The flow of fluid through a contraction (decrease in pipe diameter) results in an increase in the velocity and consequently, a pressure drop greater than the value for the equivalent straight pipe. If the contraction is sharp or sudden, the behavior of single-phase flow is as shown in Figure 1 and involves two *recirculation* regions. The first starts about 1.5 inlet pipe diameters upstream whilst the second starts at the contraction and extends up to 15 outlet pipe diameters downstream. The dissipation of energy caused by these recirculation regions means that not all the pressure drop is converted to kinetic energy (and thence recoverable at a subsequent enlargement) and reversible, and irreversible components of pressure drop must be considered. If the contraction is being used to create kinetic energy from pressure, it is necessary to employ a more gradual change in diameter so as to eliminate or minimize recirculations and thus losses.

Lighthill (1986), in discussing the calculation of pressure drop through a generalized contraction, points out that it is not possible to use a momentum balance as there is an unknown reaction force from the walls to be accounted for. An energy balance gives a computable expression for the (reversible) pressure drop. To account for both the reversible and irreversible components of pressure drop, a balance is carried out from upstream to the minimum flow area point at the *vena contracta* (with no irreversibility) and a second balance downstream where most of the dissipation occurs. The combination of these results in

where
is the mass flux; ρ, the density; S, the area ratio between upstream and downstream pipes; and C_{c}, the contraction coefficient (the ratio of areas of the *vena contracta* and the outlet pipe.) Equations for the contraction coefficient in terms of S are given by, for example, Benedict (1980) and Chisholm (1986).

In gas/liquid flow, the sudden contraction can act as a homogenizer mixing the phases and making their velocities more equal. For annular flows, the contraction can cause an increase in the proportion of liquid travelling as drops.

For two-phase pressure drop, Chisholm (1983) provides an equation equivalent to (1), derived using a separated flow approach. Comparison with experimental data shows that the homogeneous version of this equation gives the best results. However, there is recent evidence, Schmidt (1993), that the vena contracta does not always occur. The pressure profile shown in Figure 2 provides confirmation of this as it lacks the characteristic minimum seen in Figure 1, which is characteristic of the vena contracta.

#### REFERENCES

Benedict, R. P. (1980) *Fundamentals of Pipeflow*, Wiley-Interscience, New York.

Chisholm, D. (1983) *Two-Phase Flow in Pipelines and Heat Exchangers*, Pitman Press Ltd., Bath, England.

Lighthill, J. (1986) *An Informal Introduction to Theoretical Fluid Mechanics*, Oxford University Press, Oxford.

Schmidt, J. (1993) *Berechnung und Messung des Druckabfalls uber plötzliche scharfkantige Rohrerweiterungen und -verengungen bei Gas*/Dampf-Flüssigkeitsströmung, VDI-Forschungsheft.

#### References

- Benedict, R. P. (1980)
*Fundamentals of Pipeflow*, Wiley-Interscience, New York. - Chisholm, D. (1983)
*Two-Phase Flow in Pipelines and Heat Exchangers*, Pitman Press Ltd., Bath, England. DOI: 10.1080/01457638508939624 - Lighthill, J. (1986)
*An Informal Introduction to Theoretical Fluid Mechanics*, Oxford University Press, Oxford. DOI: 10.1063/1.2811466 - Schmidt, J. (1993)
*Berechnung und Messung des Druckabfalls uber plÃ¶tzliche scharfkantige Rohrerweiterungen und -verengungen bei Gas*/Dampf-FlÃ¼ssigkeitsstrÃ¶mung, VDI-Forschungsheft.