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 3 m below the surface. Find the resultant hydrostatic force F on the gate.
d = 3 mH = 2 mb = 1.5 mρ = 1000 kg/m³g = 9.81 m/s²
F = kN
2.Hydrostatic forceHard
A vertical rectangular gate 1 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 = 1.5 mb = 1 mρ = 1260 kg/m³g = 9.81 m/s²
ycp = m
3.Continuity & BernoulliMedium
Water flows through a horizontal pipe that narrows from 75 mm to 40 mm. In the wide section the pressure is 260 kPa and the velocity is 1.75 m/s. Ignoring losses, find the pressure p2 in the narrow section.
d1 = 75 mmd2 = 40 mmp1 = 260 kPav1 = 1.75 m/sρ = 1000 kg/m³
p2 = kPa
4.Continuity & BernoulliHard
A horizontal venturi meter (inlet 250 mm, throat 125 mm) carries water. A mercury U-tube manometer connected between the inlet and the throat shows a level difference of 280 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 = 250 mmd2 = 125 mmΔh = 280 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 m/s through a pipe 200 m long with an inside diameter of 75 mm. Use the Darcy friction factor f = 0.018 and g = 9.81 m/s². Find the friction head loss hf.
V = 2 m/sL = 200 mD = 75 mmf = 0.018g = 9.81 m/s²
hf = m
6.Pipe friction & head lossHard
Two open reservoirs are joined by a pipe 200 m long with an inside diameter of 200 mm and a Darcy friction factor f = 0.025. 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 = 200 mD = 200 mmf = 0.025Kentrance = 0.5Kbend = 0.3 (×3)Kvalve = 2Kexit = 1.0
Q = L/s
7.Pumps & pump powerMedium
A pump delivers 40 L/s of water (ρ = 1000 kg/m³) against a head of 40 m. The pump efficiency is 70% and the electric motor driving it has an efficiency of 88%. Use g = 9.81 m/s². Find the electrical power Pel the motor draws, in kW.
Q = 40 L/sH = 40 mηpump = 70%ηmotor = 88%g = 9.81 m/s²
Pel = kW
8.Pumps & pump powerHard
A pump moves 100 L/s of water (ρ = 1000 kg/m³) from an open lower reservoir to an upper tank whose water surface is 14 m higher. The pipe is 300 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. Find the head H the pump must add.
Q = 100 L/sΔz = 14 mL = 300 mD = 250 mmf = 0.018ΣK = 8
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 3:1 (horizontal:vertical). Water flows uniformly 0.5 m deep. Manning n = 0.03 and the bed slope is S = 0.0005. Find the flow rate Q in m³/s.
b = 2.5 my = 0.5 mz = 3n = 0.03S = 0.0005
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 0.6 m/s when flowing just full. Find the minimum slope S (in m/m).
D = 375 mmn = 0.013V = 0.6 m/s
S = m/m
3Solutions
1.Hydrostatic force
Answer: F = 117.7 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 = 3 + 2/2 = 4 m
A = H·b = 2·1.5 = 3 m²
Resultant force
F = ρ·g·yc·A = 1000·9.81·4·3 = 117.7 kN
Answer
F = 117.7 kN
2.Hydrostatic force
Answer: ycp = 3.308 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 + 1.5/2 = 3.25 m
A = H·b = 1.5·1 = 1.5 m²
Resultant force
F = ρ·g·yc·A = 1260·9.81·3.25·1.5 = 60.26 kN
Centre of pressure
Pressure grows with depth, so the resultant acts below the centroid.
Ic = b·H³/12 = 1·1.5³/12 = 0.2813 m⁴
ycp = yc + Ic/(yc·A) = 3.25 + 0.2813/(3.25·1.5) = 3.308 m
Answer
ycp = 3.308 m
3.Continuity & Bernoulli
Answer: p2 = 242.6 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 = 1.75·(75/40)² = 6.152 m/s
½ρ(v1² − v2²) = ½·1000·(1.75² − 6.152²)/1000 = −17.39 kPa
p2 = p1 + ½ρ(v1² − v2²) = 260 + (−17.39) = 242.6 kPa
Answer
p2 = 242.6 kPa
4.Continuity & Bernoulli
Answer: Q = 102.3 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)·9.81·0.28 = 34610 Pa
Throat velocity
Continuity and Bernoulli together give the ideal throat velocity.
β = 125/250 = 0.5
v2 = √(2·34610/(1000·(1 − 0.0625))) = 8.593 m/s
Actual flow rate
Q = Cd·A2·v2 = 0.97·0.01227·8.593 = 0.1023 m³/s = 102.3 L/s
Answer
Q = 102.3 L/s
5.Pipe friction & head loss
Answer: hf = 9.786 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) = 2²/(2·9.81) = 0.2039 m
Darcy–Weisbach
hf = f·(L/D)·V²/(2g) = 0.018·(200/0.075)·2²/(2·9.81) = 9.786 m
Answer
hf = 9.786 m
6.Pipe friction & head loss
Answer: Q = 62.86 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 = 0.025·200/0.2 = 25
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 = (25 + 4.4)·V²/(2·9.81)
V = √(2·9.81·6/29.4) = 2.001 m/s
Flow rate
Q = V·π·D²/4 = 2.001·0.03142 = 0.06286 m³/s = 62.86 L/s
Answer
Q = 62.86 L/s
7.Pumps & pump power
Answer: Pel = 25.48 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 = 1000·9.81·0.04·40 = 15.7 kW
Shaft power
Pshaft = 15.7/0.7 = 22.42 kW
Electrical power
Pel = 22.42/0.88 = 25.48 kW
Answer
Pel = 25.48 kW
8.Pumps & pump power
Answer: H = 20.26 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 = 0.1/0.04909 = 2.037 m/s
V²/(2g) = 0.2115 m
Head losses
hL = (f·L/D + ΣK)·V²/(2g) = (21.6 + 8)·0.2115 = 6.261 m
Pump head
Both surfaces are open and practically at rest, so pressure and velocity terms drop out.
H = Δz + hL = 14 + 6.261 = 20.26 m
Answer
H = 20.26 m
9.Open-channel flow (Manning)
Answer: Q = 0.7449 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 + 3·0.5)·0.5 = 2 m²
P = b + 2y·√(1 + z²) = 2.5 + 2·0.5·√(1 + 3²) = 5.662 m
R = A/P = 0.3532 m
Velocity
V = (1/0.03)·0.35322/3·0.00051/2 = 0.3724 m/s
Flow rate
Q = A·V = 2·0.3724 = 0.7449 m³/s
Answer
Q = 0.7449 m³/s
10.Open-channel flow (Manning)
Answer: S = 0.001429 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 = (0.013·0.6/0.2064)² = 0.001429
Answer
S = 0.001429
That is a fall of 1.429 m per kilometre.
ExamProblems · examproblems.com/courses/hydraulic-engineering