Models constant growth

KiMM is given the subscript, MM, to remind us that it reflects Michaelis-Menten enzyme kinetics as distinguished from KiM used above to model microbial growth kinetics (see Monod cases above). Note, is the same as KE in Box 12.2 when it s value represents the reciprocal of the equilibrium constant for the binding step. 

Constant Crystal Growth Model. In this instance, crystals have an inherent constant growth rate, but the rate from crystal-to-crystal varies. The modeling of this phenomenon must be accomplished by use of probability transform techniques due to the presence of a growth rate distribution. The complete solution for the population density yields a semilogarithmic population density plot that is concave upwards similar to size-dependent growth (Berglund and Larson 1984). Since it is relatively difficult to handle, a moment approximation was developed for an MSMPR crystallizer (Larson et al. 1985). 

Assuming that the multipher at p in the Eq. (56) left-hand is constant, the constant c in this equation can be estimated from the known bonndaty conditions of the irreversible aggregation model [52], This model supposes growth at p decreasing, that corresponds to the tendency, which the Eq. (56) expresses. Assuming also that Dj, 1.60 corresponds to p 0.7 [52], let us obtain the following simple relationship [55] ... 

The Avrami model was originally derived for the study of kinetics of crystallization and growth of a simple metal system, and further extended to the crystallization of polymer. Avrami assumes the nuclei develop upon cooling of polymer and the number of spherical crystals increases linearly with time at a constant growth rate in free volume. The Avratni equation is given as follow ... 

As it is known [24], solid component fiaction enhancement (namely sueh eomponent of amorphous phase a clusters are as regards devitrificated loosely paeked matrix) results to elastic constant growth. This enhancement can be described quantitatively within the frameworks of percolation theory (see the Eq. (3.7)), but in Ref. [21] the more simple variant was chosen, namely, the polymers network connectivity model [25]. Then the elasticity modulus E value is determined as follows [25] ... 

The oscillations do, however, point toward a problem with the growth term, as this would allow an uncertainty in one radial interval to flow quickly onto aU intervals. In the discretized model the growth term is, apart from a few constants, determined by the finite difference approximation. As a resnlt, the sensitivity of the model to both forward and backward instead of centred finite difference approximations can be tested and the most robust approach should be adopted. Notice that the forward approximation relies on populations f and i-i-l and the backward relies on populations i - 1 and i. The fonr, in the forward and backward approximation, replaces the eight in the denominator of the centred difference approximation as the interval has been halved in size. 

To derive the space function, we assume isothermal and isobaric conditions so that the reactivity of growth remains constant. We will successively study both isotropic and radial anisotropic models of growth. 

Here comes iht first important assumption made in the Kolmogoroffs model (i). the growth rate uXO/cms" must not depend on the size of the crystals, i.e. it could be a function of the current time moment t but not of the birth moment u. Apparently, a constant growth rate Vi is the simplest case in which this requirement is fulfilled. 

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