Pressure exerted by an ideal gas

[PainterGuy]

Hi again,

Everything is kept the same other the length of pipe in this setup. The water enters from the left and exit on the right end. In Fig. 1 the length of pipe is 1 km and in Fig. 2 it is 2 km. Do you think that the length of pipe would affect the height of water column in Tower 1, Tower 2 and Tower 3 in Fig. 2?

Moreover, in my book the Bernoulli's equation is given as P1 + ½ ρv₁^2 + ρgh₁ = P2 + ½ ρv₂^2 + ρgh₂ where P1 and P2 are pressure. If h₁ and h₂ are the same then the factors ρgh can be neglected. In the view of this equation, I don't see why the height of p1 and p3 in this picture should differ. Could you please guide me? Thanks.

 

[jbriggs444]

Moreover, in my book the Bernoulli's equation is given as P1 + ½ ρv₁^2 + ρgh₁ = P2 + ½ ρv₂^2 + ρgh₂ where P1 and P2 are pressure. If h₁ and h₂ are the same then the factors ρgh can be neglected. In the view of this equation, I don't see why the height of p1 and p2 in this picture should differ. Could you please guide me? Thanks.

https://en.wikipedia.org/wiki/Bernoulli's_principle:

"The following assumptions must be met for this Bernoulli equation to apply:[9]

• the flow must be steady, i.e. the fluid velocity at a point cannot change with time,
• the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
• friction by viscous forces has to be negligible."

Can you think of an assumption above which is not always realistic?

 

[Mister T]

What really troubles me is that if a molecule makes collisions with other molecules on its journey back and forth, it would affect the force exerted on wall drastically.

When two identical molecules have a perfectly elastic collision they simply trade momenta. So the molecule that you called molecule_1 in your later post switches places, in effect, with the one you called molecule_2 and they go on their merry way, as if there were no collision at all and they had traded identities. It has no effect on the average force exerted on the wall.

 

[PainterGuy]

Thank you.

All of those three assumptions are ideal.

But in this setup do you think that the length of pipe would affect the height of water column in Tower 1, Tower 2 and Tower 3 in Fig. 2? Everything is kept the same other the length of pipe in this setup. The water enters from the left and exit on the right end. In Fig. 1 the length of pipe is 1 km and in Fig. 2 it is 2 km.

I do understand that if the location of Tower 3 is moved to point A in Fig 2, there will be be further decrease in the height. Thanks.

 

[jbriggs444]

All of those three assumptions are ideal.

Right. In real life, one or all of them may be violated. The effects of the violation may or may not be negligible. It depends on how much error you are willing to neglect.

But in this setup do you think that the length of pipe would affect the height of water column in Tower 1, Tower 2 and Tower 3 in Fig. 2?

It depends on how much error you are willing to neglect. And pipe diameter. And pipe length. And flow rate.

 

[PainterGuy]

Thank you.

In both cases, Fig 1 and Fig 2, error tolerance is the same. Also the pipe diameter and flow rate is the same in both cases. The only difference is length - in Fig 1 it's 1 km and in Fig 2 it's 2 km. In comparative terms, would it affect the height of water column in Tower 1, Tower 2 and Tower 3 in Fig. 2 compared to Fig 1? Thank you.

 

[jbriggs444]

In both cases, Fig 1 and Fig 2, error tolerance is the same. Also the pipe diameter and flow rate is the same in both cases. The only difference is length - in Fig 1 it's 1 km and in Fig 2 it's 2 km. In comparative terms, would it affect the height of water column in Tower 1, Tower 2 and Tower 3 in Fig. 2 compared to Fig 1? Thank you.

There is no way to know from the limited information that you have provided.

Given the flow rate that you have not specified, the pipe diameter that you have not specified and the viscosity of water which you can look up, how much pressure drop would the Poiseuille equation call for?

https://en.wikipedia.org/wiki/Hagen–Poiseuille_equation

 

[PainterGuy]

Thank you.

The Hagen–Poiseuille equation is ΔP=(8μLQ)/(πR⁴) and ΔP is is the pressure difference between the two ends. I think that "the two ends" don't refer to the two ends of pipe. Could ΔP be defined as the pressure difference between two points along the pipe? Thanks.

 

[jbriggs444]

The Hagen–Poiseuille equation is ΔP=(8μLQ)/(πR⁴) and ΔP is is the pressure difference between the two ends. I think that "the two ends" don't refer to the two ends of pipe. Could ΔP be defined as the pressure difference between two points along the pipe?

Yes.

You seem strangely resistant to providing numbers or doing calculations. Say you drop three diameters off of each end of your one kilometer pipe. Can you apply Poiseuille's equation to that length?

 

[PainterGuy]

Thank you.

Is it okay? Please let me know. Thanks.

 

[PainterGuy]

Hi @jbriggs444,

Could you please confirm if the calculation is correct? Thank you.