted} = X__{feed} * 0.16 * 0.75 * (0.5 - 0.17)$ (11.9)

Dissolved ammonia excreted via the gills of fish follows similar daily pattern as the gastric evacuation rate (GER). GER is described for cold water and warm water fish by He and Wurtsbaugh (1993) and Richie et al. (2004), respectively. The excretory pattern can be well simulated with a sine function. The ammonia excretion can be calculated from Eq. (11.10):

$X__{NH__x-N,excreted}=X_{N,excreted}[g]*(sin(\frac{2\pi}{1440})+1)$ (11.10)

**Fig. 11.9** Representation of the mass flows (Sankey chart) of feed ingredients and excretory products for a fish consuming 1000 g of feed assuming an FCR of 1

A variety of models describing RAS having different levels of complexity can be found in the literature. Very complex models are available for specific aspects, such as the interaction of soluble gases and alkalinity (Colt 2013) or the description of the microbial community (Henze et al. 2002). More practical models for the mass balance of RAS are published by Sánchez-Romero et al. (2016), Pagand et al. (2000), Wik et al. (2009) and Weatherley et al. (1993). All models provide information on excretory mass flows and/or nutrient flows in dependence of time and location in the process chain. Such models provide a base for the simulation of the coupling of RAS and HP. The most important dissolved matter in RAS modelling is total ammonia nitrogen (TAN). Besides TAN the chemical (COD) and biological (BOD) oxygen demand, the total suspended solids (TSS) and dissolved oxygen concentration need to be considered. However, different notations in the scientific literature make it sometimes hard to read, to convert and to implement the information into models. In the following, notations as recommended by Corominas et al. (2010) will be used. TAN will be rewritten as XsubNHx-N/sub and nitrate nitrogen will be expressed as XsubNO3-N/sub.

The model as it is described in the following is only valid for the RAS presented in Fig. 11.8. Other possible process chains for RAS are discussed in Sect. 11.3 of this chapter. For the mathematical depiction of physical systems, the following assumptions were made:

(a) Density of water is assumed to be constant.

(b) Tank and reactor are assumed to be well mixed.

(c) Tank and reactor volume are assumed to be constant.

(d) Process water flow is always greater than zero.

The assumption of a well-mixed tank and reactor leads to a mass balance equation for continuous stirred-tank reactor (CSTR) as described by Drayer and Howard (2014) in Eq. (11.11). It must be mentioned that diffusive processes can usually be neglected in RAS calculations because of a typically high process water flow rate. For a multi-tank RAS, the following holds:

Accumulation = Inflow - outflow + generation - reduction

$V__i{\dot x}__i=Q__{in}x__{i,in}-Q__{out}x__{i,out}+x__{i,gen}-x__{i,red}$ (11.1)

$j=\begin{cases} n, & i=1\ i-1, & i\ne1 \end{cases}$ (11.1)

In the above given equation $n$ represents the number of tanks in the System, ${\dot x}__i$ is the change of concentration of a given substrate x in a volume given by $V__{i.}$. The process water flow into the tank or reactor is represented by $Q{in}$. $V__i$ is the volume of the component where the process water flow $Q__{in}$ is entering in. The process water flow $Q__{in}$ came from a component having the volume $V__j$.

The conversion of XsubNHx-N/sub into XsubNO3-N/sub in nitrifying biofilters takes place on the surface area A [msup2/sup] available on the bio-carriers in the nitrification reactor (Rusten 2006). The available bioactive surface in the nitrification is calculated by multiplying the volume of the reactor with the volume-specific active surface of the bio-carriers AsubS/sub [msup2/sup ⋅ msup-3/sup]. The total bioactive surface is calculated (Eq. 11.12) from the relative filling fsubbc/sub of the nitrification reactor which usually is 0.6 (for details, see Rusten 2006).

A = Vsubnitrification/sub ⋆ AsubS/sub ⋆ fsubbc/sub (11.2)

The total daily TAN microbial conversion μsubmax/sub g dsup-1/sup was calculated by multiplying the specific TAN conversion (nitrification) rate, NHxsubconversion-rate/sub [g msup-2/sup dsup-1/sup], with the total active surface area, A [ msup2/sup ], of the bio-carriers. Values for TAN conversion in different types of nitrifying biofilters can be found in literature. For moving bed biofilm reactors (MBBR), values are reported by Rusten (2006). This rate is valid for certain process conditions, and it is assumed that the bacteria biofilm is fully developed over the whole.

$μ__{mm} = A^*NHx__{Conversion-rate}$ (11.13)

The total mass of NHsubx/sub converted into NOsub3/sub-N can subsequently be calculated with a Monod kinetic (Eq. 11.14). For this the NHsubx-N/sub concentration, XsubNHx-N,2/sub [g ⋅ 1sup-1/sup], in the volume of the nitrification reactor (MBBR) Vsub2/sub, is needed.

$\frac{d}{dt}X__{NH__x-N,2}=-\mu__{max}*(\frac{X__{NH__x-N,2}}{K__s+X__{NH__{x-N,2}}})*\frac1{V__2}$ with $K__s=\frac{\mu_{max}}2$ (11.4)

$\frac{d}{dt}X__{N0__x-N,2}=+\mu__{max}*(\frac{X__{NH__x-N,2}}{K__s+X__{NH__{x-N,2}}})*\frac1{V__2}$ with $K__s=\frac{\mu_{max}}2$ (11.4)

Given Eqs. (11.9, 11.10, 11.11, 11.12, 11.13 and 11.14), the following statespace model (combining fish-nitrification) results

[\frac{dx(t)}{dt}=A^*X+B^*u+n]

$X=\begin{bmatrix} X__{NH__{x}-N,1} \X__{NH__{x}-N,2}\X__{NO__{3}-N,1}\X__{NO__{3}-N,2} \end{bmatrix}$ $u=\begin{bmatrix} X__{NH__x-N,\text{excreted}} \ 0\{Q__{Exc}}^{*}X__{NH__x-N,\text{hydroponics}}\0 \end{bmatrix}$ $n=\begin{bmatrix}0\ -\frac{\mu__{max}*[X] 2}{Ks+[X]2} *\frac1{V2} \ 0 \ +\frac{\mu_{max}*[X]

$A=\begin{bmatrix} -\frac{Q}{V__1}-\frac{Q__{Exc}}{V__1}&\frac{Q}{V__1}&0&0\\frac{Q}{V__2}&-\frac{Q}{V__2}&0&0\ 0 &0&-\frac{Q}{V__1}-\frac{Q__{Exc}}{V__1}&\frac{Q}{V__1} \ 0&0&\frac{Q}{V__2}& -\frac{Q}{V__1} \end{bmatrix}$

$\times B = \begin{bmatrix} \frac1{V__1}&0&0&0\0&\frac1{V__2}&0&0\0&0&\frac1{V__1}&0\0&0&0&\frac{1}{V__2} \end{bmatrix}$

(11.5)

ExampleIn this example, a theoretical RAS with V

reactor = 1300 l and Vtank = 6000 l is simulated.All simulations had a daily feed input of 2000 g/day with 500 g protein/kg feed (Eq. 11.8). The daily TAN excretion was assumed to be a sine curve (Eq. 11.9). Active surface of the bio-carriers AsubS/sub is 300 [msup2/sup msup-3/sup], and the relative filling of the reactor fsubbc/sub is 0.6. Specific TAN conversion rate, NHxsubconversion-rate/sub, is 1.2 [g msup-2 -/supd], and the biofilm is supposed to be fully developed (Eqs. 11.11 and 11.12). The state-space representation (Eq. 11.14) was implemented in MATLAB Simulink. The Example showcases the importance of mass flow for nutrient concentrations in coupled systems (Fig. 11.10 and 11.11).