Diffusion spherical geometry

With today s R R, spectophotometers make up only a small portion of total measurement error, assuming they are diffuse spherical geometry and halogen flash-lamp source units. If you have a good spectrophotometer, then the majority of test error originates from sampling and test preparation. 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case The transition from Equation (11.50) to Equation (11.51) seems to require the step that dVsIAi =d Vs/Ai] = dzs- This is not correct in general. Is the validity of Equation (11.51) hmited to situations where Ai is actually constant ... 

There are, however, obvious limitations. It is not possible to make a very small spherical electrode, because the leads that connect it to the circuit must be even much smaller lest they disturb the spherical geometry. Small disc or ring electrodes are more practicable, and have similar properties, but the mathematics becomes involved. Still, numerical and approximate explicit solutions for the current due to an electrochemical reaction at such electrodes have been obtained, and can be used for the evaluation of experimental data. In practice, ring electrodes with a radius of the order of 1 fxm can be fabricated, and rate constants of the order of a few cm s 1 be measured by recording currents in the steady state. The rate constants are obtained numerically by comparing the actual current with the diffusion-limited current. 

Apparently, the most important consequence of the dSS approach is the simplification of the expressions for the flux. /]n, as compared with the semiinfinite diffusion case. Indeed, for a given c, the steady-state flux in spherical geometry is [45] ... 

This situation, as discussed in the last section, closely resembles that of the droplet diffusion flame, in which the oxygen concentration approaches zero at the flame front. Now, however, the flame front is at the particle surface and there is no fuel volatility. Of course, the droplet flame discussed earlier had a specified spherical geometry and was in a quiescent atmosphere. Thus, hD must contain the transfer number term because the surface regresses and the carbon oxide formed will diffuse away from the surface. For the diffusion-controlled case, however, one need not proceed through the conductance hD, as the system developed earlier is superior. 

Careful readers might notice that diffusion in garnet is three-dimensional with spherical geometry, and should not be treated as one-dimensional diffusion. Section 5.3.2.1 addresses this concern. 

Example 3.5 Consider three-dimensional diffusion in a solid sphere of radius a with spherical geometry (meaning concentration depends only on r with 0 < r < a). The initial and boundary conditions are... 

SLIDER is a Fortran IV computer program for investigating the diffusion of a single fission-product isotope in a multilayered spherical fuel particle. This code enables one to compute, on the basis of Fick s law of diffusion, the transient and steady-state fission product concentrations and releases in multilayered spherical geometry. 

The analysis of the ZLC desorption curve involves solving Fickian diffusion equation with appropriate initial and boundary conditions (9). The solution of the desorption curve for spherical geometry is given as ... 

This allows one to draw two conclusions. First, the theorem of the impossibility of a dynamo with vn = 0 for arbitrary magnetic Reynolds number Rm is valid only for plane and spherical geometries. Second, the increment of a dynamo which is possible in other geometries will be determined by the magnetic diffusion and, consequently, such dynamos are effective only for small magnetic Reynolds numbers. 

Rajagopalan and Luss (1979) developed a theoretical model to predict the influence of pore properties on the demetallation activity and on the deactivation behavior. In this model the change in restricted diffusion with decreasing pore size was included. Catalysts with slab and spherical geometry composed of nonintersecting pores with uniform radius but variable pore lengths were assumed. The conservation equation for diffusion and first-order reaction in a single pore of radius rp is... 

FIGURE 17.2 Dissolution pro Jes of ne (open circles) and coarse ( lied circles) hydrocortisone (Lu et al., 1993). Simulated curves were drawn using spherical geometry without (a) and with (b) a time-dependent diffusion-layer thickness and cylindrical geometry without (c) and with (d) a time-dependent diffusion-layer thickness. Error bars represent 95% 6I=(= 3). (Reprinted from Lu, A. T. K., Frisella, M. E., and Johnson, K. C. (1993pharm. Res., 10 1308-1314. Copyright 1993. With permission from Kluwer Academic Plenum Publishers.)... 

If tj - 1 the reaction is not, or not significantly, influenced by pore diffusion. If tj pore diffusion is the sole dominating rate-limiting step. For the determination of Tj, the combined diffusion and reaction equation has to be solved. With a sequential model of the two rate phenomena, diffusion and reaction, and with the assumption of spherical geometry and validity of the Michaelis-Menten equation for the en2yme kinetics, r = kcat[E] [S]/(JCM + [S]), Eq. (5.58) results. 

Special attention should be paid to spherical geometry, since the mathematical treatment of spherical microelectrodes is the simplest and exemplifies very well the attainment of the steady state observed at microelectrodes of more complex shapes. Indeed, spherical or hemispherical microelectrodes, although difficult to manufacture, are the paragon of mathematical model for diffusion at microelectrodes, to the point that the behavior of other geometries is always compared against them. 

For spherical geometry when diffusion coefficients of species O and R are identical, the following analytical and explicit expression for the concentration profiles of species O and R can be obtained,... 

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. 

In practice, the amount of solid molecules on the surface being exposed to the solution is difficult or even impossible to quantify. Instead, the solid surface area to solution volume ratio is often used to quantify the amount of solid reactant. Therefore, experimentally determined second-order rate constants for interfacial reactions have the unit m s h As the true surface area of the solid is very difficult to determine, the BET (Brunauer-Emmett-Teller) surface area is fte-quentiy used. The maximum diffusion-controlled rate constant for a particle suspension containing pm-sized particles is ca 10 m s and for mm-sized particle suspensions the corresponding value is I0 m s h Unfortunately, the discrepancy between the true surface area and the BET surface area and the non-spherical geometry of the solid particles makes it impossible to exactly determine the theoretical diffusion-controlled rate constant. 

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... 

For diffusion in cylindrical or spherical geometries the one-dimensional, steady-state form of Eq. 1.3.10 simplifies to... 

Assuming that the autoxidation of S(1V) in the pores of the porous hybrids is isothermal, we can write the convective-diffusion equation in spherical geometry for Oa depletion as... 

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