Question 2 of 7: Problem 2 — Particle characteristics, chemistry of solutions and gases

Question 2: Problem 2 — Particle characteristics, chemistry of solutions and gases (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) Settleable particle removal, and the interference of particles with disinfection (9 marks)

Part 1 — two key engineering principles governing the removal of settleable particles.

Principle A: removal in an ideal sedimentation basin is governed by the overflow rate, not by detention time or depth. The surface loading or overflow rate is defined as

where Q is the flow and $A_s$ the plan surface area of the basin. Hazen's ideal-basin analysis shows that a discrete particle entering at the water surface will just be captured if its terminal settling velocity equals $v_o$, and that every particle with $v_s \ge v_o$ is removed regardless of where it enters, while particles slower than $v_o$ are removed only in the fraction $v_s/v_o$. Depth cancels out of the derivation entirely. The design consequence is direct and often counter-intuitive: a basin that is under-performing is enlarged in plan area, or is helped by lamella plates and tube settlers that multiply the effective projected settling area within the same footprint, whereas making it deeper achieves nothing for discrete particles. The settling velocity itself follows Stokes' law in the laminar regime,

which shows the quadratic dependence on particle diameter and the inverse dependence on viscosity. Both are exploited in practice: chemical coagulation and flocculation deliberately grow d before the particle reaches the clarifier, and because μ rises about 50 % between 20 °C and 4 °C, cold Canadian winter operation is the governing design case for primary clarifiers.

Principle B: the ideal-basin result is only realised if the hydraulics are quiescent and short-circuiting is suppressed. Hazen's theory assumes uniform horizontal plug flow, and every departure from it — a jetting inlet, density currents driven by a few tenths of a degree of temperature difference between the influent and the basin contents, wind-driven surface circulation, or an over-loaded effluent weir drawing a rising current — carries solids straight to the outlet in a fraction of the nominal detention time. The engineering measures that enforce quiescence are therefore as important as the area: inlet baffles and diffuser walls to dissipate the entering momentum, a length-to-width ratio of roughly 4:1 or greater so the flow settles into a plug, effluent weir loading rates held below about 250 m³/m·d, and continuous sludge collection so that accumulated sludge does not reduce the effective volume or go septic and float. Tracer testing that returns a $t_{10}/\bar{t}$ ratio well below unity is the diagnostic signature that these measures have failed.

Part 2 — two key engineering principles by which suspended particles compromise disinfection.

Principle C: particles physically shield embedded organisms from the disinfectant. Disinfection is a surface phenomenon: a chemical oxidant must diffuse to and react with the cell, and ultraviolet light must actually strike it. A bacterium or virus adsorbed to or occluded within a floc particle of a few tens of micrometres is protected on both counts. The oxidant is consumed at the particle exterior before it can penetrate to the core, and UV is absorbed and scattered by the particle so that the interior receives a fluence orders of magnitude below the reactor average. The observable consequence is the well-documented “tailing” of the disinfection curve: Chick–Watson kinetics, $\ln(N/N_0) = -k C^n t$, predicts log-linear inactivation without limit, but a real effluent containing particles flattens out at a residual survivor population of particle-associated organisms that no practical increase in dose will reach. Adding contact time or dose past that point buys nothing; the only effective remedy is upstream solids removal, which is why tertiary filtration is mandatory ahead of any effluent reuse disinfection duty.

Principle D: particles and their associated organic matter exert a disinfectant demand and destroy the transmittance the design depends on. Before any inactivation credit can be earned, the applied oxidant must first satisfy the effluent's demand — reduced species, nitrite, sulphide, ferrous iron and the biodegradable organic fraction carried on the solids all consume chlorine or ozone. The residual C available for the CT product is what remains afterwards, so a rise in effluent TSS silently erodes CT even at a constant applied dose, and the extra chlorine consumed reacts with the organic matter to form trihalomethanes and haloacetic acids. The equivalent penalty for an ultraviolet system is the ultraviolet transmittance: UV dose is the product of intensity and exposure time, and intensity decays through the water according to Beer–Lambert absorption, so a fall in UVT from 65 % to 55 % at 254 nm can cut the delivered dose by a third and force additional banks of lamps into service. This is why effluent turbidity ahead of disinfection is itself a regulated surrogate parameter, and why filtered drinking water in Canada must meet 0.3 NTU in 95 % of measurements before the disinfection credits in the GCDWQ multi-barrier framework may be claimed.

(ii) Hardness of the lake water and its classification (6 marks)

Given. Average concentrations of the divalent cations in a lake water sampled near a rock quarry, and the atomic weights the question directs be used:

Find. The total hardness expressed in mg/L as CaCO₃, and the classification of the water as soft, moderately hard or hard.

Approach. Hardness is a measure of the total charge carried by the polyvalent metallic cations, so it is expressed on an equivalent basis with calcium carbonate as the arbitrary reference compound. Each ion is converted by the ratio of the equivalent weight of CaCO₃ to its own equivalent weight, and the converted contributions are then summed.

1. Build the calcium carbonate reference. Using the printed atomic weights, $$M_{\text{CaCO}_3} = 40 + 12 + 3(16) = 100\ \text{g/mol}$$ Carbonate carries two units of charge, so the equivalent weight is $$EW_{\text{CaCO}_3} = \frac{100}{2} = 50\ \text{g/eq}$$ This value of 50 is the constant that appears in every hardness conversion.
2. Compute the equivalent weight of each cation. Each of the three ions is divalent, so its equivalent weight is half its atomic weight: $$EW_{\text{Ca}} = \frac{40}{2} = 20 \qquad EW_{\text{Mg}} = \frac{24}{2} = 12 \qquad EW_{\text{Fe}} = \frac{56}{2} = 28$$ all in g/eq. The question prints all three atomic weights explicitly, so these printed integers are used rather than the more precise textbook values of 24.31 and 55.85; the difference is well under 1 % and does not affect the classification.
3. Convert each cation to the CaCO₃ basis. The general conversion is $$C_{\text{as CaCO}_3} = C_{\text{ion}} \times \frac{EW_{\text{CaCO}_3}}{EW_{\text{ion}}}$$ Applying it to calcium gives $$C_{\text{Ca, as CaCO}_3} = 80 \times \frac{50}{20} = 200\ \text{mg/L as CaCO}_3$$ For magnesium the multiplier is the much larger 50/12 = 4.167, giving $70 \times 4.167 = 291.7$ mg/L as CaCO₃, and for ferrous iron the multiplier is 50/28 = 1.786, giving $20 \times 1.786 = 35.7$ mg/L as CaCO₃.
4. Sum to obtain the total hardness. The conventional definition of total hardness counts calcium and magnesium, which in almost all natural waters account for essentially the whole of it: $$TH_{\text{Ca+Mg}} = 200 + 291.7$$ $$\boxed{TH_{\text{Ca+Mg}} = 492\ \text{mg/L as CaCO}_3}$$ Ferrous iron is nevertheless a divalent metallic cation and contributes to hardness on the strict definition, so the inclusive total is also reported: $$TH_{\text{total}} = 492 + 35.7 = 528\ \text{mg/L as CaCO}_3$$
5. Classify the water. On the three-band scale implied by the question's own wording — soft below 75, moderately hard from 75 to 150, hard above 150 mg/L as CaCO₃ — both totals fall decisively in the hard band. On the four-band scale used in the Guidelines for Canadian Drinking Water Quality, which adds a very hard category above 180 mg/L as CaCO₃, the water is very hard. Either way it also exceeds the 500 mg/L as CaCO₃ level at which the GCDWQ aesthetic objective notes that consumers find the water unacceptable, so softening would be warranted for a municipal supply.

The internal structure of the answer carries the engineering message. Magnesium is reported at a lower mass concentration than calcium, 70 against 80 mg/L, yet contributes almost 50 % more hardness — 292 against 200 mg/L as CaCO₃ — because its equivalent weight is only 12 against calcium's 20, so each milligram of magnesium carries 1.67 times as much charge. A hardness calculation performed on a mass basis rather than an equivalent basis inverts the relative importance of the two ions and is simply wrong. The split also matters operationally, because the magnesium fraction is what forces the excess-lime step in lime–soda softening: calcium carbonate precipitates at about pH 10.3, but magnesium hydroxide requires pH 11 or above, so a water in which magnesium dominates costs considerably more lime and produces more sludge than its total hardness alone would suggest.

(iii) Aeration in activated sludge, and the influence of temperature on aerator sizing (5 marks)

Aeration in an activated-sludge system performs two distinct duties that must both be satisfied, and the larger of the two governs the equipment.

The first duty is oxygen supply. The heterotrophic biomass oxidises carbonaceous BOD and the autotrophic nitrifiers oxidise ammonia, and both use dissolved oxygen as the terminal electron acceptor. The oxygen requirement is set by a stoichiometric balance over the aeration basin — approximately the BOD removed less the oxygen equivalent of the biomass wasted, plus 4.57 kg O₂ per kilogram of ammonia nitrogen nitrified — and typically amounts to 1.0 to 1.5 kg O₂ per kilogram of BOD applied. The second duty is mixing: the mixed liquor suspended solids must be kept in suspension and in intimate contact with the substrate, which imposes a floor on the air delivered of roughly 20 to 30 m³ of air per 1000 m³ of tank volume per minute for diffused systems, or about 20 W/m³ of power input for mechanical aerators. In a lightly loaded plant, or in a plant operating at a small fraction of its design flow in its early years, mixing rather than oxygen demand determines the minimum number of blowers that must run.

The transfer itself follows two-film theory. The oxygen transfer rate delivered under actual process conditions is

in which $K_L a$ is the volumetric mass-transfer coefficient, $C^{*}_{\infty,T}$ the saturation concentration of dissolved oxygen at the operating temperature, $C_L$ the dissolved oxygen maintained in the basin (normally 1.5 to 2.0 mg/L), and $\alpha$ and $\beta$ the correction factors for the effect of wastewater constituents on the transfer coefficient and on saturation respectively. The driving force is the concentration difference in parentheses; everything about temperature acts through that term or through the coefficient.

Temperature acts in three directions, two of which are adverse, and the net effect is that summer governs.

First, the saturation concentration falls sharply as the water warms — from about 11.3 mg/L at 10 °C to 9.09 mg/L at 20 °C and 7.56 mg/L at 30 °C. Since the basin must still be held at 2 mg/L, the available driving force $(C^{*} - C_L)$ collapses from 9.3 to 5.6 mg/L over that range, a loss of 40 % of the transfer potential. Second, the biological oxygen demand rises with temperature, because microbial reaction rates follow a modified Arrhenius relation $k_T = k_{20}\theta^{(T-20)}$ with θ between 1.04 and 1.08 for carbonaceous removal and endogenous respiration — so a 10-degree rise from 20 to 30 °C increases the oxygen consumption rate by 50 % or more. Third, working weakly in the opposite direction, the transfer coefficient itself improves with temperature at $\theta = 1.024$, which recovers only about 24 % over the same 10 degrees, far too little to offset the other two.

The design conclusion follows directly. Because supply capability falls while demand rises, the peak summer mixed-liquor temperature is the governing case for the number and size of blowers or surface aerators, and installed capacity must be sized at that condition — typically the maximum-month load at the maximum expected liquid temperature. Winter is not thereby irrelevant: cold temperatures slow the nitrifiers severely, so a plant required to nitrify year-round in a Canadian climate must carry a longer solids retention time in winter, which means a higher MLSS inventory and therefore a higher mixing floor and a greater endogenous demand. The practical outcome is a wide required turndown, and the standard response is multiple units — commonly three or four blowers sized so that any one may be out of service — with variable-frequency drives or inlet guide vanes and dissolved-oxygen-based automatic control, rather than one or two large machines throttled inefficiently. Sizing on the annual average temperature is a classic error that leaves the plant short of oxygen every August, when effluent quality is most closely watched because receiving-water assimilative capacity is at its own seasonal minimum.