Solution of the Coagulation Equation

Noting that the last term in this equation is the total number of particles (the subscript t on N(t) can be omitted) 

The double summation on the right-hand side is equal to N2 t). So (13.64) becomes 

At t — xc, A/(xf) = No- Thus, xf is the time necessary for reduction of the initial number concentration to half its original value. The timescale shortens as the initial number concentration increases. Consider an initial population of particles of about 0.2 pm diameter, for which K = 10 x 10 10 cm3 s 1. The coagulation timescales for No I04cm-3 and 106cm 3 are 

Continuous Coagulation Equation We now consider the solution to the continuous coagulation equation (13.61) assuming a constant coagulation coefficient K(q, v) = K, assuming that vo 0  

In order to solve (13.72) we also need to know N(t). We have seen that the total number concentration N(t) for any aerosol distribution assuming constant coagulation coefficient is given by (13.66). Substituting this expression for N(t) into (13.72), we find that the continuous coagulation equation becomes 

For t = Tc, N(Xc) = No and therefore the characteristic timescale Zc is the time necessary for the reduction of the initial number concentration to half its original value. The timescale decreases as the total number concentration of available particles increases. For example, for a typical atmospheric A o = iO cm and K = 5 x 10 cm s , the time necessary is approximately 55 h. However, if there are A o = 10 cm , the timescale is reduced to approximately 33 min. 

SO the number concentration of each -mer decreases as In the short time limit 

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DYNAMICS OF AEROSOL POPULATIONS 12.3.7 Solution of the Coagulation Equation... 

Solution of the Coagulation Equation by the Laplace Transform The Laplace transform of n v,t) is by definition... 

Galkin, V.A., On stability and stabilization of solutions of the coagulation equation. Diff. Urav. 14, (1978), 1863-1874. 

Figure 7.8 Sdt -prcscrving particle size distribution for Brownian coagulation, Tlie Ibnn is appaw-imatcly lognormal. The re.sult obtained by solution of the ordinary integrodiffereniial equation for the continuous spectrum is compared with the limiting solution of Hidy and Lilly (1965) for the discrete spectrum, calculated from the discrete form of the coagulation equation. Shown also are points calculated from analytical solutions for the lower and upper ends of the distribution (Friedlandcr and Wang. 1966).

Megaridis, C. M. Dobbdjs, R. A. 1990 A bimodal integral solution of the dynamic equation for an aerosol undergoing simultaneous particle inception and coagulation. Aerosol... 

Note that, as opposed to the coagulation of bubbles in a laminar flow, coagulation in a turbulent flow is impossible in the absence of molecular forces, since Dt- at<5=r a )—>0. Therefore the integral term entering the solution of the diffusion equation has a nonintegrable singularity that can be eliminated only by the introduction of a molecular attraction force growing as at > 0. 

The existence of a self-similar solution for the coagulation equation has been addressed by the following Lushnikov (1973), Ziflf et al (1983), van Dongen and Ernst (1988). 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

In addition to the aluminum chemical dissolution, other important chemical processes that occur in an electrochemical cell are the speciation of the coagulants. Once the aluminum is dissolved (chemically or electrochemically) different species can be formed, depending on the pH of the solution. For the case of aluminum, the reactions involved are shown below (4.56)-(4.60). In this set of equations, it has also been included the carbonate/bicarbonate equilibria (4.61)-(4.62) and ionization of water, due to its high influence on the calculation of the pH value (4.63). 

The second term of this equation represents the loss of particles in the volume size range v + dv resulting from coagulation of particles of volume v and volume . The term K(v, ) is the combined coagulation coefficient for tljese two particles. Thus this equation gives an expression for the net rate of change of particles whose volumes lie between v and v + dv. Solution of the equation with appropriate initial conditions gives the number of particles of volume v + dv at any time t. 

The complexes are FeOH and Fe(OH)3. Also note that the OH ion is a participant in these reactions. This means that the concentrations of each of these complex ions are determined by the pH of the solution. In the application of the above equations in an actual coagulation treatment of water, conditions must be adjusted to allow maximum precipitation of the solid represented by Fe(OH)2(j). To allow for this maximum precipitation, the concentrations of the complex ions must be held to the minimum. The values of the equilibrium constants given above are at 25°C. 

Xiong, Y. Pratsinis, S. E. 1993 Formation of agglomerate particles by coagulation and sintering - part I. A two-dimensional solution of the population balance equation. Journal of Aerosol Science 24, 283-300. 

For additional solutions of the discrete and continuous coagulation equations, the interested reader may wish to consult Drake (1972), Mulholland and Baum (1980), Tambour and Seinfeld (1980), and Pilinis and Seinfeld (1987). 

In (13.83), a stable particle has been assumed to have a lower limit of volume of wo-From the standpoint of the solution of (13.83), it is advantageous to replace the lower limits vo of the coagulation integrals by zero. Ordinarily this does not cause any difficulty, since the initial distribution n(v, 0) can be specified as zero for v < vo, and no particles of volume v < vo can be produced for t > 0. Homogeneous nucleation provides a steady source of particles of size vq according to the rate defined by Jo(t). Then the full equation governing n(v, t) is as follows ... 

Pilinis, C., and Seinfeld, J. H. (1987) Asymptotic solution of the aerosol general dynamic equation for small coagulation, J. Colloid Interface Sci. 115, 472-479. 

Williams, M. M. R. (1984) On some exact solutions of the space- and time-dependent coagulation equation for aerosols, J. Colloid. Interface Sci. 101, 19-26. 

FIGURE 12.12 Solution of the continuous coagulation equation for an exponential initial distribution given by (12.97). 

Enukashvili, I. M., On the Solution of the Kinetic Coagulation Equation, Izv. Geophys. Ser. English Transl. Bull. Acad. Sci. USSR, No. 10, 944-948 (1964a). 

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