9.3 Mathematical Formulas: Speed, Flow Rate (GPM) & Application Rate (GPA)

Mathematical Formulas: Speed, Flow Rate (GPM) & Application Rate (GPA)

Precision pesticide application requires combining mechanical calibration with mathematical calculations. On the South Carolina commercial applicator examination, applicators must solve practical word problems determining forward travel speed, required nozzle orifice sizes, broadcast and banded application rates, and field surface areas.

---

1. Master Formula Reference Table

---

2. Ground Speed Calculation (MPH)

Tractor speedometers are frequently inaccurate due to tire slippage, wheel wear, and varying soil conditions. True ground speed must be determined in the field over a measured distance under actual working conditions.

Mathematical Basis:

• $1\text{ Mile} = 5,280\text{ feet}$.
• $1\text{ Hour} = 60\text{ minutes} = 3,600\text{ seconds}$.
• Traveling at $1\text{ MPH} = \frac{5,280\text{ ft}}{60\text{ min}} = 88\text{ feet per minute}$.

$\mathbf{\text{MPH} = \frac{\text{Distance (feet)} \times 60}{\text{Time (seconds)} \times 88}} \quad \text{or} \quad \mathbf{\text{MPH} = \frac{\text{Distance (feet)}}{\text{Time (seconds)}} \times 0.6818}$

Worked Example 1: Calculating Ground Speed

Problem: An applicator stakes out a 300-foot course in a tilled field. With the sprayer loaded half-full, the tractor travels the course in 41 seconds. What is the operating travel speed in miles per hour (MPH)?

Step-by-Step Solution:

1. Identify given values: $\text{Distance} = 300\text{ ft}$, $\text{Time} = 41\text{ sec}$.
2. Apply formula:

$\text{MPH} = \frac{300 \times 60}{41 \times 88} = \frac{18,000}{3,608} = 4.9889\text{ MPH}$

1. Answer: The travel speed is 5.0 MPH.

---

3. The Master Application Rate Formula (GPA)

The fundamental engineering formula connecting nozzle flow rate, equipment speed, and nozzle spacing is:

$\mathbf{\text{GPA} = \frac{\text{GPM} \times 5,940}{\text{MPH} \times W}}$

Origin of the Constant 5,940:

$\text{Constant} = \frac{43,560\text{ sq ft/acre} \times 60\text{ min/hr}}{5,280\text{ ft/mile} \div 12\text{ in/ft} \times 1} = \frac{2,613,600}{440} = 5,940$

• GPA = Gallons Per Acre.
• GPM = Nozzle output in Gallons Per Minute (measured from one nozzle).
• MPH = Ground speed in Miles Per Hour.
• W = Nozzle spacing in inches on the boom (or band width for a single-nozzle sprayer).

Worked Example 2: Calculating Sprayer GPA

Problem: A boom sprayer has nozzles spaced 20 inches apart. The applicator operates the rig at 4.5 MPH. A flow check shows each nozzle delivers 0.35 GPM at 35 PSI. What is the application rate in Gallons Per Acre (GPA)?

Step-by-Step Solution:

1. Identify given values: $\text{GPM} = 0.35$, $\text{MPH} = 4.5$, $W = 20\text{ inches}$.
2. Apply formula:

$\text{GPA} = \frac{0.35 \times 5,940}{4.5 \times 20} = \frac{2,079}{90} = 23.1\text{ GPA}$

1. Answer: The sprayer application rate is 23.1 Gallons Per Acre.

---

4. Required Nozzle Flow Rate Determination (GPM) & Tip Sizing

When setting up a sprayer for a specific crop recommendation, the applicator must determine the required nozzle flow rate (GPM) to select the correct nozzle tip size from a manufacturer catalog.

$\mathbf{\text{GPM} = \frac{\text{GPA} \times \text{MPH} \times W}{5,940}}$

Worked Example 3: Selecting Nozzle Tip Size

Problem: An agricultural label recommends applying a herbicide at a broadcast rate of 20 GPA. The grower plans to spray at 5.0 MPH with a boom featuring 30-inch nozzle spacing. What nozzle flow rate (GPM) is required, and which standard nozzle tip size should be selected?

Step-by-Step Solution:

1. Identify given values: $\text{GPA} = 20$, $\text{MPH} = 5.0$, $W = 30\text{ inches}$.
2. Apply formula:

$\text{GPM} = \frac{20 \times 5.0 \times 30}{5,940} = \frac{3,000}{5,940} = 0.505\text{ GPM}$

1. Answer: The required nozzle output is approximately 0.50 GPM. The applicator should select an 8005 or 11005 nozzle tip (which delivers 0.50 GPM at standard 40 PSI).

+-----------------------------------------------------------------------------+
|                   STANDARD TEEJET NOZZLE NUMBERING CODE                     |
|                                                                             |
|   Example: [ 110 ] [ 04 ]                                                   |
|              |       |                                                      |
|              |       +---> Flow Rate: 0.40 GPM at 40 PSI                    |
|              +-----------> Spray Fan Angle: 110 degrees                     |
|                                                                             |
|   Example: [ 80 ] [ 02 ]                                                    |
|              |       |                                                      |
|              |       +---> Flow Rate: 0.20 GPM at 40 PSI                    |
|              +-----------> Spray Fan Angle: 80 degrees                      |
+-----------------------------------------------------------------------------+

---

5. Broadcast vs. Band Application Mathematics

Banding involves applying pesticide in a narrow strip directly over the crop row rather than across the entire field surface. Because only a fraction of each field acre is treated, banding substantially reduces pesticide volume and chemical costs.

$\mathbf{\text{Banded Rate per Field Acre} = \text{Broadcast Rate} \times \frac{\text{Band Width (inches)}}{\text{Row Spacing (inches)}}}$

Worked Example 4: Banded Rate Calculation

Problem: A grower is planting cotton on 36-inch rows and plans to apply a pre-emergence herbicide in a 12-inch band over each row. The broadcast label rate is 15 Gallons Per Acre (GPA). How many gallons of spray solution are needed per field acre?

Step-by-Step Solution:

1. Identify given values: $\text{Broadcast Rate} = 15\text{ GPA}$, $\text{Band Width} = 12\text{ in}$, $\text{Row Spacing} = 36\text{ in}$.
2. Apply formula:

$\text{Banded Rate} = 15\text{ GPA} \times \frac{12\text{ inches}}{36\text{ inches}} = 15 \times \frac{1}{3} = 5.0\text{ Gallons per Field Acre}$

1. Answer: The sprayer will apply 5.0 gallons of spray solution per field acre, representing a 66.7% chemical reduction compared to broadcast spraying.

---

6. Land Area Calculations for Irregular Treatment Sites

Before mixing chemicals, applicators must calculate the exact surface area of the target site.

+-----------------------------------------------------------------------------+
|                         GEOMETRIC AREA FORMULAS                             |
|                                                                             |
|   [RECTANGLE / SQUARE]   ---> Area = Length x Width                         |
|                                                                             |
|   [RIGHT / ANY TRIANGLE] ---> Area = (Base x Height) / 2                    |
|                                                                             |
|   [CIRCLE]               ---> Area = pi x r^2  (~ 3.1416 x radius^2)        |
|                                                                             |
|   [TRAPEZOID]            ---> Area = [(Parallel Side a + Side b) / 2] x H   |
+-----------------------------------------------------------------------------+

• Acreage Conversion: Always divide total square feet by 43,560 to convert to acres:

$\text{Acres} = \frac{\text{Square Feet}}{43,560}$