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Applications
Published in William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain, of Physical Systems, 2020
William G. Gray, Anton Leijnse, Randall L. Kolar, Cheryl A. Blain
In this example, the balance of mass for a fluid at the microscale will be derived as a mathematical expression of the observation that, under non-nuclear conditions, mass is neither created nor destroyed. Often, fluid mechanics texts present the control volume approach, an approach that first balances fluxes and accumulation on a cubic element and then examines the balance statement as the size of the element goes to zero [e.g. Bird et al., 1960; White, 1979]. This method is often referred to as the Eulerian approach. As an alternative here, the theorems presented in Chapters 7 and 8 are used as tools to derive the balance of mass at the microscale, an approach often referred to as the Lagrangian approach [e.g. Whitaker, 1968; John and Haberman, 1980]. While the following derivation is certainly not unique to this work, it serves as a good introductory illustration of the use of integration and averaging theorems as powerful, yet simple, tools in engineering analysis.
Particle Deposition and Reentrainment
Published in Ko Higashitani, Hisao Makino, Shuji Matsusaka, Powder Technology Handbook, 2019
Manabu Shimada, Shuji Matsusaka, Hiroaki Masuda
The Eulerian method is typically applied to submicron-sized or smaller particles dominated by fluid flow motion, Brownian diffusive motion, and migration by external forces. This method uses the following convection-diffusion equation to describe the spatial and temporal changes in particle number concentration per unit fluid volume, n, as follows: dndt=∇⋅{ρfD∇(n/ρf)−(u+v)n}+Q
Eulerian Integral Methods
Published in Lorin R. Davis, Fundamentals of ENVIRONMENTAL DISCHARGE MODELING, 2018
Eulerian methods look at a fixed control volume and the changes in properties of the fluids that pass in and out of the volume. Equations are expressed in space derivatives rather than with time as in Lagrangian methods. The UDKHG* is a version of UKDHDEN4,13 that has the option of displaying the output graphically on the computer screen. It uses a Eulerian approach and integral methods to convert the governing partial differential equations to a series of ordinary differential equations. This method has been found to produce very good approximations to fluid behavior as long as boundary effects do not enter the problem.14,15 Integral methods require that appropriate distribution profiles be assumed for velocity, temperature, and concentration. To accommodate merging of multiple plumes, the UDKHG program uses 3/2 power law profiles as explained in Chapter 1.
Performance analysis of a gas cyclone with a dustbin inverted hybrid solid cone
Published in Aerosol Science and Technology, 2023
E. Dehdarinejad, F. Parvaz, S. H. Hosseini, G. Ahmadi, K. Elsayed
The governing equations for gas flow and particle trajectory analysis are reported in supplementary materials and the previous works by authors (Dehdarinejad et al. 2023; Parvaz et al. 2017a, 2018). Several researchers (Dehdarinejad and Bayareh 2021, 2022a; Dehdarinejad, Bayareh, and Ashrafizaadeh 2022; Izadi et al. 2020; Parvaz et al. 2017b, 2020; Vahedi et al. 2018a) showed that the RSM turbulence model predicted the gas cyclone flows with reasonable accuracy. Therefore, the RSM turbulence model is used in the present simulations. The finite volume-based Ansys-Fluent software is used for solving the governing equations. In all simulations, it is assumed that the flow is isothermal and incompressible. The motion of particles in fluid dynamics is a complex phenomenon that requires a combination of Eulerian and Lagrangian approaches. Therefore, the Eulerian-Lagrangian approach is used to address this issue. The numerical schemes for the airflow field were previously employed by Kaya and Karagoz (Kaya, Karagoz, and Avci 2011; Karagoz and Kaya 2007). The details of discretization schemes used in the present simulations are presented in Table 1.
Two-fluid model with variable particle–particle restitution coefficient: application to the simulation of FCC riser reactor
Published in Particulate Science and Technology, 2020
Hanbin Zhong, Juntao Zhang, Shengrong Liang, Yuqin Zhu
Gas–solid fluidized beds are widely used in the chemical industry due to their excellent gas–solid contact and favorable heat and mass transfer characteristics. In recent years, with the rapid development of computational ability, computational fluid dynamics (CFD) method has been increasingly employed as an efficient tool to investigate the complex hydrodynamic behaviors of gas–solid fluidized beds. Typically, there are two different CFD methods for modeling gas–solid fluidized beds, i.e. Eulerian–Lagrangian method and Eulerian–Eulerian method. Instead of tracking the motion of individual particles as in the Eulerian–Lagrangian method, the Eulerian–Eulerian method describes the solid phase and the gas phase as fully interpenetrating continua, and consequently, the Eulerian-Eulerian based CFD models are capable of handling multiphase systems containing large numbers of particles with low computational cost (Zhong, Liang, et al. 2016). The Eulerian–Eulerian method, which is also named as the two-fluid model (TFM) for one solid phase or multi-fluid model (MFM) for two or more solid phases, has been widely used in the CFD simulation of gas–solid fluidized bed reactors.
Characterization, testing, and optimization of load aggregation methods for ground heat exchanger response-factor models
Published in Science and Technology for the Built Environment, 2019
Matt S. Mitchell, Jeffrey D. Spitler
Wentzel (2005) describes another type of load aggregation procedure that differs from the static method. The author used the method to model a foundation wall using dynamic thermal networks, and not a GHE. Claesson and Javed (2012) later simplified the method and applied it to modeling GHE. The method could be characterized as an Eulerian approach, which in a fluid flow application tracks the flow through a fixed control volume rather than tracking individual packets or particles of fluid. In our case, we predefine the bins and allow energy to move through the bins. Hence, we term the method “dynamic” since the loads move relative to the aggregation bins. This work has taken the frame of reference to be oriented around the load values, not the aggregation bins. This is the reasoning for labeling the Yavuzturk and Spitler (1999) type methods “static” and the Claesson and Javed (2012) type methods “dynamic.” In the case of the static method, the bins move relative to the current simulation time, but the loads themselves are fixed within the bins, whereas the dynamic method bins are fixed relative to the current simulation time, but the loads move within the bins.