Analysis of the microscopic morphology of CCPsA collecting plate is placed in the combustion chamber to collect CCPs, and then S-4800 SEM and EDS are used to analyze the microstructure of the CCPs. According to the differences in the morphology of CCPs, they can be divided into the following categories:
(1)
Spherical agglomerates, as shown in Fig. 3. The agglomerate exhibit good spherical shape with a particle size of approximately 380 μm, and is widely distributed in the CCPs of containing aluminum propellant35. They are typical products formed by the agglomeration of aluminum particles. Analysis of the agglomerates using an Energy Dispersive X-ray Spectrometer (EDS) reveals that the interior of the agglomerates mainly consists of aluminum, oxygen, nitrogen, and carbon, with mass fractions of 50.29%, 24.90%, 16.68%, and 5.62% respectively. It can be observed that the surface of the agglomerates is primarily composed of aluminum oxide.Figure 3SEM image of a spherical agglomerate.
(2)
The broken aggregates, as shown in Fig. 4. During the agglomeration process of aluminum particles, liquid aluminum gradually merges together to form molten droplets, and a hard aluminum oxide shell is formed on the surface of the droplets. With the heating from the diffusion flame, the expansion coefficient of liquid aluminum is larger than that of the aluminum oxide shell, causing the liquid aluminum to expand when heated and leading to the rupture of the aluminum oxide shell36. As shown in Fig. 4, the internal structure of the broken agglomerate exhibits irregular patterns. Using an Energy Dispersive X-ray Spectrometer (EDS) to analyze the composition of the agglomerate at different positions, shown in Table 2. The elemental compositions of spectra picture 1 and spectra picture 4 are similar, with the ratio of aluminum to oxygen being close to each other, totaling about 70%. The nitrogen and carbon content is relatively low, indicating that position 1 and position 4 are mainly composed of aluminum oxide. On the other hand, the aluminum content in position 2 and position 3 exceeds 50%, with oxygen content around 30%, suggesting that position 2 and position 3 are primarily composed of aluminum, thereby demonstrating that the interior of the aluminum agglomerate is mainly made up of liquid aluminum.Figure 4SEM image of broken agglomerate.Table 2 composition of CCPs.
(3)
Connected aluminum aggregates, as shown in Fig. 5. The diameter of the left agglomerate is approximately 225 μm, while the diameter of the right agglomerate is approximately 256 μm. The formation of these agglomerates may have occurred when the temperature of the burning flame decreased before they completed their fusion, causing the aluminum oxide shells between them to not melt due to insufficient heat. As a result, the agglomeration process was interrupted, and after cooling down, these types of agglomerates were formed.Figure 5SEM image of two agglomerates connected.
(4)
Aluminum oxide particles,as shown in Fig. 6. The combustion products of aluminum oxide particles are dispersed at varying heights, have smooth surfaces, typically appear in clusters, and exhibit relatively complete spherical shapes with sizes not exceeding 2 μm. During the agglomeration process of aluminum particles, micron-sized aluminum particles first vaporize and evaporate into aluminum vapor under the heating of the diffusion flame, which then reacts with oxygen in the environment. Some of the oxidized products of aluminum vapor condense on the surface of aluminum droplets to form oxide cap structures, while the rest of the aluminum vapor condenses directly in the environment to form submicron or even nanoscale aluminum oxide particles37,38.Figure 6SEM image of alumina particle.
Microstructure of condensed combustion products of different formulations propellantIn order to investigate the microstructure of condensed combustion products from different formulations of NEPE propellants, electron scanning microscopy was used to observe the solid combustion products from four NEPE propellant formulations (JF-1, JF-2, JF-3, and JF-4) at 3 MPa and 7 MPa, as shown in Figs. 7, 8, 9,10. It can be seen that the condensed combustion products of these four NEPE propellant formulations are mainly composed of spherical aluminum agglomerates and irregularly shaped carbon agglomerates. At 3 MPa, the spherical aluminum agglomerates in the condensed combustion products of these four NEPE propellant formulations have larger sizes, and the agglomerates’ shells exhibit irregular spherical structures. Additionally, more aluminum agglomerates with broken shells can be observed in the condensed combustion products. At 7 MPa, the size of the spherical aluminum agglomerates in the condensed combustion products decreases compared to 3 MPa. The agglomerates’ shells become smoother, and the spherical shape becomes more perfect. In the condensed combustion products at 7 MPa for JF-3 and JF-4, a large number of carbon agglomerates were found. This may be due to the increased particle size of AP, which leads to uneven heat transfer within the propellant composition and incomplete combustion of components with high carbon content such as binders and plasticizers. Moreover, the excessive burning rate in the high-pressure environment could cause insufficient combustion of the binder and plasticizer. However, due to the limitations of scanning electron microscope images, it is difficult to determine the influence of propellant formulations on the size of condensed combustion products based solely on the microstructure of the four NEPE propellant formulations. Further quantitative analysis of the particle size of condensed combustion products is needed.Figure 7SEM images of the condensed combustion products of JF-1 propellant.Figure 8SEM images of the condensed combustion products of JF-2 propellant.Figure 9SEM images of the condensed combustion products of JF-3 propellant.Figure 10SEM images of the condensed combustion products of JF-4 propellant.Crystal phase composition analysis of condensate combustion productsTo investigate the combustion reaction mechanism of NEPE propellants and determine the changes in surface crystal structure of solid combustion products, X-ray diffractometry was employed to analyze the CCPs of JF-1, JF-2, JF-3, and JF-4 propellant at 3 MPa. Figure 11 shows the the diffraction patterns of the CCPs of the four propellants. Compared with the standard card, the CCPs of the four propellant samples mainly contain Al2O3 and Al. The aluminum particles burned to form α-Al2O3 and γ-Al2O3, indicating that under the current experimental conditions, the aluminum particles did not burn completely.Figure 11XRD patterns of the CCPs of NEPE propellants with different formulations.According to Fig. 11, it can be observed that as the particle size of Al increases, the intensity of the Al diffraction peak also increases. This indicates a decrease in the relative content of Al2O3 in the solid combustion products with an increase in the particle size of Al in the samples. Finer Al particles have higher reactivity, allowing them to generate a higher concentration of oxidizing gas in a given time and promote the combustion of Al particles. Furthermore, with an increase in the particle size of AP, the intensity of the Al diffraction peak gradually increases, while the intensity of the Al2O3 diffraction peak slightly decreases. This suggests an increase in the relative content of Al in the CCPs with an increase in AP particle size. This is because smaller AP particles have higher reactivity, resulting in a lower concentration of oxidizing gas generated in a given time, thereby suppressing the combustion of Al particles.Particle size analysis of the CCPsIn order to study the particle size distribution of the CCPs from different formulations of NEPE propellants, the CCPs were collected by combustion experiments at 1 MPa and 3 MPa for JF-1, JF-2, JF-3, and JF-4 propellant. The particle size distribution of the CCPs was analyzed using a laser particle analyzer within a range of 0.01–1000 μm.Due to the fact that the initial particle size of aluminum in propellants is not a fixed value but rather a range of particle sizes, it can lead to situations where the diameter of individual aluminum particles is similar to or the same as the diameter of small-sized initial aggregates. This makes it difficult to distinguish whether the particles are unaggregated aluminum particles or small-sized initial aggregates31. Therefore, it is necessary to define a cutoff diameter for aggregates, denoted as Dcut. Only when the particle diameter on the burning surface of the propellant is larger than Dcut, it will be considered as an aggregate. Particles with diameters smaller than Dcut are regarded as individual aluminum particles that have not undergone the aggregation process. The studies conducted by Sambamurthi7, Jackson31, and Gallier33, all utilized the parameter (Dcut) for aggregates to differentiate whether the particles on the burning surface are aggregates or not. The cutoff diameter for aggregates is related to the initial average particle size of aluminum in the propellant12.The particle size distribution of the at 1 MPa of JF-1 propellant is shown in Fig. 12, and the particle size measurement results are presented in Table 3. From Fig. 8, the particle size distribution of all particles in the solid combustion products can be observed, including the unaggregated aluminum particles on the burning surface of the propellant (particles with a diameter smaller than Dcut). The entire particle size distribution of the CCPs from the JF-1 propellant follows a log-normal distribution. The peak of the percentage of particle count is in the diameter range of 144–211 μm, accounting for 56.61% of the total number of particles. It is worth noting that the initial particle size of aluminum in the JF-1 propellant is 3 μm, whereas only 10% of the particles in the solid combustion products have a diameter smaller than 145 μm. This indicates that under the experimental conditions, there is significant aggregation of aluminum particles in the NEPE propellant.Figure 12Particle size distribution of CCPs at 1 MPa of JF-1 propellant.Table 3 Size distribution of the CCPs of NEPE propellants of different formulations (μm).The particle size distribution of the CCPs at 1 MPa of JF-1 propellant is shown in Fig. 13, and the particle size measurement results are shown in Table 3. For JF-1 propellant, the particle size distribution of condensed phase combustion products at 3 MPa is also roughly normal. The peak curve of particle number percentage ranges from 86.4 μm to 186 μm, and d(0.5) is 132 μm, which is 36.2% lower than that at 1 MPa. It can be seen that the agglomeration degree of JF-1 propellant during combustion at 3 MPa is lower than that at 1 MPa.Figure 13Particle size distribution of the CCPs at 3 MPa of JF-1 propellant.Figure 14 shows the particle size distribution of the CCPs of JF-2 propellant at 1 MPa, and the particle size measurement results are shown in Table 3. The particle size distribution of the condensed combustion product of JF-2 propellant under 1 MPa is similar to that of JF-1 propellant. The total number of particles of JF-2 propellant in the diameter range of 144–211 μm is 59.35%, and the peak value of particle distribution is also very close to that of JF-1 propellant, but the initial particle size of aluminum particles of SP-3 propellant is 30 μm. Therefore, it can be seen that although the particle size distribution of the CCPs of the two groups of NEPE propellants is similar, the agglomeration degree of the aluminum particles of the CCPs of JF-1 propellants is higher.Figure 14Particle size distribution of the CCPs at 1 MPa of JF-2 propellant.Figure 15 shows the particle size distribution of the CCPs at 3 MPa of JF-2 propellant, and the particle size measurement results are shown in Table 3. The peak curve of JF-2 propellant particle number percentage is located in the range of 45.6–86.4 μm, and d(0.5) is reduced by 65.6% compared with that in 1 MPa, and 47.3% compared with that in JF-1 propellant. It can be seen that when the initial particle size of Al particles in NEPE propellant increases from 3 to 30 μm, the degree of aluminum agglomeration decreases under high pressure. This may be caused by the large difference between the burning rate of the two propellants when p = 3 MPa, while the burning rate of JF-1 propellants is not much different from that of JF-2 propellants at 1 MPa, and the initial aluminum particle size of JF-2 propellants is larger, which makes it easier to form large-sized aggregates. The particle size distribution of the CCPs of the two propellants at 3 MPa is similar.Figure 15Particle size distribution of the CCPs at 3 MPa of JF-2 propellant.According to the aluminum agglomeration pocket model3, the Al particles within the propellant are located within the “pockets” formed by AP particles. When the AP particle size in the propellant remains constant, the size of the pockets formed by AP also remains constant. Therefore, as the particle size of Al decreases in the propellant, the concentration of Al within the pockets increases, leading to the formation of larger aluminum aggregates and more pronounced agglomeration phenomena. On the other hand, smaller-sized Al particles possess higher reactivity and exhibit better compatibility with AP in the propellant. Under the same experimental conditions, NEPE propellants containing smaller-sized Al particles exhibit higher thermal conductivity efficiency within their systems. This results in a stronger thermal reaction and an increased release of heat during the combustion reaction. Consequently, a greater number of Al particles undergo shell rupture due to phase transition under the effect of thermal feedback. Simultaneously, more adjacent ruptured Al particles experience agglomeration phenomena. As a result, the degree of agglomeration is higher, leading to an increase in the size of the solid combustion products after combustion.The particle size distribution of the CCPs at 1 MPa of JF-3 and JF-4 propellants is shown in Figs. 16 and 17, and the particle size measurement results are presented in Table 3. The percentage of particle count peaks for the solid combustion products of both JF-3 and JF-4 propellants follow a normal distribution. For JF-3 propellant, the peak of the percentage of particle count is in the diameter range of 11.2–21.2 μm, accounting for 76.08% of the total number of particles. Among them, 50% of the particles have a diameter smaller than 17.7 μm, which is a 91.5% decrease compared to the d(0.5) value of JF-1 propellant. For JF-4 propellant, the peak of the percentage of particle count is in the diameter range of 4.03–7.64 μm, accounting for 81.68% of the total number of particles. Among them, 50% of the particles have a diameter smaller than 5.05 μm, which is a 97.6% decrease compared to the d(0.5) value of JF-1 propellant. It’s worth noting that JF-1, JF-3, and JF-4 propellants all have an initial particle size of 3 μm for Al particles. Under the experimental conditions, as the particle size of AP particles in the NEPE propellant increases from 200–300 μm to 300–400 μm and 400–500 μm, the particle diameter distribution of the CCPs gradually decreases, and the degree of Al particle agglomeration decreases. When the AP particle size is in the range of 400–00 μm, the reduction in the peak of the particle size distribution for the is particularly significant compared to NEPE propellant containing 200–300 μm AP particles.Figure 16Particle size distribution of the CCPs at 1 MPa of JF-3 propellant.Figure 17Particle size distribution of the CCPs at 1 MPa of JF-4 propellant.The particle size distribution of the CCPs at 3 MPa of JF-3 propellant and JF-4 propellant is shown in Figs. 18 and 19, and the particle size measurement results are presented in Table 3. For JF-3 propellant, the peak of the percentage of particle count is in the diameter range of 4.03–7.64 μm, accounting for 81.68% of the total number of particles. This represents a 95.4% decrease compared to the d(0.5) value of JF-1 propellant. For JF-4 propellant, the peak of the percentage of particle count is in the diameter range of 3.12–5.92 μm, accounting for 79.66% of the total number of particles. This represents a 96.5% decrease compared to the d(0.5) value of JF-1 propellant. It can be observed that the degree of agglomeration for JF-3 propellant and JF-4 propellants at 3 MPa is lower compared to that at 1 MPa. Additionally, increasing the size of AP particles leads to a decrease in Al agglomeration in NEPE propellants.Figure 18Particle size distribution of the CCPs at 3 MPa of JF-3 propellant.Figure 19Particle size distribution of the CCPs at 3 MPa of JF-4 propellant.With the increase of AP particle size in the propellant, the burning rate of AP/Al, an important component of NEPE propellant, will be increased, and the stay time of aluminum particles on the propellant combustion surface will be reduced. Meanwhile, the heating time of Al particles will be shortened, so the melting degree of Al particles and the rupture of the oxide film on the surface of Al particles will be reduced. As a result, a relatively small number of Al particles adhere to each other to form smaller agglomerated particles. On the other hand, AP particles with smaller particle size have larger specific surface area and higher reactivity, so AP particles with smaller particle size can decompose and produce higher product concentration per unit time, and the increased product concentration can increase the contact area between oxidizing gas products and unburned Al particles and Al vapor, so the degree of combustion reaction becomes stronger and the heat release increases. The increase of heat release further accelerates the rupture of the Al2O3 shell on the surface of Al particles, exposing more molten Al elements in the oxidation shell of Al particles, increasing the contact chance between adjacent Al particles, and eventually forming larger sized agglomerated particles. Therefore, the increase of AP particle size in NEPE propellants will reduce the particle size of the CCPs to a certain extent.Al agglomeration modelThe combustion process of solid propellants is highly complex, and the microstructure and combustion environment have a significant impact on the combustion of Al particles. Therefore, it is challenging to accurately describe the physical processes of Al particle agglomeration and fusion between agglomerates using mathematical models. In early research, various models were proposed to describe the agglomeration behavior of Al particles in solid propellants, including empirical models12,26,27,28, pocket models29,30,31, physical models32,33,34, and random packing models24,25. Among them, the pocket model has gained more attention due to its simplicity and reasonable description of agglomerate properties. The main concept of the pocket model is that the region between adjacent AP particles forms a “pocket” where all the Al particles agglomerate on the burning surface to form an agglomerate. The pocket model can provide good predictions of agglomerate size. However, the pocket model is still limited by its assumption of an imprecise propellant structure, which reduces its reliability. Therefore, further research is necessary to improve the agglomeration models for particles.The establishment of Al agglomeration modelBy performing electron microscopy scanning on the solid propellant surface, the distribution of AP particles on multiple surfaces can be obtained. Figure 20 shows the SEM image of AP distribution on the surface of SP-2 propellant. In the agglomeration model discussed in this section, the distance D between adjacent AP particles in the two-dimensional SEM image is approximated as the size of the three-dimensional “pocket” region. Figure 21 illustrates the concept of the pocket model, where the circular region surrounded by adjacent AP particles is considered as the pocket range in this model. The distance D between AP particles is approximated as the size of the pocket. Within the pocket, Al particles and ammonium nitrate explosive are encapsulated, and the aluminum particles contained in the pocket undergo agglomeration during the combustion process to form aluminum agglomerates. Through SEM images of the propellant surface, the distance between AP particles can be obtained. In Fig. 16, spherical-shaped AP particles and hemispherical pits with diameters similar to AP can be clearly observed. These pits are formed when a propellant specimen is cut, and some AP particles are taken away by the other half of the propellant surface, leaving behind on the propellant surface. When calculating the pocket diameter, these pit locations also represent AP particles, allowing for a more accurate estimation of the pocket size based on the actual situation.Figure 20SEM diagram of AP distribution on SP-2 propellant surface.Figure 21Schematic diagram of the pocket model.Due to the fact that the distance D between adjacent AP particles measured from the SEM image of the propellant includes a portion of the AP particles, the actual diameter Dpocket of the pocket should be smaller than D. In this paper, according to the reference17, the measured diameter D is multiplied by a correction factor γ, so Dpocket can be expressed as:$${D}_{\text{pocket}}=\gamma D$$
(1)
where γ is the correction factor introduced to consider the spatial influence of AP particles within the diameter range Dpocket39. The value of γ is related to the diameter Dpocket of the pocket and the diameter DAP of the AP particles. As D increases and DAP decreases, γ approaches 1, indicating a higher spatial occupancy of AP particles within the pocket. Conversely, when Dpocket is smaller and DAP is larger, γ approaches 0, indicating a lower spatial occupancy of AP particles within the pocket. So γ can be expressed as:$$ \gamma = (1 – Y_{{{\text{AP}}}} )^{1/3} $$
(2)
where YAP is the volume fraction of AP particles.In JF-2 propellant, according to the mass fraction and density of AP, Al, CL-20 and HMX, the volume fraction of each component in the propellant can be calculated as follows:$$ Y_{{{\text{AP}}}} = \frac{{\varepsilon_{{{\text{AP}}}} \times \rho_{{\text{P}}} }}{{\rho_{{{\text{AP}}}} }} $$
(3)
$$ Y_{{{\text{Al}}}} = \frac{{\varepsilon_{{{\text{Al}}}} \times \rho_{{\text{P}}} }}{{\rho_{{{\text{Al}}}} }} $$
(4)
$$ Y_{{\text{CL – 20}}} = \frac{{\varepsilon_{{\text{CL – 20}}} \times \rho_{{\text{P}}} }}{{\rho_{{\text{CL – 20}}} }} $$
(5)
$$ Y_{{{\text{HMX}}}} = \frac{{\varepsilon_{{{\text{HMX}}}} \times \rho_{{\text{P}}} }}{{\rho_{{{\text{HMX}}}} }} $$
(6)
where εAP, εAl, εHMX and εCL-20 are the mass fractions of each component, and ρAP, ρAl, ρHMX, ρCL-20 and ρp are the densities of each component and the propellant, respectively.If the region between adjacent AP particles is considered as the pocket range, and the pockets only contain Al, HMX, and CL-20, assuming that the Al, HMX, and CL-20 particles within the pocket region are uniformly distributed with plasticizers and binders filling the space between them, without considering the randomness of Al, HMX, and CL-20 particles, then the volume fractions of Al, HMX, and CL-20 particles in each pocket region can be expressed as:$$ Y_{{\text{pocket,Al}}} = \frac{{Y_{{{\text{Al}}}} }}{{1 – Y_{{{\text{AP}}}} }} $$
(7)
$$ Y_{{\text{pocket,HMX}}} = \frac{{Y_{{{\text{HMX}}}} }}{{1 – Y_{{{\text{AP}}}} }} $$
(8)
$$ Y_{{\text{pocket,CL – 20}}} = \frac{{Y_{{\text{CL – 20}}} }}{{1 – Y_{{{\text{AP}}}} }} $$
(9)
According to Eqs. (7), (8) and (9), the number of HMX, CL-20 and Al particles in each pocket can be expressed as follows:$$ N_{{{\text{HMX}}}} = \frac{{D_{{{\text{pocket}}}}^{3} \times Y_{{\text{pocket,HMX}}} }}{{D_{{{\text{HMX}}}}^{3} }} $$
(10)
$$ N_{{\text{CL – 20}}} = \frac{{D_{{{\text{pocket}}}}^{3} \times Y_{{\text{pocket,CL – 20}}} }}{{D_{{\text{CL – 20}}}^{3} }} $$
(11)
$$ N_{{{\text{Al}}}} = \frac{{(D_{{{\text{pocket}}}}^{3} – D_{{{\text{HMX}}}}^{3} \times N_{{{\text{HMX}}}} – D_{{\text{CL – 20}}}^{3} \times N_{{\text{CL – 20}}} )Y_{{\text{pocket,Al}}} }}{{D_{{{\text{Al}}}}^{3} }} $$
(12)
where DHMX, DCL-20 and DAl represent the average particle size of HMX, CL-20 and Al particles in the propellant.Therefore, the total mass of aluminum particles within each pocket range can be expressed as:$$ m_{{{\text{Al}}}} = N_{{{\text{Al}}}} \times \frac{\pi }{6} \times D_{{{\text{Al}}}}^{{3}} \times \rho_{{{\text{Al}}}} $$
(13)
Assuming that all the aluminum particles contained in the pocket only aggregate to form a larger aggregate, the aggregate particle diameter Dagg formed is:$$ D_{{{\text{agg}}}} = \left( {\frac{{6 \times m_{{{\text{Al}}}} }}{{\pi \times \rho_{{\text{Al,l}}} }}} \right)^{\frac{1}{3}} $$
(14)
where ρAl,l is the density of liquid aluminum, 2.35 × 103 kg/m3. It can be seen from previous studies that the interior of the aluminum aggregate is composed of liquid aluminum19.In previous studies, it was found that the burning rate of propellant is one of the important factors influencing aluminum agglomeration40. Therefore, in this section’s model, the influence of burning rate on agglomeration needs to be considered. The empirical estimate of burning rate in the empirical model proposed by Duterque28 is \(2.42/\dot{r}\). While Liu11 provided an empirical estimation for the burning rate:\(2.69/\dot{r}\). Due to the complexity of the agglomeration model and the lack of a precise mathematical model for the effect of burning rate on agglomerate particle size, fitting mathematical models can be obtained by collecting a large amount of experimental data, which are reasonably accurate. In this paper, adopting the modified burning rate coefficient method proposed by Duterque22, and considering the influence of combustion process on agglomeration process. The modified aggregate diameter can be expressed as:$$ D_{{\text{agg,1}}} = \left( {2.24/\dot{r}} \right)D_{{{\text{agg}}}} $$
(15)
After the initial aggregate is formed and until it is removed from the propellant burning surface, multiple aggregates usually come close to each other and fuse to form larger aggregates. After taking this factor into account, it is introduced into the model. In order to facilitate calculation, coefficient a is added as the influence factor. Assuming that the initial aggregate i and the aggregate j undergo a secondary agglomeration, the size of the new aggregate formed can be expressed as:$$ D_{agg,2} = a^{3} \sqrt {\left( {D_{agg,1,i} } \right)^{3} + \left( {D_{agg,1,j} } \right)^{3} } $$
(16)
where a is the coefficient, which is affected by the agglomeration physical process, such as the process of merging three or more aggregates on the propellant burning surface, or further growth near the propellant surface before ejection, etc. For different solid propellants, the value of a is different. In this study, the value of a is 1.25.In order to facilitate calculation, assuming that the sizes of the two aluminum aggregates that undergo secondary agglomeration are equal, the size of the new agglomeration after secondary agglomeration can be expressed as:$$ D_{agg,2} = a^{3} \sqrt {2\left( {D_{agg,1} } \right)^{3} } $$
(17)
In summary, the initial size of aluminum agglomerates on the propellant combustion surface and the size of new agglomerates after secondary agglomerations can be predicted according to Eqs. (15) and (17).The verification of Al agglomeration modelIn this section, the test data of JF-1 propellant are compared with that of the prediction model to verify the reasonableness of the model. The relevant parameters of JF-1 propellant are shown in Table 4. Under the conditions of 1–8 MPa, the experimentally measured burning rate is shown in Table 5, and it can be seen that the burning rate increases as the pressure increases. As the pressure increases, the surface convective heat flux and the radiant heat flux increase, resulting in an increase of the burning41. The agglomeration size model of propellant was calculated under 1.0–8.0 MPa.Table 4 Related parameters of JF-1 propellant.Table 5 The burning rate of SP-2 propellant under different pressures.In order to verify the reliability of the agglomeration size model established in this paper, the prediction results of the earlier empirical model and the pocket model are compared with the model proposed in this paper.Empirical model proposed by Salita23:$$ D_{{{\text{Sa}}}} = \frac{869}{{r(Y_{{{\text{AP}}}} + Y_{{{\text{Al}}}} )^{2} }} $$
(18)
Empirical model proposed by Liu11:$$ D_{{{\text{Liu}}}} = \frac{{2690(Y_{{{\text{Al}}}} + Y_{{{\text{RDX}}}} + 1)}}{{r(Y_{{{\text{Al}}}} + Y_{{{\text{RDX}}}} )\left( {\frac{{D_{{{\text{Al}}}} }}{50} + 1} \right)}} $$
(19)
Pocket model proposed by Cohen12:$$ D_{{{\text{Cohen}}}} = \left( {\frac{{\rho_{{{\text{AP}}}} Y_{{{\text{Al}}}} }}{{\rho_{{{\text{Al}}}} Y_{{{\text{AP}}}} }}} \right)^{\frac{1}{3}} D_{{{\text{AP}}}} $$
(20)
Based on Cohen pocket model, Duterque28 proposed a rule of influence of combustion rate on agglomeration size, which can be expressed as:$$ D_{{{\text{Duterque}}}} = \frac{2.42}{r}D_{{{\text{Cohen}}}} + 80.26 $$
(21)
To validate the reliability of the agglomeration model proposed in this paper, the empirical models proposed by Salita and Duterque were compared with the Dagg,1 model and the Dagg,2 model. Figure 22 shows the comparison between experimental results and different agglomeration models.Figure 22Experimental and model calculation values of agglomerate size.From Fig. 22, it can be seen that the predicted values of the Dagg,1 model are closer to the results of the empirical model and the pocket model compared to the experimental measurements. However, there is a significant difference between the predicted values and the experimental measurements. On the other hand, the predicted agglomerate size from the Dagg,2 model shows better agreement with the experimental values, with a relative error within 10%. This may be attributed to the significant occurrence of secondary agglomeration in the aluminum agglomerates of the propellant being studied, which is not considered in the empirical model proposed by Salita and the pocket model proposed by Duterque. As a result, the predicted values of these two models are closer to the predictions of the Dagg,1 model. This further confirms that the secondary agglomeration prediction model (Dagg,2 model) provides a more reasonable description of aluminum agglomeration in the NEPE propellant used in this experiment. It can accurately predict the size of aluminum agglomerates in the NEPE propellant.