W 12 L 21 STEADY MASS FLOW

STEADY MASS FLOW



The momentum relation developed earlier for a general system of mass provides

us with a direct means of analyzing the action of mass flow where a change of

momentum occurs.



One of the most important cases of mass flow occurs during steady flow

conditions where the rate at which mass enters a given volume equals the rate at

which mass leaves the same volume.



Analysis of Flow Through a Rigid Container



Consider a rigid container, shown in section in Fig. (12.1 a), into which mass

flows in a steady stream at the rate m’ through the entrance section of area A1.

Mass leaves the container through the exit section of area A2 at the same rate.









Figure (12.1)





The velocity of the entering stream is v1 normal to A1 and that of leaving stream

is v2 normal to A2, if ρ1 and ρ2 are the respective densities of the two streams,

conservation of mass requires that



1A 1 1   2 A 2  2  m '



To describe the forces which act, we isolate either the mass of fluid within the

container or the entire container and the fluid within it.

We would adopt the second approach when the forces external to the container

are desired.



If the vector sum of the external force system is denoted by ΣF, which includes

1. Forces exerted on the container by attachment to other structures.

2. Forces acting on the fluid within the container at A1 and A2 due to static

pressure.

3. Weight of fluid and structure.



So





F  G

Incremental Analysis



Figure (12.1 b) illustrates the system at time t when the system mass is that of

the container, the mass within it, and an increment Δm about to inter during time

Δt.



At time t + Δt the same total mass is that of the container, the mass within it, and

an equal increment Δm which leaves the container in time Δt.



G  mv 2  mv 1  mv 2  v 1   mv



Division by Δt and passage to the limit yield





 m  dm

G  m ' v   F where m '  lim  

t 0 t

  dt

Note:

1. Body is rigid.

2. Mass of fluid in container is constant.

3. Must account for all external forces acting on the system.

4. And free-body diagram must be correct.





Angular Momentum in Steady-Flow Systems



The resultant moment of all external forces about some fixed point O on or off the

system, Fig. (12.1 a), equals the time rate of change of angular momentum of the

system about O.



M O  m'  2 d2  1d1  Velocities are in single plane.





M O  m' d2 xv 2  d1xv1  Velocities are not in the same plane.