Convective-Diffusion-Length Model

The quantitative and qualitative analysis presented in Section 20.2.1 demonstrates that the finite-diffusion-layer model provides an inadequate representation for the impedance response associated with a rotating disk electrode. The presentation in Section 20.2.2 demonstrates that a generic measurement model, while not providing a physical interpretation of the disk system, can provide an adequate representation of the data. Thus, an improved mathematical model can be developed. 

The one-term convective-diffusion model consisting of the first term in equation (11.97) 

The three-term convective-diffusion model provides the most accurate solution to the one-dimensional convective-diffusion equation for a rotating disk electrode. The one-dimensional convective-diffusion equation applies strictly, however, to the mass-transfer-limited plateau where the concentration of the mass-transfer-limiting species at the surface can be assumed to be both uniform and equal to zero. As described elsewhere, the concentration of reacting species is not uniform along the disk surface for currents below the mass-transfer-limited current, and the resulting nonuniform convective transport to the disk influences the impedance response.  

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As described in Sections 20.2.1 and 20.2.2, the quality of the regressions can be assessed to varying degrees of success by inspection of plots. The Nyquist or complex-impedance-plane representation given in Figure 20.13 reveals the difference between the finite-diffusion-length model and the models based on numerical solution of the convective-diffusion equation, but cannot be used to distinguish the models based on one-term, two-term, and three-term expansions. 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

For simplicity, we present here a two-dimensional, isothermal version of the convective-diffusive lattice-gas model, appropriate for liquid and vapor phases of a single-species system in a microcapillary. Consider a rectangular slab A oi N = Lx X Ly sites r = (x, y) on a square lattice, with lattice constant a and unit vectors ei,2 = + 1,0)a, and 63,4 = (0) 1) - The boundary layers and B2 ait y = 0 and y = (Ly — l)a are adjacent to solid walls Wi and W2 ait y = — a and y = Ly a, respectively. We will assume periodic boundary conditions in the y direction. Sites are assigned spin variables S, = + 1, representing occupancy by a single particle species of mass n, or a vacancy at site r, respectively. Furthermore, assume that this closed microcapillary system between the two walls contains a fixed number of particles, and that the system is isothermal, as if each site v/ere in contact with a heat reservoir at temperature T. Let us define a fundamental timestep At = z. At any given time, we will assume a velocity cUf, defined at each site, where c = a/i is a unit velocity, and u, is a dimensionless velocity field measured in fractions of the unit velocity. For the remainder of this paper, length and time will be expressed in units of a and x, respectively. 

In diffusion-controlled adsorption models, one assumes that there is no activation energy barrier to the transfer of surfactant molecules between the subsurface and the surface [85]. Thus diffusion is the only mechanism needed in establishing adsorption equilibrium. The time required for the molecules to transfer from the bulk to the subsurface is much longer than the time required for equilibration between the surface and the subsurface. On the contrary, if the adsorption or desorption rate at the interface is slow or comparable to the diffusion rate, the adsorption process is significant. This model is called the mixed-kinetic adsorption model. This condition may depend not only on the properties of the system but also on the diffusion length and possibly on convection conditions. The diffusion-controlled model of Eqs. (3) and (4) have been given by Fainerman et al. [86,87]. 

A fundamental difference exists between the assumptions of the homogeneous and porous membrane models. For the homogeneous models, it is assumed that the membrane is nonporous, that is, transport takes place between the interstitial spaces of the polymer chains or polymer nodules, usually by diffusion. For the porous models, it is assumed that transport takes place through pores that mn the length of the membrane barrier layer. As a result, transport can occur by both diffusion and convection through the pores. Whereas both conceptual models have had some success in predicting RO separations, the question of whether an RO membrane is truly homogeneous, ie, has no pores, or is porous, is still a point of debate. No available technique can definitively answer this question. Two models, one nonporous and diffusion-based, the other pore-based, are discussed herein. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

Step 3. Transport within a catalyst pore is usually modeled as a one-dimensional diffusion process. The pore is assumed to be straight and to have length The concentration inside the pore is ai =ai(l,r,z) where I is the position inside the pore measured from the external surface of the catalyst particle. See Figure 10.2. There is no convection inside the pore, and the diameter of the pore is assumed to be so small that there are no concentration gradients in the radial direction. The governing equation is an ODE. 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

The reaction plane model with heterogeneous reactions was discussed at length for acid-base reactions in the previous section. The same modeling technique, of confining the reactions to planes, can be applied to micelle-facilitated dissolution. As with the acid-base model, one starts with a one-dimensional steady-state equation for mass transfer that includes diffusion, convection, and reaction. This equation is then applied to the individual species i, i.e., the solute, s, the micelle, m, and the drug-loaded micelle, sm, to yield... 

The effectiveness of the internal O2 transport by diffusion or convection depends on the physical resistance to movement and on the O2 demand. The physical resistance is a function of the cross-sectional area for transport, the tortuosity of the pore space, and the path length. The O2 demand is a function of rates of respiration in root tissues and rates of loss of O2 to the soil where it is consumed in chemical and microbial reactions. The O2 budget of the root therefore depends on the simultaneous operation of several linked processes and these have been analysed by mathematical modelling (reviewed by Armstrong... 

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