Thermowell resonance

Thermowells have been known to fail due to induced vibrations from the fluid flowing past it. The problem is in general confined to the flow of gases as their high velocities lead to higher vortex shedding frequencies and the low mass and viscosities do not provide any damping to the thermowells.
In general, if this phenomenon is likely to happen, you should design the thermowell in such a way that the maximum Strouhal frequency (fs) is not higher than 70% or 75% of the thermowell natural resonance frequency (ft).
Our problem is the determination of the natural resonance frequency and to this end I propose the formula below. If it looks complicated I apologise, but I wanted to know what it was for tapered thermowells hence the added complications (and if you think that's complicated, you ought to see the rest of the calculations).
I should also mention the contribution from Lloyd Williams-Wynn to whom I gave the ungrateful task of checking the maths and who picked up the deliberate error I had left in the middle (well OK , I admit it was not deliberate) and who, despite all this, still remains on talking terms with me.

Thermowells Natural Frequency - 1st Mode of Vibration
Strouhal Frequency
The vortex shedding frequency is a function of the Strouhal number, Ns, which is defined as :
Ns = fs.D/V
Where fs is the shedding frequency, D is the diameter, V is the fluid velocity.
Ns is relatively constant over a large range of Reynolds numbers (hence the reason why vortex meters exist).
Perry's Chemical Engineering Handbook reads:
Ns=0.19 for all Re between 500 & 10000 (Refer 5-181 page 5-55 5th Edition)

The formula has been calculated by Rayleigh's approximation method. It is expected that the answer will not exceed the true frequency of resonance by more than one or two percent.
It has been compared with the theoretical value for a parallel thermowell and the agreement was better than 0.1% .
Those interested in the derivation of the formula can collect a zip file containing a copy of the source documents (PDF) plus a bonus Excel 5.0 spreadsheet.

Whilst every care was taken when calculating the above formula, the author (who developed the formula during a period of residency at a lunatics' asylum) assumes no responsibility for the information contained herein.


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