Mixing Two Fluids at Different Pressures

Depends on how your VFD for the brine tank reacts and what the back pressure controller is set at, neither of which you mention.

If the back pressure controller is set at 10bar, which seems to be the case, then 10bar is what it will be and the VFD for the Brine will ramp up accordingly to control on level in the tank.

You might not have shown it, but there should be a NRV between the mixing point and the kettle reboiler otherwise the Brine will flow backwards if the pump stops...

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Agree, pressure will remain at backpressure PIC setpoint when condensate pump fails, assuming there is a check valve on the condensate feedline upstream of mixing point. Condensate boot level at kettle reboiler then rises beyond boot LIC setpoint.

Both scenarios are possible depending on pump type. Depends on what the pump speed is controlling, volume through velocity or volume through displacement.

assuming the brine pump is centrifugal the system is overcontrolled
First - Pressure at mixing tee is controlled by PIC loop.
Second - VFD changes pump head and flowrate simulteneously. That means in respond to level change LIC changes brine pump speed, pump speed changes pump flowrate and head delivered, pump head changes pump outlet pressure. Neglecting hydraulic losses changes - LIC loop controls pressure at mixing tee.

That means that fluctuations of brine level in the vessel will cause unpredictable fluctuation of PIC loop. Based on my experience with similar cases after some fighting against the design flaw personnel will disable PIC loop and start controlling PCV opening manually.

My opinion is VFD is redundant here. Note that actual VFD hydraulic curves are uncontrollable and depend on wearing of pump internals and change over time. Note that there is also a dependence of brine concentration on system hydraulic profile.

Curious to know how the reboiler is protected from an overfilling and the condensate pump from a dry run. Can you please comment?
Can you share what is brine vessel LT sensor type?

Thanks to everyone for your comments.