Sunday, December 3, 2023

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=ρUDμ 
  • Re = Inertial force / Viscous force
  • Determines flow is Laminar, Turbulent, or Transient Flow.



Fourier Number




Fo=αtL2
  • 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=hLKs=L/Ks1/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=hLKf=hΔTKfΔT/L=qconvqcond

  • 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=να=µCpK
  • Ratio of momentum diffusivity to thermal diffusivity
  • Determines ratio of fluid/thermal Boundary layer thickness
  • Pr1/3=δfδt



Grashof Number




Gr=gβΔTL3ν2
  • Ratio of natural convection buoyancy force to viscous force
  • GrRe2<<1 forced convection
  • GrRe2>>1 Natural convection
  • GrRe21 mixed convection

Peclet Number


Pe=RePr=ULα
  • 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=hρUCp=NuPe=NuRePr

  • 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=GrPr=gβΔTL3αν
    • It is product of Gr and Pr.
    • Determines natural convection boundary layer is laminar or turbulent.

      Jakob Number


      Ja=Cp(TsTsat)hfg
      • Ratio of sensible heat to latent heat absorbed (or released) during the phase change process.


      Bond Number


      Bo=gL2Δρσ
      • 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=U2gL
        • Ratio of Inertia force to Gravitational force
        • Often the term Froude number is used for the ratio ugL.

        Euler Number

        Eu=ΔpρU2
        • Ratio of pressure force to inertia force.
        • Used for analyzing fluid flow dynamics problems in which the pressure difference, are interest

          Weber number


          We=ρU2Lσ
          • 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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