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.


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

Write the equation in the functional form.

X is a dependent variable.
It depends upon independent variables X_{1}, X_{2}, X_{3}, …, X_{n},
the equation becomes
X = F(X_{1}, X_{2}, X_{3}, …, X_{n}).

Write the above equation in the exponential form
X = C X_{1}^{a }X_{2}^{b} X_{3}^{c}……Xn ^{m }where C is a
dimensionless constant and a, b, c, m are
arbitrary dimensionless exponents.


Express each variable in terms of fundamental units.

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

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

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.

 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 =nm=7–4=3
Say π_{1}, π_{2} and π_{3}
each π term contains (m+1) variables. These m repeating variables repeat in
each π term.
Salient features of the repeating variables

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

Do not select a dependent variable as a repeating variable.

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

Repeating variables should not have the same dimensions.

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 (nm) 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=nm=74=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 nondimensionless 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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