In this screencast we will discuss the set
up of solving pipe flow based problems.
We start here with the mechanical energy equation,
which is just a conservation of energy balance
between two points.
This is also known as the extent of Bernoulli
equation and was used for the study of average
velocity flow for in-compressible fluids.
Now if we talk about steady non-changing flow
between the two points, which we call fully
developed flow.
Then these alpha terms drop out.
There is no difference between them, and if
we talk about this flow, which there is no
energy loss due to viscous dissipation.
So this term drops out.
In the resulting terms should look like something
you have seen before the Bernolli equation,
but as we move forward and discuss losses
accruing in pipes and trying to solve for
certain parameters that were not given.
We need to account for these other terms.
So this whole screencast is going to work
on just working through this equation, and
defining each of the terms.
So this loss term hL stands for the head loss.
It is sometimes first introduced as hL, and
in this form of the equation with dimensions
of Length.
This can easily be compared to head pressures
generated from pumps.
Now you see in this equation I don't have
a shaft head term, which gives us an idea
of the amount of energy being added or removed
to our fluid by a pump or taken out by a turbine.
So we are assuming that is not present, but
it is easy to compare when using these kind
of dimensions.
Now other times the equation maybe rewritten
to look as the follow.
So the second form of the equation we have
taken the specific weight of the fluid marked
here as gamma, and multiplied it through.
So now what that does is changes the dimensions
of each term.
So as before the dimensions were length now
there are pressure.
So depends on how you want to represent each
term in the energy losses that are associated
with the pipe system.
Either way this equation is going to be our
friend when determining maybe the flow rate,
pressure differences between two points, or
the diameter of a pipe that we may use for
a piping system.
Now other screencasts will demonstrate how
to determine each one through examples, but
here lets just become familiar with this equation
and each term in it.
So the equation is an energy balance between
two points.
So we have the energy associated with point
1 and set equal to the energy at point 2.
Now this extra term, our losses term is added
to the second point, because we are say that
energy is loss flowing from point 1 to point
2.
By added that energy that is lost to the energy
at point 2.
We satisfy the energy balance.
So this term the energy loss term.
Is broken down into 2 terms.
The major head losses and the minor head losses.
Now again all these losses are due to viscous
dissipation of the fluid that occurs through
the pipe.
So our major head losses is associated with
the actual friction between the pipe walls
and the fluid.
So this is represented with the friction factor
f, and is written as the following way.
Now I will go through these in more details
in just a second.
The minor head loss has to do with the energy
losses through each of the pipe components,
and so the pipe components each have a specific
loss coefficient that is associated with them.
Something that has been empirically determined,
and so when we sum these up again it becomes
a function of the flow that is going through
these components.
So looking back at the overall energy balance
the first terms on both sides are our pressure
energy, or flow energy of our fluid.
Now the pressure P is typically going to be
in gauge pressure.
Not always, but when we use gauge pressure.
If it is open atmosphere we can say that the
pressure is equal to 0 and drop that term
out of our equation.
Now if we use absolute pressures we need to
make sure that we use absolute on both sides.
Since the difference between them is really
the critical part.
Now gamma is the specific weight of our fluid.
So this can be temperature dependent.
Really the specific weight is just the density
of the fluid times the acceleration due to
gravity.
Now these terms are often combined to make
the calculations easier and can be found in
some tables.
Rather then having to plug in two different
numbers.
Now the second terms on both sides are the
kinetic energy components of the fluid flow.
Now this term alpha is called the kinetic
energy perimeter, and it is derived from analyzing
the velocity profile as it passes through
a control surface for the specified control
volume.
Now basically alpha just corrects for our
use of an average velocity in our energy balance
if the velocity profile is not uniform.
So without getting into more details alpha
will be equal to 1 for uniform flow, and therefore
drops out of the equation.
It will always be greater then 1.
We will say it is 1.08 for turbulent, and
equal to 2 for laminar or parabolic flow.
If we have fully developed flow alpha would
be the same for both points, and you can lump
those 2 terms together.
In most pipe flow problems we will have turbulent
flow.
So we will simplify our expression and just
use 1 for highly turbulent flow, and not have
to worry about this alpha term.
Now the velocity is v1 and v2.
Are the respected at points 1 and points 2
and maybe different on the geometry of the
pipes at those points, and the last variable
we have in those terms is g, which is the
acceleration due to gravity.
It is a constant.
Something you would look up and just make
sure you use the right value depending on
the unit system being used.
Third energy term in both sides is Z.
It is due to the Potential energy or the elevation
change in our piping system.
Simply put both Z values or heights based
on the same reference points.
It maybe that we choose 1 point to represent
an elevation of 0.
Meaning that if we say that Z1 is equal to
0.
Now Z2 will be with respect to Z1.
So that allows us to possibly drop out 1 of
the variables from this equation.
So at this point you are probably wondering
how does Z a dimensions of length represent
an energy.
So if you think of potential energy.
We think of mgh.
So know if we do this energy on a per volume
basis.
Since we are looking at our system, and we
are looking at an energy balance between 2
points.
We are doing this over a control volume.
So our control volume we divided by v, Now
we have mgh, and as we said before rho, g
is equal to gamma the specific weight, and
we have divided that through our equation.
So that all we have left is h.
So our Z term is just the height of our fluid
with respect to a reference plane and therefore
does represent our potential energy.
So the last part of this equation is our hL
or our losses term.
As mentioned both major and minor losses contribute
to this term.
So as we mentioned major losses are due to
friction of the pipe walls.
Due to the no slip cause between the fluid
and the pipe surface.
Wall shear stress results in an energy loss
of the fluid.
Thus the surface roughness and flow profile
plays a significant role on how much energy
is loss to friction.
Basically experiments in some dimensional
analysis later gave us this relationship correlating
the major losses due to friction based on
our pipe flow.
f is our dimensionless perimeter called the
friction factor, which is a function of the
surface roughness, which is epsilon, and the
flow characteristics of our fluid.
We represent as the Re number.
There is no theoretical analysis to prove
the relationship between the friction factor
and the surface roughness in Re number and
an excessive amounts of experiments in dimensionless
analysis has shown correlation between these
3 variables.
Allowing us to find 1 given the other 2.
Now this relationship was made graphically
available in the form in the Moody diagram
shown here.So in another screencast we go
through the Moody diagram in detail.
So if you trying to calculate it, and rather
then read of the graph you could use the Colebrook
formula or the Haaland equation.
That you can look up if need be.
Now also part of the major head loss is this
term here, which is l.
It is the total length of the pipe between
point 1 and 2.
Also D is our diameter of the pipe.
The other variables should be familiar both
velocity and the acceleration due to gravity.
Now the last part to go through is the minor
loss term.
Minor losses are associated with energy losses
due to flow through the pipe components such
bins, junctions, valves, flow meters.
Again theoretical analysis this has been to
difficult to nail down.
The losses associated with each of these components.
However experimental results in dimensional
analysis of a system has again provided correlation
for estimating these losses.
So the most common method is to use a dimensionless
parameter called kL, which is known as the
loss coefficient.
Now were as these losses are also a function
of the fluid properties and flow characteristics.
Most fluid problems we look at out turbulent.
So the only factor really is the geometry
of the componet.
Correlation between entrance and exit flow
conditions, contractions, expansions, pipe
coefficients, non-circular conduits are all
well document and can be looked up at appropriate
tables.
So that is each of the terms associated with
the energy balance.
So I want to show show you the most common
form of this equation may look like when we
analyze it in a pipe system.
So here I have taken both sides of the equation,
rearranged and equal to 0.
We might be calculating the pressure difference.
So maybe we are looking for p1-p2, and we
have information with respect to the rest
of the equation.
So this is the most straight forward type
of problem.
So we would be given a flow rate.
Usually repesented by Q. Maybe we are given
a pipe diameter.
So that allows us to solve for the average
velocity,and once we have the average velocity
now we can look up loss coefficients for each
of the components in our system.
That gives us everything we need for our minor
losses.
If we know the length we can get everything
we need for our major losses except for the
friction factor, but we also know we can solve
for our friction factor given information
about the pipe and the Re number, which is
based on the average velocity as well as our
diameter in fluid properties.
So then we can calculate everything with our
major losses.
At that point it is just a matter of knowing
the elevation difference and the difference
in velocities of points 1 to point 2.
Now typically when we do this one point might
have a negible velocity in comparison to the
other one.
So one of the velocities will drop out, and
then we will have 3 of the terms with respect
to the velocity through out the piping system.
So then you just plug in all your values and
solve.
You can also imagine that you might have to
solve for Q the flow rate.
Or the pipe diameter D, based on the pressure
drop.
Now this becomes a little bit more tricky
because it becomes a more reiterate solution
problem.
We will go over this in another screencast
demonstrating different examples of these
pipe flow problems.
