ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF HEAT CONDUCTIVITY FOR DAMAGED THERMAL INSULATION OF PIPELINES

Abstract

For heat supply systems of Ukraine, the determination and forecasting of thermal energy losses during the transportation of heat carrier is an urgent problem. The thermal lines of the heating networks are long and its insulation is damaged. Installation of heat energy metering devices at all sources and all consumers (i.e. buildings) without exception allows determining and forecasting real heat losses in the heat network, but this is a difficult task, not all consumers and sources are actually covered by heat consumption accounting. The task of determining heat losses is also relevant for energy management systems of heat supply systems, energy companies and industry. The solution to the problem is proposed by a separate mathematical modeling of the temperature state of the section of the damaged insulating layer with the determination of the heat flow through it. It is proposed to solve the problem by the method of analytical solution of the differential equation of heat conduction with boundary conditions of the third kind. With the uniform distribution of characteristic damages along the total length of the pipeline, knowing the limits of damage influence, the share of insulation damage and the number of damages on the pipeline, it is possible to determine the real heat flow from the outer surface of the pipelines, including coefficient of increase of heat losses on the section of the heat pipe in relation to those determined by regulatory documents. Traditionally, one-dimensional models or determinations of the two-dimensional temperature field and actual heat fluxes in a cross-section to the axis of a characteristic damaged section of insulation are considered. But this does not take into account the two-dimensionality of the temperature field of the damaged layer of insulation along the length (that is, along the axis) of the pipeline. Therefore, the purpose of this work is to develop a methodology for determining heat losses through pipelines, taking into account the damage to their insulation and the distribution of characteristic damages along the length. The following simulation was carried out for one of the areas. Also, experimental and numerical studies using the finite difference method in combination with the method of running variable directions for similar models confirmed the coincidence of the analytical solution of the proposed model and the finite difference model within the permissible error.