in this problem we are looking at gas
flow through the blades of a gas turbine
and we're interested in the power and
the force that is given to a blade by
the flow of gas so we're looking at a
turbine like represented in this picture
with incoming flow like so and outgoing
flow like so all the while the turbine
blade is moving in this direction over
here so there are three velocities yeah
the point of this exercise is to work
with relative velocities and see what
happens to our control volume when we
change points of view there are two
points of view you could adopt to
describe those three velocities one is
the point of view of the photographer
standing next to the turbine like we
represented in this picture here and you
are taking the photo in this point of
view the blade is moving relative to you
and you can see it on the bottom left
down here the blade would be moving at
400 meters per second like so from that
point of view you would see the gas
moving at 800 meters per second like
so and the gas would then leave at 300
meters per second like so you could also
see it from a different point of view
you could jump on the blade and move at
400 meters per second together with a
blade if you did this then you would see
the velocity here coming in the gas
coming in at 400 meters per second and
leaving at 600 meters per second like so
yeah you would obtain those vectors here
400 meters per second and 600 meters per
second by taking the inlet velocity and
outlet velocity here and subtracting to
them the blade velocity the math to do
so is relatively trivial it's just a
little bit tedious to calculate so we
have the components given here in this
problem to make things a little bit
easier so what's the force and what's
the power given to the blade well to
answer the question about the force we
need to move -let me change pages- we need to
write a momentum balance equation
 and this
looks like this we have a net force
that's applying on the control volume
it is equal to minus the sum of every
time the mass flow multiplied by the
velocity this is here for the internet
and then plus the same thing for the
outlet like so and here I would have
here the mass flow local mass flow
multiplied by the velocity vector
so which velocities and which mass flows
do we take which point of view do we adopt?
well there are two control volumes
you could take to describe those three
velocities the first control volume is
control volume that surrounds the blades
as a representative here on the bottom
left and so in this case you would have
let's say we have a box that surrounds
the blades like this you would have two
velocities incoming and outgoing you
would have the incoming velocities here
at 800 meters per second and the
outgoing velocity at 300 meters per
second so this would be here v2 and this
would be here V1 like this but you
also have to think that you are not
moving the blades are moving relative
to you and so as the blades are passing
through your control volume they
are carrying an unknown amount of mass
flow with them and so you have coming in
through your control volume this
velocity coming in let's call it v3 here
and that velocity here outgoing let's
call it v4 here attached to these two
velocities is an unknown amount of mass
flow
luckily the mass flow m dot v 4 and the
mass flow m dot V 3 are the same this is
because of the symmetry situation that
we have as we described the blades which
are just turning around the axis and
each having the same mass flow attached
to them so from the point of view of the
photographer when you describe the
control volume you have four velocities
two incoming velocities and two outgoing
velocities the same situation described
from the point of view of the blade
would look like so this is the
situation is represented on the right
right there
so we look like this you would have here
a box the same box surrounding the
blades see if I can draw this yes and
then you would have now two vectors you
would have 400 meters per second like so
and not going at 600 units per second
like so this is v1 and this is v2 like
this there is no more incoming and
outgoing velocities from the bottom or
the top because you are moving together
with the blades so no blade is entering
or leaving your control volume. which of the
two control volumes you take is up to
you. both of them will give you the same
result however my tip is to always
attach the control volume to the object
that is moving because this makes the
visualization of flows much easier you
have fewer tricks and traps to fall into
with these incoming and outgoing
velocities through your control volume
so I'm going to adopt here the control
volume shown on the right and solve the
problem with that control volume if you
have nothing to do on a rainy Saturday
afternoon then maybe you can try on the
left to use this control volume and you
will see well get exactly the same
result so let's work out the math now
for for the control volume on the right
we have the mass flow sorry the net
force is equal to the mass flow
multiplied by vector v2 minus vector V1 each of those vectors has two
components so we could write this as m
dot x V 2 X minus V 1 X and then V
2 y minus V 1 Y which numbers do we put
into this well we know that the mass
flow is 1.7 kilograms per second and
then we now have to work out the
components of velocity V 2 X is the X
component of that velocity here and so I
have to take the 623 meters per second
and squish them down with an angle of
63.4
degrees and so I did here
and I put then the cosine of the angle
sixty three point four degrees and so
this is V 2 x. To this I subtract V 1 X and
V 1 X would be the X component of that
velocity 400 squished down with an angle
of thirty seven point six so this is now
four hundred fifty three point eight
times the COS of thirty seven point six
degrees. on the y component that we take
now, this V 2 y here this would be a
negative number
times the sine of the same
angle sixty three point four degree -
here in the Y Direction positive Y
direction four hundred fifty three point
eight times the sine of thirty seven
point six degrees like so and if you
work out the math this should give you
here [cut] and now this would give you a final
result of minus 136 twenty seven and
this will be the Newtons this is our
net force
how to represent now this net force as
it applies on this control volume well
you have in the x-direction negative a
little bit 100 Newtons and in the
y-direction negative a large number
1000 and so this would be here in this
direction it would be a force that
applies like so this would be F net as a
vector something like this now with this
net force here F net applying to the
fluid we can compute the force that's
applying to the blade and if the fluid
has a force that is pushing in this
way this would be here F net then we
have applying on the blade the
opposite force which would be then here
a vector like so and this would be the
force of the fluid on the blade - what
is the power now that's applying on the
blade by the fluid the power is the work
done per second and the work done per
second will be here force due to the
fluid on the blade like so dot the
velocity of the blade this is a dot
product of two vectors it's a number
and this would be quantified as power in
watts W and this looks now like the
component of F that's along velocity
blade and let's have a look at velocity blade
it's moving with this direction here
like so this is the velocity of the
blade and so we take here the Y
component of this force applied along
this velocity this is then here
FBy as a number yes multiplied by
the velocity of the blade VB and this is
the y component of f FB which is then
minus minus 1018 like so and so I get
[cut] 9 multiplied by the velocity of the
blade which is 442 meters per second
here yeah this gives me here 6
times 10 to the power 5
watts which computes as six hundred and
twenty seven point seven kilowatts this
is the power given to a single blade how
big is this well a lot six hundred kilowatts
this is probably around something
like 500 horsepower this is the power of
a very powerful truck and this is the
power given to the by the flow to a
blade that's about this big you could
hold in your hand so blades, turbine
blades are extremely powerful some of
that power will have to be given back to
the compressor but some of that power is
extracted through the shaft and given
out to the machine turbine engines are
generally extremely powerful. So this is
how you use the momentum balance
equation to compute force and
power when things are moving.
