Nonisothermal rate data

For nitrogen on 4A, Figure 1, the nonisothermal rate data despite the heating could be represented reasonably by usual isothermal Ficks law equations (I), if D/R is taken as 7.5 X 10" min" Thus, the value of D/R calculated from the isothermal equation was 50% larger than that used to derive the nonisothermal curve in Figure 1. Here the temperature maximum occurs at low amounts adsorbed, and the increased rate owing to increased diffusivities is nearly compensated by the decreased equilibrium adsorption at the observed temperatures. Propane... 

In a recent study Wang and Hofmann (1999) have stressed the importance of nonisothermal rate data. From a simple theoretical analysis they conclude that kinetic and transport data obtained under isothermal conditions in a laboratory reactor cannot logically be used to simulate any other type of reactor. This is because of the behavior of the Lipschitz constant L, which is a measure of the sensitivity of the reaction to different models. It tells us how any two models would diverge at the end of a reactor under different thermal conditions of operation. It is therefore a useful criterion for selecting the best model. It has been shown that L is different for different reactor models ... 

Kinetic Expressions. In this study, we have analyzed nonisothermal TGA data using the Chen-Nuttall equation, the widely accepted Coats-Redfern equation, and the Anthony-Howard equation. These equations are derived from simple rate expressions. The basic single reaction kinetic equation for the decomposition of a solid has been presented by Blazek (24) as... 

Rate equations based on concentration dependence (reaction order) Under some conditions, the rate characteristics of solid-state processes can be expressed through a concentration-type dependence. For example, if the decomposition of a large number of small crystallites is controlled by an equally probable nucleation step at each particle, then this is a first-order process (Al, FI). These rate equations are also used in nonisothermal kinetic analyses of rate data in the form g(a) = kt and are therefore included in Table 5.1. 

A kinetic analysis based on the Coats-Redfern method applied nonisothermal TGA data to evaluate the stability of the polymer during the degradation experiment. Of the different methods, the Coats-Redfern method has been shown to offer the most precise results because gives a linear fitting for the kinetic model function [97]. This method is the most frequent in the estimation of the kinetic function. It is based on assumptions that only one reaction mechanism operates at a time, that the calculated E value relates specifically to this mechanism and that the rate of degradation, can be expressed as the basic rate equation (Eq. 5.3). This method is an integral method that assumes various... 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

In the kinetic analysis of the experimental data with an autoclave, the non-linear least square method was used to estimate the rate constants under nonisothermal conditions. The simulation of liquefaction calculated by substituing the estimated values into the rate equations showed good agreement with experimental values. 

Back in the 1960s you could sell all the TML you could produce, so there was a large incentive to increase production rates. A research program was initiated to increase TML production. The first step was to go to the laboratory and obtain good kinetic data. The normal laboratory run in a Parr bomb operated at constant temperature, and yields were typically higher than those obtained in the pilot plant reactor and in the plant reactor. Nonisothermal experiments were performed to follow the same temperature trajectory as seen in the plant, and these produced similar yields. 

The proposed form of data presentation became highly popular and opportune in gas chromatography. Up to the present, some thousand references to the Kovat s work [1] have been known. The RI values are proportional to the free energies of sorption this is their thermodynamic interpretation. Further development of the RI concept was aimed at its application to nonisothermal conditions of gas chromatographic (GC) analysis. For linear temperature programming regimes (which are characterized by two variables initial temperature, Tq, and rate of its increase, r, deg X min ), the linear relationship (3) does not hold. In some partial cases, other linear dependence seems more precise for the retention time approximation ... 

In this chapter we are concerned only with the rate equation for the i hemical step (no physical resistances). Also, it will be supposed that /"the temperature is constant, both during the course of the reaction and in all parts of the reactor volume. These ideal conditions are often met in the stirred-tank reactor (see-Se c." l-6). Data are invariably obtained with this objective, because it is extremely hazardous to try to establish a rate equation from nonisothermal data or data obtained in inadequately mixed systems. Under these restrictions the integration and differential methods can be used with Eqs. l-X and (2-5) or, if the density is constant, with Eq. (2-6). Even with these restrictions, evaluating a rate equation from data may be an involved problem. Reactions may be simple or complex, or reversible or irreversible, or the density may change even at constant temperatur (for example, if there is a change in number of moles in a gaseous reaction). These several types of reactions are analyzed in Secs. 2-7 to 2-11 under the categories of simple and complex systems. 

Advantages of Nonisothemal Prediction. In both isothermal and nonisothermal predictions of degradation rate at room temperature from data obtained at elevated temperatures, degradation data that include higher overall degradation yield more reliable estimates. Therefore, the difference in the precision of the estimates between isothermal and... 

Parameter Estimation. The kinetic parameters of the model given above that must be estimated for model identification include kg, g, Eg, kb, b, Eb,j. Parameter estimation for this type of model is quite difficult because the parameters appear nonlinearly, the nucleation rate parameters enter only in the boundary condition, and availability of accurate data is limited. Certainly a model that describes the behavior of a nonisothermally operated crystallizer is needed if the temperature is to be manipulated, but there have been only a few studies of the effect of temperature on crystallization processes (Kelt and Larson 1977 Randolph and Cise 1972 Rousseau and Woo 1980). For isothermal crystallization, the terms involving Eg and Eh are absorbed into kg and kb, and only kg, g, kb, b, and i need to be estimated. 

Fig. 2 Experimental uptake curves for CO2 in 4A zeolite crystals showing near isothermal behavior in large (34 and 21.5 Jim) crystals (D 9 x 10 cm s at 371 K and 5.2 X 10 cm s at 323 K). The solid lines are the theoretical curves for isothermal diffusion from Eq. 2 with the appropriate value of Ddr. The uptake curves for the small (7.3 jim) crystals show considerable deviation from the isothermal curves but conform well to the theoretical nonisothermal curves with the values of Dc estimated from the data for the large crystals, the value of p calculated from the equilibrium data, and the value of a estimated using heat transfer parameters estimated from uptake rate measurements with a similar system under conditions of complete heat-transfer control. The limiting isothermal curve is also shown by a continuous line with no points. From Ruthven et al. [8]...

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