Figure 14.56 shows a siphon, which is a device for removing...

Figure 14.56 shows a siphon, which is a device for removing liquid from a container. Tube $A B C$ must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at $A .$ The liquid has density 1000 $\mathrm{kg} / \mathrm{m}^{3}$ and negligible viscosity. The distances shown are $h_{1}=25 \mathrm{cm}, d=$ $12 \mathrm{cm},$ and $h_{2}=40 \mathrm{cm} .$ (a) With what speed does the liquid emerge from the tube at $C ?$ (b) If the atmospheric pressure is 1.0 $\times 10^{5} \mathrm{Pa}$ what is the pressure in the liquid at the topmost point $B ?$ (c) Theoretically, what is the greatest possible height $h_{1}$ that a siphon can lift water?

Figure 14.56 shows a siphon, which is a device for removing liquid from a container.
Tube $A B C$ must initially be filled, but once this has been done, liquid will flow through the tube until the liquid surface in the container is level with the tube opening at $A .$ The liquid has density 1000 $\mathrm{kg} / \mathrm{m}^{3}$ and negligible viscosity. The distances shown are $h_{1}=25 \mathrm{cm}, d=$ $12 \mathrm{cm},$ and $h_{2}=40 \mathrm{cm} .$ (a) With what speed does the liquid emerge from the tube at $C ?$ (b) If the atmospheric pressure is 1.0 $\times 10^{5} \mathrm{Pa}$ what is the pressure in the liquid at the topmost point $B ?$ (c) Theoretically, what is the greatest possible height $h_{1}$ that a siphon can lift water?

Key Concepts

Siphon Operation

A siphon is a fluid system that uses the gravitational potential energy difference between two fluid reservoirs, coupled with atmospheric pressure, to move a liquid along a continuous tube. It functions once the tube is initially filled, and flow continues as long as the exit is lower than the source, without the need for any extra energy input.

Bernoulli's Principle

Bernoulli's principle states that for an incompressible, non-viscous fluid flowing along a streamline, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant. This principle allows for the analysis of how changes in elevation or cross-sectional area affect the velocity and pressure of the fluid in the siphon.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure that develops in a fluid due to the weight of the fluid above a given point. It increases with depth and is critical in understanding the pressure differences within a siphon, especially when evaluating the pressure at various points along a vertical segment of a fluid column.

Atmospheric Pressure and Maximum Siphon Lift

The maximum siphon lift concept relies on the fact that a siphon operates based on the difference between atmospheric pressure and the pressure in the highest part of the siphon. Since atmospheric pressure provides the driving force, there is a theoretical limit on the height to which a liquid can be raised before the pressure drop causes cavitation, typically determined by the liquid’s density and the ambient atmospheric pressure.

Transcript

00:01 In this problem, we told that we have a siphon draining liquid from a container.

00:05 So we have our container here.

00:07 We've set our reference vertical position at the present level.

00:13 The siphon starts a distance d down from that, rises to a distance h1 above that, and then ends at a distance d plus h2 below the surface of the water.

00:30 So the first thing we want to do is figure out what the velocity is in the siphon.

00:36 Then we want to figure out what the pressure is at the top of the siphon.

00:41 And then we want to figure out what the maximum distance is from point a to point b to get fluid to flow.

00:48 Start out, this is going to be a problem dealing with bernoulli's equation.

00:54 So we can write between a reference point and point c, so p .0, plus 1 half row v0 squared plus row g y not equals and it equals what conditions at the exit so vc plus 1 half row vc squared plus row g y c so what do we got at zero here well we have our pressures are both atmospheric velocity at the surface is zero the height at the surface is zero.

01:39 So we simply get that vc equals square root of minus 2 gyc.

01:52 That's the square root of 2 times 9 .8 meters per second squared.

02:00 Yc is minus, so we cancel the minus out here.

02:05 And yc is d plus h2, so that's 0.

02:14 Plugging our numbers in we get that flow rate is 3 .2 meters per second.

02:22 So now we want to figure out what the pressure at point b is.

02:27 So again we can write bernoulli's equation between the surface and point b.

02:33 So again we have some things.

02:35 The velocity at the surface is zero.

02:37 The height of the surface of the water is zero.

02:40 We know the pressure is atmospheric there...