Death function

Therefore, the conservation equation plays an important role in the population balance by placing limits on the population. In general, these conservation equations are sometimes coupled into the definitions of the birth and death functions, B and D, in the population balance, thus assuring both the conservation of a property and the balance of the population simultaneously, without the necessity of two separate differential equations—population balance and property conservation—to be solved simultaneously. Such birth and death formulations will be discussed in Chapter 4 for grinding. 

The birth fimction for conservation of volume must be consistent with the death function because each particle death results in the birth of smaller particles, resulting from commimition. When particles of size L break, they produce a suite of daughter particles, called the primary progeny, with a size distribution p(x, L). This function applies... 

At this point we must use the birth and death functions described in Section 4.2.4.4 and solve this equation for the population in the mill. Then using the population balance over the classifier equation (4.57), the product population, mp(L), can be determined from the population inside the grinding mill, m(L), as follows ... 

In this chapter, the fundamentals of classification and comminution of ceramic powders have been described. Comminution is described by birth and death functions in a population balance. These birth and... 

The birth and death functions are assumed for simplidly to be zero. The solution to this differential equation is given by... 

The birth and death functions now have the same units as the population balance. To attempt a solution, an integral or continuous approach will be used in place of this discrete summation. This suggests that there is a continuous distribution of particle sizes (i.e., the sizes of interest for the population balance are much larger than that of singlets, doublets, etc.). Some key substitutions for this integration are necessary ... 

Substitution of these relationships into the discrete form of the birth and death functions yields... 

An analytical solution to this integro-partial differential equation is not possible without some simplifying assumptions. In the sections that follow, anal5d ical solutions are presented for particle growth in a CSTR and batch precipitation reactors. For systems in which shear is the dominant collision mechanism and not Brownian difhision, the birth and death functions can be rewritten in terms of the mean shear rate, y, as follows [104]. 

The first term in this partial differential equation describes the temporal change of the population tj the second term describes the atomistic growth of the particles (which assumes that G is independent of particle size r), and finally the last two terms account for the birth and death of particles of size r by an aggregation mechanism. The birth fimction describes the rate at which particles enter a particle size range r to r + Ar, and the death function describes the rate at which the particles leave this size range. In the case of continuous nucleation, an additional birth rate term is used for the production of atoms (or molecules) of product by chemical reaction. In this case, the size of the nuclei are the size of a single atom (or molecule) and the rate of their production is identical to the rate of chemical reaction, kfi, where C is the reactant concentration, giving... 

The birth and death functions predict the importance of particulate aggr ation on the final particle size distribution. The key concepts for the development of these two functions come from Smoluchowski s rapid flocculation theory, which was derived in Section 6.6.3 ... 

Appl5ing the new birth and death functions given in equations (7.53) and (7.55), respectively, the governing differential equation for the population of particles, rj(r, t), becomes... 

Verbascoside is also able to induce apoptosis [24]. The apoptosis is the programmed cell death functioning to conserve tissue homeostasis. It is... 

Birth function for particles Death function for particles Zero constant in ASL model in Eq. (4.33) Constant in ASL model in Eq. (4.33) Constant in ASL model in Eq. (4.33)... 

Hogstedt, G. 1983. Adaptation unto death function of fear screams. Am, Nat., 121, 562—570. 

We shall now turn our attention to the birth and death functions associated with breakage and aggregation processes. 

Since the birth and death functions are now identified the complete population balance equation for an aggregating population is given by substituting for h in the right-hand side of (2.7.9) — h sls calculated from... 

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