Modelling of upwards gas-liquid annular and churn flow with surfactants in vertical pipes

Based on our earlier experimental work on the effect of surfactants on air-water flow in vertical pipes with internal diameters of 34 mm, 50 mm, and 80 mm, we create a mechanistic annular flow model for the pressure gradient. The major effect of the addition of surfactants is the formation of foam. We model the formation of foam and its impact on the flow. In the model we consider a gas core and a film at the wall, which consists of a layer of liquid at the wall and a layer of foam between the liquid layer and the gas core. We do not consider entrainment in the model. We developed four closure relations in order to solve the model: (i) for the density of the foam, (ii) for the viscosity of the foam, (iii) for the interfacial friction between the gas and the film, and (iv) for the thickness of the liquid layer at the wall. Subsequently, we solve for the film thickness that yields the imposed liquid flow rate. Comparing the experimental results for the pressure gradient to the results from the model, we observe that in most cases the model can predict the pressure gradient within 25%. Furthermore, the model is able to predict the onset of downwards flow in the film. Therefore, it can predict the transition between annular flow and churn flow. We show that the effect of five different surfactants on the flow is equal, apart from a scaling factor of the concentration, which means that the model can be applied for many different types of surfactants. The scaling factor is an input parameter to the model, which needs to be determined in a small scale experiment.