Kinds Of Boundary Conditions Research Articles

Analytical solution of two-dimensional unsteady heat conduction in multi-layered cylindrical sectors applied to thermal plug and abandonment of oil wells

In this work, a hybrid analytical and numerical solution for transient heat conduction across a composite cylindrical sector is presented. A two-dimensional domain consisting of a multi-layer circular sector of angle φ was investigated (φ<2π). The Separation of Variables Method (SVM) was applied to solve the partial differential equation with non-homogeneous boundary conditions of the first, second and third kinds prescribed in the radial direction. Homogeneous boundary conditions of first and second kinds were arbitrated in the angular direction. A spatial time-independent source term gi(r,θ) was considered. The radial eigenvalues problem for the (r,θ) domain returns only real quantities and depends implicitly on the angular eigenvalues. Results for time dependent temperatures using the Separation of Variables Method were compared with numerical results, showing good accuracy. A second set of results was developed to investigate boundary conditions, material properties and the thermal source power required to rise temperature levels (mainly around the mid-angle φ/2) high enough to promote melting of certain layers of materials. These results might be useful for investigating a novel technology for the decommissioning of oil wells using thermal sources, often referred to in the literature as Thermal Plug and Abandonment (TP&A).

Thermal experimental and numerical heat transfer analysis of a solid cylinder in longitudinal direction

The analysis of heat transfer in solid bodies with orthogonal geometries and knowledge thereof, is of vast importance in different fields of engineering and research. An important field of study is the thermal analysis in machine-parts that in most cases are designed and shaped with orthogonal geometries. Nevertheless, due to the high complexity and the cost that thermal experiments represent, FEM analysis and numerical solutions are used to foresee thermal fields on these components. These methodologies are certainly reliable, although may vary from real experiments. On that account, this paper presents a thermal experimental test in a solid cylinder of length and , made of ISO C45 steel that emulates a machine-part (cylindrical parts as shafts, fasteners and the like). The temperature fields along the longitudinal direction were analyzed in steady and transient state under homogeneous boundary conditions of the first kind (prescribed temperatures at the boundaries). The three solutions, experimental, numerical and FEM simulations were compared with the purpose of validating the results obtained by each method of solution respectively.

Study of the features of modeling microwave vacuum evaporation

Evaporation is the most common concentration method. The complete reuse of latent thermal energy by means of mechanical recompression is the most technologically efficient method of operation of evaporators. This method requires much less electricity to compress the secondary steam compared to the heat of the secondary steam. In this scheme, the energy released during steam condensation can be used to heat the initial solution, which makes it possible to achieve the reuse of thermal energy at the level of 95%. Thus, during the the evaporation process implementation, when energy is supplied by electromagnetic microwave radiation, conditions are created for the initiation of vaporization in the entire liquid volume, which will avoid product overheating in the thermal boundary layer due to the absence of such. This allows to get a product of high concentration, without the taste of “cooking”, without changing color and flavor. When modeling these processes, boundary conditions of the third kind are replaced by boundary conditions of the second kind. Considering the proposed hypothesis, a process model can be considered, assuming the presence of internal energy sources evenly distributed in the product volume. This opens up the possibility of using deterministic mathematical models, provided that the corresponding initial and boundary conditions are determined. As a result of generalization of the experimental modeling results, a model in a criterion form has been obtained, which allows calculating a microwave vacuum-evaporation apparatus of periodic action for solutions containing polar molecules. The model in the range of dimensionless pressure 2 ≤ P ≤ 22, and at the level of dimensionless heat of the phase transition 1 ≤ R ≤ 4.56 provides accuracy with a maximum deviation of ± 8%.

Temperature distribution in an elliptical body without internal heat sources under boundary conditions of the third kind with partial adiabatic isolation

The article deals with the distribution of the temperature field in an elliptical body without internal heat sources. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found when moving to an elliptic coordinate system. The author has obtained an analytical solution for the distribution of the temperature field in a body with an elliptical cross section of infinite length at a given ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions.

Determination of the temperature field of the thermoelectric element when it cools down

In this paper, the question of finding the temperature field of a rectangular thermoelectric element under boundary conditions of the third kind with an adibatically-isolated side is considered. Based on the Fourier method, the distribution of the temperature field of the thermoelectric element is obtained. The solution itself is obtained in the form of a series containing trigonometric and exponential functions. The reliability of the obtained result is confirmed by the fact that one of the special cases leads the problem to a problem with boundary conditions of the first kind, when the surface temperature is constant. The presented analysis of the main characteristics of thermoelectric converters makes it possible to determine the criteria for selecting materials and temperature conditions in order to increase the efficiency of such converters.

Temperature distribution in an elliptical body without internal heat sources under boundary conditions of the third kind with partial adiabatic isolation

The article deals with the distribution of the temperature field in an elliptical body without internal heat sources. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found when moving to an elliptic coordinate system. The author has obtained an analytical solution for the distribution of the temperature field in a body with an elliptical cross section of infinite length at a given ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions.

The temperature of the elliptical cross section under boundary conditions of the first and third kind

This paper provides a solution to the heat transfer problem for a hollow elliptical cylinder under boundary conditions of the first and third kind. In this paper, a solution is given for the distribution of the temperature field in a field with an elliptical cross section under boundary conditions of the third kind. The solution is obtained by considering equidistant surfaces.

ТЕМПЕРАТУРНОЕ ПОЛЕ ОДНОРОДНОЙ КВАДРАТНОЙ ОБЛАСТИ С ДВИЖУЩИМИСЯ БЕЗ УСКОРЕНИЯ СМЕЖНЫМИ СТОРОНАМИ ПРИ ГРАНИЧНЫХ УСЛОВИЯХ ПЕРВОГО РОДА

A square area with homogeneous thermal and physical characteristics, deformed preserving 2-D similarity, is investigated. At the initial moment of time, two adjacent sides start moving respectively towards the abscissa and ordinate axes with constant speed while remaining equidistant to the other two adjacent sides (the fixed and moving sides are kept at different constant temperatures). A nonlinear initial boundary value problem with boundary conditions of the first kind and special coordinates immobilizes the moving boundary of the area into a fixed one with the corresponding transformation of the initial boundary value problem for the fixed boundaries with respect to the multiplicative variable of two unknown functions, which are defined by additional initial boundary values. These were solved by the successive application of integral sine transformations on pseudo-space variables. This enables the solution of the original problem to be notated analytically using special quadratures. The computational experiment proved the correctness of the solution and the absolute fulfillment of the initial conditions. The results also illustrate the adequacy of the qualitative calculations for the heating process of a quadratic area with moving adjacent boundaries. This approach can be applied to the differently directed motion of adjacent boundaries, to uniformly retarded or uniformly accelerated motion. Considering that Fourier's and Fick's laws are mathematically similar, the solution and its generalization are of practical importance in describing mass transfer processes, such as crystallization or dissolution.

Influence of air temperature on thaw depth of a road foundation

The expected thaw depth of the foundation soil is one of the parameters influencing the selection of technical solutions of road design in the permafrost area. The purpose of this research is a quantitative assessment of the degree of influence of air temperature on thaw depth of the road foundation soil in the permafrost area. A standard formula to calculate thaw depth of melting bodies of flat symmetry obtained through a solution of the one phase Stefan problem at boundary conditions of the first kind was used for the analysis. An algorithm using an analytical formula to assess the influence of the main initial parameters and accuracy of their determination on the resulting value, the thaw depth of the soil, is proposed. The results of the calculations are presented in a graphical form displaying the influence of average air temperature and the accuracy of its setting on the thaw depth of the road foundation soil. It was determined that the degree of change in thaw depth is non-linear and depends on both the average air temperature and on the accuracy with which it is set. For every average air temperature value there is a specific measure of required accuracy for thermal calculations so that the error of thaw depth determination does not exceed the permitted error. A 3D chart showing the expected percentage error in thaw depth forecasting depending on the average air temperature and the accuracy of its setting for an array of initial values was created.

The local entropy rate of production at boundary conditions of the third kind

The local entropy rate of production at boundary conditions of the third kind

Heat exchange between the fuel element of a nuclear reactor made of uranium oxide and its shell under boundary conditions of the fourth kind

The work is devoted to the heat exchange of a nuclear reactor. Attention was paid to the mathematical modeling of the temperature field of the fuel element and the thin gas layer. It is assumed that the thermal contact between the solid–interlayer system is ideal. Methods of differentiation, integration and approximate methods were used for numerical modeling. The dependence of the thermal conductivity of uranium oxide on temperature was also taken into account. Based on the obtained expression, the linear heat transfer coefficient and linear thermal resistance were also found.

Heat exchange between the fuel element of a nuclear reactor made of plutonium dioxide and its shell under boundary conditions of the fourth kind

The work is devoted to the heat exchange of a nuclear reactor. Attention was paid to the mathematical modeling of the temperature field of the fuel element and the thin gas layer. It is assumed that the thermal contact between the solid–interlayer system is ideal. Methods of differentiation, integration and approximate methods were used for numerical modeling. The dependence of the thermal conductivity of plutonium dioxide on temperature was also taken into account.

Determination of the temperature field of a spherical arc

The work is devoted to mathematical modeling of the plasma temperature field. To do this, it was represented as a conducting sphere under boundary conditions of the third kind. The simulation of the thermal process is based on the numerical integration of the heat equation. This takes into account the change in the thermal conductivity of the plasma and convective heat transfer. Also, based on the obtained result, an analysis of the behavior of the temperature field was carried out.

Heat exchange between the heat-generating element of the circular section of a nuclear reactor and its shell under boundary conditions of the fourth kind

The work is devoted to the heat exchange of a nuclear reactor. Attention was paid to the mathematical modeling of the temperature field of a heat-generating element of circular cross-section and surrounding it with a thin gas layer. It is assumed that the thermal contact between the solid–interlayer system is ideal. Differentiation and integration methods were used for numerical modeling.

Stability and convergence of the locally one-dimensional scheme A. A. Samarskii, approximating the multidimensional integro-differential equation of convection-diffusion with inhomogeneous boundary conditions of the first kind

Изучена первая начально-краевая задача для многомерного (по пространственным переменным) интегро-дифференциального уравнения конвекции-диффузии. Для приближенного решения поставленной задачи предложена локально-одномерная схема А. А. Самарского с порядком аппроксимации $O(h^2+\tau)$. Исследование единственности и устойчивости решения проводится с помощью метода энергетических неравенств. Получены априорные оценки решения локально-одномерной разностной схемы, откуда следуют единственность решения, непрерывная и равномерная зависимость решения от входных данных, а также сходимость решения схемы к решению исходной дифференциальной задачи со скоростью, равной порядку аппроксимации разностной схемы. Для двумерной задачи построен алгоритм численного решения, проведены численные расчеты тестовых примеров, иллюстрирующие полученные в работе теоретические результаты.

Calculation of the temperature field of a hollow isotropic elliptical body

In this paper, the problem of heat transfer is solved for a hollow elliptical cylinder, taking into account the asymmetry of heat transfer on surfaces. In this case, the boundary conditions are boundary conditions of the first and third kind. In this paper, a solution is given for the distribution of the temperature field in a field with an elliptical cross-section under boundary conditions of the third kind. The solution is obtained by considering equidistant surfaces.

Analysis of Free-Convection Hydrothermal Fields in a Rectangular Cavity under Thermal Boundary Conditions of the Second Kind

Analysis of Free-Convection Hydrothermal Fields in a Rectangular Cavity under Thermal Boundary Conditions of the Second Kind

Improvement of thermal-hydraulic efficiency of mining power equipment through the application of porous freon steam generators with high heat conductivity

In the article, various areas of technical application of porous freon steam generators in mining power equipment are described, and explanations why freon coolants can provide a positive energy effect in such facilities are given. The paper presents results of calculations of thermal-hydraulic efficiency of the porous once-through tubular steam generators with freon-12 as a model working fluid in the the laminar flow area and with boundary conditions of the first kind. The smooth-wall cylindrical channels with different diameters were used as the reference surfaces to be compared. The following mode and design parameters were taken as a calculation base: the liquid temperature and pressure on the saturation line at the entry into the channel were: Ts0= 110 °С; P0s = 39,9·105 N/m2.; temperature heads, i.e. a difference between the wall temperature and temperature of the liquid at the entry into the channel were: ΔT=Tw-Ts0 = 1 °C; 2 °C; 3 °C; 4 °C; 5 °C; the Reynolds numbers at the entry into the channel were: Reo = 100; 200; 500; 1000; 2000; 2300; the channel porosities were: θ = 0.7; 0.75; 0.8; 0.85; 0.9. The porous material was metal felt with the copper fiber diameter of 200 microns. The channel diameters were: d =3·10-3 m; 4·10-3 m; 5·10-3 m; 6·10-3 m; 7·10-3 m. On the basis of the performed computational studies, it was concluded that for the conditions of the same mass flow rates of the coolant, with laminar flow, and the same channel diameters, it is possible to achieve a significant reduction in the length of the porous once-through steam generator in comparison with the length of the smooth-wall once-through steam generator. Due to the significantly shorter length, differential pressure for pumping the coolant can be several tens of percent less in porous evaporation channels than in the similar smooth-walled channels. This computational study also made it possible to establish main regularities in dynamics of the energy efficiency coefficients and their dependence on the model mode and design parameters. It was shown, that positive dynamics of the efficiency coefficients of porous steam generators occurs with decrease of the channel diameter and temperature head, as well as with increase of the Reynolds number in the investigated region of coolant laminar flow.

Heat exchange in fuel rods at different cross sections

The work is devoted to the process of heat exchange in fuel rods of different cross-sections. It considers a stationary process without internal heat sources under boundary conditions of the first kind. The nature of the change in the temperature field during the transition from a cylindrical to an elliptical section is shown. A formula was also obtained for finding the temperature in a fuel element formed by elliptical surfaces of co-focal ellipses

Cooling of an infinite rectangular plate with an adibatically insulated side

The work is devoted to the issues of non-stationary heat transfer. The article presents a solution for the distribution of the temperature field in an infinite rectangular plate with an adiabatically isolated side. As a result, an analytical expression of the plate temperature distribution is obtained in the form of a series containing trigonometric and exponential functions. The paper also considered special cases when the internal thermal resistance of thermal conductivity is greater and when the external resistance of heat output is less. Special cases were interpreted physically. One of the special cases leads the problem to a problem with boundary conditions of the first kind, when the surface temperature is constant, which indicates the reliability of the results obtained.