Page 215 - Modelling in Transport Phenomena A Conceptual Approach

7.4. CONSERVATION OF MOMENTUM 195

ut. In the case of an accelerating sphere an additional force, called fluid inertia
force, acts besides the gravitational, the buoyancy, and the drag forces. This force
arises from the fact that the fluid around the sphere is also accelerated from the
rest, resulting in a change in the momentum of the fluid. The rate of change of
fluid momentum shows up as an additional force acting on the sphere, pointing
in the direction opposite to the motion of the sphere. This additional force has a
magnitude equal to one-half the rate of change of momentum of a sphere of liquid
moving at the same velocity as the solid sphere. Therefore, Eq. (7.41) is written
in the form

linear momentum of a sphere ) = ( Gravitational
Time rate of change of
force
- ( force ) - ( force ) - ( Fluid inertia
Buoyancy
Drag
force
and can be expressed as

where pp and Dp represent the density and diameter of the solid sphere, respec-
tively, and p is the fluid density. Simplification of Eq. (7.43) gives

(7.44)

The friction factor f is usually given as a function of the Reynolds number, Rep,
defined by
Rep = - (7.4-5)
DPV P
P
Therefore, it is much more convenient to express the velocity, v, in terms of Rep.
Thus, Eq. (7.44) takes the form
D; dRep 3
-
(PP + 0.5 PI -h-z fa?$ (7.4-6)
where Ar is the Archimedes number defined by Eq. (4.3-6). Note that when the
particle reaches its terminal velocity, i.e., d Rep /dt = 0, Eq. (7.46) reduces to Eq.
(4.3-4). Integration of &. (7.46) gives

(7.4-7)

A friction factor - Reynolds number relationship is required to carry out the
integration. Substitution of the Turton-Levenspiel correlation, Eq. (4.%10), into