Specific features of kinetics evaluation in calorimetric studies of runaway reactions

09504230(95)00018-6

J. Loss Prev. Process Ind. Vol. 8, No. 4, pp. 22%235, 1995 Copyright 0 1995 Elsevier Science Ltd Printedin Great Britain. All rights reserved 0950-4230195 $10.00 + 0.00

Specific features of kinetics evaluation in calorimetric studies of runaway reactions A. Kossoy and E. Koludarova Russian Scientific Centre St Petersburg, Russia

‘Applied

Chemistry’,

14 Dobrolubov

Avenue,

197198

This paper is dedicated to the problem of the adequacy of the kinetics evaluation methods used in calorimetric investigations of reaction kinetics. The problem is especially significant for adiabatic calorimetry because application of usual methods may lead to obtaining noncorrect kinetic models and hence to serious mistakes in hazard assessment of runaway reactions. The essence of the problem is considered by the method of mathematical simulation. The basic features and advantages of the appropriate method are discussed on the basis of real experimental data-processing for kinetics evaluation. Keywords: calorimetry;

kinetics; thermal

safety

One of the most important stages in chemical reaction investigations is the creation of a kinetic model of a reaction based on experimental data. Such kinetic models may then be used for many important purposes, such as optimization of the technological process, assessment of its thermal safety, safety analysis of conditions of storage and transportation of a chemical product, etc. The reliability of the results obtained is defined by the quality of the models applied. Therefore, one of the basic methodological problems of kinetic studies is the choice of an adequate method for model creation, i.e. the method of kinetics evaluation. The applicability of one method or another depends on the type of kinetic experiment used, and, specifically, on the way in which the variables influencing the reaction split up into factors and responses. The following definitions’,* are used: ?? factors

are the input variables or parameters defining the influence of the environment on the reacting system ?? responses are the state variables which are being measured during the experiment and which characterize the reacting system’s behaviour. For many types of experiment, variables can be separated into factors and responses without any difficulties. In particular, temperature exerting a very strong influence on the reaction rate is usually considered as a factor. Almost all methods of kinetic evaluation are based on this assumption, e.g. the widely known method of linearization using coordinates InW versus l/T, and some non-linear methods. This group of methods will be referred to as T-methods. Calorimetric methods of chemical reaction investigation, which are the main methods for thermal safety

research, occupy a special place from the point of view of kinetics evaluation due to two aspects of such methods. 1. Heat generation due to reaction changes the reactant temperature and, in principle, this variable cannot be considered as a factor. This is particularly true for adiabatic calorimetry being used for investigation of runaway reactions. 2. For many kinds of calorimetric methods, temperature is the only response which is measured directly; the heat generation rate, the traditional response for calorimetry, is then calculated on the basis of measured temperature. Examples are differential thermal analysis, adiabatic and reaction calorimetry, many kinds of scanning calorimeters, etc. It should be noted that the problem of the influence of heat evolution due to the reaction on the reactant temperature is also valid for devices which provide direct measurements of heat generation rate (heat flux or compensation calorimeters). In some rare cases, it is possible to minimize the deviation of the reactant temperature from the environment temperature by applying special measures (heat dilution, intensive heat removal, use of microsamples, etc.). Only in these conditions can the reactant temperature be considered as a factor, and application of T-methods will be well-founded. In all other cases, preliminary analysis of the applicability of the T-methods is required. However, despite the difficulties described, Tmethods are always used at present for kinetic evaluation in calorimetry without any pre-testing=. Therefore, this article pursues two main goals:

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Kinetics evaluation in runaway reactions: A. Kossoy and E. Koludarova ?? to

show that such an approach may lead to obtaining non-correct reaction kinetics, especially in those cases when adiabatic calorimetry is used for investigation ?? to discuss the basic features of a more suitable method for kinetics evaluation.

Peculiarities of kinetic evaluation in calorimetric studies Splitting up the variables in calorimetric experiments First let us examine the mathematical model of a calorimeter. Usually its reaction cell may be considered as a well-stirred batch reactor. Therefore, the model of the calorimetric cell is represented by the system of ordinary differential equations of chemical kinetics (1) and heat balance (2): da - = f(a)kO exp dt c,m, %

= m, z

- XS (T, - T,)

(2)

with the corresponding initial conditions: t = 0, (Y = 0, T, = TsO,T, = T,, and usually T,, = T,,. We consider here a simple one-stage reaction and neglect the influence of the cell walls, but the results obtained will remain applicable for reactions of any kind and for a more detailed model of the cell. Depending on the heat exchange intensity of the reactant with the environment, equations (1) and (2) will describe all the variety of thermal modes of the calorimetric cell. (The term thermostat is used below instead of environment.) Case I: XS = 03. It can be easily seen from equation (2) that the sample temperature is always equal to the thermostat temperature, i.e. T, = T,, and it does not depend on heat generation inside the cell. In this case, temperature is the factor and the heat generation rate is the response, and the reaction behaviour will be described by equation (1) with the additional condition that T, = T,. Obviously, any of the T-methods may be applied for kinetics evaluation.

Sensitivityof the model of the cell to parameter errors The essence of the kinetics evaluation is estimation of the parameters, i.e. determination of the set of model parameters (parameter vector) which ensures the best fitting experimental data. If the exact kinetic model of a reaction were defined, and experimental data were measured without errors, parameter estimation would give the exact value of the parameter vector regardless of the method of kinetics evaluation used. However, in practice, we are able to formulate only an approximate hypothesis of the model structure, and there are errors in the measurements of experimental data. Therefore, as a result of kinetics evaluation, we can only obtain an estimate of the parameter vector instead of its exact value. Evidently, equations (1) and (2) are sensitive to parameter errors and the level of sensitivity depends very strongly on the thermal mode of the calorimetric cell. When x = ~0and hence T, = T,, an error in any of the kinetic parameters will affect the reaction rate only. In contrast, when x < 00, an error in the same parameter will influence not only the reaction rate but also T,, due to heat generation. Because the reaction rate depends on T, exponentially, the influence of the error in the parameter will be significantly magnified. The lower the value of x, the larger this magnification will be. It reaches its maximum in adiabatic conditions. Suppose, for instance, that due to an error in the parameter, the calculated reaction rate is slightly less than the true one. This means that self-heating of the cell and, correspondingly, the reaction rate will progress more slowly. Therefore, the induction time for the simulated process will be larger than that for the real one. Because of the strong non-linearity of equations (1) and (2), the only way to illustrate these phenomena quantitatively is to use the method of mathematical simulation. We took as an example an adiabatic calorimeter cell in which the nth order exothermic reaction takes place

$ = (1 -

d_T,_ 1 dQ dt

Case 2: XS < CQ.According to equation (2), heat generation causes deviation of the reactant temperature from the thermostat temperature. The extent of this deviation is defined by the value of x and may vary from several degrees for isoperibolic mode to several hundred degrees for adiabatic conditions (,$ = 0). For certain types of calorimeters, reactant temperature is the only thermal response which can be measured directly during the experiment, and attempts to interpret temperature as a factor will lead to more or less serious mistakes. Therefore, for kinetics evaluation, it is necessary to consider the complete non-linear equations system (equations (1) and (2)) and use appropriate methods of non-linear optimization, termed generalized methods.

230

(u)“kOexp( - &)

-c, dt

S

z

=

Qm$

(1’)

(2’)

The behaviour of the system has been simulated using the KINETICS MK program9 for c, = 2 kJ kg-’ K-l; T,(O) = TsO= t = 0, a = 0, initial conditions: 403.15 K; and the following vector of kinetic parameters: E = 125 600 J mol-l, k0 = 1.78 x 1O1’s-l, n = 1, Q= = 500 kJ kg-l. The results of simulation of such a system (Figure la, b; curves 1) will be used below as quasi-experimental data (reference data): Q(t) = Qr(t) and T,(t) = T,(t). Consider the response of the reacting system to the disturbance introduced due to a small error energy activation in determination of the E,, = E 5 6E; 6E = 630 J mol-l, for the following two cases.

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min

1

300.0 0 i250.0 ::

2b

50.0 Time, min Figure 1 Response of the reacting (a) heat generation systems

curves;

system (b) temperature

in adiabatic mode to small disturbances due to variation of the activation energy: profiles; (1) responses of reference system; (la, lb, Za, 2b) responses of disturbed

Cuse I! The sample temperature, T,, is interpreted as a factor and is equal to the measured temperature T,. In this case, it is appropriate to use equation (1’) for calculation of heat generation in the cell:

2 =Qm(l -

cr)%, exp

(1”)

The results of the simulation are shown in Figure la (curves la, lb; index a corresponds to Ed = E - 6E, b corresponds to Ed = E + 6E). Case 2. The response of the reacting system to the disturbance is simulated on the basis of both equations

(1’) and (2’). Corresponding calculated curves 2a and 2b are shown in Figure la and b. Obviously, in the first case, the parameter error causes relatively small deviations from the reference at the initial stage of the process with low and moderate self-heat rates, whereas, in the second case, the same error leads to significant alteration of the temperature profile of the reaction (Figure lb) and consequently the time dependency of heat generation (Figure la). Attention must be paid to one important peculiarity of the behaviour of the disturbed system. If the disturbances (parameter errors) are small enough, then the temperature profiles of the disturbed and reference systems prove to be practically similar but shifted

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along the time axis. Therefore, simulation of the adiabatic process on the basis of non-correct estimation of the parameter vector (even if the errors in separate parameters are small) will lead first of all to erroneous determination of the induction times. Example of kinetics evaluation based on real data Now let us compare the results of kinetic evaluation

obtained using both possible approaches. For this purpose, we used real experimental data obtained using the accelerating rate calorimeter (ARC) for the thermal decomposition of product D (cp = 1.373; c, = 2 kJ kg-i K-l). In both cases, the KINETICS MK program has been used for parameter estimation. Case 1. Using a T-method. According to the features of the method applied, the measured sample temperature, T(t), is regarded as a factor while calculating the reaction rate (as in equation (1”)). At the same time, T(t), being the only response of the adiabatic calorimeter, is used to estimate the parameter vector, O(E, kO, Q_, n), ensuring the best coinciding calculated, T,(t), and experimental, T(t), temperature curves. This process model may be written in the usual form for adiabatic studies:

7

=e

(1 -

a)”

k,exp

s

T - TO (y=Tf_

The objective function, SS, for determining the sought-for estimate of the parameter vector, 8,,, is as follows: SS=~[(~)j-(d$))~

i=1,2...N (3)

where

(yji

and

(Tji

correspond

to the

calculated and measured values of the self-heating rate at t = ti, N is the total number of experimental points and SS(&) is minimum. The procedure for estimation of the vector 0,, (l) based on the non-linear least squares method results in the following parameter values: E = 157622 J mol-‘; k = 1.631 x 1019s-‘. n = 1.332; Qm =242 kJ kg-l, a\d the value of the’objective function was: SS(@)) = 2.07 x 105. As shown in Figure 2, the vector 0::) ensures satisfactory description of experimental data. However, if, using t?$), we simulate the behaviour of the adiabatic system on the basis of equations (1’) and (2’), the calculated temperature profile (curve 3 in Figure 2) will be far from the experimental one due to the influence of the parameter errors. Case 2. Using a generalized method. The same objective function (equation (3)) is used for estimation;

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the sample temperature, T,, is determined as the solution of equations (1’) and (2’) representing the complete model, but, in the framework of the applied approach, the reaction rate depends on the same calculated temperature, T,. Therefore, T, is really the result of simulating the adiabatic process. The measured temperature, T(t), i.e. the experimental response, is used only for calculating the ss value. The resultant parameter estimates (vector f@) are as follows: E = 157788 J mol-l; kO= 1.646 x 1019 s-l; n = 1.339; Qm = 242 kJ kg-‘, and lead to the objective function value &~ (*)) = 1.96 x 105. Figure 3 illustrates the degree of correspondence between experimental and calculated temperature curves. Comparison of the vectors @,l) and 822) shows that the corresponding parameters are very close to each other; values of the objective functions also differ insignificantly. However, 0,‘,2)ensures reliable simulation of the adiabatic mode whereas using e$,‘) leads to an entirely erroneous result. Thus, the procedure of parameter estimation based on applying the generalized method proves to be sensitive to the influence of parameter error on the temperature profile, and therefore allows correct and reliable parameter estimates to be obtained. Application of any T-method results in false estimates. According to the specific response of the adiabatic system to a small variation of the kinetic parameters, temperature profiles calculated using f3$) and @) are similar to each other, but shifted along the time axis (see Figure 4a). Usually the plot of log(dT/dt) versus l/T are used in adiabatic studies of reaction kinetics for graphic data presentation and data analysis. This plot does not allow representation of the progress of the process over time, and both calculated temperature curves and the experimental one are practically coincident (Figure 4b). Therefore, it is impossible to recognize the uselessness of the vector 8,, cl). This is one of the reasons why the inadequacy of the Tmethods has not been discovered before now despite wide application of adiabatic data for kinetics evaluation. Another important reason is that the performance of estimation procedures based on generalized methods is impossible without appropriate software because of the mathematical complexity of the problem. However, until very recently, there was no such commercial software available. The KINETICS MK program which has been created at the Russian Scientific Centre ‘Applied Chemistry’ as a part of software for thermal safety research may be considered as an example of the required software.

Conclusions The results presented in this article allow the following conclusions to be made. 1. Special attention must be paid to the choice of an adequate method of kinetics evaluation when calorimetry is used for reaction kinetics studies. In particular, application of generalized methods may

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-

,’ Y ;;roo

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Time, Figure 2 Results of kinetics evaluation using a T-method: heat generation

reactions:

rain

(1) experimental

data; (2) calculated heat generation

curve; (3) simulated

curve for adiabatic mode

160

140

80

60 100

Time,

min

200

150

Figure 3 Results of kinetics evaluation using a generalized method: (1) experimental

data; (2) calculated

250 heat generation

curve for

adiabatic mode

be recommended in adiabatic investigations because they ensure that more reliable kinetic models are obtained which are suitable for hazard assessment of runaway reactions. 2. The results of simulation of the reaction in adiabatic mode are very sensitive to small errors in the kinetic parameters. One of the main origins of these errors is inaccuracy of the experimental data used for kinetic evaluation. This makes high

demands on the organization of adiabatic experiments. The possible sources of the experimental errors should be revealed and appropriate measures for their correction must be taken.

Acknowledgements The authors are very grateful to J.-L. Gustin and J. Fillion for helpful discussions and for putting experimental data used in this work at their disposal.

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80.0

4 60.0

e

VI

E

4 240.0

P 2 i

20.0

0

Time,

0.01

’

I 75

261

.0

man

1 I 100 125 Temperature, C

I 150

I 175

Figure4 Self-heating rate of an adiabatic system: (a) using axes dT/dt versus t; (b) using axes log(d77dt) versus -1000/T; (1) experimental data; (2) calculated curve using parameter vector 8.. (2); (3) calculated curve using parameter vector tlsd (I)

8 Ahmed, M., Fisher, H. G. and Janeshek, A. M. in ‘International

References 1 Gorsky, V. G. ‘Planning of Kinetic Experiments’, Nauka, Moscow, 1984 2 Zaldivar, J. M. in ‘Safety of Chemical Batch Reactors and Storage Tanks’, eds A. Benuzzi and J. M. Zaldivar, Kluwer Academic Publishers, Dordrecht, 1991, pp 201-226 3 Gustin, J. L. in ‘Safety of Chemical Batch Reactors and Storage Tanks’, eds A. Benuzzi and J. M. Zaldivar, Kluwer Academic Publishers, Dordrecht, 1991, pp 311-354 4 Mentel, J. and Anderson, H. Thermochim. Acta 1991, 187, 121-132 5 Anderson, H. Thermochim. Acta 1992, 203, 515-518 6 Anderson, H. and Mentel, J. J. Thermal Anal. 1994,41,471-481 7 Kohlbrand, H. T. in ‘International Symposium on Runaway Reactions’, Cambridge, Massachusetts, USA, 1989, p 86

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Symposium on Runaway Reactions’, Cambridge, Massachusetts, USA, 1989, p 331 9 Kossoy, A. A., Benin, A. I., Smykalov, P. Yu. and Kazakov, A. N. Thermochim. Acta 1992, 203, 77

Nomenclature Specific heat capacity (kJ kg-’ K-‘) Activation energy (J mol-‘) Number of an experimental point Pre-exponential factor (s-l) Mass (kg) Order of a reaction Specific heat production (kJ kg-‘) Specific heat effect of a reaction (kJ kg-‘)

C

E I0 m n

Q Q-

8, Number

4

Kinetics R S T W

Gas constant (J mol-’ K-‘) Surface of heat exchange (m*) Time (s) Temperature (K) Rate of a reaction

a cp

Degree of conversion Thermal inertia

I

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reactions:

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and E. Koludarova

Heat transfer coefficient (W m-* K-‘)

Subscripts P sl

Values of parameters for the environment Value of a variable at the end of a process Sample Initial value of a variable or parameter

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