Mathematical description of birch bark impregnation and heating process of birch bark with toluene

. On the basis of differential equations of heat and mass conductivity and given boundary conditions the mathematical model describing the process of heating and birch bark impregnation with an extractant was developed. Experimental facilities for impregnation and determination of mass conductivity coefficient are created. Kinetic dependence of average concentration of toluene in birch bark during impregnation is obtained. The temperature dependence of toluene mass conductivity coefficient through birch bark was obtained on the setup for determination of mass conductivity. The analysis of kinetic curves of birch bark impregnation with toluene at different temperatures shows a considerable influence of temperature on the impregnation rate. The discrepancy between the calculated and experimental data does not exceed 13 %. The rational temperature of impregnation, which is 110 °С, has been determined. Optimum time of birch bark impregnation - 10 minutes - has been determined. Developed mathematical model of the process of impregnation and heating of wood particles can be applied to organization of calculations of the process of extraction of biologically active substances from vegetable raw materials.


Introduction
Impregnation and heating of wood biomass are the most important stages of extraction technology. By changing the lignocellulosic complex and increasing the available surface area, they accelerate the process of extraction of biologically active substances [1-8, 11, 13, 15, 17].
The impregnation process is characterized by the penetration of the extractant into the raw material. Movement of the solvent front deep into the structure is possible due to pressure difference between outside and inside of the material caused by capillary forces or excessive medium pressure, and diffusive movement of extractant molecules or ions through cell cavities [2, 4, 9-14, 16, 18].
Processes of wood impregnation for the purpose of fire and bioprotection are well studied and their mathematical description is presented.
The capillary method of impregnation means the result of wetting the surface with impregnating composition by applying a solution with a brush, dipping and spraying wood in the dry or dried state of wood [15][16][17][18][19]. Thus, the paper by Kolesnikov G.N. et al. presents a mathematical model based on the logistic approach for the aspen wood impregnation process by dipping into the impregnating liquid at +20 °С [5].
For pressure impregnation, a pressure difference is created in relation to the internal pressure in the wood [21].
Analysis of literary sources showed that mathematical models are developed for impregnation of coarse wood in order to improve its performance properties. The most common are flame retardant, antiseptic, frost-resistant and water-repellent impregnation of wood. However, mathematical description of impregnation processes before wood extraction is not described, so the relevance of this issue is obvious.
Mathematical description of the process of birch bark impregnation and heating with toluene is necessary for determining rational regimes of birch bark impregnation before extraction. Studies of the extraction process showed that toluene is an effective extractant for birch bark and it is moreover convenient to carry out impregnation by dipping. The use of toluene as an extractant is explained by its ability to extract betulin from birch wood without impurities [6].
The purpose of this work is mathematical description of the process of birch birch bark impregnation with toluene, experimental studies and checking the adequacy of the developed mathematical model.

Research methodology
Mathematical description and subsequent modeling of birch bark impregnation with toluene was based on theoretical provisions and laboratory tests. Birch birch bark plates 2 -2.5 mm thick, 100 mm wide and 100 mm long were taken as raw material samples for the research.
The process of impregnation was carried out by immersion in a bath of toluene in the setup shown in Fig. 1.  The experimental setup consists of a bath 1, a bath lid 2, a reflux condenser 3, temperature sensors 4,5, a temperature regulator 6, a buffer tank with sand 7, an induction furnace 8. The experiment was carried out as follows. Bathtub 1 was set into the buffer tank 7 with induction plate 8. Immediately the bath 1 was immersed into the sand, which provided a uniform heating of the extractant. Then toluene was poured into the bath and the heating process began. The temperature of the impregnation process was controlled by the temperature regulator 6 with sensors 4,5, which recorded the value of the temperature of the sand and toluene. At a given temperature of the process 4 plates were immersed in toluene, removed one after another at 1, 3, 6 and 10 minutes of soaking and placed in a cellophane bag which was sealed and weighed. Determination of toluene content in birch bark was performed by sampling and then drying them to a constant weight in a desiccator at 105 °C.
The impregnation process was carried out at temperatures of 60 °C, 80 °C, 100 °C and 110 °C.
The data obtained were used to determine the change in the average concentration of toluene in birch bark during impregnation by the formula: where: mt -current mass of toluene in birch bark, kg; m -current mass of birch bark in the process of impregnation, kg; mb -mass of dry birch bark, kg.
To mathematically describe the process of impregnation and heating of birch bark sheets, the diffusion equations of heat and mass transfer for a one-dimensional plate are used: ( The local change in toluene concentration in birch bark (2) is determined by the change in concentration due to the concentration gradient (the first term in the right-hand side) and due to the temperature gradient (the second term in the right-hand side). Differential equation (3) describes the local change in temperature of birch bark across the cross section of the material.
Initial conditions of equations (2) and (3) The conditions at the boundary of the impregnated plate are determined by the properties of the extractant. Under the accepted impregnation conditions we can write: where: Тн -initial temperature of the birch bark plate, 0 С; Т* -extractant temperature; С* -concentration of toluene in the extract.
The joint solution of equations (2) and (3), at boundary conditions (4) and (5), describes the processes of impregnation and heating of the birch bark plate.
The mass-conductivity coefficient am reflects the intensity of thermal motion of molecules and represents the amount of substance transported per unit time through a unit surface at a concentration gradient equal to one [7].
The mass-conductivity coefficient of birch bark during impregnation can be determined from the relation: where: m -change in mass of the birch bark plate over time , kg; В -thickness of birch bark, m; F -surface area of the birch bark sample, m 2 ; C -concentration difference of toluene on the opposite planes of birch bark, kg/kg;  -time period of the study,sek.
To determine the time index and calculate the coefficient of mass conductivity, the setup presented in Fig. 2  Installation consists of an upper container with toluene 1, thermostat with temperature regulator 2, birch bark sample 3, toluene concentration sensor 4, lower container 5.
Installation works as follows. A plate of birch bark 3 is placed between upper 1 and lower 5 containers. Toluene heated to the set temperature from thermostat 2 is poured into the upper tank 1. To maintain the temperature the toluene is circulated by the circulation pump located in thermostat 2. Lower tank 5 is empty and contains a concentration sensor 4 which registers toluene vapors. The heated toluene impregnates the sample. The time during which the toluene will be on the opposite surface of the birch bark is recorded. The concentration difference of birch bark at the opposite ends of the sample is taken to be 1 kg/kg.

Results
As a result of the laboratory tests and calculations, the temperature dependence of the mass conductivity coefficient of birch bark impregnated with toluene was obtained (Fig. 3). Analysis of the curve shows that the mass conductivity coefficient increases parabolicly with increasing temperature. Its maximum value is observed at toluene boiling point of 110 °С. In addition, earlier studies [1] showed that birch birch contains up to 38% betulin, which dissolves well at toluene boiling point.
Processing of experimental data obtained the calculated dependence of the coefficient of mass-water content on the temperature for the temperature range under study: am=6,5 10 -9  T 2 -7,3510 -7 T+4,7110 -5 .
This dependence can be used when simulating the impregnation process with the help of differential equations (2) and (3). Figures 4 and 5 show the results of mathematical modeling in the form of dynamic curves of birch bark temperature and toluene concentration by section during impregnation process at 110 °С. The choice of temperature is conditioned by toluene boiling point, at which the maximum mass-conductivity and, accordingly, the minimum time of birch bark impregnation are achieved. Analysis of the distribution of the temperature field shows that by the 10th minute of the process, the temperature in the center of the birch bark exceeds 80 ° C. The presented data allowed us to obtain a calculated kinetic curve of the average concentration of toluene in birch bark in the process of impregnation, presented in Fig. 6. The points in the figure show the experimental data, and the lines show the calculated data. Analysis of the curves shows that intensive birch bark impregnation takes place within 10 minutes. Further the rate of birch bark impregnation with the extractant decreases noticeably.
A slight increase in the current mass of the samples after 10 minutes is caused by the replacement of betulin in birch bark by toluene, which has a higher density. Thus, it is reasonable to limit the impregnation process to 10 minutes. The discrepancy between the calculated and experimental data does not exceed 13 %, which proves the adequacy of the developed model.
On the basis of differential equations of heat and mass conductivity and given boundary conditions a mathematical model has been worked out to describe the process of heating and birch bark impregnation with an extractant.
To physically simulate the process of birch bark impregnation with an extractant experimental plants for impregnation and determination of the coefficient of mass conductivity were created.
Kinetic dependence of average concentration of toluene in birch bark in the process of impregnation was obtained on the experimental unit for impregnation. The temperature dependence of toluene mass conductivity coefficient through birch bark was obtained on the mass conductivity setup.
The discrepancy between the calculated and experimental data does not exceed 13 %. The simulation of the process of heating and impregnation of birch bark with toluene revealed a rational temperature of impregnation, which is 110 °С. At that optimum time of birch bark impregnation is 10 minutes.
Developed mathematical model of the process of impregnation and heating of wood particles can be used when organizing calculations of the process of extraction of biologically active substances from plant raw materials.