PFR-Segment - Plug Flow Reactor Segment

Individual compartment within a tanks-in-series model representing plug flow behavior, enabling accurate simulation of substrate gradients and selector effects.

Overview

The PFR-Segment represents a single compartment in the tanks-in-series modeling approach for plug flow reactors. By dividing a long aeration basin into multiple CSTR segments, ErgoPlanner can accurately simulate the concentration gradients characteristic of plug flow while maintaining computational efficiency.

When to Use

• Modeling conventional activated sludge basins
• Simulating selector zones at reactor inlet
• Step-feed aeration basin design
• Oxidation ditch modeling (with recirculation)
• Any long narrow reactor with L:W > 5:1

Key Characteristics

• Discrete approximation of continuous plug flow
• Concentration gradient between segments
• Each segment is a fully-mixed CSTR
• Connected in series with flow between segments
• Enables variable conditions (DO, mode) per segment

Theory

Tanks-in-Series Model

A real plug flow reactor with dispersion is modeled as N equal-volume CSTRs:

$$\frac{C_{out}}{C_{in}} = \left(\frac{1}{1 + k\tau_i}\right)^N = \frac{1}{(1 + k\tau/N)^N}$$

Where:

• N = number of segments
• tau_i = HRT per segment = tau_total/N
• k = first-order rate constant

As N approaches infinity, this approaches ideal plug flow: $$\lim_{N \to \infty} \left(1 + \frac{k\tau}{N}\right)^{-N} = e^{-k\tau}$$

Relationship to Dispersion

The number of tanks relates to the dispersion number: $$N \approx \frac{1}{2d}$$

Where d = D/(uL) is the dimensionless dispersion number.

Dispersion Number (d)	Equivalent N	Flow Regime
0.01	50	Near plug flow
0.02	25	Low dispersion
0.05	10	Moderate
0.1	5	High dispersion
>0.25	2	Near complete mix

Peclet Number

Alternative characterization: $$Pe = \frac{uL}{D} = \frac{1}{d} = 2N$$

Mass Balance per Segment

General Mass Balance

For segment i with volume Vi: $$V_i \frac{dC_i}{dt} = Q_{i-1}C_{i-1} - Q_iC_i + r_{C,i}V_i + Q_{feed,i}C_{feed}$$

At steady state: $$C_i = \frac{Q_{i-1}C_{i-1} + Q_{feed,i}C_{feed} + r_{C,i}V_i}{Q_i}$$

Substrate Profile

For Monod kinetics in segment i: $$\frac{dS_i}{dt} = \frac{Q}{V_i}(S_{i-1} - S_i) - \frac{\mu_{max}}{Y} \times \frac{S_i}{K_s + S_i} \times X_i$$

Biomass Profile

$$\frac{dX_i}{dt} = \frac{Q}{V_i}(X_{i-1} - X_i) + \mu_{max} \times \frac{S_i}{K_s + S_i} \times X_i - b \times X_i$$

Design Equations

Segment Volume

For equal-volume segments: $$V_i = \frac{V_{total}}{N}$$

For variable volumes (e.g., selector): $$V_1 < V_2 < ... < V_N$$

Segment HRT

$$\tau_i = \frac{V_i}{Q_i}$$

Total HRT: $$\tau_{total} = \sum_{i=1}^{N} \tau_i$$

Number of Segments Selection

Application	Recommended N	Rationale
Screening calculation	3-5	Quick estimate
Design modeling	8-12	Good accuracy
Detailed simulation	15-20	High precision
Selector modeling	First 2-3	Capture gradient

Concentration at Segment Exit

For first-order kinetics: $$C_N = C_0 \times \prod_{i=1}^{N} \frac{1}{1 + k\tau_i}$$

Parameters

Segment Geometry Parameters

Parameter	Unit	Default	Range	Description
Segment index	-	1	1-N	Position in series
Volume	m3	V_total/N	variable	Segment volume
Length	m	L_total/N	variable	Segment length
Cross-section area	m2	calculated	-	W x D
L:W ratio (total)	-	10	5-20	Plug flow indicator

Hydraulic Parameters

Parameter	Unit	Default	Range	Description
Flow rate	m3/d	Q_in	-	Through segment
HRT	hours	total/N	0.5-3	Per segment
Velocity	m/h	Q/A	50-150	Horizontal velocity
Step feed fraction	-	0	0-0.5	Feed to this segment

Process Parameters

Parameter	Unit	Default	Range	Description
Operating mode	-	aerobic	aerobic/anoxic/anaerobic	Process condition
DO setpoint	mg/L	2.0	0-4.0	Target DO
MLSS	mg/L	inherited	2000-6000	From upstream
Temperature	C	15	10-25	Operating temp

Biokinetic Parameters

Inherited from parent PFR configuration - see CSTR for full ASM parameter tables.

Selector Zone Modeling

Purpose

The first 1-3 segments can function as a selector:

• High F:M ratio at inlet
• Kinetic selection for floc-formers
• Suppression of filamentous organisms
• Improved sludge settleability

Selector Types

Type	DO Level	Contact Time	Mechanism
Aerobic	>2 mg/L	10-30 min	Kinetic selection
Anoxic	<0.5 mg/L	15-45 min	Metabolic selection
Anaerobic	0 + no NO3	30-60 min	Bio-P + selection

F:M in Selector

$$F:M_{selector} = \frac{Q \times S_0}{V_{selector} \times X} = 3-6 \text{ kg BOD/kg MLSS/d}$$

Much higher than overall F:M (typically 0.2-0.4).

Configuration

Example Configuration

{
  "type": "pfrSegment",
  "id": "aeration-segment-3",
  "parameters": {
    "parentPfr": "aeration-basin-1",
    "segmentIndex": 3,
    "totalSegments": 10,
    "volume": 500,
    "mode": "aerobic",
    "doSetpoint": 2.0,
    "stepFeedFraction": 0.0,
    "aeration": {
      "kla": 12,
      "alpha": 0.6,
      "diffuserType": "fineBubble",
      "coverage": 0.12
    },
    "biokinetics": {
      "model": "ASM1",
      "inheritFromParent": true
    },
    "connections": {
      "inlet": "aeration-segment-2",
      "outlet": "aeration-segment-4"
    }
  }
}

Complete PFR with Segments

{
  "type": "pfr",
  "id": "aeration-basin-1",
  "parameters": {
    "totalVolume": 5000,
    "segments": 10,
    "length": 100,
    "width": 10,
    "depth": 5,
    "selectorSegments": 2,
    "segmentConfiguration": [
      {
        "index": 1,
        "volumeFraction": 0.08,
        "mode": "anoxic",
        "doSetpoint": 0.3
      },
      {
        "index": 2,
        "volumeFraction": 0.08,
        "mode": "anoxic",
        "doSetpoint": 0.3
      },
      {
        "index": "3-10",
        "volumeFraction": 0.105,
        "mode": "aerobic",
        "doSetpoint": 2.0
      }
    ],
    "stepFeed": {
      "enabled": true,
      "locations": [1, 4, 7],
      "fractions": [0.5, 0.3, 0.2]
    }
  }
}

Worked Example

Given

• Total flow: 15,000 m3/d
• BOD: 200 mg/L
• NH4-N: 35 mg/L
• Target: BOD <15, NH4-N <2 mg/L
• Model as 8 segments
• First 2 segments as anoxic selector

Segment-by-Segment Analysis

Step 1: Overall Design

• Total HRT = 8 hours
• Total volume = 15,000 x 8/24 = 5,000 m3
• Per segment = 5,000/8 = 625 m3

Step 2: Selector Sizing (Segments 1-2)

• Selector HRT = 2 x (8/8) = 2 hours
• Selector F:M target = 4 kg/kg/d
• Volume = 15,000 x 0.2 / (4 x 3.5) = 214 m3/segment

Adjust: Segments 1-2 at 400 m3 each, Segments 3-8 at 700 m3 each.

Step 3: Concentration Profile (simplified)

Segment	Mode	S_BOD (mg/L)	S_NH4 (mg/L)	DO (mg/L)
Inlet	-	200	35	0.5
1	Anoxic	160	34	0.3
2	Anoxic	130	33	0.3
3	Aerobic	80	28	2.0
4	Aerobic	50	22	2.0
5	Aerobic	30	15	2.0
6	Aerobic	20	9	2.0
7	Aerobic	15	5	2.0
8	Aerobic	12	2	2.0

Step 4: Oxygen Demand per Segment

Higher at inlet aerobic segments (3-4) due to BOD:

Segment	OD (kg O2/d)	% of Total
3	450	22%
4	380	18%
5	320	15%
6	280	13%
7	350	17%
8	320	15%

(Segments 7-8 higher due to nitrification peak)

Step 5: Tapered Aeration

Distribute diffusers proportionally to OD profile.

Dynamic Simulation

State Variables per Segment

For ASM1: 13 state variables per segment

• S_I, S_S, X_I, X_S, X_BH, X_BA, X_P, S_O, S_NO, S_NH, S_ND, X_ND, S_ALK

Integration Method

Each segment solved sequentially:

1. Calculate inlet conditions from upstream segment
2. Apply biokinetic reactions
3. Solve mass balance for each state variable
4. Pass outlet to next segment

Time Step Considerations

$$\Delta t < \tau_i / 10$$

For HRT of 1 hour per segment: dt < 6 minutes

Troubleshooting

Issue	Cause	Solution
Poor outlet quality	Insufficient segments	Increase N to 12-15
Numerical instability	Time step too large	Reduce dt
Selector ineffective	F:M too low	Reduce selector volume
Uneven DO profile	Poor aeration distribution	Adjust per-segment aeration
Short-circuiting	Mixing issues	Add baffles (increase N)
Excessive sludge	High inlet F:M	Distribute feed (step-feed)

Comparison: N Segments vs CSTR

N Segments	Relative to Single CSTR
1	Identical
3	85% effluent S
5	75% effluent S
10	65% effluent S
20	58% effluent S
Infinity (PFR)	50% effluent S

For first-order kinetics at ktau = 2

Related Topics

• PFR - Plug Flow Reactor
• CSTR - Continuously Stirred Tank Reactor
• Anoxic-Zone - Anoxic Zone
• Selector-Design
• Step-Feed-Activated-Sludge

References

1. Levenspiel, O. (1999). Chemical Reaction Engineering, 3rd Ed. Wiley.
2. Henze, M. et al. (2008). Biological Wastewater Treatment. IWA Publishing.
3. WEF (2018). Design of Municipal Wastewater Treatment Plants, MOP 8.
4. Tchobanoglous, G. et al. (2014). Wastewater Engineering, 5th Ed. McGraw-Hill.