American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 03, April, 2022 122 | P a g e MATHEMATICAL DESCRIPTION OF THE VEGETABLE DRYING PROCESS Jumaev Botir Assistant, Tashkent Chemical Technological Institute Samadov Otabek Assistant, Tashkent Chemical Technological Institute Tuxtauev Shuxrat Qudratovich Sen.Teacher, Tashkent Chemical Technological Institute Shomirov B. Рost.Doctoral. Tashkent State Technical University Annotation A mathematical model of IR-convective drying of fruits is constructed. The temperature gradient is expressed by the ratio of the time cyclic effect of infrared heating on the material, experimentally justified by optimizing a given coefficient of air flow and raw materials. The analytical form allows us to investigate the influence and interaction of nonlinear effects. Keywords: а mathematical model, IR-convective drying, raw materials, analytical form, optimization. Introduction Certain results are achieved in the processing of fruits grown in the country, the development of semi-finished and ready-made food products, saving energy resources for products, creating resource-saving drying technologies that ensure the production of high-quality products. The combination of these tasks, including the development of dry food technology with drying methods using electromagnetic waves in the IR ranges of energy transmission, drying according to energy and temperature dependence in order to save energy and shorten the duration of the process, the use of optimal pretreatment Parameters for dried raw materials research aimed at accelerating the drying process by heating and convection, are of great importance. In order to properly understand and calculate the mechanism of the drying process, it is necessary to know the heat and mass transfer properties of materials and the degree of their impact on the heating time and the set temperature. In agroengineering, chemical technology, biology and medicine, tubes with filter walls are often found, where the external environment affects the flow of impulses inside the American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 03, April, 2022 123 | P a g e tubes. An example would be medicine, where treatments are carried out using a hemodialysis machine. Examples can also be the tasks of soil irrigation using water filtration pipelines [1-3]. The amount of liquid evaporated from the surface increases with increasing temperature, as well as the flow of liquid from their channels moving also towards the free surface increases with increasing temperature (Fig. 1.). Fig. 1. Elementary microchannel in the form of a thin-walled tube. M is the power of the juice flow in the microchannel; P is the juice pressure inside the microchannel; RVN is the juice pressure around the tube; γ (t) is the filtration coefficient of the channel wall depending on the properties of the drying material. We analyze an experiment to study the dependence of steady-state flow on temperature (Fig. 2). Fig. 2. Diagram of the laboratory experiment The volume made of heat-resistant plastics is placed inside the microwave oven. There is a flat layer of pumpkin inside the plastic. Its horizontal surface is (20x20) cm2, and the layer thickness is 2.5 cm. When the microwave is turned on, i.e. the MV furnace, the layer is uniformly heated. In the field of temperature stabilization, for each individual process we have, within 2 hours, evaporation from the surface of the pumpkin [4,5]. The moisture flow from the inner layers is directed upwards. The change in mass Δm is determined from the formula Δmg=k(x_2-x_1) American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 03, April, 2022 124 | P a g e where, Δm is the change in the mass of the layer, p is the acceleration of gravity, k is the stiffness coefficient, x_2-x_1 is the change in the equilibrium point (4) The steady–state regime is determined experimentally. At 600C, named M (600), at 600C, named M (600), etc. got a schedule: Theoretically, it can be shown using the stationary equation 𝐴 { 𝑃𝑥 = −𝜀𝑢 2 𝑢𝑥 = 𝛾(1 − 𝑃вы 𝑃 } (1) 𝑀 = [𝑀0 − 𝑃 3(𝛾𝑃в𝑙𝑛𝑃 − 𝑃)] 1 3 (2) If, we enter 𝑃 = 𝑃0 + 𝑃 `, 𝑃` 𝑃0 ≪ 1 then from (2) we have an arbitrary formula: i.e. the graph and its. The curvature is explained by the nonlinear mechanism of dependence. Let's write the law of fluid conservation for these tasks 𝑝𝑡 + 𝑝𝑢𝑥 = 𝑓(𝑃0 − 𝑃вн, 𝑡) = 𝑓(𝑃 − 𝑃вн, 𝑡) (3) where p - is the density of the liquid in the tube; u - is its hydrodynamic velocity; P0 is the pressure of the liquid inside the tube; Phn is the external pressure around the tube. Now let's write the equations of motion 𝑝𝑢𝑡 + 1 𝑐2 𝑝𝑥 = −𝐹 where F - is a function of hydrodynamic drag forces, in particular: 𝑝𝑢𝑡 + 𝑝𝑥 = −𝜀𝑝U2 (4) Assuming that, the equation of state P=P(p), has the form Р= е2∙р were: е2 = 𝑑𝑃 𝑑𝑝 , we obtain from (3) and (4) by entering φ=ln P/p_0 where, p0 - is the density without motion [3,4] { 𝑢𝑡 + 𝑐 2𝜑𝑥 = −𝜀𝑢2 𝜑𝑡 + 𝑢𝑥 = 𝐹 ∙ 1 𝑃 (5) Let in (3) we have a replacement: { 𝐴 = 𝑢 + 𝑐𝜑 = 𝑢 + 𝑒𝑙𝑛 𝑃/𝑝0 𝐵 = 𝑢 − 𝑐𝜑 = 𝑢 − 𝑒𝑙𝑛 𝑃/𝑝0 𝜁 = 𝑥 − 𝑐𝑡 𝜉 = 𝑥 + 𝑐𝑡 𝐴пр, 𝜀=0, 𝐹=0 = 𝐴0(𝜁) 𝐵пр, 𝜀=0, 𝐹=0 = 𝐵0(𝜉) (6) For A and B we get a rewrite (6) { 𝐴𝑡 + 𝑐𝐴𝑥 = −𝜀𝑢2 + 𝐹 𝑝 𝐵𝑡 − 𝑐𝐵𝑥 = −𝜀𝑢2 − 𝐹 𝑝 (7) In system (7), the linear solution in the absence of filtering is written as: 𝐴 = 𝐴(𝑥 − 𝑐𝑡) и 𝐵 = 𝐵(𝑥 − 𝑐𝑡) American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 03, April, 2022 125 | P a g e For nonlinear and for F≠0, we have, by adding both equations and subtracting the second from the first, after the transition from A=A(ζ,ξ), B=B(ζ,ξ) the following: { 𝑑(𝐴+𝐵) 𝑑𝑡 = −2𝜀𝑈2 𝑑(𝐴−𝐵) 𝑑𝑡 = − 2𝐹 𝑝 (8) from (6) we have that А+В=2u, 𝐴 − 𝐵 = 2𝑐𝜑 = 2𝑐 ln 𝑃 𝑝0 from system (8) we have { 𝑑𝑢 𝑑𝑡 = −𝜀𝑢2 𝑑𝑝 𝑝 = 𝐹 𝑐 (9) for integration, consider an example 𝐹(𝑝, 𝑡) = 𝑓(𝑝 − 𝑝ВН) ∙ 𝛾(𝑡) We assume that 𝑓 = 𝑝 − 𝑝ВН = 𝑐2(𝑝 − 𝑃ВН 𝑐2 ) By for the second equation, we have 𝑑𝑝 𝑝 = 𝑐𝛾(𝑡) ∙ (𝑝 − 𝑝ВН 𝑐2 ) 1 𝑝 or 𝑑𝑝 𝑝 − 𝑝ВН 𝑐2 = 𝛾(𝑡)𝑑𝑡 or 𝑝 = 𝑝ВН с2 + (𝑝𝑡=0 − 𝑃ВН 𝑐2 )𝑒∫ 𝛾 𝑡 0 𝑑𝑡 (10) for the second equation we have solutions for the hydrodynamic velocity 𝑢 = 𝑢|𝑡=0 1+𝜀𝑡𝑢|𝑡=0 (11) for pressure we have, multiplying (8) by c2 𝑝 = 𝑝ВН + [𝑝𝑡=0 − 𝑝 ВН]𝑒∫ 𝛾 𝑡 0 𝑑𝑡 (12) for the expense, we have 𝑀 = 𝑝𝑠𝑢 = 𝑠 𝑢|𝑡=0 𝑐2 [𝑝ВН − (𝑝𝑡=0 − 𝑝 ВН)𝑒∫ 𝛾 𝑡 0 𝑑𝑡] 1 1+𝜀𝑡𝑢|𝑡=0 ; (13) We have obtained an exact solution (3) for a specific form if the source function 𝐹 = 𝛾 ∙ 𝑓(𝑝 − 𝑝ВН) - is given within our constraints (13). American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 03, April, 2022 126 | P a g e Generally speaking, if 𝑃−𝑃ВН 𝑃ВН ≪ 1, then we can decompose (13) into a Taylor series, and get 𝑓 = 𝑓|𝑃=𝑃ВН + 𝛼 ∙ (𝑝 − 𝑝 ВН where 𝛼 = 𝑑𝑓 𝑑(𝑃−𝑃0) | 𝑃=𝑃0 , its solution is similar to (11), (12) The resulting solution can also be applied to the problems of pulse stimulation of filtration when p=p_0 (t) and p=p_0 (ζ,ξ,0). Microchannels where there is a low velocity, i.e. u≪c, has a sufficiently large ε, in addition, ε depends inversely on the diameter of the tube.When the juices move with the direction to the surface, cracks or crack-like channels are observed. Sometimes, such cracks are created artificially in order to accelerate drying. And it is called "scalding". Therefore, the problem of neglect with nonlinearity remains open. Solution (10-12) allows us to evaluate its influence of the above-described nonlinearities. The solution γ(t) may have different signs depending on the change in the pressure gradient between the fluid pressure in the tube p, and between the external pressure 𝑝виaround the tube.The value of γ(t), still depends on the presence of pores on the walls, i.e. from their area per unit length of the tube. In conclusion, we note that the analytical form allows us to study the influence and interaction of nonlinear effects. References 1. К.Т.Норкулова, П.М. Матякубова, М.И. Мамасалиева. Новые интегральные схемы сушки. // EUROPE, SCIENCE AND WE. International Conference 2020 Praha, Czech Republic Conference Proceedings-101-102 с 2. Бозоров О.Ш., Маматкулов М. Нелинейные бегущие волны в трубопроводах // Вестник ТашГТУ, 2008. - №4.- С.28-30. 3. Норкулова К.Т., Маматкулов М.М., Жумаев Б.М., Шайзаков Б.А. Расчет вынуждающих сил для колебаний дольки продуктов при использовании скоростных прерывателей. // Проблемы энерго- и ресурсосбережения. – Ташкент. 2019. №3. С.125-126. 4. Norkulova K.T., Mamatqulov M.M., Jumaev B.M. The task of stabilizing drying with the help of accumulation material in case of overheating of the chamber under the action of sunlight. Chemical Technology. Control and Management: Vol. 2018: Iss.4. P.16-20. 5. O.B. Samadov, Sh.K. Tukhtaev, A. Zh. Choriev, K.O. Dodaev, M. Ch. Tultabaev. Investigation of changes in pumpkin moisture during infrared convective drying // Bulletin of the Kazakh University of Technology and Business. Kazakhstan, No. 4, 2019.-p.47-51. American Journal of Interdisciplinary Research and Development ISSN Online: 2771-8948 Website: www.ajird.journalspark.org Volume 03, April, 2022 127 | P a g e 6. Khamidovna N. S. On The Use of Infinitive Groups in German and Uzbek Languages //Texas Journal of Multidisciplinary Studies. – 2021. – Т. 1. – №. 1. – С. 202-204. 7. Sofiboyeva PhD G. M. DEVELOPING PUPILS’LOGICAL THINKING ABILITY IN THE STUDY OF GEOMETRIC MATERIALS IN MATHEMATICS OF PRIMARY SCHOOLS //Central Asian Journal of Education. – 2021. – Т. 6. – №. 1. – С. 1-9. 8. Sofiboyeva G. DEVELOPING IMAGINATION ABOUT SPACE OF PRIMARY SCHOOL STUDENTS IN THE LEARNING PROCESS //International Scientific and Current Research Conferences. – 2021. – С. 4-8. 9. Nazikhovna G. Y. PROGRAMMING AND ROBOTICS BASED IN STEAM LEARNING //American Journal of Interdisciplinary Research and Development. – 2022. – Т. 2. – С. 58-87. 10. Yunusova G. N. THE PROGRAM FRONT PAGE-PROGRAM OF MAKING WEB PAGE AND E-BOOK //Scientific Bulletin of Namangan State University. – 2020. – Т. 2. – №. 3. – С. 230-233.