Numerical calculation of heat and mass transfer processes in the adsorbers of the air purification system

Article History: Received: 11 January 2021; Accepted: 27 February 2021; Published online: 5 April 2021 Abstract:In this work, a series of parametric calculations was made for the processes of sorption and desorption of water vapor in relation to the conditions of the process of sorption air purification. A mathematical model was developed that adequately reflects the processes of heat and mass transfer in the adsorption unit at all stages of the adsorption cycle. Algorithms for solving problems and programs for calculating heat and mass transfer processes in an adsorption regenerated installation are obtained, results of parametric calculations of heat and mass transfer processes at each stage of the adsorption cycle and for the entire cycle as a whole are obtained.


Introduction
The main objective of research to improve adsorption air purification systems is to reduce energy costs. In addition, as long operating experience has shown, the urgent task is to create an automatic control system, which is necessary to control and optimize the operating modes of the installation. In most cyclic adsorption plants, the issue of increasing the output characteristics is also relevant. In particular, one of the requirements for the predrying unit is to ensure, as possible, a lower dew point temperature of the air at the outlet from the unit (not higher than minus 40 C) over the entire range of operating parameters of the system. This requirement is primarily associated with the need to carry out more efficient air purification from carbon dioxide.
The solution of the set tasks requires a large amount of theoretical and experimental work [1][2][3][4][5][6][7]. Due to the large number of parameters that determine the efficiency of the drying process, a complete cycle analysis is largely determined by the availability and implementation of reliable mathematical models and calculation algorithms in engineering practice. On their basis, it is possible both to take into account the features of the flow of heat and mass transfer processes at each of the steps of the cycle, and their interaction during the transition from the previous step to the next [7][8][9][10][11][12][13][14]. Numerical experiments on predicting the behavior of an adsorption system with different organization of cycle steps make it possible not only to reasonably determine the technological parameters of the process, but in many cases also to reduce the amount of experimental research.
As a result of sorption of water vapor, heat is released, the specific value of which per unit mass of the sorbent is comparable in order to the heat of condensation of water. On the contrary, during desorption, energy is absorbed.

Modeling heat and mass transfer in adsorption processes
For the numerical solution of heat and mass transfer problems, the system of differential equations is replaced by a system of difference equations. Integration of the system of equations using conventional numerical methods leads to an excessively large amount of computations, since it is necessary to simultaneously integrate all 2N (for some fixed value of N) equations at once, while moving one step in time. To reduce computational costs, in this case, a numerical method of splitting into physical processes is used. The essence of this method lies in the fact that various equations of the system are solved numerically one by one at each time step using finite difference methods.
In this case, the system is integrated in the following sequence. First, the mass transfer equations for all (N-1) components are integrated. When moving to the next time layer, the densities of eachi-th( ( ) (1) Quasilinear first-order partial differential equation general solution of the form ( ) 0 (2) The first integrals of the system will be (4) To determine the specific type of function, it is necessary to set the boundary and initial conditions. Initialconditions: ( ) x t (5) where -parameter. Substituting condition (4) into system (2), (3) and excluding the parameter  from it, we have: Substituting now expressions (3), (4) into (6), a solution is obtained that satisfies the initial condition (5): Here, as before, the expression in parentheses after 0  is the function argument ( ) We also set theboundary conditionsin the parametric form: (8) where -parameter, and ( ) ( ) t 1 1    = -a function describing changes in concentration over time at the entrance to the layer. Following the same procedure for excluding a parameter using expressions for the first integrals (3), (4), a solution is obtained in the following form: (9) with the simultaneous setting of the initial and boundary conditions, the solution consists of two parts, each of which is determined by relations (7) or (9) depending on which side of the front of the flow moving with speed and separating the regions of the initial and boundary conditions, the considered section is found.
:    calculated by (7) When ( ) :    -calculated by (9) (10) To test the program for solving the transport equation, the following initial and boundary conditions were set: A -the amplitude of the input concentration fluctuations, 2 1 T its half-period. Comparison of numerical calculations with analytical ones obtained by formulas (7), (9) in accordance with (10) showed satisfactory agreement.
Another way to test the program: checking the fulfillment of the mass conservation law written in integral form for the layer as a whole.
Integrating this equation over the coordinate from 1 x to 2 x and in time from 0 to t , we get: values at the entrance to the layer, 2 2 ,  U at the exit from the layer. Relation (11) represents the balance of mass: the amount of a substance in a layer in a free and bound state is equal to the difference in the masses of a substance supplied to the layer and exited from it along with the air flow.

Conclusions
The use of the obtained expressions to control the operation of the program has a significant advantage in comparison with the previously considered method: the program can be checked in the process of solving the problem and in the most general setting without idealizing the problem.Thus, the studies carried out have confirmed the correctness of the physical and mathematical model, which makes it possible to use it in the future to analyze the process of sorption air purification.

Acknowledgements
This work was carried out with the financial support of RFBR, project No. 20-08-00880