Head Loss and Pressure Rise During Gradual Expansion

[mech-eng]

Homework Statement

,

There is an Head Loss question, two pipes join and expanding parts 30 degrees from the horizantal.

Homework Equations

The Attempt at a Solution

Before attemting a solution I try to understand what a₁=a₂=1.06 is?

Would you like to give some information about it?

Thank you.

Source: Fluid Mechanics, Fundamentals and Applications by Çengel/Cimbala

 

[Chestermiller]

What does your book say about the definition of $\alpha$?

 

[mech-eng]

I understand it was in a earlier chapter and it is kinetic energy correction factor but why is the velocity of narrow part, namely 7 m/s, is used while calculating irreversible head loss in the expansion section because the velocity is 3.11 m/s in that section?

Thank you.

 

[mech-eng]

Is there anyone who can explain why the V1, the velocity of narrow section is used instead of the velocity V2, the velocity of wide section. And we should note that the head losses happen in the wide section or when passing to it.

Thank you.

 

[Chestermiller]

I understand it was in a earlier chapter and it is kinetic energy correction factor but why is the velocity of narrow part, namely 7 m/s, is used while calculating irreversible head loss in the expansion section because the velocity is 3.11 m/s in that section?

Thank you.

The 0.07 is an empirical factor, presumably developed for a diverging section of very long extent. I know that, for a laminar viscous flow, in such a situation, only the flow near the narrow cross section determines the pressure drop for the diverging section. This is not a laminar viscous flow but apparently the same kind of situation prevails. So the use of the loss coefficient K would seem to capture this effect, and only includes the velocity at the narrow cross section. It neglects the fact that the diverging section straightens out after the larger cross section.

Chet

 

[mech-eng]

The 0.07 is an empirical factor, presumably developed for a diverging section of very long extent. I know that, for a laminar viscous flow, in such a situation, only the flow near the narrow cross section determines the pressure drop for the diverging section. This is not a laminar viscous flow but apparently the same kind of situation prevails. So the use of the loss coefficient K would seem to capture this effect, and only includes the velocity at the narrow cross section. It neglects the fact that the diverging section straightens out after the larger cross section.

Chet

But what does a divergin section refer to for this context? Is it narrow section?

Thank you.

 

[Chestermiller]

But what does a divergin section refer to for this context? Is it narrow section?

Thank you.

The diverging section is the part of the pipe between the small cross section and the large cross section, going from small to large.

 

[mech-eng]

It neglects the fact that the diverging section straightens out after the larger cross section.
Chet

What does "diverging section straightens out after ..." refer to? I cannot fix the meaning of "straightens out". Would you please explain it another way?

Thank you.

 

[SteamKing]

What does "diverging section straightens out after ..." refer to? I cannot fix the meaning of "straightens out". Would you please explain it another way?

Thank you.

You've got two straight pipes, with diameters D1 < D2, connected by a diverging conical fitting. The flow in pipe with diameter D1 is turbulent with straight streamlines, as is the flow in the pipe with diameter D2 after the fitting, like so:

​

The phrase 'straightens out' refers to what happens to the streamlines in the fluid after they pass downstream of the diverging fitting and a new velocity profile is established in the pipe with diameter D2.

By convention, the minor loss K_L in the diverging fitting is based on the flow velocity V1 in the pipe with diameter D1 as it enters the fitting, rather than the velocity V2 as it exits.