Interaction analysis of the electrotechnological system “emitter-material” in the process of heating and drying of food plant raw materials

The article considers the heat treating process of edible vegetable raw materials in the system “emitter - material” as the simplest case of heating and drying a homogeneous and isotropic body. It is accepted that the cooling conditions, the ambient temperature and the heat transfer coefficient in time remain constant, and there are no internal heat sources. Based on the study of the fundamental works of famous scientists in the field of food drying, the problem of determining the time constant of raw material heat treatment is solved. This parameter during heat treatment directly depends only on the physical properties of the material, the cooling process on its surface, geometric shape and body size. Knowing the values of the heating time constant, it is possible to determine the time and speed of heating the material to a given temperature.

Currently, the prospects for the development of electrical technology in the food industry and agriculture show the widespread use of drying plants operating on the principle of using electric energy converted into thermal radiation energy for processing edible vegetable raw materials.
When heating and drying wet materials with IR rays, radiant energy is converted into heat, and the phenomena of heat and mass transfer develop both outside the material -in the working chamber of the drying unit and inside the material [1,3,4].
The "emitter -material" system is regarded as the simplest case of heating a homogeneous and isotropic body. The analysis of the interaction of the "emitter -material" system is based on the joint solution of the equations of heat balance and heat transfer taking into account the dynamics of heating. Calculations in radiant heat transfer between bodies must be carried out, assuming that the radiating surfaces are gray and their radiation is diffuse with a constant density on isothermal parts of the surface.
The effect of IR heating on the irradiated materials is manifested in a number of effects -in heating the material, in removing moisture from the material and in the loss of heat into the environment due to IOP Conf. Series: Earth and Environmental Science 548 (2020) 062006 IOP Publishing doi:10.1088/1755-1315/548/6/062006 2 convective and radiant heat transfer. Based on the energy conservation law, the heat balance equation for the time interval dτ has the form: where dQsupthe amount of heat supplied to the material, J; dQheatthe amount of heat spent on heating the material, J; dQspthe amount of heat spent on phase transformations, J; dQconvheat loss resulting from convective heat transfer between the material and the environment, J; dQradheat loss resulting from radiant heat transfer between the material and surrounding surfaces, J. The determination of the components of the heat balance is presented below [1,3,19]. The amount of heat supplied to the material: where Аthe radiation absorption coefficient of the material; Ethe surface density of the radiation flux, W/m 2 ; Fоthe area of the irradiated surface of the material, m 2 . The ability of a material to absorb infrared rays depends on its optical properties and the radiation wavelength, and the surface flux density from the emitters depends on the temperature of the emitters Temit, the distance between the emitters S and the distance between the emitters and the object ho.
The amount of heat spent on heating the material: where The amount of heat spent on the evaporation of moisture: where qminitial intensity or rate of evaporation of the substance, kg/(m 2 •s); rthe specific heat of evaporation (vaporization), J/kg; Fmatthe total surface area of the material, m 2 . Heat losses resulting from convective heat transfer between the material and the environment: where αconvaverage convective heat transfer coefficient, W/(m 2 •K); t and tetemperature of the material and the environment (air), °С. Decisions to determine the heat transfer coefficient αconv in convective heat transfer, described by a system of differential equations and uniqueness conditions with a large number of variables, run up against serious difficulties. These difficulties can be solved by the theory of similarity [21,22].
Using the theory of similarity prof. P.D. Lebedev [3] proposed an equation for determining the heat transfer coefficient during heat transfer, suitable for any method of supplying heat to a material that covers the entire drying process: where Nu = αl / λthe Nusselt number (dimensionless heat transfer coefficient) characterizing the intensity of the convective heat transfer process; α -с the average heat transfer coefficient, W/(m 2 •K); lthe characteristic linear size of the evaporation surface, m; λthe coefficient of thermal conductivity, W/(m•°C); A и nconstants depending on the number Re; Re = wl / νthe Reynolds number (criterion of the heat carrier motion mode), characterizing the hydrodynamic conditions of the process; wthe heat carrier velocity, m/s; νthe kinematic viscosity coefficient, m 2 /s; K = te / twthe modified Guchman criterion, determines the increase in the heat transfer coefficient due to turbulization of the air flow by the vapors formed at the surface of the material; twthe temperature of the wet thermometer, °C; θ = tem / tea parametric criterion that determines an increase in the heat transfer coefficient by reducing the thickness of the boundary layer with increasing temperature of the emitter surface during infrared heating; tememitter temperature,°С; ω / ωca parametric criterion that takes into account a decrease in the heat transfer coefficient with a decrease in the moisture content of the material during the falling drying speed; ωthe moisture content of the material during the falling drying rate; ωccritical humidity of the material; p / Ba criterion that takes into account the conditions of complex heat and mass transfer during vacuum drying of materials; pthe pressure of the medium in the working chamber, Pa; Bbarometric pressure, Pa. In drying plants for edible vegetable raw materials, the drying process occurs at atmospheric pressure, then p = B and the criterion p / B = 1.
Heat losses resulting from radiant heat transfer between the irradiated material and the surrounding surfaces where Tmaterial temperature, K; Тstemperature of surrounding surfaces, K; cred = εredc0 -the reduced emissivity of the two-body system, W/(m 2 •K 4 ); εred -reduced coefficient of thermal radiation of a system of two bodies; c0 -the emissivity of a completely black body, c0 = 5.67 W/(m 2 •K 4 ); φ12 = dQrad/dQem.matthe average angular coefficient of radiation of the material; dQem.mat-the amount of heat emitted by the material, J.
For the general case, when two bodies are arbitrarily located in space, the reduced coefficient of thermal radiation εred is determined using the coefficient of thermal radiation εmat of the material and εs of the surrounding surfaces: where φ21 -average angular emissivity of surrounding surfaces. Since the irradiated material is inside the drying chamber, for this case (φ12 = 1 and φ21 < 1) , taking into account the reciprocity of the angular coefficients φ12Fmat = φ21Fs, where Fsthe surface area of the surrounding surfaces, formula (9) takes the form: Formula (11) can be represented as follows: For practical conditions prof. P.D. Lebedev recommends taking the value of the total heat transfer coefficient α in the range from 18.6 to 23.2 W/(m 2 •K) [3].
After substituting the individual components in equation (1), obtain: Divide each term of the obtained equation (13) by αFdτ and obtain: Denote [3,19]: where Тhthe heating time constant, s; Fthe ratio of the total surface area and its irradiated part; tss the steady-state temperature of the material (at dt / dτ = 0). Then equation (15) can be written as: After solving the equation: where tinitmaterial temperature at the initial time at τ = 0,°С.  (20) shows that at τ → or practically (with an error of no more than 5%) for τ ≥ (3÷4) Тh a balance is established between the amount of heat absorbed by the material and the heat loss to the environment. This moment corresponds to the steady temperature of the heated material, with t = (0.95÷0.98)tss.
The expression obtained from equation (20) for determining the time of heating the body to any temperature t in the interval from tinit to tss is as follows: The heating rate in the process of supplying heat to the material is determined by the expression: During the heat treatment of edible vegetable raw materials, the heating rate must be limited in order to avoid damage to the heated materials.
An analysis of the interaction of the electrotechnological system "emitter -material" in the process of heating and drying food vegetable raw materials [1,3,19] showed that the heating time constant Th, which determines the value of the time and heating rate, is among the most important parameters in heat treatment.
If one takes М = Vρ (V is the volume of the material, m 3 , ρ is its density, kg/m 3 ), then expression (16) can be written: As can be seen from expression (23), the heating time constant Th is completely determined only by the cooling conditions at the interface between the material and the medium, the physical properties of the material and its geometric shape and dimensions. The heating time constants for carrots [23], Jerusalem artichoke [24], beets [25], apples [26], lingonberry, black currants and sea buckthorn [27] were determined using the above method.
In conclusion, it should be noted that theoretical calculations for the current state of the theory, technique and technology of infrared heating would not be convincing enough, nevertheless, there are some difficulties consisting in insufficiently accurate data on the parameters included in the calculation formulas. Therefore, there is a need for experimental studies of the heating process.