Growth equations, higher order

Equations (5.44) and (5.45) give only the leading-order growth of A and T. The higher-order terms become more important as the magnitude of the departure from the bifurcation point (fi — /i ) increases and then, in general, we need to compute the size of the limit cycles. 

Equations which govern the higher cumulant growth coefficients K n > 3, can also be derived using this method [14] by evaluating the coefficients of higher order derivatives of V x,t). 

After the small, higher order terms in (1.53)-(1.55) are neglected, these equations reduce to (1.49) - (1.51). In [40, 41] (1.53)-(l-55) were used to calculate the growth rate of small axisymmetric perturbations of an infinite jet (thread) of viscous Newtonian liquid and of a semi-infinite inviscid jet. For the infinite viscous jet (thread) the following result was obtained... 

Reactions that are catalyzed by a product can often be treated with Equation (12). Many growth and decay processes also give sigmoidal curves. Higher-order au-tocatalytic reactions may likewise be fitted into this scheme. For reactions of the type... 

Even with the rapid progress currently underway in the modeling of velocity fields for fully turbulent flow, the real objective remains the process result. The critical physics lies in interactions between equations of motion and scalar transport and the kinetics of reactions, crystal precipitation and growth, and other core processes. These are coupled higher order sets of equations that need to be solved simultaneously in a truly rigorous solution. 

For the two crystallization reactions, however, this simple treatment is not valid. In particular, a higher order of reaction is required. It was therefore decided to examine the theories of nucleation and growth to see if a suitable equation could be derived therefrom. 

In order to study the growth of particles with temperature, anatase powder (preheated to 150°C) was heated for a period of 3 h at 400, 600, 800 and 1000°C. Marked increase in particle size was noticed in the 600-1000°C region, as indicated by the photo-micrographs. The specific surface area (B.E.T.) of anatase heated at 400°C was 55 m2/g and decreased markedly for samples heated to higher temperatures. The crystallite size normal to the (101) and (110) reflecting planes of anatase and rutile samples was calculated by measuring the X-ray diffraction line-widths of the samples heated at 200, 400, 600, 800 and 1000°C for 3 h. The Scherrer equation corrected for instrumental line-broadening by Warren s equation was employed for the calculation.16 The line-width of the sample heated at 1000°C was taken as the reference. The crystallite size increases rapidly after 600°C (fig. lb). The transformation of pure anatase also starts only above 600°C. 

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