Tribochemical Purification of Gases. I. The Process Model

A mathematical model of the sorption process in a tribochemical reactor with a stirrer, where monolithic granules of alkali-earth metals mutually rub in the media of the flow gas, has been built. The products of reaction of the metal with the gas impurity are continuously removed mechanically from the surface of the granules, creating new metallic regions. As a result the total area of the sorbing surface appears to be by orders of magnitude larger than that of the traditional getter materials of the same mass in the existing sorption technologies. It follows from the model that the gas purification process obeys the linear law at a constant rotation speed of the stirrer. This dependence results in a great simplification of the process control. It also makes possible re-placement of a periodic mode of operation by a continuous one where the sorption capacity of the consumed chemisorbent attains theoretical limit.


Introduction
Powders, films or gas-permeable porous bodies of alkali-earth metal alloys are effective chemisorbents [1]- [6]. They react with active gases at room temperature and capture them in a form of nonvolatile chemical compounds. Not only the surface of the metal is available for these reactions with gases but its volume as well thus providing a great advantage of the alkali-earth metal alloys over adsorbents with respect to the specific sorption capacity [7] [8] [9]. To underline this peculiarity of the sorption behavior of the reactive alloys it was suggested to call them "reactants" and to single them out into a separate class of getter mate-

Reactive Sorber
The reactive sorber is a triboreactor for removal of impurities from a gas flow by cast granules of the getter reactant at stirring [11]. This sorption process differs from the usual arrangement of tribochemical reactions [12] [13] [14] [15]. Due to high chemical activity and small mechanical strength of alkali-earth metal reactants there is no need for subjecting reactant granules to impingement attack. In order to maintain a reaction between the metal and the gas at the required level, a slight mixing of the granules with a stirrer is enough.
In this relation we have to notice that while in conventional mechanochemistry one of the central questions of study is the question about quantitative relation between the reaction rate and the consumed power, in consideration of sorption gas purification technologies it is more convenient to shift the attention to the dependence of the form ( ) , where c is the concentration of gas impurity in the outlet product and t is the time of the tribochemical treatment.

The Problem Statement
The design of the sorber is shown in Figure 1.
The initial gas is fed from above into the triboreactor 1. Then it passes through the column of stirred with the stirrer 2 granules of reactant 4 and exits from the column in the purified form below the dividing mesh 5. The formed on the surface of the granules products of the reaction between the gas impurity and the metal fall at the mutual rubbing of the granules and are taken away with the gas in the form of submicron and/or micron particles through the openings in the mesh 5 into the waste collector 7. The problem is to define how the impurity concentration c in the outlet gas flow changes with time.
Two factors are to the greater degree than the others responsible for the run and the specific character of the sorption process in the given method. The first one is the shape of the blades of the stirrer 2 ( Figure 1) that creates an upward-directed constituent in the trajectory of the moving granules. This constituent K. Chuntonov et al. leads to the averaging of the parameters of the reactant along the height of the column. The second factor is the number of rotations ω of the stirrer per time unit. Increasing ω from zero to a certain value it is possible to make the sorber with the getter reactant go through all the possible gas purification regimes from the case with the motionless mass of the granules, the surface of which is covered with the layer of growing products [16] to the extreme variant of the tribochemical treatment of granules, when their entire surface is free from the products of reaction.
Here we will be interested in the intermediate regime of gas purification, when part of the surface of the granules is covered with the products of reaction and part is free from them. The given choice is grounded in the fact that the rate of capturing gases by the metal surface is by orders of magnitude higher than by the surface with the products of reaction, where the process kinetics is limited by diffusion through the layer of these products. Even when the share of the metallic regions γ is small the sorption contribution of the rest of the surface can be neglected. And this fact has a decisive importance: changing the rotation speed ω Journal of Materials Science and Chemical Engineering of the stirrer it is possible to control the number γ and thus the entire sorption process.

Mathematical Model
Let the sorber be filled with cast granules of reactant Me forming a cylindrical column of purification material with radius R and height L. The gas, which is fed from above, contains impurity Y, the concentration of which is c . The gas flow rate at the entrance into the column of granules Me is equal to v . Then the gas flows along the voids between the granules of radius r and reacts with Me forming the solid product MeY.
In the column of the reactant Me let us single out an elementary layer bounded by the cross-sections 1 x and 2 where ( ) , c t x is the fraction of the impurity molecules in the gas phase, B k is the Bolzmann constant, P is the gas pressure, ε-is the porosity of the column of granules Me, and T is the temperature. Then the equation for numerical balance of molecules Y will be written in the form where Y J is the flow of molecules Y, entering the layer dx at the boundary 1 x and exiting at the boundary 2 x , and Y Q is the rate of sorption of the impurity in the middle cross-section of the elementary layer. Basing on the results of the analysis of gas flows in the tubes with granular reactants [16] it is possible to write down Equation (2) in a more compact form In Equation (3) the uncertainty in respect to the value Y Q , which is responsible for the "chemical sink" of the impurity Y is reserved. In order to disclose Y Q it is necessary to formulate the sorption law for the reactant in the conditions of stirring the granules Me. According to the said in the previous chapter let us assume that capturing of the molecules of Y occurs practically only in the regions of the surface, which are free from the products of reaction. This assumption will be knowingly fulfilled if we consider the number of molecules Y, captured by the elementary layer during the time dt, to be equal to the product , where 0 k is the coefficient of sorption of the impurity by the pure surface Me, and the surface area Me S of the granules of the elementary layer is . In this case we get for the "sink" summand in Equation (3) Here 0 c is the concentration of the impurity in the entering into the sorber gas, Me ρ is the density of the reactant Ме, where ( ) L t is the current height of the column Ме. As the result, the mathematical model of the sorption processes taking place in the reactor with the stirrer can be written down as a system of equations In this system of Equations (7)-(9) the processes of sorption / mass transfer are described by Equation (7). The law of the decrease of the granule size is presented by Equation (8), where the coefficient b has a different form (Appendix, formula A5) than in Equation (5). The correlation between the height of the column and the size of the granules is given in Equation (9).

Model Analysis
Let us introduce the dimensionless variables and functions: time Then Equations (7)-(9) acquire the following form: The problem (10) Further, to solve Equation (11) it is necessary to evaluate the integral Substituting into (14) the value of ( ) l τ according to Equation (12) we come to The analytical solution of Equation (15) can be given implicitly Plots in Figure 2 show numerically calculated changes of impurity concentration  in the gas flowing through the column of stirred granules.
The more intensive the gas purification process, the steeper the curve u falls to 0. It follows from the formula (13) that it is possible to enhance the intensity of the gas treatment by increasing the values of γ and/or This is exactly what Figure 2 demonstrates: from the two curves u the dash-line one, which corresponds to the granules of the smaller size, is closer to the vertical axis. This is evident as with the decrease of the granules radius the specific surface area of the reactant grows and together with it the reaction kinetics in the system Me/Y grows as well.
While Figure 2 shows how the concentration u of the impurity in the gas flow decreases with the approximation to the coordinate 1 ξ = , i.e. to the exit from the column of granules Ме, Figure 3 shows how the concentration u of the impurity grows with time τ in the cross section 1 ξ = .
It is seen also that ( ) ,1 u τ continues to stay close to zero for a fairly long time after which it starts to increase rapidly. The moment of time с τ when ( ) ,1 u τ attains the limit c u of the maximum acceptable impurity concentration Y in the gas product is critical for the production processes using the sorber as the source of pure gas. Two ( ) ,1 u τ curves for different values of γ depending on the stirring rate ω are compared in Figure 4. As may be expected, the degree of gas purification is the higher the larger the share of the free surface of the granules; however at this the consumption of the mechanical energy for the activation of the reactant increases. Finally, Figure 5 provides information on the process kinetics from the dimensional viewpoint.

Discussion
Let us consider the consequences of the model, which can be useful for the fur- Our analysis is built on the comparison of metallic chemisorbents of two different classes, traditional getters adsorbents on the basis of transition metals and getters reactants on the basis of alkali-earth metals. Sorption capacity q of adsorbents is limited by the surface area, which is available for gases. After the surface is saturated with gas the process stops (the passivation state). In this passivated state the getter appears to be by the moment when its installation in the working place is completed. Therefore, it has to be activated in vacuum or in inert medium by heating. At this procedure the atoms of the sorbed gas dissolve in the volume of the getter material purifying in this way its surface and restoring its ability to react with the next portion of gas [17].
One of the most spread methods of production of super pure gases is purifica- The specific sorption capacity of getters adsorbents in the mentioned size range is very small, approximately by three orders of magnitude lower than the limit, which could be expected taking into account the stoichiometry of the products forming the passivation film. The given fact has provided the impetus for substitution of getters adsorbents for powder reactants in gas purifiers [23].
However, the success of this substitution appeared to be partial: although the sorption capacity of the purification material increased at this by more than an order of magnitude it still remained sufficiently lower than its theoretical limit. Journal of Materials Science and Chemical Engineering Further, while in the air the adsorbents just passivate the powder reactants inflame and burn. For this reason the need has been recognized to develop special equipment for transporting reactive powders from vacuum conditions into the gas purifier [23]. The other side of the problem is related with unloading the purifier and utilization of the waste material containing large amount of unreacted metal.

The Slowing Layer Δh
The answer to the question why such active material as reactant Me does not have enough time to react with the impurity to completion during the time that the gas passes through the gas purifier, is given by the kinetics of the given process. Alkali-earth metals and their alloys in the interaction with gases follow the parabolic law [9] [17] [24]. As the reaction front moves into the volume of the material the rate of capturing gases decreases. Sooner or later it decreases to such an extent that if we are talking about the production process its continuation does not make sense and it should be stopped. The layer MeY of the certain thickness h ∆ , which we call here a slowing layer, answers this moment. We can widen the notion of the slowing layer understanding under it such a layer Δh, which during some operational period of time with the exposure of the granules in the air will increase in the thickness only by a negligibly small value compared with the initial value Δh. It is possible to estimate the order of magnitude of the value Δh for reactants with the help of granulometric and sorption data on Ba-alloys [3] [8] [25] and on ternary alloys Ca-Li-Mg [9]. Using this data we come to the ratio 2 m The accordingly calculated results for η at 5 m h ∆ = µ are collected in Table  1 and depicted in Figure 6.  It follows from Table 1 that the powder area with particles of the reactant of the size 20 m r ≥ µ is not suitable for being used in gas purifiers because the share η of the unconsumed Me in this case is large. At the same time the data in this table provides the data to such easy and convenient way of filling tribochemicals reactors with purification material as charging them in the air with monolithic particles, the size of which satisfy the condition 1.5 mm r ≥ . The latter fact gives the reactants sufficient advantages over the adsorbents in the economic respect. Traditional getters on the basis of high melting transition metals before being introduced into the gas purifier undergo a long production path of the multistage treatment of the initial metal or alloy to become high porous and permeable for gases structure. In this context filling the sorber with monolithic particles or cast granules of the reactant Me in the air during one or few minutes looks maximally simple. Such a reactant is in the essence raw material, which without passing through the number of the preliminary production stages directly turns into an active gas sorbent on the instant when the stirrer starts rotating in the sorber.

Ideal Sorption Technology
The physical and chemical properties of the reactants are very different from the properties of the other chemisorbents, so adjusting reactants Me to the current production processes would require with sufficient expenses. At the same time big advantages of the reactants over adsorbents both in sorption and in cost respect make it quite natural to set an objective of developing a gas purification technology, which will be initially oriented towards getters reactants.
The best solution appears to be in an integration of getter reactants with the tribochemical reactor, where the first ones serve as consumable materials and the second one performs permanent activation of the getter. Due to such integration the highest level of simplification of the production procedures, their extreme profitability and efficiency can be reached. Let us demonstrate this.
Low mechanical strength of alkali-earth metals and the cover layers of MeY in combination with the brittleness of the latter explains easy rubbing of the gra-Journal of Materials Science and Chemical Engineering nules. Due to this property it is possible to remove, layer after layer, the outgrowths of MeY, which is forming on the surface of the granules, with minimal power consumption and in this way to provide unimpeded access of gases to Me.
As the result the total sorption surface of the reactants appears to be by orders of magnitude higher than the surface area of any getters adsorbents of the same mass.
The principle difference of tribochemical gas purification from the current methods and its undoubted advantage is the controllability of the sorption process: the intensity of reactions between the chemisorbent and the gas in the sorber is regulated by rotating speed ω of the stirrer, which determines the share of the free from MeY surface of the granules. As it is shown by the model, at  In the given situation, it is possible to reduce the value of c η without increasing the energy costs only by radical restructuring of production conditions.
As it is seen from Figure 8 this possibility exists: if in position (b), which corresponds to the moment of time t, we fill the sorber with granules Me to the initial level L 0 without interrupting the working process, then the critical moment t c Journal of Materials Science and Chemical Engineering will take place later by the mentioned value t , i.e. at c t t + . Then, after this time ( c t t + ) passes, it is possible to repeat this procedure again and again turning in this way the sorber from the apparatus of periodical operation into a continuously operating apparatus. It is preferable to carry out this kind of additional feed of the column Me, starting from small values of t, for example, as soon as the initial level of the column L 0 goes down by the value a little more than 2r 0 .
The   In the given publication we do not provide a detailed description of the design of the sorber of continuous operation, however it is seen from the above said that the new solution looks ideal: the content of the unreacted share of the reactant is equal to the estimated 0.8%; unloading of the waste of this composition from the sorber is safe as well as filling the sorber in the air with granules of Me with the size 1.5 mm r ≥ , all that under the condition that the technological requirement to the gas product in the form of c u u ≤ is fulfilled.
Tribochemical gas purification by alkali earth reactants has no limitations. It is applicable in manufacturing of super pure gases, in recycling of rare gases, in purification of natural gases from moisture, mercury vapor, nitrogen, sulfur-and oxygen containing impurities, etc.

Conclusions
1) The mathematical model of the sorption process in the flow tribochemical reactor with a stirrer has been built for the systems gas/solid, where the solid is monolithic granules of reactant Me consisting of alkali-earth metals or their alloys.
2) According to the solutions of the model the main characteristics of the processes of mass transfer in the sorber at const ω = linearly depend on time and this fact sufficiently simplifies the control of the production process.
3) The problem of maximizing the sorption efficiency of the reactants Me in the processes of production of pure gases is solved differently in different applications. In tribochemical reactors while the purifying gas flows from one gas tank into another, the share η of losses of Me is set by the diameter of the openings in the dividing mesh and in the case when these openings have the size of 0.2r 0 the sorption result is close to the theoretical limit being inferior to it by less than 1%. 4) In the production systems using the gas of the purity c u u ≤ , the purification process can be optimized by adjusting the speed ω to a value, at which the total expenses for the material Me and for the mechanical work are minimal.
5) The slowing layer MeY of the thickness Δh fatally decreases the sorption kinetics and together with it the production efficiency of getter reactants in flow sorption columns of gas purifier type but this layer is useful in tribochemical reactors as it fulfills a protective function during the filling the reactor with reactant Me in the air.
6) An ideal solution to the problem of sorption efficiency is the substitution of the tribochemical reactor of periodic operation for the reactor of continuous operation. In the latter the total share η of losses of reactant Me decreases with time to the level lower than 1% even at small speeds of ω.