Particle Growth
The particle growth source that is added to the number density of each bin is used to describe the flux of particles across the boundaries to adjacent size intervals resulting from a continuous change in size. With finite differences the growth source term can be approximated as
(3-162)
where fi (SI unit: 1/(m3·s)) is the growth flux at the right-side boundary of interval i and Gi is the growth rate (SI unit: m/s). The most basic way of expressing the boundary fluxes is with a first-order upwind approach (Ref. 3 and Ref. 4)
(3-163)
As we will see later, this way of expressing the fluxes is needed for the first and last size intervals. For the inner intervals we turn to higher order approaches, namely Weighted Essentially Non-Oscillatory (WENO) schemes. If we assume that the mapping between cell average and boundary values to be linear, we can construct a number of polynomials, q, to approximate the number density at the boundaries. The WENO scheme relies on combining a number of these polynomials to get a convex combination of lower order approximations on the form
(3-164)
where W1, W2,... WN denotes weights that fulfill
(3-165)
Using this, we can write the boundary fluxes as (Ref. 5)
(3-166)
where the order of the polynomial is given by 2r-1. A polynomial, qi, describing the number density at the right-side boundary of interval i can be written as
(3-167)
The weights in Equation 3-166 are written as (Ref. 5)
(3-168)
where S is a measure of smoothness
(3-169)
The constants Crm,j,i can be calculated using (Ref. 6)
(3-170)
while the constants crm,i can be derived by ensuring that they fulfill Equation 3-171 and Equation 3-172 (Ref. 5).
(3-171)
(3-172)
Equation 3-173 shows the summarized interval boundary fluxes.
(3-173)
For the growth rate, Gi, it is possible to use a custom expression or a predefined transport controlled expression available as
(3-174)
where ka and kv are the area and volume shape factors, Sh is the Sherwood number, Ds (SI unit: m2/s) is the solute species diffusion coefficient, Mp (SI unit: kg/mol) is the particle molar mass, ρp (SI unit: kg/m3) is the particle density, c (SI unit: mol/m3) is the species concentration in the solution and c* (SI unit: mol/m3) is the equilibrium concentration of the species. Equation 3-174 is based on the mass transport equation where the driving force is the concentration difference between the bulk solution, c and the concentration at the particle surface, assumed to be c*.