Eulerian-Lagrangian framework

The coal particles can be tracked as parcels in an Eulerian-Lagrangian framework. Discrete phase model (DPM) are used to define the injected particles that enter the reactor. In the case of INCI, simulation values for axial velocity of -1.732 m/s and a radial velocity of -l.Om/s of the particles must be provided. If agglomeration is neglected, a maximum particle diameter of 0.1 mm, a mean diameter of 0.09 mm, and a minimum diameter of 0.001 mm are assumed according to a Rosin-Rammler-Sperling-Bennett distribution with a spread parameter of = 0.688 in 10 individual groups for fluid-bed coal (see also Section 3.12.3.3). Particles can be treated as nonspherical with a shape factor of 0.85. 

For canonical turbulent flows (Pope 2000), the flow parameters required to complete the CRE models are readily available. However, for the complex flow fields present in most chemical reactors, the flow parameters must be found either empirically or by solving a CFD turbulence model. If the latter course is taken, the next logical step would be to attempt to reformulate the CRE model in terms of a set of transport equations that can be added to the CFD model. The principal complication encountered when following this path is the fact that the CRE models are expressed in a Lagrangian framework, whilst the CFD models are expressed in an Eulerian framework. One of the main goals of this book... 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

The modeling frameworks used are Eulerian, Lagrangian, or a combination... 

We begin with a general survey of atmospheric inverse methods. In all cases, the broad goal is to use concentration measurements in the air, together with information about atmospheric flow, to infer sources and sinks of entities at the earth s surface. Since the key concentration observations are remote from the surface sources and sinks, this entire class of methods relies explicitly or implicitly on an atmospheric mass or molar balance for the entity being measured, within a specified control volume. Such a balance can be either in an Eulerian framework, in which the control volume is fixed in space, or in a Lagrangian framework, in which the control volume moves with the flow. Considering the Eulerian framework first, the molar balance for a scalar entity can be written informally as... 

We begin our discussion with the simplest modeling representation—the box model—and derive the governing equations in both Eulerian and Lagrangian frameworks. 

Models seldom express the conservation relations in a Lagrangian framework. The chain rule of calculus is used to convert to an Eulerian framework. 

The geometrical flexibility of the VOF scheme can be significantly improved if in its formulation, instead of using a fixed framework, a combination of a Lagrangian-Eulerian approach is adopted. The most common approach to develop such a combined framework is the application of the Arbitrary... 

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

The study of mixing effects on chemical reactions has been an active area of research since the pioneering papers of Danckwerts (1958) and Zwietering (1959). The topic has become a part of classical Chemical Reaction Engineering and has been discussed in textbooks (Froment and Bischoff, 1990 Levenspiel, 1999 Westerterp et al., 1984) and review articles (Villermaux, 1991). Historically, this study has progressed in two parallel branches, based on the Lagrangian and Eulerian frameworks of description, respectively. 

Lagrangian- or arbitrary Lagrangian-Eulerian (ALE) methods [36] [37], and were first reported by Hirt et al. [68]. In this book this particular framework is termed the general Lagrangian approach, see Fig 1.1(C). This approach also involves quite complex mathematics and is thus hardly ever used in chemical reactor analysis. 

The model derivation given above using the Liouville theorem is in many ways equivalent to the Lagrangian balance formulation [83]. Of course, a consistent Eulerian balance formulation would give the same result, but includes some more manipulations of the terms in the number balance. However, the Eulerian formulation is of special interest as we have adopted this framework in the preceding discussion of the governing equations of classical fluid dynamics, chap 1. 

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