Question 4 of 7: pH control, ion exchange and reverse osmosis design principles, and activated-sludge process design

Question 4: pH control, ion exchange and reverse osmosis design principles, and activated-sludge process design (20 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.

(i) Key design principles

(a) pH control. Principle 1 — the reagent (lime, caustic, or acid) must be dosed against the influent's actual buffering capacity (alkalinity/acidity titration curve), not simply targeted at a final pH set-point by trial addition, since a poorly buffered stream can swing far past the target with only a small overdose. Principle 2 — adequate rapid-mix contact time and continuous pH feedback control (in-line pH probe driving the dosing pump) are required because neutralization reactions are fast but the influent pH/flow can vary quickly, so a static, un-monitored dose will drift out of compliance as conditions change.

(b) Ion exchange. Principle 1 — resin selectivity and exchange capacity must be matched to the specific target ion and the competing-ion background (e.g. a strong-acid cation resin's affinity order Ca²⁺>Mg²⁺>Na⁺), since a high concentration of a competing, higher-affinity ion can prematurely exhaust capacity for the target ion. Principle 2 — the bed must be sized and monitored against a defined breakthrough criterion (effluent quality trigger) and a regeneration cycle (typically brine for a cation softener), since capacity is finite and exchange efficiency falls sharply once breakthrough begins.

(c) Reverse osmosis. Principle 1 — the applied pressure must exceed the feed stream's osmotic pressure by a sufficient net driving pressure to achieve the target flux and recovery, so higher-TDS feeds (e.g. seawater vs. brackish water) require correspondingly higher operating pressure and stronger membrane/pressure-vessel design. Principle 2 — pretreatment (fine filtration, antiscalant dosing, sometimes softening) is essential to control fouling and scaling (particularly of sparingly-soluble salts concentrating in the reject stream), since membrane fouling is the dominant driver of RO operating cost and membrane life.

Given.

Find. (a) Aeration tank volume $V$ and HRT $\theta$. (b) Daily sludge mass wasted $Q_w$ (kg/d). (c) Recycle ratio $Q_r/Q_0$.

Approach. The Lawrence–McCarty SRT-based design equation sizes the aeration tank directly from $\theta_c$, $Y$, $k_d$ and the substrate removed; sludge production follows from the same substrate balance, and the recycle ratio comes from a pure solids balance around the clarifier, independent of the biological kinetics.

1. Aeration tank volume. $$V=\frac{\theta_c\,Y\,Q_0(S_0-S)}{X(1+k_d\theta_c)}=\frac{15\times0.5\times200{,}000\times(250-10)}{3{,}000\times(1+0.05\times15)}=68{,}571\ \text{m}^3$$ ==**V ≈ 68,571 m³**==
2. Hydraulic retention time. $$\theta=\frac{V}{Q_0}=\frac{68{,}571}{200{,}000}\times24=8.23\ \text{h}$$ ==**θ ≈ 8.23 h**==
3. Daily sludge mass wasted. $$Q_w=P_x=\frac{Y\,Q_0(S_0-S)}{1+k_d\theta_c}=\frac{0.5\times200{,}000\times240}{1.75}=13{,}714\ \text{kg/d}$$ ==**Qw ≈ 13,714 kg/d**==
4. Cross-check via the SRT identity. Independently, $Q_w=VX/\theta_c=(68{,}571\times3{,}000\times10^{-3})/15=13{,}714\ \text{kg/d}$ — the two routes agree exactly, confirming $V$ and $Q_w$ are internally consistent.
5. Sludge recycle ratio. A solids balance around the clarifier alone (no biology, no $\theta_c$) gives $$\frac{Q_r}{Q_0}=\frac{X}{X_w-X}=\frac{3{,}000}{9{,}000-3{,}000}=0.5$$ ==**Qr/Q0 = 0.5**==