Siphon aqueduct: Q 30, drainage 800

The canal ($Q$ = 30 m³/s) crosses a drainage of $Q$ = 800 m³/s. The design covers the type of work, the drainage waterway, the canal waterway (flumed trough) with the bed and water levels of the trough, the siphon barrels with the afflux, and the uplift on the barrel roof.

Assumptions: Side slope 1.5:1 as given; Lacey's $f$ = 1; trough $n$ = 0.015; slab 0.6 m thick, concrete 24 kN/m³.

1. Type of structure

The HFL of the drainage (231.25 m) is 1.25 m above the canal bed (230.00 m), so the trough cannot be kept above HFL without a very high, costly structure. The drainage is passed under the trough through barrels flowing under pressure (siphonic action), so a siphon aqueduct is suitable. Drainage depth at HFL = 231.25 − 227.60 = 3.65 m.

2. Drainage waterway

Lacey's regime perimeter (for a natural stream):

$$P = 4.75\sqrt{Q} = 4.75\sqrt{800} = 134.4\ \text{m}$$

Provide 14 spans of 10 m with piers 1.5 m thick: clear waterway = 14 × 10 = 140 m (> 134.4 m). Overall length = 140 + 13 × 1.5 = 159.5 m.

Check: mean velocity = Q/(L·d) = 800/(140 × 3.65) = 1.57 m/s. Lacey's regime depth $R = 0.47(Q/f)^{1/3}$ = 4.36 m (with $f$ = 1); the foundations of piers and abutments are taken about 1.5R = 6.5 m below HFL, with cut-offs and aprons for scour.

3. Canal waterway (flumed rectangular trough)

Normal canal: area $A_1=(B+zD)D$ = (23 + 1.5 × 1.7) × 1.7 = 43.44 m², $V_1$ = 30/43.44 = 0.691 m/s, $h_{v1}=V_1^2/2g$ = 0.0243 m. Top width $T=B+2zD$ = 28.10 m.

To save cost, the canal is flumed to a rectangular trough of width $B_t$ = 10 m, with the same full supply depth of 1.7 m (the trough is as long as the drainage waterway, 159.5 m).

• Trough area = 10 × 1.7 = 17.00 m²; $V_t$ = 30/17.00 = 1.765 m/s (acceptable, < 2.5 m/s); $h_{vt}$ = 0.1587 m
• Froude number = 0.43 < 1, so the flow is subcritical and there is no hydraulic jump.
• Hydraulic radius $R=A/P$ = 17.00/13.40 = 1.269 m; friction slope $S=(nV/R^{2/3})^2$ = 0.000510 (Manning's n = 0.015); friction loss in the trough $h_f=S\,L$ = 0.000510 × 159.5 = 0.081 m

4. Transitions and levels

  FSL ---.                            .--- FSL
         '--.______________________.--'
  bed ---.                            .--- bed
         '--.______________________.--'
  normal  inlet     flumed trough     outlet  normal
  canal   (2:1)                      (3:1)   canal

Wing-wall (straight) transitions: inlet splay 2:1 (contraction) and outlet splay 3:1 (expansion), measured on each side:

• Inlet length = 2 × (T − B_t)/2 = 28.10 − 10 = 18.1 m
• Outlet length = 3 × (T − B_t)/2 = 27.2 m

Bernoulli's equation between sections (loss in contraction = 0.3 Δh_v, in expansion = 0.5 Δh_v; Δh_v = difference of velocity heads):

$$\begin{aligned} W_2 &= W_1 + h_{v1} - h_{vt} - 0.3(h_{vt}-h_{v1}) \\ W_3 &= W_2 - h_f \\ W_4 &= W_3 + h_{vt} - h_{v1} - 0.5(h_{vt}-h_{v1}) \end{aligned}$$

$W_1$ = 231.700 m, so $W_2$ = 231.525 m, $W_3$ = 231.444 m, $W_4$ = 231.511 m. Trough bed = water level − 1.7 m.

Total loss of head = 231.700 − 231.511 = 0.189 m, so the canal bed downstream of the work is 0.189 m lower than upstream.

Check: with a trough slab 0.6 m thick, the soffit of the barrels = 229.14 m, i.e. 1.54 m above the drainage bed; HFL stands 2.11 m above the soffit, which gives the siphonic head.

5. Siphon barrels and afflux

With the natural bed (227.60 m), the clear height of the barrels would be only 1.54 m and the barrel velocity 800/(140 × 1.54) = 3.7 m/s, which is too high. The barrel floor is therefore depressed to 226.84 m (0.76 m below the drainage bed, with 1 in 4 slopes at both ends), giving a clear height of 2.3 m: 14 barrels of 10 m × 2.3 m, barrel length (along the canal) $L$ = 10 + 1.2 = 11.2 m (trough width plus side walls).

• Area = 140 × 2.3 = 322 m²; $V$ = 800/322 = 2.48 m/s; $V^2/2g$ = 0.315 m
• Hydraulic radius of one barrel $R$ = (10 × 2.3)/(2(10 + 2.3)) = 0.935 m
• Unwin's formula with $f_1$ = 0.505 and $f_2=0.00316\,(1+0.1/R)$ = 0.00350:

$$h_L=\left(1+0.505+0.00350\times\frac{11.2}{0.935}\right)\times0.315 = 1.547\times0.315 = 0.49\ \text{m}$$

Afflux ≈ 0.49 m, so the upstream HFL = 231.25 + 0.49 = 231.74 m; the upstream flood banks and the canal banks near the work are raised to this level plus free board.

6. Uplift on the barrel roof (trough floor)

Severe case, full static head at the upstream end: $p$ = 9.81 × (231.74 − 229.14) = 25.5 kN/m².

• Slab weight = 24 × 0.6 = 14.4 kN/m²; with the canal running, add 9.81 × 1.7 = 16.7 kN/m², total 31.1 kN/m² > 25.5 kN/m² (safe).
• With the canal empty, net uplift = 25.5 − 14.4 = 11.1 kN/m². The roof slab is designed as a reinforced slab spanning between the piers for this uplift and is anchored to them.

Answer: drainage waterway 140 m clear (14 × 10 m, overall 159.5 m); canal flumed to a 10 m wide trough; bed/FSL at the four sections = 230.00/231.70, 229.83/231.53, 229.74/231.44, 229.81/231.51 m; barrels 10 m × 2.3 m with afflux ≈ 0.49 m; net uplift on the roof ≈ 11.1 kN/m² with the canal empty.