I suspect that pressure pulses from the input point (pump) will trvel at the speed of sound inside the liquid until the critical temperature/pressure point where steam is created. The oscillation of pressure from the sound wave will cause and oscillation of the phase change and this energy exchange may be helpful, or destructive.
There is a similar condition in gas where it is flowing - naturally through a tube and heats a "membrane" of heater input. It is called Rijke's tube resonance.
https://en.wikipedia.org/wiki/Rijke...e tube is a,an excellent example of resonance.
A very clever guy (Joan Lluch) I met on another thread developed this with the type of radiant heat burner he introduced into his boiler for a steam loco - designed for such a gas burner.
https://www.mylargescale.com/threads/gas-burner-making-high-pitch-flute-like-noise.81946/
He resolved the issue by changing the air-fuel mix, such that the right conditions did not occur for the Rijke's tube resonance to occur. I think we made the flame front thicker? - I.E. When the oscillating gas can transit from hot-zone to cold-zone and back again across the heating element, it sounds - and resonates very loudly. When the cold to hot interface is too thick for the gas to travel across and back it cannot sound. I.E. the zone is thicker than the wavelength of the resonant frequency at the temperature and pressure of gas.
In a flash boiler, there is a zone where the temperature does not rise as the water is converting to steam. If this zone is very short, compared to the tube length, the Rijke's tube resonance could occur as the pressure pulsations (at the SAME frequency) stimulate the oscillation of this zone - if the zone is much shorter than the wavelength of the resonant frequency in the gaseous steam. (I think?).
The analogy being, that if the system's resonant frequency is stimulated by pump pulsation frequency, then a noise will sound - easily picked-up by the simplest sensor - the ear. As long as a resonant sound does not occur, then all is OK.
"SOUNDS SIMPLE" - - or am I being too simplistic?
When resonance occurs, (a mechanic once told me) it is the machine shouting "Help! Turn me off - before I explode!" And as long as we heed such warnings life should be good. Think of it this way - hit a bell and the sound is resonant - a single note. As a Tympanist I used to pick up a note from the band/orchestra and sing it close to the drum-head. when re-tuning to a different note. When truly in-tune the drum-head would sing back. - So loud I once had a conductor tell me to stop tuning at that quiet point in the music as he could hear it! - It was Out-of-tune to the key being played at that time. So we agreed where I should "sing quietly" to the drum, so it's "loud reply" didn't affect the music. Resonance is always LOUD as it absorbs virtually all the energy available as re-broadcasts it on a single frequency.
If not relevant or useful, just ignore this.
I can also explain why running two pumps at a SAME rotational speed - one with 2 pistons and one with a multiple of 2 times the pistons gives a higher frequency, but it still resonates at a numerically equivalent frequency. NORMAL practice in industry is to either add a damper or something to change the resonant frequency of the system, not necessarily change the driver, I.E. the pump. You plan on using variable speed to vary the flow, so if the resonant frequency of the BOILER is unchanged, you'll still hit a resonant frequency some way when the pulsations from the pump match it. A 2 piston pump may hit resonance of 16000Hz at 8000rpm, but a 9-piston pump will hit the same 16000Hz resonance at ~1778rpm. A different pump speed, but the same resonance. Change the resonance to 16Hz, and you won't run a pump slow enough to hit that resonant frequency. Similarly if the resonance of the boiler is too high for pump stimulation of resonance.
Damping, other than by addition of a tuned resonator (Helmholtz resonator), is different, as it absorbs the energy between each input pulse, so there is nothing left when the next pulse arrives. A Helmholtz resonator - tuned to prevent pump pulsations at the boiler resonant frequency from reaching the boiler, would be placed on a Tee-connection, located between the pump and boiler, such that the energy of pulsations - at resonant frequency - all goes into the resonator - and out again - such that the pulsations on the main-line do not go into the boiler. Like adding a large or suitably sized Capacitor on the output of a noisy electrical feed to eliminate the noise the is seen by equipment further along the line.
Tap the boiler (at pressure and filled with water) and listen to the note it "rings". Use your phone with software to tell you the frequency of that note - or compare to a piano - and then you'll know when resonance will occur, unless the phase change to steam is sufficient damping to eliminate of change the resonance. (The opposite of the Rijke's tube condition).
If it doesn't "ring" then the water, stress from pressure or something is either damping the noise, or raising the resonant frequency above the range of your hearing. (Speed of sound in water and metal is much higher than in air, so very high resonant frequencies). In this case, the pulsations from the pump (in the audible frequency range) are too slow to reach the boiler resonant frequency.
Also, if you start with a COPPER boiler that is annealed, it is likely to dampen the resonances in the metal, but after some use the copper work hardens, and the hard copper "rings" much better, as the internal material structure does not dampen the resonance. It then becomes prone to resonance fatigue failures. Tin is used for Organ pipes, so the pipe-metal resonance (very low, well damped) does not interfere with the air-column resonance (Audible range). But Tubular Bells are hard metal to "ring" at the metal-length resonance (Undamped, Audible range oscillation), tuned to the tube of air column length, so that resonates and amplifies the sound created by the walls of the ringing tube.
Some of the above may be useful, some inaccurate, but it is "what I know", so please correct me if I am wrong and ignore what does not affect this problem. (Is there really a resonance problem? - Or just a "measuring instrument resolution" issue?).
K2