# Estimation of Thermal Radiation in Bed Mattresses
## 4 Calculation of Radiation from Flames
### Thermal Radiation Estimation Applied to ISO 24678-7  
Two thermal radiation estimation methods described in ISO 24678-7 (the Mudan-Croce formula and the Shokri formula) were applied to the model equation using the combustion area and the heat emission data obtained by the experiment. The equation is as follows:  
\[\text{D}=\sqrt{\frac{4A_{s}}{\pi}}, \tag{15}\]  
where \(A\)s is the area of a pool fire source (m\({}^{2}\)), and D is the equivalent diameter of the flame (m).  
\[\dot{Q}=\Delta H_{c}\dot{m}^{\prime\prime}A_{s}, \tag{16}\]  
where \(\dot{Q}\) is heat release rate (kW), \(\Delta H_{c}\) is the heat of combustion of fuel (J/g), and \(\dot{m}^{\prime\prime}\) is the mass burning rate per unit area (kg/m\({}^{2}\cdot\)s).  
\[\dot{m}^{\prime\prime}=\dot{m}^{\prime\prime}_{\infty}\Big{(}1-e^{-kD}\Big{)}, \tag{17}\]  
where \(\dot{m}^{\prime\prime}_{\infty}\) is the mass burning rate per unit area of a sufficiently large pool (kg/m\({}^{2}\cdot\)s), and \(k\) is the radiation absorption coefficient of a flame from various fuels (m\({}^{-1}\)).  
(1) Mudan-Croce formula  
This model was proposed to estimate the thermal radiation generated from liquid fuel. For the flame height, the flame height data obtained in [PERSON] wood experiment was used; it can be applied within a pool fire diameter of 1-60 m. The emissive power equation is presented below, and the experimental results and calculated values are shown in Figure 9. Regarding the calculated values at an early stage of the flame, for 100 s, the emissive power is 0.9 kW/m\({}^{2}\) at HB 0 mm, 2.7 kW/m\({}^{2}\) at HB 215 mm, and 4.6 kW/m\({}^{2}\) at HB 515 mm, indicating that as the mattress installation height increased, the thermal radiation tended to increase. The maximum values were 11.5 kW/m\({}^{2}\) (240 s), 12.0 kW/m\({}^{2}\) (220 s), and 12.4 kW/m\({}^{2}\) (200 s) for HB 0 mm, 215 mm, and 515 mm, respectively, showing that it reached the maximum value earlier as the installation height increased. However, this confirmed a substantial difference between the calculated values and the experimental values.  
\[\text{E}=E_{max}e^{-\beta D}+E_{s}(1-e^{-\beta D}) \tag{18}\]  
where \(E_{max}\) is the emissive power of the luminous region of the flame (=140 kW/m\({}^{2}\)), \(E_{s}\) is the emissive power of smoke (=20 kW/m\({}^{2}\)), and \(\beta\) is the radiation absorption coefficient of the flame, considering the diameter as the characteristic length (=0.12 m\({}^{-1}\)).  
(2) Shokri's Formula  
The method based on [PERSON]'s formula is based on [PERSON]'s flame height experiment and assumes that the flame has a cylindrical shape with a surface close to a blackbody and a homogeneous radiator on average. This method is effective when the heat flux  
Figure 9: Comparison of calculated values by Mudan–Croce formula with the experimental values: (a) HB = 0 mm; (b) HB = 215 mm; (c) HB = 515 mm.  
is greater than 5 kW/m\({}^{2}\) and is mainly used for LNG and JP-5 fuel and a combustion diameter of 1-50 m. Figure 10 shows the comparison of the experimental results. At the early stage of the flame, for 100 s, the emissive power is 0.9 kW/m\({}^{2}\) at HB 0 mm, 2.5 kW/m\({}^{2}\) at HB 215 mm, and 4.2 kW/m\({}^{2}\) at HB 515 mm, indicating that as the mattress installation height increased, the thermal radiation tended to increase. Moreover, the calculated values became more gradual as the mattress installation height increased, and a large difference was observed between the calculated values and the experimental values.  
\[\text{E}=58\Big{(}10^{-0.00823D}\Big{)} \tag{19}\]