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 intonestage and twostage heating.
Onestage
heating of the furnace
The onestage
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 onestage heating
takes place at constant thermal tension q_{ furnace.} = t_{ furnace.}  t_{0} = const and at variable thermal flow q_{1}
= var. When regarding a onestage heating of furnace, thermal substitution scheme of onelayer 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 Kirhoffs 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 R_{2}:
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 R_{1} 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_{R}_{l}_{1},
q_{R}_{l}_{2} q_{Rkl2 }can
also be determined; they are expressed through the flows and resistances or
through the temperatures (for example q_{Rkl1}
= q_{1}, 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
onestage 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
onestage indirect heating likens the loading of an imperfect condenser
(condenser with leakage) .
Twostage heating.
It is characteristic
of the twostage 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 socalled 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 q_{1} = 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} = q_{1} = 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 twostage heating q_{CI}_{¥} and q_{nt}_{¥} = q_{n}_{¥} = q_{1}(R_{1} + R_{2})
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 onestage 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 = t_{1}, q_{nt} = q_{n} < q_{n}_{¥}. When placing these values in the expression
for q_{nt} and the same
mode with regard to t_{1}, 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} = q_{1}.R_{1}
+ 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 onestage and
twostage heating.
Table 1


t_{o},
t_{bT},
t_{bH},
t_{C},
t_{H}
temperatures as follows: initial,
internal (to the heater), external (to the environment), median (in the
middle of the wall), nominal. P_{nach},
P_{ furnace}, P_{px} powers as follows: initial,
installed, idle move q_{R},
q_{C}, thermal flows through the thermal resistance and capacity. t
time l
coefficient of thermal conductivity. 
Designations used 
TRANSITION PROCESS OF HEATING OF ELECTRICRESISTANCE
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.}
T_{0}. The deduction of the equations is done on the basis of substitution
thermal schemes built on the basis of electrotechnical 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


R_{S1} R_{Sn} ; R_{T1}
R_{Tn} thermal resistances of the wall and the body respectively N heater with thermal flows q_{S}
and q_{T} to the wall and to the body respectively R_{klS}; R_{klT}; R_{klOS};
 resistances of convection and radiation to the wall, the body and
environment t_{S1}  t_{S2} ; t_{T1}  t_{T2} moment temperatures
in the layers of the wall and the body C_{1} C_{n} ; C_{T1}
C_{Tn} thermal capacities of the wall and the heated body 
Substitution thermal chain twolayer wall and a body
divided into a layer and a center with heater 
Designations used 
In the case of a
twolayer 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 nth 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,
twodimensional and threedimensional 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, twodimensional or threedimensional. 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: onestage and twostage
heating.