Thursday, 10 May 2018

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
δ → Thermal boundary layer thickness



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