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 1.5 m wide sits in the wall of a tank of water, with its top edge 1 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 1 mH = 1 mb = 1.5 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 glycerin, with its top edge 2.5 m below the surface. Find the depth ycp of the centre of pressure below the surface.
d = 2.5 mH = 2 mb = 2 mρ = 1260 kg/m³g = 9.81 m/s²
ycp = m
3.Continuity & BernoulliMedium
Water flows through a horizontal pipe that narrows from 150 mm to 100 mm. In the wide section the pressure is 260 kPa and the velocity is 2.75 m/s. Ignoring losses, find the pressure p2 in the narrow section.
d1 = 150 mmd2 = 100 mmp1 = 260 kPav1 = 2.75 m/sρ = 1000 kg/m³
p2 = kPa
4.Continuity & BernoulliHard
A horizontal venturi meter (inlet 100 mm, throat 50 mm) carries water. A mercury U-tube manometer connected between the inlet and the throat shows a level difference of 170 mm. The discharge coefficient is 0.97. Find the flow rate Q. Use ρ = 1000 kg/m³ for water, 13 600 kg/m³ for mercury and g = 9.81 m/s².
d1 = 100 mmd2 = 50 mmΔh = 170 mmCd = 0.97ρHg = 13 600 kg/m³g = 9.81 m/s²
Q = L/s
5.Pipe friction & head lossMedium
Water flows at a mean velocity of 2.8 m/s through a pipe 150 m long with an inside diameter of 250 mm. Use the Darcy friction factor f = 0.03 and g = 9.81 m/s². Find the friction head loss hf.
V = 2.8 m/sL = 150 mD = 250 mmf = 0.03g = 9.81 m/s²
hf = m
6.Pipe friction & head lossHard
Two open reservoirs are joined by a pipe 150 m long with an inside diameter of 100 mm and a Darcy friction factor f = 0.02. The water surface of the upper reservoir is 7 m above that of the lower one. The minor losses are a square-edged entrance (K = 0.5), 3 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 = 7 mL = 150 mD = 100 mmf = 0.02Kentrance = 0.5Kbend = 0.3 (×3)Kvalve = 2Kexit = 1.0
Q = L/s
7.Pumps & pump powerMedium
A pump with an efficiency of 60% delivers 150 L/s of water (ρ = 1000 kg/m³) against a head of 54 m. Use g = 9.81 m/s². Find the shaft power Pshaft needed, in kW.
Q = 150 L/sH = 54 mη = 60%g = 9.81 m/s²
Pshaft = kW
8.Pumps & pump powerHard
A pump moves 120 L/s of water (ρ = 1000 kg/m³) from an open lower reservoir to an upper tank whose water surface is 38 m higher. The pipe is 150 m long with an inside diameter of 250 mm and a Darcy friction factor f = 0.018; the minor-loss coefficients add up to ΣK = 8 (entrance and exit included). Use g = 9.81 m/s². The pump efficiency is 72%. Find the shaft power Pshaft in kW.
Q = 120 L/sΔz = 38 mL = 150 mD = 250 mmf = 0.018ΣK = 8η = 72%
Pshaft = kW
9.Open-channel flow (Manning)Medium
A concrete storm sewer with an inside diameter of 900 mm flows just full (as an open channel, not under pressure). Manning n = 0.013 and the slope is S = 0.0015. Find the flow rate Q in m³/s.
D = 900 mmn = 0.013S = 0.0015
Q = m³/s
10.Open-channel flow (Manning)Hard
A rectangular grass-lined earth channel 4 m wide must carry 6 m³/s in uniform flow at a depth of 2 m. Manning n = 0.03. Find the required bed slope S (in m/m).
Q = 6 m³/sb = 4 my = 2 mn = 0.03
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 22.07 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 + 1/2 = 1.5 m
A = H·b = 1·1.5 = 1.5 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·1.5·1.5 = 22.07 kN
Answer
F = 22.07 kN
2.Hydrostatic force
Answer: ycp = 3.595 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 = 2.5 + 2/2 = 3.5 m
A = H·b = 2·2 = 4 m²
Resultant force
F = ρ·g·yc·A = 1260·9.81·3.5·4 = 173 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) = 3.5 + 1.333/(3.5·4) = 3.595 m
Answer
ycp = 3.595 m
3.Continuity & Bernoulli
Answer: p2 = 244.6 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 = 2.75·(150/100)² = 6.188 m/s
½ρ(v1² − v2²) = ½·1000·(2.75² − 6.188²)/1000 = −15.36 kPa
p2 = p1 + ½ρ(v1² − v2²) = 260 + (−15.36) = 244.6 kPa
Answer
p2 = 244.6 kPa
4.Continuity & Bernoulli
Answer: Q = 12.75 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.
Pressure difference from the manometer
Δp = (13600 − 1000)·9.81·0.17 = 21013 Pa
Throat velocity
Continuity and Bernoulli together give the ideal throat velocity.
β = 50/100 = 0.5
v2 = √(2·21013/(1000·(1 − 0.0625))) = 6.695 m/s
Actual flow rate
Q = Cd·A2·v2 = 0.97·0.001963·6.695 = 0.01275 m³/s = 12.75 L/s
Answer
Q = 12.75 L/s
5.Pipe friction & head loss
Answer: hf = 7.193 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 head
V²/(2g) = 2.8²/(2·9.81) = 0.3996 m
Darcy–Weisbach
hf = f·(L/D)·V²/(2g) = 0.03·(150/0.25)·2.8²/(2·9.81) = 7.193 m
Answer
hf = 7.193 m
6.Pipe friction & head loss
Answer: Q = 15.69 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 = 0.02·150/0.1 = 30
ΣK = 0.5 + 3·0.3 + 2 + 1.0 = 4.4
Energy equation
Both surfaces are open to the air and practically at rest, so all of Δz is lost in the pipe.
7 = (30 + 4.4)·V²/(2·9.81)
V = √(2·9.81·7/34.4) = 1.998 m/s
Flow rate
Q = V·π·D²/4 = 1.998·0.007854 = 0.01569 m³/s = 15.69 L/s
Answer
Q = 15.69 L/s
7.Pumps & pump power
Answer: Pshaft = 132.4 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 = 1000·9.81·0.15·54 = 79.46 kW
Shaft power
Pshaft = Ph/η = 79.46/0.6 = 132.4 kW
Answer
Pshaft = 132.4 kW
8.Pumps & pump power
Answer: Pshaft = 71.49 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 = 0.12/0.04909 = 2.445 m/s
V²/(2g) = 0.3046 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (10.8 + 8)·0.3046 = 5.726 m
Pump head
H = Δz + hL = 38 + 5.726 = 43.73 m
Shaft power
Pshaft = ρ·g·Q·H/η = 1000·9.81·0.12·43.73/0.72 = 71.49 kW
Answer
Pshaft = 71.49 kW
9.Open-channel flow (Manning)
Answer: Q = 0.7011 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 = 0.6362 m², R = D/4 = 0.225 m
Velocity
V = (1/0.013)·0.2252/3·0.00151/2 = 1.102 m/s
Flow rate
Q = A·V = 0.6362·1.102 = 0.7011 m³/s
Answer
Q = 0.7011 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.0005062 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·y = 4·2 = 8 m²
P = b + 2y = 4 + 2·2 = 8 m
R = A/P = 1 m
Solve Manning for S
S1/2 = n·Q/(A·R2/3) = 0.03·6/(8·1) = 0.0225
S = 0.0225² = 0.0005062
Answer
S = 0.0005062
That is a fall of 0.5062 m per kilometre.
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