in the analysis of internal flows
major losses are internal losses
connected to street pipes but
what if our geometry is not straight
how can we estimate the head losses
this kind of flows are more complex
and require a different approach to
analyze them
for common components of everyday use
in piping systems a simple approach
based on empirical formulations can be
used
the hair losses for non-straight pipes
are generally referred as
minor losses
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all the components that are categorized
under the minor losses
can be grouped into different categories
the pipe features such as entrance
and exit geometries as well as
cross-sectional area changes the
restrictions
that are flow nozzles orifice plates
screens grids and louvers
the fluid system components such as
elbows
tees branches bulbs
and similar components the flow through
all these
different components can become very
complex
however we can estimate the pressure
drop
through the component in a similar way
as done
for the straight pipe flow
what we can do is to set specific
inlet and outlet locations
for the component and correlate the
pressure drop
to the reynolds number and the geometric
features
of the component in this way
each part is treated like a black box
in a 1d system model
once we have the experimental or
numerical data
to build the empirical relations
we can quickly estimate the pressure
drop
based on the geometry of the component
the 1d model allows us to analyze
a system of different connected
components
and estimate the overall pressure drop
given by a combination of these parts
simply taking into account the pressure
drop
that each component will add to the
system
this is the essence of the 1d
system model this model is really
similar to the one
used to analyze an electrical network
where the pipes are the electrical wires
and the losses can be represented as
resistors
and the pressure and free flow rate
are the equivalent of voltage and
electrical current flow
for minor losses we can use a model
based on a k factor or loss coefficient
so if we analyze for example
a pipe entrance where the fluid is
disturbed
as it enters into the pipe we can see
that this creates a pressure variation
in the fluid
and also generates minor losses
if we analyze the pressure at different
locations
a b and c we can see
that the pressure in point a drops
as we reach point b this because the
fluid
accelerates as it enters the pipe
then the fluid slows back down as we
reach location c
to the velocity it will maintain along
the rest of the pipe
the deceleration will create a pressure
rise
in the flow what we see here
in the graph is the ideal condition
where the flow will slow down without
any loss
in reality the actual operating
condition
will be a lower pressure allocation c
this is caused by the pressure losses
that can be estimated as a percentage
of the kinetic energy the pressure
variation
and the hair loss equation can be
expressed in this form where
k l is the loss coefficient
or k factor there is a function
of geometry and reynolds number
the loss coefficient gives us an
estimate
of how far we are from the ideal
conditions
due to viscous losses
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now let's analyze different components
that generate manure losses in
particular
we are going to analyze the key factor
for
entrances sudden contractions and
expansions
pipe bands and other miscellaneous
pipe components for each component
we need to estimate the pressure drop
based
on the kinetic energy so we need to
properly choose
the reference velocity to compute the
manner losses
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in the next examples the red arrow with
a red
v will indicate the location where the
reference velocity
should be estimated for the component
analyzed
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first we start with the entrance losses
reservoirs are commonly attached to
pipes
to let the fluid flow through its final
destination point
as the flow moved from the tank
or reservoir to the connected pipe it
accelerates
through the pipe and it adjusts
to the pipe shape when the entrance
region
presents sharp corners flow separation
will likely
to occur because the flow momentum
will not let the flow perform such a
sharp turn
reported here is a section of the
numerical results for an entrance region
overlaid on our sketch the colors
represent the velocity magnitude
where the blue is the lowest value and
the red
is the highest one you can see that the
flow
accelerates and the color moves from
blue to red the flow is accelerating as
it
enters the pipe and then later it starts
decelerating
reaching its final velocity the darker
blue region near the corner represents
the region where we have
flow separation there
a little bubble of fluid is just
recirculating
and not moving downstream this minor
loss
can be reduced if we smooth out the
corners
in this case we can see that the fluid
will present
smoother streamlines and it will enter
the pipe
without generating any flow separation
reducing
the pressure loss the loss coefficient
for entrance region can be related to
the diameter of the pipe
and the radius of curvature the larger
the radius of curvature the lower the
loss coefficient will be
similar to the entrance region is the
sudden contraction
in pipes and channels we often encounter
this component
when larger pipes are connected to
smaller pipes
the contraction works in a similar way
as the entrance region
where we can have separated flow
the flow initially accelerates and then
the celerix in the smaller pipe the loss
coefficient
is related to the ratio of the
cross-sectional
areas of the two pipes the closer
are the areas the smaller the loss
coefficient
will be
now if we swap the order of the larger
pipe and smaller 5
we obtain a sudden expansion for sudden
expansions
we can perform a simple analysis to
obtain
and derive an expression for the loss
coefficient using
the simple conservation equations
as the flow exit the smaller pipe forms
a jet structure
and on its sides the flow will be
recirculating
then the jet spreads up to cover the
entire section
of the larger pipe now
let's see how we can analyze this
component
we first define a control volume that is
bounded
by location 2 and 3 and the side walls
of the larger pipe
the velocity of the fluid entering the
control volume
is v1 while the fluid exits
the domain at velocity v3
also we assume that the pressure along
location 2
is constant we can then apply
the continuity the momentum and the
energy equation to the control volume
and have these simple formulations
combining these three equations and
solving for the head loss
we can obtain the simple relation for
the loss coefficient
there is a direct relation connected to
the ratio
of the two cross-sectional areas
here we can look at the streamlines
of the numerical results overlapped on
the bottom half
of the schematics and we can see that
the flow
initially maintains its velocity
and then it slowly opens up
reaching the larger pipe walls
a component that is largely used in pipe
fittings
is the bend bends allow
the flow to change the direction
and avoid obstacles however
they add additional head losses compared
to simple straight pipes
due to different phenomena looking at a
set
of streamlines to the bend you can see
that the flow separates from the most
internal side
this because there the curvature radius
is the smallest for the geometry
the flow separation then causes
additional effects
the secondary flow describes a motion
of the fluid on the plane normal
to the primary flow direction the
primary
and secondary motions happen
simultaneously
as the flow passes through the band
the plot shown here reports the trend
of the friction factor as a function of
the curvature radius
and the pipe diameter the curves
represent the behavior of different
pipes
having different surface roughness the
chart
shows that relatively large values of
curvature radio
can reduce the friction factor however
this can be limited by the effective
space
available in confined spaces
other common components that can be
found
in pipe networks are 180 degrees
elbows to turn backward the flow
t junctions to split or mix flow streams
and valves to regulate and control
the flow rate through the pipes
here in this table i reported different
common components
and their relative friction factor
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you