Artem Yankovets

Faculty: physico-metallurgical
Speciality: Thermal power plants

Theme of master's work:

Methods of solution of direct tasks of heat conduction. Methods of research solutions of direct tasks of heat conduction and evaluation of their accuracy.

Scientific adviser: Oleksandr Markin

Summary of research and developments

Introduction (Motivation)

A topical issue is that the nonstationary mass exchange is becoming increasingly important in the engineering study in connection with the rapid development of new technology. Because properly heat transfer is a prerequisite for a safe and accident-free thermal units, research in the field of unsteady heat transfer are relevant to many industries, including - for the thermal power, where teplosilovoe equipment operates at very high pressures and temperatures, and often - with the unsteady heat treatment. The task of studying the regularities of development of spatial nonstationary temperature fields in bodies of different geometry closely associated with the solution of parabolic differential equations of heat conduction with a variety of boundary conditions. This task is much harder, if considered transient heat transfer, taking into account the finite speed of propagation of heat.

The aim of this work is to find solutions to tasks of direct non-thermal conductivity, based on information on the thermal state, which is determined by the temperature field of the object; definition of the causal characteristics of the heat exchange process in the body: boundary conditions and parameters, initial conditions, thermophysical properties, internal sources of heat and conductivity, as well as the geometry of the body or system.

The practical value of direct tasks of nonstationary heat conduction is the method of solution of tasks to determine the parameters of design and design calculations of the modes of heat exchange equipment.

Personal contribution to the implementation of all phases of this work in developing and carrying out calculations using the methods for the solution of the task. In the formation stages of experimental studies, as well as direct participation in conducting the experiments, processing of results, with the issuance of all necessary recommendations and opinions on this work.

The essence of how to address the direct tasks of heat conduction and methods solutions of direct tasks of heat conduction and evaluation of their accuracy.

Boundary conditions (BC)
Thermal conditions at the borders of the body are called boundary conditions (BC) and consists of two main elements: the external (surface) heat sources and heat transfer conditions between the source and the surface of the body.
It should be emphasized that the boundaries of the body are all of its external boundary (surface), and any internal boundaries (surface)before separating the body from the other solid state or from the cavity (cavity). The internal cavity may be closed or penetrating, and contain a gas or liquid.
There are four kinds of BC.
If you know body surface temperature , then the BU I type.
If the intensity of the heat flow from outside the body, the BU II type. According to the Basic Law of thermal conductivity, heat flow is equal to:

                                                                                                                                                (1)


here mean coefficient of thermal conductivity of the body, a temperature gradient refers to the point of the body located in close proximity to the body surface, on which the index points . Here and below, the SU expressed on the surface of a semiinfinite body and plate . For other forms of phone should take the appropriate value of the coordinates of the surface, for example, for a cylinder and a ball (origin at the center of the body).
If temperature environment (liquid or gas), circumfluent body, and the law of heat transfer between the environment and the surface of the body, the BU III kind. According to Newton's law, heat flows from circumfluent environment, directly proportional to the temperature difference of the environment and the body surface:

                                                                                                                            (2)

BU IV kind occurs if the body is in contact with another body , having different thermal characteristics. Contact on the surface of bodies should be so good, so that the temperature of contact points are the same:


                                                                                     .                                                                  (3)


The equation of thermal balance on the border is the following:


                                                                                                                              (4)


Thermal regimes

Nonstationary thermal mode - body temperature varies over time.
inconsistent mode is:
1. irregular - the temperature function is relatively complicated and time coordinates, and much depends on the initial temperature distribution;
2. regular - the temperature function is relatively simple, and little depends on the initial temperature distribution;
3. kvazi usual heat treatment - the value function of temperature are recurrent, or remain unchanged on a moving heat source.

The usual heat treatment - the temperature of the body over time will not change.
past regime are:
1. equilibrium - the temperature of the body at all points is constant and equal (on the borders of the body has no heat, that is, or has a perfect thermal insulation, or has a complete alignment of body temperature with the temperature of the environment);
2. nonequilibrium - the temperature at each point of the body is constant, but varies (the algebraic sum of heat flows at the borders of the body is equal to zero).

In some cases, to face the challenges applying the principles of equivalence and reciprocity , but the basic method for solving the nonstationary heat conduction, is the principle of superposition (overlay).

BASIC principle of superposition (BPS)
The practical side of the elementary principle of superposition (BPS), is that while independent of each other the actions of individual heat sources, located on the border of the body or inside it, can be considered effect of each source separately, and the final heat to find the effect, piling algebraically of all sources. In addition, you can force a single source is defined as the sum of action of any combination of sources, located on the same site and in the amount of the same temperature or intensity as the original source.
Using this principle opens up great opportunities, but, unfortunately, is not universal. In the above formulation of the principle of superposition of the application is limited. In connection with the simplicity of its application, it is called a basic principle of superposition (BPS).

The principle of superposition COMPLEX (PSC)
The principle of superposition of the complex (PSC) refers to only the source type and can be read.
In determining the action type of heat source accepted that all other sources of type have a temperature equal to zero, and the sources of the type inactive. Otherwise, all rules are valid PEC on an algebraic addition of all sources and the possibility of decomposition of each source to the elements. It remains in force and the requirement of linearity of the conditions unambiguously.
It should be noted that the zero temperature of the source type does not mean that they do not work.
Sources type located just on the borders of the body, so the SAR should be applied, when the sphere of influence of any source of income to the boundary of the body, which is the location of the source .

The basic rule of task solving method of superposition
Distribution of heat in solids can be expressed as to the many sources of heat. Thus, the initial thermal state (initial condition) is instantaneous internal heat sources, and the boundary conditions - continuous external source of heat. The decision of tasks of heat conduction can be reduced to the consideration of propagation of heat from the sources.
How to solve complex tasks? The difficulty of their decisions typically arises from the fact that the initial conditions (NU) or PG are complex, at first glance, sometimes even confusing. The method of superposition allows the solution to each task to solve a few simple tasks. It should be guided by the following rule.
A solution of the complex initial or boundary conditions can be represented as a sum of solutions to other tasks with any other NIs and PG, but the algebraic sum of heat sources, ie FC and BC in these tasks for each point of the body at any time, including beginners, should be equal to the specified values of heat sources in the original task.


Application of various principles obschefizicheskih an important opportunity for the calculation and analysis of the thermal treatment of solids and become a new method of solution of thermal tasks.

The equivalence principle
The essence of the principle of equivalence is that the replacement of one of the conditions of uniqueness, which defines the phenomenon, other conditions of the unique, does not alter the course of events in any location, covered by this phenomenon, the replacement leads to the identity tasks, not to simulate the phenomenon.
With respect to the thermal challenges of the principle of equivalence that the replacement of a single-condition does not affect the temperature of the body - the course of temperature at all locations remained unchanged.
The equivalence principle suggests the possibility of equivalent heat sources and heat resistance, and thermal characteristics, the geometric shape and size of the body.

The principle of reciprocity
The principle of reciprocity can be applied in solving some tasks of heat conduction, which means that if the heat source , located at point 1, point 2 is the change in temperature , then, if the move to a point source, 2 at point 1 will be the same temperature change .
t should be emphasized that the mutual points of rate of change of temperature the same, but temperature gradients are different, so you need to remember that the transition to a mutual problem is not a conversion to an equivalent task - the temperature fields are different.

Main planned results

Combining methods superposition, EQUIVALENCE and reciprocity is the best at finding the exact solution of the direct task of heat conduction. For a consistent calculation, the system functions is enough to have an algorithm or software solutions to the task of direct computer and on several occasions to use them and adjusted terms of uniqueness. Such computer programs, which search for the solution of the problem is reduced to the repeated treatment of the same block (algorithm) is the most rational from the perspective of their organization, debugging and possible adjustments in dealing with various practical tasks.

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