Losses Due to Sudden Enlargement

• If the cross-section of a pipe with fluid flowing through it, is abruptly enlarged (Fig. 14.2a) at certain place, fluid emerging from the smaller pipe is unable to follow the abrupt deviation of the boundary.
  
  
• The streamline takes a typical diverging pattern (shown in Fig. 14.2a). This creates pockets of turbulent eddies in the corners resulting in the dissipation of mechanical energy into intermolecular energy.

Basic mechanism of this type of loss

• The fluid flows against an adverse pressure gradient. The upstream pressure p₁ at section a-b is lower than the downstream pressure p₂ at section e-f since the upstream velocity V₁ is higher than the downstream velocity V₂ as a consequence of continuity.
  
  
• The fluid particles near the wall due to their low kinetic energy cannot overcome the adverse pressure hill in the direction of flow and hence follow up the reverse path under the favourable pressure gradient (from p₂ to p₁).
  
  
• This creates a zone of recirculating flow with turbulent eddies near the wall of the larger tube at the abrupt change of cross-section, as shown in Fig. 14.2a, resulting in a loss of total mechanical energy.
  
  
• For high values of Reynolds number, usually found in practice, the velocity in the smaller pipe may be assumed sensibly uniform over the crosssection. Due to the vigorous mixing caused by the turbulence, the velocity becomes again uniform at a far downstream section e-f from the enlargement (approximately 8 times the larger diameter).

Fig 14.2   (a)   Flow through abrupt but finite enlargement
(b)   Flow at Infinite enlargement  (Exit Loss)

• A control volume abcdefgh is considered (Fig. 14.2a) for which the momentum theorem can be written as

(14.20)

where A₁, A₂ are the cross-sectional areas of the smaller and larger parts of the pipe respectively, Q is the volumetric fllow rate and p’ is the mean pressure of the eddying fluid over the annular face, gd. It is known from experimental evidence, the p’ = p_1.

• Hence the Eq. (14.20) becomes

(14.21)

• From the equation of continuity

(14.22)

• With the help of Eq. (14.22), Eq. (14.21) becomes

(14.23)

• Applying Bernoulli's equation between sections ab and ef in consideration of the flow to be incompressible and the axis of the pipe  to be horizontal, we can write

	(14.24)

where h_L is the loss of head. Substituting (p₂ −p₁) from Eq. (14.23) into Eq. (14.24), we obtain

(14.25)

• In view of the assumptions made, Eq.(14.25) is subjected to some inaccuracies, but experiments show that for coaxial pipes they are within only a few per cent of the actual values.

Back to original slide - "Losses due to geometric changes"

Exit Loss

• If, in Eq.(14.25), , then the head loss at an abrupt enlargement tends to . The physical resemblance of this situation is the submerged outlet of a pipe discharging into a large reservoir as shown in Fig.14.2b.
  
  
• Since the fluid velocities are arrested in the large reservoir, the entire kinetic energy at the outlet of the pipe is dissipated into intermolecular energy of the reservoir through the creation of turbulent eddies.
  
  
• In such circumstances, the loss is usually termed as the exit loss for the pipe and equals to the velocity head at the discharge end of the pipe.

Back to original slide - "Losses due to geometric changes"