Temperature integral

We must remember that T in equation (6.161) is the equilibrium melting temperature. Integration of this equation will give an equation that relates melting temperature to activity. Separating variables and integrating... 

Reducer R XNOjmaX (%) T Width at half conversion Tso (°C) Tioo (°C) Efficiency (temperature integrated... 

If ArH° is independent of temperature, integration of this equation from T, to T2 yields... 

Commercial dextrins are specifically the oligomers of starch. White dextrins, so called because of their visual appearance, are produced from a 30-40% suspension under the mildest possible hydrolysis conditions (79-120°C for 3-8 h in 0.2-2% H2S04 or HC1). Yellow dextrins and British gums are the partial hydrolysates at higher time-temperature integrals. Maltodextrins, dextrose equivalent20 5-19, derive from controlled enzyme or acid partial hydrolysis of gelatinized corn starch. The 20-24 dextrose equivalent hydrolysates tire com syrups (Appi, 1991). 

For combustion reactions Levenspiel (4) gives the constant temperature integration for reaction and gas or ash diffusion controlled cases. The integration of the pyrolysis kinetics will be demonstrated in the following section. 

For H2/ at temperatures below 1000 K, the rotational constant B = 59.4 cm 1 is so large that integration is invalid, and one must sum directly the leading terms of the two sums. The same holds for D2, where B = 29.9 cm-1. For the heavier homonuclear diatomics, like 02 (B = 1.437 cm ) or N2, B is so small that the high-temperature integration of Problem 5.3.11 works well, and the difference between ortho and para states becomes experimentally indistinguishable. 

Extrapolating from the low-temperature integrated NMR spectra provides an independent measure of the equilibrium constant within the temperature range at which line shape analysis is carried out. This simplifies the line shape analysis to just one dynamic parameter, say kr, since k = Kkx. 

VARIABLE HEAT-TRANSFER COEFFICIENT. If the heat-transfer coefficient varies with temperature, one can assume that the complete exchanger consists of a number of smaller exchangers in series and that the coefficient varies linearly with temperature in each of these sections. When the last five conditions listed in the preceding section hold and the overall coefficient varies linearly with temperature, integration of Eq. (17) gives... 

Calculated effective diffusivities are tabulated in Table II. This problem was solved on the computer by the "shooting method" for a range of temperatures. Integrations were by the 4th order Runge-Kutta algorithm. The system equations are summarized below. 

Two different types of enzymatic time-temperature integrators are described. The first, under the tradename of I-point, is based on a lipase-catalyzed hydrolysis reaction (125). The lipase is stored in a nonaqueous environment containing glycerol. The indicator contains two components that are mixed when the indicator is activated. The operating principle is as follows Upon activation, two volumes of reagents are mixed with each other. Lipase is thereby exposed to its substrate, here a triglyceride. At low temperatures there will be almost no hydrolytic reaction. As the temperature increases, hydrolysis accelerates and protons are liberated. A pH indicator is dissolved in the system. The indicator is selected to shift color after a certain amount of acid has been liberated by the enzyme-catalyzed process. Since the catalytic activity is influenced both by temperature and time, this indicator strip is said to be a time-temperature integrator. 

The second time-temperature indicator is based on the use of horseradish peroxidase in liquid and solid paraffin (126). The enzyme is deposited onto non-porous glass beads md mixed with melted paraffin containing the substrate. The suspension is mixed well and quickly cooled in a dry ice/acetone bath. When the enzyme is stored in solid paraffin, the activity is extremely slow. But when the temperature increases, the paraffin may melt and thereby make the enzyme reaction a millionfold faster than in the solid hydrocarbon phase. This time-temperature integrator is based on the same concept as the I-point, but here the... 

In order that the equation may receive a more general meaning, let us first make its contents clearer by means of a graphical construction. Assuming for simplicity that q does not vary with the temperature, integration of... 

This is the fundamental equation relating heats of reaction to temperature. Integration gives ... 

The use of equation (5.4) requires evaluation of the temperature integral, variable x = E. /RT is introduced so that ... 

Zsako [29] has suggested sub-classification of integral methods on the basis of the means of evaluation of the temperature integral in equation (5.4). The three main approaches are the use of (i) numerical values of/>(x) (ii) series approximations for p(x) and (iii) approximations to obtain an expression which can be integrated. 

Tables of values of the integral p x) have been provided [75,76]. Much attention has been directed towards finding suitable approximations for the above temperature integrals [29,58,77-79]. Gorbachev [80] and Sest [49,81] have suggested that there is little value in tiying to find more accurate approximations considering the experimental imcertainties in the original a, T data. Representative examples of the series suggested for approximating (x) are given in Table 5.3. (see also references [34,82,83]).

Flynn [34] has emphasized the importance of using accurate calculated data for the temperature integral in determining the magnitudes of E, and A fi om non-isothermal measurements. He points out that modem computer methods make the use of approximations unnecessary. 

The use of derivative methods avoids the need for approximations to the temperature integral (discussed above). Measurements are also not subject to cumulative errors and the often poorly-defined boundary conditions used for integration [74], Numerical differentiation of integral measurements normally produces data which require smoothing before further analysis. Derivative methods may be more sensitive in determining the kinetic model [88], but the smoothing required may lead to distortion [84],... 

Smith, S.E., Orta-Ramirez, A., Ofoli, R.Y., Ryser, E.T., and Smith, D.M. 2002. R-phycoerythrin as a time-temperature integrator to verify the thermal processing adequacy of beef patties. Journal of Food Protection 65 814-819. 

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