## 1 Introduction  
One of the commonly used grain cleaning methods is application of air-sieve grain machines. These devices are designed to clean the grains of the main crop from impurities due to differences in their size and aerodynamic properties. According to the technological scheme, pneumatic separation ducts (usually of the vertical type) are the primary stage of cleaning the grain mixture from light impurities which additionally regulate the loading of the sieves of grain cleaning machines. Separation of the maximum amount of impurities in the pneumatic separation ducts causes reduction of the load on the sieves, which improves the quality of separation--the efficiency of the pneumatic separation depends on the productivity and quality of the cleaning of the sieves and technological performance of the entire grain cleaning machine [1,2]. The segregating of the components within the grain mixture, consisting of grains of the main crop and impurities, in the pneumatic separation ducts is due to the action of gravity forces and their interaction with the airflow. Depending on the magnitude of the airflow force and the gravitational force, the impurity particles can be redistributed and move both down and up [3]. As a result of the release of particles of lightweight impurities from the grain layer and their subsequent transportation in the aspiration system, their separation occurs. In this case, the purified grain of the main crop moves down according to a predetermined trajectory. The conditions that complicate the operation of pneumatic separation ducts are [3; 4; 5; 6; 7; 8]: uneven airflow in the work area; the method of feeding the grain mixture to the working area; heterogeneity of the grain environment; properties of the layers of grain environment; similarity of physico-mechanical, including aerodynamic, properties of impurities and grains of the main crop. The cleaning efficiency depends also on the properties of grain mixtures, in particular [5; 8] the thickness of the mixtures layer, the porosity of the grain environment, and the distribution of lightweight impurities according to the thickness of the layer. Existing methods of intensifying the processes of cleaning grain mixtures in pneumatic separation ducts are focused on the increase in the size of the work area; assembly of additional devices that facilitate the redistribution of light particles and the direction of airflow; and cleaning in several stages [5; 6; 8; 9]. On the one hand, increasing the size of the work zone of the ducts leads to an increase in productivity, on the other hand, to reduction of the quality of the mixture cleaning. The additional devices used include equipment that generates an electric field [7] or feeding the grain mixture into the work area of the duct in several layers [9]. However, most of the innovations, despite the increase in productivity and quality of cleaning of grain mixtures, have significant energy costs. Energy costs for post-harvest grain treatment and seed preparation are 5-10% under favorable conditions, and 17-22% in areas with high humidity. Among different types of grain cleaning machines based on pneumatic separation ducts, the highest specific energy consumption is in precleaning machines, 0.12-0.44 kWh\(\cdot\)Mg\({}^{-1}\); universal air-sieve machines, 0.86-1.61 kWh\(\cdot\)Mg\({}^{-1}\); vibro-pneumatic separators, 1.88-3 kWh\(\cdot\)Mg\({}^{-1}\); and triers, 0.22-0.29 kWh\(\cdot\)Mg\({}^{-1}\)[10]. A promising way to increase the efficiency of the process of cleaning grain mixtures in vertical ducts is to preliminarily stratify the mixture on a permeable pitched scaly surface [11; 12]. Dynamics of this process are similar to those in the use of air-permeable surfaces of pneumatic tables, where significant parameters are surface angles, airflow rate, size and shape of air-permeable surface holes, amplitude and frequency of surface vibration, and mixture layer thickness [13].