Fluids problem. Prefer that you use the conservation of mass, momentum or energy

Question

Fluids problem. Prefer that you use the conservation of mass, momentum or energy. Or a combination of those.
A hydroponic garden uses a 10-meter long perforated piping system, as shown, to deliver water at 20 C Pump 10 m The pump delivers water at 75 kPa (gage) at the entrance, while the other end is closed. In general, the pressure near the closed end of a manifold is surprisingly high, and there will be too much flow through the holes at that end. One remedy is to vary the hole size along the pipe axis. You are to design the piping system to make the discharge flow rate as uniform as pos- sible along the pipe axis. It is desired that the pipe have a nominal diameter as close to 5 cm as possible, and the spacing of the holes along the axis must be uniform. You are constrained to pick hole sizes that correspond to comunercially (numbered) metric drill bit sizes available in a machine shop. Is there an optimuun uuumber of holes? Comunent on the economics of your design.

Explanation / Answer

Consider control volumes around segments
containing individual holes and apply the Reynolds Transport
Theorems for mass and momentum, and Bernoulli's equation
to determine the pressure variation along the pipe and the
velocity of the jets emerging from the pipe holes. The mass
balance gives a linear decrease in the velocity along the pipe
since Q/50 volume per unit time is removed by each of the
50 holes. The momentum balance does not give the pressure,
but does give the longitudinal force on the pipe walls, in case
you needed this information. Bernoulli's equation shows
that the pressure increases downstream because the velocity
decreases. The velocity through each jet therefore increases
along the pipe (vj = (2 pj / rho )^1/2 , where vj is the jet
velocity, rho is the fluid density, and pj is the pressure for
the jth jet). The volume flow rate for each jet should be Q/50
which is vj x pi Dj^2 / 4 . Thus the jet hole diameter Dj may
be calculated from the pressure pj along the pipe, and your
assumed value of Q. Make sample calculations as you proceed
to be sure your computer or calculator program is working.
The new PowerMacs have Excel 5, which can easily do the
calculations and plots you need (339 EBII).
It is possible to apply Bernoulli's equation from the entrance
of the pipe to the jth segment. Thus,
pj = rho [ po/rho + (1/2)(Q/A)^2 (j/50)[2 - j/50]]
for j=1 to 50.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.