DIMENSIONLESS NUMBERS

CLASS NOTES FOR ENGINEERS  

 Convention equations in terms of

non-dimensionless numbers reduce

the number of variables. This makes the

equations convenient and easy to handle.

It reduces the time and level of difficulty.

Dimensionless numbers do not have

any units. Use Reynolds number

to know laminar, turbulent or transition

flow. In every dimensionless number,

Numerator is always inertia force and

Denominator is some other force.

NUSSELT NUMBER

Nusselt number describes the mechanism of convection. Convective heat transfer is greater than conduction across the same fluid stationary layer.

 Nu=Convective Heat Flux in a moving fluid/conductive Heat Flux in the same stationary fluid

Nu= hl/k_f

The higher the Nu number, the more is the HT by convection. Nusselt number is higher for a turbulent flow than a laminar flow.

REYNOLD NUMBER

It describes the type of flow/laminar/transition/turbulent.

Re=Inertia force / viscous force

Procedure to find Reynolds number for a plate and pipe is

Plate       Re=ρvl/µ

Pipe        Re= ρvd/ µ

(i) When Re is low, viscous forces are more significant and the flow is laminar.

(ii)As Re increases, inertia forces are more predominant and the flow is turbulent.

CRITICAL REYNOLDS NUMBER

Reynolds number where the laminar region ends. Its value is 5 x 10⁵for a horizontal plate. Otherwise its value is different for different geometries.

GRASHOFF’S NUMBER

Gr= Buoyant force x inertia force/ (viscous force)²

Gr= L³g β ΔT/ν²

Where

L is the length of the plate

g is acceleration due to gravity

β = 1/T_av

ΔT  is the temperature difference

ν is the kinematic viscosity

PRANDTL NUMBER

Pr= momentum diffusivity/Thermal diffusivity

Pr =µc_p/k_f

Take Pr=0.7 for gases if not given

Take Pr = 10 for water if not given

Data on Prandtl  number

Fluid                                                      Range of Pr Number

Liquid metals                                        0.004 to 0.030

Gases                                                     0.7 to 1.0

Water                                                    1.7 to 13.7

Light organic fluids                              5 to 50

Oils                                                        50 to 100000

Glycerin                                               2000 to 100000

Prandtl number best describes the relation between velocity boundary layer thicknesses to thermal boundary layer thickness. (δ_th=δ Pr ^-1/3). When Pr=1 means rate of momentum transfer and heat transfer are same.

 

RAYLEIGH NUMBER

Ra=Re Pr

BIOT NUMBER

Bi= convective HT in moving fluid/conductive HT in a solid surface

Bi = hl/k_s

Use of Biot number exists in the study of

(i)                1- D unsteady conductive HT

(ii)              In fins of varying Cross section

FOURIER NUMBER

Fo= αt/L_c²

Where    L_c is characteristic length = surface area/ volume

Firstly L_c = δ/2 for a plane wall

Secondly L_c= r/4 for a cylinder

Thirdly L_c= r₀/3 for a sphere

Fourthly L_c= a/6 for a cube

Use of Fourier number exists in the analysis of

(i) 1- D unsteady state conduction

(ii) In fins of varying cross section

STANTON NUMBER

St=Nu/ (Re Pr)

 Reynolds, Colburn  and Prandtl Analogies use Stanton number in convection.

FROUDE NUMBER

The Froude Number is a dimensionless number. It is the ratio of inertia force on an element of fluid to the weight of the fluid element. Mathematically

Fr= (Inertia force/gravitational force)^0.5  

Fr=v/ (gl)^0.5

^{WEBER NUMBER}

W_b= Inertia force/ Surface tension

W_b =ρv²l/σ

EULER NUMBER

The Euler Number (Eu) is a dimensionless number. It gives a relationship between a local pressure drop & the kinetic energy of flow per unit volume. Eu is zero for a perfect frictionless flow. It gives the energy losses in the flow.

Eu = pressure force/inertia force

Eu=ρv²/p

PHYSICAL SIGNIFICANCE OF PRANDTL NUMBER

 Prandtl number signifies the thickness of thermal boundary layer.  It also connects with the thickness of hydro boundary layer.

firstly if  Pr =1, δ =δ_th

Secondly if Pr <1 then δ < δ_th

Thirdly if Pr >1 then δ > δ_th

 PHYSICAL SIGNIFICANCE OF ‘GRASHOFF’S NUMBER

Grashoff’s number (Gr) is the ratio of the buoyancy to viscous force acting on a fluid layer.  Used in free convection. Its value indicates whether the Buoyancy force is more predominant or  the viscous force. Used where Buoyancy force is more predominant. Grashoff’s number is similar to to the Reynolds number in forced convection. Used in fluid dynamics and heat transfer.

 

Gr = βg Δt l³/ ϑ²

Buoyancy force is due to density difference. The temperature difference causes the density difference. Greater is the temperature difference, greater will be density difference. Greater will be the Grashoff’s number.

Physical Significance of Nusselt number

More is the Nusselt number, more is the heat transfer by convection than conduction. Value of Nusselt number  as one represents a case of heat transfer by pure conduction. Nusselt number increases as velocity of flow increases. Nusselt number for a turbulent flow is much higher than that for a laminar flow.

Physical Significance of Reynolds Number

Reynolds number classify the type of flow as laminar, transition and turbulent. It gives the effect of viscosity on velocity of flow. More is the viscosity, lesser is the velocity of flow and vice-versa.

Physical Significance of Rayleigh’s number

This depends on the Buoyancy driven flow. It is equal to the product of Grashoff’s number and Prandtl number. This governs free or natural convection. Its value is large, normally 10⁶ to 10⁸. Rayleigh critical value is 1708. Below this value, there is no flow and hence no convection.

Physical significance of Biot Number

It is the ratio of internal resistance to external resistance. It means for small value of Biot number, internal resistance is less. The temperature gradient is less.

Develop the model of a prototype with the help of these numbers. A prototype means an actual object. A prototype can be a big object like an airplane, a dam and a turbine. A model in case of a big object is  small. Make the model and test it for geometric, kinematic and dynamic similarity in terms of dimensionless numbers. These numbers help to achieve predicted performance for the actual prototype. Then make the prototype and put to practical use. Hence a model can be smaller or bigger than the actual object (Prototype).

TABLE: Dimensionless numbers

Sr. No.	Dimensionless number	Definition	Symbol	Formula	Practical application
1.	Reynolds number	Inertia force/viscous force	Re	ρ  v L / µ	Flow of viscous fluids in pipes and over a flat plate
2.	Froude number	(Inertia force/Gravity force)0.5	Fr	v/(Lg)0.5	Flow in open channels
3.	Euler number	(Inertia force/pressure force)0.5	Eu	v/(P/ρ)0.5	Flow in pipes/ tubes
4.	Weber number	(Inertia force/Surface tension force)0.5	We	v/(σ/ρL)0.5	Flow through capillary tube
5.	Mach number	(Inertia force/Elastic force)0.5	M	(v/(σ/ρ))0.5	High speed flow of fluids
6.	Schmidt number	momentum diffusivity/mass diffusivity	Sc	μ / ρD	Schmidt number is analogous to Prandtl number

Every dimensionless number in fluid flow is the ratio of inertia force to some other  force as described below. There are in all six forces. Accordingly, there are five dimensionless numbers in fluid mechanics.

1. Reynolds number
It is a ratio of inertia force to viscous force.
Inertia force = mass x acceleration

                     = ρ Vol v/t =  ρ (Vol/t) v

                     = ρ A v v= ρ A v²
Viscous force = Shear stress x area

                       = τ x A = µdu/dy A = µ (v/L) A
Re =  ρA v² / µ (v/L) A = ρ v L / µ

Reynolds number signifies that viscous forces play a significant role or the fluid is viscous.

It analyzes different types of flow namely Laminar, Turbulent, or Transition.
(i) When Viscous forces are dominant, it is a laminar flow

(ii) If Inertial forces are dominant, it is a Turbulent flow.
Practical applications of viscous flow are
(a) Low velocity fluid flow around the airplanes and automobiles
(b) Motion of fluid around a completely submerged submarine
(c) Flow in a high speed centrifugal compressor
(d) In-compressible fluid flow in a small diameter pipe (capillary tube)

2. Froude number

It is square root of the ratio of inertia force to gravity force.
Fr =(Fi / Fg) ^0.5 = ( ρA v²/m g)^0.5 = v/ (Lg)^0.5

Froude number is applicable where gravity forces play a significant role.

Practical applications where gravity force play a significant role are

(a)  Spillway of a dam
(b) Notches and weirs
(c) Fluid flow in open channels
(d) Motion of a fluid around a ship in the ocean

3. Euler number

It is square root of ratio of inertia force to pressure force.
Eu =(Fi / Fp )^0.5 = (ρA v²/p A)^0.5 = v/(P/ρ)^0.5
It is applicable where pressure forces are predominant.

Such practical applications are
(a) Flow through pipes, orifices, mouthpieces and sluice gates
(b) Water hammer phenomenon in a pen-stock
(c) Pressure rise due to sudden closure of valves

4. Weber number

It is the square root of the ratio of inertia force to surface tension force.
We = (Fi / Fs)^0.5 = (( ρA v²/σ L)^0.5 = v/(σ/ρL)^0.5
Its practical applications are
(a) Flow in capillary tube
(b)  Blood flow in veins

5. Mach number

Square root of ratio of inertia force to the elastic force.

M = (Fi /Fe)^0.5 = ((ρA v²/σ x A)^0.5 = v/( σ/ρ)^0.5

σ/ρ = velocity of sound in the fluid

Practical applications for elastic force are

(a) Compressible fluid flow problems at high velocities
(b) High velocity fluid flow in pipes
(c) Fluid motion across high speed projectiles and missiles

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