How would you go about calculating the pressure of water in a sealed
vessel or tank. If the tank is 1.5metres in diameter and 2.5metres
high and full of water and you then drain some water from the bottom
of the tank through an enclosed system of pipes what would the
pressure of water be acting on the head of water in the tank??

Trick question, the water won?t actually drain out unless it gurgles.
Needs to be deeper than about 10 meters. You'll get a slight dribble
but that's only because of dissolved air and slight expansion.
Diameter is irrelevant.

Best

I agree with guzzi-ga.  If the tank were completely full  - no
airspace at the top - and what you describe as closed system of pipes
- presumably precluding that air could get in the tank (guzzi's
gurgling), then the water would not drain.
The question:  "... what would the pressure of water be acting on the
head of water in the tank?"  I do not understand.  I would have
thought it would be the other way around, the head of water (2.5 m)
creating pressure  - but in a closed container with no airspace and no
way for air to enter, ... ?

The diameter of the vessel is not important.  The pressure is
calculated by the equation P=density of fluid*gravity*height of the
fluid.

water just like any liquid has a partial pressure (pp)which is temp dependent
i dont have the numbers but the (pp) of water is equal to sea level
air pressure (14.7 lbs/in^2) at a temp of 100 deg (C) now in the tank
the water is being forced up into the tank by the pressure of the air
acting on the exit of the tube
the water has an opposing downward force equal to apx 1200gm per tank
depth in meters.when the depth equals around 10 meters the force of
the water at the bottom of the tank will equal the air pressure acting
on it.at this point the water pressure at the top will be very small
when it drops below waters (pp) at that temp the  water will boil out
creating a gas space above.boiling will lower the temp of the water
which will lower its (pp) which will cause it to cease boiling.
so i guess the answer to your question is the space above the water in
the tank will be whatever it needs to be to keep the water depth equal
to the hight a coloum of water will rise due to air pressue (apx 33 ft
or 10meters)

I understood from the question that there was no air at the top of the
tank, hence  guzzi's  and my comments.  As long as there is none, the
water just won't drain out, unless air is allowed into the tank to
replace it.
I believe that that is the same as saying that there is no pressure  -
 But I coud sure be wrong.

A Water barometer can be made (had one in the stairwell of my school),
the same as a mercury barometer, but over ten meters high.  A 10 m
column of water, like a 39 inch column of mercury is so heavy that it
"hangs" at that level, creating a vacuum above it in the closed tube,
but the water  - like the mercury -  does not flow out.  A little
would when a lower pressure phase came and the Barometer didn't have
and up-turn to collect it,  but just that little.

You're point is well made, but this is just a typical engineering
question that this kid probably needed for homework.  It was probably
already due and never checked back for an answer.