Question 3 of 5: Solids Retention Time, Surface Overflow Rate, and Anaerobic Sludge Stabilization

Question 3: Solids Retention Time, Surface Overflow Rate, and Anaerobic Sludge Stabilization (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.

Part (a) — Solids retention time and surface overflow rate (15 marks). These two parameters are the biological and the physical halves of the same design. SRT decides what organisms live in the tank; SOR decides whether they can be kept out of the effluent. A plant fails if either is wrong, and the two are coupled, because the clarifier is what returns the solids that SRT presumes.

Solids retention time (sludge age, mean cell residence time, $\theta_c$) is the average time a particle of biomass spends in the system: the mass of solids under aeration divided by the mass of solids leaving the system per day.

$$\theta_c = \frac{V X}{Q_w X_w + (Q - Q_w) X_e}$$

where $V$ is the aeration-tank volume, $X$ the mixed-liquor suspended solids, $Q_w$ and $X_w$ the wasted flow and its concentration, and $X_e$ the effluent solids. Because a well-run clarifier holds $X_e$ to 10 or 20 mg/L, the second term is usually small and the working form is $\theta_c \approx V X / (Q_w X_w)$. The essential point is that SRT is controlled only by wasting: it is the one process variable an operator sets deliberately each day, by deciding how much sludge to remove. Hydraulic retention time, $\tau = V/Q$, is fixed by the flow and cannot be chosen.

Its significance is that SRT selects the microbial community. An organism can only persist if its net growth rate exceeds its washout rate, that is if $\theta_c > 1/(\mu - k_d)$. Heterotrophs oxidising carbonaceous BOD grow quickly and are retained at 3 to 5 days. Ammonia-oxidising autotrophs grow far more slowly, and slower still in the cold.

Given. Ammonia oxidisers at 12 °C with $\mu_{\max} = 0.35$ d⁻¹ and $k_d = 0.05$ d⁻¹, with a safety factor of 2.5 to absorb diurnal load peaks. Find. The washout and design sludge ages.

$$\theta_{c,\min} = \frac{1}{\mu_{\max} - k_d} = \frac{1}{0.35 - 0.05} = 3.33\ \text{d}, \qquad \boxed{\theta_{c,\text{design}} = 2.5 \times 3.33 = 8.3\ \text{d}}$$

This single calculation explains the whole ladder of activated-sludge variants: high-rate plants at 1 to 3 days for carbon removal only; conventional plants at 5 to 10 days; nitrifying plants at 10 to 20 days, longer in cold Canadian winters; extended aeration at 20 to 30 days, where the sludge is aerobically digested in the aeration basin itself and produces little waste. SRT also governs sludge quantity and quality: a long SRT means more endogenous decay, therefore less waste sludge, a lower food-to-microorganism ratio and better settling, but also higher oxygen demand and a risk of pin floc and dispersed growth if pushed too far. A short SRT gives more, younger, more active sludge that settles poorly and may be dominated by filamentous organisms at very low F/M. SRT is the master variable of the process.

Surface overflow rate is the clarifier's design hydraulic loading: the flow crossing the tank divided by the plan area of its settling zone,

$$\mathrm{SOR} = \frac{Q}{A_s} \quad [\mathrm{m^3/m^2 \cdot d}]$$

Dimensionally it is a velocity, and this is its whole meaning: SOR is the settling velocity of the slowest particle that is still completely removed in an ideal basin. The derivation is worth stating because the conclusion is counter-intuitive. In an ideal rectangular basin of length $L$, width $B$ and depth $H$, a particle entering at the surface is captured if the time it takes to travel the length exceeds the time it takes to fall the depth:

$$\frac{L}{v_h} \ge \frac{H}{v_s}, \qquad v_h = \frac{Q}{BH} \ \Rightarrow\ v_s \ge \frac{QH}{BHL} = \frac{Q}{BL} = \frac{Q}{A_s}$$

$$\boxed{v_c = \frac{Q}{A_s} \quad \text{— the depth cancels}}$$

Removal in an ideal settler is governed by surface area, not by volume or depth, and particles slower than $v_c$ are removed in the ratio $v_s/v_c$, which is also depth-free. Coupling this to Stokes' law converts an overflow rate into a particle size: a 60 µm floc of specific gravity 1.03 at 15 °C settles at

$$v_s = \frac{g(\mathrm{S.G.} - 1)\rho_w d^2}{18\mu} = \frac{9.81 (0.03)(999.1)(60 \times 10^{-6})^2}{18 (1.139 \times 10^{-3})} = 5.2 \times 10^{-5}\ \text{m/s} = 4.5\ \text{m/d}$$

so a basin at 4.5 m³/m²·d would just capture it, and every faster-settling particle as well. Typical secondary-clarifier design values are 16 to 32 m³/m²·d at average flow with a peak ceiling of 40 to 48. Two qualifications keep the theory honest. First, an activated-sludge clarifier does not experience discrete settling but zone settling, so the design must also satisfy a solids loading rate, $\mathrm{SLR} = (Q + Q_R)X/A_s$, typically 4 to 6 kg/m²·h and never above 8 at peak — and note the asymmetry, that SOR excludes the return flow because only the plant flow crosses the weirs, whereas SLR includes it because every kilogram of solids must be thickened however it arrived. Second, although depth cancels in the ideal analysis, real tanks are built 3.5 to 4.5 m deep to provide sludge-blanket storage during peak flows, to resist density currents and scour, and to give the flocculent settling that the ideal model ignores. The engineering conclusion follows directly from the algebra: if a clarifier is overloaded, adding a parallel unit halves the SOR, whereas deepening the existing one does not change it at all.

Part (b) — Mechanism of anaerobic sludge stabilization (10 marks). Anaerobic digestion stabilises primary and waste-activated sludge by converting its degradable organic fraction to methane and carbon dioxide in the absence of oxygen, in a consortium of bacteria and archaea working in an obligate four-stage sequence.

Hydrolysis comes first: extracellular enzymes secreted by fermentative bacteria break particulate and polymeric material — proteins, lipids, cellulose and the cell walls of the waste activated sludge — into soluble amino acids, long-chain fatty acids and sugars. For municipal sludge this step is rate-limiting for the whole process, which is why thermal hydrolysis, ultrasonic and chemical pre-treatments target it. Acidogenesis follows: fast-growing fermentative bacteria convert those soluble monomers to short-chain volatile fatty acids (acetic, propionic, butyric), alcohols, hydrogen and carbon dioxide. Acetogenesis then converts the propionate, butyrate and alcohols to acetate, hydrogen and CO₂. This step is thermodynamically unfavourable unless the hydrogen partial pressure is kept very low, which is achieved only by the hydrogen-consuming methanogens working alongside — an obligate syntrophy that is the reason a digester upset propagates so quickly. Finally, methanogenesis is carried out by strictly anaerobic archaea by two routes, the acetoclastic route accounting for roughly 70 percent of the methane produced:

$$\mathrm{CH_3COOH} \rightarrow \mathrm{CH_4} + \mathrm{CO_2}, \qquad \mathrm{4H_2} + \mathrm{CO_2} \rightarrow \mathrm{CH_4} + \mathrm{2H_2O}$$

The methanogens are the slow, fastidious members of the consortium: they grow at a fraction of the rate of the acidogens, are inhibited below pH 6.6, and are poisoned by oxygen, sulphide, ammonia above roughly 3000 mg/L and heavy metals. The classic failure mode follows directly — overload or a cold shock lets the fast acid-formers outrun the slow methanogens, volatile fatty acids accumulate, alkalinity is consumed, pH falls, and the methanogens are inhibited further in a self-reinforcing "sour digester". The control response is to reduce loading and restore alkalinity, and the routine warning indicator is the volatile-acid-to-alkalinity ratio, held below about 0.25.

Standard mesophilic practice is 35 °C, complete mixing, a solids retention time of 15 to 20 days at a volatile-solids loading of 1.6 to 4.8 kg VS/m³·d, achieving 45 to 60 percent volatile-solids destruction; thermophilic operation at 55 °C is faster and gives better pathogen kill but is less stable. The benefits are the reason digestion is near-universal for plants above a few thousand cubic metres per day: sludge mass and volume are roughly halved, odour and putrescibility are largely eliminated, pathogens are substantially inactivated so the biosolids can meet land-application criteria, dewaterability improves, and the biogas — 60 to 70 percent methane — is a genuine energy return. At the typical yield of 0.35 m³ of methane per kilogram of COD destroyed, a digester destroying 1200 kg COD/d produces $0.35 \times 1200 = 420$ m³ of methane per day, about 15 000 MJ/d at methane's lower heating value of 35.8 MJ/m³, which is commonly enough to heat the digesters themselves and export the balance.