Short Notes: Dimensional Analysis | Short Notes for Mechanical Engineering PDF Download

 Page 1


Dimensional Analysis  
Dimensional analysis is a mathematical technique which makes use of the study of 
dimensions as an aid to the solution of several engineering problems. It deals with 
the dimensions of the physical quantities is measured by comparison, which is 
made with respect to an arbitrarily fixed value.
Length L, mass M and Time T are three fixed dimensions which are of importance 
in fluid mechanics. If in any problem of fluid mechanics, heat is involved then the 
temperature is also taken as fixed dimension. These fixed dimensions are called 
fundamental dimensions or fundamental quantity.
Secondary or Derived quantities are those quantities which possess more than one 
fundamental dimension. For example, velocity is defined by distance per unit time 
(L/T), density by mass per unit volume (M/L3) and acceleration by distance per 
second square (L/T2). Then the velocity, density and acceleration become as 
secondary or derived quantities. The expressions (L/T), (M/L3) and (L/T2) are 
called the dimensions of velocity, density and acceleration respectively
Dimensional Analysis
Page 2


Dimensional Analysis  
Dimensional analysis is a mathematical technique which makes use of the study of 
dimensions as an aid to the solution of several engineering problems. It deals with 
the dimensions of the physical quantities is measured by comparison, which is 
made with respect to an arbitrarily fixed value.
Length L, mass M and Time T are three fixed dimensions which are of importance 
in fluid mechanics. If in any problem of fluid mechanics, heat is involved then the 
temperature is also taken as fixed dimension. These fixed dimensions are called 
fundamental dimensions or fundamental quantity.
Secondary or Derived quantities are those quantities which possess more than one 
fundamental dimension. For example, velocity is defined by distance per unit time 
(L/T), density by mass per unit volume (M/L3) and acceleration by distance per 
second square (L/T2). Then the velocity, density and acceleration become as 
secondary or derived quantities. The expressions (L/T), (M/L3) and (L/T2) are 
called the dimensions of velocity, density and acceleration respectively
Dimensional Analysis
Q u a n tity S ym bol D im e n sio n s
M ass m M
Length i L
lime t T
Temperature T e
Velocity u L T -1
Acceleration a L T " 2
Momentum/lmpulse m v M L T "1
F o rce F M L T " 2
E nergy - W ork W M L2 T2
Power P M L2 T3
Moment of F o rce M ml2 t2
Angular momentum - M L2 Tl
Angle
n
M 'W
Angular Velocity Q ) T1
Angular acceleration a T 1
Area A L 2
Volum e V L 3
F irst Moment of Area Ar L 3
S econd Moment of Area 1 L 1
D ensity
P
ML3
S pecific heat-Constant P ressure C p L 2r 2e -1
E la stic M odulus E M L_ 1 T2
F lexural R igidity E l M L3 T2
S hear M odulus G M L_ 1 T2
Torsional rigidity G J M L3 T2
Stiffness k m t2
Angular stiffness
T/n
ml2 t2
Flexibiity l/k M_ 1 T 2
Vorticity - r 1
C irculation - l 2 t ‘
V iscosity
P
ML^T1
Kinem atic V iscosity T L 2 T1
Diffusivity - L 2 Tl
Friction coefficient
f/p
M 'W
Restitution coefficient M 'W
S pecific heat- 
Constant volum e
C ,r L 2r 2 01
Dimensional homogeneity
• Dimensional homogeneity means the dimensions of each term in an equation 
on both sides are equal. Thus if the dimensions of each term on both sides of 
an equation are the same the equation is known as the dimensionally 
homogeneous equation. •
• The powers of fundamental dimensions i.e., L, M, T on both sides of the 
equation will be identical for a dimensionally homogeneous equation. Such 
equations are independent of the system of units.
• Let us consider the equation V = u + at 
Dimensions of L.H.S = V= L/T = L T -1 
Dimensions of R.H.S = LT1 + (LT2 ) (T)
= LT'1 + LT'1
Page 3


Dimensional Analysis  
Dimensional analysis is a mathematical technique which makes use of the study of 
dimensions as an aid to the solution of several engineering problems. It deals with 
the dimensions of the physical quantities is measured by comparison, which is 
made with respect to an arbitrarily fixed value.
Length L, mass M and Time T are three fixed dimensions which are of importance 
in fluid mechanics. If in any problem of fluid mechanics, heat is involved then the 
temperature is also taken as fixed dimension. These fixed dimensions are called 
fundamental dimensions or fundamental quantity.
Secondary or Derived quantities are those quantities which possess more than one 
fundamental dimension. For example, velocity is defined by distance per unit time 
(L/T), density by mass per unit volume (M/L3) and acceleration by distance per 
second square (L/T2). Then the velocity, density and acceleration become as 
secondary or derived quantities. The expressions (L/T), (M/L3) and (L/T2) are 
called the dimensions of velocity, density and acceleration respectively
Dimensional Analysis
Q u a n tity S ym bol D im e n sio n s
M ass m M
Length i L
lime t T
Temperature T e
Velocity u L T -1
Acceleration a L T " 2
Momentum/lmpulse m v M L T "1
F o rce F M L T " 2
E nergy - W ork W M L2 T2
Power P M L2 T3
Moment of F o rce M ml2 t2
Angular momentum - M L2 Tl
Angle
n
M 'W
Angular Velocity Q ) T1
Angular acceleration a T 1
Area A L 2
Volum e V L 3
F irst Moment of Area Ar L 3
S econd Moment of Area 1 L 1
D ensity
P
ML3
S pecific heat-Constant P ressure C p L 2r 2e -1
E la stic M odulus E M L_ 1 T2
F lexural R igidity E l M L3 T2
S hear M odulus G M L_ 1 T2
Torsional rigidity G J M L3 T2
Stiffness k m t2
Angular stiffness
T/n
ml2 t2
Flexibiity l/k M_ 1 T 2
Vorticity - r 1
C irculation - l 2 t ‘
V iscosity
P
ML^T1
Kinem atic V iscosity T L 2 T1
Diffusivity - L 2 Tl
Friction coefficient
f/p
M 'W
Restitution coefficient M 'W
S pecific heat- 
Constant volum e
C ,r L 2r 2 01
Dimensional homogeneity
• Dimensional homogeneity means the dimensions of each term in an equation 
on both sides are equal. Thus if the dimensions of each term on both sides of 
an equation are the same the equation is known as the dimensionally 
homogeneous equation. •
• The powers of fundamental dimensions i.e., L, M, T on both sides of the 
equation will be identical for a dimensionally homogeneous equation. Such 
equations are independent of the system of units.
• Let us consider the equation V = u + at 
Dimensions of L.H.S = V= L/T = L T -1 
Dimensions of R.H.S = LT1 + (LT2 ) (T)
= LT'1 + LT'1
= LT"1
Dimensions of L.H.S = Dimensions of R.H.S = LT"1 
Therefore, equation V = u + at is dimensionally homogeneous 
Uses of Dimensional Analysis
• It is used to test the dimensional homogeneity of any derived equation. 2. It is 
used to derive
• It is used to derive the equation.
• Dimensional analysis helps in planning model tests.
Methods of Dimensional Analysis
• If the number of variables involved in a physical phenomenon is known, then 
the relationship among the variables can be determined by the following two 
methods.
Rayleigh''s Method
• RayleigfT's method of analysis is adopted when a number of parameters or 
variables is less (3 or 4 or 5).
• If the number of independent variables becomes more than four, then it is very 
difficult to find the expression for the dependent variable
Buckingham''s (11- theorem) Method
• If there are n - variables in a physical phenomenon and those n-variables 
contains'm' dimensions, then the variables can be arranged into (n-m) 
dimensionless groups called 1 1 terms.
• If f (XI, X2, X3,...... Xn) = 0 and variables can be expressed using m
dimensions then, f (111, 112,113,...... nn - m) = 0 Where, 1 1 1 , 112,113,........are
dimensionless groups.
• Each n term contains (m + 1) variables out of which m are of repeating type 
and one is of non-repeating type.
• Each n term being dimensionless, the dimensional homogeneity can be used 
to get each 1 1 term.
Method of Selecting Repeating Variables
• Avoid taking the quantity required as the repeating variable.
• Repeating variables put together should not form a dimensionless group.
• No two repeating variables should have same dimensions.
• Repeating variables can be selected from each of the following properties
° Geometric property - Length, Height, Width, Area 
° Flow property - Velocity, Acceleration, Discharge 
° Fluid property - Mass Density, Viscosity, Surface Tension
Model Studies
• Before constructing or manufacturing hydraulics structures or hydraulics 
machines tests are performed on their models to obtain desired information
Page 4


Dimensional Analysis  
Dimensional analysis is a mathematical technique which makes use of the study of 
dimensions as an aid to the solution of several engineering problems. It deals with 
the dimensions of the physical quantities is measured by comparison, which is 
made with respect to an arbitrarily fixed value.
Length L, mass M and Time T are three fixed dimensions which are of importance 
in fluid mechanics. If in any problem of fluid mechanics, heat is involved then the 
temperature is also taken as fixed dimension. These fixed dimensions are called 
fundamental dimensions or fundamental quantity.
Secondary or Derived quantities are those quantities which possess more than one 
fundamental dimension. For example, velocity is defined by distance per unit time 
(L/T), density by mass per unit volume (M/L3) and acceleration by distance per 
second square (L/T2). Then the velocity, density and acceleration become as 
secondary or derived quantities. The expressions (L/T), (M/L3) and (L/T2) are 
called the dimensions of velocity, density and acceleration respectively
Dimensional Analysis
Q u a n tity S ym bol D im e n sio n s
M ass m M
Length i L
lime t T
Temperature T e
Velocity u L T -1
Acceleration a L T " 2
Momentum/lmpulse m v M L T "1
F o rce F M L T " 2
E nergy - W ork W M L2 T2
Power P M L2 T3
Moment of F o rce M ml2 t2
Angular momentum - M L2 Tl
Angle
n
M 'W
Angular Velocity Q ) T1
Angular acceleration a T 1
Area A L 2
Volum e V L 3
F irst Moment of Area Ar L 3
S econd Moment of Area 1 L 1
D ensity
P
ML3
S pecific heat-Constant P ressure C p L 2r 2e -1
E la stic M odulus E M L_ 1 T2
F lexural R igidity E l M L3 T2
S hear M odulus G M L_ 1 T2
Torsional rigidity G J M L3 T2
Stiffness k m t2
Angular stiffness
T/n
ml2 t2
Flexibiity l/k M_ 1 T 2
Vorticity - r 1
C irculation - l 2 t ‘
V iscosity
P
ML^T1
Kinem atic V iscosity T L 2 T1
Diffusivity - L 2 Tl
Friction coefficient
f/p
M 'W
Restitution coefficient M 'W
S pecific heat- 
Constant volum e
C ,r L 2r 2 01
Dimensional homogeneity
• Dimensional homogeneity means the dimensions of each term in an equation 
on both sides are equal. Thus if the dimensions of each term on both sides of 
an equation are the same the equation is known as the dimensionally 
homogeneous equation. •
• The powers of fundamental dimensions i.e., L, M, T on both sides of the 
equation will be identical for a dimensionally homogeneous equation. Such 
equations are independent of the system of units.
• Let us consider the equation V = u + at 
Dimensions of L.H.S = V= L/T = L T -1 
Dimensions of R.H.S = LT1 + (LT2 ) (T)
= LT'1 + LT'1
= LT"1
Dimensions of L.H.S = Dimensions of R.H.S = LT"1 
Therefore, equation V = u + at is dimensionally homogeneous 
Uses of Dimensional Analysis
• It is used to test the dimensional homogeneity of any derived equation. 2. It is 
used to derive
• It is used to derive the equation.
• Dimensional analysis helps in planning model tests.
Methods of Dimensional Analysis
• If the number of variables involved in a physical phenomenon is known, then 
the relationship among the variables can be determined by the following two 
methods.
Rayleigh''s Method
• RayleigfT's method of analysis is adopted when a number of parameters or 
variables is less (3 or 4 or 5).
• If the number of independent variables becomes more than four, then it is very 
difficult to find the expression for the dependent variable
Buckingham''s (11- theorem) Method
• If there are n - variables in a physical phenomenon and those n-variables 
contains'm' dimensions, then the variables can be arranged into (n-m) 
dimensionless groups called 1 1 terms.
• If f (XI, X2, X3,...... Xn) = 0 and variables can be expressed using m
dimensions then, f (111, 112,113,...... nn - m) = 0 Where, 1 1 1 , 112,113,........are
dimensionless groups.
• Each n term contains (m + 1) variables out of which m are of repeating type 
and one is of non-repeating type.
• Each n term being dimensionless, the dimensional homogeneity can be used 
to get each 1 1 term.
Method of Selecting Repeating Variables
• Avoid taking the quantity required as the repeating variable.
• Repeating variables put together should not form a dimensionless group.
• No two repeating variables should have same dimensions.
• Repeating variables can be selected from each of the following properties
° Geometric property - Length, Height, Width, Area 
° Flow property - Velocity, Acceleration, Discharge 
° Fluid property - Mass Density, Viscosity, Surface Tension
Model Studies
• Before constructing or manufacturing hydraulics structures or hydraulics 
machines tests are performed on their models to obtain desired information
about their performance.
• Models are a small scale replica of actual structure or machine.
• The actual structure is called prototype.
Similitude
• It is defined as the similarity between the prototype and its model. It is also 
known as similarity. There three types of similarities and they are as follows.?
c
Geometric similarity
• Geometric similarity is said to exist between the model and prototype if the 
ratio of corresponding linear dimensions between model and prototype are 
equal, i.e.
L V _ h P ^
h m H m
where Lr is known as scale ratio or linear ratio.
Kinematic Similarity
• Kinematic similarity exists between prototype and model if quantities such at 
velocity and acceleration at corresponding points on model and prototype are 
same.
(vjp = (Vi)p _(yt)p
(Vi)m (V 2)m (V 3)m ............ '
Where Vr is known as velocity ratio 
Dynamic Similarity
• Dynamic similarity is said to exist between model and prototype if the ratio of 
forces at corresponding points of model and prototype is constant.
( f j p _ (FX)p _ ( Fl) p p
(Fi)m ( ,Fz)m. (pa)m
Where Fr is known as force ratio.
Dimensionless Numbers
Following dimensionless numbers are used in fluid mechanics.
• Reynolds'^ number
• Froude"s number
• Euler"s number
• Weber"s number
• Mach number
Reynold's number
It is defined as the ratio of inertia force of the fluid to viscous force.
Page 5


Dimensional Analysis  
Dimensional analysis is a mathematical technique which makes use of the study of 
dimensions as an aid to the solution of several engineering problems. It deals with 
the dimensions of the physical quantities is measured by comparison, which is 
made with respect to an arbitrarily fixed value.
Length L, mass M and Time T are three fixed dimensions which are of importance 
in fluid mechanics. If in any problem of fluid mechanics, heat is involved then the 
temperature is also taken as fixed dimension. These fixed dimensions are called 
fundamental dimensions or fundamental quantity.
Secondary or Derived quantities are those quantities which possess more than one 
fundamental dimension. For example, velocity is defined by distance per unit time 
(L/T), density by mass per unit volume (M/L3) and acceleration by distance per 
second square (L/T2). Then the velocity, density and acceleration become as 
secondary or derived quantities. The expressions (L/T), (M/L3) and (L/T2) are 
called the dimensions of velocity, density and acceleration respectively
Dimensional Analysis
Q u a n tity S ym bol D im e n sio n s
M ass m M
Length i L
lime t T
Temperature T e
Velocity u L T -1
Acceleration a L T " 2
Momentum/lmpulse m v M L T "1
F o rce F M L T " 2
E nergy - W ork W M L2 T2
Power P M L2 T3
Moment of F o rce M ml2 t2
Angular momentum - M L2 Tl
Angle
n
M 'W
Angular Velocity Q ) T1
Angular acceleration a T 1
Area A L 2
Volum e V L 3
F irst Moment of Area Ar L 3
S econd Moment of Area 1 L 1
D ensity
P
ML3
S pecific heat-Constant P ressure C p L 2r 2e -1
E la stic M odulus E M L_ 1 T2
F lexural R igidity E l M L3 T2
S hear M odulus G M L_ 1 T2
Torsional rigidity G J M L3 T2
Stiffness k m t2
Angular stiffness
T/n
ml2 t2
Flexibiity l/k M_ 1 T 2
Vorticity - r 1
C irculation - l 2 t ‘
V iscosity
P
ML^T1
Kinem atic V iscosity T L 2 T1
Diffusivity - L 2 Tl
Friction coefficient
f/p
M 'W
Restitution coefficient M 'W
S pecific heat- 
Constant volum e
C ,r L 2r 2 01
Dimensional homogeneity
• Dimensional homogeneity means the dimensions of each term in an equation 
on both sides are equal. Thus if the dimensions of each term on both sides of 
an equation are the same the equation is known as the dimensionally 
homogeneous equation. •
• The powers of fundamental dimensions i.e., L, M, T on both sides of the 
equation will be identical for a dimensionally homogeneous equation. Such 
equations are independent of the system of units.
• Let us consider the equation V = u + at 
Dimensions of L.H.S = V= L/T = L T -1 
Dimensions of R.H.S = LT1 + (LT2 ) (T)
= LT'1 + LT'1
= LT"1
Dimensions of L.H.S = Dimensions of R.H.S = LT"1 
Therefore, equation V = u + at is dimensionally homogeneous 
Uses of Dimensional Analysis
• It is used to test the dimensional homogeneity of any derived equation. 2. It is 
used to derive
• It is used to derive the equation.
• Dimensional analysis helps in planning model tests.
Methods of Dimensional Analysis
• If the number of variables involved in a physical phenomenon is known, then 
the relationship among the variables can be determined by the following two 
methods.
Rayleigh''s Method
• RayleigfT's method of analysis is adopted when a number of parameters or 
variables is less (3 or 4 or 5).
• If the number of independent variables becomes more than four, then it is very 
difficult to find the expression for the dependent variable
Buckingham''s (11- theorem) Method
• If there are n - variables in a physical phenomenon and those n-variables 
contains'm' dimensions, then the variables can be arranged into (n-m) 
dimensionless groups called 1 1 terms.
• If f (XI, X2, X3,...... Xn) = 0 and variables can be expressed using m
dimensions then, f (111, 112,113,...... nn - m) = 0 Where, 1 1 1 , 112,113,........are
dimensionless groups.
• Each n term contains (m + 1) variables out of which m are of repeating type 
and one is of non-repeating type.
• Each n term being dimensionless, the dimensional homogeneity can be used 
to get each 1 1 term.
Method of Selecting Repeating Variables
• Avoid taking the quantity required as the repeating variable.
• Repeating variables put together should not form a dimensionless group.
• No two repeating variables should have same dimensions.
• Repeating variables can be selected from each of the following properties
° Geometric property - Length, Height, Width, Area 
° Flow property - Velocity, Acceleration, Discharge 
° Fluid property - Mass Density, Viscosity, Surface Tension
Model Studies
• Before constructing or manufacturing hydraulics structures or hydraulics 
machines tests are performed on their models to obtain desired information
about their performance.
• Models are a small scale replica of actual structure or machine.
• The actual structure is called prototype.
Similitude
• It is defined as the similarity between the prototype and its model. It is also 
known as similarity. There three types of similarities and they are as follows.?
c
Geometric similarity
• Geometric similarity is said to exist between the model and prototype if the 
ratio of corresponding linear dimensions between model and prototype are 
equal, i.e.
L V _ h P ^
h m H m
where Lr is known as scale ratio or linear ratio.
Kinematic Similarity
• Kinematic similarity exists between prototype and model if quantities such at 
velocity and acceleration at corresponding points on model and prototype are 
same.
(vjp = (Vi)p _(yt)p
(Vi)m (V 2)m (V 3)m ............ '
Where Vr is known as velocity ratio 
Dynamic Similarity
• Dynamic similarity is said to exist between model and prototype if the ratio of 
forces at corresponding points of model and prototype is constant.
( f j p _ (FX)p _ ( Fl) p p
(Fi)m ( ,Fz)m. (pa)m
Where Fr is known as force ratio.
Dimensionless Numbers
Following dimensionless numbers are used in fluid mechanics.
• Reynolds'^ number
• Froude"s number
• Euler"s number
• Weber"s number
• Mach number
Reynold's number
It is defined as the ratio of inertia force of the fluid to viscous force.
Nr6= Fj/Fv
Froude's Number (Fr)
• It is defined as the ratio of square root of inertia force to gravity force.
Model Laws (Similarity laws)
Reynold's Model Law
• For the flows where in addition to inertia force, the similarity of flow in the 
model and predominant force, the similarity of flow in model and prototype 
can be established if Re is same for both the system.
• This is known as Reynold"s Model Law.
• Re for model = Re for prototype
• (NRe)m = (NRe)p
m = ( T ) „
P m vm D m 
P p vpD p _
M m ^
Up
P rV rPr _ ^
M r
Applications
• In the flow of incompressible fluids in closed pipes.
• The motion of submarine completely under water.
• Motion of airplanes.
Froude's Model Law
• When the force of gravity is predominant in addition to inertia force then 
similarity can be established by Froude"s number.
• This is known as Froude"s model law.
Applications
• Flow over spillways
• Channels, rivers (free surface flows).
• Waves on the surface.
• Flow of different density fluids one above the other
F r= V F i/F g
• (Fr)m = (Fr)p
P