Radial node Volume

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

A10 - BIO Cell A10 contains the name dr, which refers to the nondimensional radial dimension between radial nodes (i.e., the control-volume width). Cell BIO computes the value of dr as =1 / (npoints - 1). 

If the variation of the solid diffusion coefficient with Uthium concentration is significant, then the diffusion equation is nonlinear and the above simplification does not apply. For an electrode composed of spherical particles, a pseudo-two dimensional approach is required, in which the radial diffusion equation (Equation 17) is solved at each mesh point across the porous electrode. A set of radial nodes is then required to compute the radial solid concentration profile at each linear position in the electrode. Note that Eiquation 17 is derived using the gradient in chemical potential, and assumes only that volume changes are negligible and that aU current is carried by electrons in the solid phase. The chemical diffusion coefficient, D used in Equation 17 is related to the binary diffusion coefficient derived from the Stefan-MaxweU equations, V, (also aGled the binary interaction parameter), by the relationship presented earlier (Equation 13) for concentrated solutions ... 

The spatial domain is divided into discrete volumes defined by a mesh. The values of the independent variable, f are given at the mesh points, or nodes, by fj. The value of the dependent variable w in the volume surrounding the node is presumed to be represented by the value at the node, wj. The volume surrounding each node extends midway to the neighboring node that is, the radial extent of the volume extends from fj-1/2 to 77+1/2, where fj+i/2 = fj + /7+1). 

To approximate scalar grid cell variables at the staggered w-velocity grid cell center node point, arithmetic interpolation is frequently used. The radial velocity component is discretized in the staggered t -grid cell volume and need to be interpolated to the w-grid cell center node point. The derivatives of the w-velocity component is approximated by a central difference scheme. When needed, arithmetic interpolation is used for the velocity components as well. 

An important distinction arises here between Cartesian and spherical coordinates the probability density can be inconsistently defined if we re not cautious. The probability density of a wavefunction R r) is k(r), but this gives the probability per unit volume of the particle being at some particular value of r. Often, however, we are not interested in any specific direction from the nucleus, and we want to know the density at some value of r added up over all the angles. In that case, the function we want is the radial probability density, equal to R i rYr because the volume sampled by the radial wave-function increases, like the surface area of a sphere, as r. Looking at it another way, when we integrate over the probability density, we add a factor of in the volume element. That is the same factor we include when we write the radial probability density. As shown in Fig. 3.11, the nodes that appear in... 

Radial probability functions of the hydrogen atom, (The factor is included because the radial volume element is r dr.) The number of nodes increases with both n and 1. 

WSRC has developed a special code, FLOWTRAN-TF, based on the conservation of mass, energy, and momentum to account for two-phase flow, heat transfer effects, and cross-rib gap flows in assembly subchannels. The heat conduction models developed for FLOWTRAN-FI have been incorporated in FLOWTRAN-TF. Each subchannel coolant node has radially adjacent fuel surface temperature nodes to accommodate the heat transfer in the cell.. Rib fin effects are also handled in the same manner as they are in FLOWTRAN-FI. In order to initiate the computation, an air void fraction must be assigned to the computational cell above the fuel. This is done by assuming an air void volume in the first (top) axial cell as adjusted by an experimentally determined partitioning factor. Results from FLOWTRAN-TF have been shown to be relatively insensitive to the value assigned. Two-phase flow across the ribs is modeled by the application of an assumed partition factor based on values given in the literature. 

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