Dimensionless Numbers in Fluid Thermal Engineering

Important dimensionless numbers used in fluid mechanics and heat transfer are given below:-

Dimensionless Number	Expression	Significance
Reynolds Number	$Re =\frac{ρUD}{μ}$	Re = Inertial force / Viscous force Determines flow is Laminar, Turbulent, or Transient Flow.
Fourier Number	$Fo = \frac{αt}{L^2}$	It is a measure of heat conducted through a body relative to heat stored. Larger the Fo, faster propagation of heat through a body It can also be viewed as current time to the time taken to reach steady state
Biot Number	$Bi = \frac{hL}{K_s} = \frac{L/K_s}{1/h}$	Ratio of Conductive resistance with in the body to Convection resistance at the surface of the body Bi ≤ 0.1, Lumped system analysis (assumes a uniform temperature distribution throughout the body) is applicable
Nusselt Number	$Nu = \frac{hL}{K_f} = \frac{h ΔT}{K_fΔT/L} = \frac{q_{conv}}{q_{cond}}$	Ratio of convective HT to conductive HT coefficient across the boundary layer  The larger the Nu, the more effective the convection.  Used to calculate heat transfer coefficient h
Prandtl Number	$Pr = \frac{ν}{α}= \frac{µC_p}{K}$	Ratio of momentum diffusivity to thermal diffusivity Determines ratio of fluid/thermal Boundary layer thickness ${Pr}^{1/3} = \frac{δ_f}{δ_t}$
Grashof Number	$Gr = \frac{gβΔTL^3}{ν^2}$	Ratio of natural convection buoyancy force to viscous force $\frac{Gr}{Re^2} << 1  \implies$ forced convection $\frac{Gr}{Re^2} >> 1  \implies$ Natural convection $\frac{Gr}{Re^2} ≈1  \implies$ mixed convection
Peclet Number	$Pe = Re*Pr = \frac{UL}{α}$	Ratio of convective to diffusive heat transport in a fluid Used to determine plug flow/perfect mixing (CSTR) continuous flow model validity
Stanton Number	$St = \frac{h}{ρUC_p} = \frac{Nu}{Pe} = \frac{Nu}{RePr}$	Ratio of heat transferred to the fluid to the heat capacity of the fluid. Used to characterize heat transfer in forced convection flows.
Rayleigh Number	$Ra = Gr*Pr = \frac{gβΔTL^3}{αν}$	It is product of Gr and Pr. Determines natural convection boundary layer is laminar or turbulent.
Jakob Number	$Ja = \frac{C_p(T_s - T_{sat})}{h_{fg}}$	Ratio of sensible heat to latent heat absorbed (or released) during the phase change process.
Bond Number	$Bo = \frac{gL^2Δρ}{σ}$	Ratio of gravitational force to surface tension force Used to characterize the shape of bubble or drops moving in a surrounding fluid.
Froude Number	$Fr = \frac{U^2}{gL}$	Ratio of Inertia force to Gravitational force Often the term Froude number is used for the ratio $\frac{u}{\sqrt{gL}}$.
Euler Number	$Eu = \frac{Δp}{ρU^2}$	Ratio of pressure force to inertia force. Used for analyzing fluid flow dynamics problems in which the pressure difference, are interest
Weber number	$We = \frac{ρU^2L}{σ}$	Ratio of Inertia force to surface tension force. Used for analyzing fluid flow dynamics problems in which surface tension is important

Nomenclature:

μ → viscosity of fluid

ν → kinematic viscosity of fluid 

ρ → density of fluid

U → characteristic velocity scale

D → characteristic Diameter = 4A/P

α → thermal diffusivity of fluid 

t  → time

L → characteristic length scale 

Cp → specific heat at constant pressure

h → heat transfer coefficient

K → thermal conductivity of fluid

Kf → thermal conductivity of fluid

Ks → thermal conductivity of solid 

g → gravitational acceleration 

β → volumetric thermal expansion coefficient 

ΔT → characteristic temperature difference 

Tsat → saturation temperature

Ts → surface temperature 

hfg → latent heat of condensation

Δρ → difference in density of the two phases 

σ   → surface tension

Δp → characteristic pressure difference of flow
δf  → Fluid boundary layer thickness
δt  → Thermal boundary layer thickness

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