Diffusion 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... 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

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... 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

Although the foregoing example in Sec. 4.2.1 is based on a linear coordinate system, the methods apply equally to other systems, represented by cylindrical and spherical coordinates. An example of diffusion in a spherical coordinate system is provided by simulation example BEAD. Here the only additional complication in the basic modelling approach is the need to describe the geometry of the system, in terms of the changing area for diffusional flow through the bead. 

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... 

This is Fick s second law of diffusion, the equation that forms the basis for most mathematical models of diffusion processes. The simple form of the equation shown above is applicable only to diffusion in one dimension (x) in systems of rectangular geometry. The mathematical form of the equation becomes more complex when diffusion is allowed to occur in more than one dimension or when the relationship is expressed in cylindrical or spherical coordinate geometries. Since the simple form shown above is itself a second-order partial differential equation, the threat of added complexity is an unpleasant proposition at best. 

Equation (1) is the one-dimensional form of Fick s first law in Cartesian coordinates. In cylindrical and spherical coordinates, the form of Fick s first law for radial diffusion is... 

Diffusion in a sphere may be more common than that in a cylinder in the pharmaceutical sciences. The example we may think of is the dissolution of a spherical particle. Since convection is normally involved in solute particle dissolution in reality, the dissolution rate estimated by considering only diffusion often underestimates experimental values. Nevertheless, we use it as an example to illustrate the solution of the differential equations describing diffusion in the spherical coordinate system [1],... 

A sphere is assumed to be a poorly soluble solute particle and therefore to have a constant radius rQ. However, the solid solute quickly dissolves, so the concentration on the surface of the sphere is equal to its solubility. Also, we assume we have a large volume of dissolution medium so that the bulk concentration is very low compared to the solubility (sink condition). The diffusion equation for a constant diffusion coefficient in a spherical coordinate system is... 

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... 

Model calculations have demonstrated that active cells are surrounded by zones containing substrate concentrations lower than those of the bulk liquid [12-14], This concentration gradient results from the dynamic interplay between the rates of substrate uptake and diffusion through the diffusion layer surrounding the cell (see [15] for details). Boone et al. [13] developed a model using spherical coordinates that allows calculation of the diffusive substrate flux to a suspended spherical cell. In their model calculations, the cell surface concentration was set to arbitrary values between zero and about half of the bulk concentration. It... 

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

A3.3 Three-dimensional diffusion using spherical coordinates with constant D... 

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 ... 

Guy et al. [5] derived an equation for the diffusion-controlled release of a drug from a sphere, radius r, by applying the three-dimensional form of Fick s second law of diffusion after transformation to spherical coordinates. This equation can be rearranged as ... 

Diffusion Through a Stagnant Fluid (Spherical Coordinates). 200... 

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... 

Diffusion into a sphere represents a three-dimensional situation thus we have to use the three-dimensional version of Fick s second law (Box 18.3, Eq. 1). However, as mentioned before, by replacing the Cartesian coordinates x,y,z by spherical coordinates the situation becomes one-dimensional again. Eq. 3 of Box 18.3 represents one special solution to a spherically symmetric diffusion provided that the diffusion coefficient is constant and does not depend on the direction along which diffusion takes place (isotropic diffusion). Note that diffusion into solids is not always isotropic, chiefly due to layering within the solid medium. The boundary conditions of the problem posed in Fig. 18.6 requires that C is held constant on the surface of the sphere defined by the radius ra. 

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 ... 

Fick s second law defines the behavior of a diffusing chemical in space over time. Fick s second law is derived from Fick s first law and the equation of continuity for a solute. For simplicity, we derive Fick s second law in 1-D coordinates. This can readily be extended to multiple dimensions or to spherical coordinates [22]. 

Boundary value problems in cylindrical and spherical coordinates have an inherent singularity at x = 0. These problems can be tackled using Maple s inbuilt midpoint methods. For example, diffusion of a substrate in an enzyme catalyzed reaction.[6] The governing equation for the dimensionless concentration is... 

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, < >)... 

Figure 9.6. Idealized view of (a) spherical gas bubbles in a liquid, (h) liquid droplets in a gas, and (c) cylindrical gas jets in a liquid. Diffusion in bubbles, drops, and jets may be modeled by solving the diffusion equations for cylindrical and spherical coordinates.

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