AUTOCORRELATION TREATMENT OF THERMOGRAVIMETRIC DATA WITH THE COCHRANE-ORCUTT METHOD

AUTOCORRELATION TREATMENT OF THERMOGRAVIMETRIC DATA WITH THE COCHRANE-ORCUTT METHOD

V. F. C. LINS ¹ and E. M. PAULA e SILVA ²

¹ Departamento de Engenharia Química, Universidade Federal de Minas Gerais

² Departamento de Engenharia Metalúrgica, Universidade Federal de Minas Gerais

CEP 30.160-030 - Fax: [phone]. Phone: [phone] - Belo Horizonte, MG - Brazil

(Received: November 17, 1996; Accepted: May 14, 1997)

ABSTRACT - The interaction of an Fe-Mn-Al-Si-C alloy with the environment was evaluated. The alloy showed good resistance to air and oxygen oxidation, especially at temperatures of 600 and 700º C for air oxidation and at 600ºC in oxygen atmosphere. Oxidation rates for pure oxygen were lower than those measured for air. Due to autocorrelation identified in the results, it was impossible to employ the usual procedures of Regression Analysis. Kinetic data were then analyzed by the statistical method of Cochrane and Orcutt. A parabolic kinetic behavior was observed for the Fe-Mn-Al-Si-C alloy oxidation process in the air at temperatures of 600, 700 and 800ºC and in an oxygen atmosphere at 900ºC. On the other hand, the TGA results for oxygen oxidation at 600 and 700ºC and for air oxidation at 900ºC adjusted well to the linear kinetic curve. Finally, the cubic kinetic curve was selected to represent the oxidation of the alloy in oxygen at 800ºC.

KEYWORDS: Fe-Mn-Al-Si-C alloy, oxidation kinetics, thermal analysis.

INTRODUCTION

The Fe-Mn-Al-Si-C alloys are important structural materials due to their resistance to oxidation (Lins and Paula e Silva, 1988) and to wear and to their high strain hardening rate (Queiroz and Paula e Silva, 1983).

Precise knowledge of the characteristics of the interaction of the Fe-Mn-Al-Si-C alloy with the environment is essential for its industrial uses but in the information in the literature on the corrosion resistance of the alloy is incomplete and sometimes contradictory; in addition, the thermogravimetric results of oxidation are usually studied through the linear regression analysis (Montgomery and Peck, 1982) and autocorrelation is not accounted for. This practice leads to error in the activation energy calculation and in the determination of the appropriate kinetic model.

The aim of this work was to determine the oxidation kinetics of a Fe-Mn-Al-Si-C alloy in the air and an oxygen atmosphere, in the 600-900ºC temperature range. Thermogravimetric experiments were performed and the Cochrane-Orcutt Method (Cochrane and Orcutt, 1949) was used in the autocorrelation treatment.

MATERIALS AND METHODS

An Fe-31.8 Mn-6.09Al-1.60Si-0.40C (wt%) alloy was produced on a pilot scale in an industrial plant. The alloy was smelted in anatmospheric industrial furnace. The average weight of the ingots was 43.6.10³ g and the average alloy density was 6.85.10⁶ g/m³.

The alloy was submitted to a homogenizing treatment at 1000ºC for 24 hours, followed by water quenching. Then, the material was heated to 1150ºC for 4 hours and forged in the 920-1130ºC temperature range. After a thermal treatment at 950ºC for 2 hours, the bar was cut in to samples of (15x 20).10^-3m. The samples were submitted to a standard surface treatment that included mechanical polishing, immersion in pure acetone and drying.

The thermogravimetric experiments were performed on a RIGAKU thermal balance. The alloy was oxidized using an isothermal process during variable time periods. The temperatures and times studied are presented in Table 1.

The gas pressure for both pure oxygen and synthetic air (with 20 % oxygen) was 1 atm. Each experiment was done at least in duplicate. The oxidation kinetics was determined through tests using the following models:

D m = b ₁ t + e (linear) (1)

D m ² = b ₁ t + e (parabolic) (2)

D m ³ = b ₁ t + e (cubic) (3)

D m = b ₁ log(t) + e (logarithmic I) (4)

1/ D m = b ₁ log(t) + e (logarithmic II) (5)

Oxidation in air Oxidation in oxygen Temperature (ºC) Time (h) Temperature (ºC) Time (h) 612 616 618 695 696 698 796 800 805 900 900 - 2.0 2.0 2.0 2.0 2.0 1.8 16.6 12.4 25.7 30.7 38.0 - 610 610 610 710 710 720 810 810 810 900 900 900 26.0 26.0 26.0 10.0 22.0 26.0 17.8 26.0 26.0 18.0 26.0 26.0

Oxidation in air		Oxidation in oxygen
Temperature (ºC)	Time (h)	Temperature (ºC)	Time (h)
612 616 618 695 696 698 796 800 805 900 900 -	2.0 2.0 2.0 2.0 2.0 1.8 16.6 12.4 25.7 30.7 38.0 -	610 610 610 710 710 720 810 810 810 900 900 900	26.0 26.0 26.0 10.0 22.0 26.0 17.8 26.0 26.0 18.0 26.0 26.0

Table 1: Conditions of the thermogravimetrics tests

The testing of these models was done using the Cochrane-Orcutt Method.

STATISTICAL ANALYSIS

The experimental data obtained for each temperature studied, for both air and oxygen, were submitted to a statistical methodology consisting of:

I) Exploratory analysis of the data

II) Model adjustment (equations 1 to 5) by linear regression analysis

III) Use of the methodology proposed by Cochrane and Orcutt

The exploratory analysis of the data was helpful in the preliminary studies. It aided in the visualization of sample behavior and descriptive statistics.

The second step consisted of the estimation of model parameters by linear regression. The statistical software Minitab, version 7.2, was used for the analysis.

The evaluation of model adequacy consisted of (Montgomery and Peck,1982):

• Graphics analysis: plot of residuals versus fitted values, normal probability of residuals and plot of residuals in time sequence;

• Statistical tests: Durbin-Watson -- detection of autocorrelation of residues -- and Shapiro Wilks -- verification of the assumption of normality of errors;

• Detection of influential observation: Cook’ s distance measurement.

In all the combinations of temperatures and oxidant medium, the observations showed a serial correlation. The mass increase at a given time t was correlated with the value at a previous time, violating the model assumption that the errors are not correlated. The iterative methodology developed by Cochrane and Orcutt allowed the incorporation of the correlation into the model.

THE COCHRANE-ORCUTT METHOD

The fundamental assumptions in linear regression are that the error terms e _i have a mean of zero and constant variance and are uncorrelated [E ( e _i) = 0, V ( e _i) = s ², and E ( e _{ie j} ) = 0]. For purposes of testing hypotheses and constructing confidence intervals, we often add the assumption of normality, so that the e _i are NID (0,s ²). Some applications of regression involve regressor and response variables that have a natural sequential order over time. Such data are called time series data. The assumption that errors are uncorrelated or independent for time series data is often not appropriate. Usually the errors in time series data exhibit serial correlation, that is, E (e _{ie i+j}) ¹ 0. Such error terms are said to be autocorrelated.

There are several sources of autocorrelation. Perhaps the primary cause of autocorrelation in regression problems involving time series data is failure to include one or more important regressors in the model.

When the apparent autocorrelation in the errors cannot be removed by adding one or more new regressors to the model, it is necessary to explicitly recognize the autocorrelative structure in the model and devise an appropriate parameter estimation method. There are a number of estimation procedures that can be used. The method described by Cochrane and Orcutt (1949) will be presented.

Consider the simple linear regression model with first-order autocorrelated errors. Suppose we transform the response variable so that:

Y ^, _t =Y _t - r Y _t-1 (6)

Substituting y_t and y_t-1, the model becomes:

y ^, _t = b ₀ + b ₁ x _t + e _t - r ( b ₀ + b ₁ x _t-1 + e _t-1 ) (7)

= b ₀ ( 1- r ) + b ₁ (x _t - r x _t-1 ) + e _t - r e _t-1 (8)

= b ^, ₀ + b ^, ₁ X ^, _t + a _t (9)

where b ^,₀ = b ₀ (1-r ), b ^,₁ = b ₁, x^,_t =(x_t - r x_t-1), and a_t = e _t -r e _t-1.

Note that the error terms a_t in the reparameterized model are independent random variables. Therefore, by transforming the regressor and response variables, we produce a model that satisfies the usual regression assumptions and ordinary least squares can be used. Unfortunately, the reparameterized model cannot be used directly because the new regression and response variables x^,_t and y^,_t are functions of the unknown parameter r . However, the first order autoregressive process (e _t = r e _t-1 + a_t) can be viewed as a regression through the origin. Thus r can be estimated by obtaining the residuals e_t from an ordinary least squares regression of y_t on x_t, and then regressing e_t on e_t-1.

The least square estimate of r that results is:

(10)

Using this estimate of r , the transformed regressor and response variables are obtained:

x_t = x_t - x_t-1 (11)

y_t = y_t - y_t-1 (12)

and ordinary least squares are applied to the transformed data. The Durbin-Watson test should be applied to the residuals from the reparameterized model. If this procedure indicates that the residuals are uncorrelated, then no additional analysis is required. However, if positive autocorrelation is still indicated, then another iteration is necessary. In the second iteration r is estimated with new residuals obtained by using the regression coefficients from the reparameterized model with the original regressor and response variables. This iterative procedure may be continued as necessary until the error terms in the reparameterized model are uncorrelated.

RESULTS AND DISCUSSION

The mean rates of oxidation in the air and oxygen are presented in Table 2.

It can be seen that the alloy showed good resistance to air and oxygen oxidation. In fact, the oxidation rates were smaller than the conventional oxidation limit of 3.1 g/m²h (Fontana, 1986). The oxidation rates for pure oxygen were lower than those measured for air. Oxidation stabilization trends were not observed in an oxygen atmosphere. On the other hand, oxidation in the air of the Fe-Mn-Al-Si-C alloy stabilized after 50 minutes at 600 and 700ºC.

For each temperature the most appropriate kinetic equation was chosen for the process.

OXIDATION IN THE AIR AT 600ºC

Since all samples showed a homogeneous behavior, modeling used the average mass gain (with 55 observations). Table 3 presents the values of the correlation coefficient (R²) in a transformed scale for each model.

The parabolic behavior was selected and, using the Cochrane-Orcutt Method, the variables were transformed producing the following equation:

m ^* = -0.0004 + 0.0017 t ^* (13)

Since the first term in the equation was statistically not significant, the final parabolic equation is:

D m ² = 0.0017t (14)

OXIDATION IN THE AIR AT 700ºC

The study was done with the average mass gain, with 17 observations. The Cochrane-Orcutt Method showed that the parabolic model was the appropriate one (Table 4).

In this particular case, an influential observation was identified and removed. The equation obtained was:

D m ² = -0.0047 + 0.00229t (15)

OXIDATION IN THE AIR AT 800ºC

In the first step autocorrelation was detected in all models, but after the application of the Cochrane-Orcutt Method, autocorrelation was still present only in the cubic model. Then a new calculation of the transformed cubic equation was done. The final values of R² are shown in Table 5.

An influential observation in the linear and parabolic models was identified and removed. Then, new values of R² were obtained (Table 6).

The parabolic behavior was selected and the resulting equation was:

D m ² = 0.101 + 0.0151t (16)

OXIDATION IN THE AIR AT 900ºC

The observations were read at identical time intervals according to the Cohrane-Orcutt Method. The best agreement was found for the linear behavior. The transformed equation was:

m ^* = 0.883 + 0.00814 t ^* (17)

The null hypothesis relative to the autocorrelation parameter could not be accepted and the model could be represented by the following equation:

D m = 4.95 + 0.00814t (18)

OXIDATION IN OXYGEN

For some temperatures each sample must be modeled separately. Table 7 shows the models obtained for each temperature.

Temperature (ºC) Rates of oxidation in air (g/m2h) Rates of oxidation in oxygen atmosphere (g/m2h) 600 700 800 900 0.884 0.700 0.443 2.659 0.136 0.225 0.472 0.885

Temperature (ºC)	Rates of oxidation in air (g/m2h)	Rates of oxidation in oxygen atmosphere (g/m2h)
600 700 800 900	0.884 0.700 0.443 2.659	0.136 0.225 0.472 0.885

Table 2: Mean rates of oxidation in air and oxygen

Model Correlation Coefficient -R 2 (%) Linear Parabolic Cubic Logarithmic I Logarithmic II 68.1 91.6 48.1 16.2 67.5

Model	Correlation Coefficient -R 2 (%)
Linear Parabolic Cubic Logarithmic I Logarithmic II	68.1 91.6 48.1 16.2 67.5

Table 3: Correlation coefficient in a transformed scale for each model

Model Correlation Coefficient - R 2 (%) Linear Parabolic Cubic Logarithmic I 87.6 97.2 90.1 84.8

Model	Correlation Coefficient - R 2 (%)
Linear Parabolic Cubic Logarithmic I	87.6 97.2 90.1 84.8

Table 4: Correlation Coefficients

Model Coefficient of Determination - R 2 (%) Linear Parabolic Cubic Logarithmic I Logarithmic II 88.7 86.5 71.8 58.2 20.0

Model	Coefficient of Determination - R 2 (%)
Linear Parabolic Cubic Logarithmic I Logarithmic II	88.7 86.5 71.8 58.2 20.0

Table 5: Correlation Coefficients

Model Correlation Coefficient - R 2 (%) Linear Parabolic 89.4 88.0

Model	Correlation Coefficient - R 2 (%)
Linear Parabolic	89.4 88.0

Table 6: Correlation Coefficients

Temperature (ºC) Model 600 700 800 900 Linear Linear Cubic Parabolic

Temperature (ºC)	Model
600 700 800 900	Linear Linear Cubic Parabolic

Table 7: The models selected for each temperature

CONCLUSIONS

The Fe-Mn-Al-Si-C alloy showed good oxidation resistance in the air and oxygen, especially at temperatures of 600 and 700ºC for air oxidation and at 600ºC in an oxygen atmosphere. The oxidation rates for pure oxygen were lower than those measured for air. The oxidation stabilization trend was not observed in the oxygen atmosphere. On the other hand, the oxidation process in an air atmosphere of Fe-Mn-Al-Si-C stabilized after 50 minutes, at 600 and 700ºC.

Due to an autocorrelation identified in the results, it was not possible to work with the usual procedure of Regression Analysis and kinetic data were analyzed by the statistical method of Cochrane and Orcutt. A parabolic kinetic behavior was observed for the oxidation processes in the air at temperatures of 600, 700 and 800ºC, and in an oxygen atmosphere at 900ºC. On the other hand, TGA results for oxygen oxidation at 600 and 700ºC and for air oxidation at 900ºC adjusted well to a linear kinetic curve.

The cubic kinetic model was selected using the statistical method to represent the oxidation of the Fe-Mn-Al-Si-C alloy in oxygen at 800ºC.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of the Fundação de Amparo à Pesquisa do Estado de Minas Gerais, FAPEMIG.

NOMENCLATURE

a_t Error term in the reparameterized model, dimensionless

E Mean of the error terms, dimensionless

e_t Residuals, dimensionless

D m Mass gain of the alloy, g

R² Correlation coefficient, %

t Time, h

t^* Reparameterized time variable, h

V Variance, dimensionless

X- Regressor variable, h

X^' Reparameterized regressor variable, h

Y- Response variable, g

Y'Reparameterized response variable, g

Greek letters

b ₀ First term of the Y_t equation, g

b ^,₀ First term of the Y_t^, equation, g

b ₁ Coefficient of the X_t in the Y_t equation, g/h

b ₁^, Coefficient of the X_t^, in the Y^,_t equation, g/h

e _t Error term relative to time t, dimensionless

e _I Error term relative to the i observation, dimensionless

e _j Error term relative to the j observation, dimensionless

r Autocorrelation parameter, dimension-less

The least square estimate of r , dimensionless.

s ² Variance, dimensionless

• Cochrane, D. and Orcutt, G.H., Applications of Least Square Regression to Relationships Containing Autocorrelates Error Term., J.Amer.Statist Assoc., 44, 32-61 (1949).
• Fontana, M.G., Corrosion Engineering. McGraw-Hill Inc. (1986).
• Lins,V.F.C. and Paula e Silva, E.M., Características Estruturais e Cinéticas da Oxidaçăo ao Ar de Liga Fe-Mn-Al-Si-C. Metalurgia ABM, 44, 1265-8 (1988).
• Queiroz, G.C.G. and Paula e Silva, E.M., Correlaçăo entre Propriedades Mecânicas e Parâmetros Microestruturais de um Aço Austenítico do Sistema Fe-Mn-Al-Si-C. Seminário sobre Metalurgia Física e Tratamentos Térmicos, 4, 187-197 (1983).
• Montgomery, D.C. and Peck, E.A., Introduction to Linear Regression Analysis. John Wiley and Sons Inc. (1982).

Publication Dates

• Publication in this collection
  09 Oct 1998
• Date of issue
  June 1997

History

• Accepted
  14 Mar 1997
• Received
  17 Nov 1996