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Dimensional Analysis
Dimensional analysis
Dimensional analysis is a mathematical tool
often applied in physics,
chemistry, and
engineering to simplify a problem by reducing the number of variables to the
smallest number of "essential" parameters. Systems which share these parameters
are called similar and do not have to be studied separately.
The dimension of a
physical quantity is the type of
unit
needed to express it. For instance, the dimension of a
speed is
distance/time and the dimension of a
force is mass×distance/time². In mechanics, every dimension can be expressed
in terms of distance (which physicists often call "length"), time, and mass, or
alternatively in terms of force, length and mass. Depending on the problem, it
may be advantageous to choose one or the other set of
fundamental units. Every unit is a product of (possibly fractional)
powers of the fundamental units, and the units form a
group under multiplication.
In the most primitive form, dimensional analysis is used to
check the correctness of algebraic derivations: in every physically meaningful
expression, only quantities of the same dimension can be added or subtracted.
The two sides of any equation must have the same dimensions. Furthermore, the
arguments to
exponential,
trigonometric and
logarithmic
functions must be
dimensionless numbers, which is often achieved by multiplying a certain
physical quantity by a suitable constant of the inverse dimension.
The above mentioned reduction of variables uses the
Buckingham π-theorem as its central tool. This theorem describes how every
physically meaningful equation involving n variables can be
equivalently rewritten as an equation of n-m dimensionless
parameters, where m is the number of fundamental units used.
Furthermore, and most importantly, it provides a method for computing these
dimensionless parameters from the given variables, even if the form of the
equation is still unknown.
Two systems for which these parameters coincide are called
similar; they are equivalent for the purposes of the equation, and the
experimentalist who wants to determine the form of the equation can choose the
most convenient one.
The π-theorem uses
linear algebra: the space of all possible physical units can be seen as a
vector
space over the
rational numbers if we represent a unit as the set of exponents needed for
the fundamental units (with a power of zero if the particular fundamental unit
is not present). Multiplication of physical units is then represented by vector
addition within this vector space. The algorithm of the π-theorem is essentially
a
Gauss-Jordan elimination carried out in this vector space.
A typical application of dimensional analysis occurs in
fluid dynamics. If a moving fluid meets an object, it exerts a force on the
object, according to a complicated (and not completely understood) law. The
variables involved are: the speed, density and viscosity of the fluid, the size
of the body, and the force. Using the algorithm of the π-theorem, one can reduce
these five variables to two dimensionless parameters: the drag coefficient and
the
Reynolds number. The original law is then reduced to a law involving only
these two numbers. To empirically determine this law, instead of experimenting
on huge bodies with fast flowing fluids (such as real-size airplanes in
wind-tunnels), one may just as well experiment on small models with slow
flowing, more viscous fluids, because these two systems are similar.
Consider
Einstein's well-known equation E = mc². As stated above, the two sides of
any equation must have the same dimensions. We can check this as follows.
-
E is
energy, which
has units of mass × length² / time². (This is because energy =
force × length,
and force = mass ×
acceleration, and acceleration = length / time².)
-
m is
mass, which is a
unit on its own.
-
c is
speed, which
has units of length / time.
-
The left-hand side, E, therefore has units of mass ×
length² / time².
-
The right-hand side, mc², has units of mass × (length /
time)² = mass × length² / time².
-
The two sides therefore have the same dimensions.
-
Barenblatt, G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics",
Cambridge University Press, 1996
-
Bridgman, P. W., "Dimensional Analysis", Yale University
Press, 1937
-
Langhaar, H. L., "Dimensional Analysis and Theory of
Models", Wiley, 1951
-
Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949,
42(6)
-
Porter, "The Method of Dimensions", Methuen, 1933
-
Boucher and Alves, Dimensionless Numbers, Chem. Eng.
Progress, 1960, 55, pp.55-64
-
Buckingham, E., On Physically Similar Systems:
Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345
-
Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140,
167-177
-
Rayleigh, Lord, The Principle of Similitude, Nature 1915,
95, pp. 66-68
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Silberberg, I. H. and McKetta J. J., Jr., Learning How to
Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6),
p.101; (7), p. 129
-
Van Driest, E. R., On Dimensional Analysis and the
Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34,
March
-
Perry, J. H. et al., "Standard System of Nomenclature for
Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944,
40, 251
-
Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am.
Soc. Mech. Engrs., 1944, 66, 671
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