Formula Sheet: Flow Systems

Table of Contents
1. Fluid Properties
2. Fluid Statics
3. Continuity Equation
4. Bernoulli Equation
5. Reynolds Number and Flow Regimes
6. Head Loss in Pipes
7. Pumps and Pump Systems
8. Open Channel Flow
9. Pipe Networks
10. Compressible Flow
11. Drag and Lift
12. Dimensional Analysis and Similitude
13. Flow Measurement
14. Non-Newtonian Fluids
15. Boundary Layer Theory
16. Hydraulic Transients
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Fluid Properties

Density and Specific Weight

• Density: \[\rho = \frac{m}{V}\] where ρ = density (kg/m³ or lbm/ft³), m = mass, V = volume
• Specific Weight: \[\gamma = \rho g\] where γ = specific weight (N/m³ or lbf/ft³), g = gravitational acceleration (9.81 m/s² or 32.2 ft/s²)
• Specific Gravity: \[SG = \frac{\rho_{fluid}}{\rho_{water}}\] where SG = specific gravity (dimensionless), ρ_water = 1000 kg/m³ or 62.4 lbm/ft³

Viscosity

• Dynamic (Absolute) Viscosity: \[\tau = \mu \frac{du}{dy}\] where τ = shear stress (Pa or lbf/ft²), μ = dynamic viscosity (Pa·s or lbf·s/ft²), du/dy = velocity gradient
• Kinematic Viscosity: \[\nu = \frac{\mu}{\rho}\] where ν = kinematic viscosity (m²/s or ft²/s)

Fluid Statics

Pressure

• Pressure at Depth: \[P = P_{atm} + \gamma h\] where P = absolute pressure, P_atm = atmospheric pressure, h = depth below free surface
• Gage Pressure: \[P_{gage} = P_{absolute} - P_{atmospheric}\]
• Hydrostatic Pressure: \[P = \rho g h\]

Continuity Equation

Conservation of Mass

• General Form: \[\dot{m}_1 = \dot{m}_2\] where ṁ = mass flow rate (kg/s or lbm/s)
• Mass Flow Rate: \[\dot{m} = \rho A V\] where A = cross-sectional area (m² or ft²), V = average velocity (m/s or ft/s)
• For Incompressible Flow: \[A_1 V_1 = A_2 V_2\] or \[Q_1 = Q_2\] where Q = volumetric flow rate (m³/s or ft³/s)
• Volumetric Flow Rate: \[Q = A V\]

Bernoulli Equation

Energy Equation for Ideal Flow

• Bernoulli's Equation (along streamline): \[\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2\] where P/γ = pressure head, V²/2g = velocity head, z = elevation head
  Assumes: steady, incompressible, frictionless flow along a streamline
• Alternative Form (using pressure): \[P_1 + \frac{1}{2}\rho V_1^2 + \rho g z_1 = P_2 + \frac{1}{2}\rho V_2^2 + \rho g z_2\]
• Total Head: \[H = \frac{P}{\gamma} + \frac{V^2}{2g} + z\]

Modified Bernoulli Equation

• With Pump and Head Loss: \[\frac{P_1}{\gamma} + \frac{V_1^2}{2g} + z_1 + h_p = \frac{P_2}{\gamma} + \frac{V_2^2}{2g} + z_2 + h_L\] where h_p = head added by pump, h_L = head loss due to friction
• 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 = head extracted by turbine

Reynolds Number and Flow Regimes

Reynolds Number

• For Pipe Flow: \[Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}\] where Re = Reynolds number (dimensionless), D = pipe diameter
• Flow Regime Classification:
  • Laminar flow: Re <>
  • Transitional flow: 2000 ≤ Re ≤ 4000
  • Turbulent flow: Re > 4000

Head Loss in Pipes

Major Losses (Friction)

• Darcy-Weisbach Equation: \[h_f = f \frac{L}{D} \frac{V^2}{2g}\] where h_f = head loss due to friction (m or ft), f = Darcy friction factor (dimensionless), L = pipe length
• Pressure Drop: \[\Delta P = f \frac{L}{D} \frac{\rho V^2}{2}\]

Friction Factor

• Laminar Flow (Re <> \[f = \frac{64}{Re}\]
• Turbulent Flow - Colebrook Equation: \[\frac{1}{\sqrt{f}} = -2.0 \log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}}\right)\] where ε = absolute roughness (m or ft), ε/D = relative roughness
  Note: This is an implicit equation; typically solved using Moody diagram or iterative methods
• Smooth Pipes (Turbulent) - Blasius Equation (Re <>⁵): \[f = \frac{0.316}{Re^{0.25}}\]
• Swamee-Jain Approximation (Turbulent): \[f = \frac{0.25}{\left[\log_{10}\left(\frac{\varepsilon/D}{3.7} + \frac{5.74}{Re^{0.9}}\right)\right]^2}\] Valid for: 10^-6 < ε/d=""><>^-2 and 5000 < re=""><>⁸

Minor Losses

• Loss Coefficient Method: \[h_m = K \frac{V^2}{2g}\] where h_m = minor head loss, K = loss coefficient (dimensionless)
• Equivalent Length Method: \[h_m = f \frac{L_e}{D} \frac{V^2}{2g}\] where L_e = equivalent length of pipe
• Total 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: \[K = \left(1 - \frac{A_1}{A_2}\right)^2 = \left(1 - \frac{D_1^2}{D_2^2}\right)^2\] Based on upstream velocity V₁
• Sudden Contraction: \[K = 0.5\left(1 - \frac{A_2}{A_1}\right)\] Based on downstream velocity V₂
• Pipe Entrance (Sharp-edged): K = 0.5
• Pipe Entrance (Rounded): K ≈ 0.04 to 0.2
• Pipe Exit: K = 1.0
• Globe Valve (fully open): K ≈ 10
• Gate Valve (fully open): K ≈ 0.2
• 90° Elbow (standard): K ≈ 0.9
• 45° Elbow: K ≈ 0.4

Pumps and Pump Systems

Pump Power and Efficiency

• Water Horsepower (Hydraulic Power): \[P_{hydraulic} = \gamma Q h_p = \rho g Q h_p\] where P_hydraulic = hydraulic power (W or hp), h_p = total head added by pump
• Water Horsepower (US units): \[WHP = \frac{Q \times h_p \times SG}{3960}\] where Q = flow rate (gpm), h_p = head (ft), result in hp
• Brake Horsepower: \[BHP = \frac{WHP}{\eta_p}\] where BHP = brake horsepower (power input to pump), η_p = pump efficiency
• Motor Power: \[P_{motor} = \frac{BHP}{\eta_m}\] where η_m = motor efficiency
• Overall Efficiency: \[\eta_{overall} = \eta_p \times \eta_m\]

Pump Head

• Total Dynamic Head (TDH): \[TDH = h_{static} + h_f + h_m + h_{velocity}\] where h_static = static head difference, h_velocity = velocity head difference
• Static Head: \[h_{static} = z_{discharge} - z_{suction}\]
• Net Positive Suction Head Available (NPSHA): \[NPSH_A = \frac{P_{suction}}{\gamma} + \frac{V_{suction}^2}{2g} - \frac{P_{vapor}}{\gamma}\] where P_vapor = vapor pressure of fluid
• NPSHA (Alternative Form): \[NPSH_A = \frac{P_{atm}}{\gamma} - h_{suction} - h_{f,suction} - \frac{P_{vapor}}{\gamma}\] where h_suction = suction lift (elevation difference)
• Cavitation Criterion: \[NPSH_A > NPSH_R\] where NPSH_R = required NPSH (provided by manufacturer)

Affinity Laws for Pumps

• Flow Rate Relationship: \[\frac{Q_2}{Q_1} = \frac{N_2}{N_1}\left(\frac{D_2}{D_1}\right)^3\] where N = pump speed (rpm), D = impeller diameter
• Head Relationship: \[\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^2\left(\frac{D_2}{D_1}\right)^2\]
• Power Relationship: \[\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^3\left(\frac{D_2}{D_1}\right)^5\]
• For Constant Impeller Diameter:
  • Q ∝ N
  • H ∝ N²
  • P ∝ N³

Specific Speed

• Specific Speed (US units): \[N_s = \frac{N \sqrt{Q}}{H^{0.75}}\] where N_s = specific speed, N = pump speed (rpm), Q = flow rate (gpm), H = head (ft)
  Used for pump selection and classification
• Specific Speed (SI units): \[N_s = \frac{N \sqrt{Q}}{H^{0.75}}\] where N = speed (rpm), Q = flow rate (m³/s), H = head (m)

Pumps in Series and Parallel

• Pumps in Series: Same flow rate, heads add \[Q_{total} = Q_1 = Q_2\] \[H_{total} = H_1 + H_2\]
• Pumps in Parallel: Same head, flow rates add \[Q_{total} = Q_1 + Q_2\] \[H_{total} = H_1 = H_2\]

Open Channel Flow

Flow Parameters

• Flow Rate: \[Q = A V\] where A = cross-sectional area of flow
• Hydraulic Radius: \[R_h = \frac{A}{P}\] where P = wetted perimeter
• Hydraulic Depth: \[D_h = \frac{A}{T}\] where T = top width of water surface

Manning's Equation

• Manning's Equation (SI units): \[V = \frac{1}{n} R_h^{2/3} S^{1/2}\] where V = mean velocity (m/s), n = Manning's roughness coefficient, S = slope of energy grade line (m/m)
• Manning's Equation (US units): \[V = \frac{1.486}{n} R_h^{2/3} S^{1/2}\] where V = velocity (ft/s), R_h = hydraulic radius (ft)
• Flow Rate Form: \[Q = \frac{1.486}{n} A R_h^{2/3} S^{1/2}\] (US units)
• Flow Rate Form (SI): \[Q = \frac{1}{n} A R_h^{2/3} S^{1/2}\]

Froude Number

• Froude Number: \[Fr = \frac{V}{\sqrt{g D_h}}\] where Fr = Froude number (dimensionless), D_h = hydraulic depth
• For Rectangular Channel: \[Fr = \frac{V}{\sqrt{g y}}\] where y = depth of flow
• Flow Classification:
  • Subcritical flow: Fr < 1="" (tranquil,="">
  • Critical flow: Fr = 1
  • Supercritical flow: Fr > 1 (rapid, shooting)

Specific Energy

• Specific Energy: \[E = y + \frac{V^2}{2g} = y + \frac{Q^2}{2gA^2}\] where E = specific energy (measured relative to channel bottom), y = depth of flow
• Critical Depth (Rectangular Channel): \[y_c = \left(\frac{q^2}{g}\right)^{1/3}\] where q = Q/b = discharge per unit width, b = channel width
• Minimum Specific Energy: \[E_{min} = \frac{3}{2} y_c\] Occurs at critical depth

Weirs and Flumes

• Rectangular Sharp-Crested Weir (Francis Formula): \[Q = 3.33 (L - 0.2 H) H^{3/2}\] where Q = flow rate (ft³/s), L = length of weir crest (ft), H = head above crest (ft)
• Rectangular Weir (Simplified): \[Q = C L H^{3/2}\] where C = weir coefficient (≈ 3.33 for US units)
• Triangular (V-Notch) Weir: \[Q = C H^{5/2}\] where C depends on notch angle θ
• 90° V-Notch Weir: \[Q = 2.5 H^{5/2}\] (US units: Q in ft³/s, H in ft)

Pipe Networks

Series Pipes

• Continuity: \[Q_1 = Q_2 = Q_3 = ... = Q\]
• Total Head Loss: \[h_{L,total} = h_{L1} + h_{L2} + h_{L3} + ...\]

Parallel Pipes

• Equal Head Loss: \[h_{L1} = h_{L2} = h_{L3} = ...\] Between junction points
• Flow Division: \[Q_{total} = Q_1 + Q_2 + Q_3 + ...\]

Hardy Cross Method

• Loop Correction: \[\Delta Q = -\frac{\sum h_L}{\sum (n h_L / Q)}\] For networks with loops, where n = exponent in head loss equation (typically 2 for turbulent flow)
• Head Loss in Loop: \[\sum h_L = 0\] around each closed loop

Three-Reservoir Problem

• Junction Condition: Flow into junction equals flow out
• Energy Balance: All pipes meet at common pressure at junction

Compressible Flow

Speed of Sound and Mach Number

• Speed of Sound (Ideal Gas): \[c = \sqrt{k R T}\] where c = speed of sound (m/s or ft/s), k = specific heat ratio (γ = c_p/c_v), R = specific gas constant, T = absolute temperature
• Speed of Sound (Air, approximate): \[c = \sqrt{k g R T}\] where R = gas constant = 287 J/(kg·K) for air
• Mach Number: \[M = \frac{V}{c}\] where M = Mach number (dimensionless)
• Flow Classification:
  • Subsonic: M <>
  • Sonic: M = 1
  • Supersonic: M > 1
  • Incompressible assumption valid for: M <>

Stagnation Properties

• Stagnation Temperature: \[T_0 = T \left(1 + \frac{k-1}{2} M^2\right)\] where T₀ = stagnation (total) temperature, T = static temperature
• Stagnation Pressure (Isentropic): \[\frac{P_0}{P} = \left(1 + \frac{k-1}{2} M^2\right)^{k/(k-1)}\] where P₀ = stagnation pressure
• Stagnation Density: \[\frac{\rho_0}{\rho} = \left(1 + \frac{k-1}{2} M^2\right)^{1/(k-1)}\]

Isentropic Flow Relations

• Temperature Ratio: \[\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(k-1)/k}\]
• Density Ratio: \[\frac{\rho_2}{\rho_1} = \left(\frac{P_2}{P_1}\right)^{1/k}\]
• Critical Area Ratio: \[\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{k+1}\left(1 + \frac{k-1}{2}M^2\right)\right]^{(k+1)/[2(k-1)]}\] where A* = area at sonic condition (M = 1)

Normal Shock Relations

• Downstream Mach Number: \[M_2^2 = \frac{M_1^2 + \frac{2}{k-1}}{\frac{2k}{k-1}M_1^2 - 1}\] where subscript 1 = upstream (before shock), subscript 2 = downstream (after shock)
• Pressure Ratio Across Shock: \[\frac{P_2}{P_1} = 1 + \frac{2k}{k+1}(M_1^2 - 1)\]
• Temperature Ratio: \[\frac{T_2}{T_1} = \frac{\left[1 + \frac{2k}{k+1}(M_1^2-1)\right]\left[2 + (k-1)M_1^2\right]}{(k+1)^2 M_1^2}\]
• Density Ratio: \[\frac{\rho_2}{\rho_1} = \frac{(k+1)M_1^2}{2+(k-1)M_1^2}\]
• Stagnation Pressure Ratio: \[\frac{P_{02}}{P_{01}} = \left[\frac{(k+1)M_1^2}{2+(k-1)M_1^2}\right]^{k/(k-1)} \left[\frac{k+1}{2kM_1^2-(k-1)}\right]^{1/(k-1)}\] Note: P₀₂ < p₀₁="" (loss="" across="">

Drag and Lift

Drag Force

• Drag Equation: \[F_D = C_D \frac{1}{2}\rho V^2 A\] where F_D = drag force, C_D = drag coefficient (dimensionless), A = projected (frontal) area
• Friction Drag (Skin Friction): Parallel to surface, due to viscous shear
• Pressure Drag (Form Drag): Due to pressure distribution, dominates for bluff bodies

Lift Force

• Lift Equation: \[F_L = C_L \frac{1}{2}\rho V^2 A\] where F_L = lift force, C_L = lift coefficient, A = planform (wing) area

Terminal Velocity

• Terminal Velocity (Falling Object): \[V_{terminal} = \sqrt{\frac{2mg}{C_D \rho A}}\] where m = mass, g = gravitational acceleration
  Occurs when drag force equals weight

Dimensional Analysis and Similitude

Important Dimensionless Numbers

• Reynolds Number: \[Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}\] Ratio of inertial forces to viscous forces
• Froude Number: \[Fr = \frac{V}{\sqrt{g L}}\] Ratio of inertial forces to gravitational forces
• Mach Number: \[M = \frac{V}{c}\] Ratio of flow velocity to speed of sound
• Weber Number: \[We = \frac{\rho V^2 L}{\sigma}\] where σ = surface tension
  Ratio of inertial forces to surface tension forces
• Euler Number: \[Eu = \frac{\Delta P}{\rho V^2}\] Ratio of pressure forces to inertial forces

Dynamic Similitude

• Geometric Similarity: \[L_r = \frac{L_p}{L_m}\] where subscript p = prototype, subscript m = model, L_r = length scale ratio
• Kinematic Similarity: \[V_r = \frac{V_p}{V_m}\] Velocity scale ratio
• Dynamic Similarity: Requires equality of relevant dimensionless numbers between model and prototype
• Reynolds Similarity: \[Re_m = Re_p\] \[\frac{V_m L_m}{\nu_m} = \frac{V_p L_p}{\nu_p}\]
• Froude Similarity: \[Fr_m = Fr_p\] \[\frac{V_m}{\sqrt{g L_m}} = \frac{V_p}{\sqrt{g L_p}}\]

Flow Measurement

Venturi Meter

• Flow Rate (Ideal): \[Q = A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho\left(1 - \left(\frac{A_2}{A_1}\right)^2\right)}}\] where subscript 1 = upstream section, subscript 2 = throat
• Flow Rate (Actual): \[Q = C_d A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho\left(1 - \left(\frac{A_2}{A_1}\right)^2\right)}}\] where C_d = discharge coefficient (typically 0.95-0.99)

Orifice Meter

• Flow Rate: \[Q = C_d A_0 \sqrt{\frac{2\Delta P}{\rho\left(1 - \beta^4\right)}}\] where A₀ = orifice area, β = d/D = diameter ratio, d = orifice diameter, D = pipe diameter
  C_d typically 0.60-0.65

Pitot Tube

• Velocity Measurement: \[V = \sqrt{\frac{2(P_0 - P)}{\rho}}\] where P₀ = stagnation pressure, P = static pressure
• With Coefficient: \[V = C \sqrt{\frac{2\Delta P}{\rho}}\] where C ≈ 1.0 for properly designed tubes

Flow Nozzle

• Flow Rate: \[Q = C_d A_n \sqrt{\frac{2(P_1 - P_2)}{\rho\left(1 - \left(\frac{A_n}{A_1}\right)^2\right)}}\] where A_n = nozzle throat area
  C_d typically 0.96-0.99

Non-Newtonian Fluids

Rheological Models

• Newtonian Fluid: \[\tau = \mu \frac{du}{dy}\] Viscosity μ is constant
• Power Law (Ostwald-de Waele): \[\tau = K \left(\frac{du}{dy}\right)^n\] where K = consistency index, n = flow behavior index
  • n < 1:="" pseudoplastic="" (shear="">
  • n = 1: Newtonian
  • n > 1: dilatant (shear thickening)
• Bingham Plastic: \[\tau = \tau_0 + \mu_p \frac{du}{dy}\] where τ₀ = yield stress, μ_p = plastic viscosity
  Requires minimum stress to initiate flow

Boundary Layer Theory

Boundary Layer Thickness

• Laminar Boundary Layer (Flat Plate): \[\delta \approx \frac{5x}{\sqrt{Re_x}}\] where δ = boundary layer thickness, x = distance from leading edge, \[Re_x = \frac{\rho V x}{\mu}\]
• Turbulent Boundary Layer: \[\delta \approx \frac{0.37 x}{Re_x^{1/5}}\]
• Transition Reynolds Number: \[Re_{x,crit} \approx 5 \times 10^5\] Transition from laminar to turbulent boundary layer

Drag on Flat Plate

• Laminar Flow Drag Coefficient: \[C_{f} = \frac{1.328}{\sqrt{Re_L}}\] where C_f = skin friction coefficient, \[Re_L = \frac{\rho V L}{\mu}\] and L = plate length
• Turbulent Flow Drag Coefficient: \[C_{f} = \frac{0.074}{Re_L^{1/5}}\] Valid for 5 × 10⁵ <>_L <>
• Total Drag Force: \[F_D = C_f \frac{1}{2}\rho V^2 A_s\] where A_s = wetted surface area (both sides if applicable)

Hydraulic Transients

Water Hammer

• Pressure Wave Velocity: \[c = \sqrt{\frac{K/\rho}{1 + (K/E)(D/t)}}\] where c = wave speed (m/s or ft/s), K = bulk modulus of fluid, E = elastic modulus of pipe material, D = pipe diameter, t = pipe wall thickness
• Joukowsky Equation (Rapid Valve Closure): \[\Delta P = \rho c \Delta V\] where ΔP = pressure rise, ΔV = velocity change
  Valid when closure time <>
• Pressure Head Rise: \[\Delta h = \frac{c \Delta V}{g}\]