Open Channel Flow and Hydraulic Machines

What to remember

• Open channel flow has a free surface at atmospheric pressure. Gravity drives it. Uniform flow is found with Manning's or Chezy's equation. The Froude number decides subcritical, critical or supercritical flow.
• Specific energy E = y + V²/2g has a minimum at critical depth. A hydraulic jump is the change from supercritical to subcritical flow, with large energy loss.
• Hydraulic machines: turbines convert water energy into shaft work (Pelton = impulse, Francis and Kaplan = reaction). Pumps do the reverse. Specific speed groups machines by type.

1. Geometry and uniform flow

• Wetted perimeter P is the length of channel boundary in contact with water. Hydraulic radius R = A/P. Hydraulic depth D = A/T, with T the top width.
• For a rectangle of width b and depth y: A = by, P = b + 2y. For a trapezoid with side slope m horizontal to 1 vertical: A = (b + my)y, P = b + 2y√(1 + m²), T = b + 2my.
• Uniform flow has constant depth, velocity and area along the channel. Bed slope S₀ = water surface slope = energy slope.
• Chezy's formula: V = C√(RS). C is the Chezy coefficient.
• Manning's formula: V = (1/n)·R^(2/3)·S^(1/2). n is the roughness coefficient (small for smooth lined channels, larger for earth channels with weeds). Relation: C = R^(1/6)/n.
• Discharge: Q = A·V.
• Conveyance K = A·R^(2/3)/n, so Q = K√S.

2. Best hydraulic section

• A section of given area that carries the maximum discharge has the smallest wetted perimeter, or equivalently the largest hydraulic radius.
• Rectangular: b = 2y and R = y/2.
• Trapezoidal: R = y/2, and the most efficient trapezoid is half of a regular hexagon (side slope about 60° to the horizontal, m = 1/√3). For any side slope m, b = 2y(√(1 + m²) − m).
• Triangular: side slopes at 45° to the vertical, R = y/(2√2).
• Circular: maximum discharge occurs at about 0.95D depth. Maximum velocity occurs at about 0.81D.
• The best section is not always the cheapest, because of excavation depth, lining and maintenance. Canals in Andhra Pradesh fed from the Godavari and Krishna barrages are designed with Manning's formula.

3. Specific energy and critical flow

• Specific energy E = y + V²/2g = y + q²/(2gy²) for a rectangular channel with discharge per unit width q.
• For a given q, E has a minimum at critical depth. For a given E, two depths exist (alternate depths), one subcritical and one supercritical.
• Rectangular channel: yc = (q²/g)^(1/3), Vc = √(g·yc), Emin = 1.5·yc. At critical depth the velocity head equals yc/2.
• General critical flow: Q²T/(gA³) = 1.
• Froude number: Fr = V/√(gD). Fr < 1 is subcritical (tranquil), Fr = 1 is critical, Fr > 1 is supercritical (rapid).
• In subcritical flow, a disturbance travels upstream. In supercritical flow it cannot. Control in subcritical flow is from downstream.
• Critical slope is the bed slope that gives uniform flow at critical depth. A slope is mild if normal depth yn > yc, steep if yn < yc.

4. Gradually varied flow and hydraulic jump

• Depth changes slowly along the channel because of weirs, gates, changes of slope or sections. The dynamic equation is dy/dx = (S₀ − Sf)/(1 − Fr²).

• Hydraulic jump: a sudden rise of water surface when supercritical flow changes to subcritical flow. It dissipates energy and is used below spillways and weirs, in stilling basins.
• Conjugate depths in a rectangular channel: y₂/y₁ = ½(√(1 + 8Fr₁²) − 1).
• Energy loss in the jump: hL = (y₂ − y₁)³/(4y₁y₂).
• A jump forms only when Fr₁ > 1. If Fr₁ is between 1 and about 1.7, it is a weak undular jump. A steady, well-formed jump needs Fr₁ of about 4.5 to 9.

5. Impact of jets

• Force by a jet on a fixed vertical plate: F = ρaV².
• Force on a moving plate moving away at speed u: F = ρa(V − u)².
• For a series of plates (a wheel), the whole jet is used: F = ρaV(V − u).
• For a symmetrical curved vane turning the jet through 180° (ideal), the force is twice that on a flat plate, and ideal maximum efficiency is reached at u = V/2.

6. Turbines

• Pelton wheel: jet velocity V₁ = Cv√(2gH). The buckets are split-cup shaped. Bucket speed u = φ√(2gH) with speed ratio φ about 0.45. The deflector and spear (needle) valve govern the discharge.
• Francis turbine: water enters the spiral casing, passes through guide (wicket) gate vanes, then through the runner. Wicket gates control the flow.
• Kaplan turbine: both guide vanes and runner blades can be adjusted, so efficiency stays high over a wide range of load.
• Draft tube: a gradually widening tube at the runner outlet. It lets the turbine sit above tail-race level and recovers kinetic energy as pressure. It is used with reaction turbines only.
• Efficiencies: hydraulic ηh = runner power/water power. Mechanical ηm = shaft power/runner power. Overall η = ηh × ηm.
• Unit quantities: unit speed Nu = N/√H, unit discharge Qu = Q/√H, unit power Pu = P/H^1.5.
• Specific speed (turbine): Ns = N√P/H^(5/4), with P in kW, H in m, N in rpm. It is the speed of a geometrically similar turbine producing 1 kW under 1 m head. Specific speed rises in the order Pelton, Francis, Kaplan.
• Cavitation is the formation and collapse of vapour bubbles when pressure falls to vapour pressure. It causes pitting, noise and vibration. It is most likely at the runner outlet of reaction turbines. Thoma's cavitation factor σ = (Hatm − Hvap − Hs)/H.
• Reservoir-based hydel stations on large rivers use medium-head and high-head turbines, usually Francis type for medium heads.

7. Pumps

• A centrifugal pump adds energy to water through a rotating impeller. Main parts: impeller, casing (volute), suction pipe with foot valve, delivery pipe.
• Priming means filling the pump and suction pipe with water before starting, because a centrifugal pump cannot lift air.
• Manometric head Hm = static head + friction losses + delivery velocity head. Power input P = ρgQHm/η.
• Specific speed (pump): Ns = N√Q/Hm^(3/4). Low Ns means radial flow, high head and low discharge. High Ns means axial flow (propeller pump) and low head, high discharge.
• Pumps in series add heads (same discharge). Pumps in parallel add discharges (same head).
• NPSH (net positive suction head) is the available suction head above vapour pressure. NPSH available must exceed NPSH required to avoid cavitation.
• A reciprocating pump is a positive displacement pump. Theoretical discharge for a single-acting pump is Q = ALN/60. Slip = (Qth − Qact)/Qth. An air vessel smooths the flow and reduces friction head and acceleration head.
• Multistage centrifugal pump: impellers in series for high heads.

8. Worked examples

Example 1 (Manning). Rectangular channel b = 4 m, y = 2 m, n = 0.025, S = 1/10000. A = 8 m², P = 8 m, R = 1 m. V = (1/0.025)(1)(0.01) = 0.4 m/s, Q = 8 × 0.4 = 3.2 m³/s.

Example 2 (critical flow). If the critical depth in a rectangular channel is 1.2 m, minimum specific energy = 1.5 × 1.2 = 1.8 m, and Vc = √(9.81 × 1.2) ≈ 3.43 m/s.

Example 3 (jump). Conjugate depths y₁ = 0.5 m and y₂ = 1.0 m. Loss = (0.5)³/(4 × 0.5 × 1.0) = 0.125/2 = 0.0625 m. Check: y₂/y₁ = 2 gives Fr₁ = √3.

Example 4 (specific speed). N = 300 rpm, P = 8100 kW, H = 81 m: H^1.25 = 243, √P = 90. Ns = 300 × 90/243 ≈ 111.

Example 5 (pump). Q = 0.05 m³/s, Hm = 20 m, η = 0.8: P = 1000 × 9.81 × 0.05 × 20/0.8 = 12,262 W ≈ 12.3 kW.

Exam traps

• Best rectangular section is b = 2y, but best trapezoid is half a hexagon. Do not mix the two.
• Critical depth gives minimum specific energy for a given discharge, but maximum discharge for a given specific energy.
• Emin = 1.5 yc is for rectangular channels only.
• Hydraulic jump is supercritical to subcritical, never the reverse.
• Pelton is an impulse turbine. Francis and Kaplan are reaction turbines.
• A draft tube is used only with reaction turbines.
• Kaplan has adjustable runner blades. A propeller turbine has fixed blades.
• Pump specific speed uses discharge Q, and turbine specific speed uses power P.
• Pumps in series add head, in parallel add discharge.

One-liners

• 1. Manning's equation: V = (1/n)R^(2/3)S^(1/2).
• 2. Hydraulic radius = area/wetted perimeter.
• 3. At critical flow, Froude number is 1.
• 4. Rectangular critical depth: (q²/g)^(1/3).
• 5. Hydraulic jump dissipates energy and is used in stilling basins.
• 6. Loss in a jump is (y₂ − y₁)³/(4y₁y₂).
• 7. Pelton wheel: high head, low discharge, jet at atmospheric pressure.
• 8. Kaplan turbine: low head, high discharge, adjustable blades.
• 9. Speed ratio of a Pelton wheel is about 0.45.
• 10. Cavitation causes pitting and vibration in turbines and pumps.
• 11. A centrifugal pump must be primed before starting.
• 12. Slip in a reciprocating pump is the difference between theoretical and actual discharge.

Practice questions

1. The hydraulic radius of an open channel section is defined as

   1. wetted perimeter divided by flow area
   2. flow area divided by wetted perimeter
   3. top width divided by flow area
   4. flow area divided by top width
   Answer

   B. flow area divided by wetted perimeter

   R = A/P.

2. Which expression is Manning's formula for velocity in an open channel?

   1. V = (1/n) R^(1/2) S^(2/3)
   2. V = (1/n) R^(3/2) S^(1/3)
   3. V = n R^(2/3) S^(1/2)
   4. V = (1/n) R^(2/3) S^(1/2)
   Answer

   D. V = (1/n) R^(2/3) S^(1/2)

   Manning: V = (1/n)R^(2/3)S^(1/2).

3. The best hydraulic rectangular section has bed width equal to

   1. half the depth of flow
   2. the depth of flow
   3. twice the depth of flow
   4. three times the depth of flow
   Answer

   C. twice the depth of flow

   For b = 2y, the wetted perimeter is least for the given area and R = y/2.

4. At critical flow in an open channel, the Froude number equals

   1. 0
   2. 1
   3. 0.5
   4. 2
   Answer

   B. 1

   Fr = V/√(gD) = 1 at critical depth.

5. A hydraulic jump forms when the flow changes from

   1. supercritical to subcritical
   2. laminar to turbulent
   3. subcritical to supercritical
   4. uniform to steady
   Answer

   A. supercritical to subcritical

   The flow decelerates abruptly from Fr > 1 to Fr < 1.

6. The Pelton wheel is classified as a

   1. radial-flow reaction turbine
   2. mixed-flow reaction turbine
   3. tangential-flow impulse turbine
   4. axial-flow reaction turbine
   Answer

   C. tangential-flow impulse turbine

   The jet strikes buckets at atmospheric pressure.

7. The Kaplan turbine is

   1. a radial-flow turbine with fixed blades
   2. a mixed-flow impulse turbine
   3. an impulse turbine with split buckets
   4. an axial-flow reaction turbine with adjustable runner blades
   Answer

   D. an axial-flow reaction turbine with adjustable runner blades

   Adjustable blades keep efficiency high over varying loads.

8. The main purpose of a draft tube in a reaction turbine is to

   1. increase the speed of the jet
   2. let the runner sit above tail-race level and recover kinetic energy as pressure
   3. reduce the number of blades
   4. protect the penstock from water hammer
   Answer

   B. let the runner sit above tail-race level and recover kinetic energy as pressure

   The diverging tube reduces exit velocity and creates suction at the runner outlet.

9. Cavitation in hydraulic machines mainly results in

   1. reduction of friction
   2. increase in head
   3. pitting of surfaces, noise and vibration
   4. increase in efficiency
   Answer

   C. pitting of surfaces, noise and vibration

   Collapse of vapour bubbles damages metal surfaces.

10. Priming is necessary before starting a

    1. centrifugal pump
    2. Pelton wheel
    3. Francis turbine runner
    4. reciprocating pump
    Answer

    A. centrifugal pump

    A centrifugal pump cannot pump air, so the casing and suction pipe must be filled with water.

11. Which turbine is most suitable for very high head and low discharge?

    1. Kaplan turbine
    2. Bulb turbine
    3. Propeller turbine
    4. Pelton wheel
    Answer

    D. Pelton wheel

    Impulse turbines suit high head with small flow.

12. The specific speed of a centrifugal pump is defined as

    1. N√H / Q^(3/4)
    2. N√P / H^(5/4)
    3. N√Q / Hm^(3/4)
    4. N√Q / Hm^(5/4)
    Answer

    C. N√Q / Hm^(3/4)

    Pump specific speed uses discharge; turbine specific speed uses power.

13. For a rectangular channel with discharge q per unit width, the critical depth is

    1. (q²/g)^(1/2)
    2. (q²/g)^(1/3)
    3. (q/g)^(1/2)
    4. (g/q²)^(1/3)
    Answer

    B. (q²/g)^(1/3)

    yc = (q²/g)^(1/3) from Fr = 1.

14. In a rectangular channel, the minimum specific energy equals

    1. 2 times the critical depth
    2. critical depth
    3. 0.5 times the critical depth
    4. 1.5 times the critical depth
    Answer

    D. 1.5 times the critical depth

    Emin = yc + yc/2 = 1.5yc.

15. Arranged in increasing order of specific speed, the turbines are

    1. Pelton, Francis, Kaplan
    2. Francis, Pelton, Kaplan
    3. Kaplan, Francis, Pelton
    4. Pelton, Kaplan, Francis
    Answer

    A. Pelton, Francis, Kaplan

    Specific speed rises as head falls and discharge rises.

16. The most efficient trapezoidal channel section is

    1. half of a regular hexagon
    2. half of a regular octagon
    3. a square with vertical sides
    4. a triangle with a 30° side slope
    Answer

    A. half of a regular hexagon

    R = y/2 and side slope is 60° to horizontal.

17. In supercritical flow a small surface disturbance

    1. cannot travel upstream
    2. travels upstream at higher speed
    3. travels only across the channel
    4. vanishes immediately in all directions
    Answer

    A. cannot travel upstream

    The flow velocity is greater than the wave celerity √(gD).

18. A channel is rectangular with bed width 4 m and depth 2 m. Its hydraulic radius is

    1. 1.5 m
    2. 1 m
    3. 2 m
    4. 0.5 m
    Answer

    B. 1 m

    A = 8 m², P = 4 + 4 = 8 m, R = 1 m.

19. Using Manning's formula with n = 0.02, R = 1 m and S = 0.0004, the mean velocity is

    1. 0.5 m/s
    2. 2 m/s
    3. 0.1 m/s
    4. 1 m/s
    Answer

    D. 1 m/s

    V = (1/0.02) × 1 × 0.02 = 1 m/s.

20. A best-section rectangular channel is to have a flow area of 8 m². The flow depth is

    1. 2.83 m
    2. 1 m
    3. 2 m
    4. 4 m
    Answer

    C. 2 m

    A = b·y = 2y² = 8, so y = 2 m and b = 4 m.

21. The minimum specific energy in a rectangular channel is 1.5 m. The critical depth is

    1. 1.5 m
    2. 1.0 m
    3. 2.25 m
    4. 0.75 m
    Answer

    B. 1.0 m

    yc = Emin/1.5 = 1.0 m.

22. Critical depth is 0.9 m. Taking g = 10 m/s², the critical velocity in a rectangular channel is

    1. 9 m/s
    2. 1.5 m/s
    3. 4.5 m/s
    4. 3 m/s
    Answer

    D. 3 m/s

    Vc = √(g·yc) = √9 = 3 m/s.

23. Water flows at 6 m/s at a depth of 0.9 m in a rectangular channel (g = 10 m/s²). The Froude number and flow type are

    1. 0.5, supercritical
    2. 2, subcritical
    3. 2, supercritical
    4. 0.5, subcritical
    Answer

    C. 2, supercritical

    Fr = 6/√(10 × 0.9) = 6/3 = 2 > 1.

24. Supercritical flow of depth 0.4 m with Froude number √3 forms a hydraulic jump. The conjugate depth is

    1. 1.6 m
    2. 0.6 m
    3. 0.8 m
    4. 1.2 m
    Answer

    C. 0.8 m

    y₂/y₁ = ½(√(1 + 8 × 3) − 1) = ½(5 − 1) = 2.

25. For a jump with conjugate depths 0.4 m and 0.8 m in a rectangular channel, the energy loss is

    1. 0.05 m
    2. 0.032 m
    3. 0.16 m
    4. 0.5 m
    Answer

    A. 0.05 m

    hL = (0.8 − 0.4)³/(4 × 0.4 × 0.8) = 0.064/1.28 = 0.05 m.

26. A jet of area 0.001 m² and velocity 10 m/s strikes a fixed vertical plate normally (ρ = 1000 kg/m³). The force is

    1. 10 N
    2. 50 N
    3. 1000 N
    4. 100 N
    Answer

    D. 100 N

    F = ρaV² = 1000 × 0.001 × 100 = 100 N.

27. The same jet (0.001 m², 10 m/s) strikes a single plate moving away at 4 m/s. The force is

    1. 24 N
    2. 36 N
    3. 60 N
    4. 100 N
    Answer

    B. 36 N

    F = ρa(V − u)² = 1000 × 0.001 × 36 = 36 N.

28. A Pelton wheel has net head 80 m and Cv = 0.98. With g = 10 m/s², the jet velocity is

    1. 39.2 m/s
    2. 40 m/s
    3. 78.4 m/s
    4. 19.6 m/s
    Answer

    A. 39.2 m/s

    V = Cv√(2gH) = 0.98 × √1600 = 0.98 × 40 = 39.2 m/s.

29. For a Pelton wheel under 80 m head (g = 10 m/s²) and speed ratio 0.45, the bucket speed is

    1. 36 m/s
    2. 9 m/s
    3. 18 m/s
    4. 20 m/s
    Answer

    C. 18 m/s

    u = 0.45 × √(2gH) = 0.45 × 40 = 18 m/s.

30. A turbine runs at 500 rpm, develops 1600 kW under 16 m head. The specific speed is

    1. 1000
    2. 625
    3. 500
    4. 250
    Answer

    B. 625

    Ns = 500 × 40/16^1.25 = 20,000/32 = 625.

31. A turbine has hydraulic efficiency 0.9 and mechanical efficiency 0.95. The overall efficiency is

    1. 95%
    2. 92.5%
    3. 90.5%
    4. 85.5%
    Answer

    D. 85.5%

    η = ηh × ηm = 0.9 × 0.95 = 0.855.

32. A centrifugal pump delivers 0.04 m³/s against a manometric head of 25 m with overall efficiency 0.8 (ρ = 1000, g = 10). The input power is

    1. 8 kW
    2. 12.5 kW
    3. 20 kW
    4. 10 kW
    Answer

    B. 12.5 kW

    P = ρgQHm/η = 1000 × 10 × 0.04 × 25/0.8 = 12,500 W.

33. Consider the statements about open channel flow. 1. In uniform flow the bed slope, water surface slope and energy slope are equal. 2. In gradually varied flow the depth is constant along the channel. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Gradually varied flow means the depth changes slowly along the channel.

34. Consider the statements. 1. In subcritical flow a disturbance can travel upstream. 2. Supercritical flow has a Froude number greater than 1. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Subcritical: Fr < 1; supercritical: Fr > 1.

35. Consider the statements for a given discharge in an open channel. 1. At critical depth the specific energy is a minimum. 2. At critical depth the discharge for a given specific energy is a minimum. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    For a given specific energy, discharge is maximum at critical depth.

36. Consider the statements about a hydraulic jump. 1. It forms when flow changes from supercritical to subcritical. 2. It dissipates a large amount of energy. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Jumps are used in stilling basins to destroy energy.

37. Consider the statements about turbines. 1. A Pelton wheel runner works at atmospheric pressure with a free jet. 2. A Pelton wheel normally uses a draft tube. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    A draft tube is used with reaction turbines.

38. Consider the statements about turbines. 1. The Francis turbine is an impulse turbine. 2. The Kaplan turbine has adjustable runner blades. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Francis is a reaction turbine.

39. Consider the statements about pumps. 1. Centrifugal pumps in series increase the discharge at the same head. 2. Centrifugal pumps in parallel increase the head at the same discharge. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    Series adds heads; parallel adds discharges.

40. Consider the statements about hydraulic machines. 1. Cavitation occurs when local pressure falls to the vapour pressure of the liquid. 2. Cavitation improves the efficiency of a turbine. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    Cavitation lowers efficiency and damages parts.

41. Consider the statements. 1. A draft tube is used with Pelton and Francis turbines alike. 2. A draft tube converts kinetic energy at the runner outlet into pressure. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Draft tubes are used only with reaction turbines.

42. Consider the statements about open channel sections. 1. For the best rectangular section R = y/2. 2. For the best trapezoidal section R = y/2. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both best sections have hydraulic radius equal to half the depth.

43. Consider the statements about a hydraulic jump. 1. A jump occurs when subcritical flow changes to supercritical flow. 2. Energy is conserved across a jump. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    The jump is supercritical to subcritical with a significant energy loss.

44. Which pair of turbine and typical application is correctly matched?

    1. Pelton wheel – low head with large discharge
    2. Kaplan turbine – very high head
    3. Francis turbine – impulse action
    4. Kaplan turbine – low head with large discharge
    Answer

    D. Kaplan turbine – low head with large discharge

    Kaplan is an axial-flow turbine for low heads.

45. Which pair of Froude number range and flow type is correct?

    1. Fr < 1 – supercritical
    2. Fr > 1 – supercritical
    3. Fr = 1 – subcritical
    4. Fr = 0 – critical
    Answer

    B. Fr > 1 – supercritical

    Fr > 1 is supercritical flow.