Infinite adsorption rate

Statement of the results in the case of an infinite adsorption rate 10... 

Table 16 Comparison between the volume concentrations, c ° and (1/H) c dz for the case of an infinite adsorption rate k = +oo at the time f = 5,755 s...

Figure 10 Case of an infinite adsorption rate k = +oo Comparison between concentration obtained using our effective problem (eff), average of the section of the concentration from the original problem (pbreel3) and the concentration coming from the simple average (moy) at t = 863 s.

This treatment, however, cannot give product concentration since the model employs over-simplified assumptions such as infinite adsorption rate. More detailed nonequilibrium models must be employed in order to obtain relations between product concentration or yield and operating conditions. 

Here we concentrate our attention to the case when the adsorption rate constant k is infinitely large. 

According to their analysis, if is zero (practically much lower than 1), then the liquid-film diffusion controls the process rate, while if tfis infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the so-called mechanical parameter represents the ratio of the diffusion resistances (solid and liquid film). The authors did not refer to any assumption concerning the type of isotherm for the derivation of the above-mentioned criterion it is sufficient to be favorable (not only rectangular). They noted that for >1.6, the particle diffusion is more significant, whereas if < 0.14, the external mass transfer controls the adsorption rate. 

Figure 3. Infinite-bath rates of adsorption for hydrochloric acid...

Figure 14. Infinite-bath rate of adsorption for hydrochloric acid at 50°C.

Equation [2.8.2] can be used to estimate the order of magnitude of diffusion-controlled initial adsorption. From [2.8.2] we have 1 = 0](r/c)VD]. For typical values of r = 0( 10" mol m ), c = 0( 10" M = 0.1 mol m ). and D = O(10" m s, see table 1.6.4), it is found that = 0(0.1 s). Later in the process the adsorption rate decreases because the surface is no longer a perfect "sink" the gradient in concentration, which drives the process, goes down and the diffusion layer becomes thicker, see [1.6.5.27]. Although in principle it tEikes an infinite time to attain saturation, in practice monolayer completion may be expected over seconds to tens of seconds, because in practice transport to the surface is enhanced by convection. Maintaining quiescent conditions over long times is experimentally difficult. 

Equation 6.87 shows that the lumped mass transfer coefficient in the solid fihn driving force model is related to the film mass transfer coefficient, the intraparticle diffusion, and the adsorption rate constant. This equation is a more complete form of Eq. 6.57b. It reduces to that equation when the adsorption-desorption kinetics is infinitely fast. 

For an isotherm with a Langmuir shape, if the column is initially loaded at some low concentration, c q, (ciow = 0 if the column is clean) and is fed with a fluid of a higher concentration, C] (see Figure 18-IZA), the result will be a shock wave. The feed step in adsorption processes usually results in shock waves. Experiments show that when a shock wave is predicted the zone spreading is constant regardless of the column length (a constant pattern wave). With the assunptions of the solute movement theory (infinitely fast rates of mass transfer and no axial dispersion), the wave becomes infinitely sharp (a shock) and the derivative dq/dc does not exist. Thus, the Aq/Ac term in the denominator of Eq. f 18-141... 

In tiying to model dynamic adsorption mathematically and predict the shape and size of the MTZ, assumptions must be made about the equilibrium model, molecular diffusion, axial-flow dispersion, and isothermality. The simplest model is isothermal equilibrium operation with infinite diffusion rate and negligible dispersion, that is, a stoichiometric front. 

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

The BET approach is essentially an extension of the Langmuir approach. Van der Waals forces are regarded as the dominant forces, and the adsorption of all layers is regarded as physical, not chemical. One sets the rates of adsorption and desorption equal to one another, as in the Langmuir case in addition, one requires that the rates of adsorption and desorption be identical for each and every molecular layer. That is, the rate of condensation on the bare surface is equal to the rate of evaporation of molecules in the first layer. The rate of evaporation from the second layer is equal to the rate of condensation on top of the first layer, etc. One then sums over the layers to determine the total amount of adsorbed material. The derivation also assumes that the heat of adsorption of each layer other than the first is equal to the heat of condensation of the bulk adsorbate material (i.e., van der Waals forces of the adsorbent are transmitted only to the first layer). If it is assumed that a very large or effectively infinite number of layers can be adsorbed, the following result is arrived at after a number of relatively elementary mathematical operations... 

Brunauer, Emmett, and Teller extended the Langmuir theory to multimolecular layer adsorption [8]. They related the condensation rate of gas molecules onto an adsorbed layer and the evaporation rate from that layer for an infinite number of layers. The linear form of the relationship is called the BET equation ... 

Figure 12. Modeling and measurement of oxygen surface diffusion on Pt. (a) Model I adsorbed oxygen remains in equilibrium with the gas along the gas-exposed Pt surface but must diffuse along the Pt/YSZ interface to reach an active site for reduction. Model II adsorbed oxygen is reduced at the TPB but must diffuse there from the gas-exposed Pt surface, which becomes depleted of oxygen near the TPB due to a finite rate of adsorption, (b) Cotrell plot of current at a porous Pt electrode at 600 °C and = 10 atm vs time. The linear dependence of current with at short times implies semi-infinite diffusion, which is shown by the authors to be consistent only with Model II. (Reprinted with permission from ref 63. Copyright 1990 Electrochemical Society, Inc.)...

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... 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

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