Takács Double-Exponential Settling Model

Advanced settling velocity model for secondary clarifier design and simulation

Overview

The Takács model is a double-exponential settling velocity function that accurately describes both hindered settling at high concentrations and dispersed settling correction at low concentrations.

Reference: Takács et al. (1991) "A Dynamic Model of the Clarification-Thickening Process" Water Research, Vol. 25, No. 10, pp. 1263-1271

Mathematical Model

Settling Velocity Equation

$$v_s = V_0 \cdot e^{-r_h \cdot X} - V_0' \cdot e^{-r_p \cdot X}$$

Where:

• $v_s$ = Settling velocity (m/d)
• $V_0$ = Maximum practical settling velocity (m/d)
• $V_0'$ = Maximum theoretical settling velocity for dispersed settling (m/d)
• $r_h$ = Hindered settling parameter (L/mg)
• $r_p$ = Discrete/dispersed settling parameter (L/mg)
• $X$ = Solids concentration (mg/L)

Solids Flux

$$G = v_s \cdot X$$

Where:

• $G$ = Gravity settling flux (kg/m²/d)

Total Flux (with underflow)

$$G_{total} = G_{gravity} + G_{underflow} = v_s \cdot X + v_u \cdot X$$

Where:

• $v_u$ = Underflow velocity (m/d) = Q_u / A

Parameters

Parameter	Symbol	Unit	Default	Typical Range	Description
Max settling velocity	V₀	m/d	474	200-600	Maximum velocity at low concentration
Max dispersed velocity	V₀'	m/d	250	150-350	Correction for low-concentration settling
Hindered settling coeff	rₕ	L/mg	0.000576	0.0003-0.001	Controls high-concentration settling decay
Dispersed settling coeff	rₚ	L/mg	0.00286	0.001-0.006	Controls low-concentration correction
Non-settleable fraction	fₙₛ	-	0.00228	0.001-0.005	Fraction that won't settle

SVI Correlation

Takács parameters can be estimated from Sludge Volume Index (SVI):

rₕ = 0.000005 × SVI  (L/mg)
rₚ = 0.000025 × SVI  (L/mg)

SVI (mL/g)	Settling Quality	rₕ (L/mg)	rₚ (L/mg)
80	Good	0.0004	0.002
120	Average	0.0006	0.003
200	Poor	0.001	0.005

API Endpoints

Calculate Settling Velocity

POST /api/v1/clarifier/settling-velocity

Request:

{
  "concentration": 3500,
  "svi": 120
}

Response:

{
  "velocity": 54.2,
  "velocityPerHour": 2.26,
  "concentration": 3500,
  "parameters": {
    "v0": 474,
    "v0Prime": 250,
    "rh": 0.0006,
    "rp": 0.003,
    "fns": 0.00228
  }
}

Calculate Solids Flux

POST /api/v1/clarifier/solids-flux

Request:

{
  "concentration": 3500,
  "underflowVelocity": 12.5,
  "svi": 120
}

Get Takács Parameters from SVI

GET /api/v1/clarifier/takacs-parameters?svi=120

Generate Flux Curve

POST /api/v1/clarifier/flux-curve

Request:

{
  "svi": 120,
  "underflowVelocity": 12.5,
  "minConcentration": 500,
  "maxConcentration": 15000,
  "numberOfPoints": 50
}

Perform State Point Analysis

POST /api/v1/clarifier/state-point-analysis

Request:

{
  "surfaceArea": 800,
  "sideWaterDepth": 4.0,
  "influentFlow": 10000,
  "mlss": 3500,
  "svi": 120,
  "rasRatio": 0.5,
  "wasFlow": 100
}

Run Steady-State Simulation

POST /api/v1/clarifier/steady-state

TypeScript Client

import { clarifierApi } from '@/flow/services/simulation/api/clarifierApi';

// Calculate settling velocity
const result = await clarifierApi.calculateSettlingVelocity({
  concentration: 3500,
  svi: 120
});

// Perform state point analysis
const analysis = await clarifierApi.performStatePointAnalysis({
  surfaceArea: 800,
  sideWaterDepth: 4.0,
  influentFlow: 10000,
  mlss: 3500,
  svi: 120,
  rasRatio: 0.5
});

// Run simulation
const simulation = await clarifierApi.runSteadyState({
  surfaceArea: 800,
  sideWaterDepth: 4.0,
  influentFlow: 10000,
  mlss: 3500,
  svi: 120,
  rasRatio: 0.5,
  includeLayerProfile: true
});

State Point Analysis

State point analysis determines if a clarifier is operating safely by comparing the operating flux to the limiting flux.

Key Metrics

Metric	Description	Design Target
Safety Factor	LimitingFlux / OperatingFlux	> 1.2
Available Capacity	(Limiting - Operating) / Limiting × 100	> 20%
Risk Assessment	"Safe", "Warning", "Critical"	Safe

Risk Categories

Safety Factor	Risk Level	Recommended Action
≥ 1.5	Safe	Normal operation
1.2 - 1.5	Warning	Monitor closely
< 1.2	Critical	Reduce loading or increase capacity

10-Layer Model

The secondary clarifier is discretized into 10 horizontal layers for simulation:

Layer 0: Effluent (overflow)
Layer 1-2: Clarification zone
Layer 3: Feed layer (MLSS inlet)
Layer 4-8: Thickening zone
Layer 9: Underflow (RAS/WAS)

Mass Balance

For each layer: $$\frac{dX_i}{dt} = \frac{G_{in} - G_{out} + G_{feed}}{h_i}$$

Design Validation Criteria

Criterion	Limit	Units
Overflow Rate (SOR)	≤ 32	m³/m²/d
Solids Loading Rate (SLR)	≤ 150	kg/m²/d
SVI	≤ 150	mL/g
HRT	≥ 2	hours
Safety Factor	≥ 1.2	-

Best Practices

1. Parameter Selection: Use SVI correlation for preliminary design; calibrate with settling tests for detailed modeling
2. Simulation: Run to steady state before interpreting results
3. Validation: Check mass balance closure (< 5% error)
4. Design Margin: Target safety factor ≥ 1.5 for conservative design

Related Topics

• Model-Selection-Architecture - Basic vs Advanced calculation modes
• Secondary-Clarifier - Process unit overview
• State-Point-Analysis - Detailed theory

Technical Reference

Backend Files

File	Purpose
Domain/Simulation/Models/Clarifier/TakacsParameters.cs	Takács model parameters
Domain/Simulation/ProcessUnits/Separators/SecondaryClarifier.cs	10-layer clarifier model
Application/DTOs/Clarifier/*.cs	API request/response DTOs
Application/Services/Clarifier/SecondaryClarifierService.cs	Calculation service
API/Controllers/ClarifierController.cs	REST API endpoints

Frontend Files

File	Purpose
flow/services/simulation/api/clarifierApi.ts	TypeScript API client