Flame speed enhancement of solid nitrocellulose monopropellant coupled with graphite at microscales

Improvement and control of the combustion wave propagation velocities of solid monopropellants are a crucial step in realizing improved performance in solid-propellant micro-thrusters.¹ Micro-propulsion systems are designed to provide small amounts of thrust ranging from micro-newtons to a few milli-newtons that are used for precise altitude and orbit corrections, drag compensations, and small impulse maneuvers.² A variety of micro-propulsion systems have been proposed: cold gas thruster,^3,4 bi/mono-propellant thruster,^5–9 colloid thruster,^10–13 vaporizing liquid thruster,¹⁴ micro ion thruster,¹⁵ pulsed plasma thruster,¹⁶ Hall effect thruster,^17–20 laser plasma thruster,²¹ field-emission electric propulsion thruster,^22–25 and solid-propellant thruster. The solid-propellant thruster concept is based on burning a monopropellant stored in the micro-combustion chamber. The combustion products are then accelerated in the nozzle giving the required thrust.²⁶ The solid-propellant microthrusters offer several advantages over other types of micro-thrusters, e.g., less possibility of fuel leakage, better system miniaturization due to absence of valves, fuel lines or pumps, no moving parts, and high specific impulse.

In addition to their use in micro-thrusters, microcombustion systems have also been proposed to achieve highly efficient thermal-electric energy conversion. Thermoelectric (TE) systems directly convert thermal energy into electrical energy by manipulating the flow of charge carriers through electrically conducting materials.²⁷ The TE systems have a variety of applications including civilian applications such as air-conditioning systems and also defense related applications such as night-vision systems for maneuverable aircraft.²⁷ Choi et al.²⁸ were the first to report flame speed enhancements in a solid monopropellant by coupling TNA (trinitramine) with MWCNTs (multi-wall carbon nanotubes) at nanoscales. The flame speed enhancements up to 10⁴ times the bulk value were reported. The TNA-MWCNT systems can generate very high specific power (7 kW/kg). Furthermore, a greater thermoelectric performance (150 mV vs 60 mV peak voltage) was obtained by Walia et al.^29–32 in which the carbon-nanotubes were replaced by thermally conductive oxides (alumina) and thermoelectric materials (Bi₂Te₃, Sb₂Te₃, MnO₂, and ZnO) and nitrocellulose (NC) was used as the solid fuel.

The previously mentioned studies^28–32 focused on thermal-to-electrical power generation, while the enhanced flame propagation speeds were not necessarily of primary interest, but a result of the use of thermally conductive thermoelectric materials. The purpose of the present paper is to understand the coupling between the chemical reactions (combustion) and the heat transport (both within the propellant and between the propellant and the conductive substrate), as well as its effect on the flame speeds and the oscillatory nature of the combustion waves. The burning rate of solid propellants is controlled by the heat transport (from burned to unburned material) and thus depends on the thermal conductivity of the propellants.^33,34 Most solid propellants have a thermal conductivity in the range of 0.1−1 W/m-K,³⁵ which is much lower than that of metals (a few hundred W/m-K) or carbon-based materials (as high as a few thousand W/m-K). In this study, graphite sheets were chosen as the thermally conductive base for the propellants because of their high thermal conductivity and good thermal and phase stability properties at high temperatures.³⁶ Additionally, they are also readily available and have lower manufacturing costs than carbon nanotubes. Lastly, a major drawback associated with solid-propellant micro-thrusters is their inability to provide variable thrust. However, the combustion wave propagation speed is shown to be dependent on the ratio of the fuel to the graphite thickness. Thus, by varying the fuel-to-graphite thickness, different flame speeds can be obtained within the same system.

The flame speed enhancement of nitrocellulose, when coupled to highly conductive graphite sheets, was studied in this work. The average flame speeds were measured as a function of the ratio of the fuel to the graphite layer thickness. An optimum thickness ratio was found corresponding to maximum enhancement. Additionally, a theoretical model, which couples 1-D energy conservation equations along with a one-step chemical reaction, was developed to better understand the flame speed enhancements and the oscillatory nature of the combustion waves. The effect of the non-dimensional inverse adiabatic temperature rise (⁠$β)$⁠, the ratio of the thermal diffusivity of the graphite to that of the fuel (α₀), and the ratio of the fuel to the graphite thickness (R) on the flame speeds as well as on the period and the amplitude of the oscillatory combustion waves were determined and compared to the experimental observations. Lastly, the mechanisms for the oscillatory combustion waves were discussed.

For the solid fuel, NC, C₆H₈(NO₂)₂O₅, was selected because of its high enthalpy of combustion and wide use as an energetic polymeric binder in nitrocellulose-based propellants in solid rocket motors. However, NC is difficult to ignite at room temperature, and a primary igniter is usually required. Sodium azide (NaN₃), which has a much lower activation energy (40 kJ/mol)³⁰ as compared to NC (110–150 kJ/mol),³⁰ was used as a primary igniter to achieve ignition of NC at room temperature through the use of a low-power ignition source (a resistive heating wire). For the thermally conductive base, the graphite sheets were chosen for the reasons mentioned earlier.

Millipore NC filters with 0.21 g/filter (N8645, Sigma Aldrich) were dissolved in acetonitrile, CH₃CN (260457-1L, Sigma Aldrich) at a concentration of 6% NC by weight. The solution was then drop-casted onto a graphite sheet (1168-1755-ND, Digi-key), which rested on the top of a thermally insulating glass slide. The acetonitrile was evaporated at room temperature and pressure leaving an adhesive coating of nitrocellulose behind. After the acetonitrile had been evaporated, the NaN₃ salt (S2002-5G, Sigma Aldrich) was dissolved in deionized water and then drop-casted onto the NC fuel layer. Given the low-power nature of the nichrome ignition wire, relative to other reported means of ignition,^28–32 higher concentrations of the NaN₃ solution (up to 10% by weight) were needed to achieve successful ignition. The sample was then left to dry for approximately 24 h. The nitrocellulose membranes were composed of 90% nitrocellulose (10.9–11.2 wt. % nitrogen) and 10% cellulose acetate. As observed in the experiments, these small quantities of cellulose acetate did not affect the burning behavior of the NC fuel. A strip of graphite 20 μm thick, with a length of 2.5 cm, and a width of 0.5 cm was used as the substrate. The fuel layers ranging from 20 μm to 170 μm were deposited on top of the graphite sheet.

A high-speed video camera (Phantom v7.3) and an infrared camera (FLIR-SC6100) were used to capture the ignition and flame propagation process. Fig. 1 shows the spatial plots of the luma profiles at different times. An algorithm was developed to track the brightest peak within the flame. These peaks determine the rate at which the reaction zone travels across the fuel layer. Such a method explicitly assumes that the hottest portions of the flame are very near the reaction zone. The uncertainties of the spatial and the time measurements were estimated to be less than 2% and 6%, respectively. Thus, the uncertainty in the flame speed measurements was approximately 6.5%.

The ignition of the prepared samples was achieved by a resistive heating nichrome wire, which was placed in a transverse manner at one end of the sample, as shown in Fig. 2. The ends of the nichrome wire were connected to a power source that supplied 10 A of current at 5 V. Such a setting was deemed sufficient to ignite the NC/NaN₃ layer consistently. The potential difference applied by the nichrome wire causes the NaN₃ crystals to react violently and produce visible sparks. Note that the electrical activation of the NaN₃ crystals occurs almost immediately upon the application of the potential difference; well before the heating of the nichrome wire. Alternatively, if the potential difference was not able to ignite the NaN₃ crystals, then the heat produced from the Joule heating of the nichrome wire was able to ignite the NaN₃ crystals. In both cases, the ignition of the NaN₃ crystals produces enough energy to initiate the combustion of the NC fuel layer.

Given that the length of the samples is at least five times the width (both of which are much larger than the thickness of the samples) and that the flame fronts obtained are almost planar, it is reasonable to approximate the NC-graphite system by a one-dimensional model. The 1-D model presented below was first developed by Mercer et al.^37,38 to study flame propagation in solid fuel combustion with heat loss to the ambient, which was then further modified by Choi et al.²⁸ to include the heat coupling between a solid fuel and a thermally conductive base. Assuming that the fuel undergoes complete combustion and that the reaction rate is governed by first-order Arrhenius kinetics, the mass balance equation is given by

In the above equation, R_u is the universal gas constant (8.314 J K⁻¹ mol⁻¹), ρ_f is the fuel density, A is the Arrhenius constant, E is the fuel activation energy, and w is the mass fraction of the unburned fuel. The graphite and the fuel energy conservation equations (Eq. (2)) are coupled through an interface conductance term (G_o), which when divided by the thickness of the fuel (graphite) layer gives the coefficient of heat loss (gain) by the fuel (graphite)

In the above equations, ρ_g is the graphite density, c_pf is the fuel specific heat capacity, c_pg is the graphite specific heat capacity, d_f is the fuel thickness, d_g is the graphite base thickness, k_f is the fuel thermal conductivity, k_g is the graphite thermal conductivity, and Q is the heat of combustion. Table I lists the values of the parameters used in Eqs. (1) and (2). Temperature measurements, using the FLIR-SC6100, showed that the maximum temperature of the graphite sheets during the combustion wave propagation was 940 K. Thus, for all of the above listed properties, average values over the temperature range of 300–940 K were used. Moreover, the Arrhenius constant was varied to match the bulk NC speed. The fuel thickness effect is taken into account through the interface conductance term (G_o/d_f), which determines the amount of heat loss from the fuel.

Following Choi et al.,²⁸ Eqs. (1) and (2) were non-dimensionalised to identify the important non-dimensional parameters that govern the nature of the solutions. The non-dimensional fuel temperature, graphite temperature, time, and distance are given by

Substituting these non-dimensional variables into Eqs. (1) and (2) gives the non-dimensional equations

These equations were numerically solved using the COMSOL Multiphysics^® software package using a time-adaptive technique. The absolute convergence for the non-dimensionalised temperatures and the mass fraction was set to 10⁻⁴ and 10⁻⁶, respectively, with the non-dimensional time steps ranging from 100 to 150 and the non-dimensional mesh size being 5 (40 000 grid points). The total non-dimensional simulation time and domain size was 1 000 000 and 200 000, respectively. The four important non-dimensional parameters on which the solution depends can be identified as:$β=(cpfE)/(QRu$⁠), $α0=αg/αf$⁠, $γf=(Goβ)/(dfρfcpfA$⁠), and $γg=(Goβ)/(dgρgcpgA).$ $αg$ and $αf$ are the graphite and the fuel thermal diffusivity, respectively. The effect of the absolute values of $γf$ and $γg$ on the flame propagation is negligible for values greater than 10⁻³ (corresponding to G_o > 10⁷W/m² K), similar to what was observed by Choi et al.²⁸ However, their relative value $γfγg=ρgcpgρfcpfR$ has a strong affect on the nature of the solutions observed. This non-dimensional ratio depends only on the ratio of the fuel to the graphite thickness (with ρc_p for both fuel and graphite being kept constant), and thus it is through this parameter that the effect of R is incorporated. A parametric study was performed to show the effect of $β$⁠, $α0$⁠, and R on the flame propagation speed and the oscillatory nature of the combustion waves.

Scanning electron microscopy (SEM) was used to examine the deposition thickness variation, uniformity, surface features, as well as how the nitrocellulose layer was adhered to the graphite sheet. Such information is vital to understand the observed combustion behavior of the NC/graphite system. Fig. 3(a) shows the top view of the deposited fuel on the graphite sheet. The NC fuel layer is quite smooth, which is consistent with the adhesive nature of the NC coating. However, the thickness and the adhesion of the deposited fuel vary slightly along the length of the sample, as can be seen in Fig. 3(b).

To vary the ratio of the fuel-to-graphite thickness, multiple layers (2–8) of NC ranging from 20 to 170 μm were deposited onto the 20-μm graphite base. Fig. 4 shows the total average thickness of the fuel sample as a function of the number of fuel layers drop-casted. Around 20% variation in the deposited thicknesses is observed for all the cases. In some cases, air or cellulose acetate was observed to be trapped between the multiple NC layers due to the drop-casting method used. This may lead to an increase in the contact resistance and thus decreases the effect of the high thermal conductivity of graphite. We will discuss this later.

For solid propellants, the flame does not propagate at a steady speed but has an oscillatory profile associated with it. Mercer et al.,^37,38 based on the 1-D modeling, proposed that there are three types of combustion regimes (type I, type II, and type III) possible depending on the particular values of β and the amount of heat loss present during the flame propagation. A type I combustion wave is obtained when β < 5 (irrespective of the heat loss value) or when 5 < β < 6.5 (with low values of heat loss) and is characterized by the appearance of a steady combustion wave (constant speeds) with no oscillations. For intermediate values of heat loss and for 5 < β < 6.5, a type II combustion wave is obtained, whereas a type III combustion wave is obtained for high values of heat loss. A type II wave is characterized by the appearance of peaks of similar magnitudes, whereas a type III combustion wave is characterized by the appearance of peaks of varying magnitudes. For β > 6.5, a type II is obtained if the heat loss is low, and a type III wave is obtained for intermediate to high values of heat loss. Similar types of combustion regimes were identified in this study with β, α₀, and R as the governing parameters (Figs. 8, 10, and 11). This will be discussed in detail in Sec. III C.

Fig. 5 shows the measured flame speed profiles as a function of distance travelled from the ignition wire, for the fuel-to-graphite thickness ratio of R = 3.5. A type III combustion regime is obtained with an average period of 0.068 s (for the present experiments involving NC and graphite, the values of β and α₀ are 10.36 and 5300, respectively). The time period for a type III wave is defined as the time between two successive peaks. Oscillatory velocity profiles with similar time periods were also reported by Walia et al.²⁹ 

Fig. 6 shows the nitrocellulose average flame speeds as a function of the fuel thickness for both the coupled (using the graphite sheets) and the uncoupled case. The speeds were calculated by averaging the instantaneous flame speeds, as shown in Fig. 5. The graphite thickness was kept constant at 20 μm. For the coupled case, an optimal fuel thickness was obtained for a thickness around 120 μm (R = 6.0) for which the average flame speed was 7.68 cm/s (around 14 times the bulk speed of 0.54 cm/s). For a given number of fuel layers drop-casted (i.e., for each R), some variations in the average flame speeds were observed that could be attributed to the non-uniform deposition or the separation of the fuel layer from the graphite base, which was observed in the SEM images. In contrast, for the uncoupled case, a monotonic decrease in the average flame speed with the fuel thickness was observed, where the flame speed approached the bulk flame speed of NC of 0.54 cm/s at a thickness of 350 μm and higher. The bulk flame speed value is within the range of pure NC burning rates as reported by Zhang.⁴² At low fuel thicknesses, however, deviations from the bulk flame speeds were observed. Further work is needed to better understand the effect of the microscale fuel thickness on the flame propagation speed.

In the following, we will discuss the effect of three parameters (β, α₀, and R) on both the average flame speeds and the oscillations of the velocity profiles. Fig. 7 shows the measured and predicted average flame speeds as a function of R. For the 1-D modeling, the flame speed is calculated by tracking the reaction front, which is assumed to be located at the point at which the extent of reaction (w) is 0.5. The agreement between the 1-D calculations and the experimental data is good for thin-layered depositions (R < 7). The calculated optimum thickness is around R = 5.5, corresponding to 110 μm, which agrees well with the experimentally obtained optimum thickness of 120 μm. However, for thicker depositions (R > 7, 140 μm), a significant deviation is observed. Such a deviation is likely attributed to the deposition method used. Since the deposition was performed in multiple layers, the additional contact resistance between each deposited layer decreases the net thermal conductivity of the sample in the perpendicular direction. In some cases, air was observed between the layers (using the SEM images), which further increases the contact resistance and thus decreases the net thermal conductivity of the sample. As a result, the measured flame speeds are lower than the theoretical values for thicker depositions.

Fig. 8 shows the predicted flame speed profiles as a function of distance travelled for various values of R. Obviously, the peak and the period of the oscillations depend on R. For R = 2.0 and for a distance travelled less than 2.5 cm, only a single peak is obtained signifying the period of oscillation being >0.1 s. Similar observations were made in the experiments where, for R < 2, only a single peak was obtained. For R = 3.5, a type III combustion wave results, which has peaks of 35 cm/s and 15 cm/s. The period also varies along the propagation direction. Two different periods are obtained with the values being 0.057 s and 0.102 s. As the thickness ratio R is increased above 3.5, the two peak magnitudes and the period values approach each other. At R = 6.5, a type II combustion wave is obtained with a peak flame speed of 22 cm/s and a period of 0.035 s. A qualitative comparison of the oscillation characteristics can be made with the experimental observations. The oscillation periods obtained from the 1-D calculations agree well, within the order of magnitude, with the experimental oscillation period of 0.05 s averaged over all the samples ignited for R = 3.5, 6.5, and 8.5.

Fig. 9 shows the measured and predicted peak flame speeds as a function of R. The 1-D modeling agrees well with the experimental data, and a similar periodic trend is observed. For R = 3.5, a peak flame speed of 32 cm/s was measured, which agrees well with the predicted peak flame speed of 35 cm/s. In contrast to Fig. 7 for the average flame speeds, the optimum thickness range is shifted towards R = 3.5 instead of R = 6.5. This is due to the increased heat loss to the graphite as the fuel thickness is decreased, leading to a transition from a type II wave to a type III wave. Thus, although the average flame speed is lower for R = 3.5, there are instances during the wave propagation when the heat coupling between the graphite and the fuel is such that, at some locations, the instantaneous flame speed is higher.

The effect of β (the non-dimensional inverse adiabatic temperature rise) on the oscillations of the velocity profiles is shown in Fig. 10. The value of α₀ was kept fixed at 5300 corresponding to the present experimental conditions. As β is increased, the average flame speed decreases but the amplitude and the period of the oscillation increase, similar to the trend observed by Abrahamson⁴³ (the amplitude is defined as the difference between the peak and the average flame speed). Since β is proportional to $cpf/Q,$ an increase in β implies either a decrease in the heat of combustion or an increase in the heat capacity of the fuel, both of which decrease the adiabatic flame temperature and thus reduce the average flame speeds. For β = 5, negligible oscillations are observed, and a type I combustion wave is obtained, which is consistent with the results of Mercer.^37,38 As β is increased, a transition from type I to type II combustion wave occurs, and at β = 9, the sinusoidal oscillations are obtained. Further increases in β result in a transition to a type III combustion wave.

The effect of α₀ (the ratio of the thermal diffusivity of the graphite to that of the fuel) on the oscillations of the velocity profiles is shown in Fig. 11. The value of β was kept fixed at 10.36 corresponding to the NC fuel. As α₀ is increased, the net thermal conductivity of the sample in the propagation direction increases, which results in flame speed enhancement. Moreover, since β is kept constant at 10.36, for very low values of α₀, a type II combustion wave is obtained, but for higher values of α₀ a type III combustion wave is obtained. The oscillation period, which can be seen in Fig. 11, varies very slightly with α₀. This is in contrast to the effect of β (Fig. 10) where the oscillation period increases by almost 10³ times as β is increased from 5 to 10.36. Thus, the period can be regarded as a property of the fuel depending only β and not on α_0.

The oscillation behavior observed in the experiments and predicted by the 1-D modeling can be explained using the excess enthalpy concept, which was first proposed by Shkadinskii⁴⁴ for pure solid propellant combustion. Here, the effect of adding a thermally conductive layer on the excess enthalpy was considered. The amount of total excess enthalpy available per unit surface area of the reaction front can be calculated as follows:⁴⁴ 

In the above equation, h₁ and h₂ are the specific enthalpies of the unreacted materials and the combustion products, respectively, T_i is the initial temperature (300 K), and L is length of the sample. Fig. 12 shows that the total amount of excess enthalpy available for the coupled fuel/graphite system is much greater than that available for the pure NC fuel case. This is due to the thermal coupling between the fuel and the graphite layer, which increases the net thermal conductivity (thermal transport) in the propagation direction resulting in a much wider preheating zone. Thus, the thermal base conducts the heat released by the exothermic reaction from the burned portions of the fuel to the unburned portions of the fuel over a greater distance (100 times more than the pure fuel case), which results in the flame speed enhancement. Moreover, Fig. 12 also shows that the reaction zone, as represented by ∂w/∂t, is much wider for the coupled system compared to the pure fuel case.

The local amount of excess enthalpy available in each system (both the pure fuel and the coupled fuel/graphite case) varies as the flame propagates, thus giving rise to the oscillations as will be explained next. The curves corresponding to ∂w/∂t and (1 − w) show a sharp variation and are much narrower as compared to the heated layer, which is characteristic of gasless combustion.⁴⁴ The thickness of the heated layer varies as k_eff/(cp_fb),⁴⁴ where b is the instantaneous flame speed, and k_eff is the net effective thermal conductivity of the sample (k_f < k_eff < k_g) that governs the amount of heat conducted to the unreacted portions of the fuel. Thus, for a given system, a steeper temperature profile results in a higher flame speed. Moreover, the greater the thermal conductivity of the sample, the greater will be the available excess enthalpy. Initially, when the flame speed is low (e.g., due to insufficient preheating), corresponding to lowest flame speeds in Fig. 8 (R = 3.5), significant excess enthalpy is available, which heats the unreacted portions of the fuel. Due to the heat flow from the reaction zone to the unreacted material, a decrease in the reaction temperature occurs, which further lowers the flame speed. This loss in heat is somewhat compensated by the heat flow from the hot combustion products, and thus a plateau in the flame speed is observed. The heating of the unreacted material continues until the temperature of the unreacted material reaches a point at which rapid ignition of the fuel occurs and a significant enhancement in the flame speed is achieved, thus the reason for the appearance of the first ignition peak in the velocity profile. The reaction temperature also increases and becomes greater than the average reaction temperature. This rise in temperature is then halted due to the heat flow from the reaction zone to both the combustion products and the unreacted material. Thus, a decreasing trend in the flame speed is observed thereafter. Again, the flame speed decreases until the temperature of the unreacted materials is such that a rapid ignition occurs, thus giving rise to the second ignition peak (Fig. 8, R = 3.5). The strength of the peaks depends on the maximum temperature to which the unreacted materials are heated before a rapid ignition occurs, which in turn depends on the locally available excess enthalpy.

Fig. 8 shows that, for R = 3.5, the oscillation period (0.057 s) between the first ignition peak (marked by a) and the second ignition peak (marked by b) is less than the oscillation period (0.102 s) between the second ignition peak (marked by b) and the first ignition peak of the next cycle (marked by a′). Thus, after the onset of the first ignition, the fuel is heated for a lesser time as compared to that after the second ignition, which results in the magnitude of the second ignition peak to be less than that of the first peak. The second ignition marks the end of one cycle after which the pattern is repeated giving rise to multiple peaks. The number of peaks obtained depends on the total heat reserve available when the first ignition occurred and the higher the heat reserve or the local excess enthalpy available initially, the greater will be the number of peaks observed within one whole period, which is consistent with the conclusions of Shkadinskii.⁴⁴ 

The flame speed enhancement in a solid NC monopropellant is shown to occur when NC is coupled to a highly conductive graphite sheet. The thickness of the graphite sheet was kept fixed at 20 μm, while the thickness of the deposited NC fuel varied from 20 to 170 μm. The high thermal conductivity of the graphite facilitates heat transfer from the reaction zone to the unburned portions of the fuel, which sustains the propagating exothermic reaction front. The flame speed enhancement depends on the fuel-to-graphite thickness ratio. An optimum ratio for the highest flame speed was found to be around 6 for which flame speed enhancements up to 14 times the bulk value were obtained. Moreover, the reaction front does not propagate in a uniform manner but has an oscillatory structure associated with it.

A simple 1-D model that couples the energy conservation equations along with one-step chemistry was developed to predict the flame speeds and to explain the nature of the oscillations observed in the velocity profiles. The predicted flame speeds and the optimum fuel-to-graphite thickness ratio are in good agreement with the experimental data. The model also reveals how the amplitude and the temporal period of the oscillations depend on the fuel-to-graphite thickness ratio, and a similar trend as that observed in the experiments was obtained. Adding a highly conductive graphite base enhances the thermal transport between the burned and the unburned portions of the fuel and results in much wider reaction and preheat zones. Lastly, the major cause for the emergence of the locally oscillating combustion waves was identified and is attributed to the variation of the local excess enthalpy as the flame front propagates. The absolute value of the total amount of excess enthalpy available determines the magnitude (amplitude) of the oscillations, whereas the variation of the local excess enthalpy within each system governs the frequency of those oscillations.

This work was supported by the Air Force Office of Scientific Research (AFOSR).