A substantial element of the technological process of operation of the electro-resistance furnaces is the precise reading of the temperature in the complex furnace - detail - environment. The introduction of the description of the measurement element in the system of differential equations aims at exploration of its work under certain conditions and its interaction with the other elements on the thermal chain.
MOVIE: thermocouples simulation
Modeling with system differential equation SDE (Matlab)
Inclusion of measurement devices (thermocouple, sensors of the temperature) in the system of differential equations, similarly to the rest of the elements of the thermal chain, requires adding of own equations – both basic and describing the links with the other elements. A thorough examination requires placement of thermal sensors in the six directions. The relatively small values of their thermal parameters allows their availability without influence upon the process of heating (shown below). These sensors should also be added in the nodes of the system which are in relation (radiation exchange) with the sensors of the temperature(thermal pairs or thermal resistors). The analysis is made by using the formed three-dimensional model which describes the system furnace-detail.
In this case three resistances of radiation should be added: to the wall, to the heater and to the heated body. The link of the first thermal sensor with the heater is described similarly to the other resistances of radiation the coefficients y(*) being replaced with the relevant indexes depending on their location in the matrix.
AlfaTrm = 0.0000000567* 0.7*(( y(44)+273)+(y(50)+273))*((y(44)+273)^2+(y(50)+273)^2);
RklTrm = 1/(AlfaTrm * 0.0157) ;
In this case the thermal sensor is in the fiftieth column of the matrix (y(50)), and the relevant heater of the side in which it is located in the forty-fourth column (y(44)). The resistances of radiation to the wall and to the body are described similarly:
AlfaTrm= 0.0000000567* 0.7*(( y(1)+273) +(y(50)+273))*((y(1)+273)^2 +(y(50)+273)^2);
RklTrm = 1/(AlfaTr3_1 * 0.0157);
AlfaTrm= 0.0000000567* 0.7*(( y(25)+273)+(y(50)+273))*((y(25)+273)^2+(y(50)+273)^2);
RklTrm = 1/(AlfaTr4_1 * 0.0157);
For the equations of the thermal sensors is written:
Trd1 = basic equation including: -((1/((RTr11+RklTr2_1)*CTr1)) link with the heater + (1/((RTr11+RklTr3_1)*CTr1)) link with the wall + (1/((RTr11+RklTr4_1)*CTr1)) link with the body + (1/((RTr12+Rkl1)*CTr1))); to the environment TrN1 = (1/((RTr11+RklTr2_1)*CTr1)); link with the heater TrS1 = (1/((RTr11+RklTr3_1)*CTr1)); link with the wall TrT1 = (1/((RTr11+RklTr4_1)*CTr1)); link with the body NTr1 = (1/((RTr11+RklTr2_1)*CN1)); link of the heater with the thermal sensor is added in the basic equation of the relevant heater NG13 = -(NG11+NG12+(1/((RTr11+RklTr2_1)*CN1)) ) ; STr1 = (1/((RTr11+RklTr3_1)*C11)); link of the wall with the thermal sensor is added in the basic equation of the first layer of the relevant wall S1(1) = - ((1/((RklS1+R11)*C11))+(1/((R13+R17)*C11))+(1/((R14+R22)*C11))+ +(1/((R12+R44)*C11))+(1/((R15+R56)*C11))+(1/((R16+R65)*C11))+ + (1/((RTr11+RklTr3_1)*C11)) ) ; TTr1 = (1/((RTr11+RklTr4_1)*Ct11)); link of the body with the thermal sensor is added in the basic equation of the first layer of the relevant side of the body: T1(1) = -((1/((RklT1+Rt11)*Ct11))+(1/((Rt13+Rt17)*Ct11))+(1/((Rt14+Rt42)*Ct11))+ +(1/((Rt12+Rt24)*Ct11))+(1/((Rt15+Rt55)*Ct11))+(1/((Rt16+Rt66)*Ct11))+ +(1/((RTr11+RklTr4_1)*Ct11)) ) ;     After the effected changes the equations acquire the form shown here.
It is necessary to add 7 new matrices to include the elements of the system of differential equations describing the operation of the thermal sensors:
basic matrix of the thermal sensors: 1. 1A – square matrix, the basic equations are situated in diagonal. link of the thermal sensors with the other elements: 2. T2S – matrix describing the link of the thermal sensors with the walls. Number of lines – number of the thermal sensors, number of columns N*6 (number of the layers of the wall by six walls for the three-dimensional model) 3. T2T – matrix describing the link of the thermal sensors with the body. Number of lines – number of the thermal sensors, number of columns Nt*6+1 (number of the layers of the body by six for the three-dimensional model plus the center of the body) 4. T2N – matrix describing the link of the thermal sensors with the heaters. Number of lines–number of the thermal sensors, number of columns number of heaters. According to the accepted assumptions the heaters are always six in number due to which the matrix obtained is square. links of the remaining elements of the thermal chain with the thermal sensors: 5. ST2 – matrix describing the link of the walls with the thermal sensors. Number of lines - N*6, number of columns – number of the thermal sensors 6. TT2 – matrix describing the link of the body with the thermal sensors. Number of lines - N*6+1, number of columns - number of the thermal sensors 7. NT2 – matrix describing the link of heaters with the thermal sensors. Number of lines - 6, number of columns - number of the thermal sensors The described matrices are added to the basic matrix yp: yp = [ StA + StT + StN + ST2 TqS + TqA + TqN + TT2 NaS + NaT + NaA + NT2 T2S + T2T + T2N + T2A ];     The remaining elements of yp preserve the accepted values but some changes are needed concerning the matrices NaS, NaT, NaA, StN, TqN, NaA. The final form of the file describing all matrices is shown here.
The system of differential equations for a three-dimensional model of the system furnace–detail thus described is used for simulation of the process of heating in which observed is the influence and the behavior of the included measurement devices.
Figures 1-4 shows graphs of the process of establishing of the temperature, and detail the location of the thermal sensors in relation to the other components of the thermal chain at the beginning and at the end of the process. The description is done for four layers of a furnace with detail divided into four layers and center, six heaters and six thermal sensors.
Transient process: thermal sensor toward the heated body
Numerical methods, Runge-Kutta
Beginning of the process: thermal sensor toward the layers of the furnace
The question how the added thermal sensors influence the process of establishing the temperature is relevant. Physically, the metal cable passing through the isolation of the furnace to the environment should the looked at as a thermal short circuit. The comparison of the process with thermal sensors and without them shows that for the discussed example no difference can be seen. Some differences are possible in the process of heating in some cases with certain data for the system but the investigations made in this direction does not present interest for the topic discussed here.
Modeling with FEM
REGULATION OF THE PROCESS OF HEATING
In the simulations that follow the temperature obtained from thermal sensors is used as a criterion for regulation of the process of heating of the furnace. The purpose is obtaining results which give a description of the thermal picture in the mode of regulation. This way the investigation of the process through computer simulation gives a maximally precise idea of the practical operation of the whole electro-technological complex.
At the beginning of this project, when discussing the linear models of description, comparison is made of the operation of the separate methods for numerical solving of differential equations in MatLab program. The consecutive examination of the thermal chain of an elementary node (wall with a heater) in a three-dimensional model has enabled the observation of the behavior of the computing process when new elements are added to the system of equations. According to the obtained data after introduction of the heaters as an independent element, due to the small thermal capacity in comparison with this of the walls of the furnace, the number of iterations of the computing process in some methods is increased by several thousands. After adding the description of the thermal sensors in system of DE, the processor time necessary for the realization of one simulation is increased even more and along with it increased is the necessary RAM. According to the data obtained from the linear substitution schemes the number of iterations increases quite slightly in method ode15s.
As is seen from the obtained graphs showing the temperature difference, the method applied is not precise in the process of regulation. A similar result is confirmed when discussing the three-dimensional model without heaters as well as when discussing the processes of regulation.
The attempts to use method ode45 for the same file cannot be realized at this moment. In order to shorten the computing time the discussed three-dimensional model could be presented as linear one where taken into account is one three-layer wall, a body divided into three layers with a center, a heater and a thermal sensor. The equations and the matrix are simplified to the form shown here. This reducing does not aim at obtaining precise data and could be used as illustration of the processes in the mode of regulation in relation to different criteria. Figures 1, 2 shows the graphs obtained in regulation toward the temperature of the heater the condition being that is should not exceed 800oC. Method ode45 is used.
Establishing of the temperature at 800 degrees centigrade.      Part of the graph presenting the operation of the heater
Heater, walls and thermal sensor.      Heater, body and thermal sensor
Body, thermal sensor and heater.      Wall of the furnace thermal sensor and heater
Regulation in relation to the temperature difference in the body. Method ode23.
Heating of furnace without conditions
Difference in the temperature of the body up to 100oC         Difference in the temperature of the body up to 200oC
Establishing of the process with control of the temperature in the center of the body
Establishing of the process with control of the temperature of the outward layer of the body.
Beginning of the process of heating of the body.
Another interest presents the model of regulation in which are used several conditions: controlling the temperature in the center of the body and the temperature difference between the surface and the center. The condition is that the temperature of the surface should not exceed 450oC and the temperature difference should be up to 200oC - figure 9.
Process of heating.
When discussing the three-dimensional model, graphs have been given showing the processes when the furnace is switched off. The realization of the same process obtained by a system of differential equations used in the previous examples aims to show the possibilities for regulation of the temperature of cooling. Figure 10 A shows the process of heating in which preserved is the conditions from the previous simulation, switching off the furnace and then swathing it on. As it can be seen in figure 11 A, at the moment of switching off the temperature difference surface – center of the detail reaches up to -293.38oC the minus sign showing that the thermal flow is directed from the center to the surface of the body. Bearing in mind the condition that the difference should not exceed 200oC it becomes apparent that such an incontrollable mode of cooling is undesirable. Figures 10B and 11B present a controllable process of cooling in which the difference of 200oC is preserved.
Cycle of heating, cooling and reheating
Temperature difference in the body
The removal of the thermal sensors and heaters from the description of the system furnace–detail has enabled shortening of the computing time and RAM needed for performing of computer simulations. The solutions presented above describe a linear chain. The data used allow its replacement with a three-dimensional substitution scheme. With the accepted assumption that the process should be regarded without heaters and thermal sensors this replacement little influences the computing process. The system of differential equations of a three-dimensional model is presented here, as well as the matrix for realization of the process. The description of this model aims both to examine the thermal picture and to present the description of the system in a different way. Thus, when solving practical problems, a description of the system of differential equations which is optimal for the concrete case can be chosen:
Figure 12 presents the processes of opening and closing of the door. The process of placement of a cold body in the heated furnace is examined here
Opening the door
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