Height of a fluid in a small tube

In summary, the conversation discusses a system of two tubes connected together, with a fluid of uniform density flowing through them. The goal is to find the height of a small tube branching off the larger one. After using Bernoulli's equation and the continuity equation, it is determined that the pressure and velocity of the fluid are related to the height of the small tube. The faster the fluid flows, the lower the height of the small tube becomes. A negative height would indicate a vacuum pump.
  • #1
Ampere
44
0

Homework Statement



An upper tube of cross sectional area A, atmospheric pressure, and height h1 is connected to a lower tube with cross sectional area A/2 (and height 0). A fluid of uniform density rho flows through the system, with velocity v measured in the upper tube. A small open tube branches up off the lower one, with a column of fluid of height h2 inside it. Find h2.

(See attached diagram.)

Homework Equations



Bernoulli's equation,

P1 + pgh1 +1/2*pv12 = constant

Continuity equation,

A1v1 = A2v2

The Attempt at a Solution



Applying Bernoulli's equation between the top of fluid of height h2 and the left endpoint of the system, I got:

(point 1 = left endpoint, point 2 = top of the column with height h2)

P1 = atmospheric
P2 = atmospheric
KE1 = 1/2*rho*v2
KE2 = 0
PE1 = rho*g*h1
PE2 = rho*g*h2

Therefore h2 = h1 + v2/(2g). Is this right? I would think that a faster moving fluid pushes h2 up even higher (with h2 = h1 when v=0).

The continuity equation tells us that the speed of the fluid inside the smaller tube is 2v. But does this even matter?
 

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  • #2
Welcome to PF!

Hi Ampere! Welcome to PF! :smile:
Ampere said:
Applying Bernoulli's equation between the top of fluid of height h2 and the left endpoint of the system …

No, you can't do that … Bernoulli's equation only applies along a streamline

and the streamline doesn't go up that little tube, does it? :wink:
The continuity equation tells us that the speed of the fluid inside the smaller tube is 2v. But does this even matter?

Yes, because that is in the streamline!
 
  • #3
Thanks. So in that case you can find the pressure right under the small tube (with height h2 - did this below). But what equation do I use to travel outside the streamline?

P1 = atmospheric
P2 = unknown
KE1 = 1/2*rho*v2
KE2 = 1/2*rho*(2v)2
PE1 = rho*g*h1
PE2 = 0

So P2 = Patm + rho*g*h1 - 3/2*rho*v2
 
  • #4
Hi Ampere! :smile:
Ampere said:
Thanks. So in that case you can find the pressure right under the small tube (with height h2 - did this below). But what equation do I use to travel outside the streamline?

The fluid in the thin vertical tube is stationary, so just use pressure difference = ρgh :wink:

(technically, you can use any line in a stationary fluid as a streamline … but that doesn't mean you can then tag it onto a "real" streamline in a moving part of the fluid :wink:)
 
  • #5
So I get

dP = rho*g*h2
P2 - Patm = rho*g*h2

and

h2 = h1 - 3v2/(2g)

So the faster the fluid flows, the lower the height h2 is? Is there actually a flow rate which would make h2 = 0, or even less than 0?
 
  • #6
Ampere said:
So the faster the fluid flows, the lower the height h2 is?

Bernoulli's equation does mean that when the speed is faster the pressure is lower. :smile:
Is there actually a flow rate which would make h2 = 0, or even less than 0?

work it out! :wink:
 
  • #7
I found that the speed at which h2=0 is √(2gh1/3).

So if h1 = 1m, then v = 2.556 m/s. What if I had v = 3 m/s, or even 6 m/s? What would be the physical meaning of a negative height?

For example, if we go with h1 = 1m and v = 6 m/s, we get h2 = -4.51m. Huh?
 
  • #8
Ampere said:
What would be the physical meaning of a negative height?

It would mean you have a vacuum pump! :smile:
 
  • #9
Ah so air would be sucked into the tube along with the fluid? Interesting!

Thanks, this makes sense now.
 

1. What is the height of a fluid in a small tube?

The height of a fluid in a small tube is determined by several factors, including the type of fluid, the diameter and length of the tube, and the force of gravity. It can be calculated using the equation P = ρgh, where P is the pressure of the fluid, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid.

2. How does the height of a fluid in a small tube affect the pressure?

The height of a fluid in a small tube has a direct relationship with the pressure of the fluid. As the height of the fluid increases, the pressure also increases. This is because the weight of the fluid increases with height, causing an increase in pressure at the bottom of the tube.

3. What is the relationship between the height of a fluid in a small tube and the volume of the fluid?

The height of a fluid in a small tube is inversely proportional to the volume of the fluid. This means that as the height of the fluid increases, the volume of the fluid decreases. This relationship is due to the fact that the total volume of the fluid must remain constant, and as the height increases, the cross-sectional area of the tube decreases, resulting in a decrease in volume.

4. How does the height of a fluid in a small tube change when the tube is tilted?

When a small tube containing a fluid is tilted, the height of the fluid on one side of the tube will be higher than the other side. This is because the force of gravity acts perpendicularly to the surface of the fluid, causing it to rise higher on the side where the tube is tilted. The difference in height between the two sides can be calculated using trigonometry.

5. How can the height of a fluid in a small tube be measured accurately?

The height of a fluid in a small tube can be measured accurately using a variety of methods, such as a graduated cylinder or a burette. These devices have markings that allow for precise measurement of the fluid's height. Additionally, using a digital or analog pressure gauge can also provide an accurate measurement of the fluid's height by measuring the pressure inside the tube.

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