INDIRECT HEATING OF THE BODIES – ASYMMETRIC.

Heating of electric resistance furnaces designed for periodic operation from a heated condition.

The resistance furnaces designed for periodic operation from heated condition undergo a starting mode (mode of heating) before they can go on to the operating mode (mode of loading). This mode is accompanied by a typical transition process which is asymmetric indirect heating of the bodies. The transition process starts from the moment when the furnace joins the network and finishes with the establishing of the mode of idle move. The process can be divided intone-stage and two-stage heating.

One-stage heating of the furnace

The one-stage mode of heating characterizes with instant increase of the temperature of the heaters from the starting to the nominal value (the value which is established artificially by the thermal regulator of the furnace) after which it remains constant. Practically, such a mode is hard to achieve in the electric furnaces; it is more usual in steam transfer lines, gas furnaces and others. Discussing it is predominantly of theoretical interest. The power also increases in an instance to a value lower or equal to the nominal power but after that it continuously decreases inclining to the power of idle move (an average power is meant since the full power acts at impulses). Therefore the one-stage heating takes place at constant thermal tension q _furnace. = t _furnace. - t₀ = const and at variable thermal flow q₁ = var. When regarding a one-stage heating of furnace, thermal substitution scheme of one-layer wall is used. The same chain in isometric form gives an idea about the distribution of the elements in the wall.

According to the first Kirhoff’s law applied to a thermal chain, as to electric scheme of the first form, the thermal flow through the wall divides in two compounds: through the thermal capacity and resistance:

The last equation presents the expression for the thermal balance. After its transformation it is written in the form:

Temporal constant of the chain:

When solving the equation of thermal balance at initial conditions for t = 0, q_C = 0 and for t = ¥, q_C = q_c¥ the development is obtained of temperature deviation in the material of the wall (upon C_T) and the thermal tension upon R₂:

The latter expression is the established value of q_C which can be easily obtained from q_C¥ = B/A, or form an established mode. The thermal tension upon R₁ e q_R1 = q_p - q_C = t_p - t_cp. Its established value is :

It is easy to determine the thermal flows and the quantitative heat:

The thermal drops in the individual resistances q_Rkl1, q_Rl1, q_Rl2 q_Rkl2 can also be determined; they are expressed through the flows and resistances or through the temperatures (for example q_Rkl1 = q₁, R_kl1 = t_p – t_pob.bt. and so on). It is easy to determine the development of the temperatures t _pob.bt. t_cp. t _pob.bt. The solution can be brought through the equation: q_p = q_R1 + q_c where:

Equation is obtained from which it is determined:

The analysis so far has been built on the accepted condition that the furnace is heated from its cold condition. But heating often starts from a heated condition, i.e. from initial heating q_H. In this new initial condition it is natural:

So that one-stage heating to occur it is necessary, at assigned R and C the condition P _furnace. ³ q_1H to be observed for t = 0. In an established mode C_T is loaded and ceases taking part in the process. In the thermal process participate only the resistances R_i, with which valid are the deduced above equations for t = ¥. Thus described, the one-stage indirect heating likens the loading of an imperfect condenser (condenser with leakage) .

Two-stage heating.

It is characteristic of the two-stage processes of heating that at the moment of switching on the furnace into the network the thermal flow increases instantly to the nominal value and after that it maintains constant. The temperature in the furnace increases gradually up to the end of the so-called 1^st stage, i.e. until the temperature in the furnace reaches its nominal value, then the thermal regulator is activated and maintains it artificially. From this moment starts the 2^nd stage, during which the temperature is constant and the power decreases and inclines to the power of idle move. Therefore, the 1^st stage develops at constant thermal flow q₁ = const and variable temperature t_nt = var, and with the 2nd stage it is t_n = const and q_n = var. For the regarded case the same thermal substitution schemes are used.

First stage.

From the scheme taking into account that q_R1I = q₁ = P _furnace. = const, of which follows:

When solving the last equation with initial (starting) conditions: for t = 0, q_cI = 0 and for t = ¥, q_cI = q_cI¥ we obtain:

It should be mentioned that in the process of two-stage heating q_CI¥ and q_nt¥ = q_n¥ = q₁(R₁ + R₂) are not reached since the development of q_CI and of q_nt is limited artificially by the thermal regulator after the first stage. The temperatures, the thermal flows and the quantitative heat are obtained similarly to the one-stage heating:

The solution can also be brought through the equation q_nt = q_CI + q_R1I, where:

At the end of the first stage t = t₁, q_nt = q_n < q_n¥. When placing these values in the expression for q_nt and the same mode with regard to t₁, obtained is the time of duration of the 1^st stage:

Second stage.

At the beginning of the second stage the furnace is in a heated condition – the regulator has practically established q_n = const

for the thermal flows and the quantity of heat

It is necessary to examine the speed with which the thermal deviation q_C changes at the end of the 1^st stage and at the beginning of the 2^nd stage. For this purpose determined should be the derivatives of:

by placing q_n = q₁.R₁ + q_Cik in the expression q_CII; this is valid for the beginning of the 2^nd stage. In the result we obtain:

From the last equation we can see that the speeds with which q_C changes by the end of the first and at the beginning of the second period are equal. This means that the substance of the wall manifests its thermal inertness – it is inclined to preserve its thermal condition after the change in the mode, i.e. it is inclined to get heated during the second stage too, the way it was heated during the first stage.

In Table 1 are placed the figures presenting the regarded processes of one-stage and two-stage heating.

Table 1

		to, tbT, tbH, tC, tH – temperatures – as follows: initial, internal (to the heater), external (to the environment), median (in the middle of the wall), nominal. Pnach, P furnace, Ppx – powers – as follows: initial, installed, idle move qR, qC, – thermal flows through the thermal resistance and capacity. t – time l – coefficient of thermal conductivity.
One-stage heating	Two-stage heating	Designations used

TRANSITION PROCESS OF HEATING OF ELECTRIC-RESISTANCE FURNACES

The process of heating refers to the transition process of establishing of the temperature, i.e. the process taking place from the beginning of the thermal influence upon a given object until a constant temperature is reached. The description consists in deduction of the equations in differential form for a scheme with concentrated parameters. The differential equations (DE) are consequent to the laws of thermal transfer in established mode solved as function of time. The concretization of the problematics for the resistance furnaces intended for periodic operation in a heated condition leads to discussion of the condition of the installation in starting mode to idle move. The model of heating depends on the thermal resistances R_l, capacity C_T, construction of the furnace, the installed power P _furnace, the nominal temperature T_furnace, the overheating q _furnace. = T_furnace. – T₀. The deduction of the equations is done on the basis of substitution thermal schemes built on the basis of electro-technical analogy. The suggested schemes in Table 1 are analogous to the ones suggested when discussing the laws of thermal transfer, the difference in the used designations aiming at presenting the process in a more general form:

Table 1

		RS1 – RSn ; RT1 – RTn – thermal resistances of the wall and the body respectively N – heater with thermal flows qS and qT to the wall and to the body respectively RklS; RklT; RklOS; - resistances of convection and radiation to the wall, the body and environment tS1 - tS2 ; tT1 - tT2 – moment temperatures in the layers of the wall and the body C1 – Cn ; CT1 – CTn – thermal capacities of the wall and the heated body
Substitution thermal chain – two-layer wall with heater	Substitution thermal chain – two-layer wall and a body divided into a layer and a center with heater	Designations used

In the case of a two-layer wall the analysis is done in the following order:

Thermal flow q:

The system of equations is deduced consecutively for each layer:

General formula for the n-th layer:

The equations are deduced in the form:

Temperature of the heater:

B1 is the disturbing influence:

The system of DE in this form finds application in the development with description of the transition process and its realization through numerical methods for solving of differential equations. For this purpose the equations are placed in a matrix whose mechanism of making is explained when discussing the linear, two-dimensional and three-dimensional substitution schemes. The way of recording of the last equations is convenient for illustrating the initialization of the elements of the matrix. In the regarded example of two walls it is necessary to make a square matrix 2X2 with elements a11, a12, a21 and a22 and a matrix column with elements b1 and b2:

The case of heating with included heated body in the thermal chain is defined by analogy. The thermal flow q from the heater divides in two directions, figure 2 (Table 1).

The needed volume of equations depends on the constructive parameters of the furnace: number of layers of the wall and the heated body as well as on the model of the substitution scheme: linear, two-dimensional or three-dimensional. Regardless of the choice of substitution scheme the equations are made in the given way. The analysis done so far aims to present the transition process in a form convenient for operation, for making a system of differential equations when carrying out the research. Most of the literature dealing with the problematic discussed here heating is regarded as two cases: one-stage and two-stage heating.