# Entropy Generation during Head-On Interaction of Premixed Flames with Inert Walls within Turbulent Boundary Layers

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## Abstract

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## 1. Introduction

_{2}turbulent mixing layer using high-fidelity three-dimensional Large Eddy Simulations (LES). Safari et al. [16] analysed the implications of local entropy generation rates on combustion modelling of non-premixed flames in the context of LES. The available literature on thermodynamic exergy analysis and exergy balance in combustion systems until the first half of the decade of 2000 was reviewed by Som and Datta [17]. Subsequently, the advancements in high-performance computing have enabled exergy analysis of turbulent combustion processes based on LES and Direct Numerical Simulations (DNS). Farran and Chakraborty [18] analysed entropy generation characteristics in statistically planar turbulent premixed flames for different turbulence intensities, heat release parameters and characteristic Lewis numbers using three-dimensional DNS data. They demonstrated that the regime of combustion does not have a significant influence on the augmentation of entropy generation rate in comparison to the unstretched laminar flames. Chakraborty [19] utilised the DNS data to propose models for the mean contributions of entropy generation due to viscous action, thermal diffusion, molecular mixing and chemical reaction in terms of dissipation rate of kinetic energy and scalar dissipation rate in the framework of RANS. All of these aforementioned analyses for entropy production in combustion processes were carried out in configurations without the influence of walls. Recent LES analyses [20,21,22] for non-reacting wall-bounded flows with heat transfer revealed that the presence of a wall and the associated boundary layer have significant influence on entropy generation rates. Therefore, it can be expected that the presence of a wall will have a significant influence on the entropy generation rate in premixed flames due to three-way interactions between wall, fluid flow and chemical processes within the boundary layer, but insufficient information is available in the literature for exergy analysis for turbulent premixed FWI. To address this gap in the existing literature, the current analysis uses three-dimensional DNS data of statistically planar turbulent premixed flames propagating across the turbulent boundary layer towards a chemically inert wall to analyse the entropy generation mechanisms during head-on interactions of the flame with the wall. These entropy generation mechanisms have been analysed for both isothermal and adiabatic boundary conditions at the wall to understand the influence of thermal boundary conditions on the entropy generation and also on the second-law efficiency. The main objectives of this analysis are: (1) to demonstrate the evolution of the relative contributions of different entropy generation contributions at different stages of head-on interaction of turbulent premixed flames with the wall, (2) to demonstrate the impact of thermal boundary condition on the entropy generation rate characteristics during head-on interaction of turbulent premixed flames with the wall and (3) to provide physical explanations for the observations made in the context of objectives 1 and 2.

## 2. Mathematical Framework

## 3. Numerical Implementation

## 4. Results and Discussion

## 5. Conclusions

- It has been found that the contribution of the viscous action to the overall entropy generation rate remains negligible at all stages of head-on interaction for both isothermal and adiabatic boundary conditions.
- The simulation results reveal that the mean entropy generation rates by chemical reaction, thermal diffusion and species diffusion remain comparable within the flame when it remains sufficiently away from the wall. This is found to be consistent with previous findings for turbulent premixed flames without walls [18].
- The percentage shares of entropy generation rates by viscous action, chemical reaction, thermal diffusion and molecular mixing in turbulent premixed flames away from the wall remain also comparable to that in the unstretched laminar premixed flame under globally adiabatic conditions. However, the entropy generation due chemical reaction decreases during FWI for both isothermal and adiabatic boundary conditions. The reaction rate drops at the wall due to consumption of the reactants for adiabatic boundary condition, which leads to a reduction of the entropy generation rate due to chemical reaction during FWI. By contrast, the reaction rate vanishes at the wall in the isothermal case due to flame quenching, which also leads to a reduction in entropy generation due to chemical reaction.
- The entropy generation due to thermal diffusion during advanced stages of FWI remains relatively stronger in the case of isothermal wall boundary condition than in the adiabatic case due to the high temperature gradient induced by flame quenching in the isothermal wall case. By contrast, the mean values of entropy generation due to thermal diffusion and molecular mixing remain small close to the wall even when the flame interacts with the wall under adiabatic boundary condition. This behaviour arises because the wall-normal components of the gradients of temperature and species vanish at the wall.
- The differences in entropy generation in response to thermal wall boundary conditions affect the overall thermodynamic irreversibility generation and second-law efficiency of the FWI process in turbulent boundary layers.
- The second-law efficiency has been found to increase during advanced stages of FWI because of the reduced entropy generation due to chemical reaction and molecular mixing.
- It is worth noting that the present analysis has been conducted for simple chemistry and transport as this is the first study which analysed entropy generation statistics during FWI within turbulent boundary layers. This study revealed that the entropy generation statistics is significantly affected by flame quenching in the case of isothermal walls. In addition, this study reveals for the first time that the wall boundary condition, significantly affects the entropy generation rate in FWI principally due to differences in thermal diffusion and chemical reaction contributions to entropy generation at the wall and its vicinity.
- The present findings indicate that the thermal condition prevailing at the combustor wall can significantly affect the thermodynamic performance of the combustor and, thus, the wall cooling needs to be optimized from the point of view of ensuring structural integrity and minimization of thermodynamic irreversibility in practical applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Head-on interaction with isothermal (

**top**) and adiabatic (

**bottom**) wall boundary conditions at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (

**top**to

**bottom**). The isosurface coloured in peach represents $c=0.5$. The instantaneous normalised vorticity magnitude $\Omega =\sqrt{{\omega}_{i}{\omega}_{i}}\times h/{u}_{\tau ,NR}$ is shown on the $x-y$ plane at $z/h=4$. The green surface denotes the wall.

**Figure 3.**Variations of normalised mean density $\overline{\rho}/{\rho}_{0}$, Favre-averaged non-dimensional temperature $\tilde{\theta}=(\tilde{T}-{T}_{0})/\left({T}_{ad}-{T}_{0}\right)$, Reynolds averaged reaction rate $\overline{{\dot{w}}_{c}}\times {\delta}_{th}/{\rho}_{0}{S}_{L}$ with the normalised wall-normal distance $y/h$ for isothermal (

**left**) and adiabatic (

**right**) wall boundary conditions at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (

**top**to

**bottom**). The background is coloured by the values of $\tilde{c}$.

**Figure 4.**Variations of Reynolds averaged normalised values of different entropy generation contributions (i.e., $\left\{{\overline{S}}_{1},{\overline{S}}_{2},{\overline{S}}_{3},{\overline{S}}_{4},\overline{{S}_{gen}^{\prime \prime \prime}}=\left({\overline{S}}_{1}+{\overline{S}}_{2}+{\overline{S}}_{3}+{\overline{S}}_{4}\right)\right\}\times h/{\rho}_{0}{c}_{p0}{u}_{\tau ,NR}$) with the normalised wall-normal distance $y/h$ for isothermal (

**left**) and adiabatic (

**right**) wall boundary conditions at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (

**top**to

**bottom**). The background is coloured by the values of $\tilde{c}$.

**Figure 5.**Variations of $\overline{\left|\nabla c\right|}\times {\delta}_{th}$ and $\overline{\left|\nabla \theta \right|}\times {\delta}_{th}$ with the normalised wall-normal distance $y/h$ for isothermal (

**left**) and adiabatic (

**right**) wall boundary conditions at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (

**top**to

**bottom**). The background is coloured by the values of $\tilde{c}$. The non-dimensional temperature gradient $\overline{\left|\nabla \theta \right|}$ in the isothermal case is divided by 2.0 to bring it to the same scale as $\overline{\left|\nabla c\right|}$ at $t/{t}_{f}=14.70$.

**Figure 6.**Variations of the normalised wall heat flux magnitude ${\overline{\mathsf{\Phi}}}_{w}={q}_{w}/\left[{\rho}_{0}{c}_{p0}{u}_{\tau ,NR}\left({T}_{ad}-{T}_{0}\right)\right]$ with time for the isothermal wall boundary condition.

**Figure 7.**Variations of percentage shares of different entropy generation mechanisms (i.e., $P{S}_{i}={{\displaystyle \int}}_{{V}_{f}}{S}_{i}dV/{{\displaystyle \int}}_{{V}_{f}}{S}_{gen}^{\prime \prime \prime}dV\times 100\%$ for $i=1,2,3,4$) to the overall entropy generation rate ${{\displaystyle \int}}_{{V}_{f}}{S}_{gen}^{\prime \prime \prime}dV={{\displaystyle \int}}_{{V}_{f}}\left({S}_{1}+{S}_{2}+{S}_{3}+{S}_{4}\right)dV$ within the flame brush at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (from

**left**to

**right**) where the first (second) bar of each pair represents the isothermal (adiabatic) boundary conditions. The corresponding percentage shares for a 1D unstretched globally adiabatic laminar premixed flame are also shown in the above figures (far

**right**) for the sake of comparison.

**Figure 8.**Variations of the augmentations of entropy generation in comparison to the unstretched laminar flame ${Q}_{T1},{Q}_{T2},{Q}_{T3},{Q}_{T4}$ and ${Q}_{T}$ at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (from

**left**to

**right**) where the first (second) bar of each pair represents the isothermal (adiabatic) boundary conditions.

**Figure 9.**Variations of the second-law efficiency ${\eta}_{II}$ at $t/{t}_{f}=4.20$, $10.50$, $14.70$ (from

**left**to

**right**) where the first (second) bar of each pair represents the isothermal (adiabatic) boundary conditions. The second-law efficiency ${\eta}_{II}$ for a 1D unstretched globally adiabatic laminar premixed flame is also shown (far

**right**) in the above figures for the sake of comparison.

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**MDPI and ACS Style**

Ghai, S.K.; Ahmed, U.; Chakraborty, N.; Klein, M.
Entropy Generation during Head-On Interaction of Premixed Flames with Inert Walls within Turbulent Boundary Layers. *Entropy* **2022**, *24*, 463.
https://doi.org/10.3390/e24040463

**AMA Style**

Ghai SK, Ahmed U, Chakraborty N, Klein M.
Entropy Generation during Head-On Interaction of Premixed Flames with Inert Walls within Turbulent Boundary Layers. *Entropy*. 2022; 24(4):463.
https://doi.org/10.3390/e24040463

**Chicago/Turabian Style**

Ghai, Sanjeev Kr., Umair Ahmed, Nilanjan Chakraborty, and Markus Klein.
2022. "Entropy Generation during Head-On Interaction of Premixed Flames with Inert Walls within Turbulent Boundary Layers" *Entropy* 24, no. 4: 463.
https://doi.org/10.3390/e24040463