Environmental engineer's mathematics handbook

ENVIRONMENTAL ENGINEER’S MATHEMATICS HANDBOOK......Page 1
Preface......Page 3
Acknowledgments......Page 4
Contents......Page 5
Contents......Page 0
Part I Fundamental Computation and Modeling......Page 18
1.2 CONVERSION FACTORS......Page 19
1.3 CONVERSION FACTORS: PRACTICAL EXAMPLES......Page 29
1.3.1 Weight, Concentration, and Flow......Page 30
1.3.2 Water/Wastewater Conversion Examples......Page 32
1.3.3 Temperature Conversions......Page 38
1.4.1 Conversion from Parts per Million to Micrograms per Cubic Meter......Page 40
1.6 CONCLUSION......Page 42
2.2 BASIC MATH TERMINOLOGY AND DEFINITIONS......Page 46
2.3.1 Sequence of Operations - Rules......Page 47
2.3.2 Sequence of Operations - Examples......Page 48
2.4 PERCENT......Page 49
2.5 SIGNIFICANT DIGITS......Page 53
2.6 POWERS AND EXPONENTS......Page 55
2.7 AVERAGES (ARITHMETIC MEAN)......Page 56
2.8 RATIO......Page 58
2.9 DIMENSIONAL ANALYSIS......Page 62
2.11 GEOMETRICAL MEASUREMENTS......Page 68
2.11.1.1 Perimeter and Circumference......Page 69
2.11.1.2 Area......Page 72
2.11.1.3 Volume......Page 75
2.12.1 Force and Pressure......Page 79
2.12.2.1 Static Head......Page 80
2.12.2.6 Head/Pressure......Page 81
2.13 REVIEW OF ADVANCED ALGEBRA KEY TERMS AND CONCEPTS......Page 86
3.2 MEDIA MATERIAL CONTENT......Page 88
3.2.1 Material Content: Liquid Phases......Page 90
3.3 PHASE EQUILIBRIUM AND STEADY STATE......Page 93
3.4.1 Solving Equilibrium Problems......Page 94
3.4.2.1 Ideal Gas Law......Page 95
3.4.2.2 Dalton's Law......Page 96
3.5 CHEMICAL TRANSPORT SYSTEMS......Page 98
3.6 A FINAL WORD ON ENVIRONMENTAL MODELING......Page 99
REFERENCES......Page 100
4.2 ALGORITHMS: WHAT ARE THEY?......Page 101
4.3 EXPRESSING ALGORITHMS......Page 102
4.4 GENERAL ALGORITHM APPLICATIONS......Page 103
4.5 ENVIRONMENTAL ENGINEERING ALGORITHM APPLICATIONS......Page 104
4.7 SCREENING TOOLS......Page 105
SUGGESTED READING......Page 106
Part II Fundamental Science and Statistics Review......Page 107
5.2 FUNDAMENTAL CHEMISTRY......Page 108
5.2.1 Density and Specific Gravity......Page 109
5.2.2.1 The Water Molecule......Page 112
5.2.2.2 Water Solutions......Page 113
5.2.2.3 Concentrations......Page 114
5.2.2.5 Colligative Properties......Page 116
5.2.2.6 Colloids/Emulsions......Page 117
5.2.2.7 Water Constituents......Page 118
5.2.2.8 Simple Solutions and Dilutions......Page 125
5.2.2.9 Chemical Reactions......Page 128
5.2.2.10 Chemical Dosages (Water and Wastewater Treatment)......Page 133
5.3.1.2 Weight of Water......Page 139
5.3.1.3 Weight of Water Related to the Weight of Air......Page 140
5.3.1.5 Gauge Pressure......Page 141
5.3.1.7 Discharge......Page 142
5.3.1.8 The Law of Continuity......Page 143
5.3.2 Basic Pumping Calculations......Page 144
5.3.2.1 Pumping Rates......Page 145
5.3.3 Calculating Head Loss......Page 146
5.3.5 Calculating Horsepower and Efficiency......Page 147
5.3.5.2 Pump Efficiency and Brake Horsepower (bhp)......Page 148
SUGGESTED READING......Page 151
6.3 BASIC STATISTICAL TERMS......Page 152
6.4.1 Loading Calculation......Page 153
6.4.3 30-Day Average Calculation......Page 154
6.4.4 Moving Average......Page 155
6.4.5 Geometric Mean......Page 156
6.4.5.2 Nth Root Calculation Method......Page 157
6.5 STANDARD DEVIATION......Page 158
6.6 CONCLUSION......Page 160
Part III Math Concepts: Air Pollution Control......Page 161
7.1 INTRODUCTION......Page 162
7.1.1.1 Ground-Level Ozone......Page 163
7.1.1.5 Carbon Monoxide (CO)......Page 164
7.2 GASES......Page 165
7.2.1 The Gas Laws......Page 166
7.2.1.1 Boyle's Law......Page 167
7.2.1.3 Gay-Lussac's Law......Page 168
7.2.1.5 The Ideal Gas Law......Page 169
7.2.1.6 Composition of Air......Page 170
7.4 POLLUTION EMISSION MEASUREMENT PARAMETERS......Page 171
7.5 STANDARD CORRECTIONS......Page 172
REFERENCES......Page 173
8.2 ABSORPTION......Page 174
8.2.2 Equilibrium Solubility and Henry's Law......Page 177
8.2.3 Material (Mass) Balance......Page 179
8.2.4.1 Packed Tower Absorber Diameter......Page 183
8.2.4.2 Sizing the Packed Tower Absorber Height......Page 186
8.2.4.3 Sizing the Plate (Tray) Tower......Page 190
8.2.4.4 Theoretical Number of Absorber Plates or Trays......Page 192
8.3 ADSORPTION......Page 194
8.3.2 Adsorption Forces - Physical and Chemical......Page 195
8.3.3.1 Isotherm......Page 196
8.3.3.3 Isobar......Page 197
8.3.4 Factors Affecting Adsorption......Page 198
8.3.4.3 Gas Velocity......Page 199
8.3.4.4 Bed Depth......Page 200
8.3.4.6 Contaminants......Page 203
8.4.1.2 Residence Time......Page 204
8.4.1.4 Oxygen Requirement......Page 205
8.4.1.7 Heat......Page 206
8.4.2 Incineration Example Calculations......Page 207
8.5.1 Contact Condenser Calculations......Page 210
8.5.2 Surface Condenser Calculations......Page 212
REFERENCES......Page 217
9.1.1 Interaction of Particles with Gas......Page 218
9.1.2 Particulate Collection......Page 219
9.2.3 Sedimentation Diameter......Page 220
9.3 FLOW REGIME OF PARTICLE MOTION......Page 221
9.4.1 Gravity Settlers......Page 227
9.4.2 Gravity Settling Chamber Theoretical Collection Efficiency......Page 228
9.4.3 Minimum Particle Size......Page 230
9.4.4.1 Factors Affecting Cyclone Performance......Page 234
9.4.6.1 Collection Efficiency......Page 239
9.4.6.2 Precipitator Example Calculations......Page 241
9.4.7 Baghouse (Fabric) Filters......Page 247
9.4.7.2 Baghouse Example Calculations......Page 248
REFERENCES......Page 258
10.1.1 Wet Scrubbers......Page 259
10.2 WET SCRUBBER COLLECTION MECHANISMS AND EFFICIENCY (PARTICULATES)......Page 260
10.2.2 Impaction......Page 261
10.2.4 Diffusion......Page 262
10.2.5.1 Johnstone Equation......Page 263
10.2.5.2 Infinite Throat Model......Page 264
10.2.5.3 Cut Power Method......Page 270
10.2.5.4 Contact Power Theory......Page 271
10.2.5.5 Pressure Drop......Page 275
10.4.1 Scrubber Design of a Venturi Scrubber......Page 276
10.4.2 Spray Tower......Page 284
10.4.3 Packed Tower......Page 286
10.4.4 Packed Column Height and Diameter......Page 290
REFERENCES......Page 295
Part IV Math Concepts: Water Quality......Page 297
11.1 BALANCING THE "AQUARIUM"......Page 298
11.1.1 Sources of Stream Pollution......Page 299
11.2 IS DILUTION THE SOLUTION?......Page 300
11.3 DISCHARGE MEASUREMENT......Page 301
11.4 TIME OF TRAVEL......Page 302
11.5 DISSOLVED OXYGEN (DO)......Page 303
11.5.1 DO Correction Factor......Page 304
11.6 BIOCHEMICAL OXYGEN DEMAND......Page 305
11.6.2.1 Unseeded BOD Procedure......Page 306
11.6.2.2 Seeded BOD Procedure......Page 307
11.7 OXYGEN SAG (DEOXYGENATION)......Page 308
11.8 STREAM PURIFICATION: A QUANTITATIVE ANALYSIS......Page 309
REFERENCES......Page 313
12.1 INTRODUCTION......Page 314
12.3.1.1 Volume......Page 316
12.3.1.3 Mean Depth......Page 317
12.4.2 Energy Budget Model......Page 321
12.4.5 DeBruin-Keijman Equation......Page 322
REFERENCES......Page 323
13.1 GROUNDWATER AND AQUIFERS......Page 324
13.2.1 Aquifer Porosity......Page 326
13.2.4 Transmissivity (T)......Page 327
13.3 GROUNDWATER FLOW......Page 328
13.4 GENERAL EQUATIONS OF GROUNDWATER FLOW......Page 329
13.4.2 Steady Flow in an Unconfined Aquifer......Page 330
REFERENCES......Page 331
14.2 BASIC CONCEPTS......Page 332
14.2.2 Density and Specific Gravity......Page 334
14.2.3 Force and Pressure......Page 336
14.2.5 Head......Page 338
14.2.5.5 Pressure/Head......Page 339
14.3 FLOW/DISCHARGE RATE: WATER IN MOTION......Page 340
14.3.1 Area/Velocity......Page 342
14.4.1 Bernoulli's Equation......Page 343
14.5 CALCULATING MAJOR HEAD LOSS......Page 346
14.6.3 Critical Flow......Page 347
14.6.4.2 Hydraulic Depth......Page 348
14.7 OPEN-CHANNEL FLOW CALCULATIONS......Page 349
REFERENCES......Page 350
15.1 INTRODUCTION......Page 351
15.2.1.1 Well Drawdown......Page 352
15.2.1.2 Well Yield......Page 354
15.2.1.3 Specific Yield......Page 355
15.2.1.5 Deep-Well Turbine Pump Calculations......Page 356
15.2.3 Vertical Turbine Pump Calculations......Page 357
15.3 WATER STORAGE......Page 362
15.3.1 Water Storage Calculations......Page 363
15.3.2 Copper Sulfate Dosing......Page 364
15.4.1 Coagulation......Page 365
15.4.2 Mixing......Page 366
15.4.4.1 Chamber and Basin Volume Calculations......Page 367
15.4.4.2 Detention Time......Page 369
15.4.4.3 Determining Dry Chemical Feeder Setting (Pounds per Day)......Page 370
15.4.4.5 Determining Chemical Solution Feeder Setting (Milliliters per Minute)......Page 371
15.4.5 Determining Percent of Solutions......Page 372
15.4.5.2 Determining Percent Strength of Mixed Solutions......Page 374
15.4.6 Dry Chemical Feeder Calibration......Page 375
15.4.6.1 Solution Chemical Feeder Calibration......Page 376
15.4.7 Determining Chemical Usage......Page 378
15.4.7.1 Paddle Flocculator Calculations......Page 379
15.5.1 Tank Volume Calculations......Page 380
15.5.2 Detention Time......Page 381
15.5.3 Surface Overflow Rate......Page 383
15.5.4 Mean Flow Velocity......Page 384
15.5.5 Weir Loading Rate (Weir Overflow Rate)......Page 385
15.5.6 Percent Settled Biosolids......Page 386
15.5.7 Determining Lime Dosage (Milligrams per Liter)......Page 387
15.5.9 Determining Lime Dosage (Grams per Minute)......Page 391
15.5.10 Particle Settling (Sedimentation)......Page 392
15.5.11 Overflow Rate (Sedimentation)......Page 396
15.6.1 Flow Rate through a Filter (Gallons per Minute)......Page 398
15.6.2 Filtration Rate......Page 401
15.6.3 Unit Filter Run Volume (UFRV)......Page 403
15.6.4 Backwash Rate......Page 405
15.6.5 Backwash Rise Rate......Page 406
15.6.6 Volume of Backwash Water Required (Gallons)......Page 407
15.6.7 Required Depth of Backwash Water Tank (Feet)......Page 408
15.6.8 Backwash Pumping Rate (Gallons per Minute)......Page 409
15.6.9 Percent Product Water Used for Backwashing......Page 410
15.6.10 Percent Mud Ball Volume......Page 411
15.6.11 Filter Bed Expansion......Page 412
15.6.12 Filter Loading Rate......Page 413
15.6.13 Filter Medium Size......Page 414
15.6.14 Mixed Media......Page 415
15.6.15 Head Loss for Fixed Bed Flow......Page 416
15.6.16 Head Loss through a Fluidized Bed......Page 417
15.6.17 Horizontal Washwater Troughs......Page 419
15.6.18 Filter Efficiency......Page 420
15.7.1 Chlorine Disinfection......Page 421
15.7.2 Determining Chlorine Dosage (Feed Rate)......Page 422
15.7.3 Calculating Chlorine Dose, Demand, and Residual......Page 423
15.7.4 Breakpoint Chlorination Calculations......Page 425
15.7.5 Calculating Dry Hypochlorite Feed Rate......Page 427
15.7.6 Calculating Hypochlorite Solution Feed Rate......Page 430
15.7.7 Calculating Percent Strength of Solutions......Page 431
15.7.9 Calculating Percent Strength Using Liquid Hypochlorite......Page 432
15.8 CHEMICAL USE CALCULATIONS......Page 433
15.8.1 Chlorination Chemistry......Page 434
REFERENCES......Page 436
Part V Math Concepts: Wastewater Engineering......Page 437
16.2 PRELIMINARY TREATMENT CALCULATIONS......Page 438
16.2.2 Screenings Removal Calculations......Page 439
16.2.3 Screenings Pit Capacity Calculations......Page 440
16.2.6 Grit Removal Calculations......Page 442
16.2.7 Grit Channel Velocity Calculation......Page 444
16.2.7.1 Required Settling Time......Page 445
16.2.7.3 Velocity of Scour......Page 446
16.3.2 Surface Loading Rate (Surface Settling Rate/Surface Overflow Rate)......Page 447
16.3.3 Weir Overflow Rate (Weir Loading Rate)......Page 448
16.3.4 Primary Sedimentation Basins......Page 449
16.4.1 Percent Total Solids (% TS)......Page 451
16.5 TRICKLING FILTER CALCULATIONS......Page 452
16.5.2 Hydraulic Loading......Page 453
16.5.3 Organic Loading Rate......Page 455
16.5.5 Recirculation Flow......Page 456
16.5.6 Trickling Filter Design......Page 457
16.6 ROTATING BIOLOGICAL CONTACTORS (RBCS)......Page 458
16.6.2 Hydraulic Loading Rate......Page 459
16.6.3 Soluble BOD......Page 460
16.6.4 Organic Loading Rate......Page 462
16.6.7 RBC Performance Parameter......Page 463
16.7.2 Moving Averages......Page 464
16.7.3 BOD or COD Loading......Page 465
16.7.5 Food-to-Microorganism Ratio (F/M Ratio)......Page 466
16.7.6 Gould Biosolids Age......Page 469
16.7.7 Mean Cell Residence Time (MCRT)......Page 470
16.7.8 Estimating Return Rates from SBV60 (SSV60)......Page 472
16.7.9 Biosolids (Sludge) Volume Index (BVI)......Page 473
16.7.12 Biosolids Waste Based Upon Mass Balance......Page 474
16.7.13 Aeration Tank Design Parameters......Page 476
16.7.14.1 Complete Mix with Recycle......Page 477
16.7.15.1 Process Design and Control Relationships......Page 479
16.7.15.3 Oxygen Requirements......Page 480
16.8 OXIDATION DITCH DETENTION TIME......Page 481
16.9.2 Treatment Pond Process Control Calculations......Page 482
16.9.2.2 BOD Loading......Page 483
16.9.2.4 BOD Removal Efficiency......Page 484
16.9.3 Aerated Ponds......Page 485
16.10.2 Chemical Feed Rate......Page 486
16.10.3.2 Chlorine Demand......Page 488
16.10.4 Hypochlorite Dosage......Page 489
16.10.5 Chemical Solutions......Page 491
16.10.6 Mixing Solutions of Different Strengths......Page 493
16.10.8 Solution Chemical Feeder Setting, GPD......Page 494
16.10.10 Chemical Solution Feeder Setting, Milliliters per Minute......Page 496
16.10.11 Chemical Feed Calibration......Page 497
16.10.12 Average Use Calculations......Page 500
16.11.1 Process Residuals......Page 501
16.11.3 Primary Clarifier Solids Production Calculations......Page 502
16.11.4 Secondary Clarifier Solids Production Calculation......Page 503
16.11.6 Biosolids Pumping......Page 504
16.11.9 Biosolids Production (Wet Tons per Year)......Page 505
16.11.10 Biosolids Pumping Time......Page 506
16.12.2.1 Estimating Daily Sludge Production......Page 508
16.12.2.3 Solids Loading Rate, Pounds per Day per Square Foot......Page 509
16.12.4 Air-to-Solids Ratio......Page 510
16.12.6 Centrifuge Thickening Calculations......Page 511
16.13.2.1 Volatile Solids Loading, Pounds per Square Foot per Day......Page 512
16.13.2.3 pH Adjustment......Page 513
16.13.3 Aerobic Tank Volume......Page 514
16.13.4.2 Volatile Acids-to-Alkalinity Ratio......Page 515
16.13.4.5 Volatile Matter Reduction (Percent)......Page 516
16.13.4.7 Gas Production......Page 517
16.14.3 Plate and Frame Press......Page 519
16.14.3.2 Net Filter Yield......Page 520
16.14.4.1 Hydraulic Loading Rate......Page 521
16.14.5 Solids Loading Rate......Page 523
16.14.7 Flocculant Dosage......Page 524
16.14.8 Total Suspended Solids......Page 525
16.14.9.1 Filter Loading......Page 526
16.14.11 Vacuum Filter Operating Time......Page 527
16.14.12 Percent Solids Recovery......Page 528
16.14.14.2 Solids Loading Rate......Page 529
16.14.14.3 Biosolids Withdrawal to Drying Beds......Page 530
16.15.2 Plant Available Nitrogen (PAN)......Page 531
16.15.3 Application Rate Based on Crop Nitrogen Requirement......Page 532
16.15.6 Site Life Based on Metals Loading......Page 533
16.16.1 Composting Calculations......Page 534
16.16.1.2 Compost Site Capacity Calculation......Page 535
16.17.1 The Wastewater Lab......Page 536
16.17.3 Composite Sampling Procedure and Calculation......Page 537
16.17.4.1 BOD5 ( Unseeded)......Page 538
16.17.5 BOD 7-Day Moving Average......Page 539
16.17.6.1 Moles......Page 540
16.17.6.2 Normality......Page 542
16.17.7 Settleability (Activated Biosolids Solids)......Page 543
16.17.8 Settleable Solids......Page 544
16.17.9 Biosolids Total Solids, Fixed Solids, and Volatile Solids......Page 545
16.17.10 Wastewater Suspended Solids and Volatile Suspended Solids......Page 547
16.17.11 Biosolids Volume Index (BVI) and Biosolids Density Index (BDI)......Page 549
REFERENCES......Page 550
Part VI Math Concepts: Stormwater Engineering......Page 551
17.1 INTRODUCTION......Page 552
17.2 STORMWATER TERMS AND ACRONYMS......Page 553
17.3 HYDROLOGIC METHODS......Page 558
17.3.1 Precipitation......Page 560
17.3.1.2 Intensity-Duration-Frequency (I-D-F) Curves......Page 561
17.3.1.3 SCS 24-H Storm Distribution......Page 562
17.3.1.4 Synthetic Storms......Page 563
17.3.1.5 Single Event vs. Continuous Simulation Computer Models......Page 564
17.5 RUNOFF AND PEAK DISCHARGE......Page 565
17.6.1 The Rational Method......Page 566
17.6.1.2 Limitations......Page 567
17.6.1.3 Design Parameters......Page 568
17.6.2.3 Design Parameters......Page 570
17.6.3.1 Limitations......Page 572
17.6.3.3 Design Parameters......Page 573
17.6.4.3 Design Parameters......Page 580
17.6.5.1 Limitations......Page 581
17.6.5.3 Design Parameters......Page 582
17.7.2 Allowable Release Rates......Page 583
17.7.4 Graphical Hydrograph Analysis — SCS Methods......Page 584
17.7.4.1 Procedure......Page 585
17.7.5.1 Information Needed......Page 587
17.7.6 Graphical Hydrograph Analysis, Modi.ed Rational Method Critical Storm Duration......Page 589
17.7.6.1 Information Needed......Page 591
17.7.7 Modified Rational Method, Critical Storm Duration — Direct Solution......Page 592
17.7.7.1 Storage Volume......Page 593
17.7.7.2 Rainfall Intensity......Page 594
17.7.7.4 Information Needed......Page 596
17.7.8.1 Storage Volume Calculations......Page 600
17.7.9.1 Retention Basins - Water Quality Volume......Page 602
17.7.9.2 Extended-Detention Basins - Water Quality Volume and Orifice Design......Page 603
17.7.9.3 Extended-Detention Basins - Channel Erosion Control Volume and Orifice Design......Page 607
17.7.10.1 Information Needed......Page 609
Procedure 1......Page 625
Procedure 2......Page 631
Procedure......Page 635
17.8 CONCLUSION......Page 641
REFERENCES......Page 642