Study Notes/Civil Engineering/Hydraulics & Water Resources

Hydraulics & Water Resources

Fluid mechanics, pipe flow, and open channel hydraulics

1. Fluid Properties

Fundamental Properties

Properties that define fluid behavior and response to applied forces.

Mass Density (ρ)

ρ = m/V

Water: ρ = 1000 kg/m³

Specific Weight (γ)

γ = ρg = W/V

Water: γ = 9.81 kN/m³

Specific Gravity (SG)

SG = ρ_fluid/ρ_water

Dimensionless ratio

Specific Volume

v_s = 1/ρ = V/m

Volume per unit mass

Viscosity

Dynamic Viscosity (μ)

τ = μ(dv/dy)

Newton's Law of Viscosity

Units: Pa·s or N·s/m²

Kinematic Viscosity (ν)

ν = μ/ρ

Ratio of dynamic to density

Units: m²/s

Surface Tension & Capillarity

Surface Tension (σ)

Force per unit length at liquid surface

Water at 20°C: σ ≈ 0.073 N/m

Capillary Rise

h = 4σ cos θ / (γd)

θ = contact angle, d = tube diameter

Compressibility

E_v = -Δp / (ΔV/V) = -Δp / (Δρ/ρ)

Bulk Modulus of Elasticity

Water: E_v ≈ 2.2 GPa (relatively incompressible)

2. Hydrostatics (Fluid at Rest)

Pressure Variation with Depth

In a static fluid, pressure increases linearly with depth due to fluid weight.

p = γh = ρgh

Pressure at depth h below free surface

p_abs = p_atm + p_gauge

Pressure Measurement

Manometer

p₁ + γ₁h₁ = p₂ + γ₂h₂

Pressure balance at same elevation

Pressure Head

h = p/γ

Height of fluid column equivalent to pressure

Hydrostatic Force on Plane Surfaces

Inclined Plane Surface

F = γh̄A = γȳ sin θ A

h̄ = vertical depth to centroid, ȳ = inclined distance to centroid

Center of Pressure (y_p)

y_p = ȳ + I_c/(ȳA)

I_c = moment of inertia about centroidal axis

Common Moments of Inertia

Shape	Area	I_c
Rectangle (b × h)	bh	bh³/12
Triangle (base b, height h)	bh/2	bh³/36
Circle (diameter d)	πd²/4	πd⁴/64

Buoyancy

Archimedes' Principle

F_B = γV_displaced

Buoyant force equals weight of fluid displaced

Acts through centroid of displaced volume (center of buoyancy)

Stability of Floating Bodies

MB = I/V_displaced - GB

• M = metacenter, B = center of buoyancy, G = center of gravity
• Stable if M is above G (positive MB)
• Unstable if M is below G (negative MB)

3. Fluid Dynamics Fundamentals

Types of Flow

Based on Time

Steady: Properties don't change with time
Unsteady: Properties vary with time

Based on Space

Uniform: Same velocity at all points
Non-uniform: Velocity varies spatially

Reynolds Number

Re = ρVD/μ = VD/ν

Ratio of inertial to viscous forces

Pipe Flow:

• Laminar: Re < 2000

• Transition: 2000 < Re < 4000

• Turbulent: Re > 4000

Open Channel:

• Laminar: Re < 500

• Transition: 500 < Re < 2000

• Turbulent: Re > 2000

Continuity Equation

Q = A₁V₁ = A₂V₂

For incompressible, steady flow

Q = AV

Discharge = Area × Velocity

Bernoulli's Equation

p₁/γ + V₁²/2g + z₁ = p₂/γ + V₂²/2g + z₂

For ideal fluid: steady, incompressible, frictionless flow along streamline

Head Components:

• Pressure Head: p/γ (energy due to pressure)
• Velocity Head: V²/2g (kinetic energy)
• Elevation Head: z (potential energy)
• Total Head: H = p/γ + V²/2g + z

Energy Equation (with losses)

p₁/γ + V₁²/2g + z₁ = p₂/γ + V₂²/2g + z₂ + h_L

h_L = head loss due to friction and minor losses

Momentum Equation

ΣF = ρQ(V₂ - V₁)

Force = Rate of change of momentum

Used for calculating forces on pipe bends, jets, etc.

4. Pipe Flow

Darcy-Weisbach Equation

h_f = fLV²/(2gD)

f = friction factor, L = pipe length, D = diameter

Friction Factor (f)

Laminar Flow (Re < 2000)

f = 64/Re

Hagen-Poiseuille equation

Turbulent Flow

Use Moody diagram or:

• Colebrook equation (implicit)

• Swamee-Jain (explicit approximation)

Hazen-Williams Formula

V = 0.8492C_HWR^0.63S^0.54

Commonly used for water distribution (SI units)

C_HW: New cast iron = 130, Old cast iron = 100, PVC = 150

Minor Losses

h_m = KV²/2g

K = loss coefficient

Fitting	K Value
Sharp-edged entrance	0.5
Rounded entrance	0.04
Exit loss	1.0
90° elbow (regular)	0.3
90° elbow (long radius)	0.2
Gate valve (fully open)	0.2
Globe valve (fully open)	10
Check valve	2.5

Pipes in Series and Parallel

Series

Q = Q₁ = Q₂ = Q₃

h_L = h_L1 + h_L2 + h_L3

Same flow, head losses add

Parallel

Q = Q₁ + Q₂ + Q₃

h_L = h_L1 = h_L2 = h_L3

Flows add, same head loss

5. Open Channel Flow

Open Channel vs Pipe Flow

Open channel has free surface exposed to atmosphere. Driving force is gravity (slope), not pressure difference.

Geometric Properties

Wetted Perimeter (P): Length of channel boundary in contact with water
Hydraulic Radius (R): R = A/P (flow efficiency)
Hydraulic Depth (D_h): D_h = A/T (T = top width)
Slope (S): S = h_f/L = sin θ ≈ tan θ (for small angles)

Manning's Equation

V = (1/n)R^2/3S^1/2

Q = (1/n)AR^2/3S^1/2

n = Manning's roughness coefficient (SI units)

Channel Type	n Value
Glass, PVC	0.010
Smooth concrete	0.012
Finished concrete	0.015
Earth channel (clean)	0.022
Natural stream (clean)	0.030
Natural stream (weeds)	0.050

Froude Number

Fr = V/√(gD_h)

Ratio of inertial to gravitational forces

Fr < 1: Subcritical (tranquil) - Deep, slow flow

Fr = 1: Critical flow

Fr > 1: Supercritical (rapid) - Shallow, fast flow

Specific Energy

E = y + V²/2g = y + Q²/(2gA²)

Energy per unit weight relative to channel bottom

Critical Depth (rectangular channel):

y_c = (Q²/gb²)^1/3 = (q²/g)^1/3

q = Q/b = unit discharge

Hydraulic Jump

Sequent Depth Relationship (rectangular channel)

y₂/y₁ = ½(√(1 + 8Fr₁²) - 1)

Energy loss: ΔE = (y₂ - y₁)³/(4y₁y₂)

6. Weirs and Orifices

Orifice Flow

Q = C_dA√(2gH)

C_d = discharge coefficient (typically 0.60-0.65)

H = head above center of orifice

Orifice Coefficients:

• C_c: Contraction coefficient = A_jet/A_orifice
• C_v: Velocity coefficient = V_actual/V_theoretical
• C_d: Discharge coefficient = C_c × C_v

Rectangular Weirs

Suppressed (Full Width)

Q = C_d(2/3)b√(2g)H^3/2

Francis formula:

Q = 1.84bH^3/2 (SI)

Contracted

Q = 1.84(b - 0.2H)H^3/2

Account for end contractions

Two contractions: subtract 0.1H each side

V-Notch (Triangular) Weir

Q = C_d(8/15)√(2g)tan(θ/2)H^5/2

θ = notch angle

For 90° V-notch:

Q = 1.38H^5/2 (SI, with C_d = 0.58)

Cipolletti Weir

Trapezoidal weir with 4:1 side slopes (compensates for contractions)

Q = 1.86bH^3/2 (SI)

Broad-Crested Weir

Q = C_db√g(2/3H)^3/2

Critical flow occurs at crest

Q ≈ 1.7bH^3/2 (SI, approximate)

7. Water Supply Systems

Water Demand

Design Periods (Typical):

• Source of supply: 25-50 years
• Transmission mains: 25-30 years
• Distribution system: 25 years
• Treatment plant: 15-25 years
• Pumps & equipment: 10-15 years

Per Capita Consumption

Classification	lpcd (liters/capita/day)
Rural communities	60-100
Small towns	100-150
Cities (residential)	150-200
Metro areas	200-300

Demand Variations

Average Daily Demand: Q_avg = Population × per capita consumption
Maximum Daily Demand: Q_max,day = 1.5 to 2.0 × Q_avg
Maximum Hourly Demand: Q_max,hr = 2.0 to 3.0 × Q_avg
Fire Demand: Q_fire = 100√P (P = population in thousands, Q in liters/min)

Storage Requirements

Components of Storage:

• Equalizing storage: 25-30% of max daily demand
• Fire reserve: Fire flow × duration (typically 4-6 hours)
• Emergency storage: 25% of average daily demand

Distribution System

Grid/Loop System

• Multiple supply paths
• More reliable (redundancy)
• Lower head loss
• Higher cost

Tree/Dead-End System

• Single supply path
• Less reliable
• Lower cost
• Used for small communities

8. Pumps and Pumping Systems

Total Dynamic Head (TDH)

TDH = H_s + H_d + h_f + V²_d/2g

H_s = static suction lift, H_d = static discharge head

h_f = friction losses, V²_d/2g = velocity head at discharge

Pump Power

Water Power (Output)

P_w = γQH = ρgQH

Theoretical power delivered to water

Brake Power (Input)

P_b = P_w/η

Power required at pump shaft

Pump Efficiency

η = P_w/P_b = γQH/P_b

Typical centrifugal pump: η = 60-85%

Specific Speed

N_s = NQ^1/2/H^3/4

N = rpm, Q = flow rate, H = head

Low N_s (500-1500): Radial flow - high head, low flow

Medium N_s (1500-4000): Mixed flow

High N_s (4000-15000): Axial flow - low head, high flow

Net Positive Suction Head (NPSH)

NPSH_a = p_atm/γ - H_s - h_f,s - p_v/γ

NPSH_a = Available, p_v = vapor pressure

NPSH_a > NPSH_r (to avoid cavitation)

Affinity Laws

For same pump, variable speed:

Q₂/Q₁ = N₂/N₁

Flow ∝ Speed

H₂/H₁ = (N₂/N₁)²

Head ∝ Speed²

P₂/P₁ = (N₂/N₁)³

Power ∝ Speed³

Pumps in Series and Parallel

Series

Same Q, heads add

H_total = H₁ + H₂

Used for high head applications

Parallel

Same H, flows add

Q_total = Q₁ + Q₂

Used for high flow applications

Key Takeaways for CE Board Exam

Must-Know Formulas

✓ Pressure: p = γh
✓ Bernoulli: p/γ + V²/2g + z = constant
✓ Continuity: Q = AV
✓ Darcy: h_f = fLV²/2gD
✓ Manning: V = (1/n)R^2/3S^1/2
✓ Weir: Q ∝ H^3/2 (rectangular)
✓ Pump power: P = γQH

Critical Concepts

✓ Reynolds number: Re = VD/ν
✓ Froude number: Fr = V/√(gD)
✓ Critical depth and specific energy
✓ Pipe series vs parallel
✓ Minor loss coefficients (K values)
✓ NPSH and cavitation prevention
✓ Affinity laws for pumps

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1. Fluid Properties

Fundamental Properties

Mass Density (ρ)

Specific Weight (γ)

Specific Gravity (SG)

Specific Volume

Viscosity

Dynamic Viscosity (μ)

Kinematic Viscosity (ν)

Surface Tension & Capillarity

Surface Tension (σ)

Capillary Rise

Compressibility

2. Hydrostatics (Fluid at Rest)

Pressure Variation with Depth

Pressure Measurement

Manometer

Pressure Head

Hydrostatic Force on Plane Surfaces

Inclined Plane Surface

Center of Pressure (yp)

Common Moments of Inertia

Buoyancy

Stability of Floating Bodies

3. Fluid Dynamics Fundamentals

Types of Flow

Based on Time

Based on Space

Reynolds Number

Continuity Equation

Bernoulli's Equation

Head Components:

Energy Equation (with losses)

Momentum Equation

4. Pipe Flow

Darcy-Weisbach Equation

Friction Factor (f)

Laminar Flow (Re < 2000)

Turbulent Flow

Hazen-Williams Formula

Minor Losses

Pipes in Series and Parallel

Series

Parallel

5. Open Channel Flow

Open Channel vs Pipe Flow

Geometric Properties

Manning's Equation

Froude Number

Specific Energy

Critical Depth (rectangular channel):

Hydraulic Jump

6. Weirs and Orifices

Orifice Flow

Orifice Coefficients:

Rectangular Weirs

Suppressed (Full Width)

Contracted

V-Notch (Triangular) Weir

Cipolletti Weir

Broad-Crested Weir

7. Water Supply Systems

Water Demand

Design Periods (Typical):

Per Capita Consumption

Demand Variations

Storage Requirements

Components of Storage:

Distribution System

Grid/Loop System

Tree/Dead-End System

8. Pumps and Pumping Systems

Total Dynamic Head (TDH)

Pump Power

Water Power (Output)

Brake Power (Input)

Pump Efficiency

Specific Speed

Net Positive Suction Head (NPSH)

Affinity Laws

Center of Pressure (y_p)