Gas Radiation (Leckner Model)

📝 Configuration

🔴 Temperatures

Gas Temp ($T_g$) [°C]:

Wall Temp ($T_s$) [°C]:

💨 Gas Mixture Pressures

CO₂ Partial Press ($p_c$) [bar]:

H₂O Partial Press ($p_w$) [bar]:

Total Pressure ($P_t$) [bar]:

Mean Beam Length ($L$) [m]:

Key Equations:

Leckner Emissivity Fit:
$\varepsilon_0 = \exp\left[ \sum \sum c_{ij} (\ln t_g)^i (\ln pL)^j \right]$

Pressure Broadening:
$\varepsilon_g = \varepsilon_c C_c + \varepsilon_w C_w - \Delta\varepsilon_{overlap}$

📊 Results

Configure pressures and temperatures and click Calculate.

📘 Calculation Methodology & Theory

Mathematical Model & Theory

Participating gases (such as $CO_2$ and $H_2O$) emit and absorb radiation in discrete spectral bands. Unlike solids, they do not act as black or gray bodies. The total gas mixture emissivity ($\varepsilon_g$) is modeled using Leckner's bivariate correlations (1972) as a function of temperature ($T_g$) and partial pressure-length parameter ($p_g L_e$):

$$\varepsilon_0(T_g, p_g L_e) = \exp \left[ \sum_{i=0}^3 \sum_{j=0}^2 c_{ij} \left( \ln\left(\frac{T_g}{1000}\right) \right)^i \left( \ln(p_g L_e) \right)^j \right]$$

Corrections for total system pressure are applied via scaling factors $C_c$ and $C_w$:

$$\varepsilon_g = C_c \varepsilon_c + C_w \varepsilon_w - \Delta\varepsilon$$

where $\Delta\varepsilon$ accounts for band overlap between the species. The gas absorptivity ($\alpha_g$) for radiation from a black wall at $T_s$ is given by:

$$\alpha_g = \alpha_c + \alpha_w - \Delta\alpha$$ $$\alpha_c = C_c \varepsilon_c \left( T_s, p_c L_e \frac{T_s}{T_g} \right) \left( \frac{T_g}{T_s} \right)^{0.65}$$

Worked Engineering Example

Problem Statement:
A combustion gas mixture at $T_g = 1000\text{ K}$ is contained inside a chamber. The partial pressures are $p_c = 0.1\text{ bar}$ for $CO_2$ and $p_w = 0.1\text{ bar}$ for $H_2O$. Total pressure is $p = 1.0\text{ bar}$, and mean beam length is $L_e = 1.0\text{ m}$. Bounding wall is at $T_s = 600\text{ K}$. Calculate total gas emissivity and absorptivity.

Step-by-step Solution:
1. Calculate pressure-path parameters ($L_e = 100\text{ cm}$):
$$p_c L_e = 0.1 \times 100 = 10\text{ bar}\cdot\text{cm}$$ $$p_w L_e = 0.1 \times 100 = 10\text{ bar}\cdot\text{cm}$$ 2. Evaluate Leckner's correlation for $CO_2$ at $T_g = 1000\text{ K}$:
$$\varepsilon_{c,0} \approx 0.0860$$ Since $p = 1.0\text{ bar}$, pressure correction $C_c = 1.0 \implies \varepsilon_c \approx 0.0860$. 3. Evaluate Leckner's correlation for $H_2O$ at $T_g = 1000\text{ K}$:
$$\varepsilon_{w,0} \approx 0.1120$$ With pressure correction $C_w \approx 1.25 \implies \varepsilon_w \approx 0.1400$. 4. Find overlap correction $\Delta\varepsilon$ at $\zeta = \frac{p_w}{p_c+p_w} = 0.5$:
$$\Delta\varepsilon \approx 0.0070$$ 5. Calculate total emissivity:
$$\varepsilon_g = 0.0860 + 0.1400 - 0.0070 = 0.2190$$ 6. Compute absorptivity $\alpha_g$ by evaluating at surface temperature $T_s = 600\text{ K}$:
$$\alpha_c \approx 0.1740, \quad \alpha_w \approx 0.1380$$ $$\alpha_g = 0.1740 + 0.1380 - 0.0060 = 0.3060$$
Final Result:
The total gas emissivity is $0.2190$ and the gas absorptivity is $0.3060$.

Standard Assumptions & References

Assumptions: The gas is at local thermodynamic equilibrium. Bounding walls are black. Monatomic and symmetric diatomic gases (e.g. $N_2, O_2$) do not emit or absorb radiation. Scatering by soot/particles is neglected.

References:

Leckner, B. Spectral and Total Emissivity of Water Vapor and Carbon Dioxide, Combustion and Flame, Vol. 19, pp. 33-48, 1972.
VDI Heat Atlas, 2nd Edition, Springer-Verlag.
Modest, M. F. Radiative Heat Transfer, Academic Press.

💨 Gas Radiation (Leckner Correlation)