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8/14/2019 Corey Becker Final Capstone Paper
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movements we will decrease excessive saturation. Before explaining our steps we
will list some important assumptions.
Assumptions
First, we will be ignoring the small effect that friction may play on the
pressure and velocity of the water in the irrigation pipes. Friction would obviously
have some effect but without a way to measure this small effect, we are forced to
remove it from calculations.
Second, air and wind resistance could also have a large impact on the
distribution of water. Every person who has ever seen a sprinkler on a windy day
can see the effect it has on the droplets of water. However, without the necessary
background information such as average wind speed and air density/elevation we
are forced to assume that the farmer can adjust our model to account for wind
interference.
Third, the problem did not provide an angle of dispersion from the sprinkler
head. There are several variations of sprinklers that can spray water at just about
every angle. As you can see from the calculations we provide further down, the
angle of the water leaving the sprinkler has a significant impact on the radius of the
spray zone. For this project we will be assuming that the sprinkler sprays water at
several angles which results in our fourth assumption.
Fourth, we will be assuming that the field is flat. Water may have a tendency
to run towards lower spots before being absorbed but without a topographical map
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we have to assume that the field is level. This means that at the exact moment
that the water touches soil it will be absorbed.
Developing The Model
We began by calculating spray velocitiesdirectly dependent on the number of
sprinklers. Taking the pressure and volume of water moving through the 10cm
pipes we were able to calculate the velocity at which the water left the nozzle head.
In the instances of having 4 or less sprinklers, we found velocities that were
unrealistic (range of 22.1 88.4m/s),and therefore they will not be considered. We
also did not consider the drag on the water droplets as they fly through the air,
which is something that would have a quite substantial effect on the droplets when
the velocity and distance becomes very large.
We alsohad to consider the tools we had to work with. In the most abstract
sense, our problem is to water a rectangular region with circles. When circles are
placed together, there are large gaps created since circles can only be adjacent at a
single point. Therefore, since we have to cover the entire rectangular region, there
is going to be overlap between circles within the rectangle and wasted water that
doesnt land within the rectangle. Thus, we must concede that collateral overlap
is inevitable and must simply try and minimize it. The sprinklers were connected by
these 20m segments of pipe which we assumed to be straight. This piping obstacle
also kept us from being more creative with our sprinkler layouts. We constructed
layouts that were symmetrical to minimize extra moving caused by irregularities in
the sprinkler system. We wanted the fewest number of moves so we wanted to
cover the most field as possible. In a previous paper we created a few 2-
dimensional models of possible sprinkler systems.
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There are still gaps where water will not be dispersed, so we decided that we
should shift the grid downward and to the right. The reasoning behind the move is
to minimize the amount of area that is oversaturated with water, thus we tried to
move the overlap areas to spots that was not an overlap area in the first grid. The
resulting grids with both moves are shown in the picture below:
Through extensive trial and error we were able to conclude that this system
was most effective due to its low number of moves (just 1) and low amount of
wasted water. The next hurdle was to regulate the system so that it would properly
water the ground at a rate no higher than 0.75cm/hour and no less than2cm every 4
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days. In our previous paper we assumed every point within the spray radius
received the same amount of water so we simply divided the flow rate by the area
of the field that was covered. In this case, however, we will build a function to
represent the water distribution dependent upon the distance from the sprinkler
head. Assuming that the 150L of water flowed from the sprinkler head evenly 360
degrees around and 90 degrees from horizontal to vertical, we can begin by finding
the rate of water (R), from an section of spray zone ( by ).
2R**2
If we then equate this to its corresponding landing zone ( ri*ri*j):
2R**2*ri*ri*j
=2R*2*ri*ri
lim,=2R*2*ri*ri=2R2*r*ddr
After performing some algebraic and trigonometric manipulation we substitute for
the following:
f'(r)=R2rV04a2-r2m/s
This function represents the rate of change of water distribution based on the radius
or distance from the center. Therefore, the integral of this function will give us the
amount of water at the specified point.
fr=R2-aV02*lnV02a+V04a2-r2r
Given this equation we can then apply it to our two-dimensional layout. In order to
mesh a polar coordinate equation with a Cartesian coordinate layout/field we give
first divided the field into 2400 square meters and assign them a value based on
the distance from the sprinkler head. Then the MATLAB program uses the function
above to give each area a Z-value based upon its distance from the sprinkler. This
can be seen in the attached code below. Then we made a few adjustments to
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create a more accurate model. First we leveled off the Z-values (amount of water)
at the extreme maximums since we assume that the equation isnt perfect and the
amount of water when r approaches zero does not really approach infinity. The
leveled areas can be seen in the illustration where the red areas are actually
approaching infinity according to the function. These maximums were given values
of (.0000005 m^3 per second)
Obviously, this model doesnt come close to reaching all points on the field so we
must move it. Below we move it once.
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Again, there are large valleys where the field is obviously receiving much less water
so we add two more movements (two positions in x-direction and two in the y-
direction equals four possible locations).
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This illustration suggests that moving the sprinklers three times makes the system
much more effective. However, there are still some low spots highlighted in blue.
We will have to calculate these values and determine if they reach the minimum of
2cm per every four days as outlined above.
m^3 per section per second 1.0678E-07cm^3 per section per second 0.10678cm^3 per section per minute 6.4068cm^3 per section per hour 384.408cm^3 per section per day 9225.792cm^3 per section over fourdays 36903.168cm received by each cm^2 perfour days 3.6903168
This last value is the amount of water received by the driest point on the field over
four continuous days of watering. Since anything over 2cm would be wasted we
can divide 2 by 3.69 to get .542. This means that the driest spot on the field will
become fully watered after 2.17 days or 52.03 hours. Also, given that there are four
positions, this results in roughly 13hrs per position. Logistically, this would work
well since the farmer could position the sprinkler once per day at 7pm and it would
be done in the morning at 8am. One movement per day seems like an efficient
model as far as the time spent moving the system. However, the other element we
mentioned was wasted water. In our previous paper we concluded that any water
over 2cm per four days is wasted since we assumed that this 2cm was ideal. A
weakness of this model is the fact that we rounded values to create a more practical
distribution. When attempting to calculate wasted water, however, this fudging of
numbers prevents us from finding how much water would land in the maximum
peaks. Based on our rough estimates the peaks that occur at sprinkler positions
receive about 9.365cm per four day cycle (13hrs per day). Considering the
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maximum is 72cm (.75cm per hour), we feel that 9.365 is a very respectable value
even if its not perfectly accurate.
Strengths & Weaknesses
The strongest part of our method is that it requires a very small amount of
oversight by the person in charge of irrigation. There only requires three moves in a
span of 4 days, and every portion of the 80m x 30m field issufficiently watered.
There also only needs to be 8 sprinklers for the entire 2400m2 field. The farmer can
set the sprinklers up at dusk and let them run overnight and wake up to a well-
watered field. Also, the movement pattern is very simple. The farmer just moves
the frame ten feet in one direction each day so that it completes a square.
Technically, the famer would probably want to move it a fourth time before watering
the first day of the second cycle so hes not watering the same area two days in a
row but our model assumed the soil absorbed 100% so this was not important to
our design.
Another strength that we felt should be outlined is the limited amount of
wasted water. While there were questions as to what the maximum points were, it
is clear that over the majority of the field, no spots are receiving remotely close to
.75cm per hour.
That leads us to our first weakness. While the equation for our model may
have given us and accurate estimate for 99% of the field, at every sprinkler position
the amount of water approached infinity. Even for the smallest of areas this would
conflict with our parameters given in the problem. We decided to round off the
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values because we knew the equation was not perfect and the small error shouldnt
ruin the entire model. Even though the value was chosen arbitrarily, we still feel
that its an accurate representation of the water distributed to the respective points
on the field.
The biggest weakness is brought about bythe lack of information. We had to
assume many things in order to proceed with the problem and finish in a timely
manner. If there was more context or goals, the grid system may have taken on a
different shape. With simply the goal of needing to water a field with 2cm of water
over 4 days but no faster than 0.75cm / hour, other things may not be to the liking
of a farmer. We have many spots that are watered twice as much as other parts,
and there are even regions that are watered four times as much as others. This
may cause dissatisfaction for a customer. Many of the assumptions made were fine,
but others may have a bigger impact on the situation. The drag on the water
droplets as they are flying through the air is quite substantial, especially at higher
speeds. Also, the whole system was assumed to be ideal and suitable for the
conditions that were being thrown at it. I very much doubt that this system would
really be able to shoot water out of a 0.6cm diameter opening at almost 90 m/s, and
also that the sprinkler would be able to handle such a flow of water. The possibility
of rain was also ignored but after some contemplation we figured that a simple rain
measurement could be subtracted from the four day value and then the farmer
could alter the time spent watering accordingly.
We did find that the acceptable range of watering thefield was between 2cm
and 72cm based off of the problem state. Our range of distributions was between 2
and 9.365cm for 52.03hours. We believe that this is one of the most important
parts of the project because there will be collateral overlap if it is desired to water
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everything, and we triedto minimize that as much as possible. It is because of this
that we believe that we have successfully solved the problem with respect to its
requirements.
Code:n=8; %number of sprinklersv0=88.41941283/n; %velocity of water-func of #sprinklersg=9.81; %acceleration due to gravitya=(v0^2)/g;w=.0003125/n;C=w/(pi^2);x0=40;
y0=15;x=0:1:80;y=0:1:30;[X,Y]=meshgrid(x,y);R=zeros(31,81);Z=R;
for l=0:20:20 for h=5:20:65 for i=1:81 for j=1:31
R(j,i)=sqrt((i-h)^2+(j-l)^2)+eps;Z(j,i)=Z(j,i)-((1*(C*(-1/a)*log(abs((a+sqrt(a^2-
(R(j,i)^2)))/(R(j,i))))))/4)+eps; if Z(j,i)>(5e-007)Z(j,i)=(5e-007);
end end end endend
for l=10:20:30
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for h=5:20:65 for i=1:81 for j=1:31
R(j,i)=sqrt((i-h)^2+(j-l)^2)+eps;Z(j,i)=Z(j,i)-((1*(C*(-1/a)*log(abs((a+sqrt(a^2-
(R(j,i)^2)))/(R(j,i))))))/4)+eps; if Z(j,i)>(5e-007)
Z(j,i)=(5e-007); end end end endend
for l=0:20:20 for h=15:20:75 for i=1:81 for j=1:31
R(j,i)=sqrt((i-h)^2+(j-l)^2)+eps;Z(j,i)=Z(j,i)-((1*(C*(-1/a)*log(abs((a+sqrt(a^2-
(R(j,i)^2)))/(R(j,i))))))/4)+eps; if Z(j,i)>(5e-007)
Z(j,i)=(5e-007); end end end endend
for l=10:20:30 for h=15:20:75 for i=1:81 for j=1:31
R(j,i)=sqrt((i-h)^2+(j-l)^2)+eps;Z(j,i)=Z(j,i)-((1*(C*(-1/a)*log(abs((a+sqrt(a^2-
(R(j,i)^2)))/(R(j,i))))))/4)+eps; if Z(j,i)>(5e-007)
Z(j,i)=(5e-007); end end end endend
%meshz(X,Y,Z)surfc(X,Y,Z)%pcolor(X,Y,Z)
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S OURCES & S OFTWARE MatLab, The MathWorks, Inc.
Microsoft Excel
Physicss for Scientists & Engineers, Serway & Beichner.
Heidenreich, Jacob PhD. for assistance with model distribution of water.