Method for controlling furnace temperature of multi-zone heating furnace

A method for controlling furnace temperatures of a multi-zone heating furnace having a feedback control system for detecting and inputting a furnace temperature and a furnace temperature setting device or a combustion control computer of the type for determining an optimum furnace temperature for each slab and outputting it as a set point to said feedback control system, the method comprising the steps of determining, under predetermined limit conditions which are required for the operations of the furnace, a furnace temperature pattern or a slab temperature rise curve which may minimize the overall flow rate of fuel, heating the furnace along the furnace temperature pattern or the slab temperature rise curve which is determined, reducing, in response to the instruction for suspending the discharge of slabs from the furnace, the flow rate of fuel to be supplied to the furnace to a predetermined level, and heating the furnace along a new slab temperature rise curve which is obtained by shifting the steady state slab temperature rise curve by a slab discharge suspension period, thereby making the slab temperature when the slab discharge is resumed equal to the slab temperature when the slab discharge suspension instruction is received.

BACKGROUND OF THE INVENTION 
The present invention relates to a method for controlling the furnace 
temperature of a multi-zone heating furnace for heating metallic steel 
slabs or the like. 
So far the setting on a furnace temperature control device has been so far 
made in general by the manual setting by an operator, and as a set point 
the value which was obtained from the experiences in operations of 
furnaces has been used. When the furnace temperature is set, firstly it is 
required that the average slab temperature is within a desired range when 
a slab is discharged, and secondly the slab must be heated in such a way 
that the temperature difference between the surfaces and interior of the 
slab may become minimum. The means which most simply satisfies these 
demands is considered to heat the slab as rapidly as possible in the 
initial stage of heating and thereafter soaking the slab at a 
substantially constant temperature. With the prior art manual operations, 
it has been difficult to switch finely the furnace temperature set points 
in response to the conditions of rolling lines so that there has been a 
tendency for using the simple furnace temperature setting of the type 
described above. However, this method has a defect that the furnace 
temperature set point becomes higher as a whole, thus resulting in the 
increase in a fuel unit. 
Recently in view of energy savings, there has been proposed a method for 
using a computer for determining an optimum furnace temperature set point 
in response to the distribution of slabs in the furnace and automatically 
setting this set point. As a result, it has become possible to use the 
relationship between the furnace temperature and the slab temperatures as 
a heating model, thereby predicting the heating conditions of slabs. Thus 
the accuracy with which the slab discharge temperature is controlled has 
been improved. 
For instance, Japanese Patent Application Kokai No. 52-117818 proposes a 
method wherein the furnace temperature in each heating zone is assumed, 
the loss of heat of the furnace and the deviation of a predicted discharge 
temperature of a slab from a set discharge temperature at said assumed 
furnace temperature are obtained, and the furnace temperature in each 
heating zone is so determined and set that the sum of the product of said 
thermal loss and a first coefficient and the product of the square of said 
deviation and a second coefficient may become minimum. 
The prior art methods described above are common in that the relationship 
between the furnace temperature and slab temperature is used. However, the 
relationship between the furnace temperature and fuel flow rate has not 
been directly considered. No consideration has been given to whether the 
furnace temperature pattern which is obtained can minimize the fuel. 
Meanwhile, a rolling line for rolling heated metallic steel workpieces or 
slabs is forced frequently to suspend its operations due to the exchanges 
of rolling rolls, operation troubles or the like. In this case, when the 
normal operations continue in the heating furnace, there is a fear that 
overheating of slabs results in loss of thermal energy. Therefore 
countermeasures are needed such as lowering the furnace temperature for 
some period until the resumption of the operations of the rolling line, 
thereby preventing the slabs from being overheated and contemplating the 
savings of fuel. Furthermore it is needed to guarantee accuracy of the 
slab discharge temperature after the resumption of the operations of the 
rolling line. In one example of the prior art furnace temperature controls 
in the case of the suspension of the operations of the rolling line, the 
furnace temperature set point is lowered in such a way that the 
temperatures of slabs close to the discharge outlet or exit of the furnace 
may be maintained constant, and prior to the resumption of the discharge 
of slabs, the furnace temperature is recovered to that when the slab 
discharge was suspended. With this method, the temperatures of the slabs 
spaced apart from the outlet of the furnace are deviated from the set 
points when the slab discharge is resumed, and it takes a time before the 
furnace temperature control returns to the steady state, resulting in the 
decrease in productivity. In addition, it is not necessarily true that all 
conventional heating furnaces have a plurality of zones in which 
temperatures can be controlled independently of each other. Since 
temperature control in each zone is not effected, the furnace temperature 
control in case of the suspension of the operations of the rolling line is 
not satisfactory. 
SUMMARY OF THE INVENTION 
One of the objects of the present invention is to set an optimum furnace 
temperature so as to improve a fuel unit of the furnace. 
Another object of the present invention is to provide a method for 
controlling the zone temperatures in such a way that all the slabs 
existing in all the zones in the furnace may not be overheated or 
underheated, thereby improving the slab discharge temperature, avoiding 
the decrease in productivity and enabling to effect the furnace 
temperature control adapted to save fuel. 
According to a method of the present invention, the overall flow rate of 
fuel obtained from the heat balance model within a furnace is used as an 
evaluation function, a furnace temperature pattern for minimizing the 
evaluation function under the slab discharge conditions and the furnace 
operation conditions is determined, and this is output as an optimum set 
point of a furnace temperature control device. 
Furthermore according to a method of the present invention, the 
temperatures of all slabs are computed based upon the detected value of 
the furnace temperature during the furnace control when the rolling is 
carried on smoothly the furnace temperature control is so effected that 
these values correspond to a predetermined temperature rise pattern, and 
when the rolling operations are suspended, furnace temperature control is 
so effected that these values may coincide with a suspension pattern which 
takes into consideration of the slab temperature accuracy for its 
discharge in the case of the resumption of the slab discharge, whereby the 
conditions for heating all the slabs in the furnace may be adapted for 
operations.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
The thermal phenomena in a reheat furnace are expressed in terms of the 
transfer of heat by radiation and convection. The thermal transfer between 
slabs, flames and furnace walls are governed by heat transfer by 
radiation, while the thermal transfer due to the flow of gas is governed 
by the heat transfer by convection. FIG. 1 shows the heat transfer 
conditions in steady state of a three-zone continuous heat furnace. I is a 
first zone of the heat furnace; II is a second zone; and III is a third 
zone. A recuperator 11 functions to preheat air to be charged into the 
heat furnace 1. Q.sub.f (1), Q.sub.f (2) and Q.sub.f (3) represent heat 
(Kcal/H) supplied to the first, second and third zones, respectively of 
the heat furnace; Q.sub.A (1), Q.sub.A (2) and Q.sub.A (3) denote latent 
heat (Kcal/H) of the air charged into the first, second and third zones, 
respectively; Q.sub.L (1), Q.sub.L (2) and Q.sub.L (3) indicate thermal 
losses (Kcal/H) escaping from the furnace walls and skids in the first, 
second and third zones, respectively; Q.sub.I shows latent heat (Kcal/H) 
of charged slabs; Q.sub.o is latent heat (Kcal/H) of discharged slab; and 
Q.sub.gas indicates latent heat of exhaust gases in Kcal/H. Eq. (1) is 
held between them. 
##EQU1## 
Let V denote the overall fuel flow rate (Nm.sup.3 /H). Then 
##EQU2## 
EQU Q.sub.gas =V.multidot..gamma..multidot.(T.sub.gas -T.sub.R) (4) 
where 
H.sub.L : heating value (Kcal/Nm.sup.3); 
A.sub.r : air-fuel ratio; 
T.sub.PRE : temperature (.degree.C.) of preheated air; 
T.sub.R : room temperature (.degree.C.); 
C.sub.A : specific heat of air (Kcal/Nm.sup.3 .degree.C.) 
.gamma.: heating value (Kcal/Nm.sup.3 .degree.C.) of a unit of fuel at the 
change of temperature by a unit of exhaust gases; 
T.sub.gas : temperature (.degree.C.) of exhaust gases. Let 
EQU Q.sub.L (I)=Q.sub.L 
and substituting Eqs. (2)-(4) into Eq. (1), provides 
##EQU3## 
The efficiency of the exhaust heat recovery device (recuperator) 11 is 
represented by the following equation. 
##EQU4## 
wherein T'.sub.gas : gas temperature at the inlet to the recuperator. In 
this case, the temperature T.sub.PRE of preheated air may be expressed by 
the following equation as a function of T.sub.gas. 
##EQU5## 
where C: a constant relating to the generation of exhaust gases. 
Substituting Eq. (7) into Eq. (5), we have 
##EQU6## 
where 
EQU C.sub.1 =H.sub.L +.gamma..multidot.T.sub.R (9) 
##EQU7## 
using Eq. (6), Eq. (8) may be expressed in the following equation (8)'. 
##EQU8## 
In Eq. (8)' the latent heat Q.sub.o when the slab is discharged, the heat 
loss Q.sub.L of the furnace and the temperature of exhaust gases at the 
inlet to the recuperator T'.sub.gas can be uniquely determined once the 
temperature distribution in the furnace is determined. That is, Eq. (8) or 
Eq. (8)' is the equation showing the relationship between the furnace 
temperature distribution and fuel flow rate. As a result, for a given 
furnace temperature distribution, the magnitude of its corresponding fuel 
flow rate can be evaluated. 
In general, various conditions are imposed upon the operation of heat 
furnaces. Eqs. (11)-(15) show examples of limiting conditions. 
EQU .theta..sup.(1).sub.out .ltoreq..theta..sub.out 
.ltoreq..theta..sup.(2).sub.out (11) 
EQU .DELTA..theta..sub.out .ltoreq..DELTA..theta..sup.(1).sub.out (12) 
EQU T.sub.1 .ltoreq.T.sub.1MX (13) 
EQU T.sub.2 .ltoreq.T.sub.2MX (14) 
EQU T.sub.3 .ltoreq.T.sub.3MX (15) 
where 
.theta..sub.out : average temperature of slab, 
.theta..sup.(1).sub.Out, .theta..sup.(2).sub.out : allowable minimum and 
maximum temperatures of slab, 
.DELTA..theta..sub.out : temperature difference between the surfaces and 
center of slab, 
.DELTA..theta..sup.(1).sub.out : allowable error of said temperature 
difference, 
T.sub.1, T.sub.2, T.sub.3 : temperatures in the respective zones; 
T.sub.1MX, T.sub.2MX, T.sub.3MX : upper limits of temperatures in the 
respective zones. 
.theta..sup.(1).sub.out, .theta..sup.(2).sub.out, and 
.DELTA..theta..sup.(1).sub.out are determined by the operativity or 
workability in rolling while T.sub.1MX, T.sub.2MX and T.sub.3MX are 
determined by the capacity of the heat furnace. The furnace temperature 
pattern which satisfies the equations (11)-(15) as described above and 
makes fuel flow rate minimum is an optimum furnace temperature pattern. 
Next referring to FIG. 2, the method for fuel minimization in accordance 
with the present invention will be described. 
First, in step 201 N different sets of furnace temperature distributions 
EQU II.sub.i =(T.sub.1i, T.sub.2i, T.sub.3i) 
are selected from the range wherein the limit condition equations (13)-(15) 
are satisfied. Next at step 202 the temperature of discharged slab 
.theta..sub.out and the temperature difference .DELTA..theta..sub.out 
between the surfaces and center of the slab are computed for each of the 
furnace temperature distributions. Since [RENZOKU KOHEN KANETSURO NI OKERU 
DENNETSU JIKKEN TO KEISAN HOHO] (Experiments and Computations of Heat 
Transfer in Continuous Steel Slab Heating Furnaces) TOKUBETSU HOKOKUSHO 
NO. 11 (Special Report No. 11), SHADAN HOJIN NIPPON TETSUKKOU KYOKAI, 
Showa 46.5 describes that this computation is solved by the difference 
approximation of heat transfer equations, the computation method is not 
explained. 
When the average temperature .theta..sub.out of the discharged slab is 
obtained in the manner described above, the heat Q.sub.o (Kcal/H) in the 
case of the discharge of slabs can be obtained from the following 
equation. 
EQU Q.sub.o =.theta..sub.out .multidot.C.sub.s .multidot.M.sub.s (16) 
where 
C.sub.s : specific heat (Kcal/kg .degree.C.) of slab, 
M.sub.s : weight of slab (Kg). 
And the heat loss in each furnace zone can be obtained from the following 
equation depending upon the temperature of this zone. 
##EQU9## 
where .alpha..sub.I and .beta..sub.I are constants dependent upon the 
construction of a heat furnace and are previously determined. Furthermore 
T'.sub.gas .degree.C. is a function f(T.sub.1) of the furnace temperature 
T.sub.1 in the first zone (for instance T'.sub.gas =.xi.T.sub.1, .xi.: 
constant) and can be easily obtained when T.sub.1 is determined. The 
values of these Q.sub.o, Q.sub.L and T'.sub.gas are substituted in Eq. 
(8)' so that the fuel flow rate V (Nm.sup.3 /H) for each of N sets of 
furnace temperature distributions under consideration is computed in step 
203. 
In step 204, of the V which have been obtained for each of N sets of 
patterns the maximum V.sub.max is selected, and in step 205 the furnace 
temperature distribution II.sub.A corresponding to V.sub.max which has 
been obtained in step 204 is corrected in the manner to be described 
below. That is, the center of gravity II.sub.c of the sets of (N-1) 
distributions except the furnace temperature distribution II.sub.A is 
obtained, and II.sub.A is changed to the next point II'.sub.A. Let 
EQU II.sub.c =(T.sub.c1, T.sub.c2, T.sub.c3) 
##EQU10## 
Then 
##EQU11## 
If the new furnace temperature distribution II'.sub.A fails to satisfy 
Eqs. (13)-(15), II".sub.A which satisfies the conditions as described 
above can be obtained by decreasing the value of .delta.. In step 206, 
next the temperature .theta..sub.out of the discharged slab and the 
temperature difference .DELTA..theta..sub.out between the surfaces and 
center of the slab and the fuel flow rate V for the new furnace 
temperature distribution, or furnace temperature pattern are obtained in 
the manner described above. In step 207, it is checked whether or not the 
temperature of the discharged slab, temperature difference of the slab and 
furnace temperature distribution satisfy the conditions (11)-(15). If they 
fail, the furnace temperature distribution is corrected again in step 205 
so as to satisfy these conditions. The re-correction is cycled until all 
the conditions (11)-(15) are established. Upon completion of this 
computation, the maximum flow rate of fuel is selected and its furnace 
temperature distribution is corrected. In this manner, the fuel flow rate 
is sequentially reduced and converges to an optimum value. That is, the 
standard deviation S of a set of N flow rates of fuel is obtained by the 
following equation, and when S becomes less than a preset value .epsilon., 
the flow rate of fuel is taken as being converged (step 208). 
##EQU12## 
In this case, the number of N sets of furnace temperature patterns are 
averaged and provide an optimum pattern. In this manner, a furnace 
temperature pattern is determined which satisfies the slab discharging 
conditions as well as the furnace operation conditions and which reduces 
the fuel consumption to a minimum. 
Next referring to FIG. 3, the control method in accordance with the present 
invention will be described in conjunction with a three-zone reheat 
furnace. 1 is a three-zone heating furnace, and I is called a preheating 
zone, II called a heating zone, and III called a soaking zone. 2a-2c are 
temperature sensors respectively for respective zones; 3a-3c are burners 
in respective zones each for supplying the mixture of fuel and air to be 
burned; 4a-4c are means for controlling the flow rates of fuel; 5a-5c are 
furnace temperature control means. A furnace temperature setting means 6 
in accordance with the present invention has a furnace temperature pattern 
generating means 7, a means 8 for computing an expected slab discharge 
temperature, a decision means 9 for making a decision whether or not a 
furnace temperature distribution or pattern generated by the furnace 
temperature pattern generating means 7 is an optimum furnace temperature 
pattern and a furnace temperature setting output 10 which outputs to the 
furnace temperature control means 5a-5c in terms of the furnace 
temperature setting values T.sub.1p, T.sub.2p and T.sub.3p the furnace 
temperature pattern which is decided as an optimum by the decision means 
9. 
The furnace temperature setting means 6 accomplishes the processing in 
accordance with the flow chart shown in FIG. 2. In the furnace temperature 
setting means 6, when the heating conditions such as schedules of slab 
charging and discharging and thickness of slabs are changed, the 
computation for an optimum furnace temperature pattern for said heating 
conditions is effected. That is, the furnace temperature pattern 
generating means 7 generates a set of N different furnace temperature 
patterns II.sub.i (i=1.about.N) which are within the range which satisfies 
the furnace operation conditions (13)-(15). The slab discharging 
temperature computation means 8 receives these furnace temperature 
patterns as input and computes an average temperature 
.theta..sub.out.sup.(i) and a temperature difference 
.DELTA..theta..sub.out.sup.(i) of the discharged slab for respective 
furnace temperature patterns. The decision means 9 receives as input the 
slab discharge temperature .theta..sub.out.sup.(i) and the temperature 
difference .DELTA..theta..sub.out.sup.(i) which are computed by the 
computation means 8 for computing the expected temperature of discharged 
slab and receives as input the corresponding furnace temperature patterns 
II.sub.i from the furnace temperature pattern generating means 7 so as to 
compute the flow rates Vi of fuel based upon the equation (8)'. The 
deicision means 9 selects the furnace temperature pattern II.sub.i for the 
flow rate of fuel which becomes maximum among the number of N fuel flow 
rates Vi (i=1.about.N), and commands the furnace temperature pattern 
generating means 7 to correct the furnace temperature pattern II.sub.i. In 
response to the command from the decision means 9, the furnace temperature 
pattern generating means 7 corrects the furnace temperature pattern based 
on the equation (18). However, this correction is carried out until the 
limit conditions (13)-(15) are satisfied. The slab discharge temperature 
computation means 8 computes the discharged slab temperature 
.theta..sub.out.sup.(j) and the temperature difference 
.DELTA..theta..sub.out.sup.(j) for the corrected furnace temperature 
pattern. The decision means 9 receives as input these II.sub.i, 
.theta..sub.out.sup.(j) and checks whether or not these values satisfy the 
limit conditions (11) and (12). If these values do not satisfy the limit 
conditions (11) and (12), the decision means 9 demands the furnace 
temperature pattern generating means 7 re-correction of the furnace 
temperature pattern. The furnace temperature pattern generating means 
accomplishes the fine correction of the furnace temperature pattern in the 
vicinity of II.sub.j ', and the decision means 9 again checks whether or 
not the corrected discharge temperature .theta..sub.out.sup.(j) and the 
corrected temperature difference .DELTA..theta..sub.out.sup.(j) satisfy 
the limit conditions (11) and (12). In this manner, the furnace 
temperature pattern generating means 7 generates new furnace patterns 
II.sub.j which are within the range in which the limit conditions 
(11)-(15) are satisfied. When the new furnace temperature pattern II.sub.j 
is selected, the decision means 9 obtains the flow rate V.sub.j of fuel 
which corresponds to the selected furnace temperature pattern, thereby 
forming pairs of the remaining (N-1) flow rates V.sub.i (i=1.about.N, i=j) 
of fuel. Next the decision means selects the maximum of the number of N 
sets of fuel flow rates and corrects again the furnace temperature pattern 
corresponding to it. 
When the processing described above is repeated, the number of N flow rates 
of fuel converge to one value. In this case, the number of N furnace 
temperature patterns II.sub.i (i=1.about.N) approaches one furnace 
temperature pattern II.sub.p, so that II.sub.p is decided as an optimum 
furnace temperature pattern. Whether or not the flow rates of fuel 
converge to a preset value is judged by whether or not the standard 
deviation S of the number of N flow rates of fuel which are expressed by 
Eq. (19) becomes less than a predetermined value S.sub.min. 
When the zone temperature set values T.sub.1p, T.sub.2p and T.sub.3p which 
are determined in the furnace temperature setting means 6 are output to 
the furnace temperature control means 5a-5c as set values, the furnace 
temperature control means 5a-5c determine the set values V.sub.1p, 
V.sub.2p and V.sub.3p of the flow rate of fuel depending upon the 
deviations betwween the zone temperatures T.sub.1, T.sub.2 and T.sub.3 
which are sensed by the furnace temperature sensors 2a-2c and the set 
values T.sub.1p, T.sub.2p and T.sub.3p. The set values are output to the 
fuel flow rate control means 4a, 4b and 4c as set values. The fuel flow 
rate control means 4a, 4b and 4c adjust the opening degrees of valves in a 
fuel supply system (not shown) in response to the respective set values, 
thereby changing the amounts of fuel to be supplied. Thus the temperatures 
in the respective zones are controlled to the values which are determined 
by the furnace temperature setting means 6. 
According to the embodiment of the present invention which has been 
described so far above, slabs are heated under the furnace temperature 
pattern which satisfies the slab discharging conditions as well as the 
furnace operating conditions and which makes the fuel flow rate minimum. 
The embodiment described above is related to the three-zone heating 
furnace, but the present invention is not limited to the three-zone 
heating furnace and can be equally applied regardless of the number of 
zones. In the above embodiment, the furnace temperature control method in 
accordance with the present invention has been explained in terms of the 
relationship between furnace temperature and flow rate of fuel. However, a 
predetermined releationship can be established between the furnace 
temperature and the slab temperature as will be described below, so that 
it is possible to control the flow rate of fuel by heating the slab in 
such a way that the slab temperature follows a temperature increase 
pattern. FIG. 4 shows the overall thermal balance in a heating furnace. 
V.sub.f (1), V.sub.f (2) and V.sub.f (3) are flow rates (Nm.sup.3 /H) of 
fuel supplied to the first, second and third zones; V.sub.A (1), V.sub.A 
(2) and V.sub.A (3) are flow rates (Nm.sup.3 /H) of air charged into the 
first, second and third zones; Q.sub.L (1), Q.sub.L (2) and Q.sub.L (3) 
are thermal losses (Kcal/H) escaping from the furnace walls and skids in 
the first, second and third zones, respectively; .theta..sub.1, 
.theta..sub.2 and .theta..sub.3 are temperatures (.degree.C.) of slabs 
discharged out of the first, second and third zones, respectively; Q.sub.s 
(1), Q.sub.s (2) and Q.sub.s (3) are latent heat (Kcal/H) of slabs 
discharged out of the first, second and third zones, respectively; 
T.sub.1, T.sub.2 and T.sub.3 are furnace temperatures (.degree.C.) in the 
first, second and third zones, respectively; and T.sub.out is a 
temperature (.degree.C.) of exhaust gases. A predetermined relationship is 
established between the furnace temperature T and slab temperature .theta. 
by denoting the furnace temperatures T.sub.1, T.sub.2 and T.sub.3 as T and 
the slab temperatures .theta..sub.1, .theta..sub.2 and .theta..sub.3 as 
.theta. and solving the heat transfer equations by the difference 
approximations as described above. From this relationship, the heat to be 
transferred to the slab from the furnace can be expressed by the following 
equation. 
##EQU13## 
where .phi..sub.CG : overall heat absorptivity, 
.alpha..sub.c : coefficient of heat transfer by convection (Kcal/m.sup.2 H 
.degree.C.), 
Qs: heat transferred to slab (Kcal/m.sup.2 H), 
A: heat receiving surface area of slab (m.sup.2). Since Eq. (21) is held 
true in all zones, the total sum of Qs becomes equal to (Qo-Q.sub.I) in 
Eq. (1). That is, 
##EQU14## 
where n: number of zones. By substituting Eq. (22) in Eq. (5), it becomes 
possible to determine an optimum slab temperature rise pattern. 
Conversely, when the optimum furnace temperature Ti is determined and the 
time elapsed from the inlet to the respective zone to the present position 
of the slab is denoted by .DELTA..tau., the latent heat of the slab after 
.DELTA..tau. is expressed by the following equation by the application of 
Eq. (21). 
##EQU15## 
where h: thickness of slab (m); 
Ho: latent heat of the slab at the outlet of each zone (Kcal/kg); 
Hi: latent heat of slab at the inlet to each zone (Kcal/kg). 
The average temperature .theta. of the slab is expressed by the following 
equation. 
##EQU16## 
where Cm: specific heat of slab (Kcal/kg .degree.C.) 
Therefore an optimum furnace temperature pattern for a slab may be obtained 
from Eq. (24). It is preferable in view of the limitations imposed upon 
the discharge temperature that after the slab has been charged into the 
heating furnace, the temperature control is effected by controlling the 
furnace temperature based upon the slab temperature rise pattern which is 
obtained in the manner described above. Therefore the method for 
dynamically controlling the furnace temperature based upon an optimum slab 
temperature rise pattern will be described hereinafter. 
FIG. 5 shows one example of the furnace temperature control methods which 
have been employed so far in the case of the suspension of a rolling line 
(Iron and Steel Engineer, Sept. 1972, P43-56). When the rolling line is 
suspended, the furnace temperature set point is once lowered so that the 
temperature of a slab sensed by a radiation type temperature sensor 
disposed in the vicinity of the discharge outlet of the furnace, the 
sensor detecting the slab temperature immediately below it, may be 
maintained at a constant level (ABC in FIG. 5). When the rolling is 
started again prior to re-discharging of the slab, the furnace temperature 
control for heating the slab to a desired discharge temperature is started 
and the furnace temperature is recovered to the temperature prior to the 
suspension (CDE in FIG. 5). However such method has the following defects 
or problems; (i) the temperature of the slab disposed immediately below 
the temperature sensor may be maintained constant, but other slab 
temperatures change during the suspension; that is, when slab discharge is 
started again, other slabs are overheated or underheated so that there is 
a tendency that the discharge precision of the slab temperature is lowered 
after the charging has been started again; (ii) since the furnace 
temperature control is started after the operation of the rolling line has 
been started again, it takes a time before it is heated to a set point so 
that the starting of rolling is delayed and consequently productivity is 
lowered; (iii) in general, the heating furnace has a plurality of zones, 
and the temperatures in the respective zones may be operated 
independently. However, there is left a problem how to set the 
temperatures in the respective zones and control them. Unless this problem 
is solved, the furnace temperature control in the case of the suspension 
of the rolling line goes wrong. 
FIG. 6 shows a three-zone heating furnace 1 to which is applied the furnace 
temperature control method of another embodiment of the present invention. 
The heating furnace 1 has a preheating zone, heating zone and soaking zone 
and heats slabs 2 therein. The furnace temperatures in the respective 
zones are detected by means of temperature sensors 2a, 2b and 2c. The flow 
rates of fuel supplied to the respective zones are controlled by control 
means 4a, 4b and 4c. In response to the conditions (for instance, rolling 
troubles, suspension of rolling) of a rolling line or to a production 
schedule, an operator instructs the suspension of the slab discharge to a 
combustion control computer 100 through an operating desk 20. The 
combustion control computer 100 has a unit 101 for computing the 
temperature distribution within the heating furnace 1, a unit 102 for 
computing the slab temperature within the heating furnace 1, a slab 
temperature rise pattern generating unit 103, a suspension temperature 
rise pattern generating unit 104 and a unit 105 for computing a furnace 
temperature set point. The furnace temperature distribution computation 
unit 101 receives as input the furnace temperatures T.sub.1, T.sub.2 and 
T.sub.3 which are detected by the furnace temperature sensors 2a, 2b and 
2c, respectively, in the respective zones, and effects the filtering 
treatments of the furnace temperatures in order to remove detection noise. 
For the sake of simplicity, let T.sub.1, T.sub.2 and T.sub.3 are 
substituted by T.sub.0, and the filtering treatments are effected. Then 
the new furnace temperature T is expressed by the following equation. 
EQU T=.alpha.T.sup.(-1) +(1-.alpha.)T.sub.0 (25) 
where 
a: filtering constant (In general, value of the order of 0.7 is used), 
T.sup.(-1) : the furnace temperature after the previous filtering 
treatment. 
As T.sub.0, the furnace temperatures T.sub.1-3 measured in the respective 
zones are subjected to the filtering treatments. In the following 
description, the furnace temperature refers to the furnace temperature 
which has been subjected to the filtering treatment as described above. 
The slab temperature computation unit 102 computes the present slab 
temperature based upon the thermal conduction computation within the slab 
and with the slab temperature in the previous computation as a starting 
point. The thermal conduction within the slab can be expressed by a 
three-dimensional partial differential equation, but it is known in the 
report published by Nippon Tekko Kyokai that the slab temperature can be 
computed by one-dimensional difference equations such as equations (26), 
(27) and (28) to such a degree of precision which is not objectional in 
practice as described above. When a slab is divided into m slices in the 
direction of its thickness, the interior temperature .theta.'.sub.n, upper 
surface temperature .theta.'.sub.1 and bottom surface temperature 
.theta.'.sub.m of the slab may be expressed by the following equations. 
##EQU17## 
where n: 2 (m-1) 
##EQU18## 
Eq. (26) represents the interior temperature of the slab; Eq. (27) 
represents the upper surface temperature of the slab; and Eq. (28) 
represents the bottom surface temperature of the slab. 
m: total number of divisions in the direction of the thickness of the slab; 
.theta..sub.n : temperature (.degree.C.) of the divided point n before 
.DELTA..tau. time; 
.theta.'.sub.n : present temperature of the divided portion n (.degree.C.); 
k: thermal conductivity (Kcal/m H .degree.C.); 
c: specific heat (Kcal/kg .degree.C.); 
.DELTA.x: distance between the divided portions, m; 
.rho.: specific weight Kg/m.sup.2 
##EQU19## 
.phi..sub.CG : overall heat absorption coefficient. where .theta. is heat 
transferred from the furnace to the slab. Assume that the slab temperature 
be computed at a certain time. In order to obtain the temperature after a 
time interval .tau., this time interval is divided by .DELTA..tau. from 
said certain time and the computation of the temperature rise during the 
time interval .tau. is carried out. That is, the slab temperature after 
.tau. may be obtained by the computations of N=.tau./.DELTA..tau. times. 
Let the computation interval .tau..sub.o of the slab temperature. Then the 
present slab temperature can be obtained by the computation of .tau..sub.o 
/.DELTA..tau. times with the previous slab temperature being a starting 
point. When the present slab temperature is computed, the slab temperature 
after a predetermined time interval can be predicted easily. In order to 
predict the slab temperature after a time interval .tau..sub.p, the slab 
temperature can be obtained by the computation of .tau..sub.p 
/.DELTA..tau. times when the slab is heated at the average furnace 
temperature of T.sub.F for a time interval .tau..sub.p. Therefore in order 
to heat the slab to a desired temperature after a time interval 
.tau..sub.p, it suffices to predict the temperature after .tau..sub.p by 
varying the furnace temperature T.sub.P many times so as to obtain the 
furnace temperature T.sub.F which results in a slab temperature most close 
to a set point and to set this temperature as a new furnace temperature. 
This will be described in more detail hereinafter. The processing in the 
slab temperature rise pattern generating unit 103 comprises the steps of 
predicting a time of passage, or a time interval during which a slab stays 
in the furnace (to be referred to as "a heating time") when this slab is 
charged into it, and selecting among previously preparted standard 
temperature rise patterns one pattern which corresponds to the predicted 
heating time. The prediction of the heating time is carried as follows. 
Assume that two slabs are continuously discharged out of the heating 
furnace. Then the time interval .tau..sub.M from the time when the 
preceding slab has been rolled to the time when the succeeding slab has 
been rolled can be expressed by the following equation. 
EQU .tau..sub.M (i)=.tau..sub.R (i)+.tau..sub.P (i) (29) 
where 
.tau..sub.R (i): predicted value of rolling time of the slab which is 
located in the furnace at the i-th from the discharge opening; 
.tau..sub.P (i): rolling interval between continuous workpieces to be 
rolled. The rolling time prediction value .tau..sub.R (i) is determined by 
the size of a slab so that the rolling time of each slab can be determined 
by the previous computation of the relationship between the size of a slab 
and the rolling time when the slab is charged into the furnace. TABLE 1 
shows one example of the size and rolling time of slabs. Meanwhile the 
intermission time .tau..sub.P (i) between the workpieces to be rolled is 
determined depending upon a production schedule. That is, given the 
production T/H or ton per unit time of a series of slabs .tau..sub.P (i) 
is expressed by the following equation. 
TABLE 1 
______________________________________ 
Relationship between size and 
rolling time of slabs 
Size (m) 
Length Thickness Rolling Time (minutes) 
______________________________________ 
6 .times. 0.2 1.5 
6 .times. 0.25 1.8 
6 .times. 0.30 2.1 
8 .times. 0.2 2.0 
8 .times. 0.25 2.4 
8 .times. 0.30 2.7 
______________________________________ 
EQU .tau..sub.P (i)=Wi/(T/H)-.tau..sub.R (i) (30) 
where 
Wi: weight (kg) of slab i. Furthermore, in the case of the periodic changes 
of rolling rolls, slab discharge is stopped so that such periodic 
suspension time is added to or substituted into .tau..sub.P (i). When 
.tau..sub.M (i) is computed for a slab in the furnace in the manner 
described above, a heating time R of a newly charged slab may be obtained 
from the following equation. 
##EQU20## 
where N: number of slabs which are scheduled to be discharged out of the 
furnace prior to the charging of a new slab. Next the temperature rise 
pattern selection is carreid out in the manner to be described below. The 
temperature rise pattern is a slab temperature rise pattern for heating a 
slab to a set discharge temperature for a length of heating time with a 
minimum quantity of fuel when a heating time of the slab and the set 
discharge temperature of the slab are given. The decision of the 
temperature rise pattern is made in a trial and error method in an actual 
furnace. As with the embodiment described above, based upon the combustion 
model of a furnace, the furnace temperature pattern which results in a 
minimum flow rate of fuel is determined, and the temperature rise pattern 
is determined in the form of slab temperature rise curve when the heating 
furnace is operated according to this furnace temperature pattern. FIG. 7 
shows a temperature rise pattern when a body of soft steel is heated to 
1250.degree. C. for three hours. This temperature rise pattern is stored 
in the slab temperature rise pattern generating unit 103 as function of 
position in the furnace. That is, the heating furnace is equally divided, 
and the slab temperatures on the divided points are obtained from FIG. 7 
and stored. Various slab temperatures obtained depending upon the 
combinations of set discharge temperature, heating time and kinds of steel 
are prepared as table. Alternatively, it may be possible to incorporate 
the above temperature rise pattern computation unit into an on-line 
computer so that the temperature rise pattern is computed immediately 
before a slab is charged and the slab temperatures at the divided points 
are stored in the slab temperature rise pattern generating unit 103. 
Next the processing in the furnace temperature set point computation unit 
105 is carried out in the following manner. In the furnace temperature set 
point computation unit, first the position of a slab after a predetermined 
time .tau..sub.P is obtained. The temperature of the temperature rise 
pattern at this position is obtained and is set as set point temperature 
.theta. of the slab. Furthermore the temperatures .theta. of the 
respective slabs after .tau..sub.P time are predicted in the slab 
temperature computation unit 102 on the assumption that the present 
furnace temperature be maintained. Next in the furnace temperature set 
point computation unit 105, the following evaluation functions for 
respective zones are computed. 
##EQU21## 
where N.sub.I : number of slabs in each zone; 
W: weighting function. The weighting function Wi is determined depending 
upon the position of the slab within the zone. In general, it is 
frequently desired that the temperature of the slab in the vicinity of the 
discharge side of the zone be more closer to a set point than the 
temperature of the slab at the entrance to the zone. Therefore the 
weighting function is determined in such a way as to increase from the 
entrance to the exit. The evaluation function J is a quantity which 
represents the magnitude of deviation from a set point of the slab within 
the zone after .tau..sub.P time. The evaluation function A is a quantity 
which represents whether the deviation is toward the positive or negative 
direction. When J is greater than a previously selected quantity 
.epsilon., the furnace temperature T' which is used in the prediction 
computation is corrected by the following equations. 
##EQU22## 
Next, the expected slab temperatures are obtained again as furnace 
temperature T' used in the prediction computation, and J and A are 
computed. When such process are repeated so that J becomes less than 
.epsilon., the furnace temperature T' is decided as optimum value of the 
furnace temperature from the present time to a time point after 
.tau..sub.P. 
Next the processing in the combustion control computer 100 is carried out 
in the manner to be described below when there is a stop command of an 
operator. In this case, it does not necessarily follow that the discharge 
of slabs from the furnace be immediately stopped. In general, discharge 
stop is previously instructed in such a way that when a slab which is 
scheduled to be discharged out of the furnace reaches the exit, the 
discharge of this slab is stopped. Therefore when the suspension of the 
slab discharge or the suspension schedule is input through the operation 
desk 20 to the computer 100, the information that the discharge is 
suspended when which of the slabs in the furnace is discharged and how 
long the discharge is suspended is instructed. 
When such instruction or command is given by the operator, the suspension 
temperature rise pattern generating unit 104 inputs this suspension time 
and a slab No which starts the suspension. 
When the suspension of the rolling line or the suspension schedule is 
keyed-in by the operator, the processing in the suspension temperature 
rise pattern generating unit 104 is carried out in the following manner. 
When rolling suspension (schedule) is keyed-in from the operator operation 
desk 20, the heating time of the slab becomes longer by a suspension time. 
When the slab is heated under the present conditions, overheating results 
with the resultant cause for loss of energy. Therefore the furnace 
temperature must be lowered. Furthermore, in order to effecting smoothyl 
the heating after the discharge of slabs has been resumed, it is 
preferably that when resumed, the slab temperature is recovered to the 
conditions immediately before the suspension. 
According to the present invention, when the slab discharge suspension (or 
suspension schedule) is keyed-in, the flow rate of fuel to the heating 
furnace is throttled and is returned to a high level again at a certain 
timing based upon a predicted value of the furnace temperature so that the 
slab temperature is controlled to a value immediately prior to the 
suspension when the discharge is resumed. 
FIG. 8 shows the slab temperature rise curve relative to the heating time. 
The curve a.fwdarw.b.fwdarw.c shows the slab temperature rise pattern 
before the discharge suspension (or suspension schedule) is keyed in from 
the operator's operation desk 20. And the curve a'.fwdarw.b'.fwdarw.c' 
indicates the imaginary temperature rise pattern after the discharge 
suspension. Assume that the slab temperature rise is effected along the 
curve a.fwdarw.b.fwdarw.c after the resumption of the slab discharge. 
Then, when the discharge suspension instruction is input at the point b on 
the curve a.fwdarw.b.fwdarw.c, the slab is heated up along the curve 
b'.fwdarw.c' which is obtained by shifting the curve a.fwdarw.b.fwdarw.c 
by .tau..sub.ST when the discharge is resumed. When the discharge is 
suspended at the point b on the curve a.fwdarw.b.fwdarw.c, the temperature 
set point of the slab is lowered to a value along the curve 
d.fwdarw.b'.fwdarw.c' so that the flow rate of fuel for combustion purpose 
is throttled. As a result, the slab temperature is lowered along one of 
the curves b.fwdarw.f.fwdarw.g, b.fwdarw.f'.fwdarw.g', 
b.fwdarw.f".fwdarw.g" and so on. In this case, the degree of throttling 
the flow rate of fuel; that is, lowering of the furnace temperature is 
different from one zone to another. That is, in the soaking zone, the slab 
is already in the uniformly heated state so that the furnace temperature 
is not varied over a wide range, but in the preheating zone the difference 
between the slab temperature and the furnace temperature is greater so 
that the loss of heat escaping through a stack to the outside is great. As 
a consequence, it is preferably from the energy saving view point to lower 
the furnace temperature when the discharge is suspended. Therefore the 
slab temperatures are lowered along different lowering curves in the 
respective zones. Prior to the resumption of the slab discharge, the slabs 
are heated from the intersections f, f' and f" of said lowering curves and 
the imaginary temperature rise pattern a'.fwdarw.b'.fwdarw.c' along the 
curve a'.fwdarw.b'.fwdarw.c'. When the discharge is resumed, the slab 
temperatures at the points b and b' are the same. 
The suspension temperature rise pattern computation unit 104 carries out 
said processing according to the processing flow shown in FIG. 9. 
First, when the suspension is keyed-in, the average velocity V of slabs in 
the furnace after the resumption of the discharge is obtained by the 
following procedure. Let I denote the slab located most close to the exit 
when slab discharge is suspended. A time interval .tau..sub.K (I) from the 
time of the discharge of the slab I to a time when the slab located at the 
k-th from the slab I is discharged is computed by the following equation. 
##EQU23## 
where .tau..sub.M (j) is the discharge time pitch associated with the j-th 
slab from the slab I. Let Ko denote the slab when the .tau..sub.K (I) is 
most close to the suspension time .tau..sub.ST. The relative distance 
X.sub.L between the slabs I and Ko in the furnace may be easily computed 
fom the following equation. 
EQU X.sub.L =.vertline.P(I)-P(Ko).vertline. (36) 
where P(j) is the distance in meters between the exit and the j slab. 
Therefore the average value V of the tracking velocity in the furnace 
after the resumption of the discharge can be obtained from the following 
equation. 
EQU V=X.sub.L /.tau..sub.K (I) (37) 
When the discharge suspension schedule is instructed, it is the slabs to be 
discharged subsequent to the indicated slab that are influenced by the 
suspension. Therefore the change of the temperature rise pattern to be 
described below is effected only for the slabs subsequent to the indicated 
slab. 
It is assumed that during the suspension period .tau..sub.ST the slab is 
moved in the furnace at said velocity V to the present position when the 
discharge is resumed. Then each slab is located closer to the entrance by 
X.sub.L when the suspension is started than the present position. The 
imaginary displacement during the discharge suspension is called the 
imaginary movement. Let .tau..sub.ss denote time interval after the 
discharge suspension. Then the imaginary position of the slab which is the 
position spaced apart by P(i) from the entrance; that is, the imaginary 
distance P'(i) is obtained from the following equation. 
##EQU24## 
As described above, the slab temperature rise pattern is given as function 
of the position in the furnace so that the slab temperature rise pattern 
during the discharge suspension corresponding to this imaginary position 
can be determined. This temperature rise is called imaginary temperature 
rise. According to the concept of the imaginary movement, the slab reaches 
the present position P(i) when the discharge is resumed, so that when the 
discharge is resumed, the slab heating conditions prior to the discharge 
suspension can be reproduced by heating the slab along the imaginary 
temperature rise curve during the discharge suspension. 
The imaginary temperature set point Q.sub.ss (i) of the i-th slab after the 
time period .tau..sub.ss from the discharge suspension is given by the 
following equation. 
EQU Q.sub.ss (i)=f[P'(i)] (39) 
During the discharge suspension period, the processing in the furnace 
temperature set point computation unit 105 is carried out in the following 
manner. 
First, as in the case of the steady state, the computation for predicting 
the slab temperature .tau..sub.PR after the discharge suspension is 
carried out with the suspension (or suspension schedule) instruction 
input. However, the following value T(M) is used as the M-th zone 
temperature after the prediction time .tau..sub.PR. 
EQU T(M)=To(M)-.DELTA.T(M) (40) 
where 
To(M): furnace temperature (.degree.C.) immediately before the discharge 
suspension; 
To(M): corrected value (a constant value) of the furnace temperature 
(.degree.C.). Furthermore in the respective zones, the following average 
value A.sub.M is computed with .theta.(i) being as predicted value of the 
i-th slab temperature. 
##EQU25## 
where N.sub.M : number of slabs in the M-th zone. A.sub.M is an index 
representing whether the predicted temperatures of all slabs in the M-th 
zone are above or below as a whole the imaginary temperature rise pattern. 
As far as A.sub.M is positive, the furnace temperature set point is fixed 
to T(M) (M=1, . . . , 4). Since the furnace temperature T(M) is constant, 
the furnace temperature control for correcting the slab temperature is in 
fact suspended. When A.sub.M becomes zero or negative, the furnace 
temperature control is so effected as to heat the slab along the 
temperature rise pattern Q.sub.ss (I). This control system is the same 
with the processing in the furnace temperature set point computation unit 
105 prior to the suspension operator's key-in. Because of the furnace 
temperature control the furnace temperature rises again so that when the 
discharge is resumed, the slab temperature reaches a set temperature 
immediately before the discharge suspension. 
After the discharge has been resumed, the original control for heating 
along the temperature rise pattern is executed. 
The effects of the embodiment described above are shown in FIGS. 10a, 10b 
and 10c. In the figures, part of the response of the furnace is shown when 
a trouble occurs in the rolling line so that the operator inputs the 
emergency discharge suspension instruction. FIG. 10a shows the variation 
in furnace temperature in the preheating zone and soaking zone. FIG. 10b 
shows the temperature of the discharged slab. FIG. 10c shows the whole 
fuel supplied to the furnace in terms of the ratio with respect to the 
value immediately prior to the discharge suspension. With the suspension 
key-in, the flow rate of fuel is drastically reduced and consequently the 
furnace temperature drops gradually. Prior to the discharge resumption, 
the flow rate of fuel is rapidly increased and the furnace temperature is 
recovered. It is seen that the slab discharge temperature after the 
resumption of slab discharge is slightly lower than a set point or 
substantially equal to a set point. As described above, according to the 
present invention it should be noted that it is possible to carry out the 
stable furnace operations even when the operator inputs the discharge 
suspension (or discharge schedule) so that the scheduling of the discharge 
is changed. Furthermore, according to the present invention, the flow rate 
of fuel can be economized during the suspension period of slab discharge, 
and the slab discharge temperature may be controlled with an extremely 
higher degree of accuracy.