gde-momentofsizedistributionfunction-momentevolutionequation-quadraturesolution

"The research start with finding model that represent aerosol particles transport in the cooperative from source of concentration."

Further study: Control size and shape of aerosol particles.

To be able to control the size and shape and thus the properties of the particles
produced, it is important to have detailed knowledge of the physical processes
involved in particle formation, therefore modelling is needed.

Aerosol dynamics is described by an equation termed the general dynamic
equation (GDE) [Friedlander 2000]. This equation gives the evolution in time of
the size distribution function:

Where:
n is the number density function,
t the time,
xdot the velocity vector,
u and v are particle volume,
β is the collision frequency function,
J the nucleation rate and
kdep the deposition flux.

The terms on the left-hand side of the equation give the temporal and spatial changes in the number density function. The first term on the right-hand side describes condensation, the second and third terms coagulation and the fourth term nucleation. The last term on the right-hand side accounts for losses due to deposition. Since the GDE is a nonlinear, partial integro-differential equation, it is difficult to solve.

Actually, the transport of the nanoscale particles dispersed throughout the fluid is governed by the aerosol general dynamic equation (GDE).

The GDE describes particle dynamics under the influence of various physical and chemical phenomena—convection, diffusion, coagulation, surface growth, nucleation, and other internal/external forces.

From a practical standpoint such a system of equations cannot be solved explicitly except for very small particle sizes—less than 1000 molecular or “monomer” units. To overcome this and other issues, a nodal model is employed to describe the particle size distribution in time and space (S.C. Garrick et al., 2006).

Otherwise, (Ulrika Backman as disertation VTT Processes, 2005) write that the evolution of an aerosol distribution in non-uniform temperature and flow fields requires solution of the GDE at each point in space. There is a need to formulate aerosol problems such that the GDE is of the same form as the other transport equations:

A convenient form for describing the aerosol distribution is by using moments of the size distribution function:

Based on these moments, we can generate a moment evolution equation:

Complete model, solution?. Not yet. There is still a problem with this general form.

We can't close the problem since we have integrals of functionals which we can't evaluate simply in terms of moments.

We use quadrature points and weights to evaluate and close the moment form of the equations.