corey becker final capstone paper

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)^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.

corey becker final capstone paper

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