Introduction

Natural gas (NG) has a higher hydrogen-to-carbon ratio than the other hydrocarbon fuels. Therefore, NG usage in transportation sectors may be beneficial for the reduction of carbon dioxide and particulate matter emissions, and it is as such considered as an alternative transportation fuel. However, NG engines encounter problems in application, such as lower engine efficiency [1] and higher combustion instability [2] due to its low flame speed. The addition of H2, a fuel with fast flame propagation, was considered as a method to solve these problems [3]. Therefore, combustion and emissions characteristics of engines fueled with NG and H2 blends have been widely studied [4,5,6,7]. Gong et al. [8] pointed that with H2 addition, the ignition delay in natural gas engines was shortened and the engine-out emissions were reduced, but Verma and Prasad [9] found that the knock tendency of a compressed natural gas engine increased with H2 addition at high loads.

Thermal efficiency is an important parameter for engine performance evaluation and is conventionally calculated based on the first law of thermodynamics. However, the first law of thermodynamics is limited to energy quantity analysis, without considering the energy quality change in energy conversion processes. Recently, the second-law thermodynamic analysis has been applied by some researchers to study the energy conversion in thermodynamic systems [10, 11], from which the energy quality changes could be evaluated. By applying the second law of thermodynamics to engine efficiency analysis, Rakopoulos and Michos [12] concluded that the combustion irreversibility is a primary source for exergy destruction. Therefore, it is significant to understand the energy quality degradation mechanism in combustion processes, based on which methods for efficiency improvement might be explored. Recently, Zhang et al. [13, 14] attempted to elucidate the exergy destruction sources in fuel auto-ignition and flame propagation [15, 16], based on an analysis combining thermochemistry and chemical kinetics. It was found that exergy destruction in fuel combustion might be reduced by proper fuel blending or charge preparation.

There are two types of NG engines based on the mixture preparation methods, i.e., the premixed charge spark-ignited NG engines and diesel/NG dual-fuel engines, the latter of which features the mixing-controlled combustion. These distinct combustion organization methods require researchers to identify exergy destruction characteristics in both premixed and non-premixed CH4 flames. Wang et al. [17] investigated the entropy generation in premixed methane/hydrogen flames using a multi-step kinetic model, and they found that with H2 addition, the total exergy destruction decreased because of the reduced entropy generation from thermal conduction. Besides, Emadi and Emami [18] studied the effects of H2 addition on the irreversibility in turbulent non-premixed methane flames, and they indicated that H2 enrichment increased the second-law combustion efficiency and reduced the total entropy generation. Chen et al. [19] investigated the influence of the effective equivalence ratio and H2 enrichment in fuel mixtures on the entropy generation in lean methane/air counterflow non-premixed flames and found that the total entropy generation increased with H2 fraction and effective equivalence ratio.

Although there already were some studies aiming to figure out the exergy destruction characteristics in different non-premixed CH4/H2 flames, they were mainly focused on parametric studies and the physical/chemical mechanisms underlying exergy destruction in non-premixed flames have not been well identified. In this study, the thermochemical and chemical kinetic analysis was combined to explain the exergy destruction mechanism in non-premixed counterflow flames of methane and hydrogen mixtures. The contributions of different sources to exergy destruction in the context of changed H2 percentage were studied, with the influence of each parameter on exergy destruction being elucidated.

Methodology

The one-dimensional counterflow non-premixed flames were simulated at atmospheric pressure using the GRI 3.0 Mech [20] and the CHEMKIN PRO software [21]. This mechanism has been widely validated against the experimental measurements, such as flame propagation [22], soot formation [23] and auto-ignition tendency [24]. Further, Cheng et al. [25] and Padilla et al. [26] compared the experimentally measured mole fractions of primary species, peak temperature and temperature profiles in methane counterflow flames with the calculated results using the GRI 3.0 Mech, and the comparison showed that the experimental and computational results are in good agreement. As such, GRI 3.0 Mech was used for the calculation in this study. For all the studied conditions, the unburned gas temperature and the environmental temperatures were held at 300 K and 298 K, respectively. Equal flow velocities at the fuel and oxidizer nozzle exits were set, and the distance between the two nozzles was set as 3 cm. The thermodynamic and chemical data used for exergy destruction calculation, e.g., temperature, pressure, mole fractions of species and reaction rates, could be obtained in the calculation results.

Exergy destruction is caused by entropy generation, and the entropy generation in flames includes four irreversible sources, namely heat conduction, mass diffusion, viscous dissipation and chemical reactions. Therefore, the total entropy generation is calculated by Eq. 1 [27]:

$$\dot{S}_{\text{g}} = - \frac{{\dot{q}_{\text{c}} \cdot \nabla T}}{{T^{2} }} - \frac{1}{T}\mathop \sum \limits_{i} \left[ {\dot{m}_{\text{i}} \cdot \left( {s_{\text{i}} \nabla T + \nabla \mu_{\text{i}} } \right)} \right] + \frac{\tau :\nabla u}{T} + \mathop \sum \limits_{i} \frac{{f_{\text{i}} \cdot \dot{m}_{\text{i}} }}{T} - \frac{{\mu_{\text{i}} \cdot \dot{\omega }_{\text{i}} }}{T}$$
(1)

where \(T\) is the temperature, \(\dot{q}_{\text{c}}\) is the flux of heat conduction, \(\dot{m}_{\text{i}}\) is the mass flux of the ith species, \(s_{\text{i}}\) is the specific entropy of the ith species, \(\tau\) is the friction stress, \(u\) is the velocity, \(f_{\text{i}}\) is the body force of the ith species, \(\mu_{\text{i}}\) is the chemical potential of the ith species and \(\dot{\omega }_{\text{i}}\) is the molar production rate of the ith species. The terms on the right side of Eq. 1 represent the entropy generation rates induced by heat conduction, mass diffusion, viscous dissipation, body force and chemical reactions, respectively.

According to the Fourier’s law,

$$q_{\text{c}} = - \lambda \nabla T$$
(2)

where \(\lambda\) is the thermal conductivity.

According to the Fick’s law,

$$\dot{m}_{\text{i}} = - \rho D_{\text{i}} \nabla Y_{\text{i}}$$
(3)

where \(\rho\) is the mass density of the gas, \(D_{\text{i}}\) is the diffusion coefficient of the ith species and \(Y_{\text{i}}\) is the mole fraction of the ith species.

The gravity is the only body force considered here, and thus,

$$\mathop \sum \limits_{i} f_{\text{i}} \cdot \dot{m}_{\text{i}} = \mathop \sum \limits_{i} g \cdot \dot{m}_{\text{i}} = g \cdot \mathop \sum \limits_{i} \dot{m}_{\text{i}} = 0$$
(4)

Therefore, the one-dimensional form of Eq. 1 could be expressed as Eq. 5:

$$S_{\text{g}} = \frac{\lambda \nabla T \cdot \nabla T}{{T^{2} }} + R\mathop \sum \limits_{i} \frac{{\rho D_{\text{i}} }}{{X_{\text{i}} }}\nabla Y_{\text{i}} \cdot \nabla X_{\text{i}} + \frac{\tau :\nabla u}{T} - \frac{{\mu_{\text{i}} \cdot \dot{\omega }_{\text{i}} }}{T}$$
(5)

where \(R\) is the gas constant and \(X_{\text{i}}\) is the mole fraction of the ith species.

According to Nishida et al. [28], the entropy generation from viscous dissipation is negligible compared with those from the other sources. Therefore, the entropy generation due to viscous dissipation is not considered in the following study.

After calculating the entropy generation, the exergy destruction rate can be presented as Eq. 6 [29]:

$$\dot{I}_{\text{destruction}} = T_{0} \dot{S}_{\text{g}}$$
(6)

where \(T_{0}\) is the environment temperature, i.e., 298 K.

Therefore, the exergy destruction rates from heat conduction, mass diffusion and chemical reactions can be calculated as follows:

$$\dot{I}_{\text{cond}} = T_{0} \lambda \frac{1}{{T^{2} }}\left( {\frac{{{\text{d}}T}}{{{\text{d}}x}}} \right)^{2}$$
(7)
$$\dot{I}_{\text{diff}} = T_{0} \mathop \sum \limits_{k = 1}^{k} \frac{{\rho R_{\text{gi}} D_{\text{i}} }}{{X_{\text{i}} }}\frac{{{\text{d}}Y_{\text{i}} }}{{{\text{d}}x}}\frac{{{\text{d}}X_{\text{i}} }}{{{\text{d}}x}}$$
(8)
$$\dot{I}_{\text{chem}} = - T_{0} \mathop \sum \limits_{i} \frac{{\mu_{\text{i,j}} \dot{\omega }_{\text{i,j}} }}{T}$$
(9)

where \(R_{\text{gi}}\) is the specific gas constant of the ith species, \(\mu_{\text{i,j}}\) and \(\dot{\omega }_{\text{i,j}}\) are the chemical potential and the mole production rate of the ith species in the jth reaction.

The chemical potential in Eq. 9 can be expressed as:

$$\mu_{\text{i,j}} = - Ts_{\text{i,j}} \left( {T_{\text{i}} } \right) + h_{\text{i,j}} \left( {T_{\text{i}} } \right) + RT{ \ln }\left( {X_{\text{i}} P_{\text{i}} } \right)$$
(10)

where \(s_{\text{i,j}} \left( {T_{\text{i}} } \right)\) and \(h_{\text{i,j}} \left( {T_{\text{i}} } \right)\) are the specific entropy and enthalpy of the ith species in the jth reaction, respectively, and \(T_{\text{i}}\) and \(P_{\text{i}}\) are the temperature and partial pressure of the ith species.

Fuel chemical exergy injected from nozzles changes with changed H2 blending percentage. Therefore, it is not appropriate to directly compare the absolute exergy destruction amount in different fuels. To clearly compare the exergy destruction behaviors in different CH4/H2 flames, normalized exergy destruction was used in this study. Specifically, the fuel chemical exergy can be calculated by Eq. 11:

$$E_{\text{burned}} = \left( {1 - \varphi_{\text{un}} } \right) \cdot \rho_{\text{fuel}} \cdot u_{\text{in}} \cdot e_{\text{fuel}}$$
(11)

where \(\varphi_{\text{un}}\) is in the fraction of unburned fuel, \(\rho_{\text{fuel}}\) is the fuel density, \(u_{\text{in}}\) is the fuel flow velocity, and \(e_{\text{fuel}}\) is the specific chemical exergy. The normalized exergy destruction can be calculated as Eq. 12:

$$\overline{I}_{\text{destruction}} = \frac{{\dot{I}_{\text{destruction}} }}{{E_{\text{burned}} }}$$
(12)

where \(\overline{I}_{\text{destruction}}\) is the normalized exergy destruction rates.

As described above, exergy destruction characteristics in non-premixed counterflow flames of CH4/H2 blends and air were analyzed based on thermodynamics and chemical kinetics. Specifically, the exergy destruction in flames of 100%CH4 and 50%CH4–50%H2 was compared.

Results and discussion

To explain the exergy destruction in non-premixed counterflow flames, the CH4/air flame is first taken as the baseline case, in which the flow velocities at both nozzle exits are set as 15 cm/s. As shown in Fig. 1, the total normalized exergy destruction rate curve has three peaks, the first and the third of which are caused by heat conduction, while the second is mainly induced by chemical reactions. In addition, all the normalized exergy destruction rate profiles of heat conduction, mass diffusion and chemical reactions feature double peaks. It is shown that the highest peak is caused by chemical reactions and located at the flame center, followed by the peaks from heat conduction and mass diffusion. It is also observed that the second peak of the normalized exergy destruction rate from chemical reactions is located at the flame downstream and much lower than the first peak, due to the much lower reaction rates at the flame downstream. Additionally, the two peaks of normalized exergy destruction rates from heat conduction occur at the flame edges, where the temperatures are low but temperature gradients are high, and this as such leads to high exergy destruction rates by heat conduction, according to Eq. 7. Finally, for the normalized exergy destruction rates from mass diffusion, the first peak is insignificant, and the location of the second peak is close to the flame front, where the mole fractions of species change greatly, thus causing high peak exergy destruction rate from mass diffusion, according to Eq. 8.

Fig. 1
figure 1

Normalized exergy destruction rates and temperature profile in the non-premixed CH4/air counterflow flame: P = 1 atm, Vinlet = 15 cm/s

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The contributions to exergy destruction from different sources in the flames of CH4 and the 50%CH4–50%H2 blendare shown in Fig. 2. It is observed that heat conduction contributes the most to the exergy destruction in both flames, followed by chemical reactions and mass diffusion in the CH4 flame, and by mass diffusion and chemical reactions in the flame of the 50%CH4–50%H2 blend. With H2 addition, the contribution of heat conduction and mass diffusion is promoted by 1.81% and 0.92%, while that of chemical reactions is reduced by 1.72%, causing the total exergy destruction to increase by about 1%.

Fig. 2
figure 2

Contributions to exergy destruction from each source in the flames of CH4 and a CH4/H2 blend: P = 1 atm, Vinlet = 15 cm/s

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To explain this phenomenon, the normalized exergy destruction rates in the flames of CH4 and the 50%CH4–50%H2 blend are compared in Fig. 3. It is shown that with H2 addition, the flame region is widened and the peak temperature is raised because H2 is more active and has higher diffusivity. The double peaks of the normalized exergy destruction rates from heat conduction are almost unaffected, possibly because the temperature profiles and temperature gradients are not significantly changed at flame edges. Therefore, the promoted contribution from heat conduction is mainly caused by the widened flame region.

Fig. 3
figure 3

Normalized exergy destruction rates from each source in the flames of CH4 and a CH4/H2 blend, Vinlet = 15 cm/s

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Additionally, the first peak of the normalized exergy destruction rate from mass diffusion does not change much, but the second peak is slightly reduced with H2 addition. However, considering the widened flame region, the overall contribution of mass diffusion to exergy destruction is promoted. Based on Eq. 8, the exergy destruction rate from mass diffusion is dominated by the mole fraction gradient of species. To clearly identify the exergy destruction from mass diffusion, the mole fractions of primary species, e.g., CH4, H2, O2, N2, H, O, H2O and CO2, are plotted in Fig. 4. Also, the contributions of these species to the two peaks of the exergy destruction from mass diffusion are listed in Tab. 1. For the flame of 100%CH4, it is shown that CH4 and N2 are the dominant species in the first peak of exergy destruction by mass diffusion, as their mole fraction gradients are higher than the other species. However, in the second peak, the species H2, H and O2 are more influential. When considering the whole flame domain, H2, O2, H2O, CH4 and N2 contribute most to the exergy destruction from mass diffusion, as they are the main reactants/products or the main air component. With H2 addition, the contributions of H2, H and H2O to exergy destruction from mass diffusion are promoted, whereas those of the carbon-containing species are reduced. According to Fig. 4, in the first peak, the mole fraction gradients of CH4 and N2 in the flame of CH4 are nearly equal to those of CH4, H2 and N2 in the flame of the blend, and thus, the first peak of exergy destruction rate in both flames is almost the same. In the second peak, with H2 addition, the change of species mole fraction exists in a wider region, and therefore, the peak mole fraction decreases, leading to the reduced peak normalized exergy destruction rates (Table 1).

Fig. 4
figure 4

Mole fraction distributions of primary species in the flames of CH4 and a CH4/H2 blend, Vinlet = 15 cm/s

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Table 1 Contributions of primary species to the exergy destruction from mass diffusion in the flames of CH4 and a CH4/H2 blend, Vinlet = 15 cm/s
Full size table

As shown in Fig. 3, for the normalized exergy destruction rates from chemical reactions, the first peak is reduced to one-fifth and the second peak is reduced to about two-thirds with H2 addition, causing reduced contribution of chemical reactions to exergy destruction. To further study this phenomenon, the contributions of top reactions to the two exergy destruction peaks of both fuels are listed in Tab. 2. These reactions contribute over 60% to the exergy destruction from chemical reactions, and therefore, the analysis of these reactions can help understand the exergy destruction from chemical reactions. It is obvious that in the first peak, the reactions involving carbon-containing species are more influential, while the second peak is dominated by the H2-O2 system reactions (Table 2).

Table 2 Contributions of primary reactions to the exergy destruction from chemical reactions in flames of CH4 and the 50%CH4–50%H2 blend, Vinlet = 15 cm/s
Full size table

As shown in Fig. 4, H2 addition leads to the decreased mole fraction of CH4, and as such, most carbon-containing reactions are suppressed. This reduces the first peak of the exergy destruction rate from chemical reactions. In addition, H2 addition results in an increased mole fraction of H2 and H2O, and as such, the reactions such as R1, R6 and R9, in which H2 and H2O are products, are suppressed. Therefore, the first peak of the normalized exergy destruction rate from chemical reactions is reduced. In the second peak of chemical reactions, H2 addition causes the increased mole fractions of H and H2 and also promotes the production of H2O. Some reactions are enhanced with H2 addition, such as R16, R17, R21 and R24, but the others are suppressed, such as R15, R18, R19, R20 and R22. As a result, the overall magnitude of the second peak of the exergy destruction rate from chemical reactions is negligibly changed.

Conclusions

The second-law thermodynamic analysis on non-premixed counterflow flames of CH4/H2 mixtures was numerically conducted. The test fuels were 100%CH4 and 50%CH4–50%H2, respectively. The sources causing exergy destruction in the flames included heat conduction, mass diffusion and chemical reactions. The main conclusions are listed as follows:

  1. 1.

    The total normalized exergy destruction rates feature three peaks, the first and the third of which are mainly caused by heat conduction and the second of which is dominated by chemical reactions. The normalized exergy destruction rates from heat conduction, mass diffusion and chemical reactions at most conditions show two peaks.

  2. 2.

    With hydrogen addition, the peaks of the normalized exergy destruction rate from heat conduction do not change much. For the exergy destruction peaks by mass diffusion, the first peak is almost unchanged because it is dominant by the unburned mixture. However, the second peak from mass diffusion is reduced, as the peak of the species mole fraction gradient decreases. The first peak of normalized exergy destruction rate from chemical reactions is reduced because the main reactions of carbon-containing species are suppressed, but the second peak is only slightly changed.

  3. 3.

    Heat conduction contributes the most to exergy destruction in non-premixed flames. With an increased hydrogen addition, the contributions of heat conduction and mass diffusion to exergy destruction increase, whereas the contribution of chemical reactions decreases. As a result, the total exergy destruction increases by about 1%.