--- title: "Cross drainage: Q 40, 45 cumec drainage" subject: "Irrigation and Drainage Engineering" chapter: 8 key: "semester_6/irrigation_and_drainage_engineering:cd-q40-bed30-fsd16-syphonic" source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/cd-q40-bed30-fsd16-syphonic site: BCT Engineering --- # Cross drainage: Q 40, 45 cumec drainage Irrigation and Drainage Engineering · Chapter 8: Cross-Drainage structures - Asked in: 2072 Magh, 6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/cd-q40-bed30-fsd16-syphonic - Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-8 ## Question Design a suitable cross drainage (water way, bed levels of different section and design of transitions) works if the following data at the crossing of a canal and drainage are given. Canal: Q = 40 m³/s, Bed width = 30 m, FSD of canal = 1.6 m, Bed level = 206.4 m, Side slope = 1½ H:1V. Drainage: Q = 45 m³/s, HFL = 207 m, Bed level = 204.5 m, General ground level = 206.50 m. ## Short answer Siphon aqueduct with drainage waterway 36 m clear (6@6 m spans). Canal flumed to 14 m wide trough, depth 1.6 m; inlet 20.8 m (2:1), outlet 31.2 m (3:1). Critical levels: bed/FSL: 206.40/208.00, 206.23/207.83, 206.21/207.81, 206.27/207.87 m. ## Answer The canal ($Q$ = 40 m³/s) crosses a drainage of $Q$ = 45 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:** Lacey's $f$ = 1; trough $n$ = 0.015; slab 0.6 m. #### 1. Type of structure HFL of the drainage (207.00 m) is above the canal bed (206.40 m) by 0.60 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 = 207.00 − 204.50 = 2.50 m. #### 2. Drainage waterway Lacey's regime perimeter (for a natural stream): $$ P = 4.75\sqrt{Q} = 4.75\sqrt{45} = 31.9\ \text{m} $$ Provide **6 spans of 6 m** with piers 1.0 m thick: clear waterway = 6 × 6 = **36 m** (> 31.9 m). Overall length = 36 + 5 × 1.0 = **41.0 m**. Check: mean velocity = Q/(L·d) = 45/(36 × 2.50) = 0.50 m/s. Lacey's regime depth $R = 0.47(Q/f)^{1/3}$ = 1.67 m (with $f$ = 1); 1.5R = 2.5 m below HFL gives RL 204.5 m, which is only at the natural bed and above the depressed floor under the barrels (204.11 m, see below), so scour does not govern; the foundations of piers and abutments are taken at least 1.5 m below the depressed floor (about RL 202.6 m), with cut-offs and aprons. #### 3. Canal waterway (flumed rectangular trough) Normal canal: area $A_1=(B+zD)D$ = (30 + 1.5 × 1.6) × 1.6 = 51.84 m², $V_1$ = 40/51.84 = 0.772 m/s, $h_{v1}=V_1^2/2g$ = 0.0303 m. Top width $T=B+2zD$ = 34.80 m. To save cost, the canal is flumed to a rectangular trough of width $B_t$ = **14 m**, with the same full supply depth 1.6 m (the trough is as long as the drainage waterway, 41.0 m). - Trough area = 14 × 1.6 = 22.40 m²; $V_t$ = 40/22.40 = 1.786 m/s (acceptable, < 2.5 m/s); $h_{vt}$ = 0.1625 m - Froude number = 0.45 < 1, flow is subcritical, so there is no hydraulic jump. - Hydraulic radius $R=A/P$ = 22.40/17.20 = 1.302 m; friction slope $S=(nV/R^{2/3})^2$ = 0.000504 (Manning's n = 0.015); friction loss in trough $h_f=S\,L$ = 0.000504 × 41.0 = 0.021 m #### 4. Transitions and levels ```text 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 = 34.80 − 14 = **20.8 m** - Outlet length = 3 × (T − B_t)/2 = **31.2 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): $$ \begin{aligned} W_2 &= W_1 + h_{v1} - h_{vt} - 0.3(h_{vt}-h_{v1}) \\ W_3 &= W_2 - h_f \\ W_4 &= W_3 + h_{vt} - h_{v1} - 0.5(h_{vt}-h_{v1}) \end{aligned} $$ $W_1$ = 208.000 m, so $W_2$ = 207.828 m, $W_3$ = 207.807 m, $W_4$ = 207.874 m. Trough bed = water level − 1.6 m. | Section | Location | Bed level (m) | FSL (m) | |---|---|---|---| | 1 | Normal canal, start of inlet transition | 206.400 | 208.000 | | 2 | End of inlet transition = start of trough | 206.228 | 207.828 | | 3 | End of trough = start of outlet transition | 206.207 | 207.807 | | 4 | End of outlet transition = normal canal | 206.274 | 207.874 | Total loss of head = 208.000 − 207.874 = 0.126 m, so the canal bed downstream of the work is 0.126 m lower than upstream. Check: with trough slab 0.6 m thick, soffit of the barrels = 205.61 m, i.e. 1.11 m above the drainage bed; HFL stands 1.39 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 204.11 m (1.5 m clear height) with a 1 in 4 slope at both ends. **Answer:** drainage waterway 36 m clear (6 × 6 m, overall 41.0 m); canal flumed to 14 m wide trough; bed/FSL at the four sections = 206.40/208.00, 206.23/207.83, 206.21/207.81, 206.27/207.87 m. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Top width T = 34.80 m and trough width 14 m. With a 2:1 inlet splay measured on each side, what is the inlet length? A. 20.8 m, since 2 x (34.80 - 14)/2 = 20.8 m B. 10.4 m, since it is half of (T - Bt) C. 41.6 m, since it is 2 x (T - Bt) D. 31.2 m, since this uses the 3:1 splay Answer: A. Each side contracts (34.8 - 14)/2 = 10.4 m, and at 2:1 the length is 2 x 10.4 = 20.8 m; the 3:1 outlet gives 31.2 m. (Key point: Transition length) 2. For Q = 45 m3/s, P = 4.75 sqrt(45) = 31.9 m. Which provision satisfies the design? A. 5 spans of 6 m = 30 m clear B. 6 spans of 6 m = 36 m clear, which is more than 31.9 m C. A 14 m wide waterway equal to the trough D. 31.9 m as one span with no piers Answer: B. Clear waterway must not be less than the Lacey perimeter; 36 m exceeds 31.9 m. (Key point: Drainage waterway size) 3. Flumed to 14 m, the trough has V = 1.786 m/s and Froude number 0.45. Why is this acceptable? A. Fr is above 1, so supercritical flow keeps the trough self-cleaning B. V is above 2.5 m/s, which prevents silting C. V is below 2.5 m/s and Fr is below 1, so flow is subcritical with no hydraulic jump D. Because the trough width equals the canal bed width Answer: C. Velocity within the permitted limit and subcritical flow mean no jump and a safe flume. 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