Solver library

Basic solver provides interfaces to the simulation system and to units.

Currently in Dyssol, you can develop your own agglomeration solver and add it to the solver library. Please refer to Solver development for detailed information.

The solver can also be added to a unit as a parameter.

Agglomeration solvers

Solvers are applied to calculate the birth and death rate of agglomerates according to different mathematical models. In Dyssol, 3 types of solver are available:

Cell average

Fast Fourier transformation (FFT)

Fixed pivot

The applied equations in all solvers are listed as follows.

\[\frac{\partial n(v,t)}{\partial t} = B_{agg}(n,v,t) - D_{agg}(n,v,t)\]

\[B_{agg}(n,v,t) = \frac{1}{2}\,\beta_0\,\textstyle \int\limits_{0}^{v} \beta(u,v - u)\,n(u,t)\,n(v-u,t)\,du\]

\[D_{agg}(n,v,t) = \beta_0\,n(v,t)\, \textstyle \int\limits_{0}^{\infty}\,\beta(v,u)\,n(u,t)\,du\]

Note

Notations:

\(v,u\) – volumes of agglomerating particles

\(n(v,t)\) – number density function

\(B_{agg}(n,v,t)\), \(D_{agg}(n,v,t)\) – birth and death rates of particles with volume \(v\) caused due to agglomeration

\(\beta_0\) – agglomeration rate constant, dependent on operating conditions but independent from particle sizes

\(\beta(v,u)\) – the agglomeration kernel, see section Kernels.

\(t\) – time

Cell average solver

This solver calculates the birth rate \(B_{agg}(n,v,t)\) and death rate \(D_{agg}(n,v,t)\) using a cell average technique.

Note

solid phase and particle size distribution are required for the simulation. Equidistant volume grid for particle size distribution must be used.

FFT solver

This solver calculates the birth rate \(B_{agg}(n,v,t)\) and death rate \(D_{agg}(n,v,t)\) based on a separable approximation of the agglomeration kernel and a subsequent fast Fourier transformation.

The agglomeration kernel \(\beta (v,u)\) is represented in a separable form with the separation rank \(M\) as:

\[\beta (v,u) = \sum\limits_{i=1}^{M} a_i(v)\,b_i(u)\]

Then birth and death rate terms are transformed to following equations with \(\psi_i = a_i\,n\) and \(\varphi_i = b_i\,n\).

\[B_{agg}(n,v,t) = \frac{1}{2}\,\textstyle \int\limits_{0}^{v} \psi_i(v - u,t)\,\varphi_i(u,t)\,du\]

\[D_{agg}(n,v,t) = \psi_i(v,t)\, \textstyle \int\limits_{0}^{\infty}\varphi_i(u,t)\,du\]

The separation turns the birth rate agglomeration integral into a convolution form \(\varphi_i * \psi_i\), which after piecewise constant discretization, is computed based on the convolution theorem

\[\varphi_i * \psi_i = IFFT(FFT(\varphi_i) \odot FFT(\psi_i))\]

applying the direct and inverse fast Fourier transformation (\(FFT/IFFT\)) and the elementwise product \(\odot\).

Note

List of kernels separation equations

Kernel	Separation equation
Constant	\(\beta(u,v) = \sum\limits_{i=1}^1 a_i(v)\,b_i(u),\) \(a_1 (v) = 1, b_1 (u) = 1\)
Sum	\(\beta(u,v) = \sum\limits_{i=1}^2 a_i(v)\,b_i(u),\) \(a_1 (v) = v, a_2 (v)= 1, b_1 (u) = 1, b_2 (u)= u\)
Brownian	\(\beta(u,v) = \sum\limits_{i=1}^3 a_i(v)\,b_i(u),\) \(a_1 (v) = \sqrt{2}, a_2 (v) = v^{\frac{1}{3}}, a_3 (v)= v^{-\frac{1}{3}}\) \(b_1 (u) = \sqrt{2}, b_2 (u) = u^{-\frac{1}{3}}, b_3 (u)= u^{\frac{1}{3}}\)
Product, Shear, Peglow, Coagulation, Gravitational, Kinetic energy, Thompson	Approximated by a rank-M separable function \(\beta (v,u) \approx \sum\limits_{i=1}^{M} a_i(v)\,b_i(u)\) using adaptive cross approximation

Note

solid phase and particle size distribution are required for the simulation. Equidistant volume grid for particle size distribution must be used. Grid for particle size distribution must start from 0.

Fixed pivot solver

This solver calculates the birth rate \(B_{agg}(n,v,t)\) and death rate \(D_{agg}(n,v,t)\) using a fixed pivot technique.

Note

solid phase and particle size distribution are required for the simulation.