# Numerical Investigations of Micro-Scale Diffusion Combustion: A Brief Review
## 3 Numerical Studies of Micro-Diffusion Flames  
The numerical studies of micro-diffusion flames have emerged with the aim of better understanding the physical phenomena of diffusion combustion at micro scales, where it is challenging to collect such comprehensive data through experimental investigations. [PERSON] et al. performed numerical simulations to investigate the impacts of the fuel velocity on micro flame structures in laminar methane-air combustion [10]. In their simulations, they used both one-step and detailed reaction mechanisms and were able to capture the flame heights, temperature and the methane and oxygen concentration distributions in an acceptable agreement with the experimental measurements for different inlet velocities. Through simulation results, they could capture a linear correlation between the flame height and the methane flow rate, that was consistent with the observations in the experimental data as represented in top plot of Figure 1. The contours illustrated in the bottom of this figure compare the simulation results (flow vectors (a), temperature contours (b), methane and oxygen concentrations (c) and reaction contours (d)) for two different methane inlet velocities. This work also reported on the limited buoyancy effect on the flow field (the order of \(10^{-2}\) m/s), when compared with the minimum fuel velocity of 1 m/s, and a flame temperature difference of 700 K (temperature difference between the flame edge at the vicinity of the flame base and the adiabatic temperature). Such predictions were not as exact near the flame edge, where the impacts of wall quenching and detailed finite-rate chemical kinetics become more significant. Subsequently, [PERSON] et al. conducted a numerical study to investigate the impacts of the burner size on the extinction behaviour of methane diffusion flames at micro scales [PERSON] et al. (2019). They used a convective-diffusive heat and mass transport model, where the combustion reaction was assumed by a one-step, irreversible, exothermic reaction. In order to take the near-extinction effects into account, they selected and validated specific kinetics constants. Using the Damkohler principle, they concluded that \"there is the minimum burner diameter for the micro flame to exist\" [PERSON] et al. (2019). As represented in the left diagram of Figure 2, the normalized flame height and radius (both are normalized by the burner diameter) and the scalar dissipation rate (SDR) at the maximum heat release rate location under the same Reynolds (Re) number, are plotted against the burner diameters. The SDR, or diffusive transport process rate, \(D(dZ/dn)^{2}\), defined by the derivative of the mixture fraction, \(Z\), based on inert gas normal to the flame surface, \(n\), and multiplied by a non-dimensional diffusion coefficient, \(D\), showed to be smaller than the chemical reaction rate (CRR) for \(d>0.1\) mm, and about the same order of the chemical reaction rate for \(d=0.1\) mm. It is shown that for \(d<0.1\) mm, the SDR, flame height and flame radius decrease rapidly with the increase of the burner diameter. However, for \(d>0.1\) mm, all three parameters experience a more smooth transition with the change of the burner diameter. Reducing the burner diameter further toward the limits, the SDR exceeded the chemical reaction rate and led to extinction. This work illustrates that the the crossover between the SDR and the CRR can be defined by a critical Damkohler (\(Da_{cr}\)) number, which is representative of the ratio of the diffusion time and the chemical time, i.e., the flame is controlled by diffusion or chemical process, in case of \(Da>>Da_{cr}\) number or \(Da<<Da_{cr}\) number, respectively. Investigating the impacts of different inlet methane velocities in three different burner diameters (\(d=0.0375\), \(0.05\) and \(0.075\) mm) on the flame  
Figure 1: Comparison between experimental and numerical simulations of a micro diffusion flame using methane and air by [PERSON] et al. [PERSON] et al. (2019).