Adsorption diffusion equation

The last two assumptions are the most critical and are probably violated under field conditions. Smith et al. (3) found that at least a half-hour was required to achieve adsorption equilibrium between a chemical in the soil water and on the soil solids. Solution of the diffusion equation has shown that many volatile compounds have theoretical diffusion half-lives in the soil of several hours. Under actual field conditions, the time required to achieve adsorption equilibrium will retard diffusion, and diffusion half-lives in the soil will be longer than predicted. Numerous studies have reported material bound irreversibly to soils, which would cause apparent diffusion half-lives in the field to be longer than predicted. 

For modeling of the electrode reaction coupled with adsorption of both forms of the redox couple (2.146), the diffusion equations (1.2) and (1.3) have to be solved for conditions given by (2.148) to (2.152) completed with the following boundary conditions ... 

If a gas such as ammonia or CO2 (phase 1) is absorbing into a liquid solvent (phase 2), the resistance R2 is relatively important in controlling the rate of adsorption. This is also true of the desorption of a gas from solution into the gas phase. Usually R2 is of the order 10 or 10 sec. cm. h though the exact value is a function of the hydrodynamics of the system consequently various hydrodynamic conditions give a variety of equations relating R2 to the Reynolds number and other physical variables in the system. For the simplest system where the liquid is infinite in extent and completely stagnant, one can solve the diffusion equation... 

Sorption relates to a compound sticking to the surface of a particle. Adsorption relates to the process of compound attachment to a particle surface, and desorption relates to the process of detachment. Example 2.2 was on a soluble, nonsorptive spiU that occurred into the ground and eventually entered the groundwater. We will now review sorption processes because there are many compounds that are sorptive and subject to spills. Then, we can examine the solutions of the diffusion equation as they apply to highly sorptive compounds. 

Let us assign Sp to be the mass of chemical sorbed to particles per mass of solids contained in our control volume and C to be the concentration of the compound in solution. Then, our source term in the diffusion equation is equal to the rate of change of mass due to adsorption and desorption per unit volume, or... 

In groundwater and soil pollution problems, there is sometimes discussion of fast sorption and slow sorption, where the local equilibrium assumption would not be valid. How would you formulate a diffusion equation to deal with both the fast and slow forms of adsorption and desorption ... 

It should be noted here that while in catalytic systems the rate is based on the moles disappearing from the fluid phase - dddt, and the rate has the form ( —ru) = f(k, C), in adsorption and ion exchange the rate is normally based on the moles accumulated in the solid phase and the rate is expressed per unit mass of the sohd phase dqldt where q is in moles per unit mass of the solid phase (solid loading). Then, the rate is expressed in the form of a partial differential diffusion equation. For spherical particles, mass transport can be described by a diffusion equation, written in spherical coordinates r ... 

Exchange of trace components The equations for adsorption (diffusion) can be equally applied in the case of isotopic exchange (exchange of isotopes) with minor changes. The same equations can be also be used in the case of the exchange of trace components of different valences (Helfferich, 1962). This is the case where the uptake or release of an ion takes place in the presence of a large amount of another ion in both the solid and liquid phase. In such systems, the amounts removed ate so small that the concentrations in both phases are practically constant, and thus in turn the individual diffusion coefficients also remain unaffected. Moreover, the rate-controlling step is the diffusion of the trace ion. 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

Limiting cases of the general convective-diffusion equation are often helpful. If the time dependence is ignored, i.e., 5C/0t = 0 (for example, at low bulk protein concentration, at long times, and/or when the rate of adsorption is much greater than the transport to the surface), then we have... 

When the Bom, double-layer, and van der Waals forces act over distances that are short compared to the diffusion boundary-layer thickness, and when the e forces form an energy hairier, the adsorption and desorption rates may be calculated by lumping the effect of the interactions into a boundary condition on the usual ccm-vective-diffusion equation. This condition takes the form of a first-order, reversible reaction on the collector s surface. The apparent rate constants and equilibrium collector capacity are explicitly related to the interaction profile and are shown to have the Arrhenius form. They do not depend on the collector geometry or flow pattern. 

To determine the net rale of adsorption of particles suspended in a fluid that is flowing over the collector, one may then solve the usual convective-diffusion equation subject to a reversible first-order reaction as the boundary condition, provided the diffusion boundary layer is much thicker than the interaction boundary layer. 

As an example, suppose that a fluid containing particles is in contact with a rotating disk. What will be the net rate of adsorption onto the collector disk Levich (1962) neglected radial variations and solved the usual convective-diffusion equation, taking the concentration as cf at (he disk surface and as cB far away from the disk. If c< is eliminated from his solution by imposing Equation (11) at this surface, with ] = —D dc/dy =B, the result for the net adsorption rate will be... 

The experimental method used in TEOM for diffusion measurements in zeolites is similar to the uptake and chromatographic methods (i.e., a step change or a pulse injection in the feed is made and the response curve is recorded). It is recommended to operate with dilute systems and low zeolite loadings. For an isothermal system when the uptake rate is influenced by intracrystalline diffusion, with only a small concentration gradient in the adsorbed phase (constant diffusivity), solutions of the transient diffusion equation for various geometries have been given (ii). Adsorption and diffusion of o-xylene, / -xylene, and toluene in HZSM-5 were found to be described well by a one-dimensional model for diffusion in a slab geometry, represented by Eq. (7) (72) ... 

The effect of using the sodium adsorption isotherm (equation 6) to determine the apparent diffusion coefficient can be seen by... 

The problem of mass transfer from a moving Newtonian fluid to a swarm of prolate and/or oblate stationary spheroidal adsorbing particles under creeping flow conditions is solved using a spheroidal-in-cell model. The flow field through the swarm was obtained by using the spheroid-in-cell model proposed by Dassios et al. [5]. An adsorption - 1st order reaction - desorption scheme is used as boundary condition upon the surface of the spheroid in order to describe the interaction between the diluted mass in the bulk phase and the solid surface. The convective diffusion equation is solved analytically for the case of high Peclet numbers where the adsorption rate is also obtained analytically. For the case of low Pe a non-... 

Also the choice of the electrostatic model for the interpretation of primary surface charging plays a key role in the modeling of specific adsorption. It is generally believed that the specific adsorption occurs at the distance from the surface shorter than the closest approach of the ions of inert electrolyte. In this respect only the electric potential in the inner part of the interfacial region is used in the modeling of specific adsorption. The surface potential can be estimated from Nernst equation, but this approach was seldom used In studies of specific adsorption. Diffuse layer model offers one well defined electrostatic position for specific adsorption, namely the surface potential calculated in this model can be used as the potential experienced by specifically adsorbed ions. The Stern model and TLM offer two different electrostatic positions each, namely, the specific adsorption of ions can be assumed to occur at the surface or in the -plane. 

Due to the effects of molecular size and shape and pore structure on the kinetics, the model cannot be used for general predictive purposes. In practice, in order to predict PAC adsorption, a series of experiments must first be carried out using the compound of interest, the activated carbon to be applied, and the water in which it is to be used. Equilibrium parameters, determined from the Freundlich adsorption isotherm equation, are used as input into a computer-based HSDM, which uses the method of least squares to minimize the difference between the experimental kinetic data points and the HSDM fit of the data [10]. When the best fit is achieved, the resultant kinetic parameters (liquid film mass transfer coefficient, k(, and the surface diffusion coefficient, DJ can then be used for the prediction of adsorption behavior under different conditions. 

Equation 2.20 is the advection-dispersion (AD) equation. In the petroleum literature, the term convection-diffusion (CD) equation is used, or simply diffusion equation (Brigham, 1974). When a reaction term is included, the term advection-reaction-dispersion (ARD) equation is used elsewhere. When the adsorption term is expressed as a reaction term, the ARD equation is as discussed later in Section 2.4. Several solutions of Eq. 2.20 have been presented in the literature, depending on the boundary conditions imposed. In general, they are various combinations of the error function. When the porous medium is long compared with the length of the mixed zone, they all give virtually identical results. 

We have previously written an expression for j n in Eq. (2-150), but this expression is in terms of the local bulk concentration evaluated at the interface, c, and thus to determine c we would need to solve bulk-phase transport equations. We will not pursue that subject here. However, when we use this material to solve flow problems, we will consider several cases for which it is not necessary to solve the full convection-diffusion equation for c. We will see that the concentration of surfactant tends to become nonuniform in the presence of flow -i.e., when u n and u v are nonzero at the interface. This tendency is counteracted by surface diffusion. When mass transfer of surfactant to and from the bulk fluids is added, this will often tend to act as an additional mechanism for maintenance of a uniform concentration T. This is because the rate of desorption from the interface will tend to be largest where T is largest, and the rate of adsorption largest where T is smallest. 

The present study reports the measurements of intracrystalline diffusion and adsorption equilibrium for ethanol, propanols and butanols from aqueous solution in silicalite using a modified HPLC technique. The unique feature of the present work is the use of a mathematical model with a nonlinear adsorption isotherm equation to obtain the intracrystalline diffusivity and adsorption isotherm parameters. The adsorption equilibrium data for alcohols from aqueous solution in silicalite measured by the conventional batch method are also reported and compared with the results measured by the HPLC technique. 

In more complicated models both equations have to be generalised by coupling surface and bulk convective diffusion and hydrodynamics. The situation is finely balanced since the motion of the surface has an effect on the formation of the dynamic adsorption layer, and vice versa. Adsorption increases in the direction of the liquid motion while the surface tension decreases. This results in the appearance of forces directed against the flow and retards the surface motion. Thus, the dynamic layer theory should be based on the common solution of the diffusion equation, which takes into account the effect of surface motion on adsorption-desorption processes and of hydrodynamics equations combined with the effect of adsorption layers on the liquid interfacial motion (Levich 1962). 

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