
ENERGY ANALYSIS OF ERF 
One of the principal questions when designing ERF concerns defining of the energy characteristics of the
system furnacedetail. The builtup model of a thermal installation enables an analysis of the losses and of
the coefficient of efficiency.
Figure 1
Figure 2
Figure 4(A) – shows heating of the ERF with a heated body in the process of regulation in relation to conditions (Figure 1), where in the period t1t2 the heaters are switched off and the door is opened. The same figure (A) shows the alteration of the accumulated energy in the first layer of one of the walls (chart 1) and the first layer of the door (chart 2). The latter, in the period t1t2 (Figure 4(B)), is cooled bilaterally towards the environment, which explains the rapid decrease. On closing the door and switching on the heaters, the simulation process runs with taking into account the thermal release between the walls due to which the energies become equal shortly after the moment t2. Figure 5 – (A, chart 1) shows the losses in the external layer of the body at a side oriented towards the door. According to the given data, the decrease in the accumulated energy is proportionate to the decrease of the temperature figure 4(A)– chart 3. Chart 2 shows the total losses in the period t1t2 through the remaining five walls of the ERF. The rapid decrease is due to the fact that the basic thermal flow moves through the open door towards the environment (figure 5(B) chart 2). In the same figure, chart 1 is shown the increase of the summary losses towards the environment Figure 4
Heating of ERF by opening the door  A, Losses  B. Figure 5 Energy analysis of ERF in a regulatable mode in a technological mode of opening the door A: 1  losses in a layer of the body, 2  losses towards the environment; A: 1, 2 – losses towards the environment and through the open door.
Furnace heat field  FEM
The given threedimensional models enable us to explore the behavior of a particular electroresistance furnace, when heating
bodies which have different parameters. Determining a nomenclature of details depending on their temporal constant of heating
is done through realization of the stimulation process for each detail. In order to visually present the operative area of a
threedimensional model of ERF, determining the functional dependency between the values of time t, temperature
, and the temporal constant T.
Plains a_{1}, a_{2}, a_{3}, assign the necessary criteria:
Figure 6
Figure 7
a_{1} – maximum temperature,
Charts c_{2}–c_{5} show the process of heating of ERF, loaded with various bodies, with temporal constant T_{2}–T_{5}. Chart c_{1} presents the transitional process for an empty furnace. Points A_{1}–A_{5} show the transitional process reaching the maximum possible temperature, i.e. they are obtained from the crossing of straight lines c_{1}–c_{5} with the surface _{1}. On reaching the maximum temperature, the process continues only through control by the furnace regulator. The spatial figure thus obtained from points T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}, A_{7} determines the operative area of the ERF under consideration. Charts c_{n}, lying outside the figure present the transitional process, obtained when heating bodies whose temporal constant does not permit their usage in the furnace under consideration. The presented spatial chart of the operative area of the ERF aims to visually present the operative range of the furnace. This can be achieved by making a data base or an application program using data from the simulation of a particular complex furnace–detail. A software like this, attached to the technical documentation of an installation, can be used to check its technical potentials. 

When studying the process the following principle have been accepted and the corresponding sequence of operations:  the body is divided into n independent layers, connected to each other through resistances of thermal conductivity;  the theoretical latent quantity of heat necessary only for the process of melting of the material in each layer:  the current temperature in the body layers is controlled and compared through a logical operator to the temperature of melting. When the two temperatures become equal in the corresponding layer, the current value artificially remains equal to the temperature of melting;  from the moment of retaining the temperature starts the calculation of the accumulated quantity of heat:
which, through accumulation, increases from the moment when the temperature reaches the point of melting to the moment
of its equalizing with Q_{eff i};
 when the quantities of heat become equal the solution is freed again from the imposed limitation and the temperature is obtained from the system of differential equations, but with another value of the specific heat responsible for the new state – the state of melt. There exist another variant of solution in which the body is assumed as a sole element and the procedure, described above, is applied to the whole volume (mass). The solution is obtained in a simplified way, it is also correspondent to the physical nature of the process, but the preciseness relatively decreases. This approach is represented through a solution of the process which is shown in figure 1.
Figure 1 Simulation of a process of melting the heated body.
The surface of the body (chart 2) reaches T_{tt} at the moment t_{1}, and its center does so at the moment t_{2}.
In the interval t_{1}  t_{2}, determining the time of the detail melting, the temperature does not exceed T_{tt}.
In the MatLab program the simulation of the process of melting is carried out through fulfilling the additional conditions, as follows:  each iteration of the computing process is checked for reaching ttt on the surface of the body, and if the result is positive the latter is established at the temperature of melting.  consecutively, the same condition is checked for each layer of the detail.  when ttt is reached in the center of the body, the logical conditions included in the file of the system of differential equations are ignored by the computing process. The simulation of the process of melting offers an opportunity for checking the preliminary calculation of the time for melting (t2), during the stage of designing the installation. where: P_{eff} – effective capacity, average for the whole interval of operation, c_{lat} – latent heat of the phase transition.
Figure 2 Simulation of a process of melting Differential Equations
Figure 3 Simulation of a process of melting FEM
Figure 4 Simulation of a process of melting FEM

COOLING OF THE BODIES 
The process of cooling of a body heated to a given temperature starts from its placing in a medium with a
lower temperature and goes on until both temperatures become equal. The process examines the analysis of the thermal exchange
when a heated body is removed from a furnace and is placed in air, water or other medium. The thermal chain in this case is
made up of the thermal resistances and capacity of the body and the resistance of convection and radiation of the environment.
Figure 1 COOLING OF THE BODIES  3D Model  Matlab
The thermal resistance Rkl to the environment is the sum of the coefficients of radiation
and convection. The convective thermal exchange is determined through the criteria of
Nusselt (Nu), Grasshoff (Gr), Pekle (Pe), Raynolds (Re), Prandtl (Pr) as presented below,
and their complete description is given
here.
The values K, n, C are accounted depending on the geometry of the body and its location
in the environment. This TABLE shows the data on given geometric forms in natural convection.
The accepted constant determining the temperature of the environment is characteristic of all analyses.
In some of these cases this leads to incorrect results because of the incorrectness of the accepted condition:
when an element of "horizontal surface" is streamlined by a fluid coming out of "vertical surface" the initial temperature
should have a higher value than the temperature of environment.
The parameters as function of the temperature are accounted in this TABLE. The functional dependency necessitates that the date be updated for each iteration during the process of computing. The coefficient of radiation to the environment is determined by: AlfaL = 0.0000000567* Eps *(( y(1)+273)+( 273 + TayOS))*((y(1)+273)^2+(273 + TayOS) ^2 ); The value of Rklm according to the Matlab syntax is obtained: Rkl = 1/( (AlfaK+AlfaL) *S) ; The file made with this description is shown here. The matrix yp is made up only of the differential equations describing the body due to lack of other elements in the thermal chain. Removed are also the disturbing influences from the heaters their function being performed in this case by the equations describing the influence of the temperature of the environment: Tos1 = (1/((RklT1+Rt11)*Ct11)) *TayOS ; TauOS  temperature of the environment (a constant value). After the transformations are made, the matrix assumes the form shown here. According to the data in TABLE the side walls of the body are accepted as vertical surfaces and the upper and lower side as horizontal surfaces with the thermal flow directed upward and, respectively, directed downward. The graphs obtained in realization of the process of cooling of the body heated up to 1500^{0}C and the temperature of the environment 20^{o}C are placed in Figures 2, 3 . Figure 3 presents the change of the resistances Rkl in the different sides of the body. Figure 2
Modelling convective heat transfer
Forced convective flux

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