Closed Systems( Bend Pipes)

The 26th week
2. Closed Systems( Bend Pipes)
Consider a pipe bend with a constant cross section lying in the horizontal plane and turning through an angle of .
Flow round a pipe bend of constant cross-section
Step in Analysis:
Draw a control volume
Decide on co-ordinate axis system
Calculate the Total force
Calculate the Pressure force
Calculate the Body force
Calculate the Resultant force
1. Control Volume

The control volume is draw in the above figure, with faces at the inlet and outlet of the bend and encompassing the pipe walls.
2. Co-ordinate axis system
It is convenient to choose the co-ordinate axis so that one is pointing in the direction of the inlet velocity. In the above figure the x-axis points in the direction of the inlet velocity.
3. Calculate the Total force
In the x-direction:
F_Tx=?Q(u_2x-u_1x )
u1x= u1
u2x= u2 cos ?
F_Tx=?Q(u_2 cos?-u_1x )

In the y-direction:
F_Ty=?Q(u_2y-u_1y )
u1y= u1sin?=0
u2y= u2 sin ?
F_Ty=?Q(u_2 sin?)

4 Calculate the pressure force
FP = pressure force at 1- pressure force at 2
FPx = p1 A1 cos 0 - p2 A2 cos ?= p1 A1 - p2 A2 cos ?
FPy = p1 A1 sin 0 - p2 A2 sin ?= - p2 A2 cos ?


5. Calculate the Body force
There are no body forces in the x or y directions. The only body force is that exerted by gravity (which acts into the paper in this example - a direction we do not need to consider).
Calculate the Resultant force