Rates of Adsorption into Adsorbent Particles

The isotherms given by Equations 11.1, 11.3, and 11.4, or other types of isotherms, can be used to calculate the equilibrium concentrations of adsorbates in fluid and solid phases in batch and fixed-bed adsorption processes discussed below. 

The apparent rates of adsorption into adsorbent particles usually involve the resistances for mass transfer of adsorbate across the fluid film around adsorbent particles and through the pores within particles. Adsorption perse at adsorption sites occurs very rapidly, and is not the rate-controlling step in most cases. 

we consider a case where an adsorbate in a liquid is adsorbed by adsorbent particles. If the mass transfer across the liquid film around the adsorbent particles is rate controlling, then the adsorption rate is given as 

In the case where the mass transfer within the pores of adsorbent particles is rate controlling, the driving force for adsorption based on the liquid phase is given 

In some cases, surface diffusion - that is, the diffusion of adsorbate molecules along the interface in the pores - may contribute substantially to the mass transfer of the adsorbate, and in such cases, the effective diffusivity may become much larger than the case with pore diffusion only. 

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The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

Relaxation studies have shown that the attachment of an ion to a surface is very fast, but the establishment of equilibrium in wel1-dispersed suspensions of colloidal particles is much slower. Adsorption of cations by hydrous oxides may approach equilibrium within a matter of minutes in some systems (39-40). However, cation and anion sorption processes often exhibit a rapid initial stage of adsorption that is followed by a much slower rate of uptake (24,41-43). Several studies of short-term isotopic exchange of phosphate ions between aqueous solutions and oxide surfaces have demonstrated that the kinetics of phosphate desorption are very slow (43-45). Numerous hypotheses have been suggested for this slow attainment of equilibrium including 1) the formation of binuclear complexes on the surface (44) 2) dynamic particle-particle interactions in which an adsorbing ion enhances contact adhesion between particles (43,45-46) 3) diffusion of ions into adsorbents (47) and 4) surface precipitation (48-50). 

The particle was divided into 30 to 100 shells for the calculation and Ficks Law was assumed to hold. Heat generated was taken as rate of adsorption times the heat of adsorption, and the latter quantity was assumed to be independent of amount adsorbed. With increasing tern-... 

We can put some realistic numbers into this equation to see how it would behave. We can take to be 0.4, which is a reasonable number for a packed bed of particles. The area per unit volume can be taken as 100 m per g ( 10 cm per g), the density of the solid is on the order of 1 g cm, and the number of sites per unit area N,-,s,tot is on the order of lO per cm, making Ci,s,toi 10 mole sites cm. (On a perfect surface there are 10 per atoms cm, so we have taken 10% of this value as the number of sites, which corresponds to one site in every 1 nm. Finally, the mass concentration of the adsorbate (if the latter is ideal) is. We use these numbers and a value of ka, which is one order of magnitude larger than kd. If the system were to come to equilibrium, then the mass uptake would go to zero. This would be the same when rate of adsorption is balanced exactly by the rate of desorption. We can compute the mass of i on the solid when this occurs as follows ... 

We have shown the analysis of a single zeolite crystal under isothermal conditions and non-isothermal conditions in Sections 10.2 and 10.3, respectively. These analyses are important to understand the rate of adsorption at the crystal level. In practice zeolite solids are available in pellet form, and these pellets are made by compressing zeolite crystals together, usually with a small percentage of binder to join the crystals together. Figure 10.4-1 shows schematically a typical zeolite pellet composed of many small zeolite crystals. These crystals are of the order of 0.1 to 1 micron, and the zeolite pellets are of the order of one millimeter. The void between the microparticles contributes to the mesopores and macropores of the particle. These pores act as conduit to transport molecules from the surrounding into the interior of the particle. Once inside the particle, molecules adsorb at the pore mouth of the micropores and thence the adsorbed species diffuse into the interior of the crystal. Micropores within the crystal provide the adsorption space to accommodate adsorbate molecules. 

We have discussed the behaviour of the bimodal solid with a linear isotherm. Now we discuss the other extreme of the isotherm, the irreversible isotherm. What we would expect in this case is that the concentration in the macropore behaves like a wave front, that is the adsorbed concentration in the region close to the pellet exterior is very close to the maximum concentration, while the region near the core is void of adsorbate in any form, either in free or adsorbed form. The position demarcating these two regions is the adsorption (wave) front position. How this wave front penetrates into the particle depends on the rate of macropore diffusion as well as the rate of diffusion into the micropore. 

Distribution of the Number of Macromolecules on the Particles In the early stage of the adsorption and/or for the low polymer concentration in solution, the particle surface coverage is relatively low. If the particle aggregation does not proceed, the collision rate of macromoleeules with dispersion particles is proportional to the product of the particle number and macromoleeule number in the imit voliune. Taking the above into account, the adsorption kinetics of monodisperse polymers on the particles and dynamics of particle number change with adsorbed polymer can be described in the first approximation by the following system of differential equations [66,67] ... 

This is our principal result for the rate of desorption from an adsorbate that remains in quasi-equihbrium throughout desorption. Noteworthy is the clear separation into a dynamic factor, the sticking coefficient S 6, T), and a thermodynamic factor involving single-particle partition functions and the chemical potential of the adsorbate. The sticking coefficient is a measure of the efficiency of energy transfer in adsorption. Since energy supply from the... 

The results of adsorption and desorption of CO mentioned above suggest that for the reaction at low temperature, the sites for relatively weakly chemisorbed CO are covered by the deposited carbon and the reaction occurs between molecularly adsorbed CO and oxygen on the carbon-free sites which are the sites for relatively strongly chemisorbed CO. Therefore, the definition of the turnover rate at 445 K remains as given in Equation 1. For the reaction at 518 K, however, this definition becomes inappropriate for the smaller particles. Indeed, to obtain the total number of Pd sites available for reaction, we now need to take into consideration the number Trp of CO molecules under the desorption peak. Furthermore, let us assume that disproportionation of CO takes place through reaction between two CO molecules adsorbed on two adjacent sites, and let us also assume that the coverage is unity for the CO molecules responsible for the LT desorption peak, since this was found to be approximately correct on 1.5 nm Pd on 1012 a-A O (1). Then, the number Np of palladium sites available for reaction at 518 K is given by HT/0 + NC0 LT s nce t ie co molecules under the LT desorption peak count only half of the available sites. Consequently, the turnover rate at 518 K should be defined as ... 

The general rate model of chromatography is the most complex of all the models used in this field. In this model, it is assumed that the mobile phase percolates through the interstitial volume between stationary phase particles, diffusion takes place from this stream into the particles and inside the pores of the stationary phase particles, where the mobile phase is stagnant, and adsorption-desorption takes place between the stagnant mobile phase within the pores and the adsorbent surface. 

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