Eulerian transport equation

While an understanding of the molecular processes at the fuel cell electrodes requires a quantum mechanical description, the flows through the inlet channels, the gas diffusion layer and across the electrolyte can be described by classical physical theories such as fluid mechanics and diffusion theory. The equivalent of Newton s equations for continuous media is an Eulerian transport equation of the form... 

The Eulerian transport equations can be cast in the generalized form ... 

The general form of the Eulerian transport equation for species c mass, can the be formulated as ... 

The balance principle applied to an Eulerian CV is thus by definition expressed as a balance of accumulation, net inflow by convection and diffusion and volumetric production. The Eulerian transport equations can be cast in the generalized form ... 

Equation (12.43) is called an Eulerian approach because the behavior of the species is described relative to a fixed coordinate system. The equation can also be considered to be a transport equation for particles when they are... 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

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... 

The PDF transport equations and closures considered up to this point have been Eulerian, e g-, -,(V. iJk x. / ) for fixed x and t. The large number of independent variables (e.g.,... 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

B B. By correctly choosing the coefficient matrices (a u, a 7, By, and B ), (6.159) can be made to correspond with the Eulerian velocity, composition PDF transport equation (6.19). However, it is important to note that /L / Thus it remains to determine how the Lagrangian notional-particle PDF /,( is related to the Eulerian velocity, composition PDF fn,0. This can be done by considering Lagrangian fluid particles. 

From the Eulerian composition transport equation, two types of processes can be identified. 

In summary, DQMOM is a numerical method for solving the Eulerian joint PDF transport equation using standard numerical algorithms (e.g., finite-difference or finite-volume codes). The method works by forcing the lower-order moments to agree with the corresponding transport equations. For unbounded joint PDFs, DQMOM can be applied... 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

In the Eulerian approach (Jankowski et al., 1996 Lou and Ridd, 1997 Holt and James, 1999 Ribbe and Holloway, 2001 Christiansen et al., 2002), a transport equation is solved for the sediment concentration. Recent studies (Kuhrts et al., 2004, 2006 Seifert et al., 2007) describe moving sedimentary material as a tracer variable in a three-dimensional circulation model. This approach combines a wave boundary layer (Grant and Madsen, 1979) and a friction layer (Smith and McLean, 1977) to compute the wave-induced and the current-induced contribution to the skin friction acting on the seabed. 

It is seen that the governing system equation (3.51) is transformed into a generic Eulerian control volume formulation (3.81) consisting of a volume integral determining the microscopic transport equations for the bulk phases and a surface integral determining the jump balance at the interface. 

A three dimensional extension of the Leibnitz rule for differentiating an integral is relevant for the derivation of the governing transport equations L In the material (Lagrangian) representation of continuum mechanics a representative particle of the continuum occupies a point in the initial configuration of the continuum at time t = 0 and has the position vector = (Ci, 2, Cs)-In this -space the coordinates are called the material coordinates. In the Eulerian representation the particle position vector in r-space is defined by r = (ri,r2,r3). The coordinates ri,r2,r3 which gives the current position of the particle are called the spatial coordinates. Let be any scalar, vector... 

DQMOM does not provide a model for the unclosed terms. It is thus a computational method for approximating an Eulerian joint PDF given its closed transport equation. In this sense, DQMOM is an alternative to the Eulerian Monte-Carlo simulations introduced in Chapter 7. 

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