CHEMICAL ENGINEERING

4 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

except in so far as this affects the buoyancy. Show that the settling velocitymustthen be
inversely proportional to the viscosity of the fluid.

Solution

If: u 0 DkdaWb c d, then working in dimensions ofM, LandT:

L/TDkLaML/T^2 bM/L^3 cM/LTd

Equating dimensions:

M: 0 DbCcCd

L: 1 DaCb 3 cd

T: 1 D 2 bd

Solving in terms ofb:

aD 1 ,cDb 1 ,anddD 1  2 b

∴ u 0 Dk 1 /dWb
b/
/^2 bwherekis a constant,

or: u 0 Dk/d
W
/^2 b

Rearranging:
du 0
/DkW
/^2 b

where (W
/^2 ) is a function of a form of the Reynolds number.

Foru 0 to be independent of ,bmust equal unity andu 0 DkW/d

Thus, for constant diameter and hence buoyant weight, the settling velocity is inversely
proportional to the fluid viscosity.

PROBLEM 1.

A drop of liquid spreads over a horizontal surface. What are the factors which will
influence:

(a) the rate at which the liquid spreads, and

(b) the final shape of the drop?

Obtain dimensionless groups involving the physical variables in the two cases.

Solution

(a) The rate at which a drop spreads, sayRm/s, will be influenced by: viscosity of the
liquid, ; volume of the drop,Vexpressed in terms ofd, the drop diameter; density of
the liquid, ; acceleration due to gravity,gand possibly, surface tension of the liquid,