Abstract
To treat better, add more workers—so it is thought. This paper shows that the added workers flow out of the exit—so performance is not monotone but has an interior peak. No new theorem or law is claimed.
First, removal rises monotonically from $0.4000$ to $0.8421$ between $1000$ and $8000$ mg/L (Section 2).
Second, but washout begins above $2500$ mg/L (Section 3).
Third, this is the core of the paper. Effective performance peaks at $\mathbf{5000}$ mg/L with $\mathbf{66.92}$ and falls to $62.21$ at $8000$ (Section 3).
Fourth, the peak is not at the point of maximum removal (Section 3).
Fifth, the boundary is the $2500$ mg/L at which washout starts (Section 3).
Sixth, the separator is whether the exit passes the workers (Section 4).
Keywords: activated sludge process, wastewater treatment, sludge concentration, settling tank, solid-liquid separation.
Scope (what is not claimed)
No new theorem or law is claimed—the activated sludge process, mixed liquor concentration and solid-liquid separation in a clarifier are all standard. No plant is designed—aeration rate, return ratio and clarifier area are not the subject. No microbiology is argued—community composition, growth rate and sludge age are not the subject. The numbers are a model—both the removal curve and the settling limit are simple forms chosen to bring out the trend, not measurements. No regulation is discussed—effluent standards and statutes are not the subject. Relation to earlier papers: Paper 446 showed that a theoretical floor moves with the demand—there too, a condition hid behind an apparently monotone quantity. Paper 300 showed that common root and distinct root can be decided—“removal” and “effective performance” are distinct roots by that criterion. Paper 417 showed that a ceiling moves with the requirement—there too, the operation meant to improve things ran into a different limit. What is added is tabulating removal and washout at five concentrations, showing the effective peak at $5000$ mg/L, giving the onset of washout as $2500$ mg/L, showing that the peak does not coincide with maximum removal, and placing the separator at whether the exit passes the workers.
1. Notation and the Question
Setting: Let the sludge concentration in the aeration tank be $X$ [mg/L]. Removal $\eta$ rises with saturation.
$$\eta(X)=\frac{X}{X+X_{0}}$$The clarifier can hold only up to $X_{c}$; the excess is washed out.
$$L(X)=\max\bigl(0,\ a\,(X-X_{c})\bigr)$$Effective performance subtracts what is lost from what is removed.
$$P(X)=100\,\eta(X)-L(X)$$Below, $X_{0}=1500$ mg/L, $X_{c}=2500$ mg/L and $a=0.004$.
Observation 1.1 That more workers means better treatment is true. $\eta$ is increasing in $X$—there is no lie in that part.
Observation 1.2 But the workers can also leave through the exit. Once the clarifier cannot hold them, the sludge itself washes out—what was added turns straight into a loss.
Question: Put the two together, and where is the best point.
2. Removal Rises Monotonically
Computation 2.1: Take removal at five concentrations.
| Concentration [mg/L] | Removal |
|---|---|
| $1000$ | $0.4000$ |
| $2000$ | $0.5714$ |
| $3000$ | $0.6667$ |
| $5000$ | $0.7692$ |
| $8000$ | $\mathbf{0.8421}$ |
Observation 2.2 It rises monotonically. From $0.4000$ to $0.8421$—an eightfold concentration buys a factor of $2.11$.
Observation 2.3 But the gain slows. $+0.1714$ from $1000$ to $2000$, $+0.0729$ from $5000$ to $8000$—saturation.
Observation 2.4 On this table alone, thicker is always better. There is no peak—so the conclusion is “thicker still”.
3. Effective Performance Peaks at 5000 mg/L
Computation 3.1: Add the washout and take the effective performance.
| Concentration [mg/L] | Washout | Effective |
|---|---|---|
| $1000$ | $0$ | $40.0000$ |
| $2000$ | $0$ | $57.1429$ |
| $3000$ | $2$ | $64.6667$ |
| $5000$ | $10$ | $\mathbf{66.9231}$ |
| $8000$ | $22$ | $62.2105$ |
Observation 3.2 The peak is at $5000$ mg/L. $66.9231$—beyond it, the curve falls.
Observation 3.3 At $8000$ it drops to $62.2105$. Even though removal is at its maximum $0.8421$—the operating point that removes best is not the best operating point.
Observation 3.4 $64.6667$ at $3000$ beats $62.2105$ at $8000$. A $2.67$-fold concentration, and performance falls—the direction of increase is not always right.
Observation 3.5 The boundary is $2500$ mg/L. No washout up to there, washout beyond—a line where the character changes lies between $2000$ and $3000$.
Observation 3.6 Note that the peak is at $5000$, not at $2500$. For a while after washout begins, the gain in removal still wins—the boundary and the optimum are different places.
4. The Separator
The separator: does the exit pass the workers or hold them.
Observation 4.1 If it holds them, more is always better. Because $\eta$ is monotone—there is no peak, and the ceiling is set by something else.
Observation 4.2 If it passes them, a peak appears. Because added workers turn into losses—the optimum is interior.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are called “treatment performance”—one a property of the tank, one of the tank paired with the clarifier.
Observation 4.4 The same shape as Paper 446. There, raising the recovery raised the theoretical floor—here, raising the concentration defeats the exit. In both, the improving operation rebounded on the other side.
Observation 4.5 So when reinforcement is discussed, look at the exit first. Investing in more workers pays only within what the exit can hold—miss where the bottleneck moved to, and only the cost rises.
Observation 4.6 To be honest—the numbers are a model. $X_{0}$, $X_{c}$ and $a$ are chosen, not measured—the location of the peak differs by plant. What does not differ is that a peak exists.
Observation 4.7 One last thing—enlarge the clarifier and $X_{c}$ rises, moving the peak to the right. The peak is not at a fixed place—it is a number for deciding which side to fix.
5. Roll Call and Closing
- Removal from $0.4000$ to $0.8421$ (Computation 2.1): monotone.
- The gain slows (Observation 2.3): saturation.
- The table alone says “thicker” (Observation 2.4): no peak in view.
- Washout from $2500$ mg/L (Observation 3.5): the boundary.
- The effective peak is $66.92$ at $5000$ (Computation 3.1): the core.
- $62.21$ at $8000$ (Observation 3.3): best removal is not best.
- $3000$ beats $8000$ (Observation 3.4): increase can be wrong.
- Boundary and optimum differ (Observation 3.6): $2500$ and $5000$.
- The separator is the exit (Observation 4.3): distinct roots by Paper 300.
- A bigger clarifier moves the peak right (Observation 4.7): which side to fix.
Closing: To treat better, add more workers—so it is thought—and on removal alone that is right. Raising the sludge concentration from $1000$ to $8000$ mg/L raises removal monotonically from $0.4000$ to $0.8421$. The gain slows, but no peak appears—$+0.1714$ from $1000$ to $2000$ against $+0.0729$ from $5000$ to $8000$ is saturation, not decline. On this table alone the conclusion is “thicker still”. But the workers can also leave through the exit—the clarifier holds only so much, and above $\mathbf{2500}$ mg/L the sludge itself washes out, so what was added turns straight into a loss. Subtracting the loss gives $40.00$, $57.14$, $64.67$, $\mathbf{66.92}$ and $62.21$, with the peak at $\mathbf{5000}$ mg/L. At $8000$ removal is at its maximum $0.8421$ and performance is down at $62.21$—the operating point that removes best is not the best operating point. $64.67$ at $3000$ beats $8000$ outright: a $2.67$-fold concentration, and performance falls. And the boundary at $2500$ is not the optimum at $5000$—for a while after washout begins, the gain in removal still wins. One thing separates them—does the exit pass the workers or hold them. If it holds them, $\eta$ is monotone and more is always better, with no peak. If it passes them, added workers turn into losses and the optimum is interior. They are distinct roots by the criterion of Paper 300, one a property of the tank and one of the tank paired with the clarifier. It is the same shape as Paper 446, where raising the recovery raised the theoretical floor: in both, the improving operation rebounded on the other side. So when reinforcement is discussed, look at the exit first—miss where the bottleneck moved to, and only the cost rises. To be honest—the numbers are a model, with $X_{0}$, $X_{c}$ and $a$ chosen rather than measured. The location of the peak differs by plant; what does not differ is that a peak exists. One last thing—enlarge the clarifier and the holding limit rises, moving the peak to the right. The peak is not at a fixed place: it is a number for deciding which side to fix.
References
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Reference
Yamagishi, Y. (2026). Add Too Many Workers and the Workers Start Leaving [Paper 447]. YOI: https://www.yamagishiyuuki.com/yoi/447
DOI: 10.5281/zenodo.22732494
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 447. This paper is a preprint and has not been peer reviewed.