Fluids
Fluids (part 12) Complete example of a Bernoulli's Equation exercise.
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- Let's say I have a horizontal pipe that at the left end of
- the pipe, the cross-sectional area, area 1, which is equal
- to 2 meters squared.
- Let's say it tapers off so that the cross-sectional area
- at this end of the pipe, area 2, is equal to
- half a square meter.
- We have some velocity at this point in the pipe, which is
- v1, and the velocity exiting the pipe is v2.
- The external pressure at this point is essentially being
- applied rightwards into the pipe.
- Let's say that pressure 1 is 10,000 pascals.
- The pressure at this end, the pressure 2-- that's the
- external pressure at that point in the pipe-- that is
- equal to 6,000 pascals.
- Given this information, let's say we have
- water in this pipe.
- We're assuming that it's laminar flow, so there's no
- friction within the pipe, and there's no turbulence.
- Using that, what I want to do is, I want to figure out what
- is the flow or the flux of the water in this pipe-- how much
- volume goes either into the pipe per second, or out of the
- pipe per second?
- We know that those are the going to be the same numbers,
- because of the equation of continuity.
- We know that the flow, which is R, which is volume per
- amount of time, is the same thing as the input velocity
- times the input area.
- The input area is 2, so it's 2v1, and that also equals the
- output area times output velocity, so it equals 1/2 v2.
- We could rewrite this, that v1 is equal to 1/2 R, and that v2
- is equal to 2R.
- This immediately tells us that v2 is coming out at a faster
- rate, and this is based on the size of the openings.
- We know, because V2 is coming out at a faster rate, but we
- also know because we have much higher pressure at this end
- than at this end, that the water is flowing to the right.
- The pressure differential, the pressure gradient, is going to
- the right, so the water is going to
- spurt out of this end.
- And it's coming in this end.
- Let's use Bernoulli's equation to figure out what the flow
- through this pipe is.
- Let's just write it down: P1 plus rho gh1 plus 1/2 rho v1
- squared is equal to P2 plus rho gh2
- plus 1/2 rho v2 squared.
- This pipe is level, and the height at either end is the
- same, so h1 is going to be equal to h2.
- These two terms are going to be equal, so we can cross them
- out-- we can subtract that value from both sides, and
- we're just left with P1.
- What's P1?
- P1 is 10,000 pascals plus 1/2 rho times v1 squared.
- What's v1?
- That's R over 2-- we figured that out up here.
- v2 times R over 2 squared is equal to P2, and that's 6,000
- pascals plus 1/2 rho times v2 squared.
- We figured out what v2 is-- v2 is 2R squared.
- Let's just do some simplification, and so let's
- subtract 6,000 from both sides, and we're left with
- 4,000 plus rho R squared over 8 is equal to 1/2 times R
- squared times 4.
- So this is 2 rho R squared.
- We could multiply both sides of this equation by 8, just to
- get rid of this in the denominator, so we would get
- 32,000 plus rho R squared is equal to 16 rho R squared.
- Subtract rho R squared from both sides of this question,
- and we get 32,000 is equal to 15 rho R squared.
- Then what's rho?
- What's the density of water?
- The density of water is 1,000 kilograms per meter cubed, so
- this is 1,000.
- Let's divide both sides by 15 times rho.
- We get R squared is equal to 32,000 divided by 15 rho-- rho
- is 1,000, so R squared is equal to 32,000 over 15,000,
- which is the same thing is 32 over 15.
- R is equal to the square root of 32 over 15, and that's
- going to be meters cubed per second.
- I get 32 divided by 15 is equal to 2.1, and the square
- root of that 1.46.
- So the answer is R is equal to 1.46 meters cubed per second.
- That is the volume of water that is either entering the
- system in any given second, or exiting the system in any
- given second.
- We can figure out the velocities, too-- what's the
- velocity exiting the system?
- What's two times that?
- It's 2.8 meters per second exiting the system, and going
- in it is half that, so it's 0.8 meters per second.
- Hopefully, that gives you-- actually, 0.7 meters per
- second-- a bit more intuition on fluids, and that's all I'm
- going to do for today.
- I'll see you in the next video, and we're going to do
- some stuff on thermodynamics.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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