Empirical formula for optical absorption by fractal aerosol agglomerates

H. Y. Chen; Magdy F. Iskander; J. E. Penner

doi:10.1364/AO.30.001547

Applied Optics
Vol. 30,
Issue 12,
pp. 1547-1551
(1991)
•https://doi.org/10.1364/AO.30.001547

Empirical formula for optical absorption by fractal aerosol agglomerates

H. Y. Chen, Magdy F. Iskander, and J. E. Penner

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H. Y. Chen, Magdy F. Iskander, and J. E. Penner, "Empirical formula for optical absorption by fractal aerosol agglomerates," Appl. Opt. 30, 1547-1551 (1991)

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Abstract

An empirical formula for calculating the optical absorption of fractal aerosol agglomerates over a broad frequency range is derived based on extensive calculations made on five smoke agglomerates having a widely varying number of particles ranging from 12 to 372 and an almost constant fractal dimension in the 1.7–1.9 range. The expansion coefficients in the empirical formula were determined by least-squares curve fiting of numerical data obtained using the volume integral equation formulation (VIEF).1 The formula satisfies the frequency dependence of the absorption at lower frequencies and up to the resonance frequency of the agglomerate. Expressions for the resonance frequency (f_o,), as well as the constant coefficients in the empiricial formula, are derived in terms of the structural dimensions, number and size of the primary particles, and the complex permittivity of the aerosol agglomerate. Numerical results obtained from the empirical formula are generally in good agreement with those calculated by the VIEF for soot particles with a magnitude of complex permittivity |ɛ^*| in the 1–7 range. The average error in the absorption calculation is <5%. The formula thus provides a simple and inexpensive method for calculating the optical absorption of aerosol agglomerates, which otherwise require complicated and expensive methods of calculation. This formula also facilitates the implementation of the accurate VIEF results in large climate codes such as the general circulation model.

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N	ɛ^*	a(μm)	A₁	A₂	A₃(× 10⁻⁴)
12	3 + j1	0.005	0.2106	1.3532	.3098
12	3 + j1	0.020	0.2047	1.2798	.3244
12	3 + j1	0.060	0.2141	1.2342	.3305
12	1.75 + j0.3	0.005	−0.0020	1.2231	.6554
12	1.75 + j0.3	0.020	−0.0030	1.1724	.7169
12	1.75 + j0.3	0.060	−0.004	1.1388	.8712
30	3 + j1	0.005	.2592	1.4275	.4629
30	3 + j1	0.020	.2557	1.3628	.4821
30	3 + j1	0.060	.2610	1.2914	.5160
30	1.75 + j0.3	0.005	0.042	1.2445	1.579
30	1.75 + j0.3	0.020	0.033	1.2209	1.965
30	1.75 + j0.3	0.06	0.032	1.1953	2.377
164	3 + j1	0.005	0.227	1.5811	1.178
164	3 + j1	0.020	0.209	1.4726	1.476
164	3 + j1	0.060	0.236	1.4255	1.673
164	1.75 + j0.3	0.005	−0.050	1.3419	7.125
164	1.75 + j0.3	0.020	−0.051	1.2952	9.043
164	1.75 + j0.3	0.060	−0.050	1.2636	1.085
295	3 + j1	0.005	0.222	1.6511	1.712
295	3 + j1	0.020	0.219	1.5808	2.171
295	3 + j1	0.060	0.216	1.4726	2.436
295	1.75 + j0.3	0.005	−0.034	1.3757	10.690
295	1.75 + j0.3	0.020	−0.049	1.3237	13.860
295	1.75 + j0.3	0.060	−0.040	1.2884	17.120
372	3 + j1	0.005	0.281	1.6748	1.987
372	3 + j1	0.020	0.296	1.5624	2.381
372	3 + j1	0.060	0.282	1.4890	2.763
372	1.75 + j0.3	0.005	−0.056	1.3891	14.910
372	1.75 + j0.3	0.020	−0.053	1.3349	16.230
372	1.75 + j0.3	0.060	−0.061	1.2984	19.030

A₁	\|ɛ^*\|
−.8184E − 01	1.4142
−.1200E − 01	2.2361
0.8745E − 01	2.8284
0.2574E + 00	3.1623
0.2660E + 00	3.6056
0.3124E + 00	4.1231
0.3677E + 00	4.2426
0.4099E + 00	4.4721
0.5228E + 00	5.0000
0.5300E + 00	5.0990
0.6290E + 00	5.3852
0.7159E + 00	5.6569
0.7576E + 00	5.8310
0.9220E + 00	6.4031
0.1123E + 01	7.0711

N	a(μm)	f₀ (VIEF)	f₀ (Eq.4)	error (%)
12	0.005	1.0 × 10¹⁶	9.24 × 10¹⁶	7.59
12	0.020	2.5 × 10¹⁵	2.31 × 10¹⁵	7.58
12	0.060	8.5 × 10¹⁴	7.74 × 10¹⁴	8.94
30	0.005	2.8 × 10¹⁶	2.74 × 10¹⁶	2.21
30	0.020	7.0 × 10¹⁵	6.85 × 10¹⁵	2.21
30	0.060	2.5 10¹⁵	2.28 × 10¹⁵	8.76
164	0.005	4.0 × 10¹⁶	3.70 × 10¹⁶	7.62
164	0.020	9.0 × 10¹⁵	9.24 × 10¹⁵	2.60
164	0.060	3.0 × 10¹⁵	3.08 × 10¹⁵	2.63
295	0.005	3.75 × 10¹⁶	3.54 × 10¹⁶	5.73
295	0.020	8.50 × 10¹⁵	8.84 × 10¹⁵	1.29
295	0.060	2.75 × 10¹⁵	2.94 × 10¹⁵	6.90
372	0.005	3.5 × 10¹⁶	3.44 × 10¹⁶	1.80
372	0.020	8.0 × 10¹⁵	8.59 × 10¹⁵	7.37
372	0.060	2.70 × 10¹⁵	2.86 × 10¹⁵	6.07

N	a(μm)	f₀₁ (VIEF)	f₀₁ (Eq. 5)	error (%)
12	0.005	1.0 × 10¹⁵	9.83 × 10¹⁴	1.69
12	0.020	2.5 × 10¹⁴	2.46 × 10¹⁴	1.68
12	0.060	8.0 × 10¹³	8.19× 10¹³	2.41
30	0.005	6.5 × 10¹⁴	6.54 × 10¹⁴	5.77
30	0.020	1.6 × 10¹⁴	1.63 × 10¹⁴	2.14
30	0.060	5.5 10¹³	5.55 × 10¹³	0.95
164	0.005	3.5 × 10¹⁴	3.51 × 10¹⁴	0.23
164	0.020	9.0 × 10¹³	8.77 × 10¹³	2.55
164	0.060	3.0 × 10¹³	2.92 × 10¹³	2.55
295	0.005	3.0 × 10¹⁴	2.91 × 10¹⁴	2.87
295	0.020	7.0 × 10¹³	7.29 × 10¹³	4.07
295	0.060	2.5 × 10¹³	2.43 × 10¹³	2.86
372	0.005	2.8 × 10¹⁴	2.72 × 10¹⁴	3.03
372	0.020	6.5 × 10¹³	6.79 × 10¹³	4.43
372	0.060	2.3 × 10¹³	2.26 × 10¹³	1.63

N	ɛ^*	a(μm)	A₁	A₂	A₃(× 10⁻⁴)
12	3 + j1	0.005	0.2106	1.3532	.3098
12	3 + j1	0.020	0.2047	1.2798	.3244
12	3 + j1	0.060	0.2141	1.2342	.3305
12	1.75 + j0.3	0.005	−0.0020	1.2231	.6554
12	1.75 + j0.3	0.020	−0.0030	1.1724	.7169
12	1.75 + j0.3	0.060	−0.004	1.1388	.8712
30	3 + j1	0.005	.2592	1.4275	.4629
30	3 + j1	0.020	.2557	1.3628	.4821
30	3 + j1	0.060	.2610	1.2914	.5160
30	1.75 + j0.3	0.005	0.042	1.2445	1.579
30	1.75 + j0.3	0.020	0.033	1.2209	1.965
30	1.75 + j0.3	0.06	0.032	1.1953	2.377
164	3 + j1	0.005	0.227	1.5811	1.178
164	3 + j1	0.020	0.209	1.4726	1.476
164	3 + j1	0.060	0.236	1.4255	1.673
164	1.75 + j0.3	0.005	−0.050	1.3419	7.125
164	1.75 + j0.3	0.020	−0.051	1.2952	9.043
164	1.75 + j0.3	0.060	−0.050	1.2636	1.085
295	3 + j1	0.005	0.222	1.6511	1.712
295	3 + j1	0.020	0.219	1.5808	2.171
295	3 + j1	0.060	0.216	1.4726	2.436
295	1.75 + j0.3	0.005	−0.034	1.3757	10.690
295	1.75 + j0.3	0.020	−0.049	1.3237	13.860
295	1.75 + j0.3	0.060	−0.040	1.2884	17.120
372	3 + j1	0.005	0.281	1.6748	1.987
372	3 + j1	0.020	0.296	1.5624	2.381
372	3 + j1	0.060	0.282	1.4890	2.763
372	1.75 + j0.3	0.005	−0.056	1.3891	14.910
372	1.75 + j0.3	0.020	−0.053	1.3349	16.230
372	1.75 + j0.3	0.060	−0.061	1.2984	19.030

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Equations (10)

Applied Optics