The adiabatic, evaporating, two-phase flow of steam and water in horizontal pipe including the critical flow was successfully described by a system of nonlinear differential equations which include different phase velocities, fluid acceleration, wall shear forces, interface shear forces, and mass and energy transfer between the phases. The solution of the differential equations by the method of Runge-Kutta was facilitated with a high-speed digital computer. Experimental data were obtained which included void fraction measurement by X-ray attenuation and showed that the theoretical equations described the complex flow phenomena to 2 10% when radial temperature gradients were small. Design charts based on the numerical solution of the theoretical equations are presented for rapid evaluation of the flow variables for the system steam-water including critical flow for inlet pressures from 30 to 150 Ib./sq.in.abs., mass flow rates per unit area from 300 to 1,000 Ib./sq. ft. sec., and L/D ratios from 25 to 1,000. (7) for the twophase flow of air and various liquids to the flow of boiling water. T h s method has found wide acceptance as it correlates two-phase 00w data over a wide range within reasonable limits. The homogenous flow model derived its name from the fact that the basic assumptions of the model were that the liquid and its vapor flowed at the same average velocity, and that the physical properties of the fluid could be taken as an average of those of the liquid and the vapor. Benjamin and Miller (6) developed t h s model to describe the flow of steam and water in cascade drain lines. It has proved valuable as an additional way to estimate twophase flow parameters by a straightforward calculation.Although not in general use a third model, the Linning model ( 8 ) , has included more of the flow variables than any previous model. In this model Linning included different phase velocities, mass acceleration, mass and energy transfer between the phases, interface shear forces, and wall shear forces. Linning found that he obtained excellent experimental verification of his theoretical equations over a limited range for the system of steam and water. However the notable disadvantages to the use of this are that a tedious numerical solution of the differential equations is required and a boundary condition at some point in the two-phase flow is necessary.Other flow models have been presented in the literature. Some of these models are modifications of the Martinelli-Nelson model; some are modifications of the homogenous flow model, and some are referred to as mixed models ( 3 ) , that is combinations of the Martinelli-Nelson and homogenous flow models. These models have had only slightly better success in describing the two-phase flow variables than the original ones. For discussions of flow models refer to the previously mentioned literature surveys and the surveys of Charvonia ( 9 ) , Jens and Leppert ( l o ) , and Rodabaugh (11).
D E R I V A T I O N AND S O L U T I O N OF T H E A N N U L A R FLOW EQUATIONSTo des...