Dimensionless Numbers & their Significance

Nomenclature:


D = diameter of pipe
D_{H =} 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
 = 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.

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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.

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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. 

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