PE Exam Exam > PE Exam Notes > Mechanical Engineering for PE > Cheatsheet: Fluid Properties
Cheatsheet: Fluid Properties
1. Density and Specific Weight
1.1 Fundamental Definitions
| Property | Definition and Formula |
|---|---|
| Density (ρ) | Mass per unit volume: ρ = m/V (kg/m³ or slug/ft³) |
| Specific Weight (γ) | Weight per unit volume: γ = ρg (N/m³ or lb/ft³) |
| Specific Gravity (SG) | Ratio of fluid density to water density at 4°C: SG = ρ/ρ_water |
| Specific Volume (v) | Volume per unit mass (reciprocal of density): v = 1/ρ (m³/kg) |
1.2 Standard Values
| Fluid | Density and Specific Weight at Standard Conditions |
|---|---|
| Water (4°C) | ρ = 1000 kg/m³ = 1.94 slug/ft³; γ = 9810 N/m³ = 62.4 lb/ft³ |
| Water (20°C) | ρ = 998 kg/m³; γ = 9790 N/m³ = 62.3 lb/ft³ |
| Seawater | ρ = 1025 kg/m³; γ = 10,050 N/m³ = 64 lb/ft³; SG = 1.025 |
| Air (STP) | ρ = 1.225 kg/m³ = 0.00237 slug/ft³; γ = 12.0 N/m³ = 0.0765 lb/ft³ |
| Mercury | ρ = 13,600 kg/m³; γ = 133,000 N/m³ = 847 lb/ft³; SG = 13.6 |
1.3 Temperature Effects
- Liquids: Density decreases with increasing temperature (thermal expansion)
- Gases: Density varies inversely with temperature according to ideal gas law
- Coefficient of volume expansion: β = -(1/ρ)(∂ρ/∂T)_P
2. Viscosity
2.1 Dynamic Viscosity (μ)
| Concept | Description |
|---|---|
| Definition | Measure of fluid resistance to shear or angular deformation |
| Newton's Law of Viscosity | τ = μ(du/dy), where τ = shear stress, du/dy = velocity gradient |
| Units | Pa·s or N·s/m² (SI); lb·s/ft² or slug/(ft·s) (English); 1 Poise = 0.1 Pa·s |
| Water (20°C) | μ = 1.002 × 10⁻³ Pa·s = 2.09 × 10⁻⁵ lb·s/ft² |
| Air (20°C) | μ = 1.81 × 10⁻⁵ Pa·s = 3.78 × 10⁻⁷ lb·s/ft² |
2.2 Kinematic Viscosity (ν)
| Property | Formula and Units |
|---|---|
| Definition | ν = μ/ρ (ratio of dynamic viscosity to density) |
| Units | m²/s (SI); ft²/s (English); 1 Stoke = 1 cm²/s = 10⁻⁴ m²/s |
| Water (20°C) | ν = 1.004 × 10⁻⁶ m²/s = 1.08 × 10⁻⁵ ft²/s |
| Air (20°C) | ν = 1.48 × 10⁻⁵ m²/s = 1.59 × 10⁻⁴ ft²/s |
2.3 Fluid Classification by Viscous Behavior
| Fluid Type | Characteristics |
|---|---|
| Newtonian Fluids | Linear relationship between shear stress and shear rate; constant viscosity (water, air, oil) |
| Non-Newtonian Fluids | Nonlinear shear stress-shear rate relationship; viscosity varies with shear rate |
| Pseudoplastic (Shear-thinning) | Viscosity decreases with increasing shear rate (blood, paint, ketchup) |
| Dilatant (Shear-thickening) | Viscosity increases with increasing shear rate (cornstarch suspension) |
| Bingham Plastic | Requires yield stress before flow begins (toothpaste, drilling mud) |
2.4 Temperature Effects on Viscosity
- Liquids: Viscosity decreases with increasing temperature (weaker intermolecular forces)
- Gases: Viscosity increases with increasing temperature (increased molecular collisions)
- Sutherland's formula for gases: μ = μ₀(T/T₀)^(3/2)[(T₀ + S)/(T + S)]
- Andrade's equation for liquids: μ = A·e^(B/T)
3. Surface Tension and Capillarity
3.1 Surface Tension (σ)
| Concept | Details |
|---|---|
| Definition | Force per unit length at liquid-gas or liquid-liquid interface; energy per unit area |
| Units | N/m (SI); lb/ft (English); also J/m² or dyne/cm |
| Water-Air (20°C) | σ = 0.0728 N/m = 0.00499 lb/ft |
| Mercury-Air (20°C) | σ = 0.484 N/m = 0.0332 lb/ft |
| Temperature Effect | Surface tension decreases with increasing temperature |
3.2 Pressure Inside Droplets and Bubbles
| Configuration | Pressure Difference Formula |
|---|---|
| Droplet (one interface) | Δp = 2σ/R, where R = droplet radius |
| Soap Bubble (two interfaces) | Δp = 4σ/R |
| Liquid Jet | Δp = σ/R |
3.3 Capillarity
| Property | Formula and Description |
|---|---|
| Capillary Rise/Depression | h = (4σcosθ)/(γd) = (2σcosθ)/(ρgr), where θ = contact angle, d = tube diameter, r = tube radius |
| Contact Angle (θ) | Angle between liquid surface and solid wall; θ < 90°="" (wetting),="" θ=""> 90° (non-wetting) |
| Water-Glass | θ ≈ 0°, capillary rise occurs (meniscus curves upward) |
| Mercury-Glass | θ ≈ 140°, capillary depression occurs (meniscus curves downward) |
4. Vapor Pressure and Cavitation
4.1 Vapor Pressure (p_v)
| Property | Description |
|---|---|
| Definition | Pressure at which liquid and vapor phases are in equilibrium at given temperature |
| Temperature Dependence | Vapor pressure increases exponentially with temperature |
| Water at 20°C | p_v = 2.34 kPa = 0.339 psi |
| Water at 100°C | p_v = 101.3 kPa = 14.7 psi (atmospheric pressure) |
| Boiling Point | Temperature at which vapor pressure equals surrounding pressure |
4.2 Cavitation
| Concept | Details |
|---|---|
| Definition | Formation and collapse of vapor bubbles when local pressure drops below vapor pressure |
| Occurrence | Pump impellers, propellers, valve openings, hydrofoils, pipe constrictions |
| Effects | Noise, vibration, reduced performance, material erosion and pitting |
| Cavitation Number | Ca = (p - p_v)/(½ρV²), where p = local pressure, V = reference velocity |
| Prevention | Increase local pressure, reduce velocities, minimize pressure drops, proper design |
5. Compressibility and Bulk Modulus
5.1 Bulk Modulus of Elasticity (E_v or K)
| Property | Formula and Description |
|---|---|
| Definition | Measure of fluid resistance to compression: E_v = -V(dp/dV) = ρ(dp/dρ) |
| Units | Pa or N/m² (SI); psi or lb/ft² (English) |
| Water (20°C) | E_v = 2.2 × 10⁹ Pa = 2.2 GPa = 3.2 × 10⁵ psi |
| Isothermal Bulk Modulus | E_v = -V(∂p/∂V)_T (constant temperature) |
| Adiabatic Bulk Modulus | E_v = -V(∂p/∂V)_s (no heat transfer; used for acoustic waves) |
5.2 Coefficient of Compressibility
| Coefficient | Formula |
|---|---|
| Isothermal Compressibility (β) | β = 1/E_v = -(1/V)(∂V/∂p)_T = (1/ρ)(∂ρ/∂p)_T |
| Units | Pa⁻¹ or m²/N (SI); psi⁻¹ (English) |
5.3 Speed of Sound
| Property | Formula |
|---|---|
| General Formula | c = √(E_v/ρ) = √(dp/dρ) |
| Ideal Gas | c = √(kRT), where k = specific heat ratio, R = gas constant, T = absolute temperature |
| Water (20°C) | c ≈ 1480 m/s = 4860 ft/s |
| Air (20°C) | c ≈ 343 m/s = 1125 ft/s |
5.4 Mach Number
| Parameter | Definition and Classification |
|---|---|
| Mach Number (Ma) | Ma = V/c, where V = flow velocity, c = speed of sound |
| Subsonic | Ma <> |
| Transonic | 0.8 < ma=""><> |
| Supersonic | 1 < ma=""><> |
| Hypersonic | Ma > 5 |
6. Ideal Gas Law and Gas Properties
6.1 Ideal Gas Equation
| Form | Equation |
|---|---|
| General Form | pV = nRT or pV = mR_specific T |
| Density Form | p = ρRT, where ρ = density, R = specific gas constant |
| Universal Gas Constant | R_u = 8314 J/(kmol·K) = 1545 ft·lb/(lbmol·°R) |
| Specific Gas Constant | R = R_u/M, where M = molecular weight |
6.2 Gas Constants for Common Gases
| Gas | Specific Gas Constant R (J/kg·K) |
|---|---|
| Air | R = 287 J/(kg·K) = 53.35 ft·lb/(lbm·°R); M = 28.97 kg/kmol |
| Oxygen (O₂) | R = 260 J/(kg·K); M = 32 kg/kmol |
| Nitrogen (N₂) | R = 297 J/(kg·K); M = 28 kg/kmol |
| Carbon Dioxide (CO₂) | R = 189 J/(kg·K); M = 44 kg/kmol |
| Hydrogen (H₂) | R = 4124 J/(kg·K); M = 2 kg/kmol |
6.3 Specific Heat Ratios
| Gas | k = c_p/c_v |
|---|---|
| Monatomic (He, Ar) | k = 1.67 |
| Diatomic (Air, O₂, N₂, H₂) | k = 1.4 |
| Triatomic (CO₂, H₂O vapor) | k = 1.3 |
6.4 Specific Heat Relationships
- c_p - c_v = R (Mayer's relation)
- c_v = R/(k - 1)
- c_p = kR/(k - 1)
- For air: c_p = 1005 J/(kg·K), c_v = 718 J/(kg·K)
7. Absolute and Gage Pressure
7.1 Pressure Definitions
| Type | Definition |
|---|---|
| Absolute Pressure (p_abs) | Pressure measured relative to perfect vacuum (zero pressure) |
| Gage Pressure (p_gage) | Pressure measured relative to atmospheric pressure |
| Vacuum Pressure | Negative gage pressure (below atmospheric) |
| Atmospheric Pressure (p_atm) | Local atmospheric pressure (reference for gage pressure) |
7.2 Pressure Relationships
| Relationship | Formula |
|---|---|
| Absolute to Gage | p_abs = p_gage + p_atm |
| Vacuum Pressure | p_vacuum = p_atm - p_abs (when p_abs <> |
7.3 Standard Atmospheric Pressure
| Condition | Value |
|---|---|
| Standard Atmosphere (Sea Level) | 101.325 kPa = 14.696 psi = 760 mm Hg = 29.92 in Hg = 1 atm = 1.01325 bar |
| Standard Temperature | 15°C = 59°F = 288.15 K = 518.67°R |
7.4 Unit Conversions
| Unit | Conversion |
|---|---|
| Pascal (Pa) | 1 Pa = 1 N/m² |
| Kilopascal (kPa) | 1 kPa = 1000 Pa = 0.145 psi |
| Bar | 1 bar = 100 kPa = 14.5 psi |
| PSI | 1 psi = 6.895 kPa = 144 lb/ft² |
| mm Hg (torr) | 1 mm Hg = 133.3 Pa = 0.0193 psi |
| in Hg | 1 in Hg = 3.386 kPa = 0.491 psi |
8. Reynolds Number
8.1 Definition and Formula
| Parameter | Formula |
|---|---|
| Reynolds Number (Re) | Re = ρVL/μ = VL/ν, where V = velocity, L = characteristic length |
| Physical Meaning | Ratio of inertial forces to viscous forces |
8.2 Characteristic Length for Different Flows
| Flow Type | Characteristic Length (L) |
|---|---|
| Pipe Flow | L = D (diameter) or D_h (hydraulic diameter) |
| Flow Over Flat Plate | L = x (distance from leading edge) |
| Flow Over Sphere/Cylinder | L = D (diameter) |
| Open Channel | L = 4A/P = D_h (hydraulic diameter) |
8.3 Flow Regime Classification
| Flow Configuration | Critical Reynolds Number |
|---|---|
| Pipe Flow: Laminar | Re <> |
| Pipe Flow: Transitional | 2300 < re=""><> |
| Pipe Flow: Turbulent | Re > 4000 |
| Flat Plate Boundary Layer | Re_crit ≈ 5 × 10⁵ |
| Flow Past Sphere | Re_crit ≈ 2 × 10⁵ (drag crisis) |
9. Non-Dimensional Numbers
9.1 Common Dimensionless Groups
| Number | Formula and Significance |
|---|---|
| Froude Number (Fr) | Fr = V/√(gL); ratio of inertial to gravitational forces; free surface flows |
| Euler Number (Eu) | Eu = Δp/(ρV²); ratio of pressure to inertial forces |
| Weber Number (We) | We = ρV²L/σ; ratio of inertial to surface tension forces; droplets, sprays |
| Mach Number (Ma) | Ma = V/c; ratio of flow velocity to sound speed; compressibility effects |
| Strouhal Number (St) | St = fL/V; ratio of oscillation frequency to flow; vortex shedding |
| Cavitation Number (Ca) | Ca = (p - p_v)/(½ρV²); measures cavitation potential |
9.2 Drag Coefficient
| Concept | Formula |
|---|---|
| Drag Coefficient (C_D) | C_D = F_D/(½ρV²A), where F_D = drag force, A = reference area |
| Sphere (Re <> | C_D = 24/Re (Stokes flow) |
| Sphere (10³ < re=""><> | C_D ≈ 0.4 to 0.5 |
10. Hydraulic Diameter
10.1 Definition
| Parameter | Formula |
|---|---|
| Hydraulic Diameter (D_h) | D_h = 4A/P, where A = cross-sectional area, P = wetted perimeter |
| Purpose | Equivalent diameter for non-circular conduits; used in Re, friction factor calculations |
10.2 Common Cross-Sections
| Geometry | Hydraulic Diameter |
|---|---|
| Circular Pipe (full) | D_h = D (diameter) |
| Rectangular Duct (a × b) | D_h = 2ab/(a + b) |
| Square Duct (a × a) | D_h = a |
| Annulus (D_o, D_i) | D_h = D_o - D_i |
| Wide Rectangular (a >> b) | D_h ≈ 2b |
| Open Channel (wide) | D_h = 4y, where y = depth |
11. Thermal Properties
11.1 Thermal Expansion
| Property | Formula |
|---|---|
| Coefficient of Volume Expansion (β) | β = (1/V)(∂V/∂T)_p = -(1/ρ)(∂ρ/∂T)_p |
| Density Change | ρ = ρ₀[1 - β(T - T₀)] |
| Ideal Gas | β = 1/T (where T is absolute temperature) |
11.2 Thermal Conductivity
| Fluid | Thermal Conductivity k at 20°C |
|---|---|
| Water | k = 0.598 W/(m·K) = 0.346 Btu/(h·ft·°F) |
| Air | k = 0.0257 W/(m·K) = 0.0149 Btu/(h·ft·°F) |
| Mercury | k = 8.4 W/(m·K) |
11.3 Prandtl Number
| Parameter | Formula and Significance |
|---|---|
| Prandtl Number (Pr) | Pr = ν/α = μc_p/k = (momentum diffusivity)/(thermal diffusivity) |
| Water (20°C) | Pr ≈ 7 |
| Air (20°C) | Pr ≈ 0.7 |
| Liquid Metals | Pr < 1="" (high="" thermal=""> |
| Oils | Pr >> 1 (low thermal conductivity) |
12. Additional Property Relationships
12.1 No-Slip Condition
- Fluid velocity at solid boundary equals boundary velocity (zero for stationary walls)
- Fundamental assumption in viscous flow analysis
- Valid for all Newtonian fluids at continuum scale
12.2 Continuum Hypothesis
- Fluid treated as continuous medium rather than discrete molecules
- Valid when Knudsen number Kn = λ/L < 1,="" where="" λ="mean" free="" path,="" l="characteristic">
- Breaks down for rarefied gases (high altitude, micro-scale flows)
12.3 Standard Temperature and Pressure (STP)
| Standard | Conditions |
|---|---|
| STP (NIST) | T = 20°C = 293.15 K; p = 101.325 kPa = 1 atm |
| STP (Historical) | T = 0°C = 273.15 K; p = 101.325 kPa = 1 atm |
| Normal Temperature and Pressure (NTP) | T = 20°C; p = 1 atm |
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