Mass transfer equation spherical coordinates

Locally Flat Description of Boundary Layer Mass Transfer in Spherical Coordinates. Since the radial coordinate r does not change much as one moves from the fluid-solid interface to the outer edge of the mass transfer boundary layer, it is acceptable to replace r by in the two-dimensional mass transfer equation and the equation of continuity ... 

The only assumption is that the physical properties of the fluid (i.e p and A.mix) are constant. The left-hand side of equation (11-1) represents convective mass transfer in three coordinate directions, and diffusion is accounted for via three terms on the right side. If the mass balance is written in dimensionless form, then the mass transfer Peclet number appears as a coefficient on the left-hand side. Basic information for dimensional molar density Ca will be developed before dimensionless quantities are introduced. In spherical coordinates, the concentration profile CA(r,6,4>) must satisfy the following partial differential equation (PDE) ... 

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 mass transfer equation could have been extracted from the rectangular coordinate entry in Table B.ll of Bird et al. (2002, p. 851) by setting = vg and Vy = Vr. Except for the factor of sin 0, the equation of continuity could have been extracted from the rectangular coordinate entry in Table B.4 of Bird et al. (2002, p. 846) by setting = ve and Vy = Vr. Hence, the locally flat description of this problem is justified, and the only remaining influence of spherical coordinates is the factor of sin 9 in the equation of continuity. The boundary conditions for Ca(y, x) are repeated here for completeness ... 

Use one sentence and qualitatively explain why factors of 1/r and appear in mass transfer equation B.10-3 in Bird et al. (2002, p. 850) for radial transport in spherical coordinates. 

In spherical coordinates, the dimensional mass transfer equation with radial diffusion and first-order irreversible chemical reaction exhibits an analytical solution for the molar density profile of reactant A. If the kinetics are not zeroth-order or first-order, then the methodology exists to find the best pseudo-first-order rate constant to match the actual rate law and obtain an approximate analytical solution. The concentration profile of reactant A in the liquid phase must satisfy... 

The expanded form of the one-dimensional mass transfer equation with radial diffusion and simple first-order kinetics in spherical coordinates, which is eqnivalent to (17-23),... 

Obvionsly, these requirements are satisfied by the mass transfer equation in spherical coordinates, given by (17-36). The parameter e governs whether the solution is given by ... 

Catalysts with Spherical Symmetry. This analysis is based on the mass transfer equation with diffusion and chemical reaction in spherical catalysts. For zeroth-order kinetics, the molar density of reactant A is equated to zero at the critical value of the dimensionless radial coordinate, iciiticai = /(A). The relation between the critical value of the dimensionless radial coordinate and the intrapellet Damkohler number is obtained by solving the following nonlinear algebraic equation ... 

In cylindrical and spherical coordinates, the diffusion term in the mass transfer equation includes a factor of 1/r when it is expanded. This cannot be evaluated numerically at r = 0, which corresponds to the center of the catalyst. 

Combining eqs. (4.1) and (4.2), and for spherical particles, the following diffusion equation, written in spherical coordinates (r), describes the mass transfer process ... 

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... 

The radial variable r is dimensionalized to isolate the Damkohler number in the mass balance. It is important to emphasize that dimensional analysis on the radial coordinate must be performed after implementing the canonical transformation from Ca to iJia- If the surface area factors of and 1/r are written in terms of as defined by equation (13-9), prior to introducing the canonical transformation given by equation (13-4), then the mass transfer problem external to the spherical interface retains variable coefficients. If diffusion and chemical reaction are considered inside the gas bubble, then the order in which the canonical transformation and dimensional analysis are performed is unimportant. Hence,... 

With no mass-transfer resistance in the continuous phase and an assurrption that the concentration of styrene at the surface of particles is negligible to C (initial value of C ) at any instance, an analytical solution for is derived fron the Fickean diffusional equation for spherical coordinates. ... 

A (hemi)spherical electrode has been applied to estimation of charge transfer kinetics, effect of electric migration, and kinetics of chemical steps. Spherical diffusion takes place through an increasing area as the radial coordinate r, measured from the center of the spherical electrode, increases. The main differential equation for three-dimensional mass transport is given in Table 3.1. On substitution of c=y r, this relationship can be reduced to the one-dimensional mass transport yielding... 

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