G. Warrensford Final

112 Pages·2016·5.32 MB·English

A MODEL OF NUTRIENT FLOW IN AN
AQUAPONICS SYSTEM USING
DIFFUSION-ADVECTION-REACTION
EQUATIONS
GABRIELLA WARRENSFORD
A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT
OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON
UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF BACHELOR OF SCIENCE
STETSON UNIVERSITY
2015
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. William Miles, for all of his help
throughout the semester. I would also like to thank Dr. Harry Price for
assistance with the chemistry concepts related to this project, and Dr. Erich
Friedman for help with Mathematica.
Contents
1 Abstract 1
2 Background 1
2.1 Advantages of Aquaponic Gardening . . . . . . . . . . . . . . 1
2.2 System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Derivation of the Model 4
3.1 Derivation of the DAR Equation . . . . . . . . . . . . . . . . . 4
3.2 Nonhomogenous Terms . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Flow Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 DAR Equations, Initial Conditions, and Boundary Conditions
for each Substance . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Estimation of Parameters 18
4.1 The Plant Uptake Function . . . . . . . . . . . . . . . . . . . 18
4.2 The Fish Excretion Function . . . . . . . . . . . . . . . . . . . 21
4.3 Additional Parameters . . . . . . . . . . . . . . . . . . . . . . 23
5 Numerical Methods 27
6 Future Plans 94
7 Appendix 96
7.1 The Nonlinear Regression for Fish Excretion Data . . . . . . . 96
7.2 Original Fish Excretion Data . . . . . . . . . . . . . . . . . . 97
7.3 The Parameter Table for Plant Uptake of Nitrogen . . . . . . 99
7.4 An Example of Restrictions on Explicit Numerical Methods . 100
7.5 The WENO Scheme . . . . . . . . . . . . . . . . . . . . . . . 102
7.6 The Third Order TVDRK Method . . . . . . . . . . . . . . . 104
References 106
1 Abstract
Aquaponics is a combination of hydroponics (growing plants in a soilless
environment) and aquaculture (raising fish in a tank). As a byproduct of
respiration, fish excrete toxic ammonia from their gills. In an aquaponics
system, the water in the fish tank is pumped to beds where plants are grown
in a soilless medium. Nitrifying bacteria colonize in the medium, and convert
ammonia to nitrite and then nitrate. The plants can readily absorb the
nitrate as a vital nutrient, and the water, which is now free of the toxic
ammonia and nitrite, is pumped back to the fish tank. In this paper, we
will attempt to simulate nutrient flow in an aquaponics system with a two-
dimensional model. For each of the three substances of interest: ammonia,
nitrite, and nitrate; a diffusion-advection-reaction equation will be used to
determine its concentration within the fish tank and plant bed as a function
of time and space. Model parameters will be estimated based on empirical
evidence, and we will discuss possible numerical methods that could be used
to solve the equations in our model.
2 Background
2.1 Advantages of Aquaponic Gardening
Aquaponicsnotonlycombineshydroponicsandaquaculture; itisanimprove-
ment upon each system. In aquaculture, solid fish excrement and ammonia
produced during respiration must be filtered out of the water and discarded.
Aquaponics makes use of these waste products as nutrients for the plants
(Bernstein 6-7). Similarly, hydroponics requires growers to use costly man-
ufactured nutrient solutions. For the same cost in fish feed, aquaponic gar-
deners can grow approximately the same number of plants as hydroponic
gardeners, as well as several pounds of fish. Moreover, water in a hydroponic
system must be thrown out periodically because chemicals accumulate to
toxic levels. Water is never discarded in an aquaponics system (Bernstein 4).
Aquaponics wastes much less water than traditional or hydroponic gardening
because the nutrient solution is continually circulated between the fish tank
and plant bed. Thus water is only lost when it evaporates or is taken up by
fish and plants (Bernstein 2). In addition, research conducted by Dr. Nick
Savidov in 2005 showed that aquaponics has a higher yield than hydroponics.
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He was able to grow 10-15% more tomatoes than are grown on average in
a hydroponics setup. Furthermore, Savidov achieved a threefold increase in
plant production within two years (Savidov 20, 24). It is also worth mention-
ing that since nutrients for the plants come from fish, aquaponics produces
organic products, while hydroponically grown plants are clearly not organic
(Bernstein 5).
Aquaponics is a novel way for the human race to stop using more natu-
ral resources than the earth can replenish. Demographers predict that the
population will increase by an average of 75 million people each year until
2050 (Bernstein 11). According to the World Wildlife Funds Living Planet
Report in 2008, the current population already exceeds the planets regener-
ative capacity by about 30 percent (Bernstein 12). As China and India, the
two countries with the largest populations, continue to raise their standard
of living to that of the United States, this shortage of resources will only get
worse.
Aquaponicseliminatesmanyaspectsofagriculture, suchasexcessiveland
and water use, that make it unsustainable. In 2009, United Nations Deputy
Secretary-General Asha-Rose Migiro projected that two-thirds of the earth’s
population will go without water within twenty years (Bernstein 15). In
the U.S., 80% of the water we use is for agriculture; hence it is easy to
see why a recirculating agricultural system would be of value (Bernstein
15). Furthermore, everyone knows that deforestation is a major problem;
over 40% of the earth’s land has been cleared for agriculture (Bernstein 18).
Thus less trees are available to turn carbon dioxide into oxygen, and thereby
reduce greenhouse gas emissions. Soilless agriculture such as aquaponics is
clearly a way to diminish this problem.
In addition, aquaponics could reduce overfishing. At least 2,048 species
of fish are extinct because of overfishing (Bernstein 18). If we farm more fish
instead of catching them, then in time maybe less species of fish would be
endangered. Moreover, fish are a much more efficient animal to farm than
most traditional farm animals. Fish convert about 67% of their food weight
to body weight gain; whereas chickens, pigs, cows, and sheep only convert
roughly 50%, 25%, 15%, and 13%, respectively (Bernstein 19). Aquaponics
has the potential to generate a large amount of nutritious food with a low
impact on the environment.
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2.2 System Design
There are many different ways to construct an aquaponics system. The three
major techniques are the deep water culture (DWC) method, the nutrient-
film technique (NFT), and media-based aquaponics. For both DWC and
NFT, water flows from the fish tank through a filter that traps solid fish
waste, and then to a sump tank that contains a biofilter for nitrification (a
sump is just a tank that holds excess water in an aquaponics system) (Design
of Aquaponic Units 64, 68). A typical biofilter would be plastic netting that
encloses PVC pipe shavings; the shavings create adequate surface area for
bacteria to colonize. The biofilter must also have an aerator because the
bacteria need oxygen to survive (Design of Aquaponic Units 46). From the
sump tank, some water is pumped back to the fish tank, and the rest goes
to the plants. The water that returns to the fish tank forces it to overflow
into the filter again. After the water flows through the compartments that
contain the plants, it also goes back to the sump tank (Design of Aquaponic
Units 64). For DWC, plants are placed in holes on a floating polystyrene raft
so that their roots are submerged in the nutrient-rich water pumped from
the sump. For NFT, plants are placed in holes that are drilled into pipes.
A small amount of water from the sump tank flows through the pipes and
bathes the roots (Design of Aquaponic Units 48, 49).
The major downside to DWC and NFT is that a mechanical filter and an
aerated biofilter are required, whereas in media-based aquaponics the media
in which the plants grow acts as a biofilter and filters solid waste (Design of
Aquaponic Units 45). Furthermore, DWC is mainly limited to small plants
like lettuce and herbs; larger plants are too heavy to float in rafts (Design of
Aquaponic Units 73). Additional aeration is also required in the plant canals
for DWC because plants take up dissolved oxygen (DO) from the water. The
water is relatively stagnant beneath the rafts, and thus does not contain
enough DO to support the plants’ needs. These are not problems in media-
based aquaponics because the flooding and draining of the bed allows oxygen
to reach plant roots and bacteria, and any plant that thrives in neutral pH
can be grown (Bernstein 110, 153). For these reasons, we choose to model
basic flood and drain media-based aquaponics, which is shown in figure 1
(Bernstein 98).
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Figure 1: Basic Flood and Drain Media-Based Aquaponics
3 Derivation of the Model
3.1 Derivation of the DAR Equation
In this model, the fish tank is represented by a rectangular region in the
first quadrant of the x-y plane, with one corner at the origin, O. Likewise,
the plant bed is represented by a rectangular region in the xˇ-yˇ plane, with
ˇ
one corner at the origin, O. The length of the fish tank in the x-direction
is l ; its height in the y-direction is h . The length of the plant bed in
F F
the xˇ-direction is l ; its height in the yˇ-direction is h . It should be noted
P P
that these heights are not the actual heights of the tank and bed, but rather
the maximum height that water ever reaches throughout the flood and drain
cycle. Symmetry is assumed for the third dimension of each compartment.
In other words, the tank and bed are rectangular boxes, and the nutrient flow
within each cross-section parallel to the x-y and xˇ-yˇ planes, respectively, are
assumed to be the same. The widths of the fish tank and plant bed are w
F
and w , respectively. Obviously, this set-up is somewhat unrealistic because
P
pipe openings typically occupy a small circular region of space; whereas our
assumptions of symmetry require rectangular pipe openings of infinitessimal
length and widths equal to the widths of the compartments that contain the
pipe openings. However, this notion is not so far-fetched when one considers
that gardeners frequently drill holes along the sides of PVC pipes to send
water to multiple locations within the plant bed or increase the DO in the
fish tank by creating turbulent flow. The pump is located at O, and is
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ˇ
connected to a pipe that opens at O. The overflow pipe has openings at
(l ,h ) in the xˇ-yˇ plane and (l ,h ) in the x-y plane. We assume that in
P P F F
the actual aquaponics setup the plant bed is above the fish tank so that
during overflow water enters the fish tank at (l ,h ) in the x-y plane.
F F
At time t = 0, the pump turns on, and water is pumped from the fish
tank to the plant bed at a rate v . When t = tˆ, the water has reached the
p
maximum desired height h in the plant bed, and water begins to flow from
P
the plant bed to the fish tank via the overflow pipe. This maintains the water
level in the plant bed at h until time t = t˜when the pump shuts off. Then
P
the water in the plant bed drains back through the pump at a rate vˆ until
p
the bed contains no water; this occurs at t = t¯. Until the pump turns back
on at t = t∗, there is no water flow between the tank and the bed (Bernstein
98).
Tomodeltheconcentrationsofammonia,nitrite,andnitrateasafunction
of time and space, we will use diffusion-advection-reaction (DAR) equations.
Diffusion, advection, and reaction are the mechanisms by which concentra-
tion changes with time. Diffusion is the tendency of solutes to move from
regionsofhighconcentrationtothosewithalowerconcentration. Thisresults
from the random movement of molecules in the fluid. When concentration is
constant throughout a solution, collisions between molecules send the same
number of particles, on average, in all directions. Thus concentration does
not change over time. However, when concentration varies with respect to
location in a fluid, the regions of higher concentration tend to expel more
particles of solute than regions of lower concentration (de Marsily 232). The
result, is an overall tendency toward uniform concentration. Advection is
the process by which chemicals dissolved in a fluid are transported by the
movement of the fluid; reaction describes the the consumption or production
of solutes by chemical reactions (de Marsily 230).
ˇ ˇ
Consider the concentration of ammonia, denoted by A = A(xˇ,yˇ,t), in the
plant bed. Since the medium consists of gravel or a similar material, it can
be treated as porous with porosity ω (Bernstein 111). Porosity is the fraction
of the plant bed through which water can flow, i.e. the portion that is not
occupied by gravel (Logan 30). We assume that ω is constant throughout the
plant bed. This is reasonable because all the pieces of gravel are similar in
size to one another. Furthermore, when fish waste starts to clog up regions
in the medium, the gardener must either clean out the plant bed or include
red wriggler worms in the bed. These worms break down solid waste, which
prevents clogging (Bernstein 178-9). We also assume that all of the spaces
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between pieces of gravel are filled with water at all times (Logan 30). Of
course, this contradicts the fill and drain action within the plant bed, but we
can always solve the DAR equation for small intervals of time for which the
decrease in water level is negligible.
The DAR equation for ammonia in the plant bed can be derived from
The Principle of Mass Balance, which states that the rate of change of the
mass within an arbitrary volume in a region is equal to the rate at which
the substance crosses the boundary of the volume plus the rate at which it is
created or distroyed within the volume (Logan 31). Let Ω denote the region
in R3 that is occupied by the plant bed. Let Ω(cid:48) be an arbitrary ball in Ω
with surface ∂Ω(cid:48). The outward unit vector that is normal to ∂Ω(cid:48) is denoted
by nˆ. Let Q(cid:126)(xˇ,yˇ,t) denote the rate of flow (mass per unit time) across ∂Ω(cid:48).
(cid:126)
In other words, Q is the flux of ammonia. Let R(xˇ,yˇ,t) be the rate at which
ammonia is produced or destroyed in Ω(cid:48). Mathematically, mass balance can
be stated as equation (1) (Logan 35).
(cid:90) (cid:90) (cid:90)
∂
ˇ (cid:126)
ωAdτ = − Q·nˆda+ Rdτ (1)
∂t
Ω(cid:48) ∂Ω(cid:48) Ω(cid:48)
The left hand side of equation (1) is the rate of change of the total mass
of ammonia within the ball. Notice that ω is included in the integrand on
the left side. This is because for each infinitessimal volume element dτ, only
a fraction of the volume (ωdτ) is available for water to flow through. Also
observe that Q(cid:126) ·nˆ is positive when ammonia is flowing out of Ω(cid:48). Therefore,
the amount of ammonia in the arbitrary ball decreases when the flux integral
is positive. Thus we must place a negative sign in front of the flux integral in
equation (1). Furthermore, even though there is no flow in the zˇ-direction,
(cid:126)
Q can be written as a three-dimensional vector with a zˇ-component that is
zero. Hence it can be dotted with nˆ. The second term on the right side is
the rate at which ammonia is consumed during nitrification within the ball
(Logan 31).
If Aˇ and ∂Aˇ/∂t are continuous over Ω(cid:48), then we can bring the partial
derivative inside the integrand in equation (1). Furthermore, using the di-
vergence theorem, the surface integral in equation (1) can be rewritten as a
volume integral. The divergence theorem is stated below (Calculus: Early
Transcendentals 1129).
The Divergence Theorem : Let E be a simple solid region and let S
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be the boundary surface of E, given with positive (outward) orientation.
(cid:126)
Let F be a vector field whose component functions have continuous partial
derivatives on an open region that contains E. Then
(cid:90) (cid:90)
(cid:126) (cid:126) (cid:126) (cid:126)
F ·dS = ∇·F dτ
S E
By observing that d(cid:126)a = danˆ, equation (1) can be rewritten as
(cid:90) ˇ (cid:90) (cid:90)
∂A
(cid:126) (cid:126)
ω dτ = − ∇·Qdτ + Rdτ
∂t
Ω(cid:48) Ω(cid:48) Ω(cid:48)
(cid:90) (cid:20) ˇ (cid:21)
∂A
(cid:126) (cid:126)
ω +∇·Q−R dτ = 0
∂t
Ω(cid:48)
ˇ
∂A
(cid:126) (cid:126)
ω +∇·Q−R = 0,(xˇ,yˇ) ∈ Ω,t ≥ 0 (2)
∂t
The last line above follows from the fact that Ω(cid:48) is arbitrary. The only
way that the volume integral can equal zero for any ball Ω(cid:48) is if the integrand
is identically zero. Also note that though the Del operator does not have the
(cid:126)
”ˇ” accent, the derivatives are with respect to xˇ and yˇ. The form of Q is
based on experimental evidence; it is the sum of the advective flux, molecular
diffusion flux, and dispersion flux. Recall that the advective flux Q(cid:126)(a) results
from particles being transported by the movement of water (Logan 32). It is
given by
Q(cid:126)(a) = V(cid:126) Aˇ (3)
p
(cid:126)
The vector field V is the velocity field of the water in the plant bed, with
p
units of length over time. As we stated, the molecular diffusion flux Q(cid:126)(m) is
due to molecules colliding with each other (Logan 32). By Fick’s Law,
Q(cid:126)(m) = −ωD(m)∇(cid:126)Aˇ (4)
The coefficient D(m) is the effective molecular diffusion coefficient in the
medium; it is only 10-70% as large as D , the usual molecular diffusion
o
coefficent. D is defined for a stagnant fluid with no boundaries. Finally,
o
the dispersion flux Q(cid:126)(d) results from the fluid traveling more quickly through
some paths in the medium than others (Logan 32). Again, according to
Fick’s Law,
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