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Saturday, 9 January 2016

DIMENSIONLESS NUMBERS AND THEIR PHYSICAL SIGNIFICANCE

We were discussing the basic concept of streamline and equipotential linedimensional homogeneityBuckingham pi theoremdifference between model and prototype and basic principle of similitude i.e. types of similarity in the subject of fluid mechanics, in our recent posts. 

In order to secure the qualitative behavior of a physical problem, we will have to understand the concept of dimensional analysis. Dimension analysis plays a very important role in determining the fluid behavior in various engineering applications such as heat transfer, fluid mechanics, Geology, chemical engineering, aeronautical engineering etc. 

We will discuss here the various dimensionless numbers in this post and simultaneously we will also note here the importance of dimensionless number in various engineering applications as listed above. 

Let us see what is a dimensionless number?

Dimensionless numbers could also be expressed as non dimensional parameters because these are basically secured from ratio of one force to another force and therefore such dimensionless numbers will not have any unit. There are various important dimensionless numbers that we will discuss here one by one.

Reynolds number

Reynolds number of a flowing fluid could be defined as the ratio of inertia force and viscous or friction force. We will be able to determine the type of flow i.e. laminar flow, transient flow or turbulent flow on the basis of Reynolds number.
  1. Fluid flow will be laminar, if Re < 2300
  2. Fluid flow will be transient, if Reynolds number is in between 2300 and 4000
  3. Fluid flow will be Turbulent, if Re > 4000
Reynolds number will be expressed by a symbol Re and will be defined mathematically as mentioned here.
Re = Inertia force / Viscous force
Reynolds Number (Re) = ρ V L / μ

Where,
ρ is density of flowing fluid, V is velocity , L is characteristic length and μ is dynamic viscosity of fluid.

If fluid is flowing through a pipe or duct, Characteristics length L will be replaced by hydraulic diameter D. Therefore Reynolds number for a flowing fluid through a pipe could be expressed as mentioned her

Reynolds Number (Re) = ρ V D / μ

Froude number

Froude number of a flowing fluid is also a dimensionless number which is basically defined as the square root of ratio of the inertia force to gravity force and it is used in engineering applications in order to express the influence of gravity over motion of fluid. 

Froude number will be expressed by a symbol Fr and will be defined mathematically as mentioned here.
Fr = [Inertia force / gravity force] 1/2
Fr = V/ (L g) 1/2

Where,
V is velocity, L is characteristic length and g is acceleration due to gravity
Let us see the three cases of Froude number as mentioned here
  1. Fr<1, Subcritical flow
  2. Fr>1, Supercritical flow
  3. Fr= 1, Critical flow

Euler number

Euler number of a flowing fluid is also a dimensionless number which is basically defined as the ratio of pressure forces to inertia forces and Euler number is quite important in determining the fluid flow dynamic problem where difference of pressure between two points are very important.

Euler number will be expressed by a symbol EU and will be defined as mentioned here
EU = P/ ρV2

Where,
V is velocity of fluid flow, P is characteristic pressure and ρ is density of fluid
Generally, Pressure difference between two points are used in determining the Euler number, therefore we can express the above equation of Euler number in term of following pattern
EU = ΔP/ ρV2

Weber number

Weber number of a flowing fluid is also a dimensionless number which indicates the ratio of inertia force to surface tension force. Weber number is quite important in determining the dominant energy between kinetic energy and surface tension energy. Weber number could also be used successfully in determining the thin film problems and also in problems of formation of droplets.

Weber number will be expressed as mentioned here
We = ρV2L/ σ

Where,
V is velocity of fluid flow, L is characteristic length, σ is surface tension and ρ is density of fluid.

Mach Numbers

Mach number is also a very important dimensionless number which is widely used in fluid flow dynamic problems where compressibility plays a very important role. Mach number of flowing fluid will be defined as the square root of ratio of the inertia force to elastic force and we can write it as mentioned here.

M = [Inertia force / Elastic force] 1/2
M = V/ (K/ρ) 1/2

In some higher speed fluid flow problems, density will be dependent over pressure and hence effect of compressibility will be quite important in such fluid flow dynamic problems. 

Mach number, in such fluid flow dynamic problems, will be used to determine the type of flow i.e. compressible or incompressible flow and will be expressed as mentioned here
M= V/C
C= (K/ρ) 1/2

Where,
V is velocity of fluid flow, C is speed of sound and K is elastic stress

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