(a) (i) Determine the volume of sludge before and after digestion and percent reduction for 500 kg of primary sludge with 5%…

(a)(i) Volume of sludge before and after digestion

Take the 500 kg as the mass of primary sludge, containing 5% solids: dry solids = 25 kg, of which volatile solids = 0.55 × 25 = 13.75 kg and fixed solids = 11.25 kg.

Before digestion: mass of sludge = 500 kg, specific gravity 1.01, so

V₁ = 500/(1.01 × 1000) = 0.495 m³.

Digestion destroys 90% of the volatile solids, leaving 0.1 × 13.75 = 1.375 kg of volatile solids; the fixed solids are unchanged. Digested dry solids = 11.25 + 1.375 = 12.625 kg. At 10% solids the digested sludge weighs 12.625/0.10 = 126.25 kg, and with specific gravity 1.04

V₂ = 126.25/(1.04 × 1000) = 0.121 m³.

Percentage reduction in volume = (0.495 − 0.121)/0.495 × 100 = 75.5%.

The reduction in solids is (25 − 12.625)/25 = 49.5%; the volume falls further because the digested sludge is also thicker and denser. If the 500 kg is read as dry solids rather than wet sludge, every mass and volume above is multiplied by 20 (V₁ = 9.90 m³, V₂ = 2.43 m³) and the percentage reduction is the same 75.5%.

(a)(ii) Rapid sand filter area

Water demand = 2,00,000 × 150 = 3 × 10⁷ litres/day = 30,000 m³/day. At a loading rate of 200 m³/day per m²,

total filter area required = 30,000/200 = 150 m².

With four filters, the area of each = 150/4 = 37.5 m² (a unit of, say, 7.5 m × 5 m).

If the design loading rate must still be met with one filter out of service for backwashing or repair, the three remaining filters must carry the whole flow: area per filter = 30,000/(200 × 3) = 50 m². Four filters of 50 m² would be provided, giving a total of 200 m² and a normal-operation loading of 150 m³/day per m².

(b) The flow mass curve and reservoir storage

A flow mass curve (Rippl diagram) is a plot of the cumulative volume of inflow at a site against time, drawn from the record of monthly or daily flows over as many years as are available. Its ordinate at any date is the total volume that has passed since the start of the record, so its slope at any point is the rate of inflow: steep during the monsoon, nearly flat in the dry season, never falling. The drawing is a rising, stepped curve made of a succession of humps, each hump one year's flow, with the wet-season rise and the dry-season flattening repeated year after year.

For a constant demand the demand is a straight line whose slope equals the demand rate, and the cumulative demand line is drawn parallel to it. To find the storage required to meet that demand without failure: from every peak of the mass curve (the start of each dry period) draw a line parallel to the demand line. Where this tangent runs above the mass curve, the inflow is falling short of the demand and the reservoir is being drawn down; the largest vertical distance between the tangent and the mass curve below it, measured before the curve climbs back to meet the tangent, is the storage that must have been in the reservoir at the peak to carry the demand through that dry period. The greatest of these intercepts, taken over all the years of record, is the required storage capacity. If a tangent from a peak never meets the curve again, the demand cannot be met from that point on and is greater than the mean flow. The same construction, run in reverse (tangents from the troughs), gives the volume of water that would spill, and drawing the demand line tangent to the curve from a given storage gives the safe yield for a reservoir of known capacity. The dry-weather flow before the first peak must be ignored, and the curve should cover the critical (driest) sequence of years.

(c) Modes of failure of a gravity dam

A gravity dam resists the water pressure by its own weight. Consider unit length of the dam, base width b, with the resultant of the vertical forces ΣV (self weight W downward, uplift U upward, ΣV = W − U) and of the horizontal forces ΣH (water pressure P = ½γwH², silt pressure, wave pressure, earthquake). The sketches for the following modes are the triangular section with these forces marked.

Overturning about the toe. The horizontal forces and the uplift produce an overturning moment about the toe, the self weight a restoring moment. Failure occurs if the resultant passes outside the base; the factor of safety is FS = ΣM(restoring)/ΣM(overturning) = W·x̄/(P·H/3 + U·x̄u), and it should be at least 1.5 (2 for reservoir full without earthquake). The sketch shows the resultant R of ΣV and ΣH striking the base at distance x̄ from the toe, the eccentricity e = b/2 − x̄.

Sliding. The horizontal thrust tends to push the dam along the base or a horizontal plane within it. The resistance is friction, and in a well-bonded dam the shear strength of the concrete or rock too. Factor of safety against sliding FS = μΣV/ΣH ≥ 1 (μ, the coefficient of friction, about 0.65–0.75); the more complete measure is the shear friction factor SFF = (μΣV + q·b)/ΣH ≥ 4 to 5, with q the permissible shear stress of the joint. Keys, a foundation inclined against the thrust and drainage to reduce uplift are the remedies.

Crushing or compression at the toe. The vertical stress at the base varies linearly, pmax/min = (ΣV/b)(1 ± 6e/b), maximum at the toe when the reservoir is full and at the heel when empty. Failure occurs if the principal stress there, σ₁ = pv sec²α − p' tan²α (α the slope of the downstream face, p' the tail-water pressure), exceeds the allowable compressive strength of the concrete or the foundation; the sketch shows the trapezoidal stress diagram and the principal stress at the toe.

Tension. If the eccentricity e exceeds b/6 the stress at the heel (reservoir full) or toe (reservoir empty) becomes tensile, pmin < 0. Concrete cracks in tension, the crack lets full water pressure in as uplift over its length, e increases further and the failure progresses. The middle-third rule, e ≤ b/6, i.e. the resultant kept within the middle third of the base, is the criterion; the elementary profile that just satisfies it with no tension and no sliding is the right-angled triangle of base width b = H/√(S − c), S the specific gravity of concrete and c the uplift factor.

Other modes are failure of the foundation (bearing failure, seepage and piping under the dam, which is why cut-off walls, grout curtains and drainage galleries are provided) and, for sections whose limiting dimension is governed by earthquake, failure by the inertia forces of the dam and the hydrodynamic pressure on its face.