Hydroformylation using dinuclear rhodium catalyst precursors

Hydroformylation using dinuclear rhodium catalyst precursors

267 Journal of Molecular Catalysis, 30 (1985) 267 - 280 HYDROFORMYLATION USING DINUCLEAR RHODIUM CATALYST PRECURSORS. PART 1. NUMERICAL METHODS FOR ...

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267

Journal of Molecular Catalysis, 30 (1985) 267 - 280

HYDROFORMYLATION USING DINUCLEAR RHODIUM CATALYST PRECURSORS. PART 1. NUMERICAL METHODS FOR THE DETERMINATION OF THE KINETIC PARAMETERS T. G. SOUTHERN Alberta Research (Canada)

Council,

(Received February 28,1984;

1901

Fifth

Street,

P.O. Box

429,

Nisku, Alto. TOC 2GO

accepted September 19,1984)

Summary

The hydroformylation of 1-hexene using [ Rh(SBut )(CO)P(OMe)Jz as the catalyst precursor selectively yields heptanal and 2-methylhexanal. The consumption of 1-hexene was determined by monitoring the residual pressure of the gaseous reagents in a constant volume reactor. In order to gain further insights into the mechanism, the data have been treated by various numerical techniques to obtain the rate of consumption of l-hexene. The factors influencing the numerical differentiation, including the choice of the polynomial expression used, are discussed. The results of the data analysis indicate first-order dependence on the substrate, the rate equation for the substrate being written as d[aldehyde] /dt = k [catalystlP [ 1-hexene], with p having a value which can only be reliably determined once the nature of, and the pathway to, the various rhodium intermediates are known.

Introduction

The homogeneous hydroformylation of alkenes using rhodium catalysts is a reaction of current industrial importance [l - 31. In the last few years, several major plants have come on stream using a mononuclear rhodium catalyst precursor. However, this complex, RhHCOLs (L = unidentate two-electron ligand) is also an extremely efficient hydrogenation catalyst, so that the selectivity has to be maintained by careful control of the reaction conditions and the relative concentrations of all components. As a potential means of improving the selectivity and the resistance of the rhodium species to poisoning by sulphur compounds [3], an investigation of the hydroformylation catalytic activity of sulphur-containing compounds has been proposed [ 41. For the hydroformylation of 1-hexene under relatively mild conditions (5 bar pressure and 80 “C), [ Rh(SBut )(CO)P(OMe),] 2 is known to be a 0304-5102/85/$3.30

@ Elsevier Sequoia/Printed in The Netherlands

268

catalyst precursor, forming only heptanal and 2-methylhexanal as products [4]. Very little is known about the nature of the active species or about the mechanism of this reaction. In order to study the kinetics of this reaction, a reactor system was designed so that the concentration of the substrate (1-hexene) could be monitored continuously. From’ a knowledge that the only reaction products are heptanal and 2-methylhexanal, the reaction stoichiometry becomes CHs(CH&CH=CH,

+ CO + Hz -%

CHs(CH&CHO

(1)

CH3(CH2)sCH=CH2

+ CO + Hz z

CH,(CH&CHCH,

(2)

;HO Hence, for every mole of hydrogen or carbon monoxide consumed, 1 mol of substrate is consumed, and so by monitoring the consumption of either gas or the total consumption, the consumption of the substrate or the total concentration of the products may be monitored. Such reactor systems have been described elsewhere [ 51. To interpret the data obtained so as to obtain the maximum amount of information concerning the mechanism, the rate constants or the orders with respect to the various reactants, the rate of consumption of the substrate must be determined. For this reason, a study of the various methods of obtaining this rate data numerically from the consumption data has been undertaken, and the r,esults of this study are reported herein.

Experimental All solvents were dried and freed of dissolved oxygen prior to use. Toluene, hexane, t-butylmercaptan, butyl-lithium and trimethyl phosphite were of commercial origin and were used without additional purification. The concentration of the butyl-lithium was determined prior to use. Carbon monoxide and hydrogen gases were obtained commercially and were of standard purity. l-Hexene was obtained from Aldrich and was passed over a column of activated alumina prior to being degassed and used. [ Rh(SBut )(CO)P(OMe),], (1)was prepared by the method of KaIck et al. [4,6]. The reactor system used [5] is shown schematically in Fig. 1 and consists of a 0.25 1 autoclave (A) with a jacketed heating system and a stirrer driven through a magnetic coupling (S), and was capable of being fed with the reactant gases at constant pressure via a shut-off and pressure regulating valve (Vi) from the reservoir (R) containing a 1:l mixture of hydrogen and carbon monoxide. The constant pressure in the reactor was measured by the pressure sensor Pz and the temperature was measured by a platinium resistahce thermometer (T). The pressure in the gas reservoir was measured

269

Fig. 3.. Schematic *diagram of the experimental unit used. A F autoclave, B = thermostatic aath, CR = chart recorder, MC = microcomputer, P1 and P2 = pressure sensors, R = gas reservoir containing 1 :l mixture of CO/Hz, S = stirrer system driving through a magnetic coupling, T = platibum resistance thermometer, VI, V2, and V3 = valves.

using an electronic pressure sensor (P,) which was monitored at regular time intervals by the microcomputer (MC) and recorded continuously by 3 chart recorder. The heating of the reactor was carried out by a thermostatic water bath (B). The reactor was charged under vacuum from a Schlenk tube with a solution of the catalyst precursor (1)(0.27 mmol) and the substrate, 1-hexene, (80.0 mmol) in toluene (30.0 ml). The Schlenk tube was washed with an additional 20.0 ml of toluene which was also added to the reactor, and subsequently heated to the preset value (80.0 “C) by opening valve VZ and closing valve Vs. Once the temperature had stabilized, the shut-off valve VI was opened permitting the gas mixture to enter the reactor at the predetermined pressure, while simultaneously starting the recording system (microcomputer and chart recorder). The microcompu~r monitored the pressure in the gas reservoir at 12.0 s intervals and stored the values both in memory and on the cartridge tape unit. These values were used to determine when the reaction had ceased, and hence when to automatic~ly cease measuring the dab. Typical experimental results are shown in Fig. 2. The stoichiometry of the reaction is such that 1 mole of substrate reacts with 1 mole of hydrogen and 1 mole of carbon monoxide to form 1 mole of products. Assuming the gases behave in a manner close to ideal and that the partial pressure of the mixture of 1-hexene and products does not change considerably, the pressure in the reservoir at any instant can

270

101 0

50 Time

Fig. 2. Pressure

in gas reservoir

100 (min.)

as a function

be related to the concentration formed (Y) by the relationship PV/BRT = y

of time.

of the substrate

or of the total

products (3)

where V is the volume of the reservoir (0.5 l), P is the pressure change in the gas reservoir, R is the gas constant, T is the temperature of the reservoir in K and y is the number of moles of substrate used. The sensitivity of the measuring system is governed by the volume of the gas reservoir and the sensitivity of the pressure sensor PI, and for the system used was in the range of 0.05 mmol of substrate.

Numerical differentiation The kinetics of the system can be described by a set of differential equations and their boundary conditions which can be best obtained by using approximate numerical methods. For the midpoint of a given time interval ti to ti + 1, the simplest approximation technique available is to define the rate as dY -N-z dt

AY At

Yi+ I -Yi ti+l-ti

Since all experimental values, yi, contain a certain amount of random or systematic error due to both experimental design and to the limitations oi the measuring systems coupled with the step size of the analog to digital converter, the results obtained by this technique could have large unpredictable errors. A method which has received some attention in the past for both reducing the random noise in the data and to obtain various derivatives

271 is that

described by Savitsky and Gulay [7]. The basis of this technique is to find some expression for which the derivative can be calculated by an analytical expression, and which closely approximates the curve in the interval under study. Johnson [8] describes such a procedure based on the use of a simple power series y =

a0

+

(qd)

+ (a*x2) + &x3) + (a+x4)+ . . . (c&J”)

where least-squares methods are applied to determine the values of the coefficients Ui. The order n for the polynomial is chosen such that for the given interval of 2m + 1 points the residual is a minimum. That is, the polynomial chosen should be a good fit to the curve under consideration within the given data treatment interval. Applying this criterion to the data set under consideration here gives a minimum in the residual for a fourth-order series using a seven point data treatment interval. Defining the data treatment interval as running from -m through zero to +m (-m < x < m), the least-squares matrix remains constant and hence its inverse need be calculated only once; thus, the approximate errors in the various coefficients ei may be readily obtained. This technique, however, has a major disadvantage in that although the data interval contains 2m + 1 points, there are 2m points for which no derivative can be obtained accurately, and of these 2m points, m occur at the immediate beginning of the data set, which is where information concerning the transformation of the precursor into the active species is to be found. This problem can be overcome if the interval used to obtain the derivative is allowed to vary from three (using a second-order power series) up to the optimum value, as the point under consideration moves through the data set, and, subsequently, to decrease as the end of the data set is reached. This approach reduces the number of points for which no derivative can be obtained to simply two, of which one is defined by the boundary conditions to oe zero (dy/dt = 0 at t = 0). The results of using this approach to numerically differentiate the data are shown in Fig. 3 as a function of the concentration Jf the substrate, for data treatment intervals of 7, 13 and 19 points. It is obvious from this figure that in order to obtain a relatively smooth differential curve, a data treatment interval containing at least 19 points is required, which is very costly in terms of time for the microcomputer (HP-85). The results do suggest, however, that for substrate concentrations between 20 and 60 mmol, the rate of consumption of the substrate is linearly dependent on its concentration. That is, there is first-order dependence on the substrate and the results for the pseudo-first-order rate constant are given in Table 1 along with their corresponding data treatment intervals. To overcome the problems of using a large data treatment interval and the concomitant risk of masking or removing fine detail [7], the computer program has been rewritten replacing the power series expression by the integrated form of the rate equation for second-order kinetics.

272

+

4 4”

SO

[Substrate]

2 ..

O0

40 [Substrate]

[Substrate]

Fig. 3. Plot of d[substrate]/dt

(in mm01 min-‘) as a function of time (in min) with d[substrate]ldt calculated using the following power series: A = 7 point, B = 13 point and C = 19 point data treatment intervals.

The results obtained using this approach are shown in Fig. 4 and the results of the linear regression analysis performed on the apparent linear zone are given in Table 1. Since there exists in this polynomial a linear relationship between l/(a - y) and t, the computer time required to differentiate the data set is greatly reduced, and the problems associated with changing data treatment intervals are eliminated.

Discussion The values on the apparent

obtained from the linear linear zone are all grouped

regression analysis performed about 0.0401(l) min-‘. These

273 TABLE

1

Pseudo-first-order ment interval Data treatment interval 5 7 9

11 13 15 17 19

rate constants

and apparent

initial

rates

as a function

of data treat-

y = a0 + alx + azxz + 03x3 + a4x4

(a-y)-‘--aa-‘=kx

Rate constant (min-‘)

Initial rate (mm01 min-‘)

Rate constant (min-‘)

Initial rate (mm01 min-‘)

0.0405(6) 0.0405(4) 0.0404( 3) 0.0404( 2) 0.0403(2) 0.0403( 2) 0.0403( 1)

3.05( 2) 3.05(2) 3.05(l) 3.05( 1) 3.046( 9) 3.044( 7) 3.043(6)

0.0403( 2) 0.0403( 1) 0.04024( 8) 0.04023(7) 0.04021(6) 0.04019(6) 0.04016(6) 0.04013(6)

3.045( 7) 3.042( 5) 3.041(4) 3.040(3) 3.039( 3) 3.038(3) 3.036(3) 3.035(3)

results indicate that a five point data treatment interval using the secondorder kinetics polynomial is approximately equivalent to a 19 point data treatment interval using the power series polynomial, and that very little additional precision in the pseudo-first-order rate constant or the apparent initial rate can be obtained by exceeding a data treatment interval of 13 for the second-order kinetics polynomial. The effects of varying the data treatment interval are shown by the changes it causes in the maximum value for the derivative and on the time at which this maximum occurs. These results are summarized in Table 2 and clearly show that for a power series polynomial the maximum rate is approximately 2.51 mm01 min- ‘, which occurs between 3.2 and 3.4 min. However, for the second-order kinetics polynomial, the results show that the maximum value of the derivative apparently decreases and the time at which this maximum apparently occurs increases as the data treatment interval increases. Hence, the end-use of the differentiation must be considered before the choice of method, and its parameters, are chosen. To obtain information about the formation of the catalytic species, the power series with a data treatment interval of 19, or the second-order kinetic polynomial with a five point data treatment interval, should be used, whereas for obtaining information about the pseudo-linear zone, the second-order kinetics polynomial with a larger data treatment interval is preferable . The linear dependence of the rate of substrate consumption upon its concentration for cc. 35 min during which 90% of the products are formed implies that the differential equation for the conversion of the substrate may be written: dy/dt = k[clP[alkene]

(6)

274

40 [Substrate]

0 0

0

0

40 [Substrate]

80

40 [Substrate]

80

Fig. 4. Plot of d[substrate/dt (in mmol min-‘) as a function of time (in min) with d[ suhstrate]/dt calculated using the integrated form of the rate equation for second-order kinetics. A = 7 point, B = 13 point and C = 19 point data treatment intervals.

where [c] is the catalyst concentration at time t and p is the order with respect to this species. For the period when [c] is approximately constant, k[clP is 0.401(l) min-‘. This expression for the rate with respect to the substrate is very similar to d[aldehyde]/dt

= k[alkenelq[Rh]“[PPn]/[PPco]

(7)

where PP is used to denote the partial pressure. This form of the rate equation has been reported previously by various authors for the hydroformylation of various alkenes using catalysts based on rhodium carbonyl species [3, 9, lo]. However, for these species, p has been reported to have a value of 1 or less and q a value of either 0 or 1 depending on the substrate.

215

TABLE 2 Maximum rate data as a function of data treatment interval Data treatment interval 5. I 9

11 13 15 17 19

y = a0 + a,x + aZx2 + a3x3 + a4x4

(a - y)-1

Maximum rate (mm01 min-‘)

Time (min)

Maximum rate (mm01 min-‘)

Time (min)

2.59( 17) 2.55( 12) 2.53(9) 2.52(6) 2.51(5) 2.51(4) 2.52(3)

3.2 3.4 3.4 3.4 3.2 3.2 3.2

2.50( 17) 2.48( 17) 2.47( 15) 2.46( 12) 2.45( 11) 2.43(9) 2.42( 9) 2.41( 10)

3.4 3.6 4.0 4.2 4.6 4.8 4.8 4.8

-a-l

= kx

The presence of the linear zone is confirmed by an analysis of the curve of -ln[alkene] against time which is linear with slope 0.03933(l) to the linear zone in the min- ‘, within the time limits corresponding rate plot. Since [alkene] is known experimentally, and dy /dt can be calculated by either using a 19 point power series treatment or preferentially using the second-order kinetic polynomial, a value which is proportional to [clp can be calculated. Figure 5 shows such a curve obtained using 76.2 mmol as the initial concentration of the substrate*, and a data treatment interval containing 19 points with the second-order kinetics polynomial. Since each point in this curve has been obtained from the results of the numerical differentiation, the estimates of the errors in the first derivative can be used to calculate approximate errors in [cl”. The errors are approximately 0.002 throughout the region when [clp is virtually constant and increase to cu. 0.005 for values at the end of the data set. An analysis of the form of this curve irrespective of the value of p shows that the concentration of the catalyst is virtually constant from ca. t = 11 min to t = 44 min. This result is borne out by the plot of -ln[alkene] uersus t, which for the same time interval can be fitted by a straight line with an R* value of greater than 0.9999 +. After this period of constant *Although 80 mmol of substrate were added to the reactor, for the period under consideration here only the substrate in solution plays a role in the kinetics. The concentration of substrate can be calculated using the data in Table 1 and is co. 76.2 mmol. This figure, however, does not take into account the total time period since there are co. 0.4 min at the immediate beginning of the run for which no reliable consumption data are available due to the stabilization time of the pressure sensor Pr. Optimization of the parameter along with the various rate constants gives a value of ca. 76.2 mmol.

tR =

N~Xiyi - XXiyi

(NZlXi' - (ZXi)2)1'2(NEyi2

- (Zyi)2)1’2

with N being the number of data pairs.

276

.“”

sb

0

Time

1



-100

(min.)

Fig. 5. Dependence of ‘catalyst’ concentration units of min-‘.

on time. In this plot k[catalystlP

has the

concentration, during which cc. 62 mmol of product are formed, some reaction now becomes predominant which rapidly reduces the catalyst concentration until it apparently approaches some final value dependent on the value of p asymptotically. The results of preliminary 31P NMR spectroscopic studies on this reaction system have already been reported [4], and indicate that 1-hexene does not react with the catalyst species (1) when in equimolar concentration or in excess at 80 “C. Under an atmosphere of hydrogen, the precursor reacts at 80 “C to form a complex which is not analogous to the iridium species formed from the iridium analogue of 1 [ll, 121 or the rhodium hydride complexes which are implicated as being active hydrogenation catalysts [ 13 - 191. This reaction is much slower when hydrogen is replaced by a 1:l mixture of Hz and CO, and the 31P NMR spectrum obtained either at the end of a run or during a run shows signals which are very similar to those attributed to a dinuclear species. Initial studies have been carried out to establish the active life of the catalyst. These show that the addition of another charge of substrate to the reactor at the end of the catalytic run when the concentration of the active species is very low causes the active species to be re-formed, and the rate to be very similar to that observed in the reaction of the initial batch [4]. This strongly suggests that the pathway between the active species and the non-active or poisoned species is reversible. Reaction schemes such as: [Precursor]

e

[Active species] @a)

11 [Non-active

species]

217

or [Precursor] e

[Active species]

11 [Poisoned species]

@b)

can generate a curve with similar characteristics. The long duration of the zone of approximate constant catalyst concentration could indicate the presence of more than one active species. The value of p can, in general, be obtained from an analysis of the experimental data over the period of time during which the active species is formed. However, for the data under consideration here, where more than one active species could be involved, only an estimate can be obtained. The data under consideration, up to a time of 45 min can be approximated by the simple model [catalyst] = 0.27(1 - eVkzt)

(9a)

dy/dt = k l[catalyst]P [ 1-hexene]

(9b)

This approximation gives*, assuming a value of 1 for p, 0.5 mm01 of product after 0.8 min, whereas the experimentally observed value is cu. 0.2 mmol. To reduce the amount of material produced within the initial 0.8 min, either t.he value of p has to be increased and/or the order for the conversion of the precursor to active species has to be changed. Changing the value of p to 2 decreases the amount produced to cu. 0.25 mmol, and to cu. 0.16 mmol forp = 3. Due to the complexity of the system, the comparison of the values of the pseudo-first-order rate constants between different runs using different amounts of catalyst precursor under identical temperature and pressure conditions cannot be used to estimate the value of p. The values obtained for the pseudo-first-order rate constant using 0.075 mmol and 0.35 mmol of cataIyst precursor with 56.0 mmol of substrate gives, respectively, 0.01676(l) and 0.0723(l) min-’ [20]. A comparison of these values with the value of 0.03933(l) min-’ obtained for the run under consideration here yields values of p in the range of 0.7 to 2, thus indicating that the maximum catalyst concentration is not linearly dependent on the concentration of the precursor and hence that the reaction pathway between the precursor and the active species involves at least one step which is not first order. This, therefore, suggests that, although a rhodium dimer is added to the reactor, during the transition from the precursor to the active species the number of rhodium atoms in at least one of the intermediates is not two. *The non-linear optimization was carried out by the use of the subroutine ZXSQR and the numerical integration with the subroutine DGEAR from the IMSL library, Ih4SL Inc., Houston, TX 77036.

278

Conclusion In order to numerically differentiate the experimental data efficiently, great care has to be taken in choosing the polynomial used to approximate the data within the given data treatment interval. The choice made will be influenced by the end-use made of the differential data, and the time interval between successive data points. Of the two polynomials considered here, the expression associated with second-order kinetics uses less computer time, and gives equivalent or more useful results than the power series. Care should be exercised so that the data set under consideration is not altered, as will happen if too large a data treatment interval is used with the second-order kinetics polynomial. The plot of dy/dt uersus [alkene] clearly shows a linear zone, indicating that the order with respect to [substrate] is unity when l-hexene is used. This result is confirmed by a plot of -ln[alkene] versus time which is also linear within the limits of the linear zone of the dy/dt plot. The data obtained from the rate data, duly corrected for the change in concentration of the substrate, strongly suggest that there may be more than one active species involved in the catalytic cycle. The rate equation for dependence on the substrate can be written d[ aldehyde]/dt = 12[catalystlP [ 1-hexene]

(16)

The value of p can only be obtained accurately by studying the chemical nature of the rhodium species in solution and how they evolve. An analysis of the experimental data shows clearly that [Rh(SBut)(CO)P(OMe)J2 is not the active species, and that during the transition between precursor and active species at least one of the intermediates does not contain two rhodium atoms. Although acceptable fits between the experimental data and the simulations using various models can be obtained, the models do not necessarily correspond to the actual mechanism but can give insights into what may be taking place and therefore help to design experiments to clarify the mechanism.

Acknowledgement The author wishes to thank Hewlett Packard (Toulouse) for the loan of the microcomputer (HP-85), the digital voltmeter (Model 3456) used as the analog to digital converter, and the interface. The author also wishes to express his thanks to Dr. A. Sanger for helpful discussions, and to Prof. Ph. Kalck and his group at E.N.S.C.T., Toulouse, France, for financial support, helpful discussions and invaluable assistance during his stay in their laboratories.

279

References 1 G. W. ParshaIl, Homogeneous Catalysis. The Application and Chemistry of Catalysis by Soluble Metal Complexes, Wiley-Interscience, New York, 1980, pp. 89 - 90. 2 P. J. Davidson, R. R. Hignett and D. T. Thompson, in C. Kemball (ed.), Catalysis, Vol. 1, The Chemical Society, London, 1977, pp. 375 - 385. 3 B. Cornils, in J. Falbe (ed.), New Syntheses with Carbon Monoxide, Springer-Verlag, Berlin, 1980, Chap. 1. 4 T. G. Southern, J. M. Frances, P. M. Pfister, A. Thorez and Ph. Kalck, J. Chem. ,Soc., Chem. Commun., (1983) 510. 5 T. G. Southern, J. M. Frances, P. M. Pflister, A. Thorez and Ph. Kalck, ‘Application des Methodes de la Micro-Informatique B la Chimie’, C.N.R.S., Journ. Sci. Nat., (Toulouse), May 1982. 6 cf. the preparation of [Rh(SR)(CO)PRs]: Ph. Kaick and R. Poilblanc, Znorg. Chem., 14 (1975) 2779. 7 A. Savitsky and M. J. F. Gulay, Anal. Chem., 36 (1964) 1627. 8 K. J. Johnson, Numerical Methods in Chemistry, Marcel Dekker, New York, 1980, pp. 273 - 278. 9 B. Heil and L. Marko, Chem. Ber., 101 (1968) 2214. 10 B. Heil and L. Marko, Chem. Ber., 102 (1969) 2238. 11 J. J. Bonnet, A. Thorez, A. Maisonnat, J. Galy and R. Poilblanc, J. Am. Chem. SOC., 101 (1979) 5940. 12 E. Guilmet, A. Maisonnat and R. Poilblanc, Organometallics, 2 (1983) 1123. 13 C. A. Tolman, P. Z. Meakin, D. L. Linder and J. P. Jesson, J. Am. Chem. Sot., 96 (1974) 2762. 14 A. J. Sivat and E. L. Muetterties, J. Am. Chem. Sot., 101 (1979) 4878. 15 E. L. Muetterties, Znorg. Chim. Acta, 50 (1981) 1. 16 R. R. Burch and E. L. Muetterties, J. Am. Chem. SOL, 104 (1982) 4258. 17 E. B. Meier, R. R. Burch and E. L. Muetterties, J. Am. Chem. Sot., 104 (1982) 2661. 18 E. L. Muetterties and M. J. Kause, Angew. Chem., Znt. Ed. Engl., 22 (1983) 135. 19 P. B. Hitchcock, S. I. Klein and J. F. Nixon, J. Organometall. Chem., 241 (1983) C9. 20 T. G. Southern, J. M. Francis, P. M. Pfister, A. Thorez and Ph. Kalch, unpublished results.

Appendix Data treatment methods The data under consideration, which were all measured at a regular time interval, consist of a pressure reading Pi and its corresponding time reading tie It is assumed that the error in the time measurements is negligible compared to the errors in the pressure readings. The procedures for blocking this type of data and applying least-squares techniques to obtain either ‘smoothed’ values or derivatives have been described elsewhere [7,8]. The approach* used in this article is based on the use of this blocking procedure except that at the ends of the data sets the data treatment interval (DTI), i.e. the data block size, is allowed to vary from an initial *Copies of the software used in this work may be obtained from the author.

280

value of three to the desired size. The flow diagram (Fig. 6) shows the approach used to vary the data treatment interval at the beginning and end of the data set.

I

DTI

= DTI

1

-

1

N=ZxDTI+l

1

Calculate smoothed value and derivative

Remove first two points from data block

I Displace data block by one data point

end of data block

Fig. 6. Data treatment flow sheet.