Conservation equations control volume formulation

This work has developed a novel numerical procedure to solve the reduced forms of the Navler-Stokes and continuity equations simultaneously. The technique is based upon a control volume formulation with a staggered grid. It is capable of considering a fully three-dimensional variation of fluid properties, while maintaining mass conservation within the plane of the fluid film for any level of solution dlscret izat ion. 

In finite-volume methods, the integral formulation of the conservation laws over a small physical control volume is discretized directly. FVM employs a conservative discretization, that is, each species is guaranteed to be conserved, even for coarse meshes. In contrast, many traditional FDMs are not conservative. For example, owing to the nonlinear nature of the constitutive flux equations of ionic species in an a priori unknown electric field, FDM is nonconservative, even when constant physical... 

The two-fluid formulation requires computationally demanding iteration between the two fluids to satisfy the boundary conditions between them. Instead, a single-fluid approach can be used, in which one set of equations is solved in the whole domain making use of appropriate physical properties of each phase. The conservation of total momentum for the body of fluid in control volume Q can be written as ... 

Formulation of a proper reactor model and incorporation of kinetic expressions for rs and rx- Using the fundamental law of conservation of mass (cf. Equs. 4.76 and 4.101), the general partial differential equation of S concentration in a control volume is... 

The mass balance equation, also referred to as the equation of continuity, is simply a formulation of the principles of the conservation of mass. The principle states that the rate of mass accumulation in a control volume equals the mass flow rate into the control volume minus the mass flow rate out of the control volume. In Cartesian coordinates (x, y, z), the mass balance equation for a pure fluid can be written as ... 

It is remarked that in the standard literature on fluid dynamics and transport phenomena three different modeling frameworks, which are named in a physical notation rather than in mathematical terms, have been followed formulating the single phase balance equations [91]. These are (1) The infinitesimal particle approach [2, 3, 67, 91, 145]. In this case a differential cubical fluid particle is considered as it moves through space relative to some fixed coordinate system. By applying the balance principle to this Lagrangian control volume the conservation equations for... 

Therefore, as basis we formulate the generalized form of the local instantaneous transport equation for the mixture mass, component mass, momentum, energy and entropy in a fixed control volume (CV) on microscopic scales, as illustrated in Eig. 1.1. This approach thus consider the conservation law of the quantity within the fixed macroscopic volume V bounded by the fixed macroscopic surface area A ... 

In the analysis that follows, the particle phase of a fluidized bed is treated in some respects as though it were a continuum, or fluid. Thus the term two-fluid model is sometimes applied to this and related formulations. A differential control volume. Figure 7.1, is defined, from which equations specifying conservation of mass and momentum for the one-dimensional vertical flow of the fluid and particle phases may be written. 

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