(a) The following information is available regarding the relationship between trap efficiency and capacity-inflow ratio for a…

(a) Useful life of the reservoir by the incremental method (20 marks)

Data. Initial capacity C₀ = 30 Mm³ and annual inflow I = 60 Mm³, so the initial capacity-inflow ratio is 30/60 = 0.5. The sediment inflow is 360 000 t/year at 1200 kg/m³, so the annual volume of sediment brought in is

360 000 × 1000/1200 = 300 000 m³/year = 0.30 Mm³/year.

The useful life ends when 80% of the initial capacity is silted, that is when 0.8 × 30 = 24 Mm³ of storage is lost and the capacity is reduced to 6 Mm³ (capacity-inflow ratio 0.1).

Method. The trap efficiency falls as the capacity shrinks, so the life is worked out in steps, each covering a fall of 6 Mm³ in capacity, which is 0.1 in the capacity-inflow ratio. In each step the mean trap efficiency is taken from the table, the annual deposit is 0.30 × (mean trap efficiency), and the time is (volume lost)/(annual deposit).

• Step 1, capacity 30 to 24 Mm³ (ratio 0.5 to 0.4): mean trap efficiency = (96 + 95.5)/2 = 95.75%. Deposit = 0.30 × 0.9575 = 0.2873 Mm³/year. Time = 6/0.2873 = 20.9 years.
• Step 2, capacity 24 to 18 Mm³ (ratio 0.4 to 0.3): mean = (95.5 + 95)/2 = 95.25%. Deposit = 0.2858 Mm³/year. Time = 6/0.2858 = 21.0 years.
• Step 3, capacity 18 to 12 Mm³ (ratio 0.3 to 0.2): mean = (95 + 93)/2 = 94%. Deposit = 0.2820 Mm³/year. Time = 6/0.2820 = 21.3 years.
• Step 4, capacity 12 to 6 Mm³ (ratio 0.2 to 0.1): mean = (93 + 87)/2 = 90%. Deposit = 0.2700 Mm³/year. Time = 6/0.2700 = 22.2 years.

Probable useful life = 20.9 + 21.0 + 21.3 + 22.2 ≈ 85 years.

As a check, if the initial trap efficiency of 96% held throughout, the life would be 24/(0.30 × 0.96) = 83 years. The real life is longer because the efficiency falls, so less sediment is trapped as the reservoir fills.

(b) Backwash rate and head loss of a sand filter (15 marks)

Data. d = 0.55 mm = 0.055 cm, G = 2.67, ν = 1.3 × 10⁻² cm²/s, depth L = 75 cm, porosity ε = 0.40, expansion 50%.

(b)(i) Backwash rate

Settling velocity of a grain. With C_D = 24/R (Stokes' law), the settling velocity is

v_s = g(G − 1)d²/(18ν) = 981 × 1.67 × 0.055²/(18 × 0.013) = 21.2 cm/s

The grain Reynolds number is R = v_s d/ν = 21.2 × 0.055/0.013 ≈ 90, much larger than 1, so Stokes' law is only an approximation here, and it overestimates v_s. It is used because the problem prescribes C_D = 24/R.

Porosity of the expanded bed. The volume of sand is constant, so (1 − ε_e)L_e = (1 − ε)L. With L_e = 1.5L, 1 − ε_e = 0.60/1.5 = 0.40, so ε_e = 0.60.

Backwash velocity. The velocity that fluidises the bed to this porosity, from the Fair and Hatch relation v_b = v_s ε_e^4.5, is

v_b = 21.2 × 0.60^4.5 = 21.2 × 0.1005 = 2.13 cm/s

Backwash rate ≈ 2.1 cm/s = 0.0213 m/s, that is about 77 m/h (1.28 m³/m²/min).

(b)(ii) Head loss through the expanded bed

In a fluidised bed the head loss equals the submerged weight of the sand per unit area, and is unchanged by the expansion:

h_L = (1 − ε)(G − 1)L = (1 − 0.40)(2.67 − 1)(0.75) = 0.60 × 1.67 × 0.75 = 0.75 m

The same result comes from the expanded bed: (1 − 0.60)(1.67)(1.125) = 0.75 m.

(c)(i) Velocity in a circular sewer as the flow varies (8 marks)

In Manning's equation v = (1/n)R^(2/3)S^(1/2), with n and the slope S fixed, the velocity depends only on the hydraulic mean depth R = A/P. As the depth of flow d in a circular sewer of diameter D changes, so does R, and so the velocity and the discharge change.

• At full depth and at half depth, R = D/4, so the velocity is the same, V_full.
• Between half depth and about 0.81D, R rises above D/4 (largest R at 0.81D), so the velocity is greater than the full-pipe value, with a maximum of about 1.14 V_full at d = 0.81D. The discharge is maximum, about 1.07 Q_full, at d = 0.95D.
• Below half depth, R falls, and so does the velocity. It is about 0.70 V_full at d = 0.25D and about 0.40 V_full at d = 0.1D.

Design consequences. Sewage flow varies through the day, from a minimum at night to a peak in the morning and evening. The sewer must give a self-cleansing velocity at low flow, so that solids do not settle, commonly about 0.6–0.8 m/s at the design peak flow and not much below 0.45 m/s at minimum flow. It must also keep the velocity at the maximum flow below the non-scouring limit that the pipe material can withstand (about 2.4–3 m/s), so that the pipe is not eroded. The gradient is chosen to satisfy both conditions.

(c)(ii) Threshold odour number (7 marks)

The threshold odour number (TON) is (A + B)/A, where A is the volume of the sample and B is the volume of odour-free water: TON = (20 + 180)/20 = 10.

A drinking water should have TON of about 3 or less; 10 is objectionable to consumers, and needs treatment (aeration, activated carbon, chlorination). Individual sensitivity to odour varies, so the test is done by a panel of at least five, preferably ten, persons in an odour-free room, using dilutions in odour-free water, with blanks, in increasing concentration. The threshold is the geometric mean of the individual thresholds.