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Yashaar, You measurements suggest that you do not have uniform temperature profiles near the HEX with hot and cold regions, the sensor located a few meters away, provides a more representative measurement due to turbulent mixing. You may actually have a problem with the flow/heat distribution...

You'll need far more detail to get a response other than start with the vendor documents and contact the supplier.

where you have long "dead times" and a single sensor, proportional only is the easiest to work with and doesn't over drive the system.

That looks like the kind of damage you'd expect in a plant that has been in service a few decades. Just replace it and make sure, that the tubing layout is designed for maintenance to take things apart without damaging them. Take alook at...

Depends on the tubing size, wall thickness, exposure to ambient contamination, craftsmanship, and if previous failures have occurred, then how often. Good luck,

Hopefully you are using slake lime, but in regards to effective control, you need to characterize your waste properties such as pH, conductivity, density, temperature, etc. Lacking that, then you need to set up stack gas monitoring. Contact your instrument suppliers for your site.

First you have to compare the inplace monitoring with an independent even if temporary measurement.

Confirm that you have the thermocouple wiring done correctionto avoid junction errors. When commisioning is done, ice bath testing of the sensor is done in the field, to confirm the remote display readings.

courtesy of an internet search https://www.miinet.com/?page=search&cid=temperature-transmitters&cdskeys=temperature

Those calculations are straight forward but require a lot more process detail than you have provided.

you may find this website helpful for various thermowell installations and designs. https://www.thermowellsoftware.com/

Welding thermowells in the pipe is not a good idea when two different metals are involved. The welds will fail. Van Stone flanges and thermowell in high temperature processes are common. Furnaces require other methods if extreme temperatures are involved.

Better to measure the displacement directly, though you can infer the velocity and displacement amplitudes from the accelerometer data. Instantaneous conversion for pure "sinusoidal vibration" is v=Omega*y, a= Omega*v with Omega=2*pi*freqency. Most advanced vibration monitors will allow you...