Welcome to the fourth episode of the Nutrium Podcast. I'm Trevor Walker and I'm Matthew Kidd.
And this week we will continue our discussions on hydraulics, this time looking at calculating
pressure drop in pipes.
Okay before we get started let's quickly recap what we covered last episode.
First off we used some pseudo physics to demonstrate that pressure is a key variable in hydraulics
as it provides the force to move a fluid from point A to point B.
Then we went on to talk about a guy named Daniel Bernoulli, who aside from making his
father insanely jealous, also developed an equation that describes the energy balance
involved in hydraulics.
This led to a discussion of flow regimes, of which there are three, laminar, turbulent
and transitional, with the flow pattern varying across the regimes.
Laminar flow is where the fluid flows along in straight lines without much mixing, eddies
or vortices.
Turbulent flow is the exact opposite to laminar, and transitional flow has elements of both
laminar and turbulent flow. We also noted that the calculation of pressure drop or more
specifically the friction factor calculation will vary depending on the flow regime. We
will discuss this in more detail later in this episode.
And lastly we introduced a dimensionless number called Reynolds number which can be used to
determine the flow regime a particular fluid flows in.
As we covered last episode, the Reynolds number is king among dimensionless numbers and measures
the ratio of momentum forces to viscous forces in a fluid.
If you're a process engineer and ever commit one dimensionless number to memory, make it
the Reynolds number, which happens to be the density times velocity times characteristic
length, i.e. the momentum of the fluid, divided by the kinematic viscosity, i.e. the viscous
forces.
Now, I say characteristic length because the Reynolds number can be calculated for a range
of different situations, like either flow in a pipe or natural convection up a wall.
For flow in a pipe, which happens to be what we are discussing today, the characteristic
length is the diameter of the pipe, which makes sense when you remember that the numerator
of the Reynolds number is the momentum of a slice of fluid and that momentum is equal
to mass times velocity.
Just to give you a little background on the Reynolds number, it was actually introduced
as a concept in 1851 by Sir George Gabriel Stokes of Stokes Law and Navier-Stokes equation
fame. Stokes was an interesting character and I'm sure we'll cover him in a future episode,
but briefly he was an Irish mathematician born in 1819 who roamed around the UK drinking
Guinness and making major contributions to the field of optics and fluid dynamics until
his death in 1903. However, as you may have noticed, it's called Reynolds number, not
Stokes number, and this is because it was popularised by another Irishman called Osborne
Reynolds around 1883 in his paper titled "An Experimental Investigation of the Circumstances
which Determine whether the Motion of Water in Parallel Channels Shall be Direct or Sinuous
and the Law of Resistance in Parallel Channels". Don't you just love paper titles?
Reynolds who happened to be one of the first professors in the UK to hold the title of
Professor of Engineering constructed an experimental rig that allowed him to
establish water flow through a transparent tube in which he would
inject dye and observe the flow patterns over a range of flow conditions.
In watching the dye Reynolds determined that the flow pattern changes significantly as
you go from a Reynolds number of around 2100, i.e. laminar flow, through to a
Reynolds number of around 4000, i.e. turbulent flow.
This observation was key to accurately calculating pressure drop as it indicated a need to characterize
things like friction factor over two distinct flow regimes.
So once we've calculated the Reynolds number of our pipe and determined whether the flow
is laminar, turbulent, or deity forbid, the transitional regime, the next step is to determine
the resistance of flow and this means calculating the friction factor.
They say there's no such thing as a free lunch and so it is with moving fluids around.
it's going to cost you some energy.
The next step is figuring out how much energy, in the form of pressure, it's going to take
you to push your fluid through the pipe to where you need it.
To do this, you're going to need a tool called friction factor.
The friction factor is a number which helps relate the flowing fluid in the pipe to how
much pressure we're going to lose from friction.
The friction factor is well understood, and there are many correlations we can use to
calculate it, unless we're in the transitional flow regime, in which case we need to get
to a sensible flow regime and figure out what the friction factor is for sure.
The friction factor is proportional to the pressure loss in the pipe, so the higher the
friction factor, the higher the pressure loss. Generally, the friction factor is highest
for low Reynolds numbers, and lowest for high Reynolds numbers. This makes sense intuitively
if you remember that for a low Reynolds number you might have a dense, slow-moving, very
viscous fluid through a thin pipe. So imagine trying to suck honey through a straw. You'll
have very low Reynolds number and you'll need a big pressure difference to move that honey.
On the flip side, to get a high Reynolds number, you have a low density, fast moving, low viscosity
fluid through a larger pipe. So that might be like sucking air through a straw, which
is going to be a lot easier. Even a theoretical pipe with perfectly smooth walls will extract
some energy as you push a fluid through it due to no-slip boundary effects.
Now there are two types of friction factor you might come across in the engineering world,
Darcy friction factor and Fanning friction factor. The Darcy friction factor might also
be called the Darcy-Westbach friction factor or the Moody friction factor.
The only difference between the two is a factor of four. The Darcy friction factor is four
times the Fanning friction factor, so converting between them is easy. The important thing is to be
clear which friction factor you're dealing with. There can be only one, so we like to deal with
the Darcy friction factor. There's a few names attached to these friction factors, and each has
been influential to the study of fluid flow. Henry Darcy and John Fanning, who our friction factors
are named after, are two 19th century engineer architect types, and both have to their credit
the building of a major water distribution system for a city. Darcy built a water distribution
system for Dijon in France, while Fanning designed the water system for Manchester in
the United States. With the importance of understanding pressure
losses to the successful design of a water system, it's not hard to see why both found
the time to work on unlocking the secrets of where all the pressure was going.
Wiesbach was a teacher of mathematics and mechanics, and refined the work of Darcy into
the Darcy-Weisbach equation, which is ultimately how we relate friction factor, the fluid,
and flowing conditions to pressure loss.
Lewis Moody was an engineer and professor.
The American Society of Mechanical Engineers, ASME, gives out an award named after him for
original papers which further the field of fluid engineering.
His most famous work is a chart, which we'll get to in a minute.
Before we can calculate the friction factor, we need to figure out how rough our pipe is.
Are we dealing with a nice smooth stainless steel or some kind of ancient foul brickwork
sewer pipe?
Roughness is a measure of the lumpiness of the surface of the material.
It is a kind of average bulk property, not something you could actually measure with
a ruler.
You can think of it as a measure of the average of the distance between the peaks and troughs
of the pipe wall material.
The values of pipe roughness for different materials have been determined experimentally
and you can look up values of roughness for various pipe materials and use them in combination with your preferred friction factor calculation methodology.
Roughness is usually quoted as absolute roughness for a certain pipe material.
You often need to convert that to relative roughness by dividing it by the pipe diameter.
The relative roughness gives you the ratio of the pipe wall roughness to the overall pipe diameter, and it's dimensionless.
When you are looking for the roughness of pipe materials, it's important to avoid imitations.
There are other relationships which calculate flow.
Manning's roughness and the Chezy coefficient are used for flow in open channels, and the
Hazen-Williams roughness is used as an alternative pressure loss calculations in pipes and ducts.
They're similar in concept to our pipe roughness, and they aim to do the same job of characterizing
the surface for the purpose of quantifying fluid flow.
But these roughness numbers are not interchangeable, and you have to be careful with your sources
to make sure you don't get them mixed up.
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Okay, when we have our flow regime and our relative roughness, we are finally ready to calculate the friction factor for our pipe.
To calculate the friction factor, we use a different method depending on the flow regime our fluid is in.
For the laminar flow regime, we use Darcy's equation, where the friction factor is equal to 64 divided by the Reynolds number.
A simple analytical equation that works every time.
In laminar flow, the fluid at the pipe wall is essentially stopped, so the pipe roughness doesn't matter, and doesn't factor into the friction factor.
You could think of the fluid as acting as a gap filler, essentially smoothing out the pipe.
For the uncooperative transitional flow regime, we examine the entrails of a goat and consult the oracle to determine the friction factor.
Other techniques include taking the turbulent friction factor, which will be on the high side of the true answer,
or using a weighted combination of the laminar and turbulent friction factors.
For the turbulent flow regime, we have options for determining the friction factor.
We can use a chart or one of many correlations.
These are all based on the painstaking measurements of many interested people
who set up hundreds of rigs and gathered tens of thousands of data points.
At the end of it all is the heroic work of Colebrook and White, who produced the Colebrook-White equation,
which brought all these data points together into one convenient equation.
Well, maybe not so convenient. It turns out the equation is implicit,
meaning the friction factor we are trying to find is required in the expression used to calculate itself,
and so we have to iterate to get an answer.
I couldn't really find anything interesting to say about Colebrook or White,
so we can assume that they were perfect engineers,
boring, lifeless people with no interesting features
except for their mindless pursuit of technical accuracy.
We salute you.
In 1944, Lewis Moody took the time to chart out the results
of the Colebrook-White equation, forming the popular Moody chart,
which shows the friction factor against Reynolds number
for various pipe roughnesses.
This is the chart we mentioned before
when we were talking a little about Moody,
And if you've ever had to study fluid dynamics, it's a certainty that you've seen this chart.
You can have a look for yourself over at the Nutrium article, "Pressure Loss in Pipe".
The Moody chart has been the backbone of most pipe flow calculations for engineers from
its creation until computational power reached a point where friction factor could easily
and usefully be calculated from a formula.
A 50-year span as one of the most useful tools for hydraulic calculations is not a bad run.
In the age of the modern computer, we don't need to look up charts anymore. If precision
is the aim of the game and time is no object, then solving the Colebrook-White equation
iteratively is your best bet. But if you're calculating the friction factor over and over
again, like you would in a segmented hydraulic model, this might slow you down too much.
Fortunately, hundreds of brave engineers and scientists have attempted to produce explicit
equations that accurately approximate the Colebrook-White equation. These are equations that don't require
us to do any iterations and allow us to speed up our calculations.
From this daunting list of approximations we have
Serg Heide's equation Chen's equation
Zygrang Sylvester equation The Harland equation
Churchill equation The Swami Jain equation
The Wood equation The Eck equation
The Jain equation A different Churchill equation
The Round equation The Barr equation
The Manadili equation The Monzon-Romeo-Royo equation
the Guda-Sonat equation, the Vantanka-Kuca-Zadek equation,
the Buseli equation, the Avki-Kargoz equation,
and the Evangelides-Papavengelou-Tzimpopovs equation,
just to name a few.
Oh man, I think I need a lie down.
Now you may say that which one you use
depends on the level of accuracy you need
weighed up against the computational power
you have available, but you're wrong,
because SirGuide's equation is the correct answer.
It gives great accuracy over a very wide range of Reynolds number and roughness values, and
is fine for almost all practical computational time constraints.
There are many other correlations that promise a higher level of accuracy, but I question
the need for higher accuracy given the uncertainty in the assumed roughness values of the pipe
material and the data that the relationship was based on.
So there you go, we have our friction factor.
If you were in laminar flow, we did simple calculations with Darcy's equation.
If you're in turbulent flow, you grabbed your trusty Moody chart or fired up the old Fortran
for your Colebrook-White or Serg-Hydes equation.
Now we can calculate the pressure drop in our pipe.
Believe it or not, but once you have determined the friction factor, the rest is relatively
easy.
For pressure loss in terms of head, which strictly speaking should actually be referred
to as head loss, we multiply the friction factor, the pipe length on diameter ratio
and velocity head.
Now velocity head is a term that will come up a fair bit when discussing pressure loss,
particularly pressure loss from fittings, which we will cover next episode.
Velocity head is a measure of the dynamic pressure and is equal to the velocity squared
divided by two times gravity.
If you're sitting there now and thinking this sounds pretty familiar, you would be right.
Velocity head is one of the components of Bernoulli's equation, which we covered last
episode.
Dynamic pressure, which is what velocity head measures, is just kinetic energy.
Framing kinetic energy in this way allows us to directly compare kinetic and potential
energy.
When we say one velocity head, what we mean is the pressure consumed to take a fluid from
rest to the flowing velocity, or how much potential energy is converted to kinetic energy
when a fluid starts flowing.
If you're a bit confused, don't worry, many people get confused when discussing dynamic
pressure. All you need to remember is that velocity head is the difference in
pressure when you change the velocity of a fluid. Now you know how to calculate
pressure drop the correct way, let's quickly look at another method you might
come across in industry. The Darcy friction factor and the pressure drop
equation are the accepted standard for calculating pipe pressure drop, but it
isn't the only method available. When you work in the water or fire systems
industries, you will find that they shy away from the Darcy friction factor and use the
Hazen-Williams coefficient we mentioned earlier.
The water industry is one of those industries that loves to develop its own simplified correlations
rather than depend on more widely applicable and accurate relationships.
So in around 1906, Alan Hazen, an American hydraulics expert, and Gardner Stuart Williams,
an American civil engineer, developed the Hazen-Williams equation, an imperial correlation
for pressure drop that does not depend on Reynolds number.
The equation itself is quite simple requiring only multiplication and division and the only
inputs you need are pipe length, flow rate, hydraulic diameter and the Hayes and Williams
coefficient.
You can see why it would have been popular back before the invention of the computer
when the alternative was to perform an iterative hand calculation of the Colebrook-White equation.
The downside is it's limited to water and does not include temperature and viscosity
corrections.
I still come across people using the Hayes and Williams equation from time to time.
It is particularly common when discussing fire water ring mains where clients will specify
various Hayes and Williams coefficients for different sections of pipe.
I personally tend to avoid it though because it is a bit too approximate for my liking
and we have many fine solutions for the Colbrick-White equation that are easily calculated in a spreadsheet
or hydraulic package.
So now you know how to calculate pressure loss in a straight pipe.
You can use your Reynolds number with your pipe roughness information to get the friction
factor, then throw it all into the Darcy Weisbach equation and figure out your pressure loss.
But what happens when you need to turn a corner?
That's something for next time when we look at pressure loss through pipe fittings and
some other tricky situations like two phase flow.
So this has been another episode of the Nutrium Podcast.
As always, we've added links to the related reading in the show notes.
If you have any topics you'd like us to cover or general suggestions for the show, please
contact us via the contact form at Nutrium.net or engineered.network.
Network.
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