Momentum Analysis of Flow Systems
Water is pumped through a piping section. The moment acting on the elbow for the cases of downward and upward discharge is to be determined.
The flow is steady and incompressible.
The water is discharged to the atmosphere, andthus the gage pressure at the outlet is zero.
Effects of water falling down during upward discharge isdisregarded.
Pipe outlet diameter is small compared to the moment arm, and thus we use average valuesof radius and velocity at the outlet.
We take the density of water to be 1000 kg/m
We take the entire pipe as the control volume, and designate the inlet by 1 and the outlet by 2. Wealso take the
coordinates as shown. The control volume and the reference frame are fixed.The conservation of mass equation for this one-inlet one-outlet steady flow system is, and
= constant. The mass flow rate and the weight of the horizontalsection of the pipe are
: To determine the moment acting on the pipe at point
, we need to take themoment of all forces and momentum flows about that point. This is a steady and uniform flow problem,and all forces and momentum flows are in the same plane. Therefore, the angular momentum equation inthis case can be expressed as
V mr V mr M
is the moment arm, all moments in thecounterclockwise direction are positive, and all in the clockwise direction are negative.
= 1 m
= 2 m
The free body diagram of the pipe section is given in the figure. Noting that the moments of allforces and momentum flows passing through point
are zero, the only force that will yield a moment about point
is the weight
of the horizontal pipe section, and the only momentum flow that will yield amoment is the outlet stream (both are negative since both moments are in the clockwise direction). Thenthe angular momentum equation about point
V mr W r M
m/skg1 N1m/s)kg/s)(4m)(45.54(2- N)m)(294.31(
V mr W r M
The negative sign indicates that the assumed direction for
is wrong, and should be reversed. Therefore,a moment of 70 N
m acts at the stem of the pipe in the clockwise direction.(
: The moment due to discharge stream is positive in this case, and the momentacting on the pipe at point
m/skg1 N1m/s)kg/s)(4m)(45.54(2 N)m)(294.31(
V mr W r M
Note direction of discharge can make a big difference in the moments applied on a pipingsystem. This problem also shows the importance of accounting for the moments of momentums of flowstreams when performing evaluating the stresses in pipe materials at critical cross-sections.
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