Models radial diffusive

To avoid unduly complicating the model, radial diffusion within the root is not allowed for. Equation (6.1) therefore reduces to ... 

The temperature counterpart of Q>aVR ccj-F/R and if ccj-F/R is low enough, then the reactor will be adiabatic. Since aj 3>a, the situation of an adiabatic, laminar flow reactor is rare. Should it occur, then T i, will be the same in the small and large reactors, and blind scaleup is possible. More commonly, ari/R wiU be so large that radial diffusion of heat will be significant in the small reactor. The extent of radial diffusion will lessen upon scaleup, leading to the possibility of thermal runaway. If model-based scaleup predicts a reasonable outcome, go for it. Otherwise, consider scaling in series or parallel. 

Hence, the macroscopic model of Bond and Hill reinterpreted the data in Figure 3.96 in terms of a cross-over between the limiting forms of mass transport, i.e. linear to radial diffusion, and not in terms of a slowing down in the heterogeneous kinetics. 

This problem, in which a radiogenic element is allowed to leak out of its host mineral as it forms, has found important applications in geochronology, particularly for the K-Ar method (Wasserburg, 1954) and the U-Pb method (Tilton, 1960 Wasserburg, 1963) with the so-called continuous loss model. The equation for radial diffusion of a radiogenic element in a sphere with radius a and uniform parent isotope concentration P = P0 at t = 0 can be written... 

Sorption can require more than a month to reach equilibrium for highly hydrophobic compounds, but can be adequately described by a radial diffusion model accompanied by the retarding influence of sorption. 

This discrepancy is consistent with a radial diffusion model since is proportional to ar (Equation 39), the time required to reach equilibrium should increase with the square of the particle diameter. Clearly if the same mixing time were used for both fine-grained and coarse-grained sorbents, the coarser material could have been far from equilibrium. 

Wu and Gschwend (78) successfully employed a radial diffusion model to describe laboratory observed sorption and desorption kinetics. Their data show that sorption and desorption rates were slower for more hydrophobic compounds and sorbents with a larger grain size in a manner consistent with the radial diffusion model. 

The parabolic diffusion model is used to indicate that diffusion controlled phenomena are rate limiting. It was originally derived based on radial diffusion in a cylinder where the chemical compound concentration on the cylindrical surface was constant, and initially the chemical compound concentration throughout the cylinder was uniform. It was also assumed that the diffusion of the compound of interest through the upper and lower faces of the cylinder was negligible. Following Crank [119], the parabolic diffusion model can be expressed as ... 

In the model, the internal structure of the root is described as three concentric cylinders corresponding to the central stele, the cortex and the wall layers. Diffu-sivities and respiration rates differ in the different tissues. The model allows for the axial diffusion of O2 through the cortical gas spaces, radial diffusion into the root tissues, and simultaneous consumption in respiration and loss to the soil. A steady state is assumed, in which the flux of O2 across the root base equals the net consumption in root respiration and loss to the soil. This is realistic because root elongation is in general slow compared with gas transport. The basic equation is... 

The processes of convection, axial diffusion, radial diffusion, and chemical reaction in the liquid and tissue layers all occur simultaneously. A rigorous approach requires solution of several simultaneous differential equations. To avoid this complexity in preliminary models, the transfer... 

The model also predicts an increase in uptake as tidal volume increases over a constant breathing period. As the breathing period increases at a constant tidal volume, the uptake also increases. In the former case, increased ventilation of peripheral airways with a high surface volume ratio increases uptake. In the latter, the period for radial diffusion is increased in every segment. 

Rate equation analyses for classical size exclusion chromatography have been based on treating the porous matrix as a homogeneous, spherical medium within which radial diffusion of the macromolecular solute takes place (e.g. (28,30,31)) or If mobile phase lateral dispersion Is considered Important, a two dimensional channel has been used as a model for the bed (32). In either case, however, no treatment of the effects to be expected with charged Brownian solute particles has been presented. As a... 

When a tube or pipe is long enough and the fluid is not very viscous, then the dispersion or tanks-in-series model can be used to represent the flow in these vessels. For a viscous fluid, one has laminar flow with its characteristic parabolic velocity profile. Also, because of the high viscosity there is but slight radial diffusion between faster and slower fluid elements. In the extreme we have the pure convection model. This assumes that each element of fluid slides past its neighbor with no interaction by molecular diffusion. Thus the spread in residence times is caused only by velocity variations. This flow is shown in Fig. 15.1. This chapter deals with this model. 

Strommen, M. R., and R. M. Kamens, Development and Application of a Dual-Impedance Radial Diffusion Model to Simulate the Partitioning of Semi-Volatile Organic Compounds in Combustion Aerosols, Environ. Sci. Technol., 31, 2983-2990 (1997). 

We will then compare the dynamics of the radial diffusion model with a first-order exchange model which gives the same half-life as the radial model (Eq. 18-37). A preview of this comparison is given in Fig. 18.7b, which shows that the linear model underpredicts the exchange at short times and overpredicts it at long times. 

The radial diffusion model can be approximated by a linear uptake model of the form ... 

The radial diffusion model and the linear sorption model are compared in Fig. 18.76. Since according to Eq. 19-76 the total mass of the chemical associated with the particle aggregate, (M(r), and the macroscopic solid-water distribution ratio, Kd(t), are linearly related ... 

The absolute value of ZFlin has its maximum at r = 0. Note from Fig. 18.76 that ZF of the radial diffusion model is even larger than ZF)in. 

We consider a spherical particle aggregate with radius r0 surrounded by a concentric boundary layer of thickness 8 (Fig. 19.16). Transport into the aggregate is described by the linear approximation of the radial diffusion model. Thus, the total flux from the particle to the fluid is given by Eq. 19-85 ... 

Figure 19.19 Comparison of the solution of the linear sorption model with the radial diffusion model. Numbers on curves show y defined in Eq. 19-86. Y is the fraction of the chemical taken up by the sphere when equilibrium is reached. After Wu and Gschwend (1988).

Explain the difference between the two types of curves shown in Fig. 19.19, which are labeled radial diffusion model and sorption model, respectively. 

In the RI model, all incident rays intersect at the center axis of the reactor tube, and Eq. 68 produces an infinite value of irradiance as r - 0. The DI model, on the other hand, proposes parallel layers of rays which are wider than the diameter of the tubular reactor and which traverse the reactor perpendicularly to its axis from all directions with equal probability. The calculated results of both models are far from reality, as found in industrial size photochemical reactors. Matsuura and Smith [107] proposed an intermediate model (PDI model, partially diffuse model, Figure 25b) in which parallel layers of rays are assumed, and the width of each is smaller than the diameter of the tubular reactor. These two-dimensional bands form by themselves radial arrangements, the center ray of each band intersecting the... 

If the radial diffusion or radial eddy transport mechanisms considered above are insufficient to smear out any radial concentration differences, then the simple dispersed plug-flow model becomes inadequate to describe the system. It is then necessary to develop a mathematical model for simultaneous radial and axial dispersion incorporating both radial and axial dispersion coefficients. This is especially important for fixed bed catalytic reactors and packed beds generally (see Volume 2, Chapter 4). 

Rounds, S.A. and J.F. Pankow. 1990. Application of a radial diffusion model to describe gas-particle sorption kinetics. Environ. Sci. Technol. 24 1378-1386. 

Steinberg et al. (1987) studied the persistence of 1,2-dibromoethane (EDB) in soils and found that low amounts of the organic were released with time, particularly if EDB had not been freshly added to the soil (Fig. 6.3). They suggested that the slow release rate was due to EDB being trapped in soil micropores where release is influenced by extreme tortuosity and/or steric restrictions. It was estimated that based on a radial diffusion model, 23 and 31 years would be required for a 50% equilibrium in EDB release to occur from two Connecticut soils. The previous studies point out that while sorption of pesticides is usually rapid and often reversible in the laboratory, extraction from field soils is extremely slow and often requires multiple extractions or even chemical dissolution of the soil matrix. 

Figure 9.6 Experimental and model-fitting results for tetrachlorobenzene and pentachlo-robenzene sorption kinetics on Iowa soils using the retarded/radial diffusion model of Wu and Gschwend (1986) Cs is the dissolved concentration in the bulk solution, and C0 and Ceq are the concentrations initially and at equilibrium, respectively. [From Wu and Gschwend (1986), with permission.]...

A 2D model requires a significant computational effort. If the radial diffusion (N r = 0) can be considered negligible the balance equations are much simpler the set of equations is of PDEs, ID of the second order. Thus, the computational requirement is also reduced. 

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