Hydrostatic force
- Pressure at depth
- p = ρgh
- Force on a vertical wall
- F = ρg·hc·A
- Centre of pressure
- yp = yc + Ic/(ycA)
- Rectangle
- Ic = bH³/12
Hydraulic Engineering: cram pack
Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
1Formulas and methods
Continuity & Bernoulli
- Continuity
- A1v1 = A2v2
- Bernoulli
- p1 + ½ρv1² + ρgz1 = p2 + ½ρv2² + ρgz2
- Flow rate
- Q = A·v
- Circle area
- A = πd²/4
Pipe friction & head loss
- Reynolds number
- Re = V·D/ν = ρ·V·D/μ
- Flow regime
- laminar below Re ≈ 2000, turbulent above Re ≈ 4000
- Mean velocity
- V = Q/A, A = π·D²/4
- Darcy–Weisbach
- hf = f·(L/D)·V²/(2g)
- Minor losses
- hm = ΣK·V²/(2g)
- Typical K values
- entrance 0.5, exit 1.0, 90° bend ≈ 0.3
- Pressure drop
- Δp = ρ·g·hf
- Two open reservoirs
- Δz = (f·L/D + ΣK)·V²/(2g)
- Free outlet
- H = (1 + f·L/D + ΣK)·V²/(2g)
Pumps & pump power
- Hydraulic power
- Ph = ρ·g·Q·H = Q·Δp
- Shaft power
- Pshaft = Ph/ηpump
- Electrical power
- Pel = Pshaft/ηmotor
- Pump head
- H = Δz + Δp/(ρ·g) + hL
- System losses
- hL = (f·L/D + ΣK)·V²/(2g)
Open-channel flow (Manning)
- Manning equation
- V = (1/n)·R2/3·S1/2
- Flow rate
- Q = A·V
- Hydraulic radius
- R = A/P (P = wetted perimeter, free surface excluded)
- Rectangle
- A = b·y, P = b + 2y
- Trapezoid (side slope z:1)
- A = (b + z·y)·y, P = b + 2y·√(1 + z²)
- Circular pipe, full or half full
- R = D/4
- Required slope
- S = (n·Q/(A·R2/3))²
2Practice problems
1.Hydrostatic forceMedium
A vertical rectangular gate 1 m wide sits in the wall of a tank of water, with its top edge 2 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 2 mH = 1.5 mb = 1 mρ = 1000 kg/m³g = 9.81 m/s²
F = kN
2.Hydrostatic forceHard
A vertical rectangular gate 1 m wide sits in the wall of a tank of water, with its top edge 4 m below the surface. Find the depth ycp of the centre of pressure below the surface.
d = 4 mH = 2 mb = 1 mρ = 1000 kg/m³g = 9.81 m/s²
ycp = m
3.Continuity & BernoulliMedium
Water drains from a large open tank through a sharp-edged orifice with a diameter of 50 mm. The water level is kept 1 m above the orifice and the discharge coefficient is 0.62. Find the flow rate Q. Use g = 9.81 m/s².
d = 50 mmh = 1 mCd = 0.62g = 9.81 m/s²
Q = L/s
4.Continuity & BernoulliHard
Water flows up through a pipe that narrows from 150 mm to 100 mm while rising 1 m. At section 1 the pressure is 160 kPa and the velocity is 2 m/s. Ignoring losses, find the pressure p2 at section 2.
d1 = 150 mmd2 = 100 mmp1 = 160 kPav1 = 2 m/sΔz = 1 mρ = 1000 kg/m³
p2 = kPa
5.Pipe friction & head lossMedium
Water (ρ = 1000 kg/m³) flows at 1.8 m/s through a horizontal pipe, 100 m long, with an inside diameter of 50 mm. Use the Darcy friction factor f = 0.015 and g = 9.81 m/s². Find the pressure drop Δp over the pipe in kPa.
V = 1.8 m/sL = 100 mD = 50 mmf = 0.015ρ = 1000 kg/m³
Δp = kPa
6.Pipe friction & head lossHard
Two open reservoirs are joined by a pipe 300 m long with an inside diameter of 250 mm and a Darcy friction factor f = 0.02. The water surface of the upper reservoir is 16 m above that of the lower one. The minor losses are a square-edged entrance (K = 0.5), 4 bends (K = 0.3 each), a valve (K = 2) and the exit into the lower reservoir (K = 1.0). Use g = 9.81 m/s². Find the flow rate Q in L/s.
Δz = 16 mL = 300 mD = 250 mmf = 0.02Kentrance = 0.5Kbend = 0.3 (×4)Kvalve = 2Kexit = 1.0
Q = L/s
7.Pumps & pump powerMedium
A pump delivers 150 L/s of water (ρ = 1000 kg/m³) against a head of 62 m. The pump efficiency is 85% and the electric motor driving it has an efficiency of 92%. Use g = 9.81 m/s². Find the electrical power Pel the motor draws, in kW.
Q = 150 L/sH = 62 mηpump = 85%ηmotor = 92%g = 9.81 m/s²
Pel = kW
8.Pumps & pump powerHard
A pump moves 80 L/s of water (ρ = 1000 kg/m³) from an open lower reservoir to an upper tank whose water surface is 27 m higher. The pipe is 600 m long with an inside diameter of 200 mm and a Darcy friction factor f = 0.022; the minor-loss coefficients add up to ΣK = 8 (entrance and exit included). Use g = 9.81 m/s². The pump efficiency is 65%. Find the shaft power Pshaft in kW.
Q = 80 L/sΔz = 27 mL = 600 mD = 200 mmf = 0.022ΣK = 8η = 65%
Pshaft = kW
9.Open-channel flow (Manning)Medium
A rectangular grass-lined earth channel is 2 m wide. Water flows uniformly 1.4 m deep. Manning n = 0.03 and the bed slope is S = 0.005. Find the flow rate Q in m³/s.
b = 2 my = 1.4 mn = 0.03S = 0.005
Q = m³/s
10.Open-channel flow (Manning)Hard
A trapezoidal grass-lined earth channel with a bottom width of 1 m and side slopes of 3:1 (horizontal:vertical) must carry 0.88 m³/s in uniform flow at a depth of 0.6 m. Manning n = 0.03. Find the required bed slope S (in m/m).
Q = 0.88 m³/sb = 1 my = 0.6 mz = 3n = 0.03
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 40.47 kN
Hydrostatic force on a plane surface: F = ρ·g·yc·A. It acts at the centre of pressure, ycp = yc + Ic/(ycA), which always lies below the centroid.
Centroid depth and area
yc = d + H/2 = 2 + 1.5/2 = 2.75 m
A = H·b = 1.5·1 = 1.5 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·2.75·1.5 = 40.47 kN
Answer
F = 40.47 kN
2.Hydrostatic force
Answer: ycp = 5.067 m
Hydrostatic force on a plane surface: F = ρ·g·yc·A. It acts at the centre of pressure, ycp = yc + Ic/(ycA), which always lies below the centroid.
Centroid depth and area
yc = d + H/2 = 4 + 2/2 = 5 m
A = H·b = 2·1 = 2 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·5·2 = 98.1 kN
Centre of pressure
Pressure grows with depth, so the resultant acts below the centroid.
Ic = b·H³/12 = 1·2³/12 = 0.6667 m⁴
ycp = yc + Ic/(yc·A) = 5 + 0.6667/(5·2) = 5.067 m
Answer
ycp = 5.067 m
3.Continuity & Bernoulli
Answer: Q = 5.392 L/s
Continuity and Bernoulli: for an incompressible fluid the flow rate is the same at every section, Q = A₁v₁ = A₂v₂. Along a streamline without losses, p + ½ρv² + ρgz is constant, so where the fluid is faster or higher its pressure is lower.
Ideal jet velocity
√(2gh) = √(2·9.81·1) = 4.429 m/s
Actual flow rate
The discharge coefficient accounts for the contraction of the jet and friction at the edge.
Q = Cd·A·√(2gh) = 0.62·0.001963·4.429 = 0.005392 m³/s = 5.392 L/s
Answer
Q = 5.392 L/s
4.Continuity & Bernoulli
Answer: p2 = 142.1 kPa
Continuity and Bernoulli: for an incompressible fluid the flow rate is the same at every section, Q = A₁v₁ = A₂v₂. Along a streamline without losses, p + ½ρv² + ρgz is constant, so where the fluid is faster or higher its pressure is lower.
Continuity and Bernoulli
Work in Pa, then divide by 1000 to get kPa.
v2 = v1·(d1/d2)² = 2·(150/100)² = 4.5 m/s
½ρ(v1² − v2²) = ½·1000·(2² − 4.5²)/1000 = −8.125 kPa
ρgΔz = 1000·9.81·1/1000 = 9.81 kPa
p2 = p1 + ½ρ(v1² − v2²) − ρgΔz = 160 + (−8.125) − 9.81 = 142.1 kPa
Answer
p2 = 142.1 kPa
5.Pipe friction & head loss
Answer: Δp = 48.6 kPa
Pipe flow: the Reynolds number Re = V·D/ν = ρ·V·D/μ tells laminar flow (Re below about 2000) from turbulent flow (above about 4000). Friction along a pipe costs the Darcy–Weisbach head loss hf = f·(L/D)·V²/(2g). Fittings, valves, the entrance and the exit add minor losses hm = ΣK·V²/(2g). Between two open reservoir surfaces the energy equation gives Δz = hf + hm, with the exit loss K = 1 included.
Head loss
hf = f·(L/D)·V²/(2g) = 0.015·(100/0.05)·1.8²/(2·9.81) = 4.954 m
Pressure drop
The pipe is horizontal and the velocity does not change, so the energy equation reduces to Δp = ρ·g·hf.
Δp = ρ·g·hf = 1000·9.81·4.954 = 48600 Pa = 48.6 kPa
Answer
Δp = 48.6 kPa
6.Pipe friction & head loss
Answer: Q = 162.3 L/s
Pipe flow: the Reynolds number Re = V·D/ν = ρ·V·D/μ tells laminar flow (Re below about 2000) from turbulent flow (above about 4000). Friction along a pipe costs the Darcy–Weisbach head loss hf = f·(L/D)·V²/(2g). Fittings, valves, the entrance and the exit add minor losses hm = ΣK·V²/(2g). Between two open reservoir surfaces the energy equation gives Δz = hf + hm, with the exit loss K = 1 included.
Loss coefficients
f·L/D = 0.02·300/0.25 = 24
ΣK = 0.5 + 4·0.3 + 2 + 1.0 = 4.7
Energy equation
Both surfaces are open to the air and practically at rest, so all of Δz is lost in the pipe.
16 = (24 + 4.7)·V²/(2·9.81)
V = √(2·9.81·16/28.7) = 3.307 m/s
Flow rate
Q = V·π·D²/4 = 3.307·0.04909 = 0.1623 m³/s = 162.3 L/s
Answer
Q = 162.3 L/s
7.Pumps & pump power
Answer: Pel = 116.7 kW
Pump power: the power given to the water is the hydraulic power Ph = ρ·g·Q·H = Q·Δp. The shaft must deliver more because the pump loses some: Pshaft = Ph/ηpump, and the motor draws Pel = Pshaft/ηmotor. The pump head H follows from the energy equation: static lift + pressure head + all friction and minor losses.
Hydraulic power
Ph = ρ·g·Q·H = 1000·9.81·0.15·62 = 91.23 kW
Shaft power
Pshaft = 91.23/0.85 = 107.3 kW
Electrical power
Pel = 107.3/0.92 = 116.7 kW
Answer
Pel = 116.7 kW
8.Pumps & pump power
Answer: Pshaft = 62.13 kW
Pump power: the power given to the water is the hydraulic power Ph = ρ·g·Q·H = Q·Δp. The shaft must deliver more because the pump loses some: Pshaft = Ph/ηpump, and the motor draws Pel = Pshaft/ηmotor. The pump head H follows from the energy equation: static lift + pressure head + all friction and minor losses.
Velocity
V = Q/A = 0.08/0.03142 = 2.546 m/s
V²/(2g) = 0.3305 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (66 + 8)·0.3305 = 24.46 m
Pump head
H = Δz + hL = 27 + 24.46 = 51.46 m
Shaft power
Pshaft = ρ·g·Q·H/η = 1000·9.81·0.08·51.46/0.65 = 62.13 kW
Answer
Pshaft = 62.13 kW
9.Open-channel flow (Manning)
Answer: Q = 4.608 m³/s
Manning’s equation (SI): for uniform flow in an open channel V = (1/n)·R2/3·S1/2 and Q = A·V, with n the roughness coefficient, S the bed slope (m/m) and R = A/P the hydraulic radius: flow area divided by wetted perimeter. Only the boundary in contact with the water counts in P, not the free surface. Rectangle: A = b·y, P = b + 2y. Trapezoid with side slopes z:1 (horizontal:vertical): A = (b + z·y)·y, P = b + 2y·√(1 + z²). A circular pipe flowing full or half full has R = D/4.
Geometry
A = b·y = 2·1.4 = 2.8 m²
P = b + 2y = 2 + 2·1.4 = 4.8 m
R = A/P = 0.5833 m
Velocity
V = (1/0.03)·0.58332/3·0.0051/2 = 1.646 m/s
Flow rate
Q = A·V = 2.8·1.646 = 4.608 m³/s
Answer
Q = 4.608 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.0009997 m/m
Manning’s equation (SI): for uniform flow in an open channel V = (1/n)·R2/3·S1/2 and Q = A·V, with n the roughness coefficient, S the bed slope (m/m) and R = A/P the hydraulic radius: flow area divided by wetted perimeter. Only the boundary in contact with the water counts in P, not the free surface. Rectangle: A = b·y, P = b + 2y. Trapezoid with side slopes z:1 (horizontal:vertical): A = (b + z·y)·y, P = b + 2y·√(1 + z²). A circular pipe flowing full or half full has R = D/4.
Geometry
A = (b + z·y)·y = (1 + 3·0.6)·0.6 = 1.68 m²
P = b + 2y·√(1 + z²) = 1 + 2·0.6·√(1 + 3²) = 4.795 m
R = A/P = 0.3504 m
Solve Manning for S
S1/2 = n·Q/(A·R2/3) = 0.03·0.88/(1.68·0.497) = 0.03162
S = 0.03162² = 0.0009997
Answer
S = 0.0009997
That is a fall of 0.9997 m per kilometre.
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