Siphon: canal 50, FSL 201.80, RCC flumed

The canal ($Q$ = 50 m³/s) crosses a drainage of $Q$ = 450 m³/s. Design is done in four steps: type of work, drainage waterway, canal waterway (flumed trough) and bed/water levels of the trough.

Assumptions: Flumed (RCC) trough width 18 m as given; $n$ = 0.015 for RCC; Lacey's $f$ = 1; slab 0.6 m; barrel $n$ = 0.015 and barrel length ≈ trough width + walls ≈ 19 m. Depth of canal = 201.80 − 200.00 = 1.80 m.

1. Type of structure

HFL of the drainage (200.50 m) is above the canal bed (200.00 m) by 0.50 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). A siphon aqueduct is suitable. Drainage depth at HFL = 200.50 − 198.00 = 2.50 m.

2. Drainage waterway

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

$$P = 4.75\sqrt{Q} = 4.75\sqrt{450} = 100.8\ \text{m}$$

Provide 11 spans of 10 m with piers 1.5 m thick: clear waterway = 11 × 10 = 110 m (> 100.8 m). Overall length = 110 + 10 × 1.5 = 125.0 m.

Check: mean velocity = Q/(L·d) = 450/(110 × 2.50) = 1.64 m/s. Lacey's regime depth $R = 0.47(Q/f)^{1/3}$ = 3.60 m (with $f$ = 1); foundations of piers and abutments are taken about 1.5R = 5.4 m below HFL, with cut-offs and aprons for scour.

3. Canal waterway (flumed rectangular trough)

Normal canal: area $A_1=(B+zD)D$ = (36 + 1.5 × 1.8) × 1.8 = 69.66 m², $V_1$ = 50/69.66 = 0.718 m/s, $h_{v1}=V_1^2/2g$ = 0.0263 m. Top width $T=B+2zD$ = 41.40 m.

To save cost, the canal is flumed to a rectangular trough of width $B_t$ = 18 m, with the same full supply depth 1.8 m (the trough is as long as the drainage waterway, 125.0 m).

• Trough area = 18 × 1.8 = 32.40 m²; $V_t$ = 50/32.40 = 1.543 m/s (acceptable, < 2.5 m/s); $h_{vt}$ = 0.1214 m
• Froude number = 0.37 < 1, flow is subcritical, so there is no hydraulic jump.
• Hydraulic radius $R=A/P$ = 32.40/21.60 = 1.500 m; friction slope $S=(nV/R^{2/3})^2$ = 0.000312 (Manning's n = 0.015); friction loss in trough $h_f=S\,L$ = 0.000312 × 125.0 = 0.039 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 = 41.40 − 18 = 23.4 m
• Outlet length = 3 × (T − B_t)/2 = 35.1 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):

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

$W_1$ = 201.800 m, so $W_2$ = 201.676 m, $W_3$ = 201.637 m, $W_4$ = 201.685 m. Trough bed = water level − 1.8 m.

Total loss of head = 201.800 − 201.685 = 0.115 m, so the canal bed downstream of the work is 0.115 m lower than upstream.

Check: with trough slab 0.6 m thick, soffit of the barrels = 199.24 m, i.e. 1.24 m above the drainage bed; HFL stands 1.26 m above the soffit, which gives the siphonic head. Since the clear height is less than 1.5 m, the drainage bed is depressed under the barrels to RL 197.74 m (1.5 m clear height) with a 1 in 4 slope at both ends.

5. Design of the siphon (drainage barrels)

• Barrels: 11 openings of 10 m clear width × 1.5 m clear height (drainage floor depressed to 197.74 m), area = 11 × 10 × 1.5 = 165 m².
• Velocity in barrels $V_b$ = 450/165 = 2.73 m/s (within the usual 2–3 m/s); $h_{vb}$ = 2.73²/19.62 = 0.379 m.
• One barrel: $R$ = 15/23 = 0.652 m; friction loss $h_{fb}=n^2V_b^2L/R^{4/3} = 0.015^2 \times 2.73^2 \times 19/0.652^{4/3}$ = 0.056 m.
• Head loss through the siphon = entry ($0.5h_{vb}$) + exit ($1.0h_{vb}$) + friction = 0.190 + 0.379 + 0.056 = 0.625 m (approach velocity neglected).
• Upstream HFL = 200.50 + 0.63 = 201.13 m, so the drainage banks and guide banks upstream are raised above this level, with free board.
• Uplift on the roof (canal dry, drainage at HFL): the maximum is just inside the entry, with pressure head = 201.125 − 1.5 × 0.379 − 199.237 (soffit) = 1.32 m, so the uplift = 9.81 × 1.32 = 12.9 kN/m². This is less than the slab weight 24 × 0.6 = 14.4 kN/m², so the roof is safe against uplift.

Answer: drainage waterway 110 m clear (11 × 10 m, overall 125.0 m); canal flumed to 18 m wide trough; bed/FSL at the four sections = 200.00/201.80, 199.88/201.68, 199.84/201.64, 199.88/201.68 m; siphon barrels 11 × (10 m × 1.5 m), V = 2.73 m/s, head loss 0.63 m (upstream HFL 201.13 m), maximum uplift on the roof 12.9 kN/m².