I was recently walking in Las Vegas when I ran into the world’s largest chocolate fountain
(E1), at the Jean Philippe Patisserie inside the Bellagio hotel.
A 27 feet tall fountain of melted chocolate!
Once the hypnotic effect of the chocolate flow ended different questions struck my mind.
How can this be designed?
How can they make sure that the fountain runs continuously for over a year?
Well, some engineering considerations can be made on the piping system that the fountain
relies on.
Fluid flowing through a pipe or a channel is subject to hydraulic losses caused by viscous
effects generally driven by cross section shape and surface roughness.
The energy losses can be described in terms of a hydraulic head that can be directly connected
to the Bernoulli's principle.
Recall that Bernoulli's equation is a representation of the conservation of energy in a fluid.
With this equation we can describe the total energy head for an incompressible flow in
a gravitational field as the sum of a pressure term, the kinetic energy and the local height
of the fluid.
The hydraulic head represents the flow energy and has units of length since can be seen
as an equivalent height of a static column of fluid.
For inviscid flows the term H is constant, while for viscous flows is not.
Let's consider a generic case, we have an inclined pipe of length L and diameter D,
with a fluid passing through it at a certain velocity and subject to a gravitational field.
If we consider two different locations along the pipe, location one and location two, we
can describe the hydraulic head at the two locations.
For both locations we have the three main terms: the pressure terms, the kinetic energy
term, and the potential term…
But we can notice that in the location two we also have in additional term H_L that represents
the head loss due to the viscous effects.
Indeed, as a flow moves through a pipe there will be a certain head loss generated by the
interaction of the fluid with them by pipe walls.
Now, for any compressible fully developed flow in a level pipe the Z terms are the same
as well as the velocity through the pipe.
This let us obtain an expression of the head loss in terms of a scaled pressure difference
between the two locations.
Head losses can be divided into two main categories: the major losses that are connected to friction
in straight pipes, and minor losses due to different components in the piping system
such as valves, bends, junctions and so on…
In this lesson we'll analyze the major losses in pipes and ducts.
Let's start our analysis considering a fully developed incompressible turbulent flow in
a circular pipe.
If we analyze the cross section of the pipe, we can define the control volume between locations
one and 2 that are at a distance L from each other.
The flow at the two locations presents a fully developed velocity profile and it's characteristics
velocity and pressure.
On the side of the control volume defined by the pipe wall we have friction and the
wall shear stress acts on the fluid.
The area of the three faces of the control volume is described by simple geometrical
relations.
Note that here the pipe is horizontal and so the gravity effects can be neglected.
We can now apply the integral form of the momentum equation to our control volume.
The momentum flux through is non zero only on the locations one and two.
For the forces acting on the control volume we have: the shear force due to the shear
stress at the walls and the pressure forces acting on the locations one and two.
For an incompressible flow the two terms on the left hand side are the same and cancel
out due to the opposite orientation of the normal vector.
The equation, then, can be simplified into a simple expression that relates the pressure
drop due to viscous effect to the wall shear stress and the pipe dimensions.
The values of pressure and shear stress presented here our surface area averaged quantities.
This because averaged values are sufficient in pipe flow analysis for engineering applications.
The expression for the pressure drop along the pipe can be substituted into the head
loss relation we just derived.
This gives us a direct expression for the head loss where we can see that for level
straight pipes the head lost is only connected to the shear stress.
This equation is valid both for laminar and turbulent flows
Let's now recall the definition for the skin friction coefficient and the Darcy friction
factor, that is just the skin friction coefficient multiplied by a factor of 4.
Combining these two relations with the shear stress we can obtain an expression of the
pressure drop along the pipe in terms of the Darcy fiction factor.
The relation was developed in the mid 19th century by Henry Darcy, a French engineer,
without using any integral analysis but only through dimensional analysis.
With this equation we can define a pipe flow based on the pipe dimensions, the flow rate
and the density of the fluid, and after determining an appropriate friction factor we can estimate
the pressure drop along the pipe.
The equation can be also recast to obtain a flow rate for a given pressure drop.
For smooth pipes, the Darcy friction factor is a function of the Reynolds number.
In real world, however, pipes are not smooth.
Each pipe has its own surface texture.
For example a new pipe can be fairly smooth while an older pipe can be corroded and rusty,
so its internal surface is rougher.
The surface texture of a pipe can be described as a sand roughness height or through an equivalent
grain roughness height.
Different surface textures can be represented by the same parameter.
In this way, we can recast the Darcy friction factor as a function of Reynolds number and
surface roughness.
Hence, the head loss can be expressed in this slightly modified formulation.
This relation is known as the Darcy- Weisbach equation and it's valid for any fully developed
steady incompressible pipe flow.
How can we estimate the friction factor and the pressure drop for any generic pipe?
In 1944 Lewis Ferry Moody, professor at Princeton University, published what is now called the
Moody diagram.
This diagram is a collection of information that connects the friction factor with Reynolds
number and relative pipe roughness for different materials.
The single chart provides all the available friction factor data and it’s a common tool
to estimate pressure loss in pipes.
Most of the correlation data presented in this chart where derived from the experimental
analyses by Nikuradse in 1933, who artificially roughened the pipes by adhering sand grains
of known sizes and then measuring the pressure drop at different flow rates.
As you can see the laminar flow is independent on the wall roughness.
This because the laminar layer extends up to the centerline of the pipe, while in turbulent
flows the laminar layer is really thin and is directly affected by the surface roughness
of the pipe.
Looking at the chart, we can also see that at high Reynolds numbers the friction factor
becomes independent on the Reynolds number.
This region is identified by the “complete turbulence” dashed line.
This is because in thin laminar layers the surface roughness dominates the near wall
flow characteristics.
The Moody chart can be summarized by a single formula valid for the entire non laminar range:
the Colebrook formula.
This relation is implicit in F and must be solved numerically.
Another researcher, Haaland, derived an explicit approximation of this equation that can be
directly applied to estimate the friction factor.
The Moody diagram and the Colebrook formula are based on experimental data and cannot
provide exact predictions.
Usually the predictions present a range of error of about 10% that in certain design
analysis can be acceptable.
The application of the Moody diagram can also be extended to non-circular pipes replacing
the diameter with the hydraulic diameter.
However this reduces further the accuracy of the predictions due to the differences
of the flow patterns between circular and not circular pipes.