CHEMICAL ENGINEERING

UNITS AND DIMENSIONS 9

and the angle of inclination,",

or: dDf
,,g,Q,"

Excluding"at this stage, there are 5 variables and with 3 fundamental dimensions there
will be 5  3 D2 dimensionless groups. The dimensions of each variable are:dDL,
DM/L^3 , DM/LT,gDL/T^2 ,QDL^2 /T, and, choosingQ, andgas the recurring
set, then:

QDL^2 /TTDL^2 /Q

gDL/T^2 LDgT^2 DgL^4 /Q^2 ,L^3 DQ^2 /g,LDQ^2 /^3 /g^1 /^3

andTDQ^4 /^3 /Qg^2 /^3 DQ^1 /^3 /g^2 /^3

DM/L^3 MDL^3 DQ^2 /g

DQ^2 

/g

Thus, dimensionless group 1:d/LDdg^1 /^3 /Q^2 /^3 ord^3 g/Q^2

dimensionless group 2: LT/MDQ^2 /^3 /g^1 /^3 Q^1 /^3 /g^2 /^3 /Q^2 

gD/Q

and the function becomes:d^3 g/Q^2 Df/Q
,"

PROBLEM 1.

Liquid flows down an inclined surface as a film. On what variables will the thickness of
the liquid film depend? Obtain the relevant dimensionless groups. It may be assumed that
the surface is sufficiently wide for edge effects to be negligible.

Solution

This is essentially the same as Problem 1.11, though here the approach used is that of
equating indices.

If, as before: dDK
a,b,gc,Qd,"e

then, excluding"at this stage, the dimensions of each variable are:dDL, DM/L^3 ,
DM/LT,gDL/T^2 ,QDL^2 /T.
Equating dimensions:

M: 0 DaCbi

L: 1 D 3 abCcC 2 dii

T: 0 Db 2 cdiii

Solving in terms ofbandcthen:

from (i) aDb

from (iii) dDb 2 c

and in (ii) 1 D 3 bbCc 2 b 4 cor:cD 1 / 3 ∴ dD 2 / 3 b