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 3 m wide sits in the wall of a tank of water, with its top edge 1.5 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 1.5 mH = 3 mb = 3 mρ = 1000 kg/m³g = 9.81 m/s²
F = kN
2.Hydrostatic forceHard
A vertical rectangular gate 2 m wide sits in the wall of a tank of seawater, with its top edge 3 m below the surface. Find the depth ycp of the centre of pressure below the surface.
d = 3 mH = 2 mb = 2 mρ = 1025 kg/m³g = 9.81 m/s²
ycp = m
3.Continuity & BernoulliMedium
A pitot-static tube in a wind tunnel reads a difference of 1400 Pa between the stagnation pressure and the static pressure. The air density is 1.2 kg/m³. Find the air speed v.
p0 − p = 1400 Paρ = 1.2 kg/m³
v = m/s
4.Continuity & BernoulliHard
Water flows up through a pipe that narrows from 100 mm to 75 mm while rising 1 m. At section 1 the pressure is 220 kPa and the velocity is 2.5 m/s. Ignoring losses, find the pressure p2 at section 2.
d1 = 100 mmd2 = 75 mmp1 = 220 kPav1 = 2.5 m/sΔz = 1 mρ = 1000 kg/m³
p2 = kPa
5.Pipe friction & head lossMedium
Water flows at 3 L/s through a pipe 50 m long with an inside diameter of 50 mm. Use the Darcy friction factor f = 0.025 and g = 9.81 m/s². Find the friction head loss hf.
Q = 3 L/sL = 50 mD = 50 mmf = 0.025g = 9.81 m/s²
hf = m
6.Pipe friction & head lossHard
Two open reservoirs are joined by a pipe 100 m long with an inside diameter of 200 mm and a Darcy friction factor f = 0.018. The water surface of the upper reservoir is 38 m above that of the lower one. The minor losses are a square-edged entrance (K = 0.5), 2 bends (K = 0.3 each), a valve (K = 10) 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 = 38 mL = 100 mD = 200 mmf = 0.018Kentrance = 0.5Kbend = 0.3 (×2)Kvalve = 10Kexit = 1.0
Q = L/s
7.Pumps & pump powerMedium
In a test, a pump delivers 50 L/s of water (ρ = 1000 kg/m³) against a head of 36 m while its shaft takes 20.8 kW. Use g = 9.81 m/s². Find the pump efficiency η in percent.
Q = 50 L/sH = 36 mPshaft = 20.8 kWg = 9.81 m/s²
η = %
8.Pumps & pump powerHard
A pump moves 75 L/s of water (ρ = 1000 kg/m³) from an open lower reservoir to an upper tank whose water surface is 7 m higher. The pipe is 600 m long with an inside diameter of 200 mm and a Darcy friction factor f = 0.015; the minor-loss coefficients add up to ΣK = 4 (entrance and exit included). Use g = 9.81 m/s². Find the head H the pump must add.
Q = 75 L/sΔz = 7 mL = 600 mD = 200 mmf = 0.015ΣK = 4
H = m
9.Open-channel flow (Manning)Medium
A trapezoidal brickwork channel has a bottom width of 4.5 m and side slopes of 3:1 (horizontal:vertical). Water flows uniformly 1.6 m deep. Manning n = 0.015 and the bed slope is S = 0.001. Find the flow rate Q in m³/s.
b = 4.5 my = 1.6 mz = 3n = 0.015S = 0.001
Q = m³/s
10.Open-channel flow (Manning)Hard
A trapezoidal brickwork channel with a bottom width of 2 m and side slopes of 3:1 (horizontal:vertical) must carry 30 m³/s in uniform flow at a depth of 1.6 m. Manning n = 0.015. Find the required bed slope S (in m/m).
Q = 30 m³/sb = 2 my = 1.6 mz = 3n = 0.015
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 264.9 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 = 1.5 + 3/2 = 3 m
A = H·b = 3·3 = 9 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·3·9 = 264.9 kN
Answer
F = 264.9 kN
2.Hydrostatic force
Answer: ycp = 4.083 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 = 3 + 2/2 = 4 m
A = H·b = 2·2 = 4 m²
Resultant force
F = ρ·g·yc·A = 1025·9.81·4·4 = 160.9 kN
Centre of pressure
Pressure grows with depth, so the resultant acts below the centroid.
Ic = b·H³/12 = 2·2³/12 = 1.333 m⁴
ycp = yc + Ic/(yc·A) = 4 + 1.333/(4·4) = 4.083 m
Answer
ycp = 4.083 m
3.Continuity & Bernoulli
Answer: v = 48.3 m/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.
Bernoulli to the stagnation point
The flow comes to rest at the tip, so its kinetic energy shows up as extra pressure.
p0 − p = ½ρv²
v = √(2·1400/1.2) = 48.3 m/s
Answer
v = 48.3 m/s
4.Continuity & Bernoulli
Answer: p2 = 203.4 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 = 2.5·(100/75)² = 4.444 m/s
½ρ(v1² − v2²) = ½·1000·(2.5² − 4.444²)/1000 = −6.752 kPa
ρgΔz = 1000·9.81·1/1000 = 9.81 kPa
p2 = p1 + ½ρ(v1² − v2²) − ρgΔz = 220 + (−6.752) − 9.81 = 203.4 kPa
Answer
p2 = 203.4 kPa
5.Pipe friction & head loss
Answer: hf = 2.975 m
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.
Mean velocity
A = π·0.05²/4 = 0.001963 m²
V = Q/A = 0.003/0.001963 = 1.528 m/s
Velocity head
V²/(2g) = 1.528²/(2·9.81) = 0.119 m
Darcy–Weisbach
hf = f·(L/D)·V²/(2g) = 0.025·(50/0.05)·1.528²/(2·9.81) = 2.975 m
Answer
hf = 2.975 m
6.Pipe friction & head loss
Answer: Q = 186.7 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 = 0.018·100/0.2 = 9
ΣK = 0.5 + 2·0.3 + 10 + 1.0 = 12.1
Energy equation
Both surfaces are open to the air and practically at rest, so all of Δz is lost in the pipe.
38 = (9 + 12.1)·V²/(2·9.81)
V = √(2·9.81·38/21.1) = 5.944 m/s
Flow rate
Q = V·π·D²/4 = 5.944·0.03142 = 0.1867 m³/s = 186.7 L/s
Answer
Q = 186.7 L/s
7.Pumps & pump power
Answer: η = 84.89 %
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 = 1000·9.81·0.05·36 = 17.66 kW
Efficiency
η = Ph/Pshaft = 17.66/20.8 = 84.89%
Answer
η = 84.89%
8.Pumps & pump power
Answer: H = 21.23 m
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 = 0.075/0.03142 = 2.387 m/s
V²/(2g) = 0.2905 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (45 + 4)·0.2905 = 14.23 m
Pump head
Both surfaces are open and practically at rest, so pressure and velocity terms drop out.
H = Δz + hL = 7 + 14.23 = 21.23 m
Answer
H = 21.23 m
9.Open-channel flow (Manning)
Answer: Q = 31.74 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 + z·y)·y = (4.5 + 3·1.6)·1.6 = 14.88 m²
P = b + 2y·√(1 + z²) = 4.5 + 2·1.6·√(1 + 3²) = 14.62 m
R = A/P = 1.018 m
Velocity
V = (1/0.015)·1.0182/3·0.0011/2 = 2.133 m/s
Flow rate
Q = A·V = 14.88·2.133 = 31.74 m³/s
Answer
Q = 31.74 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.001975 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 = (2 + 3·1.6)·1.6 = 10.88 m²
P = b + 2y·√(1 + z²) = 2 + 2·1.6·√(1 + 3²) = 12.12 m
R = A/P = 0.8977 m
Solve Manning for S
S1/2 = n·Q/(A·R2/3) = 0.015·30/(10.88·0.9306) = 0.04444
S = 0.04444² = 0.001975
Answer
S = 0.001975
That is a fall of 1.975 m per kilometre.
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