How Does the Diameter of Constriction Affect a Paint Sprayer's Operation?

[Coushander]

Homework Statement

A paint sprayer pumps air through a constriction in a 2.50-cm diameter pipe, as shown in the figure. The flow causes the pressure in the constricted area to drop and paint rises up the feed tube and enters the air stream. The speed of the air stream in the 2.50-cm diameter sections is 5.00 m/s. The density of the air is 1.29 kg/m3, and the density of the paint is 1200 kg/m3. We can treat the air and paint as incompressible ideal fluids. What is the maximum diameter of the constriction that will allow the sprayer to operate?

Correct Answer = 8.07mm

Homework Equations

ρvA = constant (any fluid)

vA = constant (incompressible fluid)

p + (1/2)ρv² + ρgy = constant

The Attempt at a Solution

Area at 2.5cm section = (pi)(r)² = (pi)(0.0125m)² = 4.91 x 10^-4m²

Pressure at bottom of 12.5cm tube = (1200 kg/m³)(9.81m/s²)(0.125m) = 1471.5 N/m² (Pa)

Pressure at bottom of tube = pressure at top of tube

1471.5 Pa = (1/2)(ρ)(v²)

v = 47.76 m/s

(constant at wide portion) vA = vA (constant at constricton)

A = A (v/v) = 5.14 x 10^-5 m²

r = 4.04 x 10^-3 m = 4.04mm

 

[Coushander]

Double r obviously. I had that thought while I was eating a sandwich.

d = 8.08mm (correct within rounding)