Chapter 5
: Equations of Motion and Mechanical Energy
Lecture 14 :
Losses Due to Sudden Contraction
An abrupt contraction is geometrically
the reverse of an abrupt enlargement (Fig. 14.3). Here also
the streamlines cannot follow the abrupt change of geometry
and hence gradually converge from an upstream section of
the larger tube.
However, immediately downstream of the
junction of area contraction, the cross-sectional
area of the stream tube becomes the minimum and less than
that of the smaller pipe. This section of the stream tube
is known as vena contracta, after
which the stream widens again to fill the pipe.
The velocity of flow in the converging
part of the stream tube from Sec. 1-1 to Sec. c-c (vena
contracta) increases due to continuity and the pressure
decreases in the direction of flow accordingly in compliance
with the Bernoulli’s theorem.
In an accelerating flow,under a favourable
pressure gradient, losses due to separation cannot take
place. But in the decelerating part of the flow from Sec.
c-c to Sec. 2-2, where the stream tube expands to fill the
pipe, losses take place in the similar fashion as occur
in case of a sudden geometrical enlargement. Hence eddies
are formed between the vena contracta c-c and the downstream
Sec. 2-2.
The flow pattern after the vena contracta
is similar to that after an abrupt enlargement, and the
loss of head is thus confined between Sec. c-c to Sec. 2-2.
Therefore, we can say that the losses due to contraction
is not for the contraction itself, but due to the expansion
followed by the contraction.
Fig 14.3 Flow
through a sudden contraction
Following Eq. (14.25), the loss of head
in this case can be written as
(14.26)
where Ac represents
the cross-sectional area of the vena contracta,
and Cc is the coefficient of contraction
defined by
(14.27)
Equation (14.26) is usually expressed
as
(14.28)
where,
(14.29)
Although the area A1
is not explicitly involved in the Eq. (14.26), the
value of Cc depends on the ratio A2/A1.
For coaxial circular pipes and at fairly high Reynolds numbers.
Table 14.1 gives representative values of the coefficient
K.
As
,
the value of K in the Eq. (14.29) tends to 0.5 as shown
in Table 14.1. This limiting situation corresponds to the
flow from a large reservoir into a sharp edged pipe, provided
the end of the pipe does not protrude into the reservoir
(Fig. 14.4a).
The loss of head at the entrance to the
pipe is therefore given by
and is known as entry loss.
A protruding pipe (Fig. 14.4b) causes
a greater loss of head, while on the other hand, if the
inlet of the pipe is well rounded (Fig. 14.4c), the fluid
can follow the boundary without separating from it, and
the entry loss is much reduced and even may be zero depending
upon the rounded geometry of the pipe at its inlet.
Fig 14.4 Flow
from a reservoir to a sharp edges pipe