Carrolls Method

Carroll (2003) provided a method for estimating the heat loss from a buried pipeline. This method begins with the basic heat transfer equation  

U - overall heat transfer coefficient, W/m2.°C A - heat transfer area, m2 AT m - logarithmic mean temperature difference (LMTD), °C 

The heat transfer is calculated from the enthalpy change for the fluid in the pipeline  

For the case of a buried pipeline, there are four resistances that contribute to the overall heat transfer coefficient (1) the convective heat transfer from the fluid to the pipe wall, (2) the conduction through the pipe wall, (3) the resistance due to the insulation, and (4) the conduction from the pipe to the soil. The overall heat transfer coefficient, U, is obtained from the following equation  

Ai - inner surface area of the pipe, m2 d0 and di - outside and inside diameters of the pipe, m t - thickness of the insulation, m kp - thermal conductivity of pipe (usually steel), W/m.°C 

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The Freeman and Carroll method has been shown by Fong and Chen to be the only one which gives satisfactory answers to known reactions, whether zero order, 1st order, 2nd order, or even higher. E)ven firactional orders of reaction may be determined. This method can be used with either DTA or TGA data. 

The following is a description of the Freeman-Carroll method applied to DTA data. Consider the following DTA peak in which AT is plotted vs time, t, a representation of which given as follows ... 

Figure 4.41 Synthesis trees for synthesis of (—)-epibatidine by the Carroll method (a) direct synthesis (b) includes epimerization recycling step (see Scheme 4.1 5).

Liu, N. A. and Fan, W. C. Critical consideration on the Freeman and Carroll method for evaluating global mass loss kinetics of polymer thermal degradation. Thermochemica Acta 1999 338 85. 

Sestak (43) compared the kinetic results calculated by five different methods for a system corresponding to the dehydration of -CaS04 0.5H2O. The five methods evaluated mathematically were (1) Freeman and Carroll (83) (2) Doyle (84) (3) Coats and Redfern (85) (4) Horowitz and Metzger (88) and (5) Van Krevelen et al. (87). From these calculations it was found that the deviations of computed values oF E did not differ by more than 10%. Thus, all the methods appear to be satisfactory for the calculation of E within the limits of accuracy required. The errors of each method due to the inaccuracy of visual deduction of values from the TG curves were also calculated. These errors, % and e (errors in calculation of E or n, respectively), were as follows (1) Freeman and Carroll method, eE = 4% and e = 12% (2) Horowitz and Metzger method, ee = 2% (when the correct value of n is assumed) (3) Doyle method. eE = 4%. However, the magnitude of this error depends primarily on the position of the point on the TG curve on which the calculations are being performed. In the case of differential methods, me most accurate data are calculated from the medium-steep parts of the curve. For the approximation method, the accuracy depends on the determination of the curve inflection point temperature. 

Three kinetic methods were evaluated by Sharp and Wentworth (76) using the thermal decomposition of calcium carbonate under various conditions. The physical state of the sample was as a pellet, a powder, or as 1 1 molar ratios with a-aluminum oxide or a-iron(III) oxide. The three methods used were Method 1, Freeman and Carroll Method II, Coats and Redfern and Method HI, Achar et al. (102). The kinetic data calculated by Methods II and III are presented graphically in Figure 2.41 and in Table 2.3. In every case a linear plot was obtained over a wide range of a with n = When these methods were applied with n — j, the range of x was less, especially in the case of Method III, which led to noticeable curvature at... 

Rajeshwar I52) determined the kinetics of the thermal decomposition of Green River oil shale kerogen by using direct Arrhenius. Freeman and Carroll, and Coats and Redfern methods. The E, A, and values are given in Table 2.7. Rajeshwar concluded that the ability to resolve multiple processes hinges on the efficacy of the particular kinetic analysis employed and is not an inherent difficulty with nonisothermal TG techniques in general. The direct Arrhenius and Coats and Redfern methods clearly indicate the presence of two reactions with distinctly different kinetic parameters. On the olher hand, the Freeman and Carroll method is handicapped at low fractional... 

Of course, it is of great interest to determine the kinetic parameters from the torque-temperature curves obtained with the MDR run in scanning mode. These parameters can be calculated from the curves shown in Figure 3.18 by using either the method based on Equation 3.11, called the Freeman-Carroll method, or the simple method based on fitting the parameters in Equation 3.19. 

The Freeman-Carroll method [21] is very well known, but tedious to use. The method based on fitting the parameters in Equation 3.19 is a method of least squares linear or nonlinear. It is very simple when the order of the overall cure reaction can be considered to be one. 

B. Carroll, Methods in Macromolecular Analysis, M. Dekker, New York, 1969ff. 

Determination of Kinetic Parameters by Freeman and Carroll Method [11 j... 

The kinetics of thermal degradation have generally been studied using isothermal and nonisothermal methods. In earlier literature, isothermal methods were mostly employed for the study of the kinetics of solid-state reactions. During the past three decades, however, nonisothermal methods, for example, the Doyle method [17, 18], Freeman and Carroll method [19], Coats and Redfem method [24], Ozawa method [20], Flynn and Wall method [21, 22], Friedman method [25], and Kissinger method [26], have received more attention. 

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