THE ONE-DIMENSIONAL ADIABATIC FLOW OF EQUILIBRIUM GAS-PARTICLE MIXTURES IN VARIABLE-AREA DUCTS WITH FRICTION

A general one-dimensional model for the steady adiabatic motion of gas -particle mixtures in arbitrarily oriented ducts with gradually varyin g cross-section and wall friction is presented. The particles are assu med to be incompressible and in thermomechanical equilibrium with a pe rfect gas phase, and the effects of their finite volume and of gravity are also taken into account. The equations of motion are written in a form that allows a theoretical analysis of the behaviour of the solut ions to be carried out. In particular, the results of the application to the model of a procedure that permits the identification and the to pological classification of the singular points of the trajectories re presenting, in a suitable phase space, the solutions of the set of equ ations defining the problem are described. This characterization of th e singular points is useful in order to overcome difficulties in the n umerical integration of the equations. Subsequently, a geometrical ana lysis is carried out which allows a study of the signs of the local va riations of the flow quantities, and shows that some unusual behaviour may occur if certain geometrical and fluid dynamic conditions are ful filled. For instance, in an upward motion it is possible to have a sim ultaneous decrease of velocity, pressure and temperature, while in a d ownward flow an increase of all these quantities may be found. It is a lso shown that conditions exist in which expansion and heating of the mixture may take place simultaneously, both in accelerating and decele rating flows. The model is applied to the study of upward motion in pa rticular ducts, having converging-diverging and constant-diverging cro ss-sections; to this end the equations are integrated numerically by u sing the Mach number as the independent variable. The results show tha t even limited variations of the duct diameter may give rise to signif icant qualitative and quantitative variations in the flow conditions i nside the duct and in the mass flow rate. Finally, an example is given of a subsonic downward flow in which a simultaneous increase of press ure, temperature and velocity occurs even in the case of a pure perfec t gas.