Nozzle flows with nonequilibrium condensation

Abstract

The compressible nozzle flow of a mixture of a condensible vapor and an inert carrier gas with nonequilibrium condensation is considered. An asymptotic method is used with respect to two disparate parameters appearing in the nonequilibrium condensation rate equation. A high activation limit and another limit consideration that signify a large nucleation time followed by a small droplet growth time are employed. Under the limits, the coupled integral rate equation is so controlled by the activation function of homogeneous nucleation over its six zones that there appears to be a one-to-one correspondence between these zones and the six characteristic condensation zones of the nozzle flow itself. The six zones are the initial growth of condensate, further growth, rapid growth, onset, nucleation with growth, and droplet growth zone. The last zone can be greatly simplified if terms that imply a reasonably sizable order of error are neglected. This results in the rectilinear, one-dimensional flow equations of a normal shock wave structured or resisted by droplet growth.

On the stability of stationary shock waves in nozzle flows with homogeneous condensation

“…It is well known that the flow passes through the throat almost isentropically so that the classical saddle point singularity of the system ͑1͒-͑7͒ occurs at the throat with M ϭ1. Nonequilibrium condensation thus occurs in the supersonic portion of the nozzle ͑for details of the condensation zones, see Clarke and Delale 8 and Delale et al 10,11 ͒. As a result, a considerable amount of heat is released in the heat addition zones.…”

“…Detailed structure of the condensation zones for both subcritical and supercritical flows can be found in Delale et al 10,11 using an asymptotic predictive method given by Blythe and Shih 7 and Clarke and Delale. 8 Two-dimensional numerical simulations of such flows were given by Schnerr 9 and Schnerr and Dohrmann. 13 If the heat addition to the flow is further increased ͑e.g., by increasing the initial relative humidity keeping appropriate reservoir conditions fixed͒, the flow becomes unsteady and different patterns of periodic flow structures with moving shock waves occur ͑e.g., see Adam and Schnerr 14 ͒.…”

“…Surveys of such flows are well documented in the literature. [1][2][3][4] Various investigations [5][6][7][8][9][10][11] have been devoted to nonequilibrium flows with homogeneous condensation in converging-diverging nozzles. In this case the flow in the condensation zones remains steady and supersonic as long as the latent heat release to the flow is below a critical value for a given nozzle geometry and reservoir conditions ͑subcritical flows͒.…”

The mathematical theory of thermal choking in nozzle flows

“…In spite of this fact Hill (1966) has carried out a detailed analysis of the physical mechanisms underlying the nature of the rate processes required for the construction of the nonequilibrium condensation rate equation. Asymptotic theories of the nonequilibrium condensation rate equation for nozzle flows (with different ordering of the double-limit process corresponding to large nucleation time followed by small droplet growth time) that yield the structure of condensation zones are already available from the work of Blythe and Shih (1976) and Clarke and Delale (1986). Results of experiments conducted in nozzle flows during the expansion of moist air and pure steam which reveal some general features about the physics of nonequilibrium condensation can also be found in the work of Wegener and Mack (1958), Pouring (1965), Barschdorff (1967), Wegener (1969, The system of equations (1)-(4) can then be solved for the flow velocity u in functional form as…”

“…Before proceeding any further, we first discuss the nature of flow in the interval Xs < x <-Xk. From the asymptotic theories of condensation [Blythe and Shih (1976) and Clarke and Delale (1986)], one can easily show that in the interval x~ <-x < Xk the condensate mass fraction g' is exponentially small everywhere except for a thin zone near Xk. In both theories in the interval Xs < x < Xk the nearly frozen…”