##### Title : Mathematical modeling of the adsorption process with variable conditions

#### Abstract:

In modern chemical technology, adsorption devices are widely used for the interaction of substances in various states of aggregation. The desire to intensify the operation of industrial adsorbers leads to an increase in the speed of phase movement, an increase in local and average temperature gradients and pressure drop in the active zones of adsorption devices. This work is devoted to the study of the process of adsorption of H2S/ ??2/N2 gas mixtures with variable pressure. Particular attention is paid to studies of a fixed bed of natural zeolite in a certain limit of pressure drop and the dependence of the flow rate of the incoming gas, which ensures the full use of the adsorption capacity of the zeolite. The basis for calculating the process of adsorption separation of gas mixtures is a mathematical model of heat and mass transfer processes, which consists of the following equations: Differential equation in partial derivatives of the material balance of the adsorbed substance ? ? ? ? ? ? ? ? ? ? ? ? ? r C r r k C r r C r i i i ( , ) ( ) ( , ) 1 2 2 ? ? ? i ?1,N (1) Differential equation in partial derivatives of the heat balance of the adsorbed substance ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Ci T q r T r C r r T a a ( ) 1 2 (2) 2 Kinetic laws of the adsorption process ? ? ? ? ? ? ? ? ? ? ? ? ? ? r a r r r a D i i i 2 2 1 1 ? (3) Differential equation of pressure change according to the Mendeleev-Chaperon equation ? ? ?? ? ? ? ? ? ? ? Ci R T T T P P 1 (4) Equilibria in the form of the Langmuir equation: ?? ? ? ? ? ? n i i m i i b P a b P a 1 1 1 1 (5) where Ci is the concentration of the adsorptive, ai is the concentration of the adsorbate, P is the pressure, r is the current radius of the adsorbent grain, τ is the adsorption time, k is the constant, T is the adsorption temperature, θ is the local temperature, R is the universal gas constant, q is the heat adsorption, α is the volumetric heat transfer coefficient; λ is the heat transfer coefficient (adsorbent grains), Ca is the heat capacity per unit volume of the adsorbent, Di is the diffusion coefficient, am is the limiting amount of adsorbate, b1 is a constant. In addition, the equilibrium non-isothermal process of adsorption in a grain of a spherical adsorbent is expressed by the equation a=ƒ(c,r). At the same time, the concept of the average concentration of adsorptive, adsorbate and temperature is used: ? ? ? R i c r dr R C 0 3 2 (? ) , ? ? R i ar dr R a 0 3 2 (? ) ? ? R Tr dr R T 0 3 2 (? ) (6) where R is the radius of the adsorbent grain; r is the current radius of the adsorbent grain; τ-time if we take: ? ? ? ? ? ? ? ? ? ? ? Di a b R L R 3? 15 , n V Va ? where Va is the volume of the adsorbent, V is the volume of the adsorbent bed of the adsorber, L is the height of the adsorbent bed, ao is the initial concentration of the adsorbate, m and b are constants. As a result of several mathematical operations, equations (1-5) are obtained:

Using the above model, consisting of equations (1-5) and solutions (7), we calculated the adsorptive concentration along the fixed bed of the adsorbent, showing that the adsorptive concentrations obtained at the outlet are controlled by the pressure drop in the fixed bed of zeolite in the adsorber, depending on from the flow rate of the incoming gas. The proposed formulas can be used in the study and calculations of industrial adsorbers in conditions with variable pressure.