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 4 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 4 mH = 2.5 mb = 1.5 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 glycerin, with its top edge 1.5 m below the surface. Find the depth ycp of the centre of pressure below the surface.
d = 1.5 mH = 2 mb = 1.5 mρ = 1260 kg/m³g = 9.81 m/s²
ycp = m
3.Continuity & BernoulliMedium
A pitot-static tube in a water pipe reads a difference of 23 kPa between the stagnation pressure and the static pressure. Find the water velocity v at the tip.
p0 − p = 23 kPaρ = 1000 kg/m³
v = m/s
4.Continuity & BernoulliHard
Water flows up through a pipe that narrows from 150 mm to 50 mm while rising 7 m. At section 1 the pressure is 260 kPa and the velocity is 1 m/s. Ignoring losses, find the pressure p2 at section 2.
d1 = 150 mmd2 = 50 mmp1 = 260 kPav1 = 1 m/sΔz = 7 mρ = 1000 kg/m³
p2 = kPa
5.Pipe friction & head lossMedium
Water flows at a mean velocity of 2.1 m/s through a pipe 1000 m long with an inside diameter of 250 mm. Use the Darcy friction factor f = 0.015 and g = 9.81 m/s². Find the friction head loss hf.
V = 2.1 m/sL = 1000 mD = 250 mmf = 0.015g = 9.81 m/s²
hf = m
6.Pipe friction & head lossHard
Two open reservoirs are joined by a pipe 300 m long with an inside diameter of 300 mm and a Darcy friction factor f = 0.02. The water surface of the upper reservoir is 6 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 = 6 mL = 300 mD = 300 mmf = 0.02Kentrance = 0.5Kbend = 0.3 (×3)Kvalve = 2Kexit = 1.0
Q = L/s
7.Pumps & pump powerMedium
A pump delivers 80 L/s of water (ρ = 1000 kg/m³) against a head of 76 m. The pump efficiency is 65% 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 = 80 L/sH = 76 mηpump = 65%ηmotor = 95%g = 9.81 m/s²
Pel = 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 23 m higher. The pipe is 600 m long with an inside diameter of 250 mm and a Darcy friction factor f = 0.02; the minor-loss coefficients add up to ΣK = 8 (entrance and exit included). Use g = 9.81 m/s². The upper tank is closed and the air above its water is at a gauge pressure of 350 kPa. Find the head H the pump must add.
Q = 120 L/sΔz = 23 mL = 600 mD = 250 mmf = 0.02ΣK = 8ptank = 350 kPa
H = m
9.Open-channel flow (Manning)Medium
A concrete storm sewer with an inside diameter of 450 mm flows just full (as an open channel, not under pressure). Manning n = 0.013 and the slope is S = 0.0008. Find the flow rate Q in m³/s.
D = 450 mmn = 0.013S = 0.0008
Q = m³/s
10.Open-channel flow (Manning)Hard
A trapezoidal brickwork channel with a bottom width of 6 m and side slopes of 3:1 (horizontal:vertical) must carry 54 m³/s in uniform flow at a depth of 1.9 m. Manning n = 0.015. Find the required bed slope S (in m/m).
Q = 54 m³/sb = 6 my = 1.9 mz = 3n = 0.015
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 193.1 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 = 4 + 2.5/2 = 5.25 m
A = H·b = 2.5·1.5 = 3.75 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·5.25·3.75 = 193.1 kN
Answer
F = 193.1 kN
2.Hydrostatic force
Answer: ycp = 2.633 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 = 1.5 + 2/2 = 2.5 m
A = H·b = 2·1.5 = 3 m²
Resultant force
F = ρ·g·yc·A = 1260·9.81·2.5·3 = 92.7 kN
Centre of pressure
Pressure grows with depth, so the resultant acts below the centroid.
Ic = b·H³/12 = 1.5·2³/12 = 1 m⁴
ycp = yc + Ic/(yc·A) = 2.5 + 1/(2.5·3) = 2.633 m
Answer
ycp = 2.633 m
3.Continuity & Bernoulli
Answer: v = 6.782 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 = √(2·23000/1000) = 6.782 m/s
Answer
v = 6.782 m/s
4.Continuity & Bernoulli
Answer: p2 = 151.3 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 = 1·(150/50)² = 9 m/s
½ρ(v1² − v2²) = ½·1000·(1² − 9²)/1000 = −40 kPa
ρgΔz = 1000·9.81·7/1000 = 68.67 kPa
p2 = p1 + ½ρ(v1² − v2²) − ρgΔz = 260 + (−40) − 68.67 = 151.3 kPa
Answer
p2 = 151.3 kPa
5.Pipe friction & head loss
Answer: hf = 13.49 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) = 2.1²/(2·9.81) = 0.2248 m
Darcy–Weisbach
hf = f·(L/D)·V²/(2g) = 0.015·(1000/0.25)·2.1²/(2·9.81) = 13.49 m
Answer
hf = 13.49 m
6.Pipe friction & head loss
Answer: Q = 155.3 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 = 0.02·300/0.3 = 20
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.
6 = (20 + 4.4)·V²/(2·9.81)
V = √(2·9.81·6/24.4) = 2.196 m/s
Flow rate
Q = V·π·D²/4 = 2.196·0.07069 = 0.1553 m³/s = 155.3 L/s
Answer
Q = 155.3 L/s
7.Pumps & pump power
Answer: Pel = 96.59 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 = 1000·9.81·0.08·76 = 59.64 kW
Shaft power
Pshaft = 59.64/0.65 = 91.76 kW
Electrical power
Pel = 91.76/0.95 = 96.59 kW
Answer
Pel = 96.59 kW
8.Pumps & pump power
Answer: H = 75.74 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 = 0.12/0.04909 = 2.445 m/s
V²/(2g) = 0.3046 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (48 + 8)·0.3046 = 17.06 m
Pressure head
p/(ρ·g) = 350000/(1000·9.81) = 35.68 m
Pump head
Both surfaces are practically at rest, so there is no velocity head term.
H = Δz + p/(ρ·g) + hL = 23 + 35.68 + 17.06 = 75.74 m
Answer
H = 75.74 m
9.Open-channel flow (Manning)
Answer: Q = 0.08064 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 = 0.159 m², R = D/4 = 0.1125 m
Velocity
V = (1/0.013)·0.11252/3·0.00081/2 = 0.507 m/s
Flow rate
Q = A·V = 0.159·0.507 = 0.08064 m³/s
Answer
Q = 0.08064 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.001003 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 = (6 + 3·1.9)·1.9 = 22.23 m²
P = b + 2y·√(1 + z²) = 6 + 2·1.9·√(1 + 3²) = 18.02 m
R = A/P = 1.234 m
Solve Manning for S
S1/2 = n·Q/(A·R2/3) = 0.015·54/(22.23·1.15) = 0.03167
S = 0.03167² = 0.001003
Answer
S = 0.001003
That is a fall of 1.003 m per kilometre.
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