Secondary Clarifier Sizing and Return Activated Sludge: 18-Env-A4 Water and Wastewater Engineering, May 2013, Q4

Question 4: Secondary Clarifier Sizing and Return Activated Sludge (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Clarifier Volume, Surface Overflow Rate and Solids Loading Rate

Given.

Given data
Quantity	Symbol	Value
Average plant (influent) flow	$Q$	15,000 m³/d
Clarifier hydraulic retention time	$t$	6.0 h
Side water depth	$h$	4.0 m
Mixed liquor suspended solids	MLSS	3000 mg/L

Find. Clarifier volume $\forall$, surface overflow rate SOR, and solids loading rate SLR.

Approach. Volume follows directly from $\forall=Q\,t$; the surface area needed for SOR/SLR follows from $A=\forall/h$; SOR uses the clear-water (overflow) flow $Q$, while SLR must use the FULL flow entering the clarifier — the combined mixed-liquor stream $Q+Q_r$ — because that is the flow actually carrying the MLSS solids into the tank; $Q_r$ is found in part (b) below and used here to complete SLR.

Clarifier volume. $$\forall=Q\,t=15{,}000\ \tfrac{\text{m}^3}{\text{d}}\times\left(\frac{6.0}{24}\ \text{d}\right)=\boxed{3750\ \text{m}^3}.$$
Surface area and surface overflow rate. $$A=\frac{\forall}{h}=\frac{3750}{4.0}=937.5\ \text{m}^2,\qquad \text{SOR}=\frac{Q}{A}=\frac{15{,}000}{937.5}=\boxed{16.0\ \text{m}^3/(\text{m}^2\cdot\text{d})}.$$
Solids loading rate, using $Q_r=15{,}000$ m³/d from part (b). The clarifier's feed is the full mixed-liquor stream $Q+Q_r$ at concentration MLSS: $$\text{SLR}=\frac{(Q+Q_r)\,\text{MLSS}}{A}=\frac{(15{,}000+15{,}000)\times3000}{937.5}=96{,}000\ \tfrac{\text{g}}{\text{m}^2\cdot\text{d}}=\boxed{96\ \text{kg/(m}^2\cdot\text{d})}.$$

Check: SOR≈16 m/d sits at the low (conservative) end of Metcalf & Eddy's typical average-flow range for secondary clarifiers following activated sludge (16–28 m/d), and SLR≈96 kg/(m²·d) is likewise near the low end of the typical 96–144 kg/(m²·d) range — both indicate a conservatively (not under-) sized clarifier. If SLR were instead computed on the influent flow $Q$ alone (a simpler but less rigorous convention some texts use), it would read 48 kg/(m²·d); the $Q+Q_r$ form is used here because it is the mass balance that actually governs solids capture in the clarifier.

Quantity	Value
Clarifier volume, $\forall$	3750 m³
Surface area, $A$	937.5 m²
Surface overflow rate, SOR	16.0 m³/(m²·d)
Solids loading rate, SLR (uses $Q_r$ from part b)	96 kg/(m²·d)

(b) Return Activated Sludge (RAS) Flow Rate

Given.

Given data
Quantity	Symbol	Value
Average plant flow	$Q$	15,000 m³/d
Mixed liquor suspended solids	MLSS	3000 mg/L
RAS (underflow) TSS concentration	$X_r$	6000 mg/L

Find. The return activated sludge flow rate $Q_r$.

Approach. Write a steady-state solids mass balance around the clarifier: everything entering with the combined flow $(Q+Q_r)$ at concentration MLSS must leave either as clarified effluent (assumed essentially solids-free, $X_e\approx0$) or as RAS underflow at concentration $X_r$.

Solids mass balance around the clarifier. $$(Q+Q_r)\,\text{MLSS}=Q\,X_e+Q_r\,X_r\ \xrightarrow{X_e\approx0}\ (Q+Q_r)\,\text{MLSS}=Q_r\,X_r.$$
Expand and isolate $Q_r$. $$Q\cdot\text{MLSS}+Q_r\cdot\text{MLSS}=Q_r\,X_r\ \Rightarrow\ Q\cdot\text{MLSS}=Q_r\left(X_r-\text{MLSS}\right)\ \Rightarrow\ Q_r=\frac{Q\cdot\text{MLSS}}{X_r-\text{MLSS}}.$$
Substitute the given values. $$Q_r=\frac{15{,}000\times3000}{6000-3000}=\frac{4.5\times10^{7}}{3000}=\boxed{15{,}000\ \text{m}^3/\text{d}}.$$

Check: this corresponds to a return-sludge ratio $Q_r/Q=1.0$ (100%), which is squarely within the typical activated-sludge RAS design range (50–150% of influent flow for conventional/complete-mix processes), and closes the loop with the SLR computed in part (a) — using this $Q_r$ there gives 96 kg/(m²·d), inside the typical range, which cross-checks that the mass balance's $X_e\approx0$ assumption is reasonable.

Quantity	Value
Return activated sludge flow rate, $Q_r$	15,000 m³/d
Recycle ratio, $Q_r/Q$	1.0 (100%)

Concept & Theory

Topic. Secondary (activated-sludge) clarifier sizing — hydraulic and solids loading, and the RAS solids mass balance.

Key relations. $\forall=Qt$; $\text{SOR}=Q/A$; $\text{SLR}=(Q+Q_r)\,\text{MLSS}/A$; clarifier solids balance $Q_r=\dfrac{Q\cdot\text{MLSS}}{X_r-\text{MLSS}}$ (with $X_e\approx0$).

Why this works. SOR governs clarification (particle settling velocity must exceed the upward/overflow velocity, which depends only on the CLEAR flow $Q$ leaving over the weir), while SLR governs thickening capacity (whether the tank can pass the FULL solids flux, which depends on the total mixed-liquor flow $Q+Q_r$ entering) — a clarifier can fail on either criterion independently, which is why both are checked. The RAS balance is just conservation of mass for solids around the tank under the standard assumption that essentially no solids escape in the clear overflow.

Common pitfall. Using $Q$ alone (instead of $Q+Q_r$) when computing SLR, which understates the true solids flux and can pass an under-sized clarifier; forgetting that $Q_r$ must be solved from part (b)'s mass balance before SLR in part (a) can be finalized, since the clarifier's total feed flow is not known until the recycle rate is set.

Source. Metcalf & Eddy, Wastewater Engineering, ch. 8 (activated sludge) & ch. 10 (secondary clarification, state-point analysis).

Question 4 of 5: Secondary Clarifier Sizing and Return Activated Sludge

Question 4: Secondary Clarifier Sizing and Return Activated Sludge (25 marks)

(a) Clarifier Volume, Surface Overflow Rate and Solids Loading Rate

(b) Return Activated Sludge (RAS) Flow Rate