in fig1 there is shown a curve for the heat loss vs . % excess air . one of the curves in fig1 shows the characteristic for the heat losses ( qc ) which are the losses due to combustibles in the flue gases . these are the losses which result from incomplete combustion , primarily from imperfect mixing . the imperfect mixing occurs in a practical furnace around the theoretical zero excess air value . also included in the category of combustibles is the particulate going up the flue ( soot ) and hydrogen . as shown in fig1 when the % excess ar increases away from zero , the heat loss due to combustibles will initially decrease rapidly with the rate of change diminishing as the value for excess air goes through the operating region . the curve then levels off to a minimum for the heat loss . fig1 also shows that stack heat losses ( qa ) due to excess air increases linearly with % excess air . these losses are due to the heat required to bring the excess air to the temperature of the exhaust gases . in this connection it should be kept in mind that approximately 80 % of air is nitrogen and it also must be brought to the exhaust temperature along with the unused oxygen in the exhaust gases . a combination of the two separate heat loss curves gives the total loss curve shown in fig1 . the minimum total loss as shown by the curve occurs where an incremental change in the losses due to excess air is equal and opposite to the incremental losses due to combustibles . thus , the condition for minimum losses to the stack can be expressed as follows : the measurements required to determine those terms can be obtained by making the measurements before and after a change in either the rate of fuel feed or the rate of air flow or both as long as there is a change in the resulting air / fuel ratio . these changes can be small perturbations instituted solely for the purpose of making the measurement , or the measurements may be made when changes naturally occur . frequently there are enough random fluctuations in the air / fuel ratio to provide the perturbations required . the control of the air flow for maximum efficiency must necessarily include other factors , such as : thus , it may be necessary to control the air flow or the air / fuel ratio at some value which is not the theoretical value set forth in equation ( 1 ) solely for the purpose of maintaining mandated pollution standards governing emissions . other factors such as safety may dictate the modification of equation ( 1 ) to provide for an offset or bias of the theoretically optimum air flow solely to gaurantee that sufficient oxygen will be available to avoid hazardous operation . to accomodate such an offset the relationship of equation ( 1 ) can be modified as follows : so that the optimizing control of this invention will adjust the air / fuel ratio to bring the losses due to combustibles only to substantial equality with the losses due to excess air instead of the equality being absolute . thus , it can be seen that the theoretical operating point for excess air , ea , obtained by using equation ( 1 ) may be eao in fig1 and the practical operating point may be eap as would be obtained by using equation ( 2 ). operating to obtain optimization in accordance with the invention requires the calculation of an optimum air / fuel ratio or an optimum oxygen or carbon monoxide set point depending on the configuration used to control air / fuel ratio . it is desirable to interpose the optimizing calculation into the air flow side of the control rather than the fuel flow side , for the fuel must generally be modified solely to control the heat requirements thus leaving the air flow as the variable for controlling the condition of the stack gases . fig2 shows one control arrangement for utilizing the optimum air / fuel ratio determined in accordance with the invention to modify the air and fuel feed rates to effect the desired control . in that figure the fuel rate demand signal frd is introduced on line 10 to the burner controls . that signal may be set by the operator or derived by any of a number of systems such as that shown in u . s . pat . no . 3 , 247 , 671 , issued to j . h . daniels on apr . 26 , 1966 , where the signal is shown on line 178 of said patent . that demand signal directly determines the fuel feed control by providing the setpoint for the fuel feed controller 12 which receives as its other input the signal on line 14 from flow transmitter 16 indicative of the measured flow of fuel . the controller 12 may be any of a number of standard controllers which can operate to vary the opening of control valve 18 as needed to cause the setpoint to be matched by the measured value of fuel flow . the air flow needed to obtain the required air / fuel ratio in accordance with this invention can be obtained by using the optimum air / fuel ratio signal on line 20 to determine the relationship between the fuel feed rate and the air flow rate . thus , as one example , the air flow in fig2 is controlled from the fuel rate demand signal on line 10 by feeding forward that signal as the setpoint , sp , for air flow controller 22 . the set point is modified by the function generator 24 whose output is then multiplied by the signal from line 20 to give the controller set point , sp &# 39 ;, on line 26 . the function generator is desired because the air / fuel ratio should be increased as load on the furnace is decreased , for there will be a decrease in fuel - air mixing . as shown , the controller 22 modifies the air flow rate as needed to cause the measured air flow determined by flow transmitter 28 to equal the set point sp &# 39 ;. as shown in fig2 the air flow is varied by adjustment of the air flow damper 29 . in some installations the air flow may be modified by adjustment of forced and induced draft fans or other means . also , in some installations the recalibration of the air flow system in accordance with the signal on line 20 may be accomplished on the measurement side by introducing the function generator and multiplier to the measurement side of the control instead of the set point side shown in fig2 . a variation of the air flow control system of fig2 is shown in fig3 where the output of the optimizing calculation provided on line 30 is representative of either the oxygen set point or the co set point and is used to obtain the air / fuel ratio signal on line 20 which can then be used as shown in fig2 . the system of fig3 is useful to adjust the air / fuel ratio in a manner to account for varying fuel quality , heat of combustion , and errors in the measuring system . in this connection the signal on line 30 provides the set point for the oxygen controller 32 , which in the alternative can be a co controller . assuming oxygen control is desired the oxygen measurement provided on line 34 is compared with the set point and the controller modifies the air / fuel ratio as represented by the signal on line 20 until the measured oxygen equals the set point . the signal on line 30 is multiplied by a signal on line 38 which is derived from a steam flow measurement on line 40 , shown as an input to the function generator 42 . the function generator serves to provide a change in the air / fuel ratio with load as represented by the steam flow , sf . fig4 a and 4b show an example of the logical steps which can be used in a digital computer to produce the signal required on line 20 of fig2 and , with modifications which will be described , the signal required for line 30 of fig3 . in fig4 a the block 50 serves to bypass the optimizing program until a meaningful change has taken place in the excess air , i . e ., the magnitude of δ % ea is greater than a value ε . the change in ea is determined by subtracting the value stored at the last update % ea ( τ ) from the present value % ea ( t ), i . e ., the measurement of oxygen is indicative of ea in the area of interest and may be substituted directly in the above equation . the relationship between ea and oxygen is expressed by the following : as has been mentioned , the random fluctuations in fuel and / or air may result in a meaningful change in excess air , however , if that is not the case perturbations can be injected as will be described in an example in connection with fig4 b where the perturbation signal fluctuates between the values of zero and δ once during every time period t . when a meaningful change in ea has been detected the present heat losses due to excess air and combustibles in the flue gases , qa ( t ) and qc ( t ) are calculated . the changes in heat losses , δqa and δqc are determined by comparing the present values to those stored at the last update , qa ( τ ) and qc ( τ ). this is accomplished in block 52 . the optimizing program is also bypassed if the expected sign changes are not observed i . e ., if as can be seen from fig1 an increase in the excess air will result in an increase in δqa and should result in a decrease in δqc . this sign check is made in block 54 of fig4 a . when the optimizing path is taken in fig4 a , a check is made in block 56 to determine if the magnitude of the change in excess air loss is greater than the weighted change in losses due to combustibles , i . e ., if the answer is ` no ` the stored value of the air / fuel ratio , afr &# 39 ;, is incremented by an amount δafr in block 68 and will result in increasing the air / fuel ratio and , therefore , excess air , ea . for practical reasons maximum , afr ( max ), and minimum , afr ( min ), limits are applied to the value of afr &# 39 ; as shown in blocks 62 , 64 , 70 , and 72 . if the answer is ` yes ` the stored value afr &# 39 ; is decremented in block 60 by δafr providing there are no constraint conditions discovered by the tests of block 58 . this will result in decreasing afr and , therefore , ea . if there are constraint conditions , e . g ., based on percent oxygen less than a minimum , co greater than a maximum , flue gas outlet temperature ts less than a minimum , or opacity due to particulate greater than a maximum , as tested for in block 58 , the value of afr &# 39 ; will be increased by δafr , as shown in block 68 , and the value of afr &# 39 ; will be checked for maximums and minimums , as mentioned before . the optimizing calculations having been made and the desired value of the air / fuel ratio determined , the values of % ea , qa , and qc must be updated as shown in block 66 . after updating , the question of the need for perturbation of the air / fuel ratio is considered in block 74 of fig4 b . if the random fluctuations of the air / fuel ratio are sufficient as determined by the operator for the required perturbation then δafr is set to zero , as shown in block 76 . otherwise , perturbations must be provided as mentioned before . this process is started by incrementing the counter as shown in block 78 . the contents of the counter are then compared in block 80 to t , the period between perturbations . if it is time for perturbations and δafr is greater than zero , as determined by the test in block 82 , then δafr is set to zero ( in block 84 ). otherwise δafr , the amount of change required in the air / fuel ratio to provide the needed purturbation , is set to δ . after δafr is determined then the counter is set to zero , as shown in block 86 . the final desired value for the air / fuel ratio , the signal required for line 20 in fig2 is then determined by adding δafr to afr &# 39 ; as shown in block 88 . the heat loss calculations for qa and qc can take a number of forms . however , for this invention only the relative changes are required and a form requiring the least calculations should be used . this is illustrated by the heat loss calculations set forth below . % h 2 -- h 2 content of the flue gas , % of total volume % o 2 -- o 2 content of the flue gas , % of total volume qco -- heating value in btu per cubic foot of co at atmospheric pressure , 328 btu / cfh w -- the volume in cubic feet occupied by 1 lb of air , 12 . 5 cfh / lb . the heat losses due to incomplete combustion for the case considering only hydrogen and carbon monoxide may be calculated as the change in heat losses from combustibles may be expressed as if carbon can be measured , e . g ., by opacity then equations ( 7 ) and ( 8 ) can be improved by adding a term kδ % opacity . if a hydrocarbon fuel is being used , the percent vaporization loss due to unburned hydrogen should be subtracted from the combustible losses . the change in heat losses due to excess air qa may be calculated as the minimum losses will occur when the rate of change in heat losses from combustibles equals the rate of change in stack heat losses due to a change in combustion air , i . e ., when the above equality may be used to calculate the present heat losses in fig4 a by defining qc and qa as follows , thus , the measurement of percent hydrogen , percent carbon monoxide , percent oxygen , the flue gas outlet temperature , and the ambient temperature are desired to provide the necessary inputs to a digital computer for carrying out the algorithm of fig4 . the previous discussion has assumed the afr was being adjusted directly , as shown in fig2 . large furnaces may have a plurality of such air / fuel control loops . the afr optimizer may be used to move all air / fuel ratios in unison or they may be moved individually . in the later case the loops are normally perturbated one at a time . the same optimizing program shown in fig4 may be used to adjust the o 2 setpoint when a trim control loop such as shown in fig3 is used . the only change required is to replace the quantities of afr with those for o 2 ( sp ). it should be noted that an increase in o 2 ( sp ) will result in an increase in afr . similarly , if a co based trim controller is used the optimizing program may be used to adjust its setpoint , co ( sp ). in this case , however , an increase in co ( sp ) will result in a decrease of afr , therefore , the increase and decrease of δafr will be reversed in the program of fig4 .