How does the diameter of a funnel affect the speed of air passing through it?

[Gerald Funk]

Homework Statement

Hello, I'm having troubles figuring out what seems to be a pretty simple physics question.
Attached is an image of the problem. what I am trying to solve is how a funnel effects the speed of air. The input velocity (v1) is 30 km/h, and I am trying to find the output velocity (v2) at the end of the funnel. The diameter of the funnel on the big end is 2.5 m (d1) and the output diameter is 2.5 cm (d2). The length of the funnel is 3 m, and the stem at the end is 10 cm (0.1 m).
I think the speed of air would speed up because its being compressed, but I am unsure of which formula to use to solve this problem. Please help me out!

Homework Equations

I didn't know which equation to use for this problem. any info helps!

The Attempt at a Solution

I didn't know which equation to use, so I haven't down any calculations

 

[SteamKing]

Homework Statement

Hello, I'm having troubles figuring out what seems to be a pretty simple physics question.
Attached is an image of the problem. what I am trying to solve is how a funnel effects the speed of air. The input velocity (v1) is 30 km/h, and I am trying to find the output velocity (v2) at the end of the funnel. The diameter of the funnel on the big end is 2.5 m (d1) and the output diameter is 2.5 cm (d2). The length of the funnel is 3 m, and the stem at the end is 10 cm (0.1 m).
I think the speed of air would speed up because its being compressed, but I am unsure of which formula to use to solve this problem. Please help me out!

Homework Equations

I didn't know which equation to use for this problem. any info helps!

The Attempt at a Solution

I didn't know which equation to use, so I haven't down any calculations

Assuming no leaks between the inlet of the funnel and the outlet, the continuity equation A₁v₁ = A₂v₂ can be used initially.

In other words, A₁ / A₂ = v₂ / v₁

A - area of the opening in m²
v - the velocity of the air flow in m/s
1 - inlet
2 - outlet

The problem is, if v₂ is greater than about 0.3 Mach (or about 100 m/s for air), then the air is compressed going through the funnel, and a more sophisticated analysis must be performed to check if the flow out doesn't exceed Mach 1, the speed of sound.

 

[Gerald Funk]

Thank you for your reply Steam King. Based on the formula, I got a v2 value equal to 300,002 km/h or 83,330 m/s. Thats way over the Mach 1, so how would the air behave in this scenario?

 

[SteamKing]

Thank you for your reply Steam King. Based on the formula, I got a v2 value equal to 300,002 km/h or 83,330 m/s. Thats way over the Mach 1, so how would the air behave in this scenario?

You can only pass so much air thru this funnel. Once the flow velocity reaches sonic velocity somewhere inside the funnel, usually at the exit, that's the maximum flow rate of air which can pass thru the nozzle.

 

[insightful]

This isn't a "free-standing" funnel with the large end facing a 30 km/hr wind, is it?
To me, that's what is implied by your diagram.
That would be a different problem.

 

[Gerald Funk]

Sorry, I forgot to add that the funnel would be fixed to a surface.

 

[insightful]

Sorry, I forgot to add that the funnel would be fixed to a surface.

OK, but is the 30 km/hr the velocity of the wind, or a given speed after entering the funnel?
I.e., is there a 2.5m diameter pipe that is forcing 30 km/hr air into the funnel? (Not really possible given the exit velocity that would require, right?)