Page 154 - Modelling in Transport Phenomena A Conceptual Approach

134 CHAPTER 5. RATE OF GENERATION

momentum into and out of the volume element, i.e.,

Rate of Rate of Summation of forces
( momentum in ) - ( momentum out acting on a system
Time rate of change of
momentum of a system ) (5.1-2)
On the other hand, for a given system, the inventory rate equation for momentum
can be expressed as
Rate of Rate of ) + ( Rate of momentum
( momentum in ) - ( momentum out generation
= ( Rate of momentum ) (5.1-3)
accumulation

Comparison of Eqs. (5.1-2) and (5.1-3) indicates that
Rate of momentum Summation of forces ) (5.1-4)
generation acting on a system
in which the forces acting on a system are the pressure force (surface force) and
the gravitational force (body force).

5.1.1 Momentum Generation As a Result of Gravitational
Force

Consider a basketball player holding a ball in his hands. When he drops the ball,
it starts to accelerate as a result of gravitational force. According to Eq.(5.1-4),
the rate of momentum generation is given by
Rate of momentum generation = Mg (5.1-5)
where M is the mass of the ball and g is the gravitational acceleration. Therefore,
the rate of momentum generation per unit volume, R, is given by
(5.1-6)

5.1.2 Momentum Generation As a Result of Pressure Force
Consider the steady flow of an incompressible fluid in a pipe as shown in Figure 5.1.
The rate of mechanical energy required to pump the fluid is given by Eq. (4.53)
as
I&’ = F’(v) = & IAP( (5.1-7)
Since the volumetric flow rate, &, is the product of average velocity, (v), with the
cross-sectional area, A, Eq. (5.1-7) reduces to
AlAPl-3’0 =O (5.1-8)