Partial singular integro-differential equations models for dryout in boilers
A two-dimensional model for the annular two-phase flow of water and steam, along with the dryout, in steam generating pipes of a liquid metal fast breeder reactor is proposed. The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for in the model. The mass exchange rate depends on the details of the flow conditions and it is calculated using some simple thermodynamic models. The change of phase at the free surface between the liquid layer and the vapour core is modelled by proposing a suitable Stefan problem. Appropriate boundary conditions for the model, at the onset of the annular flow region and at the dryout point, are stated and discussed. The resulting unsteady nonlinear singular integro-differential equation for the liquid film free surface is solved asymptotically and numerically (using some regularisation techniques) in the steady state case, for a number of industrially relevant cases. Predictions for the length to the dryout point from the entry of the annular regime are made. The influence of the constant parameter values in the model (e.g. the traction r provided by the fast flowing vapour core on the liquid layer and the mass transfer parameter 77) on the length to the dryout point is investigated. The linear stability of the problem where the temperature of the pipe wall is assumed to be a constant is investigated numerically. It is found that steady state solutions to this problem are always unstable to small perturbations. From the linear stability results, the influence on the instability of the problem by each of the constant parameter values in the model is investigated. In order to provide a benchmark against which the results for this problem may be compared, the linear stability of some related but simpler problems is analysed. The results reinforce our conclusions for the full problem.