# PIPE EXPANSIONS AND CONTRACTIONS

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```					PIPE EXPANSIONS AND CONTRACTIONS
How do we evaluate the head loss in an expansion or a contraction
along the length of a pipe?

Like other changes in the geometry of a pipe system, we consider
these in a similar manner to the other minor losses – a function of
the velocity through the system.

V2
hL = K L
2g

However, which velocity do we use? The purpose of a contraction
or expansion is to change the velocity of within the pipe – do we
use the upstream or downstream value of velocity (V)?

The answer: it depends. But generally the loss is a function of both
the upstream and downstream velocities as well as the geometry of
the expansion or contraction. Often a velocity is chosen to
represent the velocity head (usually the higher velocity) and the KL
parameter is a function of the expansion/contraction angle and the
area ratio between upstream and downstream pipes (which is the
reciprocal of the velocity ratio).

1
Sudden Contractions
Sudden contractions are when the area of the pipe reduces
suddenly along the length of the pipe (at a 90 degree angle). The
downstream velocity will be higher than the upstream velocity.
For sudden contractions we use the following equation.

V2 2
hL = K L
2g

The value for KL is determined experimentally and is a function of
A2/A1. The following figure is from your textbook (Section 8.4).
Note that when A1 is much larger than A2 we have a coefficient of
0.5, which is the same as the minor entrance loss from a reservoir
with 90 degree edges.

2
Sudden Expansions
Sudden expansions are when the area of the pipe increases
suddenly along the length of the pipe (at a 90 degree angle). The
downstream velocity will be lower than the upstream velocity. For
sudden expansions we use the following equation.

V12
hL = K L
2g

Sudden expansions are analogous to the exit loss coefficients.
With exit loss coefficients we know that all the kinetic energy
(velocity head) is dissipated because there is no velocity in the
reservoir. For sudden expansions a portion of the kinetic energy is
dissipated – the difference being the difference between the two
velocities. Consequently the value for KL can be determined
analytically:
(V1 − V2 ) 2
hL =
2g

And from continuity:
V2 A1
=
V1 A2

so:
2
   A 
K L = 1 − 1 
   A2 

3
This relationship can also be displayed graphically.

For gradual contractions the angle of contraction is something less
than 90 degrees as shown in the figure below.

Again we use the high-speed velocity to evaluate our head loss.

V2 2
hL = K L
2g

4
The losses in a contraction are due to flow separation similar to
what we see in a Venturi meter or entrance losses. Values for KL at
various angles and Area ratios are shown below.

A diffuser acts to slow down the flow velocity by gradually
increasing the area of the pipe.

Again, we use the high velocity as the reference velocity for our
V12
hL = K L
2g

The actual value for KL depends heavily on the area ratio and the
angle of the diffuser. The following figure (from your text) shows
the relationship in terms of the value of KL as the ratio to the
equivalent value of KL for a sudden expansion. It can be seen that

5
for angles greater than about 30 degrees an expansion pipe is less
efficient than a sudden expansion.

6
PUMPS REVISITED
Pumps were already discussed in the course notes (Section 9.11
“Simple Pump Systems”). However, some further explanation is
required on pumps operating with multiple stages and operating in
series or parallel.
Pumps in Series
For pumps in series, a new performance curve must be generated.
The two pump curves are added together so that for a given flow
rate the head added is the sum of the two original pump curves.
This new pump curve can be used with the system curve to
determine the operation point. The following figure from your text
(Section 12.4) illustrates the relationship.

Pumps in Parallel
When pumps are positioned in parallel one follows a similar
approach as for pumps in series except one sums up the flow rates
for the pumps in parallel for a particular head to get the new
performance curve. The figure below illustrates this (also from

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Single and Multistage Pumps
Single stage pumps have a single impeller in the pump housing, a
multistage pump has 2 or more impellers. The following figure
shows a design of a multistage pump with 6 impellers.

When considering how to design the pump one must know if the
pump h-Q diagram is representative of a single stage of the pump
or the pump as a whole. If the diagram represents only one stage,
it means that for a given flow rate one could expect to multiply the
amount of head generated for a particular flow rate by the number
of stages. That is, consider it a number of pumps in series.

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