Fluids Problem - Dimensional analysis, pipe flow, propeller

[elyttle]

Homework Statement

I have a tutorial question I have been struggling with. The problem is:

The flow through a closed, circular sectioned pipe may be metered by measuring the speed of
rotation of a propeller having its axis along the pipe central line. Derive a relation between
the volume flow rate and the rotational speed of the propeller in terms of the diameter of the
pipe and the propeller and the density and viscosity of the fluid. A propeller of 75mm diameter
installed in a 150mm pipe carrying water at 42.5 l/s is found to rotate at 20.7rps. If a geometrically similar propeller of 375mm diameter rotates at 10.9rps in air flow through a pipe of
750mm diameter, estimate the volume flow rate of the air. (The air density is 1:28kg/m3 with
viscosity of 1.93E-05 Ns/m2 as and the viscosity of water is 1:145E-03 Ns/m2).

The Attempt at a Solution

I think my main problem is just finding all the pertinent variables to start with. After that I should be able to use the Buckingham Pi theorem, form dimensionless groups the use similarity to find the value of volume flow rate for air.

The variables I can think of are:

Volume flow rate, Q
pipe diameter, D
propeller diameter, d
density, ρ
viscosity, μ
rotational speed, N

I have a feeling I should also include angular velocity or just theta for the angle but I'm can't figure it out.

 

[Chestermiller]

I think you have the right idea. Regarding angular velocity, that is proportional to rotational speed N. In my judgement, you have identified all the parameters you need.

Chet

PS, welcome to Physics Forums.

 

[elyttle]

Ok so using the six original variables I listed I formed three Pi terms:

Pi 1: Q/(N)(d³)

Pi 2: D/d

Pi 3: ρN(d²)/μ

From geometric similarity I found the scale is 1:5.

Then just using similarity of Pi 1 with the data given I found Q for air to be 2797.40 l/s.

I thought this seemed high but using similarity of Pi 3 with the given data I worked out the density of water to be 999.34 kg/m³ which makes sense. Is the flow rate right, I thought it should be more complicated.

 

[elyttle]

I got it now, I was right, it just seems high to me when flow rate is in l/s instead of m^3/s. I was really expecting the problem to be more complicated. Thanks for the help.

 

[Fluid Ireland]

Confirmation of Answer

I got it now, I was right, it just seems high to me when flow rate is in l/s instead of m^3/s. I was really expecting the problem to be more complicated. Thanks for the help.

I just finished the exact same problem and I'm having a problem with it.

The value for rotational velocity isn't squared for the answer. The value used is just N and not N^2, can anyone explain this?


It's in this book. The exercise is 5.4 and the answer is at the back.

https://www.google.ie/url?sa=t&rct=...Q20j_wcof1tuVmCon-smR1w&bvm=bv.64125504,d.ZGU