## 4 Comparison of Pressure Drop and Flow Field Distribution between the Basic and Modified Models
### Mathematical Model  
To investigate the temperature distribution as the combustion process occurs, the energy and species equations were applied and solved by ANSYS FLUENT 18.2 software [22].  
Energy equation:  
\[\
abla.\left(\overset{\rightarrow}{u}\left(\rho E+p\right)\right)=\
abla.\left( k\
abla T-\sum_{j}h_{j}\overset{\rightarrow}{j_{j}}\right)+S_{h} \tag{10}\]  
Equation (9) shows the energy transfer in a control volume. The term on the left-hand side represents the energy transfer due to convection. Two terms on the right-hand side represent net energy transfer around control surface due to conduction heat transfer and species diffusion. The last term represents source term, including chemical reaction, radiation (discrete ordinates (DO) radiation source term in this study), and so on.  
where  
\[E=h-\frac{p}{\rho}+\frac{u^{2}}{2} \tag{11}\]  
Enthalpy \(h\) is defined as follows:  
\[h=\sum_{j}Y_{j}h_{j}+\frac{p}{\rho} \tag{12}\]  
\[h_{j}=\int\limits_{T_{in}}^{T}c_{p,j}dT \tag{13}\]  
Due to combustion process with radiational effect, the temperature from the exit of gas torch is selected at value of 27 \({}^{\circ}\)C, higher than ambient air temperature of 4 \({}^{\circ}\)C. The steel plate temperature is selected at value of 4 \({}^{\circ}\)C.  
Figure 10: Comparison of velocity uniformity index at the outlet between the basic torch and the three new gas torches as a function of pressure drop.  
The species conservation equation is as follows:  
\[\
abla.\left(\rho\overset{\rightarrow}{u}\gamma_{i}\right)=-\
abla.\overset{ \rightarrow}{f}_{i}+R_{i} \tag{14}\]  
where \(R_{i}\) is the net rate of production of species by chemicals. In turbulent flows, the mass diffusion is computed as follows:  
\[\overset{\rightarrow}{f}_{i}=-\left(\rho D_{i,m}+\frac{\mu_{t}}{Sc_{t}} \right)\
abla Y_{i}-D_{T,i}\frac{\
abla T}{T} \tag{15}\]  
where \(Sc_{t}\) is the turbulent Schmidt number, which has a default value of 0.7. It should be noted that turbulent diffusion generally overwhelms laminar diffusion, and specification of the detailed laminar diffusion properties in turbulent flows is generally not necessary. For multicomponent mixing flow, the transport of enthalpy due to species diffusion can have a significant effect on the enthalpy field and should not be neglected.  
\[\
abla.\left[\sum_{i=1}^{n}h_{i}\overset{\rightarrow}{f}_{i}\right] \tag{16}\]  
The turbulent-chemistry interaction model combines turbulent flow and complex chemical kinetics. Turbulence enhances the mixing of the reactants. Chemical reactions that involve temperature rise change the density and affect the flow. Therefore, the coupling between the turbulence and chemistry plays a crucial role in turbulent reactive flows. ANSYS FLUENT provides a turbulence-chemistry interaction model based on the work of [PERSON] and [PERSON] called the eddy-dissipation model. The net rate \(R_{i,r}\) of the production of species \(i\) due to reaction \(r\) is selected from the smaller of the two equations from the following:  
\[R_{i,r}=v_{i,r}^{\prime}M_{w,j}A\rho\frac{\varepsilon}{k}\frac{Y_{R}}{v_{R,r}^ {\prime}M_{w,R}} \tag{17}\]  
\[R_{i,r}=v_{i,r}^{\prime}M_{w,i}AB\rho\frac{\varepsilon}{k}\frac{\frac{\
abla}{ P}Y_{P}}{\sum\limits_{j}^{N}v_{j,r}^{\prime\prime}M_{w,j}} \tag{18}\]  
where:  
\(v_{i,r}^{\prime}\) is the stoichiometric coefficients for reactant \(i\) in reaction \(r\)  
\(v_{i,r}^{\prime\prime}\) is the stoichiometric coefficients for product \(i\) in reaction \(r\)  
\(M_{w}\) is the molecular weight (kg/kmol)  
\(k\) is the turbulence kinetic energy rate (J/kg)  
\(\varepsilon\) is the turbulent dissipation rate (m\({}^{2}\)/s\({}^{3}\))  
\(Y_{P}\) is the mass fraction of product species, \(P\)  
\(Y_{R}\) is the mass fraction of a particular reactant, \(R\)  
\(A\) is an empirical constant equal to 4.0  
\(B\) is an empirical constant equal to 0.5  
The reaction rates are assumed to be controlled by the turbulence, so the Arrhenius chemical kinetic calculations can be neglected.