Activated Sludge (CSTR)

Aerobic biological treatment using suspended growth microorganisms in continuously stirred tank reactors

Overview

The activated sludge process is the most widely used biological treatment method for municipal and industrial wastewater. It employs suspended-growth microorganisms (activated sludge) in an aerated reactor to oxidize organic matter and nutrients. The process name derives from the "activated" nature of the bacterial biomass, which is continuously recycled from the secondary clarifier to maintain a high concentration of active microorganisms in the reactor.

The process consists of an aeration tank where wastewater is mixed with activated sludge and air/oxygen, followed by a secondary clarifier where the biomass settles and is separated from the treated effluent. A portion of the settled sludge (return activated sludge, RAS) is recycled to the aeration tank to maintain the desired biomass concentration, while excess sludge (waste activated sludge, WAS) is removed to control the system's solids retention time (SRT) or sludge age.

The activated sludge process can be configured in various flow patterns including plug flow, complete mix (CSTR), step feed, and contact stabilization. This documentation focuses on the complete-mix CSTR configuration, which provides uniform conditions throughout the reactor and is described by relatively simple mass balance equations. The process can be operated at different loading rates (conventional, high-rate, extended aeration) and can be modified for nitrogen and phosphorus removal (Modified Ludzack-Ettinger, A2O, etc.).

Theory and Equations

Basic Mode

Simple removal efficiency approach with oxygen demand estimation:

$$\text{BOD}{out} = \text{BOD} \times (1 - \eta_{BOD})$$

$$\text{TSS}{out} = \text{TSS} = \text{constant}$$

$$\text{NH}4\text{-N} = \text{NH}4\text{-N} \times (1 - \eta_{NH4})$$

Oxygen Requirement (simplified):

$$\text{O}2 = Q \times (\text{BOD}) \times f_{ox} + Q \times X_v \times k_d$$

Where:

• $\eta_{BOD}$ = BOD removal efficiency (typically 0.85-0.95)
• $\eta_{NH4}$ = Ammonia removal efficiency (typically 0.90-0.98)
• $f_{ox}$ = fraction of BOD oxidized (typically 0.5-0.6)
• $k_d$ = endogenous decay rate (1/d)
• $X_v$ = volatile suspended solids concentration (mg/L)

Sludge Production (simplified):

$$\text{WAS mass} = Q \times Y_{obs} \times \text{BOD}_{removed}$$

Where:

• $Y_{obs}$ = observed yield coefficient (typically 0.4-0.6 kg VSS/kg BOD)

Advanced Mode

CSTR Mass Balance for Substrate (BOD or COD):

For a complete-mix reactor at steady state:

$$\frac{dS}{dt} = 0 = \frac{Q \cdot S_0 - Q \cdot S}{V} - r_{su}$$

Solving for substrate removal rate:

$$r_{su} = \frac{Q(S_0 - S)}{V}$$

Monod Kinetics for Substrate Utilization:

$$r_{su} = \frac{\mu_{max} \cdot X \cdot S}{K_s + S \cdot Y}$$

Where:

• $\mu_{max}$ = maximum specific growth rate (1/d)
• $X$ = biomass concentration (MLVSS, mg/L)
• $S$ = substrate concentration (mg/L)
• $K_s$ = half-saturation constant (mg/L)
• $Y$ = true yield coefficient (kg VSS/kg BOD)

Effluent Substrate Concentration:

$$S = \frac{K_s(1 + k_d \cdot \theta_c)}{\theta_c(\mu_{max} - k_d) - 1}$$

Where:

• $\theta_c$ = solids retention time (SRT, days)
• $k_d$ = endogenous decay coefficient (1/d)

MLVSS Concentration:

$$X = \frac{\theta_c \cdot Y \cdot Q(S_0 - S)}{V(1 + k_d \cdot \theta_c)}$$

Oxygen Requirement (detailed):

$$\text{O}_2 = Q(S_0 - S) - 1.42 \cdot P_x$$

Where:

• $P_x$ = net biomass production (kg VSS/d)
• 1.42 = conversion factor for VSS to COD

Net Biomass Production:

$$P_x = \frac{Q \cdot Y \cdot (S_0 - S)}{1 + k_d \cdot \theta_c}$$

Waste Activated Sludge (WAS) Flow:

$$Q_w = \frac{V \cdot X}{\theta_c \cdot X_w}$$

Where:

• $Q_w$ = WAS flow rate (m³/d)
• $X_w$ = WAS concentration (mg/L)

Food-to-Microorganism Ratio:

$$F/M = \frac{Q \cdot S_0}{V \cdot X}$$

Volumetric Organic Loading:

$$\text{VOL} = \frac{Q \cdot S_0}{V}$$

Nitrification (if applicable):

$$r_{nit} = \frac{\mu_{max,AOB} \cdot X_{AOB} \cdot NH_4}{K_{NH4} + NH_4} \times \frac{DO}{K_{O2} + DO}$$

Where:

• $X_{AOB}$ = ammonia-oxidizing bacteria concentration
• $K_{NH4}$ = half-saturation constant for ammonia
• $DO$ = dissolved oxygen concentration

Model Parameters

Basic Mode Parameters

Parameter	Symbol	Unit	Default	Range	Description
BOD Removal	η_BOD	-	0.90	0.85-0.98	BOD removal efficiency
NH₄ Removal	η_NH4	-	0.95	0.85-0.99	Ammonia removal efficiency
TSS Effluent	TSS_eff	mg/L	15	5-30	Effluent TSS concentration
Observed Yield	Y_obs	kg/kg	0.5	0.4-0.7	Observed sludge yield
Oxygen Factor	f_ox	-	0.55	0.45-0.65	Fraction of BOD oxidized

Advanced Mode Parameters

Parameter	Symbol	Unit	Default	Range	Description
MLSS	X	mg/L	3000	1500-5000	Mixed liquor suspended solids
MLVSS	X_v	mg/L	2400	1200-4000	Mixed liquor volatile SS (0.8×MLSS)
SRT (Sludge Age)	θ_c	days	10	3-30	Solids retention time
HRT	θ	hours	6	4-24	Hydraulic retention time
Max Growth Rate	μ_max	1/d	6.0	4.0-10.0	Maximum specific growth rate
Half-Saturation	K_s	mg/L	20	5-60	Half-saturation constant
True Yield	Y	kg/kg	0.6	0.4-0.8	True yield coefficient
Decay Rate	k_d	1/d	0.06	0.04-0.15	Endogenous decay coefficient
DO Setpoint	DO	mg/L	2.0	1.5-3.0	Dissolved oxygen concentration
Temperature	T	°C	20	10-30	Wastewater temperature
Alpha Factor	α	-	0.8	0.6-1.0	Oxygen transfer correction
Beta Factor	β	-	0.95	0.90-0.98	Saturation correction

Design Criteria

Parameter	Conventional	High Rate	Extended Aeration	Units	Notes
SRT	5-15	3-5	18-36	days	Sludge age
F/M Ratio	0.2-0.5	0.5-1.5	0.05-0.15	kg BOD/kg MLVSS·d	Loading rate
MLSS	2000-3500	3000-5000	3000-6000	mg/L	Mixed liquor
HRT	4-8	2-4	18-36	hours	Detention time
Volumetric Load	0.5-1.0	1.0-2.0	0.2-0.5	kg BOD/m³·d	Organic loading
Oxygen Transfer	1.2-1.8	1.5-2.5	2.0-2.5	kg O₂/kg BOD	AOR/BOD removed
RAS Ratio	25-100%	50-150%	75-150%	% of Q	Return sludge

Worked Examples

Example 1: Basic Mode - Municipal WWTP

Given:

• Influent flow: 15,000 m³/d
• Influent BOD₅: 200 mg/L
• Influent NH₄-N: 30 mg/L
• BOD removal efficiency: 92%
• NH₄ removal efficiency: 95%
• Observed yield: 0.5 kg VSS/kg BOD
• Oxygen factor: 0.55

Calculate: Effluent quality, sludge production, and oxygen requirement

1. Effluent BOD:

   • $\text{BOD}_{out} = 200 \times (1 - 0.92) = 200 \times 0.08 = 16 \text{ mg/L}$

2. BOD removed:

   • $\text{BOD}_{removed} = 15,000 \times (200 - 16) / 1000 = 15,000 \times 184 / 1000 = 2,760 \text{ kg/d}$

3. Effluent NH₄-N:

   • $\text{NH}4\text{-N} = 30 \times (1 - 0.95) = 30 \times 0.05 = 1.5 \text{ mg/L}$

4. WAS production:

   • $\text{WAS} = 2,760 \times 0.5 = 1,380 \text{ kg VSS/d}$

5. Oxygen requirement (simplified):

   • $\text{O}_2 = 2,760 \times 0.55 = 1,518 \text{ kg O}_2\text{/d}$

Solution:

• Effluent BOD₅: 16 mg/L
• Effluent NH₄-N: 1.5 mg/L
• WAS production: 1,380 kg VSS/d
• Oxygen requirement: 1,518 kg O₂/d (≈ 63 kg/h)

Example 2: Advanced Mode - CSTR Design

Given:

• Flow rate: 10,000 m³/d
• Influent COD (S₀): 400 mg/L
• Target effluent COD (S): 40 mg/L
• Target SRT (θ_c): 10 days
• Y = 0.6 kg VSS/kg COD
• μ_max = 6.0 d⁻¹
• K_s = 20 mg/L
• k_d = 0.06 d⁻¹
• MLVSS (X): 3,000 mg/L

Calculate: Required reactor volume, F/M ratio, oxygen demand, WAS production

1. Verify effluent COD with SRT = 10 days:

   • $S = \frac{20(1 + 0.06 \times 10)}{10(6.0 - 0.06) - 1} = \frac{20 \times 1.6}{59.4 - 1} = \frac{32}{58.4} = 0.55 \text{ mg/L}$
   • This is much lower than target 40 mg/L, so use target S = 40 mg/L

2. Required reactor volume:

   • From mass balance: $X = \frac{\theta_c \cdot Y \cdot Q(S_0 - S)}{V(1 + k_d \cdot \theta_c)}$
   • Rearranging: $V = \frac{\theta_c \cdot Y \cdot Q(S_0 - S)}{X(1 + k_d \cdot \theta_c)}$
   • $V = \frac{10 \times 0.6 \times 10,000 \times (400 - 40)/1000}{3.0 \times (1 + 0.06 \times 10)}$
   • $V = \frac{10 \times 0.6 \times 10,000 \times 0.36}{3.0 \times 1.6} = \frac{21,600}{4.8} = 4,500 \text{ m³}$

3. Hydraulic retention time:

   • $\theta = \frac{V}{Q} = \frac{4,500}{10,000/24} = \frac{4,500}{417} = 10.8 \text{ hours}$

4. F/M ratio:

   • $F/M = \frac{Q \cdot S_0}{V \cdot X} = \frac{10,000 \times 400}{4,500 \times 3,000} = \frac{4,000,000}{13,500,000} = 0.30 \text{ kg COD/kg MLVSS·d}$

5. Net biomass production:

   • $P_x = \frac{Q \cdot Y \cdot (S_0 - S)}{1 + k_d \cdot \theta_c} = \frac{10,000 \times 0.6 \times (400-40)/1000}{1 + 0.06 \times 10}$
   • $P_x = \frac{10,000 \times 0.6 \times 0.36}{1.6} = \frac{2,160}{1.6} = 1,350 \text{ kg VSS/d}$

6. Oxygen requirement:

   • $\text{O}_2 = Q(S_0 - S) - 1.42 \cdot P_x$
   • $\text{O}_2 = 10,000 \times (400-40)/1000 - 1.42 \times 1,350$
   • $\text{O}_2 = 3,600 - 1,917 = 1,683 \text{ kg O}_2\text{/d}$ (≈ 70 kg/h)

7. WAS flow (assuming X_w = 10,000 mg/L in WAS):

   • $Q_w = \frac{V \cdot X}{\theta_c \cdot X_w} = \frac{4,500 \times 3,000}{10 \times 10,000} = \frac{13,500,000}{100,000} = 135 \text{ m³/d}$

Solution:

• Reactor volume: 4,500 m³
• HRT: 10.8 hours
• F/M ratio: 0.30 kg COD/kg MLVSS·d (conventional loading)
• Net sludge production: 1,350 kg VSS/d
• Oxygen requirement: 1,683 kg O₂/d (70 kg/h)
• WAS flow rate: 135 m³/d at 1% solids

Example 3: SRT Calculation from WAS Flow

Given:

• Reactor volume: 5,000 m³
• MLSS: 3,500 mg/L
• WAS flow: 150 m³/d
• WAS concentration: 10,000 mg/L
• Effluent TSS: 20 mg/L
• Flow: 12,000 m³/d

Calculate: System SRT

1. Total solids in reactor:

   • $\text{Mass in reactor} = V \times X = 5,000 \times 3.5 = 17,500 \text{ kg}$

2. Solids wasted per day:

   • $\text{WAS solids} = Q_w \times X_w = 150 \times 10 = 1,500 \text{ kg/d}$
   • $\text{Effluent solids} = Q \times \text{TSS}_{eff} = 12,000 \times 0.020 = 240 \text{ kg/d}$
   • $\text{Total wasted} = 1,500 + 240 = 1,740 \text{ kg/d}$

3. SRT calculation:

   • $\theta_c = \frac{\text{Mass in reactor}}{\text{Total wasted}} = \frac{17,500}{1,740} = 10.1 \text{ days}$

Solution:

• System SRT: 10.1 days

Assumptions and Limitations

• Assumes complete-mix (CSTR) conditions with uniform substrate and biomass distribution
• Steady-state equations do not capture dynamic behavior or diurnal variations
• Monod kinetics assumes single substrate limitation (typically carbon or nitrogen)
• Does not explicitly model nitrogen and phosphorus transformations (requires ASM models)
• Temperature effects on kinetics require Arrhenius correction factors
• Actual oxygen transfer depends on diffuser type, depth, DO, and wastewater characteristics
• RAS ratio must be set to maintain target MLSS (not calculated in basic mass balance)
• Does not account for non-biodegradable COD fractions
• Settling characteristics in secondary clarifier affect system performance
• Filamentous bulking can occur under certain operating conditions (low F/M, nutrient deficiency)

Troubleshooting

Issue	Possible Cause	Solution
High effluent BOD	Low SRT, hydraulic overload	Increase SRT, reduce flow, check DO
High effluent TSS	Poor settling, bulking	Check MLSS, SVI, F/M ratio, nutrients
Filamentous bulking	Low F/M, low DO, nutrient deficiency	Increase F/M, increase DO, add nutrients
Pin floc	High F/M, young sludge	Increase SRT, reduce F/M
Rising sludge	Denitrification in clarifier	Increase RAS rate, reduce SRT
Low DO	Insufficient aeration, high load	Increase air flow, check diffusers
Foam/scum	High SRT, detergents, fats	Control SRT, add anti-foam, skim surface
Nitrification failure	Low SRT, low pH, low DO, cold temp	Increase SRT, add alkalinity, increase DO

Related Process Units

• Primary Clarifier - Upstream solids removal reduces organic loading
• Secondary Clarifier - Critical for biomass separation and RAS
• MBR - Membrane-based alternative to secondary clarifier
• SBR - Sequencing batch reactor variant of activated sludge
• Anaerobic Digester - WAS processing

References

1. Metcalf & Eddy (2014). Wastewater Engineering: Treatment and Resource Recovery, 5th Edition, McGraw-Hill.
2. Grady, C.P.L., Daigger, G.T., Love, N.G., and Filipe, C.D.M. (2011). Biological Wastewater Treatment, 3rd Edition, CRC Press.
3. Henze, M., van Loosdrecht, M.C.M., Ekama, G.A., and Brdjanovic, D. (2008). Biological Wastewater Treatment: Principles, Modelling and Design, IWA Publishing.
4. Lawrence, A.W. and McCarty, P.L. (1970). "Unified Basis for Biological Treatment Design and Operation," Journal ASCE Sanitary Engineering Division, 96(SA3):757-778.
5. Ekama, G.A. and Marais, G.v.R. (1979). "Dynamic Behaviour of the Activated Sludge Process," Journal Water Pollution Control Federation, 51(3):534-556.
6. WEF (2018). Design of Water Resource Recovery Facilities, Manual of Practice No. 8, 6th Edition.
7. EPA (1987). Design Manual: Phosphorus Removal, EPA/625/1-87/001.

Model Implementation

TypeScript Model Location: backend/models/processes/ActivatedSludge.ts

Key Methods:

• calculateBasic() - Efficiency-based calculations
• calculateAdvanced() - CSTR mass balance with Monod kinetics
• calculateSRT() - Solids retention time from WAS flow
• calculateOxygenDemand() - Aeration requirements
• validate() - Parameter range validation

Change Log

Date	Version	Changes
2025-12-30	1.0	Initial documentation with CSTR mass balance and Monod kinetics