Modelling the Temperature State of Bodies with Internal Heat Sources

This article presents the results of the development of a numerical - analytical method for solving the problem of thermal conductivity in a plate fuel element. An unsteady temperature field inside a fuel element is investigated for a given spatial distribution of heat sources. The heat release rate is given by the quadratic function of the coordinate. Modeling the temperature state of bodies with internal heat sources allows you to study the operation of equipment in transient modes, control heating/cooling modes of elements, determine temperature stresses, etc. It is shown in the work that regardless of the power of internal sources of heat, the temperature state is stabilized at a temperature level that depends on the Pomerantsev number.

PREDICTION OF THERMAL ERRORS IN MACHINE TOOLS THROUGH DECOUPLED SIMULATIONS USING GENETIC ALGORITHM AND ARTIFICIAL NEURAL NETWORKS

Thermal errors are one of the major contributors towards positioning discrepancies in machine tools in precision machining. Along with friction and waste heat generated from production processes and internal heat sources, environmental influences around the machine tool create considerable thermal gradients followed by non-linear structural deformations. Efficient quantification of these three contributing sources of thermal errors are required in order to formulate a reliable thermal-error compensation system. The creation of all possible thermal configurations, which a machine tool could be subjected to, is experimentally infeasible and requires complex and time-consuming coupled flow and thermo-structural simulations. This paper presents a new approach in thermal error prediction by using CFD and finite element (FE) simulations to train a three-level interconnected neural network system. The first level essentially decouples flow simulations from thermo-structural simulations using optimal FE node points found using a Genetic Algorithm (GA), which significantly reduces the required training data. The boundary convection data obtained from this level is used in the second level to predict possible thermal configurations of the machine tool, after careful consideration of parameters related to internal heat sources and production processes. The third level maps these thermal configurations onto displacements on the machine tool.

Heating in Magnetar Crusts from Electron Captures

The persistent thermal luminosity of magnetars and their outbursts suggest the existence of some internal heat sources located in their outer crust. The compression of matter accompanying the decay of the magnetic field may trigger exothermic electron captures and, possibly, pycnonuclear fusions of light elements that may have been accreted onto the surface from the fallback of supernova debris, from a disk or from the interstellar medium. This scenario bears some resemblance to deep crustal heating in accreting neutron stars, although the matter composition and the thermodynamic conditions are very different. The maximum possible amount of heat that can be released by each reaction and their locations are determined analytically taking into account the Landau–Rabi quantization of electron motion. Numerical results are also presented using experimental, as well as theoretical nuclear data. Whereas the heat deposited is mainly determined by atomic masses, the locations of the sources are found to be very sensitive to the magnetic field strength, thus providing a new way of probing the internal magnetic field of magnetars. Most sources are found to be concentrated at densities 1010–1011 g cm−3 with heat power W∞∼1035–1036 erg/s, as found empirically by comparing cooling simulations with observed thermal luminosity. The change of magnetic field required to trigger the reactions is shown to be consistent with the age of known magnetars. This suggests that electron captures and pycnonuclear fusion reactions may be a viable heating mechanism in magnetars. The present results provide consistent microscopic inputs for neutron star cooling simulations, based on the same model as that underlying the Brussels-Montreal unified equations of state.

RESULTS OF SIMPLIFIED MATHEMATICAL MODELING OF LIQUID TEMPERATURE DISTRIBUTIONIN THE HEAT EXCHANGE CHANNEL OF AN ELECTRIC HEATER WITH AN INTERNAL HEAT SOURCE

Indirect electrical resistance heating systems are heat-ers with internal heat sources and are widely used in agri-culture for heating gaseous and liquid media. Such sys-tems are characterized by insufficient intensity of heat ex-change processes. This implies a large heat transfer sur-face area and significant geometric size. Earlier, an attempt was made to solve the problem of increasing the efficiency of heat transfer processes and minimizing the geometric size of the heat exchanger. For this purpose, the heat ex-change characteristics were simulated and the geometric dimensions of three heat exchange systems were deter-mined: “pipe with internal heat sources in a dielectric pipe”, “pipe with an internal heat source -a membraneof heated liquid” and “cylindrical coil -heated liquid”. The analysis of these heat exchange systems has shown that the most promising is a coil-type heat exchanger. This system has the best heat transfer characteristics and the most compact size. To confirm the correctness of the applied method of calculating the heat exchange and geometric parameters of the heat exchanger, the simulation of the temperature dis-tribution of the heated liquid in the channel of the coil heat exchanger is implemented in this work. The verification calculations carried out under the formulated assumptions, using the example of a coil heat exchanger, show that the method for determining the heat exchange and geometric parameters of heat exchangers is correct. As a result of the simulation, it has been found that the error in determining the required channel length of the coil heat exchanger, the number of turns and the height of the coil to reach the liq-uid temperature at the outlet of 75°C does not exceed 4%. A similar conclusion can be made regarding the heat ex-changers of the types “pipe with internal heat sources -heated liquid” and “pipe with an internal heat source -a membraneof heated liquid”.

CONCEPTION QUASIDERIVATIVES IN THE PROBLEMS OF MATHEMATICAL MODELING OF HEAT TRANSFER

In this paper, in closed form, the problems of determining stationary temperature fields in multilayer (flat, cylindrical and spherical) structures in the presence of discrete-continuous internal and point heat sources are solved. The one-dimensional differential equation of thermal conductivity in different coordinate systems is given through one parametric family of quasi-differential equations. It is assumed that the coefficients of the differential equation of thermal conductivity are piecewise constant functions. A system of two linearly independent boundary conditions is added to the equation, which in the general case are nonlocal. The solutions of such problems are constructive and are expressed exclusively through their initial data. The basic provisions of the concept of quasi-derivatives, the provisions of the theory of heat transfer, the theory of generalized systems of linear differential equations, elements of the theory of generalized functions are used. For the mathematical model of stationary thermal conductivity, the practical use of the concept of quasi-derivatives is illustrated, for the efficient construction, in a closed form, of solutions of boundary value problems with the most general boundary conditions. As an example, the problem of finding the critical radii of thermal insulation of multilayer hollow cylinders and spheres, taking into account the internal heat sources in the layers. Boundary conditions of the first and third kind. It is established that the value of the critical radius does not depend on the number of layers and the intensity of internal heat sources, but only on the thermal conductivity of the outer layer of the structure and the heat transfer coefficient between the structure and the environment. The formula for determining the critical radius of thermal insulation for a multilayer cylindrical and spherical structure is derived. The methods developed in this work have the prospect of further development and can be used in engineering calculations.

SIMULATION OF HEAT EXCHANGE IN A MULTILAYER FLAT STRUCTURE UNDER PERFECT THERMAL CONTACT IN FIRE CONDITIONS

The proposed work is devoted to the application of the direct method to the study of heat transfer processes in a multilayer flat structure. It is assumed that each layer is made of isotropic material of different thickness. There is an imperfect thermal contact between them, and the layers have internal heat sources. In this case, the isothermal surfaces are parallel planes, i.e the temperature changes in only one direction. On the outer surfaces of the structure there is a convective heat exchange with the environment, i.e the boundary conditions of the third kind are fulfilled. The coefficients of the thermal conductivity equation are considered to be piecewise constant with respect to the spatial coordinate. This is the first time the problem has been solved in this setting. The solution of the problem is realized by applying the method of reduction using the concept of quasi-derivatives and applying the theory of systems of differential equations with impulse action. The following is the procedure for separating Fourier variables using a modified method of eigenfunctions.Based on the physical content of the problem, the differential equation of thermal conductivity was written in the Cartesian coordinate system, but the solution scheme presented here without any fundamental difficulties extends to similar problems for multilayer bodies of basic geometric shapes by switching to appropriate coordinate systems. To illustrate the proposed method, a model example of finding the distribution of a nonstationary temperature field in a seven-layer flat structure under the influence of the hydrocarbon temperature of the fire is solved. The condition of ideal or non-ideal thermal contact is fulfilled between two adjacent layers. In addition, some layers have internal heat sources. The results of the calculations are presented in the form of a graph of temperature changes depending on timeand spatial coordinates.