ABOUT SOLUTION OF HEAT PROBLEMS IN WELDING AND CLADDING
Viktor V. Jakovlev
Uralmash, Ekaterinburg, Russia
Sergei M. Shanchurov
Ural State Technical University, Ekaterinburg, Russia
Igor V. Pershin
Ural Brabch Russian Academy of Science, Ekaterinburg, Russia
Introduction
Welding and cladding are high-temperature processes, that is why the main factor, determining the temper of physicochemical processes, is heat effect of the electric arc or another heat source on metal close to the boundaryof melting. In this connection during the mathematical modelling ofwelding and cladding it necessary to give the main attention to the calculation of heat fields, gradients etc.
Modelling can be carried out using analytical methods as well as numerical ones. Each of these methods has its advantages and disadvantages. Lets consider the solution nuances for cylindrical and massive solids using the example of modelling of heat field calculation peculiarities of the components, which can be often met in practice.
The numerical solution of such problems is rather difficult. This difficulty is determined by the great variety of factors, attendant to technological processesof welding and cladding:
-concentration of the heat source;
-a high speed of its motion;
-considerable power of the heat source;
-heterogeneity of weldment and cladding machinery, that is showed by considerable difference of thermal material characteristics (coefficients of equation that models of the process);
-presence of small inclusions (pores), the physical properties of which differ from ones of the parent metal;
-simultaneous existence of molten and solid phases;
-a composite geometry of the products and so on.
The processes of welding and cladding belong to such technological processes, that are characterized by the presence of concentrated heat sources of high power. The peculiarity of such processes is that close to the heat source the temperature rises significantly in a small area, high temperature gradients appear. The selection of the numerical method for solution of the problem of mathematical description that correspond to technological process demands accuracy and taking into account its peculiarities. There is a considerable problem of modelling the original task in three-dimensional area. With the use of standard methods of mathematical modelling and the use of classical difference schemes the presence of these factors (the presence of the small heat source and the necessity of problem solution if three-dimensional area) leads to the rising of the need in use of considerable amount of storage device and in taking much time for calculations. Sometimes it happens so that it is impossible to solve the problem with an acceptable accuracy because of limited computer resources.
That is why it is an actual task to develop the numerical methods, that allow to solve applied problems with the beforehand given accuracy in an acceptable time with the least computer resource expenses.
- Cylindrical body (numerical method)
Physical image of the problem: a concentrated heat source with a high power moves by circular helix over a surface of a cylindrical body. It is supposed to make the following conditions: the heat spread in the body takes place only by heat conduction, there are no phase and structural changes, thermal coefficients depend on the temperature.
It is impossible to use standard (classical) finite-difference methods, because in order to get the numerical solution with an acceptable accuracy it is necessary to set a differed net with a great amount of units (because it needs to set a differed net in three-dimensional variables, with a mesh width in 4-5 times less than the radius of the heat source spot). At that a subinterval of time also should be coordinated with one of area, and therefore, it (a subinterval of time) would be a rather small quantity, that will lead to much time of calculation (from several hours to several days). Because it is rather difficultto solve the whole three-dimensional problem in cylinder, it is proposed the following method of it solution. For example, we need to know the temperature in some specified point of a cylinder (this point can be either on the surface of a cylinder or inside it) during the conducting of a surfacing process. It is well known, that the most strong heat source influence shows by some limited area, the size of which is 4-5 diameters of arc spot. According to the principle of local effect, its influence out of this area almost does not differ from the influence of circular heat source with the same power. The idea of this problem solution is based on this principle.
Close to the point, in which we need to find the temperature, a three-dimensional problem is solving with the concentrated heat source on a small area (6-10 diameters of heat source spot) of a spiral trajectory going through this point.
Out of this area we will change the concentrated heat source to ring source of a corresponding power and will solve a two-dimensional problem of finding temperature field from this source in whole cylinder. In that way, the original problem about the determination of temperature field in a specified point from heating by the source of small sizes is divided in two:
1)a three-dimensional problem of finding a temperature at a specified point, when a heat source moves in a small area of circular helix close to this point;
2)a two-dimensional problem of finding a temperature field in a cylinder, when a ring source moves advancing on it.
A sum of solutions of 1st and 2nd problems gives a solution of the initialal problem.
Mathematical model: a mathematical solution of problem 1 and 2 comes to a solution of three- and two-dimensional equations of heat conductivity accordingly in cylindrical system of coordinates [1]. The heat conductivity equation coefficients are considered as a function of temperature. Boundary conditions of third kind are set at the boundary of calculated area. For example, in a three-dimensional case the equation and boundary conditions look like this:
where: is specific heat, is density, is heat conductivity, is a coefficient of convective heat transfer, is a radiation coefficient, is a cylinder radius, is a length of a cylinder, is an angular coordinate, is an unknown temperature, is time.
Function characterizes the power of a moving heat source and a rule of it’s distribution. At such boundaries where is no heat source, it is supposed to equal zero.
Equations and boundary conditions look similarly in the two-dimensional case. The finite-difference method of fractional steps was used in a numerical realization of the formulated problems [2].
The above described method of a solution of the whole three-dimensional problem allows to reduce considerably (in several thousand times) the number of units in spatial net, in that way to reduce considerably the time of calculations and required computer resources, and it allows to get a numerical solution with an acceptable accuracy for practice.
A mathematical modeling of modern technological processes ,as a rule, leads to the necessity of solution of singular perturbed boundary value problems. Earlier in the work [3] it was showed, in what way such technological problems transform to problems containing small parameter at a higher derivative, and in what cases these problems become singular perturbed. It was given a classification of such problems, there were proposed methods of construction of some special difference schemes. It was showed, that not all methods of net condensation are suitable for getting an exact numeral solution.
It is necessary to note that a method of special condensing nets was used during the solution of problems 1) and 2). The classical finite-difference methods for such problems do not allow to receive a solution with an acceptable accuracy (an approximate solution close to the heat source can differ from the exact one in several times). The usage of special difference methods allows to get a solution with beforehand given accuracy [4]. A chosen method of net condensing depends on thermal characteristics of a heating material as well as on rate of movement, power and a heat source concentration.
The conducted numerical experiments by the proposed algorithm showed that such method allows to get results with an acceptable accuracy, at that the calculations take a little time.
It is showed, that the error of the solution, received with a help of classic difference schemes, is from 50% to 100%, and the error of the solution, received using special difference schemes is from 15% to 25 % depending on problem parameters. At that the error at calculation of heat flows (a space derivatives from temperature) using standard difference schemes can reach 300 % and more.
The error of calculation of flows by the special scheme is not bigger than the error of the solution itself.
There were carried out research works on the effect of the law of heat flow density distribution in a heat spot: equidistributed, linear and normally distributed laws were studied. There was showed, if to change a linear law of density flow distribution on a normally distributed law, the value of temperature fields practically do not differ with arbitrary changing of problem parameters. If to change a linear law of density flow distribution on uniform law, the difference in calculations of temperature fields can vary from 3% to 20% depending on problem parameters.
2. Modelling of a cladding process of massive solids (analitical method)
The classic finite-difference methods of solving such problems do not always allow to receive results with acceptable accuracy ( the calculated heat flow close to the heat source can differ from the real one in tens times)
Physical image of the problem: a strip heat source moves over a surface with a given power.
Mathematical model of the problem:
The function characterizes the density of heat flow distribution edgewise of the heat line, an effective heat source power stays constant, specific heat flow tends to infinity, is a width of heat source tends to zero, is a speed of its motion. The problem solution can be evidently written as:
Lets consider a case, when the heat source power is constant, and the heat area tends to zero. The external asymptotic resolution is used far off the heat source, internal resolution is used close to the source, considering the peculiarities in behavior of solution in this area.
Lets introduce new variables: , carry out correspondingtransformations and investigate the asymptote at .
Lets factorize abreast the multiplier before the integral and the intergrand with small ε and will get the main term of the solution as:
where function is the McDonald’s function, an internal integral is calculated with the help of standard methods of numerical integration. Obviously that the main term of external resolution close to the source behaves itself as . It is not allowed to use this formula directly close to heat source, because at, and it can be understood that the temperature close to the heat source tends to infinity.
To present the solution (1) close to the heat source lets introduce new variables Lets factorize abreast the intergrand with small ε and will get formulas, with the help of which we can find the solution in close proximity to the heat source.
For example, if to look for the solution with accuracy to , we will get:
The main term of expansion has a form of:
From the formula (2) there were received exact analytical formulas for the solution with uniform and linear distribution of heat source.
With a uniform distribution of the heat source the main term looks in this way: , where
.
With a linear distribution of the heat source we will get , where
,
.
It is clear from these formulas, that when the parameter , the solution close to the heat source behaves itself as .
Carried out numeral experiments showed that at a high concentration and high power of the heat source the mistakes, received from the substitution of one kind of the heat source distribution by another one, are 5-15 % and less. In the case, when it is not allowed to ignore the heat source, then numeral error amounts to 30%.
In this way, nowadays numerical methods of heat problems solution can be used successfully as well as analytical ones, but anyway its solution requires corresponding qualification and experience.
References
- Tihonov A.N., Samarskiy A.A. The equations of Mathematical Physics. M.: Science, 1966.
- Samarskiy A.A. The Theory of Difference Schemes. M.: Science, 1989.
- Yakovlev V.V., Titov V.A., Shishkin G.I., Pershin I.V. To the problem of numerical solution of heat welding problems // Collected articles of Research Institute of Tyazhmash.
- Shishkin G.I. A net Approximations of singulat distributive elliptical and parabolic equations. Ekaterinburg, Ural Branch of Russian Academy of Sciences, 1992.
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