Functional model of energy consumption for mixing with a vertical paddle mixer

A description is given of a vertical paddle mixer with a batch action of bulk materials. Based on the obtained experimental data, part of the previously published and newly identified equations of regression of mixing unevenness, the functions of the factors of the empirical mixing intensity coefficient are obtained. They take into account the influence of technological (proportion of the control component, volume of a mixing tank and its degree of filling), operating (mixer angle of the blades) and design(number of blades, blade length) mixer parameters. Based on the interconnection of the functions describing the power consumption, the required duration of mixing and the mass of the prepared mixture, a mathematical model of specific energy costs is obtained. Graphic materials are presented that make it possible to evaluate the combined effect of the mixer parameters on the mass of the prepared mixture, the duration of mixing, power consumption, and specific energy consumption. The rational use of the mixer with a share of the control component of at least 10%. With an increase in the stirrer rotation frequency of more than 350min-1, the intensity of the reduction in mixing time slows down. Reducing a portion of the mixture reduces energy costs. In terms of energy consumption, small mixers are more efficient, but they have low productivity.


Introduction
In the implementation of production processes, the modern national economy makes extensive use of various composite materials and mixtures in construction [1], engineering [2], mining [3], chemical [4] and food industry [5]. These materials are prepared both in extruders and in a wide variety of mixers. For bulk products, drum [5], paddle [6,7], vibration [8] and other mixers are used.
One of the areas of application of mixtures is agriculture, in which various cereal [10] and fodder [11] mixtures are prepared during the production process. One of the sought-after mixtures is compound feeds and their preliminary derivatives -premixes and protein-vitamin supplements. For the preparation of these bulk mixtures, various paddle mixers are widely used [11,12].
In the study of mixers, the authors focus most often on the theoretical aspects of the completeness of the mixing process [13,14] and the energy costs involved [15]. However, from a practical point of view, functional dependencies are also of interest, comprehensively describing the process of mixing a particular mixer design according to the main indicators of the technological process. A design of a vertical paddle mixer was developed, during the study of which an experimental material was obtained, and based on it, regression expressions of specific performance indicators of the proposed mixer were established: power [16], mixing quality [17]. However, there is no comprehensive energy description of the technological process of the specified mixer based on the results.
Conversion of the regression equations for this mixer design into a functional model was not performed.
The aim of this study is to develop a functional model of the energy consumption of mixing of a vertical mixer with paddle working bodies for implementation in the MathCAD computer algebra system to establish the influence of structural, kinematic and technological parameters on the specific energy consumption of mixture formation.

Research methods
The methodology of the studies involved the development of a generalized functional model of specific energy costs for the process of mixture formation. In this case, the regression dependences previously established by the authors [16,17] was used, which describe individual indicators of the flow of the technological process of mixing in the developed mixer ( Figure 1).
The specific energy costs were modeled in the MatchCAD computer program. Using the developed program, numerical studies were carried out to establish the nature of the change in specific energy costs and determine the least energy-consuming parameters of the mixing process for specific technological tasks. The interval of change in the numerical values of the factors presented in the mathematical model corresponded to the values of the parameters of experimental studies.
In the process of experimental studies, the stirrer rotation frequency varied from 250 to 1000 min -1 , the number of blades from 3 to 8 pcs., the angle of installation of the blades from 15 to 60, the length of the blades superimposed on the blades from 15 to 75 mm, the volume of the tank from 1 up to 30 liters, the degree of filling from 0.   The components of the mixture are loaded through the loading funnel 2 with the shutter closed at the unloading tray 8. When the drive is turned on, the mixer rotates, providing both the circumferential movement of the mixed mass and the movement in the vertical plane. After preparing the primary mixture, it is possible to reload the remaining components of the mixture and mix them to obtain the finished mixture. When the shutter is opened, the finished mixture leaves the mixer through the unloading tray 8.

Results of the research
Evaluation criteria for the effectiveness of the mechanization of technological lines for the preparation of mixtures are the energy consumption for preparing a portion of the mixture and specific -per kilogram of the mixture. There are a number of restrictions on the use of technological machines. In case of non-compliance with technological requirements for quality indicators of the operation of devices, use is prohibited. Productivity of machines should correspond to productivity of a technological line. The power consumption of electric motors of the machines should not exceed the installed power of the engine for the implementation of technological process, taking into account the necessary margin.
Specific energy consumption in the preparation of the mixture [18], J/kg: wherePpower consumption of the drive of working bodies, W; T cthe duration of mixing, providing technological requirements for the quality of the mixture, s; Mthe mass of the prepared portion of the mixture, kg The mass of the prepared portion of the mixture [18], kg: whereV omixer tank volume; Ethe degree of filling the tank with material; ρfilling density of the mixture, kg/m 3 . For the design of the mixer under consideration, the power consumption is determined [16], W: where nrotational speed, min -1 ; l s = D/6blade width, m; L = (L p − l s )blade length not overlapping the blade width ls, m;L pfull blade length, m; Znumber of blades, pcs. The duration of the mixing cycle is defined as the combination of the durations of several measures (c): loading T Z , mixing T S , unloading T B , preparatory and final works T PZ for the technological cycle.
The duration of component loading (s) for the j-th mixing stage in the mixer tank depends on the performance of the particular metering devices used. For multi-component dispenser: whereK Zcoefficient taking into account preparatory and final actions during the operation;Q kproductivity of the dosing device of the k-th component, kg / s; M kthe mass of the k-th component according to the recipe of the mixture, kg; whereDkfraction of the k-th component according to the recipe of the mixture. The dependence of the mixing duration, which provides technological requirements for the quality of the mixture, remains unknown. Moreover, the dependence of the quality of the mixture on a number of parameters is known [17], including taking into account the duration of mixing of the components. The model function of the relative uniformity of the mixture (V p , %) is an exponential expression: where k`empirical components mixing intensity coefficient. Empirical components mixing intensity coefficients expressed by the function [17]: where K εempirical coefficient of influence of the degree of the tank filling; K Voempirical coefficient of the mixing tank volume; K Dkempirical ratio of the share of the control component. The indicated empirical coefficients are described by the expressions: Using the indicated expression (7), it was substantially refined by introducing additional coefficients that take into account the design of the mixer. The empirical coefficient of mixing intensity is described by the expression: whereK Tempirical coefficient of influence of technological parameters; K Kempirical coefficient of influence of design and operating parameters of the mixer; K Dkempirical ratio of the share of the control component; K Eempirical coefficient of influence of the degree of the tank filling; K Voempirical coefficient of the mixing tank volume;K empirical coefficient of influence of the blade angle; K nLempirical coefficient of influence of blades length; K nzempirical coefficient of influence of the number of blades. Additional empirical mixing intensity coefficients (factors) are expressed by functions ( Figure 2): The indicated empirical coefficients are obtained on the basis of the transformation of the below, obtained on the basis of experimental data, expressions of the unevenness of mixing. The expression of uneven mixing taking into account the stirrer rotation speed n and the installation angle of its blades , 0.01%: where nmixer rotation speed, min -1 ; blades installation angle, degrees. In this case, the correlation coefficient R=0.95211. The expression of uneven mixing nL taking into account the speed n of the mixer and the length of the blades L outside the blades, 0.01%: where L is the length of the blades outside the blades, m. The correlation coefficient R =0,96709. The expression of uneven mixing nz taking into account the speed n of the mixer and the number of blades Z, 0.01%: where L is the length of the blades outside the blades, m. The correlation coefficient R=0.98023.
where  techmixture unevenness (variation coefficient control component content in the samples taken) according to technological (or livestock farmingzootechnical) requirements.
It is impossible to express the duration of mixing from expressions (6), (9), (10), (14). However, using the capabilities of a computer program by the method of selecting values with an acceptable error, it was possible to establish the duration of mixing to achieve a given quality of the mixture. Factor analysis (according one factor) made it possible to graphically establish the effect of part of specific factors on the duration of mixing (Figure 2). The proportion of the control component in the mixture is most intensively affected -the smaller the value, the more intensive the increase in the duration of mixing. An increase in the rotation speed, the number of blades and the angle of installation of the mixing blades of the mixer reduce the duration of mixing. The increase in the mass of the feed portion (including the volume of the mixer and the degree of filling) require an increase in the duration of mixing.
Based on the expressions (1,2,3,15), using factor analysis (one factor each), the effect of technological and operating parameters on the combined effect can be established for the mass of the prepared portion of the mixture M (kg), power consumption P (W), mixing duration Ts (s) and specific energy consumption Ay (J / kg), shown in Figure 3.