Multi-velocity continuum

If the average distance between the inclusions is much less than the characteristic size of a continuous phase over which the macroscopic parameters (velocity, pressure, temperature etc.) change, then it is possible to describe the macroscopic processes in the mixture by the methods of mechanics of continuous mediums as well. To this end, the averaged or macroscopic parameters are introduced. At this point, the concept of multi-velocity continuum [6], representing a set of N continuums, can also be introduced. The number N is the number of considered phases, each of which corresponds to a certain constituent phase of the mixture and fills one and the same volume. For each of these constituent continuums we define in the usual manner the local density, which is called the reduced density... 

The differences in behaviour of different sized particles in a dilute phase (fully suspended flow) pneumatic conveying pipeline was illustrated by the use of a model. The model treated the system as a multi-phase continuum, with the conveying air as one phase and the solids as two other phases. The velocity and pressure profiles for two different particle sizes were illustrated graphically [3]. The model ran with gas and solids parameters typical of a dilute phase pneumatic conveying line. 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

For a multi-component mixture with 1, 2, 3, At-species in a fundamentally miscible solution, which involves a process of chemical reactions, we establish a framework of continuum thermodynamics theory. The mixture might be a liquid or gas however, since we mainly use a mass-averaged velocity (defined later), the liquid solution might be more appropriate. The description is Eulerian because we are considering a fluid. 

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