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.
Table 14.1
| A2/A1 | 0 | 0.04 | 0.16 | 0.36 | 0.64 | 1.0 |
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|
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- 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
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