Coagulation kernel

The Kij in Eqs. (1)—(3) is the rate constant of aggregation. The set of rate constants for all i and j, often called the coagulation kernel, is an infinite symmetric matrix with nonnegative elements. In order to keep the form of Kij as simple as possible we shall assume that... 

In any kinetic analysis, the time, t, comes naturally as an independent variable, but in Eq. (3) it is, in fact, a variable merely proportional to the real time. It can be regarded as a rescaled time with the scaling factors depending on the actual units of both coagulation kernel and concentration as well as on the type of... 

The Smoluchowski coagulation equation has effectively been applied to model random irreversible homopolymerization of the monomer types presented in Table 4. As can be seen, in all cases the coagulation kernel has the bilinear form ... 

Otto, E., et al. (1999). Log-normal size distribution theory of Brownian aerosol coagulation for the entire particle size range Part 11—Analytical solution using Dahneke s coagulation kernel. J. Aerosol Science. 30, 1, 17-34. 

X is the crossover time when the balance between the aggregation processes is established. The validity of this approach was confirmed by Montecarlo simulations of the full equation using a constant coagulation kernel and a breakup probability equal to k(i+j) where a and k were adjustable parameters [15]. Spatial fluctuations were compensated by cluster breakup and the generalized Smoluchowski equation had a critical dimension dc < 1. 

The rate at which two particles with masses mj and m- and concentrations nt and tij collide is given by np., where (3 is the coagulation kernel (34, 35). New particles of mass (m, + m() are formed at a rate of anj fifty, where a is the stickiness coefficient. If all aggregates are composed of unit particles of the same size, then m, = i m and (m, + m/ = mi+j = (i + j)m where m, is the mass of the unit particle. If no new unit particles are produced and there is no nonaggregation process making particles, the change in concentration of particles of size i is the difference between the rate at which the particles are formed by collision of smaller particles and the rate at which they are lost to formation of larger particles. 

Hydrodynamic Models. The coagulation kernels are usually calculated for solid spheres with hydrodynamic models of different sophistication. The simplest calculation uses fluid flow in the absence of any effect of either particle on the flow. This flow level is known as rectilinear flow. The next level of sophistication involves calculating the flow around one particle, usually the larger of the two interacting particles. This level of calculation is known as curvilinear flow. Further levels of sophistication can be obtained by considering the particle trajectories as affected by the interacting flow fields of the particles, as well as any attractive or repulsive forces between them. 

The coagulation kernel for the rectilinear case of differential sedimentation is... 

Hill (42) derived a curvilinear version of the turbulent shear coagulation kernel. 

Algae are not usually spheres. Among other traits, many of the diatoms have spines that may protrude from the cell for distances much greater than the size of the central cell. Such protrusions can make a cell effectively much larger and increase its coagulation kernels more than the equivalent spherical... 

Figure 8. Effect of spines on algal concentrations. Spines increase the effective radius of a particle. Coagulation kernels have been modified by assuming that they make the effective capture radius greater by the length of the spine. There are no grazers. (Reproduced with permission from reference 38. Copyright 1993...

The modeling also showed the importance of some of the underdeveloped aspects of coagulation theory, particularly with regard to the interaction of the hydrodynamics and the fractal structure of the aggregates on the coagulation kernels. Traditional coagulation theory emphasized understanding the interactions of two solid spheres. Some work has been done on the flow in and... 

The rate at which two particles with masses mi and rrij and concentrations ni and rij collide is given by where (3 is the coagulation kernel (34, 35). 

Here K(u, V) is a function called coagulation kernel or constant of coagulation. It characterizes the collision frequency for particles of volumes V — u and u and satisfies the symmetry condition K(u, V) = K(V, u). 

Violation of symmetry condition of the coagulation kernel means, that volumetric concentration of drops does not remain constant, which is equivalent to implicit introduction of sources and drains into the system, whose intensity would depend on degree and form of kernel s asymmetry. The symmetry of the kernel K V,co) has another consequence, namely, that a function K(V,to) at a fixed net volume of drops V + ro = z = const, has an extreme value at V = to. 

Finally, we can write the expression for the coagulation kernel K(V,co) of the kinetic equation (11.1). Since the diffusion flux of drops of radius R2 toward the drop of radius Ri at the unit concentration ri2o = 1 has the meaning of the kernel of coagulation, the replacement of radiuses by volumes in (13.133) gives... 

Equation (15.35) represents a nonlinear integro-differential equation. For a coagulation kernel of the form of Eq. (15.33), its solution can be obtained by numerical or approximate methods. We confine ourselves to an approximate solution since this allows us to obtain a rather simple analytical solution. In Part V,... 

To close the system of equations represented by Eq. (15.41) it is necessary to express the right-hand part in terms of moments. To this end, the coagulation kernel should have a special form (for example, the form of a homogeneous polynomial of degrees V and co), or it is necessary to accept that the distribution conforms to a certain class (for example, a logarithmic normal distribution or a gamma distribution). The first method is called the method of fractional moments, and the second one the parametric method. 

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