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12.2 Bernoulli’s equation  (Page 3/7)

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Bernoulli's principle—bernoulli's equation at constant depth

Another important situation is one in which the fluid moves but its depth is constant—that is, h 1 = h 2 size 12{h rSub { size 8{1} } =h rSub { size 8{2} } } {} . Under that condition, Bernoulli's equation becomes

P 1 + 1 2 ρv 1 2 = P 2 + 1 2 ρv 2 2 . size 12{P rSub { size 8{1} } + { {1} over {2} } ρv rSub { size 8{1} } "" lSup { size 8{2} } =P rSub { size 8{2} } + { {1} over {2} } ρv rSub { size 8{2} } "" lSup { size 8{2} } "." } {}

Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli's principle    . It is Bernoulli's equation for fluids at constant depth. (Note again that this applies to a small volume of fluid as we follow it along its path.) As we have just discussed, pressure drops as speed increases in a moving fluid. We can see this from Bernoulli's principle. For example, if v 2 size 12{v rSub { size 8{2} } } {} is greater than v 1 size 12{v rSub { size 8{1} } } {} in the equation, then P 2 size 12{P rSub { size 8{2} } } {} must be less than P 1 size 12{P rSub { size 8{1} } } {} for the equality to hold.

Calculating pressure: pressure drops as a fluid speeds up

In [link] , we found that the speed of water in a hose increased from 1.96 m/s to 25.5 m/s going from the hose to the nozzle. Calculate the pressure in the hose, given that the absolute pressure in the nozzle is 1 . 01 × 10 5 N/m 2 size 12{1 "." "01" times "10" rSup { size 8{5} } `"N/m" rSup { size 8{2} } } {} (atmospheric, as it must be) and assuming level, frictionless flow.

Strategy

Level flow means constant depth, so Bernoulli's principle applies. We use the subscript 1 for values in the hose and 2 for those in the nozzle. We are thus asked to find P 1 size 12{P rSub { size 8{1} } } {} .

Solution

Solving Bernoulli's principle for P 1 size 12{P rSub { size 8{1} } } {} yields

P 1 = P 2 + 1 2 ρv 2 2 − 1 2 ρv 1 2 = P 2 + 1 2 ρ ( v 2 2 − v 1 2 ) . size 12{P rSub { size 8{1} } =P rSub { size 8{2} } + { {1} over {2} } ρv rSub { size 8{2} } "" lSup { size 8{2} } - { {1} over {2} } ρv rSub { size 8{1} } "" lSup { size 8{2} } =P rSub { size 8{2} } + { {1} over {2} } ρ \( v rSub { size 8{2} } "" lSup { size 8{2} } - v rSub { size 8{1} } "" lSup { size 8{2} } \) "." } {}

Substituting known values,

P 1 = 1 . 01 × 10 5 N/m 2 + 1 2 ( 10 3 kg/m 3 ) ( 25.5 m/s ) 2 − ( 1.96 m/s ) 2 = 4.24 × 10 5 N/m 2 .

Discussion

This absolute pressure in the hose is greater than in the nozzle, as expected since v is greater in the nozzle. The pressure P 2 size 12{P rSub { size 8{2} } } {} in the nozzle must be atmospheric since it emerges into the atmosphere without other changes in conditions.

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Applications of bernoulli's principle

There are a number of devices and situations in which fluid flows at a constant height and, thus, can be analyzed with Bernoulli's principle.

Entrainment

People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment . Entrainment devices have been in use since ancient times, particularly as pumps to raise water small heights, as in draining swamps, fields, or other low-lying areas. Some other devices that use the concept of entrainment are shown in [link] .

Part a of the figure shows a rectangular section of a cylindrical Bunsen burner as a vertical column. The natural gas is shown to enter the rectangular column from the bottom upward. The air is shown to enter though a nozzle at the left side near the bottom part of the rectangular column and rise upward. Both air and natural gas are shown to rise up together along the length of the column, shown as vertical arrows along the length pointing upward. Part b of the figure shows an atomizer that uses a squeeze bulb in the shape of a small sphere to create a jet of air that entrains drops of perfume contained in a spherical bottomed container. The air is shown to come out of the squeeze bulb and the perfume is shown to rise up from the spherical bottomed container. Part c of the figure shows a common aspirator which contains a cylindrical tube held vertically. The tube is broader on the top and narrow at the bottom. Water is shown to enter the tube from the broader region and flow toward the narrow region. Air is shown to enter the cylindrical tube from the bottom part of the broader side and also flow toward the narrow tube. Part d of the figure shows the chimney of a water heater. Water heater is shown as a rectangular box at the bottom having a cylindrical section in the middle. The cylindrical section is broader at the bottom and narrow toward the top. Hot air is shown to rise up along the vertical section of the cylindrical tube. The chimney is conical at the bottom and rectangular upward and is shown above the rectangular water heater. The hot air enters the chimney at the conical end and rises upward. Cool air is shown to enter the chimney through the area between the rectangular section of heater and chimney from the two sides and rise up along the chimney with the hot air as shown by vertical arrows.
Examples of entrainment devices that use increased fluid speed to create low pressures, which then entrain one fluid into another. (a) A Bunsen burner uses an adjustable gas nozzle, entraining air for proper combustion. (b) An atomizer uses a squeeze bulb to create a jet of air that entrains drops of perfume. Paint sprayers and carburetors use very similar techniques to move their respective liquids. (c) A common aspirator uses a high-speed stream of water to create a region of lower pressure. Aspirators may be used as suction pumps in dental and surgical situations or for draining a flooded basement or producing a reduced pressure in a vessel. (d) The chimney of a water heater is designed to entrain air into the pipe leading through the ceiling.

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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