02 / CHAPTER 12
Dynamics studio: pump and pipe forces
Estimate support loads and reaction forces in a fountain plumbing run.
ANCHOR EQUATION
Pump support force
In symbols: R = −ΣFfluid
The support reaction balances the net force transmitted by the moving water.
Pump and pipe forces come from momentum flux
A bend or nozzle redirects flowing water, so the support must provide the force needed to change the water’s momentum. Pressure forces at cut surfaces matter alongside the velocity change.
Apply a control-volume balance
For steady flow, ΣF_ext = ṁ(v_out − v_in), where ṁ = ρQ is mass flow rate. The sum includes pressure forces, weight, and wall/support forces. Solve for the support reaction only after assigning directions to each term; the vector difference is why a 90° bend loads both axes.
Water with ρ = 1000 kg/m³ flows at Q = 0.002 m³/s, so ṁ = 2 kg/s. A 90° bend changes velocity from (3,0) to (0,3) m/s; momentum-force change is 2(−3,3) = (−6,6) N on the fluid, so the fluid pushes the bend (6,−6) N, magnitude 8.49 N.
Steady does not mean force-free
A steady control volume can have nonzero momentum flux and large forces even though conditions at each fixed point do not change with time. Include pressure at both sections unless both are exposed to the same atmospheric pressure and the gauge terms cancel.
Work it: For a nozzle reducing a 0.004 m² stream at 2 m/s to 0.001 m², use continuity to find outlet speed and momentum-force change. State which additional pressure data are needed for the wall force.