Formula Sheet: Momentum Transfer

Table of Contents
1. Fluid Properties and Definitions
2. Fluid Statics
3. Fluid Dynamics Fundamentals
4. Dimensional Analysis and Similitude
5. Laminar Flow in Pipes and Channels
6. Turbulent Flow in Pipes
7. Minor Losses in Pipe Systems
8. Flow Measurement
9. Pump and Compressor Fundamentals
10. Drag and External Flow
11. Flow Through Porous Media
12. Non-Circular Conduits
13. Open Channel Flow
14. Unsteady Flow
15. Boundary Layer Theory
16. Compressible Flow
17. Fluidization
18. Piping System Design
19. Two-Phase Flow
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Fluid Properties and Definitions

Density and Specific Gravity

• Density (ρ): Mass per unit volume \[\rho = \frac{m}{V}\] where m = mass (kg or lbm), V = volume (m³ or ft³)
  Units: kg/m³ or lbm/ft³
• Specific Weight (γ): Weight per unit volume \[\gamma = \rho g\] where g = gravitational acceleration (9.81 m/s² or 32.174 ft/s²)
  Units: N/m³ or lbf/ft³
• Specific Gravity (SG): Ratio of fluid density to reference density \[SG = \frac{\rho_{fluid}}{\rho_{reference}}\] For liquids: reference is water at 4°C (1000 kg/m³ or 62.4 lbm/ft³)
  For gases: reference is air at standard conditions

Viscosity

• Dynamic (Absolute) Viscosity (μ): Measure of internal resistance to flow \[\tau = \mu \frac{du}{dy}\] where τ = shear stress (Pa or lbf/ft²), du/dy = velocity gradient (s⁻¹)
  Units: Pa·s, cP (1 cP = 0.001 Pa·s), lbm/(ft·s), or lbf·s/ft²
  Conversion: 1 lbm/(ft·s) = 1.4882 Pa·s
• Kinematic Viscosity (ν): Ratio of dynamic viscosity to density \[\nu = \frac{\mu}{\rho}\] Units: m²/s, cSt (1 cSt = 10⁻⁶ m²/s), or ft²/s
  Conversion: 1 ft²/s = 0.0929 m²/s

Newtonian vs. Non-Newtonian Fluids

• Newtonian Fluids: Viscosity is constant and independent of shear rate (water, air, most gases and simple liquids)
• Non-Newtonian Fluids: Viscosity varies with shear rate
  • Power Law Model: \(\tau = K \left(\frac{du}{dy}\right)^n\)
    where K = consistency index, n = flow behavior index
    n < 1:="" pseudoplastic="">
    n > 1: dilatant (shear-thickening)
    n = 1: Newtonian

Fluid Statics

Pressure Relationships

• Hydrostatic Pressure: Pressure due to fluid column \[P = P_0 + \rho g h\] or \[\Delta P = \gamma h = \rho g h\] where P₀ = surface pressure, h = depth below surface
  Units: Pa, psi, or lbf/ft²
• Absolute, Gauge, and Vacuum Pressure: \[P_{absolute} = P_{gauge} + P_{atmospheric}\] \[P_{vacuum} = P_{atmospheric} - P_{absolute}\] Standard atmospheric pressure: 101.325 kPa or 14.696 psia
• Manometer Equation: For differential pressure measurement \[\Delta P = \rho_{manometric} g h\] For U-tube manometer with different fluids: \[P_1 - P_2 = g(\rho_m h_m - \rho_f h_f)\] where ρ_m = manometric fluid density, ρ_f = process fluid density

Forces on Submerged Surfaces

• Force on Horizontal Surface: \[F = P \cdot A = \rho g h A\] where A = surface area, h = depth to surface
• Force on Vertical or Inclined Surface: \[F = \rho g h_c A\] where h_c = depth to centroid of surface
• Center of Pressure: Location where resultant force acts \[h_{cp} = h_c + \frac{I_c}{h_c A}\] where I_c = second moment of area about centroidal axis, h_cp = depth to center of pressure

Buoyancy

• Buoyant Force (Archimedes' Principle): \[F_b = \rho_{fluid} g V_{displaced}\] where V_displaced = volume of fluid displaced by object
• Stability Condition: For floating objects, stable when metacenter is above center of gravity

Fluid Dynamics Fundamentals

Conservation of Mass - Continuity Equation

• General Form: \[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0\]
• Steady Flow: \[\dot{m} = \rho_1 A_1 v_1 = \rho_2 A_2 v_2 = constant\] where ṁ = mass flow rate (kg/s or lbm/s), A = cross-sectional area, v = velocity
• Incompressible Flow (ρ = constant): \[Q = A_1 v_1 = A_2 v_2 = constant\] where Q = volumetric flow rate (m³/s or ft³/s)
• Mass Flow Rate: \[\dot{m} = \rho A v = \rho Q\]

Bernoulli Equation

• Bernoulli Equation (along streamline, inviscid, incompressible, steady): \[\frac{P}{\rho} + \frac{v^2}{2} + gz = constant\] or between two points: \[\frac{P_1}{\rho} + \frac{v_1^2}{2} + gz_1 = \frac{P_2}{\rho} + \frac{v_2^2}{2} + gz_2\]
• Bernoulli Equation (head form): \[\frac{P}{\gamma} + \frac{v^2}{2g} + z = constant\] or \[\frac{P_1}{\gamma} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + z_2\] where each term represents head (m or ft):
  P/γ = pressure head
  v²/(2g) = velocity head
  z = elevation head
• Modified Bernoulli with Losses: \[\frac{P_1}{\gamma} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + z_2 + h_L\] where h_L = head loss due to friction
• Modified Bernoulli with Pump/Turbine: \[\frac{P_1}{\gamma} + \frac{v_1^2}{2g} + z_1 + h_p = \frac{P_2}{\gamma} + \frac{v_2^2}{2g} + z_2 + h_t + h_L\] where h_p = pump head added, h_t = turbine head extracted

Energy Equation

• General Mechanical Energy Balance (per unit mass): \[\frac{P_1}{\rho} + \frac{v_1^2}{2} + gz_1 + W_s = \frac{P_2}{\rho} + \frac{v_2^2}{2} + gz_2 + F\] where W_s = shaft work per unit mass, F = friction loss per unit mass
• Power Relationships: \[\dot{W} = \dot{m} W_s = \rho Q W_s\] Pump power: \[\dot{W}_p = \dot{m} g h_p = \rho Q g h_p = \gamma Q h_p\] Units: W or hp (1 hp = 550 ft·lbf/s = 745.7 W)
• Pump Efficiency: \[\eta_p = \frac{\text{Water power (hydraulic power)}}{\text{Brake power (input)}} = \frac{\gamma Q h_p}{\dot{W}_{brake}}\]

Dimensional Analysis and Similitude

Dimensionless Numbers

• Reynolds Number (Re): Ratio of inertial to viscous forces \[Re = \frac{\rho v D}{\mu} = \frac{v D}{\nu}\] where D = characteristic length (pipe diameter for pipe flow)
  For pipe flow: Re < 2100="" (laminar),="" 2100="">< re="">< 4000="" (transitional),="" re=""> 4000 (turbulent)
• Froude Number (Fr): Ratio of inertial to gravitational forces \[Fr = \frac{v}{\sqrt{gL}}\] where L = characteristic length
  Important for free surface flows and wave phenomena
• Euler Number (Eu): Ratio of pressure to inertial forces \[Eu = \frac{\Delta P}{\rho v^2}\]
• Weber Number (We): Ratio of inertial to surface tension forces \[We = \frac{\rho v^2 L}{\sigma}\] where σ = surface tension
• Mach Number (Ma): Ratio of flow velocity to speed of sound \[Ma = \frac{v}{c}\] where c = speed of sound in fluid
  Ma < 0.3:="" incompressible="" flow="" assumption="">
• Drag Coefficient (C_D): \[C_D = \frac{F_D}{\frac{1}{2}\rho v^2 A}\] where F_D = drag force, A = projected area
• Friction Factor (f): For pipe flow \[f = \frac{\Delta P}{\frac{L}{D}\frac{\rho v^2}{2}}\] See Darcy-Weisbach equation below

Buckingham Pi Theorem

• Number of dimensionless groups: \[\text{Number of } \pi \text{ groups} = n - m\] where n = number of variables, m = number of fundamental dimensions

Laminar Flow in Pipes and Channels

Laminar Flow in Circular Pipes

• Hagen-Poiseuille Equation: Pressure drop for laminar flow \[\Delta P = \frac{32 \mu v L}{D^2} = \frac{128 \mu Q L}{\pi D^4}\] where L = pipe length, D = pipe diameter
  Valid for Re <>
• Velocity Profile (laminar flow in pipe): \[v(r) = v_{max}\left(1 - \frac{r^2}{R^2}\right)\] where r = radial position, R = pipe radius, v_max = centerline velocity
• Maximum Velocity: \[v_{max} = 2v_{avg}\] where v_avg = average velocity
• Friction Factor (laminar flow): \[f = \frac{64}{Re}\] Valid for Re <>

Laminar Flow Between Parallel Plates

• Velocity Profile (both plates stationary): \[v(y) = \frac{1}{2\mu}\frac{dP}{dx}y(y-h)\] where h = distance between plates, y = distance from bottom plate
• Maximum Velocity (at centerline): \[v_{max} = \frac{h^2}{8\mu}\left(-\frac{dP}{dx}\right)\]
• Average Velocity: \[v_{avg} = \frac{2}{3}v_{max}\]

Turbulent Flow in Pipes

Darcy-Weisbach Equation

• Head Loss due to Friction: \[h_f = f \frac{L}{D} \frac{v^2}{2g}\] where f = Darcy friction factor (dimensionless), L = pipe length, D = pipe diameter
• Pressure Drop: \[\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}\] Units: Pa or psi
• Alternative form using flow rate: \[h_f = \frac{8fLQ^2}{\pi^2 g D^5}\]

Friction Factor Correlations

• Laminar Flow (Re <> \[f = \frac{64}{Re}\]
• Colebrook Equation (turbulent flow, all pipe roughness): \[\frac{1}{\sqrt{f}} = -2.0 \log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)\] where ε = absolute pipe roughness (ft or m)
  Requires iterative solution or Moody diagram
• Swamee-Jain Equation (explicit approximation): \[f = \frac{0.25}{\left[\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{5.74}{Re^{0.9}}\right)\right]^2}\] Valid for 10⁻⁶ < ε/d="">< 10⁻²,="" 5000="">< re=""><>
• Smooth Pipe Approximations:
  Blasius (Re < 10⁵):="" \[f="\frac{0.316}{Re^{0.25}}\]" von="" kármán="" (re=""> 10⁵): \[\frac{1}{\sqrt{f}} = 2.0 \log_{10}(Re\sqrt{f}) - 0.8\]
• Completely Rough Pipe: \[\frac{1}{\sqrt{f}} = -2.0 \log_{10}\left(\frac{\varepsilon/D}{3.7}\right)\] Valid when Re > 10⁶ (friction factor independent of Reynolds number)

Hazen-Williams Equation

• Hazen-Williams Equation (water flow only): \[v = 1.318 C_{HW} R_h^{0.63} S^{0.54}\] or \[Q = 0.432 C_{HW} D^{2.63} S^{0.54}\] where C_HW = Hazen-Williams coefficient (dimensionless), R_h = hydraulic radius (ft), S = slope of energy grade line (ft/ft), Q = flow rate (ft³/s), D = diameter (ft)
  Common C_HW values: 150 (new smooth pipe), 130 (new cast iron), 100 (old/rough pipe)
  Note: Limited to water at 60°F; not valid for other fluids
• Head Loss (Hazen-Williams): \[h_f = \frac{4.73 L Q^{1.85}}{C_{HW}^{1.85} D^{4.87}}\] where L and D in feet, Q in ft³/s, h_f in feet

Minor Losses in Pipe Systems

Minor Loss Equations

• Minor Loss (head loss form): \[h_m = K \frac{v^2}{2g}\] where K = loss coefficient (dimensionless), v = velocity
• Minor Loss (pressure drop form): \[\Delta P_m = K \frac{\rho v^2}{2}\]
• Total System Head Loss: \[h_L = h_f + \sum h_m = f \frac{L}{D} \frac{v^2}{2g} + \sum K \frac{v^2}{2g}\]

Common Loss Coefficients

• Sudden Expansion: \[K = \left(1 - \frac{A_1}{A_2}\right)^2\] where A₁ = upstream area, A₂ = downstream area
  Based on upstream velocity v₁
• Sudden Contraction: \[K = 0.5\left(1 - \frac{A_2}{A_1}\right)\] or approximately K ≈ 0.5 for sharp-edged entrance
  Based on downstream velocity v₂
• Entrance Loss:
  Sharp-edged: K ≈ 0.5
  Rounded: K ≈ 0.05 to 0.2
  Well-rounded: K ≈ 0.03
• Exit Loss: \[K = 1.0\] (complete loss of velocity head)
• 90° Elbow:
  Standard: K ≈ 0.9
  Long radius: K ≈ 0.6
  Square: K ≈ 1.3
• Gate Valve:
  Fully open: K ≈ 0.15
  ¾ open: K ≈ 0.9
  ½ open: K ≈ 4.5
• Globe Valve: K ≈ 10 (fully open)
• Check Valve: K ≈ 2 to 3 (swing type)
• Tee:
  Flow through run: K ≈ 0.6
  Flow through branch: K ≈ 1.8

Equivalent Length Method

• Equivalent Length (L_eq): Length of straight pipe producing same head loss as fitting \[K = f \frac{L_{eq}}{D}\] or \[L_{eq} = \frac{K D}{f}\] Total system head loss: \[h_L = f \frac{L + \sum L_{eq}}{D} \frac{v^2}{2g}\]

Flow Measurement

Venturi Meter

• Venturi Flow Equation: \[Q = C_v A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho(1 - \beta^4)}}\] where C_v = discharge coefficient (typically 0.95-0.99), A₂ = throat area, β = diameter ratio (D₂/D₁)
• Alternative form: \[Q = \frac{C_v A_2}{\sqrt{1-\beta^4}} \sqrt{2gh}\] where h = differential head

Orifice Meter

• Orifice Flow Equation: \[Q = C_o A_o \sqrt{\frac{2\Delta P}{\rho(1 - \beta^4)}}\] where C_o = orifice discharge coefficient (typically 0.6-0.65), A_o = orifice area, β = diameter ratio (D_o/D₁)
• Vena Contracta: Point of minimum area downstream of orifice \[A_{vc} = C_c A_o\] where C_c = contraction coefficient (≈0.6-0.65)

Pitot Tube

• Pitot Tube Velocity: \[v = \sqrt{\frac{2(P_{stagnation} - P_{static})}{\rho}} = \sqrt{2gh}\] where P_stagnation = total pressure, P_static = static pressure, h = differential head
  Measures local velocity at a point

Flow Nozzle

• Flow Nozzle Equation: \[Q = C_n A_n \sqrt{\frac{2\Delta P}{\rho(1 - \beta^4)}}\] where C_n = nozzle discharge coefficient (typically 0.95-0.99), A_n = nozzle throat area

Weirs

• Rectangular Sharp-Crested Weir: \[Q = C_w \frac{2}{3}\sqrt{2g} L H^{3/2}\] where C_w = weir coefficient (≈0.6-0.65), L = weir length (crest width), H = head above weir crest
  Simplified: Q = 3.33 L H^3/2 (US customary units: Q in ft³/s, L and H in ft)
• Triangular (V-notch) Weir: \[Q = C_v \frac{8}{15}\sqrt{2g} \tan\left(\frac{\theta}{2}\right) H^{5/2}\] where θ = notch angle, C_v = weir coefficient
  For 90° V-notch: Q = 2.5 H^5/2 (US customary units)

Rotameter

• Rotameter Flow Rate: \[Q = C_r A_a \sqrt{\frac{2g V_f (\rho_f - \rho)}{\rho A_f}}\] where C_r = rotameter coefficient, A_a = annular area, V_f = float volume, ρ_f = float density, ρ = fluid density, A_f = maximum cross-sectional area of float

Pump and Compressor Fundamentals

Pump Performance

• Total Dynamic Head (TDH): \[h_p = (z_2 - z_1) + \frac{P_2 - P_1}{\gamma} + \frac{v_2^2 - v_1^2}{2g} + h_L\] where subscript 1 = suction side, subscript 2 = discharge side
• Hydraulic Power (Water Power): \[\dot{W}_{hydraulic} = \gamma Q h_p = \rho g Q h_p\] Units: W or hp
• Brake Power (Shaft Power): Actual power input to pump \[\dot{W}_{brake} = \frac{\dot{W}_{hydraulic}}{\eta_p}\]
• Pump Efficiency: \[\eta_p = \frac{\gamma Q h_p}{\dot{W}_{brake}}\]
• Net Positive Suction Head Available (NPSHA): \[NPSHA = \frac{P_{suction}}{\gamma} - \frac{P_{vapor}}{\gamma} + \frac{v_{suction}^2}{2g}\] where P_vapor = vapor pressure of liquid at operating temperature
  Must have NPSHA > NPSHR (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)
  Used for pump selection: N_s < 2000="" (centrifugal),="" 2000-4000="" (mixed="" flow),="">4000 (axial flow)

Affinity Laws for Pumps

• Same impeller diameter, different speeds: \[\frac{Q_2}{Q_1} = \frac{N_2}{N_1}\] \[\frac{h_2}{h_1} = \left(\frac{N_2}{N_1}\right)^2\] \[\frac{\dot{W}_2}{\dot{W}_1} = \left(\frac{N_2}{N_1}\right)^3\]
• Same speed, different impeller diameters: \[\frac{Q_2}{Q_1} = \frac{D_2}{D_1}\] \[\frac{h_2}{h_1} = \left(\frac{D_2}{D_1}\right)^2\] \[\frac{\dot{W}_2}{\dot{W}_1} = \left(\frac{D_2}{D_1}\right)^3\]

Pumps in Series and Parallel

• Pumps in Series: Heads add at same flow rate \[h_{total} = h_1 + h_2\] \[Q_{total} = Q_1 = Q_2\]
• Pumps in Parallel: Flow rates add at same head \[Q_{total} = Q_1 + Q_2\] \[h_{total} = h_1 = h_2\]

Compressor Work

• Isentropic Compression Work (ideal gas): \[W_s = \frac{kRT_1}{k-1}\left[\left(\frac{P_2}{P_1}\right)^{(k-1)/k} - 1\right]\] where k = specific heat ratio (C_p/C_v), R = specific gas constant, T₁ = inlet temperature
• Polytropic Compression Work: \[W = \frac{nRT_1}{n-1}\left[\left(\frac{P_2}{P_1}\right)^{(n-1)/n} - 1\right]\] where n = polytropic exponent
• Isothermal Compression Work (ideal gas): \[W = RT\ln\left(\frac{P_2}{P_1}\right)\]
• Compressor Efficiency: \[\eta_c = \frac{W_{isentropic}}{W_{actual}}\]

Drag and External Flow

Drag Force

• Total Drag Force: \[F_D = C_D \frac{1}{2}\rho v^2 A\] where C_D = drag coefficient, A = projected area perpendicular to flow
• Friction Drag: Due to viscous shear stress
• Form (Pressure) Drag: Due to pressure distribution from flow separation

Drag on Spheres

• Stokes' Law (creeping flow, Re <> \[F_D = 3\pi \mu v D\] where D = sphere diameter
  Drag coefficient: \[C_D = \frac{24}{Re}\]
• Intermediate Reynolds Numbers (1 < re=""><>
  Use empirical correlations or charts for C_D vs. Re
• High Reynolds Numbers (Re > 10⁵):
  C_D ≈ 0.1 to 0.5 depending on surface roughness and turbulence

Terminal Velocity

• Terminal Velocity of Falling Sphere: When drag force equals gravitational force \[v_t = \sqrt{\frac{4gD(\rho_s - \rho_f)}{3C_D \rho_f}}\] where ρ_s = sphere density, ρ_f = fluid density
  For Stokes flow (Re < 1):="" \[v_t="\frac{gD^2(\rho_s" -="">

Drag on Other Shapes

• Flat Plate Normal to Flow: C_D ≈ 1.2
• Flat Plate Parallel to Flow (skin friction): C_D depends on Re and flow regime
• Cylinder (axis perpendicular to flow): C_D ≈ 1.0 (for Re = 10⁴ to 10⁵)
• Streamlined Body: C_D ≈ 0.04 to 0.1

Flow Through Porous Media

Darcy's Law

• Darcy's Law (groundwater flow): \[Q = -KA\frac{dh}{dL}\] or velocity: \[v = -K\frac{dh}{dL}\] where K = hydraulic conductivity (m/s or ft/s), A = cross-sectional area, dh/dL = hydraulic gradient
• Alternative form: \[v = \frac{k}{\mu}\left(-\frac{dP}{dL}\right)\] where k = permeability (m² or darcy; 1 darcy = 9.87 × 10⁻¹³ m²)

Packed Bed Flow

• Ergun Equation: Pressure drop through packed bed \[\frac{\Delta P}{L} = 150\frac{(1-\varepsilon)^2}{\varepsilon^3}\frac{\mu v_s}{D_p^2} + 1.75\frac{(1-\varepsilon)}{\varepsilon^3}\frac{\rho v_s^2}{D_p}\] where ε = void fraction (porosity), v_s = superficial velocity (Q/A), D_p = particle diameter
• Blake-Kozeny Equation (laminar flow in packed bed): \[\Delta P = 150\frac{(1-\varepsilon)^2}{\varepsilon^3}\frac{\mu v_s L}{D_p^2}\]
• Burke-Plummer Equation (turbulent flow in packed bed): \[\Delta P = 1.75\frac{(1-\varepsilon)}{\varepsilon^3}\frac{\rho v_s^2 L}{D_p}\]

Non-Circular Conduits

Hydraulic Diameter

• Hydraulic Diameter (D_h): \[D_h = \frac{4A}{P}\] where A = cross-sectional area, P = wetted perimeter
  Used in place of D for Reynolds number and friction factor calculations
• Hydraulic Radius (R_h): \[R_h = \frac{A}{P} = \frac{D_h}{4}\]

Common Geometries

• Rectangular Duct (width a, height b): \[D_h = \frac{4ab}{2(a+b)} = \frac{2ab}{a+b}\]
• Annulus (outer diameter D_o, inner diameter D_i): \[D_h = D_o - D_i\]
• Open Channel (rectangular, width b, depth y): \[D_h = \frac{4by}{b + 2y}\] Note: For open channels, use hydraulic radius R_h = by/(b + 2y) in Manning equation

Open Channel Flow

Manning Equation

• Manning Equation (US customary units): \[v = \frac{1.486}{n}R_h^{2/3}S^{1/2}\] or \[Q = \frac{1.486}{n}AR_h^{2/3}S^{1/2}\] where n = Manning roughness coefficient, R_h = hydraulic radius (ft), S = slope of energy grade line (ft/ft), A = cross-sectional area (ft²), v = velocity (ft/s), Q = flow rate (ft³/s)
• Manning Equation (SI units): \[v = \frac{1.0}{n}R_h^{2/3}S^{1/2}\]
• Typical Manning n values:
  Clean, straight channel: n = 0.012-0.015
  Natural streams: n = 0.025-0.035
  Rough channels with vegetation: n = 0.050-0.150

Channel Flow Regimes

• Froude Number (open channel): \[Fr = \frac{v}{\sqrt{gy}}\] where y = flow depth
  Fr < 1:="" subcritical="" flow="">
  Fr = 1: critical flow
  Fr > 1: supercritical flow (rapid)
• Critical Depth (rectangular channel): \[y_c = \left(\frac{q^2}{g}\right)^{1/3}\] where q = flow rate per unit width (Q/b)
• Critical Velocity (rectangular channel): \[v_c = \sqrt{gy_c}\]
• Specific Energy: \[E = y + \frac{v^2}{2g}\] Minimum specific energy occurs at critical depth

Hydraulic Jump

• Depth after Hydraulic Jump (rectangular channel): \[\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)\] where y₁ = upstream depth (supercritical), y₂ = downstream depth (subcritical), Fr₁ = upstream Froude number
• Energy Loss in Hydraulic Jump: \[h_L = \frac{(y_2 - y_1)^3}{4y_1 y_2}\]

Unsteady Flow

Pipe Networks and Water Hammer

• Water Hammer Pressure Rise (rapid valve closure): \[\Delta P = \rho c \Delta v\] where c = speed of pressure wave, Δv = change in velocity
• Wave Speed in Pipe: \[c = \sqrt{\frac{K/\rho}{1 + (K/E)(D/t)}}\] where K = bulk modulus of fluid, E = elastic modulus of pipe material, D = pipe diameter, t = pipe wall thickness
  For rigid pipe: c = √(K/ρ)
• Critical Time (Joukowsky time): \[t_c = \frac{2L}{c}\] where L = pipe length
  If valve closure time <>_c, use water hammer equation above
  If closure time > t_c, pressure rise is reduced

Boundary Layer Theory

Boundary Layer Characteristics

• Boundary Layer Thickness (δ): Distance from surface where velocity reaches 99% of free stream velocity
• Displacement Thickness (δ^*): \[\delta^* = \int_0^\infty \left(1 - \frac{u}{U}\right)dy\] where u = local velocity, U = free stream velocity
• Momentum Thickness (θ): \[\theta = \int_0^\infty \frac{u}{U}\left(1 - \frac{u}{U}\right)dy\]

Flat Plate Boundary Layer

• Laminar Boundary Layer (Re_x < 5="" ×="">
  Boundary layer thickness: \[\delta = \frac{5x}{\sqrt{Re_x}}\] where x = distance from leading edge, Re_x = ρUx/μ
• Local Skin Friction Coefficient (laminar): \[C_f = \frac{0.664}{\sqrt{Re_x}}\]
• Average Skin Friction Coefficient (laminar, 0 to L): \[C_{f,avg} = \frac{1.328}{\sqrt{Re_L}}\]
• Turbulent Boundary Layer (Re_x > 5 × 10⁵):
  Boundary layer thickness: \[\delta = \frac{0.37x}{Re_x^{1/5}}\]
• Local Skin Friction Coefficient (turbulent): \[C_f = \frac{0.059}{Re_x^{1/5}}\]
• Average Skin Friction Coefficient (turbulent, 5 × 10⁵ <>_L <> \[C_{f,avg} = \frac{0.074}{Re_L^{1/5}}\]

Compressible Flow

Speed of Sound

• Speed of Sound in Ideal Gas: \[c = \sqrt{kRT}\] where k = specific heat ratio (C_p/C_v), R = specific gas constant, T = absolute temperature

Isentropic Flow Relations

• Stagnation Temperature: \[\frac{T_0}{T} = 1 + \frac{k-1}{2}Ma^2\] where T₀ = stagnation temperature, T = static temperature, Ma = Mach number
• Stagnation Pressure: \[\frac{P_0}{P} = \left(1 + \frac{k-1}{2}Ma^2\right)^{k/(k-1)}\]
• Stagnation Density: \[\frac{\rho_0}{\rho} = \left(1 + \frac{k-1}{2}Ma^2\right)^{1/(k-1)}\]

Normal Shock Relations

• Downstream Mach Number: \[Ma_2^2 = \frac{Ma_1^2 + \frac{2}{k-1}}{\frac{2k}{k-1}Ma_1^2 - 1}\] where subscript 1 = upstream, subscript 2 = downstream
• Pressure Ratio across Normal Shock: \[\frac{P_2}{P_1} = 1 + \frac{2k}{k+1}(Ma_1^2 - 1)\]
• Temperature Ratio across Normal Shock: \[\frac{T_2}{T_1} = \frac{[1 + \frac{k-1}{2}Ma_1^2][1 + \frac{2k}{k+1}(Ma_1^2 - 1)]}{\frac{(k+1)^2}{2(k-1)}Ma_1^2}\]
• Density Ratio across Normal Shock: \[\frac{\rho_2}{\rho_1} = \frac{(k+1)Ma_1^2}{2 + (k-1)Ma_1^2}\]

Fluidization

Minimum Fluidization Velocity

• Minimum Fluidization Velocity (v_mf): Superficial velocity at which bed begins to fluidize
  At minimum fluidization, pressure drop equals weight of bed per unit area: \[\Delta P = \frac{(1-\varepsilon_{mf})(\rho_s - \rho_f)gL}{1}\] where ε_mf = void fraction at minimum fluidization, ρ_s = solid density, ρ_f = fluid density, L = bed height
• Empirical Correlation (Wen and Yu): \[Re_{mf} = \sqrt{(33.7)^2 + 0.0408Ar} - 33.7\] where Re_mf = ρ_fv_mfD_p/μ, Ar = Archimedes number = D_p³ρ_f(ρ_s - ρ_f)g/μ²

Piping System Design

System Curve and Operating Point

• System Head Curve: \[h_{system} = h_{static} + K_{system}Q^2\] where h_static = static head (elevation change), K_system = system resistance coefficient
  Operating point is intersection of pump curve and system curve

Economic Pipe Diameter

• Optimum Velocity Range:
  Water: 1-3 m/s (3-10 ft/s)
  Steam: 30-50 m/s (100-165 ft/s)
  Gases: 15-30 m/s (50-100 ft/s)
  Viscous liquids: 0.5-1.5 m/s (1.5-5 ft/s)

Two-Phase Flow

Flow Regimes

• Horizontal Flow Regimes: Stratified, wavy, slug, annular, dispersed
• Vertical Flow Regimes: Bubble, slug, churn, annular, wispy-annular

Void Fraction and Quality

• Quality (x): Mass fraction of vapor \[x = \frac{\dot{m}_{vapor}}{\dot{m}_{total}}\]
• Void Fraction (α): Volumetric fraction of vapor \[\alpha = \frac{A_{vapor}}{A_{total}}\]
• Slip Ratio (S): \[S = \frac{v_{vapor}}{v_{liquid}}\] For homogeneous flow model: S = 1

Two-Phase Pressure Drop

• Lockhart-Martinelli Correlation: Empirical method for two-phase pressure drop
  Two-phase multiplier: \[\left(\frac{dP}{dL}\right)_{TP} = \Phi_L^2 \left(\frac{dP}{dL}\right)_L = \Phi_G^2 \left(\frac{dP}{dL}\right)_G\] where Φ_L, Φ_G = two-phase multipliers for liquid and gas phases, obtained from Lockhart-Martinelli parameter X