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Rotation speed of multi-nozzle sprinkles. [ SPRNKL ]
From the author’s Engineering Collection, included in the ETSII4 module (ETI4 on the CL Library)

This program calculates the rotation speed of a custom sprinkler with N nozzles, located at different distances Ri from the center of a linear pipe-like frame, each inclined at an angle beta(i). The water intake is at the center of rotation. The flows Q and areas A are assumed to be the same for all nozzles.

The program prompts for the data values needed, such as the number of nozzles, their distances measured from the center, and orientation angles measured from the sprinkler rod. The rotation speed is obtained using the following formulas, with ro the density of the fluid, and depending on whether the resistant torque Mr is constant or proportional to the rotation speed:

Constant: w * SUM Ri^2 = (Mr/ ro Q) – [(Q/A) SUM Ri sin beta(i) ] ; i=1,2… n
Proportional: w = - [Q SUM Ri sin beta(i) ] / { A [(Mr/ro Q) + SUM Ri^2 ) } ; i = 1,2,.. n

where the negative sign denotes counter clockwise rotation. Angles are positive in the same counter clockwise direction from the reference (sprinkler arm).

The program uses the AMC_OS/X functions ARCLI and PMTK during the data entry process.

Example.

Nozzle 1 2 3 4
Ri (m) 1.5 3 -1.5 -3
beta(i) (deg) 45 45 -90 -90

Assuming that there’s no resistant torque; calculate the rotation speed of an irrigation sprinkler with 4 nozzles situated as shown in the table below. Each nozzle has an external diameter of 10 mm, and an exit flow of 7.5 l/min.

Q = 1.25 E-4 m^3/s
A = 1.25 E-4 m^2
The result is w = 0.5434 rad/s counter-clockwise.

What would it be if there was a resistant torque Mr = 1 Nm?
Density = 1 gr/cm^3 => w= 0.2519 rad/s counter-clockwise.
Reference URL's
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