The flow of fluid through an enlargement (increase in pipe diameter) results in a decrease in velocity and consequently, a pressure rise. If the contraction is sharp or sudden, the behavior of single-phase flow is as shown in Figure 1 and involves a recirculation region. This starts at the enlargement and extends about three outlet pipe diameters downstream. The dissipation of energy caused by this recirculation region means that not all the kinetic energy is converted to a pressure rise, and reversible and irreversible components of pressure drop must be considered. If the enlargement is being used to convert kinetic energy into pressure, it is necessary to employ a more gradual change in diameter to eliminate or minimize losses. This gradual increase in diameter is known as a *diffuser*. For diffusers which have a linear increase in diameter, an included angle of less than 7° is required to avoid separation of the flow and thus minimize losses.

As for a Contraction, it is not possible to use a momentum balance to calculate pressure change through a generalized enlargement since there is an unknown reaction force from the walls to be accounted for. An energy balance gives a computable expression for the pressure drop. The exception is an abrupt enlargement, where the resultant forces on the annular downstream-facing wall can be assumed (with reasonable accuracy) to be equal to the pressure at the end of the upstream pipe times the area of the annulus. For turbulent flow, the total pressure change is then

where P_{d} and P_{u} are the downstream and upstream pressures, respectively, is the mass flux, ρ, the density and s the area ratio between upstream and downstream pipes. The reversible pressure change is:

The irreversible portion is determined by difference.

For a diffuser, methods which calculate growth of the Boundary Layer are required; for example, those similar to Ghose and Kline (1978) which uses a momentum integral approach. If the diffuser angle is > 45° , it can be taken as a sudden enlargement.

downstream of the step. The length of the recirculation zone is similar to that for single-phase flow [Chouikhi et al. (1983)]. Application of a separated flow model equivalent to that used for single-phase flow has been found to do poorly in predicting pressure change [Schmidt (1993)]. However, recent work by Schmidt, which allows for a pressure on the annulus of the downstream-facing wall different from that at the upstream pipe outlet as well as other factors, gives good predictions.

For diffusers with annular two-phase flow, separation of the boundary layer has been inferred even at angles of 5° and depends on gas and liquid flow [Azzopardi (1992)].Figure 2 shows the effect of diffuser angle on pressure rise. The lower rise at 15° is probably due to the start of separation of the boundary layer.

**Figure 2. Measured pressure profiles in diffusers showing the effect of diffuser angle on pressure rise.**

#### REFERENCES

Azzopardi, B. J. (1992) Gas-liquid flows in cylindrical venturi scrubbers: boundary layer separation in the diffuser section. *The Chemical Engineering Journal*. 49: 55–64. DOI: 10.1016/0300-9467(92)85025-5

Chouikhi, S. M., Patrick, M. A., and Wragg, A. A. (1983) Two-phase turbulent wall transfer processes downstream of abrupt enlargements in pipe diameter. *Proc. Int. Conf. on Physical Modelling of Multiphase Flow*, Coventry. April 19–21. Pub BHRA. 53–66.

Ghose, S. and Kline, S. J. (1978) The computation of optimum pressure recovery in two-dimensional diffusers, *Trans A.S.M.E., Journal of Fluids Engineering.* 100: 419–426.

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