# Volume flow and speed of the flow with different cross-sections and heights

Ryker

## Homework Statement

You have a pipe and 4 points on it. Points 1 and 2 are at h = 0, with point 1 having a smaller cross-section than point 2. Next, point 3 has the same cross-section as point 2, but the pipe goes up, so that h > 0. Point 4 is level with point 3, but has an even larger cross-section.

Rate the points according to:

1. The volume flow rate,
2. The flow speed.
3. The water pressure.

## Homework Equations

The Bernoulli equation and the continuity equation.

## The Attempt at a Solution

It's a been some weeks since we've covered this, and I just wanted to make sure I have the right idea.

1. The same everywhere.
2. Point 1 > points 2 and 3 > point 4. The difference in heights doesn't influence flow speed, if the area is the same, right?
3. Points 3 and 4 > points 1 and 2, because they are at the same height.

Is this correct? Thanks in advance.

Oh, and I know this one is really easy, but sometimes I just get sort of a mental block with getting some stuff I had already done, I don't know why. It's like I need to get past it and then everything flows smoothly again

Homework Helper
Hi Ryker!
1. The same everywhere.
2. Point 1 > points 2 and 3 > point 4. The difference in heights doesn't influence flow speed, if the area is the same, right?

Right!
3. Points 3 and 4 > points 1 and 2, because they are at the same height.

You've missed something out …

have another look at Bernoulli's equation

Ryker
You've missed something out …
have another look at Bernoulli's equation
$$P\ +\ \frac{1}{2}\,\rho\,v^2\ +\ \rho\,g\,h\ =\ constant$$

So from the equation, pressure at point 1 < pressure at point 2, because the speed at point 1 is greater. Also, since point 3 is higher than point 2, but the speed is equal, pressure at point 2 > pressure at point 3. And pressure at point 4 < pressure at point 3, because the speed is lower. But how can you then infer the proper order if you don't have the exact heights and cross-sections? Or is there something else I'm missing?

Oh, and same height = same pressure then only holds for standing water, I presume?

Homework Helper
Hi Ryker!
So from the equation, pressure at point 1 < pressure at point 2, because the speed at point 1 is greater. Also, since point 3 is higher than point 2, but the speed is equal, pressure at point 2 > pressure at point 3. And pressure at point 4 < pressure at point 3, because the speed is lower. But how can you then infer the proper order if you don't have the exact heights and cross-sections? Or is there something else I'm missing?

No, I agree with you.

(I suspect that they meant to ask the question with the pipe widths the other way round, so that it was 1 > 2 > 3 > 4, and got it wrong! )
Oh, and same height = same pressure then only holds for standing water, I presume?

Well, for water at the same speed.

Ryker
No, I agree with you.

(I suspect that they meant to ask the question with the pipe widths the other way round, so that it was 1 > 2 > 3 > 4, and got it wrong! )
Well, there was actually a picture, but I couldn't copy it here (it's in a .pdf file and I can't extract it), so it was indeed as I described. This question is actually from one of the previous exams, and I guess you can't quantify it better than that then, right? Which is kind of weird, because the question does specifically ask us to rank the points and indicate any ties.
Well, for water at the same speed.
Yeah, and that I guess the mistake I made first was really silly of me. I mean, if the pressures were indeed the same, then what would we do with all of the Venturi meters?