Propeller Design Within the Overall Configuration of a Near-Space Airship

Highlights

What are the main findings?

• In the overall configuration design of a near-space airship, propeller efficiency improvement only has a limited effect on overall weight reduction.
• A differentiated daytime–nighttime flight speed is introduced to significantly reduce the nighttime energy demand, enabling up to a 29.6% reduction in total airship mass.

What are the implications of the main findings?

• An engineering method based on characteristic blade elements can provide a propeller adequately optimized for airship overall configuration design.
• The variable-speed strategy offers an effective pathway for the lightweight design of near-space airships, enhancing feasibility for long-endurance station-keeping missions.

Abstract

High-efficiency propeller design is essential for reducing the total mass of near-space airships under low-Reynolds-number conditions. This study optimizes the overall design parameters of near-space airships by integrating an efficient engineering propeller design method based on characteristic blade elements. This overall configuration yields results close to those obtained from single-objective optimization of the propellers. Through analysis of the overall configuration, it is evident that there is limited room for optimization in terms of propeller efficiency improvement. Therefore, a variable-speed strategy that accounts for different flight speeds during day and night is proposed. The variable-speed strategy achieves a 29.6% reduction in total airship mass compared to the constant-speed baseline. These findings verify that optimizing flight speeds and propeller efficiency is effective for achieving lightweight airship designs.

1. Introduction

1.1. Research Significance

Near-space airships are aircraft that float at near-space altitudes by filling the envelope with hydrogen or helium, which are lighter than air, thus enabling stable control or autonomous flight. If combined with other advanced technologies to achieve long-term station-keeping or regional cruise within near space, such airships could become another type of fixed-point platform with broader applications than satellites. They possess tremendous military and civilian value [1] in fields such as wireless communication [2,3], Earth observation [4], early-warning surveillance [5,6], electronic countermeasures [7], and navigation and positioning [8,9]. In recent years, countries around the world have regarded them as a key area of development, referring to them as strategic high technology of the new century [10]. Their characteristics—including low energy consumption, long endurance, and high safety—have attracted extensive attention.

Because this category of aircraft typically has a large scale and cannot easily achieve high-speed flight, propellers offer higher efficiency than jet engines under low-speed conditions, thereby increasing endurance time. At present, most near-space aircraft—such as High Altitude Long Endurance Demonstrator (HALE-D) and High-Altitude Sentinel—use propellers for propulsion. At near-space altitudes, the air density is only 7.2% of that at standard atmospheric conditions, while the flight speed remains relatively low. This results in low Reynolds numbers for airfoil sections, generally in the order of 10⁴–10⁵. Under such conditions, airflow separation tends to occur near the trailing edge, degrading aerodynamic performance. Meanwhile, the transonic region of the propeller blades becomes larger, and the influence of shock waves increases, further affecting propeller performance. Therefore, it is necessary to study propeller design technology for low-Reynolds-number conditions in near space and to develop high-efficiency propellers that meet the requirements of near-space airships.

Currently, near-space airships typically adopt an energy architecture consisting of solar panels and energy storage batteries. Solar panels collect solar energy during the day and convert it into electricity, part of which is used directly by onboard payloads and the propulsion system, while the remainder is stored in batteries for nighttime use. As a key component of the propulsion system, propeller efficiency directly affects the mass of the energy system. Since near-space airships are highly sensitive to mass, improving propeller efficiency under the condition of providing sufficient thrust is essential for reducing total system mass, increasing payload capacity, enabling day–night energy cycling, and achieving long-duration station-keeping missions.

Given the potential advantages of near-space airships and their broad military and civilian application prospects, various countries are continuously conducting technical research and flight experiments. High-efficiency propeller design is one of the key technologies in airship development and serves as a crucial technical foundation for achieving station-keeping flight, trajectory control, regional loitering, and point-return capabilities. Therefore, research on high-efficiency propellers for near-space airships is of significant theoretical and engineering importance.

1.2. Research Background

1.2.1. Near-Space Propellers

Common methods for propeller design include the chart method and theoretical methods. Generally, the chart method has a simple calculation process, and the design results can be obtained quickly by selecting appropriate charts according to the design requirements. However, there are currently almost no charts available for propellers under low-Reynolds-number conditions; therefore, aerodynamic theoretical methods are usually adopted for designing near-space propellers. The relatively mature methods at present include momentum theory, blade element theory, vortex theory, and strip theory [11].

Due to the special environmental conditions of near space (low air density), combined with the generally low flight speed of airships (<30 m/s), the Reynolds numbers of near-space propellers are low, making airfoil flow prone to separation and thereby reducing propeller aerodynamic efficiency [11]. Even so, many high-altitude airships or solar-powered aircraft still use propellers for propulsion, as shown in

Table 1. Propeller design results of different projects or designers.

Project or Designer	Diameter (m)	Number of Blades	Propeller Efficiency	Design Altitude (km)	Inflow Velocity (m/s)
POBAL-S airship [12]	10.36	3	78%	21	8
HiSentinel airship [13]	3	2	43%	20	10
HALE-D airship [14]	2	2	40%	18.3	10
DLR airship [15]	6	5	84%	22	-
Helions UAV [15]	2	2	86%	-	-
Zephyr UAV [16]	1	2	75%	21.7	25
Okuyama [17]	4.2	-	68%	18	16
Hu Ying [18]	5.95	3	60.3%	20	20
Nie Ying [19]	9	2	64%	21	20
Liu Peiqing [20]	6.5	3	75.8%	20	20
Wang Yufu [21]	8	2	53%	20	6
Morgado [22]	6	6	53%	16	30
Zheng [23]	7.2	3	73%	20	20
Jiao Jun [24]	6.8	2	70%	20	20
Liu Xinqiang [25]	2.5	2	73.8%	20	30
Xie Cong [26]	3.5	2	73.3%	20	20

“-” means corresponding data is not publicly unavailable.

below:

1.2.2. Overall Design Methods of Near-Space Airships

The overall design of an airship is carried out by determining the overall configuration and overall performance parameters after the user proposes specific design requirements. Through theoretical analysis, simulation calculations, and experimental verification, an airship scheme with good performance that meets the design requirements is obtained. The overall design of any aircraft needs to undergo repeated iterative calculations, and there may usually be multiple design schemes that satisfy the requirements; therefore, it is necessary to analyze and evaluate the overall performance parameters and select the most appropriate scheme. The overall design generally involves the application of multiple disciplines, requiring coordination among the needs and concerns of different disciplines. The overall design of near-space airships differs from that of conventional aircraft, and it is necessary to study overall design methods according to their characteristics.

Low-altitude airships usually design only the external shape parameters of the airship to satisfy thrust–drag balance and buoyancy–weight balance. Petrone [27] used traditional design methods to complete the overall design scheme of a near-space airship carrying fuel-powered batteries and solar cells. Thomas [28] considered the issue of energy balance during the full-day flight of an airship and developed an airship design analysis program. The power and endurance capabilities were decoupled, allowing the minimum solar panel area to be calculated while meeting mission requirements. Using wind-speed profile diagrams, the optimal flight altitude for Portland, Maine, was determined to be 18.9 km, and the corresponding design scheme was provided. Yu [29] analyzed the influence of environmental and operating parameters such as the operating latitude, pressure difference, temperature difference, helium purity, season, and wind speed on airship payload and the required size and area of solar panels. The results showed that if under conditions of relative low altitude, high latitude, strong winds and winter, the required solar panel area for a stratospheric airship could be too large, which may make the design of a long-endurance stratospheric airship difficult to achieve. Meanwhile, solar panel efficiency is the key technology controlling the area of the airship’s solar panels, and technological advances in regenerative fuel cells and propeller efficiency have significant impacts on the payload, size, and solar panel area of the airship. Knaupp [30,31,32] and others conducted studies in areas such as the energy system and thermal characteristics of airship overall design.

Liang Haoquan [33] proposed an improved response surface collaborative optimization method. It reduces the influence of response surface variations on gradient values and applies generalized multipliers and trust regions within the response surface. A framework model for overall design based on this method was proposed for addressing the overall design problem of near-space airships, and the various components of the airship and the corresponding disciplines were analyzed and summarized. The effectiveness of the method in the overall optimization design of near-space airships was verified. Wang Haifeng [34], addressing the differences in design between near-space airships and low-altitude airships, proposed an overall design method for airships based on energy balance, introduced the energy balance model, and completed the design scheme of a near-space airship. Yang Yanchu [35,36] conducted multidisciplinary optimization of the structure, aerodynamics, and other aspects of near-space airships using a genetic algorithm, and carried out case analyses according to technical requirements. Zhao Xinlu [37,38] studied the overall design methods and key technologies of the energy system of near-space airships using theoretical analysis and numerical simulation tests, completing the multidisciplinary optimization design of the near-space airship and obtaining optimized design schemes.

Li Yanping [39] analyzed and experimentally verified the solar illumination model and the theory of solar cell modules, establishing an engineering model for the illumination output of solar cell modules. Wang Dongchen [40] proposed a propulsion system design method for near-space airships based on energy balance, established surrogate models for the main design parameters of the propulsion motor and propeller, and completed the optimization design of the propulsion system. Shi Hong [41] analyzed and studied the conversion model of solar panels for near-space airships, establishing models for the direct, scattered, and reflected solar radiation received by the solar panels on the airship surface. Mei Xiaodong [42] studied the variation trends of full-spectrum solar radiation in near space with latitude, altitude, and time, and obtained the distribution characteristics of solar radiation and its seasonal variations.

Through investigation and analysis of the current research status, and in view of the existing research deficiencies, the main objective of this study is established as optimizing the design of near-space airship propellers to improve efficiency as much as possible and reduce overall mass while providing sufficient thrust. First, this paper verifies the feasibility of using an efficient engineering method to optimize the propeller in the overall airship scheme. Using this method for iterative calculations of the overall scheme, the design results show that the propeller obtained using this efficient engineering method has almost the same performance as that obtained from single-objective optimization, and the total mass of the airship is also almost the same. Therefore, this method can be used for airship propeller optimization to reduce computational effort and improve optimization efficiency. Second, this paper proposes a propeller design method considering different flight speeds during day and night, and carries out optimization design for two overall schemes: one with optimal daytime propeller efficiency and another with optimal nighttime propeller efficiency. This method makes full use of the advantage of sufficient daytime energy, increasing the daytime flight speed and reducing the nighttime flight speed while keeping the station-keeping performance unchanged. Although the increased daytime speed increases the airship’s flight drag and the daytime output power, resulting in an increase in the mass of the propulsion system and solar panels, the reduced nighttime speed lowers nighttime energy consumption and therefore significantly reduces the mass of the storage batteries. In the optimal daytime scheme, although the propeller efficiency during both daytime and nighttime is lower than that of the efficient engineering method, the total mass of the airship still decreases by 28%. In the optimal nighttime scheme, the propeller efficiency during both daytime and nighttime increases compared with the optimal daytime scheme, thereby achieving a further reduction in total airship mass.

2. Propeller Design Methods

2.1. Propeller Performance Prediction Based on Blade Element Momentum Theory

The blade element momentum theory is commonly used to calculate the performance of propellers. This theory divides the propeller blades into a finite number of blade element units, calculates the aerodynamic performance of each unit separately, and integrates the results to obtain the overall aerodynamic performance of the propeller.

The forces and velocities acting on a blade element are shown in Figure 1, where $V_{\infty }$ represents the forward speed of the propeller, $n_s$ represents the rotational angular velocity of the propeller, $2\pi n_{s}r$ represents the circumferential rotational speed of the propeller, $w_a$ stands for the axial induced velocity, $w_t$ stands for the circumferential induced velocity, ${\alpha }_i$ is the induced angle of attack, $\phi$ represents the flow angle, $\theta$ is the installation angle, $r$ is the radial position of the blade element, $R$ is the radius of the propeller, $dL$, $dD$, $dT$ and $dF$ are the aerodynamic lift, aerodynamic drag, thrust and circumferential force generated by the blade element, and $b$ is the cord length of the blade element.

According to the geometry relationship, the flow angle can be expressed as [25]:

$$\tan \phi ={\displaystyle \frac{V_{\infty }+w_a}{2\pi n_{s}r-w_t}}$$

(1)

The axial induction factor $a_a$ and the circumferential induction factor $a_t$ can be defined as follows:

$$\left\{\begin{array}{c}{a}_a={\displaystyle \frac{w_a}{V_{\infty }}}\\ a_t={\displaystyle \frac{w_t}{2\pi n_{s}r}}\end{array}\right.$$

(2)

The angle of attack of the blade element and actual airflow velocity can be expressed as:

$$\alpha =\theta -\phi$$

(3)

$$V=\sqrt{{\left(V_{\infty }+w_a\right)}^2+{\left(2\pi n_{s}r-w_t\right)}^2}=V_{\infty }{\displaystyle \frac{{\left(1+a_a\right)}^2}{\sin \phi }}$$

(4)

$$\tan \phi ={\displaystyle \frac{V_{\infty }}{2\pi n_s{r}_t}}{\displaystyle \frac{1+a_a}{1-a_t}}=\tan {\phi }_0{\displaystyle \frac{1+a_a}{1-a_t}}$$

(5)

The aerodynamic lift and drag of the blade element can be expressed as:

$$\left\{\begin{array}{c}dL={\displaystyle \frac{1}{2}}\rho V^2{C}_{L}bdr\\ dD={\displaystyle \frac{1}{2}}\rho V^2{C}_{D}bdr\end{array}\right.$$

(6)

The thrust and circumferential force of the blade element can be expressed as:

$$\left\{\begin{array}{c}dT={\displaystyle \frac{1}{2}}\rho V^2{C}_{T_V}bdr\\ dF={\displaystyle \frac{1}{2}}\rho V^2{C}_{F_V}bdr\end{array}\right.$$

(7)

According to Figure 1, the dimensionless thrust and circumferential force coefficient $C_{T_V}$ and $C_{F_V}$ can be expressed as:

$$\left\{\begin{array}{c}{C}_{T_V}=C_L\cos \phi -C_D\sin \phi \\ C_{F_V}=C_L\sin \phi +C_D\cos \phi \end{array}\right.$$

(8)

where the dimensionless lift coefficient $C_L$ and drag coefficient $C_D$ need to be determined based on the Reynolds number and the angle of attack of the airflow.

The thrust and torque of the blade element can be expressed as:

$$\left\{\begin{array}{l}dT={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{C}_{T_V}b{\displaystyle \frac{{(1+a_a)}^2}{{\sin }^2\phi }}dr\\ dM=rdF={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{C}_{F_V}b{\displaystyle \frac{{(1+a_a)}^2}{{\sin }^2\phi }}rdr\end{array}\right.$$

(9)

According to momentum theory, the above function can be expressed as:

$$\left\{\begin{array}{l}dT=4\pi r\rho V_{\infty }^2{a}_a\left(1+a_a\right)dr{K}_P\\ dM=4\pi r^2\rho V_{\infty }\left(2\pi n_{s}r\right)a_t\left(1+a_a\right)dr{K}_P\end{array}\right.$$

(10)

where $K_P$ represents the Prandtl momentum loss factor [43]. This correction factor is needed because the actual increase in fluid velocity should only be $K_P$ times the increase at the blade tip on the vortex surface. Its expression is as follows:

$$K_P={\displaystyle \frac{2}{\pi }}\arccos e^{-k_p}$$

(11)

where $k_P$ is obtained with:

$$k_p={\displaystyle \frac{N_B}{2}}{\displaystyle \frac{1-\overline{r}}{\sin {\phi }_t}}$$

(12)

where ${\phi }_t$ represents the airflow angle at the blade tip, and the dimensionless radius position is $\overline{r}=r/R$. At the beginning of the calculation, the value of ${\phi }_t$ is generally unknown. However, it can be calculated using ${\phi }_0$ as an initial value without introducing significant errors. Therefore, the formula can be written as follows:

$$k_p={\displaystyle \frac{N_B}{2}}{\displaystyle \frac{1-\overline{r}}{\sin {\phi }_0}}$$

(13)

where $\sigma$ represents the solidity of the propeller:

$$\sigma ={\displaystyle \frac{N_{B}b}{2\pi r}}$$

(14)

The axial induction factor $a_a$ and the circumferential induction factor $a_t$ can be defined as follows:

$$\left\{\begin{array}{l}{a}_a={\left({\displaystyle \frac{4K_P{\sin }^2\phi }{\sigma C_{T_V}}}-1\right)}^{-1}\\ a_t=a_a\tan \phi {\displaystyle \frac{C_{F_V}}{C_{T_V}}}\end{array}\right.$$

(15)

When calculating the performance of a propeller, the axial induction factor $a_a$ and the circumferential induction factor $a_t$ are computed using an iterative method. After calculating the $a_a$ and $a_t$ of each blade element, the total thrust and torque of the propeller can be obtained with:

$$\left\{\begin{array}{l}T={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{N}_B{\displaystyle {\int }_{R_{hub}}^{R}b\frac{{(1+a_a)}^2}{{\sin }^2\phi }{C}_{T_V}dr}\\ M={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{N}_B{\displaystyle {\int }_{R_{hub}}^{R}b\frac{{(1+a_a)}^2}{{\sin }^2\phi }{C}_{F_V}rdr}\end{array}\right.$$

(16)

where $R_{hub}$ is the radius of the hub of the propeller.

Non-dimensional coefficients such as thrust coefficient $C_T$, power coefficient $C_P$, and efficiency $\eta$ are commonly used, which are defined as follows:

$$\left\{\begin{array}{l}{C}_T={\displaystyle \frac{T}{\rho n_s^2{D}^4}}\\ C_P={\displaystyle \frac{2\pi n_{s}M}{\rho n_s^3{D}^5}}\\ \eta ={\displaystyle \frac{T{V}_{\infty }}{2\pi n_{s}M}}={\displaystyle \frac{C_T}{C_P}}\lambda \end{array}\right.$$

(17)

where λ is the advance ratio, representing the ratio between the forward speed and the rotational speed of the propeller, and its expression is:

$$\lambda ={\displaystyle \frac{V_{\infty }}{n_{\mathrm{s}}D}}$$

(18)

2.2. Propeller Performance Prediction Based on Characteristic Blade Element

The aerodynamic losses of a propeller are primarily concentrated at the blade root and tip. Since the root bears loads such as rotational centrifugal force and the bending moment of the entire propeller, it is typically thicker and rotates at a lower speed, thus failing to exhibit good aerodynamic performance. The vortex at the tip increases induced aerodynamic drag and reduces thrust. If the rotational speed is too high, shock wave losses may also occur at the tip. Therefore, the thrust is primarily generated between 0.7 R and 0.9 R of the propeller blades. The performance of the propeller is primarily determined by the cross-sections at these locations, and can be expressed using the characteristic blade element, $r_{char}$. The thrust and torque of the propeller can be calculated as follows:

$$\left\{\begin{array}{l}T={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{N}_B{\displaystyle \frac{{(1+a_a)}^2}{{\sin }^2\phi }}{\left.C_{T_V}\right|}_{r_{char}}{\displaystyle {\int }_{r_{char}-r_0}^{r_{char}+r_0}dr}\\ M={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{N}_{B}b{\displaystyle \frac{{(1+a_a)}^2}{{\sin }^2\phi }}{\left.C_{F_V}\right|}_{r_{char}}{\displaystyle {\int }_{r_{char}-r_0}^{r_{char}+r_0}rdr}\end{array}\right.$$

(19)

where $r_0$ represents the prediction parameter indicating the integration range on both sides of the characteristic blade element. If $r_{char}$ = 0.75 R is selected as the characteristic blade element for calculation, the propeller performance can be calculated using the following expression:

$$\left\{\begin{array}{l}T={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{N}_B{\displaystyle \frac{{(1+a_a)}^2}{{\sin }^2\phi }}{\left.C_{T_V}\right|}_{0.75R}·2r_0\\ M={\displaystyle \frac{1}{2}}\rho V_{\infty }^2{N}_{B}b{\displaystyle \frac{{(1+a_a)}^2}{{\sin }^2\phi }}{\left.C_{F_V}\right|}_{0.75R}·1.5R{r}_0\\ \eta ={\displaystyle \frac{4\lambda }{3\pi b}}{\displaystyle \frac{{\left.C_{T_V}\right|}_{0.75R}}{{\left.C_{F_V}\right|}_{0.75R}}}\end{array}\right.$$

(20)

This indicates that the propeller’s thrust and torque can be predicted through the aerodynamic parameters of the characteristic blade element and $r_0$. The efficiency of the propeller is independent of the prediction parameter $r_0$ and is only related to the performance at the characteristic blade element. This prediction method only requires iterative calculations of the performance at the characteristic blade element, rather than iterative calculations of the performance of each section of the blade. It can predict the propeller’s performance with less computational effort.

2.3. Propeller Performance Verification with CFD

The blade element momentum theory neglected the radial flow of the blades and the mutual interference between them. Furthermore, in the near-space environment, the propeller operates under low-Reynolds-number conditions, which imposes certain limitations on the calculation of its aerodynamic performance. Therefore, computational fluid dynamics (CFD) simulation can be employed to calculate the aerodynamic performance of the propeller.

In the CFD simulation, the Reynolds-Averaged Navier–Stokes (RANS) equation based on density and the $k-\omega$ Shear Stress Transport (SST) turbulence model are employed. The flow field for CFD simulation of the propeller is generated, as shown in Figure 2. It consists of an internal rotating domain and an external static domain. The distance between the static domain and the inlet boundary is 20 times the radius, and the distance between the static domain and the outlet boundary is 40 times the radius. The diameter of the static domain is 20 times the radius of the propeller. The rotation speed of the rotating domain is $n_s$, while the static domain is stationary. Interface boundary conditions are used between the static and rotating domains to exchange data. The static domain adopts velocity inlet boundary and pressure outlet boundary conditions, while the surface of the propeller is set to wall boundary conditions.

The rotating domain consists of 5.55 million grids, with triangular grids on the blade surface and hexahedral grids surrounding the boundary layer of the blade surface, as shown in Figure 3. Tetrahedral grids are used to fill the space between the propeller surface and the interface.

The static domain, as shown in Figure 4, comprises 1.97 million hexahedral meshes.

Finite element calculations are performed on the divided mesh, employing the pressure-based RANS equation and the k–ω SST low-Reynolds-number-modified turbulence model. The rotational motion of the propeller is calculated using a multi-reference frame (MRF) model, utilizing a pressure–velocity coupling algorithm, and the turbulence equation is discretized using a second-order upwind scheme.

Figure 5 illustrates the pressure distribution of the propeller under design point conditions. The pressure distribution on the upper surface of the blade gradually decreases from the root to the tip and from the trailing edge to the leading edge of the blade surface, with the minimum pressure occurring between 0.75 R and 0.98 R. On the lower surface of the blade, the pressure distribution gradually increases from the root to the tip and from the middle to both the trailing and leading edges, with the maximum pressure occurring between 0.74 R and 0.99 R.

The aerodynamic performance of the propeller calculated with different methods is compared in

Table 2. Prediction of the propeller performance using different methods: Computational fluid dynamics (CFD), blade element momentum theory (BEMT)-based method and characteristic blade element (CBE)-based method.

Performance	CFD	BEMT	Difference	CBE	Difference
$C_T$	0.0760	0.0766	0.79%	0.0756	−0.53%
$C_P$	0.0592	0.0576	−2.70%	0.0593	0.17%
$\eta$	0.7336	0.7598	3.57%	0.7285	−0.70%

. The accuracy of prediction based on the characteristic blade element at the design point is superior to the prediction based on the blade element momentum theory. The assumption in the characteristic blade element that the thrust is mainly generated between 0.7 R to 0.9 R is also consistent with the pressure distribution shown in Figure 5. The result of prediction based on the characteristic blade element is basically consistent with that of the CFD result.

2.4. Particle Swarm Optimization

In the particle swarm optimization (PSO) algorithm, each particle represents a solution within the feasible region, and all particles collectively form a swarm. Each particle determines its direction and speed of movement within the feasible region based on its own historical data and the historical data of the swarm, continuously searching and iterating within the feasible region until it finds the global optimum value and optimal solution of the objective function [44,45].

The iterative computation process of the particle swarm optimization algorithm is as follows:

• Firstly, initialize the parameters such as the particle swarm size $m$, iteration count $t$, inertia factor $\omega$, local acceleration factor $c_1$, and global acceleration factor $c_2$;
• Randomly generate the position $x_i^0$ and velocity $v_i^0$ of individual particles within the feasible region;

$$\left\{\begin{array}{l}{x}_i^0=\left(x_{i1}^0,x_{i2}^0,\dots ,x_{id}^0\right),x_i^0\in D,i=1,2,\dots ,m\\ v_i^0=\left(0,0,\dots ,0\right)\end{array}\right.$$

(21)

3.

Input the current position as the optimal position of the individual particle, $p_{id}^0$, and calculate the fitness of the individual particle. Compare the fitnesses to obtain the global optimal position, $p_{gd}^0$;

$$\left\{\begin{array}{l}{p}_{id}^k\in \left\{\left.x_{id}^0,x_{id}^1,\dots ,x_{id}^k\right|\min f(x_{id}^j),i=1,2,\dots ,m,j=1,2,\dots ,k\right\}\\ p_{gd}^k\in \left\{\left.p_{1d}^k,p_{2d}^k,\dots ,p_{md}^k\right|\min f(p_{id}^k),i=1,2,\dots ,m\right\}\end{array}\right.$$

(22)

4.

Calculate the velocity $v_i^k$ and position $x_i^k$ information of individual particles, and adjust them based on possible velocity $V$ and position $X$ constraints;

$$\left\{\begin{array}{l}{v}_i^k=\omega v_i^{k-1}+c_1{r}_1\left(p_{id}^{k-1}-x_i^{k-1}\right)+c_2{r}_2\left(p_{gd}^{k-1}-x_i^{k-1}\right),\hspace{1em}k=1,2,\dots ,t\\ v_i^k=\max \left\{V_{\min },\min \left\{v_i^k,V_{\max }\right\}\right\}\end{array}\right.$$

(23)

$$\left\{\begin{array}{l}{x}_i^k=x{v}_i^{k-1}+v_i^k,\hspace{1em}k=1,2,\dots ,t\\ x_i^k=\max \left\{X_{\min },\min \left\{x_i^k,X_{\max }\right\}\right\}\end{array}\right.$$

(24)

5.

Determine the optimal position of individual particles, denoted as $p_{id}^k$, and calculate the fitness of each particle. Based on the comparison of fitnesses, obtain the global optimal position, denoted as $p_{gd}^k$;

6.

Repeat steps 4 and 5. If the result meets the requirements, stop calculating and output the result.

2.5. The Single-Objective Optimization Method for Propeller Design

For near-space airships, the majority of energy is provided to the propulsion system. Therefore, it is necessary to maximize the efficiency of the propeller, reduce energy consumption, and enhance the long-duration loitering capability of the airship. The single-objective optimization problem for near-space propellers can be expressed as follows:

$$\begin{array}{ll}& \max \eta ={\displaystyle \frac{T{V}_{\infty }}{2\pi n_{s}M}}\hfill \\ \hfill s.t.& \left\{\begin{array}{ll}\hfill \theta & \in \left[{\theta }_L,{\theta }_U\right]\\ \hfill b& \in \left[b_L,b_U\right]\\ \hfill T& \ge 100 \mathrm{N}\\ \hfill V_{\infty }& =20 \mathrm{m}/\mathrm{s},\\ \hfill n& =600 \mathrm{RPM}\end{array}\right.\end{array}$$

(25)

where the optimization objective is the efficiency $\eta$ of the propeller, and the feasible range of the installation angle $\theta$ and chord length $b$ is determined in the previous section. The inequality constraint condition is that the thrust $T$ of the propeller needs to be greater than or equal to 100 N, and the equality constraint conditions are that the forward velocity $V_{\infty }$ of the aircraft is 20 m/s and the rotational speed $n$ of the propeller is 600 RPM. The optimization process employs the particle swarm optimization method. The performance of the propeller is calculated according to blade element momentum theory.

2.6. Efficient Engineering Method for Propeller Design

The aerodynamic performance of a propeller is obtained by integrating the aerodynamic performance of each blade element on the propeller blades. If the propeller blades are divided into $n$ sections along the radial direction and each section has two design variables, namely installation angle and chord length, then calculating the propeller performance requires 2$n$ design parameters, which increases the difficulty and reduces the efficiency of the optimization. Therefore, it is necessary to reasonably reduce the number of design parameters.

As mentioned in Section 2.2, the performance of a propeller at the design point can be calculated with the aerodynamic performance of the characteristic blade element. By combining the characteristic blade element performance prediction method with the Bézier curve construction method, a new engineering design method for propellers is proposed. This method aims to obtain a propeller with higher efficiency for near-space applications. By designing the installation angle and chord length of the propeller, the installation angle of each section is a smooth curve based on the installation angle of the characteristic blade element, while the chord length is a smooth curve related to the diameter. We assume that the expressions for the installation angle and chord length are as follows:

$$\theta ={\theta }_{char}+{\left(1-\overline{r}\right)}^3{\theta }_1+3\overline{r}{\left(1-\overline{r}\right)}^2{\theta }_2+3{\overline{r}}^2\left(1-\overline{r}\right){\theta }_3+{\overline{r}}^3{\theta }_4$$

(26)

$$b=\left[{\left(1-\overline{r}\right)}^4{b}_1+4\overline{r}{\left(1-\overline{r}\right)}^3{b}_2+6{\overline{r}}^2{\left(1-\overline{r}\right)}^2{b}_3+4{\overline{r}}^3\left(1-\overline{r}\right)b_4+{\overline{r}}^4{b}_5\right]D$$

(27)

where ${\theta }_{char}$ is the characteristic installation angle at 0.6 $R$ of the propeller. The parameters in the installation angle and chord length function are trained and verified using the particle swarm optimization method, and the results are shown in

Table 3. Parameter of the installation angle and chord length curve.

Parameters	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_4$	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	${\mathit{b}}_4$	${\mathit{b}}_5$
Value	65.05	−13.49	2.539	−8.608	−0.0099	0.1882	2.92 × 10−4	0.0703	0.0115

.

The number of design parameters is significantly reduced, with only ${\theta }_{char}$ left as a design parameter. The optimization of ${\theta }_{char}$ can be achieved with Formula (25) to obtain the final installation angle and chord length curve.

The performance of propellers designed using single-objective optimization and an efficient engineering method at different working conditions is compared; the results are shown in

Table 4. Efficiency of the propellers designed using single-objective optimization and an efficient engineering method.

${\mathit{V}}_{\mathit{\infty }}$ (m/s)	RPM	T (N)	Efficiency $\mathit{\eta }$
			Single-Objective Optimization	Efficient Engineering Method	Difference
10	600	100	58.20%	57.99%	−0.21%
15			69.85%	69.75%	−0.10%
20			76.69%	76.56%	−0.13%
25			80.95%	80.75%	−0.20%
30			83.71%	83.47%	−0.24%
20	500	100	76.44%	76.14%	−0.30%
	550		76.68%	76.59%	−0.09%
	600		76.69%	76.56%	−0.13%
	650		76.55%	76.52%	−0.03%
	700		76.30%	76.10%	−0.20%
20	600	50	80.77%	79.31%	−1.46%
		75	78.95%	78.77%	−0.18%
		100	76.69%	76.56%	−0.13%
		125	74.42%	74.37%	−0.05%
		150	72.22%	71.85%	−0.37%

.

For the design points listed in the table, the efficiency of the efficient engineering method proposed in this paper is lower but close to the results of single-objective optimization. For design points where the propeller rotation speed and thrust are the same, the efficiency increase of the efficient engineering method increases with the increase in flight speed. For design points where the flight speed and thrust are the same, the efficiency increase of the efficient engineering method does not vary significantly with the increase in propeller rotation speed. For design points where the flight speed and propeller rotation speed are the same, the efficiency increase of the efficient engineering method decreases with the increase in thrust value.

3. Overall Design Method for Near-Space Airships

This section provides an introduction to the overall design scheme of near-space airships. Near-space airships mainly adopt a non-rigid structure, in which helium—lighter than air—is filled into a sealed envelope formed by welded skin panels to provide buoyancy and to serve as the installation platform for propulsion, energy, payload, and other subsystems. During the daytime, the energy system collects solar energy through solar panels and stores it in energy storage batteries, which supply power to onboard payloads and the propulsion system at night, thereby enabling day–night long-endurance flight. To achieve the above goals, it is necessary to conduct iterative design based on the airship’s geometric dimensions and the “three major equilibrium equations”—the thrust–drag equilibrium equation, the energy equilibrium equation, and the buoyancy–weight equilibrium equation—to obtain an overall design scheme that meets performance requirements. According to the current development status of near-space airship technology, the main design indicators at the forefront of airship research are presented in

Table 5. Main design indicators of the airship.

Parameter	Value
Flight altitude	20 km
Flight speed	20 m/s
Mission payload	100 kg
Mission power	5 kW

.

3.1. Airship Geometric Parameters

The shape of the airship is generally formed by rotating multiple smooth curves around the longitudinal axis. The NPL model proposed by the National Physical Laboratory in the United Kingdom is widely used [46]. This model consists of two elliptical curves that share the same minor axis. The length of the long axis of the nose is $\sqrt{2}$ times that of the long axis of the tail. The geometry is shown in Figure 6.

The shape equation is:

$$\left\{\begin{array}{l}y=b\sqrt{1-{\displaystyle \frac{x^2}{a^2}}},\hspace{1em}x\ge 0\\ y=b\sqrt{1-{\displaystyle \frac{x^2}{2a^2}}},\hspace{1em}x<0\end{array}\right.$$

(28)

The surface area and volume of the airship can then be expressed as:

$$\left\{\begin{array}{l}{S}_{body}=2\pi \left(a+\sqrt{2}a\right)b=\pi ld\\ V_{body}={\displaystyle \frac{2}{3}}\pi \left(a+\sqrt{2}a\right)b^2={\displaystyle \frac{1}{6}}\pi l{d}^2\end{array}\right.$$

(29)

where $l=a+\sqrt{2}a$ is the length of the airship and $d=2b$ is the diameter.

The stabilizing tail fins of the airship have large surface areas and complex internal structures. The surface area associated with the materials used for the tail fins is typically estimated based on the volume of the airship [47]:

$$S_{tailfin}=0.065V_{body}^{{\displaystyle \frac{2}{3}}}$$

(30)

3.2. Thrust–Drag Equilibrium Equation

7.

Airship Drag

During station-keeping flight, the propulsion system must continuously provide thrust to overcome aerodynamic drag. The drag is mainly generated by the airship hull, and its drag coefficient can be expressed as [48]:

$$C_{body}={\displaystyle \frac{0.172f^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+0.252f^{-1.2}+1.032f^{-2.7}}{{\mathrm{Re}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}}$$

(31)

where $f=l/d$ is the airship fineness ratio.

Considering the drag characteristics of tail fins, experiments [48] show that the overall drag coefficient of the airship can be expressed by $C_{airship}={\displaystyle \frac{5}{3}}{C}_{Body}$. The total drag of the airship is therefore:

$$D_{airship}={\displaystyle \frac{5}{6}}{C}_{Body}\rho V_0^2{V}_{body}^{{\displaystyle \frac{2}{3}}}$$

(32)

where $V_{body}^{2/3}$ is the characteristic area represented by the airship volume.

8.

Airship Thrust

In the airship’s propulsion system, there are $N_{prop}$ propellers, each capable of providing a thrust of $T_{prop}$; therefore, the propulsion system can provide a total thrust of $T_{airship}$:

$$T_{airship}=N_{prop}{T}_{prop}$$

(33)

9.

Thrust–Drag Equilibrium

Assume the airship is at a constant flight speed, the total drag and total thrust must satisfy:

$$T_{airship}=D_{airship}$$

(34)

3.3. Energy Equilibrium Equation

To achieve long-duration day–night flight, the energy system of a near-space airship must provide energy harvesting, storage, and distribution. Solar panels convert solar radiation into electrical power during the day and store excess energy in batteries, which power the propulsion and payload systems during twilight and nighttime when solar radiation is insufficient or unavailable. Through proper energy management and allocation, the airship can maintain energy equilibrium.

• Airship Operating Power

From the thrust–drag equilibrium of the airship in the previous subsection, the required thrust $T_{prop}$ for each propeller can be obtained, and the operating power required by the propulsion system can then be expressed as:

$$P_{thrust}=N_{thrust}{P}_{thrust\_s}=N_{prop}{\displaystyle \frac{T_{prop}{V}_0}{{\eta }_{prop}{\eta }_{thrust}}}$$

(35)

where ${\eta }_{thrust}$ is the product of efficiencies of components other than the propeller, such as the motor and controller, ${\eta }_{thrust}$ is the efficiency of the propeller, $N_{prop}$ is the number of propellers, $T_{prop}$ is the thrust generated by a single propeller, and $V_0$ is the flight speed.

The total power required for airship operation is:

$$P_{airship}=P_{thrust}+P_{equipment}+P_{payload}$$

(36)

where $P_{equipment}$ is the power consumption of avionics equipment and $P_{payload}$ is the power consumption of payload equipment.

2.

Solar Power Acquisition

When the solar panels convert solar radiation, a solar irradiance model is required for computation. Although highly accurate irradiance models are already available, they are overly complex for the overall configuration design in this paper. Therefore, a simplified solar model is used here, assuming that the solar radiation intensity is a sine function of time, with 12 h of sunlight per day and the maximum radiation intensity occurring at 12:00 noon. The solar radiation intensity received by the solar panels can then be expressed in terms of the maximum irradiance $I_{max}$ and the illumination time $t$:

$$\left\{\begin{array}{ll}I(t)=I_{\max }\sin \left({\displaystyle \frac{t-6}{12}}\pi \right),\hfill & t\in \left[6,18\right)\hfill \\ I(t)=0,\hfill & t\in \left[0,6\right)\cup \left[18,24\right)\hfill \end{array}\right.$$

(37)

The usable electrical power collected from solar panels is:

$$P_{solar}(t)=I(t)S_{solar}{\eta }_{solar}{\eta }_{MPPT}$$

(38)

where $S_{solar}$ is the area of the solar panels, ${\eta }_{solar}$ is the solar energy conversion efficiency (the average efficiency including factors such as angle and weather), and ${\eta }_{MPPT}$ is the conversion efficiency of the maximum power point tracker (MPPT) of the solar panels.

3.

Energy Equilibrium and Battery Capacity

The daytime and nighttime energy variation of the airship with constant flight speed is shown in Figure 7. Under ideal conditions, the daytime energy collected by solar panels should be sufficient to fully charge the batteries, and the stored energy should also meet nighttime usage requirements without excess battery capacity.

Then the capacity of the energy-storage battery can be obtained from the remaining daytime energy $E_1$:

$$E_{cell}={\displaystyle \frac{{\eta }_{cell}}{{\eta }_{cellout}}}{E}_1={\displaystyle \frac{{\eta }_{cell}}{{\eta }_{cellout}}}{\displaystyle {\int }_{t_1}^{t_2}\left(P_{solar}(t)-P_{airship}\right)dt}$$

(39)

where ${\eta }_{cell}$ is the charge–discharge efficiency of the energy-storage battery, ${\eta }_{cellout}$ is the depth of discharge of the battery, and the time $t_1$ at which the battery begins charging is determined from the equilibrium point between the power obtained from the solar panels and the operating power of the airship:

$$P_{solar}(t)-P_{airship}=0$$

(40)

The energy $E_{cell}$ stored in the lithium battery obtained above and the nighttime energy consumption $E_2$ should satisfy the energy balance equation:

$$E_{cell}{\eta }_{cellout}{\eta }_{cell}=E_2={\displaystyle {\int }_0^{t_1}\left(P_{airship}-P_{solar}(t)\right)dt}+{\displaystyle {\int }_{t_2}^{24}\left(P_{airship}-P_{solar}(t)\right)dt}$$

(41)

3.4. Buoyancy–Weight Equilibrium Equation

• Airship Total Buoyancy

Airships gain surplus buoyancy to carry the structure and payload by being inflated with helium, which is lighter than air. The total buoyancy of an airship can be expressed in terms of its volume as:

$$L_{airship}={\rho }_{air}{V}_{body}g$$

(42)

where ${\rho }_{air}$ is the air density at 20 km altitude.

2.

Helium Mass

To maintain the shape of a non-rigid airship, the internal helium pressure must have a certain overpressure margin, typically 10–20%. Therefore, the required volume of injected helium can be expressed as:

$$V_{helium}=1.2V_{body}$$

(43)

The density of helium corresponding to the altitude of 20 km can be obtained by converting the air density at that altitude and air molar masses. Consequently, the mass of the injected helium is:

$$m_{helium}={\rho }_{air}{\displaystyle \frac{M_{helium}}{M_{air}}}{V}_{helium}$$

(44)

where $M_{helium}$ and $M_{air}$ are the molar masses of helium and air, respectively.

3.

Envelope Mass

During the manufacturing process of a non-rigid airship, the envelope sections are joined by welding. Considering the overlap of the envelope material and auxiliary materials, the corresponding airship envelope mass increases by approximately 25%. Thus, the total surface area of the airship envelope material is:

$$S_{envelope}=1.25\left(S_{body}+S_{tailfin}\right)$$

(45)

The envelope mass can be expressed by the envelope area density as:

$$m_{envelope}={\rho }_{envelope}{S}_{envelope}$$

(46)

4.

Propulsion System Mass

The propulsion system consists of a propulsion motor, a controller, and a propeller. The mass of the propeller is determined by its design parameters, while the mass of the propulsion motor and controller is usually determined by the propulsion power and the propulsion power density ${\rho }_{thrust}$:

$$m_{thrust}=N_{prop}{m}_{prop}+P_{thrust}/{\rho }_{thrust}$$

(47)

5.

Energy System Mass

The energy system is composed of solar panels, an MPPT unit, an energy manager, storage batteries, etc., and its mass can be expressed as:

$$m_{energy}=m_{solar}+m_{enequip}+m_{cell}$$

(48)

where $m_{solar}$ is the mass of the solar panels, which can be given by the solar panel area density; $m_{enequip}$ is the mass of the MPPT and energy manager, which is usually determined by the energy equipment power density; and $m_{cell}$ is the mass of the storage batteries, which is given by the battery energy density:

$$\left\{\begin{array}{l}{m}_{solar}={\rho }_{solar}{S}_{solar}\\ m_{enequip}={\rho }_{enequip}{P}_{enequip}\\ m_{cell}={\rho }_{cell}{E}_{cell}\end{array}\right.$$

(49)

6.

Structural Mass

The airship structural components are used for installing mass components such as propulsion, avionics, energy, and payload, as well as accessories like sensors and valves installed on the airship envelope. Their mass can be estimated by the following equation:

$$m_{struc}=0.25\left(m_{envelope}+m_{thrust}+m_{energy}+m_{equipment}+m_{payload}\right)$$

(50)

where $m_{equipment}$ is the mass of the avionics equipment necessary for airship flight, and $m_{payload}$ is the mass of the airship’s mounted payload equipment.

7.

Total Airship Mass

Then the total weight of the airship can be expressed as:

$$m_{airship}=m_{helium}+m_{struc}+m_{envelope}+m_{thrust}+m_{energy}+m_{equipment}+m_{payload}$$

(51)

8.

Buoyancy–Weight Balance

The total buoyancy and total mass of the airship obtained above must satisfy the buoyancy–weight equilibrium equation:

$$L_{airship}=G_{airship}=m_{airship}\mathrm{g}$$

(52)

3.5. Airship Design Process

The design process is illustrated in Figure 8. With airship geometric parameters as input, the aerodynamic drag and required propeller thrust are computed first based on the three major equilibrium equations. The efficient engineering method is then used to obtain propeller geometric parameters, based on which propeller power, efficiency, and mass can be calculated. The total operating power, solar panel area, total mass, and buoyancy are then computed. Finally, the geometric parameters $l$ and $d$ are updated iteratively until the buoyancy–weight equilibrium is satisfied, yielding the airship design scheme.

4. Propeller Design in the Overall Airship Configuration

The geometric parameters of a near-space airship are crucial to its aerodynamic performance and exert a significant influence on its overall performance. Therefore, these geometric parameters are typically selected as the input variables in optimization design. Since airships generally perform regional station-keeping missions, requirements on range are usually not imposed. Under conditions of energy equilibrium, long-duration day–night flight can be achieved, and theoretically, the endurance of the airship is “unlimited.” As a result, the total mass of the airship is usually chosen as the optimization objective. The total mass represents the complexity of the system and is closely related to cost; airships with smaller mass are easier to manufacture and assemble. Thus, the overall airship design problem can be formulated as:

$$\begin{array}{cc}& \min \hspace{1em}{m}_{airship}\hfill \\ \hfill s.t.& \left\{\begin{array}{cc}\hfill l& \in \left[l_L,l_U\right]\hfill \\ \hfill d& \in \left[d_L,d_U\right]\hfill \\ \hfill L_{airship}& \ge G_{airship}\hfill \\ \hfill T_{airship}& =D_{airship}\hfill \\ \hfill E_1& =E_2\hfill \\ \hfill V_0& =20 \mathrm{m}/\mathrm{s}\hfill \end{array}\right.\end{array}$$

(53)

where the optimization objective function is the total mass of the airship, and the airship length $l$ and diameter $d$ are the optimization variables. The inequality constraint requires the total buoyant lift $L_{airship}$ to be greater than the total weight $G_{airship}$. The equality constraints are the propeller design results at a flight speed $V_0$ of 20 m/s, as well as the thrust–drag equilibrium and energy equilibrium. The optimization process employs the Particle Swarm Optimization algorithm. Specifically, the propeller design uses a single-objective optimization method to obtain propeller A, and the efficient engineering method to obtain propeller B. A comparison is then conducted for the propellers and the overall schemes, respectively.

The parameters such as efficiency and density in the design methodology are set as shown in

Table 6. Parameter settings for overall optimization design.

Symbol	Parameter Name	Value	Unit
$N_{prop}$	Number of propellers	8	-
$P_{equipment}$	Avionics power	1000	W
$P_{payload}$	Payload power	2000	W
${\rho }_{envelope}$	Envelope area density	0.26	kg/m2
${\rho }_{solar}$	Solar panel area density	0.20	kg/m2
${\rho }_{thrust}$	Propulsion system power density	0.008	kg/W
${\rho }_{cell}$	Battery energy density	0.0036	kg/Wh
${\eta }_{thrust}$	Propulsion system efficiency	0.82	-
${\eta }_{solar}$	Average solar panel efficiency	0.15	-
${\eta }_{MPPT}$	MPPT conversion efficiency	0.96	-
${\eta }_{cell}$	Battery charge/discharge efficiency	0.95	-
${\eta }_{cellout}$	Battery depth of discharge	0.90	-

.

4.1. Comparison of Propeller Design Results Obtained by Different Optimization Methods

The variation of installation angle and chord length along the blade radius for propellers A and B in the overall airship design is shown in Figure 9. For both propellers, the installation angle decreases gradually with increasing radius $\overline{r}$, and the chord length first increases and then decreases. The maximum chord of propeller A appears near approximately 0.4 R, while that of propeller B appears near approximately 0.3 R. At all blade sections, the installation angle of propeller A is smaller than that of propeller B, and the chord length of propeller A is larger than that of propeller B.

Based on the design results, three-dimensional models of propellers A and B are constructed, as shown in Figure 10. The figure shows that the two propellers exhibit noticeable differences near the blade tip. According to the performance prediction method, their aerodynamic performance at the design point should also differ accordingly.

Figure 11 shows the surface pressure distribution of the blades at the design-point operating condition. On the upper surface of both propellers, the pressure decreases from root to tip and from trailing edge to leading edge. The minimum pressure for propeller A appears between 0.72 R and 0.98 R, while that for propeller B appears between 0.78 R and 0.98 R. The low-pressure region of propeller A is more widely distributed. On the lower surface, the pressure increases from root to tip and from the mid-chord region toward both the leading and trailing edges. The maximum pressure for propeller A appears between 0.68 R and 0.99 R, while that for propeller B appears between 0.89 R and 0.98 R.

As shown in Figure 12, the trend of the propeller thrust coefficient is consistent for both propellers, decreasing as $\lambda$ increases. When $\lambda$ < 0.57, the thrust coefficient of propeller A is higher than that of propeller B, after which it drops and becomes lower than propeller B, with the deviation gradually increasing. The variation in the power coefficient is consistent with that of the thrust coefficient: the power coefficient of propeller A is initially higher than propeller B and then becomes lower. The efficiencies of propeller A and B are almost identical when $\lambda$ < 0.85, after which the efficiency of propeller A is lower than that of propeller B.

A comparison of propeller performance at the design point is shown in

Table 7. Performance comparison of propellers A and B at the design point.

Design Method	Propeller	Thrust T (N)	Torque M (N·m)	Thrust Coefficient	Power Coefficient	Efficiency
Single-objective optimization	A	167.36	75.21	0.1266	0.1022	70.79%
Efficient engineering method	B	167.07	75.11	0.1265	0.1021	70.81%

. At a flight speed of 20 m/s and a rotational speed of 600 RPM, the propellers designed using the single-objective optimization method and the efficient engineering method have nearly identical thrust and efficiency. The total airship mass is also nearly the same. Therefore, the engineering method is suitable for use in overall configuration design because it reduces the computational burden while maintaining design accuracy.

4.2. Comparison of Overall Airship Configurations Obtained Using Different Propeller Designs

The overall airship design results obtained using single-objective propeller optimization are shown in

Table 8. Overall configuration design results using propeller A (single-objective propeller optimization).

Parameter	Value	Parameter	Value
Airship length l (m)	127.59	Propulsion system mass mthrust (kg)	403
Airship diameter d (m)	50.31	Structural mass mstructure (kg)	2487
Envelope surface area Senvelope (m2)	20,167	Envelope mass menvolope (kg)	6619
Envelope volume Venvelope (m3)	169,102	Helium mass mhelium (kg)	2467
Solar panel area Ssolar (m2)	1343	Solar panel mass msolar (kg)	269
Battery energy Ecell (Wh)	723,043	Battery mass mcell (kg)	2582
Propeller efficiency ηpropeller	70.79%	Total airship mass mtotal (kg)	14,904

:

From Figure 13, it is evident that the envelope mass accounts for the largest share—44.4% of the total mass. The battery mass, structural mass, and helium mass each account for approximately 16–17%. The remaining subsystems collectively contribute less than 5%. Therefore, the key contributors to airship mass are the envelope, batteries, structural components, and helium. The envelope and helium mass are directly determined by the design values of airship length l and diameter d. Battery mass is primarily influenced by propeller efficiency, and propeller performance is closely related to airship geometry. Thus, minimizing the total airship mass requires the detailed design and analysis of both airship geometric parameters and propeller performance.

Using the efficient engineering method, the resulting overall airship design is shown in

Table 9. Overall configuration design results using propeller B (efficient engineering method based on characteristic blade element).

Parameter	Value	Parameter	Value
Airship length l (m)	128.41	Propulsion system mass mthrust (kg)	387
Airship diameter d (m)	50.20	Structural mass mstructure (kg)	2489
Envelope surface area Senvelope (m2)	20,252	Envelope mass menvolope (kg)	6647
Envelope volume Venvelope (m3)	169,457	Helium mass mhelium (kg)	2472
Solar panel area Ssolar (m2)	1341	Solar panel mass msolar (kg)	268
Battery energy Ecell (Wh)	721,674	Battery mass mcell (kg)	2577
Propeller efficiency ηpropeller	70.81%	Total airship mass mtotal (kg)	14,917

:

The mass distribution of the subsystems in this configuration is shown in Figure 14:

As shown in Figure 14, the envelope mass still represents the largest proportion—44.6% of the total mass—and the mass distribution of other subsystems remains consistent with that of the propeller A configuration.

4.3. Influence of Propeller Performance on the Overall Airship Configuration

Various factors influence the airship mass, involving multiple subsystems and parameters. In analyzing these influencing factors, this study focuses primarily on the effects of propeller efficiency and propeller mass on total airship mass.

Figure 15 illustrates the impact of propeller mass on the masses of individual subsystems. The figure shows that propeller mass has almost no visible impact. Data analysis confirms that propeller mass only affects the propulsion system and structural mass, but since propeller mass is extremely small relative to the total airship mass, its influence on total mass is negligible. Therefore, using efficiency as the optimization objective is sufficient for meeting the performance requirements of the airship’s propellers.

Figure 16 illustrates the impact of propeller efficiency on subsystem masses. As propeller efficiency increases, the masses of all subsystems decrease. The amount of mass reduction is proportional to the improvement in efficiency. Improving propeller efficiency reduces the required propulsion power, which reduces battery energy requirements, leading to a decrease in battery mass and ultimately reducing total airship mass. The propeller efficiency achieved using the efficient engineering method already approaches that of single-objective optimization. The additional reduction in total mass achievable by using single-objective optimization for the propeller design is therefore limited.

5. Propeller Design Considering Different Daytime and Nighttime Flight Speeds

According to the previous results, the overall configuration design already employs the efficient engineering method, and the resulting propeller performance is close to the performance limit. Therefore, relying solely on propeller optimization to reduce the total airship mass has limited effectiveness. As a result, it becomes necessary to consider using different flight speeds during daytime and nighttime to improve daytime energy utilization, reduce nighttime energy consumption, and ultimately achieve a reduction in total airship mass.

5.1. Overall Configuration with Different Daytime and Nighttime Flight Speeds

Because a near-space airship remains at altitude through static buoyancy and does not require a minimum flight speed for sustained flight as solar-powered UAVs do, the airship can fly at a relatively low speed at night when energy replenishment is limited, thereby reducing nighttime energy consumption. Figure 17 shows the energy variation of the airship with different daytime and nighttime flight speed.

The storage battery capacity $E_{cell}$ and the remaining energy during the daytime $E_1$ are still given by the energy equilibrium equation. However, the total power of the airship at night differs from that during the daytime, and the available energy at night $E_2$ is expressed as:

$$E_{cell}{\eta }_{cellout}{\eta }_{cell}=E_2={\displaystyle {\int }_{t_2}^{t_1}\left(P_{airship}\left(t\right)-P_{solar}(t)\right)dt}$$

(54)

${\mathrm{P}}_{\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{p}}\left(\mathrm{t}\right)$ becoming a function of time is mainly caused by the difference in nighttime flight speed. Since the nighttime flight speed $V_{night}$ is lower than the daytime flight speed $V_{day}$, in order to ensure that the airship’s station-keeping capability is not affected and to fully utilize the advantage of relatively sufficient daytime energy by increasing the daytime flight speed, the flight speeds must satisfy the following condition to maintain a constant average speed over the entire day:

$${\displaystyle {\int }_0^{24}{V}_{0}dt}=2\left({\displaystyle {\int }_0^{t_0}{V}_{night}dt}+{\displaystyle {\int }_{t_0}^{12}{V}_{day}dt}\right)$$

(55)

$$12V_0=V_{night}{t}_0+V_{day}\left(12-t_0\right)$$

(56)

where t₀ is the time when the flight speed switches from nighttime speed to daytime speed, as shown in Figure 18.

Therefore, the overall airship optimization problem can be expressed as:

$$\begin{array}{cc}& \min \hspace{1em}{m}_{airship}\hfill \\ \hfill s.t.& \left\{\begin{array}{cc}\hfill l& \in \left[l_L,l_U\right]\hfill \\ \hfill d& \in \left[d_L,d_U\right]\hfill \\ \hfill L_{airship}& \ge G_{airship}\hfill \\ \hfill T_{airship}& =D_{airship}\hfill \\ \hfill E_1& =E_2\hfill \\ \hfill V_{average}& =20 \mathrm{m}/\mathrm{s}\hfill \end{array}\right.\end{array}$$

(57)

The optimization objective is the total airship mass. The design variables include the airship length l, diameter d, and daytime flight speed ${\mathrm{V}}_{\mathrm{d}\mathrm{a}\mathrm{y}}$. The inequality constraints remain unchanged. The equality constraint on flight speed is modified so that the average flight speed over the full day equals 20 m/s. The optimization procedure still uses the particle swarm method. The efficient engineering method is used for propeller design. The propeller optimized for maximum daytime efficiency is denoted as propeller C, and the propeller optimized for maximum nighttime efficiency is denoted as propeller D.

5.2. Comparison of Propeller Design Results Obtained by Maximizing Daytime and Nighttime Propeller Efficiency

The variation of installation angle and chord length along the radius for propellers C and D in the improved airship design is shown in Figure 19. For both propellers, the installation angle decreases with increasing radius $\overline{r}$, and the chord length first increases and then decreases. Propeller C has a larger installation angle than propeller D at all blade sections. Near the hub, propeller D has a larger chord length, while beyond approximately 0.3 R, propeller C has a larger chord length.

Based on the design results, three-dimensional models of propellers C and D are constructed, as shown in Figure 20. The models show that the two propellers have essentially similar geometries, indicating that their aerodynamic performance at the design point should also be similar.

Figure 21 shows the pressure distribution on the blade surfaces under daytime operating conditions. On the upper surface of both propellers, the pressure decreases from root to tip and from trailing edge to leading edge. The minimum-pressure region for propeller C lies between 0.72 R and 0.98 R, while that of propeller D lies between 0.80 R and 0.99 R, indicating that the low-pressure region of propeller C is more concentrated. On the lower surface, the pressure increases from root to tip and from the middle of the chord toward both the leading and trailing edges. The maximum-pressure region for propeller C lies between 0.82 R and 0.98 R, whereas propeller D shows maximum pressure between 0.65 R and 0.99 R, indicating that the high-pressure region of propeller C is also more concentrated.

Figure 22 shows the pressure distribution on the blade surfaces under nighttime operating conditions. The upper-surface pressure trends remain consistent, decreasing from root to tip and from trailing edge to leading edge. The minimum-pressure region for propeller C lies between 0.78 R and 0.98 R, while that for propeller D lies between 0.72 R and 0.99 R. On the lower surface, pressure increases from root to tip and from mid-chord toward the leading and trailing edges. The maximum-pressure region for propeller C lies between 0.58 R and 0.99 R, while for propeller D it lies between 0.76 R and 0.99 R.

As shown in Figure 23, the thrust and power coefficients of propeller C are greater than those of propeller D across the entire range of advance ratios. The daytime thrust and power coefficients of both propellers exceed their nighttime values. Propeller D exhibits higher efficiency than propeller C across the full range of advance ratios. When $\lambda$ < 0.8 at the same advance ratio, daytime and nighttime efficiencies are nearly identical. When $\lambda$ > 0.8, the daytime efficiency exceeds the nighttime efficiency.

The performance comparison of propellers C and D is shown in

Table 10. Performance comparison of propellers C and D at their design points.

Design Method	Propeller	Time of Day	Speed V (m/s)	RPM	Thrust T (N)	Efficiency
Daytime-optimized	C	Day	28.56	600	285.43	70.64%
		Night	12.14	269.9	59.44	68.42%
Nighttime-optimized	D	Day	29.73	681.7	304.52	72.23%
		Night	10.88	270	48.20	69.09%

. Propeller C operates at a lower daytime flight speed than propeller D, resulting in lower thrust requirements and correspondingly lower efficiency, indicating that the abundant daytime energy advantage is not fully utilized. At night, propeller C operates at a higher flight speed than propeller D, requiring more thrust but still exhibiting lower efficiency.

From the previously obtained comparison, under identical flight speed and rotational speed, the efficient engineering method yields a performance close to single-objective optimization near the design point. However, as thrust decreases, the deviation between the two methods increases, leading to nighttime efficiencies below 70%.

5.3. Comparison of Overall Configurations Obtained by Maximizing Daytime and Nighttime Propeller Efficiency

• Optimum Propeller Efficiency During Daytime

In the overall design process, the efficient engineering method is first used to design the propeller operating state for daytime flight, ensuring that the propeller maintains a high efficiency during the daytime operating state throughout the iterative calculation. For the nighttime flight state, the propeller designed for the daytime operating state is used to calculate the operating state that satisfies the nighttime flight drag under the nighttime flight speed condition, which then allows for the calculation of the power required for the airship operation at night.

By designing the propeller for maximum daytime efficiency, the overall airship design results are obtained as shown in

Table 11. Overall configuration using propeller C (daytime-optimized).

Parameter	Value	Parameter	Value
Airship length l (m)	107.44	Propulsion system mass mthrust (kg)	940
Airship diameter d (m)	46.53	Structural mass mstructure (kg)	1789
Envelope surface area Senvelope (m2)	15,707	Envelope mass menvolope (kg)	5157
Envelope volume Venvelope (m3)	121,813	Helium mass mhelium (kg)	1777
Solar panel area Ssolar (m2)	1652	Solar panel mass msolar (kg)	330
Battery energy Ecell (Wh)	183,058	Battery mass mcell (kg)	654
Propeller nighttime speed RPMnight	269.9	Total airship mass mtotal (kg)	10,732
Nighttime flight speed Vnight (m/s)	12.14	Propeller nighttime efficiency ηpropeller	68.44%
Daytime flight speed Vday (m/s)	28.56	Propeller daytime efficiency ηpropeller	70.64%

:

The mass distribution of different subsystems of the airship is shown in Figure 24:

As can be seen from Figure 24, when the daytime propeller efficiency is maximized, the envelope mass remains the largest proportion in the airship’s mass distribution, accounting for 48.1% of the total mass, an increase of 3.5% compared to the engineering method. The mass percentages of structural components and helium remain consistent with the engineering method, falling between 16% and 17%. The proportion of the storage battery shows a significant reduction, dropping from 17.3% in the engineering method to 6.1%. The percentages of the propulsion system and solar panels increased by 6.2% and 1.3%, respectively.

Due to the increased daytime flight speed, the airship’s flight drag increases, leading to a rise in the power requirement for the propulsion system, which in turn increases the mass of the propulsion system. The increase in the airship’s daytime power also results in an increased required area for solar panels, thereby raising the mass of the solar panels.

The significant reduction in the proportion of the storage battery is primarily achieved by capitalizing on the abundant energy supply during the daytime for high-speed flight, which allows for lower-speed flight at night without affecting the airship’s station-keeping capability. Because the flight speed is lower at night, the airship’s drag is reduced, which in turn lowers the required propeller output power. This substantially reduces the energy needed for nighttime flight, consequently decreasing the mass of the storage battery. The reduction in battery mass simultaneously leads to a reduction in the mass of the structural components. The overall decrease in total mass reduces the airship’s lift requirement, allowing the airship’s size to be smaller, which further reduces the envelope mass. The final iteration yields a design scheme with a substantially reduced total mass.

2.

Optimum Propeller Efficiency During Nighttime

The overall design scheme utilizing different flight speeds between day and night can significantly reduce the airship’s total mass. In the preceding discussion, the goal of reducing the airship’s total mass was achieved by optimizing for the best daytime propeller efficiency. Next, we will discuss whether the total mass can be further optimized by reducing nighttime energy requirements through designing the propeller operating state for optimum nighttime efficiency using the efficient engineering method based on characteristic blade elements. By designing the propeller for maximum nighttime efficiency, the overall airship design results are obtained, as shown in

Table 12. Overall configuration using propeller D (nighttime-optimized).

Parameter	Value	Parameter	Value
Airship length l (m)	106.36	Propulsion system mass mthrust (kg)	991
Airship diameter d (m)	46.26	Structural mass mstructure (kg)	1750
Envelope surface area Senvelope (m2)	15,456	Envelope mass menvolope (kg)	5074
Envelope volume Venvelope (m3)	119,160	Helium mass mhelium (kg)	1738
Solar panel area Ssolar (m2)	1716	Solar panel mass msolar (kg)	343
Battery energy Ecell (Wh)	144,536	Battery mass mcell (kg)	516
Propeller daytime speed RPMday	681.7	Total airship mass mtotal (kg)	10,490
Daytime flight speed Vday (m/s)	29.73	Propeller nighttime efficiency ηpropeller	72.23%
Nighttime flight speed Vnight (m/s)	10.88	Propeller daytime efficiency ηpropeller	69.09%

:

Based on the optimization design results, the mass composition proportions of the airship’s various systems are represented by the bar chart in Figure 25. When the nighttime propeller efficiency is maximized, the envelope mass remains the largest proportion in the airship’s mass distribution, accounting for 48.4% of the total mass, an increase of 0.3% compared to the daytime optimum scheme. The mass percentages of structural components and helium remain consistent with the efficient engineering method. The proportion of the storage battery is reduced once again, decreasing from 6.1% in the daytime scheme to 4.9%. The percentages of the propulsion system and solar panels increased by 0.6% and 0.2%, respectively.

Due to the further reduction in nighttime flight speed, the energy requirement at night decreases, leading to a reduction in the mass of the storage battery. However, the increase in daytime flight speed results in a rise in the maximum power of the propulsion system, thus increasing the mass of the propulsion system. Under the comprehensive effect of these multiple factors, the total mass of the airship is ultimately reduced.

6. Comparison of Different Design Schemes

In this paper, the propeller within the overall scheme is designed using both the single-objective propeller optimization method and the efficient engineering method based on characteristic blade elements. Furthermore, this efficient engineering method is employed to design overall schemes for optimum daytime propeller efficiency and optimum nighttime propeller efficiency. The comparison of the design scheme results is shown in

Table 13. Overall configuration design results with different daytime and nighttime flight speeds.

Parameter Name	Single-Objective Optimization	Engineering Method Based on Characteristic Blade Element		Daytime- Optimized		Nighttime- Optimized
Airship length l (m)	127.59	128.41	+0.6%	107.44	−15.8%	106.36	−16.6%
Airship diameter d (m)	50.31	50.2	−0.2%	46.53	−7.5%	46.26	−8.1%
Daytime flight speed Vday (m/s)	20	20	0.0%	28.56	+42.8%	29.73	+48.7%
Nighttime flight speed Vnight (m/s)				12.14	−39.3%	10.88	−45.6%
Propeller daytime efficiency ηpropeller	70.79%	70.81%	0.0%	70.64%	−0.2%	72.23%	+2.0%
Propeller nighttime efficiency ηpropeller				68.44%	−3.3%	69.09%	−2.4%
Propulsion system mass mthrust (kg)	403	387	−4.0%	940	+133.3%	991	+145.9%
Structural mass mstructure (kg)	2487	2489	+0.1%	1789	−28.1%	1750	−29.6%
Envelope mass menvolope (kg)	6619	6647	+0.4%	5157	−22.1%	5074	−23.3%
Helium mass mhelium (kg)	2467	2472	+0.2%	1777	−28.0%	1738	−29.6%
Solar panel mass msolar (kg)	269	268	−0.4%	330	+22.7%	343	+27.5%
Battery mass mcell (kg)	2582	2577	−0.2%	654	−74.7%	516	−80.0%
Total airship mass mtotal (kg)	14,904	14,917	+0.1%	10,732	−28.0%	10,490	−29.6%

:

It can be seen from the table that the propeller efficiency obtained using the efficient engineering method based on characteristic blade elements is almost identical to that obtained by single-objective optimization, and the difference in the airship’s total mass is also small. Therefore, this engineering method can be reliably used to design the propeller within the airship’s overall scheme. Meanwhile, the overall scheme considering different flight speeds between day and night shows a significant improvement in the airship’s total mass compared to the original design scheme, with the optimum nighttime propeller efficiency scheme showing the best improvement.

Under the daytime-optimized scenario, although the propeller efficiencies during both day and night are lower than the propeller efficiency achieved by the efficient engineering method based on characteristic blade elements, the total mass is reduced by 4185 kg, a decrease exceeding 28%. This is because the airship increased its daytime flight speed, raising the power output during the day. While this caused an increase in the mass of the propulsion system and solar panels, it reduced the power output required at night, thereby significantly decreasing the mass of the storage battery. This led to a subsequent reduction in the airship’s size and overall mass. In the nighttime-optimized scenario, the airship’s daytime speed is higher, and its nighttime speed is lower, with improvements in propeller efficiency during both periods, leading to a further reduction in the airship’s size and total mass.

7. Conclusions

Two propellers are designed using different design methods—single-objective optimization and an efficient engineering method. The aerodynamic performance and airship total mass obtained using the two propeller designs are compared. In order to further minimize the total mass of the airship, a propeller design method considering different flight speeds between day and night was proposed. This method fully utilizes the advantage of abundant daytime energy to increase the daytime flight speed and reduce the nighttime flight speed while maintaining constant station-keeping performance. Two additional propellers are designed and compared, one optimized for maximum daytime efficiency and one optimized for maximum nighttime efficiency. The analysis results are as follows:

• The propeller’s efficiency affects the total mass by influencing the mass of the storage battery, while the mass of the propeller itself has a smaller impact on the total mass.
• The propeller obtained through the efficient engineering method and the propeller obtained through single-objective optimization exhibit an almost identical performance, and the airship’s total mass is also nearly the same. Therefore, the efficient engineering method can be used to design the propeller, and can reduce the calculation load and improve optimization efficiency.
• Although the increased daytime speed raises the airship’s flight drag and increases the daytime power output, leading to an increased mass for the propulsion system and solar panels, the reduced nighttime speed lowers the nighttime energy consumption, thereby substantially reducing the mass of the storage battery.
• In the daytime-optimized scheme, although both daytime and nighttime propeller efficiencies are lower than those achieved by the constant speed design, the total airship mass still decreased by 28.0%. In the nighttime-optimized scheme, the propeller efficiencies during both day and night improved compared to the daytime-optimized scheme, resulting in a 29.6% decrease in the airship’s total mass.

8. Discussion

This paper takes the propeller of a near-space airship as the research object and conducts a study on the aerodynamic performance optimization of the propeller. A propeller design method considering different daytime and nighttime flight speeds is presented. The improved overall design method proposed in this paper has supported the development of an airship prototype, and the propeller designed using the propeller design method proposed in this paper has completed its design and passed high-altitude experimental verification.

Suggestions for future work are as follows:

• In this paper, the aerodynamic performance is mainly obtained from theoretical analysis and numerical simulations. In the next step, wind tunnel tests or real-environment experiments can be conducted to further validate the performance prediction method.
• The next step may include collaborative optimization with the airfoil geometry to further improve propeller efficiency.
• Although the overall design method is improved in this paper, the daytime and nighttime speeds are still fixed values. In the future, variable flight speeds that change with time or with the input energy during daytime operation may be considered, to fully utilize the abundant daytime energy, reduce nighttime energy demand, and ultimately achieve a further reduction in the total mass. The effect of speed variation on other systems such as the structure integrity, control system and station keeping accuracy can also be investigated in the future.
• Maintaining the average speed of daytime and nighttime at a constant value is an initial criterion to ensure that the movement and station-keeping ability of the airship is on the same level as the original design. Further research can be performed to look into whether low speeds at night are detrimental to the station keeping ability of the airship.