Estimation Method of Charged Particle Concentration in Aircraft Engine Exhaust Based on Electric Field Detection

Aircraft engines, serving as the power core of aircraft, also constitute a significant source of potential malfunctions. The charge level of exhaust particles from aircraft engines serves as a crucial indicator of engine malfunction severity. This paper introduces a method for estimating the charge of exhaust particles based on electric field detection to monitor charged particles in the exhaust of aircraft engines. Initially, a mathematical model for estimating the charge of exhaust particles based on Gauss’s theorem is presented. Subsequently, an exhaust electric field induction device is designed, and the distribution of electric field sensors is studied, providing an overall design for the exhaust particle charge measurement system. Finally, the proposed method for estimating the charge of exhaust particles is analyzed through finite element simulation. The results indicate that the introduced aircraft engine exhaust particle charge estimation method based on electric field detection exhibits high accuracy and effectiveness. It offers a novel means for monitoring engine operation faults in aviation, with the potential for application in engine ground testing to further investigate engine state characterization technologies based on exhaust particle monitoring.


Introduction
Aircraft engines serve as the powerhouse of aircraft and simultaneously represent a primary source of potential failures.Perceiving and acquiring knowledge about their operational states are crucial means for monitoring engine conditions and detecting faults promptly [1][2][3][4].However, conventional techniques for monitoring engine status often require faults to deteriorate significantly before corresponding changes can be detected.Additionally, these techniques face persistent challenges such as limitations in sensor placement and difficulties in signal localization [5][6].
A passive measurement approach for aircraft engine status monitoring, based on the electric charge of exhaust particles, presents an alternative.This method offers early warning capabilities for a range of faults, including blade rubbing, combustion chamber erosion, material loss, and performance degradation.It is characterized by advantages such as good linearity, robustness, and cost-effectiveness [7].Existing research in this domain relies on the principles of electrostatic induction to monitor abnormal particles in engine exhaust.For instance, studies [8][9] have explored the fundamental principles of charged exhaust particles and electrostatic sensing, leading to the development of probetype electrostatic sensors.Research by [10] delves into the optimization of different types of electrostatic sensors, sensitivity modeling, and the design of planar array electrostatic sensors, enabling the monitoring of abnormal particles and their locations.
2 However, the current state of aircraft engine exhaust particle monitoring techniques using electrostatic principles presents certain challenges.Notably, electrostatic sensors based on these principles need to be installed in the gas path or close to high-temperature exhaust, posing influences on the measurement object.Moreover, high-temperature resistance and installation space requirements for the probes themselves are demanding.
To address the issues associated with aircraft engine exhaust particle monitoring techniques, this paper proposes a non-intrusive soft measurement approach.This approach estimates the charge in the engine exhaust by measuring the spatial electric field in the vicinity of the aircraft engine exhaust, thus resolving inherent problems in existing technologies.Compared to traditional measurement methods where sensors are installed inside the engine, the proposed new method involves mounting sensors externally to the engine, allowing for the installation of sensor arrays.
The paper introduces a charge estimation method based on Gauss's theorem and subsequently designs a system for measuring the electric charge of exhaust particles.Finally, the proposed approach undergoes simulation verification.Simulation results indicate a small measurement error, demonstrating the effectiveness of this method in monitoring the electric charge of particles in aircraft engine exhaust.

Principle and Method of Exhaust Charge Estimation Based on Gauss's Theorem
When the aircraft engine is in operation, exhaust particles acquire a certain amount of charge through processes such as contact, adsorption, atomic ionization, and combustion chamber chemical reactions [11].In the exhaust of a healthy aircraft engine, carbon soot particles are prevalent, characterized by a relatively small and uniform charge distribution.However, when engine performance deteriorates or faults occur, such as blade abrasion, wear-related issues in engine air path components, or components reaching the end of their operational life, the exhaust may contain anomalous solid particles.These particles, typically larger than 40 micrometers, significantly exceed the size of normal particles in the air path (mainly distributed in the 5-7 nm and 20-30 nm ranges, such as carbon soot particles) [12][13].Moreover, anomalous particles exhibit intermittent distribution and carry a substantial charge.As faults worsen, the charge carried by these anomalous particles gradually increases.Therefore, measuring the charge of exhaust particles can reflect the operational status and faults of the aircraft.
The charge of exhaust particles can be estimated based on Gauss's theorem, requiring measurement of the electric field on a closed surface [14] to calculate the charge within that closed surface.
According to Gauss's theorem in physics [15], for an electrostatic field in a vacuum, the electric flux through any closed surface is equal to the algebraic sum of the charges within the closed surface divided by 0  (vacuum permittivity, which can be approximated as in air).This is expressed by the formula: where 0  is the vacuum permittivity, E v is the electric field vector, A u r is the effective area exposed to the external electric field, and S q  is the algebraic sum of the charge within the Gaussian closed surface.
If sensors are placed in the region near the aircraft engine exhaust perpendicular to the direction of the electric field lines, considering a single-direction electric field per sensor, the electric field vector can be scalarized.Through derivation, we obtain: From equation ( 2), it is evident that the induced free charge within the closed surface is related to the electric field intensity and the effective area of the surface.Therefore, by measuring the electric field and the area of the closed surface, the total charge within the closed surface can be calculated.Furthermore, the electric field sensor boasts both low cost and high precision.
Hence, the design involves creating an exhaust field induction device based on an array of electric field sensors to measure the electric field around the exhaust.Subsequently, Gauss's theorem is employed to estimate the total charge of charged particles within the exhaust field induction device.

Exhaust Electric Field Induction Device Design
Estimating the total charge inside the exhaust induction device based on Gauss's theorem requires measuring the electric field at each point on the surface and the surface area of the closed surface.The surface is divided into multiple regions, and the mean electric field value from sensors in each region, multiplied by the area, yields the electric flux for that region.Summing up all the regional electric flux values provides an estimation of the total flux for the entire closed surface.Therefore, obtaining electric field values at all points on the closed surface allows the calculation of the total charge within that closed region.
To measure the spatial electric field on the closed surface of aircraft engine exhaust, a device is designed for measuring the electric field around the exhaust flow.This device can be installed along the center of the exhaust gas flow, forming a closed surface around the exhaust.The electric field values at points on the surface are measured using an array of electric field sensors.To select an optimal electric field device configuration, a comparison was made between cylindrical and trumpet-shaped configurations regarding their sensitivity to the spatial charge generated by charged particles.
Geometric models of both cylindrical and trumpet-shaped exhaust field induction devices were established.Identical charged particles were arranged within each device, and finite element simulation was used to compare the field strength distributions of the two configurations.As shown in figure 1, compared to the cylindrical structure, the electric field distribution of the trumpet-shaped exhaust induction device exhibits more noticeable changes, higher sensitivity, and a larger measurement range.To prevent the high-speed jet exhaust from the engine gradually spreading in the air and contacting the inner wall of the sensor, affecting the test results, a trumpet-shaped exhaust electric field induction device was constructed.By designing an electric field induction device, it is possible to secure the installation position of the sensor.

Arrangement of Electric Field Sensor Array
To accurately and efficiently estimate the electric flux of the entire closed surface through the spatial electric field values at finite points, the planar array structure of electric field sensors needs to be optimized.The main factors influencing the measurement performance of the electric field sensor array are the number of sensors in each layer and the interlayer distance of the array [16][17][18].Assuming an entrance radius of 1 meter and a length of 3 meters for the exhaust field induction device, its geometric

Study on the Number of Sensors in Each
Layer of the Electric Field Sensor Array.By uniformly arranging different numbers of electric field measurement points on the sensor array plane (2, 4, 6, 8, 10, 12), the proposed charge estimation method based on Gauss's theorem was used to estimate the charge within the exhaust field induction device.The absolute value of the relative error between the estimated and actual charge amounts was calculated.The relationship between the number of sensors and the mean absolute value of the relative error charge calculation is shown in figure 2. Increasing the number of sensors in each layer of the sensor array generally led to a decreasing trend in the overall mean absolute value of the relative error in the proposed charge estimation method.The mean absolute value of the relative error in charge calculation reached a local minimum when each layer of the sensor array included 8 sensors.Considering the avoidance of electric field distortion between sensors, it is preferable to choose 8 electric field measurement points per layer.

Study on the Interlayer Distance of the Electric Field Sensor Array.
The interlayer distance of the electric field sensor array not only affects measurement errors but also, under the condition of a fixed length of the tail gas electric field induction device, increases the difficulty of subsequent data collection and processing as the distance decreases.Additionally, as the distance decreases, the number of sensors increases.. Based on the study of sensor quantity, each layer of the electric field sensor array includes 8 measurement points.The length of the exhaust gas electric field induction device is 3 meters.The research focused on the interlayer distance (0.25m, 0.5m, 1m, 1.5m) when the electric field sensor array is uniformly distributed.The study examined the average absolute value of the relative error in charge calculation and the impact on the number of sensors.
The results in figure 3 show that as the interlayer distance of the sensor array plane increases, the charge measurement error initially rises, then decreases before rising again.The measurement error reaches a local minimum when three layers of sensor array achieve the best arrangement.Furthermore, with the increase in the interlayer distance of the sensor array, the required number of sensors decreases, leading to a reduction in the overall system cost.Therefore, a three-layer sensor array plane is chosen to minimize the measurement error in charge estimation while reducing the number of sensors.

Overall Design of the Measurement System
Based on simulation results, the design of the particle charge measurement system for aircraft engine exhaust is presented in figure 4. The electric field sensors, serving as sensitive units, form a three-layer sensor array.This array performs non-contact measurements of the changes in the electric field caused by the charge carried by exhaust particles.Subsequently, the proposed charge estimation method based on Gauss's theorem is employed to calculate the charge of exhaust particles within the exhaust field induction device.This system aims to achieve soft measurement of the charged amount in engine exhaust with anomalous particles, offering the potential for online monitoring of engine health status.

Simulation Verification and Error Analysis of Exhaust Particle Charge Measurement
Following the completion of the design of the exhaust particle charge measurement system, the feasibility of the system is validated through finite element simulation.The simulation results are then applied to the proposed exhaust charge estimation method based on Gauss's theorem to calculate the total charge within the exhaust field induction device, enabling a comprehensive analysis of the measurement error.
Initially, the geometric model of the system is established, maintaining a consistent geometric model for the exhaust field induction device with uniform distribution of electric field measurement points on its surface.In the early stages of aircraft engine faults, the number of anomalous charged particles is low, and their distribution is dispersed.Consequently, a single particle with a geometric model of a spherical shape with a radius of 0.1mm is chosen for the study.The calculation domain is set as a cube with a side length of 50m (significantly greater than the radius of the charged particle).
Next, assuming a variation in the charged amount of the particle and grounding the calculation domain, tetrahedral meshing is applied.The electric field results measured by points on the exhaust field induction device are then calculated for different positions of the charged particle.The simulated electric field data is subsequently utilized in the exhaust charge estimation method based on Gauss's theorem to determine the total charge of the charged particle, followed by an analysis of measurement errors.
To analyze the measurement errors of the system, the measurement error is defined as: where, r Q represents the actual total charge of the charged particle, and c Q represents the measured total charge of the charged particle.
The measurement error graph is presented in figure 5, where the x-axis represents the set charge amount of the charged particle, and the y-axis represents the measured charge amount.The maximum measurement error does not exceed 15%, with a mean measurement error of 4.37%, demonstrating the feasibility of both the measurement system and the charged amount estimation method.

Conclusion
Current technologies for monitoring charged particles in aircraft engine exhaust are predominantly based on the principles of electrostatic induction.This paper introduces a novel approach that estimates the charge of exhaust particles by evaluating the surrounding electric field.The methodology involves optimizing the design of the exhaust field induction device, configuring the structure of the electric field sensor array, and proposing a system for measuring the charge of exhaust particles.Through simulation verification, the feasibility of the measurement system is established, with the proposed Gauss's theorem-based method demonstrating an average measurement error of 4.37%.
The exhaust particle charge measurement system developed in this study will be applied in ground tests of aircraft engine operations.It aims to monitor the charge of exhaust particles in real time under various operating conditions, thus advancing the development of engine condition characterization techniques based on exhaust particle monitoring.

Figure 1 .
Figure 1.Electric field distribution of cylindrical and trumpet-shaped configurations.
established.Different charged particles were set, and based on finite element simulation results, the impact of these two factors on the measurement error was studied.

Figure 2 .
Figure 2. Relationship between the number of sensors per layer and the mean absolute value of the relative error in charge calculation.

Figure 3 .
Figure 3. Relationship between the interlayer distance of the electric field sensor array and the measurement error, number of sensors.

Figure 4 .
Figure 4. Structure of the Measurement System.

Figure 5 .
Figure 5. Measurement error graph for particle charge measurement.