Vectorization of the time-dependent Boltzmann transport equation: Application to deep penetration problems

Cobos, A.; Poma, Ana Lucía; Alvarez, Guillermo D.; Sanz, Darı́o E.

doi:10.1016/j.radphyschem.2016.06.012

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…However, the DLA, BA, and RLA models are all based on certain assumptions without consideration about the electrostatic interactions between particles, so there are some limitations for their use in flocculation process modeling. The Boltzmann transport equation describes the particle distribution function evolution with time considering the velocity distribution and scattering direction of particles, and the mathematical function of Boltzmann equation can be expressed as eq , where y represents the average particle size of flocs, x is the reaction time, and A 1 , A 2 , x 0 , and d x are the reaction parameters.…”

Section: Resultsmentioning

confidence: 99%

Adsorption Neutralization Model and Floc Growth Kinetics Properties of Aluminum Coagulants Based on Sips and Boltzmann Equations

Wu¹,

Zhang²,

Zhou

et al. 2017

ACS Appl. Mater. Interfaces

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Single-molecule aluminum salt AlCl, medium polymerized polyaluminum chloride (PAC), and high polymerized polyaluminum chloride (HPAC) were prepared in a laboratory. The characteristics and coagulation properties of these prepared aluminum salts were investigated. The Langmuir, Freundlich, and Sips adsorption isotherms were first used to describe the adsorption neutralization process in coagulation, and the Boltzmann equation was used to fit the reaction kinetics of floc growth in flocculation. It was novel to find that the experimental data fitted well with the Sips and Boltzmann equation, and the significance of parameters in the equations was discussed simultaneously. Through the Sips equation, the adsorption neutralization reaction was proved to be spontaneous and the adsorption neutralization capacity was HPAC > PAC > AlCl. Sips equation also indicated that the zeta potential of water samples would reach a limit with the increase of coagulant dosage, and the equilibrium zeta potential values were 30.25, 30.23, and 27.25 mV for AlCl, PAC, and HPAC, respectively. The lower equilibrium zeta potential value of HPAC might be the reason why the water sample was not easy to achieve restabilization at a high coagulant dosage. Through the Boltzmann equation modeling, the maximum average floc size formed by AlCl, PAC, and HPAC were 196.0, 188.0, and 203.6 μm, respectively, and the halfway time of reactions were 31.23, 17.08, and 9.55 min, respectively. The HPAC showed the strongest floc formation ability and the fastest floc growth rate in the flocculation process, which might be caused by the stronger adsorption and bridging functions of Al and Al contained in HPAC.

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Section: Resultsmentioning

confidence: 99%

Adsorption Neutralization Model and Floc Growth Kinetics Properties of Aluminum Coagulants Based on Sips and Boltzmann Equations

Wu¹,

Zhang²,

Zhou

et al. 2017

ACS Appl. Mater. Interfaces

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“…They used the FDM and an implicit scheme to solve the frequency-dependent BTE for a square domain. Cobos et al analyzed geometry irradiated by photon beams by solving the time-dependent BTE with the time step given by a modified Courant-Friedrichs-Lewy condition [3]. Guo and Xu discretized the transient BTE with a second-order finite volume formulation and used the discrete unified gas kinetic scheme [11].…”

Section: Introductionmentioning

confidence: 99%

Transient Micro-Nano Heat Conduction Analysis Using a Combination of the Finite Difference Method, the Collocation Meshfree Method and the Discrete Ordinate Method

Zahiri¹,

Song²,

Bao³

et al. 2017

Computational Methods and Experimental Measurements XVIII

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For studying transient heat conduction problems in the micro-nano scale, the standard heat diffusion equation is no longer applicable, and the time-dependent Boltzmann transport equation (BTE) needs to be solved. Although mesh-based numerical methods such as the finite element method (FEM) and the finite volume method (FVM) are often employed to solve the BTE, the collocation meshfree method has special advantages since it does not require numerical integration. In this work, the collocation meshfree method and the discrete ordinate method (DOM) are implemented to discretize the spatial and angular domains, respectively, while the explicit finite difference method (FDM) is used for advancing in time. Such a method is used to solve the transient BTE for a square domain with prescribed temperature and adiabatic boundary conditions. Our results are consistent with transient heat transfer problems solved by FVM approach.

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Vectorization of the time-dependent Boltzmann transport equation: Application to deep penetration problems

Cited by 2 publications

References 14 publications

Adsorption Neutralization Model and Floc Growth Kinetics Properties of Aluminum Coagulants Based on Sips and Boltzmann Equations

Adsorption Neutralization Model and Floc Growth Kinetics Properties of Aluminum Coagulants Based on Sips and Boltzmann Equations

Transient Micro-Nano Heat Conduction Analysis Using a Combination of the Finite Difference Method, the Collocation Meshfree Method and the Discrete Ordinate Method

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