The first consideration in designing the hull is its shape. There are three types of hulls: displacement hulls which displace water and move through water easily and require minimum power, planing hulls which are used for achieving high speeds and semi-displacement hulls which in low speeds act as displacement hulls and at high speeds they plane. From the three types of hulls we chose a semi-displacement hull for its versatility. Typical semi-displacement hulls start with a V shape at the bow, front of the boat, and slowly flattens towards the stern, the back of the boat.

There are some important definitions we have to explain before we start designing such as:

Name	Defintions
Deadrise	The angle measurement between the bottom of the vessel and the horizontal plane on either side of the center keel, in our design the deadrise varies in degrees from bow (forward part of the hull of ship) to stern (the back part of the hull of a ship).
Keel	The part that is the lowest part of the hull and runs across the boat from bow to stern.
Beam (B)	The width of a boat or ship at its widest part of the waterline.
Length (L)	The distance from the bow to stern.
Draft (D)	The vertical distance between the boat's keel and the waterline.
Freeboard	The vertical distance from the waterline to the lowest point on the boat's deck.
Depth	The vertical distance that is the sum of draft and freeboard distances.

Floatation is achieved because of the buoyancy force which counteracts the force of the weight. The two forces act at two different points the center of buoyancy and center of gravity respectively these two points have to be aligned so as to ensure stability but more on that later. Equating those two forces we calculate the immersed volume of the boat, the first step in determining our first dimensions.

$$ W (weight) = m (mass) \cdot g(acceleration\; of\; gravity)$$ $$ BF(buoyancy\; force) = ρ (density\; of\; water) \cdot g \cdot Vim(immersed volume)$$

The mass of our boat consists of the material that the hull of the boat is made of plus the instruments it will carry. For the mass of the hull we also need to determine the thickness (t) of the hull.

There are four coefficients that are going to guide our design. These are the block coefficient (Cb), prismatic coefficient (Cp), waterline coefficient (Cw) and midship section coefficient (Cm) .

The block coefficient is the ratio of the immersed Volume to the block of volume from the dimensions of length waterline times the beam times the draft

$$Cb = \frac{Vim}{L \cdot B \cdot D} $$

The prismatic coefficient is the ratio of the immersed volume to the volume calculated from the midship cross-section times the length.

$$ Cp = \frac{Vim}{M \cdot L} $$

The waterline coefficient is the ratio of the waterplane to the rectangle beam times the length.

$$ Cw = \frac{WpL}{B \cdot L} $$

where Wpl is the water plane area

The midship section coefficient is the ratio of the cross-section area to the beam times the draft

$$ Cm = \frac{M}{B \cdot D} $$

Typical values for Cb are 0.5-1 for various boats, Cp values have to be higher than Cb, Cw 0.85-0.9 and Cm is 0.85 for fast boats.

In addition, considering that the beam to length ratio is ⅓ then by setting a value for either the length or beam and the draft then we calculate our coefficients.

As a start, the design of our boat follows a trial and error approach where we set our main dimensions and then calculate the respective coefficients. With this method, we try to optimize dimensioning of the remote control boat for our specific application. Since our primary application will revolve around lakes, the type of hull selected can be applied in lakes and coastal waters. For a more energy-efficient design, the size of the boat is as large as the electronics can be housed and is stable enough from minimal lake water disturbances (small waves created by wind).

The immersed volume (Vim) and waterplane (Wpl) have to be calculated first. In order for the four coefficients (Cp, Cb, Cw and Cm) to be determined. Due to their complex shape numerical methods have to be applied. In detail, numerical methods approximate the real value of the geometric property (immersed volume, waterplane area) by minimal error. The way this is achieved by the First Rule of Simpson. An example of the First Rule of Simpson in Fig. 01

Figure 01. A surface divided in two equal parts and the three vertical distances b1, b2, b3 required for the first simpson rule

$$ A = \frac{1}{3} \cdot h \cdot (b1 + 4 \cdot b2 + b3) $$

By extension any surface divided in equal distances can calculated area wise (Fig. 02)

Figure 02. Depicts the top half of a waterplane area divided in 5 equal parts

$$ A = \frac{1}{3} \cdot h \cdot ( b1 + 4 \cdot b2 + 2 \cdot b3 + 4 \cdot b4 + b5) $$

and a general equation

$$ A = \frac{1}{3} \cdot h \cdot \sum{1}^{n} ( b1 + 4 \cdot b2 + 2 \cdot b3 + ... + bn )$$

In ship blueprints, the length of a marine vessel is divided into an even number of equal distances (11 or 21). Our model is based on 21 divided sections to minimize error. The method by which we calculate the surface is a spreadsheet. The number of rows used is the number of sections selected earlier. The number of columns depends on the properties calculated. For the properties of the waterplane surface, we require six columns. The first column is the number of sections, the second column is the beam distances at each section, the third column is the Simpson coefficients required for numerical integration and the fourth column is the product of columns two and three. The two latter columns are used for calculating the center of flotation of the waterplane and is the geometric center of the surface. The center of flotation is calculated by the moment of the waterplane from the axis parallel to the plane and at the bow (Oo axis) of the hull divided by the area of the waterplane surface (Fig. 03). As a result, column five is the lever arm from the Oo axis, and column six is the product of columns five and four. The last step is acquiring two sums from columns four and six. These sums calculate the area and moment to axis Oo respectively. Then by dividing them we have the center of flotation from the bow of the hull.

Figure 03. Depicts the waterplane surface and the axis

Number of Section	Beam distances	Simpson coefficient	Product of Area	Lever arm for moment	Product for moment
1	b1	1	1*b1	0	0
2	b2	4	4*b2	1	1*(4*b2)
3	b3	2	2*b3	2	2*(2*b3)
4	b4	4	4*b4	3	3*(4*b4)
5	b5	2	2*b5	4	4*(2*b5)
6	b6	4	4*b6	5	5*(4*b6)
7	b7	2	2*b7	6	6*(2*b7)
8	b8	4	4*b8	7	7*(4*b8)
9	b9	2	2*b9	8	8*(2*b9)
10	b10	4	4*b10	9	9*(4*b10)
11	b11	2	2*b11	10	10*(2*b11)
12	b12	4	4*b12	11	11*(4*b12)
13	b13	2	2*b13	12	12*(2*b13)
14	b14	4	4*b14	13	13*(4*b14)
15	b15	2	2*b15	14	14*(2*b15)
16	b16	4	4*b16	15	15*(4*b16)
17	b17	2	2*b17	16	16*(2*b17)
18	b18	4	4*b18	17	17*(4*b18)
19	b19	2	2*b19	18	18*(2*b19)
20	b20	4	4*b20	19	19*(4*b20)
21	b21	1	1*b21	20	20*(1*b21)
			SUM1		SUM2

Figure 04. Depicts the spreadsheet described above.

We conclude with three equations to use the results from the waterplane spreadsheet. In the Area of the surface equation, h equals to a twentieth of the length of the boat and n equals to 21.

$$ A = \frac{1}{3} \cdot h \cdot ( b1 + 4 \cdot b2 + 2 \cdot b3 + ... + b21 )$$ $$ A = \frac{1}{3} \cdot \frac{L}{20} \cdot SUM 1 $$

The moment to axis:

$$ Mo = \frac{1}{3} \cdot \frac{L}{20} \cdot \frac{L}{20} \cdot SUM 2$$

and the center of flotation CF:

$$ CF = \frac{Mo}{A} $$

We follow a similar method for calculating the immersed volume and longitudinal and vertical centers of buoyancy. The rows of the second spreadsheet are the sections explained earlier. The columns are as follows. The first is the numbered sections from one to twenty-one. The second is the area of the transverse section. The third is the Simpson coefficients. The fourth is the product of the second and third columns. So far, these calculations serve for the immersed volume estimation. Next, the fifth column is the lever arm of the moment to the Xx axis (Fig. 05) while the sixth is the product of the fourth and fifth columns. The seventh is the distance of the center of each transverse area measured from the keel. Fig. 06. The last column is the product of the fourth and seventh columns. We end up with three sums and use them to measure the immersed volume, longitudinal and vertical centers of buoyancy. (Fig. 07)

Figure 05. Depicts the x’x axis on the waterplane surface.

Figure 06. Depicts the centroid distance di from the keel in a trapezoid surface.

Number of Section	Area of transverse sections	Simpson coefficients	Product for Volume	Lever arm from Xx axis	Product for LCB	Centroid from keel	Product for VCB
1	A1	1	1*A1	10	1*A1*10	d1	1*A1*d1
2	A2	4	4*A2	9	4*A2*9	d2	4*A2*d2
3	A3	2	2*A3	8	2*A3*8	d3	2*A3*d3
4	A4	4	4*A4	7	4*A4*7	d4	4*A4*d4
5	A5	2	2*A5	6	2*A5*6	d5	2*A5*d5
6	A6	4	4*A6	5	4*A6*5	d6	4*A6*d6
7	A7	2	2*A7	4	2*A7*4	d7	2*A7*d7
8	A8	4	4*A8	3	4*A8*3	d8	4*A8*d8
9	A9	2	2*A9	2	2*A9*2	d9	2*A9*d9
10	A10	1	4*A10	1	4*A10*1	d10	4*A10*d10
11	A11	2	2*A11	SUM4	2*A11*0	d11	2*A11*d11
12	A12	4	4*A12	1	4*A12*1	d12	4*A12*d12
13	A13	2	2*A13	2	2*A13*2	d13	2*A13*d13
14	A14	4	4*A14	3	4*A14*3	d14	4*A14*d14
15	A15	2	2*A15	4	2*A15*4	d15	2*A15*d15
16	A16	4	4*A16	5	4*A16*5	d16	4*A16*d16
17	A17	2	2*A17	6	2*A17*6	d17	2*A17*d17
18	A18	4	4*A18	7	4*A18*7	d18	4*A18*d18
19	A19	2	2*A19	8	2*A19*8	d19	2*A19*d19
20	A20	4	4*A20	9	4*A20*9	d20	4*A20*d20
21	A21	1	1*A21	10	1*A21*10	d21	1*A21*d21
			SUM3	SUM5			SUM7
				SUM6 = SUM5 -SUM4

Figure 07. Depicts the aforementioned columns. LCB and VCB stand for longitudinal and vertical center of buoyancy respectively. SUM 4 is the sum of rows one to ten and SUM 5 is the sum from rows twelve to twenty one. SUM 6 is the subtraction of SUM5 with SUM4

As a result the immersed volume is calculated

$$ Vim = \frac{1}{3} \cdot \frac{L}{20} \cdot SUM 3 $$

For the longitudinal center of buoyancy, the moment from the Xx axis is divided with the immersed volume.

$$ Mx = \frac{1}{3} \cdot \frac{L}{20} \cdot \frac{L}{20} \cdot SUM 6 $$ $$ LCB = \frac{Mxx}{Vim} $$

Similarly, the vertical center of buoyancy is calculated by dividing the moment from the keel (Mk) with the immersed volume

$$ Mk = \frac{1}{3} \cdot \frac{L}{20} \cdot SUM 7 $$ $$ VCB = \frac{Mk}{Vim} $$

The center of gravity of our boat is equally important in our calculations as the center of buoyancy. Similarly, we determine the longitudinal and vertical distances from the bow and keel respectively. This occurs because ships and boats have a plane of symmetry from bow to stern and perpendicular to the beam. Thus, the center of gravity is always at this plane and we only have to determine a longitudinal (LCG) and a vertical distance (VCG) for the center of gravity. Furthermore, boats do not have their mass uniformly distributed and as a result, we will use calculus to determine the center of gravity.

$$ CG \cdot W = \int_{}^{} x \,dx $$

In the equation above, CG is the center of gravity, W is the weight of the object and the right hand side of the equation is the integral of distance x with respect to weight. The distance x is from a reference line and in our case the two reference lines are from the bow for the LCG and from the keel for the VCG.

Two really common equations in physics are weight and density.

The equation of weight is

$$ W = m \cdot g$$

and of density

$$ rho = \frac{m}{V} $$

Combining these two equations we end up with

$$ W = g \cdot rho \cdot V $$

and differentiating

$$ dW = g \cdot rho \cdot dV $$

Then, we describe the differential volume in cartesian coordinates

$$ dW = g \cdot rho (x, y, z) \cdot \,dx \cdot \,dy \cdot \,dz $$

and substitute in the above integral, the center of gravity equation is transformed to

$$ CG \cdot W = g \cdot \iiint x \cdot \rho(x,y,z) \,dx\,dy\,dz $$

and since we want to determine two distances then

Figure 08. Depicts the axis x and z that is our reference to calculate the LCG

$$ LCG \cdot W = g \cdot \iiint x \cdot \rho(x,y,z) \,dx\,dy\,dz $$

Figure 09. Depicts the axis z and y that is our reference to calculate the VCG

$$ VCG \cdot W = g \cdot \iiint z \cdot \rho(x,y,z) \,dx\,dy\,dz $$

If the functional form of the mass distribution is unknown, we have to numerically integrate using a spreadsheet, following a similar method with the center of buoyancy. In detail, dividing the distances in small volume segments and calculating the average weight to volume value over the segments. The next step is taking the sum of average weight to volume values times the distances times the volume of the segments divided by the entire weight of the boat. In order to reduce excess calculations we will make use of previous spreadsheets to determine the volume. The area of transverse sections will be for the entire volume in the hull and the weight of the material of the hull will be determined by the density of the material times the volume of material in the segment times the acceleration of gravity (g). After calculating these two, we divide them and apply the equation ending up with the longitudinal center of gravity as a distance from the bow.

Number of segment	Length of boat per section from bow	Volume per segment	Weight of material per segment	w/v per segment	Product for LCG
1	2*L/20	v1	w1	w1/v1	((2*L/20)*(w1/v1)*v1)/W
2	5*L/20	v2	w2	w2/v2	((5*L/20)*(w2/v2)*v2)/W
3	8*L/20	v3	w3	w3/v3	((8*L/20)*(w3/v3)*v3)/W
4	11*L/20	v4	w4	w4/v4	((11*L/20)*(w4/v4)*v4)/W
5	14*L/20	v5	w5	w5/v5	((14*L/20)*(w5/v5)*v5)/W
6	17*L/20	v6	w6	w6/v6	((17*L/20)*(w6/v6)*v6)/W
7	L	v7	w7	w7/v7	((L)*(w7/v7)*v7)/W
					SUM8

$$ LCG = SUM 8 $$

The reasoning behind the seven segments is the volume calculations from twenty-one sections- every three values from the previous calculations- combined in one of the seven segments in this spreadsheet. In addition, the vertical center of gravity is calculated in the same manner but with the number of segments being four.

Number of segment	Depth of boat from keel	Volume per segment	Weight of material per segment	w/v per segment	Product for VCG
1	2*Dth/12	v1	w1	w1/v1	((2*Dth/12)*(w1/v1)*v1)/W
2	5*Dth/12	v2	w2	w2/v2	((5*Dth/12*(w2/v2)*v2)/W
3	8*Dth/12	v3	w3	w3/v3	((8*Dth/12)*(w3/v3)*v3)/W
4	Dth	v4	w4	w4/v4	((Dth)*(w4/v4)*v4)/W
					SUM9

$$ VCG = SUM 9 $$

In a final analysis, our effort in designing the hull of the RC-boat is circled around the shape and size. The shape is determined by the deadrise and the way it decreases from bow to stern. Specifically, from 76,69 degrees at the bow to 11,69 degrees. The initial dimensions tested in order for the trial and error approach to begin have to be centered around the specific application. On lakes and rivers, a boat of length 30 centimeters to 90 centimeters is a reasonable estimation for our operational needs. Then the beam follows the one third rule and ranges from 10 centimeters to 30 centimeters and a reasonable depth of 10 to 30 centimeters. Next step is estimating the draft and freeboard. After setting these variables, we can calculate the water plane area and immersed volume to define the coefficients. If the values of the coefficient are appropriate and we specify the density of the material and thickness of the hull then we can continue with calculating the centers of buoyancy and gravity.

Variable number	Variable symbol	Variable value	Unit
Length of boat	L	0.6	m
Beam of boat	B	0.2	m
Depth of boat	Dth	0.12	m
Draft of boat	D	0.07	m
Freeboard of boat	Frb	0.05	m
Τhickness of boat material	t	0.0025	m
Density of boat material	ρb	1240	kg/m3
Block coefficient	Cb	0,7008543854	0.5-1
Prismatic coefficient	Cp	0,8204197127	>Cb
Midship section coefficient	Cm	0,8542632198	0,85-0,99
Waterline coefficient	Cw	0,8648042584	0,85-0,9

After a couple runs of calculation we deduced that a length of 60 centimeters, a depth of 12 centimeters with a draft of 7 is optimal. An acceptable thickness is 25 millimeters so as to ensure rigidity of the structure while avoiding excess weight in our boat. At the same time, we focused on selecting an eco-friendly option in every part that contacts water. As a result, we chose a biodegradable 3D printing material named PLA with a density of 1,24 grams per cubic centimeter.

Figure 10. RC Boat design from all angles

Polylactic acid (PLA) is a biodegradable and bioactive thermoplastic aliphatic polyester derived from renewable sources such as corn starch, peeled tapioca roots or sugar cane. PLA is used as a raw material in desktop 3D fiber processing printers and is used in all kinds of models. Due to the very small shrinkage factor, it shows zero distortion even in large objects. This material is suitable for models that are not exposed to temperatures higher than 60°C and is completely safe and 100% ecological. This was the reason why we chose it for the main construction material of the hull of the RC Boat

Electronics and Mechanical Parts
Speed Controller	Brushless Electronic Speed Controller EZRUN MAX10
Battery	Gens ace 5300mAh 7.4V 60C 2S1P HardCase Lipo Battery
Battery Charger	Delta 6 Cell Digital Balance Charger
Remote Controller and Antenna	Redox Tx-ONE 3-channel digital proportional RC transmitter system
Servo	Servo R&D Manufacturer Power HD waterprf LW-20MG
GPS Module	GPS 6M Module by Ublox
pH Sensor	Gravity: Analog pH Sensor (For Arduino/Raspberry Pi)
Dissolved Oxygen Sensor	Gravity:Analog Dissolved Oxygen Sensor (For Arduino/Raspberry Pi)
Microcontroller	Microcontroller Arduino Mega 2560
Breadboard	Breadboard – Full-Size Bare
GSM Module	GSM/GPRS Shield
Axle	Brass , 8mm diameter 22cm length
Propeller	Plastic, 35mm diameter
Rudder	Max width: 3cm, Max length: 5cm

The placement of the individual components of the RC Boat was a key issue as we wanted to achieve the most ergonomic layout. Specifically, the motor axis is inclined to the XY plane axis by 20 degrees. It is placed 15 cm from the keel and the rudder is placed 3 cm from the stern. The placement of the sensors was also of major importance as they played a key role in the development of the vessel's engineering design. Despite the fact that the placement of the sensors is indicated to be parallel to the z-axis, we chose to be on the bow of the boat and parallel to the x-axis. However, such a decision would be energy inefficient because due to the speed that the vessel would develop we would observe increased friction of the water with the vertical cylinders. For the layout of the drive components looking at the RC Boat from the back to the front:

First, we mounted the servo on a structure in front of the rudder shaft. Next, we placed the remote control antenna and next to it the speedometer together with the brushless motor. Moving forward, we placed the motor shaft and in front of it we placed the arduino microprocessor together with the breadboard. Finally, reaching the front of the boat we installed the dissolved oxygen and pH sensors. At this point, we need to mention that for each individual part we made wooden constructions that snap together and are detachable so that we can disassemble them at any time.

RC Boat design from all angles (w/ hole)

All compartments needed for the RC Boat

Several of our initial concepts were created using the Fusion 360 software in order to meet the PDS outlined above.

Circular CFW

Hexagon CFW

Rectangle CFW

The task of designing any kind of system for an engineer is a question of dimensions. The first consideration is the shape of the CFW,i.e. rectangular, circular or any other shape. We designated a circular shape for our CFW because of its symmetry along the plane parallel to the waterline. A circular shape simplifies calculations for wind limitations because we can assume that the wind velocity vector is always the same regardless of its direction. Secondly, we have to determine the pattern of holes in the frame for the vegetation and substratum to be placed. Since we cannot mindlessly puncture holes in our frame we have to introduce a pitch p[m], meaning the distance between two centers of aligned circles (Fig. 13).

Figure 13. Depicts the definition of a pitch distance.

Three main assumptions have to be made for the pattern of holes so as to simplify calculations , first that a hole will always be placed in the center of the circular CFW so as to have a symmetric number of pitch distances from the center of the circle. (Fig. 14)

Figure 14. Depicts with two different colors the two pitch distances.

Secondly, that every hole in our CFW has the same distance from its surrounding holes meaning for our design we set only one value for the pitch. Thirdly, all holes have the same diameter. Given our assumptions, a pattern emerges that is in a diamond form because three holes that have the same distance form a triangle with three equal sides. Basic euclidean geometry states that an equilateral triangle has three equal and congruent 60 degrees angles. (Fig. 15).

Figure 15. Depicts the 60 degrees that form by assuming that all pitch distances are equal.

Given these arguments, we have to take into consideration the diameter D[m] (the straight line passing through the center of a circle) of the circular CFW , diameter of the holes do[m], pitch p[m] and number of holes N while setting the rule do < p. Then, a pattern emerges where we can only have this set of holes N {1, 7, 19, 37, 61, 91…} in our CFW. This set can also be expressed as {1, 1+6 , 1+18, 1+36 , 1+60, 1+90 , . . . ] and number one in this newly expressed set describes the hole always placed in the center as we assumed earlier. We see a mathematical pattern that will help us determine our number of holes in any CFW with the inputs being only D, do, p and output the number of holes N because of the three main assumptions we made. This occurs because of the 60 degree angle rule that we deduced earlier which states that around any hole placed in the center of the CFW, 6 holes can be placed circumferentially.(Fig. 16).

Figure 16. Depicts the possible number of holes (1, 7, 19, . . . ) in a circular CFW given the pattern we assumed.

Next step is expressing the set of holes in the CFW with a mathematical equation. In order to do that, we have to introduce two more definitions that relate with the diameter D of the CFW . The first is the number of pitch distances (np) that are in a given diameter D of the CFW .For example, if across a diameter D there are three holes placed then the number of pitch distances np is two. The second is the distance t[m], we describe it as the distance from the perimeter of the circular CFW to the perimeter of the first outer hole in a diameter D. (Fig. 17)

Figure 17. Depicts the distances t and the concept of number of pitches np

Equating D, np, p, do and t we end up with our first equation where the CFW diameter D equals to the number of pitch distances np times the pitch distance p adding the diameter of the hole do and double the distances t. The second equation is about the number of holes that could be fitted in a circular CFW and is equal to one plus the partial sum of half the number of pitch distances times six. The way this equation is deduced is based on two main assumptions: that in the center of the circular CFW a hole will always be fitted and that all holes have the same pitch distance.

$$ D = np \cdot p + do + 2 \cdot t \;\; (1)$$ $$ N = 1 + 6 * \sum_{n = 0}^{0.5np} n \;\; (2) $$

,where n is an operator for the series

The sequence of calculations is simple given D, p, do and t. We calculate np and apply it in the number of holes equation but there is a catch. Since we divide with the pitch it is highly probable that our np number will not be an even number or even worse not an integer. Thus, we have to make another consideration and round down to a number that is even and an integer. For example, say that np equals 9, then we round down to 8 because assuming 10 could lead to an error. On paper, we have a number of complete holes but in actuality some will be incomplete. The worst case scenario is a number really close to the next even number , i.e. np= 9, 83 . Although it is tempting to assume 10 we would fall prey to illogical thinking and an overall inferior design. Due to these errors with this methodology the maximum area of the CFW is not always utilized. After dealing with the task of determining patterns , we list our dimensions of the CFW which are:

Dimensions	Symbol
Diameter of CFW	D
Diameter of Holes	Do
Pitch	p
Depth of CFW	H
Length of plants	L
Max diameter of plants	Dp

We see that three more parameters are introduced, length of plants L[m] , depth of CFW H[m] and maximum diameter of plants dp[m], crucial for the next part of the hydrostatic analysis. For the length of plants we assume a maximum length that is acceptable. As for the depth of the CFW, it will determine the volume of the CFW and by extension the buoyancy of the structure. Considering the maximum diameter of plants, the rule is that dp < do.

After determining the geometric properties of the CFW we have to verify that they fulfill the hydrostatic limitations presented by the laws of physics. Two main definitions that we have to explain are the center of gravity (G) and center of buoyancy of a body (B). The first is the average location of the weight of an object while the second is the center of gravity for the volume of water which the immersed part of the object displaces. Calculating the position of these points is our first step in the hydrostatic analysis.

The center of gravity for a complex shape is tricky and our CFW is an intricate structure. Its determination is accomplished by narrowing our complex shapes to simpler ones . For example we will assume our frame holes and plants act as cylinders with diameters and lengths determined by our parameters mentioned above. The cylinder length and diameter for the holes are depth H and diameter of the hole do. For the plant part, the cylinder length and diameter are length of plants L and diameter of plant dp. Normally , we would require three distances to determine the center of gravity in three dimensions (X, Y, Z) and a center of coordinates. The latter is placed at the center of the circular CFW and in the lowest point of the frame and is characterized as keel (K), where it is explained more in depth below. The Z axis is perpendicular to the waterplane (horizontal plane parallel to the surface of water) and the depth (H) of the CFW is measured in this axis. As a result, axis X and Y are along the waterplane. Hole placement is done symmetrically between the X and Y axes and favors our design, in that the center of gravity is always in the center of the circular CFW. The reason is that the same number of holes are placed on either side of the X and Y axes. Consequently, the same weight is distributed on each side. Thus, our only distance to calculate is Gz. The formula for Center of Gravity on the z axis is the sum of all the distances of center of gravity of its shape times their individual weights divided by the sum of all weights.

$$ Gz = \frac{\int_{A}^{}z\,dz }{w} = \frac{\sum{1}^{i}(gz_i \cdot w_i)}{Wsum} $$ $$ Gz = \frac{gz(frame) \cdot w(frame) \cdot gz(holes) \cdot w(holes) + gz(plants) \cdot w(plants)}{W(CFW)} $$

Since we assumed the plants and holes as cylinders , then the gz coordinates are half the lengths of their respective cylinders. The center of gravity distance for the substratum cylinders is H/2 and for the plants is H+L/2 since the center of the X, Y, Z coordinates is on the lowest point of the frame. Then the equation above is transformed to:

$$ Gz = \frac{0.5 \cdot H \cdot w(frame) + (H + 0.5 \cdot L) \cdot w(plants)}{W(CFW)} $$

The center of buoyancy is our next parameter to calculate and it is the point where if you were to take all of the displaced fluid and hold it by that point it would remain perfectly balanced. Essentially, it's the same definition for the center of gravity but the mass involved is of the immersed part of the floating object and like the center of gravity we will set the axis at the keel (K) and make the same assumptions so as to calculate Bz. The only difference is that the plants do not contribute to the buoyancy force and are excluded from the calculations.

$$ Bz = \frac{\int_{}^{}bz\,dm }{m} = \frac{\sum{1}^{i}(bz_i \cdot m_i)}{Msum} $$ $$ Bz = \frac{m_{immersed}(frame) \cdot bzf + m_{immersed}(substratum) \cdot bzs}{M(CFW)} $$

In the next segment we explain in depth the calculations revolving around the masses of the CFW parts and the forces of weight and buoyancy.

The center of gravity and buoyancy are two points where the force of weight and buoyancy force apply respectively and they have to be aligned so as to avoid the object being toppled. To achieve CFW floatation, we have to equalize the two forces, W= m (CFW)*g where m (CFW) the total mass of the CFW and g the acceleration of gravity and Fb = ρ (CFW)*g*Vim where ρ is the density of the object and Vim the immersed volume of the object thus,

$$ W = m \cdot g = ρ \cdot g \cdot Vim $$

It has to be noted that the mass of the CFW is the sum of the masses of the frame, substratum, plants (plus the root) and the protective mesh which are determined by the dimensions discussed above through the process of finding their volume and multiplying it with the different density of its type.

$$ m(frame) = V(frame) \cdot ρ(frame) $$ $$ V(frame) = pi \cdot [(0, 5 \cdot D)^2 - N \cdot (0.5 \cdot do)^2] \cdot H $$ $$ m(substratum) = V(substratum) \cdot ρ(substratum) $$ $$ V(substratum) = N \cdot \pi \cdot [(0.5 \cdot do)^2] \cdot H $$ $$ m(plants) = V(plants) \cdot ρ(plants) $$ $$ V(plants) = N \cdot \pi \cdot [(0.5 \cdot dp)^2] \cdot L $$ $$ m(root) = RMF \cdot m(plants) $$

RMF is the root mass fraction of a plant and ranges from 0. 1 to 0. 5

$$ m(mesh) = Area\; of\; mesh \cdot KPA $$ $$ Area\; msh = msd \cdot \pi \cdot D + \pi \cdot (0.5 \cdot D)^2 $$

where msd is the depth of the mesh and KPA is kilograms per area of the material we are using and adding these three masses.

$$ M(CFW) = m(frame) + m(substratum) + m(plants) + m(mesh) $$

As noted before the buoyancy force is calculated as

$$ Fb = m(CFW) \cdot g = ρ(water) \cdot g \cdot Vim $$

Using this equation we will determine the Vim and since we have the area of frame and substratum then the depth of the immersed object dim and the bz distance

$$ dim = \frac{Vim}{Area(frame + substratum)} $$ $$ bz = 0.5 \cdot dim $$

Following these calculations we have all our necessary values to calculate the center of gravity Gz and buoyancy Bz

The next part of our analysis involves the possible rotation of our CFW through the wind velocity applied in the plants. The reason is that the plant that we are using is a firm reed and the probability for our object to heel is high. Heeling means the tendency of our object to tilt and occurs when the center of buoyancy is displaced from an outside disturbance called external heeling moment MH [N*m]. In order for this moment to be countered an opposite righting moment occurs called MS [N*m].

An equilibrium will be reached when MH equals MS. Therefore, for our structure to not heel catastrophically (sinking) then MH has to be smaller than MS. introducing, a factor of safety Ns for a given angle Φ

$$ Ns = \frac{MS}{MH} $$

when Ns is one then we reach a critical point where MS equals MH and the larger the number the safer our structure will be.

Our next task is to calculate MS and is calculated as the buoyant force times the lever arm GZ which is the distance from the line that connects the new buoyancy center to the metacenter.

Thus

$$ MS = ρ \cdot g \cdot Vim \cdot GZ $$

In figure 06 we can see Bφ as the new center of buoyancy, Nφ the metacenter and φ the angle of heel.

It is deduced that

$$ GZ = GN\phi \cdot \sin(\phi) $$

where GNφ is the distance between the points G and Nφ.

Next step is determining GNφ and for that we have to define more distances.

KB is the distance from the keel to the center of buoyancy B, where keel is the lowest point of the immersed object

BNφ is the distance from the center of buoyancy to the metacenter and KG is the distance from keel to the center of gravity

Now we can define

$$ GN\phi = KB + BN\phi - KG $$

Although KB and KG are easily calculated given our dimensions BNφ needs a further run of calculations

Such that

$$ BN\phi = BM \cdot (1 + 0.5 \cdot \tan(\phi)^2) $$

BM the distance from center of buoyancy to M initial metacenter and is calculated as follows

$$ BM = \frac{\ I \tau}{Vim} $$

Iτ :the transverse moment of inertia

Vim:the immersed volume mentioned before Iτ of a regular disk is defined as

$$ \ I \tau (disk) = 0.25 \cdot \pi \cdot (0.5 \cdot d) ^ 4 $$

but in our case the extra holes have to be subtracted from the moment of inertia.

$$ \ I \tau = \ I \tau D - N \cdot \ I \tau do $$ $$ \ I \tau = 0.25 \cdot \pi \cdot (0.5 \cdot D)^4 - N \cdot 0.25 \cdot \pi \cdot (0.5 \cdot do)^4 $$

ΙτD: the moment of inertia of the whole CFW without holes

N: number of holes

Iτdo: the moment of inertia of its single hole

A main point to be observed is that in order for such a simple calculation of the transverse moment of inertia to occur, is due to the symmetry of the pattern across the waterplane. After all these extensive calculations we can finally determine MS but we are left to calculate MH.

As mentioned above to ensure the stability of the CFW, the righting moment MS has to be larger than the heeling moment MH caused by wind velocity on the reeds. Our next step is to determine MH and we have to establish some extra assumptions. This will lead in reducing the efficiency of our design but at the same time reducing excess calculations which is beneficial. Our first assumption is that the reeds act as solid cylinders. The second assumption is that the forces applied to the CFW are not able to move it across the water This will help us both in determining the heeling moment using simple formulas from material mechanics and in describing the flow of air around the reeds using fluid mechanics.

In Fluid mechanics, when a force is applied to a solid object from a stream of a fluid there are two component forces that can describe the phenomenon , lift and drag.

In our case, lift is negligible but drag is our main consideration. The analytic calculation of drag force is difficult and its calculation demands the curves of shear stress and pressure on the distinct surface. It is a far more complicated approach than needed in dimensioning a CFW. Thus we resolve in the more widely used drag coefficient Cd.

$$ Cd = \frac{Df}{0.5 \cdot ρ \cdot U^2 \cdot A} $$

where Df:is the drag force

ρ:density of the fluid(air)

U:the velocity of said fluid

A: the cross section area which is perpendicular to the velocity vector

$$ A = L \cdot dp $$

where L: the length of our plants and dp: the maximum diameter

At the same time there are two forms of drag , friction drag and pressure drag. The first is a component force caused by the shear stress of the fluid to the wall of the solid object , the second component force is caused by the pressure that acts upon the object and is dependent on the shape of it. Thus our drag coefficient is the sum of these two forms of drag.

Introducing Cdf and Cdp for frictional and pressure drag coefficients respectively:

$$ Cd = Cdf + Cdp $$

In literature, these coefficients are already calculated for a solid cylindrical shape assuming our fluid is incompressible, the values are

$$ Cdf = \frac{5.93}{\sqrt{Re}} $$

where the Reynolds number is calculated

$$ Re = \frac{ρ \cdot U \cdot D}{μ} $$

μ is the dynamic viscosity of the fluid and Re has to be lower than 300,000 or this equation does not stand. The pressure drag coefficient is

$$ Cdp = 1.17 $$

Using the equations above we calculate the drag force Df and use it in our next step which is determining the moment of heel MH through some material mechanics principles.

In the subject of material mechanics, the main focus is given in stress analysis on beams and how they behave accordingly. In our case, the stresses involved in our given materials is not an issue that has to be analyzed, we will only use simple principles for determining our maximum moment of heel under a certain load from the drag force we calculated earlier. In addition, another assumption to take into consideration is that the velocity of wind is constant and is the same for all of our cylindrical reeds. Since, we have assumed that our reeds are a solid column and we have a constant load of drag force Df then we can deduce that the maximum moment is the sum of all the forces applied from N number of reeds times the lever arm L(length of plants).

$$ MH = N \cdot Df \cdot L $$

In conclusion, the sequence of calculations starts with two kinds of inputs.. The set of geometric variables are the initial inputs in our calculations and the second are the constant coefficients throughout the calculations. Next step is determining the number of holes and number of pitch distances followed by calculating areas, volumes and masses; Furthermore, defining centers of gravity and buoyancy while applying them in combination with fluid mechanics constants, determine heeling and righting moments. Last step is setting the angle Φ to values from 0 to 30 and establishing a corresponding factor of safety, plotting our data and evaluating our results.

Geometric properties that are firstly estimated are used in determining the number of pitch distances in a diameter and the number of holes. From Fig 18 to Fig 19

Variable Name	Variable Symbol	Value	Unit
Diameter of CFW	D	1.16	m
Diameter of holes	Do	0.03	m
Pitch	p	0.15	m
Depth of CFW	H	0.3	m
Length of plants	L	2.5	m
Max diameter of plants	Dp	0.03	m
Depth of mesh	msd	0.9	m
Distance t from perimeter of CFW	t	0.05	m

Figure 18. Depicts the initial geometric inputs and our final results from the diameter of CFW to the distance t

Variable Name	Variable Symbol	Value
Number of pitch distances in a diameter	np	6
Number of holes	N	37

Figure 19. Depicts the number of pitch distance and number of holes derived from the initial inputs of Fig 18.

Variable Name	Variable Symbol	Value	Unit
Density of frame	ρf	400	kg/m3
Density of substratum	ρs	1331.5	kg/m3
Density of plants	ρp	240	kg/m3
Density of water	ρw	997	kg/m3
Density of air	ρa	1,16992	kg/m3
Acceleration of gravity	g	9.81	m/s2
Kilogram per area	KPA	1	kg/m2
Dynamic viscosity of air	mhi	0,0000182	kg/(m*s)
Velocity of air	V	4	m/s
Reynolds number	Re	298265,3187
Root mass fraction	RMF	0.3

Figure 20. Depicts the constants that are used in all the calculations.All constants are at temperature 25 degrees celsius and 1 atmosphere in pressure units.

Then using our constants, initial inputs and calculations of FIG70 ,we determine our areas,volumes and by extension masses

Variable Name	Variable Symbol	Value	Unit
Waterplane area of frame	Afr	1,03067801	m2
Waterplane area of substratum	Asb	0,02615375884	m2
Waterplane area of plants	Apl	0,02615375884	m2
Area of mesh	Amsh	4,336654499	m2
Cross section Area	Acr	2.9	m2
Volume of frame	Vfr	0,3092034029	m3
Volume of substratum	Vsb	0.007846127652	m3
Volume of plants	Vpl	0,0653843971	m3
Mass of frame	mfr	123,6813612	kg
Mass of substratum	msb	10,44711897	kg
Mass of plants	mpl	15,6922553	kg
Mass of mesh	msh	4,336654499	kg
Mass of roots	mr	4,707676591	kg
Mass of CFW	mcfw	158,8650665	kg

Figure 21. Depicts the Areas calculated and ,corresponding volumes when we multiply with the depth H. Similarly, multiplying the volume with the densities above we end up calculating the masses. An exception is the mass of rts estimated by the RMF equation.

Next step is determining the centers of gravity and buoyancy.

Variable Name	Variable Symbol	Value	Unit
Center of gravity	CGz	0,2530786641	m
Center of buoyancy	CBz	0,06364876623	m

Figure 22. Depicts the result in the center of gravity and buoyancy

Variable Name	Variable Symbol	Value
Drag coefficient	Cd	1,180858087
Friction drag coefficient	Cdf	0,01085808693
Pressure drag coefficient	Cdp	1,17

Figure 23. Depicts the fluid mechanics constants cd,cdf,cfp. and lastly using the hydrostatic,fluid and material mechanics calculations we can estimate the heeling and righting moments.

As a result, a range of angle Φ values correspond to a linear correlation with the safety factor Ns.

Figure 24. Depicts the linear form that the angle Φ and safety factor correlate.

In our design, the critical angle Φ that our CFW can maximally tilt is 7 degrees assuming a constant wind speed of 4 meters per second and the plants on the CFW act as solid cylinders with a homogenous air flow. As the angle Φ increases so does the safety factor and assures that the CFW will not topple.

3D Design of CFW from all angles

Design Aspect	Considerations and Constraints
1. Performance	• More efficient at absorbing and storing phosphorus than conventional wetlands • Zero running costs
2. Cost	• Increased cost compared to the current prices of the conventional wetlands made of polymers • Aim to lower costs of production in the long run
3. Environment	• Reduced environmental footprint by using biomaterial, 100% natural and biodegradable • The external surface of the wetland must be waterproofed using a special coating made of an environmentally friendly substance • The construction material of the platform (mycelium) can withstand any temperature conditions
4. Life in Service	• Maximum 3 years lifespan, 6 months guarantee provided • The product will remain in it for at least 8 months after being placed in the respective water body
5. Maintenance	• Corrosion due to water and salinity significantly reduces the life cycle of the mycelium. For this reason, before applying it to the water, it will be necessary to pass 2-3 layers of insulating coating that will increase its resistance and durability • Handbook provided to describe how to properly apply the wetlands
6. Packaging	• Should be packed securely so that it is protected from transportation until installment
7. Quantity	• Should be produced in large quantities for batch production, ideally produced on a large scale, contract based, continuous production
8. Competitors	• Compared to other bioremediation methods it is less intrusive to the aquatic ecosystem as it is a biomaterial
9. Shipping	• Chartered delivery runs, using different methods depending on order size
10. Weight	• Each wetland with its platform, plants, soil and protectors will weigh approximately 158 kg
11. Aesthetics	• This is not a concern as it is not delivered to the final user
12. Geometry and Size	• Standard dimension to suit any type of water system.
13. Materials	• Mycelium is a stable and reliable building biomaterial • In combination with appropriate bio based coatings corrosion by water and salt is slowed down • Construction from local reeds and cellulose mesh to protect against birds and prevent the escape of seeds
14. Standards and Specifications	• The aim is to upgrade the current status of specific water bodies without disturbing the aquatic ecosystem
15. Installation	• Should be simple to install, instructions provided
16. Testing	• Testing of the construction material of the platform (mycelium) to investigate its mechanical properties and evaluation for its application in the specific construction • Testing to confirm our calculations regarding the buoyancy of the platform
17. Safety	• Design of constructions in the upper and lower part of the wetland to limit the rts and the flowering part of the plant in order to limit the risk of its reproduction • In the coming years, the goal is to build a more efficient and ergonomic mechanical barrier to ensure biosafety
18. Customer	• End users are private and state water management agencies • Use of wetlands by private individuals for ornamental purposes
19. Manufacturing Facility	• Indicative construction of environmental engineering • Outsourcing of various consumables including enzymes, filters from external suppliers
20. Quality and Reliability	• Quality and reliability should both be high, considering the customers are large industry companies
21. Patents	• Fully researched parts

Design Aspect	Considerations and Constraints
1. Performance	• Effective in monitoring water systems (lakes, reservoirs, rivers, etc. ) • Remotely piloted marine vessel using distributed ledger technology • Running costs should be low
2. Cost	• Lower cost of production compared to other methods of monitoring marine systems • Aim to lower costs of production in the long run
3. Environment	• Parts are able to withstand temperatures of up to 70 degrees Celsius • The non-electronic and mechanical components of the construction come from natural or environmentally friendly materials which, during their application, do not pollute the environment
4. Life in Service	• Minimum 5 years lifespan, 2-year guarantee provided • There is no limit to the frequency of its use
5. Maintenance	• Simple design so replacement parts are low cost • System should be easily opened so that parts inside can easily be checked and replaced
6. Packaging	• Should be packed securely so that it is protected from transportation until installment • Aesthetics of packaging are not important here as they do not reach the final user
7. Quantity	• Should be produced in large quantities for batch production, ideally produced on a large scale, contract based, continuous production
8. Competitors	• Compared to other tracking systems it is much cheaper and easier to use
9. Shipping	• Chartered delivery runs, using different methods depending on order size
10. Weight	• Total construction weight including electronic and mechanical components is 2.4kg
11. Aesthetics	• Despite the fact that it is not a criterion for its application, the remote-controlled ship is simple, neat and carefully designed so that it can cope with conditions that prevail in lakes, rivers, reservoirs, etc. in areas with a temperate Mediterranean climate
12. Geometry and Size	• Standard engineering design for RC Boat hull. It is listed in the mechatronic design
13. Materials	• The building materials are ecological and corrosion resistant • They are selected based on the weather conditions prevailing in the areas of application and the stresses to which they will be subjected
14. Standards and Specifications	• The goal is to build an economical and easy-to-use water monitoring system with the ability to automatically store the data in a cloud
15. Installation	• Should be simple to install, instructions provided • Easy to replace if parts need repairing
16. Testing	• Testing in Pineios river to ensure that the tracking system and the navigation system are working properly
17. Safety	• There is no risk of contamination as it is a purely technological construction without the use of microorganisms
18. Customer	• End users are private and state water management agencies
19. Manufacturing Facility	• General purpose tooling. Outsourcing specialist parts including electronic parts from external suppliers
20. Quality and Reliability	• Quality and reliability should both be high, considering the customers are large industry companies
21. Patents	• Fully researched parts