A hydroponic garden uses the 10 m long perforated pipe system shown below to del

Question

A hydroponic garden uses the 10 m long perforated pipe system shown below to deliver water at 20 degrees Celsius. The pipe is 5 cm in diameter and contains a circular hole every 20 cm. A pump delivers water at 75 kPa (gauge pressure) at the entrance, while the other end of the pipe is closed. You should know that the pressure near the closed end of the perforated "manifold" is surprisinglyhigh, and there will therefore be too much flow through the holes near the end if the holes are equally sized. 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 (numbered) metric drill-bits available to a typical machine shop.

A hydroponic garden uses the 10 m long perforated pipe system shown below to deliver water at 20 degrees Celsius. The pipe is 5 cm in diameter and contains a circular hole every 20 cm. A pump delivers water at 75 kPa (gauge pressure) at the entrance, while the other end of the pipe is closed. You should know that the pressure near the closed end of the perforated "manifold" is surprisinglyhigh, and there will therefore be too much flow through the holes near the end if the holes are equally sized. 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 (numbered) metric drill-bits available to a typical machine shop.

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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