Муҳаммад ал-хоразмий номидаги тошкент ахборот технологиялари
Download 3.44 Mb. Pdf ko'rish
|
KIBER XAVFSIZLIK MUAMMOLARI VA ULARNING (1)
|
Разделяя отрезок [a,b] на n частей можно найти, что ℎ = 𝑏−𝑎 𝑛 , 𝑥 𝑖 = 𝑎 + ℎ𝑖, 𝑖 = 0, 𝑛 (1.4) далее заменяя производные, в краевой задаче с конечно - разностными соотношениям можно найти, что 𝐴 𝑖 𝑦 𝑖+1 −2𝑦 𝑖 +𝑦 𝑖−1 ℎ 2 + 𝐵 𝑖 𝑦 𝑖+1 −𝑦 𝑖−1 2ℎ + 𝐶 𝑖 𝑦 𝑖 = 𝑓 𝑖 (1.5) 𝑎 1 𝑦 0 + 𝛽 1 𝑦 1 −𝑦 0 ℎ = 𝛾 1 (1.6) 𝑎 2 𝑦 𝑛 + 𝛽 2 𝑦 𝑛 −𝑦 𝑛−1 ℎ = 𝛾 2 (1.7) 56 Сеточные уравнения (1.5-1.7) удобно свести к следующему известному виду [2] 𝑎 𝑖 𝑦 𝑖+1 + 𝑏 𝑖 𝑦 𝑖 + 𝑐 𝑖 𝑦 𝑖−1 = 𝑓 𝑖 (1.8) 𝛼 01 𝑦 0 + 𝛽 01 𝑦 1 = 𝛾 01 (1.9) 𝛼 02 𝑦 𝑛−1 + 𝛽 02 𝑦 𝑛 = 𝛾 02 (1.10) Известно, что матрица системы уравнений (1.8-1.10) имеет трех диагональный вид, и удобно применение метода прогонки [3]. Теперь мы продемонстрируем другой способ решение конечно-разностных уравнений (1.5- 1.7). Для чего уравнение (1.5) разрешаем относительно 𝑦 𝑖 , а краевые условия относительно 𝑦 0 и 𝑦 𝑛 , соответственно. Тогда получим следующую систему 𝑦 𝑖 𝑘+1 = (𝑦 𝑖+1 𝑘 (− 𝐴 𝑖 ℎ 2 − 𝐵 𝑖 2ℎ ) + 𝑦 𝑖−1 𝑘 (− 𝐴 𝑖 ℎ 2 + 𝐵 𝑖 2ℎ ) + 𝑓 𝑖 ) , 1 ≤ 𝑖 ≤ 𝑛 − 1 (1.11) 𝑦 0 𝑘+1 = (𝑎 1 − 𝛽 1 𝑦 1 𝑘 ℎ ) (𝑎 1 − 𝛽 1 ℎ ) , 𝑖 = 0 (1.12) 𝑦 𝑛 𝑘+1 = (𝑎 2 + 𝛽 2 𝑦 𝑛−1 𝑘 ℎ ) (𝑎 2 + 𝛽 2 ℎ ) , 𝑖 = 𝑛 (1.13) где k – номер итераций. Заметим, что в отличие от метода прогонка, эту систему уравнений можно решит методам последовательных приближений. При этом, за нулевое приближение во внутренних точках принимаем тривиальные значения искомых величин. Рассмотрим следующую конкретную краевую задачу[1] 𝑦 ′′ + 2𝑥𝑦 ′ − 2𝑦 = 2𝑥 2 , 0 ≤ 𝑥 ≤ 1 (1.14) 𝑦(0) = 1 (1.15) 𝑦(1) = 0 (1.16) Численные результаты краевой задачи (1.14-1.16) полученные по итерационному методу (1.11-1.13), методу прогонки [3], а также вычисленные согласно точному решению приведён в таблице 1.1. таблица 1.1 𝑥 𝑖 𝜀 = 0.001, 𝑛 = 10 количество итераций 119 Приближенное решение Решение по методу прогонки 0 1.0000 1.0000 0.1 0.8101 0.8100 0.2 0.6404 0.6400 0.3 0.4902 0.4900 0.4 0.3602 0.3600 0.5 0.2500 0.2500 0.6 0.1603 0.1600 0.7 0.0903 0.0900 0.8 0.0400 0.0400 0.9 0.0101 0.0100 1 0.0000 0.0000 57 Как видно из таблицы численные результаты полученные двумя способами совпадают, при этом приближенное решение получено с точностью 𝜀 = 0.001. Литература 1. Г.Н.Воробьева, А.Н.Данилова. Практикум по вычислительной математики. Москва 1990. 2. В.Д.Колдаев. Численные методы и программирование. Москва 2009. 3. Б.П.Демидович, И.А.Марон, Э.З.Шувалова. Численные методы анализа. Москва 1967. 4. П.И.Монастырный. Сборник задач по методам вычислений. Москва 2000. THREE-DIMENSIONAL SIMULATION OF HEAT AND MOISTURE TRANSFER IN POROUS BODIES I.U. Shadmanov Bukhara State University, istam.shadmanov89@gmail.com Modern drying theory is based on the laws of moisture and heat motion in a material to be dried. Moisture moves in a porous body both at a drop in moisture content and a drop in temperature. The most effective motion of moisture and heat in colloidal capillary-porous bodies, which include agricultural products, is due to the correct combination of heating temperatures and the material moisture content. This condition is crucial for preserving the viability of seeds and the quality of agricultural products (grain, oilseeds, raw cotton, vegetables, etc.). A mathematical model of the thermal state of a porous body in the form of a rectangular parallelepiped was developed in [1]. The model takes into account internal heat release, heat transfer through the surfaces of a porous body into the 𝜀 = 0.001, 𝑛 = 20 количество итераций 83 𝑥 𝑖 Приближенное решение Решение по методу прогонки 𝑥 𝑖 Приближенное решение Решение по методу прогонки 0 1.00000 1.0000 0.55 -0.5401 -0.5400 0.05 0.7601 0.7600 0.6 -0.5603 -0.5600 0.1 0.5402 0.5400 0.65 -0.5604 -0.5600 0.15 0.3401 0.3400 0.7 -0.5401 -0.5400 0.2 0.1600 0.1600 0.75 -0.5000 -0.5000 0.25 0.0000 0.0000 0.8 -0.4402 -0.4400 0.3 -0.1401 -0.1400 0.85 -0.3601 -0.3600 0.35 -0.2601 -0.2600 0.9 -0.2603 -0.2600 0.4 -0.3603 -0.3600 0.95 -0.1402 -0.1400 0.45 -0.4401 -0.4400 1 0.00000 0.00000 0.5 -0.5000 -0.5000 58 ambient. To solve the problem, a modification of differential-difference equation was developed based on the straight line method. An expanded system of potential conductivity due to the equation for static pressure, based on the forces of the surface tension of fluid and the laws of capillary action, was given in the study by A.V. Lykov and Yu.A. Mikhailov [2]. Lykov A.V. has stated that during intense heating of a capillary-porous body, the drying kinetics can depend not only on the gradient of moisture transfer potential, but also on the temperature gradient and the gradient of internal pressure [3]. In [4], a mathematical model of the thermal state of the mass of porous bodies was given under conditions of its own heat release, solar radiation, and heat exchange with ambient air, the last two factors of which are of periodic nature. Statement of the problem. Given the main indices of the drying and storage processes of porous bodies, a mathematical model of heat and moisture transfer proposes the following system of differential equations: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (1) (2) t v v t T T T T u u u a a f x y z x y z u u u u T T T a a q x y z x y z = + + + + + + = + + + + + + with corresponding boundary and initial conditions: ( ) ( ) 0 , , ,0 , , T x y z T x y z = , ( ) ( ) 0 , , ,0 , , u x y z u x y z = ; (3) ( ) ( ) 1 1 1 0 oc x T T T R x = = − − − ; (4) ( ) ( ) 1 1 2 x oc x L T T T R x = = − − − ; (5) ( ) ( ) 1 1 3 0 oc y T T T R y = = − − − ; (6) ( ) ( ) 1 1 4 y oc y L T T T R y = = − − − ; (7) 0 0 z T z = = ; (8) ( ) ( ) 1 1 5 z oc z L T T T R z = = − − − ; (9) ( ) ( ) 2 2 0 0, , , oc x u u u y z x = = − − ; (10) ( ) ( ) 2 2 , , , x oc x x L u u u L y z x = = − − ; (11) 59 ( ) ( ) 2 2 0 ,0, , oc y u u u x z y = = − − ; (12) ( ) ( ) 2 2 , , , y oc y y L u u u x L z y = = − − ; (13) 0 0 z u z = = ; (14) ( ) ( ) 2 2 , , , z oc z z L u u u x y L z = = − − . (15) Here Т and u are the changes in temperature and moisture content of a porous body; 1 1 t a c = ; 2 2 v a c = ( ) 2 / m s are the coefficients of thermal diffusivity and moisture conductivity; 1 , 2 are the coefficients of thermal conductivity and moisture conductivity; is the body density ( ) 3 / kg m ; specific heat 1 с and moisture capacity 2 с ( ) ( ) / J kg K ; ( ) , , , f x y z b e − = is the intensity of internal heat release of the mass ( 1 K s − ); 1 u b c = is the heat release coefficient, which depends on the moisture content of porous bodies, so, ( ) ( ) , , , b f u x y z t = ; is the empirical parameter; ( ) 0 , , , q x y z m e − = the intensity of internal moisture sources; at constant values of the material density ( ) 3 / kg m ; is the drying coefficient ( ) 1/ sec ; 0 m is the maximum evaporation rate ( ) 2 / sec kg m . It should be noted that the evaporation rate is calculated using 0 m m e = . If to assume that 0 = then 0 m m const = = , we get a constant evaporation rate corresponding to the first drying period; 1 is the heat transfer coefficient between the mass and the air surrounding it; oc T is the ambient temperature; 1 , 2 , 3 , 4 , 5 are the coefficients to reduce the boundary condition to dimensional form; is the coefficient of sunlight absorption by the material; ( ) R is the insolation of the flow of solar radiation on the surface of the stored material; 2 is the moisture transfer coefficient between the mass and the air surrounding it; oc u is the moisture content of the ambient. The external shape of the porous body is taken as a rectangular parallelepiped. The parallelepiped is located in the first octant of the Cartesian coordinate system, and its dimensions in terms of coordinates are , , x y z L L L . Since the stated problem is described by a system of partial differential equations, it is difficult to obtain an analytical solution. 60 So, a mathematical model of the process of heat and moisture transfer in natural products under the influence of internal heat dissipation and ambient temperature, which occurs due to solar energy, is developed in the paper. Download 3.44 Mb. Do'stlaringiz bilan baham: 
|