in this problem we want to calculate the
pressure losses that occur due to
friction inside the pipe so we have a
big tank on the left a small tank on the
righta pipe guides the water to a turbine
here which is on the bottom right and
what we want to do is calculate how much
pressure loss we have due to the water
flowing through the pipe we have 800
liters of water flowing every second
through the pipe
the losses it will occur due to two
things one is the friction along the
walls so this is shear inside the pipe
because the velocity of the water on the
walls of the pipe is always zero so there is
shear applying inside the pipe and this
translates into a pressure loss between
the entrance and the exit of the pipe
and the second source of losses is the
bends we have four bends here one here
one there one there and one there in
each of those bends incurs losses
this is quantified in a similar way
so let's take care first about the
losses due to friction and to quantify
those we want to calculate the Reynolds
number this is the first thing we should
calculate when we start any problem in
fluid mechanics and to get the Reynolds
number we need to calculate the average
velocity inside the pipe and this we get
from the volume flow so let's write out
V average here as being the volume
flow curly V do not
mix it up with velocity divided by the
cross-sectional area and it happens to
be the volume flow here divided by PI R
squared or PI d squared over 4
like this and we can put in numbers
directly into this we have volume flow
is 800 liters per second but that's in
meters cube per second this is only 0.8
meters cube per second and I divide this
by pi times 1 over 4 times the diameter
squared which is 1.1 features I guess
you put this into your calculator you
should get
thing like 0.842 and this is a velocity
so it is m/s and this is average it's
not very fast flow on average but some
fluid particles will be much faster than
this some fluid particles will be much
slower okay
what is the Reynolds number very important
question because it determines how
turbulent the flow is how much
turbulence how much complexity there is
inside the flow well in this case the
Reynolds number by convention is
based on the diameter and the average
velocity so the we have Rho V average D
over mu here
and in this case we have water
whose density is 10 to the power 3
the average velocity we just calculated
is 0.8
let me pull this up
a little bit more space to write 0.842 meters per
second and I multiply this by the diameter
which is 1.1 and I divide this by
viscosity which is 10 to the power minus
5 I believe let me just check that 10 to
the power minus 5 Pascal seconds and
then we get with this we get 9.25 1
times 10 to the power 5 and this is the
Reynolds number it doesn't have any
unit Reynolds based on the diameter is
this a big Reynolds or small Reynolds
it's relatively large it's more than
a few thousand which is usually the
limit above which flow becomes turbulent
but it's not it's not rare in
engineering problems to have Reynolds
numbers that are several dozens of
millions so it's relatively large but
not extremely large okay what do we do
with this this allows us to see that the
flow is turbulent inside the pipe and
this will guide the rest of the problem
flow is turbulent and we want to
calculate is the pressure loss due to
friction which we call Delta P F here
and the convention in engineering is to
quantify this using a simple formula
which is the Delta P loss is one-half of
Rho of the V average squared multiplied
by a friction factor f multiplied by the
length and divided by the diameter of
the pipe L everything like so all those
terms we have except one which is the
friction factor friction factor depends
moderately on the Reynolds number and on
the roughness of the pipe and this is
found out using the Reynolds number and
the relative roughness as
input into one big diagram which is the
Moody diagram so let's prepare data that
we need for this and the data we need is the relative roughness and the
relative roughness is epsilon over D it
is the average size of the roughness on
the sides of the pipe divided by the
diameter of the pipe and this in our
case happens to be if I scroll back up
here we have 0.25 millimeters of average
roughness on the side of the pipe and
the pipe happens to be 1.1
meter in diameter so when I put these
numbers I have to pay a lot of attention
to the units 0.25 millimeters is 0.25
times 10 to the power minus 3 meters and
I divide this by 1.1 and this gives me
2.27 times 10 to the power
minus 4 met… no not meters,
this is a dimensionless ratio because I
divided meters by meters so this has no
unit this is epsilon over D like
this and this I need because I'm going
to read the friction factor f inside the
Moody diagram
so let's now turn to the Moody diagram
this is very big scary and somewhat hmm
inelegant diagram and and it's always
scary the first time but as the more you
use it the more you will find it simple
so let me show you what we have we have
on the Left what we want to
read out of this which is the friction
factor on the bottom is something we
want to input which is the Reynolds
number and on the right is another thing
we want to input which is the relative
roughness and we have those two values
we just calculated I reproduce them here
the Reynolds number […]
[…]
and the relative roughness two point
twenty seven times ten to the power
minus four
these are all dimensionless values on
the bottom on the right and on the left
this is a completely non dimensional
diagram which makes it extremely
interesting and useful in engineering
okay so what do we want to do here let's
find out where on the bottom scale we
are with our Reynolds number Reynolds
number is nine point two five one times
10 to the power of five and so it's
between 10 to the power 5 which is here
and 10 to the power 6 which is there and
if I look at where we are here is 1
times 10 to the power of 5, 2 times 10 to
the power of 5, 3, 4, 5, 6, 7, 8, 9, we are in
between nine and 10 times 10 to the power 5
yes so we are somewhere in between those
two lines and this guides me as to where
I would like to be on a vertical scale
up here so I'm going to draw a line here
that indicates where I'm going to read
this diagram and this line should be
relatively straight it should shoot up
like this on this diagram like that so
I'm gonna read the output of the
friction factor according to where I am
on this position here the second input I
want to have is the relative roughness
and the relative roughness will
read on the right scale and we
have two point 27 times 10 to the power
minus 4 which when I read up here sets me
somewhere in between 2 and 5 times 10 to
the power 4 so probably a bit closer to
2 so something like this position here
however these roughness lines they
curve up so I do not want to draw a line
is purely horizontal I want to draw a
line that follows up the curve of the
roughness so I'm going to draw something
like this that is going to go like that
and curve this is not a very good
drawing at all let me try to do it again
it's a bit difficult with my digital
tablet here so I'm going to move this
like so this is the line that I want to
follow and that the intersection of
those two lines here so when I am
relying on this point here let me draw a
cross here here this is the coordinate
of the point which lets me then shoot
right to the left side and read over
here the value of the friction factor
which I'm looking for and so had I
had I done this a bit more
precisely I would have landed it here
to a point which previously I had read as
0.0158 here
do not worry too much about how precise
you are reading those points what we're
generally looking for is is the order of
magnitude of the friction factor so I
want you to land around the correct
point here and not say at that point or
that point over there the precision in
that kind of problem is not critical to
us so this is how we this is how we read
the friction factor so we have a
friction factor of zero point zero one
five eight here in this case so we can
come back now to our calculation from
before and I can go back down here and
we have the Delta p f which is here the
Delta P due to friction now we can
calculate we can put in numbers in there
we have minus one half Rho is the
density of water
ten to the power of three the
average velocity happens to be we
calculated it above zero point eight
four two
and this is squared and I multiply this
by the friction factor which we just
read as being zero point zero one five
eight and then the length of the pipe
happens to be four kilometers so that's
four times 10 to the power 3, 4000 meters
and I divide this by the diameter which
is 1.1 and if I put this into my
calculator I get minus two point zero
three minus two point zero three times
10 to the power 4 and this is a pressure
a pressure difference and this is
Pascals all right so this is about
minus 0.2 bar so it's a lot less than
the pressure hydrostatic pressure
difference that we had available for the
turbine okay so this is the friction
loss this is the pressure loss due to
friction inside the pipe and on top of
this we have losses that are added
through the bends and we want to
calculate those as well so let me draw
on line here and let's calculate the
pressure loss due to the bends the
pressure loss due to the bends are
calculated in a very similar way we said
Delta P of the bends is the number of
bends which happens to be 4 in this case
let me show you that again
so as 1 2 3 & 4 bends over here number
of bends times minus 1/2 of Rho times V
squared this is the average velocity
multiplied by KL we qualify for each
kind of obstacle inside the flow
parameter called KL or the loss
coefficient this parameter as it turns
out remains relatively constant which
means if you basically put a bend inside
the pipe the pressure loss that this
bend will incur when there is flow is
roughly proportional to the square of
the velocity so this KL remains constant
approximately constant irrespective of
the flow velocity that we have in there
and so for each classical type of bendor valve or obstacle or filter inside
the flow we typically have a KL factor
that applies this allows us to calculate
the pressure losses so in putting data
into here and we have now four times minus
1/2 multiplied by the density of water
10 to the power 3 multiplied by the
velocity 0.842 this is squared and I
multiply this by the KL that we have
applying in our bends which happens to
be in this case 0.75 this is not a very
efficient bend of course the smoother
and the rounder the bend the lower that
number here and the lower the losses
that will be occurred due to the bend
and if you put numbers into this you put
this into the calculator you get minus
one point zero six times 10 to the power
3 pascal like so this is the Delta P
bends like so so we can see that the
bends incur a pressure loss that is
about you compare it to the pressure due
to friction pressure loss due to
friction about 20 times smaller than the
friction due to the walls so this is how
you calculate the pressure losses across
the entire pipe due to friction on the
walls and due to turbulence losses
through the bends of the pipe.
