Physical Process Modeling


One of the principal questions when designing ERF concerns defining of the energy characteristics of the system furnace – detail. The built-up model of a thermal installation enables an analysis of the losses and of the coefficient of efficiency.
It presents the process of cooling of a heated body in air medium, in natural convection. The way of making the thermal scheme detail – environment is described and its realization through a system of DE. The basic emphasis is on the description of the criteria concerning the thermal exchange through convection as well as their realization in the process of simulation.


  Cooling - Demo 1
  Cooling - Demo 2
  Cooling - Demo 3
  Cooling - Demo 4


    A Model Of PID Regulation
    A Model Of PID Program Regulator
    A Distribution Of The Installed Capacity In ERF
    Electro Resistance Furnace Part I
    Electro Resistance Furnace Part II
    Electro Resistance Furnace Part III

ENERGY ANALYSIS OF ERF



    One of the principal questions when designing ERF concerns defining of the energy characteristics of the system furnace-detail. The built-up model of a thermal installation enables an analysis of the losses and of the coefficient of efficiency.


Matlab-furnace heat transfer

Figure 1
A, C – without regulation of the temperature. B, D - mode of regulation.

technological process - condition

Matlab-furnace heat transfer

Figure 2
mode of regulation

technological process - condition

Matlab - furnace energy


Figure 3
Energy analysis of ERF in a non-regulatable mode
A - 1,2 – coefficient of efficiency, B – accumulated energy in one side of the layers of the furnace (5,6) and the body (1,2,3,4); C – 1 total losses, 2 – summary losses in the walls of the ERF and the environment, 3 – losses towards the environment, 4 –losses in the walls of the furnace, 5 – energy accumulated in the body.


    Figure 3 shows the results obtained from simulation of a non-regulatable mode. Figure 3(A) chart 1 presents the coefficient of efficiency, where the latter starts from the maximum and decreases with establishing of the temperature. The beginning of the process is shown in chart 2(A). In the same figure (A) charts 5,6 show the energy accumulated in one of the walls, in the two layers of the ERF, and charts 1, 2, 3, 4 the energy accumulated in the surface layer, in the first, second layer and in the center of the body respectively. The energy analysis is presented graphically (C) by charts 2 – total losses, obtained from the sum of – 3 losses towards the environment and 4 losses in the walls. The total capacity (chart 1) is the sum of the total losses and the energy in the body (chart 5).

    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 t1-t2 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 t1-t2 (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 t1-t2 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

furnace energy - Matlab solving

Figure 4

Heating of ERF by opening the door - A, Losses - B.
A: 1 - the heater’s temperature, 2, 3, 4, 5 – surface ,layers and center of the body, 6, 7 – walls of the furnace; A – 1, 2 accumulated energy in the first layer of one side and the door of the ERF respectively.


furnace energy - Matlab solving

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.

    The result from the conducted analysis enables us to account the summary losses of 7,5.107[J] without opening the door and of 8,5.107[J] with its opening. By the same methodology can be conducted an energy balance for a complete technological process with replacement of the detail in the chamber of the ERF.

Furnace heat field - FEM

heat field - finite element method

Figure 6

heat field - finite element method

Figure 7




DETERMINING THE FIELD OF APPLICATION OF A DESIGNED FURNACE WITH LIMITING CONDITIONS


The given three-dimensional models enable us to explore the behavior of a particular electro-resistance 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 three-dimensional model of ERF, determining the functional dependency between the values of time t, temperature , and the temporal constant T. Plains a1, a2, a3, assign the necessary criteria:

    a1 – maximum temperature,
    a2 - initial temperature,
    a3 – maximum time for achieving the maximum temperature;

furnace limit modeling


    Charts c2–c5 show the process of heating of ERF, loaded with various bodies, with temporal constant T2–T5. Chart c1 presents the transitional process for an empty furnace. Points A1–A5 show the transitional process reaching the maximum possible temperature, i.e. they are obtained from the crossing of straight lines c1–c5 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 T1, T2, T3, T4, T5, A1, A2, A3, A4, A5, A6, A7 determines the operative area of the ERF under consideration. Charts cn, 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.




PROCESS OF MELTING

    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:

energy analysis equation


- 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:

energy analysis equation


which, through accumulation, increases from the moment when the temperature reaches the point of melting to the moment of its equalizing with Qeff 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.

Matlab simulation

Figure 1

Simulation of a process of melting the heated body.

    The surface of the body (chart 2) reaches Ttt at the moment t1, and its center does so at the moment t2. In the interval t1 - t2, determining the time of the detail melting, the temperature does not exceed Ttt.

    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.
energy analysis equation


where: Peff – effective capacity, average for the whole interval of operation, clat – latent heat of the phase transition.

melting Matlab ode23   melting Matlab ode23

Figure 2

Simulation of a process of melting Differential Equations

melting furnace modelling

Figure 3

Simulation of a process of melting FEM

melting furnace modelling

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.

modelling convective heat

modelling convective heat

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.

convection Nusselt, Grasshoff ... convection Pekle, Raynolds ...


    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 15000C and the temperature of the environment 20oC are placed in Figures 2, 3 . Figure 3 presents the change of the resistances Rkl in the different sides of the body.

cooling numerical model

Figure 2
Establishing of the process

cooling model - thermal resistance

Figure 3
Convection resistances for the separate sides

cooling - Finite Element Method
Figure 4

cooling - Finite Element Method
Figure 5

cooling of the bodies - data
Figure 6


Modelling convective heat transfer
Forced convective flux

heated detail - Finite Element Method
Figure 7

heated detail - Finite Element Method
Figure 8

heated detail - Finite Element Method
Figure 9




Physical Process Modeling Resources:   Mathematics   Physics   Electronics   Programming   Heat transfer