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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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

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Significance:

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

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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)

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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)

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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.

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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.

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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

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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.

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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.

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