we complete our little tour of this
problem where a pipe feeds a turbine with
water flowing between two reservoirs two
tanks by trying to plot the pressure
distribution along the pipe and we want to
do this not in a quantitative way where
we would have very precise values and
positions of the curve but we want to do
this just in a qualitative way meaning
we want to see the basic shape of the
curve but not worry too much about
plotting and precisely we want to do
this in two steps the first step will be
imagining that the turbine here is shut
off and there is a valve which prevents
water from flowing and we will plot the
pressure distribution in this pipe in the
static case and then will open the
valve and let the water
flow and let the turbine extract power
and we'll plot the new curve that
applies in this case so let's start with
a hydrostatic case we start at this
point here at the entrance of the pipe
here and at this point here we are
already 51 meters below the surface of
the water so the pressure is 5 bars more
than the atmospheric pressure the
pressure here in the pipe will be
relatively high so this is at the
entrance the water is not flowing and so
as we go along the pipe here we just
swim horizontally the pressure will not
change we just have a straight curve
here until the first bend as we now go
up the pressure will drop or decrease
because our altitude decreases and so we
see a little dump here in the pressure
and then we go horizontally again so
that the pressure remains constant here
with respect to length and then now we
go down so the pressure will increase
will increase by approximately 3 bars so
the pressure will jump again here like
and then again we have a straight pipe here straight
horizontal pipe and so we have a
straight line like so when we arrive at
the turbine on one side of the valve we
have this pressure here and on the other
side of the valve we have only these
four metres altitude so that the
pressure on the other side of the
turbine here is very low and this
difference in pressure here that we have
this is the hydrostatic pressure difference
this is the basic fundamental pressure
difference that is available for the
turbine to run now if we now remove the
valve and the water is now flowing so
let me perhaps label this curve here and
this is here with no flow let me now
remove the valve let me let me erase
this valve that we have here and let the
water flow as the water flows the
pressure drop that's available to the
turbine will become less this is because
pressure losses occur inside the pipe
and they'll occur at the bends
but they also occur all along the length
as friction applies to the pipe so I
draw a second curve which starts at the
same point because we are here again at
51 meters below the surface but now as I
go along the pipe my pressure will
decrease
it will decrease progressively
linearly with distance let me perhaps
make a better case here that doesn't
match the second curve it will decrease
progressively and arrive maybe at this
point here and we will go down again
inside the pipe and then at this point
here instead of a horizontal line I
again have a line that's parallel like
so and then I jump up by the same amount
here so I jump up by the same amount
let me try to have this height being
approximately the same like so and then
again here I decrease the pressure along
the length the slope of these three
segments here should be the same and I
arrive here at lower pressure so that
now the Delta P that I have available to
the turbine this is here this is Delta P
turbine it is less than before and this
Delta P that I have here this here this
is the Delta P due to the friction
losses plus the Delta P due to the bends
over here note that I greatly
exaggerated the losses here you remember
maybe from the previous calculations we
had here the total Delta
available the hydrostatic pressure
difference was about seven bars and we
lost about
one bar of
pressure due to friction over here so
this Delta should be much smaller I
exaggerated the trend here so it becomes
much more visible so this would be here
the pressure drop here with flow the
pressure distribution with flow and this
is how you plot the pressure
distribution in a quantitative way
inside a given pipe where a water flow
is happening and the pressure drops,
across the two sides of the turbine to
extract power.
