Elbow-tap flow meter formula 1

[ypwtb]

Hello,

The following formula for an elbow-tap flow meter

W = 244 [SQ.ROOT SIGN] r * delta_p * d^3 * rho

is given here:

http://www.seilenterprise.co.kr/Korean/Technology/flowmetertypes.htm#Elbow

A few questions:
1. Is this formula accepted as accurate?
2. How is it to be parsed? I don't suppose that the density term is under the radical.
3. Where did this formula come from? I don't find it anywhere else. How is it derived?
4. Is there a formula for a short-radius elbow? I assume that "r" in the formula above is the centerline radius for a standard elbow -- r/d = 1.5.

Thanks,
ypwtb

 

[RWF7437]

1. I don't know but if it is like many hydraulic formulae it probably is empirical; i.e. it came from tests done in an hydraulic laboratory.
2. To parse it, look at Figure 2 and derive the formula for yourself. This will not only answer your question but will improve your understanding.
3. See 1, above. It comes from Newton's Laws of Motion, Energy, Mass and Momentum.
4. "r" is the radius of the elbow. You do not need to make any assumptions about that.

Good luck

 

[RWF7437]

http://www.pc-education.mcmaster.ca/instrumentation/flow.htm

 

[ypwtb]

Thanks for the replies.

I did the derivation thing. It might have been easy in my college days. I won't say how difficult it was these days.

The idealized formula for consistent units is:

m_dot = pi/4 * sqrt(R * D^3 * delta_P * rho)

Plugging in conversion factors to match the formula copied from the link in my original post, it becomes:
m_dot = 254 * sqrt(R * D^3 * delta_P * rho)

Applying a correction factor based on Reynolds number from Bloomer's book, Practical Fluid Dynamics for Engineering Applications, the constant, 244, from the formula that I questioned in my original post may be reasonable.

I am a little uneasy with the result because the formula seems to follow from analyzing a cylinder instead of a pyramidal shape. Forming a pyramidal shape within the elbow using the Theorem of Pappus seems reasonable. Applying Newton's second law to that pyramidal shape is doable, but the resulting formula is quite messy and contains factors which may be unknowable. And it makes the unwelcome suggestion that both line-pressure and pressure-port size might be noteworthy field measurements for this calculation.

Using a cylindrical shape for analysis in this situation seems to bend some rules. A cylinder sweeping through the elbow is going to encounter interference with adjacent cylinders which are doing likewise. That wouldn't happen with similar motion by a pyramidal shape. Excerpting cylinders from pyramidal shapes (to prevent all such interference) would leave voids in the flow that are unaccounted for.

I think that opting to analyze a cylindrical shape introduces an undetermined amount of error. But the resulting formula is relatively simple and convenient.

Bloomer starts with a formula published by Murdock et al. I assume that they analyzed a cylindrical element because that's the only way I could come up with their result; but I haven't seen their paper. Their work is apparently documented in the following paper: "Performance Characteristics of Elbow Flowmeters", published in 1963 or 1964 according to various citations. If anyone has access to it, I would be interested in knowing whether the results by Murdock et al were obtained by analyzing a cylindrical element.

For the moment, it appears to me that the formula was obtained by analyzing a cylindrical element and that it is believed to be adequate, when corrected for Reynolds number according to Bloomer's table.

ypwtb