Basics of Fluid Mechanics, 2014a

290 CHAPTER 9. DIMENSIONAL ANALYSIS
The averaged velocity could be a represented (there are better methods or choices)
of the energy flowing in the channel. The averaged velocity is U/2 and the velocity
derivative is dU/dl = constant = U/l. With these value of the Diss number is
Diss =
( ) 2 U
μ l
l
ρ U 3
8
= 4 μ
ρlU
(9.VII.c)
The results show that Dissipation number is not a function of the velocity. Yet, the
energy lost is a function of the velocity square E ∝ Diss μ U.
End Solution
9.2.3.2 Building Blocks Method: Constructing Dimensional Parameters
Note, as opposed to the previous method, this technique allows one to find a single
or several dimensionless parameters without going for the whole calculations of the
dimensionless parameters.
Example 9.8:
Assume that the parameters that effects the centrifugal pumps are
Q Pump Flow rate rpm or N angular rotation speed
D rotor diameter ρ liquid density (assuming liquid
phase)
B T Liquid Bulk modulus μ liquid viscosity
ɛ typical roughness of pump g gravity force (body force)
surface
ΔP Pressure created by the
pump
Construct the functional relationship between the variables. Discuss the physical meaning
of these numbers. Discuss which of these dimensionless parameters can be neglected
as it is known reasonably.
Solution
The functionality can be written as
0=f (D, N, ρ, Q, B T ,μ,ɛ,g,ΔP ) (9.VIII.a)
The three basic parameters to be used are D [L], ρ [M], and N [t]. There are nine (9)
parameters thus the number of dimensionless parameters is 9 − 3=6. For simplicity