Mathematical modeling of the process of extracting fat from insects by a screw press in the technology of obtaining feed additives

. The use of insects as raw materials for the production of feed additives is becoming an increasingly relevant and promising area of research. Nevertheless, this area requires numerous studies, including theoretical ones. The article presents the results of research on mathematical modeling of the process of obtaining feed additives in the form of fat and protein from the larva of the Black soldier fly (Hermetia illucens) in a screw press. Mathematical models have been obtained that allow theoretically determining the productivity of a screw press and its energy intensity in the processing of insect biomass. The models are based on the Navier–Stokes equation and Darcy’s law and take into account the design and kinematic parameters of the screw press and the rheological properties of the biomass of the face and the resulting fat.


Introduction
Insects are promising raw materials in feed production, having a rich amino acid composition similar to the composition of fish meal.Of the large number of insects used in feeding animals, birds and fish, silkworm pupae, mealworms, grasshoppers, larvae of houseflies and Black soldier flies (Hermetia illucens) are most often used [1][2][3][4][5][6][7][8].In comparison with other insects, the Black soldier fly (Hermetia illucens) has a number of advantages described in [9].A number of papers [10][11][12][13][14] present positive results on the use of the larva of the Black soldier fly (Hermetia illucens) as a source of animal protein and a substitute for fishmeal.Despite the wide elaboration of the topic of the use of the protein fraction of the larva in animal feeding, studies on the effect of the fat fraction of the larva in the diet of animals, birds and fish are not enough.Therefore, it is necessary to obtain the fat of the larva as a separate feed additive and to investigate its effect on the animal's body.Thus, it is advisable to consider the process of dividing the larva into fractions (protein and fat) and mathematically describe it.
To extract oil (fat) from raw materials, a screw press is most often used.The main parameters of the screw press are productivity and energy intensity, depending on many parameters.The analysis of mathematical models describing the spin process in a screw press showed that most take into account only the design parameters of the press, while not enough attention is paid to the rheological parameters of raw materials [15][16][17][18].In addition, there is insufficient research in the field of mathematical modeling of the extraction of fat from insect biomass by a screw press.Thus, the aim of the research was to create mathematical models that theoretically determine the productivity of the screw press and its energy intensity in the processing of insect biomass.

Theory of Motion of Oil-Containing Insect Biomass and the Process of its Extraction by a Screw Working Body
When considering and describing the spin process in screw devices, it is necessary to take into account the characteristic features that occur with the material as it moves from loading to output.During the compaction process, there is a change in the mass and volume ratios of the pressed material : a change in density.
In the oil filtration zone, the concentration changes due to nonlinear filtration.It is also necessary to take into account the stochastic processes of concentration changes depending on the stochastic phenomena of dispersion and agglomeration of the solid phase in the interturn channel [15][16][17][18].
The main factors affecting the efficient pressing process are the physical and mechanical properties of the structure of the compressible material and the conditions of the pressing process.From the analysis of the literature data, it was revealed that humidity, temperature, the size of the fat outlet opening, the pressure in the extracting chamber, which depends more on the size of the cross-section of the oil cake outlet, the number of screw rotations and the preparation of the pressed raw materials for extraction have a significant effect on oil extraction [15][16][17][18].
The extracting chamber of the screw press can be divided into 4 zones (Figure 1): in the loading zone I, the biomass of the larva of the Black soldier fly (Hermetia illucens) enters through the intake pipe, is captured by the turns of the screw and moves deeper into the housing, slightly compacted; in the compression zone (pressure zone) II, the biomass of the larva is compacted and pressure is created due to the coils and the outlet, creating back pressure to the coils necessary for fat extraction; fat filtration occurs in the zone III, the pressure continues to increase (filtration zone); in the zone IV, the pressure reaches the maximum value at which the filtration of fat stops, since the channels for the fat outlet are pressed and the fat extraction stops, the cake is discharged through the outlet.
Ԛ  is the volume capacity of biomass, kg/h; Ԛ  is the fat capacity, kg/h; Ԛ  is the productivity of cake, kg/h; P is the screw pressure, Pa; ω is the torque, min -1 Fig. 1.Insect biomass pressing zones in a screw press.

Determination of the Speed of Movement of Biomass Particles through the Outlet of the Screw Press
To describe the movement of the processed mass during the extrusion and extraction of fat, the Navier-Stokes equation is most often used.The pressure  1 corresponding to the beginning of fat extraction, the density of the material ρ and the law of change of its viscosity η are known.We accept the process of compaction and pressing of the material by thermal and permanent.Taking into account the accepted assumptions, the Navier-Stokes equation will look like the following (1-3) [9,19]: where  is the velocity of the pressed mass along the axis 0z, m/s;  is the pressure in the grain cylinder, Pa.
Given that the pressure along the axes 0x and 0y is zero (2), the material flow rate in the screw press is considered along the axes x and y, i.e.  = (, ).According to equation ( 1), the pressure is created only along the x axis, i.e.  = ().
Then the system of equations ( 1) -(3) will take the following form (4): Considering equation ( 4 where ∆ =  1 −  2 is the pressure in the chamber of the screw press at which the fat is pressed, Pa. Imagine equation ( 4) in the 0xy plane as a linear partial derivative of the second order (6): The diagram of the hole for the cake outlet is shown in Figure 2. The radius  2 remains constant, the radius  2 changes depending on the selected fitting for adjusting the counterflow and pressure changes in the extracting chamber.The boundary conditions are denoted as in the equation ( 6): on the walls of the hole, the speed of movement of insect biomass is zero (7): where C is the equation of the contour of the section normal to the 0z axis, which is the guide of the cylindrical surface bounding the hole.
Consider a special case of contour C, representing a circle with varying parameters along the y axis.
The contour of the hole for the exit of the cake has two sections with different diameters:  2  2 , in the plane 0xy, which is based on a circle.Therefore, the considered contour of the hole in the specified plane is represented in the form (8): Taking into account equation ( 8) and the boundary conditions presented in equation ( 7), the speed of the cake moving in the first section with a constant hole diameter  2 and length  2 will take the form (9): E3S Web of Conferences 381, 01080 (2023) https://doi.org/10.1051/e3sconf/202338101080AQUACULTURE 2022 For the second section with variable radius  2 and constant length  2 , the velocity is determined as follows (10): Then, in order to express the constant A, we substitute equation ( 9) in ( 6), we get (11)(12)(13): Then the rate of movement of the cake through the first and second sections will be described by the expression (14-15): Combining the found speeds of moving the cake, we get the average speed of moving the cake in the outlet (16):

Determination of the Speed of Movement of Biomass Fat through the Extracting Chamber of the Screw Press
The process of filtering fat through the porous protein surface of the larva is considered on the basis of the law of filtration of liquids and gases in a porous medium : Darcy's law.
The liquid filtration rate is described by the following equation (17,18): where k is the permeability coefficient of biomass, D (1 Darcy = 1 m 2 );   is the viscosity of fat, Pa•s.
where ∆ =  1 −  2 is the pressure in the chamber of the screw press at which the fat is pressed, Pa.
The pressure  1 corresponding to the beginning of fat extraction, the density of the material ρ and the law of change of its viscosity η are known.We accept the process of compaction and pressing of the material by thermal and permanent.Taking into account the accepted assumptions, we write the Darcy equation as follows ( where  is the velocity of the pressed mass along the axis 0z, m/s;  is the pressure in the grain cylinder, Pa. Considering that the pressure along the axes 0x and 0y is zero (20), the material flow rate in the screw press is considered along the axes x and y, i.e.  = (, ).According to equation ( 19), the pressure is created only along the x axis, i.e.  = ().
Then the system of equations ( 19) -( 21) will take the following form (22): Considering equation ( 22), it can be concluded that both parts of the equation separately are constant quantities, since   is a function of z, and  • ( ) is a function of x and y.Therefore, we denote the function of z as follows (23): where ∆ =  1 −  2 is the pressure in the chamber of the screw press at which the fat is pressed, Pa.
Imagine equation (22) in the oxy plane as a linear partial derivative of the second order (24): The diagram of the fat outlet opening in the screw press extracting chamber is shown in Figure 3.The radii  1 and  1 remain constant.We denote the boundary conditions to equation (24): on the walls of the hole, the speed of movement of the larva fat is (25): where C is the equation of the contour of the section normal to the 0z axis, which is the guide of the cylindrical surface bounding the hole.
Consider a special case of contour C, representing a circle with varying parameters along the y axis.
The contour of the hole for the exit of the cake has two sections with different diameters:  2  2 in the plane 0xy, which is based on a circle.Therefore, the considered contour of the hole in the specified plane is represented as (26): Taking into account equation ( 26) and the boundary conditions presented in equation ( 25), the speed of the cake moving in the first section with a constant hole diameter  2 and length  2 will take the form (27): Then, in order to express the constant A, we substitute equation ( 27) in (24) and get (28-30): Then the speed of movement of fat through the holes of the screw press extracting chamber (31): The permeability coefficient  and filtration coefficient   are related by the ratio (32) [14]: where γ is the specific gravity,   3 ⁄ .The specific gravity is equal to γ is determined by the formula (33): where  is the density of the porous medium,   3 ⁄ ;  is the acceleration of gravity,   ⁄ .Thus, we obtain the filtration rate of fat through one hole of the extracting chamber of the screw press (34): The next step is to determine the productivity of the screw press by fat and cake.

Determination of the Productivity of the Screw Press by Fat and by Cake in the Processing of Insect Biomass
Taking into account the rheological parameters of the fat-containing material and the geometric parameters of the holes of the extracting chamber and the holes for the exit of the cake, we take the expression to determine the productivity (35) [20][21][22][23][24]: where  is the productivity of the press,  ℎ ⁄ ; Q_0 is the total volumetric capacity of the extracting chamber for fat   and the outlet for the cake   .
The total volume capacity is determined by the formula (36): where Ԛ .. is the capacity of one hole of the press's extracting chamber,  ℎ ⁄ ; Ԛ .. is the capacity of the cake outlet hole,  ℎ ⁄ ;  .. is the number of holes in the press extracting chamber, pcs.;  .. is the number of holes for the exit of the cake, pcs.The design parameters of the extracting cylinder, namely, the diameter and length, are known.Based on them, we will calculate the optimal number of holes for fat exit.
The area of the circle is determined by the formula (37): The distance between the two holes of the extracting chamber along the screw is δ1 and δ2 across.Let us introduce into consideration the used area of the extracting chamber for fat output and denote by   = ℎ  •   , where ℎ  is the width of the extracting chamber,   .
The used area of the screw press extracting chamber for fat output is calculated based on the «margin from the edge» (Figure 4) according to the formula (38): where,  .. is the number of holes in the extracting chamber of the press;   is the area of the extracting chamber, m 2 ;  1 is the output radius of the fat hole, m;  1 is the distance between two holes along the area  1 , m;  2 is the distance between two holes across area  1 , m.Let us calculate the productivity of one hole of the extracting chamber of the screw press according to the formula (39): We substitute equations ( 31) and (37) in (39) and obtain the productivity of one hole of the extracting chamber (40): The total volumetric capacity for fat through the holes of the screw press extracting chamber Ԛ  , is calculated by the formula (41): Knowing the productivity of one hole of the extracting chamber (40) and the optimal number of these holes in the extracting chamber (38), we substitute the obtained equations into equation (42): where Ԛ  is the fat capacity of the screw press, kg/h;   is the area of the extracting chamber with fat outlet holes placed on it, m 2 ;  1 is the outer radius of one hole in the extracting chamber, m.;  1 is the distance between two holes along the width of the extracting chamber, m;  2 is the distance between two holes along the length of the extracting chamber, m; ∆ is the pressure difference at the inlet and outlet of the hole; Pa;   is the viscosity of fat; Pas; where  is the viscosity of the biomass, Pa•s;  1 ,  2 is the length of the exit hole for the cake, m.  2 is the constant inner radius of the exit hole for the cake, m;  2 is the variable inner radius of the exit hole for the cake, m.As a result, the total volume capacity of the press is calculated by the formula (44):

Determination of the Power of the Screw Press Required for the Extraction of Oil-Containing Insect Biomass
Along with the productivity of the screw press, its power is no less significant.Depending on the energy spent on the production of fat from the larva of the Black soldier fly, its final price will change.The calculation of the power of the screw press will allow determining the rational parameters of the drive necessary to carry out work on the screw press being developed.
The power of the screw press required for fat extraction can be calculated by the formula (45) [20][21][22][23][24]: where  is the power, kW   is the moment of force, Nm.;  is the angular velocity, с −1 .
To determine the moment of force of each turn of the screw shaft, we denote the condition under which the pressure is evenly distributed throughout the cross section evenly.Under this condition, the axial resistance force acts on each elementary pad of the screw surface of the screw shaft.
Based on the above, we determine the elementary moment of force applied to the elementary platform of the helical surface (46): where  is the elementary moment of effort;  is the pressure between the screw turns, Pa;  is the angle of inclination of the edge of the screw coil, degrees;  is the radius of curvature of the screw, m;

Conclusion
A mathematical model of the operation of a screw press has been developed based on the consideration of the process of pressing the biomass of the larva of the Black soldier fly (Hermetia illucens) as a viscous medium following the Ostwald-de Ville power law and the fat filtration process based on Darcy's law.
The dependence of the overall productivity of the screw press and the productivity of fat and the energy intensity of the pressing process on its design and kinematic parameters, taking into account the rheological and filtration properties of the pressed biomass and fat, is revealed.
The relationship between the rate of passage of fat through the openings of the extracting chamber and the capacity of the fat, as well as between the rate of passage of the cake through the outlet and the variable radius along its length, allowing choosing the values of variable radii along the length of the hole to ensure a constant speed of mass movement and pressure adjustment in the extracting chamber.
The obtained mathematical models will theoretically calculate the productivity and energy intensity of the process of extracting oil or fat from oil-containing (fat-containing) raw materials, taking into account its rheological properties.

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), we can conclude that both parts of the equation separately are constant quantities, since   is a function of z, and  • ( is a function of x and y.Therefore, we denote the function of z as follows (5): E3S Web of Conferences 381, 01080 (2023) https://doi.org/10.1051/e3sconf/202338101080AQUACULTURE 2022

Fig. 2 .
Fig. 2. Diagram of the hole for the exit of the cake -the protein part.

Fig. 3 .
Fig. 3. Diagram of a single hole for the exit of the fat of the extracting chamber of the screw press.