It is a natural extension of the analysis of the two-dimensional substitution schemes. The system furnace - detail is examined as a 3D data array. The achieved results after the computing process present the temperature picture as it would be in operating of a furnace with the indicated parameters and construction. The author is not going to elaborate on the advantages of such a virtual projecting of the elecro-technological devices but would rather let the reader appraise them. The basic aim of the analysis is to compare the results in constant and in variable parameters of the system. Theory Background Heat equation: Radiation: Conduction and convection: Created by Physical Process Modeling

# THREE-DIMENSIONAL MODEL OF SYSTEM FURNACE – HEATED BODY

 The description of the complex furnace – heated body – environment is made similarly to this of a mono-dimensional and a two-dimensional model the construction being viewed as dimensional form. The realization of the model necessitates making of square matrix with a size obtained by summing up of:         - the number of the layers of the furnace – N*6         - the number of the layers of the body – (NT–1)*6+1 (+1 – center of the body)         - the number of the heaters. For convenience, the number of the heaters always equals 6, and if in practice it is smaller the relevant power is written as equaling to zero.         When realizing the linear and two-dimensional models, the matrix yp includes all equations describing the individual nodes of the system furnace – detail. The three-dimensional model discussed here is described with more equations. According to the given description of making yp for a three-layer furnace, a body divided into three layers with a center and six heaters leads to square matrix 43X43. When accounting of Rkl as a function of the temperature for a linear model an approach is applied according which each layer of the wall is divided into three layers – two thin outward layers with thickness of the size of a millimeter and a basic layer. In a three-dimensional model and multi-layer walls, the same approach allows obtaining the temperature between the individual walls and hence also their thermal loading. The possibility for yp to be obtained as a square with a side of hundreds elements is quite a valid reason for the matrix to be divided into several matrices describing the separate elements. Nine matrices are made up as follows:
 walls: 1. StA basic matrix of the walls of the furnace 2. StT matrix of the walls determining the link with the body 3. StN matrix of the walls determining the link with the heaters heated body: 4. TqA basic matrix of the body. 5. TqS matrix of the body determining the link with the walls 6. TqN matrix of the body determining the link with the heaters. heaters: 7. NaA basic matrix of heaters 8. NaS matrix of heaters determining the link with the walls 9. NaT matrix of heaters is determining the link with the body Out of the nine matrices is made the basic matrix yp: ``` yp = [ StA + StT + StN TqS + TqA + TqN NaS + NaT + NaA ]; ``` Heat Transfer in Electro-Resistance Furnace (ERF) (3D Graphic - Matlab)
The formed system of differential equations (numerical methods: Taylor, Runge-Kutta, Adams-Bashfoth, Adams-Moulton, Griar)of the three-layer furnace and a body divided into three layers with a center is the mathematical model of the construction. The values of the thermal resistances and capacity are initialized according to the rules as described when discussing the linear model. In consistence with the way accepted above for making of yp, for the discussed system the matrices obtained are shown here. The results of the performed simulation using the method ode15s are shown in Figures 1-4. The furnace and the body have the form of a cube, and the heaters have equal powers of 3[kW].

 Figure 1 Establishing of the process Figure 2 Walls of the furnace with heater Figure 3 Layers of the body with heater Figure 4 Thermal difference surface – center of the body

The thermal difference surface – center of the body shown on Figure 4 is built as function of the number of iterations. The data obtained from the computing process determine the maximal temperature difference of 589.3oC in the body. In this case the temperature is established at 1800oC, which depends on the parameters of the materials and the installed power of the heaters. In some examples it is possible that the temperature exceed with many times the one admissible for electro-resistance furnaces. This question is discussed when examining the mode of regulation.

FEM heat transfer modeling

 HEATING   AND   COOLING
 The simulations performed here so far of the process of establishing the temperature in the system furnace – detail give an idea about the operation of a concrete furnace and can be used in designing one. The next question which presents interest is the obtaining of thermal picture in the mode of cooling. The process allows realization without change of the systems of differential equations already in use, the condition being assigned that at certain temperature the heaters should be switched off, i.e. their powers are written as equaling to zero. In figures 5, 6 are shown the graphs of simulation in which a condition is assigned in relation to the temperature of the heater – if the temperature reaches a certain value the furnace switches off. The system furnace – detail remains in this condition without any change in power or geometric parameters by the end of the process, i.e. the body cools along with the furnace without being removed from it. Figure 5 Switching off when certain temperature is reached           Layers of the body with heater Figure 6 Walls of the furnace with heater           Thermal difference surface – center of the body       When discussing the mono-dimensional chain it was assumed that the heater was evenly distributed along the whole wall which did not allow accounting of the mutual radiation of the wall with the body. This assumption is valid up to this moment – accounted are only the radiation from the heater to the body and from the heater to the wall. In the process of cooling it is necessary that the mutual radiation be accounted which will be done further on in this project. On this stage it should be pointed out that the data thus obtained are incorrect and they aim only at a general illustration of the process of heating and cooling. Discussing the process with and without accounting of the radiation wall – detail allows us to determine the influence of this element of the thermal chain.

The question of loading of the heated furnace with a cold detail concerns mostly the furnaces of uninterrupted operation. As an example it can be pointed at the operation of a conveyor electric resistance furnace in which the place of the heated body is taken by the cold detail. The thermal picture is obtained by performing simulation of the process of heating and the obtained data about the established temperature in the nodes of the system are used as initial data in the following simulation. The body is assigned an initial temperature. The process of heating in the examined case is shown in figures 7-10. As can be seen from the graphs, the beginning of the process presents the main interest where the change of the temperature in the heater is the most abrupt.

 Figure 7 Walls of the furnace with heater Figure 8 Layers of the body with heater

 Figure 9 Walls of the furnace with heater. Beginning of the process Figure 10 Layers of the body with heater. Beginning of the process

The primary benefit from the case discussed here is the analysis of the data about the thermal shock in a cold body when it is placed in a heated furnace. Figures 11, 12 presents a comparison between the graphs of thermal difference in heating of the body with the furnace and in the case discussed here.

Figure 11
The thermal shock in a cold body when it is placed in a heated furnace

Figure 12
Placement of a cold body in the heated furnace

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