Update and Erratum on the Numerical Evaluation of the Exact Coagulation Cernel for Simultaneous Brownian Motion and Gravitational Settling

The exact mathematical description of the coagulation kernel under combined Brownian motion and gravitational settling entails an infinite sum of the ratio of incomplete Bessel functions. Numerical evaluation of this sum is given by Simons and colleagues (1986) for small arguments (β< 100), and by Sajo (2008) for large arguments (100 <β⩽ 20 000) using multiple-precision arithmetic. In the latter work, the original paper in Aerosol Science and Technology (42:134–139, 2008) was printed with a typographic error in the exponent of one of the parameters for the least-squares approximation in the range (5.5 β⩽ 1110). This error is corrected here. In addition, an updated fit is provided which yields a better fidelity to the underlying data in the range of (5.5 β⩽ 2000).

In analyzing the problem of simultaneous Brownian and gravitational coagulation, Simons et al. (1986) give their analytical formula for the ratio of the sum kernel and the exact kernel as a function γ(β); where β is a dimensionless quantity that shows the relative importance of diffusion versus gravitational settling. In an aerosol where an appreciable amount of particles are observed at both ends of the particle size distribution β∈ (0, ∞) and γ∈ (1, 1.27). Simons (1986) investigated the behavior of γ(β) for β∈ (0, 100), whereas Sajo (2008) obtained values for β∈ (0, 20 000).

In the paper by Sajo (2008), for least squares fit #2 of the function γ(β), Table 2 of the original publication gives the coefficient C₁, Equation (8), as 1.1212502E-1. The exponent of this figure is incorrectly printed. The correct value is 1.1212502E-2, that is 1/10th of the value printed therein. The fit is remarkably sensitive to this parameter. Depending on the magnitude of the argument, using the incorrect coefficient may lead to significant departure from the correct values of the γ(β) function, ranging from 14% at β= 6 to almost 100% at β= 1000.

Readers interested in computational aerosol sciences may find it useful that in the range of β∈ (5.5, 2000) using the shape function given in Equation (8), a better fit is provided by the following set of parameters:

• A1 = 2.04316e-01

• A2 = −1.79707e-02

• B1 = 0

• B2 = 8.84424e + 00

• B3 = −1.83794e + 01

• B4 = 1.39340e + 01

• B5 = −3.25641e + 00

• C1 = 2.07133e-02

• E = 1.95231e-01

The statistical quality of this fit is characterized as

• R-squared = 1.00000

• Average Deviation = 3.13758e-05

• Total-sum-of-squares = 3.24133e-01

• Standard-error-of-estimate = 4.35557e-05

At any value above β= 2000, the asymptotic extension given by Equation (5) may be used.