SEP099804:36PMCONSOLR&DLIBRQRY P .2/20 EVALUATIONOF AIRHEATERPERFORMANCEANDTHE ACCURACYOF THERESULT Joseph T.Maskew, Duane C. McCoy,(CONSOLInc., R&D), BurtonL.Marker(NYSEG),and James U. Watts(DOE, FETC) Withthe increased emphasison the efficiency of fossil-fuel-fired, steam generationfacilities, the performance of ancillary equipment is becoming increasingly important.The air heater is a soutee of lost thermal efficiency in two ways -- air leakageinto flue gas side and poor heat recovery.Moreover, air inlcak makes it difftcultto determinethe exiting flue gas temperature and the performance of the air heater. This paper addressesthe issue of properly evaluating the air heater performance and the accuracyof the final result. The appendix discussesthe proceduresused to determine the individual measurementsand the uncertainty of these measurements. Background As part of the I-J.S. Department of Energys (DOE) Clean Coal Technology IVDemonstration Progrsm, New YorkState Electric &Gas Corporation (NISEG) selectedthe MillikenStation for installation of innovative SO?and NO, control technologiesand efficiency improvements. These improvements willallow utilitiesto comply withthe Clean AirAct Amendmentsof1990.The air heaterson Unit2 were replacedto improve unit efficiency as part of the demonstration program.The original air heater was a regenerativeLjungstrom unit; the replacementair heater was a low pressuredrop, high eff%cirncyheat pipe.CONSOL R&Devaluatedthe performance of the air heater and estimatedthe uncertainty in the evaluation. The American Society of Mechanical Engineers(ASME) provides a standardmethod of computing the performance of air heaters. This is performancetest code (PTC) ASME PTC 4.3. This method was specified as the standardof acceptableperformanceby warranteesof the new a.irheaters. While the AShfE code is often specified in equipmentwarrantees,itappearsto be rarely applied.Instead, performanceindicators such as the measuredeffectivenessof the air- and gas-sidesand the X-ratioare compareddirectly againstdesign values. Such comparisonsare poor substitutes for the ASME PTC which correctsfor part of the differences between test and design conditions independently of the vendors design algorithms. The algorithms, normally provided by the vendor as performancecurves and/or correlationsthat predict the outlet temperature based on inlet conditions, cannot be applied directly in the ASME code. The ASME code predicts the temperaturecorrectedto the design value while the performance curves predict the expected temperatureat operating conditions. This paper provides a method of applying the vendors performance curves to evaluatethe performancecorrectedto design as per ASTM PTC 4.3. An air heater is shown schematicallyin Figure I.Note that the ASh4F PTC 4.3 nomenclature is used in Figure 1 and throughout this paper. In the air heater,energy in the flue gas is recovered by the incoming combustion air.While normally severalair streamsarc present (primary and secondary),in this paper we examine only one section of the air heater. SEP099804:37PMCONSOLR&DLIBRARY P .a20 s----L--, FloeGa EnkdngI Figure1 AirHeater Schematic ASME PTC 4.3 calculates a totallycorrectedflue gas outlet temperature(TCFGOT), rc,$nrord, shown below (ASMESupplement as Equation 7.12): f GlSSTotd=fG15cU+tG156G+ki15&Y+lG1566-3tG15(1) t G,56A= Flue gas temperature leaving the air heater corrected for deviation from design entering air temperature, OF, lo,sJo= Flue gas temperature leaving the ah heater corrected for deviation from design entering flue gas temperature, F, f015M7(= Flue gas temperature leaving the air heater corrected for deviation from design X - ratio, F, f G,5SE= Flue gas temperature leaving the air heater corrected for deviation from design entering gas ff ow, F,and tGi5 = Measured fluegas temperature leaving the air heater, F. The PTC provides equationsfor the first two of the temperaturecorrectjons,rGIJdAand tc,,6G,but not for the other two, rca WRand rc,J h.Theselatter temperaturecorrectionsare unique to a specific air heater. Ifthese temperaturecorrections are not provided by the equipment manufacturer as algorithms (or plots), they can be estimatedby the procedurepresentedin this paper.The procedure uses design performancecurves and/or algorithms normally provided by the vendor to evaluatethe temperaturecorrectionsrequired by Equation 1.The TCFGOTis then compared to the design flue gas temperature. The computedvalue of the TCFGOT should be less than or equal to the design flue gas temperature,if the air heater is performing properly. The TCFGOT The two temperature correction factors provided by the PTC sre tGIsd,,and tns rl0 These are defined in terms of design values and of measuredresults of a standardtest of an air heater. For the deviation from the design entering air temperature,rclJM,this is: ~Gl56A= 'ABD'G14-rGIS)+rG14+G15-rR8) -1.4s) SEP099804:37PMCONSOLR&DLIBRRRY P.4/a rAsD= Design air temperatureentering the air heater, F, ro,4 = Measured flue gas temperatureentering the air heater, F,and t AB= Measured air temperatureentering the air heater, F. Similarly,for the deviation from the design entering flue gas temperature,the temperature correotion is: tGlsSG= *c14ll&S-48)+18+G14-h) (to14-tA8) where tG,4n= Design flue gastemperatureenteringthe air beater, F. X-RatioCorrectionFlueGasFlowCorrection // ~OssignX-Ratio~OssignX-Ratio Measured X-Ratio Figure2X-RatioCorrection \ Design FlueGasFlaws Measured Entering Flue Gas Flow Figure3Flue Gas Flow Correction The other two temperaturecorrectionsmust be derived from vendor design performance curves or provided by the vendorin analytical form.In the caseof the NYSEG heat pipe air heater, the vendor supplied a setof performancealgorithms to be applied thperformancefigures similar to Figures 2 and 3, shown above. Thesepredicted the performancetemperature;that is, the expected exit flue gas temperaturesfor the actual operating conditions. The algorithm was of the form: where SEP099804:38PMCONSOLR&DLIBRARY P.5/20 9= Correlationooefftcient,and fG,.Jx= Correctionfactors fordeviations fromdesign flue gas flowand fromdesign X-ratio,respectively. For easeof analysis and of estimating the uncertainty, theseplots were converted into mathematical expressionsof the form: fg=~,+h.& for the flue gas flow,and for the X-ratio: f,*a2+&.X+cS2.X2 where ~,,~,,a,,&6s= Correlationcoefficients, Fo = Flue gas flowrate, and X= X-ratiofor the air side. (5) (6) The forms of these equations agreewiththe shapesof the curvesin Figures 2 and 3.Aleast squarescorrelation or some other curvy fittingtechniquecanbeused to evaluatethe correlation constants. In the caseof the Millikenstudy, the correlation equationsagreedwith results from the plots withinthe ability to read the plots. Since the X-ratiois defined aathe weight times heat capacity ratio of the air over that of the flue gas, the X-ratiocan be approximatedas the ratio of the temperaturechangesfor the two fluids: x= WA9CpA %I4pG where cpA= Heat capacityof air, Btu I lb- F, cpG= Heat capacityof flue gas, But / lb-* F, fg5= Averageflue gasoutlettemperaturecorrectedto no-leak conditions, OF, w,,s = Weight of air exiting the air heater, lb / h, and wG]4= Weight Of flue gasenteringthe air heater,lb/h. (7) The no-leak flue gas temperature,rz,,is calculated from the measuredflue gas temperatureby: sEP099804:38PMCONSOLR&DLIBRRRY tNL Cl5= rG15+-Lb) P.WZB (8) where A,= Weight percent air leakageinto the flue gas, and t lunb= Tempetatureof the air leaking into the flue gas. In most air heaters,the majorityof the air in the flue gas is leakagefrom thehigherpressure,air- side of the air heater. The ASME defines the position oftheair leak as occurring after the flue gas exits the ai~heater, but before rc,, is measured. Thus, there can he no correction to beat transfer withinthe air heater for air leakage. In thesecases, t omb=fA8(9) andfAacan be substituted for z,,,,,~in the followingequations. However, this derivation willbe general. Substituting Equation 8 into Equation 7 yields: x= [ tG14-'0151oocpo -".[q.(tG,5-'-)I (10) OA9-[,!8) Note the ASME definitionfor the X-ratio is baaedon zero leak. If the vendor baseshis X-ratio correction curve on an X-ratio witha design leak, this plot should be correctedto zero leak before generating Equation 6. For application of Equation 1, two additional, independenttemperaturecorrections are required. These can be obtained from the vendors air heater performanceequation, Equation 4, and the associatedplots -- Figures 2 and 3.Equations similar to Equation 4 can be used to estimatethe effect of one parameter independentof the other parametersof the equation to obtain a temperature correction for that parameteralone. This is achievedby evaluating Equation 4 for the change in one parameterwhile holding the others constant. FOT the deviation from the design X-ratio,this procedureproduces the followingequation for the temperaturecorrection,tCISMR: ~G~~~~=~~~~+~~~G~SD-~G~~D~[~-~~~~D~~X]-~A~D~~~~~O~~X A.4 [ I[ CPA 1 (11) +100cpc --(k15-L6)II where t o,sn= Design flue gas temperatureleaving air heater, and fpD = De&go flue gas flowcorrection factor. For the deviation from design flow,the temperaturecorrection, to,, 6nis: rGlS,=tGl5+~~fG,5~-tGl4D~~-9~fG~fXD]-r~80~9'fg~~~[~ (12) SEP099804:38PMCONSOLR&DLIBRRRY Pa 7120 where fm = Design X - ratio correction factot. Equations 1Iand 12 apply the performanceequationsand/or curvesprovided by the vendor to evaluate the effect of the changem X-ratio and flue gas flowon the measuredtemperature. The changesfrom the design TCFGOT, the terms within the double lines (\I), are applied to the measuredflue gas outlet temperatureto provide the temperaturecorrections. ASMEPTC 4.3 specified the air heatertemperaturecorrectionsat design leak. For the NYSEG unit, the design leak was zero. This is reflected in quation10 where the X-ratioiscorrected to the design leak of zero before being applied to the calculation of the temperaturecorrection. The leak correction term, (13) is re-+redsince (1) the performance equation and factor plots were basedon a zero leak design, and (2) ASME PTC 4.3 specifies comparing the TCFGOT at the design conditions, which in this caseis zero leak.Therefore, the TCFGOT must be on the samebasis as the design. The first four terms of Equation 1 addin three leak terms. The measuredflue gas temperatureleaving the air heater, lcls, subtracts out three leak terms as this measuredvalue containsleak.Thus, the inclusion of a leak correction term in Equation 11 evaluatesthe TCFGOT by Equation 1, i,,,,, at zero leak, the design condition, as specifiedby the ASME PTC 4.3. Substituting (14) into Equation 11 and then expanding Equation 1 by substituting Equations2,3,11,and 12, along withthe air heater performance correlations (Equations 5 and 6), results in the followingrevised equation: SEP099804:39PMCONSOLR&DLIBRFIRY P . E/20 Inspection of this equation reveals that calculation of the TCFGOT requiresonly four measured ,and 2 determined values: inlet and outlet air temperatures,inlet and outlet flue gas temperatures, entering flue gas flowand the air leak.Allof the other parametersare constants. The calculated value oftbeTCXGTfrom an air heaterperformancetest must be equal to or less than the design value for optimal air heater performance. UncertaintyAnalysis The uncertainty in the calculation of the TCFGGT by Equation 15 was estimatedin support of a study ofair heater performance conductedat the MillikenStation ofNewYorkStateElectric & Gas Corporation (IWSEG)in1995 and 1996. Details of the air heaterperformanceand of the uncertainty analysis can be found in the referencedreports.23The uncertainty in the result of a calculation can normally be estimateddirectly by partial differentiation of Equation 15 with respectto each parameter, To accuratelyevaluatethe uncertainty with an explicit equation, the equation must not be significantly nonlinear. In the caseof Equation 15, the air leak introduces a significant non-linearity which invalidates this approach. Thus, a mathematicalapproximation was required to evaluate me uncertainty in the TCFGOT. Errors in measurementsare of two types: bias errors and random errors. Biases are associated withthe measuring equipment or procedureand cannot be minimized by repeatmeasurements. However, the TCFGOT temperaturecorrectionsconsist of differences and ratios.This tends to compensatefor bias errors. Random errors are reducedby repeatmeasurements.The following derivation assumesonly one test and thus representsthe maximum estimatederror. The bias and random errors are propagatedseparatelyusing Taylor seriesexpansionsfor highly nonlinear equations: SEP099804:39PMCONSOLR&DLIBRRRY P .9/20 S bnv = where (16) Af -= Incremental changein the function fwith respectto xi, &i Af -= Incremental changein the function fwith respectto x j, &ni crX,= Error in parameter i, oXj = Error in parameterj,and f= Function shown above as Equation 15. This numerical approach of estimating the uncertainty in the TCFGOT is similar to the one that Carl James4uses for estimating the uncertainty in the d&gnof a cross-flow heat exchanger. Of interest is the fact that the uncertainty in the design of a heat exchangerestimatedby Jamesis much larger than the uncertainty in the estimateof the performance. For a numerical approach, Equation 16 must approximate the surface ofthefunction as a linear segmentparallel to the true functional relationship.Withindependentparameters,only the i=jterms of Equation 16 are non- zero, simplifyingthe Taylor seriesexpansion. However, if the terms are correlatable, that is, not independent, then the sum of the crossproducts in Equation 16 is not zero and theseterms must be included in the estimate. This is discussedfurther in the appendix. This expansion is used to evaluatethe bias and random error contributions separately. The bias and random errors are summedseparatelyto form the bias error statistic and the random error statistic, and then combined to estimatethe total uncertainty by: u=p+(r.s)]~ (17) where U= Uncertaintyinterval, B = Overall bias error statistic, S = Overallrandom error statistic,and t= AppropriateStudentstvalue.(For 95 % significance,t=2.0 fora reasonable sample size.) The parametervalues used for the estimation of the uncertainty of the TCFGOT and the bias and random errore associatedwiththem are shown below in Table I.The bias and random errors were estimatedby separateerror propagation calculations for standard,multipointsampling arrays in the inlet and outlet ducts of the air heater. Thesemultipoint sampleswere used to SEP09'9804:4BPMCONSOLR&DLIBRARY P .lW20 evaluate averagetemperaturesand compositionsin the ducts. The appendixpresentsa brief discussion of this witha more detailed discussionavailable in the project reports5 As discussed in the Appendix, examination of the derivation of thesesampleerrors suggeststhat for standard, multipointtraversesof utility-scale equipment,the bias and random errors shown in Table IIfor these averagetemperaturesand compositionsare typical. Table K Valueof Parameters and TheirAssociated Bias and Random Errors Parameter AirTemperature @ Inletand Ambient AirTemperature @ outlet Flue Gas Temperature @ Inlet Flue Gas Temperature @ Outlet Flue Gas Flow AirLeak UnitValueBias ErrorRandom Error F1001.000.15 OF6446.440.74 F6806.810.81 F2852.850.35 1,000 lb/h1579.820.72 percent60.050.77 These parametersare propagatedusing Equation 16. The bias and random errors are propagated separately and summed to form the B and S componentsof Equation 17. Equation 17 is then used to estimate the overall uncertainty interval. The followingexample shows the evaluation of one of the incremental changeterms required by Equation 16.To evaluate the bias error associatedwiththe air temperatureat the inlet: 1.The TCFGOT is calculated at the basetemperature,100 OF,plus three times the bias error. 2.Then the TCFGOT is calculated at 100 F minus three times the bias error. 3.Designating these two values of the TCFGOT asf,andfp, respectively,the contribution to the bias error of the TCFGGT for the inlet air temperatureis evaluatedby the following: !iEP099804:40PMCONSOLR&DLIBRQRY P.ii/20 q=fa-f, 6.0, =31231-313.17 6.1 =LO143 (18) Allother parametersare held constantat the values shown in Table I during this calculation. Equation 15 is used to calculate the TCFGCT.Since the parametershand4 were evaluated at three times the bias error, o,, greater and three times the bias error lower than the actual value of the temperature of the inlet air, the total delta is six timesa,. That is, the difference between f. sndfp is divided by six times the bias error. To be an accurateestimateof the error, the function equation&must be relatively hearover the range of the error.That is, iffisthe value of TCFGOT at an inlet air temperatureof100 F, then if, f,-fOsfO-fp (31291-312.74)=(312.74-313.17) 0.436 = 0.422 then the assumption of linearity and, in turn, the validityof the estimateis conflrmed. This calculation is repeatedfor the other parameterslisted in Table Iand the products summed as shown in Equation 15 to produce the resulting bias and random errors shown in Table II,This is the estimate of the uncertainty from Equation 17 in the determinationof the totallycorrected flue gas outlet temperature, or TCFGOT, for an air heater. The estimateof the error in the determination of the totally corrected flue gas temperatureis +4.75 *F for the specific conditions shown in Table I.As a percentage,-2%,this uncertainty can be applied to evaluation of other air heatn3. Table II Estimate of the Uncertaintyof the TotallyCorrected Flue Gas Temperature ParameterJI&-BiasRandom_Uncertaintv TCFGOTF4.570.66zk4.75 Conclusions The ASMEPTC 4.3 provides a standardizedmethod for evaluatingthe performanceof utilityair heaters. It provides a mathematically cotr&Ctmeansof evaluating the performancewhich aids in SEP099804:40PMCONSOLR&DLIBRRRY P.12/20 minimixingdisputes between suppliers and purchaserswhen guaranteeperformauce evaluations are conducted. In operating plants, it is generally impossible to establishdesign conditions to verify performance.To overcome this, the PTC specifiesthat the measuredflue gas outlet temperaturemust be correctedfor differences from design inlet air temperature,design inlet flue gas temperature,design X-ratio,and design flue gasrate.Once thesecorrections are determined, the totally corrected flue gas outlet temperaturecan be calculatedand comparedwiththe design outlet temperature. Calculation of the first two temperaturecorrectionsis explicitly defined by the ASMEcode. The determinationof the temperaturecorrections for differences from design X-ratioand design flue gas floware left to the supplier or purchaserto determine. Normally the manufacturer willsupply the purchaserwith design performancecurves or equations,but not withthose to calculate the temperaturecorrectionsspecified by the PTC.This paper provides a method for evaluating the remaining two temperaturecorrections using the performance curves.Should the manufactureralso provide proceduresfor calculating the specified PTC temperature corrections,the results can be checkedusing the proposed procedure. This was done for the Mill&enair heaterperformancetesting with good agreementfound between the two methods. As part of the Mill&enair heater test program, the uncertainty in the ASME PTC 4.3 equation for calculating the TCFGOT was determined. Becauseof the non-linearity of the final equation, numerical approximations wete used to determinethe differentials neededfor the propagation procedure. For the examplepresented,the estimateduncertainty is 4.75 F for the TCFGOT at a 95% confidence level.This showsthat the uncertainty in the codeprocedureis relatively small, about 2% of the design outlet temperatureas expressedin degreesFahrenheit. References I.AirHeaters-Supplementto Performance Test Codefor SteamGenerating Units, Pl-C 4.1; ASME/ANSIPTC 4.3 -1974; Reaffirmed 1991, The American Society of Mechanical Engineers,New York,1968. 2.McCoy, D.C.; Bilonick, R. A.; MillikenStation Heat Pipe AirHeater Performance Uncertainty Analysis, Report preparedby CONSOL Inc., R&Dfor New YorkState Electric &Gas Corporation, Binghamton, New York, June 1995. 3.Maskew, J. T.; MillikenStation Heat Pipe Air Heater PerformanceUncertainty Analysis of TotallvCorrected GasTemneratureLeavine the AirHeater, Report preparedby CONSOL Inc., R&Dfor New YorkStateElectric &Gas Corporation, Binghsmton, New York,April1996. 4.James,C. A.;Taylor, R. F.; Hedge, B. K.;The Application of Uncertainty Analysis to Cross-Flow Heat ExchangerPerformancePredictions; Heut Transfer Engineering; 16, 4; pp 50-61; 1995. 5.McCoy, D.C.; Heat Pipe Performance-- Final Report; Final report prepared by CONSOL Inc., R&Dfor New York State Electric &Gas Corporation, Binghamton, New York,August 1998. SEP0998 04:41P,,CCNSOLR&DCIBRFlRY APPENDXX Estimationof Uncertaintyin the IndividualParameters Requiredforthe Evaluation of the ASMlE PTC 4.3 TotallyCorrectedFlue Gas OutletTemperature The uncertainty analysesdiscussedin this paper are for the American Society ofMechanical Engineering (ASME) proceduresfor testing the performanceof air heatersand, specifically, for the equation to predict the totallycorrectedflue gasoutlet temperature(TCFGOT).The estimatesof bias errors and random errors for the individual parameterswere derived for the equipment and methodology used in obtaining the data required for a test program. This test program focused on evaluating the performanceof an air heaterrecently installed in the Milliken Station of New YorkState Eleotric &Gas. The methods followed in deriving the estimatesof the uncertainty of the individual parametersarepublished by ASME.Comprehensive discussion of all of the calculations is published e1sewhere.l MillikenStation Unit2 is a 150 MW, pulverized coal-fired boiler with twin, parallel air heaters. Each air heater heats both primary and secondaryair for half of the unit in separatesectionswith the flue gas mixed before and after the tirheater. The uncertainty analysispresentedbelow contains the results for both the primary and secondarysidesof the air heater. The design of the air heater was such.that all of the air leakageoccurred at sootblowerports.Airleaked from outside of the air heater into the flue gas heating the primary air.Leakageinto the side heating the secondaryair was insignificant and was ignored in the followingevaluation. Test Procedure The general test procedure followed in the determinationof the TCFGOT was the ASME Performance Test Code (PTC) PTC 4.1 and PTC 4.34. Individual parametersrequired by the PTC 4.3 were measuredfollowinggenerally acceptedmethods,normally U. S. Environmental Protection Agency (EPA) methods. For gas velocity, EPA Method 2 was used along withEPA Method 16. The gas composition was determinedgenerally followingthe proceduresof EPA Method 3?.Since the ASME procedurebasesthe flue gas and air flow rates on the coal feedrate andgas properties, rather than on the measuredgas and air velocities, the derivation of the errors of the individualparametersis complex.However, using the coal feedrate,from calibrated feeders, and gas compositions as a base createsa common bond between the air and flue gas flows.This createsa consistentbasis for the calculations. Background Error propagation is calculatedby Taylor Seriesexpansionof the resultant function.In general, ifr=f(xa x2,.., x,. . ., xJ,then the error statistics,S,,,,for either the bias error or the random error is calculated by where SEP099804:42PMCONSOLR&DLIBRFIRY P .14/20 afal -= Partial derivatives of jwith respectto Y, (or xi),and ax,c%cj cr,, , oxj= Error with respectto xi (or xj). whenthe parametersare independent,only the i=jterms are significant.For many of the parametersexamined in this work, the parameterswere aindependentand all of the terms in Equation Alwere evaluated. Note that using a single thermocoupleto measureall of the temperaturesin the traverse of a plane in a duct createsa dependencybetween these measurements. The bias error associatedwith tbe thermocoupleis the samefor all points.Thus, it is dependent. To illustrate the calculation complexity for the estimateof the errors of the individualparan~eters,a step-by-stepcalculation of the estimateof the uncertainty for a weight averagetemperatureof a gas is shown below.The averageis basedon a traverseof an inlet (or outlet) duct.For the details of the estimation of the uncertaintiesof other parameters,refer to the final Millikenproject report8Only the errors and uncertainty for theseother evaluations are presentedhere. TemperatureTraverse UncertaintyCalculation The weightaveragetemperatureof a gas flowingin a duct is basedon a flowweighted average of the temperaturesobtained from a standardtraverseof the duct.That is, where tAiwT, T=%=j=l iA,wi ill Ai = Cross sectional area for point i,ft2, i = Traversepoint number, q= Temperahue measuredat point i, R, vi = Velocityin areaAi determinedat point i,fps, and pi= Fluid density in areaA;,lb / f?. (a The fluidvelocity is determinedby a Pitot tube measurement.The gas is assumedto behave ideallyand the velocity is constant over the entire cross-sectionalareaA,.The velocity is calculated by: I vi=85.49 cl:. Mi.T,2 1 1 P,,.Mi (A31 SW099804:42PMCONSOLR&DLIBRRRY P .15/20 CP,-PitottubeflowcoeffkzicnLdimensionless, 0,-Vclacityheadinareai,inchesW.C., P,,=Staticpressureinareai,inchesHg.absolute,and M,=GAS moleweightinCWS i.Ib/lb-mol. Similarly,the gas density is calculated: 0.04578. Mi .I,; Pi= T, (A41 Substituting the formulas for vi(Equation A3) and p,(Equation A4) into Equation A2 and simplifyingyields: Equation A5is partiallydifferentiated with respectto A,CP, dP,, M,, PI,, and T,, and the resulting partial summed as indicated in Equation Al.Equation A5 producessix sets of partial differentialequations. Ifthe denominatorof Equation A5 is set equal to Sum1 and the numerator equal to Sum2 to simplifythe resulting equations,thesepartial differentials are: Gg c~.(~..M,.P,i.7;)t.suml-cP. ,.(~~.~i).sum2(A6) -= 34 SumI dr,,= ~.(~.~,.~i.T)f.Suml-A, , Sum2 @I e, -= aw xsuml-c