Bernoulli's Theorem: Perfume Atomizer
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An old fashioned perfume atomizer operates by the user squeezing a bulb which sets up a rapid flow of air through a horizontal tube. The reduction in pressure due tothis rapid flow causes perfume to be sucked up a vertical tube from the bottle and then expelled from the horizontal tube along with the air. For simplification, we assume that immediately after the bulb is squeezed there is no pressure gradient in the horizontal tube, and that immediately outside the tube, flow ceases and the pressure is atmospheric pressure Po. Furthermore, we assume that the height of the liquid in the bottle remains constant and that capillary action effects are negligible.
1) Consider first the air flowing through the horizontal section of tubing. Use Bernolli's equation to derive the following expression for the pressure P in the horizontal tube at the top of the vertical tube:
P = Po-1/2(density^a)v^2, where v is the velocity of the air in the tube and density^a is the density of the air.
2) Consider the case where the air flows with just sufficient velocity to bring the perfume to the top of vertical pipe, but not into the horizontal section of pipe. Use bernoulli's equation to show that this velocity is given by the following equation:
v = sqrt(2gh)(density^p)/(density^a), where density^p is the density of the perfume and h is the height where the vertical pipe meets the horizontal pipe above the surface of the liquid in the bottle.
Calculate the velocity when the horizontal pipe is 5.00 cm above the surface of the liquid in the bottle, you may take the density of air to be 1.30 kg m^-3 and the density of the perfume to be that of water, 1000 kg m^-3.
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Solution Summary
1. By Bernoulli's theorem, total energy density = constant --> P + (1/2)*density^a * v^2 = Po + (1/2)*density^a * v(outside)^2. 2. If the velocity of air is just sufficient to raise up the perfume to the top of vertical pipe, by Bernoulli's theorem, energy of air = energy density of perfume at top level of vertical tube.
Solution Preview
1) By Bernoulli's theorem,
Total energy density = constant
P + (1/2)*density^a * v^2 = Po + (1/2)*density^a * v(outside)^2
Because, v(outside) = velocity of air outside the tube = 0
=> P + (1/2)*density^a * v^2 = Po
=> P = Po - ...
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