Basics of Fluid Mechanics, 2014a

9.4. SUMMARY OF DIMENSIONLESS NUMBERS 317
In this analysis, it is assumed that pressure is uniform in the cross section. This
assumption is appropriate because only the secondary flows in the radial direction (to
be discussed in this book section on pumps.). Hence, the ratio of power between the
two pump can be written as
Ẇ p
= (PAU) p
(9.XX.h)
Ẇ m (P AU) m
Utilizing equations above in this ratio leads to
P p /P m
A p /A m
{ }} { { }} {
( ) 2 ( ) { 2
Ẇ p Dp Dp
=
Ẇ m D m D m
U p /U m
(
Dp
}} ) {
=
D m
(
Dp
D m
) 5
(9.XX.i)
End Solution
Example 9.21:
The flow resistance to flow of the water in a pipe is to be simulated by flow of air.
Estimate the pressure loss ratio if Reynolds number remains constant. This kind of study
appears in the industry in which the compressibility of the air is ignored. However, the
air is a compressible substance that flows the ideal gas model. Water is a substance that
can be considered incompressible flow for relatively small pressure change. Estimate the
error using the averaged properties of the air.
Solution
For the first part, the Reynolds number is the single controlling parameter which affects
the pressure loss. Thus it can be written that the Euler number is function of the
Reynolds number.
Eu = f(Re)
(9.XXI.a)
Thus, to have a similar situation the Reynolds and Euler have to be same.
Re p = Re m Eu m = Eu p (9.XXI.b)
Hence,
U m
U p
and for Euler number
ΔP m
ΔP p
and utilizing equation (9.XXI.c) yields
= l p ρ μ p
l m ρ m μ m
= ρ m
ρ p
U m
U p
( ) 2 ( ) 2 ( )
ΔP m lp μm ρp
=
ΔP p l m μ p ρ m
(9.XXI.c)
(9.XXI.d)
(9.XXI.e)