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 2 m wide sits in the wall of a tank of water, with its top edge 0.5 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 0.5 mH = 2.5 mb = 2 mρ = 1000 kg/m³g = 9.81 m/s²
F = kN
2.Hydrostatic forceHard
A vertical rectangular gate 1.5 m wide sits in the wall of a tank of oil, with its top edge 0.5 m below the surface. Find the depth ycp of the centre of pressure below the surface.
d = 0.5 mH = 1 mb = 1.5 mρ = 880 kg/m³g = 9.81 m/s²
ycp = m
3.Continuity & BernoulliMedium
Water flows at 23 L/s through a horizontal pipe that narrows from 200 mm to 150 mm. The pressure in the narrow section is 220 kPa. Ignoring losses, find the pressure p1 in the wide section.
Q = 23 L/sd1 = 200 mmd2 = 150 mmp2 = 220 kPaρ = 1000 kg/m³
p1 = kPa
4.Continuity & BernoulliHard
Water flows up through a pipe that narrows from 100 mm to 50 mm while rising 3 m. At section 1 the pressure is 150 kPa and the velocity is 2.25 m/s. Ignoring losses, find the pressure p2 at section 2.
d1 = 100 mmd2 = 50 mmp1 = 150 kPav1 = 2.25 m/sΔz = 3 mρ = 1000 kg/m³
p2 = kPa
5.Pipe friction & head lossMedium
Water (ρ = 1000 kg/m³) flows at 1.1 m/s through a horizontal pipe, 800 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 pressure drop Δp over the pipe in kPa.
V = 1.1 m/sL = 800 mD = 50 mmf = 0.025ρ = 1000 kg/m³
Δp = kPa
6.Pipe friction & head lossHard
Two open reservoirs are joined by a pipe 150 m long with an inside diameter of 300 mm and a Darcy friction factor f = 0.018. The minor losses are a square-edged entrance (K = 0.5), 4 bends (K = 0.3 each), a valve (K = 5) and the exit into the lower reservoir (K = 1.0). Use g = 9.81 m/s². What difference in water level Δz between the reservoirs gives a flow rate of 100 L/s?
Q = 100 L/sL = 150 mD = 300 mmf = 0.018Kentrance = 0.5Kbend = 0.3 (×4)Kvalve = 5Kexit = 1.0
Δz = m
7.Pumps & pump powerMedium
A pump delivers 150 L/s of water (ρ = 1000 kg/m³) against a head of 42 m. The pump efficiency is 72% and the electric motor driving it has an efficiency of 95%. Use g = 9.81 m/s². Find the electrical power Pel the motor draws, in kW.
Q = 150 L/sH = 42 mηpump = 72%ηmotor = 95%g = 9.81 m/s²
Pel = kW
8.Pumps & pump powerHard
A pump moves 60 L/s of water (ρ = 1000 kg/m³) from an open lower reservoir to an upper tank whose water surface is 5 m higher. The pipe is 200 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 = 3 (entrance and exit included). Use g = 9.81 m/s². The pump efficiency is 75%. Find the shaft power Pshaft in kW.
Q = 60 L/sΔz = 5 mL = 200 mD = 200 mmf = 0.015ΣK = 3η = 75%
Pshaft = kW
9.Open-channel flow (Manning)Medium
A trapezoidal gravel channel has a bottom width of 1.5 m and side slopes of 2:1 (horizontal:vertical). Water flows uniformly 0.6 m deep. Manning n = 0.025 and the bed slope is S = 0.004. Find the flow rate Q in m³/s.
b = 1.5 my = 0.6 mz = 2n = 0.025S = 0.004
Q = m³/s
10.Open-channel flow (Manning)Hard
A concrete sewer with an inside diameter of 375 mm (Manning n = 0.013) must reach a self-cleansing velocity of 0.75 m/s when flowing just full. Find the minimum slope S (in m/m).
D = 375 mmn = 0.013V = 0.75 m/s
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 85.84 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 = 0.5 + 2.5/2 = 1.75 m
A = H·b = 2.5·2 = 5 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·1.75·5 = 85.84 kN
Answer
F = 85.84 kN
2.Hydrostatic force
Answer: ycp = 1.083 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 = 0.5 + 1/2 = 1 m
A = H·b = 1·1.5 = 1.5 m²
Resultant force
F = ρ·g·yc·A = 880·9.81·1·1.5 = 12.95 kN
Centre of pressure
Pressure grows with depth, so the resultant acts below the centroid.
Ic = b·H³/12 = 1.5·1³/12 = 0.125 m⁴
ycp = yc + Ic/(yc·A) = 1 + 0.125/(1·1.5) = 1.083 m
Answer
ycp = 1.083 m
3.Continuity & Bernoulli
Answer: p1 = 220.6 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.
Velocities
v1 = Q/A1 = 0.023/0.03142 = 0.7321 m/s
v2 = Q/A2 = 0.023/0.01767 = 1.302 m/s
Bernoulli
p1 = p2 + ½ρ(v2² − v1²) = 220 + ½·1000·(1.302² − 0.7321²)/1000 = 220 + 0.579 = 220.6 kPa
Answer
p1 = 220.6 kPa
4.Continuity & Bernoulli
Answer: p2 = 82.6 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 = 2.25·(100/50)² = 9 m/s
½ρ(v1² − v2²) = ½·1000·(2.25² − 9²)/1000 = −37.97 kPa
ρgΔz = 1000·9.81·3/1000 = 29.43 kPa
p2 = p1 + ½ρ(v1² − v2²) − ρgΔz = 150 + (−37.97) − 29.43 = 82.6 kPa
Answer
p2 = 82.6 kPa
5.Pipe friction & head loss
Answer: Δp = 242 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) = 0.025·(800/0.05)·1.1²/(2·9.81) = 24.67 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 = 1000·9.81·24.67 = 242000 Pa = 242 kPa
Answer
Δp = 242 kPa
6.Pipe friction & head loss
Answer: Δz = 1.704 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.
Velocity
V = Q/A = 0.1/0.07069 = 1.415 m/s
V²/(2g) = 0.102 m
Loss coefficients
f·L/D = 0.018·150/0.3 = 9
ΣK = 0.5 + 4·0.3 + 5 + 1.0 = 7.7
Total head loss
hf = 9·0.102 = 0.9181 m
hm = 7.7·0.102 = 0.7855 m
Δz = hf + hm = 1.704 m
Answer
Δz = 1.704 m
7.Pumps & pump power
Answer: Pel = 90.36 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 = 1000·9.81·0.15·42 = 61.8 kW
Shaft power
Pshaft = 61.8/0.72 = 85.84 kW
Electrical power
Pel = 85.84/0.95 = 90.36 kW
Answer
Pel = 90.36 kW
8.Pumps & pump power
Answer: Pshaft = 6.55 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 = 0.06/0.03142 = 1.91 m/s
V²/(2g) = 0.1859 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (15 + 3)·0.1859 = 3.346 m
Pump head
H = Δz + hL = 5 + 3.346 = 8.346 m
Shaft power
Pshaft = ρ·g·Q·H/η = 1000·9.81·0.06·8.346/0.75 = 6.55 kW
Answer
Pshaft = 6.55 kW
9.Open-channel flow (Manning)
Answer: Q = 2.177 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 = (1.5 + 2·0.6)·0.6 = 1.62 m²
P = b + 2y·√(1 + z²) = 1.5 + 2·0.6·√(1 + 2²) = 4.183 m
R = A/P = 0.3873 m
Velocity
V = (1/0.025)·0.38732/3·0.0041/2 = 1.344 m/s
Flow rate
Q = A·V = 1.62·1.344 = 2.177 m³/s
Answer
Q = 2.177 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.002232 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.
Hydraulic radius
R = D/4 = 0.375/4 = 0.09375 m
Solve Manning for S
S = (n·V/R2/3)² = (0.013·0.75/0.2064)² = 0.002232
Answer
S = 0.002232
That is a fall of 2.232 m per kilometre.
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