I have to do a shutdown and Start up on a set of pumps
(Centrifugal)

For Start Up, the proceedure says

Open Suction Valve, Vent Air from Pump, Start Pump
Open Discharge Valve.

So, The pump will pump its maximum head
with the discharge valve closed at zero flow right ?

What about the pressure in the suffing box
at the Mech seal ?

How do ensure the seal will be OK

rho g h = p   if I know what the max head
of the pump is at zero flow (?)

But am I right in thinking the box
pressure is a third of this pressure or similar

Basically I want to know the pressure
of the stuffing box at closed discharge.

The startup and shutdown of a centrifugal pump is specific to the user's application - one can't expect that general directions will apply 100% of the time.  That's what engineers are supposed to oversee and ensure.

A pump's proper and recommended startup depends on the type of pump, the fluid, the pump's configuration and piping, the process downstream, the flow and pressure controls installed, and the hazards involved (if any).  There may be other factors I left out, but those are just some of the main ones.  All manufacturers send you General (un-specific) startup instructions.  I presume these are the ones you allude to.  Your specific, actual application and configuration are the main and most important items to consider first.  You can startup with your discharge open; but you'll impose a heavy current draw on your electric motor and you should be ready to handle the discharge downstream.  This is the way I usually prefer to startup because it automatically "vents" the air from the pump if I direct the initial discharge back to the suction source.  However, this takes more instrumentation.  Also, you must have an automatically flooded pump - i.e., you should have a positive static head over the pump.   These are just some of the examples of what I mean by "actual application and configuration" and how they influence your decision on how to startup.  It's not like a recipe book.

What about the mechanical seal?  What do you mean to say?  The mechanical seal is there and will do its job of sealing if you subject it to its function.  The manufacturer designed it to seal and you don't explain your concerns as to why you should worry about it.  If you don't run it dry (which is Rule #1), you should not have a concern.  The pressure at the face of the seal will be characteric of that pump's design - whatever it is.  The seal is surely designed to operate at deadhead pressure - although I wouldn't subject a pump to this extreme.  Sustained operation at deadhead (as I'm sure you're aware) pressure is just bad, bad practice that puts abuse on a pump - mechanically and thermally.  Don't do this unless you want to buy more pumps.

If you want to know the pressure of the stuffing box at closed discharge, inquire with your pump manufactuer.  This is not a usual question from a user, but if you're interested in knowing, I'm sure people like Goulds and other recognized pump manufacturers will tell you.  You haven't said what kind of mechanical seal you're using so I assume you mean the process face of the seal, directly behind the impeller.

Normally, the pump fabricator does not write up the operating and startup/shutdown instructions for the pump.  This is the Owner/operator's job, since he is in control of the application and the need.   The pump fabricator furnishes "general" instructions, but these should never be applied literally to a specific application.  The fabricator normally knows little, if anything, about how you are going to use his product.  Of course, the fabricator can help you in the application - if you share all your basic data and knowledge about the application with him.  This is information over and above what is normally furnished in the Specification Sheet.  I would certainly use the fabricator as a consultant in special cases like troublesome fluids, difficult NPSHa situations, automatic prime and startup applications, non-newtonian fluids, etc.   This is just smart engineering that is available to you if you apply common sense.

Generally speaking for pump start up reasons, the initial discharge valve position is related to the type of impeller fitted to the pump. A simple radial vane or francis vane impeller draws low amps at dead head, so start pump with discharge closed. A mixed flow or axial flow impeller draws high amps at dead head, so start with discharge open. In both instances this prevents overloading of the motor during start up. It is possible to start a pump with a radial vane impeller with the discharge open providing you have a non return valve at the pump discharge, and the header or manifold is at pressure. The seal chamber pressure at dead head depends on the type of impeller fitted. An enclosed impeller with front and rear wear rings, and that also features balance holes between the rear wear ring and impeller eye, will generally give a seal chamber pressure of slightly above suction pressure. A semi open impeller, or an enclosed impeller with only a front wear ring and no balance holes, will subject the seal chamber at dead head condition to a pressure of around 60% of dead head pressure. Although as previous posts have mentioned you may have bypass or reirc lines that bleed off some pressure with the dicharge valve shut at start up, this is probably due to local operational conditions. Best advice is to follow the pump manufacturers recommendation, if in doubt or just out of interest, look at your pump curve at the absorbed HP curve at dead head. A low draw will indicate starting with valve shut and a high draw will indicate starting with valve open.

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