Particle birth-death processes

Then, since in any system accumulation is the net result of both evolution and birth/death processes, and since any latex particle can be characterized by a set of physical quantities which will fully specify a given particle or class of particles, one can obtain the following population balance equation (33) ... 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

Deep-bed filtration involves the flow of particles through randomly distributed passages thus, it tends to be stochastic in nature. The filtration process has been modeled as a pure birth process [12,13], a birth-death process [14-17], a random-walk process [18], and a stochastic diffusion process [19]. 

In Litwiniszyn s pure birth model, the entire bed is considered as on state, and the number of blocked pores in a unit volume of the bed is considered as a random variable. In his birth-death model, the number of trapped particles over the entire bed is a random variable. Fan and his co-workers have extended the pure birth and birth-death process models by incorporating... 

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

A more refined approach is based on the local description of fluctuations in non-equilibrium systems, which permits us to treat fluctuations of all spatial scales as well as their correlations. The birth-death formalism is applied here to the physically infinitesimal volume vo, which is related to the rest of a system due to the diffusion process. To describe fluctuations in spatially extended systems, the whole volume is divided into blocks having distinctive sizes Ao (vo = Xd, d = 1,2,3 is the space dimension). Enumerating these cells with the discrete variable f and defining the number of particles iVj(f) therein, we can introduce the joint probability of arbitrary particle distribution over cells. Particle diffusion is also considered in terms of particle death in a given cell accompanied with particle birth in the nearest cell. 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

For an arbitrary combined material volume element constituting a combined sub-volume Vsv[t) of the particle phase space the integral formulation of the population balance states that the only way in which the number of particles can change is by birth and death processes [95, 96, 35, 93, 94[. The system balance is thus written on the form ... 

In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes. The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., liquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical applications to assume that the particle state can be described by a finite dimensional vector. ... 

We recall the domain A t) in particle state space considered in Section 2.6, which is initially at and continuously deforming in time and space. For the present, the particles are regarded as firmly embedded in the deforming particle state continuum described in Section 2.5. The only way in which the number of particles in A t) can change is by birth and death processes. We assume that this occurs at the net birth rate of /i(x, r, Y, t) per unit volume of particle state space so that the number conservation may be written as... 

The development in this section arises from the generalization of an idea due to Kendall (1950) for a simple birth-and-death process. This generalization, accomplished by Shah et al (1977) was significant in that it provides a route to the simulation of a particulate system of arbitrary complexity and is limited only by the amount of computational power available. It is statistically exact in that the random numbers to be generated satisfy exactly calculated distribution functions from the model for particle behavior, thus allowing not only the calculation of the average system behavior but also the fluctuations about it. Furthermore, we shall see that it is free from arbitrary discretizations of time (or any other governing evolutionary coordinate) that were characteristic of the simulations of Section 4.6.1. 

For particle diameters less than 1 pm, the diiiusional transport of particles in a flowing fluid is identified as convective diffusion (Friedlander, 1977). The role of Brownian motion is considerable at smaller particle sizes. For cases where there are no birth and death processes (B = De = 0) and there is no particle growth/decay, equation (6.2.51c) may be simplified as follows ... 

Solution of the particle concentration profile in the particle concentration boundary layer from in the feed suspension liquid to the concentration on top of the cake (and equal to the concentration in the cake) requires consideration of the particle transport equation in the boundary layer. We will proceed as follows. We will first identify the basic governing differentied equations and appropriate boundary conditions (Davis and Sherwood, 1990) and then identify the required equations for an integral model and list the desired solutions from Romero and Davis (1988). However, we will first simplify the population balance equation (6.2.51c) for particles under conditions of steady state 8n rp)/dt = O), no birth and death processes (B = 0 = De), no particle growth (lf = 0) and no particle velocity due to external forces Up = 0), namely... 

The second term, F, is the loss or death term and captures the processes that decrease the concentration or density. The rate of consumption, removal or death of particles of a given type must go to zero as the density of the particles goes to zero, Ff (p) 0 as /O 0. Otherwise, the concentration p of those particles can become negative. Introducing the per capita birth rate and loss rate, we write, where convenient. 

Thanks to these equations we are now in the position to take into account agglomeration besides processes driven by supersaturation like nucleation and growth however, there are restrictions. With increasing size, aggregates and agglomerates are running the risk to be destroyed by forces due to fluid flow. Then the intraparticle forces are decisive whether a disruption of particles takes place. David and Vil-lermaux (1991) have introduced a model for the volume-based disruption rate by birth (B) and death (D) with the disruption frequency I a-L ) and the dismption probability p a. -L ) as the decisive parameter ... 

---