Kinetics growth

Under ideal batch growth conditions, the quantity of biomass, and therefore the biomass concentration will increase exponentially with respect to time and in accordance with all cells having the same probability to multiply. Thus the overall rate of biomass formation is proportional to the biomass itself where 

Here rx is the rate of cell growth (kg cell m 3 s 1), X is the cell concentration (kg cell m 3) and k is a kinetic growth constant (s 1). For a batch system, this is equivalent to 

The plot of the logarithm of cell concentration versus time will often yield a straight line over a large portion of the curve, as shown in Fig. 1.22. 

The initial time period up to I, represents a period of zero growth, which is known as the lag phase. In this period the cells synthesise enzymes and other cellular components appropriate to the particular environmental conditions of the fermentation. 

An exponential (or logarithmic) growth phase follows the lag phase, and during this period the cell mass increases exponentially. The growth rate is at a maximum during this phase, and the population of cells are fairly uniform with respect to chemical composition and metabolic composition. 

Garside (1971) defined an effectiveness factor for crystal growth that is a measure of the relative importance of diffusion and surface integration as the rate-controlling factors. The effectiveness factor is defined by the relation 

Ast) —r 1, diffusion plays a smaller and smaller role in controlling the rate. Methods for estimation rj as well as other methods to determine the relative importance of diffusion and surface integration are discussed in Nyvit et al. (1985). 

The development and operation of industrial crystallization processes can be made significantly easier if some data on the kinetics of crystal growth are available. This information can be incorporated in process models, can be used in process and crystallizer design, and can shed light on the observed behavior of the system. 

In the previous section, we reviewed a number of different crystal growth theories. These provide a theoretical basis for the correlation of experimental crystal growth data and the determination of kinetic parameters from the data to be used in models of 

55) employs a linear crystal growth velocity (length/time) with Eq. (2.56) employs a mass rate of crystal growth (raass/area time). The constants in Eqs. (2.55) and (2.56) can be related to each other through the expression 

An atom added to a roigh surface has several possible sites with different binding energies as follows  

The site marked is equivalent to the two-dimensional picture shown in F re 6.9. The eneigy marked is the same as the addition to atomically smooth surface. 

Any atom incident on a rough surface has a greater sticking probabil- 

FIGURE 619 Simplistic representation of the solid—solution interfaces in a growing ciystal (a) Atomically smooth surface and (h) rou crystal surface. Redrawn, with permission from Elwell and Scheel [28], 

Tempkin [29] and Jackson [30] characterized the roughness of a crystal surface with a surface entropy factor, a, defined as 

016451 Fastest mean generation time during autotrophic 

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A3.3.4 LATE-STAGE GROWTH KINETICS AND OSTWALD RIPENING... 

Various plasma diagnostic techniques have been used to study the SiH discharges and results have helped in the understanding of the growth kinetics. These processes can be categorized as r-f discharge electron kinetics, plasma chemistry including transport, and surface deposition kinetics. 

For those pesticides which are utilized as microbial growth substrates, sigmoidal rates of biodegradation are frequentiy observed (see Fig. 2). Sigmoidal data are more difficult to summarize than exponential (first-order) data because of their inherent nonlinearity. Sigmoidal rates of pesticide metabohsm can be described using microbial growth kinetics (Monod) however, four kinetics constants are required. Consequentiy, it is more difficult to predict the persistence of these pesticides in the environment. 

An empirical approach can also be used to relate growth kinetics to supersaturation with a power-law function of the form... 

AH the models described above indicate the importance of system temperature on growth rate. Dependencies of growth kinetics on temperature are often expressed in terms of an Arrhenius expression ... 

Many industrial crystallizers operate in a weU-mixed or nearly weU-mixed manner, and the equations derived above can be used to describe their performance. Furthermore, the simplicity of the equations describing an MSMPR crystallizer make experimental equipment configured to meet the assumptions lea ding to equation 44 useful in determining nucleation and growth kinetics in systems of interest. 

The relative nucleation growth kinetics for this system are given by the relation (adapted from Garside and Jancic, 1979)... 

Garside, J., Gibilaro, L.G. and Tavare, N.S., 1982. Evaluation of crystal growth kinetics from a desupersaturation curve using initial derivatives. Chemical Engineering Science, 37, 1625-1628. 

Nielsen, A.E. and Toft, J.M., 1984. Electrolyte crystal growth kinetics. Journal of Crystal Growth, 67, 278-288. 

Tavare, N.S. and Garside, J., 1986. Simultaneous estimation of crystal nucleation and growth kinetics from batch experiments. Chemical Engineering Research and Design, 64, 109. 

More recently, simulation studies focused on surface melting [198] and on the molecular-scale growth kinetics and its anisotropy at ice-water interfaces [199-204]. Essmann and Geiger [202] compared the simulated structure of vapor-deposited amorphous ice with neutron scattering data and found that the simulated structure is between the structures of high and low density amorphous ice. Nada and Furukawa [204] observed different growth mechanisms for different surfaces, namely layer-by-layer growth kinetics for the basal face and what the authors call a collected-molecule process for the prismatic system. 

H. Nada, Y. Furukawa. Anisotropic growth kinetics of ice crystals from water studied by molecular dynamics simulation. J Cryst Growth 169 581, 1996. 

Flow thins protective film to equilibrium thickness which is a function of both mass transfer rate and growth kinetics. Erosion corrosion rate is controlled by the dissolution rate of the protective film. 

Bubbles are formed instantaneously. This conclusion made in [33] is based on estimates taken from earlier works [37]. As seen from the above cited works by S. E. Sosin et al., this is not always true viscoelastic liquids under triaxial stretching stress are not destroyed instantly. The existence of an induction period may produce a considerable effect on foam growth kinetics upon free foaming, when pressure is lowered instantaneously from P > Pcr to P < Pcr in a melt with dissolved gas. However, it would appear that microfaults in polymer melts, which are caused by factors... 

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