## 4 Comparison of Pressure Drop and Flow Field Distribution between the Basic and Modified Models
### Boundary Conditions for Turbulent Combustion  
The selected computational domain was from the gas torch outlet to the steel plate. The hole diameters of the torch outlet and the distance from the outlet to the steel plate were denoted by d and Y, respectively. The boundary conditions and mesh model were established as shown in Figure 11. Due to the symmetry of the model and the steel plate, a quarter of the model was utilized for the simulation to reduce the computation time. Due to the quarter model, symmetric boundary conditions were selected at two surfaces, as shown in Figure 11b. A pressure outlet boundary condition was set at the two outside faces. The velocity inlet boundary condition of the jet holes was used for the LPG/air mixture. Because of the nonuniform flow at the outlet of the torch, the basic model had the highest Reynolds number of 3901 at the center area and the lowest Reynolds number of 3176 at the end of the torch. The Reynolds number of every flame jet was selected as 3483 for the modified model with uniform flow at the outlet of the torch. The other top surfaces except the LPG/air inlet were treated as ambient air inlets. A no-slip condition was selected for the wall boundary condition imposed on the impinging jet. The grid independence of the solution was checked for three different amounts of elements of 808,045, 1,829,319, and 3,792,625. The flow temperature deviation between the medium and largest amounts of elements was less than 3%. Therefore, to save calculation time for the simulation, the medium grid size of 1,829,319 was chosen. Figure 11: _Cont._  
Due to the high temperature in the combustion process, radiation losses are important [23; 24]. In order to examine the effect of radiation, the DO radiation model was used. The DO model covers the entire range of optical thickness and can be used to solve problems ranging from surface-to-surface radiation to radiation participating in combustion problems. The default boundary conditions were selected for the DO radiation model in ANSYS FLUENT 18.2 [21]. The absorption coefficient was modeled using the weighted sum of gray gases model (WSGGM). The DO equation has the following form:  
\[\
abla.\Big{(}I\Big{(}\overset{\rightarrow}{r},\overset{\rightarrow}{s} \Big{)}\overset{\rightarrow}{s}\Big{)}+(a+\sigma_{s})I\Big{(}\overset{ \rightarrow}{r},\overset{\rightarrow}{s}\Big{)}=an^{2}\frac{\sigma T^{4}}{ \pi}+\frac{\sigma_{s}}{4\pi}\underset{0}{\overset{\rightarrow}{r}}I\Big{(} \overset{\rightarrow}{r},\overset{\rightarrow}{s^{\prime}}\Big{)}\phi\Big{(} \overset{\rightarrow}{s},\overset{\rightarrow}{s^{\prime}}\Big{)}d\Omega^{\prime} \tag{19}\]  
where,  
\(\overset{\rightarrow}{r}\) is the position vector  
\(\overset{\rightarrow}{s}\) is the direction vector  
\(\overset{\rightarrow}{s^{\prime}}\) is the scattering direction vector  
\(a\) is the absorption coefficient  
\(n\) is the refractive index  
\(\sigma_{s}\) is the scattering coefficient  
\(\sigma\) is the Stefan-Boltzmann constant (5.669 \(\times\) 10\({}^{-8}\) W\(\cdot\)m\({}^{-2}\)\(\cdot\)K\({}^{-4}\))  
\(I\) is the radiation intensity  
\(T\) is the local temperature  
\(\phi\) is the phase function  
\(\Omega^{\prime}\) is the solid angle  
Equation (15) means that the net radiation intensity around a control volume (the first term on the left-hand side) is equal to the sum of the extinction (the second term on the left-hand side), emission (the first term on the right-hand side), and scattering term (the second term on the right-hand  
Figure 11: Schematic diagram of steel preheating process with gas torch, fluid domain, and steel plate. (**a**) A half model (x-y plane cut view); (**b**) three-dimensional magnified view of the quarter model; (**c**) mesh model of the quarter model. side).