Dimensionless Numbers & their SignificanceNomenclature:

D = diameter of pipeDH = Hydraulic diameterL = Length of the pipeLch = characteristic lengthR = Length through which conduction occurs.u = mean characteristic velocity of the object relative to the fluid.Vch = Characteristic velocityCp = specific heat capacity at constant pressure.k = thermal conductivityμ = dynamic viscosity of the fluid

 = density of fluid.DAB = mass diffusivityh = heat transfer coefficient.g = acceleration due to earths gravity.t = characteristic timeν = Kinematic viscosity of fluid.α = Thermal diffusivityβ = volumetric thermal expansion coefficient ( = 1/T for ideal fluids, T = absolute temperature)Ts = surface temperatureT∞ = 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 it’s a laminar flow & when Inertial forces are dominant it is a Turbulent flow.

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.

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. 

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

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

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)

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)

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.

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.

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

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.

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.