Effect of initial conditions on the inhibition process of H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonations using CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O

Abstract

The unwarranted leakage/release of hydrogen gas from metal processing, automotive, petrochemical industries, and nuclear reactors, along with its subsequent ignition and transition to detonation, could lead to catastrophic damage to both life and property. The development of practical hazard prevention and safety control systems demands an understanding of the effectiveness of the chemical inhibitors to suppress/mitigate a detonation wave under varying operational conditions. In the current study, the inhibition efficiency of chemical inhibitors under varying mixture initial conditions was investigated using numerical computations. The inhibition efficiency of trifluoroiodomethane (CF$_{\textrm{3}}$I), carbon dioxide (CO$_{\textrm{2}})$, and steam (H$_{\textrm{2}}$O) on hydrogen-oxygen/air mixtures was evaluated using a detailed chemical kinetic model for hydrogen oxidation. ZND computations were carried out over a range of initial mixture composition, pressure, and temperature. It was found that CF$_{\textrm{3}}$I is a better inhibitor than CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O at all the initial mixture conditions. However, at very high temperatures, the inhibitors CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O have a similar detonation inhibition efficiency. The inhibition efficiency of carbon dioxide and steam is comparable and significantly lower than CF$_{\textrm{3}}$I. The findings from the current work can be used to design optimized detonation safety systems over a range of practical operating conditions.

1 Introduction

Accidental explosions occur during the storage and transportation of fuels, primarily due to gas leaks and fuel spillage. In the case of highly reactive fuels such as hydrogen, the chemical reactions and species molecular diffusion rates are greater than for typical hydrocarbon fuels. Therefore, a flame accelerates rapidly, leading to fast deflagrations or even a detonation through the deflagration-to-detonation transition (DDT) phenomenon. It is well known that gaseous detonations resulting from the formation of an explosive mixture due to the uncontrolled release of flammable gas and an oxidizer are the most destructive accident scenarios. A fully developed detonation wave propagates at supersonic velocities ($>1800$ m/s), reaching overpressures of more than 20 bars. The mechanical load associated with a shock wave of such high overpressures is a major destructive factor. Therefore, Wang et al. suggested that the flame acceleration and the DDT phenomenon must be taken into consideration for safety and hazard evaluation [1].

Hydrogen is a very reactive clean fuel and is believed to be a future energy carrier. Hydrogen is seen as a universal remedy to address problems related to global warming, increased air pollution, and fossil fuel depletion. However, the primary concern for adopting hydrogen as a suitable alternative to the currently used energetic fuels is the safety aspects associated with possible gas leaks and hydrogen hazards. Due to the low ignition energy and wide explosion limit of hydrogen, several hydrogen explosion accidents have been reported in the production, storage, and refueling stations [2]. Hydrogen being a small molecule, and due to its low viscosity, it is more prone to leakages from pipeline connections as compared to larger hydrocarbons. It was found that hydrogen leaks three times and five times faster than natural gas and propane, respectively, on a volumetric basis [3]. The leakage of hydrogen from storage tanks or transport pipelines results in the formation of a large vapor cloud; the hydrogen vapor mixes with air and forms a flammable mixture before ignition occurs. Vapor cloud explosions have been widely studied in the literature due to their resemblance to practical explosion scenarios [4,5,6,7,8,9]. Oran et al. showed that such vapor clouds under favorable conditions could undergo DDT even in the absence of any confinement and could lead to loss of life and resources [7, 8]. Jiang et al. carried out experimental studies on unconfined hydrogen explosions in the presence of external turbulence. It was found that the cumulative effect of external turbulence and the inherent flame instabilities could result in flame acceleration [9]. Also, the blast wave overpressures for a partially confined detonation were found to be significantly greater than an unconfined detonation for the same amount of fuel [10]. In the case of nuclear reactors, hydrogen gas can be formed by the reaction of zirconium and steam at elevated temperatures during a severe accident. Also, hydrogen can be formed in light water-cooled reactors due to the radiolytic decomposition of water [11]. The damage reported at the Fukushima nuclear plant (2011) resulted from an intense hydrogen explosion [11]. Thus, there is a great interest in hydrogen flammability and detonability limits and their implications for safety and prevention in many applications.

The use of chemical inhibitors and diluents for detonation and fire suppression has been widely studied by researchers worldwide [12,13,14,15,16,17,18,19,20,21,22]. The chemical inhibitors, when added to a fuel-oxidizer mixture, interfere with the oxidation chemistry. The primary objective of the addition of a chemical inhibitor to a fuel-oxidizer mixture is to alter the mixture’s detonability and thus reduce detonation hazard. Halogenated hydrocarbons, especially those containing more than one halogen compound, have been found to be very effective in detonation inhibition [12]. Brominated additives such as bromotrifluoromethane or Halon 1301 (CF$_{\textrm{3}}$Br) and bromochlorodifluoromethane or Halon 1211 (CF$_{\textrm{2}}$BrCl) have been found to be excellent additives for fire suppression in a given fuel-air mixture [13]. However, their production and use in fire suppression have been banned by the Montreal Protocol because of the high ozone depletion potential of bromine [14, 15]. The bromine atoms from the brominated compounds reach the stratosphere and are a major ozone depletor. In the recent past, efforts were devoted to find a suitable replacement for Halon 1301 with similar suppression ability and also environment-friendly production and application. It was found that iodinated hydrocarbon could be a suitable replacement. Leclerc et al. [14] conducted an experimental study on methane oxidation in the presence of trifluoroiodomethane (CF$_{\textrm{3}}$I) to evaluate the inhibiting effect. It was found that CF$_{\textrm{3}}$I is as good an inhibitor as brominated compounds [14]. The addition of chemical inhibitors reduces the rate of bimolecular reactions. The addition of CF$_{\textrm{3}}$I to hydrocarbon fuels such as methane, propane, and acetylene was observed to reduce the laminar flame speed by consuming the active radicals and producing fewer reactive radicals [15]. Moen et al. evaluated the detonation inhibition efficiency of various inhibitors [16]. The experimental results showed that CO$_{\textrm{2}}$ as a diluent has a better detonation inhibition tendency than CF$_{\textrm{3}}$Br for fuel-air and fuel-oxygen mixtures. It was found that dilution with nitrogen and carbon dioxide leads to a change in the thermodynamic properties of a fuel-oxidizer mixture. The addition of these diluents reduces the thermal diffusivity and thermal conductivity of the fuel-oxidizer mixture, thereby reducing the flame speed of the mixture [17]. Kumar and Singh performed numerical computations to study the efficacy of CF$_{\textrm{3}}$I, H$_{\textrm{2}}$O, CO$_{\textrm{2}}$, and N$_{\textrm{2}}$ for suppressing H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonations. The results suggested that H$_{\textrm{2}}$O is a better inhibiting agent than CF$_{\textrm{3}}$I based on the retardant weight required to suppress a detonation wave [14]. It was also found that CF$_{\textrm{3}}$I has a detonation promotion effect at lower concentrations (< 6000 PPM), and the expected inhibition effect is observed at higher additive concentrations for H$_{\textrm{2}}$–O$_{\textrm{2}}$ mixtures. The promotion effect of CF$_{\textrm{3}}$I at lower concentrations is due to an increase in the production of H radical through I $+$ H$_{\textrm{2}} \rightarrow$ HI $+$ H, whereas at higher concentrations, the reaction, H $+$ CF$_{\textrm{3}}$I $\rightarrow$ HI $+$ CF$_{\textrm{3}}$ dominates and consumes the reactive H radical leading to detonation inhibition [18]. Thus, the inhibition effect of CF$_{\textrm{3}}$I, a potential Halon 1301 (CF$_{\textrm{3}}$Br) replacement, is well documented in the literature. However, the studies reported in the literature are specifically carried out at stoichiometric conditions and standard initial pressure and temperature. The hydrogen explosions occurring, especially in confined spaces and nuclear reactors, can have a range of initial mixture composition and pressure-temperature variations. The boiling water reactors and pressurized water reactors operate at very high initial pressures; also, partially combusted fuel-oxidizer mixtures can heat the unburned mixture and change the initial thermodynamic state of the flammable mixture [11]. Therefore, it is important to study these chemical and thermal inhibitors over a range of mixture initial conditions. However, the literature regarding this is minimal [23]. As such, no literature is available to the author’s knowledge in which a systematic study of the effect of a mixture’s initial thermodynamic state on the inhibition efficiency of halogenated inhibitors has been carried out. Therefore, in the present work, we attempted to evaluate the effect of varying initial conditions on the inhibition efficacy of detonation inhibitors.

Gaseous detonations are characterized by a cellular structure resulting from strong interactions between the incident shock, the transverse waves, and Mach stems [24]. The detonation cell size is quantified based on the average width of the detonation cell, which is the spacing between transverse waves of the same family. The detonation cell width ($\lambda )$ depends on the type of fuel and oxidizer, the fuel-oxidizer stoichiometry, the initial thermodynamic state of the reactive mixture, dilution, etc. The detonation cell width is the most prominent detonation parameter and has been extensively studied due to its relation to different detonation propagation parameters [25,26,27,28]. Researchers in the past have used simplified one-dimensional analysis (i.e., ZND detonation model) to predict the dynamic detonation parameters. It was found that the ZND reaction length ($\Delta )$ correlates linearly with the cell width of multi-dimensional detonations [29,30,31,32], i.e., $\lambda = A\Delta$. The value of A changes by more than an order of magnitude for different fuel-oxidizer-diluent mixtures [29, 30]. Further efforts were devoted at refining the correlation by developing elaborative theoretical approaches for the ratio of the cell width to the induction length. Gavrikov et al. proposed an approach to generalize the correlation ($\uplambda =A\,\times \,\Delta _{\textrm{i}})$ by considering the multi-dimensional structure of detonations. The proportionality constant A was considered to be a function of two stability parameters [dimensionless effective activation energy ($\varepsilon _{\textrm{I}})$ and a parameter describing the relation between chemical energy and the initial thermal energy of the combustible mixture ($T_{\textrm{vN}} / T_{0})]$ [29]. Ng et al. also developed a correlation where A was defined as a function of the non-dimensional stability parameter ($\chi \, =\, \varepsilon _{\textrm{I}}\times \Delta _{\textrm{i}}\,\frac{\dot{\sigma }_{\textrm{max}}}{u_{\textrm{CJ}}})$ with empirically fitted coefficients [33]. Mevel et al. proposed a correlation with $\, A\, =30\varPhi ^{-0.233}\left( 1-\, X_{\textrm{N}_{\textrm{2}}} \right) ^{0.098}(\nicefrac {P_{1}}{P_{0}})^{0.699}$ as a function of mixture equivalence ratio, nitrogen mole fraction ($X_{\textrm{N}_{\textrm{2}}})$, and the ratio of initial to standard pressure ($\nicefrac {P_{1}}{P_{0}})$; it predicted the cell width with a mean relative error of 28% for H$_{\textrm{2}}$–N$_{\textrm{2}}$O (-N$_{\textrm{2}}$) mixtures [34]. Zhang et al. proposed a correlation with A having a similar functional form as given by Mevel et al. and including the effect of argon dilution for H$_{\textrm{2}}$–N$_{\textrm{2}}$O-Ar mixtures, i.e., $A\, =25.68\varPhi ^{-0.112}\left( 1-\, X_{\textrm{Ar}} \right) ^{-1.23}{(\nicefrac {P_{1}}{P_{0}})}^{0.016}\,$ [35]. The primary difference between the two expressions of A is the exponent of the diluent term ($1-\, X_{\textrm{dil}})$. The induction length does not change much with argon dilution due to the two competing effects: an increase in the specific heat ratio ($\gamma )$ and a decrease in the total energy release [35]. However, experimental observation reveals increased cell width with increasing argon dilution. Therefore, to keep A increasing progressively, the exponent of the diluent term (argon molar fraction) was thus set to be negative [35].

Although the one-dimensional detonation model does not incorporate the entire structure of the multi-dimensional detonation, it has been historically used to predict the detonation cell width using the induction length based on detailed chemical kinetics. The use of such a simplified model can significantly reduce the computational costs and time. It allows the study of fuel oxidation chemistry over a range of initial conditions and fuel-oxidizer combinations using detailed kinetic reaction mechanisms. The addition of chemically active inhibitors increases the induction length of a ZND detonation, which indicates a loose coupling between the reaction zone and the leading shock wave and thus qualitatively represents less detonable mixtures. Thus, chemical inhibitors are critical in reducing the detonability hazards of practical fuels, and their effectiveness under varying initial conditions warrants further study.

The efficiency of detonation inhibitors can be evaluated based on various detonation parameters such as the detonation cell width, ZND induction zone length, CJ detonation velocity, and the weight of the inhibitor required [18]. Along with the detonation parameters, the economic feasibility of a particular inhibitor can also be used as a limiting parameter for ranking the effectiveness of a given inhibitor over a range of mixture conditions. However, the use of the cell width or the ZND induction zone length based on detailed chemical kinetics has been proven to be the most appropriate parameter to evaluate the efficiency of various inhibitors [12]. Therefore, in the present study, the factor by which a given inhibitor increases the induction zone length of H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonations is used as an indicative parameter of the inhibition efficiency.

In the present study, the effect of the initial thermodynamic state of the fuel-oxidizer mixture on the inhibition efficiency of CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O was evaluated. The primary objective of the present work is to study the influence of initial mixture conditions on the efficiency of detonation inhibitors using a detailed chemical kinetics model for hydrogen oxidation. The computations include varying the initial mixture composition, initial pressure, and temperature of H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I/CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures.

2 Methodology

In the present work, the one-dimensional ZND detonation model was used to study the efficacy of detonation inhibitors on H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures at different initial conditions. The modified version of the Caltech Shock and Detonation Toolbox was used for the ZND computations [36]. The chemical kinetic computations were carried out using CANTERA [37] integrated with MATLAB. The Foundational Fuel Chemistry Model – 1(FFCM-1) was used to model hydrogen oxidation. The FFCM-1 model consists of 39 species and 301 reactions [38]. The mechanism adopted by Leclerc et al. for CF$_{\textrm{3}}$I was incorporated into the FFCM-1 reaction mechanism to model CF$_{\textrm{3}}$I chemistry [14]. In our previous work [18, 39], it was highlighted that other models describing CF$_{\textrm{3}}$I chemistry (Mathieu et al. [15] and Babushok et al. [21]) were not able to capture the ignition promotion effect shown by CF$_{\textrm{3}}$I at lower concentrations. However, the CF$_{\textrm{3}}$I sub-mechanism from Leclerc et al. [14] was able to predict the ignition promotion effect of CF$_{\textrm{3}}$I at low concentrations and produced results that were in excellent agreement with previous experimental results for hydrogen-oxygen detonations [32]. The comparison between the predicted cell width and the experimentally measured cell width is shown in Fig. 1.

Fig. 1

Effect of varying CF$_{\textrm{3}}$I concentration on the detonation cell width. The filled symbols represent the experimental cell width data from Crane et al. [32], and the hollow symbols represent the predicted cell width based on the induction length correlation ($\uplambda \, =27.6\,\times \,\Delta _{\textrm{i}})$ given in [32]. The induction length was evaluated under similar conditions using the combined FFCM1-CF$_{\textrm{3}}$I model

The reaction zone in a standard ZND detonation model includes the induction zone and the recombination zone. The induction zone is dominated by two-body chain branching and radical generation reactions, whereas the recombination zone is dominated by three-body chain-terminating reactions. The induction zone length is a characteristic detonation length scale of the standard one-dimensional ZND model. It is now known that a smaller induction length is indicative of more reactive mixtures. A larger induction zone length denotes a slower reaction, and if the induction length is large enough, the reaction zone can be completely decoupled from the leading shock front. However, the shock and reaction front can propagate independently if a sufficient flammable mixture is available [18]. It must also be noted that the leading shock and the reaction zone can again become coupled and propagate as a detonation wave under favorable conditions. Thus, the increase in the induction length indicates a corresponding decrease in the detonability of a given fuel-oxidizer mixture. Therefore, in order to rank the inhibitor efficiency, the induction length is utilized as a governing parameter.

The inhibition efficiency of an inhibitor is given by the factor of increase (f) in the induction length/time with the addition of inhibitor at the same concentration levels for all the inhibitors. A higher factor of increase in the induction length or time indicates higher efficiency of a detonation inhibitor and vice versa. The factor of increase in the induction length ($f_{\Delta })$ is the ratio of induction length with inhibitor to without inhibitor for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air-inhibitor detonations. It is given by

$$\begin{aligned} f_{\Delta }= \frac{\Delta _{\textrm{i,}\, \textrm{inhibitor}}\,}{\Delta _{\textrm{i,}\, \textrm{no}\, \textrm{inhibitor}}} \end{aligned}$$

Similarly, the factor of increase in the induction time ($f_{\uptau })$ is given by

$$\begin{aligned} f_{\uptau } = \frac{\uptau _{\textrm{i,}\, \textrm{inhibitor}}}{\uptau _{\textrm{i,}\, \textrm{no}\, \textrm{inhibitor}}} \end{aligned}$$

Kumar and Singh, in their work, pointed out the dual nature of CF$_{\textrm{3}}$I addition to H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures. They observed detonation promotion at a small molar concentration of CF$_{\textrm{3}}$I addition (<1200 PPM for H$_{\textrm{2}}$–air mixtures and <6000 PPM for H$_{\textrm{2}}$–O$_{\textrm{2}}$ mixtures) [18]. Therefore, in order to overcome the ambiguity of any promotion effect, the concentration of the inhibitor addition was fixed as 50,000 PPM or 5% molar concentration. The ZND computations were carried out at these concentration levels, and no promotion effect was observed. Therefore, the inhibitor concentration of 50,000 PPM or 5% molar concentration was used for further computations.

Table 1 Critical detonation parameters ($T_{\textrm{vN}}$, $\Delta _{\textrm{i}}$, and $\uptau _{\textrm{i}}$) and the factor of increase ($f_{\Delta }$, $f_{\uptau })$ at various mixture equivalence ratios ($\varPhi )$ with and without inhibitors for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures

Fig. 2

Effect of varying initial mixture composition on the induction length and time with and without retardants (5% CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O molar concentration): (a) H$_{\textrm{2}}$–O$_{\textrm{2}}$ detonations and (b) H$_{\textrm{2}}$–air detonations at $P_{0} =$ 1 bar and $T_{0} =$ 295 K

Fig. 3

Effect of varying initial mixture composition on the factor of increase in the induction length ($f_{\Delta })$ and induction time ($f_{\uptau })$ at 5% CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O molar concentration: (a) H$_{\textrm{2}}$–O$_{\textrm{2}}$ detonations and (b) H$_{\textrm{2}}$–air detonations at $P_{0} =$ 1 bar and $T_{0} =$ 295 K

3 Results and discussion

3.1 Effect of varying equivalence ratio on the inhibition efficiency

3.1.1 H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures

The induction length/time and the corresponding factor of increase were evaluated for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures with 5% CF$_{\textrm{3}}$I addition at an initial pressure of $P_{0} =$ 1 bar, and initial temperature of $T_{0} =$ 295 K (refer to Table 1 and Figs. 2 and 3). It is known that CF$_{\textrm{3}}$I shows a chemical inhibition effect at higher concentrations, where it drastically increases the induction length/time [18]. The chemical inhibition effect of CF$_{\textrm{3}}$I is primarily due to the radical scavenging of active radicals (specifically H radical) in the combustion environment. The recombination of active radicals with the halogenated atom is the major inhibition step for all halogenated inhibitors. The reaction cycle responsible for the inhibition effect of most halogenated compounds is as follows [39]:

$$\begin{aligned}{} & {} \hbox {H }+\hbox { I }+\hbox {M }\rightarrow \hbox { HI }+\hbox { M } \end{aligned}$$

(R1)

$$\begin{aligned}{} & {} \hbox {I}_{\textrm{2}} +\hbox { O }\rightarrow \hbox { I }+\hbox { IO} \end{aligned}$$

(R2)

$$\begin{aligned}{} & {} \hbox {H }+\hbox { CF}_{\textrm{3}}\hbox {I }\rightarrow \hbox { CF}_{\textrm{3}} +\hbox { HI} \end{aligned}$$

(R3)

The above reaction cycle is responsible for the depletion of active radicals H and O in the reaction pool and its corresponding substitution with a less reactive CF$_{\textrm{3}}$ radical. The reactions (R1)–(R3) compete with the following conventional chain branching and chain propagation reactions:

$$\begin{aligned} \hbox {O}_{\textrm{2}} +\hbox { H }\rightarrow \hbox { O }+\hbox { OH} \end{aligned}$$

(R4)

$$\begin{aligned} \hbox {H}_{\textrm{2}}\,+\hbox { O }\rightarrow \hbox { H }+\hbox { OH} \end{aligned}$$

(R5)

$$\begin{aligned} \hbox {H}_{\textrm{2}} +\hbox { OH }\rightarrow \hbox { H }+\hbox { H}_{\textrm{2}}\hbox {O} \end{aligned}$$

(R6)

and thus produce an inhibition effect on H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonations.

The addition of 5% CF$_{\textrm{3}}$I to H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures increases the induction zone length/time for all the equivalence ratios (0.4–2.0). This is because the post-shock temperature, $T_{\textrm{vN}}$, which governs the chemical kinetics in the induction zone, decreases drastically with the addition of CF$_{\textrm{3}}$I. The post-shock temperature without CF$_{\textrm{3}}$I for $\varPhi =$ 0.4, 1.0, and 2.0 is 1621.6 K, 1761.6 K, and 1724.9 K, respectively, for H$_{\textrm{2}}$–O$_{\textrm{2}}$ mixtures. However, with the addition of a 5% molar concentration of CF$_{\textrm{3}}$I, $T_{\textrm{vN}}$ drops to 1459.1 K, 1605.8 K, and 1545.5 K for $\varPhi =$ 0.4, 1.0, 2.0, respectively. Similar results can be observed for H$_{\textrm{2}}$–air mixtures (refer to Table 1). Thus, the addition of CF$_{\textrm{3}}$I decreases $T_{\textrm{vN}}$ substantially, leading to a larger induction length and time. In the case of fuel-lean mixtures, the induction length and time are higher as compared to stoichiometric conditions, and the addition of CF$_{\textrm{3}}$I further increases the $\Delta _{\textrm{i}}$ and $\uptau _{\textrm{i}}$. The significant increase in the $\Delta _{\textrm{i}}$ and $\uptau _{\textrm{i}}$ for both hydrogen-oxygen/air mixtures with CF$_{\textrm{3}}$I addition potentially indicates a very weakly coupled shock-reaction zone, thereby representing a higher probability of a complete decoupling of the shock and the reaction zone. Also, it can be observed that the induction length for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures is more than an order of magnitude larger than for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures at the same addition levels. Thus, CF$_{\textrm{3}}$I is a far better inhibitor than CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O at all the equivalence ratios studied.

Fig. 4

Normalized sensitivity coefficients (top ten reactions) for (a) 5% CO$_{\textrm{2}}$ and (b) 5% H$_{\textrm{2}}$O addition to stoichiometric hydrogen-air mixtures. The initial conditions $P_0$ and $T_0$, used for the ZND computations were 1 bar and 295 K, respectively, and the post-shock conditions ($P_{\textrm{vN}}$ and $T_{\textrm{vN}})$ were used as the initial mixture state for the sensitivity analysis

The variation of $f_{\Delta }$ and $f_{\uptau }$ with equivalence ratio is illustrated in Fig. 3. The inhibition efficiency varies similarly to the induction length and time with equivalence ratio. It should be noted that the inhibition efficiency of CF$_{\textrm{3}}$I is higher in fuel-lean H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures as compared to stoichiometric and fuel-rich mixtures (refer to Table 1 and Figs. 2 and 3). This is because, in fuel-lean mixtures, the H radicals are present in small amounts. The H-radical is consumed by both the reactions (R3) and (R4). Therefore, the presence of CF$_{\textrm{3}}$I hinders the H radical consumption by (R4), thereby hindering the chain branching process and thus increasing the detonation length and time scales in fuel-lean H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures. Therefore, higher detonation inhibition efficiency can be observed for CF$_{\textrm{3}}$I in fuel-lean conditions due to the limited H radical concentration and competition between hydrogen oxidation and radical scavenging by CF$_{\textrm{3}}$I. It can also be observed that with the addition of 5% CF$_{\textrm{3}}$I, no detonation was observed at equivalence ratios less than 0.7 (ER < 0.7) for H$_{\textrm{2}}$–air–CF$_{\textrm{3}}$I mixtures. Thus, it can be concluded that the addition of CF$_{\textrm{3}}$I reduces the detonability of a given fuel-air mixture, which is beneficial for hydrogen safety.

3.1.2 H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures

The variation in the induction length and time and the inhibition efficiency of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O at 5% molar additive concentration were evaluated at various equivalence ratios for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures. Unlike CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O are known to have an inhibition tendency at all concentrations. Carbon dioxide addition inhibits detonation due to the rapid recombination of the H, O, and OH active radicals with CO and CO$_{\textrm{2}}$:

$$\begin{aligned}{} & {} \hbox {CO}_{\textrm{2}} +\hbox { O }\rightarrow \hbox { CO }+\hbox { O}_{\textrm{2}} \end{aligned}$$

(R7)

$$\begin{aligned}{} & {} \hbox {CO}_{\textrm{2}} +\hbox { H }\rightarrow \hbox { OH }+\hbox { CO} \end{aligned}$$

(R8)

$$\begin{aligned}{} & {} \hbox {CO}_{\textrm{2}} +\hbox { OH }\rightarrow \hbox { HO}_{\textrm{2}} +\hbox { CO} \end{aligned}$$

(R9)

$$\begin{aligned}{} & {} \hbox {O }+\hbox { CO }\rightarrow \hbox { CO}_{\textrm{2}} \end{aligned}$$

(R10)

$$\begin{aligned}{} & {} \hbox {H }+\hbox { CO }\rightarrow \hbox { HCO} \end{aligned}$$

(R11)

The consumption of these reactive radicals inhibits the chain branching process and thus increases the induction length and time.

Similarly, H$_{\textrm{2}}$O inhibits detonation by removing the active H radical from the reaction pool. However, for H$_{\textrm{2}}$O, only the following reaction is critical:

$$\begin{aligned} \hbox {H }+\hbox { OH }+\hbox { H}_{\textrm{2}}\hbox {O }\rightarrow 2\hbox { H}_{\textrm{2}}\hbox {O} \end{aligned}$$

(R12)

in which the H and OH radicals get recombined to form steam (H$_{\textrm{2}}$O) [18]. Thus, the reactions (R7)–(R11) for CO$_{\textrm{2}}$ and (R12) for H$_{\textrm{2}}$O compete with the chain branching and chain propagating reactions (R4)–(R6) and inhibit detonation.

Carbon dioxide and water vapor act like a suppressant or inhibitor over a wide range of fuel-air equivalence ratios, where the induction length steadily increases with the addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O (see Fig. 2). The post-shock temperature, $T_{\textrm{vN,}}$ also decreases with the addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O. The decrease in the post-shock temperature is slightly greater for CO$_{\textrm{2}}$ than H$_{\textrm{2}}$O, but it is substantially less than that for CF$_{\textrm{3}}$I. The reduction in the post-shock temperature affects the kinetics in the induction zone and governs the induction zone length and time. Thus, CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O are comparable inhibitors at all the equivalence ratios but are inferior to CF$_{\textrm{3}}$I as the increase in $\Delta _{\textrm{i}}$ is higher for CF$_{\textrm{3}}$I than CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O under the conditions tested.

In order to gain insights into the underlying reaction kinetics, temperature sensitivity analysis was performed for stoichiometric hydrogen-air detonations in the presence of 5% CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O. The initial conditions for the computations were the corresponding post-shock thermodynamic state ((a) CO$_{\textrm{2}}$: $P_{\textrm{vN}} =$ 26.9 bar, and $T_{\textrm{vN}} =$ 1438.1 K; (b) H$_{\textrm{2}}$O: $P_{\textrm{vN}} =$ 27.15 bar, and $T_{\textrm{vN}}$ $=$ 1480.4 K). It can be seen that the addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O results in similar sensitivity spectra under the given conditions. The primary chain branching reaction, $\textrm{H}\,+\,\textrm{O}_{\textrm{2}}\,\leftrightarrow \,\textrm{O}\, +\,\textrm{OH}$ exhibits the highest positive sensitivity coefficient. A slight increase in the reaction rate of this reaction can significantly increase the temperature (in the reaction zone), leading to lower induction zone length/time for both H$_{\textrm{2}}$–air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures. The reactions that exhibit a positive sensitivity coefficient are mostly chain branching or chain propagating in nature, as shown in Fig. 4. It must be observed that the reactions responsible for the inhibition effect of carbon dioxide (R7)–(R11) and steam (R12) are not among the ten most influential reactions for the given mixture under detonating conditions. Thus, it can be concluded that the inhibition effect observed for CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O when added to hydrogen-air mixtures is primarily due to the change in the post-shock temperature, $T_{\textrm{vN}}$, and the chemical reaction cycle plays a secondary role under the conditions tested.

Table 2 Critical detonation parameters ($T_{\textrm{vN}}$, $\Delta _{\textrm{i}}$, and $\uptau _{\textrm{i}})$ and the factor of increase ($f_{\Delta }$, $f_{\uptau })$ at various mixture initial pressure ($P_{\textrm{0}})$ with and without inhibitors for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures

The inhibition efficiency of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O remains relatively constant for H$_{\textrm{2}}$–O$_{\textrm{2}}$–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures at all the equivalence ratios, as shown in Table 1 and Fig. 3a. However, for H$_{\textrm{2}}$–air–CO$_{\textrm{2}}$/ H$_{\textrm{2}}$O mixtures, the inhibition efficiency is comparatively higher at fuel-lean conditions than at fuel-rich conditions. In the case of H$_{\textrm{2}}$–air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures, the factor of increase in the induction length ($f_{\Delta })$ at $\varPhi =$ 0.6 is ~7.3 and ~9.4, and at $\varPhi =$ 1.6, it is ~2 and ~1.9 for CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O respectively. Thus, H$_{\textrm{2}}$O has a better inhibiting ability for H$_{\textrm{2}}$–air detonations at off-stoichiometric fuel-lean conditions.

The inhibition efficiency of the inhibitors CF$_{\textrm{3}}$I, CO$_{\textrm{2,}}$ and H$_{\textrm{2}}$O varies considerably with the fuel-oxidizer equivalence ratio, specifically for fuel-air-inhibitor mixtures. It can be concluded that the variation in the inhibition efficiency is the result of two separate governing parameters: the post-shock temperature, $T_{\textrm{vN}}$, which changes with the addition of inhibitor, and the radical abstraction mechanism of the inhibitors, which can be quantified by evaluating the maximum H-concentration.

3.2 Effect of varying initial pressure on the inhibition efficiency

3.2.1 H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures

The induction length decreases with an increase in initial mixture pressure from 0.2 bar to 5 bar at the stoichiometric condition with and without any additives or inhibitors, as shown in Table 2 and Fig. 5a and b. The addition of CF$_{\textrm{3}}$I to H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures increases the induction length due to the suppression or inhibition effect at any pressure (refer to Table 2 and Fig. 5). Although the addition of inhibitors offers an inhibiting effect for most H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures at any pressure, the inhibition efficiency or the factor of increase in the induction length and time decreases with increasing pressure. This is shown in Fig. 6a and b for CF$_{\textrm{3}}$I -diluted stoichiometric H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures. For example, the addition of 5% CF$_{\textrm{3}}$I at an initial pressure of 0.6 bar increases the induction length for a stoichiometric hydrogen-oxygen mixture by a factor of ~15.4 compared to the case with no CF$_{\textrm{3}}$I. However, for the same concentration at a higher initial pressure of 5 bar, the induction length increases by ~11.8 compared to the case with no CF$_{\textrm{3}}$I. Similarly, the addition of 5% CF$_{\textrm{3}}$I at an initial pressure of 0.6 bar increases the induction zone length for a stoichiometric hydrogen-air mixture by a factor of ~235 compared to the case with no CF$_{\textrm{3}}$I. However, for the same concentration at a higher initial pressure of 5 bar, the induction length increases by a factor of ~91 compared to the case with no CF$_{\textrm{3}}$I. Thus, the inhibition efficiency of CF$_{\textrm{3}}$I seems to decrease with increasing pressure, but still, CF$_{\textrm{3}}$I is a far better inhibitor than CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O.

At higher initial pressures, the recombination reaction

$$\begin{aligned} \hbox {O}_{\textrm{2}} +\hbox { H }(+\hbox { M})\,\rightarrow \hbox { HO}_{\textrm{2}}\,\, (+\hbox {M}) \end{aligned}$$

(R13)

competes with the branching reaction

$$\begin{aligned} \hbox {O}_{\textrm{2}} +\hbox { H}\rightarrow \hbox { OH }+\hbox { O} \end{aligned}$$

(R14)

It causes an increase in the induction length and time scales due to a decrease in the radical pool concentration. Since detonations are already inhibited by the effects of this competition for H atoms, the addition of CF$_{\textrm{3}}$I does not have a more significant impact on the $\Delta _{\textrm{i}}$ and $\uptau _{\textrm{i}}$ as the pressure is increased.

Fig. 5

Effect of varying initial mixture pressure on the induction length and time with and without inhibitors (5% CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O molar concentration): (a) H$_{\textrm{2}}$–O$_{\textrm{2}}$ detonations and (b) H$_{\textrm{2}}$–air detonations at $\varPhi =$ 1 and $T_{\textrm{0}} =$ 295 K

Fig. 6

Effect of varying initial mixture pressure on the factor of increase in the induction length ($f_{\Delta })$ and induction time ($f_{\uptau })$ at 5% CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O molar concentration: (a) H$_{\textrm{2}}$–O$_{\textrm{2}}$ detonations and (b) H$_{\textrm{2}}$–air detonations at $\varPhi =$ 1 and $T_{\textrm{0}} =$ 295 K

Therefore, the inhibition efficiency decreases with increasing initial pressure for a particular concentration of an inhibitor like CF$_{\textrm{3}}$I. The effect of mixture initial pressure on the $\mathrm {\Delta }_{\textrm{i}}$ and $\mathrm {\tau }_{\textrm{i}}$ is similar for both H$_{\textrm{2}}$–O$_{\textrm{2}}$–CF$_{\textrm{3}}$I and H$_{\textrm{2}}$–air–CF$_{\textrm{3}}$I mixtures. The same is true for H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures without any inhibitors.

3.2.2 H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures

The addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O diluents or inhibitors to H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures increases the induction length for constant $P_{\textrm{0}}$ and $T_{\textrm{0}}$. Thus, the increase in the $\Delta _{\textrm{i}}$ and $\uptau _{\textrm{i}}$ reduces the detonability and, in turn, the detonation hazard for CO$_{\textrm{2}}$/H$_{\textrm{2}}$O diluted H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures. The addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O provides an inhibiting effect for hydrogen-oxygen and hydrogen-air mixtures at any pressure. Also, the inhibiting efficiency of these inhibitors increases rapidly with increasing pressure. This is shown in Figs. 5 and 6 for CO$_{\textrm{2}}$-diluted and H$_{\textrm{2}}$O-diluted stoichiometric H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures at $T_{\textrm{0}} =$ 295 K. For example, the addition of 5% CO$_{\textrm{2}}$ at an initial pressure of 0.2 bar increases the induction zone length for a stoichiometric hydrogen-oxygen mixture by a factor of ~1.23 compared to the case with no CO$_{\textrm{2}}$. However, for the same dilution levels of CO$_{\textrm{2}}$ at a higher initial pressure of 5 bar, the induction length increases by a factor of ~1.75 compared to the case with no CO$_{\textrm{2}}$.

Table 3 Critical detonation parameters ($T_{\textrm{vN}}$, $\Delta _{\textrm{i}}$, and $\uptau _{\textrm{i}})$ and the factor of increase ($f_{\Delta }$, $f_{\uptau })$ at various mixture initial temperatures ($T_{\textrm{0}})$ with and without inhibitors for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures

Similarly, the addition of 5% CO$_{\textrm{2}}$ at an initial pressure of 0.2 bar increases the induction zone length for a stoichiometric hydrogen-air mixture by a factor of ~1.46 compared to the case with no CO$_{\textrm{2}}$. However, for the same dilution at a higher initial pressure of 5 bar, the induction length increases significantly by a factor of ~3.8 compared to the case without CO$_{\textrm{2}}$. For mixtures diluted with H$_{\textrm{2}}$O, the addition of 5% H$_{\textrm{2}}$O at an initial pressure of 0.2 bar increases the induction zone length for a stoichiometric hydrogen-oxygen mixture by a factor of ~1.2 compared to the case with no steam. However, for the same dilution levels of steam at a higher initial pressure of 5 bar, the induction length increases significantly by a factor of ~1.9 compared to the case with no steam. Similar results were obtained for stoichiometric hydrogen-air mixtures. Thus, when compared with the no-inhibitor case, the inhibitors seem to have increased inhibition efficiency with increasing pressure (refer to Table 2 and Fig. 6a and b).

The addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O inhibitors to stoichiometric hydrogen-air mixtures increases the induction length and time at pressures greater than 1 bar ($P_{\textrm{0}}$ > 1 bar). However, at very low pressures, the induction zone lengths seem to approach each other with the addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O, see Fig. 5b. The computational results also exhibit a local minimum in the induction length with increasing initial pressure. The increase in initial pressure in the presence of inhibitors like CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O would decrease the induction length initially and attain a minimum. After minima, the induction length increases with an increase in initial pressure, as shown in Fig. 5b. However, this effect is primarily seen in H$_{\textrm{2}}$–air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures and not in H$_{\textrm{2}}$–O$_{\textrm{2}}$–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures. The results are consistent with the literature where Stamps and Tieszen observed a cell width minimum in H$_{\textrm{2}}$O-diluted hydrogen-air detonations [23].

3.3 Effect of varying initial temperature on the inhibition efficiency

3.3.1 H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures

Hydrogen-oxygen and hydrogen-air mixtures with CF$_{\textrm{3}}$I have larger induction zone lengths than those without $\text {CF}_{3}\textrm{I}$ addition. However, it can be observed that the induction length and time for CF$_{\textrm{3}}$I diluted H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures decrease with an increase in the initial temperature. In contrast, the induction length and time increase for CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O diluted H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures (refer to Table 3). Although the addition of CF$_{\textrm{3}}$I increases the induction length and time (compared to the no inhibitor case) and offers an inhibiting effect for most hydrogen-oxygen and hydrogen-air mixtures at all mixture temperatures, the inhibiting effect decreases rapidly with increasing temperature. This is shown in Figs. 7 and 9 for CF$_{\textrm{3}}$I -diluted stoichiometric H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures at $P_{\textrm{0}} =$ 1 bar.

Fig. 7

Effect of varying initial mixture temperature on the induction length and time with and without retardants (5% CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O molar concentration): (a) H$_{\textrm{2}}$–O$_{\textrm{2}}$ detonations and (b) H$_{\textrm{2}}$–air detonations at $\varPhi =$ 1 and $P_{\textrm{0}} =$ 1 bar

Fig. 8

Normalized sensitivity coefficients (top ten reactions) for stoichiometric H$_{\textrm{2}}$–O$_{\textrm{2}}$–CF$_{\textrm{3}}$I (5%) mixtures at (a) $T_{0} =$ 300 K and (b) $T_{0} =$ 500 K. The initial conditions for the sensitivity computations were the corresponding post-shock conditions ($P_{\textrm{vN}}$ and $T_{\textrm{vN}})$. Reactions that belong to the CF$_{\textrm{3}}$I sub-mechanism are marked as solid color bars

Fig. 9

Effect of varying initial mixture temperature on the factor of increase in the induction length ($f_{\Delta })$ and induction time ($f_{\uptau })$ at 5% CF$_{\textrm{3}}$I, CO$_{\textrm{2}}$, and H$_{\textrm{2}}$O molar concentration: (a) H$_{\textrm{2}}$–O$_{\textrm{2}}$ detonations and (b) H$_{\textrm{2}}$–air detonations at $\varPhi =$ 1 and $P_{\textrm{0}} =$ 1 bar

For example, the addition of 5% CF$_{\textrm{3}}$I increases the induction zone length for a stoichiometric hydrogen-oxygen mixture by a factor of ~14.8 at 300 K compared to the case with no CF$_{\textrm{3}}$I. However, the induction length decreases for the same concentration level at a higher initial temperature of 1000 K compared to the case with no CF$_{\textrm{3}}$I. Similarly, the addition of 5% CF$_{\textrm{3}}$I at an initial temperature of 300 K increases the induction zone length for a stoichiometric hydrogen-air mixture by a factor of ~227.5 compared to the case with no CF$_{\textrm{3}}$I. Again, for the same concentration level at a higher initial temperature of 1000 K, the induction length increases by ~2.5 compared to the case with no CF$_{\textrm{3}}$I. Thus, the impact of the initial temperature on the induction zone length and time is similar for both H$_{\textrm{2}}$–O$_{\textrm{2}}$–CF$_{\textrm{3}}$I and H$_{\textrm{2}}$–air–CF$_{\textrm{3}}$I mixtures, where the inhibiting efficiency of CF$_{\textrm{3}}$I was found to decrease rapidly with increasing temperature. The effect of the mixture’s initial temperature on the induction length is quite different for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures and H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures. For example, the induction length increases with an increase in initial temperature from 300 to 1100 K for hydrogen-oxygen/air mixtures in the absence of CF$_{\textrm{3}}$I. However, for hydrogen-oxygen/air–CF$_{\textrm{3}}$I mixtures, the induction length decreases with an increase in initial temperature from 300 to 1100 K.

The decrease in the inhibition effectiveness of CF$_{\textrm{3}}$I at higher temperatures is an interesting result, as the inhibition effectiveness of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O remains relatively constant with changes in initial temperature. In order to evaluate the dominant reactions responsible for the observed effect, temperature sensitivity analysis was carried out for H$_{\textrm{2}}$–O$_{\textrm{2}}$–CF$_{\textrm{3}}$I mixtures at different initial temperatures. The initial conditions for the sensitivity computations were the corresponding post-shock states ((a) $T_{0} =$ 300 K: $P_{\textrm{vN}} =$ 32.4 bar, and $T_{\textrm{vN}} =$ 1608.6 K; (b) $T_{0} =$ 500 K: $P_{\textrm{vN}} =$ 18.8 bar, and $T_{\textrm{vN}} =$ 1727.5 K). The normalized sensitivity coefficients for the top ten reactions are shown in Fig. 8. It can be observed that the consumption of H radical through the reaction $\textrm{H}\,+\, \textrm{I}\, +\, \textrm{M}\leftrightarrow \textrm{HI}\, +\, \textrm{M}$ shows the highest negative sensitivity coefficient (amongst reactions belonging to the CF$_{\textrm{3}}$I sub-mechanism). The reaction is responsible for an increase in the induction length with the addition of CF$_{\textrm{3}}$I, irrespective of the initial mixture temperature. At $T_{0} =$ 300 K (Fig. 8a), the reaction ${\textrm{CHF}}_{\textrm{3}}\, +\,\textrm{I}\,\leftrightarrow {\textrm{CF}}_{\textrm{3}}\mathrm {+HI}$ exhibits a negative sensitivity coefficient, thereby contributing to the inhibition effect. However, as the initial temperature is increased to $T_{0} =$ 500 K (Fig. 8b), the reaction exhibits a positive sensitivity coefficient. The production of H radical through the combined effect of the reactions, ${\textrm{CHF}}_{\textrm{3}}\, +\, \textrm{I}\, \leftrightarrow {\textrm{CF}}_{\textrm{3}}+\,\textrm{HI}$ and ${\textrm{CF}}_{\textrm{3}}\,+\, \textrm{HI}\,\leftrightarrow {\textrm{CF}}_{\textrm{3}}\textrm{I}+\,\textrm{H}$ results in radical proliferation, thereby reducing the induction length for H$_{\textrm{2}}$ – O$_{\textrm{2}}$ – CF$_{\textrm{3}}$I mixtures at elevated temperatures. Thus, the inhibition effectiveness of CF$_{\textrm{3}}$I diminishes at higher initial temperatures due to the change in sensitivities of the elementary reactions to the initial conditions.

3.3.2 H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures

H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures diluted with CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O have larger induction zone lengths than the same mixture without dilution. The addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O inhibits H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonations; however, it can be seen from Fig. 9a and b that the inhibition efficiency of these inhibitors is minimally affected with an increase in the initial temperature of the reacting mixture, unlike CF$_{\textrm{3}}$I diluted stoichiometric H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures. For example, the addition of 5% CO$_{\textrm{2}}$ at an initial temperature of 300 K increases the induction zone length for a stoichiometric hydrogen-oxygen mixture by a factor of ~1.3 compared to the case with no CO$_{\textrm{2}}$. However, for the same dilution at a higher initial temperature of 1000 K, the induction length increases by only a factor of ~1.1.

This represents that the inhibition efficiency of CO$_{\textrm{2}}$ decreases slightly with the rise in initial temperature. For mixtures diluted with steam, the addition of 5% H$_{\textrm{2}}$O at an initial temperature of 300 K increases the induction zone length for a stoichiometric hydrogen-oxygen mixture by a factor of ~1.3 compared to the case with no steam. However, at a higher initial temperature of 1000 K, the induction length increases only by a factor of ~1.1. Similarly, the addition of 5% H$_{\textrm{2}}$O at an initial temperature of 300 K increases the induction zone length for a stoichiometric hydrogen-air mixture by a factor of ~1.8 compared to the case with no $\textrm{H}_{2}\textrm{O}$. However, the same amount of dilution at a higher initial temperature of 1000 K increases the induction length by only a factor of ~1.2 (refer to Table 3). Thus, the inhibition efficiency of H$_{\textrm{2}}$O decreases with an increase in the initial temperature.

The addition of CO$_{\textrm{2}}$/H$_{\textrm{2}}$O seems to have quite a different effect for H$_{\textrm{2}}$–O$_{\textrm{2}}$ and H$_{\textrm{2}}$–air mixtures. The addition of CO$_{\textrm{2}}$/H$_{\textrm{2}}$O increases the induction zone length in the case of $\textrm{H}_{2}$–$\textrm{O}_{2}$ mixtures, whereas the induction length remains mostly constant for $\textrm{H}_{2}$–air mixtures. Therefore, at a higher mixture initial temperature, CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O are as good inhibitors as CF$_{\textrm{3}}$I and thus can be used for detonation inhibition under the conditions studied.

4 Conclusions

The efficiency of detonation inhibitors on H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonations was evaluated under varying mixture initial conditions using numerical computations. The inhibitors CF$_{\textrm{3}}$I, CO$_{\textrm{2,}}$ and H$_{\textrm{2}}$O at 5% molar concentration were added to H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures, and the factor of increase in the induction zone length and time was used as an indicative parameter of the inhibition efficiency of the detonation inhibitors. It was found that CF$_{\textrm{3}}$I is a far better inhibitor than CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O at all the mixture equivalence ratios studied in the current work. The inhibition efficiency of CF$_{\textrm{3}}$I is higher in fuel-lean H$_{\textrm{2}}$–O$_{\textrm{2}}$/air mixtures than in stoichiometric and fuel-rich mixtures. The addition of CF$_{\textrm{3}}$I drastically reduces the detonability of a given fuel-air mixture. It was also found that CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O have a comparable inhibition efficiency at all the equivalence ratios but are inferior to CF$_{\textrm{3}}$I in inhibiting a detonation wave under similar initial mixture conditions. In comparison to CO$_{\textrm{2}}$, H$_{\textrm{2}}$O has a better inhibiting ability for H$_{\textrm{2}}$–air detonations at off-stoichiometric fuel-lean conditions. The inhibition efficiency was found to be the result of two separate governing parameters: the post-shock temperature, $T_{\textrm{vN}}$ which changes with the addition of an inhibitor and the radical abstraction mechanism of the inhibitors. The inhibiting efficiency of CF$_{\textrm{3}}$I was found to decrease with increasing initial mixture pressure; however, it was still a better inhibitor when compared with CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O. On the other hand, the inhibition efficiency of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O increased rapidly with increasing mixture pressure. At very low initial pressures, the induction zone lengths approached each other with the addition of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O as inhibitors. The variation of induction length with initial mixture temperature showed an interesting behavior where it decreases for H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CF$_{\textrm{3}}$I mixtures and increases for undiluted and H$_{\textrm{2}}$–O$_{\textrm{2}}$/air–CO$_{\textrm{2}}$/H$_{\textrm{2}}$O mixtures. The inhibition efficiency of CO$_{\textrm{2}}$ and H$_{\textrm{2}}$O is minimally affected with an increase in the initial temperature. A key takeaway from the current study is the effect of initial temperature on the inhibition efficiency of CF$_{\textrm{3}}$I. It was found that the efficiency decreased drastically at a higher mixture initial temperature. This result is of particular importance for flame and detonation suppression in nuclear reactors where the initial flammable mixture temperature is higher, and the results from the current findings can be used to make a proper choice of inhibitor under such conditions.

The inhibiting effect of detonation inhibitors varied drastically with the initial conditions, and the same should be taken into consideration while designing fire and detonation safety systems for applications in petrochemical industries and nuclear reactors. It should be noted that for all the inhibitors studied, the inhibition efficiency is minimal at very high initial temperatures. The current study on H$_{\textrm{2}}$–O$_{\textrm{2}}$/air detonating mixtures lays the groundwork for further studies on detonation inhibition in hydrocarbon-O$_{\textrm{2}}$/air mixtures and scenarios for possible gas leaks leading to vapor cloud formation, especially under confined spaces.