Attributes

Common Attributes of the Solver Class

General Information

The following tables summarize the most important class attributes for performing PBE simulations. These attributes are defined in the _init_base_parameters method of the BaseSolver class in base_solver.py.

Note: Additional attributes may be defined in specific solver implementations.

Simulation Parameters

Attribute	Type	Default	Description
`dim`	`int`	\(-\)	Dimension of the PBE (1=1D, 2=2D, 3=3D). The 3D case has not been fully adapted yet
`t_vec`	`array-like`	`None`	Simulation time vector
`t_total`	`float`	601	Total simulation time [second]
`t_write`	`float`	100	Output time interval. If `t_vec` is not specified, use `t_total` and `t_write` to construct `t_vec`
`work_dir`	`Path`	Current working directory	Baseline path for the simulation

Agglomeration Kernel Parameters

Attribute	Type	Default	Description
`COLEVAL`	`int`	1	Case for calculation of beta (collision frequency). 1: Chin 1998, 2: Tsouris 1995, 3: Constant kernel, 4: Volume-sum kernel
`SIZEEVAL`	`int`	1	Case for implementation of size dependency. 1: No size dependency, 2: Model from Soos2007
`X_SEL`	`float`	0.310601	Size dependency parameter for Selomulya2003 / Soos2006
`Y_SEL`	`float`	1.06168	Size dependency parameter for Selomulya2003 / Soos2006
`POTEVAL`	`int`	1	Case for interaction potentials in DLVO theory. Case 2 is faster, Case 3 uses pre-defined alphas
`alpha_prim`	`array-like`	`np.ones(dim**2)`	Collision efficiency array
`CORR_BETA`	`float`	25.0	Correction term for collision frequency describing external factors influence

Breakage Kernel Parameters

Attribute	Type	Default	Description
`BREAKRVAL`	`int`	3	Case for breakage rate calculation. 1: Constant, 2: Volume-sum, 3: Jeldres 2018 Power Law, 4: Jeldres 2018 volume fraction
`BREAKFVAL`	`int`	3	Case for breakage function calculation. 1-2: Leong2023 models, 3-5: Diemer Olson 2002 models
`process_type`	`str`	`"breakage"`	Process type: `"agglomeration"`, `"breakage"`, or `"mix"`
`pl_v`	`int`	2	Number of fragments in product function of power law
`pl_q`	`int`	1	Parameter describing breakage type in product function of power law
`pl_P1`	`float`	0.01	1st parameter in power law for breakage rate (1D/2D)
`pl_P2`	`float`	1	2nd parameter in power law for breakage rate (1D/2D)
`pl_P3`	`float`	0.01	3rd parameter in power law for breakage rate (2D)
`pl_P4`	`float`	1	4th parameter in power law for breakage rate (2D)
`B_F_type`	`str`	`'int_func'`	Breakage function calculation method: `'int_func'`, `'MC_bond'`, or `'ANN_MC'`
`work_dir_MC_BOND`	`str`	`work_dir/bond_break/int_B_F.npz`	Path to MC bond breakage data file

Particle Size Distribution Parameters

Attribute	Type	Default	Description
`USE_PSD`	`bool`	`True`	Whether to use .npy file to initialize particle distribution (False = monodisperse primary particles)
`DIST1_path`	`str`	`work_dir/data/PSD_data`	Path to PSD data directory for component 1
`DIST2_path`	`str`	`work_dir/data/PSD_data`	Path to PSD data directory for component 2
`DIST3_path`	`str`	`work_dir/data/PSD_data`	Path to PSD data directory for component 3

DLVO Theory Parameters

Attribute	Type	Default	Description
`PSI1`	`float`	0.001	Surface potential component 1 [V] - NM1
`PSI2`	`float`	0.001	Surface potential component 2 [V] - NM2
`PSI3`	`float`	-0.04	Surface potential component 3 [V] - M
`A_NM1NM1`	`float`	10e-21	Hamaker constant for NM1-NM1 interaction [J]
`A_NM2NM2`	`float`	10e-21	Hamaker constant for NM2-NM2 interaction [J]
`A_MM`	`float`	80e-21	Hamaker constant for M-M interaction [J]

Hydrophobic Interaction Parameters

Attribute	Type	Default	Description
`C1_NM1NM1`	`float`	0	Short-range hydrophobic interaction parameter NM1-NM1 [N/m]
`C2_NM1NM1`	`float`	0	Long-range hydrophobic interaction parameter NM1-NM1 [N/m]
`C1_MNM1`	`float`	0	Short-range hydrophobic interaction parameter M-NM1 [N/m]
`C2_MNM1`	`float`	0	Long-range hydrophobic interaction parameter M-NM1 [N/m]
`C1_MM`	`float`	0	Short-range hydrophobic interaction parameter M-M [N/m]
`C2_MM`	`float`	0	Long-range hydrophobic interaction parameter M-M [N/m]
`C1_NM2NM2`	`float`	0	Short-range hydrophobic interaction parameter NM2-NM2 [N/m]
`C2_NM2NM2`	`float`	0	Long-range hydrophobic interaction parameter NM2-NM2 [N/m]
`C1_MNM2`	`float`	0	Short-range hydrophobic interaction parameter M-NM2 [N/m]
`C2_MNM2`	`float`	0	Long-range hydrophobic interaction parameter M-NM2 [N/m]
`C1_NM1NM2`	`float`	0	Short-range hydrophobic interaction parameter NM1-NM2 [N/m]
`C2_NM1NM2`	`float`	0	Long-range hydrophobic interaction parameter NM1-NM2 [N/m]
`LAM1`	`float`	1.2e-9	Range of short-range hydrophobic interactions [m]
`LAM2`	`float`	10e-9	Range of long-range hydrophobic interactions [m]
`X_CR`	`float`	2e-9	Alternative range criterion for hydrophobic interactions [m]

Experimental/Process Parameters

Attribute	Type	Default	Description
`c_mag_exp`	`float`	0.01	Volume concentration of magnetic particles [Vol-%]
`Psi_c1_exp`	`float`	1	Concentration ratio component 1 (V_NM1/V_M) [-]
`Psi_c2_exp`	`float`	1	Concentration ratio component 2 (V_NM2/V_M) [-]
`G`	`float`	1	Shear rate [1/s]
`V_unit`	`float`	1	Unit volume used to calculate total particle concentration

Calculated Parameters

The following parameters are automatically calculated in the _reset_params method:

Attribute	Type	Description
`t_num`	`int`	Number of time steps based on `t_vec`
`cv_1`	`float`	Volume concentration of NM1 particles [Vol-%]
`cv_2`	`float`	Volume concentration of NM2 particles [Vol-%]
`V01`	`float`	Total volume concentration of component 1 [unit/unit] - NM1
`V02`	`float`	Total volume concentration of component 2 [unit/unit] - NM2
`V03`	`float`	Total volume concentration of component 3 [unit/unit] - M
`N01`	`float`	Total number concentration of primary particles component 1 [1/m³] - NM1
`N02`	`float`	Total number concentration of primary particles component 2 [1/m³] - NM2
`N03`	`float`	Total number concentration of primary particles component 3 [1/m³] - M
`DIST1`	`str`	Full path to PSD file for component 1
`DIST2`	`str`	Full path to PSD file for component 2
`DIST3`	`str`	Full path to PSD file for component 3

DPBESolver Specific Attributes

General Information

The following attributes are specific to the DPBESolver class and are defined in the __init__ method of dpbe_base.py. These attributes extend the common BaseSolver attributes for discrete Population Balance Equation (dPBE) solving.

Discretization Parameters

Attribute	Type	Default	Description
`disc`	`str`	`'geo'`	Discretization scheme: `'geo'` for geometric grid, `'uni'` for uniform grid
`NS`	`int`	12	Grid parameter [-]
`S`	`float`	2	Geometric grid ratio (V_e[i] = SV_e[i-1]). Actual primary particle size is R[1] = ((1+S)/2)(1/3)R01
`solve_algo`	`str`	`"ivp"`	Solution algorithm: `"ivp"` uses integrate.solve_ivp, `"radau"` uses RK.radau_ii_a (debug only)
`aggl_crit`	`float`	1000.0	Upper volume limit for agglomeration (grid-based)

Material Parameters

Attribute	Type	Default	Description
`R01`	`float`	2.9e-7	Basic radius component 1 [m] - NM1
`R02`	`float`	2.9e-7	Basic radius component 2 [m] - NM2
`R03`	`float`	2.9e-7	Basic radius component 3 [m] - M3 (magnetic component)

Physical Constants

Attribute	Type	Default	Description
`KT`	`float`	4.0434e-21	Boltzmann constant × Temperature [J] (k*T at 293K)
`MU0`	`float`	4π×10⁻⁷	Permeability constant vacuum [N/A²]
`EPS0`	`float`	8.854e-12	Permittivity constant vacuum [F/m]
`EPSR`	`float`	80	Permittivity material factor [-]
`E`	`float`	1.602e-19	Electron charge [C]
`NA`	`float`	6.022e23	Avogadro number [1/mol]
`MU_W`	`float`	1e-3	Viscosity water [Pa*s]
`EPS`	`float`	Calculated	Permittivity (EPSR×EPS0) [F/m]

Process Parameters

Attribute	Type	Default	Description
`I`	`float`	1.0	Ionic strength [mol/m³] (converted from mol/L)

Computational Options

Attribute	Type	Default	Description
`JIT_FM`	`bool`	`True`	Whether to precompile FM (Formation Matrix) calculation
`JIT_BF`	`bool`	`True`	Whether to precompile BF (Breakage Function) calculation

Submodules

The following submodules are automatically instantiated:

core: DPBECore instance for core PBE functionality
post: DPBEPost instance for post-processing
visualization: DPBEVisual instance for visualization

PBMSolver Specific Attributes

General Information

The following attributes are specific to the PBMSolver class and are defined in the __init__ method of pbm_base.py. These attributes extend the common BaseSolver attributes for Population Balance Moment (PBM) method solving.

Moment Method Parameters

Attribute	Type	Default	Description
`n_order`	`int`	5	Order parameter where n_order×2 is the order of the moments [-]
`n_add`	`int`	10	Number of additional nodes [-]
`GQMOM`	`bool`	`False`	Flag for using Generalized Quadrature Method of Moments
`GQMOM_method`	`str`	`"gaussian"`	Method for GQMOM implementation
`nu`	`int`	1	Exponent for the correction in gaussian-GQMOM

Integration Tolerance Parameters

Attribute	Type	Default	Description
`atol_min`	`float`	1e-16	Minimum absolute tolerance
`atol_scale`	`float`	1e-9	Scaling factor for absolute tolerance
`rtol`	`float`	1e-6	Relative tolerance

Runtime Calculated Attributes

The following attributes are calculated during runtime:

Attribute	Type	Description
`moments_norm`	`array`	Normalized moments
`moments_norm_factor`	`array`	Normalization factors for moments
`atolarray`	`array`	Absolute tolerance array for integration
`rtolarray`	`array`	Relative tolerance array for integration
`x_max`	`float`	Maximum x-coordinate for normalization
`moments`	`array`	Calculated moments
`indices`	`array`	2D moment indices for CHY or C method

Submodules

The following submodules are automatically instantiated:

post: PBMPost instance for post-processing
quick_test: PBMQuickTest instance for quick testing
core: PBMCore instance for core PBM functionality

MCPBESolver Specific Attributes

General Information

The following attributes are specific to the MCPBESolver class and are defined in the __init__ method of mcpbe.py. These attributes extend the common BaseSolver attributes for Monte Carlo Population Balance Equation solving.

Simulation Parameters

Attribute	Type	Default	Description
`c`	`array`	`np.full(dim,0.1e-2)`	Concentration array of components
`x`	`array`	`np.full(dim,1e-6)`	(Mean) equivalent diameter of primary particles for each component [m]
`x2`	`array`	`np.full(dim,1e-6)`	(Mean) equivalent diameter of primary particles for each component (bi-modal case) [m]
`a0`	`float`	1000.0	Total amount of particles in control volume (initially) [-]
`CDF_method`	`str`	`"disc"`	Method for fragment distribution: `"disc"` uses discrete points, `"conti"` uses continuous functions
`VERBOSE`	`bool`	`False`	Whether to print detailed calculation information

Initial Condition Parameters

Attribute	Type	Default	Description
`PGV`	`array`	`np.full(dim, 'mono')`	Initial particle size distribution type for each component: `'mono'`, `'norm'`, `'weibull'`
`SIG`	`array`	`np.full(dim,0.1)`	(Relative) standard deviation of normal distribution (STD = SIG×v)
`PGV2`	`array`	`None`	Second mode distribution type for bi-modal distributions (None for mono-modal)
`SIG2`	`array`	`None`	Standard deviation for second mode in bi-modal distributions

Runtime Variables

The following variables are calculated during initialization:

Attribute	Type	Description
`V_flat`	`array`	Volume matrix where each column represents one particle/agglomerate
`v2`	`array`	Volume of primary particles (bi-modal case)
`n`	`array`	Number concentration for each component
`n2`	`array`	Number concentration for bi-modal case
`n0`	`float`	Total number of primary particles
`Vc`	`float`	Control volume
`a`	`array`	Total number of primary particles (integer)
`a2`	`array`	Total number of primary particles for bi-modal case (integer)
`a_tot`	`int`	Final total number of primary particles in control volume
`X`	`array`	Equivalent diameter calculated from total volume
`t`	`list`	Time array
`betaarray`	`array`	Beta array for collision calculations
`frag_num`	`float`	Expected number of fragments from breakage
`break_rate`	`array`	Breakage rate array
`break_func`	`array`	Breakage function array

Process Control Flags

Attribute	Type	Description
`process_agg`	`bool`	Whether agglomeration process is active
`process_break`	`bool`	Whether breakage process is active

Data Storage Arrays

Attribute	Type	Description
`V_save`	`list`	List storing volume matrices at different time steps
`Vc_save`	`list`	List storing control volumes at different time steps
`V0`	`array`	Initial volume matrix
`X0`	`array`	Initial equivalent diameter
`V0_save`	`list`	List storing initial volume matrices
`step`	`int`	Current step counter
`MACHINE_TIME`	`float`	Calculation time (set after solving)

Method Parameters

Additional parameters for breakage methods:

Attribute	Type	Description
`rel_frag`	`array`	Relative fragment sizes (1D case)
`rel_frag1`	`array`	Relative fragment sizes for component 1 (2D case)
`rel_frag3`	`array`	Relative fragment sizes for component 3 (2D case)
`cdf_interp`	`function`	Interpolation function for continuous CDF method