Finite transient diffusion spherical

As an alternative to the previous example, we can also solve the problems with inhomogeneous boundary conditions by direct application of the finite integral transform, without the necessity of homogenizing the boundary conditions. To demonstrate this, we consider the following transient diffusion and reaction problem for a catalyst particle of either slab, cylindrical, or spherical shape. The dimensionless mass balance equations in a catalyst particle with a first order... 

Transient Finite (Symmetric) Spherical Diffusion So far, we have only examined ID (Cartesian) examples of Fick s second law. Solving Fick s second law in alternative coordinate systems (e.g., for radial, spherical, 2D, or 3D problems) is not really any different. As an example, we examine here the case of transient finite spherical diffusion, which is essentially analogous to the transient finite planar diffusion problem that we just finished discussing. 

Pick s second law is a second-order partial differential equation. Solving it in order to predict transient diffusion processes can be fairly straightforward or quite complex, depending on the specific situation. In this chapter, analytical solutions were discussed for a number of cases, including ID transient infinite and semi-infinite diffusion, ID transient finite planar diffusion, and transient spherical finite diffusion as summarized in Table 4.4. In all cases, solution of Pick s second law requires the specification of a number of boundary conditions and initial conditions. 

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

Consider the transient finite spherical diffusion problem illustrated in Figure 4.15, which describes the diffusion of H2 into a spherical particle. 

FIGURE 4.15 Transient finite diffusion of Hj into a spherical particle. The initial concentration of Hj in the particle is zero [Cj (r,r = 0) = 0]. At time t = 0, the particle is exposed to a sufficiently high hydrogen gas pressure to fix the Hj concentration on the exterior surface of the particle at the solid-solubility limit, given by. As time evolves, hydrogen gradually diffuses into the particle as shown. 

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