Finite volume elements

Ewing, R., Lazaeov, R., Lin, T., Lin, Y., Mortar finite volume element approximations of second order elliptic problems, East-West J. Numer. Math. 8, 2 (2000) 93-110. 

For a finite volume element with a Gaussian intensity distribution in three dimensions, the autocorrelation can be written as... 

A more rigorous approach, however, is to directly compute the interaction between the transition densities. This was first done numerically by discretizing the transition densities into finite volume elements of a 3D grid in the transition density cube (TDC) method [84]... 

Consider an arbitrary three-dimensional enclosure of total volume V and surface area A which confines an absorbing-emitting medium (gas). Let the enclosure be subdivided (zoned) into M finite surface area and N finite volume elements, each small enough that all such zones are substantially isothermal. The mathematical development in this section is restricted by the following conditions and/or assumptions ... 

In the corresponding equation for the mass transfer rate of component i, calculated by Eq. 6.27, the film transfer coefficient kf,im is based on the overall surface area dAs of the adsorbent of all particles (Eq. 6.20) in the finite volume element (Eq. 6.28) ... 

A three dimensional turbulent flow field in unbaffled tank with turbine stirrer or 6-paddle stirrer was numerically simulated by the method of finite volume elements [80], whereas in the case of free surface the vortex profile was also determined using iterative techniques. The prediction of the velocity and turbulence fields in the whole tank and the stirrer power was compared with literature data and their own results. Of the two simulation techniques used, turbulent eddy-viscosity/zc-e turbulence model and the DS model (differential 2. order shear stress), only the latter produced satisfactory results. In particular it proved that fluctuating Coriolis forces have to be taken into account by source terms in the transport equation for the Reynolds shear stress. 

A reactor is now considered through which A is continnonsly passed and the prodncts continnonsly withdrawn with no accnmnlation. The simplest case is the plng-flow reactor (PFR) in which elements of the flnid move nnidirectionally as plugs, with no feedback from the downstream to the upstream side. Since the composition changes with distance, the material balance will hold only for a finite volume element dV with a conversion of dX and gives... 

To apply this equation to the finite volume elements, within which it is assumed that the phytoplankton concentration is uniform, it is necessary to average this reduction factor throughout the depth of the volume element Vj. Let Hoj and Hxj be the depths of the surface and bottom, respectively, of the volume element Vj. For example, if the volume element Vj extends from the water surface to the bottom of the water body, then H0j = 0 and is the water depth at the location of Vj. For the sake of simplicity, it is assumed that this is the case. If Hoj 0, a straightforward generalization of the following average is required. 

This equation gives the change of concentration in a finite volume element with time. In the approach of Barrer and Jost, the diffusivity is assumed to be isotropic throughout the crystal, as Dt is independent of the direction in which the particles diffuse. Assuming spherical particles. Pick s second law can be readily solved in radial coordinates. As a result, all information about the exact shape and connectivity of the pore structure is lost, and only reflected by the value of the diffusion constant. 

The Reynolds equations, the continuity equation, which is turned into an equation for pressure correction [10], and the transport equations for the turbulence quantities k and e, are integrated over the respective finite volume elements resulting from the discretization of the stirred tank domain. The convection and diffusion terms in the transport equations are approximated using the hybrid-scheme of Patankar [10]. The resulting algebraic equations are then solved with the aid of the commercial CFD software PHOENICS (Version 2.1). 

More technically, it is the integral of the square of the wavefunction over a finite volume element that is observable, or... 

The equation of continuity is derived by writing a mass balance over an arbitrary finite volume element AxAyAz through which a fluid is flowing, as is shown in Figure 6.1. We then take the limit as the volume element goes to zero to get the appropriate equation for any point in the spatial domain. 

Within a finite volume element given by the Eulerian simulation of the extracellular liquid phase, the velocities of the liquid phase in each spatial coordinate can be linearly approximated, which allows one to obtain an analytical solution of Eq. (3.21). The random movement caused by turbulent dispersion is superimposed on the convective flow represented by the velocity field v. In the Lagrangian... 

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