CHEMICAL ENGINEERING

UNITS AND DIMENSIONS 5

. In this event:RDf,d,
,g,. The dimensions of each variable are:RDL/T,
DM/LT,dDL, DM/L^3 ,gDL/T^2 ,andDM/T^2. There are 6 variables and 3
fundamentals and hence 6  3 D3 dimensionless groups. Taking as the recurring set,
d,
andg, then:

dL, LDd

M/L^3 ∴MD L^3 D

d^3

gL/T^2 ∴T^2 DL/gDd/gandTDd^0.^5 /g^0.^5

Thus, dimensionless group 1:RT/LDRd^0.^5 /dg^0.^5 DR/dg^0.^5

dimensionless group 2: LT/MDdd^0.^5 /g^0.^5 

d^3 D/g^0.^5 

d^1.^5 

dimensionless group 3:T^2 /MDd/g

d^3 D/g

d^2 

∴ R/dg^0.^5 Df

(

g^0.^5 

d^1.^5

,



g

d^2

)

or:

R^2

dg

Df

(

g

^2 d^3

,



g

d^2

)

(b) The final shape of the drop as indicated by its diameter,d, may be obtained by
using the argument in (a) and puttingRD0. An alternative approach is to assume the
final shape of the drop, that is the final diameter attained when the force due to surface
tension is equal to that attributable to gravitational force. The variables involved here will
be: volume of the drop,V; density of the liquid, ; acceleration due to gravity,g,andthe
surface tension of the liquid,. In this case:dDfV,
,g,. The dimensions of each
variable are:dDL,VDL^3 , DM/L^3 ,gDL/T^2 ,DM/T^2. There are 5 variables
and 3 fundamentals and hence 5  3 D2 dimensionless groups. Taking, as before,d,

andgas the recurring set, then:

dL, LDd

M/L^3 ∴MD L^3 D

d^3

gL/T^2 ∴T^2 DL/gDd/gandTDd^0.^5 /g^0.^5

Dimensionless group 1:V/L^3 DV/d^3

Dimensionless group 2:T^2 /MDd/g
d^3 D/g
d^2 

and hence: d^3 /VDf

(



g

d^2

)

PROBLEM 1.

Liquid is flowing at a volumetric flowrate ofQper unit width down a vertical surface.
Obtain from dimensional analysis the form of the relationship between flowrate and film
thickness. If the flow is streamline, show that the volumetric flowrate is directly propor-
tional to the density of the liquid.