Radial diffusion spherical coordinates

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... 

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... 

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... 

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... 

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 ... 

The following assumptions were made in formulating this model 1) there is no solute adsorption to the stationary phase, 2) the porous particles which form the stationary phase are of uniform size and contain pores of identical size, 3) there are no interactions between solute molecules, 4) the mobile phase is treated as a continuous phase, 5) the intrapore diffusivity, the dispersion coefficient and the equilibrium partition coefficient are independent of concentration. The mobile phase concentration. Cm, is defined as the mass (or moles) per interstitial volume and is a function of the axial coordinate z and the angular coordinate 0. The stationary phase concentration, Cs, is defined as the mass per pore volume and depends on z, 6 and the radial coordinate, r, of a spherical coordinate system whose origin is at the center of one of the particles. 

There are three possibilities corresponding to the dimension of the distribution. The first is a ID concentration distribution (d = 1), in which the diffusing species spreads evenly in the z directions from an initial line pulse at z = 0 on the xz plane. In this case, the variable r in (6-37) is the Cartesian variable z. The second case is a circularly symmetric distribution for c (d = 2), which evolves by diffusion on a plane from an initial compact planar pulse. In this case, r in (6 37) is the radial component of a polar (or cylindrical) coordinate system that lies in the diffusion plane. The third case is a spherically symmetric distribution corresponding to d = 3, which evolves at long times from a compact 3D pulse that diffuses outward into the frill 3D space. In this case, r is the radial variable of a spherical coordinate system. To obtain the long-time form of the distribution we must solve (6-37), but subject to the integral constraint that the total amount of the diffusing species is constant, independent of time ... 

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... 

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

Radial Velocity Profile. The equation of continuity is employed to calculate Vr for one-dimensional flow in spherical coordinates. Incompressibility is a reasonable assumption because the fluid density is not expected to change much as oxygen, for example, diffuses across the gas-Uquid interface. If vq and are negligible, one must solve... 

Answer For boundary layer mass transfer in an incompressible liquid that contacts a zero-shear interface, a previous example problem on pages 311 and 312 reveals that the relative importance of the second term on the right side of the spherical coordinate expression for radial diffusion,... 

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... 

Now the mass transfer eqnation for I a(t) with radial diffusion and chemical reaction exhibits a flat description in spherical coordinates ... 

The homogeneous diffusion model in spherical coordinates accounts for the fact that the surface area across which radial diffusion occurs increases qnadratically with dimensionless coordinate rj as one moves radially outward from the center of a spherically shaped catalyst. Once again, basic information for I a = /(i ) is obtained by integrating the dimensionless mass balance for reactant A with radial diffusion and chemical reaction... 

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),... 

In spherical coordinate system, a point z is specified hy the radial distance r from the origin, a polar angle 0, and an azimuthal angle 4>. For spherically symmetric concentration distribution there is no angular dependence and the diffusion equation becomes... 

Consider a spherical particle of radius a such as a protein or micelle, toward which small molecules or ligands diffuse (see Figure 18.6). At what rate do the small molecules collide with the sphere To compute the rate, solve the diffusion Equation (18.10) in spherical coordinates (see page 313 and problem 2, page 314). At steady state dc/dt = 0. Assume that the flow depends only on the radial distance away from the sphere, and not on the angular variables. Then you can compute c(r) using... 

Dcoop is called the cooperative difiliision coefficient. Note, ffiough Eq. 29 maffi-ematically is exactly the radial term of die diffiision equation in spherical coordinates, this equation was physically not derived fiom diffusion ffieory. Typical values of Dcoop for neutral hydrogels are lO cm /s (Gehrice 1993), and typical values for polyelectrolyte hydrogels are (510. .. 10 )cm /s (Skouri et al. 1995 Milimouk et al. 2001). 

B.2, Here, we demonstrate once more how Brownian dynamics relates to diffusive behavior, by simulating spherical particles of radius 1 mm in water at room temperature. At time f = 0, a particle is released at the origin and undergoes 3-D Brownian motion. Write a program that repeats this simulation many times and plots the radial concentration profile of particles as a function of time. It is easier to do the data analysis if you do the simulations concurrently. Then, solve the corresponding time-dependent diffusion equation in spherical coordinates and compare the results to that obtained fi om Brownian dynamics. 

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

The microparticle diffusion is treated in the same way as in the solid-core model, and it is assumed that each of these microparticles grows independent of each other according to the existing local monomer concentration. To write the mole balance for the monomer in the macroparticle in spherical coordinates, let us define Di as the effective diffusion coefficient for the macroparticle, ri as the radial length, and R Mi, as the rate of consumption of monomer at The governing equation for the macroparticle can be easily derived as... 

Figure 4. The following data were used leak radius r0 (40 micron), leak conductance F = 1 cc. sec.-1, gas pressure (air) p (20 torr), diffusion coefficient (26) for ions Di — 2 sq. cm. sec.-"1, for electrons De = 2000 sq. cm. sec.-1 (at 20 torr air). The flow velocities were assumed independent of 6 and (spherical polar coordinates). The spherically symmetrical flow pattern, which obviously overestimates the flow velocity for large values of 6 was chosen because of its simplicity. The velocity of the radially directed flow at a distance r is v = F/2irr2. The time re-...

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 geometry of the microelectrodes is critically important not only from the point of view of the mathematical treatment, but also their performance. Thus, the diffusion equations for spherical microelectrodes can be solved exactly because the radial coordinates for this electrode can be reduced to the point at r = 0. On the other hand, a microelectrode with any other geometry does not have a closed mathematical solution. It would be advantageous if a microdisc electrode, which is easier to fabricate, would behave identically to a microsphere electrode. This is not so, because the center of the disc is less accessible to the diffusing electroactive species than its periphery. As a result, the current density at this electrode is nonuniform. 

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... 

By using Vg, Sg, and p, Wheeler replaced the complex porous pellet with an assembly (having a porosity e) of cylindrical pores of radius a. To p e- diet Dg from the model the only other property necessary is the length of the diffusion path. If we assume that, on the average, the pore makes an angle of 45° with the coordinate r in the resultant direction of diffusion (for example, the radial direction in a spherical pellet), =. Jl r. Owing to pore interconnections and noncylindrical shape, this value of is not very satisfactory. Hence, it is customary to define in terms of an adjustable parameter, the tortuosity factor 5, as follows ... 

Here m is proportional to the volume of a spherical shell of liquid between the bubble wall and the radial position coordinate r, and hence, represents a Lagrangian coordinate, providing that a negligible volume of liquid is vaporized. U is therefore a measure of the heat content of the spherical shell, to within an arbitrary additive function of time, K(t) alternatively, it may be viewed as a temperature potential function. In terms of the new coordinates the diffusion equation becomes... 

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,... 

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 ... 

This equation gives the change of concentration in a finite volume element with time. In the approach of Barrer and Jost, the diffusivity is assumed to be isotropic throughout the crystal, as Dt is independent of the direction in which the particles diffuse. Assuming spherical particles. Pick s second law can be readily solved in radial coordinates. As a result, all information about the exact shape and connectivity of the pore structure is lost, and only reflected by the value of the diffusion constant. 

After passing through the film, solute then diffuses in the pores by normal diffusion (large pores), Knudsen diffusion (small pores), or surface diffusion. In polymer resins where there are no permanent pores, the solute diffuses in the polymer phase. For spherical particles with a radial coordinate r, the diffusion equation in pores is... 

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. 

Two concentric spherical metallic shells of radii a and b cm (a < b) are separated by a solid of thermal diffusivity a (cm /s). The outer surface of the inner shell is maintained at Tq°C and the inner surface of the outer shell at Ti°C. Derive the differential equatitm governing the unsteady state temperature distribution in the solid as a function of time and radial coordinate. 

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