A hydroponic garden uses a 10-m-long perforated pipe system to deliver water at

Question

A hydroponic garden uses a 10-m-long perforated pipe system to deliver water at 20°C. The pipe is 5 cm in diameter and contains a circular hole every 20 cm. A pump delivers water at 75 kPa (gage) at the entrance, while the other end of the pipe is closed. The pressure near the closed end of this perforated "manifold" is surprisingly high, and there will be too much flow through the holes near that end. One remedy is to vary the hole size along the pipe axis. Make a design analysis to pick the optimum hole size distribution that will make the discharge flow rate as uniform as possible along the pipe axis. You are constrained to pick hole sizes that correspond only to commercial metric drill-bit sizes available to the typical machine shop. It is recommended that you consult Machinery's Handbook or talk to Oleg Shargorodsky in the machine shop (basement of the MakerSpace) to determine acceptable drill-bit sizes. 20 cm Pump 10 m

Explanation / Answer

Relevant equations:-

1) bernoulli's modified energy equation for head loss:

P1/gamma + V1^2/2*g + z1 = P2/gamma + V2^2/2*g + z2 + hf(headloss)

Where P=gage pressure kpa, V= average velocity in pipe, Gamma=rho*g, z= vertical position in meters, g=gravity. rho=density, 1 and 2 denote different places of interest in center of pipe.

2) Head loss equation with respect to friction factor:

hf= f*L*V^2/(2*g*D)

where f= friction factor, L = length of pipe where flow is being analyzed, D is diameter of pipe, g= gravity,

3) V= 4Q/(pi*D^2)

where Q= flow rate in m^3/s

4) Reynolds #, (Re)= rho*V*D/mu

mu= viscosity

turbulent flow Re>2300

Laminar flow Re<2300

5) for Laminar flow, f=64/Re

for Turbulent flow, f= determined from moody chart

Assume smooth pipe curve for moody chart.

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