In recent years, works aimed at studying and designing desublimation have become especially topical. The most promising way for the solution of similar problems is the mathematical modeling of unsteady state heat and mass transfer processes occurring in the course of desublimation, as the experimental approach is expensive and takes a lot of time [1-3].

For the unsteady-state process of desublimation into vertical immersible tanks with smooth internal walls and horizontal finning, we developed the two-dimensional mathematical model, which does not use any empirical data and takes into account the motion of gas inside a tank, the desublimation of on the end walls of a tank, the elliptic shape of these walls, and the unsteady-state character of heat and mass transfer processes in contrast to the known mathematical models [4, 5].

A non-stationary mathematical model of the process of desublimation of gaseous hexafluorosilicate into vertical cylindrical containers of a gas collector is presented. The mathematical modeling of the process takes into account the movement of gas inside the tanks, the ellipticity of the end walls of the tanks, and the mobility of the phase transition boundary. The possibilities of using experimental approaches are limited and require long-term and expensive experiments to be carried out on operating equipment or laboratory facilities, so one of the promising ways to solve the production problems described above is to create mathematical models of heat and mass transfer processes occurring sublimation and desublimation of HFS. This work is devoted to mathematical modeling of the process of desublimation of gaseous HFS into vertical transport containers. The existing mathematical models of the HPS desublimation process [6, 7] are, as a rule, stationary or quasi-stationary. Their common disadvantages are the lack of consideration of heat transfer by convection and friction of the gaseous HFS against the desublimate layer during its movement along the heat exchange walls; desublimation on the end elliptical walls of heat exchange tanks; nonstationarity of heat and mass transfer processes. In this regard, the purpose of this work was to create a non-stationary mathematical model of HPS desublimation, taking into account the presence of convection and HPS desublimation.

The mathematical model was developed taking into account the following assumptions: all the heat released during the phase transition is removed by the refrigerant through the vessel wall and the desublimate layer; gaseous HFS does not contain impurities; thermal processes inside the system under consideration can be described within the framework of the traditional theory of heat conduction; the gas flow before entering the container is considered to be isentropic, so the values of the total enthalpy and entropy are set on the cut; HFS desublimation on the upper elliptical wall of the container with a branch pipe is not taken into account, since it protrudes from the thermostat filled with refrigerant, the desublimation condition is set on the remaining boundaries with the wall; the gas is considered to be polytropic, and the viscosity and thermal conductivity are neglected in the calculation of the gas phase. In mathematical modeling of the processes of heat and mass transfer occurring during the desublimation of gaseous HFS, a two-layer system was considered, where the first layer is the vessel wall material of constant thickness, and the second is the desublimate layer with a movable outer phase boundary. To calculate the heat flux, we used a one-dimensional differential equation of heat conduction for the vessel wall, a desublimate layer, and an integral law of conservation of energy to check the heat balance. The iteration-interpolation method [8] was used to numerically solve the heat conduction equations. The calculation of the temperature in the layer of solid HFS was carried out using a moving coordinate system associated with the wall-desublimate interface. In the course of solving these equations, the temperature distribution in the wall-desublimate system was determined, the heat flux and the desublimation rate of ammonium hexafluoride silicate were calculated, after which the thickness of the solid phase layer formed was calculated.

The task of this work was to create a mathematical model of the HFS desublimation process in a container with water cooling of its cylindrical surface. Containers and "capacity" were chosen as the object of modeling, because for these structures, there were reliable experimental data on HPS desublimation. A feature of the containers and "capacity" is their horizontal arrangement during the desublimation process. Their inlet valve is located in the upper part of the front end surface. The main task in building a model of the desublimation process was to determine the degree of filling of empty containers and the "capacity" of hexafluorosilicate in real time. The calculation model was based on the following assumptions:

• The process of HFS condensation occurs mainly on the cylindrical surface of the “tank” (container) irrigated with cooling water. Condensation of HFS on the end walls of the "tank" (container) due to weak heat exchange with the surrounding air is insignificant.
• The condensed HFS at each moment of time is a cylinder with axial symmetry. The axial symmetry is due to the uniform pressure distribution of the gaseous HFS inside the "tank" (container), as well as a slight change in the temperature of the cooling water as it flows laminar-wave around the side wall of the "tank" (container).
• The temperature of the outer surface of the cylindrical wall of the "tank" (container) is constant and equal to the temperature of the cooling water at the considered moment of time. The temperature of the inner wall of the condensed hexafluorosilicate is equal to the temperature of the phase transition (desublimation) of the HFS for the current pressure in the "tank" (container).
• The process of HFS desublimation in a “container” (container) is quasi-static, i.e. the temperature distribution and the thickness of the condensed HFS layer are linear and vary very little with time.

Not only the stability and efficiency of the desublimation equipment, but also the environmental safety of production often depends on the correct organization of the thermal regimes of the process of desublimation of volatile substances in surface desublimators. Thus, when obtaining uranium hexafluoride (HFS) by fluorination of uranium oxides, HFS during desublimation can accumulate on certain areas of the surface of the apparatus, where the most effective conditions for this process are formed [9]. As the solid product layer is formed, the free cross section of the desublimation apparatus decreases, and, accordingly, the gas velocity increases, the temperature of the desublimation surface changes, as well as the conditions for heat transfer from the desublimating product to the surface of the apparatus [10]. This leads to a change in the temperature of the desublimation surface, which can lead to a breakthrough of gaseous HPS through the apparatus, so gases after the fluorination stage containing HPS, oxygen and excess fluorine pass through two or more series-connected desublimators. In each subsequent apparatus along the gas flow, the desublimation surface increases. Therefore, it is relevant to determine the optimal conditions for the desublimation of HFS from a vapor-gas mixture in order to control the desublimation front in the apparatus and increase the degree of filling of the desublimator with a solid product due to its uniform distribution over the volume of the apparatus, eliminate product losses associated with inefficient operation of the desublimation surface, and prevent sudden clogging of the apparatus with solid HFS. These problems can be solved using the mathematical model of the desublimation process developed by us, which describes the mass, thermal and hydrodynamic flows inside the apparatus. This model makes it possible to determine:

• Mass flows of vapor-gas mixture and solid HFS inside the apparatus (material flows);
• Coefficients of heat and mass transfer from the gas-vapor mixture to the solid surface;
• Cooling time of the gas-vapor mixture from the initial temperature to the HFS desublimation temperature;
• Speed and mass of HFS released from the gas flow into the solid phase, per unit area of the desublimator;
• Change in heat and hydrodynamic flows inside the apparatus, occurring due to an increase in the thickness of the product layer on its walls in the process of desublimation. Calculation of devices having an annular shape or the shape of flat plates is very complex, so let's consider a mathematical model to determine the optimal thermal desublimation conditions for annular devices

The motion of gaseous HFS   was considered to be two-dimensional axisymmetric and was described by a system of integral equations for the conservation of mass, momentum, and energy.

Where C is an arbitrary closed contour that bounds the area S in the plane of variables z, r - Cylindrical coordinate system, the origin of which is placed on the axis of symmetry in the input section; t is time; p is pressure;  – gas density; e - specific internal energy

The gas flow parameters were considered in the cross sections formed by the radial and azimuthal axes at the top and bottom ends of vertical fins. The change in the calculate parameters of gas between these cross sections along the axial coordinate was assumed to be linear, thus making the considered computational region pseudo-three-dimensional. In this case, the set of equations of gas dynamics is as follows:

Due to the fact that only the temperature gradient formed in the direction perpendicular to the heat exchange surface was taken into account, a one dimensional model was used to simulate thermal processes.

The process of heat exchange between the gas and the cooling agent through the side tank wall with the formed desublimate layer was described by the heat-conductivity equations in the one-dimensional approximation.

The gas/solid phase interface is not linear. This system of equations was reduced to a dimensionless form, the solution domain was normalized. The auxiliary coordinate system was brought to the dimensionless form [11] 

The formula for calculating the heat flux through the cylindrical surface of the HFS layer during the quasi-static heat transfer process [1] has the form: 

The motion of the phase transition front in the desublimate layer was taken into account by introducing the normalized coordinate =r/r_w

The heat-conductivity equation for the de sublimate layer was written as follow:

The motion of gaseous HFS in the tank was considered to be two-dimensional axisymmetric and was described by a system of integral equations for the conservation of mass, momentum, and energy. This system of equations was reduced to a dimensionless form, the solution domain was normalized.

The developed simplified three-dimensional mathematical model of heat treatment desublimation was implemented as software Solid work. At the next stage, the collector was calculated. At the initial moment, containers with some residual pressure are connected to the collector. In the process of filling them with de sublimate, due to the increase in the thickness of the solid HFS layer, the thermal resistance of the vessel wall – de sublimate layer system increases, the heat flow from the gaseous HFS to the refrigerant decreases, as a result of which the de sublimation rate decreases and the pressure inside the container increases. The value of the temperature of the HFS phase transition from the gaseous state to the solid state depends on the pressure value. Temperature, in turn, affects the density, thermal conductivity, and heat capacity of solid HPS. At the moment when the pressure in the first tank reaches a critical value, the second tank, which is part of the collector, is connected, and so on in the same way until the entire volume of the first and then subsequent collector tanks is filled. The filled container is immediately cut off from the collector and replaced with an empty one.

This algorithm has been implemented as an application package with a user-friendly interface. As a result, we have developed a two-dimensional non-stationary mathematical model of HPS de sublimation in a reservoir consisting of several tanks, taking into account the presence of HPS convection and de sublimation on the bottom wall of the tank, as well as the ellipticity of the upper and lower walls of the tank, which allows us to calculate a collector of any capacity, consisting of tanks of various types (Figure1).

Figure 1: Distribution of velocity (a), pressure (b) and temperature (c, d) of the gas mixture of hexafluorosilicate and nitrogen, natural flow (a-c) and forced cooling (d).