Download Dimensionless numbers

Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Dimensionless numbers
SOE2156: Fluids Lecture 5
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Dimensions
Physical quantities have dimensions { i.e. units { associated
with them. Distance can be measured in
metres
millimetres
miles
light-years
but always has dimensions of Length
For an equation describing a physical situation to be true, the
two sides must be equal both numerically and dimensionally
1 elephant + 3 aeroplanes = 4 days
but
1 metre + 3 metres = 4 metres
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
In general, if
A + B + C + ::: = Z
then A : : : Z must all have same dimensions { dimensionally
homogeneous. (Use this to check equations!)
Could have dimensions for everything { force, voltage,
frequency. However SI system prescribes 6 base quantities :
Length L
Mass M
Time T
Current I
Temperature Luminous Intensity
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
All other quantities can be expressed in terms of these { force
F = ma, dimensions
[F ] = [M ][LT 2] = [MLT 2]
Dimensionless equations can be important quantities as well.
Eg. an ellipse :
a
b
gives the shape of the ellipse
b
a
b
a
Similarly in uid dynamics
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Geometric similarity
2 objects are geometrically similar if all their dimensions are in
the same proportion
B
b
a
A
Here
a b
=
A B
{ Typically this might be a real object and a scale model for
testing.
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
If 2 objects are geometrically similar, same Reynolds number
the ow patterns will be the same
as will be the forces
: : : if expressed as dimensionless numbers
If an object in a ow (e.g. a car) experiences a force F from
the ow, this force will depend on the xsect area A, and the
uid ow speed V and density . However, the quantity
C=
F =A
is dimensionless
V2
1
2
Hence C will be the same for any geometrically similar objects
at the same Reynolds number
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Other examples { for 2 geometrically similar pumps, the head
and ow coecients
KH =
gH
N 2D 2
KQ =
will be the same.
For a turbine, the power coecient
KP =
would be important
P
N D 5
3
Q
ND 3
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Reynolds numbers
V
L
Choose a characteristic length L and ow speed V :
L3 = a mass
V2
= an `acceleration'
L
2
so L3 VL = Inertial force
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Similarly, since
F
@u
@ ux V
=
x ; and
L
A
@y
@y
then the viscous force is
so
V
F = L2
L
Inertial force = V 2L2
Viscous force (V =L)L2
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Similarly, since
F
@u
@ ux V
=
x ; and
L
A
@y
@y
then the viscous force is
so
V
F = L2
L
Inertial force = V 2L2
Viscous force (V =L)L2
= VL
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
Deriving dimensionless numbers
How do we derive dimensionless numbers (sometimes called
dimensionless groups)? Like this :
1 Identify the parameters that will make up the number (i.e.
the parameters of importance in the problem) A D .
(There will be up to 4).
2 Express the dimensions of each parameter in terms of the
base dimensions M ; L; T .
3 The dimensionless number will be of the form
Aa B b C c D d = dimensionless
4 We need to nd a
d !!
Dimensionless
numbers
Dimensions
Geometric
similarity
Reynolds
numbers
Deriving
dimensionless
numbers
However, for each base dimension M ; L; T the powers
a d must sum to zero
6 This gives a set of simultaneous equations that can be
solved
NB. We end with 3 simultaneous equations for 4 unknowns {
no unique solution. This gives us an equation of the form
5
AB a
CD
i.e.
AB
CD
= dimensionless
is the dimensionless number