Preservative distribution

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

In order to make it feasible to treat heartwood to satisfactory depths, it may be necessary to incise with an open pattern prior to applying this spore suspension. Since this could be done in a single operation, it would be economically attractive. Such a system might provide a means of achieving a uniform preservative distribution, which is currently not possible with incising alone without incurring excessive damage to the wood. 

Size distributions of (he agglomerates generated by cluster-duster aggregation compu-tations may approach a self-preserving form.This is found both forthe Langevin simulations and for l andom walk on a lattice (Meakin. 1986). Direct calculations of the self-preserving distribution,s are made in the sections that follow. 

The lime required for an initially monodisperse aerosol to reach the self-preserving distribution is shown in Fig. 8.6 as a function of Df for the free molecule and continuum... 

Example Derive an expression for the change in the number density with time for the self-preserving distribution in the continuum regime. 

Figure 11.5 Size distributions of Fig. 11.3 plotted in the self-preserving form (Husar and Whitby, 1973). The curve is based on the data. The self-preserving distribution for simultaneous coagulation and growth in the free-molecule range has not been calculated from theory, so no comparison is made.

The integral tf/T dtj equals 0.89 when evaluated from the self-preserving distribution for the free molecule range. The heaviest mass deposition occurs upstream where and are largest. 

Time to Reach the Self-Preserving Distribution (SPD) 217 Problems 219 References 220... 

Other than the Ornstein-Uhlenbeck process, the remaining piece of the splitting corresponds to Newtonian constant-energy (microcanonical) Hamiltonian dynamics. Since Hamiltonian dynamics leaves invariant any function of the energy, its corresponding Fokker-Planck operator (in this case the Liouvillian, = — //) will preserve distributions that are functions of the Hamiltonian H. This implies in particular that it preserves pp, which is proportional to exp(— 6//). Thus the forward propagator associated to the Hamiltonian system automatically preserves the Gibbs distribution, and we have... 

Show that the system (8.6)-(8.7) preserves distributions with densities of the form... 

Equation (4.8) has also been solved numerically, and it was shown that Equation (4.10), the self-preserving distribution, applies weU after a sufficient time lag [43,44]. Equations (4.10) and (4.11) were also derived using Monte Carlo simulations [45-47] with a collision frequency K(g, n) expressed as... 

F. 4.12 Effect of polydispersity on the static structure factor left constant aggregation number N with log-normal distribution of primary particle size, right variation of N with p(A/ a, av) as log-normal distribution and as self-preserving distribution as obtained with DLCA processes (Eq. (4.10))... 

Fig. 4.13 Normalised scattering curve and its negative slope in a log-log plot for polydisperse DLCA aggregates = 20 nm, (T]]i = 0.3, self-preserving distribution of N with N = 200)...

The influence of polydispersity is exemplarily shown in Fig. 4.24 for DLCA aggregates. Variation in primary particle size (monodisperse or log-normal distributed with = 0.4) as well as in aggregation number (log-normal distributed and a self-preserving distribution function, Eq. (4.10)) were considered. As in the previous section, a reciprocal correlation between N and jCp v was assumed for the calculation. The normalised diffusion coefficients are plotted versus a -axis that is scaled with the effective radius of gyration Rg etc (cf- Elq- (4.59)) ... 

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