# Study on Thermal Radiation Characteristics and the Multi-Point Source Model of Hydrogen Jet Fire
## 3 Theoretical Basis of Numerical Simulation
### Optimization of Thermal Radiation Prediction Model  
Based on the multi-point source radiation model proposed by [PERSON] et al. [25; 33], the discrete weight coefficients in the model were optimized using the temperature distribution along the jet flame axis obtained from the numerical simulations, enhancing the accuracy of the prediction model for calculating the thermal radiation from the hydrogen jet flames.  
This study focused on the vertical hydrogen jet fire. In the multi-point source thermal radiation model, the flame can be considered as a collection of several particles distributed on the flame axis. The heat flux at a target point is expressed as  
\[\dot{q}_{x,y,z}=\sum_{i}\frac{w_{i}\alpha_{i}\chi_{\chi}\dot{Q}}{4\pi r_{i}^{2} }\cos\theta_{i} \tag{13}\]  
where \(w_{i}\) is the weight coefficient of the heat source point \(i\); \(\alpha_{i}\) is the transmittance of the radiation of the heat source point \(i\) in the air; \(\chi_{\chi}\) is the radiative emission coefficient of the  
Figure 4: Temperature distribution in \(y\)-direction calculated by different grid sizes.  
flame; \(\dot{Q}\) is the total heat power of the flame; \(\theta_{i}\) is the angle between the line connecting the heat source point with the target point and the horizontal plane; and \(r_{i}\) is the distance from the heat source point \(i\) to the target point.  
\(\alpha_{i}\) can be calculated by the following equation [34]:  
\[\alpha_{i}=1.06R_{i}^{-0.09} \tag{14}\]  
The emission coefficient of hydrogen was measured by experimentation and can be expressed as [12]  
\[\chi_{\rm r}=0.072\rm lg\tau_{\rm f}+0.156 \tag{15}\]  
where \(\tau_{\rm f}\) indicates flame stopping time, calculated by the following equation:  
\[\tau_{\rm f}=\frac{\rho_{\rm f}D_{\rm f}^{2}L_{\rm f}f_{\rm s}}{3\rho_{\rm e}d _{\rm e}^{2}u_{\rm e}} \tag{16}\]  
where \(\rho_{\rm f}\) is the flame equivalent density.  
According to [PERSON]'s blackbody law of thermal radiation, thermal radiation results from the radiative emission of high-temperature combustion products and is calculated by the following equation:  
\[E_{b\lambda}=\frac{c_{1}\lambda^{-5}}{e^{\frac{\sigma^{2}}{kT}}-1} \tag{17}\]  
where \(E_{b\lambda}\) is the blackbody radiation force; \(\lambda\) is the radiation wavelength; and \(T\) is the blackbody temperature.  
If the effect of the wavelength is neglected, the thermal radiation is calculated by the following equation according to the Stefan-Boltzmann law:  
\[E_{b}=\sigma T^{4} \tag{18}\]  
where \(\sigma\) is the Stepan-Boltzmann constant, equal to 5.67 \(\times\) 10\({}^{-8}\).  
Therefore, the flame temperature distribution directly affects the intensity and distribution of thermal radiation. When the flame trace is assumed to consist of \(i\) discrete points, the contribution weight of each point to the total radiation can be calculated by the following equation:  
\[w_{i}^{\prime}=\iiint_{V_{i}}T^{4}(x,y,z)dxdydz \tag{19}\]  
where \(V_{i}\) is the area encompassed by the 1500 K temperature flame surface around point source \(i\). The flame temperature distribution was obtained through simulation.  
The weight coefficients required for the multi-point source model can be obtained by normalizing \(w_{i}^{\prime}\) according to the following equation:  
\[w_{i}=\frac{w_{i}^{\prime}}{\sum\limits_{i}^{n}w_{i}^{\prime}} \tag{20}\]