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connected to a pipe that opens at Ǒ. The overflow pipe has openings at(lP , hP ) in the x̌-y̌ plane and (lF , hF ) in the x-y plane. We assume that inthe actual aquaponics setup the plant bed is above the fish tank so thatduring overflow water enters the fish tank at (lF , hF ) in the x-y plane.At time t 0, the pump turns on, and water is pumped from the fishtank to the plant bed at a rate vp . When t t̂, the water has reached themaximum desired height hP in the plant bed, and water begins to flow fromthe plant bed to the fish tank via the overflow pipe. This maintains the waterlevel in the plant bed at hP until time t t̃ when the pump shuts off. Thenthe water in the plant bed drains back through the pump at a rate v̂p untilthe bed contains no water; this occurs at t t̄. Until the pump turns backon at t t , there is no water flow between the tank and the bed (Bernstein98).To model the concentrations of ammonia, nitrite, and nitrate as a functionof time and space, we will use diffusion-advection-reaction (DAR) equations.Diffusion, advection, and reaction are the mechanisms by which concentration changes with time. Diffusion is the tendency of solutes to move fromregions of high concentration to those with a lower concentration. This resultsfrom the random movement of molecules in the fluid. When concentration isconstant throughout a solution, collisions between molecules send the samenumber of particles, on average, in all directions. Thus concentration doesnot change over time. However, when concentration varies with respect tolocation in a fluid, the regions of higher concentration tend to expel moreparticles of solute than regions of lower concentration (de Marsily 232). Theresult, is an overall tendency toward uniform concentration. Advection isthe process by which chemicals dissolved in a fluid are transported by themovement of the fluid; reaction describes the the consumption or productionof solutes by chemical reactions (de Marsily 230).Consider the concentration of ammonia, denoted by Ǎ Ǎ(x̌, y̌, t), in theplant bed. Since the medium consists of gravel or a similar material, it canbe treated as porous with porosity ω (Bernstein 111). Porosity is the fractionof the plant bed through which water can flow, i.e. the portion that is notoccupied by gravel (Logan 30). We assume that ω is constant throughout theplant bed. This is reasonable because all the pieces of gravel are similar insize to one another. Furthermore, when fish waste starts to clog up regionsin the medium, the gardener must either clean out the plant bed or includered wriggler worms in the bed. These worms break down solid waste, whichprevents clogging (Bernstein 178-9). We also assume that all of the spaces5

between pieces of gravel are filled with water at all times (Logan 30). Ofcourse, this contradicts the fill and drain action within the plant bed, but wecan always solve the DAR equation for small intervals of time for which thedecrease in water level is negligible.The DAR equation for ammonia in the plant bed can be derived fromThe Principle of Mass Balance, which states that the rate of change of themass within an arbitrary volume in a region is equal to the rate at whichthe substance crosses the boundary of the volume plus the rate at which it iscreated or distroyed within the volume (Logan 31). Let Ω denote the regionin R3 that is occupied by the plant bed. Let Ω0 be an arbitrary ball in Ωwith surface Ω0 . The outward unit vector that is normal to Ω0 is denoted by n̂. Let Q(x̌,y̌, t) denote the rate of flow (mass per unit time) across Ω0 . is the flux of ammonia. Let R(x̌, y̌, t) be the rate at whichIn other words, Qammonia is produced or destroyed in Ω0 . Mathematically, mass balance canbe stated as equation (1) (Logan 35).ZZZ R dτ(1)Q · n̂ da ω Ǎ dτ t Ω0Ω0 Ω0The left hand side of equation (1) is the rate of change of the total massof ammonia within the ball. Notice that ω is included in the integrand onthe left side. This is because for each infinitessimal volume element dτ , onlya fraction of the volume (ωdτ ) is available for water to flow through. Also · n̂ is positive when ammonia is flowing out of Ω0 . Therefore,observe that Qthe amount of ammonia in the arbitrary ball decreases when the flux integralis positive. Thus we must place a negative sign in front of the flux integral inequation (1). Furthermore, even though there is no flow in the ž-direction, can be written as a three-dimensional vector with a ž-component that isQzero. Hence it can be dotted with n̂. The second term on the right side isthe rate at which ammonia is consumed during nitrification within the ball(Logan 31).If Ǎ and Ǎ/ t are continuous over Ω0 , then we can bring the partialderivative inside the integrand in equation (1). Furthermore, using the divergence theorem, the surface integral in equation (1) can be rewritten as avolume integral. The divergence theorem is stated below (Calculus: EarlyTranscendentals 1129).The Divergence Theorem : Let E be a simple solid region and let S6

be the boundary surface of E, given with positive (outward) orientation.Let F be a vector field whose component functions have continuous partialderivatives on an open region that contains E. ThenZZ · F dτF · dS SEBy observing that d a da n̂, equation (1) can be rewritten asZZ Ǎ dτ · Q dτ R dτω tΩ0Ω0Ω0 Z Ǎ ω · Q R dτ 0 tΩ0 Ǎ ω · Q R 0, (x̌, y̌) Ω, t 0 tZ(2)The last line above follows from the fact that Ω0 is arbitrary. The onlyway that the volume integral can equal zero for any ball Ω0 is if the integrandis identically zero. Also note that though the Del operator does not have the is”ˇ” accent, the derivatives are with respect to x̌ and y̌. The form of Qbased on experimental evidence; it is the sum of the advective flux, molecular (a) resultsdiffusion flux, and dispersion flux. Recall that the advective flux Qfrom particles being transported by the movement of water (Logan 32). It isgiven by (a) V p ǍQ(3)The vector field V p is the velocity field of the water in the plant bed, with (m) isunits of length over time. As we stated, the molecular diffusion flux Qdue to molecules colliding with each other (Logan 32). By Fick’s Law, (m) ωD(m) ǍQ(4)The coefficient D(m) is the effective molecular diffusion coefficient in themedium; it is only 10-70% as large as Do , the usual molecular diffusioncoefficent. Do is defined for a stagnant fluid with no boundaries. Finally, (d) results from the fluid traveling more quickly throughthe dispersion flux Qsome paths in the medium than others (Logan 32). Again, according toFick’s Law,7

(d) ωD(d) ǍQ(5)The coefficient D(d) is the dispersion coefficient (Logan 33). Thereforethe total flux is given by V p Ǎ ωD(m) Ǎ ωD(d) ǍQ Ǎ V p Ǎ ωD (6)The coefficient D is the hydrodynamic dispersion coefficient; it is thesum of the effective molecular diffusion and dispersion coefficients (Logan equation (2) can be written as33). With this form for Q, Ǎ Ǎ] R 0 · [Vp Ǎ ωD (7) tFor now, we will assume that D is constant within the grow bed. Inreality, the component of D in the direction of flow is much larger than thecomponents of D perpendicular to the flow field (de Marsily 236). This willbe discussed in detail later when we talk about choosing reasonable parametervalues. Therefore,ωω Ǎ · (Vp Ǎ) ωD 2 Ǎ R 0 t(8)If we let φ̌ and ψ̌ be the x̌- and y̌- components of the flow field, respectively, then (Ǎφ̌) (Ǎψ̌) x̌ y̌ Ǎ φ̌ Ǎ ψ̌ φ̌ Ǎ ψ̌ Ǎ x̌ x̌ y̌ y̌ Ǎ Ǎ φ̌ ψ̌ φ̌ ψ̌ Ǎ Ǎ x̌ y̌ x̌ y̌ Ǎ) · V p Ǎ( · V p ) ( · (V p Ǎ) Thus,8(9)

Ǎ Ǎ) · V p Ǎ( · V p )] ωD 2 Ǎ R 0 [( t Ǎ Ǎ) ω 1 Ǎ( · V p ) ω 1 R D 2 Ǎ ω 1 V p · ( tω(10)Equation (10) is the DAR equation for Ammonia within the plant bed.Notice that Ǎ does not depend on ž and the ž- component of V is zerobecause there is no flow in the ž-direction. Therefore, the Laplacian andgradient in equation (10) are two-dimensional for this model. The concentrations of nitrite and nitrate in the fish tank will be represented by B̌ and Č,respectively. The DAR equations for these substances in the plant bed havethe same form as the equation for ammonia. In the fish tank, the equationsfor each substance are derived in exactly the same manner as for the plantbed, except the fish tank does not contain a porous medium. Consequently,the diffusion coefficient is assumed to be Do (though there are boundariesand the water is not still), and the entire volume in the fish tank is availableto water. Hence, the concentration of ammonia, A, in the fish tank is givenby A · V F ) R Do 2 A V F · ( A) A( t(11)where V F is now the flow field in the fish tank. The concentrations ofnitrite and nitrate in the fish tank will be represented by B and C, respectively. The DAR equations for these substances in the fish tank have thesame form as the equation for ammonia.3.2Nonhomogenous TermsThe reaction term R varies depending on the substance and its locationwithin the aquaponics system. In the plant bed, R will describe the rate ofchange of concentration due to nitrification. The chemical reaction equationsfor nitrification are given below (”Wastewater Management.” 1).N H3 O2 N O2 3H 2e N O2 H2 O N O3 2H 2e 9(12)(13)

Essentially, these equations state that ammonia (N H3 ) transforms intonitrite (N O2 ), which then transforms into nitrate (N O3 ). Ammonia is converted to nitrite mainly by the bacteria Nitrosomonas sp.; nitrite is transformed into nitrate mostly by Nitrobacter sp. bacteria (Tyson 20). TheChemical Law of Mass Action given below states that the rate at which aproduct is created (or a reactant is consumed) is proportional to the productof the concentrations of the reactants (”Reaction Kinetics.” 2).The Chemical Law of Mass Action13 : If a chemical reaction is givenbya b c,thend[b]d[c]d[a] k[a][b] ,dtdtdtwhere [u] denotes the concentration of substance u.However, we can assume that oxygen (O2 ) and water (H2 O) are at muchhigher concentrations than ammonia and nitrite. This is reasonable becauseammonia and nitrite are dissolved in a large amount of water in the plantbed. Also, the plant bed is well-oxygenated due to the flood and drain process. Hence, the percentages of oxygen and water consumed by the reactionwill be negligible compared to their total concentrations. With minimalerror, we can assume that the concentrations of oxygen and water remainconstant throughout the reaction. Therefore, these concentrations can beabsorbed into the constants of proportionality (Copeland 479). Thus therates of change of the concentrations of ammonia, nitrite, and nitrate due tonitrification are given bydǍ k1 ǍdtdB̌ k1 Ǎ k2 B̌dtdČ k2 B̌dt(14)(15)(16)The constants k1 and k2 are called the rate constants for equations (12)and (13), respectively. There are negative signs in equations (14) and (15)10

because as the concentrations of the reactants increase, the reaction speedsup, and reactants are consumed faster. The right-hand sides of equations(14), (15), and (16) are reaction terms for ammonia, nitrite, and nitrate,respectively, in the plant bed.In addition to the reaction term given by equation (16), the DAR equationfor nitrate in the plant bed will include another source (or more accuratelydrain) term P (C) that describes plant uptake of nitrate. Plants also takeup ammonia, but they take up far less ammonia than nitrate. Later, whenwe discuss parameter values, we will prove empirically that plant uptake ofammonia is negligible. Very little nitrification occurs in the fish tank becauseit does not contain a biofilter. Therefore, equations (14) - (16) do not applyin the fish tank. The only source term present in the fish tank is F (t), whichdescribes the fish ammonia excretion. We will approximate P and F laterusing experimental evidence.3.3Flow FieldsAdvection in the fish tank and plant bed is determined by the velocity fieldof the water during each stage of the flood and drain process. During the F and V P U P .period when the plant bed is filling with water, let V F U To simulate fluid flow in the fish tank, UF has a sink at the location of thepump so that the velocity vector at each point in the tank points toward the F approaches vp as the distance from the pump approachespump, and Uzero.As the distance from the pump approaches the maximum distance ofp F approaches zero. ThuslF2 h2F , we assume that U!p2 y2x x, y F vp 1 ppV F U22lF hFx2 y 2!11 vp p 2 p x, y , 0 t t̂ (17)lF h2Fx2 y 2 F is has an infiniteOf course, this creates a removable discontinuity;Upp22 magnitude at the origin, and UF vp (1 x̌ y̌ / lF2 h2F ) elsewhere.This is okay, since we are only concerned with fluid flow within the tank.When we numerically solve the DAR equations, we can exclude the origin P has a source at the origin where waterfrom our mesh. In the plant bed, U P approaches vp as the distanceflows into the bed. As in the fish tank, U11

P approaches zero asfrom the origin approaches zero, and we assume that Upthe distance from the pump approaches the maximum distance of lP2 h2P .Hence,px̌2 y̌ 2! x̌, y̌ p x̌2 y̌ 2!11p x̌, y̌ , 0 t t̂ (18) p2lP h2Px̌2 y̌ 2 P vpV P U vp1 plP2h2PDuring the period of overflow, V F has a sink at the origin and a source at(lF , hF ), where water from the overflow pipe flows in. Notice that the rateof inflow must equal the rate of outflow so that the water level in the plantbed is maintained at hp . This implies that!p22 x lF , y hF F vp p x ypV F U, t̂ t t̃(19)22(x lF )2 (y hF )2lF hFSimilarly, V P has a source at the origin and a sink at (lP , hP ). Hence,!p2 y̌ 2x̌ l x̌, hP y̌ P vp pp PV P U, t̂ t t̃(20)22(x̌ lP )2 (y̌ hP )2lP hPWhile the plant bed is draining, V F has a source at the origin where waterflows back through the pump. Similarly, V P has a sink at the origin. Therefore V P and V F approach vˆp as the distance from the origin approaches approach zero as the distance from the originzero. Again,p VP and VFpapproaches lP2 h2P and lF2 h2F , respectively. This yieldsV F vˆpV P vˆp1px2 y 21plP2 h2P1 p!lF2 h2F1 px̌2 y̌ 2 x, y , t̃ t t̄(21) x̌, y̌ , t̃ t t̄(22)!Finally, for t̄ t t , the plant bed is completely drained. Thus,V P V F 0 until the pump turns on again. These velocity fields assume12

that flow within the tank and the bed is only affected by water flowingbetween the tanks. This is fairly accurate for the plant bed; though theporosity of the medium will cause some deviation from the expected flowpattern; However, this assumption is somewhat inaccurate in the fish tank.Fish generally require highly oxygenated water, thus aerators and air stonesare often placed in the fish tank to increase the DO. This creates turbulence,which we have decided to ignore for the purposes of this model. Plots ofthe flow fields are shown in figures 2 - 4. The lengths and heights chosenfor the fish tank and plant bed in these figures are based upon physicalconsiderations that will be discussed when we estimate parameters.Figure 2: The velocity fields in the fish tank (left) and plant bed (right) for0 t t̂.13

Figure 3: The velocity fields in the fish tank (left) and plant bed (right) fort̂ t t̃.Figure 4: The velocity fields in the fish tank (left) and plant bed (right) fort̃ t t̄.14

3.4DAR Equations, Initial Conditions, and BoundaryConditions for each SubstanceNow we are ready to write out the DAR equations for each substance ofinterest. Each substance will have its own hydrodynamic dispersion coefficient in the plant bed. The dispersion coefficients for ammonia, nitrite, andPPnitrate are DA, DB, DCP , respectively. Likewise, each substance has its ownmolecular diffusion coefficient in the fish tank. The diffusion coefficients forFFammonia, nitrite, and nitrate in the plant bed are DA, DB, DCF , respectively.Therefore, the DAR equations in the Fish Tank (0 x lF , 0 y hF )for 0 t t̄ are AF 2 DA A V F t BF 2 DB B V F t C DCF 2 C V F t A( · V F ) F (t)· A(23) B( · V F )· B(24) C( · V F )· C(25)Similarly, the equations in the Plant Bed (0 x lP , 0 y hP ) for0 t t̄ are written as ǍP 2 DA Ǎ ω 1 V P t B̌P 2 DB B̌ ω 1 V P tk2 B̌) Č DCP 2 Č ω 1 V P tω 1 P (x̌, y̌, t) Ǎ ω 1 Ǎ( · V P ) ω 1 k1 Ǎ· (26) B̌ ω 1 B̌( · V P ) ω 1 (k1 Ǎ · (27) Č ω 1 Č( · V P ) ω 1 k2 B̌ · (28)When t̄ t t , Ǎ B̌ Č 0 in the plant bed because the bed isdrained (we are ignoring the small amount of water contained in the porousmedium). In the fish tank, the water is assumed to be stagnant and the fishare still excreting ammonia. Hence the concentrations of the nitrogenous15

compounds are given by AF 2 DA A F (t) t BF 2 B DB t C DCF 2 C t(29)(30)(31)Later we will show that, due to the flood-and-drain process, there aresome values of y and y̌ that are not valid during each time period. This occursbecause the DAR equations are only valid in domains that are completelyfilled with water (in the plant bed this means that every void in the porousmedium is filled with water).For each substance and each boundary in the tank and bed, we havezero-flux (Neumann) boundary conditions (BCs). This makes sense becausenone of our substances of interest are dissolving into the water from the airabove each compartment. Furthermore, material cannot diffuse or advectacross the walls of these containers, which are typically plastic or some otherimpermeable material. For 0 t t̂, we have a source at Ǒ and a corresponding drain at O. Thus we will impose a Dirichlet BC by solving theequations for the fish tank first, and then requiring that the concentrationat Ǒ equal the concentration at O. This implies that the BCs for ammoniawhen 0 t t̂ are given by A x Ǎ x̌ A y Ǎ y̌Ǎ 0 for x 0, 0 y yJ and x lF , 0 y yJ(32) 0 for x̌ 0, 0 y̌ yˇL and x̌ lP , 0 y̌ yˇL(33) 0 for y 0, 0 x lF and y yJ , 0 x lF(34) 0 for y̌ 0, 0 x̌ lP and y̌ yˇL , 0 x̌ lP(35) A for x y x̌ y̌ 0(36)where yJ (t) and y̌L (t) are the heights of the water in the fish tank andplant bed, respectively. Notice that these heights vary depending on whichpart of the flood and drain cycle we are in. These heights will be found for16

each time period when we define our computational grid in the numericalmethods section. When the water in the plant bed drains back through thepump (t̃ t t̄), the BCs are the same except we must find the solution inthe plant bed first so that we have enough information to satisfy equation(36). For the overflow period (t̂ t t̃), the BCs are the same except wemust also impose a Dirichlet BC where the plant bed overflows into the fishtank. Hence for t̂ t t̃,A(lF , yJ , t) Ǎ(lP , hP , t)(37)During the period when the plant bed is empty, the BCs are the same asthose given by equations (32) - (35), except there is no Dirichlet BC at theorigin; all BCs are zero-flux. The BCs for the DAR equations for nitrite andnitrate are the same as for ammonia during each time period. It should benoted that when we impose these BCs n

Aquaponics is a combination of hydroponics (growing plants in a soilless environment) and aquaculture (raising sh in a tank). As a byproduct of respiration, sh excrete toxic ammonia from their gills. In an aquaponics system, the water in the sh tank is pu