Mechanics of Pipe Flow Following Column Separation

When the velocity of a liquid in a pipe is decreased rapidly at a point, for example, at a rapidly-closing valve, the pressure at this point drops and may reach the vapor pressure of the liquid, thus causing the formation of a vapor column. The motion of the liquid downstream is studied to determine if collapse of the vapor column will occur, and to determine the water-hammer pressure that will be created by such a collapse. A vapor column in a straight portion of a pipe at a slope is considered. The problem is confronted in two steps: (1) The study of the motion of the vapor-liquid interface by neglecting its spreading under the influence of gravity; and (2) the study of its spreading. In the first step, it is found by a topological study of the phase plane that the occurrence of collapse of the vapor column depends on the elevation of the point of separation, the slope of the pipe, and the velocity of flow at separation. In the second step, the spreading of the interface is described by two quasi-linear partial differential equations that are hyperbolic. The nature of the flow is revealed by a special case in which these equations become homogeneous. It is found that two discontinuities of the interface may develop. The water-hammer pressure that is created may be computed by neglecting the spreading of the interface.