© Wuhan University 2026

0 Introduction

The fuel injection ring is a crucial component in the fuel supply system of micro-turbojet engines. Its primary function is to deliver fuel stably and uniformly from the fuel tank or fuel supply system to the combustion chamber of the engine^[1]. During this process, the parameters of the fuel injection ring, such as fuel flow rate, pressure, and injection angle are needed to meet the engine's requirements to guarantee the normal operation and high efficiency performance of the engine. Improper design of the distribution, size, or shape of the fuel injection holes in the fuel injection ring may lead to uneven fuel distribution during injection progress. Uneven fuel supply can result in a series of issues, such as a decrease in combustion efficiency, a decrease in engine power, an increase in fuel consumption, an increase in emissions pollution, the overheating of the engine, the aggravation of the engine abrasion, flight safety risks and so on^[2]. Therefore, it is essential to ensure the uniformity of fuel supply from the fuel injection ring to maintain the normal operation and high performance of the engine.

With regard to the issue of engine fuel injection, extensive research has been conducted^[3-4]. Do et al^[5] investigated the impact of relatively low inlet flow rate and fuel types on the fuel flow distribution at the nozzle outlet. Enagi et al^[6] studied the influence of various nozzle diameters on the combustion characteristics of liquid fuels. Tan et al^[7] adopted the scheme with unequal-diameter nozzles, in which the sizes of the nozzles were altered in the fuel injection ring to compensate for pressure loss effects. Wang et al^[8] studied the internal flow field characteristics of a single fuel nozzle in a large-scale turbojet engine, as well as the flow distribution at the nozzle outlet.

The fuel distribution uniformity of the injection ring in micro-turbojet engines directly affects combustion efficiency and engine performance. While existing research mostly focuses on large-scale engines or single-parameter optimization, research on the structural adaptability of nozzles for micro-scale engines remains insufficient. In this paper, the Leiting LT70 is used as the test prototype. By quantifying characteristic non-uniformity parameters, the coupled effect of nozzle inner diameter and inlet flow rate is systematically analyzed, which provides an accurate basis for the optimal design of injection rings for micro-turbojet engines.

1 Modeling and Simulation

1.1 Establishment of Physical Model

The geometric model of the fuel injection ring was established in SolidWorks, as shown in Fig. 1. The fuel injection ring is composed of a ring body, a fuel inlet tube, and 12 fuel nozzles equidistantly arranged along the circumferential direction of the ring. For the ring body, the inner diameter$d$ is 2.5 mm, the wall thickness is 0.5 mm, and the pitch diameter $D_w$ is 120 mm. For the fuel inlet tube, the length $L_m$ is 118 mm, the inner diameter $D_i$ is 2 mm and the wall thickness is 0.2 mm. For the nozzles, the length $L_h$ is 16 mm, the wall thickness of 0.1 mm, and the inner diameters $D_n$ range from 300 μm to 900 μm.

Fig. 1 Fuel injection ring fluid dimensional parameters

The three-dimensional geometric model was imported into ANSYS Fluent to establish the physical model for fluid computation, as shown in Fig. 2.

Fig. 2 Fuel injection ring fluid calculation physical model

1.2 Boundary Conditions and Mesh Division

In the Fluent pre-processing, the boundary conditions were set as follows: the fuel inlet pipe cross-section as a mass flow inlet, the nozzle outlet cross-section as a constant pressure outlet, and the remaining surfaces as no-slip walls. Aviation kerosene was used as the fuel, a hydrocarbon mixture with an average formula of ${\mathrm{C}}_{\mathrm{12}}{\mathrm{H}}_{\mathrm{23}}$, a density of $\mathrm{780}\mathrm{k}\mathrm{g}/{\mathrm{m}}^{\mathrm{3}}$, and a viscosity of $\mathrm{0.002}\mathrm{4}\mathrm{P}\mathrm{a}·\mathrm{s}$. The local mesh of the internal flow field of the fuel injection ring is shown in Fig. 3. Unstructured tetrahedral meshes were employed, offering good geometric adaptability and ease of mesh size control.

Fig. 3 Local mesh model

1.3 Numerical Method and Grid Independence Verification

In this study, the semi-implicit method for pressure-linked equations (SIMPLE) algorithm is adopted for pressure-velocity coupling, and the renormalization group (RNG) k-ε turbulence model is used to accommodate relatively complex flows. In numerical simulation calculations, it is assumed that the fuel is an incompressible fluid with three-dimensional, steady-state flow, and energy conversion during fuel flow is not considered. In this study, the nozzle diameters are all greater than 50 μm, which complies with the continuous medium assumption. So the continuity equation and numerical calculation methods based on the Navier-Stokes equation^[9] can be used. The equations are as follows:

Continuity Equation:

$\frac{\partial \rho }{\partial t}+\nabla \rho \cdot C=\mathrm{0}$(1)

Navier-Stokes Equations:

$\begin{array}{l}\frac{\partial \rho {\nu }_x}{\partial t}+\frac{\partial \left(\rho {\nu }_x{\nu }_x\right)}{\partial x}+\frac{\partial \left(\rho {\nu }_y{\nu }_x\right)}{\partial y}+\frac{\partial \left(\rho {\nu }_z{\nu }_x\right)}{\partial z}\\ ={\mu }_e\left(\frac{\partial {v_x}^{\mathrm{2}}}{\partial x^{\mathrm{2}}}+\frac{\partial {v_x}^{\mathrm{2}}}{\partial y^{\mathrm{2}}}+\frac{\partial {v_x}^{\mathrm{2}}}{\partial z^{\mathrm{2}}}\right)-\frac{\partial P}{\partial x}\end{array}$(2)

$\begin{array}{l}\frac{\partial \rho {\nu }_y}{\partial t}+\frac{\partial \left(\rho {\nu }_x{\nu }_y\right)}{\partial x}+\frac{\partial \left(\rho {\nu }_y{\nu }_y\right)}{\partial y}+\frac{\partial \left(\rho {\nu }_z{\nu }_y\right)}{\partial z}\\ ={\mu }_e\left(\frac{\partial {v_y}^{\mathrm{2}}}{\partial x^{\mathrm{2}}}+\frac{\partial {v_y}^{\mathrm{2}}}{\partial y^{\mathrm{2}}}+\frac{\partial {v_y}^{\mathrm{2}}}{\partial z^{\mathrm{2}}}\right)-\frac{\partial {\rm P}}{\partial y}+\rho g_y\end{array}$(3)

$\begin{array}{l}\frac{\partial \rho v_z}{\partial t}+\frac{\partial \left(\rho v_x{v}_z\right)}{\partial x}+\frac{\partial \left(\rho v_y{v}_z\right)}{\partial y}+\frac{\partial \left(\rho v_z{v}_z\right)}{\partial z}\\ ={\mu }_e\left(\frac{\partial v_z^{\mathrm{2}}}{\partial x^{\mathrm{2}}}+\frac{\partial v_z^{\mathrm{2}}}{\partial y^{\mathrm{2}}}+\frac{\partial v_z^{\mathrm{2}}}{\partial z^{\mathrm{2}}}\right)-\frac{\partial P}{\partial z}\end{array}$(4)

In these equations: $C=(v_x,{\nu }_y,v_z)$ represents the velocity vector, ${\mu }_e$ represents the effective viscosity, $\rho$ represents the fuel density, and P represents the pressure.

A pressure-based solver is adopted as the solver, and the pressure-velocity coupling calculation uses the SIMPLE algorithm. For the turbulent flow phenomena occurring during fuel flow, a turbulence model needs to be added before the solution calculation. In this study, the RNG k-ε turbulence model is adopted to accommodate relatively complex flows. When the solved residual value of the continuity equation is less than 1×10^-5, the calculation is considered converged, and the calculation results are highly accurate^[10]. Figure 4 shows the convergence history of the residuals for the continuity equation, the velocity components in all three directions, the turbulent kinetic energy (k), and the turbulent dissipation rate (ε).

Fig. 4 Iterative calculation results

Before numerical simulation calculations, grid independence verification is necessary. In this paper, a fuel injection ring with a nozzle diameter of 450 μm for numerical simulation is adopted. The inlet flow rate is set at $\mathrm{50}\mathrm{g}/\mathrm{m}\mathrm{i}\mathrm{n}$, and the grid is divided into varying numbers ranging from 350 000 to 1 200 000. The results of grid independence verification are shown in Table 1. When the number of grids reaches 603 678 or above, the maximum change in fuel inlet pressure is 0.02 kPa. To minimize the impact of the number of the grids on the accuracy of the solution, the number of grids in all numerical simulation schemes is 600 000 or above, and according to the structural differences of different fuel injection ring models, the corresponding number of grids is between 600 000 and 1 500 000.

2 Case Analysis

2.1 Parameter Settings

This study investigates the effects of nozzle inner diameter and inlet flow rate on the fuel flow distribution at the nozzle outlets of the injection ring. Seven nozzle inner diameters (300, 400, 500, 600, 700, 800, and 900 μm) and eight inlet mass flow rates (5, 10, 20, 30, 50, 100, 200, and 300 g/min) were examined, yielding a total of 56 simulation cases.

2.2 Simulation Results and Analysis

When the inlet mass flow rate remains constant at $\mathrm{50}\mathrm{g}/\mathrm{m}\mathrm{i}\mathrm{n}$, the fuel flow distribution at the outlets of the fuel injection ring nozzles is shown in Fig. 5. It can be observed that the fuel flow distribution at the outlets of nozzles with an inner diameter of 300 μm is relatively uniform. As the inner diameter of the nozzles gradually increases to 900 μm, the outlet mass flow rates of nozzles No. 1 and No. 2 are larger, while the outlet mass flow rates of nozzles No. 6 and No. 7 are smaller.

Fig. 5 Fuel distribution at an inlet mass flow rate of $\mathrm{50}\mathbf{g}/\mathbf{m}\mathbf{i}\mathbf{n}$

To better analyze and compare the uniformity of the fuel distribution at the nozzles' outlets, the normalized mass flow rate is utilized to analyze the fuel distribution. The calculation expression for this is shown in formula (5):

$\overline{M}=\frac{{\overline{m}}_i}{{\overline{m}}_q}=\frac{{\overline{m}}_i}{m_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}/\mathrm{12}}$(5)

In the formula, $\overline{M}$ denotes the normalized mass flow rate, ${\overline{m}}_i$ denotes the actual outlet mass flow rate of the nozzle, ${\overline{m}}_q$ denotes the average outlet mass flow rate, and $m_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ denotes the inlet mass flow rate.

Under different inlet mass flow conditions, the normalized mass flow distribution at the nozzle outlets is shown in Fig. 6. When the nozzle diameter of the fuel injection ring is 300 μm, the normalized mass flow rate is close to 1, as shown in Fig. 6(a). When the nozzle diameter is 600 μm, the normalized mass flow rate is slightly greater than 1, among them, the nozzles No. 1 and No. 12 have relatively large deviations, and the maximum deviation of the nozzle No. 12 is 7.1%. When the nozzle diameter is 900 μm, the deviations are larger, with a maximum deviation of 29.4%, as shown in Fig. 6(c) and (d). It can be seen that the greater the distance between the nozzle and the inlet oil pipe opening, the smaller the outlet fuel flow rate. And as the nozzle diameter increases, the unevenness of the outlet flow rate among the nozzles of the fuel injection ring increases.

Fig. 6 Results of normalized mass flow rate distribution under different conditions

To quantitatively evaluate the uniformity of the fuel distribution at the nozzle outlets, the non-uniformity parameter of nozzle outlet flow rate is introduced:

${\sigma }_m=\sqrt[]{\frac{{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{12}}}({\overline{M}}_i{-\mathrm{1})}^{\mathrm{2}}}{\mathrm{12}}}$(6)

The results of the non-uniformity parameter of nozzle outlet flow are shown in Fig. 7. As the nozzle diameter increases, the non-uniformity parameter also increases. When the nozzle diameter is 900 μm, the non-uniformity parameter reaches a maximum of 16.5%. In other words, the larger the nozzle diameter of the fuel injection ring and the smaller the inlet mass flow rate, the greater the difference in fuel flow rate at each nozzle outlet, resulting in the more uneven fuel flow distribution. When the inlet fuel mass flow rate is $\mathrm{300}\mathrm{g}/\mathrm{m}\mathrm{i}\mathrm{n}$, as the nozzle diameter gradually increases, the non-uniformity parameter of outlet flow is no more than 8.1%, indicating that when the inlet flow rate is large, the fuel flow rate difference decreases as nozzle diameter increases, and the fuel flow distribution is relatively uniform. When the nozzle diameter of the fuel injection ring is 300 μm, the maximum non-uniformity parameter of each outlet flow is 0.42%, and the inlet flow rate has a relatively small impact on the uniformity of fuel flow distribution, resulting in a relatively uniform fuel flow distribution.

Fig. 7 Results of non-uniformity parameters for different inlet flow rates and nozzle diameters

In addition, the pressure drop between the inlet and outlet of the fuel injection ring was obtained through numerical simulation, as shown in Fig. 8. As the nozzle diameter of the fuel injection ring increases, the pressure drop between its inlet and outlet rapidly decreases. When the inlet flow rate of the fuel injection ring remains constant, a smaller nozzle diameter results in a larger pressure drop. The maximum pressure drop between the inlet and outlet of a fuel injection ring with a nozzle diameter of 300 μm is 217.3 kPa, while that of a fuel injection ring with a nozzle diameter of 900 μm is 14.9 kPa at its maximum. This indicates that the pressure drop between the inlet and outlet of the fuel injection ring increases as the nozzle diameter decreases.

Fig. 8 Pressure drop across the inlet and outlet of various fuel injection rings

Considering both the fuel flow distribution at the nozzle outlet of the fuel injection ring and the pressure drop between the inlet and outlet, for a nozzle inner diameter of 500 μm, the maximum non-uniformity parameter of the outlet flow is 2.4%, and the maximum pressure drop between the inlet and outlet is 41.4 kPa. For a nozzle inner diameter of 600 μm, the maximum non-uniformity parameter of the outlet fuel flow is 4.1%, and the maximum pressure drop between the inlet and outlet is 26.2 kPa. The non-uniformity parameter of the outlet flow for a nozzle inner diameter of 600 μm does not exceed 5%, indicating a relatively uniform fuel flow distribution. Therefore, considering the uniformity of fuel distribution at the nozzle outlet of the fuel injection ring and the power consumption of the fuel pump, a fuel injection ring with a nozzle inner diameter of 600 μm is more suitable for supplying fuel to the combustion chamber of this micro-turbojet engine.

3 Conclusion

This study numerically investigated the effects of nozzle inner diameter and inlet flow rate on the fuel flow distribution of injection rings in micro-turbojet engines using the RNG k-ε turbulence model. The results show that reducing the nozzle inner diameter and increasing the inlet flow rate lead to a more uniform fuel flow distribution at the nozzle outlets. For a nozzle diameter of 300 μm, the non-uniformity parameter remains stable across varying inlet flow rates, indicating robust distribution uniformity. Conversely, larger nozzle diameters or lower inlet flow rates result in increased flow unevenness. The pressure drop between the inlet and outlet increases with decreasing nozzle diameter and increasing flow rate. Specifically, the pressure drop for a 300 μm nozzle is approximately ten times that for a 900 μm nozzle. Taking both distribution uniformity and pressure drop into account, a fuel injection ring with a nozzle inner diameter of 600 μm is optimal for the micro-turbojet engine discussed in this paper.

References

All Tables

All Figures

	Fig. 1 Fuel injection ring fluid dimensional parameters
In the text

	Fig. 2 Fuel injection ring fluid calculation physical model
In the text

	Fig. 3 Local mesh model
In the text

	Fig. 4 Iterative calculation results
In the text

	Fig. 5 Fuel distribution at an inlet mass flow rate of $\mathrm{50}\mathbf{g}/\mathbf{m}\mathbf{i}\mathbf{n}$
In the text

	Fig. 6 Results of normalized mass flow rate distribution under different conditions
In the text

	Fig. 7 Results of non-uniformity parameters for different inlet flow rates and nozzle diameters
In the text

	Fig. 8 Pressure drop across the inlet and outlet of various fuel injection rings
In the text