Thursday, December 22, 2011

Dimensionless Numbers & their Significance


Nomenclature:


D = diameter of pipe
DH = Hydraulic diameter
L = Length of the pipe
Lch = characteristic length
R = Length through which conduction occurs.
u = mean characteristic velocity of the object relative to the fluid.
Vch = Characteristic velocity
Cp = specific heat capacity at constant pressure.
k = thermal conductivity
μ = dynamic viscosity of the fluid
{\rho}\, = density of fluid.
DAB = mass diffusivity
h = heat transfer coefficient.
g = acceleration due to earths gravity.
t = characteristic time
ν = Kinematic viscosity of fluid.
α = Thermal diffusivity
β = volumetric thermal expansion coefficient ( = 1/T for ideal fluids, T = absolute temperature)
Ts = surface temperature
T∞ = Bulk Temperature










Significance:

  • Ratio of Inertial forces to viscous forces.
  • Primarily used to analyse different flow regimes namely Laminar, Turbulent, or both.
  • When Viscous forces are dominant its a laminar flow & when Inertial forces are dominant it is a Turbulent flow.
=======X=======X=======X=======X=======X======X=======X=======X====





Significance:
  • Depends only on fluid & its properties. It is also ratio of velocity boundary layer to thermal boundary layer
  • Pr = small, implies that rate of thermal diffusion (heat) is more than the rate of momentum diffusion (velocity). 
  • Also the thickness of thermal boundary layer is much larger than the velocity boundary layer.
=======X=======X=======X=======X=======X======X=======X=======X====









Significance:
  • Analogous of Prandtl number in Heat Transfer.
  • Used in fluid flows in which there is simultaneous momentum & mass diffusion
  • It is also ratio of fluid boundary layer to mass transfer boundary layer thickness.
  • To find mass transfer coefficient using Sherwood number, we need Schmidt number. 
=======X=======X=======X=======X=======X======X=======X=======X====










Significance:
  • Ratio of thermal diffusivity to mass diffusivity.
  • Fluid flow with simultaneous Heat & mass transfer by convection.
  • It is also ratio of Schmidt number to Prandtl number
=======X=======X=======X=======X=======X======X=======X=======X====
















Significance:
  • Heat transported by convection to Heat transported by conduction.
  • Product of Re & Pr for Pe(HT) & product of Re & SC for Pe(MT)
=======X=======X=======X=======X=======X======X=======X=======X====





Significance:
  • It is the ratio of  heat transferred to the fluid to the heat transported by the fluid (ratio of Nusselt number to Peclet number)
  • Used to find heat transfer in forced convection flows.
  • St(HT) = Nu/(Re.Pr) & St(MT) = Sh/(Re.Sc)
=======X=======X=======X=======X=======X======X=======X=======X====
















Significance:
A) Sherwood Number:
  • Ratio of Convective to diffusive mass transport. Used in mass transfer operations.
  • Analogous of Nusselt number in Heat transfer OR Sherwood number is Nusselt number for mass transfer.
B) Nusselt Number
  • Ratio of convective to conductive heat transfer coefficient across the boundary layer.
  • Low Nu => conduction is more => Laminar flow 
  • High Nu => convection is more=> Turbulent flow.
  • It can also be viewed as conduction resistance to convection resistance of the material.
  • Free convection: Nu = f(Ra, Pr)
  • Forced Convection: Nu = f(Re, Pr)
=======X=======X=======X=======X=======X======X=======X=======X====









Significance:
  • Ratio of Buoyancy force to viscous force in natural convection.
  • Reynolds number is used in forced convection of  fluid flow, whereas Grashof number is used in natural convection.
=======X=======X=======X=======X=======X======X=======X=======X==== 









Significance:
    • used in unsteady state (transient) heat  transfer conditions.
    • ratio of  heat  transfer resistance inside the body to  heat  transfer resistance at the surface of the body. OR ratio of  internal thermal resistance to external thermal resistance .
    • Shows the variation of temperature inside the body w.r.t to time.
    • Bi < 0.1 => heat transfer resistance inside the body is very low => inside the body conduction takes place faster compared to convection at the surface. => no temperature gradient inside the body (uniformity in temperature) vice versa implies that Temperature is not uniform throughout hte material volume.
    =======X=======X=======X=======X=======X======X=======X=======X====














    Significance:
    • It shows the presence & strength of convection in a fluid body.
    • Heat transfer by Conduction within fluid < Critical value for that fluid < Heat transfer by convection. (consequences of Ra values)
    • Product of Gr.Pr
    =======X=======X=======X=======X=======X======X=======X=======X====













    Significance:

    • Characterizes laminar flow in a conduit OR transfer of heat by streamline fluid flow in a pipe
    • In case of  mass transfer, Pr is replaced by Sc.
    =======X=======X=======X=======X=======X======X=======X=======X====

















    Significance:

    • Ratio of rate of heat conduction to the rate of heat storage.
    • Used along with Biot number to solve transient state heat transfer problems.
    • For mass transfer by diffusion, Fourier number for MT is used.
    • It can also be understood as current time to the time taken to reach steady state.
    =======X=======X=======X=======X=======X======X=======X=======X====

    13 comments:

    Share it !!!