The Irrigation Problem�

Dan Teague

NC School of Science and Mathematics

TCM 2024

https://www.youtube.com/watch?v=_MF_Jv9AvTk

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What is Our Goal?

1. Specifications for the pipe and nozzles (how far apart and where to drill the holes for the nozzles).

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• Some estimate on how uniform the distribution of water will likely be.

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• Some concerns that arose during your development of the model.

What Must We Do?

Determine a way to measure how much water each nozzle delivers on a plant that it moves across

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Determine the effect of having overlapping spray from different nozzles.

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Determine the distribution of the water over the width of the field.

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Devise a metric to assess the uniformity of the water distribution over the field.

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Cardinal Rule of Mathematical Modeling

Create the simplest version of the problem that contains the essence of the problem.

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Make reasonable assumptions (that can be modified in later iterations).

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Ignore extraneous realities.

George Box: All models are wrong; some models are useful.

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Pablo Picasso: We all know that Art is not truth. Art is a lie that makes us realize truth at least the truth that is given us to understand.

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Core Assumptions

• All nozzles send a uniform spray over a circular area with a radius of 22 feet.

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• The system moves across the field as a solid unit and at a constant rate. This rate may be changed depending on the amount of water that needs to be delivered.

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We will Ignore:

• Wind

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• Uneven ground

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• Amount of water plants need

and speed of the system

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End Conditions

Important Issues in Mathematical Modeling

Modeling vs Mathematics: The mathematical optimum is often not the real-world optimum. Sensitivity, practicality, and simplicity must be considered.

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Sensitivity Analysis: What of your assumptions is most likely to be violated in reality? What is the most important variable to get right?

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This diagram is a problem for students.

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Why?

Distribution from One Nozzle

Water proportional to sum of the chord lengths

Water Spray, Chord Length, Water Distribution

Desmos

https://www.desmos.com/calculator/mdqn3pshvg

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Very Uneven (Non-Uniform) Configurations

Water Function “Wave Front” (5 nozzles)

Desmos

Geogebra

Measures of Uniformity on [0, D]

Range Standard Deviation

Discrete Continuous

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(computed numerically)

Python Program

No separation distance produces good uniformity.

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Results differ based on the method of calculation, but all metrics show two satisfactory solutions to the uniformity problem.

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Three Good Choices for Distance

Random Normal Nozzles N(22, 0.5)

Sensitivity Analysis

Netlogo Simulation

• import numpy as np
• import matplotlib.pyplot as plt
• import statistics as stat
• xlist = []
• ylist = []
• zlist = []
• d = 5
• while d < 44:
•   def f(x):
•     if x < -22 or x > 22:
•       return 0
•     else:
•       return 2*(484-x**2)**0.5
•   def w(x):
•     return f(x)+f(x-d)+f(x+d)+f(x-2*d)+f(x+2*d)
•   x = np.linspace(0,d,1000)
•   y = np.array([w(num) for num in x])
•   z = (max(y) - min(y)) # range
•   sd = stat.stdev(y) #standard deviation
• �  xlist.append(d)
•   ylist.append(sd)
•   zlist.append(z)
•   d = d + 0.1
• �plt.scatter(xlist, ylist)
• plt.show()
• plt.scatter(xlist, zlist)
• plt.show()

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Comparison of “Optimal” Solutions

Water Function Calculated Every Foot

Measures of Uniformity

Area Mean Absolute Deviation

Discrete Continuous

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(computed numerically)

Irrigation

Distance Metric

20.04 sd

19.725 sd

19.9 Area

20.167 sd

37.22 sd

20 Area

20.161 sd