Hypersonic flight vehicle (HFV) has great potential in transportation and the military. The scramjet is a vital component of the HFV air-breathing propulsion system; therefore, the cooling of the scramjet wall is crucial to the performance of HFV. In the efficient and widely-used regenerative cooling technology, aviation kerosene can be used as a coolant to cool down the scramjet wall by absorbing waste heat in the flow process. Coke is a carbonaceous deposition that is generated on the metal wall of the cooling channel during the process [1]. In practice, the typical size of a channel is normally 1–5 mm [2]. The heat exchange capacity of the fluid and the cooling efficiency are subject to the variation of the flowing area caused by coke in micro-channels. What’s more, the accumulation of coke can even block the tube [3]. Hence, the research on coke is good for improving the safety and efficiency of the scramjet.

The coke mechanism is distinct in different temperature ranges. Under various conditions, the physical properties of coke are different. Gascoin et al. [4] found three types of coke in reactors made from different materials. The density of coke in the stainless steel reactor and the low carbon steel reactor was 560 and 1883 kg/m³, respectively, and the porosity was 70% and 6%, respectively. Ji et al. [5] found that coke of n-decane was an accumulation of microcosmical spherical particles, and pores and cracks were found on the surface of coke. The envelope density and the real density of coke were 1049 and 1498 kg/m³, and the porosity was 29.9%. Lucas et al. [6] studied the density of coke which was formed in thermal cracking, and found that the range of the density varied from 2010 to 2080 kg/m³. Tevelde et al. [7] measured four kinds of coke formed from different types of kerosene, and learned that thermal conductivity changed between 0.0005–0.05 W/(m∙K). Most of the research revealed that the essence of coke was porous media. To date, many studies on the thermal behavior of fluid in channels with porous media have been conducted. However, few of them were on supercritical fluid characterized by dramatically variable properties. Hereby, the influence mechanism of supercritical fluid flow in porous media on the mainstream of the channel is still unclear. Although related research does exist, the amounts are few, the subjects are normally supercritical CO₂ or supercritical water [8–10], and the supercritical aviation kerosene in porous coke is seldom concerned.

Considerable studies on the thermal behavior of aviation kerosene during the cracking or coking process have been carried out. For instance, Li et al. [11] found pyrolysis reduced the cooling effect of fuel in unilateral heated channels with dimples. Gong et al. [12] discovered the adverse effect of buoyancy on fuel cracking under different flow directions in vertical tubes, leading to an increase in the heat transfer coefficient in the downward flow and a decrease in the heat transfer coefficient in the upward flow. Gong et al. [13] studied heat transfer of pyrolytic RP-3 in curved channels and found that vortex structure in curved channels helped to reduce coking precursors. Compared with the research on the pyrolytic RP-3, less attention was paid to studying the effect of coking. Pan et al. [14] discovered that the coke layer had two distinct effects on heat transfer capacity in different temperature ranges. When the temperature of RP-3 approached or exceeded pseudo-critical temperature, porous coke strengthened heat transfer. On the contrary, heat transfer deterioration happened without pseudo boiling. Liu et al. [15] applied a partition algorithm to compute the fluid phase and the solid phase for coupled heat transfer and chemical reactions and found that heat transfer was augmented with inlet flow acceleration. Yuan et al. [16] experimentally probed into the role that the coke of RP-3 played inside the miniature tubes, and found that heat transfer was weakened at the peak coking region, and heat transfer enhancement occurred at high-temperature region. The result obtained from Wang et al.’s [17] research indicated that coking morphology and deposition law changed the heat convection of RP-3. The influence of coking on the heat transfer of RP-3 inside helical tubes was experimentally studied, and more coking deposits that weakened heat transfer were found on the outer tube side [18]. Feng et al. [19] established a 2D numerical model that was based on a Time-marching algorithm and used a coupled method to reveal the effect of turbulence on the pyrolysis of hydrocarbon fuel and found an enhancement of heat and mass transfer by turbulence in the core flow. Sun et al. [20] carried out large eddy simulations of thermal oxidative coking of aviation kerosene in a U-tube and discussed the combination effect of buoyancy and centrifugal force on heat transfer.

Except for coke and pyrolysis, heat transfer characteristic is also influenced by other factors, such as channel size, inlet mass flow rate, heating flux, etc. Li et al. [21] discussed the effect of the dimple on the heat transfer of kerosene, the results showed that the overall heat transfer ability is increased with the dimple effect. Jiang et al. [22] explored the impact of geometry parameters on the flow of hydrocarbon fuel, revealing that thin ribs and smaller total flow areas improved the cooling performance. Yu et al. [23] discussed heat transfer and flow of RP-3 under different heat fluxes and channel inclinations and revealed that the average heat transfer coefficient declined first, and then increased as heat flux rose and the angle of inclination decreased. Wang et al. [24] experimentally explored the effects of mass flux, inlet pressure, and inlet temperature on the unstable flow of RP-3, and proposed that the increase of mass flow contributed to heat transfer enhancement and led to the dropping of vibration amplitude of inner wall temperature. Factors that influence the heat transfer of aviation kerosene are supposed to be studied in regenerative cooling channels.

To the best of our knowledge, though much effort has been expended to explore the thermal behavior of fluid in channels with porous media, few researchers have explored the influence of porous coke on the flow and heat transfer of supercritical RP-3. Given that the morphology of coke is porous media, the effect of coke and its corresponding effect mechanism is yet to be discovered. Consequently, in this paper, the impacts of coke layer thickness, inlet mass flow rate, and wall heat flux variation on heat convection of RP-3 are discussed.

The schematic diagram of the three-dimensional model of the cylindrical regenerative cooling channel is presented in Fig. 1. The outside view of the physical model is depicted in Fig. 1a, and the y-z section of the model is shown in Fig. 1b. The model included three parts: the channel wall, the coke layer, and the fluid. The inner diameter and outer diameter were 1.8 and 2.2 mm, respectively. The length of the uniformly heated section was 600 mm, and a 150 mm adiabatic inlet section was set to ensure the full development of turbulent flow. Apart from that, there was a 150 mm adiabatic outlet section that aimed to reduce the influence of backflow. As shown in Fig. 1b, the solid channel wall on the adiabatic inlet section and the adiabatic outlet were not designed to simplify the model. The flow direction of the fluid was perpendicular to the gravity direction. The buoyancy effect of fluid was considered as well.

In the model, the fluid applied was supercritical Chinese aviation kerosene RP-3, and the wall of the regenerative cooling wall was 1Cr18Ni9Ti. As mentioned in the introduction, the coke of aviation kerosene is normally porous media. The porosity, apparent density, and permeability of the coke layer referred to Gascoin et al.’s results [4]. The properties of coke and 1Cr18Ni9Ti are illustrated in Table 1.

Aviation kerosene RP-3 is a complex mixture composed of various components. To simplify the calculation process, RP-3 was regarded as a simple substance rather than a complicated mixture. The physical properties of RP-3 were from Yu et al.’s research [25], as presented in Fig. 2.

The governing equations of RP-3 are presented as following:(1) ∇⋅(ρuf)=0 (2) ∇⋅(ρufuf)=−∇P+∇⋅τRe+ρg+Sj (3) ∇⋅(ρufef)=∇⋅(λf∇Tf)−∇⋅(Puf)+ρg⋅uf where τRe is the turbulent stress evaluated with the turbulence model. The flow of RP-3 in the regenerative cooling channel was turbulent. To model the turbulent flow of aviation kerosene, the RNG k-ε model was chosen because the model had higher accuracy and better efficiency [19,20].

The pores in the coking layer formed in the supercritical regeneration cooling channel were much smaller than the channel size, so the porous media model was used to simplify the treatment, without considering the details of the flow between pores. The porous media model was obtained by adding a momentum source term of the momentum equation to the standard flow equation. C₂ was inertial resistance coefficient, and the local thermal equilibrium model was applied to solve the heat transfer of porous media.(4) Sj=−(μαvi+C212ρ|v|vj)

To simulate the turbulent flow, an enhanced wall treatment was used in RNG k-ε model. Values of C_1e, C_2e and C_3e were 1.42, 1.68 and 0.0845, respectively.(5) ∂∂xi(ρuiε)=∂∂xj(αεμe∂ε∂xj)+C1εεk(Gk+C3εGb)−C2ερε2k−Rε+Sε (6) ∂∂xi(ρuik)=∂∂xj(αkμe∂k∂xj)+Gk+Gb−ρε−YM+Sk

α_k and α_ε represented the effective Prandtl numbers corresponding to k equation and ε equation, respectively. α_k = α_ε ≈ 1.393. μ was the effective viscosity. G_k and G_b were production of turbulence kinetic energy generated from average velocity gradient and buoyancy force. S_ε and S_m were source items.

The governing equation of the solid phase was the equation of heat conduction:(7) ∇⋅(λs∇T)=0

Unless otherwise specified, the boundary conditions were set as follows: the flow rate at the mass flow inlet was q_m = 3 g/s, and the temperature of the mass flow inlet remained at T_in = 400 K. The type of outlet was a pressure outlet of p = 5 MPa. The constant heat flux imposed on the tube wall was normally q = 400 kW/m². Coupled heat transfer occurred at the interface where the channel wall and the coke layer intersected. The junction of the coke layer and aviation kerosene was the interface boundary. Besides, the no-slip conditions were applied on other walls.

The simulation work was completed with the aid of the commercial software ANSYS FLUENT 18.0. The governing equations were discretized by the finite volume method, and the convection terms and diffusion terms were discretized by the QUICK scheme. The pressure and velocity coupling equations were solved by the SIMPLEC algorithm. When judging the convergence, the convergence criterion of the energy equation was set to 10⁻⁸, and other parameters were 10⁻⁶. The schematic diagram of the grid structure is presented in Fig. 3.

Before the model validation, the grid independence verification was conducted, and the bulk temperatures at the outlet obtained from different grids were compared in Table 2. Considering the accuracy and the efficiency, Mesh-2 was selected to complete the following simulation work.

To verify the porous model, the wall temperature of the heated aluminum tube with porous aluminum foam was calculated. The data acquired from the computational model were compared with the results from Dukhan et al.’s experiment [26] and are demonstrated in Fig. 4a. The variation trend was consistent, and the largest deviation was only 0.74%. To verify the numerical model dealing with supercritical RP-3, the inner wall temperatures obtained from Zhang et al.’s [27] experiment and our model were compared and are shown in Fig. 4b, with the maximum deviation being 4.7%. The numerical model and the method applied are reliable for the present work.

To find out the influence of the coke layer in the regenerative cooling channel on heat convection characteristics, several simulation work considering the coke layer as porous media were carried out in the manuscript. Additionally, heat transfer and flow processes with and without coke were compared under various inlet mass flow rates as well as heat flux densities. Several conclusions were reached as follows:

1. The existence of the coke layer has a nonnegligible effect on the temperature distribution and velocity of RP-3. The coke layer generally weakens the heat transfer of RP-3, but the weakening effect is not reinforced by the thickening of coke layers. When heat transfer enhancement that is caused by the flow acceleration and heat transfer deterioration that results from a thermal resistance increase reaches a balance at a certain thickness of the coke layer, the heat transfer coefficient tends to be stable. A steep rise in the local heat transfer coefficient that is led by the buoyancy effect is observed only in the channel without coking.

2. Heat transfer inside the smooth channel is less sensitive to the acceleration of the inlet mass flow, but both the average heat transfer coefficient inside the smooth channel and the coking channel increase with the flow rate. The heat transfer capacity varies slightly in the inlet flow velocity between the smooth channel and the coking channel. The increase in the inlet mass flow rate leads to an obvious rise in the pressure differential between the inlet and the outlet, but the pressure differential in the smooth channel mounts slowly.

3. The increasing heat flux makes a minor difference in the heat transfer of the coking channel and the smooth channel. The results show that the increase in the heat flux density will reduce the pressure differential between the inlet and the outlet in the smooth channel but make the pressure drop of the coking channel rise, which is due to the increased flow resistance caused by coke.

Though the effect of the coke layer on heat transfer and flow characteristics of RP-3 has been discussed in the paper, the numerical results are not the same as the actual situation. In future research, we will focus on improving the accuracy of the computational model that takes the crack of RP-3, the non-uniform thickness of the coke layer along the channel, and so on into consideration.