A siphon (Fig. P12.88) is a convenient device for removing...

A $siphon$ ($\textbf{Fig. P12.88}$) is a convenient device for removing liquids from containers. To establish the flow, the tube must be initially filled with fluid. Let the fluid have density $\rho$, and let the atmospheric pressure be $p$$_a$$_t$$_m$. Assume that the cross-sectional area of the tube is the same at all points along it. (a) If the lower end of the siphon is at a distance $h$ below the surface of the liquid in the container, what is the speed of the fluid as it flows out the lower end of the siphon? (Assume that the container has a very large diameter, and ignore any effects of viscosity.) (b) A curious feature of a siphon is that the fluid initially flows "uphill." What is the greatest height $H$ that the high point of the tube can have if flow is still to occur? $\textbf{ELEPHANTS UNDER PRESSURE.}$ An elephant can swim or walk with its chest several meters underwater while the animal breathes through its trunk, which remains above the water surface and acts like a snorkel. The elephant’s tissues are at an increased pressure due to the surrounding water, but the lungs are at atmospheric pressure because they are connected to the air through the trunk. The figure shows the gauge pressures in an elephant's lungs and abdomen when the elephant’s chest is submerged to a particular depth in a lake. In this situation, the elephant's diaphragm, which separates the lungs from the abdomen, must sustain the difference in pressure between the lungs and the abdomen. The diaphragm of an elephant is typically 3.0 cm thick and 120 cm in diameter. (See "Why Doesn't the Elephant Have a Pleural Space?" by John B. West, $Physiology$, Vol. 17:47–50, April 1, 2002.) Figure P12.88 (FIGURE CAN'T COPY)

Transcript

00:01 So a siphon has interesting properties, and we are going to apply bernoulli's equation in order to investigate this.

00:08 And if we were to apply it to the top of the tube, we have the atmospheric pressure plus half the density times the velocity in the top of the tube, i apologize, times the velocity in the top of the tube, plus the density times the gravity, times the height.

00:33 And then if we were to apply it to the bottom of the tube, we have the atmospheric pressure again, plus half times the density times the velocity sub 2 squared.

00:47 And then we don't have this pg8, this row gh term on this bottom, because it's going to be at the bottom of the tube and there isn't any potential energy in this, at the bottom of the tube.

01:07 So to further define it.

01:11 So basically, we can cancel these atmospheric pressures out, and we're trying to solve for velocity.

01:20 So in this sense, we could say that we could move this over and say that the density times the velocity sub 1 squared plus 2 row gh equals rather simply row v squared, v .2 squared.

01:55 And you see that the density is all here, so you can cancel it out.

02:00 It's present in all of the terms.

02:02 And then we see that the velocity sub 2 squared is going to be equal to the velocity initials, the velocity sub 1 squared in the first in the top of the pipe plus 2gh.

02:20 Now for a container with a large diameter and at the top of the container, this container is going to have a large diameter.

02:30 So if it's for a container with a large diameter, we can say that the velocity is going to be zero.

02:42 So for a container with a large diameter, we can say that the velocity is going to be zero meters per second.

02:51 So given this, that means that this can cancel out.

02:57 And we simply have that the velocity at the bottom of the container is simply going to be radical to gh, which is exactly the same thing as we get when we apply the conservation of energy, given that all kinetic energy transfers to all potential energy so it's the same exact equation however we arrived to this formula through the bernoulli's equation versus the conservation of energy and again for a large container that with a large diameter that velocity is going to be zero so and then so this is the answer for part a and then it's asking what is the highest what is the greatest height that i can reach...

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