Environmental engineer's mathematics handbook

PART I: FUNDAMENTAL COMPUTATION AND MODELING ...........................................1 Chapter 1 Conversion Factors and SI Units ...............................................................................3 1.1 Introduction ............................................................................................................................3 1.2 Conversion Factors.................................................................................................................3 1.3 Conversion Factors: Practical Examples .............................................................................13 1.3.1 Weight, Concentration, and Flow.........................................................................14 1.3.2 Water/Wastewater Conversion Examples.............................................................16 1.3.3 Temperature Conversions .....................................................................................22 1.4 Conversion Factors: Air Pollution Measurements...............................................................24 1.4.1 Conversion from Parts per Million to Micrograms per Cubic Meter .................24 1.4.2 Conversion Tables for Common Air Pollution Measurements............................26 1.5 Soil Test Results Conversion Factors ..................................................................................26 1.6 Conclusion...........................................................................................................................26 Chapter 2 Basic Math Operations.............................................................................................31 2.1 Introduction ..........................................................................................................................31 2.2 Basic Math Terminology and Definitions ...........................................................................31 2.3 Sequence of Operations .......................................................................................................32 2.3.1 Sequence of Operations — Rules ........................................................................32 2.3.2 Sequence of Operations — Examples .................................................................33 2.4 Percent.................................................................................................................................34 2.5 Significant Digits..................................................................................................................38 2.6 Powers and Exponents .........................................................................................................40 2.7 Averages (Arithmetic Mean)................................................................................................41 2.8 Ratio ....................................................................................................................................43 2.9 Dimensional Analysis ..........................................................................................................47 2.10 Threshold Odor Number (TON)..........................................................................................53 2.11 Geometrical Measurements..................................................................................................53 2.11.1 Geometrical Calculations .....................................................................................54 2.11.1.1 Perimeter and Circumference............................................................54 2.11.1.2 Area ...................................................................................................57 2.11.1.3 Volume...............................................................................................60 2.12 Force, Pressure, and Head Calculations ..............................................................................64 2.12.1 Force and Pressure ...............................................................................................64 2.12.2 Head......................................................................................................................65 2.12.2.1 Static Head ........................................................................................65 2.12.2.2 Friction Head.....................................................................................66 2.12.2.3 Velocity Head ....................................................................................66 2.12.2.4 Total Dynamic Head (Total System Head).......................................66 2.12.2.5 Pressure/Head ....................................................................................66 2.12.2.6 Head/Pressure ....................................................................................66 2.13 Review of Advanced Algebra Key Terms and Concepts ....................................................71 Chapter 3 Environmental Modeling .........................................................................................73 3.1 Introduction ..........................................................................................................................73 3.2 Media Material Content .......................................................................................................73 3.2.1 Material Content: Liquid Phases..........................................................................75 3.3 Phase Equilibrium and Steady State....................................................................................78 3.4 Math Operations and Laws of Equilibrium.........................................................................79 3.4.1 Solving Equilibrium Problems.............................................................................79 3.4.2 Laws of Equilibrium.............................................................................................80 3.4.2.1 Ideal Gas Law ...................................................................................80 3.4.2.2 Dalton’s Law .....................................................................................81 3.4.2.3 Raoult’s Law .....................................................................................83 3.4.2.4 Henry’s Law......................................................................................83 3.5 Chemical Transport Systems ...............................................................................................83 3.6 A Final Word on Environmental Modeling.........................................................................84 References .......................................................................................................................................85 Chapter 4 Algorithms and Environmental Engineering ...........................................................87 4.1 Introduction ..........................................................................................................................87 4.2 Algorithms: What Are They?...............................................................................................87 4.3 Expressing Algorithms.........................................................................................................88 4.4 General Algorithm Applications ..........................................................................................89 4.5 Environmental Engineering Algorithm Applications ..........................................................90 4.6 Dispersion Models ...............................................................................................................91 4.7 Screening Tools ....................................................................................................................91 References .......................................................................................................................................92 Suggested Reading..........................................................................................................................92 PART II: FUNDAMENTAL SCIENCE AND STATISTICS REVIEW .................................93 Chapter 5 Fundamental Chemistry and Hydraulics .................................................................95 5.1 Introduction ..........................................................................................................................95 5.2 Fundamental Chemistry .......................................................................................................95 5.2.1 Density and Specific Gravity ...............................................................................96 5.2.2 Water Chemistry Fundamentals ...........................................................................99 5.2.2.1 The Water Molecule..........................................................................99 5.2.2.2 Water Solutions ...............................................................................100 5.2.2.3 Concentrations.................................................................................101 5.2.2.4 Predicting Solubility........................................................................103 5.2.2.5 Colligative Properties ......................................................................103 5.2.2.6 Colloids/Emulsions..........................................................................104 5.2.2.7 Water Constituents ..........................................................................105 5.2.2.8 Simple Solutions and Dilutions ......................................................112 5.2.2.9 Chemical Reactions.........................................................................115 5.2.2.10 Chemical Dosages (Water and Wastewater Treatment)..................120 5.3 Fundamental Hydraulics ....................................................................................................126 5.3.1 Principles of Water Hydraulics ..........................................................................126 5.3.1.1 Weight of Air...................................................................................126 5.3.1.2 Weight of Water ..............................................................................126 5.3.1.3 Weight of Water Related to the Weight of Air ...............................127 5.3.1.4 Water at Rest ...................................................................................128 5.3.1.5 Gauge Pressure................................................................................128 5.3.1.6 Water in Motion ..............................................................................129 5.3.1.7 Discharge.........................................................................................129 5.3.1.8 The Law of Continuity....................................................................130 5.3.1.9 Pipe Friction ....................................................................................131 5.3.2 Basic Pumping Calculations ..............................................................................131 5.3.2.1 Pumping Rates.................................................................................132 5.3.3 Calculating Head Loss .......................................................................................133 5.3.4 Calculating Head ................................................................................................134 5.3.5 Calculating Horsepower and Efficiency.............................................................134 5.3.5.1 Hydraulic Horsepower (WHP)........................................................135 5.3.5.2 Pump Efficiency and Brake Horsepower (bhp)..............................135 References .....................................................................................................................................138 Suggested Reading........................................................................................................................138 Chapter 6 Statistics Review....................................................................................................139 6.1 Statistical Concepts ............................................................................................................139 6.2 Measure of Central Tendency............................................................................................139 6.3 Basic Statistical Terms.......................................................................................................139 6.4 DMR Calculations..............................................................................................................140 6.4.1 Loading Calculation ...........................................................................................140 6.4.2 Monthly Average Loading Calculations ............................................................141 6.4.3 30-Day Average Calculation ..............................................................................141 6.4.4 Moving Average .................................................................................................142 6.4.5 Geometric Mean .................................................................................................143 6.4.5.1 Logarithm (Log) Method ................................................................144 6.4.5.2 Nth Root Calculation Method.........................................................144 6.5 Standard Deviation.............................................................................................................145 6.6 Conclusion.........................................................................................................................147 PART III: MATH CONCEPTS: AIR POLLUTION CONTROL .........................................149 Chapter 7 Air Pollution Fundamentals ...................................................................................151 7.1 Introduction ........................................................................................................................151 7.1.1 Six Common Air Pollutants ...............................................................................152 7.1.1.1 Ground-Level Ozone.......................................................................152 7.1.1.2 Nitrogen Oxides ..............................................................................153 7.1.1.3 Particulate Matter ............................................................................153 7.1.1.4 Sulfur Dioxide (SO 2 ).......................................................................153 7.1.1.5 Carbon Monoxide (CO) ..................................................................153 7.1.1.6 Lead .................................................................................................154 7.2 Gases .................................................................................................................................154 7.2.1 The Gas Laws.....................................................................................................155 7.2.1.1 Boyle’s Law.....................................................................................156 7.2.1.2 Charles’s Law..................................................................................157 7.2.1.3 Gay–Lussac’s Law ..........................................................................157 7.2.1.4 The Combined Gas Law.................................................................158 7.2.1.5 The Ideal Gas Law..........................................................................158 7.2.1.6 Composition of Air .........................................................................159 7.3 Particulate Matter ...............................................................................................................160 7.4 Pollution Emission Measurement Parameters ...................................................................160 7.5 Standard Corrections..........................................................................................................161 References .....................................................................................................................................162 Chapter 8 Gaseous Emission Control.....................................................................................163 8.1 Introduction ........................................................................................................................163 8.2 Absorption.........................................................................................................................163 8.2.1 Solubility.............................................................................................................166 8.2.2 Equilibrium Solubility and Henry’s Law...........................................................166 8.2.3 Material (Mass) Balance ....................................................................................168 8.2.4 Sizing Packed Column Diameter and Height of an Absorber ..........................172 8.2.4.1 Packed Tower Absorber Diameter ..................................................172 8.2.4.2 Sizing the Packed Tower Absorber Height.....................................175 8.2.4.3 Sizing the Plate (Tray) Tower.........................................................179 8.2.4.4 Theoretical Number of Absorber Plates or Trays...........................181 8.3 Adsorption.........................................................................................................................183 8.3.1 Adsorption Steps ................................................................................................184 8.3.2 Adsorption Forces — Physical and Chemical...................................................184 8.3.3 Adsorption Equilibrium Relationships...............................................................185 8.3.3.1 Isotherm...........................................................................................185 8.3.3.2 Isostere.............................................................................................186 8.3.3.3 Isobar ...............................................................................................186 8.3.4 Factors Affecting Adsorption .............................................................................187 8.3.4.1 Temperature .....................................................................................188 8.3.4.2 Pressure............................................................................................188 8.3.4.3 Gas Velocity.....................................................................................188 8.3.4.4 Bed Depth........................................................................................189 8.3.4.5 Humidity..........................................................................................192 8.3.4.6 Contaminants...................................................................................192 8.4 Incineration........................................................................................................................193 8.4.1 Factors Affecting Incineration for Emission Control ........................................193 8.4.1.1 Temperature .....................................................................................193 8.4.1.2 Residence Time ...............................................................................193 8.4.1.3 Turbulence .......................................................................................194 8.4.1.4 Oxygen Requirement.......................................................................194 8.4.1.5 Combustion Limit ...........................................................................195 8.4.1.6 Flame Combustion ..........................................................................195 8.4.1.7 Heat..................................................................................................195 8.4.2 Incineration Example Calculations ....................................................................196 8.5 Condensation.....................................................................................................................199 8.5.1 Contact Condenser Calculations ........................................................................199 8.5.2 Surface Condenser Calculations.........................................................................201 References .....................................................................................................................................206 Chapter 9 Particulate Emission Control .................................................................................207 9.1 Particulate Emission Control Basics..................................................................................207 9.1.1 Interaction of Particles with Gas........................................................................207 9.1.2 Particulate Collection .........................................................................................208 9.2 Particulate Size Characteristics and General Characteristics............................................209 9.2.1 Aerodynamic Diameter ......................................................................................209 9.2.2 Equivalent Diameter ...........................................................................................209 9.2.3 Sedimentation Diameter .....................................................................................209 9.2.4 Cut Diameter ......................................................................................................210 9.2.5 Dynamic Shape Factor .......................................................................................210 9.3 Flow Regime of Particle Motion .......................................................................................210 9.4 Particulate Emission Control Equipment Calculations .....................................................216 9.4.1 Gravity Settlers ...................................................................................................216 9.4.2 Gravity Settling Chamber Theoretical Collection Efficiency............................217 9.4.3 Minimum Particle Size.......................................................................................219 9.4.4 Cyclones .............................................................................................................223 9.4.4.1 Factors Affecting Cyclone Performance.........................................223 9.4.6 Electrostatic Precipitator (ESP)..........................................................................228 9.4.6.1 Collection Efficiency.......................................................................228 9.4.6.2 Precipitator Example Calculations..................................................230 9.4.7 Baghouse (Fabric) Filters ...................................................................................236 9.4.7.1 Air-to-Filter (Media) Ratio .............................................................237 9.4.7.2 Baghouse Example Calculations.....................................................237 References .....................................................................................................................................247 Chapter 10 Wet Scrubbers for Emission Control.....................................................................249 10.1 Introduction ........................................................................................................................249 10.1.1 Wet Scrubbers.....................................................................................................249 10.2 Wet Scrubber Collection Mechanisms and Efficiency (Particulates) ...............................250 10.2.1 Collection Efficiency ..........................................................................................251 10.2.2 Impaction ............................................................................................................251 10.2.3 Interception .........................................................................................................252 10.2.4 Diffusion .............................................................................................................252 10.2.5 Calculation of Venturi Scrubber Efficiency .......................................................253 10.2.5.1 Johnstone Equation .........................................................................253 10.2.5.2 Infinite Throat Model......................................................................254 10.2.5.3 Cut Power Method ..........................................................................260 10.2.5.4 Contact Power Theory.....................................................................261 10.2.5.5 Pressure Drop ..................................................................................265 10.3 Wet Scrubber Collection Mechanisms and Efficiency (Gaseous Emissions)...................266 10.4 Assorted Venturi Scrubber Example Calculations ............................................................266 10.4.1 Scrubber Design of a Venturi Scrubber .............................................................266 10.4.2 Spray Tower........................................................................................................274 10.4.3 Packed Tower......................................................................................................276 10.4.4 Packed Column Height and Diameter ...............................................................280 10.5 Summary of Key Points.....................................................................................................285 References .....................................................................................................................................285 PART IV: MATH CONCEPTS: WATER QUALITY ............................................................287 Chapter 11 Running Waters ......................................................................................................289 11.1 Balancing the “Aquarium”.................................................................................................289 11.1.1 Sources of Stream Pollution...............................................................................290 11.2 Is Dilution the Solution?....................................................................................................291 11.2.1 Dilution Capacity of Running Waters................................................................292 11.3 Discharge Measurement.....................................................................................................292 11.4 Time of Travel....................................................................................................................293 11.5 Dissolved Oxygen (DO) ....................................................................................................294 11.5.1 DO Correction Factor.........................................................................................295 11.6 Biochemical Oxygen Demand...........................................................................................296 11.6.1 BOD Test Procedure...........................................................................................297 11.6.2 Practical BOD Calculation Procedure ...............................................................297 11.6.2.1 Unseeded BOD Procedure ..............................................................297 11.6.2.2 Seeded BOD Procedure ..................................................................298 11.7 Oxygen Sag (Deoxygenation)............................................................................................299 11.8 Stream Purification: A Quantitative Analysis....................................................................300 References .....................................................................................................................................304 Chapter 12 Still Waters .............................................................................................................305 12.1 Introduction ........................................................................................................................305 12.2 Still Water Systems ............................................................................................................307 12.3 Still Water System Calculations ........................................................................................307 12.3.1 Still Water Body Morphometry Calculations ....................................................307 12.3.1.1 Volume.............................................................................................307 12.3.1.2 Shoreline Development Index (D L ).................................................308 12.3.1.3 Mean Depth.....................................................................................308 12.4 Still Water Surface Evaporation.........................................................................................312 12.4.1 Water Budget Model ..........................................................................................312 12.4.2 Energy Budget Model ........................................................................................312 12.4.3 Priestly–Taylor Equation ....................................................................................313 12.4.4 Penman Equation................................................................................................313 12.4.5 DeBruin–Keijman Equation ...............................................................................313 12.4.6 Papadakis Equation ............................................................................................314 References .....................................................................................................................................314 Chapter 13 Groundwater...........................................................................................................315 13.1 Groundwater and Aquifers.................................................................................................315 13.1.1 Groundwater Quality ..........................................................................................317 13.1.2 GUDISW ............................................................................................................317 13.2 Aquifer Parameters.............................................................................................................317 13.2.1 Aquifer Porosity .................................................................................................317 13.2.2 Specific Yield (Storage Coefficient)...................................................................318 13.2.3 Permeability (K) .................................................................................................318 13.2.4 Transmissivity (T) ..............................................................................................318 13.2.5 Hydraulic Gradient and Head ............................................................................319 13.2.6 Flow Lines and Flow Nets .................................................................................319 13.3 Groundwater Flow .............................................................................................................319 13.4 General Equations of Groundwater Flow..........................................................................320 13.4.1 Steady Flow in a Confined Aquifer ...................................................................321 13.4.2 Steady Flow in an Unconfined Aquifer .............................................................321 References .....................................................................................................................................322 Chapter 14 Basic Hydraulics ....................................................................................................323 14.1 Introduction ........................................................................................................................323 14.2 Basic