Environmental Engineering-I (CE-602) - Unit 1 Short Notes

UNIT 1: WATER SUPPLY AND WASTEWATER ENGINEERING

1. POPULATION FORECASTING AND WATER DEMAND ESTIMATION

Population Projection Methods

• Arithmetic Increase Method

  • Assumes constant absolute growth per decade.

  • Formula: $$\displaystyle P_n = P_0 + n \cdot \bar{x} $$, where $$\displaystyle \bar{x} = \frac{\sum (P_n - P_{n-1})}{n} $$ is average increase.

  • Suitable for: Large, old cities with mature growth.

• Incremental Increase Method

  • Assumes growth rate decreases over time. Uses average of incremental increases.

  • Formula: $$\displaystyle P_n = P_0 + n \cdot \bar{x} + \frac{n(n-1)}{2} \cdot \bar{y} $$, where $\bar{y}$ is average of incremental increases.

  • Suitable for: Cities with slowing growth rate.

• Logistic Curve Method (Frequently Examined)

  • Concept: S-shaped curve representing growth with a saturation limit ($$\displaystyle P_s $$). Growth rate is proportional to $$\displaystyle P(P_s - P) $$.

  • Logistic Equation:

$$P_t = \frac{P_s}{1 + e^{a + b t}}$$

*   **Parameters to Determine**:

    1.  **Saturation Population ($$\displaystyle P_s $$)**: Extrapolated from plotted population vs. time curve or from three known populations.

    2.  **Coefficients (a, b)**: Solved using three known data points $$\displaystyle (t_0, P_0) $$, $$\displaystyle (t_1, P_1) $$, $$\displaystyle (t_2, P_2) $$.

        *   From $$\displaystyle P_t = \frac{P_s}{1 + e^{a+bt}} $$, rearrange: $$\displaystyle \ln\left(\frac{P_s}{P_t} - 1\right) = a + b t $$.

        *   Solve simultaneous equations for `a` and `b`.

*   **Past Paper Application**: Given populations at $$\displaystyle t_0=0 $$, $$\displaystyle t_1=20 $$, $$\displaystyle t_2=40 $$ years, find $$\displaystyle P_s $$, `a`, `b`, and predict $P$ at $$\displaystyle t_3=60 $$.

[!TIP] Exam Tip: For logistic curve, first plot population vs. time to visually estimate $$\displaystyle P_s $$. Use the linearized form $$\displaystyle \ln\left(\frac{P_s}{P} - 1\right) = a + bt $$ for solving coefficients.

Per Capita Water Demand

• Factors Affecting Demand:

  • Climate: Temperature, rainfall, humidity.

  • Socio-economic: Living standards, meterization, pressure.

  • System Losses: Leakage, unauthorized connections.

  • Infrastructure: Industrial/commercial activity, public facilities.

  • Water-borne diseases, fire protection needs.

• Components of Total Water Demand:

  • Domestic (drinking, cooking, bathing, sanitation)

  • Industrial/Commercial

  • Public Use (street washing, gardening, public taps)

  • Fire Demand (separate, not included in average daily)

  • System Losses (theft, leakage, measurement errors)

Fire Demand

• Importance: Critical for designing distribution mains and storage for firefighting.

• Influencing Factors: Population density, building type (height, occupancy), street width, water main pressure, fire department facilities.

• Estimation Formulas (Frequently Examined):

  1. Kuchling's Formula:

$$Q = 3182 \sqrt{P} \text{ (litres/min)}$$

where $P$ = population in thousands.

    \boxed{Q = 3182 \sqrt{P}}

2.  **Boston's Formula**:

$$Q = \frac{15120}{P^{0.1}} \text{ (litres/min)} \quad \text{or} \quad Q = 100 \sqrt{P} \text{ (m³/hr)}$$

    \boxed{Q = 100 \sqrt{P} \ (\text{m³/hr})}

3.  **National Board/Writer's Formula (US)**:

$$Q = 4637 \sqrt{P} \left(1 - 0.01 \sqrt{P}\right) \text{ (litres/min)}$$

    \boxed{Q = 4637 \sqrt{P} (1 - 0.01 \sqrt{P})}

Variations in Water Demand

• Average Daily Demand: Total annual consumption / 365.

• Maximum Daily Demand: 1.2 to 1.8 times average daily. Used for treatment plant design.

• Maximum Hourly Demand: 1.5 to 3.0 times average daily. Used for distribution system and pumping capacity design.

• Peak Factor: Ratio of maximum hourly to average hourly demand.

2. WATER SOURCES AND INTAKE STRUCTURES

Sources of Water

Infiltration Gallery: A horizontal perforated pipe laid in gravel bed below water table to collect seepage water. Used where aquifer is shallow and permeable.

Intake Structures

• Types:

  • Canal Intake: For pumped canals. Simple grating at canal head.

  • Reservoir Intake: Multi-level towers to draw water at desired depth (temperature/quality control).

  • River Intake: Tower/pump house on riverbank or submerged. Must withstand floods, scour, debris.

• Selection Factors:

  • Location (upstream of pollution, stable bank).

  • Depth (below river bed to avoid silt, above low water level).

  • Protection (against floods, ice, debris, vessels).

  • Hydrology (flow variation, sediment load).

  • Proximity to treatment plant.

3. WATER TREATMENT PROCESSES

Unit Operations in Water Treatment

Screening → Coagulation/Flocculation → Sedimentation → Filtration → Disinfection.

Coagulation and Flocculation

• Theory: Add chemicals (coagulants) to neutralize negative charges on colloidal particles, forming micro-flocs (coagulation). Gentle mixing promotes collision to form larger, settleable flocs (flocculation).

• Common Coagulants & Reactions:

  1. Aluminium Sulfate (Alum): $$\displaystyle \text{Al}_2(\text{SO}_4)_3 \cdot 18\text{H}_2\text{O} \rightarrow 2\text{Al(OH)}_3 \text{( gelatinous)} + 3\text{SO}_4^{2-} + 18\text{H}_2\text{O} $$

     • Al(OH)₃ enmeshes particles.

  2. Ferric Chloride/Sulfate: $$\displaystyle \text{FeCl}_3 + 3\text{H}_2\text{O} \rightarrow \text{Fe(OH)}_3 + 3\text{HCl} $$

  3. Chlorinated Copper: $$\displaystyle \text{CuCl}_2 + \text{Ca(OH)}_2 \rightarrow \text{Cu(OH)}_2 + \text{CaCl}_2 $$

• Factors Affecting Coagulation:

  • pH (optimal: 6.5-7.5 for alum)

  • Temperature (higher = faster)

  • Mixing intensity/duration (rapid mix for dispersion, slow mix for flocculation)

  • Coagulant dose

  • Raw water characteristics (turbidity, alkalinity)

• Coagulant Dose Calculation (considering alkalinity):

  • Alum reaction consumes alkalinity: $$\displaystyle \text{Al}_2(\text{SO}_4)_3 \cdot 14\text{H}_2\text{O} + 3\text{Ca(HCO}_3)_2 \rightarrow 2\text{Al(OH)}_3 + 3\text{CaSO}_4 + 14\text{H}_2\text{O} + 6\text{CO}_2 $$

  • 1 mg/l alum ≈ 0.5 mg/l alkalinity (as CaCO₃) consumed.

  • If raw water alkalinity < required, add lime/soda ash.

Sedimentation

• Theory: Stokes' Law (Frequently Examined)

  • Assumptions: Spherical particle, laminar flow (Re < 0.2), no wall effect, particle density >> fluid density, discrete settling.

  • Terminal Settling Velocity ($$\displaystyle V_s $$):

$$V_s = \frac{g(\rho_p - \rho) d^2}{18 \mu}$$

    where $g$ = gravity, $$\displaystyle \rho_p $$ = particle density, $\rho$ = fluid density, $d$ = diameter, $\mu$ = dynamic viscosity.

*   **Design Parameter - Overflow Rate (OFR)**: $$\displaystyle OFR = \frac{Q}{A} $$ (m³/m²/day). Set equal to $$\displaystyle V_s $$ of smallest particle to be removed (e.g., 0.3 mm sand, $$\displaystyle V_s \approx 0.5 $$ m/hr).

*   **Detention Time ($$\displaystyle t_d $$)**: $$\displaystyle t_d = \frac{V}{Q} = \frac{A \cdot D}{Q} $$, typically 2-4 hours.

• Temperature Correction:

  • $$\displaystyle V_{s2} = V_{s1} \cdot \frac{\mu_1}{\mu_2} $$ (since $\rho$ change negligible).

  • Viscosity of water decreases with temperature → $$\displaystyle V_s $$ increases.

  • Use viscosity ratio from tables or $$\displaystyle \mu \propto \frac{1}{T} $$ approx.

• Design Dimensions:

  • Length:Width (L:B) = 2:1 to 5:1.

  • Depth: 3-4.5 m.

  • Rectangular tanks common.

Filtration

• Slow Sand Filter:

  • Construction: Concrete tank, graded sand (0.3-1.0 m depth) over gravel, underdrain system.

  • Working: Water flows by gravity (~0.2 m/hr). Top 2-3 cm forms Schmutzdecke (biological layer) which does most filtration. Requires scraping when head loss increases.

  • Advantages: Simple, low cost, excellent quality, no chemicals.

  • Disadvantages: Large area, slow, frequent cleaning, skilled operation.

  • DiagramSEARCH: slow sand filter diagram schmutzdecke

• Rapid Sand Filter:

  • Construction: Similar tank, sand depth 0.6-0.9 m, gravel underdrain with strainers.

  • Working: Higher rate (5-15 m³/m²/hr). Requires backwashing (reverse flow with air scour + water) to clean. Cycle: filter 24-72 hrs, backwash 5-10 min.

  • Design Parameter: Filtration rate.

  • Backwash Calculations:

    • Backwash Velocity ($$\displaystyle V_b $$): Minimum velocity to expand bed. $$\displaystyle V_b = C \sqrt{(G-1) d} $$ (from Ergun/White formula), where $C$ depends on shape factor $\phi$ and porosity $n$.

    • Expanded Depth ($$\displaystyle D_e $$): $$\displaystyle D_e = \frac{D (1-n)}{1-n_e} $$, where $D$ = original depth, $n$ = original porosity, $$\displaystyle n_e $$ = expanded porosity.

  • DiagramSEARCH: rapid sand filter backwash air scour

• Comparison:

• Forces in Filtration: Sedimentation, straining, inertial impaction, interception, diffusion, adsorption, biological action.

Disinfection

• Methods: Chlorination (most common), Ozonation, UV radiation, Chloramines.

• Break Point Chlorination (Frequently Examined):

  • Purpose: To destroy pathogens and oxidize organic/inorganic matter, leaving free chlorine residual.

  • Curve: Plot of chlorine dose vs. residual chlorine.

    1. Initial Demand: Chlorine reacts with reducing agents (Fe²⁺, Mn²⁺, H₂S, NH₃) → no residual.

    2. Break Point: Point where all demand satisfied; residual starts to appear.

    3. After Break Point: Further chlorine appears as free residual (HOCl/OCl⁻).

  • Significance: Ensures adequate disinfection and prevents regrowth in distribution.

  • DiagramSEARCH: break point chlorination curve

Water Softening

• Soda Lime Process:

  • Reaction: $$\displaystyle \text{Ca(HCO}_3)_2 + \text{Ca(OH)}_2 \rightarrow 2\text{CaCO}_3 \downarrow + 2\text{H}_2\text{O} $$

  • $$\displaystyle \text{Mg(HCO}_3)_2 + 2\text{Ca(OH)}_2 \rightarrow \text{Mg(OH)}_2 \downarrow + 2\text{CaCO}_3 \downarrow + 2\text{H}_2\text{O} $$

  • Lime dose based on alkalinity & hardness.

  • Advantages: Cheap, simple.

  • Disadvantages: Produces large sludge; non-carbonate hardness not removed; pH adjustment needed.

• Ion Exchange Method:

  • Process: Water passed through zeolite/resin beads. Ca²⁺, Mg²⁺ exchanged for Na⁺ (or H⁺).

  • Regeneration: With concentrated NaCl (for Na⁺) or acid (for H⁺).

  • Advantages: Very soft water (< 5 mg/l); compact; no sludge.

  • Disadvantages: High capital/operational cost; produces saline wastewater; not for large municipal scale.

4. WATER DISTRIBUTION SYSTEMS

Distribution Network Layouts (Sketches Important)

Dead-end systems are simpler but less reliable.

Distribution Reservoirs

• Types:

  • Storage Reservoir: Raw water storage (impounding, off-stream).

  • Service Reservoir: Treated water storage within distribution system. Provides:

    • Balancing: Equalize hourly demand variations.

    • Emergency: Fire, breakdowns.

    • Pressure Maintenance: Elevation provides head.

  • Balancing Reservoir: Specifically for hourly demand equalization (24-hr pumping).

• Location Criteria:

  • Central to distribution area.

  • High ground for gravity flow.

  • Geologically stable, safe from contamination.

  • Proximity to water treatment plant outlet.

• Storage Capacity for Balancing Reservoir (Mass Curve Method for 24-hr Pumping):

  1. Plot cumulative pumping rate (constant if 24-hr pumping) vs. time (24 hrs).

  2. Plot cumulative demand rate (hourly variations) vs. same time.

  3. Storage Required = Vertical distance between the two cumulative curves at any point.

  4. Maximum Storage = Maximum vertical gap.

  5. Minimum Storage = Point where cumulative pumping = cumulative demand (overflow point).

  \boxed{\text{Storage} = \max\left[ \int_0^t (Q_{demand} - Q_{pump}) , dt \right]}

  DiagramCANVAS: Mass curve diagram with cumulative pumping (straight line) and cumulative demand (curved), showing storage as vertical difference

Pumps and Pumping Stations

• Types: Centrifugal (most common), reciprocating, rotary.

• Power Calculations (Example-Based):

  • Water Horsepower (WHP): $$\displaystyle WHP = \frac{Q \cdot H}{75} $$ (metric) or $$\displaystyle \frac{Q \cdot H}{3960} $$ (US), where $Q$ = discharge (LPS or GPM), $H$ = total head (m or ft).

  • Brake Horsepower (BHP): $$\displaystyle BHP = \frac{WHP}{\eta_p \cdot \eta_m} $$, where $$\displaystyle \eta_p $$ = pump efficiency, $$\displaystyle \eta_m $$ = motor efficiency.

  • Total Head ($H$): $$\displaystyle H = H_{static} + H_{friction} + H_{velocity} + H_{pressure} $$.

  • Friction Loss (Darcy-Weisbach/Hazen-Williams): $$\displaystyle h_f = f \frac{L}{D} \frac{V^2}{2g} $$.

Distribution System Appurtenances

• Valves: Gate/ sluice (on/off), check (prevent backflow), pressure reducing, air release, scour.

• Fire Hydrants: Post/ barrel type.

• Water Meters: For billing, leak detection.

• Air Valves: Release air pockets (prevent cavitation, maintain flow).

• Pressure Reducing Valves (PRV): Protect low-lying areas from high pressure.

5. WATER QUALITY STANDARDS AND ANALYSIS

Water Quality Parameters

Water Quality Standards

• BIS (IS 10500:2012): Indian drinking water standards. Specifies limits for parameters (e.g., Turbidity < 1 NTU, pH 6.5-8.5, Total Hardness < 200 mg/l CaCO₃, E. coli absent/100 ml).

• WHO Guidelines: International reference. More stringent for some chemicals.

• CPCB Standards: For effluent discharge, not drinking water.

Waterborne Diseases

Microbiological Analysis

• Coliform Index: Number of coliform bacteria per 100 ml. Total coliform indicates general pollution; Fecal coliform/E. coli indicates fecal pollution.

• Most Probable Number (MPN) Test (Frequently Examined):

  • Procedure: Serial dilution (3-tube, 5-tube, or 10-tube) in lactose broth with Durham tubes. Incubate at 35-37°C for 48 hrs. Count positive tubes (gas production). Use MPN table to estimate bacteria count per 100 ml.

  • Significance: Statistical estimate of coliform density. Standard method for water quality assessment.

6. WASTEWATER CHARACTERISTICS AND ANALYSIS

Wastewater Definition and Sources

• Definition: Used water from community/industry, carrying dissolved/undissolved solids, organic matter, pathogens.

• Sources:

  • Domestic: From sinks, toilets, bathing.

  • Industrial: Process-specific pollutants.

  • Stormwater: Runoff (carries debris, oil, metals).

  • Groundwater infiltration.

Characteristics of Sewage

Significance of Wastewater Analysis

• Determines treatment process selection and design loading (BOD, COD, TSS).

• Assesses treatability (biodegradability BOD/COD ratio).

• Evaluates environmental impact of discharge (on receiving water, soil).

• Regulatory compliance (effluent standards).

Decomposition of Organic Matter

1. Aerobic Stage: Aerobic bacteria oxidize organics → CO₂, H₂O, nitrates, stable organics. Produces BOD.

2. Anaerobic Stage (when DO depleted):

   • Acid fermentation → Volatile Fatty Acids (VFA), alcohols, CO₂, H₂.

   • Methane fermentation → CH₄, CO₂, H₂S (rotten egg smell).

   • Putrefaction → H₂S, NH₃, mercaptans ( foul odor).

   • Gases: CH₄ (50-70%), CO₂ (30-50%), H₂S, N₂.

Population Equivalent (PE)

• Concept: Expresses industrial wastewater strength in terms of number of persons contributing equivalent organic load.

• Calculation:

$$PE = \frac{\text{Industrial BOD load (kg/day)}}{\text{Standard domestic BOD load per person (kg/day)}}$$

• Standard: Often 0.06 kg BOD₅/person/day (60 g) or 0.08 kg (80 g) depending on country.

• Example: Industry with 300 kg/day BOD → $$\displaystyle PE = 300 / 0.06 = 5000 $$.

Relative Stability

• Concept: Measure of the degree of stabilization (decomposition) of organic matter in sewage, expressed as percentage.

• Determination: Rideal-Stewart Test or Modified Iodine Method.

• Application: Used to assess sewage age in treatment ponds or to determine required detention time in sedimentation tanks.

7. SEWERAGE SYSTEMS

Types of Sewer Networks

Sewer Appurtenances (Functions & Sketches Important)

• Manholes: For inspection, cleaning, junction, change of direction/gradient. Sections: chamber, benching, steps, cover.

  • DiagramSEARCH: manhole diagram parts

• Lamp Holes: Simple openings for visual inspection (no entry). Used on straight runs.

• Inspection Chambers: Similar to manholes but shallower, for domestic connections.

• Catch Basins / Gully Pots: Collect street runoff, trap silt/debris. Located at low points/kerbs.

• Flushing Tanks: Provided at dead ends to flush deposits. Manual or automatic.

• Street Inlets: Openings in gutters to admit runoff into catch basins.

Sewer Design and Hydraulics

• Variation in Sewage Flow:

  • Average Daily Flow: $$\displaystyle Q_{avg} = \text{Water supply} \times \text{Return Factor} (0.7-0.8) $$.

  • Dry Weather Flow (DWF): $$\displaystyle Q_{DWF} = Q_{avg} + \text{groundwater infiltration} $$.

  • Maximum Flow / Peak Hourly Flow: $$\displaystyle Q_{max} = Q_{DWF} \times \text{Peak Factor} (1.5-3.0) $$.

  • Storm Flow (Combined): Rational formula $$\displaystyle Q = \frac{C i A}{360} $$ (liters/sec).

• Hydraulic Design (Manning's Formula):

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

*   $$\displaystyle Q = A \cdot V $$

*   For circular sewer flowing **partially full**:

    *   $d/D$ (depth/diameter) ratio chosen (usually 0.5-0.75 for self-cleaning velocity 0.6-0.9 m/s).

    *   Use tables or charts (like Hinds' chart) to find $A$, $R$, $V$ for given $d/D$ and $n$.

• Design Parameters:

  • Minimum Velocity: 0.6 m/s (self-cleaning at DWF).

  • Maximum Velocity: 2.5-3.0 m/s (prevent scouring/settlement).

  • Diameter: Based on $$\displaystyle Q_{max} $$ and gradient.

  • L/B Ratio: Not directly applicable; governed by cover depth and gradient.

  • Minimum Cover: 0.7-1.0 m to prevent damage.

Sewer Construction

• Techniques:

  • Trenching: Open cut (common), tunnelling (under roads/railways), pipe-jacking.

  • Laying: Bed preparation (sand/lean concrete), lowering pipe, jointing (rubber gasket, cement mortar).

  • Testing: Water test (fill with water, measure loss over time), Air test (pressurize, measure pressure drop).

• Quality Control & Safety:

  • QC: Check grade (laser), alignment, joint integrity, material defects.

  • Safety: Shoring/trench boxes, sloping, atmospheric testing (H₂S, CH₄), PPE, barricading, dewatering.

8. WASTEWATER TREATMENT

Activated Sludge Process (Detailed)

• Process Flow: Primary sedimentation → Aeration Tank (mixed liquor) → Secondary Clarifier → Treated effluent. Sludge from clarifier recycled to aeration tank; excess wasted.

• Key Components:

  • Aeration Tank: Provides oxygen and mixing. Types: Plug flow (long rectangular), Complete mix (circular/oval).

  • Secondary Clarifier: Settles biomass (activated sludge). Surface loading rate ~ 100 m³/m²/day.

  • Sludge Recycle: Maintains high MLSS (2000-4000 mg/l) in aeration tank.

• Process Control Parameters:

  • Food to Microorganism Ratio (F/M): $$\displaystyle \frac{\text{Influent BOD (kg/day)}}{\text{Mass of MLVSS (kg)}} $$. Typical: 0.2-0.4 kg BOD/kg MLVSS·day.

  • Mean Cell Residence Time (SRT/θc): $$\displaystyle \frac{\text{Mass of MLVSS in system (kg)}}{\text{Mass of WASTED sludge per day (kg/day)}} $$. Typical: 5-15 days.

• Modifications:

  • Sequencing Batch Reactor (SBR): Batch operation in single tank (fill, react, settle, decant, idle).

  • Oxidation Ditches: Circular channel with surface aerators, extended aeration (SRT > 20 days).

  • Extended Aeration: Long aeration period, complete oxidation, less sludge production.

Natural Treatment Methods

• Land Treatment:

  • Slow Rate (Irrigation): Wastewater applied to crops/forests at agronomic rates. Primary treatment needed.

  • Rapid Infiltration: High-rate application to permeable soils; groundwater recharge. Requires secondary treatment.

  • Overland Flow: Application to sloped terraces; treatment by vegetation and soil. Secondary treatment needed.

• Dilution in Surface Water Bodies:

  • Concept: Discharge treated/untreated sewage into flowing river. Self-purification (deoxygenation/re-aeration) dilutes and oxidizes pollutants.

  • Limitations: Requires adequate dilution factor (stream flow >> sewage flow); downstream pollution; not acceptable for untreated sewage; depends on river self-purification capacity (Oxygen Sag Curve).

9. STREAM POLLUTION AND SELF-PURIFICATION

Oxygen Sag Curve (Frequently Examined - Sketch & Explanation)

• Sketch: Plot of Dissolved Oxygen (DO) concentration vs. Distance downstream of pollution source.

  • Curve Features:

    1. Critical Point (C): Minimum DO. Location of maximum pollution effect.

    2. Deficit (D): $$\displaystyle D = D_s - DO $$, where $$\displaystyle D_s $$ = saturation DO at that temperature.

    3. Deoxygenation Curve: Exponential decay of organic matter → oxygen consumption ($$\displaystyle L_t $$).

    4. Reaeration Curve: Exponential approach to saturation DO.

  • DiagramCANVAS: Oxygen sag curve with axes: Distance (x) vs DO (mg/l). Show: upstream DO (saturated), drop after discharge, minimum (critical point), recovery to saturation. Label: D (deficit), L (ultimate BOD), Ds (saturation DO), Ka (reaeration), Kd (deoxygenation).

[!TIP] Exam Tip: Always sketch the sag curve with all labeled components: DO, D, L, Ds, critical point, deoxygenation curve, reaeration curve.

Streeter-Phelps Equation (Frequently Examined)

• Derivation:

  • Deoxygenation rate (first-order): $$\displaystyle \frac{dL_t}{dt} = K_d (L_t - L_0) $$, where $$\displaystyle L_t $$ = BOD remaining at time t, $$\displaystyle L_0 $$ = ultimate BOD.

  • Solution: $$\displaystyle L_t = L_0 (1 - e^{-K_d t}) $$.

  • Reaeration rate: $$\displaystyle \frac{dD_t}{dt} = K_d L_t - K_a D_t $$, where $$\displaystyle D_t $$ = oxygen deficit at time t.

  • Substitute $$\displaystyle L_t $$ and solve (using integrating factor) for Deficit ($$\displaystyle D_t $$):

$$D_t = \frac{K_d L_0}{K_a - K_d} \left( e^{-K_d t} - e^{-K_a t} \right) + D_0 e^{-K_a t}$$

    where $$\displaystyle D_0 $$ = initial deficit at $$\displaystyle t=0 $$.

*   **Critical Deficit ($$\displaystyle D_{crit} $$)** and **Critical Time ($$\displaystyle t_{crit} $$)**:

$$t_{crit} = \frac{1}{K_a - K_d} \ln \left( \frac{K_a}{K_d} \right)$$

$$D_{crit} = \frac{K_d L_0}{K_a - K_d} \left( \left(\frac{K_a}{K_d}\right)^{-K_d/(K_a-K_d)} - \left(\frac{K_a}{K_d}\right)^{-K_a/(K_a-K_d)} \right) + D_0 \left(\frac{K_a}{K_d}\right)^{-K_a/(K_a-K_d)}$$

• Temperature Effects:

  • Rate constants $$\displaystyle K_d $$ and $$\displaystyle K_a $$ are temperature-dependent.

  • Use temperature coefficient $\theta$:

$$K_{T2} = K_{T1} \cdot \theta^{(T_2 - T_1)/10}$$

*   Typical $\theta$: $$\displaystyle K_d $$: 1.135 (sewage), $$\displaystyle K_a $$: 1.024 (natural water).

*   **Past Paper Application**: Given $$\displaystyle K_d $$ at 20°C, find $$\displaystyle K_d $$ at 15°C using $$\displaystyle \theta=1.135 $$.

Disposal Standards and River Classification

• Objective: Prevent oxygen sag below critical level for aquatic life (typically > 4-5 mg/l DO).

• River Classification (CPCB/India):

  • Class A: Drinking water source (without conventional treatment).

  • Class B: Outdoor bathing.

  • Class C: Drinking water after conventional treatment.

  • Class D: Wildlife, fisheries.

  • Class E: Irrigation, industrial cooling.

• Standards: Set maximum allowable BOD, DO, fecal coliform for discharge into different classes. Use Streeter-Phelps to ensure compliance.