Formula Sheet: Water Treatment

Fundamentals of Water Treatment

Flow and Detention Time

Detention Time (Theoretical) \[t = \frac{V}{Q}\]

• t = detention time (hr or min)
• V = volume of tank or basin (gal or ft³)
• Q = flow rate (gpm, gpd, or cfs)
• Units must be consistent

Hydraulic Loading Rate (Surface Loading Rate) \[q = \frac{Q}{A}\]

• q = hydraulic loading rate (gpm/ft², gpd/ft², or m/d)
• Q = flow rate (gpm, gpd, or m³/d)
• A = surface area (ft² or m²)
• Also called overflow rate for sedimentation basins

Weir Loading Rate (Weir Overflow Rate) \[WLR = \frac{Q}{L}\]

• WLR = weir loading rate (gpd/ft or gpm/ft)
• Q = flow rate (gpd or gpm)
• L = total length of weir (ft)
• Typical range: 10,000 - 20,000 gpd/ft for settling basins

Mixing and Flocculation

Velocity Gradient (G-Value) \[G = \sqrt{\frac{P}{\mu V}}\]

• G = velocity gradient (s⁻¹)
• P = power input (ft·lb/sec or watts)
• μ = absolute (dynamic) viscosity (lb·s/ft² or N·s/m²)
• V = volume of basin (ft³ or m³)
• For water at 50°F: μ ≈ 2.73 × 10⁻⁵ lb·s/ft²
• For water at 68°F: μ ≈ 2.09 × 10⁻⁵ lb·s/ft²

Power Requirement for Mixing \[P = G^2 \mu V\]

• P = power input (ft·lb/sec or watts)
• G = velocity gradient (s⁻¹)
• μ = absolute viscosity (lb·s/ft² or N·s/m²)
• V = basin volume (ft³ or m³)

Gt Product (Camp Number) \[Gt = G \times t\]

• Gt = dimensionless mixing parameter
• G = velocity gradient (s⁻¹)
• t = detention time (seconds)
• Rapid mix: Gt typically 20,000 - 60,000
• Flocculation: Gt typically 50,000 - 200,000

Power from Paddles \[P = \frac{C_D \rho A_p v^3}{2}\]

• P = power (ft·lb/sec or watts)
• C_D = drag coefficient (dimensionless, typically 1.8)
• ρ = fluid density (lb/ft³ or kg/m³)
• A_p = total paddle area (ft² or m²)
• v = relative velocity of paddle to water (ft/s or m/s)
• v ≈ 0.75 × paddle tip velocity for rotational mixers

Paddle Tip Velocity \[v_{tip} = \frac{2\pi r N}{60}\]

• v_tip = paddle tip velocity (ft/s or m/s)
• r = radius to paddle tip (ft or m)
• N = rotational speed (rpm)

Coagulation and Flocculation

Chemical Dosing

Chemical Dosage (Mass Rate) \[D = Q \times C \times 8.34\]

• D = dosage rate (lb/day)
• Q = flow rate (MGD)
• C = concentration (mg/L)
• 8.34 = conversion factor (lb·L/mg·MG)

Feed Rate (Liquid Chemical) \[F = \frac{D \times 100}{S \times SG \times 8.34}\]

• F = feed rate (gal/day)
• D = dosage (mg/L)
• S = solution strength (% by weight)
• SG = specific gravity of solution
• 8.34 = conversion factor

Dry Chemical Feed Rate \[F = \frac{Q \times D \times 8.34}{P}\]

• F = feed rate (lb/day)
• Q = flow rate (MGD)
• D = dosage (mg/L)
• P = purity (decimal fraction)
• 8.34 = conversion factor

Jar Test Relationships

Alum Dosage Scale-Up \[D_{plant} = D_{jar} \times \frac{V_{jar}}{V_{water}}\]

• D_plant = plant dosage (mg/L)
• D_jar = jar test dose (mg or mL of stock solution)
• V_jar = volume of jar test sample (L)
• V_water = volume being treated
• Assumes stock solution concentration is known

Sedimentation

Settling Velocity (Type I - Discrete Particle)

Stokes' Law (Laminar Flow, Re <> \[v_s = \frac{g(ρ_p - ρ_w)d^2}{18\mu}\]

• v_s = settling velocity (ft/s or m/s)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• ρ_p = particle density (lb/ft³ or kg/m³)
• ρ_w = water density (lb/ft³ or kg/m³)
• d = particle diameter (ft or m)
• μ = absolute viscosity (lb·s/ft² or N·s/m²)
• Valid for Reynolds number < 1="" (laminar="">

Newton's Law (Turbulent Flow, Re > 10⁴) \[v_s = 1.74\sqrt{\frac{g(ρ_p - ρ_w)d}{ρ_w}}\]

• v_s = settling velocity (ft/s or m/s)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• d = particle diameter (ft or m)
• ρ_p = particle density (lb/ft³ or kg/m³)
• ρ_w = water density (lb/ft³ or kg/m³)
• Valid for Reynolds number > 10⁴ (turbulent settling)

General Settling Equation \[v_s = \sqrt{\frac{4g(ρ_p - ρ_w)d}{3C_Dρ_w}}\]

• v_s = settling velocity (ft/s or m/s)
• C_D = drag coefficient (dimensionless)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• d = particle diameter (ft or m)
• ρ_p = particle density (lb/ft³ or kg/m³)
• ρ_w = water density (lb/ft³ or kg/m³)

Reynolds Number for Settling Particles \[Re = \frac{v_s d ρ_w}{\mu}\]

• Re = Reynolds number (dimensionless)
• v_s = settling velocity (ft/s or m/s)
• d = particle diameter (ft or m)
• ρ_w = water density (lb/ft³ or kg/m³)
• μ = absolute viscosity (lb·s/ft² or N·s/m²)

Overflow Rate and Removal Efficiency

Critical Settling Velocity (Overflow Rate) \[v_c = \frac{Q}{A} = \frac{Q}{L \times W}\]

• v_c = critical settling velocity (ft/s, gpm/ft², or m/d)
• Q = flow rate (ft³/s, gpm, or m³/d)
• A = surface area (ft² or m²)
• L = length of basin (ft or m)
• W = width of basin (ft or m)
• Particles with v_s ≥ v_c are theoretically 100% removed

Removal Efficiency (Ideal Settling) \[R = \frac{v_s}{v_c} \text{ for } v_s < v_c\]="" \[r="1.0" \text{="" for="" }="" v_s="" \geq="" v_c\]="">

• R = fractional removal (dimensionless)
• v_s = particle settling velocity (ft/s or m/s)
• v_c = critical settling velocity (ft/s or m/s)
• Assumes ideal settling conditions

Scour Velocity \[v_{scour} = \sqrt{8k\frac{(ρ_p - ρ_w)}{ρ_w}gd}\]

• v_scour = horizontal velocity causing scour (ft/s or m/s)
• k = constant (typically 0.04 for sand)
• ρ_p = particle density (lb/ft³ or kg/m³)
• ρ_w = water density (lb/ft³ or kg/m³)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• d = particle diameter (ft or m)

Sedimentation Basin Design Parameters

Length to Width Ratio

• Typical range: 3:1 to 5:1
• Helps promote plug flow conditions

Depth Requirements

• Typical depth: 10 - 15 ft (3 - 4.5 m)
• Sidewater depth for rectangular basins

Horizontal Velocity \[v_h = \frac{Q}{A_{cross}} = \frac{Q}{W \times H}\]

• v_h = horizontal velocity (ft/s or m/s)
• Q = flow rate (ft³/s or m³/s)
• A_cross = cross-sectional area (ft² or m²)
• W = width (ft or m)
• H = depth (ft or m)
• Typical range: 0.5 - 1.5 ft/min to prevent scour and short-circuiting

Filtration

Filter Loading and Headloss

Filtration Rate (Hydraulic Loading) \[F = \frac{Q}{A}\]

• F = filtration rate (gpm/ft², gpd/ft², or m/h)
• Q = flow rate through filter (gpm, gpd, or m³/h)
• A = filter surface area (ft² or m²)
• Typical range for rapid sand filters: 2 - 6 gpm/ft²
• Typical range for slow sand filters: 0.015 - 0.15 gpm/ft²

Unit Filter Run Volume (UFRV) \[UFRV = F \times t_{run}\]

• UFRV = unit filter run volume (gal/ft² or m³/m²)
• F = filtration rate (gpm/ft² or m/h)
• t_run = filter run time (min or h)
• Higher UFRV indicates better filter performance

Carmen-Kozeny Equation (Clean Bed Headloss) \[h_L = \frac{k \mu v L}{ρ_w g d^2} \times \frac{(1-\varepsilon)^2}{\varepsilon^3}\]

• h_L = headloss (ft or m)
• k = Carmen-Kozeny constant (≈ 5.0)
• μ = absolute viscosity (lb·s/ft² or N·s/m²)
• v = approach velocity (ft/s or m/s)
• L = depth of filter bed (ft or m)
• ρ_w = water density (lb/ft³ or kg/m³)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• d = grain diameter (ft or m)
• ε = porosity (dimensionless, typically 0.4 - 0.45)

Fair-Hatch Equation (Modified Carmen-Kozeny) \[h_L = \frac{1.067 C_D L v^2}{g d \varepsilon^3}\]

• h_L = headloss (ft or m)
• C_D = drag coefficient (dimensionless)
• L = depth of filter media (ft or m)
• v = approach velocity (ft/s or m/s)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• d = grain diameter (ft or m)
• ε = porosity (dimensionless)

Stratification in Mixed Media Filters \[d_1\sqrt{ρ_1 - ρ_w} = d_2\sqrt{ρ_2 - ρ_w}\]

• d₁, d₂ = grain diameters of media 1 and 2 (ft or m)
• ρ₁, ρ₂ = densities of media 1 and 2 (lb/ft³ or kg/m³)
• ρ_w = water density (lb/ft³ or kg/m³)
• Condition for equal settling velocities during backwash

Backwash Requirements

Minimum Fluidization Velocity \[v_{mf} = \frac{1.7 \times 10^{-3} d^{1.5}}{t^{0.75}}\]

• v_mf = minimum fluidization velocity (gpm/ft²)
• d = grain diameter (mm)
• t = temperature (°C)
• Empirical relationship for clean sand

Backwash Expansion \[E = \frac{L_e - L}{L} \times 100\%\]

• E = bed expansion (%)
• L_e = expanded bed depth (ft or m)
• L = initial bed depth (ft or m)
• Typical expansion: 20 - 50%

Backwash Rate \[Q_{bw} = A \times v_{bw}\]

• Q_bw = backwash flow rate (gpm or m³/h)
• A = filter area (ft² or m²)
• v_bw = backwash rate (gpm/ft² or m/h)
• Typical range: 12 - 24 gpm/ft² for sand filters

Backwash Water Volume \[V_{bw} = Q_{bw} \times t_{bw}\]

• V_bw = volume of backwash water (gal or m³)
• Q_bw = backwash flow rate (gpm or m³/min)
• t_bw = backwash duration (min)
• Typical duration: 5 - 15 minutes

Filter Media Properties

Effective Size (ES or d₁₀)

• Grain diameter at which 10% by weight is finer
• Typical for rapid sand filters: 0.45 - 0.55 mm

Uniformity Coefficient (UC) \[UC = \frac{d_{60}}{d_{10}}\]

• UC = uniformity coefficient (dimensionless)
• d₆₀ = grain size at which 60% by weight is finer (mm)
• d₁₀ = grain size at which 10% by weight is finer (effective size) (mm)
• Typical range: 1.3 - 1.7 for good filtration
• Lower UC indicates more uniform media

Porosity \[ε = \frac{V_{voids}}{V_{total}} = 1 - \frac{ρ_{bulk}}{ρ_{grain}}\]

• ε = porosity (dimensionless)
• V_voids = volume of voids
• V_total = total volume
• ρ_bulk = bulk density of media
• ρ_grain = grain density of media
• Typical range: 0.40 - 0.45 for sand

Disinfection

Chlorination

Chlorine Dose \[Cl_{dose} = Cl_{demand} + Cl_{residual}\]

• Cl_dose = total chlorine dose (mg/L)
• Cl_demand = chlorine demand (mg/L)
• Cl_residual = desired chlorine residual (mg/L)

Chlorine Feed Rate \[F = Q \times D \times 8.34\]

• F = chlorine feed rate (lb/day)
• Q = flow rate (MGD)
• D = chlorine dose (mg/L)
• 8.34 = conversion factor (lb·L/mg·MG)

Hypochlorite Solution Feed \[F_{hypo} = \frac{Q \times D \times 8.34}{C \times SG}\]

• F_hypo = hypochlorite solution feed rate (gal/day)
• Q = flow rate (MGD)
• D = dose (mg/L as Cl₂)
• C = concentration of available chlorine (% by weight)
• SG = specific gravity of hypochlorite solution
• 8.34 = conversion factor
• Sodium hypochlorite (NaOCl) typically 12-15% available Cl₂, SG ≈ 1.2

CT Values and Inactivation

CT Concept \[CT = C \times t\]

• CT = disinfection constant (mg·min/L)
• C = disinfectant residual concentration (mg/L)
• t = contact time (min)
• Required CT values depend on pathogen, pH, temperature

Contact Time (t₁₀) \[t_{10} = \frac{V}{Q} \times BF\]

• t₁₀ = time for 10% of water to pass through basin (min)
• V = basin volume (gal or ft³)
• Q = flow rate (gpm or cfs)
• BF = baffling factor (dimensionless, 0.1 - 0.7)
• BF = 0.7 for superior baffling (serpentine)
• BF = 0.3 - 0.5 for average baffling
• BF = 0.1 for poor or no baffling

Log Inactivation (Chick's Law) \[log \frac{N}{N_0} = -k \times C \times t\] \[\text{Log removal} = -log \frac{N}{N_0}\]

• N = number of organisms remaining
• N₀ = initial number of organisms
• k = rate constant (L/mg·min)
• C = disinfectant concentration (mg/L)
• t = contact time (min)
• Positive log removal indicates inactivation

Percent Inactivation \[\% \text{ inactivation} = \left(1 - \frac{N}{N_0}\right) \times 100\%\]

• N = organisms remaining
• N₀ = initial organisms
• Relationship: 1-log = 90%, 2-log = 99%, 3-log = 99.9%, 4-log = 99.99%

Ultraviolet (UV) Disinfection

UV Dose \[D_{UV} = I \times t\]

• D_UV = UV dose (mJ/cm² or μW·s/cm²)
• I = UV intensity (mW/cm² or μW/cm²)
• t = exposure time (s)
• 1 mJ/cm² = 1,000 μW·s/cm²
• Typical dose for drinking water: 40 mJ/cm²

Log Inactivation by UV \[\text{Log inactivation} = k_{UV} \times D_{UV}\]

• k_UV = inactivation rate constant for specific organism (cm²/mJ)
• D_UV = UV dose (mJ/cm²)
• Organism-specific relationship

Softening

Lime-Soda Softening

Hardness Definitions \[TH = Ca^{2+} + Mg^{2+}\] \[CH = TH - NCH\]

• TH = total hardness (mg/L as CaCO₃)
• CH = carbonate hardness (temporary hardness) (mg/L as CaCO₃)
• NCH = noncarbonate hardness (permanent hardness) (mg/L as CaCO₃)
• Ca²⁺ = calcium hardness (mg/L as CaCO₃)
• Mg²⁺ = magnesium hardness (mg/L as CaCO₃)

Carbonate Hardness \[CH = TA \text{ (when TA < th)}\]="" \[ch="TH" \text{="" (when="" ta="" ≥="" th)}\]="">

• CH = carbonate hardness (mg/L as CaCO₃)
• TA = total alkalinity (mg/L as CaCO₃)
• TH = total hardness (mg/L as CaCO₃)

Noncarbonate Hardness \[NCH = TH - TA \text{ (when TH > TA)}\] \[NCH = 0 \text{ (when TH ≤ TA)}\]

• NCH = noncarbonate hardness (mg/L as CaCO₃)
• TH = total hardness (mg/L as CaCO₃)
• TA = total alkalinity (mg/L as CaCO₃)

Lime Requirement for CO₂ Removal \[CaO = CO_2 \times \frac{56}{44}\]

• CaO = lime required (mg/L as CaO)
• CO₂ = carbon dioxide (mg/L)
• Molecular weight: CaO = 56, CO₂ = 44

Lime for Calcium Carbonate Hardness Removal \[CaO = Ca(HCO_3)_2 \times \frac{56}{100}\]

• CaO = lime required (mg/L as CaO)
• Ca(HCO₃)₂ = calcium bicarbonate hardness (mg/L as CaCO₃)
• Reaction: Ca(HCO₃)₂ + Ca(OH)₂ → 2CaCO₃↓ + 2H₂O

Lime for Magnesium Carbonate Hardness Removal \[CaO = Mg(HCO_3)_2 \times 2 \times \frac{56}{100}\]

• CaO = lime required (mg/L as CaO)
• Mg(HCO₃)₂ = magnesium bicarbonate hardness (mg/L as CaCO₃)
• Factor of 2 accounts for Mg(OH)₂ precipitation requiring higher pH
• Reaction: Mg(HCO₃)₂ + 2Ca(OH)₂ → Mg(OH)₂↓ + 2CaCO₃↓ + 2H₂O

Soda Ash for Calcium Noncarbonate Hardness \[Na_2CO_3 = CaSO_4 \times \frac{106}{100}\]

• Na₂CO₃ = soda ash required (mg/L)
• CaSO₄ = calcium noncarbonate hardness (mg/L as CaCO₃)
• Molecular weight: Na₂CO₃ = 106, CaCO₃ = 100
• Reaction: CaSO₄ + Na₂CO₃ → CaCO₃↓ + Na₂SO₄

Lime for Magnesium Noncarbonate Hardness \[CaO = MgSO_4 \times \frac{56}{100}\]

• CaO = lime required (mg/L as CaO)
• MgSO₄ = magnesium noncarbonate hardness (mg/L as CaCO₃)
• Reaction: MgSO₄ + Ca(OH)₂ → Mg(OH)₂↓ + CaSO₄

Soda Ash for Magnesium Noncarbonate Hardness (After Lime) \[Na_2CO_3 = MgSO_4 \times \frac{106}{100}\]

• Na₂CO₃ = soda ash required (mg/L)
• MgSO₄ = magnesium noncarbonate hardness (mg/L as CaCO₃)
• Treats CaSO₄ formed from lime treatment of Mg²⁺

Excess Lime for Magnesium Removal \[Excess\;CaO = (Mg_{initial} - Mg_{final}) \times \frac{56}{100}\]

• Excess CaO = additional lime (mg/L as CaO)
• Mg_initial = initial Mg hardness (mg/L as CaCO₃)
• Mg_final = desired final Mg hardness (mg/L as CaCO₃)
• Typically requires pH > 11 for effective Mg removal
• Usually add 1.5 times stoichiometric requirement

CO₂ for Recarbonation \[CO_2 = (CaO_{excess}) \times \frac{44}{56}\]

• CO₂ = carbon dioxide required (mg/L)
• CaO_excess = excess lime (mg/L as CaO)
• Used to lower pH after excess lime softening
• Reaction: Ca(OH)₂ + CO₂ → CaCO₃↓ + H₂O

Ion Exchange Softening

Exchange Capacity \[V = \frac{C \times M}{H}\]

• V = volume of water treated (gal or L)
• C = exchange capacity (grains or eq)
• M = mass of resin (lb or kg)
• H = hardness removed (grains/gal or eq/L)
• 1 grain/gal = 17.1 mg/L as CaCO₃
• Typical resin capacity: 20,000 - 30,000 grains/ft³

Salt Requirement for Regeneration \[S = \frac{H_{removed} \times V_{treated}}{E_{regen}}\]

• S = salt (NaCl) required (lb)
• H_removed = hardness removed (grains/gal)
• V_treated = volume treated between regenerations (gal)
• E_regen = regeneration efficiency (grains/lb salt)
• Typical: 0.3 - 0.5 lb NaCl per 1000 grains removed

Regeneration Brine Concentration \[C_{brine} = \frac{S}{V_{brine}}\]

• C_brine = brine concentration (lb/gal)
• S = salt required (lb)
• V_brine = volume of brine solution (gal)
• Typical: 10-15% NaCl solution (≈ 1 lb/gal)

Adsorption

Activated Carbon

Freundlich Isotherm \[q_e = K_F C_e^{1/n}\] \[log(q_e) = log(K_F) + \frac{1}{n}log(C_e)\]

• q_e = mass of adsorbate per mass of adsorbent at equilibrium (mg/g)
• C_e = equilibrium concentration of adsorbate (mg/L)
• K_F = Freundlich capacity factor (units vary)
• 1/n = Freundlich intensity parameter (dimensionless)
• Linear form used for parameter determination from log-log plot

Langmuir Isotherm \[q_e = \frac{q_m K_L C_e}{1 + K_L C_e}\] \[\frac{C_e}{q_e} = \frac{1}{q_m K_L} + \frac{C_e}{q_m}\]

• q_e = mass adsorbed per mass of adsorbent at equilibrium (mg/g)
• q_m = maximum adsorption capacity (mg/g)
• K_L = Langmuir constant related to bonding energy (L/mg)
• C_e = equilibrium concentration (mg/L)
• Linear form for parameter determination

Empty Bed Contact Time (EBCT) \[EBCT = \frac{V_{bed}}{Q}\]

• EBCT = empty bed contact time (min)
• V_bed = volume of adsorbent bed (ft³ or m³)
• Q = flow rate (ft³/min or m³/min)
• Typical range: 5 - 30 minutes for GAC contactors

Hydraulic Loading Rate for GAC \[HLR = \frac{Q}{A}\]

• HLR = hydraulic loading rate (gpm/ft² or m/h)
• Q = flow rate (gpm or m³/h)
• A = cross-sectional area (ft² or m²)
• Typical range: 2 - 10 gpm/ft²

Carbon Usage Rate \[CUR = \frac{M_{carbon}}{V_{water}}\]

• CUR = carbon usage rate (lb/1000 gal or kg/m³)
• M_carbon = mass of carbon consumed (lb or kg)
• V_water = volume of water treated (1000 gal or m³)

Membrane Processes

Reverse Osmosis and Nanofiltration

Water Flux \[J_w = K_w(\Delta P - \Delta\pi)\]

• J_w = water flux (gal/ft²·day or L/m²·h)
• K_w = water permeability coefficient (gal/ft²·day·psi or L/m²·h·bar)
• ΔP = transmembrane pressure difference (psi or bar)
• Δπ = osmotic pressure difference (psi or bar)

Osmotic Pressure (van't Hoff Equation) \[\pi = i \times C \times R \times T\]

• π = osmotic pressure (psi or bar)
• i = van't Hoff factor (dimensionless)
• C = molar concentration (mol/L)
• R = gas constant (0.0821 L·atm/mol·K or 0.08314 L·bar/mol·K)
• T = absolute temperature (K)
• For NaCl: i ≈ 2; approximate π ≈ 0.01 psi per mg/L TDS

Recovery Rate \[R = \frac{Q_p}{Q_f} \times 100\%\]

• R = recovery (% or fraction)
• Q_p = permeate (product) flow rate
• Q_f = feed flow rate
• Typical RO recovery: 75 - 85%

Rejection (Salt Rejection) \[Rejection = \frac{C_f - C_p}{C_f} \times 100\%\]

• Rejection = salt rejection (%)
• C_f = feed concentration (mg/L)
• C_p = permeate concentration (mg/L)
• Typical RO rejection: 95 - 99%

Concentration Factor \[CF = \frac{C_c}{C_f} = \frac{1}{1-R}\]

• CF = concentration factor (dimensionless)
• C_c = concentrate concentration
• C_f = feed concentration
• R = recovery (decimal fraction)

Mass Balance \[Q_f C_f = Q_p C_p + Q_c C_c\] \[Q_f = Q_p + Q_c\]

• Q_f = feed flow rate
• Q_p = permeate flow rate
• Q_c = concentrate (reject) flow rate
• C = concentration at respective stream

Microfiltration and Ultrafiltration

Flux \[J = \frac{Q}{A}\]

• J = flux (gal/ft²·day, gfd, or L/m²·h, lmh)
• Q = permeate flow rate
• A = membrane area
• Typical MF/UF flux: 50 - 150 gfd (80 - 250 lmh)

Transmembrane Pressure (TMP) \[TMP = \frac{P_{feed} + P_{concentrate}}{2} - P_{permeate}\]

• TMP = transmembrane pressure (psi or bar)
• P_feed = feed pressure (psi or bar)
• P_concentrate = concentrate pressure (psi or bar)
• P_permeate = permeate pressure (psi or bar)

Specific Flux \[SF = \frac{J}{TMP}\]

• SF = specific flux (gfd/psi or lmh/bar)
• J = flux (gfd or lmh)
• TMP = transmembrane pressure (psi or bar)
• Used to monitor membrane fouling

Aeration and Gas Transfer

Henry's Law and Gas Solubility

Henry's Law \[C = K_H \times P\]

• C = concentration of dissolved gas (mg/L or mol/L)
• K_H = Henry's law constant (mg/L·atm or mol/L·atm)
• P = partial pressure of gas (atm)
• Temperature dependent

Saturation Dissolved Oxygen \[C_s = C_{s,20} \times \theta^{(T-20)}\]

• C_s = saturation DO at temperature T (mg/L)
• C_s,20 = saturation DO at 20°C (mg/L)
• θ = temperature correction factor (typically 1.024)
• T = temperature (°C)

Oxygen Transfer

Oxygen Transfer Rate (OTR) \[\frac{dC}{dt} = K_L a (C_s - C)\]

• dC/dt = rate of change of DO concentration (mg/L·h)
• K_La = overall mass transfer coefficient (h⁻¹)
• C_s = saturation DO concentration (mg/L)
• C = DO concentration at time t (mg/L)

Integrated Oxygen Transfer (Clean Water) \[ln\frac{C_s - C_0}{C_s - C_t} = K_L a \times t\]

• C_s = saturation DO (mg/L)
• C₀ = initial DO (mg/L)
• C_t = DO at time t (mg/L)
• K_La = mass transfer coefficient (h⁻¹ or min⁻¹)
• t = time (h or min)

Standard Oxygen Transfer Rate (SOTR) \[SOTR = K_L a_{20} \times C_{s,20} \times V\]

• SOTR = standard oxygen transfer rate (lb O₂/h or kg O₂/h)
• K_La₂₀ = mass transfer coefficient at 20°C (h⁻¹)
• C_s,20 = saturation DO at 20°C, 1 atm (mg/L)
• V = volume (gal or m³)
• Standard conditions: clean water, 20°C, zero DO

Field Oxygen Transfer Rate (OTR_field) \[OTR_{field} = SOTR \times \alpha \times \beta \times \frac{C_{s,T,P} - C_L}{C_{s,20}} \times \theta^{(T-20)} \times F\]

• OTR_field = field oxygen transfer rate
• SOTR = standard oxygen transfer rate
• α = correction factor for K_La (wastewater vs clean water, typically 0.4-0.8)
• β = correction factor for C_s (salinity/surfactants, typically 0.9-0.97)
• C_s,T,P = DO saturation at field temperature and pressure (mg/L)
• C_L = operating DO level (mg/L)
• C_s,20 = DO saturation at 20°C (mg/L)
• θ = temperature coefficient (typically 1.024)
• T = temperature (°C)
• F = fouling factor (typically 0.85-0.95)

Standard Aeration Efficiency (SAE) \[SAE = \frac{SOTR}{P}\]

• SAE = standard aeration efficiency (lb O₂/hp·h or kg O₂/kW·h)
• SOTR = standard oxygen transfer rate (lb O₂/h or kg O₂/h)
• P = power input (hp or kW)

pH Adjustment and Alkalinity

Alkalinity Relationships

Total Alkalinity \[TA = [HCO_3^-] + 2[CO_3^{2-}] + [OH^-] - [H^+]\]

• TA = total alkalinity (mg/L as CaCO₃ or eq/L)
• Brackets denote concentrations (mg/L as CaCO₃ or eq/L)
• Typically bicarbonate alkalinity dominates at pH 6-9

Phenolphthalein Alkalinity Relationships

• P = 0, TA > 0: Only bicarbonate (HCO₃⁻) present
• P < ½="" ta:="" bicarbonate="" and="" carbonate="">
• P = ½ TA: Only carbonate (CO₃²⁻) present
• P > ½ TA: Carbonate and hydroxide present
• P = TA: Only hydroxide (OH⁻) present

Alkalinity Species Calculations

• HCO₃⁻ = 2(TA - P) when P < ½="">
• CO₃²⁻ = 2P when P < ½="">
• CO₃²⁻ = 2(TA - P) when P > ½ TA
• OH⁻ = 2P - TA when P > ½ TA
• P = phenolphthalein alkalinity, TA = total alkalinity
• All in mg/L as CaCO₃

Chemical Additions for pH Control

Lime (Ca(OH)₂) Addition for pH Increase \[Ca(OH)_2 = Alk_{increase} \times \frac{74}{100}\]

• Ca(OH)₂ = lime dose (mg/L)
• Alk_increase = desired alkalinity increase (mg/L as CaCO₃)
• Molecular weight: Ca(OH)₂ = 74, CaCO₃ = 100

Caustic Soda (NaOH) Addition for pH Increase \[NaOH = Alk_{increase} \times \frac{80}{100}\]

• NaOH = caustic soda dose (mg/L)
• Alk_increase = desired alkalinity increase (mg/L as CaCO₃)
• Molecular weight: NaOH = 40 (× 2 for neutralization = 80), CaCO₃ = 100

Sulfuric Acid (H₂SO₄) for pH Decrease \[H_2SO_4 = Alk_{decrease} \times \frac{98}{100}\]

• H₂SO₄ = sulfuric acid dose (mg/L as 100%)
• Alk_decrease = alkalinity to be neutralized (mg/L as CaCO₃)
• Molecular weight: H₂SO₄ = 98, CaCO₃ = 100
• For commercial acid strength, divide by purity fraction

Corrosion Control

Langelier Saturation Index (LSI)

Langelier Saturation Index \[LSI = pH - pH_s\]

• LSI = Langelier Saturation Index (dimensionless)
• pH = actual pH of water
• pH_s = pH at saturation with CaCO₃
• LSI > 0: water is supersaturated, scale-forming tendency
• LSI = 0: water is at saturation equilibrium
• LSI < 0:="" water="" is="" undersaturated,="" corrosive="">

pH at Saturation (pH_s) \[pH_s = (pK_2 - pK_s) + pCa + pAlk\]

• pH_s = pH at saturation
• pK₂ = negative log of second dissociation constant for H₂CO₃
• pK_s = negative log of solubility product for CaCO₃
• pCa = negative log of calcium concentration (mol/L)
• pAlk = negative log of alkalinity (eq/L)
• Temperature and TDS dependent; nomographs or tables used

Ryznar Stability Index (RSI)

Ryznar Stability Index \[RSI = 2pH_s - pH\]

• RSI = Ryznar Stability Index (dimensionless)
• pH_s = pH at saturation
• pH = actual pH
• RSI < 6:="">
• RSI = 6-7: stable water
• RSI > 7: increasingly corrosive

Aggressive Index (AI)

Aggressive Index \[AI = pH + pAlk\]

• AI = Aggressive Index (dimensionless)
• pH = actual pH
• pAlk = negative log of alkalinity (mg/L as CaCO₃)
• AI < 10:="" very="">
• AI = 10-12: moderately aggressive
• AI > 12: non-aggressive

Iron and Manganese Removal

Oxidation Requirements

Chlorine for Iron Oxidation \[Cl_2 = 0.62 \times Fe^{2+}\]

• Cl₂ = chlorine required (mg/L)
• Fe²⁺ = ferrous iron concentration (mg/L)
• Reaction: 2Fe²⁺ + Cl₂ + 6H₂O → 2Fe(OH)₃↓ + 2Cl⁻ + 6H⁺
• Additional chlorine needed for disinfection

Chlorine for Manganese Oxidation \[Cl_2 = 1.29 \times Mn^{2+}\]

• Cl₂ = chlorine required (mg/L)
• Mn²⁺ = manganous manganese concentration (mg/L)
• Requires pH > 9.5 for effective oxidation

Oxygen for Iron Oxidation \[O_2 = 0.14 \times Fe^{2+}\]

• O₂ = oxygen required (mg/L)
• Fe²⁺ = ferrous iron (mg/L)
• Reaction: 4Fe²⁺ + O₂ + 10H₂O → 4Fe(OH)₃↓ + 8H⁺

Potassium Permanganate for Iron Oxidation \[KMnO_4 = 0.94 \times Fe^{2+}\]

• KMnO₄ = potassium permanganate required (mg/L)
• Fe²⁺ = ferrous iron (mg/L)

Potassium Permanganate for Manganese Oxidation \[KMnO_4 = 1.92 \times Mn^{2+}\]

• KMnO₄ = potassium permanganate required (mg/L)
• Mn²⁺ = manganous manganese (mg/L)

Fluoridation

Fluoride Chemical Dosing

Sodium Fluoride (NaF) Dose \[NaF = F^- \times \frac{42}{19}\]

• NaF = sodium fluoride dose (mg/L)
• F⁻ = desired fluoride ion concentration (mg/L)
• Molecular weights: NaF = 42, F = 19

Sodium Fluorosilicate (Na₂SiF₆) Dose \[Na_2SiF_6 = F^- \times \frac{188}{114}\]

• Na₂SiF₆ = sodium fluorosilicate dose (mg/L)
• F⁻ = desired fluoride ion concentration (mg/L)
• Molecular weights: Na₂SiF₆ = 188, 6F = 114

Hydrofluorosilicic Acid (H₂SiF₆) Dose \[H_2SiF_6 = F^- \times \frac{144}{114}\]

• H₂SiF₆ = hydrofluorosilicic acid dose (mg/L as 100%)
• F⁻ = desired fluoride ion concentration (mg/L)
• Molecular weights: H₂SiF₆ = 144, 6F = 114
• Typically supplied as 20-30% solution

Sludge Production

Coagulation Sludge

Alum Sludge Production \[S_{alum} = 8.34 \times Q \times (SS_{removed} + 0.26 \times Alum_{dose})\]

• S_alum = dry sludge production (lb/day)
• Q = flow rate (MGD)
• SS_removed = suspended solids removed (mg/L)
• Alum_dose = alum dose (mg/L)
• 0.26 = stoichiometric factor for Al₂(SO₄)₃·14H₂O to Al(OH)₃
• 8.34 = conversion factor (lb·L/mg·MG)

Ferric Chloride Sludge Production \[S_{FeCl_3} = 8.34 \times Q \times (SS_{removed} + 0.66 \times FeCl_3_{dose})\]

• S_FeCl₃ = dry sludge production (lb/day)
• Q = flow rate (MGD)
• SS_removed = suspended solids removed (mg/L)
• FeCl₃_dose = ferric chloride dose (mg/L)
• 0.66 = stoichiometric factor for FeCl₃ to Fe(OH)₃

Softening Sludge

Lime Softening Sludge Production \[S = 8.34 \times Q \times (SS + 2.6 \times Ca_{removed} + 4.6 \times Mg_{removed})\]

• S = dry sludge production (lb/day)
• Q = flow rate (MGD)
• SS = suspended solids in raw water (mg/L)
• Ca_removed = calcium hardness removed (mg/L as CaCO₃)
• Mg_removed = magnesium hardness removed (mg/L as CaCO₃)
• 2.6 = factor accounting for CaCO₃ precipitate (100/40 × practical factor)
• 4.6 = factor accounting for Mg(OH)₂ precipitate (58/24 × practical factor)

Common Conversions and Constants

Unit Conversions

Flow Rate Conversions

• 1 MGD = 694.4 gpm = 1.547 cfs
• 1 cfs = 0.646 MGD = 448.8 gpm
• 1 gpm = 0.00144 MGD = 0.00223 cfs
• 1 m³/d = 0.264 gal/d = 183.5 gpd

Mass/Volume Conversions

• 1 mg/L = 1 g/m³ = 8.34 lb/MG
• 1 grain/gal = 17.1 mg/L (as CaCO₃)
• 1 ppm ≈ 1 mg/L (for dilute aqueous solutions)

Area Loading Conversions

• 1 gpm/ft² = 1,440 gpd/ft² = 40.7 m³/m²·d

Important Constants

Water Properties

• Density of water: 62.4 lb/ft³ = 1,000 kg/m³ (at 4°C)
• Specific weight: 8.34 lb/gal (at 4°C)
• Dynamic viscosity at 68°F (20°C): μ = 2.09 × 10⁻⁵ lb·s/ft² = 1.002 × 10⁻³ N·s/m²

Molecular Weights (g/mol)

• CaCO₃ = 100
• Ca(OH)₂ = 74
• CaO = 56
• Mg(OH)₂ = 58
• Al₂(SO₄)₃ = 342
• Al₂(SO₄)₃·14H₂O = 594 (alum)
• FeCl₃ = 162
• NaOH = 40
• Na₂CO₃ = 106
• H₂SO₄ = 98
• CO₂ = 44
• Cl₂ = 71