Prediction of pressure gradient and holdup in wavy stratified liquid–liquid inclined pipe flow

Pressure gradient and holdup data are presented for oil–water flow in a horizontal and inclined 0.026 m i.d. pipe (borosilicate glass, 15.5 m length and pipe inclinations of −10°, −20° and +10°). A wavy stratified flow in the laminar-turbulent regime with no dispersion whatsoever at the interface was observed. The relatively high-viscosity oil flow (280 mPa s) dominates the friction and the low Eötvös number indicates the existence of a wavy and curved interface. A new closure relation for the interfacial friction factor is suggested. Recent interfacial wave amplitude data are used for the proposition of a correlation for the interfacial friction factor based on the equivalent-sand-roughness concept. An explicit equation for the interface shape based on the constant-curvature-arc model is proposed, which is a function of the Eötvös number, holdup and contact angle. A discussion on the typical contact angles observed in liquid–liquid flow in pipes of different materials is carried out. It was found that for the slower lighter phase (oil) the effective wall friction factor is significantly lower than the single-phase friction factor, corresponding to an increase of the respective hydraulic diameter. CFD simulations provided an estimate of the cross-sectional wave shape and delivered holdup and pressure gradient results. The phenomenological model is validated against data from the literature and its predictions are compared with present data, models from the literature and CFD results. The favorable comparisons and simplicity of the proposed closure relations are promising, aiming to practical application.

► New closure relations specific for wavy stratified liquid–liquid flow.
► New equation for the interface shape as a function of Eo, holdup and contact angle.
► Discussion on the typical contact angles observed in liquid–liquid flow.
► Proposed model offers favorable comparisons and simplicity.