Function Of Number Density Research Articles

An analytic approach for nonlinear collisional fragmentation model arising in bubble column

The phenomenon of coagulation and breakage of particles plays a pivotal role in diverse fields. It aids in tracking the development of aerosols and granules in the pharmaceutical sector, coagulation or breakage of droplets in chemical engineering, understanding blood clotting mechanisms in biology, and facilitating cheese production through the action of enzymes within the dairy industry. A significant portion of research in this direction concentrates on coagulation or linear breakage processes. In the case of linear case, bubble particles break down due to inherent stresses or specific conditions of the breakage event. However, in many practical situations, particle division is primarily due to forces exerted during collisions between particles, necessitating an approach that accounts for nonlinear collisional breakage. Despite its critical role in a wide array of engineering and physical operations, the study of this nonlinear fragmentation phenomenon has not been extensively pursued. This article introduces an innovative semi-analytical method that leverages the beyond linear use of equation superposition function to address the nonlinear integro-partial differential model of collisional breakage population balance. This approach is versatile, allowing for the resolution of both linear/nonlinear equations while sidestepping the complexities associated with discretization of domain. To assess the precision of this method, we conduct a thorough convergence analysis. This process utilizes the principle of contractive mapping in the Banach space, a globally recognized strategy for verifying convergence. We explore a variety of kernel parameters associated with collisional kernels, alongside breakage and initial distribution functions, to derive novel iterative solutions. Comparing our findings with those obtained through the finite volume method regarding number density functions and their integral moments, we demonstrate the reliability and accuracy of our approach. The consistency and correctness of our method are further validated by depicting the errors between the exact and approximated solutions in graphical and tabular formats.

Explicit and approximate solutions for a classical hyperbolic fragmentation equation using a hybrid projected differential transform method

Population balance equations are widely used to study the evolution of aerosols, colloids, liquid–liquid dispersion, raindrop fragmentation, and pharmaceutical granulation. However, these equations are difficult to solve due to the complexity of the kernel structures and initial conditions. The hyperbolic fragmentation equation, in particular, is further complicated by the inclusion of double integrals. These challenges hinder the analytical solutions of number density functions for basic kernel classes with exponential initial distributions. To address these issues, this study introduces a new approach combining the projected differential transform method with Laplace transform and Padé approximants to solve the hyperbolic fragmentation equation. This method aims to provide accurate and efficient explicit solutions to this challenging problem. The approach's applicability is demonstrated through rigorous mathematical derivation and convergence analysis using the Banach contraction principle. Additionally, several numerical examples illustrate the accuracy and robustness of this new method. For the first time, new analytical solutions for number density functions are presented for various fragmentation kernels with gamma and other initial distributions. This method significantly enhances solution quality over extended periods using fewer terms in the truncated series. The solutions are compared and verified against the finite volume method and the homotopy perturbation method, showing that the coupled approach not only estimates number density functions accurately but also captures integral moments with high precision. This research advances computational methods for particle breakage phenomena, offering potential applications in various industrial processes and scientific disciplines.

Dense QCD2 with matrix product states

We study one-flavor SU(2) and SU(3) lattice QCD in (1 + 1) dimensions at zero temperature and finite density using matrix product states and the density matrix renormalization group. We compute physical observables such as the equation of state, chiral condensate, and quark distribution function as functions of the baryon number density. As a physical implication, we discuss the inhomogeneous phase at nonzero baryon density, where the chiral condensate is inhomogeneous, and baryons form a crystal. We also discuss how the dynamical degrees of freedom change from hadrons to quarks through the formation of quark Fermi seas.

Two moments preserving sectional approach for an enzymatic coagulation equation

The coagulation process has found extensive applications in monitoring the evolution of aerosol and granule preparation in pharmaceutical sciences, blood clotting in biology, and cheese manufacturing due to the enzymes in the dairy industry. Among these, modeling the cheese manufacturing process is more challenging due to three indistinguishable sub-mechanisms: (a) enzymatic proteolysis, (b) coagulation, and (c) gelation, which occurs during the enzymatic coagulation of milk. The current study focuses on developing a sectional approach based on the cell average technique for monitoring the evolution of enzyme-induced coagulation of paracasein micelles over time. The proposed technique preserves two integral properties, such as total number and total volume in the system. The mathematical formulation of the proposed technique is very simple, easy to code, and has a robust implementation on any uniform and non-uniform grids. Due to the unavailability of the analytical solutions of the number density functions, the validation of the new proposed approach is done by extracting the new series solutions through the modification of the Homotopy perturbation method [Kaur et al., J. Phys. A 52(38), 385201 (2019)] and exact integral moments for several kernels. It has been shown that the new approach not only estimates the first two integral moments accurately but also computes the second-order moment with high precision without any specific measures. Moreover, domains of varying size grids are taken into account to analyze the convergence behavior of the average-size paracasein micelles formed in the system based on the zeroth and first moments.

Strange Quark Stars: The Role of Excluded Volume Effects

We study cold strange quark stars employing an enhanced version of the quark-mass density-dependent model, which incorporates excluded volume effects to address non-perturbative QCD repulsive interactions. We provide a comparative analysis of our mass formula parametrization with previous models from the literature. We identify the regions within the parameter space where three-flavor quark matter is more stable than the most tightly bound atomic nucleus (stability window). Specifically, we show that excluded volume effects do not change the Gibbs free energy per baryon at zero pressure, rendering the stability window unaffected. The curves of pressure versus energy density exhibit various shapes—convex upward, concave downward, or nearly linear—depending on the mass parametrization. This behavior results in different patterns of increase, decrease, or constancy in the speed of sound as a function of baryon number density. We analyze the mass–radius relationship of strange quark stars, revealing a significant increase in maximum gravitational mass and a shift in the curves toward larger radii as the excluded volume effect intensifies. Excluded volume effects render our models compatible with all modern astrophysical constraints, including the properties of the recently observed low-mass compact object HESSJ1731.

A dynamic compartment model for spatially heterogeneous reactors: Scalar and Monte-Carlo particle mixing

The coupling of multidimensional population balance models with transport equations is often necessary for the simulation of spatially heterogeneous reactors. Because of numerical intractability, a compromise on both the hydrodynamics and the population description has to be made. In this work, the Monte-Carlo method is used to calculate the local integral source term due to the dispersed phase (isodense particles) and a compartment model is used to describe the fluid motion in the reactor. The number density function n(x,t,ξ) is approximated by the number of Monte-Carlo particles per compartment and their movement in the space of compartment uses a stochastic transport algorithm adapted from existing works. The numerical tool is thoroughly and successfully validated in the context of mixing and first order reaction in a 70 L stirred tank equipped with a Rushton turbine. The dynamic method makes use of time resolved CFD simulations to built time dependent flow matrices reflecting the periodic nature of the flow field. The dynamic approach is finally compared to other usual compartment model approaches in terms of complexity, accuracy and rapidity which reveals the benefit of considering the flow unsteadiness within a compartiment model approach for stirred tanks.

Measuring Galaxy Abundance and Clustering at High Redshift from Incomplete Spectroscopic Data: Tests on Mock Catalogs

The number density and correlation function of galaxies are two key quantities to characterize the distribution of the observed galaxy population. High-z spectroscopic surveys, which usually involve complex target selection and are incomplete in redshift sampling, present both opportunities and challenges to measure these quantities reliably in the high-z Universe. Using realistic mock catalogs, we show that target selection and redshift incompleteness can lead to significantly biased results, especially due to the flux-limit selection criteria. We develop a new method to correct the flux-limit effect, using information provided by the parent photometric data from which the spectroscopic sample is constructed. Our tests using realistic mock samples show that the method is able to reproduce the true stellar mass function and correlation function reliably. Mock catalogs are constructed for the existing zCOSMOS and VIPERS surveys, as well as for the forthcoming Prime Focus Spectrograph (PFS) galaxy evolution survey. The same set of mock samples are used to quantify the total variance expected for different sample sizes. We find that the total variance decreases very slowly when the survey area reaches about 4 deg2 for the abundance and about 8 deg2 for the clustering, indicating that the cosmic variance is no longer the dominant source of error for PFS-like surveys. We also quantify the improvements expected in the PFS-like galaxy survey relative to zCOSMOS and VIPERS surveys.

Modified variational iteration method and its convergence analysis for solving nonlinear aggregation population balance equation

This study proposes a novel approach based on the variational iteration method to solve the nonlinear aggregation population balance equation. The approach provides great flexibility by allowing the selection of appropriate linear operators and efficiently determining the Lagrange multiplier in the nonlinear aggregation population balance equation. The mathematical derivation is supported by conducting a detailed convergence analysis using the contraction mapping principle in the Banach space. Furthermore, error estimates for the approximate solutions are derived, thereby improving our understanding of the accuracy and reliability of the proposed method. To validate the new approach, the obtained solutions are compared with the exact solutions for analytically tractable kernels. However, for more complex physically relevant kernels including polymerization, Ruckenstein/Pulvermacher, and bilinear kernels, due to lack exact solutions, the obtained series solutions corresponding to different initial conditions are verified against the finite volume scheme (kumar et al., 2016). The outcomes illustrate that the proposed approach offers superior approximations of number density functions with fewer terms and demonstrates higher accuracy over extended time domains than the traditional variational iterative method. The new approach also has the tendency to capture the zeroth and first order moments of the number density function with high precision.

A new test of gravity – II. Application of marked correlation functions to luminous red galaxy samples

ABSTRACT We apply the marked correlation function test proposed by Armijo et al. (Paper I) to samples of luminous red galaxies (LRGs) from the final data release of the Sloan Digital Sky Survey (SDSS) III. The test assigns a density-dependent mark to galaxies in the estimation of the projected marked correlation function. Two gravity models are compared: general relativity (GR) and $f(R)$ gravity. We build mock catalogues which, by construction, reproduce the measured galaxy number density and two-point correlation function of the LRG samples, using the halo occupation distribution model (HOD). A range of HOD models give acceptable fits to the observational constraints, and this uncertainty is fed through to the error in the predicted marked correlation functions. The uncertainty from the HOD modelling is comparable to the sample variance for the SDSS-III LRG samples. Our analysis shows that current galaxy catalogues are too small for the test to distinguish a popular $f(R)$ model from GR. However, upcoming surveys with a better measured galaxy number density and smaller errors on the two-point correlation function, or a better understanding of galaxy formation, may allow our method to distinguish between viable gravity models.

The function of nano layer in enhancing the thermal conductivity of TiO2/water nanofluids

Nanoparticles have the capability to effectively improve the thermal conductivity of base fluids, thus improving the heat transfer coefficient of heat transfer systems. In this study, a non-equilibrium molecular dynamics (NEMD) method based on the Fourier law is employed to study the thermal conductivity of TiO2 (r-TiO2)/water nanofluids with temperatures ranging between303K and 333K and volume fractions in the range of1-2%. The ordered layer structure as a shell is analyzed and its influence is surveyed by calculating the number density and radial distribution function (RDF).The results revealed that a clear, solid-like nanolayer of about 0.5 nm can be observed around the nanoparticle. In this regard, the thickness of the nanolayer is less affected by variations in volume fraction and temperature. The g(r) values and the number density decreased with the increase in temperature. Additionally, the g(r) values and the number density at the level of the nanolayer were much higher compared to those at other parts. This indicates the existence of more water molecules in the nanolayer, thereby reducing contact thermal resistance and improving thermal conductivity. Macroscopically, the thermal conductivity increases with the increase in volume fraction. It was found that the increase in the volume fraction from 1%to 2%at303Kresulted in an increase in the effective thermal conductivity from1.027 and 1.042, respectively. In other words, the thermal conductivity of the nanofluid was 2.7% and 4.2% higher than that of the base liquid under the same conditions.

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.

Homotopy analysis method and its convergence analysis for a nonlinear simultaneous aggregation-fragmentation model

In this article, we focus on addressing the missing convergence analysis of the homotopy analysis method (HAM) for solving pure aggregation and pure fragmentation population balance equations [Kaur et al. (2023), J. Math. Anal. Appl., 512(2), 126166]. This technique is further extended to determine analytical series solutions for a simultaneous aggregation–fragmentation (SAF) population balance equation. The convergence analysis of the extended approach for a SAF equation is performed using the concept of contraction mapping in the Banach space. The HAM method enables us to derive recursive formulas to obtain series solutions, distinguishing it from traditional numerical approaches. One noteworthy advantage of HAM is its capability to solve both linear and nonlinear differential equations without resorting to discretization, while incorporating a convergence control parameter. Given the complex nature of the SAF equation, only a single analytical solution has been available, specifically for a constant aggregation kernel and a binary breakage kernel with a linear selection function. However, our study presents new series solutions for the number density functions, considering the combination of sum and product aggregation kernels with binary breakage kernels and linear/quadratic selection functions. These particular solutions have not been previously documented in the existing literature. To verify the accuracy and efficiency of the proposed approach, the results with the finite volume scheme [Singh et al. (2021), J. Comput. Phys., 435, 110215] for establishing the accuracy and effectiveness of the proposed approach.

Homotopy perturbation method and its convergence analysis for nonlinear collisional fragmentation equations

The exploration of collisional fragmentation pheno-mena remains largely unexplored, yet it holds considerable importance in numerous engineering and physical processes. Given the nonlinear nature of the governing equation, only a limited number of analytical solutions for the number density function corresponding to empirical kernels are available in the literature. This article introduces a semi-analytical approach using the homotopy perturbation method to obtain series solutions for the nonlinear collisional fragmentation equation. The method presented here can be readily adapted to solve both linear and nonlinear integral equations, eliminating the need for domain discretization. To gain deeper insights intothe accuracy of the proposed method, a convergence analysis is conducted. This analysis employs the concept of contractive mapping within the Banach space, a well-established technique universally acknowledged for ensuring convergence. Various collisional kernels (product and polymerization kernels), breakage distribution functions (binary and multiple breakage) and various initial particle distributions are considered to obtain the new series solutions. The obtained results are successfully compared against finite volume method [26] solutions in terms of number density functions and their moments. The error between the exact and obtained series solutions is shown in plots and tables to confirm the applicability and accuracy of the proposed method.

The novel oxy-sulfide glassy ionic conductors Na4P2S7-xOx 0 ≤ x ≤ 7: Understanding the features of static and dynamic cations

We present a 23Na nuclear spin dynamics model for interpreting nuclear magnetic resonance (NMR) spin-lattice relaxation and central linewidth data in the invert glass system Na4P2S7-xOx, 0 ≤ x ≤ 7. The glassy nature of this material results in variations in local Na+ cation environments that may be described by a Gaussian distribution of activation energies. A consistent difference between the mean activation energies determined by NMR and DC conductivity measurements was observed, and interpreted using a percolation theory model. From this, the NaNa coordination number in the sodium cation sublattice was obtained. These values were consistent with jumps through tetrahedral faces of the sodium cages for the sulfur rich glasses, x < 5, consistent with proposed models of their short range order (SRO) structures. From NMR spin-echo measurements, we determined the NaNa second moment M2 resulting from the NaNa magnetic dipole interaction of nearest neighbors. Values of M2 obtained as a function of sodium number density N were in agreement with models for uniform sodium distribution, indicating that these invert glass systems form so as to maximize the average NaNa distance. A simple Coulombic attraction model between Na+ cation and X (=S−, O−) anion was applied to calculate the activation energy. In the range 1.5 ≤ x ≤ 7, an increase in activation energy with increasing oxygen content x occurred, and was consistent with the decrease in average anionic radius, and the increase in Coulombic attraction. For small oxygen additions, 0 ≤ x ≤ 1.5, the suggested minimum at low oxygen concentration seen in the activation energies obtained from DC conductivity data is not evident in the model.

A note on homotopy perturbation approach for nonlinear coagulation equation to improve series solutions for longer times

In this work, the approach presented in a recent publication by Kaur et al. (2019) is improved. The truncated series solution derived from the existing Homotopy Perturbation method (HPM) behaves peculiarly for a longer time domain and provides accurate results only for a shorter time. The Homotopy perturbation approach and the Pade approximation are coupled to estimate the nonlinear coagulation equation to tackle this problem. This significantly improves solution quality over a longer time frame by consuming fewer terms of the truncated series. The effectiveness of the new approach is tested by deriving the new analytical solutions of the number density function for a bilinear kernel with exponential initial distributions. In addition, the new solutions for physical relevant shear, Ruckenstein/Pulvermacher and Brownian kernels corresponding to exponential and gamma initial distributions are also derived. Due to the non-availability of the analytical solutions, the verification of the new results is done against the mass conserving finite volume scheme (Singh et al., 2015) for shear stress, bilinear and Brownian kernels.

Self-similar Solution of Population Balance Equation for Aggregation with Constant Kernel

Self-similar solution of population balance equation includes the number density function which remains invariant or contains a part that is invariant. This paper describes self-similar solution of aggregation population balance model with constant kernel. Using this constant kernel, moment of the population balance system achieves the form μ_i (t)∝t^(i-1) and the aggregation population balance equation with constant kernel reduces to a first order linear ordinary integro-differential equation whose solution is exponential function with negative power.

Kinetic Theory Study of Pressure and Equation of State in a Strongly Degenerate Fermi Gas

Starting from the Fermi-Dirac distribution fuunction, Kinetic Theory prescriptions are applied to express number density and pressure as integrals, which are then solved for nonvanishing temperature. The results are then combined to arrive at an equation of state, given in closed form, yielding pressure as a function of number density.

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.

Rate of convergence of two moments consistent finite volume scheme for non-classical divergence coagulation equation

This study focuses on performing a convergence analysis of the recently developed mass based finite volume scheme [20] to simulate a Smoluchowski coagulation equation. The goal of this method is to retain the total number of particles in the system while maintaining a constant mass of the particles utilizing some weights in the formulation. The Lipschitz continuity and consistency of the MBFVS are examined and investigated thoroughly by providing theorems. The rate of convergence of the MBFVS is investigated on (a) uniform grid, (b) non-uniform smooth (geometric) grid, and (c) locally uniform grid. The numerical results in term of number density function and its moments are computed and compared with exact results to ensure the authenticity of the scheme for gelling kernels. In comparison to the sectional approaches such as fixed pivot technique (Giri and Hausenblas, 2013 [13]) and cell average technique (Giri and Nagar, 2015 [14]), this scheme is the best candidate for coupling with powerful tools such as CFD, Ansys, and gPROMS due to its simplified formulation, robust implementation on any grid and coagulation kernel, second order convergence rate and consistency with two moments.

Modeling Cell Size Distribution With Heterogeneous Flux Balance Analysis

For over two decades, Flux Balance Analysis (FBA) has been successfully used for predicting growth rates and intracellular reaction rates in microbiological metabolism. An aspect that is often omitted from this analysis, is segregation or heterogeneity between different cells. In this work, we propose an extended FBA method to model cell size distributions in balanced growth conditions. Hereto, a mathematical description of the concept of balanced growth in terms of cell mass distribution is presented. The cell mass distribution, quantified by the Number Density Function (NDF), is affected by cell growth and cell division. An optimization program is formulated in a general manner in which the NDF, average cell culture growth rate and reaction rates per cell mass are treated as optimization variables. As qualitative proof of concept, the methodology is illustrated on a core carbon model of Escherichia coli under aerobic growth conditions. This illustrates feasibility and applications of this method, while indicating some shortcomings intrinsic to the simplified biomass structuring and the time invariant approach.