x = moles of as - fired fuel per 100 moles of dry gas product ( the assumed solution “ base ”). α k = as - fired ( wet - base ) fuel constituent per mole of fuel σα k = 1 . 0 ; k = 0 , 1 , 2 , . . . 10 . b pls = molar fraction of pure limestone ( caco 3 ) required for zero cao production , moles / base . γ = molar ratio of excess caco 3 to stoichiometric caco 3 ( e . g ., γ = 0 . 0 if no effluent cao ). z = moles of h 2 o per effluent caso 4 , based on lab tests . σ = kronecker function : unity if ( α 6 + α 9 )& gt ; 0 . 0 , zero if no sulfur is present in the fuel . β = air pre - heater dilution factor , a ratio of air leakage to true combustion air , molar ratio . r act = ratio of total moles of dry gas from the combustion process before entering the air pre - heater to gas leaving ; defined as the air pre - heater leakage factor . φ act = ratio of non - oxygen gases ( nitrogen and argon ) to oxygen in the combustion air , molar ratio . a act = concentration of o 2 in the combustion air local to ( and entering ) the system , molar ratio . bbtc = energy flow delivered derived directly from the combined combustion process and those energy flows which immediately effect the combustion process , btu / hr . δh f - 77 0 = heat of formation at 77 . 0 f , btu / lbm or btu / lb - mole δh f - cal 0 = heat of formation at t cal , btu / lbm or btu / lb - mole . hhv = measured or calculated higher heating value , also termed the gross calorific value , btu / lbm af . hhvp = as - fired ( wet - base ) higher heating value , based on hhv , corrected for constant pressure process , btu / lbm af . hpr ≡ enthalpy of products from combustion ( hhv - or lhv - based ), btu / lbm af . l i = specific heat loss term for a ith process , btu / lbm af . lhv = lower heating value based on measurement , calculation or based on the measured or calculated higher heating value , lhv is also termed the net calorific value , btu / lbm af . lhvp = as - fired ( wet - base ) lower heating value , based on lhv , corrected for a constant pressure process , btu / lbm af . m ′ i = weight fraction of ith effluent or combustion air relative to as - fired fuel , —. m af ≡ as - fired fuel mass flow rate ( wet with ash ), lbm af / hr . t cal ≡ calorimetric temperature to which heating value is referenced , f . t ra ≡ reference air temperature to which sensible heat losses and credits are compared ( defined by ptc 4 . 1 ), f . t slack ≡ boundary temperature of the system effluents , commonly taken as the “ stack ” temperature , f . w fd = brake power associated with inflow stream fans ( e . g ., forced draft fans ) within the system boundary , btu / hr . w id = brake power associated with outflow stream fans ( e . g ., induced draft & amp ; gas recirculation fans ), btu / hr . the preferred embodiment for determining boiler efficiency , η b , divides its definition into two components , a combustion efficiency and boiler absorption efficiency . this was done such that an explicit calculation of the major terms , as solely impacting combustion efficiency , could be formulated . this invention teaches the separation of stack losses ( treated by terms effecting combustion efficiency ), from non - stack losses ( treated by terms effecting boiler absorption efficiency ). to develop the combustion efficiency term , the input / loss method employs an energy balance uniquely about the flue gas stream ( i . e ., the combustion process ). this balance is based on the difference in enthalpy between actual products hpr act , and actual reactants hrx act . actual , as - fired , enthalpy of reactants is defined in terms of firing correction : hrx act ≡ hrx cal + hbc . combustion efficiency is defined by terms which are independent of fuel flow . its terms are integrally connected with the combustion equation , eq . ( 19 ) discussed below . η c - hhv ≡ - hpr act + hrx act hhvp + hbc ( 10 ) this formulation was developed to maximize accuracy . typically for coal - fired units , typically over 90 % of the boiler efficiency &# 39 ; s numerical value is comprised of η c . all individual terms comprising η c have the potential of being determined with high accuracy . hpr act is determined knowing effluent temperature , complete stoichiometric balances , and accurate combustion gas , air and water thermodynamic properties . rrx act is dependent on hpr ideal , heating value and the firing correction . hbc applies the needed corrections for the reactant &# 39 ; s sensible heat : fuel , combustion air , limestone ( or other sorbent injected into the combustion process ), water in - leakage and energy inflows . . . all referenced to t cal ( detailed below ). the boiler absorption efficiency is developed from the boiler &# 39 ; s “ non - chemistry & amp ; sensible heat loss ” term , hnsl , i . e ., product sensible heat of non - combustion processes associated with system outflows . it is defined such that it , through iterative techniques , may be computed independent of fuel flow : η a ≡ 1 . 0 - hnsl - hpr act + hrx act ( 11 )  = 1 . 0 - hnsl η c - hhv  ( hhvp + hbc ) ( 12 ) hnsl comprises radiation & amp ; convection losses , pulverizer rejected fuel losses ( or fuel preparation processes ), and sensible heats in : bottom ash , fly ash , effluent dust and effluent products of limestone ( or other sorbent ). hnsl is determined using a portion of ptc 4 . 1 &# 39 ; s heat - loss method . the definition of η a allows η b of eq . ( 3c ) to be evaluated using hpr act & amp ; hrx act terms , illustrating consistency with eq . ( 1a ), explained as follows . since : hsl ≡ hpr act − hpr ideal ; [− hpr act + hrx act ]= hhvp + hbc − hsl ; the following is evident : η b - hhv ≡  [ - hpr act + hrx act hhvp + hbc ]  [ - hpr act + hrx act - hnsl - hpr act + hrx act ]  ( 13a ) =  - hpr act + hrx act - hnsl hhvp + hbc  ( 13b ) =  hhvp + hbc - hsl - hnsl hhvp + hbc  ( 13c ) =  1 . 0 - σ   losses m af  ( hhvp + hbc )  ( 13d ) =  bbtc m af  ( hhvp + hbc )  ( 13e ) where σlosses ≡ m af ( hsl + hnsl ). the energy flow delivered from the combustion process , bbtc , is m af ( hhvp + hbc ) less σlosses . equating eqs . ( 13b ) and ( 13e ) results in defining the specific energy flow delivered , bbtc / m af . since hnsl and bbtc are the same for either hhv - or lhv - based calculations , the enthalpy difference [− hpr act + hrx act ] must be identical . with a computed boiler efficiency , the as - fired fuel flow rate , m af , may be back - calculated : m af = bbtc η b - hhv  ( hhvp + hbc ) ( 15 ) assuming t cal is not known and an arbitrary thermodynamic reference temperature ( t ra ) must be used , t cal = t ra , then the practicality of any boiler efficiency method should be demonstrated through the sensitivity of the denominator of eq . ( 15 ) with its assumed reference temperature . fuel flow , bbtc , and hhvp are constants for a given system evaluation . in regards to fuel flow , the use of an arbitrary t ra is compatible with the methods of this invention provided the computed fuel flow of eq . ( 15 ) is demonstrably insensitive to a “ reasonable change ” in the thermodynamic reference temperature , t ra . by “ reasonable change ” in the thermodynamic reference temperature is meant the likely range of the actual calorimetric temperature . for solid fuels this likely range is from 68f to 95f , or as otherwise would actually be used in practicing bomb calorimeters . for gaseous fuels , whose heating values are computed , not measured , this likely range is whatever would limit the variation in computed fuel flow to less than 0 . 10 %. this invention teaches that the product η b − hhv ( hhvp + hbc ) be demonstrably constant for any reasonable range of t ra , if used . this is not to suggest that effects on η b and hr may be ignored if fuel flow is found insensitive ; the insensitivity of η b and hr must be demonstrated through the hpr ideal term , before a given t ra is justified . however , if η b is mis - evaluated through mis - application of t ra , effects on fuel flow are not proportional given the influence of the hbc term evaluated using the methods of this invention . a 1 % change in η b ( e . g ., 85 % to 84 %) caused by a change in t ra will typically produce a 0 . 2 % to 0 . 4 % change ( δm af / m af ) in fuel flow , which is considered not acceptable . further , eq . ( 15 ) also illustrates that the use of a fuel efficiency ( in which hbc ≡ 0 . 0 ), in combination with an arbitrary reference temperature is flawed : since η b = f ( t ra ), and bbtc & amp ; hhvp are constants , changes in computed fuel flow are then proportional to η b , and wrong . once fuel flow is correctly determined , stoichiometric balances are then used to resolve all boiler inlet and outlet mass flows , including effluent flows required for regulatory reporting . the computation of effluent flow is taught in u . s . pat . no . 5 , 790 , 420 , col . 22 , line 38 thru col . 23 , line 17 ; but without the benefit of high accuracy fuel flow as taught by this invention . system heat rate associated with a steam / electric power plant follows from eq . ( 15 ) in the usual manner . the effects on hr given mis - application of t ra will compound ( add ) the erroneous effects from η b and fuel flow . hr hhv ≡  m af  ( hhvp + hbc ) / power   ( 16 ) =  bbtc / ( η b - hhv   power )  ( 17 ) given the commercial importance of computing fuel & amp ; emission flows for industrial systems , and determining system heat rate consistent with these flows , accurately determining boiler efficiency is important ( upon which these quantities are based ). the determination of on - line fuel heating values , coupled to sophisticated error analysis as used by the input / loss method , demands integration of stoichiometrics with high accuracy boiler efficiency . to assist in understanding , discussed is the determination of heats of formation evaluated at t cal . by international convention , standardized heats of formation are referenced to 77f ( 25c ) and 1 . 00 bar pressure . for typical fossil combustion , pressure corrections are justifiably ignored . to convert to any temperature from 77f the following approach is used : δ   h f - t 0 = δ   h f - 77 0 + ∫ 77 t    h compound - ∑ ∫ 77 t    h elements ( 18 ) use of the 77f - base standard is important as it allows consistency with published values . consistent δh f - t 0 values for co 2 , so 2 and h 2 o allow consistent evaluations of the hpr ideal term , and the difference between the as - fired heating value plus firing correction and [− hpr act + hrx act ] . . . thus intrinsically accounting for stack losses and the vagaries of coal pyrolysis given unburned fuel . the finest compilation of heats of formulation and other properties is the so - called codata work ( cox , wagman , & amp ; medvedev , codata key values for thermodynamics , hemisphere publishing corp ., new york , 1989 ). enthalpy integrals used in eq . ( 18 ) and elsewhere herein are obtained from the work of passert & amp ; danner ( industrial eninee chemistry , process desin and develoment , volume 11 , no . 4 , 1972 ; also see manual for predicting chemical process design data , chapter 5 , aiche , n . y ., 1983 , revised 1986 ). all fluid components in the thermal system ( e . g ., combustion gases , water in the combustion effluent , moist combustion air , gaseous constituents of air ) must use a consistent dead state for thermodynamic property evaluations . preferred methods employ 32 . 018f as a uniform dead state temperature , t dead , and 0 . 08872 psia pressure , for all properties ( e . g ., the defined zero enthalpy for dry air , gaseous compounds , saturated liquid water , etc .). thermodynamic properties are evaluated in the usual manner , for example from t dead to t cal , and from t dead to t stack , thus net the evaluation from t cal to t stack . given such foundations , eq . ( 18 ) with codata , heats of combustion of gaseous fuels , given their known chemistries , may be computed for any calorimetric temperature ( e . g ., at the industrial standard of 60f & amp ; 14 . 73 psia ; see astm d1071 & amp ; gpa 2145 ). solid and liquid fuel heating values , determined by test using an adiabatic or isoperibol bomb calorimeter , are in theory referenced to 68 . 0f ( 20c ). refer to astm d271 , d1989 , d2015 & amp ; d3286 for coals ( being replaced by d5865 ), and astm d240 for liquid fuels . the 68f reference for solid fuels is rarely practiced ; typically , coal bombs are typically conducted at 82 . 5f or 95f . knowing the calorimetric temperature , if using this temperature in strict compliance with the definition of heating value , all system energies affecting boiler efficiency may then be computed . the following combustion equation is presented for assistance in understanding nomenclature used in the detailing procedures . refer to u . s . pat . no . 5 , 790 , 420 for additional details . the nomenclature used is unique in that brackets are included for clarity . for example , the expression “ α 2 [ h 2 o ]” means the fuel moles of water , algebraically α 2 . the quantities comprising the combustion equation are based on 100 moles of dry gaseous product . x  [ α 0  [ c yr  h zr ] + α 1  [ n 2 ] + α 2  [ h 2  o ] + α 3  [ o 2 ] + α 4  [ c ] + α 5  [ h 2 ] + α 6  [ s ] + α 7  [ co 2 ] + α 8  [ co ] + α 9  [ h 2  s ] + α 10  [ ash ] ] as  -  fired   fuel + b z  [ h 2  o ] in  -  leakage + [ ( 1 + β )  ( a  [ o 2 ] + a   φ act  [ n 2 ] + b a  [ h 2  o ] ) ] air + [ ( 1 - γ )  b pls  [ caco 3 ] ] as  -  fired   pls = d act  [ co 2 ] + g act  [ o 2 ] + h  [ n 2 ] + j act  [ h 2  o ] + k act  [ so 2 ] + [ e act  [ co ] + f  [ h 2 ] + l  [ so 3 ] + m  [ no ] + p  [ n 2  o ] + q  [ no 2 ] + t  [ c yp1  h zp1 ] + u  [ c yp2  h zp2 ] ] minor   components + x   α 10  [ ash ] + σ   b pls  [ caso 4 · z   h 2  o ] + [ ( 1 - σ + γ )  b pls  [ cao ] ] excess   pls + v  [ c refuse ] + [ β  ( a  [ o 2 ] + a   φ act  [ n 2 ] + b a  [ h 2  o ] ) ] air   leakage ( 19 ) eq . ( 19 ) contains terms which allow consistent study of any combination of effluent data , especially the principle “ actual ” effluent measurements d act , g act , j act , and the system terms β , φ act & amp ; r act . by this is meant that data on either side of an air pre - heater may be employed , in any mix , with total consistency . this allows the stoichiometric base of eq . ( 19 ), of 100 moles of dry gas , to be conserved at either side of the air pre - heater : dry stack gas = dry boiler = 100 moles . the following paragraphs discuss detailed procedures associated with the input / loss method of determining boiler efficiency . the firing correction is closely defined and only relates to terms correcting hrx cal . absorption efficiency , η a , is based on the non - chemistry & amp ; sensible heat loss term , hnsl , whose evaluation employs several ptc 4 . 1 procedures . hnsl is defined by the following : hnsl ≡ l β + l p + l d / fly + l d / prec + l d / ca + l r + w id / m af ( 20 ) hnsl bears the same numerical value for both higher or lower heating value calculations , as does η a . differences with ptc 4 . 1 and ptc 4 procedures include : l β is referenced to the total gross ( corrected ) higher heat input , ( hhvp + hbc ), not hhv ; the l w term is combined with the ash pit term l p ; l d / fly is sensible heat in fly ash ; l d / prec is the sensible heat in stack dust at collection ( the assumed electrostatic precipitator ), considered a separate stream from fly ash ; and l d / ca is the sensible heat of effluents from sorbent injection if used ( e . g ., caso 4 . zh 2 o and cao effluents given limestone injection ). l r and w id are discussed below . all terms of eq . ( 20 ) are evaluated relative to unity as - fired fuel . numerical checks of all effluent ash is made against fuel mineral content ( and optionally may re - normalize the fuel &# 39 ; s chemistry ). the radiation & amp ; convection factor , β r & amp ; c , is determined using either the american boiler manufacturers &# 39 ; curve ( ptc 4 . 1 ), or its equivalence may be derived based on the work of gerhart , heil & amp ; phillips ( asme , 1991 - jpgc - ptc - 1 ), or its equivalence may be based on direct measurement or judgement . the resulting l β loss is always determined using the higher heating value : l β is then applied to either lower or higher heating value efficiencies through hnsl . the coal pulverizer rejects loss term , l r , is referenced to the total gross ( corrected ) higher heating value of rejected fuel plus the firing correction , hhvp rej + hbc , given rejects contain condensed water . further , it is assumed the grinding action may result in a concentration of mineral matter ( commonly referred to as “ ash ”) in the reject , thus the fuel chemistry is renormalized based on a corrected fuel ash , α 10 - corr = f ( wf ′ ash - af ); see eq . ( 19 ). this is based on the weight fraction of ash downstream from the pulverizers ( true as - fired ), wf ′ ash - af . wf ′ ash - af derives from : the weight fraction of rejects / fuel ratio , wf rej ; ash in the supplied fuel , wf ash - sup ; and corrected heating values . for lower heating value computations , the ratio hhv rej / hhv sup in eq . ( 22a ) is replaced by lhv rej / lhv sup . the assumption of the reject loss being based on the higher heating value , although convenient for the hnsl term , implies , given the possibility of renormalized fuel chemistry , that the hrx act - lhv term must be corrected for the fuel water &# 39 ; s latent heat . this correction is described by eq . ( 22c ), applied in eq . ( 22b ) yielding a corrected lhvp . the δh l / h term is evaluated using as - fired chemistry downstream from the pulverizers , see eq . ( 39b ). within eq . ( 22c ): ξ ≡( 1 . 0 − wf ′ ash - af )/( 1 . 0 − wf ash - sup ). ξ also corrects both eq . ( 37b ) & amp ; ( 39b ). these same procedures are applicable for a fuel cleaning process where the fuel &# 39 ; s mineral matter ( ash ) is removed . the steam / air heater energy flow term , q sah , is assigned to hbc provided the system encompasses this heater , which it should as preferred . bbtc is defined in the classical manner ( e . g ., throttle less feedwater conditions , hot less cold reheat conditions ). this is best seen by equating eqs . ( 13b ) & amp ; ( 13e ), noting hpr act = hpr ideal + hsl : bbtc =  m af  [ - hpr act + hrx act - hnsl ]  ( 22d ) =  m af  [ - ( hpr ideal + hsl ) + ( hrx cal + hbc ) - hnsl ]  ( 22e ) =  m af [ - hpr ideal + hrx cal + hbc - hsl - hnsl )  ( 22f ) if eq . ( 2b ), and its hpr ideal term , is to be conserved , the right side of eq . ( 22f ) must be corrected for the total energy flow attributable to combustion : thus hbc includes the + q sah term , as must the bbtc term ( resulting in a higher fuel flow ). although ( bbtc − q sah ) is the net “ useful ” output from the system , bbtc is the total and directly derived energy flow from the combustion process applicable to η b . . . so defined such that eqs . ( 13e ) & amp ; ( 22d ) are conserved . the hsl term of eq . ( 22f ) is not explicitly evaluated , discussed below . the id fan energy flow term , w id , given that thermal energy is imparted to the gas outflow stream ( e . g ., id or recirculation fans ), the hpr act term must be corrected ( through hnsl ) such that the fuel &# 39 ; s energy term hpr ideal is again properly conserved . the coal pulverizer shaft power is not accounted as no thermal energy is added to the fuel . crushing coal increases its surface energy ; for a generally brittle material , no appreciable changes in internal energy occur . the increased surface energy and any changes in internal energy are well accounted for through the process of determining heating value . if using astm d2013 , coal samples are prepared by grinding to a # 60 sieve ( 250 μm ). inconsistencies would arise if the bomb calorimeter samples were prepared atypical of actual firing conditions . miscellaneous shaft powers are not accounted if not affecting hpr act or hrx act , i . e ., not affecting the energy flow attributable to combustion . the use of “ net ” efficiencies or “ net ” heat rates , incorporating house electrical loads ( the b xe term of ptc 4 . 1 ), is not preferred for understanding the thermal performance of systems . having evaluated hnsl , the absorption efficiency is determined from either hhv - or lhv - based parameters : η a = 1 . 0 - hnsl - hpr act - hhv + hrx act - hhv = 1 . 0 - hnsl - hpr act - lhv + hrx act - lhv ( 23 ) all unburned fuel downstream of the combustion process proper ( e . g ., carbon born by ash ) is treated by the combustion efficiency term , as are all air , leakage and combustion water terms . for accuracy considerations , stack losses ( hsl ) are not independently computed ; however to clarify , they relate for example to η c - hhv as , using ptc 4 . 1 nomenclature in eq . ( 25 ): η c - hhv = 1 . 0 - hsl hhv hhvp + hbc ( 24 ) hsl hhv =[ l g ′ + l mg + l mf + l ma + l mca + l z + l h + l co + l uh + l uhc + l uc1 + l uc2 ] hhv ( 25 ) where : the l mg term is moisture created from combustion of chemically - bound h / c fuel ; l mca is fuel moisture bound with effluent caso 4 ; l uc1 is unburned carbon in fly ash ; l uc2 is unburned carbon in bottom ash ; all others per ptc 4 . 1 . non - combustion energy flows are not included ( see hnsl ). terms of eq . ( 25 ) as fractions of ( hhvp + hbc ) or ( lhvp + hbc ), are computed after η c , by back - calculation ; they are presented only as secondary calculations for the monitoring of individual effects . combustion efficiency is determined by the following , as either hhv - or lhv - based : η c - hhv ≡ - hpr act - hhv + hrx act - hhv hhvp + hbc ( 26 ) η c - lhv ≡ - hpr act - lhv + hrx act - lhv lhvp + hbc ( 27 ) the development of the combustion efficiency term , as computed based on hpr act & amp ; hrx act and involving systematic use of a combustion equation , such as eq . ( 19 ), is believed an improved approach versus the primary use of individual “ stack loss ” terms . mis - application of terms is greatly reduced . numerical accuracy is increased . most importantly , valid system mass and energy balances are assured . of course fuel flow must compute identically from either efficiency base , thus : m af = bbtc η b - hhv  ( hhvp + hbc ) = bbtc η b - lhv  ( lhvp + hbc ) ( 30 ) such computations of fuel flow using either efficiency , at a defined t cal , is an important numerical overcheck of this invention . after hnsl is computed , as observed in eqs . ( 23 ), ( 26 ) & amp ; ( 27 ) only the three major terms hpr act , hrx act & amp ; hbc remain to be defined to complete boiler efficiency . these are defined in the following paragraphs . to fully understand the formulations comprising hpr act , hrx act & amp ; hbc , take note of the subscripts associated with the individual terms . for example , when considering water product created from combustion , n comb - h2o of eq . ( 31 ), its heat of formation ( saturated liquid phase ) at t cal must be corrected for boundary ( stack ) conditions , thus , h stack − h f - cal the enthalpies of reactants of eqs . ( 34 ) & amp ; ( 35 ) are determined from ideal products at t cal , the firing correction then applied . differences in formulations required for higher or lower heating values should also be carefully reviewed . higher heating values require use of the saturated liquid enthalpy evaluated at t cal ; lower heating values require the use of saturated vapor at t cal . water bound with effluent caso 4 is assumed in the liquid state at the stack temperature ; whereas its reference is the heating value base ( fuel water being the assumed source for z [ h 2 o ] of eq . ( 19 )). the quantities which are not so corrected are the last two terms in eqs . ( 31 ) & amp ; ( 32 ): water born by air and from in - leakage undergo no transformations , having non - fuel origins . heating values and energies used in eqs . ( 31 ) thru ( 35 ) are always associated with the system boundary : the as - fired fuel ( or the “ supplied ” in the case of fuel rejects ), ambient air and location of the continuous emission monitoring system ( cems ) and temperature measurements ( at the “ stack ”). hpr act − hhv ≡ σhpr i +[ n comb − h2o ( δ h f - cal - liq 0 + h stack − h f - cal )+ n fuel − h2o ( h stack − h f - cal )+ n lime − h2o ( h f - stack − h f - cal )+ n cair − h2o ( h stack − h g - cal )+ n leak − h2o ( h stack − h steam )] h2o n h2o /( xn af ) ( 31 ) hpr act − lhv ≡ σhpr i +[ n comb − h2o ( δ h f - cal / vap 0 + h stack − h g - cal )+ n fuel − h2o ( h stack − h g - cal )+ n lime − h2o ( h f - stack − h g - cal )+ n cair − h2o ( h stack − h g - cal )+ n leak − h2o ( h stack − h steam )] h2o n h2o /( xn af ) ( 32 ) where : hpr 1 =  enthalpy   of   non  -  water   product   i   at   the   boundary ≡  [ δ   h f - cal / i 0 + ∫ t cal t stack    h i ]  n i  n i / ( xn af )  ( 33 ) n comb - h2o =  molar   water   found   at   the   boundary   from   combustion ≡  x  ( α 0  zr / 2 + α 5 + α 9 ) - f n fuel - h2o =  molar   water   found   at   the   boundary   born   by  as  -  fired   fuel   ( as   total   inherent   and   surface   moisture ) ≡  j act - [ b a + b z + σ   b pls  z + x  ( α 0  zr / 2 + α 5 + α 9 ) - f ] n lime - h2o =  molar   water   bound   with   effluent   caso 4 ≡  σ   b pls  z n cair - h2o =  molar   water   found   at   the   boundary   born   by  combustion   air   and   air   in  -  leakage ≡  b a  ( 1 . 0 + β ) n leak - h2o =  molar   water   found   at   boundary   from   direct  in  -  leakage ≡  b z h stack - h2o =  f  ( p stack - h2o , t stack ) , where   p stack - h2o   is   water &# 39 ;  s  partial   pressure   per   wet   molar :  p amb  ( j act + β   b a ) / ( 1 . 0 + j act + β   b a ) . hrx act − hhv ≡ hhvp + hbc + hpr co2 − ideal + hpr so2 − ideal +[( α 0 zr / 2 + α 5 + α 9 )( δh f - cal / liq 0 n ) h2o / n af ]+ hrx caco3 ( 34 ) hrx act − lhv ≡ lhvp + hbc + hpr co2 − ideal + hpr so2 − ideal +[( α 0 zr / 2 + α 5 + α 9 )( δh f - cal / vap 0 n ) h2o / n af ]+ hrx caco3 ( 35 ) where : hpr co2 - ideal =  energy   of   co 2   ideal   product   from   complete  combustion   at   the   calibration   temperature . ≡  δ   h f - cal / co2 0  ( α 0  yr + α 4 + α 8 )  n co2  /  n af hpr so2 - ideal =  energy   of   so 2   ideal   product   from   complete  combustion   at   the   calibration   temperature . ≡  δ   h f - cal / so 2 0  ( α 6 + α 9 )  n so 2  /  n af hpr h2o - ideal =  energy   of   h 2  o   ideal   product   from   complete  combustion   at   the   calibration   temperature . ≡  ( α 0  zr  /  2 + α 5 + α 9 )  ( δ   h f - cal / liq 0  n ) h2o  /  n af ;  for   hhv ≡  ( α 0  zr  /  2 + α 5 + α 9 )  ( δ   h f - cal / vap 0  n ) h2o  /  n af ;  for   lhv . hrx caco3 =  energy   of   injected   pure   limestone , caco 3 ,  at   the   calibration   temperature ; use   of  δ   h f - cal / caco3 0   anticipates   heats   of   formation  associated   with   limestone   products   appearing  in   eq .  ( 33 ) . ≡  δ   h f - cal / caco3 0  b pls  ( 1 . 0 + γ )  n caco3  /  ( xn af ) hbc ≡ c p ( t af − t cal ) fuel +( q sah + w fd )/ m af +[( h amb − h cal ) air a ( 1 . 0 + β )( 1 . 0 + φ act ) n air +( h g - amb − h g - cal ) h2o b a ( 1 . 0 + β ) n h2o +( h steam − h f - cal ) h2o b z n h2o + c p ( t amb − t cal ) pls b pls ( 1 . 0 + γ ) n caco3 ]/( xn af ) ( 36 ) h g − amb − h2o = saturated water enthalpy at ambient dry bulb , t amb . ( h amb − h cal ) air = δenthalpy of combustion dry air relative to t cal . ( h g - amb − h g - cal ) h2o = δenthalpy of moisture in combustion air relative to saturated vapor at t cal . ( h steam − h f - cal ) h2o = δenthalpy of water in - leakage to system relative to saturated liquid at t cal . c p ( t amb − t cal ) pls = δenthalpy of pure limestone relative to t cal . the above equations are dependent on common system parameters . common system parameters are defined following their respective equations , eqs . ( 31 ) thru ( 36 ). further , these terms are discussed in ptc 4 . 1 and 4 , and throughout u . s . pat . no . 5 , 790 , 420 . in addition , the bbtc term , also comprising common system parameters , is determined from commonly measured or determined working fluid mass flow rates , pressures and temperatures ( or qualities ). several ptcs and “ coal ” textbooks employ simplifying assumptions regarding the conversion of heating values . for example , a constant is sometimes used to convert from a constant volume process hhv ( i . e ., bomb calorimeter ), to a constant pressure process hhvp . the following is preferred for completeness , for solid and liquid fuels , and is also applicable for lhv : δ h v / p ≡ rt cal , abs ( α 5 / 2 − α 1 )/( ξ jn af ) ( 37b ) where , in us customary units : t cal , abs is absolute temperature ( deg − r ); r = 1545 . 325 ft - lbf / mole - r ; and j = 778 . 169 ft - lbf / btu . for gaseous fuels , the only needed correction is the compressibility factor assuming ideally computed heating values : z and hhv ideal are evaluated using american gas association procedures . to convert from a higher ( gross ) to a lower ( net ) heating value use of eq . ( 39b ) is exact , where δh fg − cal / h2o is evaluated at t cal . the oxygen in the effluent water is assumed to derive from combustion air , and not fuel oxygen ( thus α 3 is not included ). δ h l / h ≡ δh fg − cal / h2o ( α 0 zr / 2 + α 2 + α 5 + α 9 ) n h2o /( ξ n af ) ( 39b ) to more fully explain this invention fig1 is presented . box 20 of fig1 represents the determination of a fossil fuel &# 39 ; s heating value , and its correction if needed for a constant pressure process using eqs . ( 37a ) & amp ; ( 37b ). if a gaseous fuel , determination of hhvp is generally a computation , establishing t cal by convention ; in north america 60 f is commonly used . if a solid or liquid fuel , whose heating value is tested by bomb calorimeter , t cal is measured and / or otherwise established as part of the testing procedure . box 22 describes the calculation of the hpr ideal term , comprising hpr co2 − ideal , hpr so2 − ideal and hpr h2o |− ideal , expressed below eq . ( 35 ) where associated heats of formation are computed from eq . ( 18 ) at t cal . box 30 describes the computation of the firing correction term , hbc , using eq . ( 36 ) as referenced to t cal box 32 represents the calculation of the uncorrected enthalpy of reactants evaluated at t cal , from eq . ( 2b ) requiring results from boxes 20 and 22 . box 40 represents the calculation of the enthalpy of reactants at actual firing conditions using eqs . ( 34 ) or ( 35 ), requiring input from boxes 30 and 32 . box 42 represents the calculation of the enthalpy of products at actual boundary exit conditions ( e . g ., stack temperature ), using eq . ( 31 ) or ( 32 ). box 44 represents the calculation of the non - chemistry & amp ; sensible heat loss term , hnsl , using eq . ( 20 ) whose procedures and individual terms are herein discussed . box 50 represents the computation of combustion efficiency , using either eq . ( 26 ) or ( 27 ), with inputs from boxes 20 , 30 , 40 , and 42 . box 52 represents the computation of boiler absorption efficiency , using either form of eq . ( 23 ), with inputs from boxes 40 , 42 and 44 . box 54 represents the computation of boiler efficiency , using either eq . ( 28 ) or ( 29 ), with inputs from boxes 50 and 52 . the following presents typical numerical results as evaluated by the ex - foss computer program , commercially available from exergetic systems , inc ., of san rafael , calif . which has now been modified to employ the methods of this invention . to illustrate the effects of mis - using calorimetric temperature table 1 presents the results of a methane - burning boiler . as observed , boiler efficiency is insensitive to slight changes in heating values provided t cal is not varied in other terms comprising η b . however , when consistently altering t cal ( as its impacts hpr ideal ), results indicate serious , and un - reasonable , error in boiler efficiency . one may not establish a reference temperature for the fuel &# 39 ; s chemical energy , at t cal , and then not consistently apply it to other energy terms . if misapplied as suggested by table 1 , errors in ri and system heat rate will be assured . use of eq . ( 15 ), given η b derives from eq . ( 10 ) & amp ; ( 11 ), demands consistency in the hpr act , hrx act and hbc terms ; the same system can not have a difference in its computed fuel flow . table 2 presents typical effects on boiler efficiency and system heat rate of mis - use of calorimetric temperatures on a variety of coal - fired power plants . the effect of such mis - use are considered un - reasonable . these computations are based on ex - foss , varying only t cal . data was obtained from actual plant conditions . table 3 lists computational overchecks of higher and lower heating value calculations , verifying that the computed fuel flow rates of eq . ( 30 ), are numerically identical . these simulations were selected from input / loss &# 39 ; installed base as having unusual complexity , based on actual plant conditions . the only changes in these simulations was input of hhv or lhv , and an ex - foss option flag ; lhv or hhv are automatically computed by ex - foss given input of the other . several modern bomb calorimetric instruments are automated to run at t cal = 95f ( 35c ). the repeatability accuracy of these instruments is between ± 0 . 07 % to ± 0 . 10 %. modern bomb calorimeters use benzoic acid powder for calibration testing . calibration results are typically analyzed using the well - known washburn corrections ( journal of physical chemistry , volume 58 , pp . 152 - 162 , 1954 ). based on these procedures , nist standard reference material 39j certification for benzoic acid makes a multiplicative correction for temperature : [ 1 . 0 − 45 . 0 × 10 − 6 ( t cal − 25 ° c .)]. such corrective coefficients ( e . g ., 45 . 0 × 10 − 6 ) were computed for a number of coals , using average chemistries for different coal ranks , and with methane . for example , a correction of 122 × 10 − 6 implies a 0 . 122 % change in hhv over 10 ° c . as observed below in table 4 , heating values with increasing fuel moisture are generally increasingly sensitive to calorimetric temperature , especially for gaseous fuels and poor quality lignite coals . effects on hhvs associated with the common coals are not great . however , the sensitivity of temperature on hpr ideal is appreciable for most ranks ; computed using ex - foss . this sensitivity demonstrates the fundamental cause for the sensitivities observed in tables 1 and 2 . the method of this invention generally causes an insensitivity in computed fuel flow when using an arbitrary reference temperature over a reasonable range . table 5 demonstrates this for several coal ranks , assuming t ra changed from 68f to 77f , and from 68f to 95f . such effects on fuel flow are additive to those associated with boiler efficiency when considering net effects on system heat rate ( system efficiency ). the results illustrated in tables 1 , 2 and 4 indicate generally un - reasonable sensitivity in computed boiler efficiency and system heat rate . considered reasonable accuracy as attainable using the methods of this invention , are δη b − hhv errors , or δη b − lhv errors , in boiler efficiency of 0 . 15 % δη b or less . considered reasonable accuracy in computed system heat rate are δhr / hr errors no greater than 0 . 25 %. considered reasonable accuracy in computed fuel flow , using eq . ( 15 ), are δm af / m af errors no greater than 0 . 10 %. this work demonstrates a systemic approach to determining boiler efficiency . it demonstrates that the concept of defining boiler efficiency in terms of the enthalpy of products ( hpr act ), the enthalpy reactants ( hrx act ) and the firing correction ( hbc ), it is believed , provides enhanced accuracy when these major boiler efficiency terms are referenced to the same calorimetric temperature . such accuracy is needed by the input / loss method , and for the improvement of fossil combustion in a competitive marketplace . the hpr act & amp ; hrx act concept forces an integration of combustion effluents with fuel chemistry through stoichiometrics .