Formula Sheet: Hydraulics

Table of Contents
1. Fluid Properties
2. Fluid Statics
3. Fluid Dynamics - Fundamental Equations
4. Pipe Flow - Head Loss
5. Pipe Systems
6. Pumps and Turbines
7. Open Channel Flow
8. Flow Measurement
9. Dimensional Analysis and Similitude
10. Water Hammer
11. Forces on Submerged Objects
12. Cavitation and Air Entrainment
13. Steady Flow in Pipes - Additional Topics
14. Unsteady Flow
15. Hydraulic Machinery - Additional
View more

Fluid Properties

Density and Specific Weight

• Density (ρ): Mass per unit volume \[\rho = \frac{m}{V}\] where m = mass (kg or slugs), V = volume (m³ or ft³)
  Units: kg/m³ (SI), slugs/ft³ (US)
• Specific Weight (γ): Weight per unit volume \[\gamma = \rho g\] where ρ = density, g = gravitational acceleration (9.81 m/s² or 32.2 ft/s²)
  Units: N/m³ (SI), lb/ft³ (US)
  Water at 4°C: γ = 62.4 lb/ft³ = 9810 N/m³
• Specific Gravity (SG): Ratio of fluid density to water density \[SG = \frac{\rho_{fluid}}{\rho_{water}} = \frac{\gamma_{fluid}}{\gamma_{water}}\] Dimensionless quantity

Viscosity

• Dynamic (Absolute) Viscosity (μ): Measure of fluid resistance to shear \[\tau = \mu \frac{du}{dy}\] where τ = shear stress, du/dy = velocity gradient
  Units: N·s/m² = Pa·s (SI), lb·s/ft² (US)
  Common unit: centipoise (cP); 1 cP = 0.001 Pa·s
• Kinematic Viscosity (ν): \[\nu = \frac{\mu}{\rho}\] where μ = dynamic viscosity, ρ = density
  Units: m²/s (SI), ft²/s (US)
  Common unit: centistoke (cSt); 1 cSt = 10⁻⁶ m²/s

Fluid Statics

Hydrostatic Pressure

• Pressure at Depth: \[p = p_0 + \gamma h\] or \[p = p_0 + \rho g h\] where p₀ = pressure at surface, h = depth below surface
  Units: Pa (SI), lb/ft² = psf (US)
• Pressure Head: \[h = \frac{p}{\gamma}\] where p = pressure, γ = specific weight
  Units: m (SI), ft (US)
• Gauge Pressure: \[p_{gauge} = p_{absolute} - p_{atmospheric}\] where p_atmospheric ≈ 101.3 kPa = 14.7 psi = 2116 psf

Hydrostatic Forces on Surfaces

• Force on Horizontal Surface: \[F = pA\] where p = pressure, A = area
  Force acts at centroid of area
• Force on Vertical Plane Surface: \[F = \gamma h_c A\] where h_c = depth to centroid of area, A = area
• Center of Pressure (Vertical Surface): \[h_{cp} = h_c + \frac{I_c}{h_c A}\] where h_cp = depth to center of pressure, I_c = second moment of area about centroid, h_c = depth to centroid
• Force on Inclined Plane Surface: \[F = \gamma h_c A\] where h_c = vertical depth to centroid
• Center of Pressure (Inclined Surface): \[y_{cp} = y_c + \frac{I_c \sin\theta}{y_c A}\] where y_cp = distance along incline to center of pressure, y_c = distance along incline to centroid, θ = angle of inclination

Buoyancy

• Buoyant Force (Archimedes' Principle): \[F_b = \gamma_{fluid} V_{displaced}\] where V_displaced = volume of fluid displaced
  Force acts upward through center of buoyancy (centroid of displaced volume)
• Floating Body Equilibrium: \[W = F_b\] where W = weight of body

Fluid Dynamics - Fundamental Equations

Continuity Equation

• Conservation of Mass (Incompressible Flow): \[Q = A_1 V_1 = A_2 V_2\] where Q = volumetric flow rate, A = cross-sectional area, V = average velocity
  Units for Q: m³/s (SI), ft³/s = cfs (US), gal/min = gpm (US)
• Mass Flow Rate: \[\dot{m} = \rho Q = \rho A V\] where ṁ = mass flow rate
  Units: kg/s (SI), slugs/s (US)

Bernoulli Equation

• Bernoulli Equation (Ideal Flow): \[\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2\] where p/γ = pressure head, V²/2g = velocity head, z = elevation head
  All terms in units of length (m or ft)
  Assumptions: steady, incompressible, frictionless flow along a streamline
• Bernoulli Equation (Energy Form): \[\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L\] where h_L = head loss between points 1 and 2 (includes friction and minor losses)
• Total Head (H): \[H = \frac{p}{\gamma} + \frac{V^2}{2g} + z\] Units: m (SI), ft (US)
• Hydraulic Grade Line (HGL): \[HGL = \frac{p}{\gamma} + z\] Elevation of water surface in piezometer
• Energy Grade Line (EGL): \[EGL = \frac{p}{\gamma} + \frac{V^2}{2g} + z = HGL + \frac{V^2}{2g}\] Total energy per unit weight

Momentum Equation

• Linear Momentum Equation: \[\sum \vec{F} = \dot{m}(\vec{V}_2 - \vec{V}_1) = \rho Q(\vec{V}_2 - \vec{V}_1)\] where ΣF = sum of external forces, V₁ = inlet velocity, V₂ = outlet velocity
• Force on Pipe Bend: Apply momentum equation in x and y directions separately
  Include pressure forces, weight of fluid, and reaction forces
• Jet Impact Force (Normal to Fixed Plate): \[F = \rho Q V = \rho A V^2\] where V = jet velocity
• Jet Impact Force (Moving Plate): \[F = \rho A (V_{jet} - V_{plate})^2\] where V_plate = plate velocity

Pipe Flow - Head Loss

Major Losses (Friction)

• Darcy-Weisbach Equation: \[h_f = f \frac{L}{D} \frac{V^2}{2g}\] where h_f = friction head loss (m or ft), f = friction factor (dimensionless), L = pipe length, D = pipe diameter, V = average velocity
  Most accurate equation for pipe friction loss
• Hazen-Williams Equation: \[h_f = \frac{4.727 L Q^{1.852}}{C^{1.852} D^{4.87}}\] (US Customary units: L in ft, Q in ft³/s, D in ft, h_f in ft)
  \[V = 0.849 C R^{0.63} S^{0.54}\] where C = Hazen-Williams coefficient (dimensionless), R = hydraulic radius (ft), S = slope of energy grade line = h_f/L
  Typical C values: 140-150 (new pipe), 100-130 (old pipe), 120 (concrete)
• Hazen-Williams (SI Units): \[V = 0.85 C R^{0.63} S^{0.54}\] where V in m/s, R in m
• Manning Equation: \[V = \frac{1.486}{n} R^{2/3} S^{1/2}\] (US Customary units)
  \[V = \frac{1}{n} R^{2/3} S^{1/2}\] (SI units)
  where n = Manning roughness coefficient, R = hydraulic radius, S = slope of energy grade line
  Commonly used for open channel flow but applicable to pipe flow
• Head Loss from Manning: \[h_f = \frac{n^2 L V^2}{2.208 R^{4/3}}\] (US Customary units)

Reynolds Number and Flow Regime

• Reynolds Number: \[Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}\] where D = pipe diameter (characteristic length)
  Dimensionless parameter determining flow regime
• Flow Regimes (Pipe Flow):
  • Laminar flow: Re <>
  • Transitional flow: 2000 < re=""><>
  • Turbulent flow: Re > 4000

Friction Factor Determination

• Laminar Flow (Re <> \[f = \frac{64}{Re}\] Friction factor independent of roughness
• Colebrook Equation (Turbulent Flow): \[\frac{1}{\sqrt{f}} = -2.0 \log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)\] where ε = absolute roughness (ft or m), ε/D = relative roughness
  Implicit equation requiring iteration or use of Moody diagram
• Swamee-Jain Equation (Explicit Approximation): \[f = \frac{0.25}{\left[\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{5.74}{Re^{0.9}}\right)\right]^2}\] Accurate approximation to Colebrook equation for 10⁻⁶ < ε/d="">< 10⁻²="" and="" 5000="">< re=""><>
• Typical Absolute Roughness (ε) Values:
  • Drawn tubing, glass: 0.000005 ft = 0.0015 mm
  • Commercial steel, wrought iron: 0.00015 ft = 0.046 mm
  • Galvanized iron: 0.0005 ft = 0.15 mm
  • Cast iron: 0.00085 ft = 0.26 mm
  • Concrete: 0.001-0.01 ft = 0.3-3 mm
  • Riveted steel: 0.003-0.03 ft = 0.9-9 mm

Minor Losses

• Minor Loss Equation: \[h_m = K \frac{V^2}{2g}\] where K = loss coefficient (dimensionless), V = velocity (typically downstream velocity)
• Total Head Loss in Pipe System: \[h_L = h_f + \sum h_m = f \frac{L}{D} \frac{V^2}{2g} + \sum K \frac{V^2}{2g}\]
• Equivalent Length Method: \[h_m = f \frac{L_e}{D} \frac{V^2}{2g}\] where L_e = equivalent length of straight pipe producing same head loss as fitting
• Common K Values:
  • Sudden expansion: K = (1 - A₁/A₂)²
  • Sudden contraction: K ≈ 0.5 (depends on A₁/A₂)
  • Sharp-edged entrance: K ≈ 0.5
  • Well-rounded entrance: K ≈ 0.05
  • Exit loss (pipe to reservoir): K = 1.0
  • 90° standard elbow: K ≈ 0.9
  • 45° standard elbow: K ≈ 0.4
  • Gate valve (fully open): K ≈ 0.15-0.2
  • Globe valve (fully open): K ≈ 10
  • Check valve: K ≈ 2-4

Pipe Systems

Series and Parallel Pipes

• Pipes in Series:
  • Same flow rate through all pipes: Q_total = Q₁ = Q₂ = Q₃
  • Total head loss is sum: h_L,total = h_L,1 + h_L,2 + h_L,3
• Pipes in Parallel:
  • Total flow is sum: Q_total = Q₁ + Q₂ + Q₃
  • Head loss is same through each branch: h_L,1 = h_L,2 = h_L,3
• Three-Reservoir Problem: Apply energy equation from each reservoir to junction point
  Apply continuity at junction: ΣQ_in = ΣQ_out
  Iterative solution typically required

Pipe Network Analysis

• Hardy Cross Method: Iterative method for solving pipe networks
  Flow correction for each loop: \[\Delta Q = -\frac{\sum h_f}{\sum \frac{n h_f}{Q}}\] where n = exponent in head loss equation (typically 2 for Darcy-Weisbach)

Pumps and Turbines

Pump Fundamentals

• Energy Equation with Pump: \[\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L\] where h_p = pump head added to fluid (m or ft)
• Pump Power (Water Horsepower): \[P_w = \gamma Q h_p = \rho g Q h_p\] where P_w = water power (hydraulic power)
  Units: W (SI), ft·lb/s (US)
  1 horsepower (hp) = 550 ft·lb/s = 746 W
• Brake Horsepower: \[P_b = \frac{P_w}{\eta_p} = \frac{\gamma Q h_p}{\eta_p}\] where P_b = brake power (shaft power), η_p = pump efficiency (decimal)
• Pump Efficiency: \[\eta_p = \frac{P_w}{P_b} = \frac{\gamma Q h_p}{P_b}\] Typical efficiencies: 60-85% for centrifugal pumps
• Motor Power Required: \[P_m = \frac{P_b}{\eta_m}\] where P_m = motor power, η_m = motor efficiency

Pump Performance

• Total Dynamic Head (TDH): \[TDH = h_p = h_{static} + h_L\] where h_static = static elevation difference + pressure head difference
• Net Positive Suction Head Available (NPSHA): \[NPSH_A = \frac{p_{atm}}{\gamma} - \frac{p_v}{\gamma} - z_s - h_{L,s}\] where p_atm = atmospheric pressure, p_v = vapor pressure of liquid, z_s = suction lift (elevation difference), h_L,s = suction line losses
  Must exceed NPSH_R (required) to avoid cavitation
• Specific Speed (N_s): \[N_s = \frac{N \sqrt{Q}}{h_p^{3/4}}\] where N = rotational speed (rpm), Q = flow rate (gpm), h_p = head (ft)
  Dimensionless parameter characterizing pump type
  Radial flow: N_s < 4200;="" mixed="" flow:="" 4200=""><>_s < 9000;="" axial="" flow:="">_s > 9000

Affinity Laws (Pump Similarity)

• At Constant Impeller Diameter: \[\frac{Q_2}{Q_1} = \frac{N_2}{N_1}\] \[\frac{h_{p,2}}{h_{p,1}} = \left(\frac{N_2}{N_1}\right)^2\] \[\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3\] where N = rotational speed (rpm)
• At Constant Speed: \[\frac{Q_2}{Q_1} = \frac{D_2}{D_1}\] \[\frac{h_{p,2}}{h_{p,1}} = \left(\frac{D_2}{D_1}\right)^2\] \[\frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^3\] where D = impeller diameter

Turbines

• Energy Equation with Turbine: \[\frac{p_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_t + h_L\] where h_t = turbine head extracted from fluid
• Turbine Power Output: \[P_t = \eta_t \gamma Q h_t\] where η_t = turbine efficiency
• Specific Speed (Turbine): \[N_s = \frac{N \sqrt{P}}{h_t^{5/4}}\] where P = power (hp), h_t = head (ft)
  Impulse (Pelton): N_s < 5;="" francis:="" 5=""><>_s < 75;="" propeller/kaplan:="">_s > 75

Open Channel Flow

Channel Geometry

• Hydraulic Radius: \[R = \frac{A}{P}\] where A = cross-sectional flow area, P = wetted perimeter
• Hydraulic Depth: \[D_h = \frac{A}{T}\] where T = top width of water surface
• Rectangular Channel:
  • Area: A = by
  • Wetted perimeter: P = b + 2y
  • Hydraulic radius: R = by/(b + 2y)
  • Top width: T = b
  where b = channel width, y = depth
• Trapezoidal Channel:
  • Area: A = y(b + my)
  • Wetted perimeter: P = b + 2y√(1 + m²)
  • Hydraulic radius: R = y(b + my)/[b + 2y√(1 + m²)]
  • Top width: T = b + 2my
  where m = side slope (horizontal:vertical)
• Triangular Channel:
  • Area: A = my²
  • Wetted perimeter: P = 2y√(1 + m²)
  • Hydraulic radius: R = y/(2√(1 + m²))
  • Top width: T = 2my
• Circular Channel (Partially Full):
  • Area: A = (D²/4)(θ - sin θ)
  • Wetted perimeter: P = Dθ/2
  • Hydraulic radius: R = (D/4)(1 - sin θ/θ)
  where D = diameter, θ = central angle in radians
  For circular pipe flowing full: R = D/4

Uniform Flow (Normal Depth)

• Manning Equation: \[Q = \frac{1.486}{n} A R^{2/3} S_0^{1/2}\] (US Customary units)
  \[Q = \frac{1}{n} A R^{2/3} S_0^{1/2}\] (SI units)
  where S₀ = channel bed slope = sin θ ≈ tan θ for small slopes
  Applies to uniform flow at normal depth
• Velocity Form of Manning: \[V = \frac{1.486}{n} R^{2/3} S_0^{1/2}\] (US Customary)
• Manning's n Values:
  • Smooth concrete, glass: n = 0.012
  • Ordinary concrete: n = 0.013
  • Corrugated metal: n = 0.024
  • Natural earth channel, clean: n = 0.022-0.025
  • Natural channel with vegetation: n = 0.03-0.05
  • Mountain streams with rocks: n = 0.04-0.05
• Normal Depth (y_n): Depth of uniform flow; found by solving Manning equation for depth
  Requires iterative solution for most channel shapes

Energy and Specific Energy

• Specific Energy: \[E = y + \frac{V^2}{2g} = y + \frac{Q^2}{2gA^2}\] where E = energy head relative to channel bottom
  For rectangular channel: E = y + Q²/(2gb²y²)
• Critical Depth (y_c): Depth at which specific energy is minimum for given discharge \[\frac{dE}{dy} = 0\] For rectangular channel: \[y_c = \left(\frac{Q^2}{gb^2}\right)^{1/3} = \left(\frac{q^2}{g}\right)^{1/3}\] where q = Q/b = discharge per unit width
• Critical Velocity: \[V_c = \sqrt{g D_h}\] For rectangular channel: V_c = √(gy_c)
• Minimum Specific Energy: \[E_{min} = \frac{3}{2} y_c\] (for rectangular channel)

Froude Number and Flow Regime

• Froude Number: \[Fr = \frac{V}{\sqrt{g D_h}}\] For rectangular channel: \[Fr = \frac{V}{\sqrt{gy}}\] Dimensionless parameter characterizing flow regime
• Flow Classification:
  • Subcritical flow: Fr < 1,="" y=""> y_c, tranquil, slow
  • Critical flow: Fr = 1, y = y_c
  • Supercritical flow: Fr > 1, y <>_c, rapid, shooting
• Wave Celerity: \[c = \sqrt{g D_h}\] For rectangular channel: c = √(gy)
  Speed at which small gravity wave propagates

Gradually Varied Flow

• Energy Equation for GVF: \[z_1 + y_1 + \frac{V_1^2}{2g} = z_2 + y_2 + \frac{V_2^2}{2g} + h_L\] where z = channel bottom elevation
• Water Surface Profile Equation: \[\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}\] where S₀ = bed slope, S_f = friction slope = energy slope
• Friction Slope: \[S_f = \frac{n^2 V^2}{2.208 R^{4/3}}\] (US Customary units; from Manning equation)
• Profile Classifications:
  • M profiles: Mild slope (y_n > y_c)
  • S profiles: Steep slope (y_n <>_c)
  • C profiles: Critical slope (y_n = y_c)
  • H profiles: Horizontal slope (S₀ = 0)
  • A profiles: Adverse slope (S₀ <>
  Number designation: 1 (y > y_n and y > y_c), 2 (between), 3 (y <>_n and y <>_c)

Hydraulic Jump

• Conjugate Depths (Rectangular Channel): \[\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)\] or \[y_2 = \frac{y_1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)\] where y₁ = upstream depth (supercritical), y₂ = downstream depth (subcritical), Fr₁ = upstream Froude number
• Energy Loss in Hydraulic Jump: \[h_L = E_1 - E_2 = \frac{(y_2 - y_1)^3}{4 y_1 y_2}\] (rectangular channel)
• Jump Classification (by Fr₁):
  • 1.0-1.7: Standing wave (undular jump)
  • 1.7-2.5: Weak jump
  • 2.5-4.5: Oscillating jump
  • 4.5-9.0: Steady jump (best performance)
  • > 9.0: Strong jump, rough surface

Weirs and Flumes

• Sharp-Crested Rectangular Weir (Suppressed): \[Q = C_w \frac{2}{3} \sqrt{2g} L H^{3/2}\] \[Q = 3.33 L H^{3/2}\] (US Customary, simplified with C_w ≈ 0.62)
  where L = weir length (ft), H = head above weir crest (ft), C_w = weir coefficient
• Contracted Rectangular Weir: \[Q = 3.33 (L - 0.2nH) H^{3/2}\] (US Customary, Francis formula)
  where n = number of end contractions (0, 1, or 2)
• V-Notch (Triangular) Weir: \[Q = C_v \frac{8}{15} \sqrt{2g} \tan\frac{\theta}{2} H^{5/2}\] For 90° V-notch: \[Q = 2.5 H^{5/2}\] (US Customary, C_v ≈ 0.58)
  where θ = notch angle, H = head above notch bottom
• Cipolletti (Trapezoidal) Weir: \[Q = 3.367 L H^{3/2}\] (US Customary)
  Side slopes 1H:4V compensate for end contractions
• Broad-Crested Weir: \[Q = C_b L y_c^{3/2} = C_b L \left(\frac{2E}{3}\right)^{3/2}\] where C_b ≈ 3.09 (US Customary), E = upstream specific energy
  Flow is critical over weir crest
• Parshall Flume: \[Q = 4 W H_a^{1.522}\] (for throat widths W = 1-8 ft, free flow, US Customary)
  where W = throat width (ft), H_a = upstream head (ft)
  Equations vary with throat width and submergence

Flow Measurement

Closed Conduit Meters

• Venturi Meter: \[Q = C_v A_2 \sqrt{\frac{2g(h_1 - h_2)}{1 - (A_2/A_1)^2}}\] \[Q = C_v A_2 \sqrt{\frac{2\Delta p/\gamma}{1 - (D_2/D_1)^4}}\] where C_v = venturi coefficient ≈ 0.98, A₂ = throat area, Δp = pressure difference, D₂ = throat diameter, D₁ = inlet diameter
• Orifice Plate: \[Q = C_d A_0 \sqrt{\frac{2g\Delta h}{1 - (A_0/A_1)^2}}\] \[Q = C_d A_0 \sqrt{\frac{2\Delta p/\gamma}{1 - (D_0/D_1)^4}}\] where C_d = discharge coefficient ≈ 0.6-0.65, A₀ = orifice area, D₀ = orifice diameter
• Flow Nozzle: Similar to orifice plate; C_d ≈ 0.95-0.99
• Pitot Tube: \[V = \sqrt{2g(h_s - h_{st})} = \sqrt{\frac{2(p_t - p_s)}{\rho}}\] where h_s = static head, h_st = stagnation head, p_t = total pressure, p_s = static pressure
  Measures point velocity

Orifices and Nozzles

• Torricelli's Theorem (Free Discharge): \[V = C_v \sqrt{2gh}\] where C_v = velocity coefficient ≈ 0.97-0.99, h = head on orifice centerline
• Discharge Through Orifice: \[Q = C_d A_0 \sqrt{2gh}\] where C_d = discharge coefficient = C_c × C_v
  C_c = contraction coefficient ≈ 0.61-0.64
  C_d ≈ 0.6 for sharp-edged orifice
• Submerged Orifice: \[Q = C_d A_0 \sqrt{2g(h_1 - h_2)}\] where h₁ = upstream head, h₂ = downstream head
• Time to Empty Tank Through Orifice: \[t = \frac{2A_t}{C_d A_0 \sqrt{2g}} \left(\sqrt{h_1} - \sqrt{h_2}\right)\] where A_t = tank cross-sectional area (constant)

Dimensional Analysis and Similitude

Dimensionless Parameters

• Reynolds Number: \[Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}\] Ratio of inertial to viscous forces
• Froude Number: \[Fr = \frac{V}{\sqrt{gL}}\] Ratio of inertial to gravitational forces
  Important in free-surface flows
• Weber Number: \[We = \frac{\rho V^2 L}{\sigma}\] Ratio of inertial to surface tension forces
  where σ = surface tension
• Mach Number: \[Ma = \frac{V}{c}\] Ratio of flow velocity to speed of sound
  where c = speed of sound in fluid
• Euler Number: \[Eu = \frac{\Delta p}{\rho V^2}\] Ratio of pressure forces to inertial forces

Similitude and Model Testing

• Geometric Similarity: \[\frac{L_m}{L_p} = \frac{D_m}{D_p} = \frac{H_m}{H_p} = L_r\] where subscript m = model, p = prototype, L_r = length ratio (scale factor)
• Kinematic Similarity: \[\frac{V_m}{V_p} = V_r, \quad \frac{t_m}{t_p} = t_r\] Velocity and time ratios must be consistent
• Dynamic Similarity: Requires equality of force ratios; key dimensionless parameters must be equal
  Re_m = Re_p and/or Fr_m = Fr_p
• Reynolds Similarity: \[\frac{V_m L_m}{\nu_m} = \frac{V_p L_p}{\nu_p}\] \[V_r = \frac{\nu_r}{L_r}\] Used when viscous forces dominate
• Froude Similarity: \[\frac{V_m}{\sqrt{g_m L_m}} = \frac{V_p}{\sqrt{g_p L_p}}\] \[V_r = \sqrt{L_r}\] (assuming g_m = g_p)
  Used for free-surface flows; most common for hydraulic models
• Scale Ratios (Froude Similarity):
  • Length: L_r
  • Area: A_r = L_r²
  • Velocity: V_r = √L_r
  • Time: t_r = √L_r
  • Discharge: Q_r = L_r^5/2
  • Force: F_r = L_r³

Water Hammer

Pressure Surge

• Wave Speed in Pipe: \[a = \sqrt{\frac{K/\rho}{1 + (K/E)(D/t)}}\] where a = wave speed (celerity), K = bulk modulus of fluid, E = elastic modulus of pipe material, D = pipe diameter, t = pipe wall thickness
  For rigid pipe: a = √(K/ρ) ≈ 4700 ft/s for water
• Simplified Wave Speed: \[a \approx \frac{4660}{1 + (D/t)(K/E)}\] (US Customary, approximate for water)
  Typical values: 3000-4000 ft/s in steel pipes
• Pressure Rise (Sudden Valve Closure): \[\Delta p = \rho a \Delta V\] or \[\Delta h = \frac{a \Delta V}{g}\] where ΔV = change in velocity (velocity reduction)
  Applies when closure time < 2l/a="" (rapid="">
• Critical Closure Time: \[t_c = \frac{2L}{a}\] If actual closure time <>_c, treat as instantaneous closure
• Maximum Pressure Head: \[h_{max} = h_0 + \frac{a V_0}{g}\] where h₀ = initial pressure head, V₀ = initial velocity

Forces on Submerged Objects

Drag and Lift

• Drag Force: \[F_D = C_D A \frac{\rho V^2}{2}\] where C_D = drag coefficient (dimensionless), A = projected area perpendicular to flow, V = free stream velocity
• Lift Force: \[F_L = C_L A \frac{\rho V^2}{2}\] where C_L = lift coefficient, A = planform area
• Typical Drag Coefficients:
  • Sphere (Re > 10⁴): C_D ≈ 0.4-0.5
  • Circular cylinder (Re > 10³): C_D ≈ 1.0-1.2
  • Flat plate (normal to flow): C_D ≈ 1.2-2.0
  • Streamlined body: C_D ≈ 0.04-0.1
  • Airfoil: C_D ≈ 0.01-0.05
  Values depend on Reynolds number and geometry

Cavitation and Air Entrainment

Cavitation

• Cavitation Number (σ): \[\sigma = \frac{p - p_v}{\rho V^2/2}\] where p = local pressure, p_v = vapor pressure of liquid
  Cavitation occurs when local pressure drops to vapor pressure
• Thoma Cavitation Parameter: \[\sigma_T = \frac{NPSH}{h}\] where h = total head
  Used for turbomachinery
• Vapor Pressure of Water: At 20°C (68°F): p_v ≈ 2.3 kPa abs = 0.34 psia
  At 40°C (104°F): p_v ≈ 7.4 kPa abs = 1.07 psia
  Increases with temperature

Steady Flow in Pipes - Additional Topics

Siphon

• Flow Rate in Siphon: Apply Bernoulli equation from surface to exit: \[V = \sqrt{2g\Delta h}\] where Δh = elevation difference between reservoir surfaces
• Pressure at Summit: Critical point for cavitation; must maintain p > p_v
  Maximum siphon height limited by atmospheric pressure and vapor pressure

Pipe Entrance and Exit Conditions

• Entrance from Reservoir: Velocity approaches zero in large reservoir
  Apply Bernoulli from reservoir surface to pipe entrance with entrance loss
• Exit to Reservoir: All kinetic energy is lost (exit loss K = 1.0)
  Pressure at exit equals reservoir pressure

Unsteady Flow

Reservoir Discharge

• Variable Head Discharge: \[\frac{dV}{dt} = Q_{in} - Q_{out}\] For tank with constant area A_t: \[A_t \frac{dh}{dt} = Q_{in} - Q_{out}\] Integrate to find h(t) or t for given head change
• Time to Drain Tank: For Q_in = 0 and Q_out through orifice: \[t = \frac{2A_t}{C_d A_0 \sqrt{2g}} \left(\sqrt{h_i} - \sqrt{h_f}\right)\] where h_i = initial head, h_f = final head

Hydraulic Machinery - Additional

Pump Selection and Operation

• System Head Curve: \[h_{system} = h_{static} + K Q^2\] where K depends on friction factor and minor losses
  Intersection with pump head curve gives operating point
• Pumps in Series: Heads add: h_total = h₁ + h₂
  Same flow through each pump
• Pumps in Parallel: Flows add: Q_total = Q₁ + Q₂
  Same head across each pump

Impulse-Momentum Applications

• Force on Pipe Reducer/Expander: Apply momentum equation in flow direction: \[F_x = p_1 A_1 - p_2 A_2 - F_{pipe,x} = \rho Q(V_2 - V_1)\] Solve for F_pipe,x (reaction force on pipe)
• Force on Pipe Bend: Apply in x and y directions separately: \[F_x = p_1 A_1 \cos\theta_1 - p_2 A_2 \cos\theta_2 - F_{pipe,x} = \rho Q(V_2 \cos\theta_2 - V_1 \cos\theta_1)\] \[F_y = p_1 A_1 \sin\theta_1 - p_2 A_2 \sin\theta_2 - F_{pipe,y} - W = \rho Q(V_2 \sin\theta_2 - V_1 \sin\theta_1)\] where W = weight of fluid in bend
• Resultant Force: \[F_R = \sqrt{F_x^2 + F_y^2}\]