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 4 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 4 mH = 1 mb = 2 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 oil, with its top edge 1 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 1 mH = 2 mb = 2 mρ = 880 kg/m³g = 9.81 m/s²
F = kN
3.Continuity & BernoulliMedium
Water flows through a horizontal pipe that narrows from 200 mm to 50 mm. In the wide section the pressure is 130 kPa and the velocity is 0.75 m/s. Ignoring losses, find the pressure p2 in the narrow section.
d1 = 200 mmd2 = 50 mmp1 = 130 kPav1 = 0.75 m/sρ = 1000 kg/m³
p2 = kPa
4.Continuity & BernoulliHard
A closed water tank has compressed air above the water at a gauge pressure of 100 kPa. Water leaves through an opening with a diameter of 50 mm, 2.5 m below the water surface, into the open air. Ignoring losses, find the flow rate Q. Use g = 9.81 m/s².
pair = 100 kPa (gauge)h = 2.5 md = 50 mmρ = 1000 kg/m³g = 9.81 m/s²
Q = L/s
5.Pipe friction & head lossMedium
Water (ρ = 1000 kg/m³) flows at 2.1 m/s through a horizontal pipe, 1000 m long, with an inside diameter of 300 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 = 2.1 m/sL = 1000 mD = 300 mmf = 0.015ρ = 1000 kg/m³
Δp = kPa
6.Pipe friction & head lossHard
A pipe 100 m long with an inside diameter of 300 mm (Darcy friction factor f = 0.02) runs from an open reservoir and discharges as a free jet into the air. The minor losses are a square-edged entrance (K = 0.5), 3 bends (K = 0.3 each) and a valve (K = 2). Use g = 9.81 m/s². How far H must the reservoir surface be above the pipe outlet for a flow rate of 80 L/s?
Q = 80 L/sL = 100 mD = 300 mmf = 0.02Kentrance = 0.5Kbend = 0.3 (×3)Kvalve = 2
H = m
7.Pumps & pump powerMedium
A pump with an efficiency of 82% delivers 80 L/s of water (ρ = 1000 kg/m³) against a head of 38 m. Use g = 9.81 m/s². Find the shaft power Pshaft needed, in kW.
Q = 80 L/sH = 38 mη = 82%g = 9.81 m/s²
Pshaft = 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 16 m higher. The pipe is 150 m long with an inside diameter of 200 mm and a Darcy friction factor f = 0.018; the minor-loss coefficients add up to ΣK = 4 (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 100 kPa. Find the head H the pump must add.
Q = 80 L/sΔz = 16 mL = 150 mD = 200 mmf = 0.018ΣK = 4ptank = 100 kPa
H = m
9.Open-channel flow (Manning)Medium
A trapezoidal grass-lined earth channel has a bottom width of 2.5 m and side slopes of 2:1 (horizontal:vertical). Water flows uniformly 1.2 m deep. Manning n = 0.03 and the bed slope is S = 0.005. Find the flow rate Q in m³/s.
b = 2.5 my = 1.2 mz = 2n = 0.03S = 0.005
Q = m³/s
10.Open-channel flow (Manning)Hard
A trapezoidal clean earth channel with a bottom width of 3.5 m and side slopes of 2:1 (horizontal:vertical) must carry 27 m³/s in uniform flow at a depth of 2.2 m. Manning n = 0.022. Find the required bed slope S (in m/m).
Q = 27 m³/sb = 3.5 my = 2.2 mz = 2n = 0.022
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 88.29 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 + 1/2 = 4.5 m
A = H·b = 1·2 = 2 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·4.5·2 = 88.29 kN
Answer
F = 88.29 kN
2.Hydrostatic force
Answer: F = 69.06 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 + 2/2 = 2 m
A = H·b = 2·2 = 4 m²
Resultant force
F = ρ·g·yc·A = 880·9.81·2·4 = 69.06 kN
Answer
F = 69.06 kN
3.Continuity & Bernoulli
Answer: p2 = 58.28 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 = 0.75·(200/50)² = 12 m/s
½ρ(v1² − v2²) = ½·1000·(0.75² − 12²)/1000 = −71.72 kPa
p2 = p1 + ½ρ(v1² − v2²) = 130 + (−71.72) = 58.28 kPa
Answer
p2 = 58.28 kPa
4.Continuity & Bernoulli
Answer: Q = 30.99 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.
Bernoulli from the surface to the jet
v = √(2·(100000/1000 + 9.81·2.5)) = 15.78 m/s
Flow rate
Q = A·v = 0.001963·15.78 = 0.03099 m³/s = 30.99 L/s
Answer
Q = 30.99 L/s
5.Pipe friction & head loss
Answer: Δp = 110.3 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·(1000/0.3)·2.1²/(2·9.81) = 11.24 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 = 1000·9.81·11.24 = 110250 Pa = 110.3 kPa
Answer
Δp = 110.3 kPa
6.Pipe friction & head loss
Answer: H = 0.7225 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 = 0.08/0.07069 = 1.132 m/s
V²/(2g) = 0.06529 m
Loss coefficients
f·L/D = 0.02·100/0.3 = 6.667
ΣK = 0.5 + 3·0.3 + 2 = 3.4
Energy equation
The outlet jet still carries its velocity head; it is not a loss but it has to be supplied.
H = (1 + 6.667 + 3.4)·0.06529 = 0.7225 m
Answer
H = 0.7225 m
7.Pumps & pump power
Answer: Pshaft = 36.37 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 = 1000·9.81·0.08·38 = 29.82 kW
Shaft power
Pshaft = Ph/η = 29.82/0.82 = 36.37 kW
Answer
Pshaft = 36.37 kW
8.Pumps & pump power
Answer: H = 31.98 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 = 0.08/0.03142 = 2.546 m/s
V²/(2g) = 0.3305 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (13.5 + 4)·0.3305 = 5.784 m
Pressure head
p/(ρ·g) = 100000/(1000·9.81) = 10.19 m
Pump head
Both surfaces are practically at rest, so there is no velocity head term.
H = Δz + p/(ρ·g) + hL = 16 + 10.19 + 5.784 = 31.98 m
Answer
H = 31.98 m
9.Open-channel flow (Manning)
Answer: Q = 11.41 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 = (2.5 + 2·1.2)·1.2 = 5.88 m²
P = b + 2y·√(1 + z²) = 2.5 + 2·1.2·√(1 + 2²) = 7.867 m
R = A/P = 0.7475 m
Velocity
V = (1/0.03)·0.74752/3·0.0051/2 = 1.941 m/s
Flow rate
Q = A·V = 5.88·1.941 = 11.41 m³/s
Answer
Q = 11.41 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.0008208 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 = (3.5 + 2·2.2)·2.2 = 17.38 m²
P = b + 2y·√(1 + z²) = 3.5 + 2·2.2·√(1 + 2²) = 13.34 m
R = A/P = 1.303 m
Solve Manning for S
S1/2 = n·Q/(A·R2/3) = 0.022·27/(17.38·1.193) = 0.02865
S = 0.02865² = 0.0008208
Answer
S = 0.0008208
That is a fall of 0.8208 m per kilometre.
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