Does engine purging in a liquid rocket engine cause dry run of the turbopump? If so, does that have any implication for the design, health, maintenance, or operation of the pump?
How do we calculate the purging pressure?
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No, it doesn't. There is always a valve between the purging circuitry and the turbopump, preventing the turbopump from going dry (which, as @GittingGud points out in a comment) is highly undesirable. The valve is closed by the same gas as the one used to purge. For a nice description, see p. 60 of Huzel and Huang, "Design of Liquid-Propellant Rocket Engines" dealing with a (hypothetical!) engine using fluorine for an oxidizer.
To answer the second part of your question: if a dry run ever occurs, the turbopump will spin up to rpms well above its specification, and eventually some material fault will make one of the impeller blades break by centrifugal force. The pieces of the blade will shatter the pump encasing, just before or at the moment the rest of the impeller flies apart because the centre of mass is not on its axis anymore. The impeller debris will fly, at very high speeds, through the various parts of your rocket. Depending on the stage the rocket is in (pre-launch, ignition, flight) this may lead to what the Brits call, tongue-in-cheekishly, accelerated self-disintegration (an explosion).
At engine burnout, the propellant remaining behind in the turbopump is part of the engine's so-called wet mass.