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.