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.


Introduction
Heat engineering is a field of science and technology dealing with the issues of obtaining and using heat.There are two types of heat useenergy and technological.The energy use of heat is based on processes that convert heat into mechanical work.The technological use of heat is based on the realization of heat for the purposeful change of physico-chemical properties during the implementation of various technological processes.Devices in which direct heat supply is used for technological purposes include various furnaces, dryers, heaters, heaters, etc.The science that studies the patterns of heat exchange between bodies is called the theory of heat transfer.
The processes of heat exchange and associated mass transfer play an exceptional role in nature and technology.Indeed, the temperature regime of the environment depends on them.The flow of the workflow in a variety of technological installations depends on them.It is not surprising, therefore, that the theory of heat transfer has developed intensively, especially in recent decades.This is due to the needs of thermal power engineering, nuclear energy, and cosmonautics.The intensification of various technological processes, as well as the creation of installations that are optimal in terms of energy consumption, is unthinkable without a deep study of the thermophysical processes that take place in these installations.
The study of heat exchange processes is currently devoted to the internal layer of work.Of particular scientific interest are the works describing modern heat exchange elements of heat exchange equipment with a detailed description of the methods of their manufacture, as well as the control of a heat exchanger with a variable heat exchange surface area.The processes of heat exchange and the mass transfer resulting from it currently play an important role both in the technical sphere and in nature.The temperature regime of the surrounding medium directly depends on this group of processes.In addition, this group of processes determines the workflow in various technological installations.This leads to the active development of the theory of heat transfer, especially during the last 10-15 years.The active development of the theory is also due to the needs of such spheres of human activity as cosmonautics, thermal power engineering, and nuclear energy.
It is not surprising, therefore, that the theory of heat transfer has developed intensively, especially in recent decades.This is due to the needs of thermal power engineering, nuclear energy, and cosmonautics.The intensification of various technological processes, as well as the creation of optimal installations from the point of view of energy consumption, is unthinkable without a deep study of the thermophysical processes that take place in these installations.In connection with the improvement of the thermal equipment of energyconsuming and producing devices, a more accurate calculation of heat transfer processes in thermal networks is required.Therefore, it seems advisable to improve the methods of calculating heat transfer in such systems.It is known that for better cooling of fuel elements (electrical conductors, rods of nuclear reactors, etc.), it is necessary to have a large heat transfer surface.An increase in the surface can be achieved either by finning, or by replacing rods of circular cross-section, which have a minimum surface of the heat sink, with rods of another cross-section, for example, oval or elliptical.Bodies with an elliptical cross-section occupy a special place.Their peculiarity lies in the fact that by manipulating the change in the length of the semi-axes of the ellipse, it is possible to obtain accurate analytical solutions to stationary thermal conductivity problems for a very wide range of shape changes: from a cylinder to a thin plate.
To increase the cooling intensity of the elements, a large surface for heat transfer is needed.The surface can be enlarged either by finning, or by replacing surfaces of circular cross-section, which have a minimum area, with other surfaces with an increased crosssection, for example, with an oval or elliptical cross-section.Therefore, works describing modern heat exchange elements of heat exchange equipment with a detailed description of the methods of their manufacture, as well as the control of a heat exchanger with a variable heat exchange surface area, are of particular scientific interest.
To date, an impressive layer of work has been devoted to the study of heat exchange processes.Of particular scientific interest are the works describing modern heat exchange elements of heat exchange equipment with a detailed description of the methods of their manufacture, as well as the control of a heat exchanger with a variable surface area of the heat exchanger.Several works [1][2][3] are devoted to the calculation of temperature fields in bodies of elliptical cross-section in the presence of internal heat sources under various conditions.This article is a continuation of the works [4][5][6][7].The main task of this work is to find the distribution of the temperature field in a hollow body with an elliptical crosssection of infinite length under boundary conditions of the first and third kind.The solution is obtained by considering equidistant surfaces.An equidistant surface is a surface whose distance at any point is constant from some given surface.Which allowed us to move away from the elliptical coordinate system.

Main Part
The main task of this work is to find the distribution of the temperature field in a body with an elliptical cross-section of infinite length under boundary conditions of the third kind without internal heat sources.In power plants, heat transfer is often observed between two media (heat carriers) through a solid wall separating them, which is called heat transfer.In this case, heat from a more heated heat carrier is transferred to the wall due to heat transfer and thermal radiation, heat exchange occurs inside the wall due to thermal conductivity, and from the opposite surface of the wall is carried out due to heat transfer to a less heated heat carrier.
Boundary conditions of this type play an important role in the theory of heat and mass transfer, since they are a mathematical formulation of the conditions of convective heat and mass transfer.
In power plants, heat exchange between two media (heat carriers) through a solid wall separating them is often found, which is called heat transfer.In this case, heat from a more heated coolant is transferred by heat transfer and thermal radiation to the wall, heat exchange occurs inside the wall due to thermal conductivity, and heat transfer is carried out from the opposite wall surface to a less heated coolant.
In the theory of thermal conductivity, the heat transfer process is understood as the thermal conductivity of the wall under boundary conditions of the third kind.The boundary condition of the third kind consists in specifying a linear combination of the transfer potential and its derivative with respect to the normal at the boundary of the region under consideration.Boundary conditions of this type play an important role in the theory of heat and mass transfer, as they are a mathematical formulation of the conditions of convective heat and mass transfer.
To find the temperature distribution, it is necessary to solve the Poisson equation, which is an elliptic partial differential equation The Poisson equation in elliptic coordinates (fig. 1) has the form As a computational domain, we take a homogeneous isotropic hollow elliptical cylinder of length H.In the space R 3 , we introduce a Cartesian coordinate system (X,Y,Z).The Z axis is compatible with the axis of symmetry of the elliptical cylinder.In the accepted coordination, the parametric equations of the inner surface Ω have the form where: a, b are the major and minor semi-axes of the ellipse in an arbitrary radial section.
Next, we assume that the calculated area is bounded by equidistant surfaces z = 0, z = h.Heat transfer occurs in a stationary mode.There are no internal heat sources (fig.2).Taking into account the assumptions made, in the accepted coordinate system, the equation of a stationary one-dimensional thermal conductivity problem for a shell in the form of a hollow elliptical cylinder will be written as where: α is the heat transfer coefficient from the inner surface, W/(m 2 K); T0 is the temperature of the cooling medium, K; λ is the coefficient of thermal conductivity of the body.
Integrating equation ( 4), we obtain a two-parameter family of primordial It follows from this that the temperature does not depend on either the heat transfer coefficient or the shape of the body, but is completely determined by the ambient temperature.

Conclusions
In this paper, a solution is given for the distribution of the temperature field in a hollow body with an elliptical cross section under boundary conditions of the first and third kind.The solution is obtained for the case of equidistant surfaces.

Fig. 3 .
Fig. 3. Radial section of an elliptical cylinder.At the same time, the surface temperature of TH is set on the upper surface z = h