Diffusion in spherical coordinates

Let us consider the solution of an unsteady diffusion problem in spherical coordinates 

Because of the last boundary condition, we have symmetrj at x = 0 and the Jacobi polynomials will be used with a = 3 because of the spherical geometry. We will use the weighting of = 1. 

The linear set of ordinary differential equations is easily solved using standard MATLAB ordinary diflFerential equation subroutines. 

As a convenient way to evaluate the accuracy of the solution, we will focus on the mass flux at the surface x = 1. The mass flux is given by 

Experience has shown that low-order weighted residual approximations work well when the solution varies smoothly over the entire domain of the trial function. When there are rapid changes over a varying portion of the domain, then the method requires high-order approximations. In that case, the finite-difference approach is more effective. Convective-diffusion problems are particularly difficult for the w eighted residual method, whereas pure conduction problems can be treated very efficiently by this approach. 

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The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

The rate of heterogeneous condensation depends on the exchange of matter and heat between a particle and the continuous phase. The extreme cases of a particle much larger or much smaller than the mean free path of the suspending gas are easy to analyze. In the continuum range (dp ip), diffusion theory can be used to calculate the transport rate. For a single sphere in an infinite medium, the steady-state equation of diffusion in spherical coordinates takes the form... 

One must consider unsteady-state radial diffusion in spherical coordinates with no chemical reaction. Tangential diffusion in the polar coordinate direction 6 is neglected. 

C.1 Dissolution of a Bubble into a Molten Polymer Matrix. Consider a molten polymer matrix with dispersed bubbles of radius = 10 pm. Calculate the dissolution time of the bubble, tdis, using Pick s second law of diffusion in spherical coordinates and moving (shrinking) boundaries. As the gas (species A) diffuses out the radius of the bubble decreases. Also,... 

For one-dimensional diffusion in Cartesian coordinates For radial diffusion in cylindrical coordinates For radial diffusion in spherical coordinates... 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

Unsteady-State Heat Conduction and Diffusion in Spherical and Cylindrical Coordinates... 

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

Analysis of drug transport in a solid tumor compartment could be represented in spherical coordinates as well as cylindrical [19] as depicted in Eq. (56). In this case, and assuming that drug diffusion occurs only in the radial direction, Eq. (53) can be written as... 

The H2O diffusion equation in spherical coordinates is as follows (Equation 4-92) ... 

It should be noted here that while in catalytic systems the rate is based on the moles disappearing from the fluid phase - dddt, and the rate has the form ( —ru) = f(k, C), in adsorption and ion exchange the rate is normally based on the moles accumulated in the solid phase and the rate is expressed per unit mass of the sohd phase dqldt where q is in moles per unit mass of the solid phase (solid loading). Then, the rate is expressed in the form of a partial differential diffusion equation. For spherical particles, mass transport can be described by a diffusion equation, written in spherical coordinates r ... 

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 ... 

In problems such as the drying of droplets or diffusion through films around spherical catalyst pellets, it is more convenient to use Eqs. (40b) and (49) in spherical coordinates. Then for steady state diffusion in the radial direction alone, one has in the absence of chemical reactions... 

Hence, assuming that the diffusion coefficient of the species considered is constant over a particular layer, one may write, in spherical coordinates, the following diffusion equation for that layer ... 

Equation 1 is given in spherical coordinates, thus assuming a spherical shape for the carbon particle, an assumption which accords reasonably well with microscopic observations of the geometry of particles of the experimental carbon. In Equation 1, C represents the H30+ activity in solution t, time r, the radial distance from the particle center D, the diffusion coefficient and S, the H30+ concentration at the surface of the carbon. For the present experiments, the equilibrium relationship between S and C is described in terms of the Freundlich expression... 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

Spheres. Consider a 5-rich sphere of /3 phase of radius R = R(t) growing in an infinite a matrix under diffusion-limited conditions as shown in Fig. 20.6. This problem can be solved by using the scaling method with r) defined by rj = r/ ADat)1/2. The diffusion equation in the a phase in spherical coordinates in rt-space (see Eq. 5.14) becomes, after transformation into 77-space,... 

This sensor uses cylindrical microelectrode geometry (Fig. 7.14) for which the diffusion-reaction reaction is written in spherical coordinates, similar to (2.24). 

We shall assume that our system is spherically symmetric so with the nabla operator in spherical coordinates, the diffusion equation may be written... 

The limiting step in the kinetics of ion exchange in the zeolite is the interdiffusion of the electrolyte ions A zi and ions of the species B [24], In the case where the solid ion-exchanger particle is spherical (see Figure 7.9) and the particle diffusion control is the rate-determining process, then Fick s second law equation in spherical coordinates is [47]... 

There are certain practical diffusion problems, which can be treated most appropriately in cylindrical or in spherical coordinates. In many cases, choosing the natural coordinate system allows for the coordinates to be separated, and one is left with the simpler problem of dealing with one-dimensional diffusion along the radial coordinate. Basically, the only technical complication which arises as compared to the one-dimensional diffusion in Cartesian coordinates treated so far, concerns the approximation of the spatial derivative of the concentration involved by the diffusion equation. 

The radial diffusion equation in spherical coordinates may be written for constant diffusion coefficient as ... 

In this section we commence our analysis of the diffusion-limited reaction of solute A. As expounded in Section IILA, we initially restrict our considerations to the scenario wherein the concentration of the droplets can be construed as dilute. In such a case, it suffices to focus on the diffusion and reaction of A in the presence of a single fluctuating sink. In view of the spherical symmetry exhibited by the problem, we formulate the transport and reaction of A in terms of a diffusion equation expressed in spherical coordinates (r, 6, < >)... 

The convective diffusion equation in spherical coordinates has the form... 

The problem can be solved effectively by converting the convection-diffusion equation into the well studied heat conduction equation by introducing the stream fimction P as a new variable. In terms of the stream function the velocity components in spherical coordinates z and 0 are,... 

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