Dimensional Analysis
Dimensional Analysis
z
This technique is useful in all branches of
science and engineering
z
Basic premises are
1.
Physical quantities have dimensions
2.
All equations developed from basic laws of physics
are dimensionally homogeneous
z
We can develop a useful theory from this
information
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Dimensions
Dimensions


1
1
z
Consider viscosity,
μ,
which has SI
units
kg/(m•sec) and USCS
units
slug/(ft •sec)
z
We write
z
[
μ
] means “dimensions of
μ
” while
M
,
L
and
T
denote
dimensions
of mass, length and time,
respectively
z
If temperature is relevant, we denote its
dimension by
Θ
z
M
,
L
,
T
and
Θ
are called
independent
dimensions
Dimensions
Dimensions


2
2
z
All dimensions can be expressed in terms of
the independent dimensions
z
For example, consider pressure,
p
, which has
dimensions of force per unit area…appealing
to Newton’s law, we have
z
Thus, for dimensional analysis purposes,
pressure has SI units of kg/(m•sec
2
) and
USCS units of slug/(ft•sec
2
)
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Dimensional Homogeneity
Dimensional Homogeneity


1
1
z
Universal equations are valid regardless of
the choice of units
z
Counterexample…
theoretical hull speed
for a sailboat (single hull, nonsurfing)
U
hull
= Sailboat speed
in knots
L
w
= Hull length at the
waterline in feet