Introduction to Agricultural Engineering Technology PDF

Harry L. Field · John M. Long Introduction to Agricultural Engineering Technology A Problem Solving Approach Fourth Edition Introduction to Agricultural Engineering Technology Harry L. Field John M. Long Introduction to Agricultural Engineering Technology A Problem Solving Approach Fourth Edition Harry L. Field John M. Long Oklahoma State University Biosystems & Agricultural Engineering Stillwater, OK, USA Department Oklahoma State University Stillwater, OK, USA ISBN 978-3-319-69678-2 ISBN 978-3-319-69679-9 (eBook) https://doi.org/10.1007/978-3-319-69679-9 Library of Congress Control Number: 2017957542 © Springer Science+Business Media, LLC 2007, 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface This text is intended for any class or reader interested in improving their knowledge about agriculture and enhancing their abilities in solving problems common to agriculture. It is an update of An Introduction to Agricultural Engineering Tech- nology by Field and Solie. This edition includes a new chapter on site-specific crop management. The topics and their treatment were selected for three reasons: (1) to acquaint individuals with a wide range of applications of engineering principles to agricul- ture, (2) to present a selection of independent but related topics, and (3) to develop and enhance the problem-solving ability of the individual. Each chapter contains educational objectives, introductory material, and exam- ple problems (where appropriate). Most chapters are self-contained and can be used independently of the others. Those that are sequential are organized in a logical order to ensure that the knowledge and skills needed are presented in a previous chapter. As principal author, I wish to express my gratitude to Dr. John Long for his technical assistance, for writing the chapter on site-specific crop management, and for reviewing the text. Stillwater, OK, USA Harry L. Field v Contents 1 Problem-Solving....................................... 1 1.1 Objectives...................................... 1 1.2 Introduction..................................... 1 1.3 Mathematical Problem-Solving....................... 1 1.3.1 Diagrams and Sketches...................... 2 1.3.2 Patterns................................. 2 1.3.3 Equations and Formulas..................... 4 1.3.4 Unit Cancellation.......................... 6 1.3.5 Intuitive Reasoning........................ 8 1.3.6 Spreadsheets............................. 9 1.3.7 Flow Charts.............................. 18 2 Significant Figures and Standard Form..................... 21 2.1 Objectives...................................... 21 2.2 Introduction..................................... 21 2.3 Uncertainty...................................... 21 2.4 Exact and Approximate Numbers...................... 22 2.4.1 Exact Numbers............................ 22 2.4.2 Approximate Numbers...................... 22 2.5 Precision....................................... 23 2.6 Accuracy....................................... 24 2.7 Significant Figures................................ 24 2.8 Rounding Numbers................................ 26 2.9 Scientific Notation and Standard Form.................. 26 3 Common Units of Measure............................... 29 3.1 Objectives...................................... 29 3.2 Introduction..................................... 29 3.3 Systems of Units.................................. 29 3.4 Distance (D)..................................... 30 3.5 Area........................................... 30 vii viii Contents 3.6 Temperature..................................... 30 3.7 Volume........................................ 30 3.8 Weight......................................... 30 3.9 Force (F)....................................... 30 3.10 Pressure........................................ 31 3.11 Time (T)....................................... 31 3.12 Velocity........................................ 31 3.13 Power (P)....................................... 31 3.14 Torque (To)..................................... 34 3.15 Horsepower (HP)................................. 34 3.16 Using Units in the Metric (SI) System.................. 37 3.16.1 Distance................................ 38 3.16.2 Area................................... 38 3.16.3 Temperature.............................. 38 3.16.4 Volume................................. 38 3.16.5 Weight and Force (Mass).................... 39 3.16.6 Pressure................................. 40 3.16.7 Time................................... 40 3.16.8 Velocity, Speed, and Acceleration.............. 40 3.16.9 Power.................................. 40 3.16.10 Torque.................................. 41 3.16.11 Horsepower.............................. 41 4 Simple Machines....................................... 43 4.1 Objectives...................................... 43 4.2 Introduction..................................... 43 4.3 Lever.......................................... 43 4.3.1 Class One Lever........................... 44 4.3.2 Class Two Lever.......................... 46 4.3.3 Class Three Lever......................... 48 4.3.4 Wheel and Axle........................... 50 4.3.5 Pulley.................................. 51 4.3.6 Inclined Plane............................ 53 4.3.7 Screw.................................. 54 4.3.8 Combining Machines....................... 55 4.3.9 Metric Problems........................... 56 5 Internal Combustion Engines............................. 59 5.1 Objectives...................................... 59 5.2 Introduction..................................... 59 5.3 Theory of Operation............................... 60 5.4 Four-Stroke Cycle................................. 61 5.4.1 Intake Stroke............................. 61 5.4.2 Compression Stroke........................ 62 5.4.3 Power Stroke............................. 62 5.4.4 Exhaust Stroke............................ 62 Contents ix 5.5 Two-Stroke Cycle................................. 63 5.5.1 Intake–Exhaust Stroke...................... 64 5.5.2 Compression Stroke........................ 64 5.6 Types of Engines................................. 64 5.7 Displacement.................................... 66 5.7.1 Compression Ratio......................... 66 5.7.2 Theoretical Power......................... 68 5.7.3 Metric Problems........................... 69 6 Powertrains........................................... 71 6.1 Objectives...................................... 71 6.2 Introduction..................................... 71 6.3 Mechanical Systems............................... 72 6.3.1 Pulleys.................................. 72 6.3.2 Sprocket Sizes............................ 74 6.3.3 Gear Sizes............................... 75 6.3.4 Speed Ratios............................. 75 6.3.5 Direction of Rotation....................... 76 6.3.6 Complex Powertrains....................... 78 6.3.7 Speed and Torque......................... 80 6.3.8 Transmission of Power and Torque............. 81 6.3.9 Power Transmission through Powertrains......... 82 6.4 Pneumatic and Hydraulic Systems..................... 82 6.4.1 Hydraulic Pressure......................... 83 6.4.2 Hydraulic Flow........................... 85 6.4.3 Metric Problems........................... 86 7 Tractors and Power Units................................ 91 7.1 Objectives...................................... 91 7.2 Introduction..................................... 91 7.3 Categories of Tractors.............................. 91 7.3.1 General Purpose........................... 92 7.3.2 Row Crop............................... 93 7.3.3 Orchard................................. 93 7.3.4 Vineyard................................ 93 7.3.5 Industrial................................ 94 7.3.6 Garden.................................. 94 7.4 Tractor Power Ratings.............................. 94 7.4.1 Engine Power............................. 95 7.4.2 Brake Power............................. 95 7.4.3 PTO Power.............................. 95 7.4.4 Drawbar Power........................... 95 7.4.5 Converting Tractor Power Ratings.............. 96 7.4.6 Lugging Ability........................... 97 7.4.7 Derating Power Units....................... 98 x Contents 7.5 Tractor Testing................................... 103 7.5.1 Principles of Testing........................ 103 7.6 Metric Problems.................................. 104 8 Machinery Calibration.................................. 107 8.1 Objectives...................................... 107 8.2 Introduction..................................... 107 8.3 Principles of Calibration............................ 108 8.3.1 Incorrect Application Rates and Patterns......... 108 8.3.2 Acceptability............................. 108 8.3.3 Five Principles of Calibration................. 109 8.4 Calibrating Fertilizer Applicators...................... 113 8.4.1 Gravity Flow Stationary Calibration............ 115 8.4.2 Broadcast Spreader Mobile Calibration.......... 116 8.5 Calibrating Grain Drills............................. 120 8.5.1 Grain Drill Stationary Calibration.............. 120 8.5.2 Mobile Grain Drill Calibration................ 123 8.6 Calibrating Row Crop Planters....................... 124 8.6.1 Row Crop Planter Stationary Calibration......... 125 8.6.2 Row Crop Planter Mobile Calibration........... 126 8.7 Air Seeders...................................... 128 8.7.1 Air Seeder Mobile Calibration................ 129 8.8 Calibrating Sprayers............................... 130 8.8.1 Sprayer Stationary Calibration................ 133 8.8.2 Mobile Sprayer Calibration................... 137 8.9 Preparing Spray Mixes............................. 138 8.10 Metric Problems.................................. 140 9 Equipment Efficiency and Capacity........................ 143 9.1 Objectives...................................... 143 9.2 Introduction..................................... 143 9.3 Efficiency....................................... 143 9.3.1 Performance Efficiency...................... 145 9.3.2 Field Efficiency........................... 149 9.4 Capacity........................................ 149 9.5 Throughput...................................... 151 9.6 Metric Problems.................................. 152 10 Economics of Agricultural Machinery...................... 155 10.1 Objectives...................................... 155 10.2 Introduction..................................... 155 10.3 Selection Criteria................................. 155 10.3.1 Company Name........................... 156 10.3.2 Cost.................................... 156 10.3.3 Repairs................................. 156 Contents xi 10.3.4 Design.................................. 156 10.3.5 Capacity................................ 157 10.4 The Optimum Machine Size......................... 157 10.4.1 Time Limitation........................... 157 10.4.2 Power Limitation.......................... 160 10.4.3 Matching Tractors and Machines............... 161 10.5 Costs of Machinery................................ 163 10.5.1 Fixed Costs.............................. 163 10.5.2 Variable Costs............................ 168 10.5.3 Annual Cost.............................. 171 10.6 Ways to Reduce Costs.............................. 173 10.6.1 Width Utilization.......................... 174 10.6.2 Time Utilization........................... 174 10.6.3 Matching Tractors and Machines............... 174 10.6.4 Reducing Original Investment................. 174 10.6.5 Increasing Annual Use...................... 174 10.6.6 Increasing Service Life...................... 175 10.7 Break-Even Use.................................. 175 10.8 Maintenance Schedules............................. 178 10.9 Metric Problems.................................. 178 11 Sound and Noise....................................... 181 11.1 Objectives...................................... 181 11.2 Introduction..................................... 181 11.3 What Is Sound................................... 181 11.4 How Sound Is Measured............................ 182 11.5 Comparing Different Sounds......................... 183 11.6 The Effect of Noise................................ 184 11.7 Determining Noise Exposure......................... 184 11.8 Exposure Guidelines............................... 186 11.9 Controlling Noise................................. 186 11.10 Metric Problems.................................. 187 12 Measuring Distance.................................... 189 12.1 Objectives...................................... 189 12.2 Introduction..................................... 189 12.3 Measuring Distances............................... 189 12.3.1 Pacing.................................. 190 12.3.2 Odometer................................ 191 12.3.3 Taping/Chaining.......................... 192 12.3.4 Stadia.................................. 200 12.3.5 Optical Range Finders...................... 202 12.3.6 Electronic Distance Measurement (EDM)........ 202 12.4 Calibration Procedure.............................. 203 12.4.1 Determining Correction Factor................ 203 12.5 Metric Problems.................................. 204 xii Contents 13 Angles and Areas...................................... 207 13.1 Objectives...................................... 207 13.2 Introduction..................................... 207 13.3 Angles......................................... 207 13.3.1 Chord Method............................ 208 13.3.2 3-4-5 Method............................. 208 13.3.3 Tape-Sine Method......................... 210 13.4 Areas of Standard Geometric Shapes................... 214 13.5 Triangle........................................ 215 13.5.1 Base and Height........................... 215 13.5.2 Three Sides.............................. 215 13.6 One Angle and Two Sides........................... 216 13.7 Rectangle, Square, and Parallelogram.................. 217 13.8 Circle or Sector.................................. 218 13.9 Trapezoid....................................... 219 13.10 Determining Areas of Irregular-Shaped Fields Using Standard Geometric Shapes................................. 220 13.11 Determining Areas of Irregular-Shaped Fields Using Trapezoidal Equations.............................. 222 13.12 Metric Problems.................................. 224 14 Land Description...................................... 229 14.1 Objectives...................................... 229 14.2 Introduction..................................... 229 14.3 Metes and Bounds................................ 229 14.4 Block and Lot.................................... 230 14.5 Rectangular System............................... 230 14.6 Metric Problems.................................. 235 15 Differential and Profile Leveling........................... 237 15.1 Objectives...................................... 237 15.2 Introduction..................................... 237 15.3 Leveling Terms.................................. 237 15.3.1 Benchmark.............................. 237 15.3.2 Backsight................................ 238 15.3.3 Height of Instrument....................... 239 15.3.4 Foresight................................ 239 15.3.5 Turning Point............................. 240 15.4 Surveying Equipment.............................. 240 15.5 Rocking a Rod................................... 241 15.6 Reading a Philadelphia Rod.......................... 241 15.7 Setting Up a Surveying Level........................ 242 15.8 Common Sources of Error........................... 243 15.9 Recording Field Notes.............................. 244 Contents xiii 15.10 Differential Leveling............................... 245 15.10.1 Error Control............................. 247 15.11 Profile Leveling.................................. 248 15.11.1 Error Control............................. 251 15.12 Using Profile Data................................ 251 15.13 Metric Problems.................................. 252 16 Weather............................................. 255 16.1 Objectives...................................... 255 16.2 Introduction..................................... 255 16.3 Areas of High and Low Pressure...................... 255 16.4 Air Masses...................................... 256 16.5 Storms......................................... 257 16.6 Hydrologic Cycle................................. 258 16.7 Rainfall Intensity, Duration, and Recurrence............. 260 16.7.1 Intensity................................ 260 16.7.2 Duration................................ 260 16.7.3 Recurrence Interval........................ 261 16.7.4 Intensity–Duration–Recurrence Interval.......... 261 16.8 Metric Problems.................................. 263 17 Water Runoff......................................... 269 17.1 Objectives...................................... 269 17.2 Introduction..................................... 269 17.3 Peak Rate of Runoff............................... 269 17.4 Rational Method of Calculating Peak Rate of Runoff....... 271 17.4.1 Runoff Coefficient (C)...................... 271 17.4.2 Rainfall Intensity (I)........................ 271 17.5 Effect of Varying Recurrence Interval.................. 275 17.6 Mixed Watersheds................................ 275 17.7 Metric Problems.................................. 276 18 Erosion and Erosion Control............................. 279 18.1 Objectives...................................... 279 18.2 Introduction..................................... 279 18.3 Soil Development................................. 279 18.4 Causes of Soil Erosion............................. 280 18.5 Two Types of Erosion.............................. 280 18.5.1 Water.................................. 280 18.5.2 Wind................................... 282 18.6 Estimating Soil Loss............................... 282 18.6.1 Rainfall Factor (R)......................... 283 18.6.2 Soil Erodibility (K)......................... 283 18.6.3 Topographic Factor (LS)..................... 283 18.6.4 Cropping and Management Factor (CP).......... 285 18.7 Erosion Control.................................. 287 18.8 Metric Problems.................................. 288 xiv Contents 19 Irrigation............................................ 289 19.1 Objectives...................................... 289 19.2 Introduction..................................... 289 19.3 The Effect of Irrigation............................. 290 19.4 Irrigation Systems................................. 290 19.4.1 Above Ground............................ 290 19.4.2 Surface................................. 290 19.4.3 Subsurface............................... 291 19.5 Depth of Water to Apply............................ 291 19.6 System Capacity.................................. 294 19.7 Seasonal Need................................... 299 19.8 Metric Problems.................................. 300 20 Handling, Moisture Management, and Storage of Biological Products............................................. 303 20.1 Objectives...................................... 303 20.2 Introduction..................................... 303 20.3 Handling....................................... 303 20.4 Augers......................................... 304 20.5 Pneumatic Conveyors.............................. 306 20.6 Sizing a Pneumatic System.......................... 307 20.7 Moisture Management.............................. 309 20.8 Adding or Removing Water.......................... 312 20.9 Storage of Biological Products........................ 314 20.10 Metric Problems.................................. 315 21 Animal Waste Management.............................. 317 21.1 Objectives...................................... 317 21.2 Introduction..................................... 317 21.3 Handling Solid Animal Waste........................ 318 21.4 Handling Liquid Waste............................. 321 21.5 Waste Treatment.................................. 322 21.5.1 Aerobic Treatment......................... 322 21.5.2 Anaerobic Treatment....................... 322 21.6 Metric Problems.................................. 323 22 Insulation and Heat Flow................................ 325 22.1 Objectives...................................... 325 22.2 Introduction..................................... 325 22.3 Insulation....................................... 326 22.4 R-Values and U-Values............................. 327 22.5 Heat Flow...................................... 330 22.6 SI Problems..................................... 331 23 Heating, Ventilation, and Air Conditioning.................. 333 23.1 Objectives...................................... 333 23.2 Introduction..................................... 333 23.3 Psychometric Chart................................ 333 Contents xv 23.4 Reading a Psychrometric Chart....................... 334 23.4.1 Dry-Bulb Temperature...................... 335 23.4.2 Wet-Bulb Temperature...................... 335 23.4.3 Dew Point............................... 336 23.4.4 Total Heat............................... 336 23.4.5 Specific Volume........................... 336 23.4.6 Relative Humidity......................... 336 23.4.7 Moisture Content.......................... 336 23.5 Heating........................................ 343 23.6 Ventilation Rate.................................. 343 23.7 Building Heat Balance............................. 345 23.8 Air Conditioning.................................. 351 23.9 Metric Problems.................................. 358 24 Selection of Structural Members........................... 359 24.1 Objectives...................................... 359 24.2 Introduction..................................... 359 24.3 Simple and Cantilever Beams........................ 359 24.4 Beam Loading................................... 360 24.5 Dimensioned Lumber.............................. 361 24.6 Size of Beam.................................... 363 24.7 SI Metric....................................... 363 25 Principles of Electricity.................................. 367 25.1 Objectives...................................... 367 25.2 Introduction..................................... 367 25.3 Electricity....................................... 367 25.4 Electrical Terms.................................. 368 25.5 Electrical Power.................................. 369 25.6 Electrical Energy................................. 370 25.7 SI Metric Customary Calculations..................... 371 26 Series and Parallel Circuits............................... 373 26.1 Objectives...................................... 373 26.2 Introduction..................................... 373 26.3 Series and Parallel Circuits.......................... 373 26.3.1 Series Circuit............................. 373 26.3.2 Parallel Circuit............................ 374 26.4 Series–Parallel Circuits............................. 376 26.5 Determining Voltage and Amperage in Circuits........... 377 26.6 Using Voltmeters and Ammeters...................... 378 26.6.1 Voltmeters............................... 378 26.6.2 Ammeters............................... 379 26.7 Grounding...................................... 381 27 Sizing Conductors...................................... 383 27.1 Objectives...................................... 383 27.2 Introduction..................................... 383 xvi Contents 27.3 Calculating Voltage Drop........................... 383 27.4 Calculating Conductor Size.......................... 384 27.5 Selecting Conductor Sizes from a Table................. 387 27.6 Circuit Protection................................. 388 27.7 Metric Problems.................................. 388 28 Electric Motors........................................ 391 28.1 Objectives...................................... 391 28.2 Introduction..................................... 391 28.3 Electric Motors................................... 391 28.4 Advantages and Disadvantages....................... 392 28.4.1 Advantages.............................. 392 28.4.2 Disadvantages............................ 393 28.5 Use and Performance Classifications................... 393 28.5.1 Type of Current........................... 393 28.5.2 Type of Enclosure......................... 394 28.5.3 Type of Bearings.......................... 394 28.5.4 Type of Mounting Base..................... 394 28.5.5 Load-Starting Ability....................... 395 28.5.6 Starting Current........................... 395 28.5.7 Reversibility............................. 395 28.5.8 Dual Voltage Potential...................... 395 28.6 Types of Motors.................................. 395 28.6.1 Split Phase............................... 395 28.6.2 Capacitor Motor........................... 396 28.6.3 Repulsion Motors.......................... 396 28.7 Overcurrent Protection............................. 397 28.8 Overload Protection............................... 397 28.9 Motor Nameplate Data............................. 398 29 Site-Specific Crop Management........................... 401 29.1 Objectives...................................... 401 29.2 Introduction..................................... 401 29.3 Global Navigation Satellite Systems................... 401 29.3.1 Accuracy................................ 402 29.3.2 Type of Receiver.......................... 403 29.4 Automation..................................... 404 29.5 Spatial Data..................................... 405 29.5.1 Soil Sampling............................ 405 29.5.2 Yield Monitors............................ 408 29.6 Variable Rate.................................... 414 29.6.1 Real-Time............................... 415 29.6.2 Post-Processed............................ 415 29.7 Metric Problems.................................. 418 Contents xvii Appendices............................................... 419 Appendix I: Conventional Unit Conversions.................... 419 Appendix II: Conventional to SI Conversions................... 420 Appendix III: Estimating Combine Losses..................... 422 Appendix IV: Efficiency, Speed Estimated Live and Repair Factors of Common Agricultural Machines.................... 422 Appendix V: Draft Parameters and an Expected Range in Drafts Estimated by the Model Parameters for Tillage and Seeding Implements........................................... 424 Appendix VI: Solid Animal Waste Production and Characteristics... 426 Appendix VII: Nutrient Utilization by Crops................... 427 Appendix VIII: Maximum Annual Application Rates for Phosphates Based on Soil Family......................... 427 Appendix IX: Insulating Properties of Various Building Materials...................................... 428 Appendix X: Moisture and Heat Produced..................... 432 Appendix XI: Allowable Fiber Stress by Species................ 432 Appendix XII: Copper Resistance (Ohms/1000 ft)............... 433 Appendix XIII: Copper Resistance (Ohms/100 m)............... 433 Appendix XIV: Copper Wire Sizes for 120 Volt, Single Phase, 2% Voltage Drop....................................... 434 Appendix XV: Copper Wire Sizes for 120 Volt, Single Phase, 2% Voltage Drop....................................... 435 Appendix XVI: Standard Test Weight and Marketable Moisture Content....................................... 436 Index................................................... 437 Chapter 1 Problem-Solving 1.1 Objectives 1. Be able to define problem-solving. 2. Be able to describe the common problem-solving methods. 3. Be able to select the appropriate method for solving a problem. 4. Understand the function and use of spreadsheets. 5. Understand the use of and common symbols for flow charts. 1.2 Introduction Problem-solving is a part of living. We face many problems on a daily basis. Some of these problems occur because of the environment, others involve people and human relations, and some require the use of math. In this chapter, we will discuss problems involving mathematical solutions and several ways these problems can be approached. 1.3 Mathematical Problem-Solving Mathematical problem-solving is the process by which an individual uses previ- ously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation. The essence of the process is the ability to use information and facts to arrive at a solution. Remember two characteristics when solving problems using mathematical processes: 1. Mathematical processes do not always provide the answer. They may just provide more information that can be used to make a more informed decision. Good decision-making requires good information. 2. Establish levels or intervals of acceptability whenever perfection is not possible or expected. When perfection is not possible, the amount of error that is acceptable must be determined. Several resources are available for determining © Springer Science+Business Media, LLC 2018 1 H.L. Field, J.M. Long, Introduction to Agricultural Engineering Technology, https://doi.org/10.1007/978-3-319-69679-9_1 2 1 Problem-Solving the acceptable level of error. These include experts, standards, manufacturers’ recommendations, comparison to other situations or machines, personal experi- ences, and others. Both of these characteristics will be explained using examples and sample problems in later chapters. One of the objectives of this chapter is to increase the reader’s knowledge of problem-solving methods. The following sections discuss seven different approaches to solving mathematical problems. These are diagrams and sketches, patterns, equa- tions and formulas, unit cancellation, intuitive reasoning, spreadsheets, and flow charts. 1.3.1 Diagrams and Sketches Some problems involve the determination of a quantity of items, such as the number of nails per sheet of plywood or the number of studs in a wall. In solving these types of problems, it usually is helpful to draw a sketch or a diagram. Problem How many posts are needed to build a fence 100 feet long with posts 10 feet apart? Solution For many people, the first response would be 10: 100 ft Posts ¼ ¼ 10 posts 10:0 ft=Post but a diagram, Fig. 1.1, shows that the correct number of posts is 11. This is an example of a situation where a wrong answer is possible if you do not interpret the problem correctly. In this example, 10 is not the number of posts. It is the number of spaces. Fig. 1.1 Number of posts 1.3.2 Patterns The solution to some problems may depend upon one’s ability to discover a pattern in an array of numbers or values. Frequently, it is convenient to examine the patterns in a sample rather than the entire population. Once a pattern is discovered and shown to be consistent for the sample, it can be used to predict the solution for the entire population. 1.3 Mathematical Problem-Solving 3 Table 1.1 Patterns in numbers, first sample Cow number Ration Child 1 2 3 4 5 6 7 8 9 10 Grain 1 N N N N N N N N N N Mineral 2 Y Y Y Y Y Hay 3 Y Y Y Silage 4 Y Y Water 5 Y Y Problem A dairy farmer has five children. Each child is responsible for one part of the daily feed ration for the family’s 100 dairy cows. The oldest is responsible for the grain, the second for the minerals, the third for the hay, the fourth for the silage, and the fifth for the water. Instead of feeding each cow, the first child decides she will not feed the cows at all that day. The second child decides just to feed every other cow, the third child feeds every third cow, and so on. Dad soon discovers how the cows were fed and needs to know which cows did not receive any feed or water. Solution When one is faced with this type of problem, it is usually helpful to set up a table. In this case, it would be very time-consuming to set up a table for all 100 cows. Instead, select a sample of the cows. If a pattern is true for the sample, there is a high probability that the pattern will be true for a large group. Determining the size of a sample is not always easy. Pick one, and if a clear pattern does not appear, increase the size until a pattern develops. We will start with the first ten cows (Table 1.1). In this sample, cows #1 and #7 did not receive any grain, mineral, hay, silage, or water. Is this enough information to establish a pattern? We will predict that the next cow that did not receive any feed or water is #11. Why? To test this prediction, the sample size must be extended to include a larger number of cows. Table 1.2 shows that the prediction was right; cow #11, along with #13, #17, and #19, did not receive any grain, minerals, hay, silage, or water. It is now safe to consider that the prediction could be used to identify all of the animals within the herd that did not receive any grain, minerals, hay, silage, or water (those animals represented by prime numbers, i.e., a number divisible only by itself and one). Table 1.2 Patterns in numbers, second sample Cow number Ration Child 11 12 13 14 15 16 17 18 19 20 Grain 1 N N N N N N N N N N Mineral 2 Y Y Y Y Y Hay 3 Y Y Y Silage 4 Y Y Y Water 5 Y Y 4 1 Problem-Solving 1.3.3 Equations and Formulas Equations and formulas are similar but different. All formulas are equations, but not all equations are formulas. Many sources consider the terms interchangeable. The following sections will explain the difference. 1.3.3.1 Equations An equation is a statement that declares two quantities as being equal by using an equal (¼) sign. Equations are useful because they show the equality relationship between the variables in a problem, such as a ¼ 2b. Determining the value for the variables to make the equation true solves the equation. The example uses a simple equation with only two variables; consequently, there are more than one set of numbers that will make the equation true, a ¼ 4 & b ¼ 2, a ¼ 8 & b ¼ 4, etc. All combinations have the same relationship: b is one half of a. This may be an appropriate answer, but other equations will have only one set of numbers that will make both quantities equal. Equations can be developed, modified, or adapted to solve any problem. Problem Determine, using an equation, the length of fence wire required to build a single wire fence around a rectangular field measuring 450 feet on the long side and 350 feet on the short side. Solution The fence forms a perimeter around the rectangular field. The solution of the problem is an equation that will solve for the perimeter of a rectangular field. The unknown quantity is the perimeter, and the known quantities are the length and width. Begin by assigning the variables L to represent the length, W to represent the width, and Pr to represent the perimeter. A rectangle has four sides, two long sides and two short sides. The equation for finding the perimeter is: Pr ¼ ðL1 þ L2 Þ þ ðS1 þ S2 Þ substituting the known values: Pr ¼ ð450 ft þ 450 ftÞ þ ð350 ft þ 350 ftÞ ¼ 900 ft þ 700 ft ¼ 1600 ft The field will require 1600 feet of wire. This equation can be used to determine the length of wire for any rectangular field. Being a rectangular field, the equation can also be written as Pr ¼ 2L + 2S. The first equation could be modified to solve for the length of wire that would be needed for a four-sided field that is not a rectangle, but the second one could not. Problem Determine, using an equation, the amount of fence wire required to build a single wire fence around a four-sided field with two long sides of 450 feet and two short sides of 350 feet and 225 feet. 1.3 Mathematical Problem-Solving 5 Solution Substituting the values: Pr ¼ ðL1 þ L2 Þ þ ðS1 þ S2 Þ Pr ¼ ð450 ft þ 450 ftÞ þ ð350 ft þ 225 ftÞ ¼ 900 ft þ 575 ft ¼ 1475 ft To solve this problem with the second equation, it must be modified. Pr ¼ 2L þ ðS1 þ S2 Þ Pr ¼ ð2 450 ftÞ þ ð350 ft þ 225 ftÞ ¼ 900 þ 575 ¼ 1475 ft This example illustrates that the ability to develop and adapt equations is a required skill when using equations. 1.3.3.2 Formula A formula is a group of mathematical symbols, representing variables, which express a specific relationship. Formulas can be treated as a recipe or set of instructions that will produce a specific result. An example is the formula used to solve for the area of a circle, A ¼ π r2. The same variables, A and r, are used each time, and the relationship of the variables is the same each time the area of a circle is calculated. Many formulas also contain a constant. The formula for the area of a circle also illustrates this principle. The variable pi (π) is a constant. There are at least two important considerations when using formulas. 1. You must substitute values with the correct units of measure for each variable. All formulas have specific units of measure for each variable. If the units are incorrect, the answer will be incorrect. The formula used to determine the application rate of a boom-type sprayer is an example. gal 5940 Flow rate ðgal=minÞ Application rate ¼ ac Speed ðmin=hrÞ nozzle spacing ðinÞ If the constant, 5940, is not included, the units in the answer will evaluate to: gal hr 1 gal 6¼ which is not correct. min min in ac Unit conversion values must be used for the formula to produce the correct answer. These unit conversion values are used each time the application rate is calculated. This formula is an example of a situation in which the unit conver- sion values that are always used each time the problem is solved are combined into a constant, 5940. When the application rate is determined using an equation that includes all of the unit conversion values, the source of the constant becomes apparent: 6 1 Problem-Solving gal gal hr 1 60 min 1 mi 12 in 43,560 ft2 ¼ ac min mi in 1 hr 5280 ft 1 ft 1 ac gal 31,363,200 ¼ ac 5280 gal ¼ 5940 ac Notice that the three-bracketed variables of the formula, flow rate, velocity, and nozzle spacing, are included in the application rate equation. It is important to remember that when using a formula, the variables must be entered with the correct unit of measure. If the unit of measure for the variable is not correct, it must be converted, or an additional unit conversion variable must be added. This will change the constant that must be used. 2. Many formulas contain equal signs and thus can be treated as equations. A situation may arise where the result of a formula is known but an input variable may be the unknown. This requires the ability to rearrange the formula to solve for the unknown value. The application rate formula has four variables: appli- cation rate, flow rate, velocity, and nozzle spacing. It is possible that you know the desired application rate but need to determine the nozzle spacing. This requires that we treat the application rate formula as an equation. Rearranging and solving for nozzle spacing in inches (nsi): 5940 Flow rate ðgal=minÞ Nozzle spacing ðnsiÞ ¼ Speed ðmi=hrÞ Application rate ðgal=acÞ Through this rearrangement, a new formula for nozzle spacing has been pro- duced. Most formulas are equations because they show the equality of two quantities, but not all equations are formulas because equations do not always represent a specific relationship. This difference is a small mathematical nuance; therefore, for the remainder of this text, the terms equation and formula are used interchangeably. 1.3.4 Unit Cancellation Patterns, equations, and formulas solve many problems, but they require a more than minimum level of algebra skills. A less intense algebraic method for solving math- ematical problems in agriculture is called the unit cancellation method. The unit cancellation method, or unit method, develops the equation needed to solve the problem by ensuring the unwanted measurement units are eliminated and only the desired unit of measure for the answer remain. Variables without units such as π can also be included. The unit cancellation method follows two mathematical principles: 1. The unit of measure associated with the number (feet, gallons, minutes, etc.) follows the same mathematical rules as the number. 2. The measurement unit of the number behaves according to the rules of fractions. 1.3 Mathematical Problem-Solving 7 For example: 2 2 ¼ 4 or 22 With units of feet the same equation is: 2 ft 2 ft ¼ 2 2 and ft ft or 4 ft2 To review the rules of fractions, study the following example: 3 4 34 3 ¼ ¼ 4 5 45 5 In this example, the 4’s in the numerator and denominator cancel out (4/4 ¼ 1). When the units of measure are included, they behave in the same way: 3 ton 4 hour 12 ton ton ¼ ¼ 0:6 4 hour 5 day 20 day day In this example, the 4’s and the units associated with them cancel out. The un-canceled units become the units for the answer. The following example shows another variation of this principle (where gal ¼ gallon and hr ¼ hour): gal 5 3 hr ¼ 15 gal hr In this example, the unit of hour in the numerator and denominator cancel each other, leaving the unit of the answer in gallons. Problem What is the weight (lb) of one pint of water? Solution If a scale and a one pint measure were available, it would be a simple task to weigh the pint of water. An alternative is to use the conversion factors found in a table of weights and measures (Appendix I) and the unit cancellation method. Note, in this example, two types of measure are used: volume and weight. The heart of the problem is to find the conversion value(s) that will convert from volume (pints) to weight (pounds). This value is the density of water. To begin, refer to Appendix I and identify conversion factors that use both volume and weight. You should find that 1 cubic foot contains 7.48 gallons and 1 gallon contains 8 pints. This is a start, but you need something more. If you also know that water weighs 62.4 pounds per cubic foot, the problem is solved with (lb ¼ pounds, gal ¼ gallons, pt ¼ pints, and ft3 ¼ cubic feet): lb 62:4 lb 1 ft3 1 gal ¼ 3 pt 1 ft 7:48 gal 8 pt lb ¼ 1:04 pt The units of pint, gallon, and cubic feet all cancel each other leaving the answer in the desired units of pound per pint. This example illustrates the principles and sequence of steps when using the unit cancellation method. 8 1 Problem-Solving Begin by writing down the correct units for the answer. Write down the equal sign (¼). Begin entering the values and their units. The first value entered should have one of the desired units of the answer in the correct position (numerator or denom- inator), even if it is a unit conversion value from Appendix I or another source. Starting with one of the units of measure in the correct position will eliminate the possibility of having the right half inverted. Enter a value that will cancel out the unwanted units, if any, in the first value entered. Continue to add variables with the appropriate units until the only units that remain are the units of the answer. If all of the units cancel except the desired answer, and the units are in the correct position, then the only possible mistake is a math error in calculating the final answer. Calculate the final numerator by multiplying all of the numerators together. Calculate the final denominator by multiplying all the denominators together. For a final answer in decimal format, divide the numerator by the denominator. The process of unit cancellation is also useful for problems requiring the development of a new unit. For example, a very common quantity in agriculture is power. Power is measured in different units, depending on whether it is electrical or mechanical. Mechanical power is measured in units of foot-pounds per minute. The solution to a problem in which a 24 ounce weight is moved 15 feet in 5 s would look like this (oz ¼ ounces, sec ¼ seconds, lb ¼ pounds, ft ¼ feet): ft lb 15 ft 60 sec 1 lb 24 oz Power ¼ min 5 sec 1 min 16 oz 1 21,600 ft lb ¼ 80 min 270 ft lb ¼ min (Note that lb. ft. is a compound unit, not feet minus pounds or feet times pounds.) This same process will work just as well for problems with units that are more complex and have more variables. 1.3.5 Intuitive Reasoning Intuitive reasoning is a process by which an individual arrives at a correct answer through insight or a hunch, usually without being able to explain the process used. The actual process depends on the individual and cannot be defined in progressive steps. Problem You ask your employees to determine how many vehicles in the parking lot need their seat covers replaced. They return with an answer of 150. Then you realize you need to know how many single seat pickups and how many cars with 1.3 Mathematical Problem-Solving 9 two seats. Your employees remember that there were the same number of cars in the lot needing seat covers as pickups; so how many car and how many pickup seat covers do you need? Solution Some people would solve this problem algebraically, but with intuitive reasoning, a series of approximations can be developed. If there were 20 cars and 20 pickups, you would need 60 seat covers [(20 1) + (20 2) ¼ 60], if 40 cars and 40 trucks 120 seat covers, and if 60 of each then 180 seat covers. The answer must be less than 60 and more than 40. The correct number of vehicles is 50. Which means you would need 50 pickup seat covers and 100 car seat covers. 1.3.6 Spreadsheets The adoption of computers has provided a very useful problem-solving method. This is the spreadsheet. A spreadsheet is a very powerful data management tool that provides a means to enter, manipulate, and plot data and also information. The features of spreadsheets programs are not all the same, but they all include several common features. Some of the common features and uses of spreadsheets are presented in the following sections. 1.3.6.1 Data Entry Before a computer can manipulate data, it must be able to locate it. Spreadsheet programs accomplish this by using a grid or array consisting of columns and rows. A single grid is usually called a sheet or page. Most spreadsheet programs use letters to identify columns and numbers to identify the rows. The junction of each row and column is called a cell. The labels used for the columns and rows give each cell a unique address or location; see Fig. 1.2. The location of the “X” in Fig. 1.2 is A1, and the cell that has the “XX” is cell E6. The number of columns and rows in a spreadsheet is limited by the memory of the computer or by the spreadsheet program. Spreadsheets can store and manipulate large amounts of information. A grid with 256 columns and Fig. 1.2 Columns and rows of a spreadsheet 10 1 Problem-Solving Fig. 1.3 Linking sheets in a workbook 10,000 rows has 2,560,000 individual cells for data or information. Cells in a spreadsheet can contain text or numbers, and both text and numbers can be manipulated. Text and numbers are typically entered into the spreadsheet via the keyboard. Spreadsheets also provide copy and paste functions and other ways to input data. Spreadsheets contain features for manipulating data. One example is the ability to link information in different sheets or pages. For example, one type of spread- sheet uses the term “workbook” to describe a spreadsheet file. Each workbook can have several pages and the data in one page can be linked to another page. In this spreadsheet, if cell B3 in sheet #1 contained the equation “¼Sheet2!C4,” that cell would display the contents of cell C4 in sheet #2 (Fig. 1.3). Although the primary purpose of spreadsheets is the manipulation of data, text can also be entered into cells to organize and label the data. Text can also be used to manipulate data. When setting up a spreadsheet, the individual should spend some time studying the data to determine the best structure of the spreadsheet. Answering questions about the number of columns, the number of rows, what cells will contain subtotals, etc., and the location of the outputs before starting the data entry will prevent problems that force a restart or the reorganization of the data. This will also make the spreadsheet easier to use and reduce the number of mistakes that will occur. Another reason for the popularity of spreadsheets is the ability to output the data in different formats. Spreadsheets can output data in charts, graphs, and text that can be displayed by the program or exported to presentation programs. 1.3.6.2 Data Manipulation Data in spreadsheets can be manipulated with any mathematical operation. The user selects the cell for the answer and identifies the location of the data and inserts or writes the equation for the desired computation. The data can be in single cells, a 1.3 Mathematical Problem-Solving 11 Fig. 1.4 Adding data in spreadsheet Fig. 1.5 Example of mathematical operation in a spreadsheet row of cells, a column of cells, or a grid of cells. The spreadsheet program and the user’s ability to use it are the only limits to the complexity of equations that can be used. For example, if the desired operation in Fig. 1.4 is finding the sum of two numbers and have the answer appear in cell D2, the user selects cell D2 and enters the equation ¼ A2 + B2 (Fig. 1.5). Note the equation starts with the “¼” sign. In most spreadsheets, starting with the “¼” sign tells the computer that a mathematical process will be performed on data from other cell(s) and the results displayed in the cell with the equal sign. This is a simple example, but all mathematical operations are entered in the same way. Functions To expedite the data manipulation process, spreadsheets also include functions. A function performs a specific operation on the input data to produce a specific output. Functions can be as simple as sum and average or as complex as square root, trigonometric functions, and many other more complex math operations. Functions are usually accessed through a menu system. Functions may be organized into categories. Not all functions will be explained in this text. Two examples will be included to show how functions are entered into a spreadsheet and how they work. The previous addition example in Fig. 1.5 can be completed using the sum function (Fig. 1.6). Fig. 1.6 Addition using the sum function 12 1 Problem-Solving Fig. 1.7 Using sum function with multiple numbers In this example, 7, the sum of 3 and 4, would appear in cell D2. The advantage of the sum function is much more apparent when larger data sets are used (Fig. 1.7). Figure 1.7 also illustrates that cells can be summed both horizontally and vertically. The sum function will also be used to add up a grid of cells. Using the sum function is much simpler than writing the equation when adding several numbers. To write the equation using the “+” operator, each cell reference must be included with a “+” in between. The equation “¼ A2 + B2 + C2 + D2 + E2 + F2” would be entered into cell H2 to sum all the numbers in row #2. The “sum” function is used by entering an ¼ into the cell where the answer is desired and then typing sum. Next enter the left parenthesis “(.” The third step is entering the beginning and ending cell references separated by a “:”. Finish by entering the right parenthesis “)” into cell H2 and hit the enter/return key (Fig. 1.7). When the same function will be used for several rows or columns of data, the appropriate fixed or relational reference must be used. This is explained in a following section. It cannot be demonstrated in this text, but inserting functions is easier than the previous description because the cells can be entered by highlighting the desired cells instead of typing the beginning and ending cell references. All of the functions are entered in a similar manner. In the second example, a logic function will be used to make a decision about the data. Logic Functions Another feature of spreadsheets is logic functions. Logic functions are expressions such as “IF,” “AND,” “FALSE,” “NOT,” “OR,” and “TRUE.” A logic function is an equation that adds decision-making capabilities. Logic functions are very useful because they are used to compare the values of cells, draw a conclusion based on the data in the cells, and express the conclusion in a cell. The “IF” logic function compares the values in a set of cells and returns a number or word based on the logic statement. For example, a flour miller must constantly monitor the percent of flour milled from the wheat because a drop in the flour yield rate indicates a problem 1.3 Mathematical Problem-Solving 13 Fig. 1.8 Spreadsheet using the “IF” function during the milling process. The “IF” function can be used to trigger an error or warning message when the yield drops below the desired rate (Fig. 1.8). In this example, the “IF” function was used to trigger the spreadsheet to insert the word “error” whenever the percent yield of flour was less than 90% and to leave the cell blank when the flour yield was 90% or more. The function used in B6 through H6 is written as ¼IF(reference cell < 90,”Error”,” “). For cell B6 the equation is written as ¼IF(B5 0.98 > 0.95). In this example breaking the problem down into parts requires less math ability than trying to determine the answer by using one equation. The basic equation is relatively simple: bu Vol ðbuÞ nr ðrevÞ R ¼ ac nr ðrevÞ A ðacÞ But when the values are included to arrive at bu/rev and rev/ac, the math is more complicated. The first variable for grain drills is determined by: Vol ðbuÞ 1 ðbuÞ W ðlbÞ ¼ nr ðrevÞ γ ðlbÞ nr ðrevÞ The second variable is determined by: nr rev 1 rev ¼ A ðacÞ 1 ft width 1 ft 1 ac π dia nu 12 in unit 12 in 43,560 ft2 8.5 Calibrating Grain Drills 123 where nu ¼ number of metering units. Putting the two parts together produces: bu W lb collected R ¼ ac lb γ nr ðrevÞ bu 1 rev 1 ft width 1 ft 1 ac π dia nu 12 in unit 12 in 43,560 ft2 The solution for the sample problem is: bu 1:03 lb R ¼ ac lb 60 25 rev bu 1 rev 1 ft 6:0 in 1 ft 1 ac π 26 9 12 in unit 12 in 43,560 ft2 0:000687 bu 1 rev bu ¼ ¼ 0:9767... or 0:98 rev 0:000703 ac ac The calibration of the drill is not completed until the uniformity of distribution is also checked. The acceptability of the distribution is determined setting upper and lower limits for each metering unit. The limits are set around the mean amount of seeds collected from the metering units. This will give: W t 1:03 lb Mean ¼ ¼ ¼ 0:1144... lb nu 9 units The distribution upper limit is: bu DLu ¼ 0:114... þ ð0:114... 0:05Þ ¼ 0:1197 or 0:12 ac The distribution lower limit is: bu DLl ¼ 0:114... ð0:114... 0:05Þ ¼ 0:1083 or 0:108 ac A comparison of these limits to the data shown in Table 8.2 indicates that although the grain drill seeding rate is acceptable, the distribution is not. The rate for metering units #8 and #9 is above the upper limit. Both metering units should be repaired before the drill is used. 8.5.2 Mobile Grain Drill Calibration During a mobile calibration, collection containers are attached to each metering unit, and the drill is driven a measured distance. Mobile calibration is more 124 8 Machinery Calibration problematic than stationary because of the problem of attaching containers to the metering units that stay in place. The calibration process is the same as the one used for fertilizer spreaders—collecting the seeds and calculating the seeding rate. Problem Determine the seeding rate for a 17–8 grain drill that traveled 50.0 feet during collection. 1.32 pounds of seeds were collected. Solution bu 1 bu 1:32 lb 43560 ft2 1 12 in ¼ ac 60 lb 50:0 ft ac 17 8 in 1 ft 689,990:4 ¼ 40,8000 bu ¼ 1:691... or 1:69 ac The drill is planting 1.69 bushels per acre. In this example the distribution pattern cannot be checked because the data was not collected for each individual metering unit. 8.6 Calibrating Row Crop Planters Row crop planters or precision planters are used to plant crops in rows while maintaining spacing between seeds within the row unlike those planted by grain drills. They use a different type of metering mechanism. Row crop planters are commonly used to plant large seeds, such as corn, soybeans, and sunflowers, but they are also used to plant small-seeded vegetable crops such as radishes. The common metering units for row crop planters are called plate, disk, and drum. A common feature of these types of metering units is that they adjust the planting rate for changes in velocity, within limits, using a mechanical drive wheel or computer controlled electric drive motor (Fig. 8.5). Fig. 8.5 Single unit of row crop planter 8.6 Calibrating Row Crop Planters 125 Plates are the tradition method for metering seeds, but for most crops, they have been replaced with disks. One style of antique plate planter was designed to plant hills of seeds, but most plate and the modern disk metering units singulate seeds. Singulating means selecting individual seeds from a volume of seeds. The plates are mounted horizontally underneath a seed hopper. The plates have cells, or notches, around the rim. As the plate rotates, the seeds are deposited in holes (cells) on the rim metering plate. As the metering plate rotates, the cells pass over a tube that connects to the furrow opener. The seeds drop from the plate and fall into the seed tube which delivers them to an open furrow in the soil. Two methods are used to change the seeding rate. A plate with a different number of cells can be used, and/or the drive train speed ratio between the drive wheel and the metering unit can be changed. Changing the speed ratio may require changing sprockets, but some planter designs use a gearbox for changing planting rate. Disk metering units also have cells or holes in the rim, but the disk is mounted vertically. Different disks are used for different sizes of seeds and types of plants. They are called air planters because two different designs are used, based on air pressure either greater than or less than atmospheric. One type uses pressurized air to hold the seed in a vertical seed plate, and when the air is shut off, the seeds drop into the opened furrow or are blown into tubes that deliver them to the opened furrow. The other type of air planter uses a vacuum to hold the seeds in the cells in the seed plate. Both types of air planters change the seeding rate by changing disks and/or changing the speed ratio of the disk drive train. The drum type of metering unit works on the same principles as the pressurized air disk planter. The primary difference is that the drum has a ring of holes for each row of plants instead of a separate metering unit for each row. Each drum usually singulates seeds for either 6 or 8 rows. The metering rate can be adjusted by changing drums and varying the speed between the drum and the drive wheel. 8.6.1 Row Crop Planter Stationary Calibration Precision row crop planters can be calibrated using the stationary method. The seeding rate is calculated by dividing the seeds planted per revolution of the drive wheel by the representative acres covered per revolution of the drive wheel, or: sd n R¼ r A nr This equation can also be written as: sd nr R¼ nr A where R ¼ seeding rate (seeds/ac); sd ¼ number of seeds; nr ¼ number of revolutions of drive wheel; A ¼ area, ft2. 126 8 Machinery Calibration Problem What is the planting rate for corn, seeds/ac, if 0.22 pounds of seed is weighed after an 18.0 inch drive wheel is turned 25 revolutions? The planter is set for 36.0 inch rows, and the seed size is 1500 seeds per pound. Solution When calibrating row crop planters using the stationary method, the area used is determined by the row spacing and the representative distance traveled: A¼wd where: A ¼ area (ft2); w ¼ row spacing (ft); d ¼ distance (ft). The pounds per acre are: lb W ðlbÞ nr ðrevÞ R ¼ ac nr ðrevÞ AðacÞ 0:22 lb 1 rev ¼ 25 rev 1 ft 1 ac 1 ft 18:0 in π 2 36 in 12 in 43,560 ft 12 in 0:22 1 0:22 lb ¼ ¼ ¼ 27:114... 25 3:245... E-4 8:1136... E-3 acre The planting rate in seeds per acre is: sd 27:114... lb 1500 sd sd R ¼ ¼ 40,672:36 or 40,700 ac ac 1 lb ac The row crop planter is planting 40,700 seeds per acre. 8.6.2 Row Crop Planter Mobile Calibration Mobile calibration of precision row crop planters can be completed the same as fertilizer spreaders and grain drills, but because they place individual seeds in the ground a different method can also be used. For each row spacing and planting rate, there is a unique spacing for the seeds in the row. The planting rate of row crop planters can be determined by operating the planter in the field and measuring the seed spacing in the row. To complete a mobile calibration of a row crop planter, seeds are added to the hopper, and the planter is lowered into the ground and driven for a short distance to ensure the system has stabilized. Next carefully uncover several seeds in a row and measure the distance between the seeds. On some models the press wheel can be disabled during this step to make seed spacing measurement easier. The average distance is used to determine the seeding rate. 8.6 Calibrating Row Crop Planters 127 Problem Determine the seeding rate, seeds/ac, for a row crop planter set for 20.0 inch rows. The in-the-row soybean spacing is 2–1/8, 2–1/4, 2–3/16, and 2–3/16 inches. Solution First convert the fractional distances to decimals: 1 2 ¼ 2:125 8 1 2 ¼ 2:25 4 3 2 ¼ 2:1875 16 Next determine the mean distance between the seeds in the row: 2:125 þ 2:25 þ 2:1875 þ 2:1875 Mean distance ¼ ¼ 2:1875 in 4 The last step is determining the seeding rate: sd 1 sd 12 in 43560 ft2 1 row 12 in R ¼ ac 2:1875 in 1 ft 1 ac 20:0 in 1 ft 6,272,640 sd ¼ ¼ 143,374:62... or 143,000 or 1:43 E5 43:75 ac This planter is placing 143,000 seeds per acre in the ground. To evaluate the acceptability of this performance, this rate must be compared to the desired rate. If the operator of a row crop planter knows the in-the-row seed spacing for the seeds per acre and the row spacing, they can do a quick check of the planter performance by uncovering and measuring the distance between several seeds. This can be determined using the unit cancellation method. Problem What should the in the row seed spacing (in/seed) be for 40,000 plants per acre when the planter is set for 28.0 inch rows? Solution Using unit cancellation: in 1 ac 43,560 ft2 144 in2 1 row ¼ seed 40,000 seed 1 ac 1 ft2 28 in 6,272,640 row-in ¼ ¼ 5:60057... or 5:60 1,120,000 seed Is this the correct answer? The units in the answer do not match the desired units. It is correct because it is common practice to drop off the unit of row in the answer and just use in/sd. When doing a few calculations, it would be appropriate and preferred to com- plete the problem as in this example by using the unit cancellation method. This 128 8 Machinery Calibration would not be true if the in-the-row spacing needed to be calculated many times. In this situation it is easier to use an equation. In Chap. 1 the source of many equations was identified as unit cancellation calculations where the unit conversion numbers were reduced to a constant. Combining the unit conversion units in the previous problem (43,560 ft2 144 in2) produces the constant 6.273 E6. The seed spacing in the row can be found by the equation: 6:273 E6 SS ¼ POP RS where SS ¼ seed spacing in the row (in); POP ¼ population, planting rate (seeds/ac); RS ¼ row spacing (in). If the seeding rate is not correct, further evaluation of the planter must be made. Because the metering unit is ground driven, the source of the error may be in the drive train ratio between the drive wheel and the metering unit. This mobile calibration method is quick and requires very few resources. An additional benefit is that other planting parameters such as the seed depth and seed orientation can be evaluated at the same time. 8.7 Air Seeders Air seeders have become the machine of choice for planting wheat and other crops in large fields. Seeders can have effective widths of 70–100 feet or more. The basic design is a combination of a seed cart containing a hopper and a bulk metering unit pulled behind a field cultivator or another secondary tillage machine that has soil- engaging tools mounted on shanks. The seed cart will have at least one but could have two or more hoppers. When multiple hoppers are used, one hopper will be for seed, and the other(s) can be used for fertilizer or other materials. Each hopper will have a single ground-driven metering unit that meters the material. Each metering unit controls the flow of material for all or a large number of soil-engaging tools. After metering the material passes through splitter manifolds until the stream is reduced down to volume for one opener (Fig. 8.6). Fig. 8.6 Air seeder 8.7 Air Seeders 129 Air seeder technology has progressed in recent years. Modern seeders combine the seed cart and soil-engaging tools into one unit. In addition the seed cart is placed in front of the soil-engaging shanks and in many machines, mounted on the back of the seed cart. The effectiveness of the soil opening devices and metering units has also been improved. Most manufacturers state that air seeders can be calibrated using either station- ary or mobile methods, but the size of these machines makes stationary calibration problematic. In addition the seeding rate for some air seeders calibrated stationary may be less than the actual rate in the field. For this reason the mobile calibration method is recommended. 8.7.1 Air Seeder Mobile Calibration Mobile calibration methods for air seeders are the same as grain drills. One difference is the number of rows that must be collected. Air seeders have a large number of seed to soil interfaces, so collecting all of the seeds will require a lot of resources. Therefore it is common to collect a few of the rows and assume they are a representative sample. Recommendations for the number of rows that should be collected vary, but 25% is a general recommendation. It is important to collect the same number of points from each splitter manifold. The appropriate number of seed hoses is disconnected from the individual seed tubes and connected to a bag or other collection device. The seeder is driven a fixed distance, 100–200 feet, and the material is weighed from each bag. With this information and the effective width of each shank, the seeding rate or application rate can be calculated. Problem Determine the seeding rate (lb/ac) for a 50.0 foot air seeder with a 10.0 inch shank spacing. The following quantities of seeds were collected from every fourth shank on a single row during 200.0 feet of travel (Table 8.3). Solution The first step is to determine the total amount of seeds collected: Total Pounds ¼ 0:34 þ 0:33 þ 0:33 þ 0:36 þ 0:34 þ 0:35 þ 0:35 þ 0:33 þ 0:34 þ0:34 þ 0:35 þ 0:35 þ 0:33 þ 0:34 þ 0:34 ¼ 5:12 lb The next step is to determine the area [Note: for calibration purposes the effective width is the number of shanks times the spacing between shanks.]: 10 in 1 ft 1 ac Area ðacÞ ¼ 16 row 200 ft row 12 in 43,560 ft2 32,000 ¼ ¼ 0:06121... or 0:061 ac 522,720 130 8 Machinery Calibration Table 8.3 Quantity of seeds Collector lbs collected during calibration of 1 0.34 air seeder 2 0.33 3 0.33 4 0.36 5 0.34 6 0.35 7 0.35 8 0.33 9 0.33 10 0.34 11 0.34 12 0.35 13 0.35 14 0.33 15 0.34 16 0.34 The air seeder applied 5.45 pounds of seeds in 0.61 acre. The last step is to determine the pounds of seed per acre: lb 5:12 lb lb R ¼ ¼ 83:934... or 83:9 ac 0:061 ac ac The air seeder is applying seeds at the rate of 83.9 pounds per acre. It is up to the owner/operator to determine if this is an acceptable level of performance. 8.8 Calibrating Sprayers Accurate calibration of spray equipment is very important because small variations in the application rate can cause chemical damage to the crop or the environment, be wasteful of materials or be ineffective. There are two important differences between the design of sprayers and that of grain drills, and row crop planters. (1) Sprayers do not use ground-driven metering units. The application rate (gal/ac) is a function of the flow rate of the nozzles (gal/min) and the velocity of the sprayer (mi/hr). The flow rate from the nozzles does not change when the ground speed changes; therefore changing the sprayer velocity changes the application rate. (2) Most agricultural sprayers use PTO or engine-driven pumps. To maintain a constant application rate, the PTO or engine speed must be constant. Ground driven pumps do exist, but they are not as common as PTO and engine driven pumps in dedicated spray equipment. The focus of this text will be on sprayers with PTO or engine equipped pumps. 8.8 Calibrating Sprayers 131 The problems associated with variability in speed and flow can be managed by using a sprayer controller. Sprayer controllers are designed with a variety of capabilities. Simple controllers may just monitor system pressure and provide switches to operate the sprayer systems. The most expensive controllers will monitor and regulate all the sprayer systems. They have the ability to the vary application rate to compensate for changes in sprayer speed or changes in flow from the pump to insure the application rate is constant. Figure 8.7 illustrates the common parts for the typical overlapping boom sprayer. Overlapping boom sprayers consists of tank, filter, pump, means to control the pressure, multiple boom sections with nozzles, and boom selection valve. The mixture in the tank flows through the filter to remove any particles that might plug the orifice in the nozzles and then to the pump. Several different types of pumps can be used, depending on the flow (gal/min) and the pressure (psi) needed by the system. From the pump the spray goes to the pressure-regulating valve. In some sprayer designs, a portion of the flow from the pump is returned to the tank for agitation. Fluid agitation is used to prevent the spray materials from separating in the tank. Mechanical agitation is used when fluid agitation will cause foaming of Fig. 8.7 Parts of a typical boom sprayer 132 8 Machinery Calibration the sprayer mix. Any excess flow produced by the pump is returned to the tank by the pressure-regulating valve. From the pressure-regulating valve, the fluid is pumped to the boom selector valve. At some point in this line, a connection is made for a pressure gage. The boom selector valve directs the flow of the mixture to the different sections of the boom. The boom selector valve can be manual, but some sprayer monitor systems control the boom sections electronically. Some sprayers have provisions for a handgun. A handgun is very useful for spraying skips or along fencerows and other obstructions. One popular selector valve design has seven positions: 1. All outlets off 2. All booms on 3. Left boom on 4. Center boom on 5. Right boom on 6. Hand gun on 7. Booms and handgun on In the typical design, the pump can be engine, power take off (pto), or hydraulic motor driven. As long as the pump is operating and the selector valve is in one of the on positions, fluid will flow. It is important to understand the effect ground speed has on the spray application rate (gal/ac). Unless the sprayer is equipped with monitoring system with variable flow rate capabilities, when it slows down, the application rate increases. This occurs because the nozzle flow rate (gal/min) is constant. When the travel speed is reduced, the same amount of material is applied to a smaller area: gal gal min ¼ ac min ac When the travel speed is reduced, the acres per minute (ac/min) are reduced. Conversely, when the sprayer travel speed increases, the application rate decreases. The effect of travel speed will become more apparent when reviewing the following problems, but changing the velocity by one mile per hour can change the applica- tion rate by several gallons per acre. For some materials this would cause serious consequences. Therefore, precise control of the sprayer speed is very important. The design shown in Fig. 8.7 is often modified to meet the demands of different types of plants or application methods. Two additional examples are shown in Figs. 8.8 and 8.9. Fig. 8.8 Nozzles arranged for banding between the rows 8.8 Calibrating Sprayers 133 Fig. 8.9 Nozzles arranged for row crop sprayer The application rate (gal/ac) of a field sprayer is controlled by three factors: 1. The speed of the sprayer (mi/hr) 2. The rate of discharge from the nozzle (gal/min) 3. The width covered by one nozzle (in.) Sprayers can be calibrated using either the stationary or mobile method. During mobile calibration collectors are placed under each nozzle, and the sprayer is driven a calculated distance at the speed it will be operated in the field. 8.8.1 Sprayer Stationary Calibration When the stationary method is used, collectors are placed under each nozzle, and the sprayer is operated for a specific time period. The specified time period can be preselected, or a recorded amount as long as it provides an adequate sample without exceeding the capacity of the collection vessel. The volume per time is converted to gallons per acre. The application rate can be calculated with the unit method or by an equation. The unit method: gal gal 60 min 1 hr 1 mile 43,560 ft2 12 in in ¼ ac min 1 hr mile 5280 ft ac 1 ft noz The bracketed variables are unit conversions. When they are combined, the source of the constant in the equation is evident: 60 min 1 mile 43,560 ft2 12 in 31,363,200 ¼ ¼ 5940 1 hr 5280 ft ac 1 ft 5280 The sprayer equation is: 5940 Q R¼ Vw where R ¼ application rate (gal/ac); 5940 ¼ unit conversion constant; Q ¼ flow rate per nozzle (gal/min); V ¼ travel speed (mi/hr); w ¼ nozzle spacing (in). 134 8 Machinery Calibration Note: Use either the flow rate (gal/min) from one nozzle and the spacing between two adjacent nozzles or the flow from all nozzles (gal/min number of nozzles) and the total width of the sprayer (w number of nozzle). In either case, the application rate (gal/ac) will be the same. Do not interchange these values. Problem Determine the application rate for a sprayer that produced 0.14 gallons of spray in 1.12 min. The nozzle spacing is 18.0 inches and the travel speed was 4.3 miles per hour. Solution 5940 Q R¼ Vw 0:14 gal 5940 ¼ 1:12 min 4:3 mi 18 in hr 742:5 gal ¼ ¼ 9:593... or 9:6 77:4 ac This sprayer produced 9.6 gallons per acre of spray. One of the methods used to adjust the application rate of a sprayer is changing the size (gal/min) of the nozzles. For this reason it is sometimes necessary to calculate the size of nozzle that should be used to apply the desired rate. The required nozzle size can be calculated by rearranging the equation to solve for nozzle size. Problem What size of nozzles (gal/min) is required for a boom-type sprayer to apply 20.0 gallons of spray per acre? The sprayer has 24 nozzles spaced 18.0 inches apart. Solution Because the application rate of field sprayers is speed-dependent, begin by selecting a reasonable speed that can be maintained in the field, and then determine the size of nozzles needed. For this problem we will use the typical speed (Appendix IV) of 6.5 mi/h. The required flow rate for the nozzles can be determined by rearranging the standard sprayer equation: gal 5940 Q gal min R ¼ ac mi V w ðinÞ hr 20:0 gal 6:5 mi gal R V w 18:0 in Q ¼ ¼ 1 ac hr min 5940 5940 2340 gal ¼ ¼ 0:3939... or 0:39 5940 min 8.8 Calibrating Sprayers 135 For this application rate and nozzle spacing, nozzles with a capacity of 0.39 gallons per minute would apply the desired 20 gallons per acre. The nozzles would be installed, and before the sprayer is used, it should be calibrated to ensure that the application rate is correct. New sprayer nozzles have small variations in the construction that can cause a sprayer to have an incorrect application rate. Small variation in the pressure at the nozzles can also cause an unacceptable error in the application rate. Assume the operator installed the 0.39 gallon per minute nozzles on the sprayer and proceeded with the calibration. Problem A container placed under all 24 nozzles of the sprayer collected 14.40 gallons of spray in 2.0 min of operation. The desired application rate was 20.0 gal/ ac. Is the sprayer accurate? Solution In this example only the total volume is known. One alternative is to determine the average flow rate per nozzle and then use the sprayer equation, but this process will not be as accurate as using the unit cancellation method and the total flow rate: gal 14:40 gal 60:0 min 1h 1 mi 43,560 ft2 R ¼ ac 2:0 min 1h 6:5 mi 5280 ft 1 ac 12:0 in 1 nozzle 1 1 ft 18:0 in 24 nozzles 451,630,080 gal ¼ ¼ 15:2307... or 15 29,652,480

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