Formula Sheet: Fluid Properties

Fundamental Fluid Properties

Density

Mass Density (ρ): \[ \rho = \frac{m}{V} \]

• ρ = mass density (kg/m³ or lbm/ft³)
• m = mass (kg or lbm)
• V = volume (m³ or ft³)

Specific Weight (γ): \[ \gamma = \rho g \]

• γ = specific weight (N/m³ or lbf/ft³)
• ρ = mass density (kg/m³ or lbm/ft³)
• g = gravitational acceleration (9.81 m/s² or 32.2 ft/s²)

Specific Gravity (SG): \[ SG = \frac{\rho_{fluid}}{\rho_{reference}} \]

• SG = specific gravity (dimensionless)
• ρ_fluid = density of fluid
• ρ_reference = density of reference fluid (water at 4°C: 1000 kg/m³ or 62.4 lbm/ft³)
• Note: SG can also be expressed as the ratio of specific weights

Specific Volume (v): \[ v = \frac{1}{\rho} = \frac{V}{m} \]

• v = specific volume (m³/kg or ft³/lbm)
• ρ = mass density

Pressure

Absolute and Gage Pressure Relationship: \[ P_{abs} = P_{gage} + P_{atm} \]

• P_abs = absolute pressure (Pa or psia)
• P_gage = gage pressure (Pa or psig)
• P_atm = atmospheric pressure (101,325 Pa or 14.7 psia at sea level)

Vacuum Pressure: \[ P_{abs} = P_{atm} - P_{vacuum} \]

• P_vacuum = vacuum pressure (always positive, measured below atmospheric)

Hydrostatic Pressure: \[ P = \rho g h \] or \[ P = \gamma h \]

• P = pressure (Pa or psf)
• h = depth below free surface (m or ft)

Viscosity

Dynamic (Absolute) Viscosity

Newton's Law of Viscosity: \[ \tau = \mu \frac{du}{dy} \]

• τ = shear stress (Pa or lbf/ft²)
• μ = dynamic (absolute) viscosity (Pa·s or lbf·s/ft²)
• du/dy = velocity gradient perpendicular to flow direction (1/s)
• Note: Common units for μ include centipoise (cP): 1 cP = 0.001 Pa·s
• Applies to Newtonian fluids only

Kinematic Viscosity

Kinematic Viscosity (ν): \[ \nu = \frac{\mu}{\rho} \]

• ν = kinematic viscosity (m²/s or ft²/s)
• μ = dynamic viscosity (Pa·s or lbf·s/ft²)
• ρ = mass density (kg/m³ or lbm/ft³)
• Note: Common units include centistokes (cSt): 1 cSt = 1 mm²/s = 10^-6 m²/s

Temperature Effects on Viscosity

Andrade's Equation (Liquids): \[ \mu = A e^{B/T} \]

• A, B = empirical constants
• T = absolute temperature (K or °R)
• Note: Viscosity of liquids decreases with increasing temperature

Sutherland's Formula (Gases): \[ \mu = \mu_0 \frac{T_0 + S}{T + S} \left(\frac{T}{T_0}\right)^{3/2} \]

• μ₀ = reference viscosity at reference temperature T₀
• S = Sutherland constant (K or °R)
• T = absolute temperature (K or °R)
• Note: Viscosity of gases increases with increasing temperature

Surface Tension and Capillarity

Surface Tension

Surface Tension Force: \[ F = \sigma L \]

• F = surface tension force (N or lbf)
• σ = surface tension (N/m or lbf/ft)
• L = length of contact line (m or ft)

Pressure Difference Across Curved Interface (Spherical Droplet): \[ \Delta P = \frac{4\sigma}{d} \]

• ΔP = pressure difference across interface (Pa or psf)
• d = droplet diameter (m or ft)

Pressure Difference Across Curved Interface (Soap Bubble): \[ \Delta P = \frac{8\sigma}{d} \]

• Note: Factor of 8 accounts for two surfaces (inside and outside)

Pressure Difference for Cylindrical Interface: \[ \Delta P = \frac{2\sigma}{r} \]

• r = radius of curvature (m or ft)

Capillary Action

Capillary Rise/Depression: \[ h = \frac{4\sigma \cos\theta}{\gamma d} = \frac{4\sigma \cos\theta}{\rho g d} \]

• h = height of capillary rise (positive) or depression (negative) (m or ft)
• σ = surface tension (N/m or lbf/ft)
• θ = contact angle between fluid and tube wall (degrees or radians)
• d = tube diameter (m or ft)
• γ = specific weight (N/m³ or lbf/ft³)
• Note: For water-glass interface, θ ≈ 0° (wetting fluid, rise occurs)
• Note: For mercury-glass interface, θ ≈ 140° (non-wetting fluid, depression occurs)

Compressibility and Elasticity

Bulk Modulus of Elasticity

Bulk Modulus (E_v or K): \[ E_v = -V \frac{dP}{dV} = \rho \frac{dP}{d\rho} \]

• E_v = bulk modulus of elasticity (Pa or psi)
• V = volume (m³ or ft³)
• P = pressure (Pa or psi)
• ρ = density (kg/m³ or lbm/ft³)
• Note: Negative sign ensures E_v is positive (volume decreases as pressure increases)

Finite Change Form: \[ E_v = -\frac{\Delta P}{\Delta V / V} \]

• ΔP = change in pressure
• ΔV/V = volumetric strain (fractional volume change)

Coefficient of Compressibility

Coefficient of Compressibility (β): \[ \beta = \frac{1}{E_v} = -\frac{1}{V} \frac{dV}{dP} \]

• β = coefficient of compressibility (Pa^-1 or psi^-1)
• Note: Reciprocal of bulk modulus

Isothermal Compressibility

Isothermal Bulk Modulus (Ideal Gas): \[ E_v = P \]

• For isothermal compression of ideal gas
• P = absolute pressure

Isentropic Compressibility

Isentropic Bulk Modulus (Ideal Gas): \[ E_v = kP \]

• For isentropic (adiabatic) compression of ideal gas
• k = specific heat ratio (c_p/c_v)
• P = absolute pressure

Speed of Sound

Speed of Sound in Fluid: \[ c = \sqrt{\frac{E_v}{\rho}} \]

• c = speed of sound (m/s or ft/s)
• E_v = bulk modulus of elasticity (Pa or psf)
• ρ = density (kg/m³ or lbm/ft³)

Speed of Sound in Ideal Gas: \[ c = \sqrt{kRT} \]

• k = specific heat ratio (c_p/c_v)
• R = specific gas constant (J/(kg·K) or ft·lbf/(lbm·°R))
• T = absolute temperature (K or °R)

Vapor Pressure

Vapor Pressure Relationship:

• P_v = vapor pressure at given temperature (Pa or psia)
• Cavitation occurs when local pressure drops below vapor pressure: P_local <>_v
• Note: Vapor pressure increases with temperature
• Note: Water boils when P_atm = P_v

Clausius-Clapeyron Equation (Approximate): \[ \ln\left(\frac{P_{v2}}{P_{v1}}\right) = \frac{h_{fg}}{R_v} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \]

• P_v1, P_v2 = vapor pressures at temperatures T₁ and T₂
• h_fg = enthalpy of vaporization (J/kg or Btu/lbm)
• R_v = specific gas constant for vapor
• T₁, T₂ = absolute temperatures (K or °R)

Ideal Gas Law

Ideal Gas Law: \[ PV = mRT \] or \[ P = \rho RT \]

• P = absolute pressure (Pa or psia)
• V = volume (m³ or ft³)
• m = mass (kg or lbm)
• R = specific gas constant (J/(kg·K) or ft·lbf/(lbm·°R))
• T = absolute temperature (K or °R)
• ρ = density (kg/m³ or lbm/ft³)

Universal Gas Constant Relationship: \[ R = \frac{R_u}{M} \]

• R = specific gas constant
• R_u = universal gas constant (8314 J/(kmol·K) or 1545 ft·lbf/(lbmol·°R))
• M = molecular weight (kg/kmol or lbm/lbmol)

Thermal Expansion

Volumetric Thermal Expansion

Coefficient of Volumetric Thermal Expansion (α_v): \[ \alpha_v = \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P \]

• α_v = coefficient of volumetric thermal expansion (1/K or 1/°R)
• V = volume
• T = temperature
• Subscript P indicates constant pressure

Volume Change Due to Temperature: \[ \Delta V = V_0 \alpha_v \Delta T \]

• ΔV = change in volume
• V₀ = initial volume
• ΔT = change in temperature

Density Change Due to Temperature: \[ \rho = \frac{\rho_0}{1 + \alpha_v \Delta T} \]

• ρ = density at new temperature
• ρ₀ = density at reference temperature
• Note: For small ΔT: ρ ≈ ρ₀(1 - α_vΔT)

Ideal Gas Thermal Expansion: \[ \alpha_v = \frac{1}{T} \]

• T = absolute temperature (K or °R)
• Valid for ideal gases at constant pressure

Fluid Classification

Newtonian vs. Non-Newtonian Fluids

Newtonian Fluids:

• Shear stress proportional to velocity gradient: τ = μ(du/dy)
• Viscosity (μ) is constant at given temperature and pressure
• Examples: water, air, most gases, simple oils

Non-Newtonian Fluids:

• Bingham Plastic: τ = τ_y + μ_p(du/dy) where τ_y = yield stress
• Pseudoplastic (Shear-thinning): apparent viscosity decreases with shear rate
• Dilatant (Shear-thickening): apparent viscosity increases with shear rate
• Thixotropic: viscosity decreases with time under constant shear
• Rheopectic: viscosity increases with time under constant shear

Power Law Model: \[ \tau = K \left(\frac{du}{dy}\right)^n \]

• K = consistency index
• n = flow behavior index
• n < 1:="" pseudoplastic="">
• n = 1: Newtonian
• n > 1: dilatant (shear-thickening)

Dimensionless Numbers Related to Fluid Properties

Reynolds Number

Reynolds Number (Re): \[ Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu} \]

• Re = Reynolds number (dimensionless)
• ρ = density (kg/m³ or lbm/ft³)
• V = characteristic velocity (m/s or ft/s)
• D = characteristic length (diameter for pipes) (m or ft)
• μ = dynamic viscosity (Pa·s or lbf·s/ft²)
• ν = kinematic viscosity (m²/s or ft²/s)
• Represents ratio of inertial forces to viscous forces

Weber Number

Weber Number (We): \[ We = \frac{\rho V^2 L}{\sigma} \]

• We = Weber number (dimensionless)
• L = characteristic length (m or ft)
• σ = surface tension (N/m or lbf/ft)
• Represents ratio of inertial forces to surface tension forces

Mach Number

Mach Number (Ma): \[ Ma = \frac{V}{c} \]

• Ma = Mach number (dimensionless)
• V = flow velocity (m/s or ft/s)
• c = speed of sound in fluid (m/s or ft/s)
• Ma < 0.3:="" incompressible="" flow="" (density="" changes=""><>
• Ma < 1:="" subsonic="">
• Ma = 1: sonic flow
• Ma > 1: supersonic flow

Cavitation Number

Cavitation Number (Ca or σ): \[ Ca = \frac{P - P_v}{\frac{1}{2}\rho V^2} \]

• Ca = cavitation number (dimensionless)
• P = local pressure (Pa or psi)
• P_v = vapor pressure (Pa or psi)
• ρ = density
• V = characteristic velocity
• Low Ca indicates high cavitation potential

Fluid Property Relationships

Density-Pressure-Temperature Relationships

General State Equation: \[ \rho = f(P, T) \]

• Density is a function of pressure and temperature
• For liquids: weak function of P and T
• For gases: strong function of both P and T

Incompressible Fluid Assumption:

• ρ = constant (independent of P and T)
• Valid when: Ma < 0.3="" or="" δp/p=""><>
• Applies to most liquid flows and low-speed gas flows

Specific Heat Relationships

Specific Heat Ratio: \[ k = \frac{c_p}{c_v} \]

• k = specific heat ratio (dimensionless)
• c_p = specific heat at constant pressure (J/(kg·K) or Btu/(lbm·°R))
• c_v = specific heat at constant volume (J/(kg·K) or Btu/(lbm·°R))
• k ≈ 1.4 for air and diatomic gases
• k ≈ 1.67 for monatomic gases

Ideal Gas Specific Heat Relationship: \[ c_p - c_v = R \]

• R = specific gas constant

No-Slip Condition

No-Slip Boundary Condition:

• Fluid velocity at solid boundary equals velocity of boundary
• At stationary wall: V_fluid = 0
• At moving wall: V_fluid = V_wall
• Fundamental assumption in viscous fluid mechanics

Standard Atmospheric Properties

Standard Atmospheric Pressure at Sea Level:

• P_atm = 101,325 Pa = 101.325 kPa
• P_atm = 14.696 psia ≈ 14.7 psia
• P_atm = 29.92 in Hg
• P_atm = 760 mm Hg (torr)
• P_atm = 1.01325 bar

Standard Temperature:

• T = 15°C = 288.15 K
• T = 59°F = 518.67°R

Water Properties at Standard Conditions (4°C):

• ρ = 1000 kg/m³ = 62.4 lbm/ft³
• γ = 9810 N/m³ = 62.4 lbf/ft³

Air Properties at Standard Conditions (15°C, 101.325 kPa):

• ρ = 1.225 kg/m³ = 0.0765 lbm/ft³
• R = 287 J/(kg·K) = 53.35 ft·lbf/(lbm·°R)
• k = 1.4