DIMENSIONAL ANALYSIS CLASS NOTES FOR ENGINEERS

DIMENSIONAL ANALYSIS

CLASS NOTES FOR

ENGINEERS

 Dimensional analysis is a tool to find

relations among physical quantities in

terms of dimensionless groups. It uses

their physical dimensions usually mass,

length, time, electric charge and temperature.

An equation must have the same dimensions

on the left and right sides (dimensional

homogeneous). Checking the dimensions

on the left and right of the equation is the

basic way of performing dimensional

analysis. It makes the study simple.

Improvements in design,

fabrication and operation

become easy.

BASICS

Dimensions

There are two types of dimensions.

((a) Primary Dimensions or Fundamental Dimensions

(i) Mass M

(ii) Length L

(iii) Time T

(iv)Temperature θ

(b) Secondary dimensions or derived dimensions

 Express quantities in terms of fundamental dimensions.

a. Geometric quantities: area, volume, moment of inertia

b. Kinematic quantities; u,v,w,ω, α,g,ν,ψ (stream function),

Γ(circulation) and Ω (vorticity)

c. Dynamic quantities:

F, ρ,μ,ζ (shear stress), w (Specific weight), p, mv,

W, E,P(power) and T(torque)

d. Thermodynamic quantities

T, p, h, s, u, k, HT, R

DIMENSIONAL HOMOGENEITY

There is an equation between dependent variable

and independent variables. Dimensional homogeneity

means that dimensions on left side is equal to the

dimensions on right hand side of the equation. It is

in terms of mass, length, time and temperature separately.

Fig. Dimensional Analysis

Dimensional analysis.

It gives a mathematical relation between various

variables of a process in dimensionless form.

 Various methods of dimensional analysis
Two methods
(a) Rayleigh’s method
(b) Buckingham’s Pi Method

 Rayleigh’s Method

It develops a relation between dependent

and independent variables in a  process.

It is on the basis of dimensional homogeneity.

The maximum total number of variables should

be five or less. It becomes very cumbersome

and complex for more variables.

1. 1. List all the independent variables that                                                                                                                                                 are likely to influence the dependent variable.

   2. Write the equation in the functional form.

     X is a dependent variable.

It depends upon independent variables X₁, X₂, X₃, …, X_n,

 the equation becomes

X = F(X₁, X₂, X₃, …, X_n).

1. Write the above equation in the exponential form

X = C X₁^a X₂^b X₃^c……Xn ^m   where C is a

dimensionless constant and a, b, c, m are

arbitrary dimensionless exponents.

1. 1. Express each variable in terms of fundamental units.

   2. By using same dimensions (dimensional homogeneity)                                                                                                                  on both sides of the equation.

   3. Obtain a number of simultaneous equations for                                                                                                                               various fundamental dimensions involving the exponents a, b, c, … m.

   4. Solve these equations to obtain the value of exponents a, b, c, m.

      v. Now put the values of exponents in the main equation.

      vi. Now make non- dimensional groups by arranging                                                                                                                   the variables on both sides of the equation.

1. NOTE: Rayleigh method is applicable for

 total 5 number of variables.

For variables more than five, number of equations are less than the unknown quantities.

Solution becomes indeterminable.

Buckingham π theorem

Buckingham π theorem is in commonly use.

It is an improvement over Rayleigh’s Method.

Rayleigh’s method does not specify the number

of dimensionless groups formed in the equation.

Buckingham theorem states

(i)  An equation has total n physical variables

(ii) express each of these in m  fundamental quantities.

(iii) The equation has p = n − m  dimensionless groups.

Simple and complex problems in fluid flow, heat transfer

and other processes are easily solvable by this method.

Let total variables be n = 7

Total number of dimensions involved                    m  = 4

Then number of dimensionless groups =n-m=7–4=3

Say π₁,  π₂  and  π₃

each π term contains (m+1) variables. These m repeating variables repeat in

each π term.

Salient features of the repeating variables

1. m repeating variables must contain all the fundamental dimensions involved in the equation.

2. Do not select a dependent variable as a repeating variable.

3. The repeating variables should not form a dimensionless group among themselves.

4. Repeating variables should not have the same dimensions.

5. Preference for choosing repeating variables are

(a) First geometric property like length, diameter, height

(b) Flow property like velocity, acceleration

(c) Fluid property like mass density, weight density, dynamic viscosity

(d) Combinations can be (l,V,ρ), (d,V,ρ), (l,V,μ) and (d,V,μ).

Finding various Pi (π) terms

Form each π term by combining m dimensional parameters with one of the remaining (n-m) variables. Each π term contains (m+1 ) variables. These m variables are repeating variables for each π term. These m variables chosen should not form a dimensionless parameter. Preferably repeating variables are
(i) Length, velocity and density
(ii) Diameter, velocity and density
(iii) Length, velocity and viscosity
(iv) Diameter, velocity and viscosity
Suppose variables are l,v,ρ,µ,k, g, & cp
N=7
M=4 L,M,T, and ϴ
No of π terms=n-m=7-4=3
According to π theorem
π1= f(π2, π3, π4)
OR
π2= f(π1, π3, π4)
OR
π3= f(π1, π2, π4)
OR
π4= f(π2, π3, π1)

  Utility of dimensional analysis.

(i) It gives a simplified theoretical solution of a flow problem by proper selection of variables in non dimensional form.
(ii) It helps in developing correlation for experimental data which help in the design of the process.

Advantages of dimensional analysis

(i) It gives a functional relationship between independent and dependent variables in dimensionless form.
(ii) Proper selection of repeating variables help in finding non-dimensionless parameters.
( iii) Draw Curves for the experimental data easily.
( iv) It gives a theoretical solution for complicated problem.

 Limitations of dimensional analysis?

(i) It does give any idea about the selection of repeating variables.
(ii) It does not complete information about the variables. It gives only how these are related.
( iii) It does give any physical explanation of the process or phenomenon.
( iv) It does give any information regarding the effect of one variable over other variables

 

 

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