Population density balance growth rate

As shown by Eq. (54), growth rate G can be obtained from the slope of a plot of the log of population density against crystal size nucleation rate B° can be obtained from the same data by using the relationship given by Eq. (57), with n° being the intercept of the population density plot. Nucleation rates obtained by these procedures should be checked by comparison with values obtained from a mass balance (see the later discussion of Eq. (66)). 

Once there is an appreciable amount of cells and they are growing very rapidly, the cell number exponentially increases. The optical cell density of a culture can then be easily detected that phase is known as the exponential growth phase. The rate of cell synthesis sharply increases the linear increase is shown in the semi-log graph with a constant slope representing a constant rate of cell population. At this stage carbon sources are utilised and products are formed. Finally, rapid utilisation of substrate and accumulation of products may lead to stationary phase where the cell density remains constant. In this phase, cell may start to die as the cell growth rate balances the death rate. It is well known that the biocatalytic activities of the cell may gradually decrease as they age, and finally autolysis may take place. The dead cells and cell metabolites in the fermentation broth may create... 

Growth and nucleation interact in a crystalliser in which both contribute to the final crystal size distribution (CSD) of the product. The importance of the population balance(37) is widely acknowledged. This is most easily appreciated by reference to the simple, idealised case of a mixed-suspension, mixed-product removal (MSMPR) crystalliser operated continuously in the steady state, where no crystals are present in the feed stream, all crystals are of the same shape, no crystals break down by attrition, and crystal growth rate is independent of crystal size. The crystal size distribution for steady state operation in terms of crystal size d and population density // (number of crystals per unit size per unit volume of the system), derived directly from the population balance over the system(37) is ... 

In unicellular organisms, the progressive doubling of cell number results in a continually increasing rate of growth in the population. A bacterial culture undergoing balanced growth mimics a first-order autocatalytic chemical reaction (Carberry, 1976 Levenspiel, 1972). Therefore, the rate of the cell population increase at any particular time is proportional to the number density (CN) of bacteria present at that time ... 

The population balance approach to measurement of nucleation and growth rates was presented by Randolph and Larson (1971, 1988). This methodology creates a transform called population density [n(L)], where L is the characteristic size of each particle, by differentiating the cumulative size distribution N versus L. shown in Fig. 4-22, where N is the cumulative number of crystals smaller than L. Per unit volume, the total number of particles, total surface area, and total volume/mass are calculated as the first, second, and third moments of this distribution. 

It will be shown in a later section that the solution of a differential population balance requires a knowledge of tbe relationship between growth rate and size of the growing ciystals Moreover, this relationship can often be deduced from the form of population density deta. A special condition, which simplifies such belances, results when all crystals in the magma grow at (he sama constant rate. Crystal-solvent systems that show this behavior ate said to follow tha AL Law proposed by McCabe.1 while systetes that do not are said to exhibit anomalous growth. 

Analysis of the PSD Data. From the population balance for a CMSMPR crystallizer operated under steady-state condition, the population density n for size-independent crystal growth is given by Equation (1), where n , G, 6 and 1 are nuclei density, growth rate, residence time of reactants and particle size, respectively. 

One method is to solve the population balance equation (Equation 64.6) and to take into account the empirical expression for the nucleation rate (Equation 64.10), which is modified in such a way that the expression includes the impeller tip speed raised to an experimental power. In addition, the experimental value, pertinent to each ch ical, is required for the power of the crystal growth rate in the nncleation rate. Besides, the effect of snspension density on the nucleation rate needs to be known. Fnrthermore, an indnstrial suspension crystallizer does not operate in the fully mixed state, so a simplified model, such as Equation 64.6, reqnires still another experimental coefficient that modifies the CSD and depends on the mixing conditions and the eqnipment type. If the necessary experimental data are available, the method enables the prediction of CSD and the prodnction rate as dependent on the dimensions of the tank and on the operating conditions. One such method is that developed by Toyokura [23] and discussed and modified by Palosaari et al. [24]. However, this method deals with the CTystaUization tank in average and does not distinguish what happens at various locations in the tank. The more fundamental and potentially far more accurate simulation of the process can be obtained by the application of the computational fluid dynamics (CFD). It will be discussed in the following section. 

The total solids content, Mj, and hence the production rate of an MSMPR crystallizer are both controlled by the feedstock and operating conditions. Equation 9.26 is, in effect, a growth rate constraint because for a given population density of nuclei only one value of G will satisfy the mass balance. The mass of crystals dM in a given size range dL is... 

Boundary conditions of the type (2.7.12) are important in crystallization where secondary nucleation, as pointed out by Randolph and Larson (1988), may be governed by the growth rate of existing particles. For example, consider a well-mixed crystallizer where the number density is only a function of the sole internal coordinate selected as particle size x as represented by a characteristic length, which should satisfy a population balance equation of the type (2.7.6). Randolph and Larson discuss a variety of nucleation mechanisms and conclude that contact nucleation is the most significant form of nucleation. Thus, the mechanical aspects of the crystallizer equipment which provide contact surfaces contribute to increased nucleation rate. When growing crystals, containing adsorbed solute on their surfaces, come into contact with other solid surfaces, nucleation is induced. The boundary condition for the formation of new nuclei in a real crystallizer is therefore considerably more complicated than that implied by (2.7.7). Instead, the boundary condition must read as... 

In the equations, according to Larson and Garside [10], j denotes the exponent for the suspension density M, i is the quotient of the exponents m (for the supersaturation s at the nucleation rate and n (for the supersaturation s at the crystal growth rate G), Ip is the dominant particle size of the population balance, fey is the volume shape factor for the crystallized mass produced, and is the quotient of the proportionality constants of the secondary nucleation rate B and k "of the crystal growth rate G. 

---