Conversion model-independent

The phase transition rate in the crystallization of polymeric materials is of the same order as the rates of the heat exchange processes accompanying crystallization. Consequently, the boundary between phases becomes spatially dispersed. This excludes the possibility of using methods based on the front transition model proposed for metals to calculate residual stresses in plastics.148 It is possible to split the general problem and to find the temperature-conversion field independently. Then, assuming that the evolution of temperature T(x,t) and degree of crystallinity a(x,t) in time t and in space (x is the radius vector of an arbitrary point in a body) is known, we can analyze the mechanical problem.143... 

When more than one set of experimental results is available, the unknown form of the conversion function f(a) or g(ar) may be eliminated by comparing measurements made at a common value of a under the two (or more) sets of different conditions. These isoconversional methods are thus model independent, or nondiscriminating methods of estimating the Arrhenius parameters [14,42,43]. 

The kinetics of the selective catalytic reduction of nitric oxides (NOx) on a proprietary high temperature catalyst with diesel as the reductant have been studied. The objective was to derive a kinetic model that can be used for real time simulation of the catalyst. In the extension, the real time simulation will be used when controlling the injeetion of reductant. This is a requirement for achieving a high efficiency and a low fuel penalty. The response time and the NOx conversion level upon transient diesel injection was found to be dependent on the temperature. At temperatures below 570 K very low or no NOx conversion was observed. Above 570 K a small conversion was observed. No direct response upon diesel injection could be distinguished and the NOx conversion was independent on the hydrocarbon concentration. As the temperature was increased the response became apparent and then faster and the conversion level gradually became more dependent on the hydrocarbon concentration. Above 700 K the response was immediate (response time less than 15 s) and the conversion level was directly dependent on the hydrocarbon concentration. It was concluded that the NOx reduction proceeds via the formation of a hydrocarbon intermediate and the successive reaction between the hydrocarbon intermediate and NOx. When this reaction mechanism was modeled mtiny features of the catalyst behaviour were reproduced. 

In the previous chapter it has been shown that single-pulse PLP experiments can be used for the model-independent determination of the chain-length dependence of termination reactions. When employing on-line conversion data from TR-SP-PLP experiments, k f can be determined directly from the monomer concentration versus time trace and its first and second derivatives, according to ... 

A new advance with regard to the instrumentation and methods available for online monitoring of heterogeneous polymerization reactions was made by using ACOMP for monitoring the evolution of multiple characteristics during polymerization. The information-rich data collected simultaneously by multiple detectors provide absolute, model-independent determination of quantities such as conversion, composition, and molar mass distribution and avoid potentially damaging effects of the reactor environment. 

In this context, the use of ACOMP for the simultaneous monitoring of the evolution of coUoid phase (monomer droplets, polymer latex particles) and solution phase (polymer and monomer) characteristics during emulsion polymerization was recently reported as a versatile characterization tool that offers absolute, model-independent determination of quantities such as conversion, composition, and molecular weight [39],... 

While 5(0 is an important parameter to be monitored, since it allows running the reaction to any degree of conversion desired, there was no model-independent route to obtaining conversion, in contrast to typical ACOMP measurements of polymerization reactions, where model-independent conversion is directly obtained from collected data. The authors proposed the use of a chemically specific detector, such as FTIR, for the direct measurement of conversion in future studies. 

Sodium alkoxides, namely, sodium methoxide, sodium ethoxide, sodium n-propoxide, and sodium iso-propoxide were synthesized and characterized by various analytical techniques. Thermal decomposition of these sodium alkoxides was studied under isothermal and non-isothermal conditions by thermogravimetric (TG) method. Non-isothermal experiments were carried out at different linear heating rates. Mass spectrometric technique was followed for identifying the evolved gases. The onset temperatures of decomposition of sodium methoxide, sodium ethoxide, sodium n-propoxide, and sodium iso-propoxide were found to be 623, 573, 590, and 545 K, respectively. These sodium alkoxides form gaseous products of saturated and unsaturated hydrocarbons and leave a mixture of sodium carbonate, sodium hydroxide, and free carbon as the decomposition residue. Activation energy and pre-exponential factor for the decomposition reactions were deduced from the TG data by model-dependent and model-independent (iso-conversion) methods. The probable decomposition mechanism of sodiiun alkoxides is described in this chapter. 

Experimental data obtained from TG is generally deconvolved to calculate kinetic parameters by either model-independent or model-dependent method. The extent of conversion, a, is extracted from TG data using the following equation. 

Model-independent method for non-isothermal experiments Model-dependent methods need prior knowledge of mathematical function of fractional decomposition, f(a) as reported in the literature [45-47]. Iso-conversion method eliminates the need of assumption of a mathematical functional form, f(a) [52-55]. 

The diad fractions for the low conversion experiments only are reproduced in Table II. The high conversion data cannot be used since the Mayo-Lewis model does not apply. Again diad fractions have been standardized such that only two independent measurements are available. When the error structure is unknown, as in this case, Duever and Reilly (in preparation) show how the parameter distribution can be evaluated. Several attempts were made to use this solution. However with only five data points there is insufficient information present to allow this approach to be used. 

In case of fast gradient (below 15 min), S could be considered constant for all the investigated molecules and wiU only have a small influence on the retention time of the compounds. Thus, the gradient retention times, of a calibration set of compounds are linearly related to the ( )o values [39]. Moreover, Valko et al. also demonstrated that the faster the gradient was, the better the correlation between t, and < )o [40]. Once the regression model was established for the calibration standards, Eq. 8 allowed the conversion of gradient retention times to CHI values for any compound in the same gradient system. Results are then suitable for interlaboratory comparison and database construction. The CH I scale (between 0 and 100) can be used as an independent measure of lipophilicity or also easily converted to a log P scale. 

Equations I and J indicate that for the temperature range of interest (640 to 830 °K), the heat capacity per unit mass is substantially independent of the conversion level. Furthermore, the temperature dependent contribution to the heat capacity will not vary much over the temperature range involved. Hence without introducing errors comparable to those inherent in the use of a one-dimensional model, we may take the heat capacity as constant at 0.250 cal/g-°K or 0.250 BTU/(lb-°F). 

The kinetics of the CTMAB thermal decomposition has been studied by the non-parametric kinetics (NPK) method [6-8], The kinetic analysis has been performed separately for process I and process II in the appropriate a regions. The NPK method for the analysis of non-isothermal TG data is based on the usual assumption that the reaction rate can be expressed as a product of two independent functions,/ and h(T), where f(a) accounts for the kinetic model while the temperature-dependent function, h(T), is usually the Arrhenius equation h(T) = k = A exp(-Ea / RT). The reaction rates, da/dt, measured from several experiments at different heating rates, can be expressed as a three-dimensional surface determined by the temperature and the conversion degree. This is a model-free method since it yields the temperature dependence of the reaction rate without having to make any prior assumptions about the kinetic model. 

By virtue of the conditions xi+X2 = 1>Xi+X2 = 1, only one of two equations (Eq. 98) (e.g. the first one) is independent. Analytical integration of this equation results in explicit expression connecting monomer composition jc with conversion p. This expression in conjunction with formula (Eq. 99) describes the dependence of the instantaneous copolymer composition X on conversion. The analysis of the results achieved revealed [74] that the mode of the drift with conversion of compositions x and X differs from that occurring in the processes of homophase copolymerization. It was found that at any values of parameters p, p2 and initial monomer composition x° both vectors, x and X, will tend with the growth of p to common limit x = X. In traditional copolymerization, systems also exist in which the instantaneous composition of a copolymer coincides with that of the monomer mixture. Such a composition, x =X, is known as the azeotrop . Its values, controlled by parameters of the model, are defined for homophase (a) [1,86] and interphase (b) copolymerization as follows... 

The simplest scenario to simulate is a homopolymerization during which the monomer concentration is held constant. We assume a constant reaction volume in order to simplify the system of equations. Conversion of monomer to polymer, Xp defined as the mass ratio of polymer to free monomer, is used as an independent variable. Use of this variable simplifies the model by combining several variables, such as catalyst load, turnover frequency, and degradation rate, into a single value. Also, by using conversion instead of time as an independent variable, the model only requires three dimensionless kinetics parameters. 

As Figure 11.26 undoubtedly demonstrates, the deviation between the same catalytic material under practically identical reaction conditions is in the range of 2% conversion (if appropriate measures are taken this error can be reduced to 0.5%). These experimental data points lead to the important verification of the above-discussed CFD modeling results and confirm the assumption of realizing identical reaction conditions over the whole reactor system independent from the position of a catalyst to be tested. By testing inert carrier material in reactor column number 8, the inertness and catalytic inactivity of the reactor steel can be proven. 

A matrix of independent variables, defined for a linear model in Eq. (29) and for nonlinear models in Eq. (43) Transpose of matrix X Inverse of the matrix X Conversion in reactor A generalized independent variable... 

Predicted value of the reactor conversion, for model i A matrix of the independent variables x,... 

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