Torricelli's law

src: i.ytimg.com

Torricelli's Law , also known as Torricelli's Theorem , is a theorem in fluid dynamics relating to the velocity of the liquid flowing out of the hole to the liquid level above the opening. The law states that the final velocity, v , from the liquid through the sharp hole at the bottom of the filled tank to the depth of h is the same as accelerating that the body (in this case a drop of water) will obtain free fall from height h , ie ${\ displaystyle v = {\ sqrt {2gh}}}$ , where g is acceleration due to gravity (9.81 N/kg near the surface of the earth). This last expression comes from equating the obtained kinetic energy, ${\ displaystyle {\ frac {1} {2}} mv ^ {2}}$ , with missing potential energy, mgh , and the completion for v . The law was discovered (though not in this form) by Italian scientist Evangelista Torricelli, in 1643. This then proved to be a special case of the Bernoulli principle.

Video Torricelli's law

Derivasi

Di bawah asumsi cairan mampat dengan viskositas yang dapat diabaikan, prinsip Bernoulli menyatakan bahwa:

${\ displaystyle {v ^ {2} \ over 2} gh {p \ over \ rho} = {\ text {constant}}}  ÂÂ$

where v is the velocity of the fluid, g is the acceleration of gravity (9,81 m/s ² ), h is the fluid level above the reference point, p is pressure, and ? is the density. Determine the opening to be at h = 0. At the top of the tank, p is equal to atmospheric pressure. v can be considered 0 because the liquid surface drops very slowly compared to the speed at which the liquid exits from the tank. At opening, h = 0 and p are atmospheric pressure again. Eliminating constants and solutions gives:

Torricelli's law can be demonstrated in spouting experiments, designed to show that in liquids with open surfaces, the pressure increases with depth. It consists of a tube with three separate holes and an open surface. Three holes are blocked, then the tube is filled with water. When it is full, the holes are blocked. The lower the jet is in the tube, the stronger it is. The speed of exit fluid further below the tube.

Ignoring other viscosities and losses, if the nozzle leads vertically upwards each jet will reach the surface level of the liquid in the container.

Maps Torricelli's law

Total time to empty the container

Pertimbangkan wadah berbentuk silinder yang berisi air hingga tinggi h sedang dikosongkan melalui tabung dengan bebas. Biarlah ketinggian air setiap saat. Biarkan kecepatan efflux menjadi ${\ displaystyle v = {dx \ over dt} = {\ sqrt {2gh}} \}  ÂÂ$

is the time it takes to free water from the height of h ₁ to h ₂ h ₁ & gt; h ₂ . This formula can be used to calibrate the water clock.

src: i.ytimg.com

Clepsydra Problem

Clepsydra is a clock that measures time based on water flow. It consists of a pot with a small hole in the bottom where the water can come out. The amount of water that comes out gives the size of time. As given by Torricelli's law, the rate of exhaust through a hole depends on the height of the water; and when the water level decreases, the discharge is not uniform. The simple solution is to keep the water level constant. This can be achieved by allowing a constant flow of water into the ship, an abundance that allows to escape from above, from other holes. Thus having a constant height, water consumption from below can be collected in another cylindrical vessel with a uniform graduation to measure time. This is the clepsydra inflow.

Or, carefully choosing the shape of the ship, the water level on the ship can be made downhill at a constant rate. By measuring the water level remaining in the vessel, time can be measured. This is an example of clepsydra outflow. Because the water flow rate is higher when the water level is higher (due to more pressure), the volume of fluid should be more than just a cylinder when the water level is high. That is, the fingers should be bigger when the water level is higher. Leave the radius ${\ displaystyle r}$ increases with water level elevation $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂhÂ\; Â\; Â\; Â\; Â\; Â}Â\; Â\; Â\; Â$ above the ${\ displaystyle a.}$ That is, ${\ displaystyle r = f (h)}$ . We want to find a radius that has a constant drop rate, ie ${\ displaystyle dh/dt = c}$ .

Pada tingkat air tertentu $            {\displaystyle         h      }        \{\backslash \; displaystyle\; h\}\;  ÂÂ$ , luas permukaan air adalah ${\ displaystyle A = \ pi r ^ {2}}  ÂÂ$ . Tingkat perubahan instan dalam volume air adalah

${\ displaystyle {\ frac {dV} {dt}} = A {\ frac {dh} {dt}} = \ pi r ^ {2} c.}  ÂÂ$

Dari hukum Torricelli, tingkat keluarnya

${\ displaystyle {\ frac {dV} {dt}} = av = a {\ sqrt {2gh}},}  ÂÂ$

Dari dua persamaan ini,

${\ displaystyle {\ begin {aligned} a {\ sqrt {2gh}} & amp; = \ pi r ^ {2} c \\\ Rightarrow \ quad h & amp; = { \ frac {\ pi ^ {2} c ^ {2}} {2ga ^ {2}}} r ^ {4}. \ end {aligned}}}  ÂÂ$

Dengan demikian, radius wadah harus berubah sebanding dengan akar kuartik tingginya, ${\ displaystyle r \ propto {\ sqrt [{4}] {h}}.}  ÂÂ$

src: slideplayer.com

Lihat juga

• Hukum Darcy
• Tekanan dinamis
• Statika fluida
• Persamaan Hagen-Poiseuille
• Teorema Helmholtz
• persamaan Kirchhoff
• Persamaan Knudsen
• persamaan Manning
• Persamaan kemiringan ringan
• Persamaan Morison
• Persamaan Navier-Stokes
• Aliran Oseen
• Hukum Pascal
• Hukum Poiseuille
• Aliran potensial
• Tekanan
• Tekanan statis
• Kepala tekanan
• Persamaan relativitas Euler
• Dekomposisi Reynolds
• Aliran Stokes
• Fungsi aliran Stokes
• Fungsi streaming
• Merampingkan, streaklines dan pathlines

src: i.ytimg.com

Referensi

src: pbs.twimg.com

Bacaan lebih lanjut

• T. E. Faber (1995). Dinamika Fluida untuk Fisikawan . Cambridge University Press. ISBN: 0-521-42969-2.
• Stanley Middleman, Pengantar Dinamika Fluida: Prinsip Analisis dan Desain (John Wiley & amp; Sons, 1997) ISBNÂ 978-0-471-18209-2
• Dennis G. Zill, Kursus Pertama dalam Persamaan Diferensial (2005)

Source of the article : Wikipedia