Patent Publication Number: US-4223385-A

Title: Control of workpiece heating

Description:
This is a continuation of application Ser. No. 184,161, filed Sept. 27, 1971, now abandoned. 
    
    
     CROSS-REFERENCE TO RELATED APPLICATION 
     Reference is made to the following concurrently filed and related patent application which is assigned to the present assignee: 
     Ser. No. entitled &#34;Digital Computer Control of Workpiece Heating&#34; and filed by C. E. Peck and W. R. Miller. This application has been abandoned. 
     BACKGROUND OF THE INVENTION 
     The application of a programmed digital computer control system and method for the control of slab heating, particularly for operation in conjunction with an associated hot strip mill, is desirable because of the potential economic saving of considerable dollars. The heating of slabs involves the use of very large amounts of heat energy and the related production of very large total tonnages of steel when integrated over a calendar period such as one year. Improvements in overall consumption of fuel and in scale loss of the metal can be realized and evaluated into dollar figures which far outweigh any investment required to appy such a computer control system. In addition, improvements in slab heating furnace operation and in better overall control of product quality can be realized. The cost advantages and savings in dollars are multiplied to a greater extent as the number of furnaces provided to match the rolling capacity of the associated rolling mills is higher in number and the resultant control actions necessaary to obtain properly heated slabs are sufficient in magnitude to go beyond the ability of a human furnace operator to handle the wide variety of situations which can arise. The availability of high speed programmed digital computer control techniques permits very fast analysis of monitored changes in operating conditions and the desired resultant corrective furnace control operations to follow changes in rolling mill operation faster and more efficiently. 
     This invention pertains to the operation of a slab heating furnace and, more particularly, to controlling the operation of a slab reheat furnace used to heat metal slabs in preparation for rolling in a hot strip rolling mill. 
     To satisfy the demands of hot strip mills, it is necessary to heat slabs to a desired temperature in continuous furnaces, which heat is supplied to each slab on both its upper and lower surfaces as it passes through predetermined heating zones of the furnace. Generally, the slabs are transported in a direction opposite to that of the flow of the heated gases, with the furnaces being divided into predetermined heating zones wherein each heating zone has its own temperature controller for the firing rate of the one or more burners in each of the heating zones. For example, in the conventional five heat zone pusher furnace, three heating zones are utilized and conventionally these are called preheat or charge, heat and soak. 
     The preheat or charge zone is most generally the largest of the three heating zones, both in terms of physical size and in amount of heat added to the slab; it has separate burner systems for the upper and lower slab surfaces. The heat zone which follows the charge zone than heats the slab to the desired rolling temperatures using a similar burner system. When the slab surfaces have reached a desired temperature, the slabs are then transported into the soak zone for the purpose of providing a uniform temperature profile distribution throughout the slab. 
     Because of the continuous nature of material passage through these furnaces, it is highly desirable to approach an operating condition wherein the steady state heated slab output of the furnace substantially matches the heated slab input requirements of the associated rolling mill. Also, it would be advantageous to provide optimum operating conditions in the slab heating furnace to provide the most economical and satisfactory operation of the whole associated process. In addition to economic advantages, improvements in furnace operation and better overall control of heated slab quality will be realized as compared to the prior art conventional and existing modes of furnace operation. 
     It is desired to provide a uniformly heated slab at the proper value of rolling temperature for a wide variety of slab sizes, mixes of grades of slabs, and rates of individual slab movement. Wide fluctuations and sudden changes in firing rates for the respective zones should be minimized in meeting the demands of the associated rolling mill. The soak time requirements in the soak zone for each slab should be set at a minimum consistent with maximum production and minimizing of skid marks, by assuring that the slab has the required total heat content before entering the soak zone area of each furnace. Minimum furnace temperatures should be provided corresponding to the slab tonnage rates being demanded by the rolling mill and the average slab thickness proceeding through each zone of each furnace. It is desired to distribute firing rates between individual firing zones for most efficient utilization of provided heat energy, and multizone continuous furnaces are designed with this capability in relation to variation in demand of the associated rolling mill. The slabs should be protected from damage due to excessive heating, scaling, and so forth when demands by the rolling mill for heated slabs change rapidly. Increased life and lower maintenance on individual furnace refractories can be realized by minizing effects of changes in demand for heated slabs. A control system and method should be provided that is flexible and expandable to permit furnace response to zone temperatures measured and operational feed back from key locations in the associated rolling mill; for example, slab temperatures measured in relation to the initial pass through an early stand of the hot strip rolling mill. The control approach should be applicable to any multizone controlled continuous type slab heating apparatus, such as a pusher, walking beam or roller hearth types of furnaces, or to combinations of these types, and also be applicable to combination fuel fired and electric furnace systems. 
     A previously filed patent applicaion, Ser. No. 705,506 and filing date Feb. 14, 1968, was filed by the same inventors to conver an earlier control system for slab reheat furnaces, which included sensing the physical movement of the slabs as well as monitoring of mill pace time and temperature sensing devices for providing temperature feedback signals related to the upper and lower surface temperatures of the slab. This application has been abandoned. 
     A digital process control computer can include a central integrated process control or setup processor operative with a software sequentially stepped instruction program which is entered into and stored within the storage memory unit of the computer, and including associated input and output equipment such as generally described in an article entitled &#34;Understanding Digital Computer Process Control&#34; by B. H. Murphy, which appeared in Automation for January 1965, pages 71 to 76, and in an article entitled &#34;Small Control Computers--A New Concept&#34; by F. G. Willard which appeared in the Westinghouse Engineer for November 1964 at pages 174 to 179. Two other articles of interest here in regard to the programming of a process control computer should also be noted; one was published in the January 1965 Westinghouse Engineer at pages 13 to 19 by Paul E. Lego and the other was published in the 1966 Iron and Steel Engineer Year Book at pages 328 to 334 by J. S. Deliyannides and A. H. Green. Each computer processor is associated with predetermined input systems not specifically shown, which typically include an input system which scans process signals representing the status of various process operating conditions, a conventional analog input system which scans and converts process analog signals and operator controlled and other input devices and systems which could include paper tape, teletypewriter and dial input apparatus. Various kinds of information are entered into the computer control system through information input devices including for the example of a slab heating furnace, the desired slab heat content H T , grade of slab being heated and so forth as well as hardware oriented programs and control programs for the programming system and so forth. The input system interfaces the computer control system with the process through the medium of measured or detected signals. To effect desired output control actions, control devices are operated directly by means of an output system or by means of analog signals derived from the output system through a digital to analog converter. One such control action outputs from the computer control system the temperature setpoint for each heating zone which are applied to the respective temperature controllers to determine the desired operation of the heating zones. The previously determined slab heat content values for each heating zone are stored in memory along with the calculated values of other selected operational parameters. A suitable output display can be provided for operation with the computer control system in order to keep the process operator generally informed about the process operation and in order to signal the operator regarding an event or condition relative to any particular furnace which may require some action on his part. The use of an on-line digital computer control system requires that one or more models relating to the controlled process be stored in the memory unit of the computer to enable predictive operation and control of the process and adaptive control of the process relative to updating information obtained from actual operation of the process. 
     SUMMARY OF THE PRESENT INVENTION 
     The control arrangement shown is operative to adjust for each furnace the zone temperature setpoint of the respective temperature controller for each heating zone, based upon updated heat content calculations of the individual slabs in the slab map for that furnace so that the slabs leaving a given zone have achieved a heat content reference value which will produce a slab suitable for subsequent rolling by the associated rolling mill when the slab is discharged from the furnace. 
     For any given temperature setpoint, each heating zone temperature controller varies the firing rate for its respective heating zone to maintain the provided temperature setpoint condition. The updated heat content calculations of the slabs in the furnace map within the computer memory will vary the provided setpoint as required in order to obtain the required heat content to each slab as it leaves each controlled firing zone. 
     In effect, this provides heat content calculation control operation in conjunction with the conventional analog setpoint temperature controller of each zone firing system. A correction in setpoint occurs when the calculated heat content deviates from the required reference heat content for any particular slab in each zone being controlled. 
     The heat content calculations are determined in relation to the following variable inputs in relation to an established mathematical model: 1. the measured effective furnace temperature adjacent the furnace wall in the controlled zone, 2. the total elapsed time that each slab is in each controlled zone, 3. the established size and location of the slab in the respective Zone, and 4. the calculated surface temperature of the slab. The heat transfer rates into a given slab are related directly to the measured true values of the respective heating zone effective furnace temperatures, and the actual heating rates obtained by empirical test observations indicate that the heating of a slab can be calibrated in terms of a dependable measurement of the effective furnace temperature. This can be done by means of thermocouples or closed end radiation tubes located adjacent to the inside surface of the furnace wall at a location outside of the direct flame and exposed to the gas envelope resulting from that flame and used to control the firing rate for the furnace zone. A mathematical model is provided to predict changes in the heat content of each slab in relation to this temperature measurement for obtaining a control operation for slab heating that is consistent and reproducible. The slab heat content control here described is not dependent on the measurement of slab surface temperature, although this would be desirable. Up to the present time, attempts to obtain consistent, reliable and reproducible measurements of the slab surface temperature have not been successful due to conditions inside the furnace at the local measuring point such as the amount of scale on the slab, and the flame characteristics, and so forth. Instead, the slab surface temperature is determined by calculation in accordance with the present teachings. 
     The following principal steps are performed in the present approach to the slab heating process control here described. First, the process control computer has stored in its memory a slab map for each heating furnace that is controlled. At any given time, this slab map shows the location and size of each slab in the respective controlled heating zones of that furnace. Also related to results from a mathematical slab heating model, the process control computer determines the heating status of each slab as related to the heat content of the slab and surface temperature of that slab at its respective location. This heating status determination establishes a time-temperature history of that slab as it progresses through the furnace. Secondly, each slab in each of the various furnaces is updated with respect to heating status and location with the map changes due to the discharge of a slab from that furnace. Thirdly, and independent of slab discharge, each slab is updated at periodic time intervals such as every three minutes for a given furnace with respect to heating status. Fourthly, based on the established time-temperature history and the known location of each slab, the process control computer adjusts the temperature setpoints of the various control heating zones to correspond to determined changing conditions. Fifthly, some of the principal changes in operating conditions which require evaluation and decisions relative to changes in temperature setpoints may be classified in the following categories, and temperature controlling setpoints are adjusted as required to meet these conditions; (a) The furnaces can deliver properly heated slabs on demand as required by the associated rolling mill. Under this condition, the demands of the rolling mill are assumed to be equal to or are less than the capacity of the combination of slab heating furnaces. The corresponding required temperature control setpoints will follow these changes in demand up to the maximum permissible capacity of the respective furnaces. The control operation will adjust temperature setpoints for variable tonnage rates, variable slab sizes and fluctuations in time intervals between delivery of each slab to the rolling mill due to variation in time required by the associated rolling mill to roll the various slabs. (b) With the furnaces operating at maximum capacity and the associated rolling mill demands being greater than furnace capacity, the furnaces are limiting the total process operation rather than the rolling mill. Slabs should not be permitted to be discharged until each slab has been properly heated, in relation to sufficient heat content, sufficient surface temperature and sufficient time in the soak zone. (c) Rolling mill operational delays as distinguished from regular mill operation changes at variable tonnage rates are analyzed and temperature control setpoints and resultant firing rates are gradually reduced. The control operation is set up to adjust temperature setpoints and resulting firing rates for unexpected delays in slab rolling of either short or long durations, and also is set up to adjust temperature setpoints on a planned basis as related to a planned delay of a known duration, especially those which are relatively long. 
     The present control system and method will evaluate operational conditions, make decisions and adjust temperature control setpoints for practically any known variations in operating conditions whether through limitation by furnaces or by the associated rolling mill or by planned or unplanned delays in the operation of the associated rolling mill. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a diagrammatic showing of a slab heating furnace operative with a process control computer; 
     FIG. 2 is a curve plot showing the division of slab heating between the charge zone and the heat zone in relation to predicted travel time for an average seven inches thick slab in the theoretical control area of the furnace heat zone. This curve plot is used to determine the heat content H H  supplied by the heat zone and the heat content H CH  supplied by the charge zone to this average slab; 
     FIG. 3 is a curve plot showing that heat content that is still needed for an average seven inches thick slab in the theoretical control area of the furnace charge zone in relation to the predicted time remaining for that slab in the charge zone. This curve plot is used to determine the temperature setpoint for the charge zone. 
     FIG. 4 is a curve plot showing the heat content that is still needed for an average seven inches thick slab in the theoretical control area of the furnace heat zone in relation to the predicted time remaining for that slab in the heat zone. This curve plot is used to determine the temperature setpoint for the heat zone; 
     FIG. 5 is a composite curve showing of the operational characteristics for a typical 80 inch modern hot strip mill. The curve is used to determine the average rolling mill cycle time interval between the demands for heated slabs and is related to the capacity of the rolling mill to handle heated slabs at this rate; 
     FIG. 6 is a schematic showing of the typical layout of a slab heating furnace including three controlled heating zones; 
     FIG. 7 is a curve plot to heat content for an average seven inches thick slab in relation to time that slab is located within a furnace of known temperature. This curve plot is used to establish the curve plots shown in FIGS. 2, 3 and 4; 
     FIG. 8 is a curve plot to illustrate the Schmidt method of determining the heat content of a slab in relation to time and effective furnace temperature as the slab moves through a furnace; 
     FIG. 9 is a curve plot to illustrate the variation in heat transfer to the surface of a slab as it travels through a furnace; 
     FIG. 10 is a curve plot similar to that shown in FIG. 2, however, for an average six inches thick slab; 
     FIG. 11 is a curve plot similar to that shown in FIG. 3, however, for an average six inches thick slab; 
     FIG. 12 is a curve plot similar to that shown in FIG. 4, however, for an average six inches thick slab; 
     FIG. 13 is a curve plot similar to that shown in FIG. 2, however, for an average 8 inches thick slab; 
     FIG. 14 is a curve plot similar to that shown in FIG. 3, however, for an average 8 inches thick slab; 
     FIG. 15 is a curve plot similar to that shown in FIG. 4, however, for an average 8 inches thick slab; 
     FIG. 16 is a curve plot similar to that shown in FIG. 2, however, for an average 9 inches thick slab; 
     FIG. 17 is a curve plot similar to that shown in FIG. 3, however, for an average 9 inches thick slab; 
     FIG. 18 is a curve plot similar to that shown in FIG. 4, however, for an average 9 inches thick slab; 
     FIG. 19 is a curve plot similar to that shown in FIG. 2, however, for an average 10 inches thick slab; 
     FIG. 20 is a curve plot similar to that shown in FIG. 3, however, for an average 10 inches thick slab; 
     FIG. 21 is a curve plot similar to that shown in FIG. 4, however, for an average 10 inches thick slab; 
     FIG. 22 is a curve plot similar to that shown in FIG. 2, however, for an average 11 inches thick slab; 
     FIG. 23 is a curve plot similar to that shown in FIG. 3, however, for an average 11 inches thick slab; 
     FIG. 24 is a curve plot similar to that shown in FIG. 4, however, for an average 11 inches thick slab; 
     FIG. 25 is a schematic showing of N furnaces discharging heated slabs onto the run out table taking those slabs into the associated rolling mill; 
     FIG. 26 illustrates the determination of the average slab thickness within the theoretical control area of the charge zone or the heat zone; 
     FIG. 27 is a showing of the system flow chart for the delay override program and the push to extract program; 
     FIG. 28 is a showing of the system flow chart for the expected delay program; 
     FIG. 29 is a showing of the system flow chart for the slab heat content, the charge zone temperatures setpoint and the heat zone temperature setpoint periodic determination program; 
     FIG. 30 is a showing of the system flow chart for the charge zone temperature setpoint determination subroutine program; 
     FIG. 31 is a showing of the system flow chart for the heat zone temperature setpoint determination subroutine program; 
     FIGS. 32A, 32B and 32C are a showing of the more detailed programmer flow chart for the delay override program and the push to extract program; 
     FIG. 33 is a showing of the more detailed programmer flow chart of the expected delay program; 
     FIGS. 34A and 34B are a showing of the programmer flow chart of the program for the update of the furnace slab heat content map included within the system flow chart shown in FIG. 29; 
     FIGS. 34C and 34D are a showing of the programmer flow chart of the program for the furnace out of service function when no updating of the slab heat content map is done and the delay logic function for controlling the heat content of the slabs during a delay time period; 
     FIGS. 34E, 34F and 34G are a showing of the programmer flow chart of the program for the manipulate data and control entry to heat content in slab, mill capacity time and slab temperature profile calculation; 
     FIG. 34H is a showing of the programmer flow chart of the program for the limit check of furnace temperature feedbacks and limit of allowable change in furnace temperature for an output temperature setpoint; 
     FIG. 35 is a showing of the programmer flow chart of the program for the mill time capacity calculation for determining the furnace cycle time in accordance with block 264 of FIG. 30; 
     FIG. 36 is a showing of the programmer flow chart of the program for the calculation of the heat content in the average slab in the theoretical control area of each of the charge zone and the heat zone; 
     FIGS. 37A, 37B, 37C and 37D are a showing of the programmer flow chart of the program for the calculation of the temperature profile for the slabs in the furnace under consideration; 
     FIGS. 38A, 38B, 38C, 38D and 38E are a showing of the programmer flow chart of the program to calculate for each of the charge zone and the heat zone the temperature setpoint based upon the average heat content of a slab in the theoretical control area thereof; and 
     FIG. 39 is a showing of the programmer flow chart of the program provided to output the temperature setpoint to each of the charge zone, the heat zone and the soak zone of furnace X under consideration. 
    
    
     GENERAL DESCRIPTION OF A PREFERRED EMBODIMENT 
     The present control system and method is operative to obtain the desired heating of each slab before permitting delivery of that slab from a furnace for rolling. For this consideration, the furnace is defined as a multi-zoned continuous type with a charge zone or preheat area where slabs are charged into the furnace and start absorbing heat, a heat zone following the charge zone in which the remaining desired amount of heat is added before the slab enters the final soak zone. It is desired that the heat content of the slab and a related surface temperature, both of which are periodically updated by the mathematical model portion of the present process control, should correspond to established reference values when reaching the exit of the heat zone and just before entering the soak zone. Sufficient time should elapse in the soak zone to achieve the desired amount of temperature equalization and to eliminate or minimize skid marks to a level acceptable for proper rolling in the associated rolling mill. In addition, the present control system and method will provide for an efficient division of the desired total heat content of each slab between that portion added in the charge zone and the remaining portion added in the heat zone; this means that a high percentage of the available energy in the heat zone should be utilized at all times, and that major variations in firing rate should preferably be confined to the charge zone whenever the furnace capacity is varied to meet the demands of the associated rolling mill. This division of heating load will give higher firing efficiency and lower overall fuel requirements for heating over a wide variety of operating rates in the respective furnaces. 
     Some of the functional operations of the control system and method are as follows: 
     I. Prediction of slab travel time required to move an additional and entering slab through each furnace. 
     II. Periodic updating of heat content of each slab in the furnace corresponding to actual elapsed times and to actual existing effective furnace temperature conditions. 
     III. Prediction of required temperature setpoints of the various heating zones temperature controllers. 
     IV. Before a slab is discharged from the furnace, certain evaluations and decisions are made as to whether it has been heated to the required heat content, surface temperature and whether it has had sufficient time in the soak zone. 
     For the prediction of slab travel time, after each discharge of a slab from a given furnace, calculations are made of the time required to roll the slabs now in each furnace as shown by the updated furnace map. These calculations are based on estimates related to the determinated rolling mill capacity or pace time, and indicate the time necessary for the slabs now entering the same furnace to travel through the furnace. 
     Superimposed on this calculated time is an arbitrarily selected smaller gap time increment which is set by the furnace operator by means of a suitable selection device. This has the effect of modifying the theoretical established mill pace estimates to values more in line with observed actual conditions. The furnace operator can introduce flexibility into the operation in this manner, by changing the gap time when he knows that certain changes are coming which are related to operation of the associated rolling mill and which cannot reasonably be anticipated or programmed as part of a regular process control pattern for predicting the normal mill capacity or demand operational rolling times. In practice, the operator will not find it necessary to change this gap time very frequently. Actual desired changes may occur only a few times in an eight hour turn. In addition, failure by the furnace operator to properly utilize gap time settings in this manner does not materially affect the functioning of the bore described control system and method since there will still be provided heated slabs suitable for rolling in the associated rolling mill. This gap time corrects for errors in predicting T CH  (rem), T H  and T H  (rem) based on an ideal mill cycle time for an average thickness slab in the theoretical control area of the furnace and the mill operating at optimum conditions of operation. The actual mill operation will probably be less than this optimum operation. The furnace operation, over a period of actual furnace operation, will be able to establish a workable guide chart of suitable gap times in relation to the movement of slabs through the furnace and the sizes of those slabs, to better determine what this gap time should be for a desired furnace operation and not result in objectionable hunting of the temperature control setpoints. Once a given furnace operation is properly tuned, and this is somewhat of a startup function, with gap times established in relation to the desired actual furnace operation, the adjustment of gap time thereafter should become rather infrequent. 
     In addition to the manual setting of gap time by the furnace operation, a computer determined gap time can be provided to be in relation to the actual slab movement through the furnace by a difference comparison between the mill capacity and the slab discharge rate, provided a known delay is not present. Also, another furnace malfunction indication that the computer can sense that this gap time should be adjusted is the occurrence of excessive changes such as 50° F., being scheduled by the heat control programs in the temperature control setpoints. The calculation of slab travel time in relation to the slab map and selection of a gap time as described above, specifically defines the time the entering slab is predicted to be in the various zones of the furnace. 
     The next step is to predict the various furnace zone temperature control setpoints. The predicted times and corresponding predicted temperature setpoints will produce a heated slab with the required heat content and temperature profile as it is discharged from the furnace. This assumes that conditions corresponding to these predicted values will be maintained as the entering slab moves through the furnace. While the slab progresses through the furnace, many deviations from the predicted times in the various zones may occur due to mill delays or changes in mill pacing. This deviation from the predicted values may be sufficient to require changes in zone setpoints. The provided control operation will evaluate changes in such conditions and adjust temperature control setpoints to achieve proper heating and desired heating of the slab. 
     In general the present control system and method are operative in relation to the determination of three selected travel times as predicted by the provided control model. These are (1) the travel time for a theoretical average thickness slab in the control area of the charge zone to move through a predetermined control portion of the charge zone of each furnace, (2) the travel time for the above slab to move through the total heat zone of each furnace and (3) the travel time for a theoretical average thickness in the control area of the heat zone to move through a predetermined control portion of the heat zone of each furnace. 
     The effect of deviation of predicted travel times from actual time due to mill pacing and due to delays when the slab heating operation is furnace limited, should be considered. If the associated rolling mill actual demand is such that the predicted time is too slow, for example the furnace gap time selection is set for too large a value, then the operation with respect to the rolling mill if it keeps proceeding in this manner will be limited by the available heated slab output of the furnaces. Also, if the predicted temperature setpoints corresponding to the predicted times are lower than required, unless furnace is already operating at capacity, then delays in the delivery of properly heated slabs to the associated rolling mill will occur due to insufficient heating. The control system and method will try to raise the provided temperature control setpoints, up to the limit of furnace capacity, but the correction may be starting from a lower level than that corresponding to the correct gap time. in this regard, the amount of setpoint correction necessary to compensate for incorrect gap time settings are normally small unless the gap time settings are considerably larger than they should be. The operator can quickly learn to avoid the latter settings as he observes the operation of the control system over a large range of heated slab production rates. When the operation does become furnace limited and results in insufficient heat content of a given slab, there can be provided an indicating light at the furnace operator&#39;s station to show this condition and it also is possible to incorporate into the overall control technique a delay of the next discharge of a slab until the desired heat content and other conditions are fulfilled, unless overriden by the furnace operator. Unless the furnaces are already operating at capacity, the corrections required for the furnace limiting conditions consist of increasing the temperature control setpoint and or decreasing the predicted time by making the furnace gap time smaller. 
     When the slab heating operation is rolling mill limited, if the actual rolling mill pacing is such that the predicted time is too fast, or the gap time is set too small, then the predicted temperature setpoints corresponding to the predicted slab travel times are higher than required. This results in the charge zone taking more of the total slab heating load than normally intended and larger deviations of the provided temperature setpoint in the charge zone. The corrections required for this condition are to lower the temperature control setpoint and/or increase the predicted time. The control system and method will lower the temperature setpoint but the correction starts from a higher travel than that corresponding to the correct gap time. 
     When there occurs unexpected delays in slab heating operation, these are related to unexpected interruptions in the flow of slabs from the combination of furnaces to the rolling mill which may arise due to troubles with the rolling mill or with the furnaces. The times involved for this type of delay may vary from a relatively short period to quite long periods depending upon the reason for the stoppage. When such an unexpected delay occurs, the control system keeps functioning to periodically update the change in heat content of each slab in the respective furnaces, and as the total heat content is gradually increasing, the control system will function to gradually reduce the zone temperature setpoints with the resultant gradual decrease in firing rates in relation to each slab being heated. After an unexpected delay occurs, the reason for the delay and the nature of the delay may require that a planned and related furnace delay control be initiated. This is desirable if a long delay is expected and a prediction can be made of the duration of that delay. During any type of delay, the control system keeps functioning to periodically update the heat content status of each slab in the respective furnaces, unless an indication is made that a given furnace will be out of service as compared to a delay in the process operation. When there occurs expected or planned delays in slab heating operation, the reason for these delays should be known and can be planned ahead in the provided furnace control schedule, such that furnace operation can be adjusted in anticipation of such a planned delay. A separate type of program is used to adjust the furnace operating conditions in relation to expected delays. The corrective action consists of lowering the zone temperature control setpoints on a predetermined functional basis related to the predicted length of the delay, also to hold these setpoints at predetermined minimums for exceedingly long delays and to re-establish proper temperature setpoints after the delay terminates. For the determination of zone temperature control setpoints, the mill pace calculation and the established gap time determine the time period that a given entering slab will be in each of the furnace heating control means. The total desired heat content H T  for proper rolling of that slab must be attained in the slab before it enters the furnace soak zone, since it is assumed that the soak zone is not used to add any additional heat content but is used to equalize temperatures by redistributing the heat energy already in the slab. The total desired heat content H T  for that slab is a precalculated and furnace operator supplied input value to be used as a reference for comparison to the calculated actual heat content H A  contained in the slab before it enters the soak zone. The total desired heat content H T  is used as a reference and is calculated from the known temperature rise of the slab necessary to attain proper rolling temperature, for example, 70° F. to 2350° F. With the times in the respective heating zones determined and with the total reference heat content H T  known, the next step is to properly proportion the increments of heat added in the various zones of each furnace to attain the total reference heat content H T , in accordance with the relationship H T  =H H  +H CH , where H H  =that portion of the heat control contributed by the heat zone and H CH  =that portion contributed by the charge zone. 
     The desirable distribution of H H  and H CH  is to utilize the heat zone capacity at some predetermined percent of its maximum firing rate, and the H H  corresponding to this, when subtracted from total heat content H T  leaves the heat content H CH  to be supplied by the charge zone. To obtain this desirable distribution, a curve such as shown in FIG. 2 is empirically established for each given slab thickness to show the value of H H  corresponding to the predicted time in hours for the slab to travel through the heat zone and utilize the capacity of the heat zone at about 80% of its maximum rating. A considerable amount of empirically obtained basic data is required to determine the values of H H  versus time shown in FIG. 2. 
     When the value of H CH  is known from derived empirical information, such as illustrated by the curve of FIG. 2, it is possible to determine the present actual heat content for each slab in the map in the charge zone, and how much additional heat content H CH  (needed) is still needed for that particular slab to attain a heat content value of H CH  when that slab reaches the exit of the charge zone. This additional heat content needed will determine the required charge zone temperature control setpoint. The location of the slab or slabs in the theoretical control area chosen to determine the amount of heat which must be added and the related temperature setpoint is usually about one half way through the charge zone. 
     The periodic calculation and updating of the heat content of each slab in the furnace permits determination of the amount of heat content in the slab as it leaves the charge zone and as it enters the heat zone. This heat content is used as a basis for determining the amount of heat to be added in the heat zone H H  (needed) during the time interval remaining in the heat zone, and the required temperature setpoint for the control of the heat zone is determined on this basis. 
     Similar updating calculations are made periodically for slabs entering the soak zone and some small adjustments of temperature setpoint can be made in the soak zone if necessary. However, the required heat content and surface temperatures are desired to be contained in the slab before it reaches the soak zone. The slab heating model and the furnace control model included in the overall process control system and method contain all of the necessary curves and equations, and so forth, to determine the desired slab heat contents and the temperature setpoints for the various furnace heating zones. These models may be readily modified as apparent to persons skilled in this art for the particular furnaces used in the slab heating process. In addition, the process control system and method have flexibility to perform optional functions where the slab heating process requires it, such as (1) Furnace temperature setpoints can be adjusted to compensate for differences in temperature which may be measured at preferred locations of the associated rolling mill. This requires reliable measurement of the slab surface temperature in the rolling mill location, and (2) Furnace temperature setpoints can be adjusted as a function of the grade of steel where temperatures can be related to grade, such as stainless steel versus low carbon grades, and so forth. 
     As stated previously, any changes in the temperature setpoints are based on: (1) Existing heat contents of each slab in furnace map as periodically updated, (2) Predicted time for entering slabs to travel through the furnace based on estimates of mill rolling time and a gap time which can be varied by the furnace operator. If each slab moved through the furnace at a rate corresponding to the predicted time intervals remaining in the various zones, then the predicted and actual conditions of heating zone temperatures would be the same and no alteration from predicted values would be required. However, actual conditions may or may not correspond to predicted conditions. Also independent of predicted values, the actual updated picture of the heat content of each slab as it progresses through the furnaces is required in order to make the correct operating decisions. The process control system and method analyzes conditions before each slab discharge, and based on the updated information corresponding to actual conditions, makes decisions and dictates actions related to the following considerations; (1) A predetermined and sufficient time is desired for each slab to remain in the soak zone, (2) The heat content of each slab at end of the heat zone and before entering the soak zone should be at a desired reference level to produce a heated slab suitable for rolling, and (3) The predicted times for each slab remaining in the charge zone and each slab remaining in the heat zone and the actual updated heat contents of slabs at certain key locations should determine any change as needed in the respective temperature setpoints. 
     DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT 
     Referring now to FIG. 1 a five zone slab reheat furnace 10 is shown having three control zones designated as the charge zone 12, the heat zone 14, and the soak zone 16. A slab enters at the charge zone 12 and moves in the direction of the arrow until it is ultimately discharged at the exit of the soak zone 16 at a predetermined and desired heat content and surface temperature. There are a plurality of slabs 18 in the furnace at any given time, as generally shown, which travel at the same rate through the furnace although this rate may change from time to time depending on conditions to be discussed later. As any particular slab passes through the series of controlled heating zones, its ultimate heat content will depend upon the time spent in each such zone and the temperature setpoint for that zone. Most modern furnaces permit the varying of temperature and heating rates within a given heating zone thus causing the heat content supplied to a slab traveling through that zone to be related to both the speed of travel of that slab through that zone as well as the effective furnace temperature of that heating zone. 
     A process control computer 20 is utilized to monitor and control the slab heating furnace operation in accordance with predetermined operating characteristics and adaptive techniques to provide a desired heat content, desired surface temperature and predetermined time in the soak zone for each slab discharged from the furnace. A number of conventional analog input-output devices provide communication between each controlled furnace and the process control computer 20. The signals from all the analog input devices are converted through suitable analog to digital converters for use in the process control computer 20 and all digital output signals from the process control computer 20 are converted through suitable digital to analog converters to provide a proper and desired operation for each slab heating furnace. 
     A slab movement device 22, such as a well known pusher mechanism, is positioned at the entry of the furnace to move the slabs through the furnace at a rate determined by the process control computer 20. Temperature sensors 24 are provided respectively in the upper and lower portions of the charge zone for sensing the effective furnace temperature and temperature feedback signals to the process control computer 20 which are indicative of the respective upper and lower effective furnace temperatures of the charge zone 12. Furnace heating is provided by well known burners 26 which are oriented in the charge zone 12 to pass the resulting hot gases in a direction opposite to that of the slab movement. The heat zone 14 of the furnace similarly has temperature sensors 28 in both the upper and lower portion of the heat zone 14 of the furnace and burners 30 to provide heat to the respective upper and lower portions of the heat zone. The final zone or &#34;soak&#34; zone 16 has a set of burners 32 in the upper portion of the soak zone and a temperature sensor 34 for sensing the effective temperature and providing a corresponding feedback signal to the process control computer 20. 
     The temperature control for each of the respective heating zones in the furnace is by respective temperature controllers which provide a reference temperature setpoint relative to each individual heating zone in accordance with the slab heat content required in that particular heating zone of the furnace. Thusly, the charge zone has a temperature controller 36 to determine the effective furnace temperature in the upper portion of the charge zone 12 and a temperature controller 38 similarly operative with the lower portion of the charge zone 12. The heat zone has temperature controllers 40 and 42 operative with the respective upper and lower portions of the heat zone 14. The soak zone 16 has a temperature controller 44 for controlling the effective furnace temperature of the heat zone 16. A temperature detector 46 is provided at the exit of the furnace and may be positioned after a chosen stand of the rolling mill to provide an indication of the surface temperature of the slab as the slab is being rolled by the rolling mill. 
     The process control computer 20 is operative to output the determined temperature setpoint references to the primary furnace control instrumentation system, i.e. the respective heating zone temperature controllers. One way of doing this, for example, is to have each temperature setpoint reference from the process control computer 20 transmitted by timed closure of suitable contacts for each setpoint position of a respective temperature controller. In this case two different sets of contacts would be required for each heating zone temperature controller, with a first set of contacts to raise the temperature setpoint and a second set of contacts to lower the temperature setpoint. Selected contact closure of the proper duration would energize a reversible stepping motor in the controller to establish the desired setpoint setting, with the contact selection and duration of closure determined by the process control computer 20. 
     The computer is operative with the mathematical model to change setpoints of the temperature controllers to make decisions as to changing temperature of the respective heating zones of the furnace in relation to such conditions as changes in the slab map as to width, thickness, and so forth, and changes in the rate at which slabs are moving through the furnace as influenced by the demands of the rolling mill for slabs, mill delays, and the like. 
     The curve plot shown in FIG. 5 is used to determine the predicted time required in the furnace for slabs to be delivered to meet the rolling mill demands as related to the operating characteristics of the particular associated rolling mill, with the rolling mill capacity, and the rolling mill pacing being some of the factors involved. A typical rolling time interval curve for a modern 80 inch hot strip mill is shown in FIG. 5. The average time interval between delivery of the average slab is derived from this curve and is related to the capacity of the rolling mill to handle slabs at this rate. In utilizing the information obtained from the curve of FIG. 5, the slab map of the furnace is surveyed by the process control computer, and the thickness of each slab in a given furnace is added together and this sum is divided by the number of slabs in the furnace in order to obtain the average slab thickness to be used in relation to the curve of FIG. 5. The width of each slab in the same furnace is added together and this sum is divided by the number of slabs in the furnace in order to obtain the average slab width to be used in relation to the curve of FIG. 5. The average mill cycle per slab is then obtained from the curve. This represents the time interval for the associated rolling mill to receive and roll successive average slabs, when the rolling mill is operated at its design capacity, and this is the same as the time interval between discharge of each slab from the combination of the furnaces. When the mill is operating near its design capacity, the ability of the furnaces to feed the rolling mill at this rate may be limited by the furnace capacity to provide properly heated slabs, for example, the new 80 inch hot strip rolling mills which are supplied by a combination of three furnaces rated at a nominal 250 tons per hour may be furnace limited on the wider or on the thicker slabs such as 60 inches wide or wider and 8 inches thick or thicker. In order to compensate for this and other unusual operating conditions, the predicted time interval between each discharge of the average slab from the combination of furnaces as determined from the curve of FIG. 5 can be altered by the use of an operator setting device which can be used to adjust and vary when necessary, the gap time between successive slab discharges from each of the furnaces. With this operator setting device, the predicted time for discharge of a heated slab from each furnace can be brought more in line with actual process operational conditions when these conditions are known by the experience of the operator to be considerably off normal. As explained previously, this gap time can also be adjusted by the computer being programmed to adjust this gap time setting to correspond to the sensed process operation conditions. It should be here noted that the predicted time and added interval of gap time are numerical values used in determining the travel time of each slab through a furnace. The actual slab discharges from the furnace may take place on demand by the rolling mill, unless the furnace operator intervenes, or unless the control method is set up to prevent discharge of a slab until the desired heat content is obtained. The sum of the predicted time as determined by the demand of the associated rolling mill and obtained form the curve of FIG. 5 and the gap time setting by the operator or computer represents the predicted time between successive discharges of the average slab from the combination of furnaces to the mill, and each furnace would have a predicted slab travel time as determined by the number of furnaces in operation. 
     To further illustrate this determination of predicted slab travel time through a furnace, assume for purpose of example, that three pusher type furnaces are feeding heated slabs to the rolling mill, and the average width of the slabs in a given furnace is 55 inches and the average thickness is 7 inches. The curve of FIG. 5 indicates a rolling mill cycle of 70 seconds between these successive slabs being delivered to the rolling mill. If three furnaces are operating, then each furnace would be required to deliver such a slab every 210 seconds. The operator may, for example, set a gap time in the order of 10 seconds in addition to the rolling mill cycle time of 30 additional seconds in addition to the furnace cycle time per slab from each furnace when using three furnaces. The time between discharge of each slab from a given furnace is then predicted to be 240 seconds. Also, assuming a five heating zone pusher type furnace with an effective hearth length of 111 feet, then the velocity of the average 55 inch wide slab through this furnace, based upon discharging a heated slab from the furnace every 240 seconds, is 68.8 feet per hour. The furnace capacity level to provide properly heated slabs as shown by FIG. 2 for an average 7 inches thick slab would appear to be limited to 61.2 feet per hour, which corresponds to a slab discharge rate of 269 seconds for each slab from the furnace if slabs are 55 inches wide or 89.3 seconds per slab to the rolling mill when using three furnaces. Thusly, in this example, the predicted time based on the rolling mill capacity is such that furnace capacity is exceeded, and if heated slabs are discharged at a rate corresponding to this rolling mill capacity for the same average size of slab, the present control system and method including the programmed slab model and furnace control model will indicate after some elapsed time that slabs are not being heated to rolling temperature with the furnaces operating at their capacity and that the actual discharge rates will have to be reduced to match the furnace capacity in order to have the slabs heated sufficiently. The provided control programs can indicate when the slabs in a given furnace do not have sufficient heat content and a discharge of those slabs can be prevented either directly or by alarming the operator to take appropriate action. The reason for using a calculated predicted time which can be readily adjusted by the furnace operator gap time setting will become more apparent as further steps in the operation of the present control system and method are described. If the rolling mill cycle plus the operator established gap time setting varies considerably from the actual available furnace capacity, either above or below the normal rating of the furnace, then the output temperature control setpoints will vary over a wider range around the desired temperature setpoint which corresponds to the actual operational conditions. These output changes above and below the actual desired temperature setpoint can be narrowed down to relatively small amounts by adjustment of this operator provided gap time. In other words, even though the predicted times may for one reason or another depart considerably from the actual time required corresponding to the existing furnace operating condition, the control programs will function to indicate the temperature setpoint required, but variations above and below this setpoint can be minimized by adjusting gap times corresponding to conditions reasonably within the capability of the furnace. In general, wide departures of predicted time from the actual time occur when the mill cycle based on mill capacity exceeds the available furnace capacity. This occurs on the wider, thicker slabs where existing installed furnace capacity is limited compared to the possible installed mill capacity. It should also be noted, that mill capacity is an ideal condition figure and many variations in everyday practical operation related to expected and unexpected mill delays reduce the ideal mill capacity to levels where the existing and installed furnace capacity can better meet the demands of the rolling mill. 
     For determining the heat content of the average slab corresponding to predicted time of that slab in a particular heating zone of the furnace, the predicted mill cycle time plus gap time previously set by the operator can be used to calculate the amount of time required to move a given slab through the heat zone 14 of the furnace. A schematic showing of the principal dimensions of a typical five zone pusher type slab heating furnace is shown in FIG. 6, with the dimensions being typical so these dimensions will be used in the following example. If the average slab, as available from the furnace slab map, is 48&#34; wide and 7&#34; thick and of sufficient length to cover about 85 percent of the furnace hearth, then from the curve on mill cycle time shown in FIG. 5, it can be determined that the rolling mill could roll this slab at a rate of one slab about every 66 seconds. With a combination of three furnaces in operation, this corresponds to 198 seconds of slab heating time per furnace. If the furnace operator sets the gap time per furnace to 42 seconds, or 14.0 seconds per slab discharged to the rolling mill, this gives the value of 198 second plus 42 seconds or 240 seconds for each furnace cycle time and this gives 80 seconds per slab discharge to the rolling mill with three furnaces in operation, because this time interval is necessary to stay within the limits of normal furnace capacity. With 240 seconds per slab and a slab width of 4 feet, the average rate of slab movement through the furnace if 60 feet per hour. Since the furnace heat zone length is 34 feet, as shown in FIG. 6, this gives the predicted total slab travel time t H  in the heat zone of 0.57 hours. From the curve plot of FIG. 2, the total amount of heat H H  added in the heat zone for a predicted time t H  of 0.57 hours is approximately 21000 B.T.U. per square foot of slab and corresponding to this, the total heat content which must be added in the charge zone H CH  is approximately 89000 B.T.U. per square foot of slab. This is assuming for this example that H H  plus H HC  =110,000 B.T.U. per square foot of slab, which corresponds to the predetermined desired total heat content H T  of the average 7 inches thick slab before the slab enters the soak zone of the furnace. This heat content H T  would be such that the slab has attained an average temperature of 2300° F. which should be suitable for rolling in the associated rolling mill. The values shown in FIG. 2 are based on utilizing the heat zone at about 80% of its maximum capacity and at an operating temperature of 2500° F. for the furnace of this example. The heat zone is maintained at this capacity for maximum efficiency in utilization of fuel, and the remaining desired heat content required in the slab will be added by the charge zone, with the effective furnace temperature of the charge zone being controlled over the range necessary to do this up to that corresponding to the maximum permissible charge zone temperature for the particular furnace involved. 
     The determination of temperature setpoint in the charge zone of the furnace is in relation to a theoretical average 7 inches thick slab in the theoretical control slab area of the charge zone. In order to determine a desired change in the controller temperature setpoint of the charge zone, it is necessary to evaluate the heat content status of a selected theoretical control slab at some predetermined location part way through the charge zone, and compare this with the desired heat content required when the slab leaves the charge zone. The theoretical control slab location and composite width of this theoretical control slab is defined in this example as 96&#34; wide and the center line of this theoretical control slab is located on half way through the charge zone as shown in FIG. 6. The determination of the time remaining for this theoretical control slab to reach the exit of the charge zone is based on the velocity of the slab movement previously determined, and specifically in this example the time remaining t CH  equals about 0.72 hours, based on a velocity of 60 feet per hour and on a length of charge zone of 43 feet. Thusly, the time remaining t CH  (remaining) for the theoretical control slab to travel from a halfway point in the charge zone to the exit of the charge zone is 0.36 hours. To determine the remaining heat content H CH  (needed) which must be added to the heat content currently existing in the theoretical control slab in order to reach the required reference value of heat content H CH  of the slab as it exits from the charge zone, it is first necessary to determine the heat content currently existing in the theoretical control slab, which is the periodically calculated heat content value stored in the slab map of the furnace and obtained by the Schmidt method to be later described. This in turn is based on the sensed effective furnace temperatures and the actual elapsed time that the slab has been in the furnace up to the location of the theoretical control slab area. For example, if the heat content calculation indicates that the heat content of the theoretical control slab at its location is 50,000 B.T.U. per square foot of slab, then the H CH  (needed) is 89,000 minus 50,000 or 39,000 B.T.U. per square foot of slab. By now going to the curve plot in FIG. 3 for H CH  (needed) of 39,000 B.T.U. per square foot of slab and for the time remaining t CH  (rem) in the charge zone of 0.36 hours, this determines the temperature control setpoint for the charge zone to be 2450° F. 
     To illustrate the effect of a delay, suppose the theoretical control slab described above does not keep moving due to a delay in the operation of the associated rolling mill, and during the delay period the updated heat content calculation becomes 61,000 B.T.U. per square foot of slab. The predicted time t CH  (rem) has not changed and the required heat content H CH  of 89,000 B.T.U. per square foot of slab at the exit of the charge zone has not changed. The value of H CH  (needed) becomes 89,000-61,000 or 28,000 B.T.U. per square foot of slab, and the time remaining t CH  (rem) is unchanged at 0.36 hours. Referring again to the curve of FIG. 3, the new required temperature setpoint is now approximately 2400° F. If the delay would continue, the heat content of the theoretical control slab at this location in the charge zone would increase, and the resulting temperature setpoint required for the charge zone would gradually decrease with this trend reversing after the slabs started moving again at the end of the delay period. 
     In relation to the curve plot shown in FIG. 4, and for the determination of temperature setpoints in the heat zone of each furnace, it is desired to control the slab heating operation in relation to the location of theoretical control slab in the heat zone of the furnace. Similar to the heat control operation for the charge zone, it is necessary to evaluate the heat content status of the theoretical control slab some predetermined location in the heat zone and compare this with the heat content required when the slab leaves the heat zone. When this slab arrives at the exit of the heat zone, it should contain the required heat content H H  before entering the soak zone, since very little heat energy is to be added in the soak zone. The location and average width of a theoretical control slab is defined in this example as 96&#34; wide and the center line of this control slab is located one third of the way through the heat zone, as shown in FIG. 6. The time remaining t H  (rem) for the control slab in the control area of the heat zone to reach the exit of the heat zone is based on the velocity of the slab movement as previously determined. Specifically, in this example the remaining time in the heat zone t H  (rem) equals about 0.57 hours, based on a slab velocity of 60 feet per hour and on a length of the heat zone of 34 feet. Thusly, the time remaining t H  (rem) for the control slab to travel from one third of the way in the heat zone to the end of the heat zone is 0.38 hours. For determining the remaining heat content H H  (needed) which must be added to the heat content currently existing in the theoretical control slab in order to reach the required reference value of heat content H T  of the slab as it exits from the heat zone, the heat content currently existing in the theoretical control slab is the calculated value obtained by the Schmidt method to be subsequently described, which in turn is based on the sensed effective furnace temperatures and the actual elapsed time that this slab has been in the furnace up to the location of the theoretical control slab area in the heat zone. Assuming that the latter calculation indicates that the heat content of the theoretical control slab is 104,000 B.T.U. per square foot of slab, and the required heat content at the exit of the heat zone is 110,000 B.T.U. per square foot of slab, then H H  (needed) is 110,000 minus 104,000 or 6000 B.T.U. per square foot of slab. From the curve of FIG. 4, for t H  (rem) of 0.38 hours and H H  (needed) of 6000 B.T.U. per square foot of slab, the temperature control setpoint for the heat zone is approximately 2490° F. Continuing the example, suppose that the control slab at its location one third of the way through the heat zone is stopped for a time period due to a delay in the delivery of heated slabs to the rolling mill; while in this position and during this delay, the updated heat content calculation gives a result of 108,000 B.T.U. per square foot of slab. The predicted time remaining in the heat zone has not changed, so the t H  (rem) is still 0.38 hours, and the required total heat content H T  of 110,000 B.T.U. per square foot of slab has not changed, and therefore H H  needed is now 110,000 minus 108,000 or 2000 B.T.U. per square foot of slab. From the curve plot of FIG. 4, the temperature setpoint for the heat zone is now determined to be a new value of approximately 2450° F. If the delay would continue, the heat content of the slab at this location would increase and the resulting setpoint would gradually decrease to a lower level, with this trend reversing as soon as the slabs started moving again at the end of the delay period. 
     Thusly, for determining changes in temperature setpoints, the three curve plots as shown in FIG. 2, FIG. 3 and FIG. 4 are used in relation to each other to determine the changes in the required temperature setpoints as furnace operating conditions vary. The curves shown are typical of those which are incorporated into the illustrated instruction program listing shown in Appendix A and the associated data files shown in Appendix B, and similar curves are derived for each incremental average control slab thickness using one inch steps, with the process control computer programs being operative to extrapolate the desired values in between these one inch steps of thickness. The derivation of each of these curve plots for any given thickness involves a substantial amount of empirical data analysis of monitored changes in the heat content of typical slabs being heated, and in general, the information contained in these curve plots is generated from a key family of curves for each slab thickness which shows the heat content of the slab as a function of time and effective furnace temperature, with a typical set of the latter family of curves being shown in FIG. 7. The heat content curves are generated from known time and temperature relationships established in relation to scheduled incremental changes in slab temperature profiles with time for various effective furnace temperature conditions. These studies were empirically made in various ways, such as the application of the Schmidt method to be later described, and the results were checked against actual operational data and actual heating rates which were obtained in the operation of actual furnaces such that the overall slab heating results provided by the present control system and method are known to be in line with actual furnace operating experience. Once the here described control programs are generated and applied to a given slab heating furnace system, the provided results can be altered if desired to fit actual performance of that furnace system by applying a readily established factor of correction to the key curves, for example, the results from the curve plots shown in FIG. 2, FIG. 3 and FIG. 4 could be multiplied by an operator supplied factor &#34;f&#34; equal to 0.998 or 1.002, to bring the predicted temperature setpoint values in line with the actual operating slab heating furnace practice and to correct where necessary any slight differences in the operation of individual furnaces to be controlled. 
     DESCRIPTION OF SCHMIDT METHOD OF DETERMINING THE HEAT CONTENT OF EACH SLAB AS IT MOVES THROUGH THE FURNACE 
     The making of decisions to change the control temperature setpoints depends upon the determination of the heat content of each slab in a given furnace as related to time in that furnace and the sensed effective furnace temperature as the slab moves through the respective heating zones of that furnace. The method employed for this determination based on the algebraic equivalent of a graphical method known as the Schmidt method, which graphical method was developed by E. Schmidt, as described in the publication by A Foppls Festschrift, p. 179, J. Springer, Berlin, in 1924. A detailed fundamental description of the method is given in the publication Industrial Furnaces, Vol. I., Appendix I, by W. Trinks, John Wiley &amp; Sons, New York. The method of determining the heat content of a workpiece positioned within a heating zone suitable for use with the present control system and method is not intended to be limited to the Schmidt method here described. If desired, a more sophisticated differential equation approach may be preferred, or any other well known analytical method of determining the time versus heat content relationship of a workpiece positioned within such a heating zone. 
     The here described method as applied specifically to slab heating is operative such that at any given point in time, &#34;t&#34;, the incremental heat added to the slab in the time interval &#34;Dt&#34; is related to the following: 
     (a) TF(°F.); the surrounding and effective furnace temperature which radiates heat to the slab. 
     (b) TS(°F.); the surface temperature of the slab. 
     (c) C. (B.T.U., FT., Thick/°F., sq. ft., hr.); the thermal conductivity of the slab material, which varies with the temperature of the slab. 
     (d) CM (B.T.U./lb., °F.); the specific heat of the slab material, which varies with the temperature of the slab. 
     (e) kT (B.T.U./sq. ft., hr., °F.); the instantaneous overall heat transfer coefficient, which measures the rate of heat transmission between the furnace and the surface of the slab. This coefficient is defined as a direct relationship between the temperature difference between the effective furnace temperature TF and the slab surface temperature TS; whereas the actual rate of heat transfer which is practically all by radiation, and is a function of the difference between the fourth power of the absolute temperature of the furnace and the fourth power of the absolute temperature of the slab surface. This heat transfer coefficient kT is a derived value, and depends upon the heat transferred by radiation and it changes with the furnace temperature and the slab surface temperature. The concept of kT as an overall heat transfer coefficient is necessary to establish and utilize key points in calculations of heat content with the Schmidt method. 
     Other nomenclature used when carrying out the calculations is summarized as follows. 
     Q (lbs./ft. 3 ) density of slab material, and is assumed to be constant as slab is heated. 
     L (Ft.) is 1/2 of the slab thickness. 
     DS(Ft.) is the width of each lamina into which the slab is divided. For good accuracy, the 1/2 slab is divided into at least 4 lamina as indicated in FIG. 8. 
     TS(KN)°F. represents the surface temperature of the slab in the new temperature profile &#34;N&#34; at the end of time increment Dt. For example, TS (K1) is the surface temperature for profile &#34;1&#34;, and so forth. 
     T(KM,I)°F. is the designation for a particular temperature in the new temperature profile at the end of time increment Dt., where t 1  =total elapsed time in hours which established the previous profile and t 2  =total elapsed time in hours to establish the next profile and Dt=t 2  -t 1 . The subscript &#34;KN&#34; represents a particular profile and the subscript &#34;I&#34; represents the location of the temperature point in the profile. For example, T(KO,O) means that &#34;KO&#34; portion of the subscript represents the temperature in the initial profile &#34;O&#34;, and the &#34;O&#34; portion of the subscript represents the location at lamina boundary &#34;O&#34;. The lamina boundaries are shown in FIG. 8. T(KO,1) represents the temperature in the initial profile &#34;O&#34; at lamina location &#34;1&#34;. T(K1,O) represents the temperature of profile &#34;1&#34; at lamina location &#34;O&#34;. T(K1,1) represents the temperature of profile &#34;1&#34; at lamina &#34;1&#34;. T(K1,2) represents the temperature of profile &#34;1&#34; at lamina location 2. T(K100,4) represents the temperature of profile &#34;100&#34; at lamina location 4, and so forth. Information for each new temperature profile corresponding to the passage of time increment &#34;Dt&#34; is based on the previous profile. The various steps to be followed and the relationships used to establish the heat content for a particular temperature profile are shown in detail by the following examples. The pattern of the calculations shown is the basis for the slab heat model program which is used to update the heat content status of each slab in the furnace as time elapses and as the slabs are moved through the furnace. 
     For the calculation of profile &#34;KO&#34;, and in reference to FIG. 8, the first step is to determine the profile interrupt ##EQU1## which equation is derived from the geometry of similar triangles shown graphically in FIG. 8. 
     M=C/kT; C=33-0.01TS(K,O); TS(K,))=50° F. 
     Orig T=50° F.; DS/2=0.0363 ft. 
     C=32.5; kT=30.7 from FIG. 9, showing kT values for various effective furnace temperatures and temperature differences between furnace and slab surface. 
     TF=effective temperature for radiation of heat to slab at entrance section to furnace. In this example TF=2040° F.; the heating zone temperature controller setpoint is set for 2500° F., but the effective furnace temperature is lower at the entering area. TF varies to indicate changes in the temperature profile of the effective furnace temperature in the area where the slab is located in the furnace, and as the slab moves further, the effective furnace temperature will approximately equal the setting of the temperature setpoint supplied to the controller. ##EQU2## Note that T(K1,1) is a point or profile &#34;K1&#34; which is derived from information available from profile &#34;KO&#34;. 
     T(KO,1)=orig T=50° F. 
     T(KO,2)=orig T=50° F. 
     T(KO,3)=orig T=50° F. 
     T(KO,4)-orig T=50° F. 
     Note that profile &#34;KO&#34; is in reality the original temperature profile of the slab before heat is applied and the calculation of T(KO,O) is a necessary construction point to derive a point on the first temperature profile &#34;K1&#34;. In other words, lamina location point &#34;O&#34; is a fictitious 1/2 lamina DS/2 extending beyond the actual slab surface. The various intercepts on lamina location point &#34;O&#34; are all necessary points used to determine the intercept of the succeeding temperature profile point on lamina &#34;1&#34;. 
     The second step is the calculation of profile &#34;K1&#34; as follows: 
     (a) Calculate T(K1,1) from information calculated above for Profile &#34;KO&#34;. ##EQU3## (b) Calculate TS(K1) TS(K1)=TF-GM, with this equation being derived from the geometry of similar triangles shown in FIG. 8. ##EQU4## C=33-0.01TS(KO)=33-0.01×50=32.5 kT=30.7, as obtained from FIG. 9 showing the kT valus. ##EQU5## TS(K1)=2040-1782×1.06=150° F. (c) Calculation of T(K1,0) 
     
         T(K1,0)-TF-G(M-DS/2) 
    
     It should be noted that this equation is derived from the geometry of similar triangles as shown graphically in FIG. 8. 
     M=C/kT; C=33-0.01TS(K1) C=33-0.01×150=31.5 
     kT=32.2 from FIG. 9, showing kT values 
     M=(31.5/32.2)=0.978 ##EQU6## T(K1,0)=2040-1928 (0.978-0.0363)=224° F. (d) Calculation of Remaining Profile &#34;K1&#34; ##EQU7## T(K1,4)=T(KO,3)=50° F. (e) Summary of &#34;K1&#34; profile 
     TS(K1)=150° F. 
     T(K1,1)=84.1° F. 
     TS(K1,2)=50° F. 
     T(K1,3)=50° F. 
     T(K1,4)=50° F. 
     (f) Calculation of Dt; the time intervals at which the temperature points on the profile are attained. 
     
         Dt=0.5(DS).sup.2 Q(CM)/2 
    
     The time interval Dt required for a given point on an existing temperature profile to reach the new temperature at the same corresponding point (at the same location of slab) can be calculated from the above Equation shown for Dt. It should be noted that the time interval is dependent upon the specific heat and the thermal conductivity of the material, both of which vary with the temperature of the slab. For this reason the time interval Dt required to establish new points on the temperature profile is a variable and the new updated profile calculated does not represent this profile was attained with a common time interval. This does not cause any problem in interpreting the result of an updated heat content calculation, since the time at which the surface temperature reaches its new calculated value is the value used for measuring elapsed time, because the calculation of heat transferred into slab is calculated from the change in surface temperature rather than using temperature changes inside the slab. The time interval Dt required to raise the surface temperature from Profile &#34;KO&#34; to Profile &#34;K1&#34; is determined as follows: 
     
         Dt=0.5 (0.0726).sup.2 ×490×(CM/C) 
    
     
         CM (Specified Heat)=0.12+(0.16/1300) TS(KO) 
    
     
         TS(KO)=50° F. 
    
     
         CM=0.12006 
    
     
         C=33-0.01 TS(KO)=33-0.01×50=32.5 
    
     
         Dt=0.5 (0.0726).sup.2 ×490×(0.12006/32.5)=00476 hrs. 
    
     The time interval required to raise the temperature of profile point T(KO,1) to corresponding point T(K1,1) is also 0.00476 hrs. since T(KO,1) is also 50° F., and so forth. 
     It is therefore noted that the time interval required to raise the temperature from one profile point to the next corresponding profile point uses CM (specified heat) and C (conductivity) values based on the temperature of the existing profile point. 
     In calculating the time intervals, 
     CM=0.12+(0.16/1300) T(KN,I), where T(KN,I) is less than 1350° F. 
     CM=0.43-(0.1/900) T(KN,I), where T(KN,I) is equal to or greater than 1350° F., and this provides for the change in state of steel as it passes from magnetic to the non-magnetic condition. 
     Also C=33-0.01 T(KN,I) 
     Following the above equations will give the time interval Dt required to update a point on the temperature profile. 
     (g) Calculation of Heat Content 
     The incremental heat content added as the temperature profile of the slab changes from an existing profile to a new profile is calculated from the equation ##EQU8## Where DH=incremental heat content added in B.T.U. per square foot of slab for a given thickness. 
     A=radiating area to slab, and since the slab is heated from both sides, A=2 sq. ft. 
     e=overall effective emissivity to radiate heat from the effective furnace wall to the surface of the slab. This is dictated by experience with the actual furnaces used, and in this case e=0.8. 
     T 1  =effective radiating temperature of furnace wall to the slab (°F.). This corresponds to the heating zone temperature setpoint setting as dictated by the furnace being used. 
     T 2  =surface temperature of slab, which was defined previously as TS(KN) (°F.). 
     Dt=time interval (hrs.) to transmit heat corresponding to DH. 
     Applying this equation to profile &#34;K1&#34;, where T 1  =2040° F. and T 2  =TS(K1)=150° F. and DT=0.00476 hrs. ##EQU9## DH=0.277 [392,000-1395] (0.00476)=515 B.T.U. 
     To clarify the determination of successive temperature profiles beyond the initial profile, the second complete profile &#34;K2&#34; will be shown in detail and in reference to FIG. 8 of the drawings. 
     (a) Calculate T(K2,1) 
     
         T(K2,1)=T(K1,0)+T(K1,2)/2=224+50/2=137° F. 
    
     (b) Calculate TS(K2) 
     
         TS(K2)=TF=GM ##EQU10## C=33-0.01 TS(K1)=33-0.01×150-31.5 kT=32.2, from FIG. 9 showing kT value 
    
     M=31.5/32.2=0.978 
     G=(2040-137/0.978+0.0363)=1875 
     TS(K2)=2040-1874×0.8=210° F. ##EQU11## C=33-0.01TS(K2)=33-0.01×210=30.90 kT=33.2, from FIG. 9 showing kT values M=(30.9/33.2)=0.931 
     G=(2040-137/0.931+0.0363)=1967 
     T(K2,O)=2040-1967 (0.931-0.0363) 
     T(K2,0)=280 
     (d) Calculate remaining profile ##EQU12## T(K2,4)=T(K1,3)=50° F. (e) Summary of Profile &#34;K2&#34; 
     TS(K2)=210° F. 
     T(K2,1)=137° F. 
     T(K2,2)=67° F. 
     T(K2,3)=50° F. 
     T(K2,4)=50° F. 
     (f) Calculation of the time interval required to go from surface temperature TS(K1) to TS(K2) 
     
         Dt=0.5(DS).sup.2 (Q(CM)/C) 
    
     
         CM=0.12+(0.16/1300) TS(K1) 
    
     CM=0.12184 
     C=33-0.01 TS(K1)=33-0.01×150=31.5 
     
         Dt=0.05 (0726).sup.2 490×(0.12184/31.5)=0.005 hrs. 
    
     (g) Calculation of Heat Content Increment ##EQU13## 
     SUMMARY ##EQU14## As each profile is determined, Dt and DH are incrementally added to for the total. Thus, at end of accumulated time t, there is a corresponding total Heat Content H. This is the method used to obtain the heat content status of the slab as it is moving through the furnace. This method when applied algebraically and programmed into the digital process control computer permits flexibility not possible when applying the basic relationships and equations associated with the conduction of heat in a solid on variable time and temperature basis. 
     Many of the variables which occur in the practical application can be introduced and their effects evaluated. These are: variation in heat transfer to the surface (kT); variation in the temperature of the heat source (TF); variation in the thermal conductivity of material with temperature (C); variation in the specific heat of the material with temperature (CM). Evaluation of these variables in combination with the use of basic relationships of radiation, surface heat transfer and conduction will result in a close check of heating rates as compared to actual slab heating furnace experience. 
     It should be noted that the variation of the temperature of the effective furnace heat source (TF) can be refined by having the computer scan a profile of various measured wall temperatures, or effective furnace temperatures adjacent the wall, along the length of the furnace particularly in the charge zone, where it is known that the temperatures at the charge zone portion of the furnace are lower than near the heat zone portion and which also vary with the rate of furnace output. Those can be weighted or averaged into values used in the Schmidt method. 
     However, it should be noted that the temperature setpoint change is applied in relation to the furnace heating zone temperature sensing thermocouple device and the furnace performance is in effect calibrated as based on the heating zone temperature measuring point. 
     DESCRIPTION OF THE CONSTANTS OF FURNACE SYSTEM REQUIRED IN THE CONTROL MODEL 
     1. Furnace Dimensions 
     The length of the various heating zones and the designations for those zones is necessary to calculate the time in the zone for a given rate of slab movement. 
     The location of the theoretical control slab must be specifically defined, with this information being here selected in accordance with the showing of FIG. 6. 
     The width of the furnace to handle the desired maximum size of slab is utilized in defining hearth area available for slab heating and for defining heating rates. 
     2. Furnace Design Capacities--Heat Content References 
     A particular furnace is designed to deliver a certain ton per hour based on a specific size of slab. The installed heat capacity in B.T.U. per hour, and the effective heating zone temperatures at the control points, must be sufficient to attain the specified tons per hour, if the design criteria are to be met. The minimum times, and the corresponding maximum velocities of slab movement, are used in the control model to correspond to the specified design capacity of the furnace. 
     If the design capacities are not realized in actual practice, then the constants in the control model must be adjusted to correspond to actual operating capacities. The alternate procedure is to apply a correction factor to the constants already in the control model. 
     A summary of the principal constants used for one typical furnace is shown in Table I below. 
     
                       TABLE I                                                     
______________________________________                                    
Summary Of Furnace Capacity - Times - Heat Contents                       
                          Approx.                                         
                          Reqd.  Approx.                                  
                          Heat   Heat                                     
                 Max.     Content                                         
                                 Content                                  
                 Fce.     At Exit                                         
                                 At Exit                                  
        Max.     Capacity of Heat                                         
                                 of     Ave.                              
        Fce.     Lbs Per  Zone   Charge Velocity                          
Slab    Capacity Sq. Ft.  2300° F.                                 
                                 Zone   Ft. Per                           
Thickness                                                                 
        Tons Per Per      Ave.   at     Hour at                           
Inches  Hour     Hour     Temp.  Capacity                                 
                                        Capacity                          
______________________________________                                    
11      235      130      174,000       31.8                              
10      250      138      158,000       37.3                              
9       262      145      142,000       4305                              
8       275      152      126,500       51.3                              
7       288      158      110,000                                         
                                 90,000 61.3                              
6       300      165       64,800       74.3                              
______________________________________                                    
 
    
     The elapsed time in each zone at furnace capacity can be calculated by using the velocity figure in the table and by using charge zone length=42.5 feet, heat zone length=33.75 feet, and soak zone length=33.75 feet. 
     The tons per hour is based on slabs of length of 33 feet, giving a hearth coverage of approximately 85% 
     The time required to heat a slab is principally a function of its thickness. The time in the heating zone and the slab width determine the average velocity of slab movement. If the slabs are not of maximum length, the tons per hour will be reduced correspondingly but heating times will remain unchanged. Heat Contents are expressed in B.T.U. per square foot of slab for a given thickness and all reference heat contents are based on this, so that variations in slab length and hearth coverage do not effect the decisions and actions evaluated in the control model related to predicting a change in the temperature controller setpoint. 
     Also, the effective furnace temperatures in the respective heating zone, at the temperature sensing thermocouple device location are chosen to be 2500° F. for the charge zone and 2500° F. in the heat zone. 
     The soak zone temperatures are assumed to be 2300° F. to 2350° F., in order to maintain the heat content already in the slab as it leaves the heat zone and enters the soak zone. 
     In general, the specific information as outlined above is for purpose of illustration, and can be utilized in the control model for any other similar multizoned and continuous furnace. The particular design capacities, zone data, and so forth as here set forth are applied for purpose of illustration in relation to the particular furnace involved. 
     DESCRIPTION OF THE DETERMINATION OF THE FAMILY OF CURVES SHOWN IN FIG. 2 
     The use of three different sets of curve plots, such as shown in FIG. 2, FIG. 3, and FIG. 4 is the basis for determining changes in temperature controller setpoints in the charge zone and heat zone of the furnace. 
     The curve of FIG. 2 is derived as follows, with a numerical example being worked out as each step is given. 
     1. It is first necessary to define a value for the total time t H  (predicted) required to travel through the Heat Zone. Assume the theoretical control slab in the heat zone is an average 7 inches thick, and the range of values for this time t H  (predicted) must lie within certain limits in order to stay within the slab heating capacity of the furnace. In order to obtain the approximate limits, the time t H  (predicted) corresponding to furnace capacity should be evaluated, and this is done from above Table I showing a summary of furnace constants the capacity of the furnace is 288 tons per hour for a 7 inch thick slab and the corresponding rate of slab movement is given as 61.3 feet per hour. The heat zone is 33.75 feet long, and therefore, the corresponding t H  (predicted) for the theoretical control slab to travel through the heat zone is 0.55 hours. Any time smaller than this value will mean that this furnace is operating at a rate greater than the specified design capacity. In general, time values of 0.55 and larger should be the practical values used to derive points on the curve for FIG. 2. On this basis and to obtain a suitable operating point, assume t H  (predicted)=0.6 hours. 
     2. Next, determine the total value of heat content H T  which should be in the theoretical control slab as it exits from the heat zone. This is another constant in the system, and from the above Table I the required heat content for a rollable 7 inches thick slab with average temperature of 2300° F. is 110,000 B.T.U. per square foot of slab. 
     3. Determine which portion of the total heat content of H T  is added in the heat zone (H H ) and which portion of H T  is added in the charge zone (H CH ), since H T  =H H  +H CH . The distribution of heat content is determined on the basis of adding the increment H H  in the heat zone by operating this heat zone at approximately 80 percent of its capacity, and supplying the remaining increment H CH  in the charge zone at capacity or less; the basic idea in predicting this distribution as shown in FIG. 2 is to utilize the heat zone near its full firing rate as much as possible, and to vary the firing rate of the charge zone to make up the additional heat content required. This has been established in actual operation of slab heating furnaces to provide for the most efficient utilization of fuel in firing the various heating zones of a given furnace. For the above example, the approximate effective furnace temperature required in the heat zone to achieve firing rates corresponding to 80 % of the heat zone capacity is taken as 2500° F. To find a point for FIG. 2 we utilize the basic heat content versus time curves for a 7 inches thick slab as shown in FIG. 7, and from the 2500° F. curve, the total time in the slab heating furnace to achieve the total heat requirement H T  of 110,000 B.T.U. per square foot of 7 inches thick slab is approximately 1.25 hours. The heat content increment added in the heat zone H H  is found by following the 2500° F. curve back down from a point starting at 1.25 hours until the value of t H  =0.6 hours is reached, that is t CH  (predicted) is 1.25-0.6=65 hours. At this point on the 2500° F. curve of FIG. 7 the heat content is already approximately 85,000 B.T.U. per square foot of slab, and therefore H H  =110,000-85,000=25,000 B.T.U. per square foot of slab. Thus for 0.6 hours t H  (predicted) a point on the heat zone curve of FIG. 2 is plotted as H H  =25,000 B.T.U. per square foot of slab, and H CH  =85,000 B.T.U. per square foot of slab and this point on the other curve is also plotted. It should be noted that H H  +C CH  =H T  or 25,000+85,000=110,000. Using this same technique the other points on the respective curves of FIG. 2 may be determined and the curves plotted as shown. 
     DESCRIPTION OF THE DETERMINATION OF THE FAMILY OF CURVES SHOWN IN FIG. 3 
     The curve plot shown in FIG. 3 is used to determine the temperature setpoints in the charge zone, with one such set of curves being used for each slab thickness spaced on one inch intervals as shown in respective FIGS. 11, 14, 17, 20 and 23. First, the curves of FIG. 2 have already been employed in relation to t H  (predicted) and t CH  (predicted) to determine H H  and H CH . When starting with a known average slab thickness mill cycle time and gap time, with the assumption made for purpose of example that the average thickness and average width of the theoretical control slab is 7&#34; thick and 48&#34; wide, the mill cycle for this slab, based on known mill capacity as shown in FIG. 5, is approximately 66 seconds per slab to the mill, or 3×66 or 198 seconds per slab discharged from each furnace, if three furnaces are in operation. Assume the gap time per furnace has been set by the furnace operator at 56 seconds, then the furnace cycle per discharged slab is 198+56 seconds or 0.0712 hours per heated slab. This corresponds to approximately 14 slabs per hour, and if slabs are 48 inches (4 feet) wide, the corresponding velocity is 14×4=56 feet per hour. The predicted time t H  (predicted) in the heat zone is then approximately 0.61 hours based on the heat zone length of 34 feet. From FIG. 2, H H  =26,000 B.T.U. per square foot of each slab and the corresponding H CH  =84,000 B.T.U. per square foot of slab. The corresponding t CH  (predicted), which is the predicted time in the charge zone, is approximately 1.26×t H  predicted; this is obtained from the known ratio of the charge zone length (43 feet) to the heat zone length (34 feet), which then gives t CH  (predicted)=0.768 hours. 
     Secondly, to calculate the elapsed time to the location of the theoretical control slab in the charge zone, include in this figure any unexpected delays which result in this slab not moving from that position in relation to predicted rate of movement. From the above calculation, the total predicted time t CH  (predicted) in the charge zone was 0.768 hours. The center line of the theoretical control slab is initially considered to be located one half way through the charge zone, and therefore, the elapsed time to center line location of the theoretical slab is one half of 0.768 hours or 0.384 hours, assuming that no additional dealays have occurred. 
     Thirdly, to calculate the total elapsed time in the charge zone, without any delays, this value is equal to t CH  (predicted) or 0.768 hours, and if there were any delays, the delay time would be added. 
     Fourthly, to calculate the time remaining t CH  (rem) in the charge zone, subtract from the total elapsed time the elapsed time to the location of the control slab, or t CH  (rem)=0.768-0.384=0.384 hours. This is an abscissa location on the curves shown in FIG. 3 t CH  (rem). 
     Fifthly, to determine the charge zone temperature required to attain H CH  (predicted) at time interval t CH  (predicted), and if there were any delays this delay time would be added to t CH  (predicted), the charge zone temperature required is found on the heat content curves of FIG. 7, using H CH  (predicted)=84,000 and t CH  (predicted)=0.768 hours, such that the heating zone temperature corresponding to those values is approximately 2420° F. 
     Sixthly, to determine the heat content of the theoretical control slab, this will correspond to a point on the extrapolated 2420° F. zone temperature curve as provided in relation to FIG. 7, with the elapsed time corresponding to the location of the theoretical slab being 0.384 hours, such that the corresponding heat content is read from the 2420° F. temperature curve of FIG. 7 to be approximately 47,000 B.T.U. per square foot of slab. 
     Seventhly, to determine the remaining heat content H CH  (needed) required in the charge zone, from H CH  subtract the heat content of the theoretical control slab found above or 84,000-47,000=37,000 B.T.U. per square foot of slab. This is the corresponding ordinate derived for plotting a point on FIG. 3, and the abscissa is 0.384 hours found previously. The points in FIG. 3 determined by successively going through the above procedure will fall along the dotted boundary line indicated by the designation, &#34;Limit Curve Furnace Capacity&#34;. 
     Other points are determined by increasing the elapsed time that the theoretical control slab is in its location, which is the equivalent of an unexpected delay. This longer time increases the heat content of the slab while the slab stays in this location, and this means that when the slabs start moving again, the remaining heat content needed H CH  (needed) is smaller and the required heating zone temperature setpoint to do this in the time remaining t CH  (rem) is also reduced to a lower value. 
     Other points along the curves are determined in a similar manner to generate the curves shown in FIG. 3 as follows: Assume the elapsed time for the theoretical control slab is increased form 0.384 hours to a new value of 0.6 hours, and then the total time in the charge zone is 0.6+0.384 or 0.984 hours. The total heat required in the charge zone remains at 84,000 B.T.U. per square foot of slab is derived previously, and by using the heat content versus time curves of FIG. 7, a determination is made of the new heating zone temperature, such that for 84,000 B.T.U. per square foot of slab and 0.984 hours in the charge zone, the zone temperature setpoint is approximately 2350° F. 
     For 0.6 hours elapsed time at the theoretical control slab location and 2350° F. zone temperature, FIG. 7 will give a heat content of the theoretical slab of 63,000 and the ordinate of the new point on FIG. 3 is then 84,000-63,000=21,000  B.T.U. per square foot of slab which is H CH  (needed), and the abscissa is 0.384 hours for t CH  (rem) and the temperature point is 2350° F. 
     By using this procedure, other points for values plotted on FIG. 3 can be determined. It should be generally noted that determination of points for FIG. 3 curves from values shown in FIG. 7 may require a larger scale for FIG. 7 than that illustrated, in order to obtain sufficient accuracy for the incremental changes being established. 
     DESCRIPTION OF THE DETERMINATION OF THE FAMILY OF CURVES SHOWN IN FIG. 4 
     The FIG. 4 set of curves is used to determine the temperature setpoints in the heat zone, with one set of curves being used for each average slab thickness spaced on one inch intervals in accordance with the respective FIGS. 12, 15, 18, 21 and 24. 
     The derivation is worked out in accordance with the following successive steps, with a related numerical example being provded to illustrate each step. 
     First, use FIG. 2 to determine t H  (predicted) and H H , and follow the same details shown in the first step of the previous FIG. 3 determination. Then, using the same slab size, and so forth, t H  (predicted)=0.61 hours and H H  =26,000 B.T.U. per square foot of slab. 
     Secondly, determine the elapsed time in the heat zone from the start of the heat zone to the location of the theoretical control slab in the heat zone, and if there is some delay it should be included in this figure. The theoretical control slab center line is initially located 1/3 of the way into the heat zone and the elapsed time to this location is therefore 1/3×0.61 hours=0.203 hours. 
     Thirdly, find the point of intersection on the heat content versus time curves of FIG. 7 where the total heat content H T  crosses the 2500° F. curve. This time represents the total time in the charge zone plus the heat zone, and if there is delay time involved, and this delay to the latter total time and find from the curves of FIG. 7 a new intersection of heat zone temperature for the same total heat content H T . H T  for the average 7 inches thick slab is 110,000 B.T.U. per square foot of slab, and the total time on the 2500° F. curve required to obtain 110,000 B.T.U. per square foot heat content is 1.25 hours. 
     Fourthly, determine the total elapsed time in the heat zone only, including unexpected delay time if any, and this is t H  (predicted) or 0.61 hours. 
     Fifthly, determine the time remaining in the heat zone, t H  (rem), and this is the difference between 0.61-0.203=0.406 hours. 
     Sixthly, determine the total heat content of the theoretical control slab in the heat zone, and this is done by tracing the heat zone temperature curve of FIG. 7 back down the curve, and starting at the intersection of 1.25 hours and 110,000 B.T.U. per square foot in FIG. 7 on the 2500° F. curve, go back down the curve a time interval of 0.406 hours. This establishes a new point of intersection at 0.844 hours (1.25-0.406), and at this point the heat content of the slab is found from 2500° F. curve of FIG. 7 is 98,500 B.T.U. per square foot of slab. 
     Finally, determine the remaining heat content needed H H  (needed), and this is equal to 110,000-98,500=11,500  B.T.U. per square foot of slab. A point on curve FIG. 4 is now determined by plotting the values for H H  (needed) and t H  (rem) found previously above, for example, 11,500 B.T.U. per square foot of slab is the ordinate and 0.406 hours in the abscissa and the heat zone temperature is 2500° F. 
     The points on FIG. 4 as above determined will follow a 2500° F. curve. Other points for other temperature curves are determined by increating the elapsed time that the theoretical slab is in its location, such as would occur during a delay, and this longer time will increase the heat content of the slab while it stays in this location. This means that when the slabs start moving again, the remaining heat content needed H H  (needed) is smaller, and the required heat zone temperatures to do this in the time remaining t H  (rem) is also reduced to a lower value. 
     For example, to determine the total time cycle including a delay, assume the elapsed time at the location of the theoretical control slab in the heat zone is increased from 0.203 hours inside the heat zone to 0.34 hours, which is an increase of 0.14 hours and the new total time is 1.25 plus 0.14 or 1.39 hours. Now to determine the new heat zone temperature, find the intersection point of 1.39 hours and heat content H T  of 110,000 in FIG. 7 and find what furnace temperature corresponds to this, with the furnace temperature parameter at this intersection point being approximately 2490° F. Then, to determine the total elapsed time in the heat zone, including the delay time, this is t H  (predicted)+delay time or 0.61 hours+0.14 hours is equal to 0.75 hours. To determine the time remaining t H  (rem) in the heat zone, this is now the difference between 0.75 hours-0.35 hours=0.41 hours, and this stays the same independent of the delay time and is equal to t H  (rem) as previously determined. To determine the heat content of the theoretical slab in the heat zone, this is done by tracing the zone temperature curve (2490°) of FIG. 7 back down the curve, starting at the intersection point at 1.39 hours and H T  =110,000, until the time corresponding to t H  (rem) or 0.41 hours is reached. This corresponds to 1.39-0.41 or 0.98 hours on curve (2490° F.) of FIG. 7, and the corresponding total heat content=103,000 B.T.U. per square foot of slab. Now, to determine the remaining heat content needed H H  (needed), this is equal to 110,000-103,000=7000 B.T.U. per square foot of slab. Another point on the curve of FIG. 4 is now determined by plotting the time remaining t H  (rem) in the heat zone and the remaining heat content needed H H  (needed), for examole, 7000 B.T.U. per square foot of slab is the ordinate and 0.41 hours is the abscissa and the heat zone temperature parameter is 2490° F. 
     Using the same procedure for other delay times will determine other points to be plotted to generate the curves of FIG. 4. It should be noted that the portion of the curves used in FIG. 7 to determine values for FIG. 4 in the heat zone involve small differences, and for sufficient accuracy these portions could be plotted on a scale of greater magnitude than FIG. 7 as now shown. 
     DESCRIPTION OF THE DETERMINATION OF THE OVERALL HEAT TRANSFER COEFFICIENT (kT) 
     The incremental changes in the temperature of the surface of the slab with time are related to the rates of heat transfer between the effective furnace temperature and the slab. The Schmidt Method previously described requires a definition of the overall instantaneous rate of heat transfer, in order to determine the distance C/kT in the graphical representation as shown in FIG. 8. Since practically all of the heat is transferred by radiation, the incremental DH heat content added for a time interval Dt by radiation is: ##EQU15## 
     Also this same increment of heat added can be expressed in terms of an overall coefficient of heat transfer 
     DH=(kT)A(TF-TS)Dt, where 
     TF=effective furnace temperature and TS=surface temperature of the slab. 
     (kT)=overall heat transfer coefficient in B.T.U. per square foot, hour, degrees fahrenheit. 
     Equating the above last two equations and solving for kT, ##EQU16## 
     The family of curves for various effective furnace temperatures as shown in FIG. 9 is based on the latter equation. 
     DEFINITIONS OF VARIOUS TERMS USED IN THE PREVIOUS DESCRIPTIONS 
     t H  (predicted)=time in hours for the slab to travel through the heat zone of the furnace as predicted by mill cycle time and gap time settings for the average slab in the furnace as determined by survey of the slabs in the determine furnace slab map. 
     t H  (rem)=time in hours remaining for the center line of the theoretical control slab to travel through the heat zone. 
     t CH  (predicted)=time in hours, for a slab to travel through the charge zone, with the slab traveling at the same velocity as that used in determining t H  (predicted). 
     t CH  (rem)=time in hours, remaining for the center line of the theoretical control slab to travel through the charge zone. 
     Dt=interval of time (hours) between determination of each new temperature profile point in the slab as defined in the Schmidt Method. 
     Mill Cycle=Time (seconds) required for the average slab (thickness and width) to be rolled in the associated rolling mill at its nominal rated capacity. It is used together with a variable gap time set by the operator to predict the travel time of slabs through the furnace. 
     Gap Time=Time (seconds) set by the furnace operator to adjust the predicted furnace operating conditions to actual demands by the rolling mill. 
     t s  =Time in hours that slab is in soak zone. 
     TF=Temperature of the effective heat radiation sources, including the furnace wall and heated gases surrounding slab, in °F.; also corresponds to furnace heating zone temperature. 
     TS=Surface temperature of slab in °F. 
     T[KN,M]=Designation for a temperature point in the slab temperature profile pattern as determined by the Schmidt Method and shown in FIG. 8. KN locates the laminar division point on the profile; N=0, 1, 2, 3, 4. The slab from its center line is divided into four laminar divisions, as shown in FIG. 8. M equals the sequence number of each complete profile for each interval of time Dt, and starts at 0 and increases by one interval as each profile is generated. 
     H H  =Heat content component of total heat content of a slab which is added in the heat zone, in B.T.U. per square foot of slab. 
     H CH  =Heat content component of total heat content of a slab which is added in the charge zone, in B.T.U. per square foot of slab. 
     H T  =Total heat content of a slab required to attain desired temperature necessary for rolling, with the approximate average slab temperature being 2300° F., in B.T.U. per square foot of slab. This is a predetermined reference value corresponding to the amount of heat which should be in the slab as it leaves the exit of the heat zone and enters the soak zone. All of the various temperature setoint determination curves are based on achieving this reference value, in the relationship of H T  =H H  +H CH . 
     H H  (needed)=Increment of needed or remaining content which is to be added in the heat zone in the time t H  (rem), as the theoretical control slab travels from its theoretical location in the heat zone to the exit of the heat zone, in B.T.U. per square foot of slab. 
     H CH  (needed)=Increment of needed or remaining heat content which is to be added in the charge zone in the time t CH  (rem), as the theoretical control slab travels from its theoretical control area location in the charge zone to the exit of the charge zone, in B.T.U. per square foot of slab. 
     ILLUSTRATIVE MINIMUM TIMES IN SOAK ZONE 
     For the typical pusher type furnace, the following minimum elapsed time in the soak zone is probably desirable. These are based on velocities of slabs which are within the furnace design capacity for a given thickness of slab. 
     
         ______________________________________                                    
                     Max. Fce.q                                           
Slab     Max. Fce.   Capacity     Time In                                 
Thickness                                                                 
         Capacity    Lbs. per Sq. Soak Zone                               
Inches   Tons/Hr.    Ft. per Hr.  Hrs.                                    
______________________________________                                    
6        300         165          0.454                                   
7          287.5       158.4      0.552                                   
8        275         152          0.657                                   
9        262         145          0.775                                   
10       250         1,38         0.905                                   
11       235         130          1.07                                    
______________________________________                                    
 
    
     DESCRIPTION OF SOME OF THE DEVICES USED IN CONJUNCTION WITH PRESENT CONTROL SYSTEM AND METHOD 
     The process control system and method here described are operative in conjunction with the already provided and conventional analog temperature controllers of the furnace combustion system with its attendant instrumentation and hardware. There are some additional devices required whose main function is to indicate and control whether or not a slab is properly heated and should be discharged from the furnace. Examples of these are given below. 
     INSUFFICIENT HEAT CONTENT--PUSHLITE 
     An indicating light will show that the required heat content has not been reached when the slab reaches the exit of the heat zone and just before it enters the soak zone. The furnace discharge push will not be permitted to take place until the desired slab heat content is obtained, and this pushbutton can be used to override the action and permit a push if the furnace operator desires to do so. One such device is provided for each furnace. An insufficient heat content in the slab will also initiate a printout. 
     INSUFFICIENT SOAK TIME--INDICATING LITE 
     An indicating light will show that the slab has not been in the soak zone for a sufficient time. One such device is provided for each furnace. An insufficient soak time will also initiate a printout. 
     NUMBER OF SLABS TO BE DISCHARGED BEFORE START OF A PLANNED DELAY (Thumbwheel or equivalent selector) 
     In relation to an anticipated or planned delay, the furnace operator can set by this device the number of slabs to to be discharged before the delay begins, or the selection can also be coded to initiate a delay immediately and can be coded to terminate a delay ahead of the scheduled termination of a delay time. One such device is provided for each furnace. 
     DURATION OF ANTICIPATED DELAY (Thumbwheel or equivalent selector) 
     The furnace operator sets the duration of the anticipated delay in tenths of hours. One such device is provided for each furnace. 
     ANTICIPATED DELAY FURNACE SELECTOR SWITCH 
     (Switch Positions) (All Furnaces) or Fce 1, Fce 2, Fce 3, and so forth. 
     ANTICIPATED DELAY INITIATE--PUSHBUTTON 
     One for each furnace. Starts the delay program. 
     GAP TIME (Seconds) Thumbwheel Switch 
     The furnace operator sets up the desired gap time interval which is added to the calculated mill cycle time for each slab in the slab map as governed by operational conditions which affect the travel time for entering slabs to go through a given furnace. 
     PROCESS CONTROLLER SELECTED &#34;OFF&#34; PREHEAT ZONE 
     PROCESS CONTROLLER SELECTED &#34;OFF&#34; HEAT ZONE 
     PROCESS CONTROLLER SELECTED &#34;OFF&#34; SOAK ZONE--EAST 
     PROCESS CONTROLLER SELECTED &#34;OFF&#34; SOAK ZONE--WEST 
     The last four items are required for each furnace but not necessarily as a separate device on an operating station; this function may be provided by a signal from contacts on mode operation switches already available on primary furnace combustion control instrumentation. When selected off, the process controller can function, perform and printout on demand, all control functions such as setpoint reference feedback or holding a slab due to insufficient heat or insufficient soak time but cannot actually control these functions on the furnace itself. 
     DIGITAL DISPLAY (TONS PER HOUR) One for all furnaces 
     DISPLAY SELECTOR (Select any furnace) 
     The last two items are optional but convenient in operation as a direct measure of furnace capacity. 
     TOP ND BOTTOM ZONES RATIO FIRING STATUS 
     No device involved, but in the primary furnace control instrumentation the zone firing has an option that provides for top and bottom zones in the charge zone and also in the heat zone sections to fire as one control unit. On this basis, the top temperature sensing units in each heating zone are the only ones used for control of the respective heating zone firing. The bottom zone firing rate is a fixed, but adjustable, ratio of the top zone firing rate in each of the heating zones (charge zone and heat zone but not in soak zone which has not bottom firing). Although separate firing control of the top and bottom sections may be provided, the usual method is to fire the top and bottom in relation to a feedback signal from a control temperature sensor device, such as a thermocouple, located in the top portion of the furnace, and set a firing ratio between the top and bottom. The reason for this practice is due to difficulties encountered in locating a control temperature sensor in the bottom area in a manner which will accurately measure and reproduce an effective furnace temperature which follows and varies directly the bottom firing rates. The status check by means of contacts in the primary control instrumentation tells the process control computer which method of firing control is being used. The following number of signals would be required. 
     (a) Charge Zone one for each furnace 
     (b) Heat Zone one for each furnace 
     ANALOG SIGNALS FOR TEMPERATURE SETPOINT CONTROL 
     The following analog type signals are required between the primary furnace control instrumentation and the process control computer. It is assumed that the signals from all primary control sensors such as temperature sensors will be converted through suitable transmitters (which are a part of the primary instrumentation) to standard D.C. voltage levels for use in the analog to digital converters at the process control computer. These analog type inputs between furnace instrumentation and process controler are as follows: 
     1. Charge zone temperature--Top--One for each furnace 
     2. Charge zone temperature--Bottom--One for each furnace 
     3. Heat zone temperature--Top--One for each furnace 
     4. Heat zone temperature--Bottom--One for each furnace 
     5. Soak zone temperature--Left Side--One for each furnace 
     6. Soak zone temperature--Right Side--One for each furnace 
     MILL DELAY - Indicating Light 
     On furnace operator station this light will be illuminated whenever there is a delay provided in the operation in any area of the mill. This light will operate in parallel to those in other pulpits and will be extinguished when all delays have been released, with one such device being provided for all furnaces. 
     FURNACE DATA DEMAND PRINTOUT BUTTON 
     This device initiates by the operator the printout on the furnace display page printer all of the data which has been classified to be printed on demand. One such device is provided for all furnaces. 
     COMPUTER NOT OPERATING - Indicating Light 
     This light notifies the furnace operator that the process control computer is not functioning to control the slab heating furnaces; it may be considered desirable to also tie this function in with alarm on the furnace annunciator since process control computer failure may not be immediately evident to the furnace operator, who may not always be located at operator&#39;s console where light is located. One such device is provided for all furnaces. 
     PRINTER MESSAGE - Indicating Light 
     This light tells furnace operator when the process control computer is outputting data to the page printer which is located in furnace control area. This light indicates each occurrence of a new message being printed. 
     ANALOG SIGNALS FOR SETPOINT CONTROL 
     The following gives a listing of analog type signals which are provided between the primary furnace control instrumentation and the process control computer. It is assumed that the signals from all primary control sensors, such as temperature sensors, will be converted through suitable transmitters (which are also a part of the primary instrumentation) to standard D.C. voltage levels for use in the analog to digital converters at the process controller. These analog type inputs between furnace instrumentation and the process control computer are as follows: 
     1. Setpoint reference- Charge zone - Top--One for each furnace 
     2. Setpoint reference- Charge zone - Bottom--One for each furnace 
     3. Setpoint reference- Heat zone - Top--One for each furnace 
     4. Setpoint reference- Heat zone- Bottom--One for each furnace 
     5. Setpoint reference- Soak zone - Top Left--One for each furnace 
     6. Setpoint reference- Soak zone - Top Right--One for each furnace 
     DESCRIPTION OF SYSTEM FLOWCHART PROGRAMS 
     It should be understood that the particular system logic flowchart programs shown in FIGS. 27 to 31 are provided merely to illustrate one suitable embodiment of the present control system and method. It should be readily apparent to persons skilled in the art of dynamic process control that many modifications can be made in relation to the selected functions and the arrangement of same as set forth in these programs. 
     There are two basic heat control operations in regard to the slab heating control programs as shown in FIGS. 27, 28 and 29. One is an operator interface as provided wherein the programs react to operator provided inputs; a first of these is the delay override interrupt for furnace X+1 and the second of these is the push to extract interrupt for furnace X as shown in FIG. 27, and a third primary operator input is the expected delay interrupt as shown in FIG. 28. The other heat control operation is periodically undertaken in regard to the rolling mill cycle time and the number of slab heating furnaces in operation, as shown in FIG. 29. The control operation is generally described in relation to three furnaces, however more or less could be controlled, each having five heat control zones. Furnace X refers to any one of furnaces 1, 2, or 3, and furnace X+1 would then be the next succeeding furnace. All three furnaces supply heated slab workpieces to an associated rolling mill, which could include a roughing mill and a finishing mill. 
     There are two basic timing operations in relation to the heat control programs; one operation is provided when the operator has direct input control over initiating or changing what is going on in the furnace operation and the other is the timing operation that is determined by a selected time base of the clock within the process control computer and is called on periodically. The computer has its own internal clock, and every three minutes, or as often as may be desired by the furnace operator, the heat control program shown in FIG. 29 makes certain periodic updates of the information stored in the computer memory in relation to the heat content status of each slab in each furnace. 
     In reference to the operator inputs shown in FIG. 27, the most important operation is probably the push to extract interrupt. When this interrupt occurs, due to the furnace operator deciding to take the next slab out of a given furnace in response to the computer telling him when to do it and after it has been determined that the rolling mill is ready for the next slab and the next slab has been heated properly, the next heated slab is removed from the furnace by a suitable mechanism operative to move at least this next slab in response to the push to extract signal from the operator. 
     At block 203 a check is made to see if the furnace X delay-pending flag is set; if the answer is no, the program continues to block 204 and the previously described normal furnace extract program. If the answer at block 203 is yes, a check is made at block 205 to see if the count equals zero; if the answer is no at block 205, then the count is set to a count of minus one at block 207 and the program then continues to block 204 and the normal furnace extract program. If the answer to the check at block 205 is yes, this indicates that the count does equal 0, and the program advances to the next block 209 of the flow chart. This indicates that furnace X is ready to start into an expected delay, so at block 209 the furnace X delay in progress flag is set, the program gets the length of the delay, and finds the established temperature curves which correspond to that particular delay length. At block 211 the temperature data for the furnace temperature output program is stored, and this part of the program shown in FIG. 27 ends. Then at block 204 a furnace control log is printed out for the furnace operator, which has information pertinent to the slab that was just extracted to indicate its total heat content, the time it was in the furnace and such information. At block 206 a calculation is made of the mill cycle time for the next slab to extract from the next succeeding furnace (X+1), and T MP  is this time that is calculated there. At block 208 a determination is made to see if the next slab out will have a proper soak time, which is another consideration taken here to see if the slab was in the soak zone a proper length of time to allow the heat to be evenly penetrating in the slab and there is a fairly uniform temperature gradient through the slab. It is well known by persons skilled in the art of slab heating furnace operation that at certain temperatures it requires so much time for the heated slab to soak properly before it is ready to be rolled, and an empirical time factor determined by the furnace operator is provided here for this purpose. 
     The three minute furnace cycle for each furnace is required when the associated rolling mill could roll a heated slab every minute, and with three furnaces in operation this requires putting a slab into and taking a slab out of each furnace once every three minutes. There can be anywhere from 30 to 100 slabs being heated in each furnace, and this is dependent upon the width of each slab. A typical slab width will be anything from about 28 inches to about 75 inches or so. If the typical furnace is 100 feet long, and only 28 inch slabs were put into the furnace, this would mean roughly 40 slabs would fit in the furnace. However, some slabs can be placed end-to-end, which permits about 80 or 90 typical slabs to go into the furnace. The present heat control is operative to handle 100 slabs in each furnace as a maximum, but it would be unusual in practice to reach that maximum. 
     The determination at block 208 to see if the next slab will have the proper soak time is based on clock time versus the computer keeping track of the real time that the slab is positioned in the soak zone. If the answer is no, then at block 210 the delay light for furnace (X+1) is lighted. The push to extract operation is for the next furnace (X+1); if a heated slab has just been extracted from furnace X, the heat control program now is looking ahead to the next furnace X+1 in relation to the mill cycle or pace time just calculated to see if the next slab out of the next furnace X+1 will have the proper soak time by the time the rolling mill is ready for it. The furnace delay light for the next furnace (X+1) is lit when there will not be the proper soak time at the time the rolling mill will request the next slab from furnace (X+1), so the operator knows there is going to be a delay and he will not be able to take that slab out of the next furnace (X+1) when the rolling mill is ready for it because it will not be soaked long enough. The heat delay light is also lighted, and this is a delay light provided for the heater operator as compared to the furnace delay light being provided for the furnace operator, since the heater operator is located in one room and the furnace operator is located in another room. To furnish a record of this delay there is printed out a log indicating that this slab is delayed and how long it is going to be delayed and why it was delayed. The delay time is stored in the display buffer at block 212 for internal bookkeeping in the computer. At the next block 214 the contents of this display buffer is outupt to the extract next slab display, which will give an indication to the furnace operator of how long the slab will be in delay. 
     At block 208, if the answer is yes and the next slab will have the proper soak time at the expiration of the mill cycle time, the program advances to block 216 to determine if the slab in the nose of the heat zone will have the proper heat content at the time of the next push. In other words, the determination here is, will the heat content of the slab at that time be equal to H T , the total heat desired plus or minus a delta H provided by the furnace operator as a tolerance factor, in accordance with the relationship H=H T  ±ΔH. This is in relation to the next slab which will leave the heating zone of the next furnace for this determination, since this is the last time to give any appreciable chance to put more heat into that slab and once it moves out of the heat zone, the furnace is not going to be able substantially to increase this heat content. This program step is looking ahead to see whether or not a heat delay should be provided to make sure that the slab is positioned there long enough to get the proper heat content. If the answer to the question in block 216 is yes, at block 214 there is provided a display of the time to extract the next slab and this particular program ends. If the answer at block 216 is no, the program goes through a heat delay procedure similar to the one that was provided in relation to the soak time. The associated lights are lit and the logs are updated as indicated at block 218 and at block 220 the delay time is stored. 
     The furnace operator is given the opportunity to override the computer program generated delay time if desired, and upon his initiation the delay override program for furnace (X+1) is started at block 230. The furnace operator may decide that he wants to get a particular slab through to the rolling mill rather than delay the rolling mill to permit the proper heating of that slab. It may be that he thinks the slabs behind the one about to leave furnace (X+1) are all heated properly, and there was something unusual about the heating of this particular slab. Another reason for providing the delay override function is for debugging a control program that is not working right, and this permits the furnace operator to proceed as he may desire. The delay override program at block 232 will cause the computer to print out the fact that for an identified slab the delay on that slab has been overridden by the furnace operator. At block 234 the delay lights are turned out for the pusher operator and for the extractor operator. At block 236, the display buffer is restored to the mill cycle or pacing time to indicate what operation is desired for rolling mill and not what operation is desired for the furnace. 
     The pusher pulpit is located at the entrance or charge end of the slab heating furnace and includes an operator pushing the slabs into the furnace, and he is not too concerned with the details of the slab heating function. The extractor pulpit is located at the exit or the other end of the furnace, and includes an operator who takes the slabs out of the furnace and he does have some consideration in relation to whether or not the slabs are ready to roll. These operators have verbal communication between each other. When some indicated delay is for the benefit of the extractor operator, it is necessary to also inform the pusher operator what is going on because he may want to put another slab into the furnace but he cannot put one in until the extractor operator takes a slab out of the furnace. 
     At block 210 there was indicated a soak time delay and then at block 218 there was indicated a heat delay, and either one of these can be overridden. If for example a soak delay was previously requested and the operator overrides the soak delay, the control program at block 238 provides a check for a heat delay. If a heat delay was needed at block 216, the delay light at block 218 was turned on. Now if the furnace operator desires to override the heat delay he would initiate the delay override interrupt at block 230, and the control program at block 238 would then ask if this is a heat delay override, and if the answer is yes and the program would advance to block 214 and then turn off. Now the control operation will be in a situation where the rolling mill operation is pacing the extraction of the next heated slab and not the furnace control operation. 
     The expected delay program shown in FIG. 28 is utilized by the furnace operator if something like a roll change in the rolling mill is desired, and this will stop the extraction of slabs from the furnaces for a given period of time such as 10 minutes to an hour. Also, it could be used when the rolling mill is shut down for a turn, and it is only going two turns a day and at the end of the second turn the rolling mill would shut down for eight hours. The furnace operator can use the expected delay program to input this length of expected delay. When the furnace operator dials in the length of an expected delay, this will result in cutting back the furnaces for the indicated length of time. The computer memory includes some predetermined and charted heat control information based on the length of time of an expected delay and how much the control system should cut back the respective furnace temperatures accordingly, and then after a predetermined time bring those temperatures back up so that the furnaces will then be ready to operate as desired at the end of the scheduled expected delay time period. In regard to slabs that have been affected by an expected delay, it is desired to more or less ignore all the data calculated and in the respective slab maps for these slabs, because they may have been setting in the furnaces for three or four hours, in which case the heating of these slabs will even out but the available data have associated with them would not be desirable to feed back into the operation of the control program models because the models are based on the continuous flow of the slabs through the furnace and if the movement of the slabs is stopped for a couple of hours the resulting data is not desired for making control program feedback modifications. 
     One of the big advantages of the computer control system and method is that when a delay happens and the flow of new slabs into the furnace and the old ones out of the furnace is stopped, soon those slabs in the furnace get heated up and after a half hour or to an hour, they probably become pretty well heated. At that time it is no longer necessary to hold a given setpoint control temperature such as 2500°-2600° F. any more, and the furnace can be cut all the way back down to a setpoint temperature of about 1800° F. or so which is generally what the present computer control system will do. Thusly, for any delay from an hour up to eight hours or longer, the furnaces could be cut back down to a temperature range of about 1800° F. and held there for some length of time related to the delay and then slowly brought back up to the desired operating temperature at a controlled time prior to the furnace it is desired to take the next slab out of the furnace. The furnaces are usually not turned off completely, because it is desired to be in condition to come back up in temperature reasonably quick, so a temperature range such as 1800° F. can be selected as the temperature that the furnace would drop down to in 50° F. steps, such as 50° F. every three or four minutes. This can be just another consideration included in the total slab heating furnace control philosophy of the present heat control programs. 
     The expected delay program shown in FIG. 28 is initiated by the operator provided input of an expected delay interrupt. When the computer control system receives the expected delay interrupt at block 300, the computer reads three pieces of input information: the furnace number, the length of the expected delay, and the number of slabs to discharge from the furnace until the start of the delay. At the next block 302, the computer memory stores the furnace number, the length of the delay, and the count number of slabs for the slab extraction program. In the next block 304 a flag is set for the furnace X delay-pending flag, and the program stops. 
     In reference to the program flowchart shown in FIG. 29, at block 248 the control program updates the slap map in the computer memory concerning the heat content and surface temperature data about each slab in furnace X. At block 250 a determination is made to see if furnace X is not on an expected delay. If the answer is yes, and it is not on such a delay, then the program proceeds through the normal calculations concerned with updating all of the slab map information the computer memory in relation to the slabs in the furnace X. If the answer at block 250 is no, and the furnace is not on an expected delay, the program proceeds to block 252 to check if new setpoints need to be output from prestored data. In this way, the temperature setpoints are raised or lowered as may be desired for an expected delay condition of furnace operation. 
     On a typical slab heating furnace there are conventional analog setpoint temperature controllers, including a manual dial that the furnace operator can set to 2400° F. and this automatically operates the controller to control a given heating zone of the furnace at a temperature of 2400° F. Thermocouples are located in the furnace so they are not directly in the flame but give a good idea what the effective furnace temperature is around the slab, and these thermocouples provide temperature feedback signals to the individual controllers. For example, the operator can manually establish the temperature control setpoint for a desired 2400° F., or the computer control system can set this setpoint for 2400°. A switch is provided for this purpose on each controller that can be used to provide for computer control or manual control. 
     Referring back to block 252 of FIG. 29, this check is made to see if new temperature control setpoints need to be output, based on a prestored data. If the furnace operation is on expected delay, this path is taken to see whether or not the respective control heating zone temperatures need to be updated based on prestored data. One consideration here is setpoint change tolerance in that the program which does the actual output of the desired temperature setpoints to the furnace heating zone controllers will not put a 300° F. step change, and instead it provides a maximum change of 50° F. per minute. So the control program cycles itself every minute, and if it is desired to make a 300° F. change, the program will actually output six 50° F. steps at one minute intervals so actually it is going to take six minutes to make that change. It is possible to come through this program and want to make another change within that six minute time period, so this block 252 is providing a check to see if new temperature setpoints are needed to meet the desired temperature within a certain tolerance such as a 25° F. tolerance or maybe a 5° F. tolerance as may be desired. This tolerance comes about from another factor in that the output to the setpoint controllers cannot tell the controller to make a 300° change and the furnace zone temperature controller can only run its motor for a given length of time, such as in intervals of a 20th of a second, so the motor is turned on and it stops a 20th of a second later or 2/20th of a second later, and so forth, or a minute later but with no greater resolution than a 20th of a second. This 20th of a second then becomes the tolerance that is available for setting the temperature controllers. This is not critical to the philosophy of the present furnace heat control operation, but this is an example of one of the practical real time control constraints that must be considered. 
     The more normal operation of the program shown in FIG. 29 is where the furnace is not on an expected delay, so the program then proceeds to the first subroutine which is indicated in block 254 as the heat content calculation and surface temperature calculation for furnace X. The previously described Schmidt method is used as the way the heat content calculation is made through an arithmetic averaging technique. This is done for each slab in the furnace, and there may be anywhere from 30 to 90 slabs in a furnace, to update the furnace map with this information so that each slab has now an updated temperature profile and updated surface temperature and an updated heat content. The program advances to block 256, which is the subroutine for calculating the control setpoint temperature in the change zone based on the heat content information just updated in the previous subroutine block 254 and based on the known physical properties of each slab. This calculation is in relation to the illustrations of FIGS. 2 and 3. It is desirable to run the respective heating zones of a slab heat furnace at a generally constant temperature control setpoint since the various operation conditions are more predictable and more controllable, and this gives a longer life out of the furnace bricks and so forth. This charge zone setpoint calculation subroutine of block 256 is described in detail in relation to the flowchart shown in FIG. 30. 
     At block 258 of FIG. 29, the program goes to the heat zone temperature setpoint calculation subroutine, described in detail in relation to the flowchart shown in FIG. 31. 
     The program shown in FIG. 29 has now determined the temperature setpoints for the charge zone and for the heat zone of furnace X. At block 260 the predetermined soak zone temperature, which is provided by the furnace operator as an input reference soak zone temperature control setpoint, is read by the computer control system. At block 262 a check is made to see if the last temperature setpoint changes were actually met by the respective furnace heating zones within the provided tolerance. If the answer is yes, then at block 264 the now established and now desired new temperature setpoints are output to the respective zone temperature controllers. It is possible that the computer control system will get busy and not be able to turn off a given output contact in a proper time interval, such that the temperature setpoint form some controller is advanced too much; at block 262, a feedback check is provided to alarm the furnace operator if this condition should ever happen. 
     If it becomes desirable to shut down the furnace for some time period, such as for 8 hours, this would be an expected delay and the program shown in FIG. 29 would not run through all the heat content, charge zone setpoint and heat zone setpoint calculations on the slabs because the furnace is not being controlled. When the expected delay terminates, the furnace is full of slabs that have not been properly heated. Thusly, a flag is set on each of the data buffers associated with those slabs to indicate that those slabs were affected by an expected delay, and these flags are checked at block 250 and if set prevents the program going through the here described calculations. It is desirable to clean out the furnace and push in new slabs after an extended expected delay, before the program operation settles down and builds up all the slab map information and goes through all the routines provided for optimum control of the furnace. 
     In reference to the charge zone temperature setpoint calculation subroutine shown at block 256 of the program shown in FIG. 29, a detailed showing of this same subroutine is provided in FIG. 30. When this program is called, at block 259 the program calculates the average thickness and the average width of the slabs in the slab map of furnace X. At block 261 the mill cycle time is calculated based on the average width and average thickness of the slabs, since a thicker slab takes more work than a thin slab and the thicker slab becomes longer when it is rolled, which mill cycle time calculation is based on the well known information generally illustrated in FIG. 5. At block 263 a check is made on the operator input of the furnace X gap time, which is an overriding time between slabs that the operator can input for many reasons, for example if he may want to have two minutes between slabs instead of one minute between slabs going to the rolling mill. At block 265 the furnace cycle time is calculated based on the furnace control model, the mill cycle time and the gap time and the number of furnaces in operation. At block 266 there are determined the slab travel time t H  (predicted) through the heat zone and the remaining slab travel time t CH  (rem) through the charge zone. At block 268 and with these predicted time, the curves of FIGS. 2 and 3 for a 7 inches average control slab thickness are used to determine in relation to the total heat content H T  the heat H H  added in the heat zone and H CH  the heat added in the charge zone. It should be here noted that t H  (predicted) is the travel time for theoretical control slab in the charge zone to travel through the heat zone, whereas t CH  (rem) is the time remaining for the theoretical control slab in the charge zone to travel through the remainder of the charge zone; the control slab in the charge zone is already partly through the charge zone and it has to go all the way through the heat zone. The total heat content H T  is used to divide up the total heat effort between these two heating zones. 
     A schematic showing of a typical slab heating furnace is shown in FIG. 6; the charge zone is 43 feet long, and in that 43 feet there is a certain heat content B.T.U. that is going to be put in the slab; the heat zone is 34 feet long, and in the heat zone there is a certain amount of heat content B.T.U. that is going into the slab. For this purpose a theoretical or control slab area is initially established as an eight foot wide section of slabs, located 211/2 feet from the entry of the charge zone or about half-way through the charge zone. 
     At block 270, shown in FIG. 30, the slab map is checked for the actual thicknesses of all slabs and slab portions within this control slab area, such that an average thickness is then established in relation to the actual widths of those slabs. At block 272 the time remaining t CH  (rem) for this control slab or representative slab in the charge zone is determined in relation to the known demands of the rolling mill and the operation of the furnaces. At block 274 the remaining heat content H CH  (needed) required in the charge zone is determined as H T  minus the sum of H H  and H CH  (actual), where H CH  (actual) is the heat content already in the control slab. Another way to say this same relationship is H CH  (needed) is equal to H CH  minus H.sub.(actual), and H CH  is H T  minus H H . The amount of time this control slab will remain in the charge zone and the amount of heat needed to be put into it is what determines, in relation to the curve information of FIG. 3, what the temperature control setpoint should be for the charge zone, and this is calculated at block 276 of the flowchart. It should be understood that the present control system and method can be fine tuned in its operation relative to a given slab heating furnace by moving the theoretical control slab location within the charge zone as well as the theoretical average dimensions of this control slab as may be desired to better represent the actual heating conditions present within the charge zone of the furnace. 
     In reference to the flowchart program in FIG. 31, which is called as the subroutine to calculate the heat zone temperature setpoint, at block 280 a check is made of the slab map for the theoretical control slab in the heat zone to determine the average thickness of same. At block 282 the time remaining t H  (remaining) for this control slab to travel to the end of the heat zone is calculated. At block 284 there is calculated the heat content H H  (needed) to obtain the desired total heat H T  in this control slab at the end of the heat zone. At block 286, using FIG. 4 and based on the H H  (needed) and the T H  (remaining), the heat zone temperature control setpoint is determined. 
     Pages 1 through 4 contain macro definitions used within the program. Pages 5 through 17 contain constants and data location used within the program. Page 18 contains a small subroutine called NEWFRM used within the program. Pages 19 through 22 contain additional constants and temporary locations used within the program. Page 23 begins up-data furnace map program. 
     For the program shown in FIGS. 27 and 28, the furnace operator has director input control over the operation of the slab heating furnace, whereas the program shown in FIG. 29 is operative off of a time clock within the control computer, and is periodically bid by the control computer. This period can be in the order of one minute to every three minutes, and for the example of three furnaces in operation, every three minutes can be selected in regard to the heat flow calculation for each furnace, with the program making periodic updates of certain information such as heat content and surface temperature in relation to each slab contained within a particular heating furnace. The more frequent this updating is done by the control computer, this should result in a better control of the furnace operation; however, the duty cycle of the control computer must be considered, and a compromise between the desire to frequently update this information on one hand as compared to the desire of minimizing the duty cycle requirement of the control computer on the other hand must be established. The desired resolution of the final control system operation is determinative of how frequently this updating of furnace slab heat treatment information is made. The operator input portion of the here described furnace heat control philosphy is probably the less important when compared to this periodic updating of the slab heat content and furnace heating zone control information. 
     In reference to the operator input flowcharts shown in FIGS. 27 and 28, probably the most important opertor input is the push-to-extract interrupt for furnace X. This relates to a furnace operator decision to take the next slab out of the furnace. If desired, the push-to-extract interrupt could be automatically provided by the computer control system upon sensing that a previous slab has entered the first stand pass of the associated rolling mill, for example, and it is now time for another slab to be supplied to the entry tables leading to the rolling mill. The operation of the here described control system is such that the control computer tells the operator to provide the push-to-extract interrupt signal for the next slab ready and available to be supplied to the rolling mill, and the control computer does not care from which furnace it is supplied. The slab map in the computer memory for each of the furnaces enable the computer to determine which slab, in the combination of the heating furnaces, is ready and should be the next selected slab to be supplied to the rolling mill. This decision is based upon the rolling mill capacity program and furnace control considerations. Not only does the rolling mill have to be ready to receive the next slab, but the next slab has to be heated properly, such that, in effect, a determination is made when both the actual extraction of the next slab is desired and the supply of that slab workpiece to the input of the rolling mill is to be provided. The provision of the push-to-extract interrupt program is provided such that the operator can make this decision when another slab is needed to be extracted from one of the available furnaces and supplied to the rolling mill. After a push-to-extract interrupt by the operator for a particular furnace, such as furnace X, is provided, the computer goes through the storage information for that particular furnace, and updates the slab map information relative to all of the remaining slabs residing in that furnace. This updating includes the information that one slab has been removed from furnace X and another slab is now entering furnace X, and updating the temperature and heat content status information for each slab in the furnace, and so forth. Known relationships are established between the temperature of each slab and the desired period of time required for that slab to be positioned in the soak zone at that temperature level, since this is a more or less empirical time factor relationship that is provided for the desired operation of the slab furnace and the subsequent rolling mill. A typical value here for this soak time may be in the order of 10 minutes for the usual desired temperature level of the slab. A rolling mill will typically take a slab about once every minute; and for this reason, a three minute periodic update for the combined operation of three slab furnaces would provide an information update for one of the furnaces about every time that a workpiece is removed from each of the furnaces. Thusly, a slab is put into and removed from each furnace about every three minutes; and the information update for that same furnace at the same time period relationship should provide a desired operation of the furnace control program. 
     GENERAL CONSIDERATIONS 
     The term effective furnace temperature as here used is in relation to the radiation and reradiation of heat energy into a slab within a given heating zone from all effective heat sources within the furnace such as the furnace walls, the flame, the hot gases surrounding the slab and so forth. The bulk of this slab heating is probably by direct radiation from the heated furnace wall, and this is why the effective furnace temperature TF is sensed by a thermocouple or total radiation device within each heating zone, which device is positioned adjacent the furnace wall in a location outside the direct flame from the burner. This effective furnace temperature TF is used in relation to the slab heating model such as shown in FIGS. 8 and 9, and empirically it should be recognized tha other heat sources such as the surrounding hot gases and the flame itself are also included. The slabs come out of the furnace soak zone in relation to how fast the rolling mill can accept those slabs, so that if the rolling mill is accepting a heated slab every minute, and only one furnace were in operation, then a slab would leave that furnace every minute and thus all the slabs would move by an amount corresponding to the width of the removed slab. Thickness is the primary consideration in relation to heat content control of the slabs, but it is also necessary to know how long it will take a given ideal or representative control slab of a determined average thickness to move through each of the charge zone and heat zone, and that becomes then a function of the width of the slabs leaving the soak zone. But with two furnaces in operation, the speed at which the slabs come out of each furnace is not just a function of how fast the rolling mill accepts slabs from each furnace, but generally the rate of slab movement would be cut in half out of each furnace because now a first slab will be supplied by one furnace, then the second slab will be supplied by the other furnace and so forth. With three furnaces in operation, the slab movement through each furnace would generally be cut into a third. 
     The mill capacity model for the rolling mill includes a rate of rolling limit which has a width factor, because wide slabs require more horsepower to roll than do narrow slabs. An additional consideration is a mill limit in relation to the rolling of 80 inch slabs as opposed to 20 inch slabs since the former would tend to heat up the drive motors more. The purpose here is to try to find out how fast the rolling mill is going to accept successive slabs, and this then influences, along with the number of furnaces in service, how long the time interval is that it will take a slab to move from one part of a given heating zone to another part of the furnace, and this involves predicting what the rolling mill is going to require. 
     In general, it is desired to get each slab up to a certain predetermined heat content by the time it gets to the entrance to the soak zone of the furnace, because no substantial heat is desired to be added to the slab in the soak zone. The soak zone allows the temperature profile within the slab to even out from being hotter at the surface and cooler at the center to being uniformly heated throughout. Thusly, at the time each slab enters a furnace, it is assigned a desired heat content upon leaving the end of the heat zone and entering the beginning of the soak zone. The slab model establishes the actual heat content of the slab as it passes through the furnace. The heat control program is operative to distribute the heating of the slabs in the most efficient manner, and will correct any deviation from this. The heat control program includes a limiting point in the control of the heat zone where the control operation has the option to delay the slab movement of the furnace if the slabs do not have as much heat as is required, and the furnace operator has the ability to override this delay operation if he so desires. When a given slab does not have 100% of the desired heat content when that slab is scheduled to leave the heat zone, the computer will alarm this situation to the operator. At the noze or end of the heat zone, the computer makes a final check to see whether or not the slab has the desired predicted value of heat content. If it does not and it is less than 100%, the heat control program will stop the slab movement and an internal flag is set to indicate this has been done and a message is printed on a typewriter for the benefit of the furnace operator; also, there is a print-out of the discrepancy to show how much heat content the slab is actually lacking. This gives the furnace operator the opportunity to see if the margins are not too great such that they should roll the slab anyway and override the computer delay and take that slab out. There is a similar time delay provided in the operation of the soak zone which is based on the consideration that the soaking function is strictly a matter of time and a determination is made as to when the slab is evenly heated in relation to how long the slab has been in the soak zone; if the furnace is being pushed too fast for some reason and with slabs of varying thicknesses being positioned throughout the furnace, these conditions are taken care of by the final override control operation in relation to time in the soak zone. For example, if the slab is set for 10 minutes in the soak zone and it has actually been there only 5 minutes when a request is made to supply a slab to the rolling mill, the heat control program delays the slab for five more minutes in the soak zone before permitting the slab discharge. The operator can override this consideration, if he ascertains that it is better off to go ahead and roll a particular slab that is not completely soaked in this regard. The operator does this overriding by simply pushing a delay override interrupt control button, and the computer then resumes normal operation as though the desired temperature, heat content and soak zone time limitations have been satisfactorily met. 
     The slab heating model is provided in part because it was not reliable to measure the temperature of the slab directly, with a device like a radiation pyrometer looking at the slab because of scale on the slab and like considerations. The slab model is based on well known slab heating technology in relation to the heating of slabs that has been going on for the last 20 years or more. The slab model is operative in relation to a predetermined control area of the slab rather than an entire slab, and the length of the slab does not come into consideration because the model makes the assumption that the slab is heated uniformly across the whole length of the slab. The heat content of the slab in relation to a theoretical control area in a given heating zone is considered, with a control volume of the slab being established by the known average thickness of the control slab area, such as a square foot of surface area. In effect the heat control program is looking down on top of the slabs in the charge zone, for example, to see a square foot of surface area and the heat control volume is established by the effective average thickness of the slabs in this theoretical control area. The known properties of the steel in this control area is considered in the slab model also, and such things as variation of thermal conductivity and specific heat as a function of temperature of the slab are considered. 
     An additional control parameter is brought into the heat control program after a slab leaves the furnace and enters the rolling mill, such as after the first pass through a roll stand, to provide an indication of the quality of the heat control operation; this would then be a feedback control signal into the furnace model to give an indication of the actual delivery slab temperature. This actual temperature sensing is done with a radiation pyrometer in the rolling mill where it is practical to get a fairly good reading of what the slab temperature is, since much of the scale has been removed from the slab surface at that location. 
     The slab heat control system can include hot metal detector for determining the movement of each slab and tracking it out of the furnace. The slab tracking operation can work in conjunction with a succession of these hot metal detectors as shown in FIG. 25 to track the slab movement out of the furnace and into the associated rolling mill. Several slabs can be actually tracked by keeping a map in the computer memory which is representative of the physical locations that each slab would have between the furnace and the rolling mill, such that a pictorial image is kept in the computer memory of the location of each slab identified by slab ID numbers and other information to tell which furnace the particular slab came out of. 
     After the first stand pass in the rolling mill, the temperature of the slab is measured because at that point a fairly clean slab surface is available, and this measured temperature is correlated with that slab temperature supposedly delivered from the furnace. If the slab did not roll as it should have rolled, this gives an indication whether or not the furnace is running cold or hot. If the furnace is running cold, the heat control program can raise the furnace temperature, and if the furnace is running hot, the heat control program can lower the furnace temperature. In this way, a feedback control is provided on top of all the heat control considerations previously discussed, such that in effect a control multiplier is provided that will be 1.000 initially. By adjusting this multiplier such as to one of 0.999 to 1.001, or even providing a greater adjustment than this as may be desired, this multiplier is going to raise or lower the furnace control temperature setpoints accordingly. 
     FIG. 6 is provided to illustrate the control slab in the theoretical control area in the furnace heating zone in relation to the predetermined physical locations in the furnace. This theoretical control slab is 8 feet wide, and this 8 feet is selected as one of the control parameters for tuning the operation of the here described control system and method. Initially, it was chosen that the theoretical control slab is 8 feet wide and the center of that slab is locationed 291/2 feet inside the charge zone. The corresponding theoretical control slab in the heat zone is 8 feet wide and location 11.3 feet inside the heat zone. These figures were chosen to give a good representation of an average grouping of slabs, because this theoretical control slab is actually composed of several slabs since no one actual slab can be 8 feet wide; this control unit slab is probably composed of parts of 2 or 3 actual slabs. The theoretical control slab has a known cross-sectional surface area and a determined average thickness, which average thickness is used in relation to the illustrated curves, for example the 7 inch thickness in the case of FIG. 2. Therefore, FIG. 6 shows the placement of the theoretical control slab in each of the charge zone and the heat zone of the furnace. The specific distances shown in FIG. 6 are not critical, and it should be understood these distances can be changed to in effect fine tune the desired control system operation. The control area dimensions and the location chosen for the theoretical slab are merely for the purpose of example, and can be varied as readily apparent to persons skilled in this art for improving the operation of the present control system and method. 
     It should be understood, that in the event slab surface temperature measuring devices are subsequently improved to become more reliable and adequate in operation, then the present control system and method is readily operative with such an actual temperature measurement for use in conjunction with the here calculated slab surface temperature. For example, in the calculation of the heat transfer coefficient KT as shown in FIG. 9, if the slab surface temperature TS can be actually measured instead of calculated by the Schmidt Method in accordance with the showing of FIG. 8, this could enable a more immediate check on the accuracy of the calculated heat content of the slab as determined by the Schmidt method. Without the benefit of such an actual slab surface temperature measurement device and operation, it becomes necessary to wait for this check until a given slab leaves the furnace and passes through the first pass rolling operation such that the actual temperature of the slab can be measured by the radiation pyrometer at that location. If the surface temperature TS could be accurately measured while the slab is still within the furnace, then the difference value (TF-TS) shown in relation to FIG. 9 could be better established and this in turn would permit a better calculation of the slab heat content in relation to the Schmidt Method as shown in FIG. 8. This would in effect improve the slab heating model operation as herein described. 
     The surface temperature of a slab, even though it could be measured accurately inside a heating zone of the furnace, is only a partial indication of the suitability of a heated slab for hot rolling. The distribution of heat within a slab and the related temperature profile between the surface and the center of a slab are different for each thickness of slab. Therefore, the total heat content corresponding to a given surface temperature at a given time is a more accurate and comprehensive measure of the actual heated status of slabs of various thicknesses and their suitability for rolling. 
     The heat content of a slab at any instant in time is defined as the summation of the heat energy corresponding to the temperature profile from the outside surface of the slabs to the inside center of the slab, as shown in FIG. 8, and assuming the slab is heated from both sides such as happens in the charge zone and the heat zone of the slab heating furnace illustrated in FIG. 1. The slab heating model predicts the complete temperature profile and resultant heat content as the slab is heated and is based on the actual elapsed time and the actual effective furnace temperature by which the slab is heated. 
     The following Table II is provided to show generally the relationship between the various curve plots of FIGS. 2 through 4 and 10 through 24 in relation to the representative or average thickness slab in the control area of the respective charge zone and heat zone of a furnace. 
     
                       TABLE II                                                    
______________________________________                                    
 Slab                                                                     
thickness                                                                 
         7&#34;      6&#34;      8&#34;    9&#34;    10&#34;   11&#34;                            
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distribution                                                              
         Fig. 2  Fig. 10 Fig. 13                                          
                               Fig. 16                                    
                                     Fig. 18                              
                                           Fig. 22                        
of heating                                                                
between                                                                   
heat zone                                                                 
and charge                                                                
zone                                                                      
temperature                                                               
         Fig. 3  Fig. 11 Fig. 14                                          
                               Fig. 17                                    
                                     Fig. 20                              
                                           Fig. 23                        
setpoiunt                                                                 
deter-                                                                    
mination                                                                  
for charge                                                                
zone                                                                      
temperature                                                               
         Fig. 4  Fig. 12 Fig. 15                                          
                               Fig. 18                                    
                                     Fig. 21                              
                                           Fig. 24                        
setpoint                                                                  
deter-                                                                    
mination                                                                  
for heat                                                                  
zone                                                                      
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     GENERAL DESCRIPTION OF INSTRUCTION PROGRAM LISTING AND RELATED DATA TABLES 
     In Appendix A there is included an instruction program listing that has been prepared to control the operation of three slab heating furnaces in accordance with the here disclosed control system and method. The instruction program listing is written in the machine language of the PRODAC P 500 digital computer system, which was first sold by Westinghouse Electric Corproation in about 1963 for real time process control computer applications. Many of these digital computer systems have already been supplied to customers, including customer instruction books and descriptive documentation to explain to persons skilled in this art the operation of the hardware logic and the executive software of this digital computer system. This instruction program listing is included to provide an illustration of one suitable embodiment of the present control system and method that has actually been prepared. This instruction program listing at the present time is a more or less development program and has not been extensively debugged through the course of practical operation for the real time control of a combination of several slab heating furnaces. It is well known by persons skilled in this art that most real time process control application programs contain some bugs or minor errors, and it usually takes varying periods of actual operation time to identify and correct and more critical of these bugs. 
     In relation to the instruction program listing set forth for illustrative purposes in Appendix A, the following general comments may assist in the understanding of same. 
     Pages 1 through 4 contain macro definitions used within the program. Pages 5 through 17 contain constants and data locations used within the program. Page 18 contains a small subroutine called NEWFRM used within the program. Pages 19 through 22 contain additional constants and temporary locations used within the program. Pages 23 to 28 contain the update for the furnace slab map. Page 29 begins the check for a furnace in service and the expected delay program which ends on page 30. Pages 30 and 31 are the expected delay temperature setpoint output calculations. Pages 31 through 36 contain some bookkeeping type of programming associated with locating information on drum tracks. Pages 37 to 44 contain some index initialization data for internal bookkeeping and to get the setup to perform the Schmidt calculation on the heat content data. Pages 45 and 46 contain a subroutine which is used in the calculations themselves to determine the mill time calculation. Pages 47 to 49 contain the subroutine to calculate the ideal zone or the theoretical control in the heat zone. Pages 51 through 54 contain the determination of the position of the slabs for the calculations used in the charge zone and heat zones. Pages 55 through 64 contain the heat calculation on each slab for all the slabs in the furnace, going through the furnace and updating each slab then going back to do the setpoint calculation for the charge zone and for the heat zone. Page 67 contains interface functions for data logging. Page 68 starts the charge zone setpoint calculation. Page 78 starts the heat zone setpoint calculation. Page 86 starts the setpoint check program against some of the actual feedbacks to determine whether or not the temperatures were met within the proper tolerance level; this is checking the actual setpoint controller feedback to make sure that the controller met the information sent to it last time, as a feedback on the controller itself and not on the furnace. Page 86 also contains an overriding safety part of the program which checks the actual thermocouple to make sure they are not too high or too low. 
     On page 86 through page 94 is a limit checking program which checks all the actual furnace temperatures against present alarm conditions and if for some reason the controller has run away when the temperature gets too hot this program will provide a check based on the thermocouple feedback and will alarm that to the operator, as a safety program. On page 95 through page 101 will be where the last setpoint temperature is met within the tolerance level. The output required setpoint changes for all zones would be on page 102 and page 103. Pages 105 through 107 are the expected delay program. Page 108 is the delay override program. Pages 109 through 117 are the push to extract interrupt program. 
     A person skilled in the art of writing computer instruction program listings, particularly for an invention such as the present control system and method for a slab heating furnace, must generally go through the following determinative steps: 
     Step One--Study the slab heating furnace and its operation to be controlled, and then establish the desired control system and method concepts. 
     Step Two--Develop the control system logic analysis, regarding both hardware and software. 
     Step Three--Prepare the system flowcharts and the more detailed programmer&#39;s flowcharts. 
     Step Four--Prepare the actual computer instruction program listings from the programmer&#39;s flowcharts. 
     This instruction program listing included in Appendix A was prepared in relation to the programmer&#39;s flowcharts shown in FIGS. 32 to 39. 
     The following Table III is provided to more clearly set forth the relationship between the instruction program listing, included in Appendix A, and the system logic flowchart programs shown in FIGS. 27, 28 and 29. 
     
                       TABLE III                                                   
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Pages of   Figure of                                                      
INstruction                                                               
           System                                                         
Program    Flowchart      Block                                           
listing    Program        Number                                          
______________________________________                                    
108        27             Delay Override                                  
109 to 117 27             Push to Extract                                 
105 to 107 28             Expected Delay                                  
23 to 28   29             248                                             
29 to 30                                                                  
30 to 31   29             252                                             
and 34                                                                    
55 to 64   29             254                                             
68 to 77   29             256                                             
           (also Fig. 30)                                                 
78 to 84   29             256                                             
           (also Fig. 31)                                                 
99 and 100 29             260                                             
95 to 101  29             262                                             
102 and 103                                                               
           29             264                                             
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     In Appendix B there are included data tables provided for use by the instruction program listing for all desired table lookup calculations. The curves shown in the various figures have been plotted into a data table form to be suitable for storage in the computer memory and operative with the computer for the necessary calculations required by the present control system and method. 
     The following general comments may assist in the understanding of these data tables. Pages 2 and 3 contain the values of the total heat content H T  needed by the various slabs to be heated, with one-half values being provided since the computer has an eighteen bit word length and the half values were more suitable for this reason. Page 3 contains operation modifiers K0, K1 and K2, as explained by the programmer&#39;s comments included in the instruction program listing. Pages 3 and 4 contain the start of the lookup table for the required soak zone times. Pages 56 to 58 contain the heat transfer coefficient kT data in accordance with FIG. 9. Pages 59 to 63 provide spare data space. Pages 64 and 65 contain interpolation data for use to later interpolate in finding data from the data tables for respective FIGS. 2, 3 and 4. Pages 66 and 67 provide spare data space. Pages 68 to 75 contain the data table corresponding to FIG. 2 and similar figures for other slab thicknesses. Pages 76 to 99 contain the data table corresponding to FIG. 3 and similar figures for other slab thicknesses. Pages 100 to 123 contain the data table corresponding to FIG. 4 and similar figures for other slab thicknesses. ##SPC1## ##SPC2## ##SPC3## ##SPC4## ##SPC5## ##SPC6## ##SPC7## ##SPC8## ##SPC9## ##SPC10## ##SPC11##