The quadrature-based moments method (QBMM) has the same advantages as the quadrature method of moments (QMOM) but uses the ζ kinetic scheme for the moments flux reconstruction to guarantee the realizability of the moments when a second-order spatial discretization scheme is used [4].
Using conventional spatial discretization schemes of a higher (than one) order in simulations of transport of moments can lead to the creation of negative (that is, non-realizable) moments [8]. The QBMM addresses this problem, thus providing an alternative to the QMOM when a higher order of accuracy is preferred for solving population balance equations. Also, the QBMM is potentially faster than the QMOM when the time step sizes are approximately the same in both solvers.
The QBMM was originally introduced by Kah-et-al [3] and then further developed and generalized by Laurent and Nguyen [4] and Passalacqua et-al [6]. It can be used for any number of moments for unstructured meshes in complex industrial applications.
The QBMM uses a similar approach to that used in the QMOM in the sense that it tracks a finite number of moments associated with a set of quadrature points that approximate a particle size distribution. However, the numerical scheme used in the QBMM is different and is based on the moments realizability criterion. Since the non-realizability can originate from both the advection and source terms, a time-split method is used to separate the advection and source contributions. Therefore, Equation 14–801 in the Fluent Theory Guide can be rewritten as:
 | (27–2) |
 | (27–3) |
where all the terms on the right-hand side of Equation 27–3 are treated in the same way as in the QMOM (see Equation 14–815 through Equation 14–819 in the Fluent Theory Guide). With this separation, the advection part is first integrated explicitly over time using the kinetic scheme for reconstructing the moments’ fluxes, and then the source terms evolution represented by an ODE is integrated over time.
Solving Equation 27–2 for a set of moments while maintaining their realizability is numerically non-trivial because the conventional spatial discretization/reconstruction and time integration do not impose a realizability criterion. In the QBMM, the realizability of moments is achieved by introducing an auxiliary set of variables named ζs to perform the advection of the moments followed by the time integration, which is carried out explicitly based on the strong stability-preserving Runge-Kutta (SSP-RK) method [4], [6], [2].
Since the time stepping in any explicit method is determined by the corresponding critical Courant number, the moments advection time step is usually smaller than that of any other quantity solved implicitly, such as continuity, momentum, and so on. Therefore, the moments advection solver takes an adequate number of sub-time steps until it is in sync with the rest of the quantities solved implicitly. The advection integration is given in detail by Passalacqua et al. [6].
The source terms are integrated once the advection integration is completed. Since the source terms depend on the local quadrature points, the source terms integration is reduced to an ODE system as shown in Equation 27–3. The quadrature points are obtained using the Wheeler algorithm, and the source contributions are accounted for using an appropriate time step.
Similar to the advection solution, the strong stability-preserving Runge-Kutta (SSP-RK) method is used as the time integration scheme since it preserves the realizability of the moments. However, the time step is determined by an embedded SSP-RK method, which involves using two SSP-RK schemes with consecutive orders of accuracy. If, when starting with an initial time step and carrying out the source integration, the difference in solution between the two methods falls below a specified threshold, the solution is considered converged. Otherwise, the time step is reduced by half, and the process is repeated until the convergence criterion is met. The error of the time integration process is defined as:
 | (27–4) |
where 
is the number of moments, 
and 
are the moments calculated using the embedded SSP-RK scheme with two orders
of accuracy, and 
is the scaled error measure calculated as:
 | (27–5) |
where 
and 
are the absolute and relative errors of the 
th moment, respectively.
During the integration, the time step value of the ODE solver is modified as follows:
 | (27–6) |
where 
, 
, and 
are user-specified parameters, and 
is a constant with a value of 2. In the absence of any spatial dependencies
among the sources, the time step for the integration in each cell is determined independently.
The source term integration is given in detail in Nguyen et al. [5]
[1] The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis. John Wiley & Sons. 1997.
[2] "Strong Stability-Preserving High-Order Time Discretization Methods". SIAM Review. 89–112. 2001.
[3] "A high order moment method simulating evaporation and advection of a polydisperse liquid spray". Journal of Computational Physics. 394–422. 2012.
[4] "Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution". Journal of Computational Physics. 309–338. 2017.
[5] "Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes". Journal of Computational Physics. 129–156. 2016.
[6] "A second-order realizable scheme for moment advection on unstructured grids". Journal of Computer Physics Communications Contents. 309–338. DOI: 10.1016/j.cpc.2019.106993. 2020.
To use the quadrature-based moments method (QBMM), follow the steps below. For the theory behind this model, see Theory.
In the Multiphase Model dialog box, enable the QBMM method (Population Balance Model tab).
Note: Alternatively, you can enable the QBMM by using the following text command:
define/models/multiphase/population-balance/model/qbmmIn the Parameters group box, specify the following parameters:
- Moments
is the number of moments.
- Kv
specifies the value for the particle volume coefficient
(as described in Particle Growth). By default, this coefficient has a value of
π/6.- Max Size
specifies the maximum size of the particle.
- Min Size
specifies the minimum size of the particle.
You can use the built-in size calculator that is available by clicking the button to obtain the recommendations as described in Size Calculator in the Fluent User's Guide.
Optionally, adjust the following QBMM parameters via the Text User Interface (TUI):
define/models/multiphase/population-balance/expert/qbmm/Enter the menu for the quadrature-based moments method (QBMM).
define/models/multiphase/population-balance/expert/qbmm/advection-cflSpecifies the Courant number for the QBMM advection. This value must be less than or equal to 0.5.
define/models/multiphase/population-balance/expert/qbmm/advection-cfl-methodAllows you to select the method for the maximum Courant number calculation from the following options:
0= flux-based method (default). The sum of the cell face fluxes and the cell volume are used to determine a cell-based timescale for the calculation of the maximum Courant number.1= averaged flux-based method. The timescale evaluated in method 0 is volume-averaged over each cell and its neighbors. This method gives a lower value of the maximum Courant number for stiff problems. You should use this option with caution.2= hybrid flux-based method. This method is a combination of methods 0 and 1 with the blending factor
:3= max flux-based method. This method is based on the maximum flux leaving a cell, the cell volume, and the number of outgoing fluxes. This method leads to very high maximum Courant numbers.4= explicit VOF solver CFL method. This option uses the same time step as the explicit VOF solver. It will reduce the maximum Courant number but may result in several unrealizable moments in the domain. This option should be used with caution and only if the time integration method for the volume fraction equation is explicit.
define/models/multiphase/population-balance/expert/qbmm/advection-int-max-stepsSpecifies the maximum number of advection integration time steps. Default: 250.
define/models/multiphase/population-balance/expert/qbmm/advection-int-schemeAllows you to select the time advection time integration from the following options:
explicit Euler
strong stability-preserving Runge-Kutta (SSP-RK) 2nd order method, aka SSP-RK2 (default)
SSP-RK 3rd order method, aka SSP-RK3
Runge-Kutta 4th order method (RK4)
define/models/multiphase/population-balance/expert/qbmm/exit-on-non-real?When enabled, the mechanisms that prevent or fix non-realizable moments are by-passed, and the simulation stops abruptly with an error message when a non-realizable moment is detected in the domain. By default, this option is disabled.
define/models/multiphase/population-balance/expert/qbmm/max-volume-fractionSpecifies the maximum moment-based volume fraction
, where 
is the third moment. (Default: 1.0.)Note: Typically, the zeroth moment represents the total number density, the second moment represents the total surface area per unit volume, and the third moment represents the total volume fraction.
define/models/multiphase/population-balance/expert/qbmm/min-volume-fractionSpecifies the minimum moment-based volume fraction, below which the moments are considered negligible. (Default: 0.0.)
define/models/multiphase/population-balance/expert/qbmm/redistribute-moments-based-on-volume-fractionWhen enabled, then for the cells in which the
-based volume fraction is greater than the specified maximum volume
fraction, the moments are redistributed using an iterative algorithm that respects both
the upper limit of the volume fraction and the moment realizability criterion. You can
select from the following options: 0= disables the redistribution.1= (default) redistributes the moments in the cells partially filled with the dispersed phase. This occurs at the end of each advection time step.2= redistributes the moments via modifying the outgoing moments fluxes. This occurs at the end of each stage (in multi-stage RK methods) of the advection time step.3= is the combination of option 1 and option 2.
define/models/multiphase/population-balance/expert/qbmm/redistribution-max-iterationSpecifies the maximum number of iterations for the moment redistribution process. (Default: 40.)
define/models/multiphase/population-balance/expert/qbmm/replace-negative-moments?When enabled (default), the negative moments in a cell are replaced with the averaged moments from the neighboring cells in which the moments are valid. This may be helpful when the simulation suffers from a few cells with unrealizable or negative moments. Alternatively, you can improve the mesh quality around such cells, or select a smaller timestep or CFL number for your simulation.
define/models/multiphase/population-balance/expert/qbmm/solve-advection?When enabled (default), the advection of moments are solved. This option should be disabled only for certain applications or debugging.
define/models/multiphase/population-balance/expert/qbmm/sort-abscissas?When enabled (default), the quadrature points are sorted out based on the abscissas from the smallest to the largest. This option has no effect on the solution, but may be useful for postprocessing purposes.
define/models/multiphase/population-balance/expert/qbmm/source-int-schemeAllows you to select the source time integration schemes for the embedded SSP-RK method described in Sources. The following options are available:
the time integration of the 1st order Euler and the error estimator of the 2nd-order SSP-RK2
(default) the time integration of the 2nd order SSP-RK and the error estimator of the 3rd-order SSP-RK2
define/models/multiphase/population-balance/expert/qbmm/source-int-time-steps-report?When enabled, Ansys Fluent reports in the console the distribution of steps for source time integration over a default number of bins for all the cells in the domain. This option is useful for debugging and for setting an error bound for the error estimator (Equation 27–4).
define/models/multiphase/population-balance/expert/qbmm/source-min-dtSpecifies the minimum allowed source integration time step. Violation of this boundary may indicate a solution divergence. (Default: 1.0e-8)
define/models/multiphase/population-balance/expert/qbmm/source-ode-err-absSpecifies the absolute error of the source time integration
in Equation 27–5. (Default: 1.0e-8)define/models/multiphase/population-balance/expert/qbmm/source-ode-err-relSpecifies the relative error of the source time integration
in Equation 27–5. (Default: 1.0e-6)define/models/multiphase/population-balance/expert/qbmm/source-ode-factorSpecifies the source time integration factor for error estimation
in Equation 27–6. (Default: 0.9)define/models/multiphase/population-balance/expert/qbmm/source-ode-factor-maxSpecifies the maximum source time integration factor for error estimation
in Equation 27–6. (Default: 2.0)define/models/multiphase/population-balance/expert/qbmm/source-ode-factor-minSpecifies the minimum source time integration factor for error estimation
in Equation 27–6. (Default: 0.5)define/models/multiphase/population-balance/expert/qbmm/update-moments-from-zetas?When enabled (default), the moments are updated from the
set of variables. This option improves the stability of the QBMM
solver.define/models/multiphase/population-balance/expert/qbmm/verbosity-levelSpecifies how much information should be displayed during the simulation. The following options are available:
0= no information is printed in the console1= minimal information is printed about the last advection sub-time step and the source integrations2= minimal information is printed about each advection sub-time step and the source integrations3= in addition to the information printed for option 2, prints realizability status in each cell (for debugging only)4= detailed information is printed (for debugging only)
define/models/multiphase/population-balance/expert/qbmm/zero-moment-minusSpecifies the lower threshold above which the moments are considered to be negligible. This parameter should be set to a small negative number. The default value is -1e-12. If non-realizable moments are generated during the simulation, you can decrease this parameter (even by a few orders of magnitude, for example, -1e-9).
define/models/multiphase/population-balance/expert/qbmm/zero-moment-plusSpecifies the upper threshold below which the moments are considered to be negligible. This parameter should be set to a small positive number. The default value is 1e-12. If non-realizable moments are generated during the simulation, you can increase this parameter (even by a few orders of magnitude, for example, 1e-9).
define/models/multiphase/population-balance/expert/qbmm/zero-zeta-minusSpecifies the lower threshold above which the auxiliary quantities are considered to be negligible. This parameter should be set to a small negative number. The default value is -1e-12. If non-realizable moments are generated during the simulation, this parameter can be lowered by a few orders of magnitude (for example, -1e-9).
define/models/multiphase/population-balance/expert/qbmm/zero-zeta-plusSpecifies the upper threshold below which the auxiliary quantities are considered to be negligible. This parameter should be set to a small positive number. The default value is 1e-12. If non-realizable moments are generated during the simulation, this parameter can be increased by a few orders of magnitude (for example, 1e-9).
