CHEMICAL ENGINEERING

UNITS AND DIMENSIONS 11

Thus, dimensionless group 1:QT/L^2 DQ^1 /^3 /
^1 /^3 g^2 /^3 /^4 /^3 /
^4 /^3 g^2 /^3 DQ
/

dimensionless group 2:υLDυ^2 /^3 /

^2 /^3 g^1 /^3 or, cubingDυ^3

2 g/^2

and: υ^3
2 g/^2 DfQ
/

This may be written as: υ^3
2 g/^2 DKQ
/n

For streamline flow,υ/Q^1 /^3 ornD 1

and hence: υ^3
2 g/^2 DKQ
/,υ^3 DKQ/
gandυDKQ/
g^1 /^3

As the resistance to heat transfer is attributable to the thermal resistance of the conden-
sate layer which in turn is a function of the film thickness, then:h/k/υwherekis the
thermal conductivity of the film and sinceυ/ 1 /^3 ,h/k/^1 /^3 , that is the coefficient is
inversely proportional to the one third power of the liquid viscosity.

PROBLEM 1.

A spherical particle settles in a liquid contained in a narrow vessel. Upon what variables
would you expect the falling velocity of the particle to depend? Obtain the relevant
dimensionless groups.
For particles of a given density settling in a vessel of large diameter, the settling velocity
is found to be inversely proportional to the viscosity of the liquid. How would this depend
on particle size?

Solution

This problem is very similar to Problem 1.13, although, in this case, the liquid through
which the particle settles is contained in a narrow vessel. This introduces another variable,
D, the vessel diameter and hence the settling velocity of the particle is given by:uD
fd,
,,D,
s,g. The dimensions of each variable are:uDL/T,dDL, DM/L^3 ,

DM/LT,DDL, (^) sDM/L^3 ,andgDL/T^2. With 7 variables and 3 fundamental
dimensions, there will be 7  3 D4 dimensionless groups. Takingd,
and as the
recurring set, then:
dL, LDd
M/L^3 , MD L^3 D
d^3
M/LT, TDM/L D
d^3 /dD
d^2 /
Thus: dimensionless group 1:uT/LDu
d^2 /dDdu
/
dimensionless group 2:D/LDD/d
dimensionless group 3: (^) sL^3 /MD (^) sd^3 /
d^3 D (^) s/

and dimensionless group 4:gT^2 /LDg
^2 d^4 /^2 dDg
^2 d^3 /^2
Thus: du
/DfD/d
s/
g
^2 d^3 /^2 
In particular,du
/DKg
^2 d^3 /^2 nwhereKis a constant.