The Partially Stirred Reactor (PaSR) model is carried out for the ammonia-air combustion system by means of stochastic modeling, namely by solving the transport equation for the joint Probability Density Function (PDF). The turbulent mixing is accounted for by the Linear Mean-Square Estimation (LMSE) mixing model. Notwithstanding the simplified nature of the PaSR modeling, the transported-PDF method enables capturing the effect of mixing frequency on the combustion system, especially the NOx emission. Since the chemical source term is in a closed form in the transported-PDF method, it allows us to apply different chemical mechanisms to explore, whether the set of elementary reactions that are identified as important for the prediction of NOx in the PaSR model is sensitive to the choice of chemical mechanisms. Furthermore, the effect of the residence time in the PaSR model has also been studied, and compared with those in the Perfectly Stirred Reactor (PSR) model (infinite large mixing frequency). Moreover, since the ammonia under oxygen enrichment shows some similar combustion behaviors in terms of e.g. laminar burning velocity as the ammonia under hydrogen enrichment, how large the difference of thermo-kinetic states (e.g. temperature and NOx emission) predicted by PaSR models and in laminar premixed flame configuration is also investigated. A further discussion focuses on the effect of thermal radiation, where the radiative heat loss roles in the prediction of NOx for the turbulent simulation is examined. By using the optically thin approximation model, it is shown that the thermal radiation exhibits little effect on the considered combustion systems within a typical turbulent time-scale.
1 Introduction
Ammonia is a zero-carbon fuel and has been considered as the next possible source for power supply and energy storage (Valera-Medina et al. 2018, 2021). Many studies involving ammonia combustion systems have been carried out over the last decade. Among them are for example the development of ammonia chemical kinetics (Shrestha et al. 2021; Gotama et al. 2022; Nakamura et al. 2017; Dai et al. 2020), numerical and experimental determination of laminar burning velocity (Shrestha et al. 2021; Rocha et al. 2021; Wang et al. 2021), forced ignition processes (Lhuillier et al. 2020; Fernández-Tarrazo et al. 2023; Lesmana et al. 2022; Yu et al. 2023), extinction and flammability limit (Lee and Kwon 2011; Ichimura et al. 2019; Richter et al. 2023) and many others. Unlike the traditional hydrocarbon fuels, the ammonia mixture has some special combustion properties such as low flammability, low laminar burning velocity, high ignition energy requirement and so on Valera-Medina et al. (2018, 2021). Furthermore, the ammonia system suffers from high nitrogen oxides (NOx) emissions in combustion, and the NOx emission is closely correlated with other combustion properties such as flame stability which is important for real-life applications. For example, under lean conditions, NOx emissions were prohibitive while stability was restricted to a very narrow equivalence ratio range. As shown in Valera-Medina et al. (2019); Mashruk et al. (2022); Rocha et al. (2019), ammonia-hydrogen-air combustion system close to lean conditions lead to prohibitive NOx emissions while stability was compromised because of a very narrow equivalence ratio range, and rich conditions would be more suitable for practical applications.
Moreover, the investigation of the turbulence effect on the ammonia combustion system is more practice-oriented, and the increasing number of literature focuses on the turbulent ammonia flame. For example, the effect of Karlovitz number on the turbulent premixed flame structure and turbulent burning velocity has been reported in Fan et al. (2022). In Abdelwahid et al. (2023); Chi et al. (2023); Tian et al. (2023) the effect of the molecular transport model on the flame structure and species prediction has been intensively studied, showing that the differential diffusion including thermal diffusion (Soret effect) has a large impact on the NO production and turbulent flame speed. However, DNS or LES further complicates the study on the turbulence-chemistry interaction since one must use detailed molecular transport model which increases the computational cost or the averaged reaction rate must be accurately modeled/solved or the CFD grid must be fine enough to resolve the flame structure. Various physical variables could have an impact on the chemical kinetics, which makes the investigation more complicated.
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In order to explore the turbulence-chemistry interaction for ammonia combustion systems, a stochastic modeling for the Partially Stirred Reactor (PaSR) model developed by Correa (1993) and Chen (1997) is used, a PDF-method-based models where the temperature and composition concentrations are described by a joint PDF. While the conventional Perfectly Stirred Reactor (PSR) model considers infinite fast mixing process and can provide valuable insights into the chemical kinetics, the PaSR model has the advantage that it allows to investigate the interaction between chemistry and finite-rate mixing and subsequently provides useful information about the effect of mixing processes on the chemical kinetics (Correa 1993; Cannon et al. 1998). Moreover, since the Monte-Carlo method has been used for the numerical simulation of PaSR model, the chemistry can be solved in an exact manner without any modeling.
The objective of this work will focus on several interesting questions, that provide a better understanding of the ammonia combustion system under the interaction with mixing processes:
How large is the impact of mixing frequency on the ammonia system? Whether the influence of mixing frequency is similar to those on the conventional hydrocarbon fuels (Cannon et al. 1998; Correa 1993). In addition, to the author’s knowledge, there is still limited investigation on the statistical properties of the ammonia system. How the statistics change with mixing frequency is a very important question. The understanding of the statistics is especially important for those who want to use the presumed PDF method in the turbulent simulation (Pope 1985; Borghi 1988; Bray et al. 1989).
How large is the impact of residence time on the ammonia system, especially on the NOx emission. Whether the influence of residence time in the PaSR model is the same as the one in the conventional PSR model. Moreover, it is of great interest to know which elementary reactions are important for the NOx emission, and whether these important reactions remain unchanged for different residence times. Such information helps us to better understand chemical kinetics and provides useful information for the development of chemical mechanisms.
Since there are various chemical mechanisms developed for ammonia systems that have been validated against common combustion properties such as ignition delay times and laminar burning velocities, whether the difference on the predicted NOx emission under the consideration of finite-rate mixing process in the PaSR models behaves similar as the difference predicted by the PSR model. If the difference in the NOx emission is observed, whether the important elementary reactions for the prediction of NOx emission are the same even for different chemical mechanisms.
Whether the statistics (or the PDFs) of thermo-kinetic quantities (e.g. temperature, species concentration) are similar for different chemical mechanisms?
Besides the physical variables (mixing frequency and residence time) and chemical kinetic models, thermal radiation has been proven to have a large impact on the structure of turbulence flames. As shown in various works such as Dos Santos et al. (2016); Xu et al. (2021); Giordano and Lentini (2001); Poitou et al. (2012); Cao et al. (2007), while the thermal radiation causes only slightly decrease of mean temperature for the turbulent flames (around 20 - 50 K) and almost no differences for major species, the prediction of minor species or pollutants (e.g. NOx and soot) can be largely influenced by considering the radiative heat loss. Although the turbulence-radiation interaction has been intensively investigated for various hydrocarbon systems (Cao et al. 2007; Ilbas 2005; Giordano and Lentini 2001), there are limited studies about this topic for the ammonia combustion system. To the author’s knowledge, only Nakamura et.al. in Nakamura and Shindo (2019) investigated intensively the thermal radiation effect on the ammonia-hydrogen-air system based on the simple optical thin approximation model (OTM) for unstrained laminar premixed flame configurations, confirming that the radiative heat loss has a strong effect on the laminar flame speed and the NOx emission in the high temperature regimes. However, the turbulence-radiation interaction for the ammonia-air system is still not investigated. Therefore, the novelty of this work regarding thermal radiation would be focused on understanding the impact of thermal radiation on the flame quantities?
2 Stochastic Modeling and Simulation of the Partially Stirred Reactor (PaSR)
2.1 Stochastic Modeling
The considered PaSR model is schematically illustrated in Fig. 1. The computational cell is split into three steps: one describing a through-flow process, one where reactions take place, and another characterized by only mixing. The final species concentration of the cell is determined from the mass exchange, characterizing the turbulence intensity.
Fig. 1
Schematic illustration of PaSR model
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Suppose we have the thermo-kinetic state vector \(\vec {\phi }\) including scalars specific enthalpy h (in unit J/kg), pressure p (in unit Pa) and mass fractions of i-th species \(Y_i\):
where \(n_{s}\) is the number of species involved in the chemical reactions. According to the Chen (1997), the stochastic modeling of the PaSR can be achieved by solving the transport equation for the joint Probability Density Function (PDF). The PDF applied in the PaSR modeling is the single-point joint scalar PDF \(f_{\vec {\phi }}(\vec {\psi },t)\), and the resulting joint PDF equation for the PaSR is Chen (1997)
where \({\mathcal {R}}_\alpha\) is the source term of \(\alpha -\)th species in \(\vec {\phi }\), and \(\varepsilon _{\alpha \beta }\) the cross-scalar dissipation rate. In this equation, there exists three important time-scales: one is the residence time \(\tau _\text {res}\), second one the mixing time-scale \(\tau _\text {mix}\) in the term of cross-scalar dissipation rate, and the third one the chemical time-scale \(\tau _\text {reac}\) determined by the eigenvalue of the Jacobian matrix of the chemical source term (Maas and Pope 1992).
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The first two terms on the right hand side of Eq. 2 represent the effect of the through-flow and the chemical reaction processes on the joint PDF. Both terms can be solved in a detailed way, and the simulation can be performed with detailed chemical kinetics without additional modeling and computing limitations. The third term on the right hand side of Eq. 2 stands for the effect of the micro-scale mixing process on the joint PDF, which needs to be modeled.
The transported PDF equation can be computed efficiently by using the Monte-Carlo particle method (Pope 1985), where the PDF \(f_{\vec {\phi }}(\vec {\psi },t)\) is represented by the \(N_p\)-number of notional particles. Each notional particle k includes the information of specific enthalpy, pressure and mass fractions of \(i-\)th species in the thermo-kinetic state vector \(\vec {\phi }^{(k)} = (h^{(k)},p^{(k)},Y_i^{(k)})^\text {T}\). Based on the \(\vec {\phi }^{(k)}\), the ensemble-averaged values of thermo-kinetic state can be determined as
and the root mean square (r.m.s.) of the fluctuation, \(\vec {\phi }^{\prime } = \vec {\phi }^{(k)} - \langle \vec {\phi } \rangle\), can be calculated as
where \({\dot{\omega }}_i\) and \(W_i\) are the net production rate and molar mass of species i, and \(\rho\) is the density of the gas mixture. The symbol \({\dot{q}}_\text {rad}^{(k)}\) stands for the radiative heat loss rate per unit mass (J/(kg\(\cdot\)s)):
The term \({\mathcal {M}}\) is a crucial issue in the simulation of transported PDF equation, since its modeling can largely affect the numerical prediction (Chang and Chen 1996; Cao et al. 2007). According to Pope (1985), the term of cross-scalar dissipation rate in Eq. 2, in which scalar diffusion, conditional upon the scalar value, must be modeled in the way, such that the decay of the scalar fluctuation should admit \(\text {d}\langle \vec {\phi }^{\prime 2} \rangle / \text {d} t = -2C_\phi \omega _\text {mix} \langle \vec {\phi }^{\prime 2} \rangle\), where \(C_\phi\) is the mixing model parameter and \(\omega _\text {mix}\) the mixing frequency. An explicit approximation of this term is the Linear Mean-Square Estimation (LMSE) (Dopazo and O’Brien 1974), a mixing model is one of the most simple but commonly used mixing models, which can give the correct decay of scalar fluctuation. For this model, the evolution of the thermo-kinetic states of the particle k is expressed via an ordinary differential equation
in which \(\langle \vec {\phi } \rangle\) the mean value of the thermo-kinetic states of \(N_p\) particles.
Note that although the LMSE is simple and gives reasonable prediction for various cases (Cao et al. 2007; Raman and Pitsch 2007), the principle deficiency of LMSE model is the fact that the shape of the scalar PDF remains unchanged due to the absence of mean scalar gradient and thus never relaxes to Gaussian distribution (Celis and Silva 2015; Fox 2003). Therefore, many other advanced mixing models have been developed. Among them are e.g. the Euclidean Minimum Spanning Trees (EMST) model (Subramaniam and Pope 1998), the Multiple Mapping Conditioning (MMC) model (Cleary and Klimenko 2009) and the shadow-position conditioning mixing model (SPMM) (Pope 2013).
2.2 Numerical Implementation
The numerical simulation has been performed by using the self-written PaSR4Comb package, a MatLab-based code coupled with Cantera (Goodwin et al. 2022) for the simulation of PaSR for combustion system, which is also available for download in Github via (Yu 2023). The numerical simulation follows the procedure described in Chen (1997), namely a fractional step scheme with a time step \(\Delta t\) is used to determine the PDF equation for the PaSR (c.f. Eq. 2), and particles are assumed to have the same weight. For each time step, the following three steps are performed:
(i)
Through-flow: For this step, \(N_\text {replaced}\) notional particles are randomly selected from the whole stochastic ensemble, and replaced by the notional particles with thermo-kinetic states of incoming mixture properties \(\vec {\phi }_\text {inlet}\). The \(N_\text {replaced}\) is calculated via
Note that if \(N_\text {replaced}\) is less than one, the \(N_\text {replaced}\) is saved and accumulated until it reaches one and then replaces the particle accordingly.
(ii)
Molecular mixing process: The mixing process will be performed for the time step \(\Delta t\). For the mixing model, the model parameter \(C_\phi\) must be given a-priori. In the present work, only the LMSE mixing model will be used. For the LMSE model, the thermo-kinetic state after mixing can be calculated in a deterministic way and the analytical solution reads:
Chemical reaction process: During this reaction process step, the thermo-kinetic states of the notional particles are evolved due to chemical reaction with the time step \(\Delta t\) following the ordinary differential equation (ODE)
The initial condition for this chemical reaction process is the thermo-kinetic states of the notional particles after the mixing process. In the code, the chemical source term is evaluated through the Cantera (Goodwin et al. 2022), and the ODE system is computed by using in-built ode15s, a variable-step variable-order (VSVO) solver in Matlab for the numerical integration of stiffness ODE (Shampine and Reichelt 1997).
For the starting point, all particles inside the PaSR at \(t=0\) have the thermo-kinetic states under equilibrium conditions.
2.3 Validation of the PaSR4Comb Package
The PaSR4Comb (Yu 2023) has been validated against various existing simulation results in different works such as Cannon et al. (1998); Correa (1993); Chen (1997). Before continuing our discussion on the ammonia-air combustion system, we shortly show in this section the validation of the implemented algorithm applied in PaSR4Comb package.
Figure 2 shows the comparison of ensemble-averaged temperature \(\langle T \rangle\) and NO emission \(\langle\)NO\(\rangle\) versus mixing frequencies from Cannon et al. (1998) (in lines) and from PasR4Comb code (in symbols). The GRI 2.11 mechanism is used and the CH\(_4\)-air mixture with three different fuel/air equivalent ratios is considered. The results from PasR4Comb code are based on exact identical model and numerical parameters from Cannon et al. (1998). We notice that there are almost no observable differences between both results, showing the correctness of PaSR4Comb code.
Fig. 2
Dependence of ensemble-averaged temperature \(\langle T \rangle\) and NO emission \(\langle\)NO\(\rangle\) on mixing frequencies \(\omega _\text {mix}\). Mixture: methane-air. Mechanism: GRI 2.11. Lines: simulation results from Cannon et al. (1998). Symbols: simulation results using PaSR4Comb code
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3 Combustible Gas System
In the present work, we focus on the stoichiometric 70%NH\(_3\)-30%H\(_2\)-air combustion system. This 70%-30% (vol) ammonia-hydrogen blend has been investigated in many works for different purposes such as spark ignition engines and gas turbines (Chiong et al. 2021; Mashruk et al. 2022; Xiao et al. 2017; Valera-Medina et al. 2019), showing that 30% (vol) hydrogen in an ammonia mixture could produce high power as methane and provides wide operational range with considerably stable combustion property (Mashruk et al. 2022; Chiong et al. 2021). Figure 3 shows the tendency of adiabatic temperature and NOx concentration against the residence time \(\tau _\text {res}\) for a PSR model, which corresponds to the PaSR model with \(\omega _\text {mix} \rightarrow \infty\). Also, four different chemical mechanisms developed for the ammonia-hydrogen-air system are applied. Here, the PSR calculation is consistent with the model proposed in Glarborg et al. (1986).
Fig. 3
Tendency of adiabatic temperature and NOx concentration against the residence time \(\tau _\text {res}\) for a PSR model. Blue: Nakamura-2017 (Nakamura et al. 2017); Red: Stagni-2020 (Stagni et al. 2020); Yellow: Shrestha-2021 (Shrestha et al. 2021); Purple: Gotama-2022 (Gotama et al. 2022)
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As expected the adiabatic temperature at a steady state in PSR drops as residence time becomes shorter due to shorter available reaction time, which is consistent with many works such as Hottel et al. (1958); Lu and Law (2005); Rutar et al. (1998); Turns (1996). The behavior of NOx (including NO and NO\(_2\)), on the other hand, is more complicated, with maxima exhibited at a certain residence time (more explanation in Sec. 4.3). Furthermore, we notice that the temperature profiles predicted by different mechanisms are similar, while the predicted NOx concentrations can be significantly different. However, the tendency is the same for all tested mechanisms, namely the NOx concentration has a peak at a certain residence time.
It is also observed that the extinction residence time (very left point), below which no burning solution can be obtained, is almost similar even with different chemical mechanisms. This is because all four mechanisms have been well validated against the experimental measurement of ignition delay time.
In the following, we select the Shrestha-2021 mechanism for our further discussion about the influence of mixing frequency and residence time because of the following two reasons:
It has been validated for different purposes such as ignition delay times and laminar flame speeds intensively against experiment measurements (Shrestha et al. 2021).
This mechanism was also proven to give an accurate prediction for the oxygen-enriched ammonia-air system (Shrestha et al. 2021) in terms of laminar flame speed. Later, the effect of hydrogen addition and oxygen addition in the ammonia-air mixture for the PaSR model will be examined.
However, the influence of both mixing frequency and residence time on the NOx concentration predicted by the other three detailed mechanisms will also be discussed shortly, showing that the obtained results and observation are not affected by the chemical mechanisms.
Moreover, if not specified, the adiabatic system is considered, which means a zero radiative heat loss in the equation (\(q_\text {rad}=0\) in Eq. 7). However, the effect of the radiation on the predicted ensemble-averaged quantities, especially on the NOx emission which is very sensitive to the temperature, will be still investigated. As reported in Nakamura and Shindo (2019); Zheng et al. (2023), the thermal radiation can change the flame structure, species concentrations and minimal ignition energy for the ammonia-hydrogen-air combustion system. The radiation model used in the present work follows the one from Nakamura and Shindo (2019), namely the optically thin approximation model (OTM), where the volumetric radiative heat loss \({\dot{Q}}_\text {rad}\) in J/(m\(^3\cdot\)s) is given as:
here \(\sigma =5.669\cdot 10^{-8}\)W/(m\(^{2}\cdot\)K\(^{4}\)) is the Stefan-Boltzmann constant. T is the local temperature and \(T_\text {b}\) is the background temperature (\(T_\text {b}=T_\text {ub}\)). k is the total Planck mean absorption coefficient, and determined as \(k = \sum p_i a_{p,i}\), where \(p_i\) is the partial pressure of radiating species i in and \(a_{p,i}\) the Planck mean absorption coefficient of radiation species i. Following the model in Nakamura and Shindo (2019), the H\(_2\)O, NO, N\(_2\)O and NH\(_3\) are the four most important radiating species in the ammonia-hydrogen-air system, and their \(a_{p,i}\) values can be found in Nakamura and Shindo (2019). Figure 4 shows the dependence of \(a_{p}\) on the temperature range \(T \in [300\text { K},2500\text { K}]\) for all involved four radiating species.
Fig. 4
Planck mean absorption coefficients of radiation species in ammonia-hydrogen-system against temperature ranges
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4 Results: Effect of PaSR Model Parameters
In this section, the effect of PaSR model parameters, the mixing frequency \(\omega _\text {mix}\) and the residence time \(\tau _\text {res}\), on the thermo-kinetic quantities of the combustion system at the stochastic steady state will be discussed. Note that throughout the whole work, the mixing frequency \(\omega _\text {mix}\) is assumed to be equal for all thermo-kinetic states.
We consider the configuration that has only one stream describing the stoichiometric premixed mixture (70%NH\(_3\)–30%H\(_2\)-air) flows into the reactor with temperature \(T_\text {in}=300\) K. The pressure of the whole system is controlled unchanged with \(p=1\) bar.
The LMSE mixing model with the standard mixing model parameter \(C_\phi = 2.0\) will be used. Moreover, the marching time step \(\Delta t=1\times 10^{-6}\) s and 800 notional particles (\(N_p=800\)) are applied, and a further decrease of \(\Delta t\) and a further increase of \(N_p\) do not alter the ensemble-averaged thermo-kinetic quantities at stochastic steady state.
4.1 Convergence to Stochastic Steady State
Figure 5 shows the typical convergence of the ensemble-averaged temperature \(\langle T \rangle\), NOx emission \(\langle\)NOx\(\rangle\) and mole fractions of NH\(_3\)\(\langle x_\text {NH3} \rangle\) and HNO \(\langle x_\text {HNO} \rangle\) for different mixing frequencies \(\omega _\text {mix}\). The dependence of the ensemble-averaged values of different quantities at the stochastic steady-state on the mixing time will be discussed in the next section. The main observation here is that the proposed algorithm described above according to Chen (1997) converges to a stochastic steady state for every case as well as for the ammonia-hydrogen combustion system. Note that the x-axis describes the non-dimensional time \(\omega _\text {mix} \cdot t\). According to Chen (1997), for a given residence time, the time to reach the stochastic steady state could be correlated inversely to the mixing frequency by using the LMSE model. Therefore, we observe that for different mixing frequencies, the steady states reach about the same \(\omega _\text {mix} \cdot t\) point if they start from the same initial conditions.
Fig. 5
Convergence of the ensemble-averaged temperature \(\langle T \rangle\), NOx emission \(\langle\)NOx\(\rangle\) and mole fractions of NH\(_3\)\(\langle x_\text {NH3} \rangle\) and HNO \(\langle x_\text {HNO} \rangle\) for different mixing frequencies \(\omega _\text {mix}\). \(\tau _\text {res} = 1\) ms. Mixture: 70%NH\(_3\)+30%H\(_2\)+air. Adiabatic system
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4.2 Effect of Mixing Frequency
4.2.1 Tendency Against the Mixing Frequency
As previously shown in Cannon et al. (1998); Correa (1993) for methane-air combustion system, the PaSR results tend to their PSR results in the limit of infinite large mixing frequency (\(\omega _\text {mix} \rightarrow \infty\)) and one obtains extinction results for small mixing frequency. Furthermore, the fluctuation of thermo-kinetic states decreases with increasing turbulent frequency, and it approaches zero for \(\omega _\text {mix} \rightarrow \infty\). These observations have also been observed for the considered ammonia-hydrogen-air system, which is clearly shown in Fig. 6.
It shows the influence of mixing frequencies \(\omega _\text {mix}\) on the ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) for the case with residence time \(\tau _\text {res}=1\) ms. Note that although the discussion is based on one certain residence time, the conclusion is also valid for other residence times, which can be observed in Fig. 9 including different residence times till extinction limit. It should be mentioned at first that the curves end at the extinction mixing frequency \(\omega _\text {mix}^\text {E}\) (the very left point), below which combustion system is quenched and no burning solution is obtained (all particles have the temperature same as the temperature of incoming cold particles). Furthermore, if the mixing frequency is infinitely large (\(\omega _\text {mix} \rightarrow \infty\)), the results tend to the PSR limit.
Fig. 6
Influence of mixing frequencies \(\omega _\text {mix}\) on the ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) (left y-axis) and their r.m.s. of the fluctuation (right y-axis). \(\tau _\text {res} = 1\) ms. Mixture: 70%NH\(_3\)+30%H\(_2\)+air. Adiabatic system
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A clear observation is that the thermo-kinetic quantities decrease monotonically with decreasing mixing frequency, which is consistent with those shown in Cannon et al. (1998); Correa (1993) for methane-air combustion system. With the decrease in mixing frequency, the mixing intensity between the incoming cold particles and hot particles inside the reactor decreases, leading to a lower temperature of cold particles after mixing and a longer time for ignition. Therefore the averaged values of thermo-kinetic quantities decrease due to increasing incomplete mixing (smaller mixing frequency). If the mixing frequency is below the extinction mixing frequency \(\omega _\text {mix} < \omega _\text {mix}^\text {E}\), the temperature of incoming cold particles increased after mixing is too low, so that the ignition temperature cannot be reached and these particles cannot be successfully ignited to support for further combustion.
Moreover, it is observed that the r.m.s.(\(T^{\prime }\)) and r.m.s.(\(\text {NO}_\text {x}^{\prime }\)) increase with decreasing mixing frequency. This is because the disparity between low-temperature and high-temperature particles becomes larger when the mixing frequency decreases. In other words, if the mixing frequency is larger, all particles are relaxed to or close to their mean values \(\langle \phi \rangle\), so that all particles have the same or similar values after mixing, leading to smaller fluctuation and thus smaller corresponding r.m.s.. In the PSR limit case where \(\omega _\text {mix} \rightarrow \infty\), the corresponding r.m.s. values tend to vanish, since all the particles are perfectly mixed and have the same thermo-kinetic states.
4.2.2 Statistics of Thermo-Kinetic States
Since the mixing frequency has a significant impact on the ensemble-averaged values of thermo-kinetic states, it is worth checking the influence of the mixing frequency on the PDF of thermo-kinetic states from the notional particles. Figure 7 shows the PDFs of temperature and NOx at statistical steady state for different mixing frequencies for 1 ms residence time as an example. Since the LMSE is a deterministic mixing model in which all thermo-kinetic states of the particles are relaxed to their mean values, the predicted temperatures and NOx follow a single trajectory without scatter, which is also observed in Chang and Chen (1996).
Fig. 7
Scatter plots and PDFs of temperature and NOx at statistical steady state. Residence time: \(\tau _\text {res}=1\) ms. Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system. Three different turbulent frequencies from Fig. 6 are selected
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It is noticed from Fig. 7 that for a large mixing frequency (red and yellow bars here), only one peak in the PDF of temperature and NOx is observed, corresponding to the decreasing r.m.s. of the fluctuations. As the mixing frequency becomes smaller (blue bars here), we observe one PDF peak at the high temperature and a noticeable PDF at the low temperature. Furthermore, we observe two peaks in the PDF of NOx as well: one corresponds to the existence of cold particles with low values of NOx; the other to the existence of hot particles with high values of NOx. This information would be helpful for the statistical modeling of turbulent ammonia-hydrogen combustion systems based on the presumed PDF method, in which the shapes of the PDF must be presumed for the solution of transported-PDF method (Pope 1985; Borghi 1988; Bray et al. 1989). While the commonly used presumed \(\beta -\)PDF (Landenfeld et al. 2002; Mukhopadhyay et al. 2015; Salehi and Bushe 2010; Wang et al. 2016) can be reasonable for flows with larger mixing frequencies where only one PDF peak is observed, it can be inaccurate for the prediction of NOx for flows with smaller mixing frequencies because the \(\beta -\)function cannot combine a peak for burnt gases with an intermediate peak between the unburnt fresh gas and burnt gas (Poinsot and Veynante 2005) such as the PDF of NOx here.
4.2.3 Statistical Independence of Thermo-Kinetic Variables
Fig. 8
Joint PDF of \(x_\text {NH2}\) and \(x_\text {NNH}\) (left figure) and joint PDF of \(x_\text {N2}\) and \(x_\text {NNH}\) (right figure) for different mixing frequencies \(\omega _\text {mix}\). \(\tau _\text {res} = 1\) ms. Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system
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In the simulation based on the presumed PDF method, the multivariate joint PDF must be presumed if multi-dimensional progress variables are used (Landenfeld et al. 2002; Wang et al. 2016). For example, the joint PDF for composition is PDF(\(\vec {\phi }\)) = PDF(\(\phi _1,\phi _2,\cdots ,\phi _j\)), the assumption of statistical independence is usually employed (Poinsot and Veynante 2005; Shrotriya et al. 2020; Landenfeld et al. 2002), leading to the joint PDF being equal to the product of the single PDFs as PDF(\(\vec {\phi }\)) = PDF(\(\phi _1\)) \(\cdot\) PDF(\(\phi _2\)) \(\cdot\)\(\cdots\)\(\cdot\)PDF(\(\phi _j\)).
Although such statistical independence assumption is numerically simple to implement, it is shown that such assumption may give inaccurate prediction (Marchisio et al. 2001; Gerlinger 2003). Therefore, it is worth investigating the PDF based on two variables for the ammonia-hydrogen system, whether different thermo-kinetic variables are statistical independent.
In Fig. 8 the joint PDF of \(x_\text {NH2}\) and \(x_\text {NNH}\) (left figure) and the joint PDF of \(x_\text {N2}\) and \(x_\text {NNH}\) (right figure) are presented. These three species are selected because, according to Chen et al. (2021), a large amount of NH\(_2\), dissociated from NH\(_3\), will be consumed to NNH and further to major product N\(_2\). From the figure we observe again that for large mixing frequency (here yellow bars), the peak of joint PDF occurs in the hot particles regime (marked with pink circles), indicating that most of the NH\(_2\) is transferred to major product N\(_2\) (Chen et al. 2021). However, if the mixing frequency is low (here blue bars), we observe a small PDF peak in the cold particles regime (marked with green circles), indicating that in the statistical steady state a certain amount of produced NH\(_2\) cannot be further transferred to NNH and N\(_2\) due to low mixing intensity between cold particles and hot particles. This study tells us that for small mixing frequency, it may be inaccurate to simplify the joint PDF to a product of involved one-variable PDFs (here e.g. PDF(\(x_\text {NH2},x_\text {N2}\))\(\ne\)PDF(\(x_\text {NH2}\))\(\cdot\)PDF(\(x_\text {N2}\))), since these variables are strongly dependent on each other. In this case, the construction of presumed multi-dimensional PDF becomes much more difficult (Poinsot and Veynante 2005; Pope 1985; Lockwood and Naguib 1975).
4.3 Effect of Residence Time
In this part, the effect of residence time \(\tau _\text {res}\) on the PaSR prediction will be discussed. Figure 9 shows the dependence of ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) on the residence time for different mixing frequencies under the consideration of 70%NH\(_3\)–30%H\(_2\)-air adiabatic system. The PSR results are also presented as comparative model, corresponding to the PaSR with \(\omega _\text {mix} \rightarrow \infty\).
Fig. 9
Dependence of ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) on the residence time for different mixing frequencies. Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system
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4.3.1 Tendency Against the Residence Time
It is clearly observed that all the PaSR results (with different \(\omega _\text {mix}\)) are below the PSR results, and they end up with different extinction residence time \(\tau _\text {res}^E\) (the very left point), below which no stable burning solution can be obtained. And for the smaller \(\omega _\text {mix}\), \(\tau _\text {res}^E\) becomes longer. This is easy to understand because the disparity of the thermo-kinetic states of notional particles increases with decreasing \(\omega _\text {mix}\), therefore more particles are with lower temperatures compared to those under \(\omega _\text {mix}\) and consequently need longer time to be ignited. Hence a longer residence time is required for cold particles to be ignited with sufficient long time at low \(\omega _\text {mix}\).
Furthermore, we observe here that for \(\omega _\text {mix}=1\times 10^5\) (Hz), the PaSR results and PSR results for both temperature and NOx emission shown here are almost overlap, with the difference only at short \(\tau _\text {res}\) regime. This suggests that the PSR model can be used to describe the turbulence-chemistry-interaction for the combustion simulation, if the the thermo-kinetic states in the flow field tend to be mixed in a highly homogeneous manner, which is also discussed and investigated in Minamoto and Swaminathan (2015); Chen et al. (2017); De et al. (2011). On the other hand, with the decreasing residence time, the PaSR results further depart from the PSR results, indicating that the PSR is not suitable for the turbulence-chemistry interaction if the mixing frequency is small.
While the temperature decreases monotonically with decreasing residence time, the NOx concentration shows non-monotonic behaviors against residence time, and a peak at a certain residence time for all cases. According to the reaction pathway analysis in Chen et al. (2021); Singh et al. (2022), under the hydrogen enrichment, NO is the primary component of NOx which is also several orders of magnitude higher than the NO\(_2\). Furthermore, the NOx can be both produced from e.g. educt NH\(_3\) and consumed to produce other species such as N\(_2\)O and N\(_2\). Therefore, the level of NOx concentration results from the competition between the consumption time-scale and the production time-scale. For the PaSR model, at long residence time, the consumption time-scale for the NOx is sufficiently long, leading to a smaller amount in the stochastic steady state, because NOx can be further consumed. With decreasing residence time, the production time-scale for NOx is longer than its consumption time-scale, and NOx which is produced from NH\(_3\) cannot be further consumed. Therefore, the NOx concentration concentration increases and reaches a peak at a certain residence time. If the residence time further decreases, the production time-scale of NOx is comparable short so that NOx cannot be produced from educt. In this way, the concentration of NOx decreases with decreasing residence time again, until the extinction residence time reaches and the combustion system quenches.
4.3.2 Statistics of NOx
In this part, it is interesting to see how the statistics in terms of PDF will be changed with residence time. Therefore, in Fig. 10 we show the PDFs of NOx at statistical steady state for two different residence times \(\tau _\text {res}=0.65\) ms and \(\tau _\text {res}=5\) ms. The selection of both residence times is based on the reason that the corresponding predicted \(<\text {NO}_\text {X}>\) have almost the same values (c.f. Fig. 9), but they have totally different PDFs (c.f. Fig. 10).
Fig. 10
PDFs of NOx at statistical steady state for two different residence times. Mixing frequency: \(\omega _\text {mix}=1\times 10^4\) Hz. Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system
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For a long residence time (here \(\tau _\text {res}=5\) ms), the PDF with one peak is observed, showing that new incoming cold particles have enough time to be ignited after mixing and most of the particles have the same thermo-kinetic states. For this case, a commonly used \(\beta\)-function with suitable parameter choice is reasonable to describe the PDF of the NOx with one peak.
However, the PDF of the NOx for short residence time (here \(\tau _\text {res}=0.65\) ms) deviates from the one for long residence time significantly. For short residence time, the PDF of NOx has two peaks: one with a high peak is similar to the one for long residence time; the other one with a low peak does not exist noticeably for long residence time. This low peak comes from the new incoming cold particles, which are ignited after the mixing process and NOx can be produced from NH\(_3\) only in a small amount due to insufficient reaction time (short residence time). For this case, a combination of two normal distributions must be applied to capture both PDF peaks.
From this short discussion, we can clearly see that although the averaged NOx concentration can be similar for different residence times, statistics (PDFs) can be totally different. This is attributed to its complicated chemical nature, which makes the prediction of NOx in the turbulent case more challenging.
4.3.3 Sensitivity of Chemical Reactions
Since the statistics of the thermo-kinetic states (especially NOx prediction) at long or short residence times can be different, it is of great interest to know whether the residence time will alter important chemical reactions for the NO emission, and whether these important chemical reactions are the same for both PSR and PaSR models. In order to answer such questions, a sensitivity analysis has been performed. The relative sensitivity \(S^\text {rel}\) used in this work describes the normalized sensitivity of NO concentration \(f_\text {NO}\) with respect to the reaction rate of one elementary reaction:
Note that we use the brute-force method to evaluate the sensitivity, namely multiplying the \(k_r\) with factor 2. The main focus is on the NO species since it is the primary component of NOx. According to the definition of \(S^\text {rel}\), a positive sensitivity signifies an increase of \(f_\text {NO}\) with increased reaction rate of one elementary reaction, while a negative sensitivity signifies a decrease of \(f_\text {NO}\) with an increased reaction rate. Figure 11 shows the relative sensitivity of NO to reaction rates for PSR and PaSR with two different residence times. The \(\tau _\text {res}=0.8\) ms corresponds to the case near extinction, and the \(\tau _\text {res}=4.0\) ms corresponds to the case with almost perfect mixing. In this figure, only the elementary reactions with \(\text {abs}(S_\text {NO}^\text {rel}) > 0.01\) are presented. For both residence times, the PSR and PaSR models show the same most sensitive elementary reactions. For all considered models and conditions, the NH+OH=HNO+H shows the largest and most positive sensitivity values. According to reaction pathflow in Chai et al. (2021), the HNO is the main species producing NO species, and the faster this reaction is, the more NO will be produced. The NH+NO=N\(_2\)O+H is the second most sensitive reaction, and this reaction is mainly responsible for the consumption of NO, which explains the negative sensitive values.
Fig. 11
Relative sensitivity of NO to reaction rates for PSR and PaSR with two different residence time. \(\omega _\text {mix} = 1\times 10^4\) (Hz). Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system
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4.4 Effect of Different Chemical Mechanisms
Since the chemical mechanisms give different results when using the PSR models (c.f. Figure 3), it is necessary to check whether these differences are also observed when using the PaSR model. In order to address this concern, we show in Fig. 12 the influence of mixing frequencies \(\omega _\text {mix}\) on the ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) predicted by four different chemical mechanisms. In this case, the residence time is kept as \(\tau _\text {res}=1\) ms. We notice immediately, as consistent with the observation from Fig. 3, the averaged temperatures predicted by these four different mechanisms are very similar while the averaged NOx can be significantly different. Gotama-2022 (purple lines) shows the highest prediction of NOx while the Stagni-2020 (red lines) and Shrestha-2021 (yellow lines) show similar values, and the NOx predicted by Gotama-2022 can be almost two times larger than predicted by Stagni-2020 and Shrestha-2021.
Fig. 12
Influence of mixing frequencies \(\omega _\text {mix}\) on the ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) predicted by four different chemical mechanisms. \(\tau _\text {res} = 1\) ms. Mixture: 70%NH\(_3\)+30%H\(_2\)+air. Adiabatic system. Blue: Nakamura-2017 (Nakamura et al. 2017); Red: Stagni-2020 (Stagni et al. 2020); Yellow: Shrestha-2021 (Shrestha et al. 2021); Purple: Gotama-2022 (Gotama et al. 2022)
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Although the predicted averaged-NOx can be significantly different by using different chemical mechanisms, Fig. 13 shows that the shapes of the PDF(NOx) are in a similar manner under the same conditions (here \(\omega _\text {mix}=5\times 10^3\) Hz and \(\tau _\text {res} = 1\) ms). All the PDF(NOx) show two peaks with one low peak in the low NOx concentration regime and the other high peak in the high NOx concentration regime.
Fig. 13
PDFs of NOx at statistical steady state for four different chemical mechanisms. Mixing frequency: \(\omega _\text {mix}=5\times 10^3\) Hz. Residence time \(\tau _\text {res} = 1\) ms. Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system. Blue: Nakamura-2017 (Nakamura et al. 2017); Red: Stagni-2020 (Stagni et al. 2020); Yellow: Shrestha-2021 (Shrestha et al. 2021); Purple: Gotama-2022 (Gotama et al. 2022)
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In Fig. 14 we show further the sensitivity of NOx with respect to the chemical reaction rate for these four different mechanisms. This figure ensures that 10 most sensitive elementary reactions for each chemical mechanisms are included. Overall these chemical mechanisms show some same reactions such as NH+HO=HNO+H and NH+NO=N\(_2\)O+H, which are sensitive to the NO concentration. However, Nakamura-2017 mechanism shows that the NH+OH=N+H\(_2\)O is not important while the reaction rates of NH\(_2\)+NO=N\(_2\)+H\(_2\)O and NH\(_2\)+H=NH+H\(_2\) have large impact on the NO concentration prediction, the other three mechanisms show a weaker impact of these two reactions. Such difference needs to be investigated in more details, if more experimental measurement of NOx concentration is available for the validation of chemical mechanisms.
Fig. 14
Relative sensitivity of NO to reaction rates for PaSR with different chemical mechanisms. \(\omega _\text {mix} = 1\times 10^4\) (Hz), \(\tau _\text {res} = 1\) ms. Mixture: 70%NH\(_3\)–30%H\(_2\)-air. Adiabatic system. Blue: Nakamura-2017 (Nakamura et al. 2017); Red: Stagni-2020 (Stagni et al. 2020); Yellow: Shrestha-2021 (Shrestha et al. 2021); Purple: Gotama-2022 (Gotama et al. 2022)
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5 Results: Effect of Hydrogen or Oxygen Enrichment on the PaSR Modeling
So far we focus on the ammonia-air combustion system enriched by 30% hydrogen (70%NH\(_3\)–30%H\(_2\)-air). As shown in Shrestha et al. (2021), it is found that the 70%NH\(_3\)-30%H\(_2\)-air flame and NH\(_3\)-30%O\(_2\)/70%N\(_2\) flame have similar laminar flame speed. Therefore, it is interesting to see whether both combustion systems have also similar behaviors in the PaSR models, especially the NOx concentration. This section begins with a short review on the laminar premixed flame for both systems. Then the PaSR prediction of dependence on the mixing frequency will be discussed. The main results of this part are related to the third subsection, where the effect of thermal radiation on the combustion system for both mixtures will be discussed.
5.1 A-priori Brief Comparison for Laminar Premixed Flame
Figure 15 shows the comparison of laminar burning velocity (LBV) for the stoichiometric ammonia-air with hydrogen enrichment (red) and with oxygen enrichment (blue). Note that the validation of applied Shrestha-2021 mechanism for ammonia-hydrogen-air and ammonia-oxygen-air in terms of ignition delay times and laminar burning velocity against the experimental measurements can be found in Shrestha et al. (2021). We notice from this figure that both H\(_2\) and O\(_2\) have positive effect on the LBV, which is consistent with those reported in many works such as Mei et al. (2019); Li et al. (2015); Sun et al. (2023); Ichikawa et al. (2015). Furthermore, it is observed that indeed the stoichiometric 70%NH\(_3\)–30%H\(_2\)-air combustion system and stoichiometric NH\(_3\)–30%O\(_2\)-70%N\(_2\) combustion system have very similar LBV.
Fig. 15
Comparison of laminar burning velocity (LBV) for the stoichiometric ammonia-air with hydrogen enrichment (red) and with oxygen enrichment (blue). \(T_\text {ub}=300\) K and \(p=1\) bar. Adiabatic system
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Figure 16 compares further the adiabatic flame temperature \(T_\text {ad}\) (left) and NOx emission (right) for the stoichiometric ammonia-air with hydrogen enrichment (red) and with oxygen enrichment (blue). It is consistent with those shown in Mei et al. (2019); Li et al. (2015); Sun et al. (2023) that the adiabatic flame temperature \(T_\text {ad}\) increases monotonically with increasing H\(_2\) and O\(_2\) contents in the ammonia-air mixtures, and the O\(_2\) has more stronger effect on \(T_\text {ad}\) than the H\(_2\). Desipte the higher \(T_\text {ad}\) due to O\(_2\) enrichment, the corresponding NOx emission of ammonia-air mixture under O\(_2\) enrichment is much higher than it under H\(_2\) enrichment.
Fig. 16
Comparison of adiabatic flame temperature \(T_\text {ad}\) (left) and NOx emission (right) for the stoichiometric ammonia-air with hydrogen enrichment (red) and with oxygen enrichment (blue). \(T_\text {ub}=300\) K and \(p=1\) bar
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5.2 Comparison for PaSR Case
Figure 17 compares the ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) (left y-axis) and their r.m.s. of the fluctuation (right y-axis), if the 70%NH\(_3\)–30%H\(_2\)-air (red) and NH\(_3\)–30%O\(_2\)-70%N\(_2\) (blue) mixtures are applied. The major two observations can be addressed here:
The decrease of mixing frequencies leads to decrease of both \(\langle T \rangle\) and \(\langle\)NOx\(\rangle\), as explained above. And the extinction mixing frequency \(\omega _\text {mix}^\text {E}\) for both gas mixtures are close to each other. However, as consistent with the observation from Fig. 16, the NH\(_3\)–30%O\(_2\)-70%N\(_2\) gas mixture gives higher values of both \(\langle T \rangle\) and \(\langle\)NOx\(\rangle\). This means that for the consideration of turbulence mixing process, the NH\(_3\)–30%O\(_2\)-70%N\(_2\) system has higher combustion temperature, but also higher levels of NOx emission.
The decrease of mixing frequency leads to increase of both r.m.s. of the \(\langle T \rangle\) and r.m.s. of the \(\langle\)NOx\(\rangle\), as explained above. And their values are similar against the variation of mixing frequency. This indicates that the mixing frequency has the similar effect on the decay of fluctuations for both combustion systems.
Fig. 17
Influence of mixing frequencies \(\omega _\text {mix}\) on the ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) (left y-axis) and their r.m.s. of the fluctuation (right y-axis) for 70%NH\(_3\)–30%H\(_2\)-air (red) and NH\(_3\)–30%O\(_2\)-70%N\(_2\) (blue) mixtures. \(\tau _\text {res} = 1\) ms. Adiabatic system
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5.3 Effect of Thermal Radiation on Both Combustion Systems
In this part, we will focus on the effect of thermal radiation on the studied combustion system. In order to clarify the effect of the thermal radiative heat loss from the combustion system, we introduce the so-called "radiant fraction" \(f_\text {rad}\), describing the radiative heat loss normalized by the total heat release rate as:
where \({\dot{\omega }}_i\) and \(h_i\) are the net production rate and enthalpy for species i. Furthermore, we define the Importance of Radiating Species IRS as:
For the ammonia-hydrogen-air system, the radiating species are H\(_2\)O, NO, N\(_2\)O and NH\(_3\) according to Nakamura et al. (2017). And it is shown in Nakamura and Shindo (2019) that the predicted NOx emission by considering the radiative heat loss results in the decrease of NOx emission at high temperature regime and almost no change at low temperature regime (more explanation in Nakamura and Shindo (2019)). For the PaSR, the averaged temperature of the system decreases monotonically with decreasing mixing frequency (see Sect. 4.2). Therefore, it is expected that at low mixing frequency where the averaged temperature is low, the effect of radiation would be small, which is confirmed in Fig. 18(left).
Fig. 18
Comparison of ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) for adiabatic condition (solid lines) and radiative condition (dashed lines) for \(\omega _\text {mix} = 3\times 10^3\) (Hz) and residence time \(\tau _\text {res}=1\) ms. \(T_\text {ub}=300\) K and \(p=1\) bar
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There the time development of averaged temperature and NOx emission under adiabatic and radiative conditions is compared for 70%NH\(_3\)–30%H\(_2\)-air system under \(\omega _\text {mix} = 3\times 10^3\) (Hz) and residence time \(\tau _\text {res}=1\) ms over a 6 ms (\(6 \tau _\text {res}\)) time interval. Note that for both conditions, the initial conditions are exactly the same. It is observed that the profiles for both adiabatic and radiative conditions develop in almost the same manner, and the effect of thermal radiation at small mixing frequency is not observed. To understand this, we plot in Fig. 18(right) the radiant fraction \(f_\text {rad}\)(right) the radiant fraction \(f_\text {rad}\) and the importance of radiating species (IRS) for all four involved species. We observe that the radiative heat loss is only around 0.02% of the total HRR from 2 ms, which is a very small amount, and this results in a negligibly small influence of thermal radiation on the flame quantities for \(\omega _\text {mix} = 3\times 10^3\) (Hz). A further look into the IRS indicates that the H\(_2\)O, as a major product, contributes over 95% to the radiative heat loss, then is the NH\(_3\) with around 3% contribution.
In Fig. 19 comparison of ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) for adiabatic condition (solid lines) and radiative condition (dashed lines) for a larger mixing frequency \(\omega _\text {mix} = 1\times 10^5\) (Hz) with residence time \(\tau _\text {res}=1\) ms is presented. Again time domain of 6 ms (\(6 \tau _\text {res}\)) is shown. We observe that for both mixture compositions, the decrease of temperature caused by thermal radiation is only around 5 K or 6 K, and the decrease of NOx concentration is also small, namely for 70%NH\(_3\)–30%H\(_2\)-air only around 1.5% decrease and for NH\(_3\)–30%O\(_2\)-70%N\(_2\) only around 2.5% decrease. The radiant fractions for both systems, which are presented in Fig. 20, show that the radiative heat loss is only around 0.2% of the total HRR. These results show that at least for the considered system here, the thermal radiation has only minor effect on the combustion system.
Fig. 19
Comparison of ensemble-averaged temperature \(\langle T \rangle\) and NOx emission \(\langle\)NOx\(\rangle\) for adiabatic condition (solid lines) and radiative condition (dashed lines) for \(\omega _\text {mix} = 1\times 10^5\) (Hz) and residence time \(\tau _\text {res}=1\) ms. \(T_\text {ub}=300\) K and \(p=1\) bar
Fig. 20
Comparison of radiant fraction \(f_\text {rad}\) for the stoichiometric ammonia-air with hydrogen enrichment (red) and with oxygen enrichment (blue). \(T_\text {ub}=300\) K and \(p=1\) bar
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6 Conclusions
In the present work, the Partially Stirred Reactor (PaSR) model is used to investigate the turbulence-chemistry interaction for the ammonia-air combustion system. The PaSR is based on a stochastic modeling by solving the transported PDF equation, and the effect of turbulence mixing is considered by means of the LMSE mixing model. The transprted-PDF method allows us to include the effect of turbulent frequency, and to investigate the effect of chemical mechanisms since the chemical source term is in a closed form. A comprehensive investigation of different physical and chemical parameters is carried out, and various main observations can be concluded briefly here:
The effect of the mixing frequency on the ammonia-hydrogen-air system is consistent with those on the hydrocarbon system, namely a decrease in mixing frequency leads to a decrease of thermo-kinetic quantities (e.g. temperature and NOx concentration) and an increase of their fluctuation. The statistics analysis shows that the PDF of the NOx at low turbulent frequency has two peaks while at high turbulent frequency only one. Further analysis shows that the PDFs of different species are closely correlated with each other, and the assumption about statistical independence which is usually used in the presumed PDF method is not valid.
The effect of residence time on the PaSR model is similar to the one on the PSR model. While the averaged temperature decreases with shorter residence time monotonically, the NOx concentration shows a non-monotonic dependence on the residence time due to the reaction pathway NH\(_3 \rightarrow\)NOx\(\rightarrow\)N\(_2\). Further statistics analysis showed that the shapes of the NOx PDFs can be totally different for different residence times. While the PDF(NOx) at long residence time can be approximated by a \(\beta -\)function, the PDF(NOx) at short residence time must be approximated by e.g. combining two normal distributions.
Sensitivity analysis of NOx with respect to the chemical reaction rate displays that the elementary reactions which are sensitive according to the PSR model are also the most sensitive ones according to the PaSR model, independent of residence times.
Four different chemical mechanisms have been applied and compared for PaSR case, indicating that the predicted NOx can be significantly different. However, the statistics analysis shows that the shapes of PDF(NOx) are in a similar manner by using these four different mechanisms. Further sensitivity analysis of NO with respect to the chemical reaction rates shows that for all these chemical mechanisms the most sensitive reactions such as NH+OH=HNO+H are the same.
The effect of oxygen enrichment and hydrogen enrichment on the PaSR model prediction is discussed. It is noted that although 30%-O\(_2\) enrichment has a similar effect as 30%-H\(_2\) enrichment in terms of laminar flame speed, the 30%-O\(_2\) enrichment leads to higher averaged temperature and NOx concentration.
The numerical simulation showed that the thermal radiation gas had little effect on the system, regardless of the mixing frequency.
At the end, it is worth pointing out that all conclusions are obtained and discussed based on the assumption that the mixing frequencies are equal for all thermo-kinetic states. However, various chemical species may have different mixing frequencies due to e.g. the small-scale-turbulence/scalar interactions, which may be of great interest for future research.
Acknowledgements
The idea of this paper comes during the preparation of the joint project "Research of combustion reaction kinetics of a multi-component e-fuel for efficient utilization".
Declarations
Conflict of interest
The authors declare that they have no conflict of interest.
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