Finite Volume Fraction Research Articles

Existence and uniqueness of normal shock waves in gas-particle mixtures

A mixture of a gas and small solid particles is considered which, far upstream, is in a constant equilibrium state, and moves with a constant velocity. The existence of shock waves is investigated in the four possible cases, namely for frozen flow, for two kinds of partly frozen flow, and for equilibrium flow. It is shown that, in all these cases, compressive shocks may exist, if the upstream velocity exceeds the velocity of sound appropriate to the type of flow. Rarefaction shocks are impossible in each case. Moreover, it is shown that the downstream values of the flow parameters are determined uniquely, and the direction of their change is given. Only rather general assumptions concerning the behaviour of the gas are needed. The paper takes into account the influence of the finite particle volume fraction unlike most previous papers on the topic.

Some effects of finite particle volume on the dynamics of gas-particle mixtures

In the analysis of two-phase flows in which a gas carries along a large number of small particles, the volume of the particles is customarily assumed to be negligible. This assumption often is well justified, but if either the mass fraction of the particles or the gas density is sufficiently high, the particle volume fraction may become significant. It is thus important to establish the conditions at which this parameter should be included in a flow analysis. The particles may be considered as incompressible by comparison with the gas, so that a finite particle volume fraction appears as an additional variable in the basic equations. Consequences of a finite particle volume are examined for isentropic changes of the mixture, for frozen and equilibrium speed of sound, and for both the frozen flow immediately behind a shock wave and the equilibrium flow that is established further downstream. The errors that would result from neglecting the particle volume range from insignificant to large. For example, even for gas-particle density ratios as low as 10 ~, the equilibrium flow velocity behind a shock front is quite sensitive to changes of the volume of the particles if they represent more than one-half of the mass of the mixture.

Free diffusion measured by biological assay in multilayered cells: I. Tables of the average concentration in successive layers and the effect of assay errors

The expressions for free diffusion from an initial infinitely sharp boundary were considered in terms of the type of concentration measurement made and its error. The percentage error in the computed diffusion coefficient D will be less than the percentage concentration error if c c o < 0.1 and h/ (2 Dt ) > 0.8 . Here c o is the initial concentration below the boundary, t the time and h the height of the measurement plane above the boundary. If the concentration can be measured where c c o ≈ 10 −5 , then a 100% error in c c o causes only a 10% error in D. A table is given of the integral of the error function complement (ierfc x) to seven figures for 0.05 increments in the argument to 2.55 and six figures to 5.40. The average relative concentration c c o of successive, finite volume fractions was computed from ierfc x for a multilayered, double-infinite cell intended for nonoptical diffusion measurements. Using this table and the method of reflection, an appropriate table can be constructed for any finite cell. A special case of such restricted diffusion is presented for a five-layer cell in which the initial solution occupies the lowest layer. For both the double-infinite and the five-layer cells the weighting factor for each entry in the table of c c o is given for use in obtaining a weighted mean diffusion coefficient from all the data from several fractions in one experiment or replicate experiments.