3.2.14. Absorptivity Model

The absorption coefficient submodel calculates a spectrally averaged total absorptivity value for a homogeneous ( in thermodynamic state and composition ) mixture of gaseous ${\rm CO}_2$ , ${\rm H}_2{\rm O}$ , and soot particles. It should be recognized that this model does not account for either the presence of volatilized hydrocarbon molecules nor for the spectral line broadening effects of ${\rm N}_2$ gas. The following implicit assumptions are made:

Thermodynamic equilibrium between soot and gas phase.

Homogeneous mixture over length scale of interest ( cf. input 1 )

Individual ( non agglomerated ) spherical soot particles with diameter much smaller than the radiation wavelength (Rayleigh scattering).

Absorptivity of the soot varies inversely with radiation wavelength.

The following quantities are required:

Length scale indicating the optical path length of interest, $L_{cell}$ in centimeters.

Mixture temperature, T, in Kelvin.

Total mixture pressure, $p_{mix}$ , in bar.

Partial pressures of the ${\rm CO}_2$ and ${\rm H}_2{\rm O}$ gaseous components, $p_{co2}$ , $p_{h2o}$ , in bar.

Soot volume fraction, $X_{soot}$ .

The absorptivity model generates the following output:

Spectrally averaged absorptivity, $\alpha$ , in ${\rm cm}^{-1}$ .

The absorptivity is based on empirical correlations for the total emittance of a homogeneous, isothermal mixture with a given optical path length. The correlations used in this model are based on empirical data covering a range of optical path lengths, temperatures, soot concentrations and pressures:

$1\ {\rm cm} \le L_{cell} \le 10^3\ {\rm cm}$

$600 K \le T \le 2400 K$

$10^{-8} \le X_{soot} \le 10^{-5}$

$0.1\ {\rm bar} \le p_{co2},\ p_{h2o} \le 1\ {\rm bar}$

The absorptivity values provided by the equations in this model are accurate to within 10% - 30% of their value with greater accuracy at higher temperatures, path lengths, and concentrations.

3.2.14.1. Theory

The total ( e.g. integrated over all wavelengths ) absorptivity of a homogeneous ( in composition and temperature ) thickness $L_{cell}$ layer of ${\rm CO}_2$ gas, ${\rm H}_2{\rm O}$ gas, and soot particles may be expressed in terms of the total emittance of the layer

(3.484) $\alpha = - {1 \over {L_{cell}}} \log \left( 1 - \kappa \right) ,$

where $\alpha$ is the total absorptivity and $\kappa$ is the total emittance. The total emittance of the mixture may be expressed in terms of the total emittance of the soot and gas phase (Siegel and Howell [9], Eq. (13-145)),

(3.485) $\kappa = \kappa_{soot} + \kappa_{gas} - \kappa_{soot} \kappa_{gas} ,$

where $\kappa_{soot}$ and $\kappa_{gas}$ are the total emittance of the soot and gas phase respectively as if the other phase were not present.

To evaluate the absorptivity within a given control volume, the layer length, $L_{cell}$ , is taken to be the geometric path length through the cell. This assumption ( cf. assumption 2 ) implies that the mixture composition and temperature are uniform within the given cell. For convenience, the hydraulic diameter may be used for the layer thickness (in three dimensions),

(3.486) $L_{cell} = 2 \left[ {3 \over 4} {V \over \pi} \right]^{1/3} ,$

where $V$ is the cell volume. Alternatively, Tezduyar [66] proposes a more expensive length scale for finite element grids,

(3.487) $L_{cell} = 2 \hat{s} \cdot \left( \sum_{i=1}^{n_e} \nabla \phi_i \right) ,$

where, $L_{cell}$ is the path length through the element in direction $\hat{s}$ , and $\phi_i$ is the finite element basis.

3.2.14.2. Emittance Model

The KAMELEON fire code ( Holen, et al. [52] ) employs the work of Felske and Tien [67] to provide the emittance of a mixture of ${\rm CO}_2$ , ${\rm H}_2{\rm O}$ , and soot particles. Assuming the absorptivity of the soot phase varies inversely with wavelength (Rayleigh scattering theory), a closed form expression may be obtained for the total emittance of the soot phase,

(3.488) $\kappa_{soot} = 1 - {{15} \over {\pi^4}} \Psi^{(3)} \left[ 1 + {{c X_{soot} T L_{cell}} \over {C_2}} \right] ,$

where, $X_{soot}$ is the soot volume fraction, $T$ is the temperature, $C_2 = 0.01438769$ m-K is the second Planck constant, and $c = 7.0$ ( Felske and Charalampopoulos [68] suggest $c = 5.0$ ). The pentagamma function $\Psi^{(3)}(x)$ is given by Abramowitz and Stegun [69],

(3.489) $\Psi^{(n)} \left( z \right) = {{{\rm d}^{n+1}} \over {{\rm d}z^{n+1}}} \log \left[ \Gamma \left( z \right) \right] = \left( -1 \right)^{n+1} \int_0^{\infty} {{t^n e^{-zt}}\over{1-e^{-t}}}{\rm d}t, \qquad n=1,2,3,\ldots$

(3.489) may be evaluated by the series expansion (Abramowitz and Stegun [69]),

(3.490) $\Psi^{(3)} \left( z \right) = 6 \sum_{k=0}^{\infty} {1 \over {\left( z + k \right)^4}} ,$

and by the seven-term asymptotic expansion,

(3.491) $\Psi^{(3)} \left( z \right) = {2 \over {z^3}} + {3 \over {z^4}} + {2 \over {z^5}} - {1 \over {z^7}} + {4 \over {3 z^9}} - {3 \over {z^{11}}} + {10 \over {z^{13}}} + \ldots$

(3.491) is accurate to within 1% of the value given by (3.490) for $z > 1.6$ and accurate to within 0.1% of the value given by (3.490) for $z > 2$ . A plot of the pentagamma function and the asymptotic expansion are provided in Figure 3.5 for reference.

Fig. 3.5 Pentagamma function and asymptotic expansion

The emittance of the gas phase is given by Leckner [70]. Leckner’s model is relatively involved and assumes that the path length, $L_{cell}$ , is given in centimeters, the temperature, $T$ , is given in Kelvin, and the pressure, $p$ , is given in bars. Leckner also defines a reference temperature, $T_{\circ} = 273$ K, and pressure, $p_{\circ} = 1$ bar, for reduction purposes. Two additional quantities used by Leckner are the scaled temperature, $\theta = T/1000{\rm K}$ and the logarithm of the optical path length, $\lambda_\nu = \log_{10} \left( p_\nu L_{cell} \right)$ where the subscript $\nu$ represents one of the species ${\rm CO}_2$ or ${\rm H}_2{\rm O}$ . These quantities are summarized in Table 3.8.

The emittance of the gas phase (cf. (3.485)) is the sum of the ${\rm CO}_2$ and ${\rm H}_2{\rm O}$ contributions less a correction factor which accounts for overlap in the ${\rm CO}_2$ and ${\rm H}_2{\rm O}$ absorption bands,

(3.492) $\kappa_{gas} = \kappa_{h2o} + \kappa_{co2} - \Delta \kappa ,$

where the species emittance at a given partial pressure and temperature is expressed in terms of a scale emittance, $\kappa_{\nu,\circ}$ .

(3.493) ${{\kappa_\nu} \over {\kappa_{\nu,\circ}}} = \exp \left( - \xi \left( \lambda_{max} - \lambda_{\nu} \right)^2 \right) \left( {{A P_E + B} \over {P_E + A + B - 1}} - 1 \right) + 1$

Table 3.9 summarizes the quantities on the right hand side of (3.493). The scale emittance, $\kappa_{\nu,\circ}$ , for both species is given by the expressions

(3.494) $\log \left( \kappa_{\nu,\circ} \right) = a_0 + \sum_{i=1}^M a_i \lambda_{\nu}^{i} ,$

(3.495) $a_i = c_{i0} + \sum_{j=1}^N c_{ij} \theta^{j} ,$

where the coefficients $a_i$ and $c_{ij}$ are given in Table 3.10 and Table 3.11 for ${\rm CO}_2$ and ${\rm H}_2{\rm O}$ respectively. (Leckner provides several alternative listings for the coefficients for calculating the total emittance of ${\rm CO}_2$ . The values listed in Table 3.10 are the values employed in the KAMELEON-II-FIRE program (1994).)

The effect of the overlap correction factor in (3.492) is relatively small so Leckner [70] employed an approximate expression obtained from emittance data for a total pressure of 1 bar and temperatures between 1000K and 2200K:

(3.496) $\Delta \kappa = \left( {{\zeta} \over {10.7 + 101 \zeta}} - 0.0089 \zeta^{10.4} \right) \left( \log_{10} \left[ \left( p_{co2} + p_{h2o} \right) L_{cell} \right] \right)^{2.76} ,$

where,

(3.497) $\zeta = {{p_{h2o}} \over {p_{h2o} + p_{co2}}} .$

The following observations are made to clarify the range of applicability of the absorptivity submodel specifically for hydrocarbon combustion applications. The absorptivity model does not account for the presence of volatilized hydrocarbon molecules which may have strong absorption bands in the infrared region. The VULCAN/KAMELEON fire code (Holen, et al. [52]) accounts for the presence hydrocarbon molecules by treating hydrocarbon molecules in the same manner as the ${\rm CO}_2$ and ${\rm H}_2{\rm O}$ product species ( cf. the partial pressure submodel ). This is a convenient although questionable assumption which provides for a zeroth order treatment of absorption by hydrocarbon molecules.

Table 3.8 Parameters used in Leckner’s gas phase emittance model.
Quantity	Definition
Temperature units, $\left[ T \right]$	Kelvin
Path length units, $\left[ L_{cell} \right]$	centimeters
Pressure units, $\left[ p \right]$	bar
Reference temperature, $T_{\circ}$	273 K
Reference pressure, $p_{\circ}$	1 bar
Scaled path length, $\lambda_\nu$
Scaled temperature, $\theta$	T/1000K

Table 3.9 Species-specific parameters used in (3.493)
Quantity	${\rm CO}_2$	${\rm H}_2{\rm O}$
Equivalent pressure, $P_E$	$P_E = p_{mix} \left( 1 + 0.28 \displaystyle{{p_{co2}} \over {p_{mix}}} \right)$	$P_E = p_{mix} \left( 1 + 1.49 \displaystyle{{p_{co2}} \over {p_{mix}}} \sqrt{\displaystyle{{T_{\circ}} \over T}} \right)$
	for $T > 700K$
	$\lambda_{max} = \log_{10} \left( 0.225 \theta^2 \right)$
Maxima location, $\lambda_{max}$		$\lambda_{max} = \log_{10} \left( 13.2 \theta^2 \right)$
	for $T < 700K$
	$\lambda_{max} = \log_{10} \left( 0.054 \theta^{-2} \right)$
Coefficient, $\xi$	$\xi = 1.47$	$\xi = 0.5$
Coefficient, $A$	$A = 1.0 + 0.1 \theta^{-1.45}$	$A = 1.888 - 2.053 \log_{10} \theta$
		$\theta = 2.145$ if $T < 750K$
Coefficient, $B$	$B = 0.23$	$B = 1.1 \theta^{-1.4}$

Table 3.10 Coefficients $C_{ij}$ for calculating the scale total emittance of CO2 from (3.494) and (3.495), (valid for T > 400K).
i	j (N=4)
(M=3)	0	1	2	3	4
0	-3.9781	2.7353	-1.9882	0.31054	0.015719
1	1.9326	-3.5932	3.7247	-1.4535	0.20132
2	-0.35366	0.61766	-0.84207	0.39859	-0.063356
3	-0.080181	0.31466	-0.19973	0.046532	-0.0033086

Table 3.11 Coefficients $C_{ij}$ for calculating the scale total emittance of ${\rm H}_2{\rm O}$ from (3.494) and (3.495), (valid for T > 400K).
i	j (N=2)
(M=2)	0	1	2
0	-2.2118	-1.1987	0.035596
1	0.85667	0.93048	-0.14391
2	-0.10838	-0.17156	0.045915