>> Dr. John Biddle: 1 minus little R over
big R quantity squared. That's [Inaudible].
So now we know if it's [Inaudible] into the
[Inaudible] will be parabolic. The next step
we did last time was to get the flow rate
Q. We integrated over [Inaudible] times the
differential area. We talked about that last
time what that was. It was [Inaudible]. We
have different area. Like this. There's our
[Inaudible]. There's another area. We've got
a circular area, [Inaudible] squared. We've
got the surface area and we got the differential
area where we have the pipeline [Inaudible].
This is the radius R. Circumference times
DR and DR is [Inaudible]. Circumference 2
Pi R. So the differential area of 2 Pi R,
DR. Okay so anyway we integrated that from
the centerline, R equals zero out to the outside
radius. Here's a picture of the outside radius
goes to here, capital R. When you do that,
you end up with this equation. This is this
equation. Okay, I think that's about where
we left off last time. Now the next step.
Yeah I think I did this too last time. [Inaudible]
equals 2B. Yeah okay. So what we said last
time was [Inaudible] of B equal Q over A.
We put that Q up there, divided by the area,
Pi big R squared and we get the average velocity,
LPD squared over 32 UL. Note when we did this
last time we compared that to BC. That the
average velocity just BC over 2. Okay, that's
I think where we stopped off last time. Now
we're going to solve this guy here for Delta
B. We're going to divide this by one-half
row B squared. That back in chapter three,
yeah, chapter three. That was called the dynamic
pressure. One row B squared. So divide that
by one-half row B squared. And when you do
that, this is what you end up with. [Inaudible]
is row BD over U. Now here's Delta B multiplied
to by one-half row B squared over two. That's
that one. There's Delta B. This thing, this
was the definition. This is the definition.
We define F by this equation. This F is called
the friction factor. So if we want to find
the pressure drop in a pipe, we can take the
length, divide by the pipe diameter, multiply
it by the density of the fluid flowing, multiply
it by the average velocity squared, divide
it by 2 and times a factor, which we call
a friction factor. Now if it's [Inaudible]
you can see in [Inaudible] what F is. There's
F. So far [Inaudible] F equals 64 over [Inaudible].
Now friction factor [Inaudible]. So it's important
to know. And again, friction factor is something
we've invented, we define it. Why did we invent
it? Because it makes it easy to calculate
the pressure drop in the pipe with this concept
of a friction factor. And [Inaudible] is very
simple. It's 64 divided by the [Inaudible].
Now we can also divide this by row G. So divide
this guy by row G. So our row G is Gamma so
Gamma, Delta B, over Gamma is equal to FL
over D. The rows cancel out. The G downstairs.
B squared over 2 G. Units are feet, are meters.
And we know call that the head loss due to
friction. The head loss due to friction. And
where do we use that? That's an equal sign
there. We use that of course in the energy
equation from chapter five. Okay? The 
energy equation, chapter five. There it is.
Now let's just review that equation again.
It's sometimes called the modified Beroulli,
except now we allow, compared for Bernoulli's equation, 
we allow many other things to happen. First
that's the head by the pump. This is the head
developed by the turbine. This L stands
for the length of a piece of pipe. This is
the head loss due to pipe friction. This is
called the minor losses due to elbows and fittings and valves. They're all this is called the
head form of the energy. Because every terms
has units of feet or meters. This HL now,
we used to say this was called the losses.
Just called losses, L-O-S-S-E-S. We call it
that. Now we've got a way to find the
losses. There it is. I got to find though
the friction factor. That's the key. I know
how long the pipe is. I know it's diameter.
It still gives me the flow rate. I can find
the velocity. But it comes down to finding
F. So that's what it amounts to. We have to
find F and if we have [Inaudible] it's very
easy. There's the equation for F. Okay, let's
work an example problem using F. Otherwise
the reason why we write the energy equation
like that, on the right hand side of the board
is because everything on the left hand side
is energy that is coming into the fluid. The
subscript one means entering. A pump does
what? Adds energy. So the fluid brings in
energy. The pump adds more energy. Now where
does the energy go? Some of it goes up with
the fluid leading. There it is. A turbine
can take energy out. That's what a turbine
does. Losses of course reduce the energy available.
[Inaudible] of course reduce the energy available.
So that's why it's nice to write that equation
in that form. Because you could then talk
about it in words and not just in symbols
like that. Okay, so our example then horizontal
pipe. What if we had a horizontal pipe, a
horizontal pipe of constant diameter? A horizontal
pipe of constant diameter in compressible
flow? Continuity, think compressible. Z1,
Z2, the same. Horizontal pipe, Z1, Z2. Okay,
we're left with what? P1 is clearly P2. Yeah,
pressure is high coming in. Delta P divided
M equal if there's no pump. If there's no
turbine to bind. If we neglect minor losses,
gone. Then and only then do we get that guy.
Okay. That's where it comes from down. Just
so you know where it came from. That equation
there is valid for what again? Incompressible
horizontal pipe. No pumps, no turbines, neglect
minor losses. If that's not true, go back
up there. That H of L. This same thing. But
you don't have this equation. See this equation
right here? H of L is equal Delta Gamma, that's
where it comes from. Just be careful when
you use that guy. That thing is always true.
That thing is always true. This thing is sometimes
true. When it's sometimes? Well here we go
again. No pumps, no turbines, neglect minor
losses. An incompressible horizontal pipe.
So this example is incompressible horizontal
pipe or specific 8.5. Okay velocity, [Inaudible]
velocity. 13 centimeter diameter type. At
a flow rate, Q, 0.2 to 0 cubic meters per
second. I want to find head loss in 100 meter
length of pipe. Okay. There it is, head loss.
Head loss, H of L. It's horizontal, incompressible,
constant diameter. There it is. No pumps,
no turbines, no minor losses, H of L. Delta
P over again, which is F over D, B squared
over G. First of all, pretty much always in
a fluid problem, flow in a pipe or a tube
or a duct, always calculate the [Inaudible]
number. You have to know that to start with.
VD over new. Okay, let's get the velocity
up here. The velocity is Q over A. So we have
0.020 divided by the area, Pi over 4B squared.
Diameter 15 centimeters. Velocity 1.13 meters
per second. Okay, 1.13. Now if you want to,
you could use--, we have an equation which
was in terms of Q for the [Inaudible] number.
You could do that too. That's best a shortcut
way to do it but I found the velocity first
[Inaudible] later on. We'll see. Times the
diameter, diameter is 15 centimeters. Divided
by [Inaudible] given, 6 times 10 over. So
the [Inaudible] 283. Less than 2100, so 
it's [Inaudible]. Okay, so now we know, yeah
okay [Inaudible] now I can get the friction
factor. Friction factor, 64 over [Inaudible]
number. Friction factor, .226. Now I plug
it into the Delta P equation. F226, the length
of the pipe, 100 meters. The diameter 
of the pipe, .15 meters. B1.13 squared divided
by 2G, 9.83 meters. What I did to [Inaudible].
Okay, so forward flow. That's how we get that.
Now we take a little more difficult one. Because
most of the flows in the real world are not
laminar, most of them are turbine. So now
you got to figure out the way to get the turbine
pressure down the pipe. Okay. We'll let's,
by the way this equation is still going to
hold whether the flow is turbine. The only
thing we can do is this guy right here. Because
he came from here and he came from here and
he came from here. And it came from Newton's
Law this constant. So this is the one you
cannot use for turbine flow. But this equation
is still true. This equation is still true.
Except now we need F for turbine flow. Let's
just review real quick laminar flow. Laminar
flow pretty much if you follow a fluid particle
as it moves down in a laminar flow field,
it pretty much is straight. Now if you go
to a turbine flow field, if you greatly saturate
it, just to show you on the board, that's
what it does. See the word laminar comes from
lamina. What does lamina mean? Lamina means
less surfaces. A laminate flooring for your
kitchen or whatever, flat surface. A textbook,
these pages are like lamina, they can slide
over each other very smoothly. They slide,
that's laminar. So the fluid motion, you pretty
much go in a U direction. X direction. The
U velocity. And they don't [Inaudible] they
go straight like this. They don't hit each
other too much. Now when you get the turbine
flow, you get a little B component of velocity.
U goes this way. V goes normal to it. You
get a little B component. So this molecule
starts to bounce around like this as they
go down. They bouncing like this and of course
two cars on a freeway, one going like this,
one going like this, guess what happens? Metal
on metal. Friction slows the car down. Guess
what happens to molecules in the water? They
bump into each other. They exchange momentum
and friction develops and they start to slow
down. This is rougher. It's got more reaction
so that's what's happening here. There is
a B component of velocity in turbine flow
as opposed to a nice, smooth flow in laminar
flow. Okay let's get on turbine flow. Turbine
flow. To divvy up the values they didn't go
to any theoretical variation like we did for
laminar flow. No it's very, very difficult,
nearly impossible. So what they did is they
took pipes and it's described in here. They
took pipes and they--, different diameters
and they glued sand on the inside surface
of the pipe. Now they glued very precise diameter
sand particles not random sand particles.
Very precise diameter sand particles onto
the inside of a pipe to go to a rougher and
rougher pipe. And then they made the flow
turbulent and then they looked at the results
of this pipe roughness. So they artificially
roughened the inside surface of a pipe with
sand particles and glued [Inaudible]. But
anyway, that's what they did. Now they didn't
get an equation. What they did was is they
took all the data and different people massaged
it and worked with it and they ended up with
something called a Moody Diagram. Now here's
what a Moody Diagram plotted. On the Y-axis
they are plotting the friction factor, F.
Now let me give you one and then I'll just
go put it on the board. So if you don't want
to copy it, don't copy it. Anyway, that's
the friction factor. And this is the Reynolds
number. So what I will do is give it to you
now so you can look at it. Okay, [Inaudible].
So this Moody, I'm just going to sketch on
the board again, mention some things about
it. We're plotting F on the X-axis. It starts
down there as 0.008 and it goes up to 0.10.
This is 0.01. Yeah, okay. 2, 3, 4, 5, 6, 7,
8, 9. That's 0.01. Rows number 10 squared.
I'll just start at 10 squared. [Inaudible],
4th, 5th, 6th. Laminar flow. There are probably
about 2100. Here's 100. Now this is a long,
long axis first of all. Long, long axis. Both
axes are long. So here's 100, 1,000, 10,000,
100,000, million. So laminar flow of 2100.
2,000, 10,000, 2,000, 3,000, 4, 5, 6, 7, 8,
9, 10 up there. 2100. In that range in laminar
flow the friction factor is 64 or the Reynolds
number. When you plot that on long, long axes,
that's about one. Plot it on long, long axes,
guess what you get? A straight line. Okay,
there it is. Here down to about .03. So there's
a line at FD equals 64 over the Reynolds number.
Laminar flow. But now you see another family
of lines and they start out up here at about
.003 and this is 4, roughly 4,000. So this
4,000 and these lines go something like this.
I'm just going to sketch them right here.
And these lines are labeled on the right hand
axes. You can either say little E or epsilon.
Take your choice. I'll use little E. E over
D. The relative roughness. E over D, these
lines are E over D. This guy is called the
pipe roughness. This one of course is the
pipe diameter. Where do you get the pipe roughness?
In the little legends. In the little legend
it says the pipe is cast iron or galvanized
iron or drawn tubing or concrete. Here's the
value of little E. So they give you the value
of little E for different pipe materials.
Notice concrete. Really, really rough compared
to a wrought iron type. Of course concrete
is rough. [Inaudible]. It feels rough. Now
taking on a PVC pipe, [Inaudible] PVC pipe.
Wow, it's [Inaudible]. Copper, wow is it [Inaudible].
This line here is labeled a smooth pipe line.
PVC can be approximated most of the time as
smooth. Drawn copper tubing could be approximated
most of the times as smooth. This is a little
bit beyond that they hydraulically smooth.
But normally in this class, if you hear something
like PVC or copper, look at drawn tubing.
Oh my gosh. What are there? Six zeroes in
five feet? Yeah, it's really pretty smooth.
So yeah we're going to assume a pipe like
that is smooth. Yeah. So how did they get
those lines like that? Don't forget they got
this one over here. They got this one from
theory. These guys they didn't get from theory.
They got them during what? Gluing sand particles
very carefully on the inside of a tube. And
the bigger the sand particles were, the rougher
the tube was. This is high roughness. This
is no roughness. So we want to get F in turbine
flow to put this equation right here. Or up
here. Then we have to get the relative roughness
divided by the diameter. Find what line we're
on. Get the Reynolds number, go up here with
the Reynolds number until you find out what
line you're on. Then go across horizontal
until you get to the F axis and that's the
value of the friction factor. Okay. Let's
see just for your own information, I gave
you three examples on the bottom of that page
with the Moody Chart on it. So make sure you're
reading the Moody Chart regularly. One is
for galvanized iron pipe, one is for a riveted
seal pipe, the other one is for a drawn tubing
pipe. And I got the points labeled there to
show you on the Moody Chart where those points
are, to give you the friction factor in the
left hand side of the page there. So that's
examples of how you get the friction factor.
Okay, this is called [Inaudible]. You can
see where we start to be flat around here.
You can draw a dash line here. This is where
it starts to be flat. Everything to the right
of that dash line, these lines are horizontal.
Horizontal. This region is labeled in our
textbook completely turbulent. The region
to that side of the line, the region up here
to here is labeled transition. Transition
region between these two dash lines. Transition
region. The region from here to here of course,
laminar. The region between these two dots
from here to here, critical. So there are
four regions of the Moody Diagram. Four regions
of the Moody Diagram. You wouldn't need the
Moody Diagram for laminar flow because you
don't really use the Moody Diagram for laminar.
You do it mathematically 64 divided by Reynolds
number. As long as you read in the log chart.
But where you need it is over here of course
or you could find some perfect equation that
might work but you get that as a function
for different things. Okay, so let's say you're
in the laminar region. It depends on well,
on let's say you're in the complete [Inaudible]
turbulent region. That depends on E over D
only. Finding one pipe right here. That diamond
cap won't change along here where it's horizontal.
And what's the region called where it's horizontal?
The completely turbulent region. Now take
the transition. That depends on, okay, now
it depends on Reynolds and E over D. It depends
on both things, the Reynolds number and the
relative roughness. E over D is called the
relative roughness. It's dimensions. F is
dimensionless. The Reynolds number is dimensionless.
This is a dimensionless chart. We engineers
love dimensionless charts and fluid mechanics
and heat transfer, wow, they're must be 25
or 30 dimensionless parameters. The no salt
number, the Stanton number, the [Inaudible]
number, the Reynolds number. Blah, blah, blah.
The friction factor, the relative roughness,
those are all dimensionless parameters. So
this is a dimensionless chart plotted on long
log paper. Sometimes people have a lot of
difficult reading a long log axis. If this
is 10 squared and this is 10 and this is 10
cubed, okay, find 105. Reynolds number 105.
Because if you can't read this axis, you're
not going to get the F value right. 105, what
this one? 100. What's that number? 10. What's
that number? 1,000. Now don't say this. I
think that's 500. No, no. Here's what it is.
If you take the distance from here to here.
Take a ruler and measure this distance, and
take 70 percent of it. You got that? 70 percent
of it. Don't think linearly, think logarithmically.
That's how the skills make it. There's 500.
It's not in the middle. No, it's 70 percent
of the distance between my two fingers. That's
where you put the 5 times 10 squared. That
by the way, is 5 times 10 squared. What is
9 times 10 squared? There. Where is 2 times
10 squared? There. What's that? 200. What
is 105? Right there and boy you better read
those guys right. It's not rocket science.
You know, it's not calculus. No. Not [Inaudible].
No. It's how you read a log shield reasonably
accurate. So be careful. Some people when
they look like this they really have a problem
reading those scales. Okay, let's go back
here again. In the series, oh this big time
series on smooth pipe, very steep curve. Very
steep curve. If you [Inaudible], some people
think okay, 110,000, 1.1 times 10 to the fifth,
there's two-tenths to the fifth. Okay, right
there. It's right there. So be careful. It
can really get you if you're not careful.
Alright. I didn't do the critical region yet.
I didn't do critical region. No, I didn't
do it and I'm not going to do it. Look at
the Moody Chart. What do you see there? I
see a blank region. There's nothing there.
The lines end. That's right. The lines end.
What does that mean? It means don't even attempt
to get a friction factor in the critical region.
Well what if I design something and it's in
the critical region? I say well you should
have been an accountant then. [Inaudible].
But you're not an engineer. No engineer designs
something to operate in the critical region
and guess why? Because to the left is laminar
flow and to the right is turbulent flow and
since we don't know where we are, your neck
is on the chopping block. Don't say well I
think it might be laminar or you know, I'm
pretty sure it's turbulent. I was pretty sure
I think. It means you are in trouble. No it
just means engineers don't design anything
to operate in that region. Now we know we've
got to go past it to get from no flow when
you turn the pump on, no flow. You're going
to be turbulent. We go past it but we don't
sit there. We don't design something to sit
there. Because we don't know what's happening.
For instance, the flow can be laminar pipe
in a factory, in a plant in a pipe and then
the [Inaudible] and over there 20 feet away,
is a big compressor. A big compressor. The
compressor doesn't operate continuously. It
goes on with the pressure. The air goes down
in the line. The compressor kicks on. The
flow is forced to vibrate. Guess what happens?
You're running the risk of tripping back laminar
flow into turbulent flow just because the
compressor went on. Which means you the engineer
don't know what is going to happen. So the
lesson is, don't design anything near that
region. Stay away. Say I'm going to make it
exactly 2,100. Oh man. You're [Inaudible].
Don't do that. That's not a magic number.
Four textbooks, our [Inaudible] textbook says
2,300. Some textbooks say 2,000. Some textbooks
say 2,100. There's no magic number. It's around
2,100 plus or minus a couple hundred. So you
know, don't think it as magically as 2,100.
It's going to kick up to critical. No, no,
no. Is 4,000 magic? No it's not magic. Now
here is the rule in our class, yes magic number.
If you get it on an exam, 2,050, okay it's
laminar flow. You get 4,001, it's turbulent
flow. So in our class, yeah, there are magic
numbers. In the real world, no, they're not
magic, they're negotiable. Where's your pipe?
Inside the plant, outside the plant. How close
is it to a diesel engine? Blah, blah, blah.
Things like that. Okay. So that's our turbulent.
Here's what it depends on. Now there are various
fit equations to fit that data. Some are more
accurate than others. We're going to use one
equation that fits that data. And it's called
modified Colburn equation. So this is for
our entire non-linear and non-laminar flow.
You could use the Colburn formula. It's called
a modified Colburn. Sometimes called a [Inaudible]
equation but we'll call it a modified Colburn.
1 over square root F. Looks like you can put
them in the chart on some kind of computer
code obviously. You got to [Inaudible] equation.
Minus 1.8, log base 10. V over D, 3.7. Let's
just make it simpler. So you can use this
to calculate F. It still takes calculation
obviously. And that works anywhere to the
right of my piece of paper. And to the right
of my piece of paper. Here from V of D, it's
a smooth pipe then of course the roughness
is zero. Okay. And our textbook tells you
how to read it here. Okay. A word of caution
is in order concerning the use of the Moody
Chart or the equivalent formulas. Because
of various inherent inaccuracies involved
on certainty and relative roughness, on certainty
experimental data to produce a Moody Chart,
the use of several place accuracy in pipe
flow problems is usually not justified. As
a rule of thumb, a ten percent accuracy is
the best expected. That's the best. It goes
downhill from there. So the best you can get
by reading a Moody Chart or using the equation
is within ten percent of the actual one. And
of course that's only for brand new pipe.
The pipe has been in the line for five years,
all bets are off. The roughness has increased
dramatically because of hard water deposits
on the inside of the pipe for instance. Your
pipes in your home, hard water in the western
region of the U.S., oh yeah, what does that
do? The deposits calcium makes the surface
really rough and it makes the opening smaller
and smaller with time as you get more and
more calcium. So yeah that's a big problem.
This is only for designing new pipes. Okay.
Now look at the friction factor for laminar
flow. Is that equation valid for oil? Of course
it is. Water? Yeah. Air? Uh huh. Carbon dioxide?
Yeah. A winch pipe? Sure. Six-inch pipe? Yeah.
Rectangular duct, HPC duct? Uh huh. 30 foot
water pipe? Yeah. Okay? Is it valid for PVC?
Yeah. Is it valid for cast iron? Uh huh. Concrete?
Yeah. You get the point. There's not a special
equation for different diameters. For different
flow rates? Flow rate two gallon per minute?
Ten gallon per minute? 30 gallons per minute?
Does it work for all of that? Yeah it works
for all of that. No matter what the pipe is
made out of. Well here's a Reynolds number.
Here's what the Reynolds number is. First
of all it's a function of the pipe diameter.
Second of all, it's a function of a flow rate.
Third of all, it's a function of what the
fluid is. Those are the three things that
the friction factor depends on. So there's
only one equation for everything. Now you
go to turbulent flow. Now I add one more parameter.
This dimension I said, E over D to that one.
So now you tell me what kind of pipe it is.
Stainless steel. Okay. I get E over D. Tell
me it's flow rate. 100 gallons per minute.
Okay, I got a V now. Two-inch pipe. Got it.
What it's carrying? Water. Got it. So I know
everything I need. What kind of pipe it is,
what's it's diameter, what's the flow rate,
what's the fluid? Based on that, I can go
over here and I can find the friction factor.
Is there 10,000 pieces of paper? Let's see.
I'm trying to find stainless steel. Oh here
it is. Now I'm trying to find stainless steel,
too much diameter. Oh there's a Moody Chart
for that right there and there. No, there's
not 10,000 sheets of paper. One for each fluid,
one for each diameter. One for each flow rate.
No, no, no, no. Guess how many sheets of paper
I need to give you the friction factor for
any pipe material, any pipe flow rate, any
fluid in the pipe? I'll tell you. One. I wish
I invented the diagram. I love the sound of
that. No I'm sorry. I couldn't do it. Preston
Moody did that back then. So yeah, now people
don't realize that. Are you kidding me? One
sheet of paper for friction factor, for everything
us engineers can imagine? That's why everybody
graduates from fluid class better know the
Moody Diagram. Everybody who graduating from
a ME 218. [Inaudible] materials what should
they know about a certain diagram?
>> [Inaudible]
>> Dr. John Biddle: The what?
>> [Inaudible]
>> What's the one with the circle?
>> Dr. John Biddle: You got it. You got it.
Are there equations for doing that? Yes there
are. Why do we still teach the more [Inaudible]?
Because we engineers love the visual impact
of a diagram like that. And why would you
use the equations? Because the equation came
from that. So number one, you should learn
how to read this. And this is just then what
you put onto the Moody Diagram to solve for
that. But that diagram. It would be interesting.
I just love that diagram. You know, that's
the way we are. We're that kind of people.
Now the manuals say oh that I love that equation.
I know, I love them too. They're good people.
But they're not engineers okay? But we love
other things. That's why I love that Moody
Diagram. Everything is on that thing. [Inaudible].
That's called a dimensionless diagram. We
engineers love dimensionless numbers. You
don't plot the flow right here, the velocity,
you don't plot the diameter, it's all divided
by different things. Here and here. And F,
dimensionless. That equation right there.
It's a dimensionless equation. Nothing has
dimensions. Okay. So that's the diagram that
you want to use to get the friction factors
for homework and other problems. You'll have
one on the final exam. So you'll have that
in your answer. Okay, I think a good stopping
point. Next time I'm going to bring in. Here
it is. We're going to bring the [Inaudible].
Next time we're going to discuss the minor
losses.
