CHEMICAL ENGINEERING

UNITS AND DIMENSIONS 3

Thus the power number is a function of the Reynolds number to the powerm.In
factNPis also a function of the Froude number,DN^2 /g. The previous equation may be
written as:

P/D^5 N^3 DkD^2 N

/m

Experimentally: P/N^2

From the equation, P/NmN^3 ,thatismC 3 D2andmD 1

Thus for the same fluid, that is the same viscosity and density:

P 2 /P 1 D^51 N^31 /D^52 N^32 DD 12 N 1 /D^22 N 2 ^1 or:P 2 /P 1 DN^22 D^32 /N^21 D^31 

In this case,N 1 DN 2 andD 2 D 2 D 1.

∴ P 2 /P 1 D 8 D^31 /D^31 D 8

A similar solution may be obtained using the Recurring Set method as follows:

PDfD,N,

,,fP,D,N,

,D 0

UsingM, LandTas fundamentals, there are five variables and three fundamentals
and therefore by Buckingham’stheorem, there will be two dimensionless groups.
ChoosingD, Nand as the recurring set, dimensionally:

DL

NT^1

ML^3

]

Thus:

[LD

TN^1

M L^3 D

D^3

First group, 1 ,isPML^2 T^3 ^1 P
D^3 D^2 N^3 ^1 

P

D^5 N^3

Second group, 2 ,isML^1 T^1 ^1 
D^3 D^1 N^1 

D^2 N

Thus: f

(

P

D^5 N^3

,

D^2 N

)

D 0

Although there is little to be gained by using this method for simple problems, there is
considerable advantage when a large number of groups is involved.

PROBLEM 1.

It is found experimentally that the terminal settling velocityu 0 of a spherical particle in
a fluid is a function of the following quantities:

particle diameter,d; buoyant weight of particle (weight of particleweight of displaced
fluid),W; fluid density, , and fluid viscosity,.

Obtain a relationship foru 0 using dimensional analysis.
Stokes established, from theoretical considerations, that for small particles which settle
at very low velocities, the settling velocity is independent of the density of the fluid