**Nongray EWB and WSGG Radiation Modeling in Oxy-Fuel Environments**

Osama A. Marzouk<sup>1</sup> and E. David Huckaby<sup>2</sup>

<sup>1</sup>*U.S. Department of Energy, National Energy Technology Laboratory; and West Virginia University Research Corporation* <sup>2</sup>*U.S. Department of Energy, National Energy Technology Laboratory USA*

#### **1. Introduction**

492 Computational Simulations and Applications

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According to a recent U.S. Greenhouse Gas Emissions Inventory (1), about 42% of 2008 CO2 (a greenhouse gas) emissions in the U.S were from burning fossil fuels (especially coal) to generate electricity. The 2010 U.S. International Energy Outlook (2) predicts that the world energy generation using coal and natural gas will continue to increase steadily in the future. This results in increased concentrations of atmospheric CO2, and calls for serious efforts to control its emissions from power plants through *carbon capture* technologies. Oxy-fuel combustion is a *carbon capture* technology in which the fossil fuel is burned in an atmosphere free from nitrogen, thereby reducing significantly the relative amount of N2 in the flue-gas and increasing the mole fractions of H2O and CO2. This low concentration of N2 facilitates the capture of CO2. The dramatic change in the flue composition results in changes in its thermal, chemical, and radiative properties. From the modeling point of view, existing transport, combustion, and radiation models that have parameters tuned for air-fuel combustion (where N2 is the dominant gaseous species in the flue) may need revision to improve the predictions of numerical simulations of oxy-fuel combustion.

In this chapter, we consider recent efforts done to revise radiation modeling for oxy-fuel combustion, where five new radiative-property models were proposed to be used in oxy-fuel environments. All these models use the weighted-sum-of-gray-gases model (WSGGM). We apply and compare their performance in two oxy-fuel environments. Both environments consist of only H2O and CO2 as mixture species, and thus there is no N2 dilution, but the environments vary in the mole fractions of these two species. The first case has a CO2 mole fraction of 65%, whereas the second has a CO2 mole fraction of 90%. The former case is more relevant to what is referred to as *wet flue gas recycle (wet FGR)* where some flue gas is still recirculated into the furnace, but after to act as coal carrier or diluent (to temper the flame temperature). On the other hand, the second case is more relevant to what is referred to as *dry flue gas recycle (dry FGR)* where some flue gas is still recirculated into the furnace but after a stage of H2O condensation. This increases the CO2 fraction in the recycled flue gas (RFG) and consequently in the final flue gas leaving the furnace and the boiler of the plant.

To highlight the influence of using an air-fuel WSGGM (a model with parameters were developed for use in air-fuel combustion) in oxy-fuel environments, the air-fuel WSGGM of

then Equation (4) can be re-written as

respective spectral quantities.

and the radiative source becomes

that belongs to the *i*

source =

exist where much fewer RTEs are solved to resolve the spectrum.

*d Itot*

is what acts to augment the radiation. Therefore, the RTE of the *i*

*d Ii*

through its full-spectrum averaging. In that approach, the RTE becomes

*d s* <sup>=</sup> *<sup>s</sup>*<sup>ˆ</sup> • ∇*Itot* <sup>=</sup> *kgray*

source = *kgray*

*d s* <sup>=</sup> *<sup>s</sup>*<sup>ˆ</sup> • ∇*Ii* <sup>=</sup> *ki*

*th* spectral fraction. The source term is

*i ki* 

source = ∑

 ∞ *η*=0

In the most comprehensive approach, known as the line-by-line (LBL) approach (4), the spectrum is divided into high-resolution intervals where *k<sup>η</sup>* is approximately constant over each interval, and an RTE per direction is solved for each interval. Then, the total radiative intensity and the total radiative source term are obtained from spectral integration of the

Nongray EWB and WSGG Radiation Modeling in Oxy-Fuel Environments 495

The spectral absorption coefficient for gaseous species is known to vary rapidly and it is far from being a smooth function of *η*. This is due to the fact that radiation from a hot gas (e.g., a flame) is absorbed by combustion gases only at wavenumbers at which electrons can be excited to the next discrete energy level. Therefore these gases are radiatively-transparent at certain portions of the spectrum, but become radiatively-active at other portions (8). The LBL approach for solving the radiation problem is not practical in real combustion simulations, where such approach would involve hundreds of thousands of RTEs. Alternative approaches

Of course the extreme case is to solve a single RTE per direction for the entire spectrum, assuming constant properties over the entire spectrum. This approach is referred to as *gray*. This simplifies the calculations greatly, but completely loses the spectral character of radiation

As a compromise between the formidable LBL approach and the too-coarse gray approach, we apply two other approaches where spectral variation is accounted for, but with a much lower resolution than the LBL. These approaches are the nongray WSGGM and the box model based on the EWBM. In either approach, for each direction a small number of RTEs solved, each of which covers a fraction of the spectrum where the linear absorption coefficient is considered to be constant, and where the fraction of the total blackbody radiation over that spectral portion

where the quantity *ai* is the fraction of the total (i.e., spectrally-integrated) blackbody radiation

In the box/EWB, the spectrum partitioning is based on modeled band structure that reflects the presence of the vibration-rotation or pure-rotation bands of the emitting/absorbing species. In the nongray WSGGM, no direct partitioning of the spectrum is done, and each of

*ai Ib*,*tot* − *Ii*

*k<sup>η</sup> G<sup>η</sup>* − 4*π k<sup>η</sup> Ib*,*<sup>η</sup>*

*dη* (6)

*Ib*,*tot* <sup>−</sup> *Itot* (7)

*Gtot* <sup>−</sup> <sup>4</sup> *<sup>π</sup> Ib*,*tot* (8)

*th* fraction is

*Gi* <sup>−</sup> *ai* <sup>4</sup> *<sup>π</sup> Ib*,*tot* (10)

(9)

Smith et al. (1982) is included as the sixth WSGGM. The WSGG solutions are accompanied by solutions using the more-rigorous exponential wide band model (EWBM) approach and the spectral line-base weighted-sum-of-gray-gases model (SLW) approach. All the solutions presented here are nongray, meaning that the radiative properties of the emitting/absorbing mixture vary across the spectrum and multiple radiative transfer equations (RTEs) are solved per spectrum. The total pressure is 1 atm (101 325 N/m2).

#### **2. Mathematical description**

The spectral radiative transfer equation (RTE) along a path *s* (with a unit vector *s*ˆ) in an emitting/absorbing medium is (3; 4)

$$\frac{d\ I\_{\eta}(\mathbf{s},\boldsymbol{\eta})}{d\mathbf{s}} = \hat{\mathbf{s}} \bullet \nabla I\_{\eta} = k\_{\eta}(\mathbf{s},\boldsymbol{\eta}) \left( I\_{b,\eta}(\mathbf{s},\boldsymbol{\eta}) - I\_{\eta}(\mathbf{s},\boldsymbol{\eta}) \right) \tag{1}$$

where *η* is the wavenumber (its SI unit is 1/m), *I<sup>η</sup>* is the spectral radiative intensity (its SI unit is W/m2 1 <sup>m</sup> steradian), *Ib*,*<sup>η</sup>* is the blackbody radiative intensity, and *<sup>k</sup><sup>η</sup>* is the spectral linear radiative absorption coefficient (its SI unit is 1/m). From a molecular view, when *k<sup>η</sup>* is uniform along a path, 1/*k<sup>η</sup>* is the mean free path traveled by a photon until it is absorbed by an electron (3; 4). From a continuum view, and from Equation (1), it can also be viewed as simply the fraction of radiation pencil absorbed over a distance of 1 meter (5). The blackbody radiative intensity, or the Planck function, (*Ib*,*η*) depends on the wavenumber (or wavelength), local temperature, and the refractive index of the medium. This dependence has the following form:

$$I\_{b, \eta}(s, \eta) = \frac{2 \ln c\_0^2 \eta^3}{n^2 \left(\exp\left(\frac{h c\_0 \eta}{k\_B T}\right) - 1\right)}\tag{2}$$

where *n* is the refractive index of the medium (being unity for vacuum), *h* is the Planck constant (in SI units, *<sup>h</sup>* <sup>=</sup> 6.6261 <sup>×</sup> <sup>10</sup>−<sup>34</sup> J-s), *<sup>c</sup>*<sup>0</sup> is the speed of light in vacuum (in SI units, *<sup>c</sup>*0= 299 792 458 m/s), *kB* is the Boltzmann constant (in SI units, *kB* <sup>=</sup> 1.3807 <sup>×</sup> <sup>10</sup>−<sup>32</sup> J/K), and *T* is the temperature. When Equation (2) is integrated over the entire spectrum, we obtain the total blackbody radiation intensity, which depends only on the medium type (through its refractive index) and the local temperature, as follows:

$$I\_{b,tot}(\mathfrak{n}, \mathfrak{T}) = n^2 \sigma \,\mathrm{T}^4/\pi \tag{3}$$

where *<sup>σ</sup>* is the Stefan-Boltzmann constant (in SI units, *<sup>σ</sup>* = 5.67×10−<sup>8</sup> W/m2-K4). In modeling, the thermal effect of radiation appears in the energy equation through a radiative source term (its SI unit is W/m3), which takes the following form:

$$\text{source} = \int\_{\eta=0}^{\infty} \left( \int\_{4\pi} k\_{\eta} \, I\_{\eta} d\Omega - 4\pi \, k\_{\eta} \, I\_{b,\eta} \right) d\eta \tag{4}$$

where Ω is the solid angle (in steradian). This source term is negative when the radiation has cooling effect on the medium, as in flames and reacting flows (6; 7). Defining the spectral direction-integrated incident radiation

$$\mathcal{G}\_{\eta} \equiv \int\_{4\pi} I\_{\eta} d\Omega \tag{5}$$

then Equation (4) can be re-written as

2 Numerical Simulations / Book 1

Smith et al. (1982) is included as the sixth WSGGM. The WSGG solutions are accompanied by solutions using the more-rigorous exponential wide band model (EWBM) approach and the spectral line-base weighted-sum-of-gray-gases model (SLW) approach. All the solutions presented here are nongray, meaning that the radiative properties of the emitting/absorbing mixture vary across the spectrum and multiple radiative transfer equations (RTEs) are solved

The spectral radiative transfer equation (RTE) along a path *s* (with a unit vector *s*ˆ) in an

where *η* is the wavenumber (its SI unit is 1/m), *I<sup>η</sup>* is the spectral radiative intensity (its SI unit

<sup>m</sup> steradian), *Ib*,*<sup>η</sup>* is the blackbody radiative intensity, and *<sup>k</sup><sup>η</sup>* is the spectral linear radiative absorption coefficient (its SI unit is 1/m). From a molecular view, when *k<sup>η</sup>* is uniform along a path, 1/*k<sup>η</sup>* is the mean free path traveled by a photon until it is absorbed by an electron (3; 4). From a continuum view, and from Equation (1), it can also be viewed as simply the fraction of radiation pencil absorbed over a distance of 1 meter (5). The blackbody radiative intensity, or the Planck function, (*Ib*,*η*) depends on the wavenumber (or wavelength), local temperature,

<sup>0</sup> *<sup>η</sup>*<sup>3</sup>

*k<sup>η</sup> Iηd*Ω − 4*π k<sup>η</sup> Ib*,*<sup>η</sup>*

 *h c*<sup>0</sup> *η kB T* − 1

*Ib*,*η*(*s*, *η*) − *Iη*(*s*, *η*)

*Ib*,*tot*(*n*, *T*) = *n*<sup>2</sup> *σ T*4/*π* (3)

*Iηd*Ω (5)

(2)

*dη* (4)

(1)

*d s* <sup>=</sup> *<sup>s</sup>*<sup>ˆ</sup> • ∇*I<sup>η</sup>* <sup>=</sup> *<sup>k</sup><sup>η</sup>* (*s*, *<sup>η</sup>*)

and the refractive index of the medium. This dependence has the following form:

*Ib*,*η*(*s*, *<sup>η</sup>*) = <sup>2</sup> *h c*<sup>2</sup>

where *<sup>σ</sup>* is the Stefan-Boltzmann constant (in SI units, *<sup>σ</sup>* = 5.67×10−<sup>8</sup> W/m2-K4).

 4*π*

*G<sup>η</sup>* ≡ 4*π*

source term (its SI unit is W/m3), which takes the following form:

cooling effect on the medium, as in flames and reacting flows (6; 7). Defining the spectral direction-integrated incident radiation

 ∞ *η*=0

source =

*n*2 exp

where *n* is the refractive index of the medium (being unity for vacuum), *h* is the Planck constant (in SI units, *<sup>h</sup>* <sup>=</sup> 6.6261 <sup>×</sup> <sup>10</sup>−<sup>34</sup> J-s), *<sup>c</sup>*<sup>0</sup> is the speed of light in vacuum (in SI units, *<sup>c</sup>*0= 299 792 458 m/s), *kB* is the Boltzmann constant (in SI units, *kB* <sup>=</sup> 1.3807 <sup>×</sup> <sup>10</sup>−<sup>32</sup> J/K), and *T* is the temperature. When Equation (2) is integrated over the entire spectrum, we obtain the total blackbody radiation intensity, which depends only on the medium type (through its

In modeling, the thermal effect of radiation appears in the energy equation through a radiative

where Ω is the solid angle (in steradian). This source term is negative when the radiation has

per spectrum. The total pressure is 1 atm (101 325 N/m2).

*d I<sup>η</sup>* (*s*, *η*)

refractive index) and the local temperature, as follows:

**2. Mathematical description**

is W/m2 1

emitting/absorbing medium is (3; 4)

$$\text{source} = \int\_{\eta=0}^{\infty} \left( k\_{\eta} \, G\_{\eta} - 4 \pi \, k\_{\eta} \, \text{I}\_{b,\eta} \right) d\eta \tag{6}$$

In the most comprehensive approach, known as the line-by-line (LBL) approach (4), the spectrum is divided into high-resolution intervals where *k<sup>η</sup>* is approximately constant over each interval, and an RTE per direction is solved for each interval. Then, the total radiative intensity and the total radiative source term are obtained from spectral integration of the respective spectral quantities.

The spectral absorption coefficient for gaseous species is known to vary rapidly and it is far from being a smooth function of *η*. This is due to the fact that radiation from a hot gas (e.g., a flame) is absorbed by combustion gases only at wavenumbers at which electrons can be excited to the next discrete energy level. Therefore these gases are radiatively-transparent at certain portions of the spectrum, but become radiatively-active at other portions (8). The LBL approach for solving the radiation problem is not practical in real combustion simulations, where such approach would involve hundreds of thousands of RTEs. Alternative approaches exist where much fewer RTEs are solved to resolve the spectrum.

Of course the extreme case is to solve a single RTE per direction for the entire spectrum, assuming constant properties over the entire spectrum. This approach is referred to as *gray*. This simplifies the calculations greatly, but completely loses the spectral character of radiation through its full-spectrum averaging. In that approach, the RTE becomes

$$\frac{d\,I\_{tot}}{ds} = \\$ \bullet \,\nabla I\_{tot} = k\_{\text{gray}} \left( I\_{b,tot} - I\_{tot} \right) \tag{7}$$

and the radiative source becomes

$$\text{source} = k\_{gray} \left( G\_{tot} - 4 \,\pi \, I\_{b,tot} \right) \tag{8}$$

As a compromise between the formidable LBL approach and the too-coarse gray approach, we apply two other approaches where spectral variation is accounted for, but with a much lower resolution than the LBL. These approaches are the nongray WSGGM and the box model based on the EWBM. In either approach, for each direction a small number of RTEs solved, each of which covers a fraction of the spectrum where the linear absorption coefficient is considered to be constant, and where the fraction of the total blackbody radiation over that spectral portion is what acts to augment the radiation. Therefore, the RTE of the *i th* fraction is

$$\frac{d}{ds}\frac{I\_l}{s} = \hat{\mathbf{s}} \bullet \nabla I\_l = k\_l \left( a\_l \, I\_{b,tot} - I\_l \right) \tag{9}$$

where the quantity *ai* is the fraction of the total (i.e., spectrally-integrated) blackbody radiation that belongs to the *i th* spectral fraction. The source term is

$$\text{source} = \sum\_{i} k\_i \left( G\_i - a\_i 4 \,\pi \, I\_{b,tot} \right) \tag{10}$$

In the box/EWB, the spectrum partitioning is based on modeled band structure that reflects the presence of the vibration-rotation or pure-rotation bands of the emitting/absorbing species. In the nongray WSGGM, no direct partitioning of the spectrum is done, and each of

η **[1/cm]**

λ **[**μ**m]**

**10<sup>0</sup> 10<sup>1</sup> 10<sup>2</sup>**

**<sup>2000</sup> <sup>4000</sup> <sup>6000</sup> <sup>8000</sup> <sup>10000</sup> <sup>0</sup>**

**Eb [W/m2/(1/cm)]**

**k [1/m]**

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

Nongray EWB and WSGG Radiation Modeling in Oxy-Fuel Environments 497

**0**

linear absorption coefficient and at 1 500 K for two oxy-fuel environments

**35% H2O 65% CO2**

Fig. 1. Spectra (versus wavenumber) of the blackbody emissive power and the box/EWB

**10 -4 x Eb [W/m2**

**k [1/m]**

**0**

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

**0**

Fig. 2. Spectra (versus wavelength) of the blackbody emissive power and the box/EWB

box/EWB approach serves to provide a benchmark solution to compare with. In the nongray WSGG approach, Equation (9) is still solved as was the case in the box model, but the physical interpretation and the evaluation of the *ki* and *ai* are very different. The WSGG approach (5; 16; 19–21) is based on the presence of *N* hypothetical gray gases; *N* − 1 are absorbing/emitting, and one is clear (no radiative emission or absorption) to represent the presence of spectral windows. Each absorbing/emitting gray gas has a constant *ki*, and the clear gas has *k*0=0. The fractions *ai* are cast as a polynomial of temperature only. The parameters of a WSGGM are the *ki* and the polynomial coefficients for each absorbing/emitting gray gas. There are (*N* − 1) × (*M* + 1) model parameters for *N* gray gases and a polynomial order *M*. The parameters for a single total pressure and a single gas composition (H2O and CO2 partial pressures) are calculated through an optimization process.

linear absorption coefficient and at 1 500 K for two oxy-fuel environments

**2**

**4**

**6**

**8**

**10**

**/**μ**m]**

η **[1/cm]**

λ **[**μ**m]**

**10<sup>0</sup> 10<sup>1</sup> 10<sup>2</sup>**

**<sup>2000</sup> <sup>4000</sup> <sup>6000</sup> <sup>8000</sup> <sup>10000</sup> <sup>0</sup>**

**Eb [W/m2/(1/cm)]**

**0**

**10 -4 x Eb [W/m2**

**0**

**2**

**4**

**6**

**8**

**10**

**10% H2O 90% CO2**

**/**μ**m]**

**10**

**20**

**30**

**40**

**50**

**<sup>60</sup> 10% H2O, 90% CO2**

**10**

**20**

**30**

**40**

**50**

**<sup>60</sup> 35% H2O, 65% CO2**

**k [1/m]**

**k [1/m]**

**0**

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

the so-called fractions is a hypothetical collection of noncontiguous intervals of the spectrum having the same value of the spectral absorption coefficient. In the following subsections, we describe further the box/EWB model and the nongray WSGG model.

#### **2.1 Box/EWB model**

In the general box model, the erratic spectral profile of *k<sup>η</sup>* is idealized as a piecewise-constant function, with constant *k<sup>η</sup>* values over a range of *η*. This value can be zero over intervals of spectrum where no absorption is occurring (called the *windows*). In the present work, a piecewise-constant function of *k<sup>η</sup>* is calculated using the exponential wide band model, which idealizes each vibration-rotation band of H2O or CO2 as well as the far-infrared pure-rotation band of H2O according to the block approximation (9). A block is formed between the edges of each idealized band. There are 6 vibration-rotation bands of CO2, four vibration-rotation bands of H2O, and a pure-rotation band of H2O. The number of blocks varies depending on the width of each idealized band; which in turn depends on the fractions of H2O and CO2 in the medium, its temperature, and its total pressure.

We have used a model with 22 blocks that cover the wavenumbers from *η*=0 to 100 000 1/cm. This corresponds to wavelengths from *λ*=0.1 *μ* m to ∞. Such range is wide enough to handle thermal radiation (10). Consequently, 22 RTEs per direction are solved to resolve the spectrum. This range covers more than 99.99% of the area under the Planck function at 1 500 K. The band equivalent widths are computed using the Edwards-Menard 3-regime expressions (11; 12) for the vibration-rotation bands, and using the Fleske-Tien theoretical expression (13; 15) for the pure-rotation band. The parameters for the vibration-rotation bands are those in (14) and for the pure-rotation bands are those in (15). Relating this approach to Equation (9), each *ki* is a block value and each *ai* is the fraction of the Planck function over that block. The box/EWB approach requires the specification of a mean pathlength (some characteristic length for radiation) for the problem, which is approximated as 3.6 times the volume divided by the surface areas (3). For the 12×12×40 m rectangular enclosure we consider here, this value is 9.3913 m. This length was also used to obtain *ki* for each block from its calculated emissivity.

Figure 1 shows the idealized spectra of *k<sup>η</sup>* for the two gas compositions studied in this work (i.e., 65% CO2 with 35% H2O, and 90% CO2 and 10% H2O) at a constant temperature of 1 500 K. The corresponding blackbody emissive power (*Eb*,*<sup>η</sup>* = *π Ib*,*η*) is superimposed in each plot. The corresponding spectra using the wavelength *λ* as the spectral variable are given in Figure 2.

Whereas both CO2 and H2O are radiatively-active, as some H2O is replaced by CO2 (moving from wet recycle to dry recycle), the absorption/emission of the mixture decreases (17; 18). The full listing of the linear absorption coefficients and blackbody weights of each block for both oxy-fuel environments are given in Appendix 7, Tables 9 and 10.

#### **2.2 WSGGM**

Despite the large reduction in the number of calculations when switching from the LBL approach to the box/EWB approach, it is still desirable to attain further reduction in the number of RTEs to be solved for the entire spectrum when performing complex combustion simulations as they involve many physical and chemical phenomena other than radiation. The WSGGM has enjoyed great popularity (4) and is utilized here as a *more-practical* approach for complex combustion modeling, whereas the aforementioned more-expensive 4 Numerical Simulations / Book 1

the so-called fractions is a hypothetical collection of noncontiguous intervals of the spectrum having the same value of the spectral absorption coefficient. In the following subsections, we

In the general box model, the erratic spectral profile of *k<sup>η</sup>* is idealized as a piecewise-constant function, with constant *k<sup>η</sup>* values over a range of *η*. This value can be zero over intervals of spectrum where no absorption is occurring (called the *windows*). In the present work, a piecewise-constant function of *k<sup>η</sup>* is calculated using the exponential wide band model, which idealizes each vibration-rotation band of H2O or CO2 as well as the far-infrared pure-rotation band of H2O according to the block approximation (9). A block is formed between the edges of each idealized band. There are 6 vibration-rotation bands of CO2, four vibration-rotation bands of H2O, and a pure-rotation band of H2O. The number of blocks varies depending on the width of each idealized band; which in turn depends on the fractions of H2O and CO2 in

We have used a model with 22 blocks that cover the wavenumbers from *η*=0 to 100 000 1/cm. This corresponds to wavelengths from *λ*=0.1 *μ* m to ∞. Such range is wide enough to handle thermal radiation (10). Consequently, 22 RTEs per direction are solved to resolve the spectrum. This range covers more than 99.99% of the area under the Planck function at 1 500 K. The band equivalent widths are computed using the Edwards-Menard 3-regime expressions (11; 12) for the vibration-rotation bands, and using the Fleske-Tien theoretical expression (13; 15) for the pure-rotation band. The parameters for the vibration-rotation bands are those in (14) and for the pure-rotation bands are those in (15). Relating this approach to Equation (9), each *ki* is a block value and each *ai* is the fraction of the Planck function over that block. The box/EWB approach requires the specification of a mean pathlength (some characteristic length for radiation) for the problem, which is approximated as 3.6 times the volume divided by the surface areas (3). For the 12×12×40 m rectangular enclosure we consider here, this value is 9.3913 m. This length was also used to obtain *ki* for each block from its calculated

Figure 1 shows the idealized spectra of *k<sup>η</sup>* for the two gas compositions studied in this work (i.e., 65% CO2 with 35% H2O, and 90% CO2 and 10% H2O) at a constant temperature of 1 500 K. The corresponding blackbody emissive power (*Eb*,*<sup>η</sup>* = *π Ib*,*η*) is superimposed in each plot. The corresponding spectra using the wavelength *λ* as the spectral variable are given in

Whereas both CO2 and H2O are radiatively-active, as some H2O is replaced by CO2 (moving from wet recycle to dry recycle), the absorption/emission of the mixture decreases (17; 18). The full listing of the linear absorption coefficients and blackbody weights of each block for

Despite the large reduction in the number of calculations when switching from the LBL approach to the box/EWB approach, it is still desirable to attain further reduction in the number of RTEs to be solved for the entire spectrum when performing complex combustion simulations as they involve many physical and chemical phenomena other than radiation. The WSGGM has enjoyed great popularity (4) and is utilized here as a *more-practical* approach for complex combustion modeling, whereas the aforementioned more-expensive

both oxy-fuel environments are given in Appendix 7, Tables 9 and 10.

describe further the box/EWB model and the nongray WSGG model.

the medium, its temperature, and its total pressure.

**2.1 Box/EWB model**

emissivity.

Figure 2.

**2.2 WSGGM**

Fig. 1. Spectra (versus wavenumber) of the blackbody emissive power and the box/EWB linear absorption coefficient and at 1 500 K for two oxy-fuel environments

Fig. 2. Spectra (versus wavelength) of the blackbody emissive power and the box/EWB linear absorption coefficient and at 1 500 K for two oxy-fuel environments

box/EWB approach serves to provide a benchmark solution to compare with. In the nongray WSGG approach, Equation (9) is still solved as was the case in the box model, but the physical interpretation and the evaluation of the *ki* and *ai* are very different. The WSGG approach (5; 16; 19–21) is based on the presence of *N* hypothetical gray gases; *N* − 1 are absorbing/emitting, and one is clear (no radiative emission or absorption) to represent the presence of spectral windows. Each absorbing/emitting gray gas has a constant *ki*, and the clear gas has *k*0=0. The fractions *ai* are cast as a polynomial of temperature only. The parameters of a WSGGM are the *ki* and the polynomial coefficients for each absorbing/emitting gray gas. There are (*N* − 1) × (*M* + 1) model parameters for *N* gray gases and a polynomial order *M*. The parameters for a single total pressure and a single gas composition (H2O and CO2 partial pressures) are calculated through an optimization process.

Ref. *N* T-poly Num. sets *T*ˆ(*K*) T range (K) PL range Training data (22) 4 quadratic 2 1 200 500−2 500 0.01−60 bar-m SNBM (23) (22) 5 quadratic 2 1 200 500−2 500 0.01−60 bar-m SNBM (23) (24) 5 quadratic N/A 1 200 500−2 500 0.01−60 bar-m SNBM (23) (25) 4 linear 3 1 1 000−2 000 0.005−10 atm-m empirical

Nongray EWB and WSGG Radiation Modeling in Oxy-Fuel Environments 499

(27) 5 cubic 7 1 200 500−3 000 0.001−60 atm-m EWBM (9; 14;

(28) 4 cubic 5 1 300−3 000 0.001−10atm-m EWBM (9; 14;

<sup>c</sup> *Kp*,*<sup>i</sup>* are expressed as linear functions of the H2O/CO2 molar ratio, and *bij* as quadratic

<sup>f</sup> at compositions: H2O:CO2 = 11.1%:88.9%, 20%:80%, 33.3%:66.7%, 42.9%:57.1%, 50%:50%, —66.7%:33.3, 80%:20 by mole (3 others sets are given but not for oxy-fuel environments) <sup>g</sup> at compositions: H2O:CO2 <sup>=</sup> <sup>→</sup> 0:0 (diluent is N2), 10%:10% (80% N2), 20%:10% (70% N2),

In coupled combustion simulations, different sub-models interact and thus it becomes difficult to examine the independent response of a particular sub-model. It is advantageous to isolate the radiation modeling when examining different solution approaches, which is what we have followed here. The two test problems to be presented in this section correspond to a stagnant homogeneous isothermal gas mixture. Only the radiative intensity is allowed to vary, thereby eliminating cross-model interactions which could make it difficult to judge the performance of the performance of the particular radiation model from the simulation results. Since our primary goal is to study the performance of the different oxy-fuel WSGG models when used in oxy-fuel environments, we considered two idealized oxy-fuel product gas compositions. Both environments have an atmospheric total pressure, which is also the sum of the partial pressures of H2O and CO2 (thus, no N2 dilution, which is relevant to oxy-fuel operations). The only difference between the two environments is the gaseous composition, which is summarized in Table 2. In both environments, the CO2 mole fraction is higher than the fraction of the H2O. However, the second environment features dominance of CO2 (9 times

Test case H2O mole fraction CO2 mole fraction Total pressure (atm) Temperature (K) case 1 35% 65% 1 1 500 case 2 10% 90% 1 1 500

<sup>d</sup> at compositions: H2O:CO2 = 10%:10% (80% N2), 33%:66%, and 10%:90% by mole <sup>e</sup> implied from the empirical correlation used for the training data

Table 1. Summary of the 6 WSGG models considered here (5 oxy-fuel and 1 air-fuel)

<sup>a</sup> at compositions: H2O:CO2 = 11.1%:88.9% and 50%:50% by mole

H2O), which is more relevant to dry-recycle oxy-fuel operations.

Table 2. Summary of the 2 studied oxy-fuel environments

<sup>b</sup> *SNBM* is statistical narrow band model

—0:→ 0, 0%:100% (diluent is N2) by mole.

functions of it

**3. Test cases**

correlation (26)

15)

15)

The optimization requires a set of emissivities for a range of temperatures and pathlengths at these total pressure and gas composition.

When used to calculate the total emissivity (either during the model coefficient optimization process or for evaluating the total emissivity with fixed model coefficients), the WSGGM returns a weighted sum of individual emissivities of the hypothetical absorbing/emitting gray gases, i.e.

$$\epsilon\_{tot} = \sum\_{i=1}^{N-1} a\_i(T) \left( 1 - \exp\left[ -K\_{p,i} PL \right] \right) \tag{11a}$$

$$\text{where}\quad a\_i(T) = \sum\_{j=1}^{M+1} b\_{ij} \text{ (}\mathcal{T}/\hat{\mathcal{T}}\text{)}^{j-1} \tag{11b}$$

where *tot* is the total emissivity (dimensionless), *PL* is pressure-pathlength, *L* is the mean pathlength, *Kp*,*<sup>i</sup>* are the pressure absorption coefficients for the *N* − 1 absorbing/emitting gray gases, *ai* are the blackbody weights for these absorbing/emitting gray gases, *bij* are the coefficients for a polynomial of degree *M* in *T*/*T*ˆ, and *T*ˆ is a scaling temperature that aids in the minimization process.

When the WSGGM is used to perform nongray calculations for use in Equation (9), the weights *ai* are also evaluated from the temperature polynomial in Equation (11b); the *i th* linear absorption coefficient is evaluated as

$$k\_i = K\_{p,i} \text{ P} \tag{12}$$

where *P* is the sum of the partial pressures of H2O and CO2 (in units consistent with those of *Kp*,*i*). A total of *N* RTEs are solved per direction to resolve the spectrum. In the WSGG models considered here, *N* takes the value of 4 or 5, which is a considerable reduction in computations compared to the box/EWB procedure described in subsection 2.1.

Table 1 compares the characteristics of the WSGG models which we consider. The first five WSGG models have been optimized for oxy-fuel combustion, whereas the last was developed for air-fuel combustion. Its inclusion in the study is a method to estimate the errors in radiation modeling when applying air-fuel WSGG models in oxy-fuel combustion simulations.

All models shown in Table 1 have mode parameters at finite sets of gas compositions, except for the 2011 model of Johansson et al. (24) where the model parameters are expressed as continuous functions of the molar ratio H2O/CO2. We perform piecewise-linear interpolation/extrapolation using the molar ratio H2O/(H2O+CO2) as an independent variable to apply the model at arbitrary gas compositions (18; 26). Marzouk and Huckaby (18) compared this technique to the piecewise-constant technique and recommended the former based on gray radiation modeling of non-isothermal media. The full listing of the linear absorption coefficients and blackbody weights for the gray gases of the 6 WSGG models for both oxy-fuel environments is given in Appendix 7, Tables 11-16. Notice that for either oxy-fuel environment, the clear-gas weight (*a*0) in the air-fuel WSGGM (28) is higher than its counterpart in all the oxy-fuel WSGG models. This acts to reduce the radiative participation of the gaseous mixture.


<sup>a</sup> at compositions: H2O:CO2 = 11.1%:88.9% and 50%:50% by mole

<sup>b</sup> *SNBM* is statistical narrow band model

<sup>c</sup> *Kp*,*<sup>i</sup>* are expressed as linear functions of the H2O/CO2 molar ratio, and *bij* as quadratic functions of it

<sup>d</sup> at compositions: H2O:CO2 = 10%:10% (80% N2), 33%:66%, and 10%:90% by mole <sup>e</sup> implied from the empirical correlation used for the training data

<sup>f</sup> at compositions: H2O:CO2 = 11.1%:88.9%, 20%:80%, 33.3%:66.7%, 42.9%:57.1%, 50%:50%, —66.7%:33.3, 80%:20 by mole (3 others sets are given but not for oxy-fuel environments)

<sup>g</sup> at compositions: H2O:CO2 <sup>=</sup> <sup>→</sup> 0:0 (diluent is N2), 10%:10% (80% N2), 20%:10% (70% N2), —0:→ 0, 0%:100% (diluent is N2) by mole.

Table 1. Summary of the 6 WSGG models considered here (5 oxy-fuel and 1 air-fuel)

### **3. Test cases**

6 Numerical Simulations / Book 1

The optimization requires a set of emissivities for a range of temperatures and pathlengths at

When used to calculate the total emissivity (either during the model coefficient optimization process or for evaluating the total emissivity with fixed model coefficients), the WSGGM returns a weighted sum of individual emissivities of the hypothetical absorbing/emitting gray

> *ai*(*T*)

where *tot* is the total emissivity (dimensionless), *PL* is pressure-pathlength, *L* is the mean pathlength, *Kp*,*<sup>i</sup>* are the pressure absorption coefficients for the *N* − 1 absorbing/emitting gray gases, *ai* are the blackbody weights for these absorbing/emitting gray gases, *bij* are the coefficients for a polynomial of degree *M* in *T*/*T*ˆ, and *T*ˆ is a scaling temperature that aids in

When the WSGGM is used to perform nongray calculations for use in Equation (9), the

where *P* is the sum of the partial pressures of H2O and CO2 (in units consistent with those of *Kp*,*i*). A total of *N* RTEs are solved per direction to resolve the spectrum. In the WSGG models considered here, *N* takes the value of 4 or 5, which is a considerable reduction in computations

Table 1 compares the characteristics of the WSGG models which we consider. The first five WSGG models have been optimized for oxy-fuel combustion, whereas the last was developed for air-fuel combustion. Its inclusion in the study is a method to estimate the errors in radiation modeling when applying air-fuel WSGG models in oxy-fuel combustion

All models shown in Table 1 have mode parameters at finite sets of gas compositions, except for the 2011 model of Johansson et al. (24) where the model parameters are expressed as continuous functions of the molar ratio H2O/CO2. We perform piecewise-linear interpolation/extrapolation using the molar ratio H2O/(H2O+CO2) as an independent variable to apply the model at arbitrary gas compositions (18; 26). Marzouk and Huckaby (18) compared this technique to the piecewise-constant technique and recommended the former based on gray radiation modeling of non-isothermal media. The full listing of the linear absorption coefficients and blackbody weights for the gray gases of the 6 WSGG models for both oxy-fuel environments is given in Appendix 7, Tables 11-16. Notice that for either oxy-fuel environment, the clear-gas weight (*a*0) in the air-fuel WSGGM (28) is higher than its counterpart in all the oxy-fuel WSGG models. This acts to reduce the radiative participation

weights *ai* are also evaluated from the temperature polynomial in Equation (11b); the *i*

1 − exp [−*Kp*,*<sup>i</sup> PL*]

*ki* = *Kp*,*<sup>i</sup> P* (12)

*<sup>j</sup>*−<sup>1</sup> (11b)

(11a)

*th* linear

*tot* =

where *ai*(*T*) =

compared to the box/EWB procedure described in subsection 2.1.

*N*−1 ∑ *i*=1

*M*+1 ∑ *j*=1 *bij T*/*T*ˆ

these total pressure and gas composition.

gases, i.e.

the minimization process.

simulations.

of the gaseous mixture.

absorption coefficient is evaluated as

In coupled combustion simulations, different sub-models interact and thus it becomes difficult to examine the independent response of a particular sub-model. It is advantageous to isolate the radiation modeling when examining different solution approaches, which is what we have followed here. The two test problems to be presented in this section correspond to a stagnant homogeneous isothermal gas mixture. Only the radiative intensity is allowed to vary, thereby eliminating cross-model interactions which could make it difficult to judge the performance of the performance of the particular radiation model from the simulation results. Since our primary goal is to study the performance of the different oxy-fuel WSGG models when used in oxy-fuel environments, we considered two idealized oxy-fuel product gas compositions. Both environments have an atmospheric total pressure, which is also the sum of the partial pressures of H2O and CO2 (thus, no N2 dilution, which is relevant to oxy-fuel operations). The only difference between the two environments is the gaseous composition, which is summarized in Table 2. In both environments, the CO2 mole fraction is higher than the fraction of the H2O. However, the second environment features dominance of CO2 (9 times H2O), which is more relevant to dry-recycle oxy-fuel operations.


Table 2. Summary of the 2 studied oxy-fuel environments

radiative source term (in kW/m3) along the 12×40 vertical midplane (the symmetry plane midway between the two vertical side walls separated by a distance of 12 m). Due to the symmetry of the problem, this plane should be identical to the horizontal symmetry plane. Next, the 1D profiles of this radiative source term along the centerline of the enclosure (i.e., the 40-m longitudinal line passing through the geometric center of the 12×12 cross-section of the enclosure) are presented in subsection 4.4. In these profiles, we also include published

Nongray EWB and WSGG Radiation Modeling in Oxy-Fuel Environments 501

The SLW approach (originally proposed by Denison and Webb (31)) is a more-rigorous implementation of the WSGGM. The individual gray gases now have a physical meaning and direct mathematical relationship with the absorption spectrum (in terms of the absorption cross-section, whose SI unit is m2/mol). The range of the absorption coefficient is divided into segments, each of which represents an absorbing/emitting gray gas. In addition, there is one clear gas (as in the WSGG approach). The segmentation of the range of absorption coefficient is typically done such that their logarithmic values are equally spaced. For each segment (i.e., each absorbing/emitting gray gas), a logarithmic average absorption cross-section *Ci* is assigned, and the corresponding blackbody weight *ai* is evaluated to be the fraction of the Planck function that belongs to the range of absorption coefficient of the segment

the absorption cross-section by the species molar concentration (its SI units is kmol/m3). The exact implementation of this method would require the processing of a high-resolution spectrum (which incurs the processing of millions of spectral data points at high combustion or flue temperatures), the computations are highly simplified by utilizing a fitted hyperbolic tangent function for the cumulative distribution of the absorption cross-section, which is

Several different approaches have been developed to apply the SLW method to multicomponent gas mixtures. The approaches are derived using different assumptions and vary in computational cost and accuracy. For the SLW solution we include here, the absorption cross-section domains of H2O and CO2 were individually discretized into 20 logarithmically-spaced intervals between 3×10−<sup>5</sup> <sup>m</sup>2/mol and 120 m2/mol for H2O, and between 3×10−<sup>5</sup> <sup>m</sup>2/mol and 600 m2/mol for CO2. The analytical expressions for the absorption-line blackbody distribution functions of H2O (32) and CO2 (33) were used to compute the blackbody weights of each gray gas. The multiplication method (34) was used to handle the presence of a mixture. Implied in this method, is the assumption that the absorption cross-sections of H2O and CO2 are statistically independent. The number of RTEs per direction was 21 (one RTE per each of the 20 gray gases plus an RTE for the clear gas). The SLW calculations were performed using the T4 angular quadrature (35), and using the same spatial resolution we employed for the other two approaches (namely 27×27×82) and using

Unlike the box/EWB and WSGG solutions, in which we use the angular finite-volume method for treating the angular dependence of radiation, the SLW solutions were obtained using the discrete-ordinate method. Whereas both methods have some similarity, the angular finite-volume method conserves the radiative energy (4) and thus is considered a more accurate method for handling the directional dependence of radiation. In addition, the analytical fits for ALBDF of H2O and CO2 are based on an extension of an old version (1991/1992) of the spectral database HITRAN (36). This database was assembled for a (low) temperature of 296 K and thus when applied at high temperatures the absorption of the

known as the absorption-line blackbody distribution function (ALBDF) (32; 33).

*th* gray gas. The linear absorption coefficient for a species is related to

results (25) using the SLW approach.

a similar angular resolution (128 directions).

represented by the *i*

The geometry of both problems is a large rectangular enclosure, with dimensions 12×12×40 m. The medium temperature is 1 500 K. The temperature of the walls is kept at 750 K, with an emissivity of 0.725. This configuration was proposed by Krishnamoorthy et al. (25) to roughly represent the dimensions of a full-scale 300 MW front-wall-fired, pulverized-coal, utility boiler (29). The domain is discretized with a uniform mesh of 27×27×82 cells, resulting in a total of 59 778 hexahedral cells.
