Determination of Temperature Field in a Solid Circular Cylinder in the Fire

Abstract

The proposed work is devoted to the application of the direct method to the study of heat transfer processes in a solid homogeneous cylindrical structure in the presence of convective heat exchange with the environment, that is, the boundary conditions of the third. It is assumed that the law of change of ambient temperature, which washes the surface layer of the structure, is an arbitrary function of time. This function is uniformly distributed over the surface such that the isotherms inside the structure are coaxial cylindrical surfaces, that is, the temperature field inside the cylinder depends only on the radius r and the time τ.This problem is solved by applying the reduction method, when the original problem is represented by the sum of two unknown but related functions. It is found that the solution of the corresponding quasistationary problem does not depend on the radius r and is equal to the law of change of the ambient temperature.In the following, the corresponding inhomogeneous problem is solved, the main stage of which is the eigenvalue problem obtained after applying the Fourier variable method. As a standard procedure, a characteristic equation is obtained and the eigenvalues are determined and their own functions are constructed. The Fourier method was used to construct a nonuniform problem solution.It is found that when at the time τ = 0 the initial distribution of the structural temperature field and the ambient temperature coincide, the solution of the original problem is greatly simplified.To illustrate the proposed method, a model example of finding the distribution of the temperature field in a reinforced concrete homogeneous column of a circular cross section under the conditions of the temperature of the hydrocarbon fire is solved. The results of the calculations are presented as a graph of temperature change as a function of time. The numerical implementation of the method was performed using the Maple 13. computer algebra system. It should be noted that the first 30 roots of the characteristic equation were used to achieve the result with the given accuracy. The results obtained are directly applicable in a number of applications.