5.2 Radiative heat transfer
5.2.2 Rosseland diffusion approximation
In the Rosseland diffusion approximation, the volume averaged divergence of the radia- tive heat flux is expressed as [Modest, 2013]
h∇·~q00radi= ∇·(−krad∇hTi) (5.4)
and the volume averaged radiative heat flux is
h~q00radi=−krad∇hTi (5.5) where radiative conductivity krad is a function of the effective refractive index of the
porous particle, the local temperature, and the Rosseland-mean extinction coefficient [Modest, 2013], krad = 16n 2 effσhTi 3 3βtrR (5.6)
where neff is the effective refractive index, σ = 5.6704×10−8W m−2K−4 and is the
Stefan–Boltzmann constant, and βtrR is the Rosseland-mean extinction coefficient. The
§5.2 Radiative heat transfer
refractive indices of the solid and fluid phases,
neff = (1−φ)ns+φnf (5.7)
When the solid phase contains both CaCO3and CaO, the real part of the refractive index
of the solid phase used to evaluate the radiative conductivity of the resulting participating medium is modeled as a concentration averaged combination of the real parts of the refractive indices for CaCO3 and CaO.
ns= (1−X)nCaCO3+XnCaO (5.8)
The Rosseland-mean extinction coefficient is defined as [Modest, 2013]
1 βtrR = Z ∞ λ=0 1 βtrλ dIbλ dT dλ Z ∞ λ=0 dIbλ dT dλ (5.9)
where Ibλ is the spectral blackbody intensity. The gray-band approximation of Eq. (5.9),
evaluated with 120 bands, is used in Eq. (5.6) to obtainkrad.
Boundary conditions At the center of the particle, the Rosseland diffusion approxi- mation does not affect the boundary condition given by Eq. (4.26). The approximation replaces the radiative flux termqrad00
in the conservation of energy equation boundary conditions at the particle surface for either considered heating mode given by Eqs. (4.27) and (4.29) in Section 4.4.2. keff ∂hTi ∂r − q00rad r=rp = (keff+krad) ∂hTi ∂r r=rp (5.10)
No further changes to the surface boundary condition for the sweep gas controlled heating mode given by Eq. (4.29) are needed.
For the directly irradiated heating mode, the net radiative flux into the particle in Eq. (4.27) includes terms for absorbed solar irradiation, absorbed radiation emitted by the distant enclosure walls, and the radiation emitted by the particle surface.
q00net=αeff,solarq00surf+αeff,wσTw4 −εeff,pσ
hTi|r=r
p
4
(5.11) whereαeff,solar is the effective absorptivity for solar radiation, q00surf is the time-periodic
radiative flux incident at the particle surface defined by Eq. (4.31),αeff,w is the effective
absorptivity for radiation emitted by the enclosure walls,Tw is the temperature of the
distant enclosure walls, andεeff,pis the effective emissivity for the particle surface.
Regions of CaCO3, CaO, and pore space at the surface are treated as optically
5 Closure models for intra-particle radiative transfer and mass diffusion
radiative properties of the individual species of the system. A flowchart of the process for calculatingαeff,solar,αeff,w, andεeff,pis shown in Fig. 5.3. The notation convention for
surface radiative properties used in this work follows that of Modest [2013]. Absorptivity and emissivity variables without superscript denote quantities averaged over all incoming and outgoing directions, respectively. For reflectivity, a prime superscript denotes a directional variable, and a hemisphere superscript denotes a hemispherically averaged variable. When two superscripts are used, the first refers to incoming directions and the second refers to outgoing directions. Reflectivity variables without superscripts denote quantities averaged over all incoming and outgoing directions.
As shown in Fig. 5.3, the spectral, directional–hemispherical reflectivityρ0λ of bulk
CaCO3, bulk CaO, and the fluid phase are first evaluated from their respective refractive
indices and then combined into an effective spectral, directional–hemispherical reflectivity
ρ0λ,eff, where each constituent is treated as optically discrete. The effective spectral,
directional–hemispherical reflectivityρ0λ,eff is next averaged over all incident directions
to yield the effective spectral, hemispherical reflectivityρλ,eff. Finally, the total effective
absorptivity αeff and emissivity εeff are calculated by averaging the effective spectral,
hemispherical reflectivity over the appropriate incident or emitted spectra. For further reading on the evaluation of effective radiative properties of the surface of a CaO looping sorbent particle, the reader is directed to Yue and Lipi ´nski [2015a].
Calcium carbonate refractive index mλ,CaCO3 Calcium oxide refractive index mλ,CaO Calcium carbonate spectral, directional– hemispherical reflectivity ρ0 λ,CaCO3 Calcium oxide spectral, directional– hemispherical reflectivity ρ0 λ,CaO Effective spectral, directional– hemispherical reflectivity ρ0 λ,eff Spectral, hemispherical reflectivity ρλ,eff Total reflectivity for the
incident spectrum ρeff, Tsrc= 5777 K orTw
Total absorptivity αeff
Total reflectivity for the emitted spectrum ρeff, Tsrc = hTir=rp
Total emissivity εeff
Figure 5.3: Flowchart for calculating effective absorptivity and emissivity
The complex refractive indices for bulk CaCO3 and CaO are used directly in the
§5.2 Radiative heat transfer
CaCO3and CaO, respectively. The fluid phase is modeled as radiatively non-participating,
thus, ρ0λ,f is zero. The reflectivities of CaCO3, CaO, and the fluid phase are combined
using the local reaction extent and porosity of the surface to yield the effective spectral, directional–hemispherical reflectivity for the surface ρ0λ,eff according to the following
equation,
ρλ0,eff = (1−φ)(1−X)ρλ0,CaCO3 +Xρ0λ,CaO+φρ0λ,f (5.12)
whereφandX are the surface porosity and local reaction extent, respectively, taken as
the porosity and local reaction extent of the outermost volume element, respectively. The effective spectral, directional–hemispherical reflectivity ρ0λ,eff is then averaged
over all incident directions yielding the effective spectral, hemispherical reflectivityρλ,eff.
ρλ,eff = 1 π
Z
2π
ρ0λ,effcosθdΩ (5.13)
whereθis the polar angle andΩis the solid angle. The effective spectral, hemispherical
reflectivityρλ,eff is then averaged over a blackbody spectrum. If the spectrum represents
incident irradiation, the average yields the total effective absorptivityαeff for that spec-
trum; if the spectrum represents emitted radiation, the average yields the total effective emissivityεeff. αeff,src = Z ∞ λ=0(1−ρλ,eff)Ebλ,srcdλ σTsrc4 (5.14) εeff,src = Z ∞ λ=0(1−ρλ,eff)Ebλ,srcdλ σTsrc4 (5.15) whereEbλ is the spectral blackbody emissive power and Tsrcis the temperature of the
incident or emitted blackbody spectrum.
Solar irradiation, radiation emitted by the enclosure walls and incident on the par- ticle surface, and emitted radiation from the particle surface are modeled to have the spectra of a blackbody at 5777 K, the wall temperatureTw, and the surface temperature
hTi|r=rp, respectively. Effective absorptivity of the particle surface therefore varies with surface composition, while effective emissivity varies with both surface composition and temperature.