BROWN ROBERTSON.pdf

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964 L. F. BROWN and B. A. ROBINSON H 6

reviews e.g., Coats and Redfern, 1963; Falconer and Schwars, 1984) describe many techni ques.All of these approaches depend upon a linear temperature ramp for their validity. There arehazards in these methods. When equations are linearized or regions empha sized, data are weightedimplicitly, and weighting of data containing error can distort result s and lead to incorrectconclusions (Kittrell, Hunter, and Watson, 1965). If the system is followed continuously, as inmass spectrometric analyses of carrier gases from gas-solid systems, this may not present adifficulty. If limited data are taken, the problem might be serious.Nonlinear regression uses information in the TPR spectrum to the fullest extent possible; areaction-rat e expression is assumed, and its parameters come from the regression. The varianceshould approximate that predicted from data accuracy, and no trends sh ould appear betweenpredicted and measured TPR curves. If these conditions are fulfilled, the assumed reaction-rateexpression is consistent with the da+a and the parameters are evaluated as well as possible.Without numerical testing, it is unknown whe ther Kittrell's problem is present in the systems weare studying. Linear an d nonlinear regressions are used in the numerical simulations presentedhere. and results are compared to see their relative worth for our systems.

NUMERICAL HODELLNGThe Fundamental Equation for TPR Experiments.Both TPR and isothermal kinetics experiments are modeled numerically. For any nonisothermal ex-periment with an n-th order reaction and an Arrhenius temperature depandance,

Jcrs A ITremElnTt .co c To

A constant temperature- rise rate and changing the independent variable from t to T givef(cr) 5 - t _I Tre-E'RT dT ,

1)

in which 8 t (dT/dt). In this equation, the function f(Cr) = Xn (Cr/Co) for a first-orderreaction, and equals cc; - c;-n )/(1-n) for a non-first-orde r reaction. Equation (2) is usedbelow in the two overall methods of solution to study the TPR kinetics experiment. However, notethat the nonlinear least-squares technique does not require a linear temperature rise. sincenumeric81 integration for any time-temperatu re history is performed easily.

Nonlinear Regression.Both nonlinear and linear reg ression were used in different aspects of the study. A first-orderand a second-order expression were chosen for modeling a constant-den sity A + productsreaction. The conditions for both TPR and isothermal experiment simulations for the nonlinearregression tests are Listed in Table 1. Exact conversions which should have been observed forthese conditions were calculated from Eqn. (2). To simulate random measu rement error in the eon-centration data, we set Gaussian distributions of errors around t he -exact+' concentratio ns.Error uas assumed to be a fraction of the exact concentration . and the absolute errors declinedas the concentration decreased. Standard deviations ranged from 0.01 to 0.05.

TABLE 1 Experimental Conditions Assumed for Numerical Mode lingRate expressions: =A = - Ae -E/RTC A =A - - Ae'E/RTCiActivation energies: 104.6 kJ/mol 104.6 kJ/molPre-exponential factors: A = 2 x 1010 min-L

ACo- 2 x 1010 min-1

TPR Initial temperature: 273 KRate of temperature rise: 1 K/minexperiments Final temperature: 573 KIsothermal Temperature levels: 438, 453, AND 468 KexperimentsNumber of data points in any experiment series: 9Standard deviations of random error in data: 0.01 - 0.05


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H 6 Kinetics analysis of liquid phase reactions 965

Nonlinear regression was used to disc*iminate amdng different rate exprassians to see which wasmost appropriate for the data, and to determine activation energies and pre-exponential factorsfor first and second-order reactions. 1n the nonlinear reqression, a search was made in whichthe activation energy and the pre-exponential factor of the rate constant wets varied in erder tominimiee the sum of the squared differences between the concentrations predicted by using theseparameters and those in the synthetic data.

Linear Regression.

Linear reqression was used to analyee effects on parameter precision of error level, sample fre-quency, tamparatura-rise rata, and ranges of kinetics parameters. Simplification of Ecm. (2) canbe made using a series expansion of the integral (Coatas and Redfern, 1964). For large values ofE/RT, the equation becomes

(3)

Plotting In I-f(c,)/T:] versus l/Tr should yield a straight line, since .&I (AR,'pE)[l-_(2RT/E)]is essentially constant over any reasonable temperature range for our systems. From this plot,the activation energy may be obtained from the linear least-squares slope of -E/R. By using thedata point at or nearest c/co - 0.5, rather than the intercept, the temperature and concantra-tion can be used in Eqn. (31 to obtain the best value of A (Sohn and Kim, 1980).

CHEMICAL EXPERIMENTSThe first reaction we used to test the TPR technique experimentally was the alkaline hydrolysisof ethyl acetate:

CH3Cooc2H5 + on- + cn3coo- + C2H5OH . (4)Experimental procedures are described in detafl in Robinson (7985). A high pressure, high tam-perature stirred autoclave reactor was used for the kinetics experiments. Ethyl acetate andethanol were analyzed using gas chromatography. For the isothermal experiments, the temperaturewas achieved with a heating jacket which brought the reactor up to the operating point, and thenwas controlled to within fZ*C. In the TPR experiments, the reactor was adjusted to its maximumhaatup rate, which was acceptable for our purposes. A linear temperature ramp was not achieved,but this was not necessary since nonlinear least-squares regression was used to analyze the data.The rate law is first order in ethyl acetate (EA) and hydroxide concentrations (Kirby, 1972):

%A = -k2[OH-][EA] -Although a second-order reaction, we are interested in dilute aqueous reactions, so EoH 1 remainsessentially constant in our systems. With a rate constant equal to k2[OU-1, a first-order ex-pression could characterize the reaction rate. For the sodium bicarbonate-acetic acid buffersystem used, [OH-] is a strong function of temperature due to the temperature dependence of thewater and carbonic acid dissociation constants (Robinson, 1985). Since this temperature functionapproximately follows an Arrhenius temperature behavior, the product k2EOR-] retains thecharacter of a pseudo-first order rate constant and the reaction may be treated with the methodssuggested in this paper. When comparinq the isothermal and TPR experiments, where the rate con-stants were measured at different values of [OH-], the second-order rate constant is used.

NDWXRICAL MODELING RESULTSNonlinear Least-Squares Raqression.Discrimination among rate expressions. Table 2 gives results of discriminating between reaction-rate exnressions bv nonlinear least-squares regression. It presents four cases using syntheticfirst-order reaction data -- two with error containing standard deviations of 0.01 and two witherror containing standard deviations of 0.05. Wodals assuming both a first-order and a second-order reaction were used to predict the data, and the sums of the squared differences between thepredicted values and the data were evaluated. Optimal values of activation energy and pre-exponential factor were calculated and used for the predictions. i.e., those values which gavethe minimum possible sum of squarad differences (minimum sum of squared residuals).Table 2 confirms the truism that aa data accuracy improves, it is easier to discriminate amongdifferent models. In the TPR experiments whose data have an error standard daviatlon of 0.01, afirst-order reaction model fits both sets of data much better than does a second-order model.The minimum sums of squared residuals for the second-order reaction model are 26 and 47 times amgreat as the sums for the first-order model. This is proper, since a first-order reaction wasused to generate the data. When the data's error standard deviation is 0.05, the differences insums of squared differences are much less, differing by factors of only 2.3 and q.4.


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9 L.F. BROWN and B. A. ROBINSON H-6

ErrOrImposedon Data0.01

0.01

0.01

0.01

0.05

0.05

0.05

0.05

TABLE 2 D%scriminating Between Rate ExpressionsType of Assumed Calculated CalculatedExperiment Reaction Activation Pre-Exponential

Order Energy,kJ/mol Factor, min-lTPR ? 106.5 3.245 x lOlo

2 157.3 3.658 x 1Ol6TPR 1 102.5 1.151 x 1010

2 150.8 6.685 x 1015Isothermal 1 104.2 1.813 x lOlo

2 104.1 2.663 x lOloIsothermal 1 105.3 2.379 x 1010

2 105.6 3.949 x 1010TPR 1 113.0 1.811 x 1011

2 168.9 7,234 Y 101'TPR 1 107.7 4.661 x lOlo

2 162.9 1.674 x 1017Isothermal 1 103.7 1.547 x 1010

2 103.1 1.926 x 10Isothermal 1 108.8 6.149 x 10

2 109.6 1.162 x 10

For isothermal experiments, differences in the sums of squared residuals are significantly

sum ofSquared

Residuals2.643 x lo-*6.863 x lO-31.886 x 10-48.870 x 1O-31.826 x lO-42.138 x lo-'2.900 x 10-52.004 x lo-'2.844 x 1O-36.615 x lO-38.810 x 10-31.258 x lO-22.245 x lO-32.227 x lo-'4.789 x 10-32.713 x lo-'

greakcr . FaCtOrS are 117 mvZi 691 for two cases with an error standard deviation of 0.01 in thedata. and 9.9 and 5.6 for two cases with an error standard deviation of 0.05. Thus when isother-mal experiments were used. in all cases it was possible to distinguish which of the two modelsfit the data better. For the TPR experiments, it miqht not be possible to distinguish thewlity of fit of the two models in the higher-error situation.Sums of the squared residuals in Table 2 should be of the order 0.0004 when the error standarddeviation is 0.01. and of the order 0.01 when the standard deviation is 0.05. The valuea werecalculated from the generated data's standard deviation from the exact canversionsr observedsums of residuals should be less than this because optimal values ofmodels closer to the generated data than to the exact values.residuals reported in Table 2 are consistent with this.

the parameters would createMinimum sums of squared

Similar results occurred when a second-order reaction generated data, and first-order and second-order models predicted results. Here, of course, second-order models gave the superior fit.

TABLE 3 Parameter PrecisionFrom TPR Experiments From

Std. Dav. of Error in Data 0.01 0.05t?o* of cases 100 100True B, kJ/mol 104.6 104.6/Mean of Calc. E, k.l/mol 104.6 106.6Std. Dev. in Calc. E, kJ/wl 1.59 6.90

Isothermal Experiments0.01 0.05100 100

104.6 104.6104.6 104.70.71 3.77

True k473, min -1 0.05608 0.05608 0.05608 0.05608Mean of Calc. k473, min-' 0.05603 0.05690 0.05610 0.05619Std. mv. in Calc. k473, min-l 0.00070 0.00326 0.00055 0.00276

Parameter estimation. Table 3 presents the results of parameter cstimtion studies for the TPRand isothermal-e xperiment approaches. In contraat to the rate-expression discrimination c-parison. the two techniques qave almost equivalent results for the precision of parameterestimation. The standard deviations in the calculated values of the activation energy from theTPR experiments were at most about twice those from the isothermal experiments. and somewhatbelow that at the higher error levels. Standard deviatiene in the calculated rate constants fromthe TPR experiment.8 ranged only from 18 to 27 percent higher than those from the steady-stateexperiments. The rate constants rom both methods are quite close to the true values7 this saysthat TPR can be used effectively to make accurate kinetics measurements in a single experiment.Again. the use of data generated by a second-order reaction gave snalogoue results.


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H 6 Kinetics analysis of liqui d phase reactions 967

Linear Least-sqtlarcs Reqreesion.

TABLE 4 comparison of Nonlinear and Linear Regression ResultsError Type of Calculated calculated calculatedImposed Regression Activation Pre-Exponential k -1on Data Rnergy,kJ/mcl FaCtOr, min-l 4,3. n n0.01 nonlinear 106.5 3.245 x lOlo 0.05667linear 104.1 1.767 x lOlo 0.056300.01 nonlinear 102.5 1.151 x 1010 0.05529linear 103.0 1.302 x lOlo 0.055480.05 nonlinear 113.0 1.811 x 1011 0.0 5926

linear 111.8 1.383 x 101 0.06261

0.05 nonlinear 107.7 4.661 x 10 0.059601 inear 100.2 7.443 x to9 0.06427

True values 104.6 2.000 x 1010 0.05608

Comparison of nonlinear ana linear regression resultr. Table 4 presents four results from non-linear n linear least-squares reqressicns on the ~lame apta. In all oa*e*, results are close,and the difficulty noted by Kittrell and mentioned earlier is not present here. For the twocasea where the error standard deviation was 0.01, the activation energies were within 2.2 and0.4 percent of each other. When the standard deviation was 0.05, energies were within 1.1 and8.7 percent. Rate constants at 473 K calculated using parametere from regressions were within0.7 and 0.4 percent when the standard deviation was 0.01, ana within 5.6 and 7.8 percent when thestandard deviation wae 0.05. The higher the error level, the more results from the two ap-proaches diverged. Nevertheless, resulte from the two typee of regression were always close.Nonlinear regression carried cut on TPR data treats a responee surface with a Steep valley. Ifstarting values are outside the valley, the Levenberg-Ma rquordt algorithm cm cause divergence,mak+ng nonlinear rsgreesicn nmre difficult to use. We used linear regression when carrying outevaluations which reguired Large numbers of synthetic runs. since result8 from the nonlinear andlinear reqressione are quite close. conclusions from linear reqreesion calculat ions would bequalitatively correct, and quantitative ly change only slightly if nonlinear reqreaeion were used.The next paragraphs present the effect on calculated reaction parameters of data error, samplingfrequency, temperatureria e rate, pre-exponentia l factor, and activation energy. Uany eimula-tions were performed, e-q., each mean and standard deviation of the activation energy c ame from1000 simulated TPR runs- Table 5 lists conditions for each case. ana Table 6 qives means anddeviations of the activation energy and the first-order rate constants at several temperatures .

case1

23456789

101112

cl

0.010.020.030.040.050.070.050.050.050.050.05O-05

TABLE 5 Values Used for Parameter-Eval uation TeatsA, min -1 R, kJ/mcl Cmin CIMX 19, K/min At., min2 x 1010 104.6 0.03 0.95 1 102 x 1 1 104.6 0.03 0.95 1 102 x 1010 104.6 0.03 0.95 1 102 x 1010 104.6 0.03 0.95 1 102 x 1010 104.6 0.03 0.95 1 102 x 1010 104.6 0.03 0.95 1 102 x 1019 104.6 0.03 0.95 1 52 x 1019 104.6 0.03 0.95 1 22 x 1010 104.6 0.03 0.9 0.1 1002 x 1010 104.6 0.03 0.9 1 102 x 1010 104.6 0.03 0.9 10 11.7 x 101' 167.4 0.03 0.9 1 5

Error level. Caaee l-6 s ummarrze the results of varying the standard deviation of error. As ex-pected, the etandara deviation of activation energies in creased wit h increasing deviation in thedata error (Table 6). A systematic deviation occurred in mean activation energy, where the cal-culated E wae generally Icwer than the true value. the difference increasing with increaeinq c.It is ascribed to using linear instead of nonlinear reqresaion.


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968 L. F. BROWN and B. A. ROBINSON H-6

case

123456789

101112

TABLE 6 Statistical Results of Parameter-Evaluation TestsAverage E. Std. Dev. of E. Rate Constants, min-l

k.I/mol

(104.6)104.2101..7100.498.797.595.0

100.0102.999.299.6

100.0160.3

(167.4)

kJ/slOl

2.935.567.669.83

11.714.88.364.9010.19.049.62

14.0

423 K 473 KAvg. Std. Dev. Avg. Std. Dev.

(2.42~10-~)2.47~10-~2.60~10~~2.68~10-~2.78~10-~2.87~10-~3.04x10-32.67~10-~2.49x10-32.40~10-~2.75x10-33.22~10-~4.61x10-'

(3.65~10-~)

1.31x10-42.80x10-43.86x10-*5.17x10-46.43x10-48.65x10-'*4.47x10-42.80~10-~1.86x10-44.96x1O-41.31x10-41.44x10-4

(5.61~10-~)5.63~10-~5.50x10-25.46x10-25.38x10-~5.32~10-~5.22~10-~5.39x10-25.46x10-24.99x10-25.46~10-~6.04x10-~5.37~10-~

(5.58~10-~)

2.36~10-~4.05XlQ_35.94x10-37.32~10-~8.78x10-3l.O8xlO-*6.72x10-24.62~10-~1.61~10-~7.04x10-36.75~10-~6.10x10-3

Numbers in parentheses are true values of the rate coustaut. Those at the top referto cases 1-11. Those at the bottom refer to Case 12 only.1000 simulations used in each case.

The error level in a calculated rate constant may be significantly less than that in the activa-tion energy or pre-exponential factor. A compensation effect exists; when a low value of E iscalculated, it is offset by a low value of A, and the two low values tend to compensate for oneanothet when calculating rate constants in the temperature range where the data were taken (Table6). This effect has bee observed before, and methods have been proposed to treat differentaspects of it (e.g., Himmelbla , 1970).Sample frequency. Sample frequency has a marked effect on the accuracy of the results, as seeby comparing; Cases 5. 7, and 8. Huch more accurate values of E and the calculated rate con-s-taut* are obtained when -the time between taking samples is cut from 10 to 5 and then to 2minutes. For example, changing the sampling frequency from 1 sample/l0 min to 1 sample/l millcan increase the accuracy in E more than a decrease in the etror standard deviation from 0.05to 0.02. Sample frequency is the most easily changed of the variables we evaluate here. so thisfinding is significant.Temperature ramp parameter 8. Cases 9-11 show that the accuracy did not chauge significantlyover two orders of magnitude in 8. These calculations assumed that the same number of datapoints fell within the range of concentration detectability Cmin to C,,. To achieve this ex-perimentally or in a simulation. sampling frequency must be changed by the ssme factor as 8.Different kinetics parameters. Results of equal accuracy can be obtained for different values ofactivation energy and pte-exponential factor provided the correct sampling frequency is chosen.The higher the activation energy, the more frequently samples must be taken to achieve the sameaccuracy, all else being equal (cases 10 and 12). In TPR, s reaction with a higher activationenergy will proceed from a concentration of 1 to 0 over a smaller temperature range.

EXPERIMENTAL RESULTSTo test TPR experimentalLy for acquiring liquid-phase kinetics parameters, t o preliminary ethyl-acetate hydrolyses were carried out. The purpose was to check the capabilities of the apparatusand to see if numerical techniques we had available could analyze the dats satisfactorily. Theexperiments were successful in these goals, and the data displayed interesting properties.In the temperatures where concentrations changed within limits of detectable differences from 1.0and 0.0. six data points were obtained in one run, and five in the second. The temperature-riserates were ot constant, and nonlinear regression was used to calculate optimal values of activa-tion energy ahd pre-exponential factor- The reaction was carried out under pressure, whichallowed operation above the normal bailing point of the solution. If the maximum operating tem-perature had been LOO-C, solution of Eqn. (3) indicates that a 8 of the order of 1O-4 K/minwould have been necessary and a TPR run would have needed over a month. At our conditions, a rrequired about four hours.


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Kinetics analysis of l iquid phase r eactionsThe activation energies and preexponential f ctors obtained from the regressions are presentedin Table 7. Also reported are kinetics parameters from isothermal experiments carried out byRobinson (19851, and parameter values calculated from the correlation of David and Villarmaux(t9SO), baaed on previous ethyl-acetate hydrolyai8 studies.

SourceTABLE 7 Reeulta of TPR and Isothermal Experiments

Activation Pr-eesponential Calc. rate Calc. rateenergy E, factor A, constant kk.J/mol dm3/(mo1)(s) at 430 I

con8tantbkdm3/~rnOl)(8~

at 400 It,dm3/SrnOl~(S)

TPR expt. En-72 70.0 1.44 x 1010 45.2 10.4TPR cxpt. EA-13 52.7 1.12 x 108 44.3 14.7Isothe-1 expte, 40.9 3.05 x 106 32.8 f3.9David-Villermaux 49.0 4.36 8 to7 48.6 17.6correlationa. Approximate mid-point temperature of TPR experiments.b. Approximate mid-point temperature of isothermal experiment

The observed differences in activation energy for the TPR experiments is almost certainly anotherexhibition of the compensation effect oauaed by data error. Given the error level, there slereinsufficient numbers of samples in these preliminary experiments to give reliable values of IX-tivation energy and pre-exponential factor. It is encouraging that calculated fate constants inthe critical temperature range of the TPR experiment chow such good agreement with those calou-lated from previous studies. The results of testing the effect of sample frequency on accuracyof results indicates that good values of reaction parameters can be obtained using this techniqueat the level of data error we now have. It appear8 almost certain that when TPR experiment8 areproperly carried out and the proper amount of data are taken, they will qivc reliable values ofthe rate parameters.

DISCUSSION AND COUCLUSIONSBoth numerical simulations and experimental test8 show that T PR is a practical method for deter-mining reaction-rate expressions and kinetics parameters (A and E) for liquid-phase reactions. Aseries of isothermal experiment8 may be slightly more accurate for a qiven number of data points,but the virtues of experimental epeed and simplicity should often make TPR the preferred method.Operation of the experimental system under pressure can extend the temperature range betweenfreezing and boiling and make.it possible to obtain TPR spectra within reasonable time periods.Nonlinear regression should be used for the analysis to obtain the mOBt accurate result, and alsoto eliminate the restrictive experimental reguirement of having to produce a constant rate oftemperature rise. For example, the natural temperature rise of our reactor could be used for ourTPR experiment8 without having to implement a sophisticated temperature control scheme to achieve8 linear ramp temperature increase. Nonlinear regression also.makeS it possible to perform TPRfor reactions with larger heat generation.Linearized equations are valuable in preliminary attempts to find the proper reaction-rateexpression. They also can be used in selecting starting values of the expression's parameters inthe nonlinear regression.The experimental variable with the most marked effect on the TPR technique ie the number ofsamples taken in the region where the reactant concentration is measurably below 1.0 andmeasurably above 0.0. This is determined by the sampling frequency. A large'increase in ac-curacy of the kinetic8 parameters and calculated rate constant8 may be obtained simply byanalyzing more samplea.The insensitivity of results to the temperature-ramp parameter 6 is encouraginq. Any achiev-able temperature-rise rate between 0.1 and 10 K/min appears acceptable, and even with anonlinear temperature rise we would not expect p to fall outside this range.The compensation effect, noticed by other investigators when analysins isothermal data, was alsoobserved in TPR. Techniques to minimioe this remain to be d@velaped for TPR.

ACKNOWLEDGMENTSThis work we8 carried out under the auepicss of the U.S. Department of Energy, and the govern-ments of zapan and the Federal Republic of Germany. Steve A. Birdsell's disousaions and graphswere very helpful, a8 were discussions with the University of Colorado's Professor David Clough.


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ACCmaxCminC o=*Eklk2nRtAteTT0TB=Q

L. F.BROWN and B. A. ROBINSONNOMENCLATURE

Pre-exponential factor of rate constant. s-1Concentration, mol/m3Maximum reactor concentration included in interpreted samples. mol/m3Minimum raactor concentration included in interpreted samples, mol/m3Concentration of reactant in reactor at time t 5 0, mol/m3Concentration of reactant in reactor at time t, mo1/m3Activation energy in rate constant, J/nwlFirst-order reaction rate conetant, s-lsecond-order reaction-rate constant, m'/(mol)(e)Order of reaction. dimensionlessUniversal gae constant, .Y/fmol)IX)Time, sInterval betken taking TPR eamples for analysis, eTemperature, KTemperature in reactor at time t = 0, KTemperature in reactor at time t, XRate of reactor temperature rise, K/sStandard deviation of error in relative concentrationa, dimensionless

REFERENCES

H 6

Coats, A. W., and J. P. Redfern (1963). Thermogravimatric Analysis: A Review. Analyst, 9, 906-924.

Coats, A. W., and J. P. Redfern (1964). Kinetic Parameters from Thermogravimetric Data. II. J.-Polym. Sci., Part B, 2, 917-920.Dankiericz, J., and K. wieczorek-Ciurowa (1979). Kinetic Study of the The-l Dehydration of

Syngenite K2Ca(S04)2.H20 Under Non-Isothermal Conditions. J. Thermal Anal., z, 11-24.David, R., and J. Vil\lermaux (1980). comlnents onr A nixing Model for a mntinuous Flow Stirred

Tank Reactor. Ind. Eng. Chem. Fundam., 19, 326-327.FalcOner, J. L., and J. A. Schwarz (1983). Temperature-Programmed Desorption and Reaction:

Applicationa to Supported Catalysts. Catal. Rev.-Sci. Eng., 25. 147-227.Himmelblau, D. M. (1970). Process Anslyeis by Statistical Methods, pp. 194-196. Wiley, New York.Kirby. A. J. (1912). Hydrolyeis and Formation of Ester6 of Organic Acids. In Comprehsnrive

Chemical Kinetics (C. A. Bamford and C. F. 32. Tipper, eds.), pp. 57-207. Elrevirr, New York.Kittrell, J. R., W. 0. Hunter, and C. C. Watson (1965). Nonlinear Least Squares Analysis of

Catalytic Rate Models. AIChE J., 11, 1051-1057.Koch, E. (1977). Non-Isothermal Reaction Analysis, Chap. 3, Non-Isothermal Reaction Kinetics of

Elemantury Reactions. pp. 90-126. Academic Prees, New York.Robinson, B. A. (1985). Nonreactive and Chemically Reactive Tracers: Theory and Applications.

Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA.Robinson, B. A.. J. W. Tester, and L. F. Brown (1984). Using Chemically Reactive Tracers to

Determine Temperature Charaateristics of Geothermal Reservoirs. bans. Geothermal ReSOufChSCouncil, 8_, 337-341.Sohn, H. Y., and S. K. KLm (1980). Intrinsic Kinetics of the Reaction between Oxyqen and

Carbonaceous Residue in Retorted Oil Shale. Ind. hg. Chem. Process yes. rev.. 3 550-555.

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