Advanced Civil Engineering Handbook › Chapter 9

Chapter 9 of 14 · 40 formulas · 82 one-liners

Environmental Engineering — formulas & one-liners

Water supply, water quality and treatment, sewerage and sewage treatment, stream pollution, air and noise pollution and solid waste management, following the CPHEEO manuals, IS 10500:2012, IS 2470, CPCB norms and the 2016 waste-management rules.

9.1 Water demand & population forecasting

Arithmetic increase method

P_{n} = P_{0} + n \overset{x}{ˉ}

P₀ = last known population; n = number of decades; x̄ = average increase per decade

Gives low values; suits old, large, saturated cities

Geometric increase method

P_{n} = P_{0} (1 + \frac{r}{100})^{n}

r = average (geometric mean) percentage growth per decade; n = decades

Gives high values; suits young, fast-growing towns

Incremental increase method

P_{n} = P_{0} + n \overset{x}{ˉ} + \frac{n ( n + 1 )}{2} \overset{y}{ˉ}

x̄ = average increase per decade; ȳ = average incremental increase (may be +ve or −ve)

Gives values between arithmetic and geometric methods

Logistic curve – saturation population

P_{s} = \frac{2 P _{0} P _{1} P _{2} - P _{1}^{2} ( P _{0} + P _{2} )}{P _{0} P _{2} - P _{1}^{2}}

P₀, P₁, P₂ = populations at three equally spaced times t₀, t₁, t₂

S-shaped growth curve

Goodrich formula (peak demand)

p = 180 t^{- 0.10}

p = demand as % of annual average; t = period in days (1/24 to 365)

Gives 180% for a day, 148% for a week, 128% for a month

Fire demand (CPHEEO)

Q = 100 P

Q = fire demand in kL/day; P = population in thousands

Recommended for communities above 50,000

CPHEEO per capita supply for towns with piped supply but no sewerage is 70 lpcd.
CPHEEO per capita supply for cities with piped supply where sewerage exists or is planned is 135 lpcd.
CPHEEO per capita supply for metropolitan and mega cities with sewerage is 150 lpcd.
CPHEEO peak factor for water supply is 3.0 for population up to 50,000, 2.5 for 50,000–2 lakh and 2.0 above 2 lakh.
Design period recommended by CPHEEO is 30 years for distribution pipes, intake works and mains, and 15 years for treatment units, pumps and service reservoirs.
Rural water supply under the Jal Jeevan Mission is planned at 55 lpcd (earlier 40 lpcd).

9.2 Water quality & IS 10500:2012

Hardness from Ca and Mg

T H = Ca^{2 +} \times \frac{50}{20} + Mg^{2 +} \times \frac{50}{12}

TH = total hardness (mg/L as CaCO₃); Ca, Mg in mg/L; 50, 20, 12 = equivalent weights

Equivalent weight of CaCO₃ = 50

Carbonate & non-carbonate hardness

C H = min (T H, Alk), N C H = T H - C H

CH = carbonate (temporary) hardness; NCH = non-carbonate (permanent) hardness; Alk = total alkalinity as CaCO₃

IS 10500:2012 pH range for drinking water is 6.5–8.5 with no relaxation.
Turbidity: acceptable limit 1 NTU, permissible limit 5 NTU (IS 10500:2012).
Total dissolved solids: acceptable 500 mg/L, permissible 2000 mg/L.
Total hardness as CaCO₃: acceptable 200 mg/L, permissible 600 mg/L.
Chloride: acceptable 250 mg/L, permissible 1000 mg/L.
Sulphate: acceptable 200 mg/L, permissible 400 mg/L.
Fluoride: acceptable 1.0 mg/L, permissible 1.5 mg/L; excess causes dental and skeletal fluorosis.
Nitrate: 45 mg/L with no relaxation; excess causes methaemoglobinaemia (blue-baby syndrome).
Free residual chlorine: minimum 0.2 mg/L, permissible up to 1 mg/L (IS 10500:2012).
Lead 0.01 mg/L, mercury 0.001 mg/L and cadmium 0.003 mg/L are limits with no relaxation.
Acceptable limit of arsenic in drinking water is 0.01 mg/L.
Total coliform and E. coli shall not be detectable in any 100 mL sample of drinking water.

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9.3 Sedimentation & coagulation

Stokes' law – settling velocity

v_{s} = \frac{g ( G _{s} - 1 ) d ^{2}}{18 ν}

g = 9.81 m/s²; G_s = specific gravity of particle; d = diameter (m); ν = kinematic viscosity (m²/s)

Valid for laminar settling, Re < 1 (d up to about 0.1 mm)

Surface overflow rate

v_{o} = \frac{Q}{A} = \frac{Q}{B L}

Q = flow (m³/d); A = plan area of tank (m²); B, L = width and length

Particles with v_s ≥ v_o are fully removed in an ideal tank

Removal of a slower particle (ideal tank)

r = \frac{v _{s}}{v _{o}}

r = fraction removed for particles with v_s < v_o

Detention time

t_{d} = \frac{V}{Q}

V = tank volume (m³); Q = flow (m³/h)

Velocity gradient (Camp)

G = \frac{P}{μ V}

P = power input (W); μ = dynamic viscosity (N·s/m²); V = volume (m³); G in s⁻¹

Removal efficiency of an ideal settling tank depends only on surface overflow rate, not on depth or detention time.
Discrete particle settling is Type I, flocculent settling Type II, zone (hindered) settling Type III and compression settling Type IV.
Flocculators are designed for G of about 10–75 s⁻¹ with Gt of about 10⁴–10⁵ and detention of about 10–40 minutes.
Optimum coagulant dose is found by the jar test.
Alum, Al₂(SO₄)₃·18H₂O, is the most common coagulant and works best in the pH range of about 6.5–8.5.

9.4 Filtration, disinfection & softening

Uniformity coefficient

C_{u} = \frac{D _{60}}{D _{10}}

D₆₀, D₁₀ = sizes finer than which 60% and 10% (by mass) of the sand lies; D₁₀ = effective size

Chick's law of disinfection

ln \frac{N _{t}}{N _{0}} = - k t

N₀, N_t = organisms initially and after time t; k = rate constant

Slow sand filters work at about 100–200 L/h/m², while rapid sand filters work at about 3000–6000 L/h/m².
Effective size of sand for slow sand filters is about 0.2–0.3 mm.
Rapid sand filter media have an effective size of about 0.45–0.70 mm and a uniformity coefficient of about 1.3–1.7 (CPHEEO).
In a slow sand filter purification is mainly biological, by the 'schmutzdecke' layer on top of the sand; bacterial removal is about 98–99%.
Chlorine reacts with water to form hypochlorous acid (HOCl), which is about 80 times more effective than the hypochlorite ion (OCl⁻).
Break-point chlorination: beyond the break point, any added chlorine remains as free residual chlorine.
Minimum contact time for chlorine before supply is 30 minutes.
Fresh bleaching powder contains about 30–35% available chlorine (commonly taken as 33%), and it loses strength on storage.
In lime–soda softening, lime removes carbonate hardness and soda ash removes non-carbonate hardness.
Zeolite (base-exchange) softening can produce water of zero hardness; the bed is regenerated with brine (NaCl).
The Nalgonda technique for defluoridation uses alum, lime and bleaching powder.

9.5 Distribution systems & storage

Hazen–Williams formula (SI)

V = 0.849 C R^{0.63} S^{0.54}

V = velocity (m/s); C = Hazen–Williams coefficient; R = hydraulic radius (m); S = slope of energy line

Water hammer pressure rise (sudden closure)

Δ p = ρ c V

c = pressure-wave celerity (m/s); V = velocity destroyed (m/s)

Applies when closure time ≤ 2L/c

The dead-end (tree) system is cheap and simple but water stagnates at the dead ends.
The grid-iron system gives supply from more than one direction, so there are no dead ends and fewer shutdowns.
CPHEEO minimum residual pressure in distribution mains is 7 m for single-storey, 12 m for two-storey and 17 m for three-storey buildings.
Balancing storage of a service reservoir is found by the mass curve method or by the analytical (hourly demand) method.
The Hardy Cross method is used to analyse looped pipe networks.

9.6 Sewage quantity & sewer design

Babbitt peak factor

\frac{Q _{ma x}}{Q _{a v g}} = \frac{5}{P ^{0.2}}

P = population in thousands

Rational method (storm runoff)

Q = \frac{C I A}{360}

Q = peak runoff (m³/s); C = runoff coefficient; I = rainfall intensity (mm/h) for duration = t_c; A = area (ha)

Manning's formula

V = \frac{1}{n} R^{2/3} S^{1/2}

n = Manning's roughness; R = hydraulic radius (m); S = bed slope

R = D/4 for a circular sewer running full or half full

Self-cleansing velocity (Camp–Shields)

V_{s} = \frac{8 β}{f} (G_{s} - 1) g d

β = sediment constant (≈0.04 to start motion, up to 0.8 for full cleansing); f = Darcy friction factor; G_s = specific gravity of particle; d = particle size

Sewage flow is commonly taken as 80% of the water supplied (CPHEEO).
CPHEEO (2013) sewage peak factor is 3.0 up to 20,000 population, 2.5 for 20,000–50,000, 2.25 for 50,000–7.5 lakh and 2.0 above 7.5 lakh.
Minimum velocity in sanitary sewers should be 0.6 m/s at present peak flow and 0.8 m/s at design (ultimate) peak flow (CPHEEO 2013).
Maximum velocity in sewers is generally limited to 3.0 m/s to prevent scouring.
Sanitary sewers are designed to run at most 0.8 full at ultimate peak flow (CPHEEO 2013).
Minimum diameter of a public sewer is 150 mm.
Manning's n for concrete sewers is commonly taken as 0.013.
A circular sewer running half full has the same velocity as when running full but half the discharge.
In a circular sewer the maximum velocity occurs at about 0.81 depth and the maximum discharge at about 0.94 depth.

9.7 BOD, COD, DO & stream pollution

BOD exerted in t days (base e)

B O D_{t} = L_{0} (1 - e^{- k t})

L₀ = ultimate first-stage BOD; k = rate constant (base e, per day)

BOD exerted in t days (base 10)

B O D_{t} = L_{0} (1 - 1 0^{- K_{D} t})

K_D = rate constant (base 10)

k = 2.303 K_D

Temperature correction of BOD rate

k_{T} = k_{20} θ^{T - 20}

T = temperature (°C); θ ≈ 1.047

BOD by dilution method

B O D = \frac{D _{1} - D _{2}}{P}

D₁, D₂ = DO of diluted sample initially and after incubation (mg/L); P = decimal fraction of sample in the mix

Streeter–Phelps oxygen deficit

D_{t} = \frac{k _{d} L _{0}}{k _{r} - k _{d}} (e^{- k_{d} t} - e^{- k_{r} t}) + D_{0} e^{- k_{r} t}

k_d = deoxygenation constant; k_r = reaeration constant (base e); D₀ = initial DO deficit

Critical DO deficit

D_{c} = \frac{k _{d}}{k _{r}} L_{0} e^{- k_{d} t_{c}}

t_c = time to reach critical (minimum DO) point

Self-purification constant f = k_r/k_d

Population equivalent

P E = \frac{BOD load of waste (kg/d)}{per capita BOD (kg/d)}

Per capita BOD commonly taken 0.08 kg/d in textbooks

Standard BOD is the 5-day BOD at 20 °C; in India BIS also specifies a 3-day BOD at 27 °C.
For K_D = 0.1 per day (base 10), BOD₅ is about 68% of the ultimate first-stage BOD.
Nitrification (second-stage BOD) generally begins after about 8–10 days at 20 °C.
COD is measured by refluxing the sample with potassium dichromate in strong sulphuric acid, and COD is normally greater than BOD for the same sample.
Saturation DO of fresh water at 20 °C is about 9.1 mg/L, and it falls as temperature rises.
CPCB general standards for discharge to inland surface water: BOD (3-day, 27 °C) 30 mg/L, COD 250 mg/L, TSS 100 mg/L, pH 5.5–9.0.
CPCB Class A water (drinking after disinfection only) requires DO ≥ 6 mg/L and BOD ≤ 2 mg/L.
CPCB Class B water (outdoor bathing) requires DO ≥ 5 mg/L and BOD ≤ 3 mg/L.

9.8 Sewage treatment & sludge

Food-to-microorganism ratio

\frac{F}{M} = \frac{Q S _{0}}{V X}

Q = flow (m³/d); S₀ = influent BOD (mg/L); V = aeration tank volume (m³); X = MLSS (mg/L); F/M in per day

Mean cell residence time (sludge age)

θ_{c} = \frac{V X}{Q _{w} X _{r} + ( Q - Q _{w} ) X _{e}}

Q_w = waste sludge flow; X_r = MLSS of return/waste sludge; X_e = effluent suspended solids

Sludge volume index

S V I = \frac{V _{30} \times 1000}{X}

V₃₀ = settled sludge volume after 30 min (mL/L); X = MLSS (mg/L); SVI in mL/g

Recirculation ratio

\frac{Q _{r}}{Q} = \frac{X}{X _{r} - X}

Q_r = return sludge flow; X_r ≈ 10⁶/SVI (mg/L)

NRC trickling filter efficiency (first stage)

E = \frac{100}{1 + 0.4432 \frac{W}{V F}}

E = BOD removal (%); W = BOD load (kg/d); V = filter volume (m³); F = recirculation factor

Recirculation factor

F = \frac{1 + R}{( 1 + 0.1 R ) ^{2}}

R = recirculation ratio Q_r/Q

Sludge volume vs moisture

\frac{V _{2}}{V _{1}} = \frac{100 - p _{1}}{100 - p _{2}}

p₁, p₂ = moisture contents (%) before and after; V₁, V₂ = sludge volumes

Velocity through bar screens should not exceed about 0.8–1.0 m/s at peak flow so that screenings are not forced through.
Grit chambers remove inorganic particles of about 0.2 mm size (specific gravity 2.65) at a horizontal velocity of about 0.3 m/s.
Primary sedimentation removes roughly 30–35% of BOD and 50–65% of suspended solids.
Conventional activated sludge plants work at F/M of about 0.2–0.4 per day, MLSS 1500–3000 mg/L and HRT 4–8 h.
Extended aeration works at a low F/M of about 0.05–0.15 per day and an HRT of about 18–36 h.
An SVI of about 50–150 mL/g indicates good settling; above about 200 mL/g indicates sludge bulking, often due to filamentous organisms.
High-rate trickling filters use recirculation, while low-rate filters normally do not; Psychoda (filter) flies are a nuisance in low-rate filters.
Septic tank (IS 2470 Part 1): detention commonly 24 h, minimum width 750 mm, minimum liquid depth 1.0 m and length 2–4 times the width.
Oxidation (stabilisation) ponds are about 1.0–1.5 m deep and depend on symbiosis: algae release oxygen and bacteria release CO₂.
Mesophilic sludge digestion occurs at about 30–38 °C and thermophilic digestion at about 50–57 °C.
Digester gas contains about 65–70% methane, the rest being mainly CO₂.

9.9 Air, noise & solid waste

Gaussian plume – ground-level concentration

C (x, y, 0) = \frac{Q}{π u σ _{y} σ _{z}} exp (- \frac{y ^{2}}{2 σ _{y}^{2}}) exp (- \frac{H ^{2}}{2 σ _{z}^{2}})

Q = emission rate (g/s); u = wind speed at stack height (m/s); σ_y, σ_z = dispersion coefficients (m); H = effective stack height (m)

Includes ground reflection

ppm to µg/m³ (25 °C, 1 atm)

C_{μg / m^{3}} = \frac{C _{pp m} \times M \times 1000}{24.45}

M = molecular weight (g/mol); 24.45 L = molar volume at 25 °C

ESP collection efficiency (Deutsch–Anderson)

η = 1 - e^{- w A / Q}

w = drift velocity (m/s); A = collecting plate area (m²); Q = gas flow (m³/s)

Sound pressure level

L_{p} = 20 lo g_{10} \frac{p}{p _{0}}

p = rms sound pressure; p₀ = 20 µPa

Sound intensity level L = 10 log(I/I₀), I₀ = 10⁻¹² W/m²

Adding sound levels

L = 10 lo g_{10} \sum 1 0^{L_{i} /10}

L_i = individual levels in dB

Dry adiabatic lapse rate is about 9.8 °C per km; the standard environmental lapse rate is about 6.5 °C per km.
Looping plumes occur under super-adiabatic (unstable) conditions, coning under neutral conditions and fanning under an inversion.
India's AQI (2014) covers eight pollutants: PM10, PM2.5, NO₂, SO₂, CO, O₃, NH₃ and Pb.
AQI bands: Good 0–50, Satisfactory 51–100, Moderate 101–200, Poor 201–300, Very Poor 301–400, Severe 401–500.
NAAQS 2009: PM10 is 60 µg/m³ annual and 100 µg/m³ (24 h); PM2.5 is 40 µg/m³ annual and 60 µg/m³ (24 h).
NAAQS 2009 (industrial, residential, rural and other areas): SO₂ and NO₂ are 50 and 40 µg/m³ annual respectively, and both are 80 µg/m³ for 24 h.
Two equal noise sources together are 3 dB louder than one alone.
For a point source, sound level drops by 6 dB for every doubling of distance; for a line source it drops by 3 dB.
Noise Rules 2000 limits Leq dB(A) day/night: industrial 75/70, commercial 65/55, residential 55/45, silence zone 50/40.
SWM Rules 2016 require segregation at source into three streams: biodegradable (wet), non-biodegradable (dry) and domestic hazardous waste.
Under SWM Rules 2016, non-recyclable waste with calorific value of 1500 kcal/kg or more shall not be landfilled but used for energy recovery or co-processing.
Landfill composite liner (SWM Rules 2016): 1.5 mm HDPE geomembrane over 90 cm compacted clay with permeability not more than 10⁻⁷ cm/s.
Optimum C/N ratio for composting is about 25–30.
Under the Plastic Waste Management Rules, plastic carry bags must be at least 120 µm thick (from 31 December 2022).
Bio-Medical Waste Rules 2016 colour coding: yellow for anatomical waste, red for contaminated recyclable plastics, white puncture-proof containers for sharps and blue for glassware.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.