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Cheatsheet: Momentum Transfer
1. Fluid Properties and Statics
1.1 Fundamental Properties
| Property | Definition and Formula |
|---|---|
| Density (ρ) | Mass per unit volume; ρ = m/V (kg/m³) |
| Specific Gravity (SG) | SG = ρ_fluid/ρ_water; ρ_water = 1000 kg/m³ at 4°C |
| Dynamic Viscosity (μ) | Resistance to shear deformation; units: Pa·s or cP (1 cP = 0.001 Pa·s) |
| Kinematic Viscosity (ν) | ν = μ/ρ (m²/s); 1 St = 10⁻⁴ m²/s |
| Surface Tension (σ) | Force per unit length at interface; units: N/m or dyne/cm |
1.2 Fluid Statics
| Concept | Formula |
|---|---|
| Hydrostatic Pressure | P = P₀ + ρgh; h = depth below surface |
| Pressure at Depth | dP/dz = -ρg (z positive upward) |
| Manometer (U-tube) | ΔP = ρgh; h = height difference |
| Force on Submerged Surface | F = ρgh_c A; h_c = depth to centroid |
| Buoyant Force | F_b = ρ_fluid V_displaced g (Archimedes principle) |
2. Fluid Flow Fundamentals
2.1 Flow Classification
| Type | Description |
|---|---|
| Laminar Flow | Re < 2100="" for="" pipe="" flow;="" smooth,="" layered=""> |
| Turbulent Flow | Re > 4000 for pipe flow; chaotic, mixing motion |
| Transitional Flow | 2100 < re="">< 4000;="" unstable=""> |
| Steady Flow | Properties at a point do not change with time |
| Uniform Flow | Properties do not change along streamline |
2.2 Reynolds Number
| Configuration | Formula |
|---|---|
| Pipe Flow | Re = ρVD/μ = VD/ν; D = pipe diameter, V = average velocity |
| Non-Circular Duct | Re = ρVD_h/μ; D_h = 4A/P (hydraulic diameter) |
| Flow Past Sphere | Re = ρVd/μ; d = sphere diameter |
| Flow Past Cylinder | Re = ρVD/μ; D = cylinder diameter |
2.3 Continuity Equation
- Mass conservation: ṁ₁ = ṁ₂ or ρ₁A₁V₁ = ρ₂A₂V₂
- Incompressible flow: A₁V₁ = A₂V₂ = Q (volumetric flow rate)
- Differential form: ∂ρ/∂t + ∇·(ρV) = 0
3. Bernoulli Equation and Energy Balance
3.1 Bernoulli Equation
| Form | Equation |
|---|---|
| Standard Form | P₁/ρg + V₁²/2g + z₁ = P₂/ρg + V₂²/2g + z₂ + h_L |
| Energy per Unit Mass | P₁/ρ + V₁²/2 + gz₁ = P₂/ρ + V₂²/2 + gz₂ + losses |
| Ideal (Frictionless) | P/ρg + V²/2g + z = constant along streamline |
3.2 Extended Bernoulli (Mechanical Energy Balance)
- P₁/ρ + V₁²/2 + gz₁ + W_p = P₂/ρ + V₂²/2 + gz₂ + F
- W_p = pump work per unit mass (J/kg)
- F = friction losses per unit mass (J/kg)
- For pumps: W_p = gH_p; H_p = pump head (m)
- For turbines: W_t = gH_t (work extracted)
3.3 Head Terms
| Head Type | Expression |
|---|---|
| Pressure Head | h_p = P/ρg (m) |
| Velocity Head | h_v = V²/2g (m) |
| Elevation Head | h_z = z (m) |
| Total Head | H = P/ρg + V²/2g + z (m) |
4. Pipe Flow and Friction Losses
4.1 Darcy-Weisbach Equation
| Parameter | Expression |
|---|---|
| Head Loss | h_f = f(L/D)(V²/2g); f = friction factor |
| Pressure Drop | ΔP = f(L/D)(ρV²/2) |
| Friction Factor (Laminar) | f = 64/Re for Re <> |
4.2 Friction Factor Correlations
4.2.1 Turbulent Flow
- Colebrook equation: 1/√f = -2log₁₀(ε/3.7D + 2.51/(Re√f)); ε = roughness
- Swamee-Jain (explicit): f = 0.25/[log₁₀(ε/3.7D + 5.74/Re⁰·⁹)]²
- Smooth pipes (Blasius): f = 0.316/Re⁰·²⁵ for Re <>
- Fully rough: 1/√f = -2log₁₀(ε/3.7D)
4.2.2 Moody Chart Usage
- Horizontal axis: Reynolds number (Re)
- Vertical axis: Friction factor (f)
- Curves: Relative roughness (ε/D)
- Laminar zone: f = 64/Re line
- Transition zone: 2100 < re=""><>
4.3 Minor Losses
| Component | Loss Coefficient (K) |
|---|---|
| Sudden Expansion | K = (1 - A₁/A₂)²; h_L = K(V₁²/2g) |
| Sudden Contraction | K ≈ 0.5(1 - A₂/A₁); h_L = K(V₂²/2g) |
| 90° Elbow (threaded) | K ≈ 0.9 |
| 45° Elbow (threaded) | K ≈ 0.4 |
| Gate Valve (open) | K ≈ 0.15 |
| Globe Valve (open) | K ≈ 10 |
| Entrance (sharp) | K ≈ 0.5 |
| Entrance (rounded) | K ≈ 0.05 |
| Exit | K = 1.0 |
- Minor loss formula: h_L = K(V²/2g)
- Equivalent length: L_eq = KD/f
- Total head loss: h_L,total = h_f + Σh_L,minor
5. Laminar Flow Analysis
5.1 Hagen-Poiseuille Equation
| Parameter | Expression |
|---|---|
| Volumetric Flow Rate | Q = (πD⁴ΔP)/(128μL) |
| Average Velocity | V_avg = (D²ΔP)/(32μL) |
| Maximum Velocity | V_max = 2V_avg (at centerline) |
| Velocity Profile | v(r) = V_max[1-(r/R)²]; parabolic distribution |
| Shear Stress | τ = μ(dv/dr) = -(ΔP/L)(r/2) |
| Wall Shear Stress | τ_w = (4μV_avg)/R = (ΔPD)/(4L) |
5.2 Laminar Flow Between Parallel Plates
- Velocity profile: v(y) = (ΔP/2μL)y(h-y); y from bottom plate
- Maximum velocity: V_max = (ΔPh²)/(8μL) at y = h/2
- Average velocity: V_avg = (2/3)V_max
- Flow rate per unit width: q = (ΔPh³)/(12μL)
6. Turbulent Flow and Boundary Layers
6.1 Turbulent Velocity Profile
| Model | Expression |
|---|---|
| Power Law | v/V_max = (y/R)^(1/n); n ≈ 7 for turbulent flow |
| One-Seventh Power Law | v/V_max = (y/R)^(1/7) |
| Velocity Ratio | V_avg/V_max ≈ 0.817 for turbulent flow |
6.2 Boundary Layer Concepts
- Boundary layer thickness (δ): Distance where v = 0.99V_∞
- Laminar BL on flat plate: δ/x ≈ 5/√Re_x; Re_x = ρV_∞x/μ
- Turbulent BL on flat plate: δ/x ≈ 0.37/Re_x^(1/5)
- Transition: Re_x,crit ≈ 5×10⁵
- Displacement thickness: δ* = ∫₀^∞(1-v/V_∞)dy
- Momentum thickness: θ = ∫₀^∞(v/V_∞)(1-v/V_∞)dy
7. Drag and Lift Forces
7.1 Drag Force
| Concept | Formula |
|---|---|
| Drag Force | F_D = C_D A (ρV²/2); A = projected area |
| Drag Coefficient | C_D = F_D/(A ρV²/2); dimensionless |
| Skin Friction Drag | Due to viscous shear at surface |
| Form Drag | Due to pressure distribution (separation) |
7.2 Drag Coefficients for Common Shapes
| Object | C_D (Approximate) |
|---|---|
| Sphere (Re > 10⁴) | 0.4-0.5 |
| Sphere (Stokes, Re <> | C_D = 24/Re; F_D = 3πμVd |
| Cylinder (long) | 0.9-1.2 |
| Flat Plate (perpendicular) | 1.1-1.3 |
| Flat Plate (parallel) | 0.001-0.01 (laminar to turbulent) |
| Streamlined Body | 0.04-0.1 |
7.3 Terminal Velocity
- Balance: F_D = (ρ_p - ρ_f)Vg; ρ_p = particle density
- Sphere terminal velocity: V_t = √[(4gd(ρ_p-ρ_f))/(3C_Dρ_f)]
- Stokes regime (Re < 1):="" v_t="">
7.4 Lift Force
| Parameter | Expression |
|---|---|
| Lift Force | F_L = C_L A (ρV²/2) |
| Lift Coefficient | C_L = F_L/(A ρV²/2); function of angle of attack |
8. Flow Measurement
8.1 Pitot Tube
- Measures stagnation pressure: P_stag = P_static + ρV²/2
- Velocity: V = √[2(P_stag - P_static)/ρ]
- Combined with static tap to measure velocity
8.2 Venturi Meter
- Flow rate: Q = A₂√[2(P₁-P₂)/(ρ(1-(A₂/A₁)²))]
- With discharge coefficient: Q = C_d A₂√[2(P₁-P₂)/(ρ(1-(A₂/A₁)²))]
- C_d ≈ 0.95-0.99 for venturi
8.3 Orifice Plate
- Flow rate: Q = C_d A_o√[2ΔP/(ρ(1-β⁴))]; β = d_o/D
- C_d ≈ 0.6-0.65 for sharp-edged orifice
- ΔP = pressure drop across orifice
8.4 Flow Nozzle
- Similar to venturi, more compact
- Q = C_d A_n√[2ΔP/(ρ(1-(A_n/A₁)²))]
- C_d ≈ 0.95-0.99
8.5 Rotameter
- Variable area flowmeter
- Float height proportional to flow rate
- Balance: F_drag + F_buoy = W_float
9. Non-Newtonian Fluids
9.1 Rheological Models
| Fluid Type | Constitutive Equation |
|---|---|
| Newtonian | τ = μ(dv/dy); viscosity μ constant |
| Bingham Plastic | τ = τ₀ + μ_p(dv/dy); τ₀ = yield stress |
| Power Law | τ = K(dv/dy)ⁿ; K = consistency index, n = flow index |
| Pseudoplastic (Shear Thinning) | n < 1;="" apparent="" viscosity="" decreases="" with="" shear=""> |
| Dilatant (Shear Thickening) | n > 1; apparent viscosity increases with shear rate |
9.2 Apparent Viscosity
- Power law: μ_app = K(dv/dy)^(n-1)
- Bingham: μ_app = τ₀/(dv/dy) + μ_p
9.3 Pipe Flow (Power Law)
- Generalized Reynolds number: Re_gen = ρV^(2-n)D^n/(K8^(n-1))
- Laminar if Re_gen <>
- Hagen-Poiseuille modified: Q = (πD³/4)[(ΔPD)/(4KL)]^(1/n) × [n/(3n+1)]
10. Pump and Compressor Fundamentals
10.1 Pump Performance
| Parameter | Expression |
|---|---|
| Pump Head | H_p = (P₂-P₁)/ρg + (V₂²-V₁²)/2g + (z₂-z₁) + h_L |
| Pump Power (Hydraulic) | P_hyd = ṁgH_p = ρgQH_p (W) |
| Brake Power | P_brake = P_hyd/η_pump |
| Pump Efficiency | η = P_hyd/P_brake |
| Specific Speed | N_s = NQ^(1/2)/H^(3/4); N in rpm, Q in gpm, H in ft |
10.2 Affinity Laws (Pumps)
| Relationship | Formula |
|---|---|
| Flow Rate | Q₂/Q₁ = (N₂/N₁)(D₂/D₁)³ |
| Head | H₂/H₁ = (N₂/N₁)²(D₂/D₁)² |
| Power | P₂/P₁ = (N₂/N₁)³(D₂/D₁)⁵ |
- N = rotational speed (rpm); D = impeller diameter
- Same pump, same fluid density
10.3 Net Positive Suction Head (NPSH)
- NPSH_available = (P_atm - P_vapor)/ρg + z_s - h_L,suction
- NPSH_required = manufacturer specification
- Cavitation prevention: NPSH_available > NPSH_required
- z_s = elevation of fluid surface above pump centerline (negative if below)
10.4 System Curve and Operating Point
- System head: H_sys = H_static + KQ²; K depends on friction
- Operating point: Intersection of pump curve and system curve
- Pumps in series: heads add at same Q
- Pumps in parallel: flows add at same H
11. Dimensional Analysis and Similitude
11.1 Buckingham Pi Theorem
- If n variables with m fundamental dimensions, then (n-m) dimensionless groups
- Select m repeating variables (containing all dimensions)
- Form π groups by combining remaining variables with repeating variables
11.2 Common Dimensionless Groups
| Group | Definition |
|---|---|
| Reynolds Number | Re = ρVL/μ = inertial/viscous forces |
| Froude Number | Fr = V/√(gL) = inertial/gravity forces |
| Euler Number | Eu = ΔP/(ρV²) = pressure/inertial forces |
| Weber Number | We = ρV²L/σ = inertial/surface tension forces |
| Mach Number | Ma = V/c = flow velocity/sound velocity |
| Drag Coefficient | C_D = F_D/(ρV²A/2) |
| Friction Factor | f = ΔP/(L/D)(ρV²/2) |
11.3 Similarity Requirements
- Geometric similarity: Length ratios constant (L_model/L_prototype = λ)
- Kinematic similarity: Velocity ratios constant at corresponding points
- Dynamic similarity: Force ratios constant; requires matching π groups
- Reynolds similarity: Re_model = Re_prototype
- Froude similarity: Fr_model = Fr_prototype (free surface flows)
12. Compressible Flow Basics
12.1 Speed of Sound and Mach Number
| Parameter | Expression |
|---|---|
| Speed of Sound | c = √(kRT) for ideal gas; k = c_p/c_v |
| Mach Number | Ma = V/c |
| Subsonic | Ma <> |
| Sonic | Ma = 1 |
| Supersonic | Ma > 1 |
12.2 Isentropic Relations (Ideal Gas)
- P/ρ^k = constant; T/P^((k-1)/k) = constant
- Stagnation temperature: T₀/T = 1 + [(k-1)/2]Ma²
- Stagnation pressure: P₀/P = [1 + [(k-1)/2]Ma²]^(k/(k-1))
- Stagnation density: ρ₀/ρ = [1 + [(k-1)/2]Ma²]^(1/(k-1))
- For air: k ≈ 1.4
12.3 Choked Flow
- Occurs when Ma = 1 at throat of converging-diverging nozzle
- Mass flow rate maximum at choked condition
- ṁ_max = A*P₀√(k/(RT₀)) × (2/(k+1))^((k+1)/(2(k-1)))
- Critical pressure ratio: P*/P₀ = (2/(k+1))^(k/(k-1)) ≈ 0.528 for air
13. Open Channel Flow
13.1 Flow Classification
| Parameter | Definition |
|---|---|
| Hydraulic Radius | R_h = A/P; A = flow area, P = wetted perimeter |
| Hydraulic Depth | D_h = A/T; T = top width |
| Froude Number | Fr = V/√(gD_h) |
| Subcritical Flow | Fr < 1;="" slow,="" deep=""> |
| Critical Flow | Fr = 1; minimum specific energy |
| Supercritical Flow | Fr > 1; fast, shallow flow |
13.2 Manning Equation
- V = (1/n)R_h^(2/3)S^(1/2); n = Manning roughness coefficient
- Q = (A/n)R_h^(2/3)S^(1/2); S = slope of energy grade line
- SI units: V in m/s, R_h in m
- Manning n values: concrete (0.012), earth (0.025), gravel (0.029)
13.3 Specific Energy
- E = y + V²/2g; y = flow depth
- Minimum E at critical depth: y_c = (q²/g)^(1/3) for rectangular channel
- q = Q/b = discharge per unit width
- Critical velocity: V_c = √(gy_c)
13.4 Hydraulic Jump
- Transition from supercritical to subcritical flow
- Sequent depths (rectangular): y₂/y₁ = (1/2)[-1 + √(1+8Fr₁²)]
- Energy loss: ΔE = E₁ - E₂ = (y₂-y₁)³/(4y₁y₂)
14. Piping System Design
14.1 Equivalent Length Method
- Total head loss: h_L = f(L_total/D)(V²/2g)
- L_total = L_pipe + ΣL_eq,fittings
- L_eq = KD/f for each fitting
14.2 Economic Pipe Diameter
- Balance capital cost (larger D) vs. operating cost (pumping)
- Genereaux equation: D_opt = 0.363Q^(0.45)ρ^(0.13) (inches, gpm)
- Velocity guidelines: 3-10 ft/s for water in pipes
14.3 Series and Parallel Pipes
14.3.1 Pipes in Series
- Same flow rate: Q_total = Q₁ = Q₂ = Q₃
- Head losses add: h_L,total = h_L1 + h_L2 + h_L3
14.3.2 Pipes in Parallel
- Flow rates add: Q_total = Q₁ + Q₂ + Q₃
- Same head loss: h_L1 = h_L2 = h_L3
14.4 Pipe Networks (Hardy Cross Method)
- Iterative method for solving looped networks
- Assume initial flow distribution satisfying continuity
- Correction: ΔQ = -Σh_L/(nΣh_L/Q) around each loop
- n = 2 for turbulent flow (h_L ∝ Q²)
15. Navier-Stokes Equations
15.1 Conservation Equations
15.1.1 Continuity (Incompressible)
- ∇·V = ∂u/∂x + ∂v/∂y + ∂w/∂z = 0
15.1.2 Momentum (Navier-Stokes)
- ρ(∂V/∂t + V·∇V) = -∇P + μ∇²V + ρg
- x-component: ρ(∂u/∂t + u∂u/∂x + v∂u/∂y + w∂u/∂z) = -∂P/∂x + μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²) + ρg_x
15.2 Simplified Forms
| Condition | Simplification |
|---|---|
| Steady Flow | ∂/∂t = 0 |
| Inviscid (Euler Equation) | μ = 0; ρ(∂V/∂t + V·∇V) = -∇P + ρg |
| Fully Developed Pipe Flow | ∂u/∂x = 0; flow in x-direction only |
16. Turbomachinery
16.1 Euler Turbomachine Equation
- Specific work: W = U₂V_θ2 - U₁V_θ1
- U = rω = tangential velocity of rotor
- V_θ = tangential component of absolute velocity
- Power: P = ṁW = ṁ(U₂V_θ2 - U₁V_θ1)
16.2 Velocity Triangles
- V = absolute velocity (relative to stationary frame)
- W = relative velocity (relative to rotating frame)
- U = blade velocity = rω
- Vector relation: V = W + U
- V_r = radial component; V_θ = tangential component
16.3 Reaction vs. Impulse
| Type | Characteristic |
|---|---|
| Impulse Turbine | Pressure change only in nozzle; P₁ = P₂ across rotor |
| Reaction Turbine | Pressure change across rotor; reaction = ΔP_rotor/ΔP_total |
17. Two-Phase Flow
17.1 Flow Regime Definitions
- Void fraction: α = V_gas/(V_gas + V_liquid)
- Quality: x = ṁ_vapor/ṁ_total
- Slip ratio: S = V_gas/V_liquid
17.2 Flow Patterns (Horizontal Pipe)
- Stratified: Gas above, liquid below
- Wavy: Interface waves at low velocities
- Slug: Intermittent liquid plugs and gas pockets
- Annular: Liquid film on wall, gas core in center
- Dispersed/Mist: Liquid droplets in continuous gas
17.3 Pressure Drop Models
- Homogeneous model: Treat as single phase with mixture properties
- ρ_m = αρ_g + (1-α)ρ_l; μ_m = αμ_g + (1-α)μ_l
- Separated flow models: Track phases separately
- Lockhart-Martinelli parameter: X² = (ΔP_L/ΔP_G)
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