## State of the

The first predictive equation of the dynamic behavior of the filters was proposed by Bohart and Adams in 1920 . In their work, Bohart and Adams  developed an equation based on a "chemical kinetic" model type, that is, assuming that the adsorption kinetic was the controlling step of the process (instantaneous diffusion), for the case of irreversible isotherm and neglecting the accumulation on the fluid phase. The obtained equation can be expressed as :

Ce exp

where C is the solute concentration at the axial position z, CE is the solute concentration at the bed inlet, k1 is the adsorption kinetic constant, t is the time, We is the adsorbent capacity, p is the bed density and u0 is the superficial velocity.

Since then several authors have developed equations for the prediction of the breakthrough time (tb) of the filters , namely the Danby equation , the Klotz equation , the Wheeler-Jonas equation [6, 7] and the Yoon-Nelson equation [8, 9] given respectively by:

kD CE

WeW We pb .( CE - Cb ) tb =—e---e—b~ln —E-b (4)

QCE kWJCE

QCE kYNCEQ

In these equations L and A represent the bed length and cross section area, Cb is the breakthrough concentration, Q is the volumetric flowrate, a and W are the adsorbent superficial area and weight. Re and Sc are respectively the Reynolds and Schmidt numbers.

The Danby, Wheeler-Jonas and Yoon-Nelson equations are equivalent and can be obtained directly from the Bohart and Adams equation assuming that exp

and merely modifying the definition of each respective adsorption kinetic constant.

The equation known as the Klotz equation is empirical and is based on the work of Mecklenburg. This author developed an equation assuming that until the breakthrough time, the solute accumulation on the fluid phase is negligible, that is, that all the solute entering the bed is retained on the adsorbent. Through this hypothesis he defined a new variable designated critical bed depth . Klotz  reintroduced the Mecklenburg equation using the Gamson, Thodos and Hougen correlation for the calculation of the critical bed depth .

From all these equations, the most frequently employed in the correlation of filter breakthrough times obtained experimentally is the Wheeler-Jonas equation [11-18].

In the derivation of the above mentioned equations it was always considered that the toxic compound is removed only by physical adsorption. However, it is known that for the case of some compounds with low molecular weight, such as those with military interest (HCN, CNCl, etc.), an efficient elimination can not be achieved with only physical adsorption. The adsorbent (activated carbon) is then normally impregnated with metals such as copper, chromium and silver that react with the toxic compounds after adsorption  enhancing the filter protective capacity.

The first models that consider systems where adsorption is coupled with reaction were developed in the scope of chromatograph reactors [20-22]. Afterwards, Schweich et al.  studied these systems through an equilibrium model with linear adsorption and instantaneous reversible equation.

Loureiro et al. [24, 25], using also an equilibrium model analyzed a system with Langmuir adsorption and irreversible reaction of several orders.

Friday  developed the first model with coupled adsorption and reaction for application on individual protection filters, and more specific for the system ASC activated carbon/cyanogen chloride. This author considered isothermal operation, plug flow, external (film) and internal (LDF) resistances and Langmuir adsorption followed by irreversible second-order reaction between the adsorbed CNCl and the impregnated metals.

A similar study was performed by Graceffo et al.  but, contrarily to Friday, these authors consider that the impregnated metals do not react with the adsorbate but act as a catalyst that deactivates with time. The reaction rate is expressed as function of the fluid phase CNCl concentration and the Freundlich isotherm is used to represent the adsorption equilibrium.

Chatterjee and Tien [28, 29] proposed a model in which they assume that the reaction between CNCl and the impregnated metals is instantaneous and only after the metals are totally consumed is the solute adsorbed. As the toxic compound is being removed, the adsorbent particle is seen as being composed of two distinct zones. In the first one, the metal has already been fully used-up and adsorption is taking place while the second zone is completely free from adsorbate which is reacting at the frontier. The radial position of this frontier decreases with time (shrinking core model).

Friday's model was later modified by Soares et al.  in order to include the effect of axial dispersion. The authors concluded that axial dispersion affects significantly the breakthrough time of the filters and should not be neglected. The effect of axial dispersion is important not only due to the shallow bed length, but also because the threshold toxic concentration is very low. The simulation results showed that this parameter should be carefully evaluated.

The models were then further extended by Ribeiro and Loureiro  that considered different filter geometries, that is, the fixed bed operating in either axial or radial (inner and outer) flow. Mathematical models of increasing complexity were solved for each geometry and the influence of the dimensionless parameters on the breakthrough time assessed. It was shown that the filter breakthrough time is significantly dependent on the model considered and on the different parameters values. Comparing the results between geometries they concluded that the one with axial flow was better, as it was with this one that longer breakthrough times were attained in the simulations with the models that best represent reality. The same authors validated the model results with experimental results . Due to the high risks associated with the compounds that are eliminated by the impregnated activated carbons by physical adsorption followed by chemical reaction no experimental data was obtained for such systems. Instead, the validation was done using results taken from the literature. They observed that, using an optimized value for the reaction kinetic constant, the prediction of the experimental breakthrough times was globally reasonable. Experimental data was then obtained for systems where only adsorption occurs and used to validate the mathematical models. They chose as adsorbents three activated carbons (BPL, ASC and ASC-Teda) and as adsorbates a volatile organic compound, methyl tert-butyl ether (MTBE), and water vapor. This last adsorbate was also studied as it is usually present in common use of individual protection filters either pre-adsorbed and/or as humidity in the in flow stream. The results obtained with MTBE in the BPL were reasonably simulated by the more complex model. The breakthrough curves determined in the impregnated carbons suggested the presence of a higher internal resistance in this kind of carbons. The simulations that took into account this effect reproduced better these experimental results. The water vapor adsorption tests point to the presence of an increasing internal resistance at larger concentrations, even higher than that considered with the inclusion of the adsorption isotherm derivative in the calculation of the homogeneous effective diffusivity. It was verified that simulations which consider a lower value for the effective diffusivity reproduce satisfactorily all the experimental results of sequential saturation at higher concentrations and even the 0-80% tests.

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