INTRODUCTION

The paper presents an experimental study of the dependence of the speed of flame spread over a fuel film on different substrates in an oxygen-enriched environment. The objectives of the study are to clarify the possibility of flame spread with a high speed of the order of the normal speed of the corresponding homogeneous stoichiometric composition and to generalize the experimental data obtained.

At flame speeds of about 1 m/s, the substrate–fuel system may become thermally thick. In this case, the flame speed is independent not only of the substrate properties, but also on the fuel layer thickness. That is, the data obtained can be used for pools of any depth and for spills of flammable liquids in corresponding environments.

Flame spread over liquid fuel surfaces in air has been the subject of numerous studies. Flame spread at fuel temperature above and below the flash point has been studied [1–7]. The dominant mechanism of heat transfer into the pre-flame zone is the convection of the liquid phase due to thermocapillarity [3]. It has been shown that a decrease in flame speed in the case of a shallow pool is due primarily to the viscosity of the liquid and secondarily to heat loss into the vessel wall [5]. Thus, heat convection in the liquid phase is the main form of heat transfer for flame spread over liquid fuels at temperatures below the ignition temperature, whereas heat conduction in the liquid phase and flame radiation play a secondary role. The object of study in most of the papers cited above are complex practical fuels, but pure substances have also been studied. In these studies, the fuel layer thickness is basically such

that the physical properties of the walls of fuel tanks (pools) can be neglected.

Another limiting case is flame spread over rather thin films of liquid fuels on substrates. Such systems are encountered in practice. These are porous media wetted with liquid fuel, whose free space contains air. This problem is of practical interest since a wetted inert porous medium can become fire and explosion hazardous [8, 9]. Porous media are also often used as an explosion-proof filler for fuel tanks of various vehicles [10], i.e., where the porous medium is always wetted.

Scientific interest in the problem of flame spread in a porous medium is due to the possibility of implementing various steady combustion modes: heterogeneous detonation [11], the high-speed mode [12], and the vaporization–diffusion mode [13, 14]. The flame speed in the corresponding processes is 400–1050 m/s [11], 0.1–10 m/s [12], and 4–10 cm/s [14].

Of practical interest are the cases of flame spread with speeds in the range of 0.1–10 m/s. Flame spread over ann-octane film in three porous media was investigated in [14]. Two media were highly porous with a fraction of free space of about 0.98, and one medium was low-porosity and filled with steel balls with a fraction of free space of 0.4. The highly porous medium was open-porosity polyurethane foam consisting of thin polyurethane fibers and a porous filler based on aluminum foil. The oxidizer was air at different initial pressures. With an increase in air vapor pressure, the concentration of n-octane went beyond the lean limit, and under these conditions, the flame propagated in highly porous media. It was found that the speed of flame spread beyond the lean limit was the same in these two environments despite the differences in structure and physical properties. In [15], the effect of the substrate on the speed and limits of flame spread was studied using a model system consisting of a fuel film on a thin flat substrate, which simulated a foil porous medium.

The goal was achieved by varying the parameters of the substrate and fuel and the geometry and orientation of samples relative to the acceleration vector due to gravity. In the propagation of a diffusion flame above the surface of a condensed (solid or liquid) material, there are two limiting cases: thermally thick and thermally thin systems [16]. Thermally thin systems consisting of a fuel and a substrate are of particular interest. In this case, the flame speed should not depend on the velocity of the opposed oxidizer flow. The system of a fuel film on a thin metal substrate satisfies the thermal thinness condition. A number of steady and unsteady modes of flame spread over such systems has been determined in experimental studies [17].

According to the propagation mechanism, they can be divided into free convection and low-speed flame modes. In the free convection mode, the flame spreads upward. Rising combustion products heat the fuel film, which evaporates and burns to form a new portion of products. The flame speed in this case depends on the angle of inclination of the substrate to the horizontal, weakly depends on the properties of the fuel (saturated hydrocarbons with 10 to 16 atoms: C10H22–C16H34), and reaches maximum values of about 0.3 m/s for the vertical position of the substrate [15, 17]. In the free convection mode, the flame speed is practically independent of the properties of the fuel, in particular, on the difference \(T_{st}-T_0\), where \(T_{st}\) is the temperature at which the vapor of the liquid fuel form a stoichiometric mixture with air, and \(T_0\) is the ambient temperature.

In the low-speed mode, the flame speed on thermally thin substrates is about 0.02 m/s. The flame speed does not depend on the angle of inclination of the substrate to the horizontal in the range from downward spread to spread in a horizontal plane. It has been shown [18] that in the low-speed mode, unlike in the free convection mode, the flame speed depends on the fuel properties. The greater the number of carbon atoms in the saturated hydrocarbon, the lower the speed. The effect of the thermal diffusivity of the substrate on the flame speed has been studied [19]. It has been shown that the higher the thermal diffusivity of the substrate, the higher the speed. For a system consisting of a substrate with a high thermal conductivity and a fuel film on it, the rate-determining stage of flame spread is heat transfer through the substrate from combustion products to the cold zone with the fuel film. A mathematical model of the flame spread mechanism has been proposed [20] which takes into account heat transfer through the substrate. It has also been experimentally shown [20] that when estimating the flame speed over fuel on copper substrates, the heat flux due to thermocapillarity can be neglected. It has been found that the flame speed increases with decreasing difference \(T_{st}\,-\,T_0\). In [21], saturated hydrocarbons with different numbers of atoms were also used as a fuel. The temperature \(T_{st}\) was reduced by adding methane to air in concentrations below the lean limit. The more methane in air, the less liquid fuel needs to be vaporized to obtain the stoichiometric concentration and the lower the temperature \(T_{st}\) for mixtures of fuel vapor with methane. Increasing the methane concentration led to an increase in the flame speed according to the equation for speed from [20]. It should be noted that when \(T_{st}\) is varied in this manner, the thermal effect of combustion of a unit volume of the stoichiometric mixture practically does not change. In [21], new experimental evidence was obtained for the validity of the method proposed in [20] to estimate the flame speed in a thermally thin system. In addition, it was first shown experimentally that in a thermally thin system, the flame speed does not depend on the velocity of the opposed oxidizer flow. However, the hypothesis [21] of the flame spread limit in this system is consistent with experiment only when accounting for the fuel film flow under the action of a temperature gradient along the substrate. A study [22] of the effect of the initial temperature on the speed of flame spread over a liquid fuel film on metal substrates has also shown that the model proposed in [20] for calculating the flame speed agrees satisfactorily with experiment, but the flame spread limit results from the impossibility of providing sufficient thickness of the evaporating fuel film because of the fuel runaway forward due to the thermocapillary effect. Thus, experiments have indicated that thermocapillary flow in thermally thin systems does not affect the flame speed, but affects the flame spread limit in terms of the oxidizer flow velocity and the initial temperature of the system.

Flame spread over fuel films on thin flat substrates with low thermal conductivity has been studied [23]. It has been shown that such a system is thermally thick. It has been concluded that the speed of flame spread over thin substrates with low thermal conductivity depends on the liquid temperature gradient near the leading edge of the flame. The temperature gradient is determined not only by the thermophysical properties of the fuel and substrate and their thickness, but primarily by the derivative of the temperature dependence of the surface tension coefficient \(\sigma\) (\(d\sigma /dT\)) and the dynamic viscosity \(\mu\) of the liquid fuel. The role of the substrate reduces to heat losses, which are determined by its heat capacity and thickness.

Flame spread over a thin liquid layer in a flat quartz channel with an opposed oxidizer flow has been investigated [24, 25] using as oxidizer oxygen–nitrogen mixtures in the range from air to pure oxygen. It has been shown experimentally that the speed of flame spread over an n-butanol layer of about 1 mm ranges from 2 m/s without oxygen flow to 4 m/s at an opposed oxygen flow velocity of 3 m/s. This system is thermally thick.

It has been shown [25] that when the flame propagates over a liquid layer about 0.4 mm thick, a wave moving with the flame speed is formed ahead of the flame front due to the thermocapillary effect in the liquid. The wave crest height correlates with the flame speed: it first increases from a height of 0.2 mm at speeds up to 10 cm/s and then drops abruptly to zero from a height of 0.6 mm at speeds above 20 cm/s. It has been shown that the temperature gradients in the liquid must be unrealistically high in order for the wave crest to move with the flame speed at high flame speeds.

Thus, for thermally thick systems, the mechanism determining the flame speed in the range below 0.1 m/s must include thermocapillary phenomena. And in the range of speeds above 0.1 m/s, the flame speed is almost independent of the thermal diffusivity of the substrate and the thermocapillary flow.

The purpose of this study is to clarify the possibility of flame spread at a sufficiently high speed of the order of the normal speed of the corresponding homogeneous stoichiometric composition with an increase in the oxygen content in the oxidizing environment. A feature of this formulation of the problem is that increasing the oxygen content will lead, on the one hand, to an increase in \(T_{st}\) and, on the other hand, to an increase in the thermal effect of a unit volume of the mixture. This will increase the temperature of combustion products \(T_{b}\). However, at a given oxygen content, the temperatures of combustion products \(T_{b}\) of stoichiometric mixtures will be close for hydrocarbons of the same homologous series.

It follows from this brief review that concepts and theories describing flame spread over a liquid fuel surface at high speeds are not available in the literature. The main objective of this work is to draw the attention of researchers to the theoretical description of flame spread over the surface of liquid fuels in environments more aggressive than air. A particular objective of this work is to present experimental data on flame spread under these conditions for the purpose of verifying future theories. The paper also presents the results of experimental data processing in the form of dependences of flame speed on the parameters that determine, in our opinion, the process of flame spread over a liquid at high speed.

EXPERIMENT

In our experiments, we measured the average speed of flame spread over thin fuel films (about 10 \(\mu\)m thick) in environments with different oxygen content in the range from air to pure oxygen. Copper foil 45 and 60 \(\mu\)m thick and glass 3 mm thick were used as substrates for films. A number of normal alkanes C\(_{n}\)H\(_{2n + 2}\), \(n = 10\), 11, 12, 13, and 16, and the alcohols n-butanol (C4H9OH) and n-pentanol (C5H11OH) were used as fuel. All experiments were carried out at atmospheric pressure and room temperature \(T_{0} = 22\pm 2\)°C and the flash point of all fuels was above this value.

Fig. 1
figure 1

Experimental setup: (1) high-pressure mixer; (2) valve; (3) lower glass plate on which the experiments were performed; (4) copper foil strip; (5) upper shorter glass plate; (6) video camera.

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The setup was a square tube composed of glass strips 3 mm thick (Fig. 1). The tube was located horizontally. The tube was 1 m long with an inner cross-section of \(50\times 50\) mm, the foil was located at a height of 25 mm, and the fuel was applied to the top side of the foil. The air–oxygen mixture was supplied from a high-pressure mixer 1 on the left through a valve 2. The top of the tube at the end opposite to the oxidizer inlet has an open part 4 cm long, through which the oxidizing gas comes out during purging and ignition is carried out by a torch of a propane–air mixture. The experiment was carried out as follows. A fuel layer about 10 \(\mu\)m thick was applied to the bottom plate of the tube 3or the top surface of the copper substrate 4. Before applying the layer, the top plate of the tube was removed. Then the top plate, which was 4 cm shorter than the bottom plate, was put in place, the valve 2 opened, and air was displaced by a tenfold volume of the oxidizer flowing through the open top of the tube. After the valve 2 was closed, the fuel film was ignited. Thus, unlike in [24, 25], flame spread occurs in the absence of opposed oxidizer flow. The flame spread process was recorded with a video camera 5 located on top.

RESULTS

In the structure of the flame spreading over a copper substrate in air, the following characteristic elements can be distinguished: the flame front with an even edge located near the substrate surface and a short wedge-shaped region of liquid under the flame. The flame edge is an even straight line perpendicular to the parallel side edges of the substrate. Ahead of the flame edge, there is a thickening of the liquid film with a length of 10–15 mm. Behind the flame edge, under the glow zone with a characteristic size of about 2 cm, there is a wedge-shaped liquid region with a length 1–10 mm, which is short for saturated hydrocarbons and longer for alcohols. The flame is blue. The flame speed is constant with an accuracy of 2% [15]. Increasing the oxygen content increases the flame length and the length of the fuel film under the flame. For heavier hydrocarbons, the flame turns white. At a high oxygen content, the flame edge becomes uneven, and a leading point can be identified whose speed of propagation is not constant. The flame near the leading point is blue, and afterburning of the film behind the leading point occurs in the form of individual white tongs. Below, by the flame speed is meant the average speed of the leading point.

Fig. 2
figure 2

Flame speed over films of the unsaturated hydrocarbonsn-decane (C10H22) (1),n-undecane (C11H24) (2),n-dodecane (C12H26) (3),n-tridecane (C13H28) (4), and n-hexadecane (C16H34) (5) on a copper substrate versus the volume concentration of oxygen in the oxidizer.

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Figure 2 shows the dependence of the average flame speed over films of saturated hydrocarbons and alcohols on a copper foil substrate 45 \(\mu\)m thick on the oxygen concentration. It can be seen that the higher the oxygen concentration, the higher the flame speed. The higher the molecular weight of hydrocarbon, the lower the flame speed at the same oxygen concentration.

A similar dependence for alcohols is shown in Fig. 3.

Fig. 3
figure 3

Flame speed over films of n-butanol (C4H9OH) (1) andn-pentanol (C5H11OH) (2) on copper substrate versus volume concentration of oxygen in the oxidizer.

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It follows from Figs. 2 and 3 that the flame speed as a function of oxygen concentration changes by almost two orders of magnitude for alcohols and by more than an order of magnitude for n-decane and n-undecane.

Fig. 4
figure 4

Flame speed over films of n-alkanes and alcohols versus volume concentration of oxygen in the oxidizer on glass substrates (\(h_{s} = 5\) mm; points with odd numbers) and copper substrates (\(h_{s} = 45\) \(\mu\)m; points with even numbers) for butanol (points 1 and2), pentanol (points 3 and 4), n-decan (points5 and  6),n-undecane (points 7 and 8), andn-dodecane (points9 and  10).

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Figure 4 shows the dependences of the flame speed over films ofn-alkanes and alcohols on the oxygen concentration in the oxidizer on glass and copper substrates. The data on flame spread over copper substrates are taken from Figs. 2 and 3 at the same oxygen concentrations at which flame spread over glass substrates was observed. At a volume concentration of oxygen below 0.5, flame over fuel films on glass substrates does not spread. The flame speed over glass substrates is basically the same as over copper substrates.

Several experiments were carried out with a copper substrate 60 \(\mu\)m thick. The flame speed in oxygen was as follows: 1.85 m/s (2.25 m/s) for n-butanol, 1.10 m/s (1.43 m/s) forn-decane, and 0.62 cm/s (0.95 m/s) forn-undecane; figures in brackets indicate the speed on a copper substrate 45 \(\mu\)m thick. This comparison of speeds indicates that the substrate has an influence. The thicker the substrate with high thermal conductivity, the higher the heat loss to it and the lower the flame speed.

DISCUSSION

Comparison of flame speeds over films of various fuels on a thicker copper substrate (60 \(\mu\)m) with speeds on a thinner substrate (45 \(\mu\)m) with the same thermal diffusivity of the substrates (1.1 cm2/s) indicates that even at the high flame speeds, heat loss to the substrate affects the flame speed. The thicker the substrate, the greater part of the heat flux from the flame front is consumed for its heating and the lower the flame speed. If the thermal diffusivity of the substrate is low like that of glass (of the order 0.01 cm2/s), then the heat loss is lower, despite the larger substrate thickness (5 mm), and the flame speed is higher.

Experimental data on flame speeds were obtained in a wide range from 2 to 265 cm/s on substrates 45 \(\mu\)m and 5 mm thick. It is obvious that in some cases, the substrate–fuel system is thermally thin, and in others, it is thermally thick. The flame spread mechanism will be different in these two cases. The thermal thinness criterion of the system is the ratio of the characteristic heating times of the system in the direction of flame spread and in the perpendicular direction and is expressed as follows: the characteristic heating time of the fuel layer in the direction perpendicular to the flame spread (\(\tau_1 =h^2/\varkappa_1\)) must be less than the characteristic time of heat transfer along the substrate (\(\tau_2=\varkappa_2/u^2\)) in the direction of flame spread. Here \(h\) is the film thickness, \(\varkappa\) is the thermal diffusivity, and \(u\) is the flame speed. The characteristic heating time of a fuel film with a thickness \(h_f =10\) \(\mu\)m is \(\tau _f = h_f^2/ k_f = 10^{-10}/(0.8\cdot 10^{- 7}) \approx 0.001\) s. For the system of a copper substrate 45 \(\mu\)m thick and a fuel, the thermal diffusivity is \(10^{-4}\) m2/s, and for the glass–fuel system, it is \(6\cdot 10^{-7}\) m2/s. For the copper substrate, the characteristic heating time is \(\tau _{s1}= h_{s1}^2/k_{s1} =4.5^2\cdot 10^{- 10}/(1.1\cdot 10^{- 4})\approx 2\cdot 10^{- 5}\) s, and for a glass substrate 5 mm thick, it is \(\tau _{s2} = h_{s2}^2/k_{s2} =5^2\cdot 10^{- 6}/(6\cdot 10^{- 7})\approx 5\) s. The characteristic heating time ahead of the flame front on the copper substrate is \(\tau _2 =\varkappa/u^2\approx 10^{- 4}/u^2\) [s]. It can be seen that for speeds below 0.1 m/s, the system with a copper substrate is thermally thin: \(\tau_2> 10^{- 2}\gg\tau _{f}\) and \(\tau_2\gg\tau _{s1}\). For the glass–fuel system, \(\tau_2 =\varkappa /u^2\approx 6\cdot 10^{- 7}/u^2\) and \(\tau_2 < 3\cdot 10^{- 5}\) s for the entire range of flame speeds, i.e., the system is thermally thick: \(\tau_2\ll\tau_{f}\) and \(\tau_2\ll\tau_{s2}\). At speeds above 30 cm/s, \(\tau_2 < 0.001\) s, i.e., \(\tau_2<\tau_f\), the thermal thinness condition is not satisfied even for the fuel layer; therefore the flame speed at these speeds should depend on the substrate properties only slightly.

When the flame spreads over a liquid fuel film, the leading edge of the flame, i.e., the leading front which is the most protruding in the direction of flame spread and the closest to the liquid phase, is the line ahead of which the ratio of the fuel and oxidizer concentrations is stoichiometric. This line is an analogue of the triple point in diffusion flames [26]. On the rest of the diffusion flame surface, only the ratio of the fuel and oxidizer fluxes is stoichiometric. In our opinion, physicochemical processes near this line are of decisive importance in the mechanism of flame spread at high speed. Therefore, we will try to relate the flame speed with the parameters of a homogeneous stoichiometric mixture of a given fuel with oxidizer.

Fig. 5
figure 5

Flame speed (circles) over a n-butanol film on glass, the normal speed (triangles) of a homogeneous stoichiometric mixture of butanol and oxidizer, and flame temperature (squares) versus oxygen content in a mixture with nitrogen.

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Consider how the speed of flame spread over a fuel film correlates with the normal speed and flame temperature of a homogeneous stoichiometric mixture of the corresponding fuel usingn-butanol as an example. The normal speed \(S_{u}\) and the flame temperature \(T_{b}\) were calculated according to [27–29]. Figure 5 shows the dependences of the flame speed \(u\) over an n-butanol film on glass, the normal speed \(S_{u}\) of the homogeneous stoichiometric mixture, and the flame temperature \(T_{b}\) on the oxygen content in the mixture with nitrogen. Note that according to calculations [27–29], the normal speed of the homogeneous mixture with air is 41 cm/s, and the flame speed over ann-butanol film on a copper substrate 45 \(\mu\)m thick is 2.5 cm/s. Under these conditions, the flame does not spread over glass.

It can be seen from Fig. 5 that the calculated equilibrium temperature of the flame in the range of oxygen volume concentrations in which the flame spreads over the fuel film (0.5–1.0) changes only slightly (2833–3054 K). The normal speed increases from 163 to 265 cm/s. With an increase in the oxygen concentration in the same range, the flame speed over the fuel film increases from 41 to 236 cm/s, i.e., it increases faster than the normal speed of the homogeneous stoichiometric mixture. This suggests a difference between the mechanism of flame spread over fuel films and the mechanism of laminar flame spread in homogeneous mixtures.

It has been shown [25] that at flame speeds above 30 cm/s, there is no thickening of the fuel film ahead of the flame front. Thus, due to the absence of motion of the fuel film and the weak dependence of the flame speed on substrate parameters (thickness and thermal conductivity) the mechanism of flame spread at high flame speeds should be different from the mechanisms described above [20, 23]. Identification of characteristic features of this mechanism and generalization of the obtained experimental results is the subject of further discussion.

Consider what changes during flame spread when the oxygen concentration is increased. During flame spread over the surface of a condensed (liquid or solid) material, the beginning of the flame front (tip) is located where a stoichiometric mixture of fuel vapor with oxidizer (triple point) is formed [20, 26]. With an increase in the oxygen concentration in the oxidizer, the normal speed of the stoichiometric mixture increases. However, at a given oxygen concentration, it remains approximately the same for saturated hydrocarbons regardless of their number in the homologous series. The approach of the flame tip to the surface is limited by a dead zone \(d\), which depends on the Peclet number based on the normal speed and thermal diffusivity \(\varkappa\) gas: \(\mathop{\rm Pe}\nolimits _{\rm cr}=S_{u}d/\varkappa\). The Peclet number should be approximately the same for all mixtures. Thus, with an increase in the proportion of oxygen, the normal flame speed increases and the value of \(d\) decreases in inverse proportion to \(S_{u}\). Apparently, the position of the flame tip at this distance is stable. Suppose that the flame tip has moved closer to the liquid fuel surface and is located at a distance less than the critical one. In this case, the heat flux into the liquid phase and the heat loss from the flame front increase and the temperature and normal speed of the flame decrease. The flame tip will move away from the surface of the liquid phases. Suppose that the tip has moved away a distance greater than the critical distance. In this case, behind the flame tip, at a distance from the liquid fuel less than the distance from the liquid surface to the flame tip, there will be a stoichiometric concentration of the fuel vapor. Since the distance is greater than the critical one, the flame will be able to move to this region slightly behind but closer to the liquid fuel.

As a result, with an increase in the oxygen content, the flame approaches the fuel film and the gas-phase temperature gradient increases toward the fuel and the substrate due to the approach to the surface as well as due to an increase in the flame temperature. This leads to an increase in the heat flux into the condensed phase, which, in turn, should increase the heating of the fuel film and substrate and enhance fuel vaporization. The enhancement of fuel vaporization should increase fuel diffusion in the direction of flame spread and the flame speed. In addition, when the oxygen concentration is increased, a higher concentration of fuel vapor is required to obtain a stoichiometric concentration of the fuel. This process should somewhat slow down the increase in the flame speed. In general, an increase in oxygen concentration leads to an increase in flame speed.

For combustion of a substance of the form C\(_{n}\)H\(_{m}\)O\(_{k}\) using as oxidizer a mixture of \(b\) volumetric parts of oxygen with \(1-b\) parts of nitrogen, the equation of the chemical reaction is written as

$$\mathrm{C}_n\mathrm{H}_{m}\mathrm{O}_{k} +\Big(n+\frac{m}{4}-\frac{k}{2}\Big) \Big[\mathrm{O}_2+\frac{1-b}{b}\,\mathrm{N}_2\Big]{}$$
$${}\to n\mathrm{CO}_2+ \Big(\frac{m}{2}-k\Big) \mathrm{H}_2\mathrm{O}+\Big(n+\frac{m}{4}-\frac{k}{2}\Big) \,\frac{1-b}{b}\,\mathrm{N}_2.$$

In this case, the expression for stoichiometric fuel concentrations in volume fractions \(c_{st}\) can be written as

$$c_{st} = \frac{1}{1 +(n + m/4 - k/2)(1 + (1 - b)/b)}{}$$
$${}= \frac{b}{b + n + m/4 -k/2};$$

for alcohols (\(m = 2n + 2\) and \(k = 1\)), it can be written as

$$c_{st} = b / (b + 1.5n),$$

and for saturated hydrocarbons (\(m = 2n + 2\) and\(k = 0\)), as

$$c_{st} = \frac{1}{1 +(n + m/4 -k/2) (1 + (1 - b)/b)}{}$$
$${}=\frac{b}{b + (3n + 1)/2}.$$

It can be seen that the volume concentration of hydrocarbon in a stoichiometric mixture increases with increasing oxygen concentration.

It follows from Figs. 2 and 3 that neither in the region of low speeds nor in the region of high speeds does the general dependence of the flame speed on the volume concentration of oxygen exist.

Fig. 6
figure 6

Flame speed versus volume concentration of fuel:n-butanol (1),n-pentanol (2),n-decane (3),n-undecane (4), andn-dodecane (5); point 6 refers to the data of [30] for fuels1–3 and 5.

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Fig. 7
figure 7

Flame speed versus mass concentration of fuel (a) and versus the thermal effect of a unit volume of a stoichiometric mixture (b): (1) n-butanol; (2) n-pentanol; (3) n-decane; (4) n-undecane; (5) n-dodecane.

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Figure 6 shows the dependence of the flame speed on a glass substrate on the stoichiometric volume concentration of fuel \(c_{st}\) at high flame speeds. The points in the figure form two groups. In both groups, the dependences are close to linear. Moreover, unsaturated (and heavy in this case) hydrocarbons have high speeds at lower concentrations, and the dependence of speed on concentration for them is steeper than for alcohols. The scatter of experimental points within each group is relatively small. The occurrence of two groups of dependences may be due to the significantly different molecular weight as well as to the slightly different thermal effects of combustion per unit volume of the stoichiometric mixture. To eliminate the influence of the molecular weight, we plotted the dependences of the flame speed on the concentration of equilibrium fuel vapor \(\rho _{st}\) at the temperature \(T_{st}\) corresponding to this concentration and on the thermal effect of combustion of a unit volume of the mixture \(Q_{ st}\). These dependences are presented in Fig. 7. The fuel concentration was calculated from the known temperature dependences of the equilibrium vapor pressure represented as the Antoine equation [31] \(\log p = a - b/(c + t)\), and the thermal effect of unit volume \(Q_{st}\) (MJ/m3) was calculated from the molar calorific value \(q\) for each fuel known from [31]:

$$\rho _{st} = \frac{c_{st} p_\mathrm{atm} M}{RT_{st}},\qquad Q_{ st} = \frac{c_{st} p_\mathrm{atm} }{RT_{st}}\,q.$$

Here \(p_\mathrm{atm}\) is the atmospheric pressure, \(M = 14n + 2~+~16k\) is the molecular weight of hydrocarbon, and \(R\) is the universal gas constant.

Figure 7 shows the same dependences of the flame speed over films of various fuels on a substrate as in Fig. 6 but on different parameters: on the mass concentration of fuel in the stoichiometric mixture in Fig. 7a and on the thermal effect of combustion of a unit volume of the mixture in Fig. 7b. It can be seen that the correlation of the flame speed with the mass concentration of fuel or the thermal effect of a unit volume of the stoichiometric mixture brings together the dependences for different groups of fuel. It should be noted that the thermal effect of combustion of saturated hydrocarbons per unit mass of fuel is almost the same—about 44.5 MJ/kg, and for alcohols, it is as follows: 36.9 MJ/kg for n-butanol and 38.5 MJ/kg for n-pentanol. Therefore, for unsaturated hydrocarbons, Fig. 7b is just a change in scale along the abscissa in Fig. 7a. However, for alcohols there was a divergence of the points that were initially on the same straight line in Fig. 7a. Obviously, this is due to the difference in thermal effect per unit volume of the mixture. Although the thermal effect of n-butanol is lower than that of n-pentanol, the speed curve is shifted toward higher speeds at lower thermal effect of combustion of the stoichiometric mixture. This may be due to the lower heat loss from the flame tip for heating the fuel to a temperature that provides a stoichiometric concentration of fuel vapor. It is seen that at similar thermal effects, the speed is the higher the lower the hydrocarbon number and the lower \(T_{st}\). Thus, an increase in \(T_{st}\) significantly reduces the flame speed, which indicates an increase in heat consumption for fuel heating, which increases with increasing \(T_{st}-T_{0}\).

Consider what changes during flame spread if the oxygen concentration is the same, but the molecular weight of the fuel increases. If the oxygen concentration is the same, the normal speed of a stoichiometric mixture and the temperature of combustion products will be similar for different hydrocarbons. Therefore, the heat flux from the flame tip into the condensed phase will not depend on the hydrocarbon number. The hydrocarbon number determines the temperature at which a stoichiometric fuel vapor-to-oxidizer ratio is obtained. The higher the number (the higher the molecular weight) of hydrocarbon, the higher this temperature. Heating of liquid fuel requires more heat. At constant heat flux, this implies an increase in the heating time and hence a decrease in the flame speed. It should also be noted that the amount of heat for vaporization required to obtain a stoichiometric mixture decreases somewhat with increasing hydrocarbon number. Therefore, a decrease in speed with an increase in the hydrocarbon number indicates that this decrease is due not to fuel vaporization but to heat losses for heating the liquid fuel–substrate system. At high flame speeds, heat losses are significant only into the liquid phase. This is indicated by the fact that the flame speed is independent of the thermal conductivity of the substrate even for thin fuel layers (10 \(\mu\)m), and is also independent of the fuel film thickness. For films 10 \(\mu\)m and 1 mm thick [24], the flame speed over corresponding fuels is the same. This gives confidence that this is also true for larger thicknesses of the fuel layer. Thus, data on the flame speed obtained on thin fuel layers (in a given range of speeds) will also be representative for larger thicknesses of fuel layers, and the experimental procedure is much simpler.

CONCLUSIONS

• It has been shown experimentally that the speed of flame spread over fuel on a substrate increases with an increase in oxygen concentration. The flame speed increases faster than the normal speed of the corresponding homogeneous stoichiometric mixture.

• As the flame speed increases, the effect of heat loss to the condensed phase decreases.

• For equal thermal effects of combustion of a unit volume of the stoichiometric mixture, the flame speed over a fuel film is the lower the higher the value of \(T_{st}-T_0\).

• For flame speeds above 0.3 m/s, i.e., for the cases where the speed weakly depends on the substrate properties, the flame speed depends linearly on both the volumetric stoichiometric concentration of the fuel of a given homologous series and the calorific value of a unit volume of the stoichiometric mixture.

• The speed of flame spread over a single-component fuel is determined by the following parameters: chemical formula, dependence of the fuel vapor pressure on temperature, thermal conductivity, heat capacity, density, heat of combustion of a unit volume of the stoichiometric mixture, and the difference \(T_{st}-T_0\).