Extended Quadrature Method Of Moments Research Articles

Modelling of the leaching process by the extended quadrature method of moments

An approach based on the extended quadrature method of moments (EQMOM) has been developed to solve the population balance equation written for the evolution of shrinking-core particles in transient leaching processes. It has been shown that the EQMOM can naturally address the dependency of the shrinkage-core model on the unreacted core size and outer size of particles. The approach has been validated by comparing its predictions with either analytical or reference solutions for several test cases. Moreover, the approach has been employed to simulate an experimental leaching process taken from the literature. The obtained results confirm the significance of considering the particle size distribution in the simulation of the leaching process, hence the importance of solution methods such as that presented in this work. Finally, it has been shown that the proposed approach is also useful for the simulation of the dissolution process with shrinking particles.

Large Eddy Simulation of Soot Formation in a Real Aero-Engine Combustor Using Tabulated Chemistry and a Quadrature-Based Method of Moments

Abstract Considering the increasingly stringent targets for aircraft emissions, computational fluid dynamics (CFD) is becoming a viable tool for improving future aero-engine combustors. However, predicting pollutant formation remains challenging. In particular, directly solving the evolution of soot particles is numerically expensive. To reduce the computational cost but retain detailed physical modeling, quadrature-based moments methods can be efficiently employed to approximate the particle number density function (NDF). An example is the recently developed split-based extended quadrature method of moments (S-EQMOM), which enables a continuous description of the soot particles' NDF, essential to consider particle oxidation accurately. This model has shown promising results in laminar premixed flames up to turbulent laboratory scale configurations. However, the application to large-scale applications are still scarce. In this work, the S-EQMOM model is applied to the Rolls-Royce BR710 aero-engine combustor to investigate the soot evolution process in practically relevant configurations. For this, the soot model is embedded into a high-fidelity simulation framework, consisting of large eddy simulation for the turbulent flow and mixing and the flamelet-generated manifold method for chemistry reduction. An additional transport equation for polycyclic aromatic hydrocarbons is solved to model their slow chemistry and the transition from the gaseous phase to the solid phase. Simulations are performed for different operating conditions (idle, approach, climb, takeoff) to validate the model using experimental data. Subsequently, the results are analyzed to provide insights into the complex interactions of hydrodynamics, mixing, chemistry, and soot formation.

Further developments of the extended quadrature method of moments to solve population balance equations

Developing numerical methods to solve polydispersed flows using a Population Balance Equation (PBE) is an active research topic with wide engineering applications. The Extended Quadrature Method of Moments (EQMOM) approximates the number density as a positive mixture of Kernel Density Functions (KDFs) that allows physical source terms in the PBEs to compute continuous or point-wise form according to the moments. The moment-inversion procedure used in EQMOM has limitations such as the inability to calculate certain roots even if it is defined, absence of consistent result when multiple roots exist or when the roots are nearly equal. To address these limitations, the study proposes a modification of the moment-inversion procedure to solve the PBE based on the proposed Halley-Ridder (H-R) method. Although there is no significant improvement in the extent of variability relative to the mean of the tested shape parameter σ values, an increase in the number of floating point operations (FLOPS) is observed which the proposed algorithm responds in limitations mentioned above. The total number of FLOPS for all the kernels used for the approximation increased by around 30%. This is an improvement towards the development of a more reliable and robust moment-inversion procedure.

3D CFD-PBM simulation of gas–solid bubbling beds of Geldart A particles with sub-grid drag correction

The importance of bubble dynamics in fluidized beds is underscored by the efforts undertaken over the past decades through experimental and numerical studies. Nonetheless, the bubble dynamic evolution and its effects on sub-grid closures were neglected in previous coarse-grid CFD models, particularly for Geldart A particles. In this article, a population balance model with improved kernels was employed to describe the bubble dynamic evolution in three-dimensional bubbling beds, whereby the EMMS drag model was closedto be incorporated into an Eulerian continuum model . Furthermore, the Extended Quadrature Method of Moments (EQMOM) was utilized to reconstruct the number density function of bubble size once their lower moments were obtained. The simulation results showed that the hydrodynamics including bed expansion characteristics, solids concentration and bubble size distribution can be well predicted by the model, indicating its capability to capture the heterogeneous structures in three-dimensional bubbling beds of fine particles.

Soot particle size distribution reconstruction in a turbulent sooting flame with the split-based extended quadrature method of moments

The Method of Moments (MOM) has largely been applied to investigate sooting laminar and turbulent flames. However, the classical MOM is not able to characterize a continuous particle size distribution (PSD). Without access to information on the PSD, it is difficult to accurately take into account particle oxidation, which is crucial for shrinking and eliminating soot particles. Recently, the Split-based Extended Quadrature Method of Moments (S-EQMOM) has been proposed as a numerically robust alternative to overcome this issue [Salenbauch et al., “A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth, and oxidation,” J. Aerosol Sci. 128, 34–49 (2019)]. The main advantage is that a continuous particle number density function can be reconstructed by superimposing kernel density functions (KDFs). Moreover, the S-EQMOM primary nodes are determined individually for each KDF, improving the moment realizability. In this work, the S-EQMOM is combined with a large eddy simulation/presumed-probability density function flamelet/progress variable approach for predicting soot formation in the Delft Adelaide Flame III. The target flame features low/high sooting propensity/intermittency and comprehensive flow/scalar/soot data are available for model validation. Simulation results are compared with the experimental data for both the gas phase and the particulate phase. Good quantitative agreement has been obtained especially in terms of the soot volume fraction. The reconstructed PSD reveals predominantly unimodal/bimodal distributions in the first/downstream portion of this flame with particle diameters smaller than 100 nm. By investigating the instantaneous and statistical sooting behavior at the flame tip, it has been found that the experimentally observed soot intermittency is linked to mixture fraction fluctuations around its stoichiometric value that exhibits a bimodal probability density function.

Zero-flux approximations for multivariate quadrature-based moment methods

The evolution of polydisperse systems is governed by population balance equations. A group of efficient solution approaches are the moment methods, which do not solve for the number density function (NDF) directly but rather for a set of its moments. While this is computationally efficient, a specific challenge arises when describing the fluxes across a boundary in phase space for the disappearance of elements, the so-called zero-flux. The main difficulty is the missing NDF-information at the boundary, which most moment methods cannot provide. Relevant physical examples are evaporating droplets, soot oxidation or particle dissolution.In general, this issue can be solved by reconstructing the NDF close to the boundary. However, this was previously only achieved with univariate approaches, i.e. considering only a single internal variable. Many physical problems are insufficiently described by univariate population balance equations, e.g. droplets in sprays often require the temperature or the velocity to be internal coordinates in addition to the size.In this paper, we propose an algorithm, which provides an efficient multivariate approach to calculate the zero-fluxes. The algorithm employs the Extended Quadrature Method of Moments (EQMOM) with Beta and Gamma kernel density functions for the marginal NDF reconstruction and a polynomial or spline for the other conditional dimensions. This combination allows to reconstruct the entire multivariate NDF and based on this, expressions for the disappearance flux are derived. An algorithm is proposed for the whole moment inversion and reconstruction process. It is validated against a suite of test cases with increasing complexity. The influence of the number of kernel density functions and the configuration of the polynomials and splines on the accuracy is discussed. Finally, the associated computational costs are evaluated.

Application of extended quadrature method of moments for simulation of bubbly flow and mass transfer in gas‐liquid stirred tanks

In the present paper, two gas‐liquid stirred tanks, one agitated by a radial impeller and another by an axial impeller, are modelled using the open‐source computational fluid dynamic (CFD) package OpenFOAM (open source field operation and manipulation). The combined effect of the bubble break‐up and coalescence in the tank is considered by a population balance model (PBM) called extended quadrature method of moments (EQMOM). The three‐dimensional simulation is made using a multiple reference frame (MRF), a well‐established method for the modelling of mixers. Dispersed gas and bubble dynamics in the turbulent flow are modelled using the Eulerian‐Eulerian approach (E‐E) with mixture k‐epsilon turbulent model and the modified Tomiyama drag coefficient for the momentum exchange. The model is developed to predict the spatial distribution of gas phase fraction, Sauter mean bubble diameter (), number density function (NDF), dissolved oxygen (DO) evolution, and flow structure. The numerical results are compared with experimental data and a fair agreement is achieved. The results of the axial impeller are discussed based on four impeller rotational speeds with different volumetric mass transfer coefficients.

Comparisons of methods for reconstructing particle size distribution from its moments

The reconstruction of particle size distribution (PSD) from a finite number of its moments is extremely important for many industrial operations, especially in dynamic multi-phase flows in chemical engineering where the composition of discrete phases evolves with time and space. For example, control of the fuel droplet PSD in diesel engines has a significant influence on the flame structure. Different reconstruction methods have been proposed in the literature during the last decades. However, most of them have only been tested with particular and simple cases. This paper describes and evaluates three typical reconstruction methods: parameter fitting method with prescribed distributions, extended quadrature method of moments (EQMOM) and maximum entropy (ME). The corresponding mathematical formulations of these methods are described. A number of test cases are devised where different types of PSDs are adopted so that a comprehensive evaluation of these methods (advantages and limitations) can be achieved. In this paper, a new approach for estimating the particle size range is proposed to improve the reconstruction performance of the ME method. Furthermore, a coupled moment projection method (MPM) and the revised ME approach for particle dynamic simulation is introduced and compared with EQMOM for a combined particle process of inception, growth, coagulation and fragmentation. Results suggest that the revised ME method is very efficient with a wide range of applicability. High accuracy can be achieved with this revised ME method as long as sufficient moments are adopted. By contrast, the parameter fitting method has a narrow applicability and EQMOM suffers from the oscillation problem when more than one kernel function are used. The findings observed in this work can be used as a good guide for choosing a proper PSD reconstruction technique.

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.

A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth and oxidation

Several method of moments (MOM) models were developed recently to describe the evolution of soot particle populations. However, especially soot oxidation is still very challenging in MOM, as pointwise information of the underlying soot particle number density function (NDF) is usually unknown and thus, the prediction of smallest particles that oxidize completely is not easy. The recently proposed Extended Quadrature Method of Moments (EQMOM) (Yuan, Laurent, & Fox, 2012) has the potential to resolve this issue providing a continuous NDF reconstruction based on the set of transported moments. While its general suitability for soot was already demonstrated in the literature, the EQMOM moment inversion procedure reveals several numerical difficulties.In this work, we propose an EQMOM modification called split-based EQMOM that builds upon the general idea of EQMOM to represent the NDF by a weighted sum of superimposed, non-negative, continuous kernel density functions (KDFs), avoiding, however, the numerical problems of its original version. The model is based on a split of the entire NDF into a sum of overlaying density functions, whose evolution is governed by individual population balance equations (PBEs). These PBEs are derived and implemented in a novel Monte Carlo (MC) framework which is applied to simulate two fuel-rich burner-stabilized laminar premixed flames with different sooting behaviours. Results are compared to a classical MC model to demonstrate the suitability. Next, the MC data is used to assess the suitability of lognormal, gamma and inverse Gaussian distributions to approximate the NDF shape by employing the Wasserstein metric as criterion. This information is finally used to formulate an improved EQMOM soot model, which is then evaluated for both fuel-rich and oxidizing conditions. For the latter, the two-stage burner experiments of Echavarria, Jaramillo, Sarofim, & Lighty (2011) is considered, where particle oxidation is the dominant process. It is demonstrated that the proposed MOM model allows an accurate and numerically robust description of soot formation, growth and also oxidation.

Application of the extended quadrature method of moments as a multi-moment parameterization scheme for raindrops sedimentation

In numerical weather prediction models, previous approaches have employed bulk parameterization schemes based on presumed-number-density-functions or quadrature methods of moments (QMOM). In the present work, a new parameterization based on the extended quadrature method of moments (EQMOM) introduced in Yuan et al. (2012) is applied to the case of pure sedimentation of rain drops (one-dimensional “rain-shaft” test case). In EQMOM, the drop size distribution is represented by a weighted sum of kernel density functions, combining elements of quadrature and presumed functional form methods. In cloud microphysics, moment parameterization is frequently based on Gamma distributions, which guided the choice of the kernel shape employed here. EQMOM allows inclusion of a number of prognostic moments in the method (e.g. M(0)-M(6)), which improves flexibility in the representation of a continuous size distribution. QMOM and EQMOM up to order 3 were applied in two drop sedimentation test cases previously presented in the literature, in which initial states consist of different cloud heights and drop size distribution shapes. Results were compared to a spectral reference model using a number of transported bin sizes showing good agreement. The analysis was focused in the sedimentation induced errors obtained by the different approaches. In QMOM, size sorting due to different fall velocities generates step patterns in the moment profiles. With EQMOM, on the other hand, these artifacts are significantly suppressed. Furthermore, predictions of the number concentration, total liquid content, radar reflectivity, mass mean diameter and rain rates are shown to be greatly improved when EQMOM is employed. Quantitatively, EQMOM is capable of reducing global error measures by nearly one order of magnitude, when compared to results obtained by previous methods in a common benchmark, showing the great potential of the method in the field of meteorology.

New developments of the Extended Quadrature Method of Moments to solve Population Balance Equations

Population Balance Models have a wide range of applications in many industrial fields as they allow accounting for heterogeneity among properties which are crucial for some system modelling. They actually describe the evolution of a Number Density Function (NDF) using a Population Balance Equation (PBE). For instance, they are applied to gas–liquid columns or stirred reactors, aerosol technology, crystallisation processes, fine particles or biological systems. There is a significant interest for fast, stable and accurate numerical methods in order to solve for PBEs, a class of such methods actually does not solve directly the NDF but resolves their moments. These methods of moments, and in particular quadrature-based methods of moments, have been successfully applied to a variety of systems. Point-wise values of the NDF are sometimes required but are not directly accessible from the moments. To address these issues, the Extended Quadrature Method of Moments (EQMOM) has been developed in the past few years and approximates the NDF, from its moments, as a convex mixture of Kernel Density Functions (KDFs) of the same parametric family. In the present work EQMOM is further developed on two aspects. The main one is a significant improvement of the core iterative procedure of that method, the corresponding reduction of its computational cost is estimated to range from 60% up to 95%. The second aspect is an extension of EQMOM to two new KDFs used for the approximation, the Weibull and the Laplace kernels. All MATLAB source codes used for this article are provided with this article.

Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications

In order to optimize and design new bubbly flow reactors, it is necessary to predict the bubble behavior and properties with respect to the time and location. In gas-liquid flows, it is easily observed that the bubble sizes may vary widely. The bubble size distribution is relatively sharply defined, and bubble rises are uniform in homogeneous flow; however bubbles aggregate, and large bubbles are formed rapidly in heterogeneous flow. To assist in the analysis of these systems, the volume, size and other properties of dispersed bubbles can be described mathematically by distribution functions. Therefore, a mathematical modeling tool called the Population Balance Model (PBM) is required to predict the distribution functions of the bubble motion and the variation of their properties. In the present paper, two rectangular bubble columns and a water electrolysis reactor are modeled using the open-source Computational Fluid Dynamic (CFD) package OpenFOAM. Furthermore, the Method of Classes (CM) and Quadrature-based Moments Method (QBMM) are described, implemented and compared using the developed CFD-PBM solver. These PBM tools are applied in two bubbly flow cases: bubble columns (using a Eulerian-Eulerian two-phase approach to predict the flow) and a water electrolysis reactor (using a single-phase approach to predict the flow). The numerical results are compared with measured data available in the scientific literature. It is observed that the Extended Quadrature Method of Moments (EQMOM) leads to a slight improvement in the prediction of experimental measurements and provides a continuous reconstruction of the Number Density Function (NDF), which is helpful in the modeling of gas evolution electrodes in the water electrolysis reactor.

An open-source quadrature-based population balance solver for OpenFOAM

The extended quadrature method of moments (EQMOM) for the solution of population balance equations (PBE) is implemented in the open-source computational fluid dynamic (CFD) toolbox OpenFOAM as part of the OpenQBMM project. The moment inversion procedure was designed (Nguyen et al., 2016) to maximize the number of conserved moments in the transported moment set. The algorithm is implemented in a general structure to allow the addition of other kernel density functions defined on R+, and arbitrary kernels to describe physical phenomena involved in the evolution of the number density function. The implementation is verified with a set of zero-dimensional cases involving aggregation and breakage problems. Comparison to the rigorous solution of the PBE provides validation for these cases. The coupling of the EQMOM procedure with a CFD solver to address aggregation and breakage problems of non-inertial particles is validated against experimental measurements in a Taylor-Couette reactor from the literature.

Droplet breakage and coalescence in liquid–liquid dispersions: Comparison of different kernels with EQMOM and QMOM

Droplet coalescence and breakage in turbulent liquid–liquid dispersions is simulated by using computational fluid dynamics (CFD) and population balance modeling. The multifractal (MF) formalism that takes into account internal intermittency was here used for the first time to describe breakage and coalescence in a surfactant‐free dispersion. The log‐normal Extended Quadrature Method of Moments (EQMOM) was for the first time coupled with a CFD multiphase solver. To assess the accuracy of the model, predictions are compared with experiments and other models (i.e., Coulalogou and Tavlarides kernels and Quadrature Method of Moments [QMOM]). EQMOM and QMOM resulted in similar predictions, but EQMOM provides a continuous reconstruction of the droplet‐size distribution. Transient predictions obtained with the MF kernels result in a better agreement with the experiments. © 2016 American Institute of Chemical Engineers AIChE J, 63: 2293–2311, 2017

Modeling soot oxidation with the Extended Quadrature Method of Moments

Modeling the oxidation of soot particles in flames is a challenging topic both from a chemical point of view and regarding the statistical treatment of the evolution of the soot number density function (NDF). The method of moments is widely-used for the statistical modeling of aerosol dynamics in various applications, and a number of different moment methods have been established and successfully applied to the modeling of soot formation and growth. However, a shortcoming of existing moment methods is the lack of an accurate, numerically robust, and computationally efficient way to treat soot oxidation, especially regarding the prediction of the particle number density. In this work, the recently developed Extended Quadrature Method of Moments (EQMOM) is integrated with a physico-chemical soot model and combined with a treatment for particle removal by oxidation. This leads to a modeling framework for the simulation of coupled inception, growth, coagulation, and oxidation of soot in flames. In EQMOM, the moment equations are closed by reconstructing the soot NDF with a superposition of continuous kernel functions. Various standard distribution functions can be used as kernel functions, and the algorithm has been implemented here using gamma and lognormal distributions. It is shown that and discussed why gamma distributions are more suitable as kernel functions than lognormal distributions in order to accurately predict soot oxidation. The integrated model is validated by comparisons with analytical solutions for the NDF, results from Monte Carlo simulations of soot formation and oxidation in flames, and experimental data.

Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes

The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended quadrature method of moments (EQMOM) based on the work of Yuan et al. [52] and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al. [30]. For both methods, the closure employs a continuous reconstruction of the number density function of the particles from its moments, thus allowing evaluation of all the unclosed terms in the moment equations, including the negative flux due to the disappearance of particles. Here, new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method. Moreover, robust and accurate numerical methods are developed, ensuring the realizability of the moments. The robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy. EQMOM and TSM are compared to a sectional method for various simple but relevant test cases, showing their ability to describe accurately the fine-particle population with a much lower number of variables. These results demonstrate the efficiency of the modeling and numerical choices, and their potential for the simulation of real-world applications.

Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry

We consider the simplest member of the hierarchy of the extended quadrature method of moments (EQMOM), which gives equations for the zeroth-, first-, and second-order moments of the energy density of photons in the radiative transfer equations in slab geometry. First we show that the equations are well-defined for all moment vectors consistent with a nonnegative underlying distribution, and that the reconstruction is explicit and therefore computationally inexpensive. Second, we show that the resulting moment equations are hyperbolic. These two properties make this moment method quite similar to the attractive but far more expensive $M_2$ method. We confirm through numerical solutions to several benchmark problems that the methods give qualitatively similar results.

Bivariate extensions of the Extended Quadrature Method of Moments (EQMOM) to describe coupled droplet evaporation and heat-up

In this study, four Eulerian moment methods based on the Extended Quadrature Method of Moments (EQMOM) (Yuan, Laurent, & Fox, 2012) are applied to model the evaporation of droplets. EQMOM enables the number density function (NDF) to be reconstructed from a set of moments of a droplet ensemble. Knowledge of the NDF allows us to describe the disappearance flux of the smallest droplets accurately, which is a critical issue in moment methods (Fox, Laurent, & Massot, 2008; Yuan et al., 2012). As EQMOM is restricted to univariate distributions, initially only the droplet size is considered as an internal coordinate. The description is then extended to the bivariate models mono-thermal Extended Conditional Quadrature Method of Moments (mt-ECQMOM), whose formulation is based on the semi-kinetic approach (Vié, Laurent, & Massot, 2013) and Extended Conditional Quadrature Method of Moments (ECQMOM, Marchisio & Fox, 2013). Based on ECQMOM and the general principle of a sectional method, a third bivariate approach is proposed, called the Extended Conditional Sectional Method of Moments (ECSQMOM). The resulting three bivariate approaches allow us to describe the coupling of heat-up and evaporation using the internal coordinates droplet diameter and temperature.The results based on EQMOM, mt-ECQMOM, ECQMOM and ECSQMOM are evaluated against a direct numerical solution of the population balance equation (PBE) investigating a test setup based on experimental studies for iso-octane sprays. Results indicate that in the limiting case of a mono-thermal distribution, the ECQMOM model implementation referenced above is not applicable to describe evaporating sprays accurately, whereas the mt-ECQMOM leads to much better results. ECSQMOM is also found to yield good results, even if the accuracy of mt-ECQMOM is not reached. However, since ECSQMOM offers the potential to capture distributions with nonzero conditional variance, it can be considered a suitable approach to describe the evolution of droplet distribution in sprays in future applications.

An extended quadrature-based moment method with log-normal kernel density functions

An extended quadrature method of moments (EQMOM) with log-normal kernel density functions is developed in this work, and applied to the solution of a population balance equation (PBE) for aggregation and breakup, coalescence, and condensation problems. The cases with one and two kernel density functions are studied analytically, and the existence of an analytical solution is shown. A numerical procedure based on the work of Yuan et al. (2012) is adopted to address cases with a larger number of kernel density functions. Results for the reconstructed number density function (NDF), the time evolution of the zero-order moment and of the mean particle size are compared with those obtained from the rigorous solution of the PBE reported by Vanni (2000) for the cases of aggregation and breakup. A problem concerning coalescence and one regarding condensation, both with analytical solution, are also examined. The results obtained with the proposed approach are compared to those provided by EQMOM with gamma kernel densities. Satisfactory results were obtained for the reconstructed distribution. Excellent agreement was observed between the rigorous solution and the approximated one for the time evolution of the total number density and the mean particle size.