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do not memorize it like this since it is long and people will be intimidated by it at first
memorize its as this, with only one side of the equation, knowing that the opposite side is the exact same, but measurments of each pressure term are taking under different circumstances.
so when you look at bernoulli's equation, think of it as a conservation of pressure through a pipe analogous to conservation of energy. The 1/2pv^2 term is like the kinetic energy term. The pgz term is like the potential energy term in the kinematics problems we already learned. In those kinematics type problems where, for example, we have a ball dropped from a certain height. The kinetic energy + potential energy is equal to the total energy of the ball. Ke + pe = e. And according to that equation, before the ball is dropped, it has all potential energy, but when we drop it, the potential energy decreases since it is converted into kinetic energy, whose term increases. But the total energy of the ball is still the same! Thats how i think of a fluid in a pipe. Instead of a conservation of energy, its a conservation of pressure.
Okay, so imagine a pipe with two different cross-sectional areas. Maybe a question asks us what the p2 pressure is at that point2. You can use
to calculate this. Since the pipes are at the same height, at points 1 and 2 we can just cancel the
pgz terms. All that is left is the "kinetic pressure" term and the "intrinsic" pressure term, p. Mcat will give us a velocity of the fluid at either point 1 or point 2. Using that velocity we can calculate the velocity of the fluid at the other point using the equation for volumetric flow rates in a pipe,
v1a1 = v2a2. Mcat will also give you the "intrinsic" pressure at one point, so in this case they would give us p1 since we are solving for p2. So now just set the two sides of bernoulli's equation equal to each other since there is a conservation of total pressure throughout the pipe. Then just plug in the numbers and do the math and you will get the pressure,
p2.
the reason i think its useful to think about it as a conservation of pressure, is for when they ask conceptual questions about changes in the intrinsic pressure or velocity of the fluid. If they ask about if they decrease the radius of a pipe, then you know that they decreased the area of the pipe compared to earlier sections of the pipe where the radius was bigger. Using
v1a1 = v2a2, you will see that since the area was decreased at this point, then the velocity will increase! If velocity increases then the "kinetic pressure" term,
1/2pv^2, will also increase. Since that term increased, according to bernoulli's equation which is about conversation of pressure, then that means the "intrinsic" pressure, p, had to decrease.
p +
1/2pv^2 +
pgz = constant
this is just like a kinematics/gravity problem i was talking about earlier. The ball might not have as much potential energy as it falls, the kinetic energy has increased when the ball falls, but the total energy of the ball is still conserved. Their relationship shows that as one term changes, the other term will change in the opposite way so that they sum to the total energy of the ball. Just like that, the total pressure of a fluid in a pipe is conserved.
One problem i keep seeing is where they ask you water levels in small tubes above certain points in a pipe like this:
see how in the middle of the pipe where the area is smaller, where the velocity is really high, the amount of water in the tube above it is less than when the fluid is moving slower. This is because it increases the "kinetic pressure" term in bernoulli's equation, so the "intrinsic" pressure term, p, has to decrease as a result. Since the intrinsic pressure is less in the small part of the pipe, not as much fluid is going to be forced up, that is why it is lower than at the other two points in the tube. The pgz term has no effect since the whole pipe is at relatively the same height. If these changes took place at different heights then you would have to involve the pgz term.
I think you should memorize bernoulli's and the velocity equation. Hopefully this helped you understand it a lil better.
v1a1 = v2a2
btw: Most of the time, bernoulli's can be simplified to this
because the measurements in the pipe are being taken at the same height so the pgz terms can drop out.
