The numerical algorithm for calculating temperature fields of the pneumatic tires during vulcanization

In the article discussed the mathematical formulation and numerical algorithm for solving the problem of calculating the temperature field in the process vulcanizing of the product, whose the thermal characteristics are depended on the temperature. As a mathematical model considered the system of differential equations of heat conduction, taking into account the change in the coefficients of thermal conductivity and heat density in multilayer product of the temperature. The system of equations is solved for a given initial distribution of temperature and for a given (time-dependent) temperatures on the border of the product to the press-mold and to the diaphragm. On the border of the contacts of adjacent layers are given the condition of continuity of temperature and heat flux. Change of the thermal conductivity from the time is approximated by linear functions. The activation energy of the vulcanization process is determined on the basis of experimental data obtained in the control test samples using a reometer. Considering the function representing the corresponding integrals of the thermal conductivity, the original system of differential equations is transformed to an equivalent system of differential equations convenient for constructing numerical algorithms for solving the problem. The resulting system of partial differential equations derived using the method of finite-difference approximation is replaced by a system of algebraic equations. Solution of the system of algebraic equations is carried out under the scheme explicit difference approximation. In the article calculated the temperature field for the tire at given initial and boundary conditions. Stability and accuracy of the numerical algorithm for solving the problem is demonstrated by the calculations performed with different sampling step along the time and space coordinates. Assessment of the degree of completion of the process is carried out by calculated equivalent time for temperature values, accepted as equivalent. The developed algorithm is an important part of the algorithm for solving the problem of determining the optimal mode of curing, providing the required quality products at the lowest cost, which is very important in conditions of continuous growth in energy prices.