Cheatsheet: Hydraulics

1. Fluid Properties

1.1 Basic Properties

1.2 Vapor Pressure

• Pressure at which liquid vaporizes at given temperature
• Water at 20°C: 2.34 kPa absolute; at 100°C: 101.3 kPa absolute
• Cavitation occurs when local pressure drops below vapor pressure

2. Fluid Statics

2.1 Pressure Relationships

2.2 Manometry

2.3 Forces on Submerged Surfaces

2.3.1 Horizontal Plane Surfaces

• F_R = γh_cA; h_c = depth to centroid of area
• Force acts at centroid of area

2.3.2 Vertical and Inclined Plane Surfaces

• F_R = γh_cA; h_c = vertical depth to centroid
• Center of pressure: y_cp = y_c + I_xc/(y_cA); I_xc = moment of inertia about centroid
• Center of pressure always below centroid (except for horizontal surfaces)

2.3.3 Common Centroid and Moment of Inertia Values

2.4 Buoyancy

• Archimedes' Principle: F_B = γ_fluidV_displaced
• Buoyant force acts upward through center of buoyancy (centroid of displaced volume)
• Floating body: Weight = Buoyant force
• Stability: Metacenter must be above center of gravity for stable floating

3. Fluid Dynamics Fundamentals

3.1 Flow Classification

3.2 Reynolds Number

• Re = ρVD/μ = VD/ν; V = velocity, D = characteristic length (diameter for pipes)
• Dimensionless ratio of inertial to viscous forces
• Pipe flow: Laminar Re < 2000;="" turbulent="" re=""> 4000
• Open channel: Laminar Re < 500;="" turbulent="" re=""> 2000

3.3 Continuity Equation

• Conservation of mass: ṁ = ρ₁A₁V₁ = ρ₂A₂V₂
• Incompressible flow: Q = A₁V₁ = A₂V₂; Q = volumetric flow rate
• Q (m³/s) = A (m²) × V (m/s)
• 1 m³/s = 1000 L/s = 15,850 gpm

3.4 Energy Equation (Bernoulli)

3.4.1 Bernoulli Equation

• Ideal flow (frictionless): P₁/γ + V₁²/(2g) + z₁ = P₂/γ + V₂²/(2g) + z₂
• Terms: P/γ = pressure head; V²/(2g) = velocity head; z = elevation head
• Total head = sum of all three heads

3.4.2 Energy Equation with Losses

• P₁/γ + V₁²/(2g) + z₁ + h_p = P₂/γ + V₂²/(2g) + z₂ + h_L + h_t
• h_p = pump head added; h_t = turbine head extracted; h_L = head loss
• Pump power: P = γQh_p/η; η = pump efficiency

3.5 Momentum Equation

• ΣF = ṁ(V₂ - V₁) = ρQ(V₂ - V₁)
• Vector equation; apply component-wise
• Forces include pressure, weight, reaction forces
• Used for pipe bends, nozzles, jets impacting surfaces

4. Pipe Flow

4.1 Head Loss - Major Losses

4.1.1 Darcy-Weisbach Equation

• h_L = f(L/D)(V²/2g); f = friction factor, L = pipe length, D = diameter
• Most accurate method for head loss calculation

4.1.2 Friction Factor Determination

4.1.3 Moody Diagram

• Graphical representation of friction factor vs Reynolds number and relative roughness (ε/D)
• Used when iterative Colebrook solution is impractical

4.1.4 Hazen-Williams Equation

• V = 1.318C_HWR_h^0.63S^0.54 (SI: use coefficient 0.849)
• h_L = 10.67L Q^1.85/(C_HW^1.85D^4.87) [SI units: m, m³/s]
• C_HW values: New cast iron = 130; PVC = 150; Old pipes = 100-120; Concrete = 120-140
• Valid only for water at 60°F; limited to turbulent flow

4.1.5 Manning Equation

• Used for open channels but applicable to pipes flowing full
• V = (1.49/n)R_h^2/3S^1/2 (US); V = (1/n)R_h^2/3S^1/2 (SI)
• n = Manning roughness coefficient; R_h = hydraulic radius = A/P

4.2 Pipe Roughness Values

4.3 Minor Losses

4.3.1 Minor Loss Formula

• h_m = K(V²/2g); K = loss coefficient
• Total head loss: h_L,total = h_L,major + Σh_m

4.3.2 Common K Values

4.3.3 Equivalent Length Method

• L_e/D values convert fittings to equivalent pipe length
• Use in Darcy-Weisbach: h_L = f[(L + ΣL_e)/D](V²/2g)

4.4 Pipe Systems

4.4.1 Series Pipes

• Same flow rate through all pipes: Q = Q₁ = Q₂ = Q₃
• Total head loss: h_L,total = h_L,1 + h_L,2 + h_L,3

4.4.2 Parallel Pipes

• Same head loss across all pipes: h_L = h_L,1 = h_L,2 = h_L,3
• Total flow: Q_total = Q₁ + Q₂ + Q₃

4.4.3 Three-Reservoir Problem

• Identify junction point where pipes meet
• Flow direction determined by hydraulic grade line
• Continuity at junction: ΣQ_in = ΣQ_out

4.4.4 Hardy Cross Method

• Iterative method for pipe networks
• Correction factor: ΔQ = -Σ(h_L)/(nΣ|h_L|/Q)
• Apply to each loop until convergence

5. Pumps and Pump Systems

5.1 Pump Fundamentals

5.1.1 Key Definitions

5.1.2 Pump Efficiency

• η = (Power out)/(Power in) = (γQH)/(Power input)
• Pump efficiency range: 60-85% for centrifugal pumps
• Wire-to-water efficiency = η_pump × η_motor

5.2 Net Positive Suction Head (NPSH)

5.2.1 NPSH Available

• NPSH_A = P_atm/γ + z_s - h_L,suction - P_v/γ
• z_s = elevation of source relative to pump centerline (negative if below)
• P_v = vapor pressure of liquid

5.2.2 NPSH Required

• NPSH_R = manufacturer specification from pump curve
• Requirement: NPSH_A ≥ NPSH_R + safety factor (1-2 m)
• Cavitation occurs when NPSH_A <>_R

5.3 Pump Performance Curves

• Head-capacity (H-Q) curve: head vs flow rate
• Efficiency curve: efficiency vs flow rate (peaks at best efficiency point)
• Power curve: brake horsepower vs flow rate
• Operating point: intersection of pump curve and system curve

5.4 System Curves

• H_system = H_static + KQ²; K = resistance coefficient
• Static head independent of flow; friction losses proportional to Q²
• Operating point found where pump curve intersects system curve

5.5 Pump Affinity Laws

5.5.1 Same Pump, Different Speed

• Q₂/Q₁ = N₂/N₁
• H₂/H₁ = (N₂/N₁)²
• P₂/P₁ = (N₂/N₁)³
• N = rotational speed (rpm)

5.5.2 Geometrically Similar Pumps

• Q₂/Q₁ = (D₂/D₁)³(N₂/N₁)
• H₂/H₁ = (D₂/D₁)²(N₂/N₁)²
• P₂/P₁ = (D₂/D₁)⁵(N₂/N₁)³
• D = impeller diameter

5.6 Specific Speed

• N_s = N√Q/H^0.75; N in rpm, Q in gpm, H in ft
• Dimensionless parameter characterizing pump type
• Radial flow (centrifugal): N_s = 500-4000
• Mixed flow: N_s = 4000-9000
• Axial flow: N_s = 9000-15000

5.7 Pumps in Series and Parallel

5.7.1 Series Configuration

• Same flow rate through both pumps
• Total head = H₁ + H₂
• Used for high-head applications

5.7.2 Parallel Configuration

• Same head across both pumps
• Total flow = Q₁ + Q₂
• Used for high-capacity applications

6. Open Channel Flow

6.1 Channel Geometry

6.1.1 Geometric Parameters

6.1.2 Common Channel Shapes

6.2 Flow Classification

6.2.1 Froude Number

• Fr = V/√(gD_h); measures ratio of inertial to gravitational forces
• Fr < 1:="" subcritical="" flow="" (tranquil,="" slow,="" controlled="" by="" downstream="">
• Fr = 1: Critical flow
• Fr > 1: Supercritical flow (rapid, shallow, controlled by upstream conditions)

6.2.2 Critical Depth

• Rectangular: y_c = (q²/g)^1/3; q = Q/b = discharge per unit width
• General: y_c where specific energy is minimum for given Q
• At critical depth: V = √(gD_h)

6.3 Uniform Flow

6.3.1 Manning Equation

• Q = (1.49/n)AR_h^2/3S^1/2 (US customary)
• Q = (1/n)AR_h^2/3S^1/2 (SI)
• S = bed slope = sin(θ) ≈ tan(θ) for small angles
• n = Manning roughness coefficient

6.3.2 Manning n Values

6.3.3 Normal Depth

• Depth at which uniform flow occurs for given Q, n, S, and channel geometry
• Solve Manning equation iteratively for y_n
• Flow is subcritical if y_n > y_c; supercritical if y_n <>_c

6.4 Specific Energy

6.4.1 Specific Energy Definition

• E = y + V²/(2g) = y + Q²/(2gA²)
• Energy per unit weight relative to channel bottom

6.4.2 Specific Energy Curve

• For constant Q, E vs y has minimum at critical depth
• Two possible depths (alternate depths) for given E > E_min
• Upper limb: subcritical flow (high depth, low velocity)
• Lower limb: supercritical flow (low depth, high velocity)

6.4.3 Minimum Specific Energy

• E_min occurs at critical depth
• Rectangular: E_min = 1.5y_c

6.5 Gradually Varied Flow

6.5.1 Classification of Flow Profiles

• Mild slope (M): y_n > y_c (subcritical normal flow)
• Critical slope (C): y_n = y_c
• Steep slope (S): y_n <>_c (supercritical normal flow)
• Horizontal (H): S = 0
• Adverse (A): negative slope

6.5.2 Common Flow Profiles

• M1: Backwater curve; y > y_n > y_c; caused by downstream obstruction
• M2: Drawdown curve; y_n > y > y_c; approaches critical depth downstream
• S1: Subcritical on steep slope; y > y_c > y_n
• S2: Supercritical on steep slope; y_c > y > y_n

6.5.3 Direct Step Method

• Δx = (E₂ - E₁)/(S₀ - S_f); S₀ = bed slope, S_f = friction slope
• S_f = (nV/1.49R_h^2/3)²
• Calculate distance between known depths

6.6 Hydraulic Jump

6.6.1 Sequent Depths

• Rectangular: y₂/y₁ = 0.5(√(1 + 8Fr₁²) - 1); Fr₁ = Froude number before jump
• Sudden transition from supercritical to subcritical flow
• Significant energy loss

6.6.2 Energy Loss in Jump

• ΔE = E₁ - E₂ = (y₂ - y₁)³/(4y₁y₂)
• Used for energy dissipation below spillways and gates

6.7 Weirs

6.7.1 Sharp-Crested Weirs

6.7.2 Broad-Crested Weirs

• Q = C_dL√(2g)H^3/2; C_d = 0.5-0.6
• Critical depth occurs on weir crest

6.8 Culverts

6.8.1 Inlet Control

• Flow controlled by inlet geometry and headwater
• Culvert flows partially full
• Q = C_dA√(2gh); h = headwater depth

6.8.2 Outlet Control

• Flow controlled by tailwater, barrel roughness, and length
• Culvert may flow full
• Use energy equation with friction losses

7. Hydraulic Machinery

7.1 Turbines

7.1.1 Turbine Types

7.1.2 Turbine Power

• P_out = ηγQH; η = turbine efficiency (80-95%)
• P_out in Watts if γ in N/m³, Q in m³/s, H in m

7.1.3 Specific Speed

• N_s = N√P/H^5/4; N in rpm, P in hp, H in ft
• Pelton: N_s <>
• Francis: 5 <>_s <>
• Kaplan: N_s > 100

7.2 Water Hammer

7.2.1 Pressure Surge

• ΔP = ρcΔV; c = wave speed, ΔV = velocity change
• c = √(E_v/ρ) for rigid pipe; c ≈ 1000-1400 m/s for water
• Elastic pipe: c = √(E_v/ρ)/(1 + (E_vD)/(eE))^1/2; e = pipe wall thickness, E = pipe elastic modulus

7.2.2 Pressure Wave Period

• T = 4L/c; L = pipe length
• Maximum pressure if valve closure time <>

8. Flow Measurement

8.1 Venturi Meter

• Q = C_dA₂√[2g(P₁ - P₂)/γ]/√(1 - (A₂/A₁)²)
• C_d = 0.96-0.99 (high accuracy)
• Low head loss

8.2 Orifice Meter

• Q = C_dA₀√[2g(P₁ - P₂)/γ]/√(1 - (A₀/A₁)²)
• C_d = 0.6-0.65
• High head loss, low cost

8.3 Pitot Tube

• V = √[2g(P_stagnation - P_static)/γ]
• Measures point velocity
• P_stagnation at tube opening; P_static from side taps

8.4 Parshall Flume

• Open channel flow measurement device
• Q = CH_aⁿ; C and n depend on flume throat width
• H_a = head at specified upstream location
• Common: 3-inch throat, Q = 0.992H_a^1.547 (Q in cfs, H in ft)

8.5 Flow Over Weirs

• See Section 6.7 for formulas
• Commonly used for flow measurement and rating curves

9. Dimensional Analysis and Similitude

9.1 Buckingham Pi Theorem

• Reduces n variables to (n - m) dimensionless groups
• m = number of fundamental dimensions (usually 3: M, L, T)
• Select m repeating variables, form pi groups with remaining variables

9.2 Common Dimensionless Numbers

9.3 Model Similitude

9.3.1 Types of Similarity

• Geometric: Length ratio constant; (L_m/L_p) = constant
• Kinematic: Velocity ratio constant; (V_m/V_p) = constant
• Dynamic: Force ratio constant; (F_m/F_p) = constant

9.3.2 Reynolds Similarity

• Re_m = Re_p; viscous forces dominant
• V_m/V_p = (L_p/L_m)(ν_m/ν_p)
• Used for pipe flow, submerged bodies

9.3.3 Froude Similarity

• Fr_m = Fr_p; gravitational forces dominant
• V_m/V_p = √(L_m/L_p)
• Used for open channel flow, spillways, ships

9.3.4 Scale Ratios for Froude Similarity

• Length: L_r = L_m/L_p
• Velocity: V_r = √L_r
• Time: T_r = √L_r
• Discharge: Q_r = L_r^5/2
• Force: F_r = L_r³

10. Unsteady Flow

10.1 Gradually Varied Unsteady Flow

10.1.1 Continuity Equation

• ∂A/∂t + ∂Q/∂x = q; q = lateral inflow per unit length

10.1.2 Momentum Equation

• ∂Q/∂t + ∂(Q²/A)/∂x + gA∂y/∂x + gAS_f - gAS₀ = 0
• S₀ = bed slope; S_f = friction slope

10.2 Saint-Venant Equations

• Continuity and momentum equations form hyperbolic partial differential system
• Solved numerically using method of characteristics or finite difference schemes

10.3 Surge Wave Speed

• Positive surge (moving into still water): c = √(g(y₁ + y₂)y₂/(2y₁))
• Small disturbance wave speed: c = √(gy)

11. Groundwater Hydraulics

11.1 Darcy's Law

• Q = KiA; K = hydraulic conductivity, i = hydraulic gradient = dh/dL, A = cross-sectional area
• v = Ki; v = Darcy velocity (apparent velocity)
• V_s = v/n; V_s = seepage velocity (actual pore velocity), n = porosity

11.2 Hydraulic Conductivity

11.3 Well Hydraulics

11.3.1 Confined Aquifer (Steady State)

• Q = 2πKb(h₂ - h₁)/ln(r₂/r₁); b = aquifer thickness
• Transmissivity: T = Kb

11.3.2 Unconfined Aquifer (Steady State)

• Q = πK(h₂² - h₁²)/ln(r₂/r₁)
• Dupuit assumptions: flow horizontal, hydraulic gradient = slope of water table

11.3.3 Well Drawdown

• s = drawdown = h₀ - h; h₀ = initial water table elevation
• Cone of depression forms around pumping well
• Radius of influence: distance where drawdown becomes negligible

11.4 Theis Solution (Transient Flow)

• s = (Q/4πT)W(u); W(u) = well function (tabulated)
• u = r²S/(4Tt); S = storativity, t = time since pumping started
• Used for pump test analysis

12. Hydraulic Structures

12.1 Dams and Spillways

12.1.1 Spillway Types

• Overflow (ogee): shaped to match nappe profile, high efficiency
• Side channel: flow parallel then perpendicular to crest
• Chute: steep channel conveys water down slope
• Morning glory (shaft): circular crest with vertical shaft

12.1.2 Spillway Design Flow

• Q = CLH^3/2; C depends on crest shape and approach conditions
• Ogee spillway: C ≈ 4.0 (US units with Q in cfs, L in ft, H in ft)

12.2 Stilling Basins

• Energy dissipation structure using hydraulic jump
• Basin length ≈ 4-5 times sequent depth
• USBR Types I-IV for different Froude numbers and applications

12.3 Gates and Valves

12.3.1 Sluice Gates

• Q = C_dbh√(2gy₁); b = gate width, h = gate opening, y₁ = upstream depth
• C_d = 0.6-0.7
• Vena contracta forms downstream

12.3.2 Control Valves

• Flow controlled by valve loss coefficient K
• See Section 4.3.2 for K values

12.4 Pipe Outlet Protection

• Riprap sizing: d₅₀ = C(V²/2g); C = 0.02-0.04 for various configurations
• Energy dissipators prevent scour at pipe outlets