CHEMICAL ENGINEERING

12 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

For particles settling in a vessel of large diameter,u/ 1 /.Butu// 1 /^2 nand,
whennD1,n/ 1 /. In this case:

du

/DKg

^2 d^3 /^2 

or: du/d^3 andu/d^2

Thus the settling velocity is proportional to the square of the particle size.

PROBLEM 1.

A liquid is in steady state flow in an open trough of rectangular cross-section inclined at
an angle"to the horizontal. On what variables would you expect the mass flow per unit
time to depend? Obtain the dimensionless groups which are applicable to this problem.

Solution

This problem is similar to Problems 1.11 and 1.12 although, here, the width of the trough
and the depth of liquid are to be taken into account. In this case, the mass flow of liquid
per unit time,Gwill depend on: fluid density, ; fluid viscosity, ; depth of liquid,h;
width of the trough,a; acceleration due to gravity,gand the angle to the horizontal,".
Thus:GDf
,,h,a,g,". The dimensions of each variable are:GDM/T, DM/L^3 ,
DM/LT,hDL,aDL,gDL/T^2 and neglecting"at this stage, with 6 variables
with dimensions and 3 fundamental dimensions, there will be 6  3 D3 dimensionless
groups. Takingh,
and as the recurring set then:

hL, LDh

M/L^3 , MD L^3 D

h^3

M/LT, TDM/L D

h^3 /hD

h^2 /

Thus: dimensionless group 1:GT/MDG
h^2 /
h^3 DG/h

dimensionless group 2:a/LDa/h

dimensionless group 3:gT^2 /LDg

^2 h^4 /^2 hDg

^2 h^3 /^2

and: G/hDfa/hg
^2 h^3 /^2 

PROBLEM 1.

The resistance force on a spherical particle settling in a fluid is given by Stokes’ Law.
Obtain an expression for the terminal falling velocity of the particle. It is convenient to
express experimental results in the form of a dimensionless group which may be plotted
against a Reynolds group with respect to the particle. Suggest a suitable form for this
dimensionless group.
Force on particle from Stokes’ LawD 3 du;where is the fluid viscosity,dis the
particle diameter anduis the velocity of the particle relative to the fluid.