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Dimensional Analysis
Author: John M. Cimbala, Penn State University
Latest revision, 29 August 2007
Primary Dimensions (Review)
•
There are seven
primary dimensions
. All other dimensions can be formed by combinations of these. The
seven primary dimensions are, along with their symbols:
mass
length
time
temperature
current
amount of light
amount of matter
{m}
{L}
{t}
{T}
{I}
{C}
{N}
•
The primary dimensions of variables in an experiment or analysis can be used to our advantage –
to reduce
the required amount of effort
.
Dimensional Homogeneity
•
We state the
law of dimensional homogeneity
as “
Every additive term in an equation must have the same
dimensions.
”
•
Example: The total energy (
E
) of a system is composed of internal energy (
U
), kinetic energy (KE), and
potential energy (PE), i.e.,
.
KE
PE
EU
=+ +
•
Let’s look at the primary dimensions of each term in this equation:
{ } { } { }
energy
force length
E
==
⋅
{ } { }
22
mL /t
E
=
{} { }
{}
energy
mass
energy
mass
Um
u
⎧⎫
=
⎨⎬
⎩⎭
{ } { }
mL /t
U
=
2
2
2
1
length
mass
2t
KE
mV
i
m
e
⎧
⎫
⎨
⎬
{ } { }
mL /t
KE
=
{}{ }
2
length
mass
length
time
PE
mgz
⎧
⎫
⎨
⎬
{ } { }
mL /t
PE
=
•
The law of dimensional homogeneity is the basis for the useful technique of dimensional analysis, which we
discuss next.
Dimensional Analysis and the Method of Repeating Variables
•
Dimensional analysis is a simple, powerful tool that is useful in
all
disciplines (but unfortunately, is usually
taught only in fluid mechanics).
•
The
goal
of dimensional analysis is to
reduce the number of independent variables in a problem.
•
We accomplish this by converting all dimensional variables into
nondimensional parameters
, called “Pi’s”,
and given the symbol
Π
(upper case Greek letter pi).
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 Spring '08
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