Dimensional Analysis


This article defines a word and clarifies its application and usefulness in scale studies.

Definition


Kirk-Othmer Definition

From Kirk-Othmer Encyclopedia of Chemical Technology, (c) John Wiley & Sons, Inc:

Dimensional analysis is a technique that treats the general forms of equations governing natural phenomena. It provides procedures of judicious grouping of variables associated with a physical phenomenon to form dimensionless products of these variables; therefore, without destroying the generality of the relationship, the equation describing the physical phenomenon may be more easily determined experimentally. It guides the experimenter in the selection of experiments capable of yielding significant information and in the avoidance of redundant experiments, and makes possible the use of scale models for experiments. The method is particularly valuable when the problems involve a large number of variables. On such occasions, dimensional analysis may reveal that, whatever the form of the inaccessible final solution, certain features of it are obligatory. The technique has been utilized effectively in engineering modeling (1–7).

Mechanical Engineers Rules Of Thumb Definition

From Rules of Thumb for Mechanical Engineers:
Often in fluid mechanics, we come across certain terms, such as Reynolds number, Randtl number, or Mach number, that we have come to accept as they are. But these are extremely useful in unifying the fundamental theories in this field, and they have been obtained through a mathematical analysis of various forces acting on the fluids. The mathematical analysis is done though Buckingham’s Pi Theorem. This theorem states that, in a physical system described by n quantities in which there are m dimensions, these n quantities can be rearranged into (n-m) nondimensional parameters. Table 1 gives dimensions of some physical variables used in fluid mechanics in terms of basic mass (M), length (L), and time (T) dimensions.
Such dimensionless ratios will compare physical variables such as
  • Force F
  • Discharge Q
  • Pressure P
  • Acceleration a
  • Density rho
  • Specific weight gamma
  • Dynamic viscosity mu
  • Kinematic viscosity v
  • Surface tension theta
  • Bulk modulus of elasticity K
  • Gravity g

Tables of Exponents


Table of Exponents of Dimensions for Electromagnetic Quantities. From Kirk-Othmer Encyclopedia of Chemical Technology, (c) John Wiley & Sons, Inc, "Dimensional Analysis" v8 p.584:


Absolute
Gravitational
Engineering
Quantity
m
l
t
f
l
t
f
m
l
t
acceleration
0
1
-2
0
1
-2
0
0
1
-2
angular acceleration
0
0
-2
0
0
-2
0
0
0
-2
angular velocity
0
0
-1
0
0
-1
0
0
0
-1
area
0
2
0
0
2
0
0
0
2
0
angular momentum
1
2
-1
1
1
1
0
1
2
-1
density, mass
1
-3
0
1
-4
2
0
1
-3
0
energy, work
1
2
-2
1
1
0
1
0
1
0
force
1
1
-2
1
0
0
1
0
0
0
frequency
0
0
-1
0
0
-1
0
0
0
-1
length
0
1
0
0
1
0
0
0
1
0
linear acceleration
0
1
-2
0
1
-2
0
0
1
-2
linear momentum
1
1
-1
1
0
1
0
1
1
-1
linear velocity
0
1
-1
0
1
-1
0
0
1
-1
mass
1
0
0
1
-1
2
0
1
0
0
moment of inertia
1
2
0
1
1
2
0
1
2
0
power
1
2
-3
1
1
-1
1
0
1
-1
pressure
1
-1
-2
1
-2
0
1
0
-2
0
stress
1
-1
-2
1
-2
0
1
0
-2
0
surface tension
1
0
-2
1
-1
0
1
0
-1
0
time
0
0
1
0
0
1
0
0
0
1
viscosity, absolute
1
-1
-1
1
-2
1
1
0
-2
1
viscosity, kinematic
0
2
-1
0
2
-1
0
0
2
-1
volume
0
3
0
0
3
0
0
0
3
0
Key: f = force, l = length, m = mass, t = time


Table of Exponents of Dimensions for Electromagnetic Quantities. From Kirk-Othmer Encyclopedia of Chemical Technology, (c) John Wiley & Sons, Inc, "Dimensional Analysis" v8 p.585:

Quantity
l
m
t
q
l
m
t
E
l
m
t
m
charge
0
0
0
1
'3/2
'1/2
-1
'1/2
'1/2
'1/2
0
'-1/2
capacitance
-2
-1
2
2
1
0
0
1
-1
0
2
-1
current
0
0
-1
1
'3/2
'1/2
-2
'1/2
'1/2
'1/2
-1
'-1/2
electric field intensity 1
1
-2
-1
'-1/2
'1/2
-1
'-1/2
'1/2
'1/2
-2
'1/2
electric potential difference 2
1
-2
-1
'1/2
'1/2
-1
'-1/2
'3/2
'1/2
-2
'1/2
electric flux
0
0
0
1
'3/2
'1/2
-1
'1/2
'1/2
'1/2
0
'-1/2
electric flux density
-2
0
0
1
'-1/2
'1/2
-1
'1/2
-1.5
'1/2
0
'-1/2
inductance
2
1
0
-2
-1
0
2
-1
1
0
0
1
magnetic field intensity
-1
0
-1
1
'1/2
'1/2
-2
'1/2
'-1/2
'1/2
-1
'-1/2
magnetic flux
2
1
-1
-1
'1/2
'1/2
0
'-1/2
'3/2
'1/2
-1
'1/2
magnetic flux density 0
1
-1
-1
-1.5
'1/2
0
'-1/2
'-1/2
'1/2
-1
'1/2
magnetomotive force 0
0
-1
1
'3/2
'1/2
-2
'1/2
'1/2
'1/2
-1
'-1/2
permeability
1
1
0
-2
-2
0
2
-1
0
0
0
1
permittivity
-3
-1
2
2
0
0
0
1
-2
0
2
-1
resistance
2
1
-1
-2
-1
0
1
-1
0
0
-1
1
Key: l = length, m = mass, t = time, q = charge, E (epsilon) = permittivity

Dimensional Analysis And Similitude

Most of these nondimensional parameters in fluid mechanics are basically ratios of a pair of fluid forces. These farces can be any combination of gravity, pressure, viscous, elastic, inertial, and surface tension forces. The flow system variables from which these parameters are obtained are: velocity V, the density p, pressure drop Ap, gravity g, viscosity p, surface tension Q, bulk modulus of elasticity K, and a few linear dimensions of l.
These nondimensional parameters allow us to make studies on scaled models and yet draw conclusions on the prototypes. This is primarily because we are dealing with the ratio of forces rather than the forces themselves. The model and the prototype are dynamically similar if (a) they are geometrically similar and (b) the ratio of pertinent forces are also the same on both.

Nondimensional Parameters Used in Fluid Mechanics

The following five nondimensional parameters are of great value in fluid mechanics.

Reynolds Number

Reynolds number is the ratio of inertial forces to viscous forces: This is particularly important in pipe flows and aircraft model studies. The Reynolds number also characterizes different flow regimes (laminar, turbulent, and the transition between the two) through a critical value. For example, for the case of flow of fluids in a pipe, a fluid is considered turbulent if R is greater than 2,000. Otherwise, it is taken to be laminar. A turbulent flow is characterized by random movement of fluid particles.

See Reynolds Number.

Froude Number

Froude number is the ratio of inertial force to weight: This number is useful in the design of spillways, weirs, channel flows, and ship design.

Weber Number

Weber number is the ratio of inertial forces to surface tension forces. This parameter is signifcant in gas-liquid interfaces where surface tension plays a major role.

Mach Number

Mach number is the ratio of inertial farces to elastic forces: where c is the speed of sound in the fluid medium, k is the mtio of specific heats, and T is the absolute temperam. This parameter is very important in applications where velocities are near OT above the local sonic velocity. Examples are fluid machineries, aircraft flight, and gas turbine engines.

Pressure Coefficient

Pressure coefficient is the ratio of pressure forces to inertial forces: This coefficient is important in most fluid flow situations.