What is the diameter of each hole in the shower head?

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SUMMARY

The diameter of each hole in a low-flow showerhead with 18 nozzle holes, a volume flow rate of 4.00 L/min, and a maximum water speed of 2.70 m/s can be calculated using fluid dynamics equations. The volumetric flow rate (Q) is converted to cubic meters per second as 6.6667E-5 m³/s. The average velocity through each hole is determined to be 1.35 m/s, and the viscosity of water is noted as 8.90E-4 Pa·s. The solution requires dividing the total flow rate by the number of holes to find the volumetric throughput rate per hole.

PREREQUISITES
  • Understanding of fluid dynamics principles
  • Familiarity with the equations of continuity and flow rate
  • Basic knowledge of viscosity and its effects on fluid flow
  • Ability to perform unit conversions (L/min to m³/s)
NEXT STEPS
  • Study the derivation of the equation Q=Av_max for fluid flow
  • Learn about the effects of viscosity on flow rates in pipes
  • Explore the relationship between nozzle diameter and flow rate in showerheads
  • Investigate the application of Bernoulli's principle in fluid dynamics
USEFUL FOR

Students in physics or engineering courses, mechanical engineers, and anyone involved in designing or analyzing fluid flow systems, particularly in plumbing and showerhead design.

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Homework Statement


In a low-flow showerhead that contains 18 nozzle holes, the volume flow rate of the water is 4.00 L/min and the water maximum speed is 2.70 m/s.

What is the diameter of each hole? Do not ignore viscosity of water.

Homework Equations


Q=Av_max
Q=((pi)r^2)v_max
Q=(pi(r^4)/8nl)*(deltaP)
v_max=2Q/(pi(r^2))
d=2r

3. The Attempt at a Solution
Given:
Q:4L/min = 6.6667E-5 m^3/s
v_max=2.7 m/s
v_average=v_max/2
v_average=1.35 m/s
n(vescosity):8.90E-4

I am not quite sure where to start, I used the v_max equation, but I got the wrong answer. I know at some point I may need to divide by 18 for one nozzle.
 
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What is the volumetric throughput rate per hole? What is the average velocity through each hole? How is the average velocity per hole related to the volumetric throughput rate per hole?
 

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