Radial flux and spherical coordinates

This equation shows that, at constant growth rate, the more incompatible the elements, the longerrit takes for steady-state to establish. We therefore can expect kinetic disequilibrium beween mineral and liquid to be more conspicuous for incompatible than for compatible elements. 

In the case of flux with spherical symmetry, i.e., with no dependence on the latitude and longitude, gradient and Laplacian operators must be expressed as a function of the radial distance r to the origin 

Let us write the Laplacian explicitly as a sum of second-order derivatives relative to 

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The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

The flux of particles is in the opposite sense to the direction of the concentration gradient increase. Equation (6) is Fick s first law, which has been experimentally confirmed by many workers. D is the mutual diffusion coefficient (units of m2 s 1), equal to the sum of diffusion coefficients for both reactants, and for mobile solvents D 10 9 m2 s D = DA + jDb. The diffusion coefficient is approximately inversely dependent upon viscosity and is discussed in Sect. 6.9. As spherical symmetry is appropriate for the diffusion of B towards a spherically symmetric A reactant, the flux of B crossing a spherical surface of radius r is given by eqn. (6) where r is the radial coordinate. The total number of reactant B molecules crossing this surface, of area 4jrr2, per second is the particle current I... 

The kinetic term, v, is a function of the kinetic parameters vector P and the particle substrate and product concentrations, cs and cP, respectively. Ds and DP are the corresponding effective diffusion coefficients and r is the particle coordinate (in the case of spherical geometry it is the radial distance). Parameter n depends on the geometry of the biocatalyst particle and is 0,1,2 for a plate, a cylinder and a sphere, respectively. Since concentrations on the particle surface are assumed to be identical with bulk concentrations, boundary conditions do not include the influence of external mass transfer. Solving the above differential equations, the observed reaction rate in the packed bed is evaluated from the rate of substrate flux to the particle or of product flux from the particle... 

For three-dimensional (spherical) tumors, most investigators have considered tumors to be avascular and have used models developed for spheroids (e.g., McFadden and Kwok, 1988 Van Osdol et al., 1993). Theoretically, these models are analogous to the Krogh cylinder model, except for the boundary conditions and the use of spherical instead of radial coordinates. Therefore, they do not provide significant insights into the role of vasculature and transvascular flux. 

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