An Improved Climbing Strategy for High-Altitude Fast-Deploy Aerostat Systems

Abstract

Due to the restrictions associated with the actual deployment time, the flight performance of traditional aerostat systems in the climbing process needs to be improved to reduce the climbing time and the horizontal movement. This paper presents a scheme comprising a dual-balloon system, including an assisting system and a station-keeping system. In this study, a thermal and dynamic model for an aerostat system in the climbing course was established. To verify the theoretical model, flight experiments including traditional and improved aerostat systems were conducted. The performance of the improved aerostat system was compared with that of the traditional aerostat system. In addition, in this paper, the effects of helium mass in the tow balloon and payload mass on the climbing performance and equilibrium height of the improved aerostat system are discussed in detail. The results demonstrate that larger tow balloon volume does not guarantee better performance. With a fixed payload mass, equilibrium height initially rises sharply with helium mass but soon plateaus. Compared to traditional zero-pressure balloons, the dual-balloon system cuts ascent time by two-thirds. The proposed conceptual design and theoretical model could be a pathway towards achieving rapid deployment in high-altitude dual-balloon systems.

1. Introduction

High-altitude aerostat systems are scientific platforms that carry out different tasks and fulfill different roles, ranging from contributing to scientific data collection to being used as reconnaissance devices and near-space radars [1,2]. In last 20 years, various countries have established projects and carried out research related to aerostat systems, with such activities being especially evident in developed countries in Europe and America, greatly promoting the development of aerostat systems [3].

Generally, a high-altitude aerostat system mainly includes a near-space airship and a high-altitude balloon [4,5]. The airship has a propulsion and control system that permits it to be flown to a desired loitering location and to be maintained in that location for a period of time; however, these airships have some shortcomings, as they are complex systems that are inconvenient to control [6]. For a high-altitude balloon, station-keeping flight is one of the key considerations because the vehicle needs to have the ability to fly at high altitude for an extended period of time [7]. Many studies carrying out high-altitude scientific experiments on high-altitude balloons have been conducted. Studies on high-altitude aerostat systems focus on energy and management, thermal issues, the control system design, and the station-keeping capability.

As alluded to above, high-altitude aerostat systems are capable of achieving prolonged station-keeping flight at high altitude [8,9]. Many investigations on the dynamic performance and trajectory of such systems have been published. Rotter [10] analyzed the influence of balloon volume and gross inflation during the ascent, finding that the randomness of gross inflation has the strongest effect on the climbing rate uncertainty. Liu et al. [11] proposed a novel numerical model including a dynamic model and a thermal model to investigate the thermal and dynamic characteristics of scientific balloons in both climbing and floating conditions. The dynamic model was solved numerically by using a computer program developed with Matlab/Simulink to calculate the velocity and trajectory. Sushko et al. [12] studied high-altitude balloon dynamics by analyzing the effect of variation in latex balloon volume on the buoyant lift of a balloon. Extending the in-air mission duration increases the platform’s utility in atmospheric research. Palumbo et al. [13] developed a new software tool to predict the flight trajectory (horizontal and vertical motions) and thermal behavior of high-altitude zero-pressure balloons. The software tool was successfully used for flight prediction and showed an error in trajectory forecast of less than 1%.

Fast-deploy aerostat systems’ climbing performance requirements are higher than those for classical sounding balloons. They require a shorter climbing time, in addition to closer horizontal movement. Zhang et al. [14] investigated the effects of launch conditions on the flight performance of a high-altitude balloon during climbing phase. They concluded that a larger inflating quantity value is required to increase rising speed based on film strength safety. Öznur et al. [15] investigated a new model that proved to be better at predicting the ascent and floating at constant altitude for longer periods. It also improved descent process prediction and the prediction of the balloon’s 3D motion. The horizontal movement of a high-altitude balloon depends on wind speed and direction, as well as altitude, all of which are uncontrollable [16]. Christopher et al. [17] studied the performance of a passive balloon guidance system and the effects of wind data and line of sight on balloon velocities in the ascent and descent stages. Lloyd et al. [18] investigated the ensemble wind and overall weather conditions for balloon mission locale and for well-behaved weather patterns. Furthermore, they studied and optimized a methodology to enable throughput of day-of-launch winds into time- and location-dependent go/no go decision criteria within an hour.

The aim of implementing a faster phase velocity is to improve the fixed area coverage and implement rapid flight mission deployment [19]. In this study, a double-balloon system was designed to improve phase performance. However, there are a few existing studies on the dynamics and rising characteristics of double-balloon systems. The double-balloon system model used in this study was inspired by the tethered balloon model. While discussing conventional, common aircraft reported in the literature, Waghela [20] presented a tether-and-sail passive balloon system model to analyze system performance, focusing on factors such as tether location and sail mass. Nock [21] explored the physics of operation, aerodynamic and system performance, and concept of operations of a winged balloon guidance system. Based on data from prototype test flights and experimental data, the winged balloon guidance system proved to provide the best and most efficient technology available for guiding high-altitude balloons.

The above-mentioned scholars and specialists numerically investigated the climbing performance of high-altitude aerostat systems. Over the last few decades, to prove the practicability and reliability of the high-altitude aerostat system, some research institutions have carried out some experiments. For example, National Aeronautics and Space Administrations (NASA) studied the design approach used to enhance super-pressure balloon deployment at heavier payloads and higher altitudes [22]. The French space agency CNES (Centre National d’Etudes Spatiales) supported a significant scientific ballooning program and outlined a new command control system for big zero-pressure stratospheric balloons [23,24]. The Japan Aerospace Exploration Agency (JAXA) carried out a balloon campaign from Alice Springs in Australia and aimed to observe astrophysical gamma rays with a high angular resolution using an emulsion optical telescope [25]. In spite of these existing studies, experimental investigations of the climbing performance of double-balloon systems in the near-space environment are still rare.

The purpose of the present study was to conduct experimental and numerical investigations of the climbing performance of a high altitude dual-balloon system. In this paper, a numerical model is developed to investigate the effects of the tow balloon, helium mass, and payload mass on the climbing performance of an aerostat system. The method used to research the equilibrium height of the aerostat system was developed to estimate the initial lift gas inflation so as to ensure that the balloon ascends at a reasonable climbing rate and remains in the floating area for a prolonged period of time.

2. High Altitude Dual-Balloon System

To achieve long-endurance regional station-keeping, the flight altitude needs to be controlled in the quasi-zero wind layer, as shown in Figure 1. The mass of the lifting gas inside the balanced balloon directly affects its ascent speed and final equilibrium height.

In order to quickly reach the mission altitude when the equilibrium height of the balloon is constant, traditional balloons usually adopt the strategy of injecting excessive lifting gas and discharging excess gas after reaching the designated height. This strategy requires preparing a set of exhaust devices and corresponding control units suitable for the height of the near space. Moreover, this strategy can result in prolonged altitude oscillations and significant horizontal displacement due to the limited deflation speed of traditional deflation devices for zero pressure balloons. As shown in Figure 1, the buckle release unit under the tow balloon was unlocked, and the balance balloon was dropped after reaching the balance height. After releasing the balance balloon, the tow balloon continued to accelerate upwards due to the remaining buoyancy. After losing the upward pulling force from the tow balloon, the balance balloon oscillates at the designed equilibrium height due to the combined effect of motion inertia and aerodynamic drag. At the same time, the tow balloon carried the release unit and continued to rise until it exploded, then descended slowly and opened the parachute after reaching the parachute opening speed.

Therefore, this study designed a rapid deployment dual-balloon system. A tow balloon is added above the balance balloon to accelerate the ascent speed of the system. And the balance balloon only needs to achieve height balance at a specific height. This strategy can effectively reduce the required ascent time and horizontal displacement during the ascent process.

3. Theoretical Analysis

3.1. Atmospheric Model

The atmospheric model is categorized into three parameters: pressure, P; temperature, T; and density, ρ. These parameters are calculated from sea level to an altitude of 32 km [26].

$$P=\left\{\begin{array}{ll}{P}_{SL}\cdot W^{5.2559}\hfill & 0\le H\le 11 \mathrm{km}\hfill \\ 1.1953\times {10}^{-1}{P}_{SL}\cdot W\hfill & 11\le H\le 20 \mathrm{km}\hfill \\ 2.5158\times {10}^{-2}{P}_{SL}\cdot W^{-34.1629}\hfill & 20<H\le 32 \mathrm{km}\hfill \end{array}\right.$$

(1)

$$T=\left\{\begin{array}{ll}288.15\cdot W\hfill & 0\le H\le 11 \mathrm{km}\hfill \\ 216.65\hfill & 11\le H\le 20 \mathrm{km}\hfill \\ 216.65\cdot W\hfill & 20<H\le 32 \mathrm{km}\hfill \end{array}\right.$$

(2)

$$\rho =\left\{\begin{array}{ll}{\rho }_{SL}\cdot W^{4.2559}\hfill & 0\le H\le 11 \mathrm{km}\hfill \\ 1.5898\times {10}^{-1}{\rho }_{SL}\cdot W\hfill & 11\le H\le 20 \mathrm{km}\hfill \\ 3.2722\times {10}^{-2}{\rho }_{SL}\cdot W^{-35.1629}\hfill & 20<H\le 32 \mathrm{km}\hfill \end{array}\right.$$

(3)

where the parameter $W$ is introduced to simplify the calculation of the atmospheric parameters [27] and represents different functions in each segment. The sea level values are selected as reference values, and the formula for calculating atmospheric parameters is denoted using the subscript $S L$.

$$W=\left\{\begin{array}{ll}1-{\displaystyle \frac{H}{44.3308}}\hfill & 0\le H\le 11 \mathrm{km}\hfill \\ \exp ({\displaystyle \frac{14.9647-H}{6.3416}})\hfill & 11\le H\le 20 \mathrm{km}\hfill \\ 1+{\displaystyle \frac{H-24.9021}{221.552}}\hfill & 20<H\le 32 \mathrm{km}\hfill \end{array}\right.$$

(4)

In Equations (1)–(4), the gravitational potential altitude $H$ is introduced to replace the geometric altitude $Z$, and the gravitational field is taken into account, along with the influence of altitude [28]. The conversion between them expressed as

$$H=Z/(1+Z/R_0)$$

(5)

where $R_0$ is the Earth’s radius.

3.2. Constrained-Volume Zero-Pressure Balloon Geometry

The expansion of the balloon as a function of altitude can safely be assumed to be governed by the ideal gas law. According to the ideal gas law, the state of an amount of gas is determined by its pressure, volume, and temperature according to the following equation [29]:

$$P_{gas}{V}_{balloon}=n{R}_{gas}{T}_{gas}$$

(6)

where $P_{gas}$ is the pressure of the lifting gas, $V_{balloon}$ is the volume of the balloon, $n$ is the amount of substance of the lifting gas, $T_{gas}$ is the absolute thermodynamic temperature of the lifting gas, and $R_{gas}$ is the ideal gas constant of the lifting gas [30].

The lift gas pressure is described as

$$P_{gas}=P_{air}+\Delta P$$

(7)

where $\Delta P$ is the differential pressure between the lifting gas and external air. For a constrained-volume system, the approximate average differential pressure due to buoyancy in a zero-pressure design can be found via the following [31]:

$$\Delta P={\omega }_1\cdot b_{buo}\cdot V_{balloon}^{1/3}$$

(8)

where $w_1=0.517$, and $b_{buo}$ is the specific buoyancy:

$$b_{buo}=\left({\rho }_{air}-{\rho }_{gas}\right)\cdot g$$

(9)

where ${\rho }_{gas}$ is the density of the lifting gas in the balloon. The volume of the balloon $V_{balloon}$ at altitude can be found using the following equation:

$$V_{balloon}={\displaystyle \frac{m_{gas}}{M_{gas}}}\cdot R_{gas}\cdot {\displaystyle \frac{T_{gas}}{P_{gas}}}$$

(10)

where $R_{gas}$ is the ideal gas constant of the lifting gas, $m_{gas}$ is the lifting gas mass, and $M_{gas}$ is the molar mass of the lifting gas.

The balloon diameter (top view in Figure 1) can be calculated according to the volume of the balloon, as shown in Figure 2 [31]:

$$d_{top}=1.383\cdot V_{balloon}^{1/3}$$

(11)

The projected area of a balloon varies with the solar elevation angle [31], ${\alpha }_{ELV}$, which can be described as follows:

$$A_{proj}={\displaystyle \frac{\pi }{4}}\cdot d_{top}^2\cdot \left({\zeta }_1+{\zeta }_2\cdot \cos \left(\pi -2\cdot {\alpha }_{ELV}\right)\right)$$

(12)

For a constrained-volume zero-pressure balloon, ${\zeta }_1=0.9125$, and ${\zeta }_2=0.0875$ [32].

3.3. Thermal and Dynamic Models

Before investigating the thermal and dynamic models, it is necessary to make some fundamental assumptions:

(1)

During the dual-balloon system’s climbing process, the length of the connecting rope cannot remain unchanged, because of the changing rope tension. The geometric deformation of the aerostat system is neglected for the purpose of simplification. Therefore, the effect of the elastic deformation of rope is not considered.

(2)

The motion of the upper balloon will have a certain impact on the aerodynamic characteristics of the balloon below. In order to reduce the impact, the length of the connecting rope between two balloons was designed to be greater than five times the diameter of the balloon. Therefore, in this paper, the aerodynamic effect between the two balloons is ignored.

(3)

Compared with the direct solar irradiance and reflected radiation on the balloon, the radiation between the tow balloon and the balance balloon is relatively small, and the envelope temperatures of the two balloons are similar. The thermal effect between the two balloon envelopes is ignored.

The variation in ambient temperature and pressure at different altitudes is the key point that describes the reasons for the variation in the climbing velocity. The dynamic equation of motion represents a variation in the climbing rate according to two major terms: the first term is a buoyant force; the second is a drag force [33]. The following section demonstrates the variation in these forces, as well as their influence on the climbing rate. For this case, Newton’s second law of motion may be written as follows:

$$(m_G+m_{\mathrm{add}}){\displaystyle \frac{d^{2}z}{d{t}^2}}=B-m\cdot g+D_z$$

(13)

where $B$ is the total buoyancy of a balloon [34], $m_G$ is the total mass of the balloon system, $g$ is the gravitational acceleration, and $m_{\mathrm{add}}$ is the added mass. $D_z$ is the aerodynamic drag of the system in the vertical direction. For our model, the Z-axis is oriented in the vertical direction and upward is the positive direction. The X-axis points to east, Y-axis points to north east.

The added mass of a balloon can be obtained as follows:

$$m_{\mathrm{add}}=C_V\cdot {\rho }_{air}(z)\cdot V_{balloon}(z)$$

(14)

where $C_V$ is the added mass coefficient with a value of 0.5 [35].

The aerodynamic drag of the system in the vertical direction, $D_z$, on the system can be determined using the following:

$$\begin{array}{c}{D}_z = D_{z\_TB}+D_{z\_BB}\\ D_{z\_TB}=-{\displaystyle \frac{1}{2}}\cdot {\rho }_{air}\left(z_{TB}\right)\cdot \left|v_{\mathrm{Z}\_TB}\right|\cdot v_{\mathrm{Z}\_TB}\cdot C_{D\_TB}\cdot A_{eff\_TB}\\ D_{z\_BB}=-{\displaystyle \frac{1}{2}}\cdot {\rho }_{air}\left(z_{BB}\right)\cdot \left|v_{\mathrm{Z}\_BB}\right|\cdot v_{\mathrm{Z}\_BB}\cdot C_{D\_BB}\cdot A_{eff\_BB}\end{array}$$

(15)

where $v_{\mathrm{Z}\_TB}$ and $v_{\mathrm{Z}\_BB}$ are the vertical velocities of the tow balloon and balance balloon relative to the air. $C_{D\_TB}$ and $C_{D\_BB}$ are the drag coefficients of the tow balloon and the balance balloon in the vertical direction, respectively [36].

$$\begin{array}{l}{v}_{\mathrm{Z}\_TB}=v_{\mathrm{Z}\_\mathrm{DB}}-v_{\mathrm{Z}\_\mathrm{wind}\_TB}\\ v_{\mathrm{Z}\_BB}=v_{\mathrm{Z}\_\mathrm{DB}}-v_{\mathrm{Z}\_\mathrm{wind}\_BB}\end{array}$$

(16)

where $v_{\mathrm{Z}\_\mathrm{DB}}$ is the vertical velocity of the dual-balloon system in the inertial coordinate system, and $v_{\mathrm{Z}\_\mathrm{wind}}$ is local wind speed in the vertical direction.

The drag coefficient of a balloon can be obtained as follows:

$$C_{D\_*}=\left\{\begin{array}{cc}0.5+24/\left(\mathrm{Re}+0.01\right)& \mathrm{Re}\le 4.5\times {10}^5\\ 0.3& \mathrm{Re}>4.5\times {10}^5\end{array}\right.$$

(17)

As shown in Figure 3, the high-altitude dual-balloon system designed by the authors is composed of two main systems: the assisting system and the station-keeping system. In the initial design process, the net buoyancy of the tow balloon is greater than the total weight of the assisting system, and the net buoyancy of the balance balloon is slightly greater than or equal to the total weight of the station-keeping system. Therefore, the tow balloon always pulls the station-keeping system upward. The total buoyancy of the high-altitude fast-deploy aerostat system can be defined as

$$\begin{array}{c}B=B_{TB}+B_{BB}\\ B_{TB}=V_{TB}\cdot \left({\rho }_{air\_TB}-{\rho }_{gas\_TB}\right)\cdot g\\ B_{BB}=V_{BB}\cdot \left({\rho }_{air\_BB}-{\rho }_{gas\_BB}\right)\cdot g\end{array}$$

(18)

where $B_{TB}$ and $B_{BB}$ are the buoyancy provided by the tow balloon in the assisting system and the balance balloon in the station-keeping system, respectively. $V_{TB}$ and $V_{BB}$ are the volumes of the tow balloon and the balance balloon, respectively. The volumes can be obtained using the following:

$$\begin{array}{l}{V}_{TB}={\displaystyle \frac{m_{gas\_TB}}{M_{gas\_TB}}}\cdot R_{gas\_TB}\cdot {\displaystyle \frac{T_{gas\_TB}}{P_{gas\_TB}}}\\ V_{BB}={\displaystyle \frac{m_{gas\_BB}}{M_{gas\_BB}}}\cdot R_{gas\_BB}\cdot {\displaystyle \frac{T_{gas\_BB}}{P_{gas\_BB}}}\end{array}$$

(19)

where $m_{gas\_*}$ is the lifting gas mass, and $M_{gas\_*}$ is the molar mass of the lifting gas. In Equations (18) and (19), the subscripts of $gas\_TB$ and $gas\_BB$ represent the lifting gas in the tow balloon and balance balloon, respectively.

$$\begin{array}{l}{P}_{gas\_TB}=P_{air}+\Delta P_{gas\_TB}\\ P_{gas\_BB}=P_{air}+\Delta P_{gas\_BB}\end{array}$$

(20)

where $\Delta P_{gas\_TB}$ and $\Delta P_{gas\_BB}$ can be calculated according to Equation (8); ${\rho }_{gas\_TB}$ and ${\rho }_{gas\_BB}$ are the density of the lifting gas in the tow balloon and the balance balloon, respectively.

The total mass, $m_G$, on the high-altitude dual-balloon system mainly includes the mass of the tow balloon, the release unit, the balance balloon, and the payload.

$$m_G=m_{TB}+m_R+m_{BB}+m_{pay}$$

(21)

The balloon system is only affected by drag in the horizontal direction, as shown in Figure 4, and its differential equation can be expressed as

$$\left(m_G+m_{\mathrm{add}}\right)\cdot {\displaystyle \frac{d^{2}x}{d{t}^2}}=D_x=D_{x,TB}+D_{x,BB}$$

(22)

$$\left(m_G+m_{\mathrm{add}}\right)\cdot {\displaystyle \frac{d^{2}y}{d{t}^2}}=D_y=D_{y,TB}+D_{y,BB}$$

(23)

where $x,y$ are the displacement in the horizontal plane, and $D_x$, $D_y$ are the drag of the balloon system on the horizontal plane.

The horizontal drags can be obtained using the following:

$$\begin{array}{l}{D}_{x,TB}=0.5\cdot {\rho }_{air\_TB}\cdot \left|v_{x\_TB}\right|\cdot v_{x\_TB}\cdot C_{D\_TB,x}\cdot A_{eff\_TB}\\ D_{x,BB}=0.5\cdot {\rho }_{air\_BB}\cdot \left|v_{x\_BB}\right|\cdot v_{x\_BB}\cdot C_{D\_BB,x}\cdot A_{eff\_BB}\end{array}$$

(24)

$$\begin{array}{l}{D}_{y,TB}=0.5\cdot {\rho }_{air\_TB}\cdot \left|v_{y\_TB}\right|\cdot v_{y\_TB}\cdot C_{D\_TB,y}\cdot A_{eff\_TB}\\ D_{y,BB}=0.5\cdot {\rho }_{air\_BB}\cdot \left|v_{y\_BB}\right|\cdot v_{y\_BB}\cdot C_{D\_BB,y}\cdot A_{eff\_BB}\end{array}$$

(25)

The relative velocities between the two balloons and the local wind speed in the horizontal direction are the main factors influencing the horizontal drags, which can be expressed as follows:

$$\begin{array}{l}{v}_{x\_TB}=v_x-v_{wind\_TB, x}\\ v_{x\_BB}=v_x-v_{wind\_BB, x}\\ v_{y\_TB}=v_y-v_{wind\_TB, y}\\ v_{y\_BB}=v_y-v_{wind\_BB, y}\end{array}$$

(26)

where $v_{wind\_*, *}$ is local wind speed in the horizontal direction.

The expansion and compression of the lifting gas and the internal free convection interaction with the envelope are the main factors affecting the lifting gas temperature’s rate of change. It is possible to write the following equation:

$${\displaystyle \frac{d{T}_{gas}}{dt}}={\displaystyle \frac{q_{in,gas}}{M_{gas}\cdot c_v}}+\left(\gamma -1\right)\cdot T_{gas}\cdot \left({\displaystyle \frac{1}{m_{gas}}}\cdot {\displaystyle \frac{d{m}_{gas}}{dt}}-{\displaystyle \frac{1}{V_{gas}}}\cdot {\displaystyle \frac{d{V}_{gas}}{dt}}\right)$$

(27)

where $\gamma =c_p/c_v$, and $\gamma -1=R_{gas}/c_v$. $c_v$ is the specific heat at constant volume of the lifting gas. $R_{gas}$ is the specific gas constant of the lifting gas. $q_{in,gas}$ is the internal convective heat load, the expression of which is based on what was reported in Refs. [37,38].

Regarding Equation (27), the influence of the change in lifting gas mass in the balloon during flight is considered when calculating the gas temperature. The changes in gas mass inside the balloon are mainly caused by gas leakage and actively opening the valve to vent. The mass change is formulated on the flow through a duct or valve modified with the lifting gas diffusion.

$${\displaystyle \frac{d{m}_{gas}}{dt}}=-A_{valve}\cdot c_{disch}\cdot \sqrt{2\cdot \Delta P_{valve}\cdot {\rho }_{gas}}-M_{gas}\cdot D_{real}\cdot {\displaystyle \frac{P_{gas}}{P_{air}}}\cdot {\displaystyle \frac{\left(C_{in}-C_{out}\right)}{d_{env}}}\cdot A_{balloon}$$

(28)

where $A_{valve}$ is the cross-sectional area of the valve. $c_{disch}$ is the discharge coefficient. $\Delta P_{valve}$ is the differential pressure across the area interface. The effective diffusion coefficient, $D_{real}$, in the mass transfer equation is calculated by considering Knudsen and molecular diffusions, while $C_{in}$ and $C_{out}$ are the inside and outside concentrations of the lifting gas [39].

The envelope temperature differential equation can be expressed as

$$m_{env}\cdot c_{env}\cdot {\displaystyle \frac{d{T}_{env}}{dt}}=q_{in, env}$$

(29)

The heat-flux terms $q_{in, env}$ in Equation (29) can be given as

$$q_{in, env}=q_D+q_S+q_{albedo}+q_{IR}+q_{conv}$$

(30)

where $q_D$ is the absorbed direct sunlight heat. The absorbed direct solar radiation includes the heat absorbed by the outer surface of the envelope and the heat absorbed by the inner surface through the envelope. The effective absorption of the envelope will be improved due to multiple reflection [40].

$$q_D={\alpha }_{env}\cdot A_{proj}\cdot I_{sun}\cdot {\tau }_{atm}\cdot \left(1+{\displaystyle \frac{{\tau }_{env}}{1-{\gamma }_{env}}}\right)$$

(31)

where ${\alpha }_{env}$ is the absorption coefficient of the envelope material with respect to sunlight, and $A_{proj}$ is the projected balloon area. $I_{sun}$ is the direct solar irradiance. ${\tau }_{atm}$ is the transmissivity of a solar beam through the atmosphere, which is modified by influencing factors, namely the high- and low-altitude atmospheric conditions. ${\tau }_{env}$ and ${\gamma }_{env}$ are the transmittance and reflectance of the envelope material with respect to sunlight.

$q_S$ is the absorbed scattered radiation, which can be expressed as

$$q_S={\alpha }_{env}\cdot {\kappa }_S\cdot A_{surf}\cdot I_{sun}\cdot {\tau }_{atm}\cdot \left(1+{\tau }_{env}\cdot \left(1+{\gamma }_{env}\right)\right)$$

(32)

where ${\kappa }_S$ is the atmospheric scattering empirical coefficient; $A_{surf}$ is the exposed area of a balloon.

$q_{albedo}$ is the ground albedo radiation, and its intensity is mainly affected by the direct solar radiation intensity and the average albedo of the ground.

$$q_{albedo}={\alpha }_{env}\cdot A_{surf}\cdot V_F\cdot I_{albedo}\cdot \left(1+{\tau }_{env}\cdot \left(1+{\gamma }_{env}\right)\right)$$

(33)

where $I_{albedo}={\epsilon }_G\cdot I_{sun}\cdot \sin \left({\alpha }_{ele}\right)$; ${\epsilon }_G$ is the ground albedo; ${\alpha }_{ele}$ is the sun elevation angle; and $V_F$ is the angle coefficient between the balloon surface and Earth surface, related to the flight altitude.

$q_{IR}$ is the absorbed infrared radiation heat, which mainly includes ground infrared radiation, $q_{IR,E}$; infrared radiation from the sky, $q_{IR,sky}$; and infrared radiation self-glow from the interior, $q_{IR,env}$.

$$q_{IR}=q_{IR,E}+q_{IR,sky}+q_{IR,env}$$

(34)

$$\begin{array}{l}{q}_{IR,E}={\alpha }_{IR}\cdot A_{surf}\cdot I_{IR,E}\cdot V_F\cdot \left(1+{\tau }_{IR}/\left(1-r_{IR}\right)\right)\\ q_{IR,sky}={\alpha }_{IR}\cdot A_{surf}\cdot I_{IR,sky}\cdot \left(1-V_F\right)\cdot \left(1+{\tau }_{IR}/\left(1-r_{IR}\right)\right)\\ q_{IR,env}={\alpha }_{env}\cdot \epsilon \cdot A_{surf}\cdot T_{env}^4\cdot \left(1+{\alpha }_{IR}/\left(1-r_{IR}\right)+{\tau }_{IR}/\left(1-r_{IR}\right)\right)\end{array}$$

(35)

$I_{IR,sky}={\epsilon }_g\cdot \sigma \cdot T_{sky}^4\cdot {\tau }_{atm\_IR}$, $T_{sky}$ is the effective temperature of sky [41].

$q_{conv}$ represents the convective heat loads on the envelope, which involves transfers between the atmosphere and the exterior envelope and the lifting gas with the interior envelope. The external convection is partitioned into free convection and forced convection.

$$q_{conv}=h_{ex}\cdot A\cdot \left(T_{env}-T_{atm}\right)+h_{in}\cdot A\cdot \left(T_{env}-T_{gas}\right)$$

(36)

The external convective heat transfer coefficient of zero-pressure balloons between the film and external atmosphere can be calculated as follows [42]:

$$\begin{array}{l}{h}_{ex}={(h_{free-ex}^3+h_{forced-ex}^3)}^{1/3}\\ h_{free-ex}={\displaystyle \raisebox{1ex}{$N{u}_{atm}\cdot k_{atm}$}\!\left/ \!\raisebox{-1ex}{$L_0$}\right.}\\ h_{forced-ex}={\displaystyle \raisebox{1ex}{$k_{atm}$}\!\left/ \!\raisebox{-1ex}{$d_{top}$}\right.}\cdot (2+0.41\cdot {\mathrm{Re}}_{atm}^{0.55})\end{array}$$

(37)

In Equation (37), $h_{free\text{-}ex}$ and $h_{forced\text{-}ex}$ are the free convective heat transfer coefficient and forced convective heat transfer coefficient. $N{u}_{atm}$ is the free convection Nusselt number, $k_{atm}$ is the thermal conductivity of atmosphere [41]. ${\mathrm{Re}}_{atm}$ is the Reynolds number of the external atmosphere.

$${\mathrm{Re}}_{\mathrm{atm}}={\displaystyle \raisebox{1ex}{$v_{\mathrm{DB}}\cdot {\rho }_{\mathrm{atm}}\cdot d_{top}$}\!\left/ \!\raisebox{-1ex}{${\mu }_{\mathrm{atm}}$}\right.}$$

(38)

where $v_{\mathrm{DB}}$ is the relative velocity between the dual-balloon system and the atmosphere [37], ${\rho }_{gas}$ is the density of atmosphere, and ${\mu }_{gas}$ is the dynamic viscosity of atmosphere.

The internal natural convective heat transfer coefficient can be defined as

$$h_{in}=0.13\cdot k_{gas}\cdot {\left({\displaystyle \raisebox{1ex}{${\rho }_{gas}^2\cdot g\cdot \left|T_{env}-T_{gas}\right|\cdot {\mathrm{Pr}}_{gas}$}\!\left/ \!\raisebox{-1ex}{$T_{gas}\cdot {\mu }_{gas}$}\right.}\right)}^{{\displaystyle \frac{1}{3}}}$$

(39)

where $k_{gas}$ is the thermal conductivity of internal lifting gas, ${\rho }_{gas}$ is the density of the internal lifting gas, $T_{gas}$ is the average temperature of the internal lifting gas, and ${\mathrm{Pr}}_{gas}$ and ${\mu }_{gas}$ are the Prandtl number and dynamic viscosity of the internal lifting gas [31].

3.4. The Equilibrium Height of the Aerostat System

At the equilibrium height, the vertical velocity of the aerostat system is zero. The following equation can be obtained:

$$B_{BB}=m_G\cdot g$$

(40)

$$V_{BB}\cdot \left({\rho }_{air}-{\rho }_{gas\_BB}\right)\cdot g=\left(m_{TB}+m_R+m_{BB}+m_{pay}\right)\cdot g$$

(41)

A flow chart showing the process of calculating the equilibrium height of the aerostat system can be seen in Figure 5. After the release of the tow balloon, aerostat system needs to control the flight height to the designed equilibrium height by deflating and releasing ballast in different mission seasons and flight latitudes.

4. Experimental Procedure

In order to study the climbing performance of the high-altitude dual-balloon system and verify the applicability of the theoretical model, two experimental schemes were designed, as shown in Figure 6. The constrained volume zero-pressure balloon system includes two parts: a sounding balloon with a safety valve and the available payload, as shown in Figure 6a. The dual-balloon system is composed of four sections, including the tow balloon, release unit, balance balloon, and available payload, from top to bottom, as shown in Figure 6b. In the dual-balloon system, the tow balloon is the conventional sounding balloon, and the balance balloon is the sounding balloon with the safety valve. When the dual-balloon system reaches the desirable floating altitude, the release device is activated, and the tow balloon flies away with the release device. The balance balloon with the available payload reaches the balance height after height adjustment. In these experiments, a radiosonde based on Beidou/GPS is fixed in the payload cabin. The radiosonde is mainly composed of a temperature sensor, a humidity sensor, a pressure sensor, a data acquisition module, a Beidou/GPS receiving antenna, a Beidou/GPS navigation module, a transmitter module, and a transmitting antenna. Combined with ground observation equipment, the flight state of the aerostat system and environment condition can be monitored.

In this study, three groups of experiments were designed. The design altitudes of Experiment 1 and Experiment 2 were designed to be the same in order to analyze the influence of the tow ball on climbing performance. Experiment 3 was designed to explore the maximum flight altitude of the dual-balloon system and verify the dynamic model of the dual-balloon system established in this paper. Some basic parameters of the experimental aerostat systems are listed in

Table 1. Design index of the experimental aerostat system.

Parameters	Experiment 1 (Dual-Balloon)	Experiment 2 (Constrained Volume Balloon)	Experiment 3 (Dual-Balloon)
Design altitude (km)	26.7	26.7	42
Tow balloon weight (kg)	0.7	-	0.8
Balanced balloon weight (kg)	1.5	1.5	2.45
Maximum diameter of Balanced balloon (m)	19.8	19.8	26.8
Payload weight (kg)	0.65	0.65	0.65
Helium gas purity	99.99%	99.99%	99.99%
Release vehicle weight (kg)	0.3	-	0.3
Connection string between tow balloon and balance balloon (m)	45	-	45
Connection string between balance balloon and payload (m)	12	12	12
Launch time	9:30	9:30	18:30
Launch site	Huaihua City	Huaihua City	Huaihua City

.

Table 2. Main parameters of electronic sounding instrument.

Parameters	Index
Operating frequency band (mHz)	1675 ± 3
Transmission power (mW)	400
Data acquisition rates (/s)	1
Temperature measurement range (°C)	−80~40 ± 0.3
Pressure measurement range (hPa)	10~1050 ± 1.5

listed the main parameters of the sounding instrument used in the flight test. The XGL-1 digital electronic sounding instrument was used in the flight test.

As shown in Figure 7, in the zero-pressure balloon test flight, the primary goal was quick balloon deployment. The detailed design and physical layout of the balloon was generated based on the requirements defined for a rapidly climbing balloon system, employing the weather radar to collect the requisite data. To collect the necessary dynamics data, the payload would require the use of accelerometers, inclinometers, temperature sensors, and cameras.

The balloon was designed to float at a reasonable altitude, in the zero wind layer with a suspended load. The launch and ascent were within the predicted ranges; horizontal movement began to occur after the balloon launched because of the atmospheric athwart wind. The balloon was fully deployed, and the payloads started working effectively after the separation between the zero-pressure balloon and the tow balloon after the balloon reached the predetermined altitude; the covered area of observation equipment with a 45 degree viewing angle is shown in Figure 7. Before the termination of the balloon mission, the balloon continued to move in a certain horizontal direction. In this test flight, we hoped that the balloon would prove large enough to carry an appropriately sized payload to a reasonable float altitude.

Figure 8 characterizes aspects of the atmospheric wind environment. The wind direction data and altitude measurements were obtained from weather balloons released before the test flight. The atmosphere of our planet is a very dynamic environment with great fluctuations in temperature, pressure, density, and wind speeds; the atmospheric conditions have a significant impact on the design of stratospheric balloons. Wind profile, to a large degree, depends on the location, time of the year, and altitude. The magnitude and direction of wind varies with the altitude. In general, wind velocity can be divided into the meridional and zonal [11]. For the meridional component, the south direction wind is defined as positive, and for the zonal component, the west direction wind is defined as positive. A sample of the data is shown in Figure 8; the wind data covered an altitude of 0 to 27.5 km, and a diagram of wind velocity and wind direction as a function of altitude over the test flight site is also shown.

5. Verification of Theoretical Model

The dynamic model of the balloon considering the thermal effect was verified by comparison with NASA analysis data (real flight data and ThermTraj model) and Sherif Saleh’s analysis data [16]. The maximum flight altitude of the reference balloon is 36.7 km, and the maximum volume is 66,375 m³. The payload weight, helium weight, and gross mass are 196.8, 69.2, and 381 kg, respectively. As shown in Figure 9, the results of several comparative models, including the model established in this paper, are very similar.

For the validation of our numerical methods, we compared our numerical simulation results with experimental results of Experiment 3, as shown in Figure 10, Figure 11 and Figure 12. The experimental results are represented using different symbols with different conditions, while the numerical simulation results are represented by continuous lines with different colors. In the numerical simulation of this study, the environmental temperature and air pressure outside the dual-balloon at different heights were provided by experimental test results. The balloon trajectories in the climbing phase and float phase can be seen in Figure 10. By comparing numerical simulation results with experimental results, the variation in balloon flight altitude with flight time distributions was almost consistent.

Figure 11 represents the real climbing velocity and predicted ascent speed in the climbing phase and the float phase, respectively. The real flight data and the expected data are almost close to each other at the desired float altitude and the troposphere climbing altitudes, because the climbing velocity tends to zero. The discrepancy at some points may be caused by external environmental factors.

Figure 12 illustrates the meridional displacement and zonal displacement in the climbing phase and float phase, respectively. The numerical simulation results were calculated based on the wind field data of the test site. It can be seen from the figure that the zonal displacement of the balloon gradually increases with time, while the meridional displacement first gradually increases in the negative direction. After a period of time, due to the influence of the wind direction, the meridional displacement gradually increases in the positive direction. The experimental flight trajectories in the meridional and zonal directions are basically consistent with the simulation results. The above comparison between the climbing phase and floating phase of the balloon shows that the numerical simulation results are in good agreement with the experimental results, which proves that the model adopted in this paper is both reasonable and fairly accurate.

For the validation of our numerical methods in different flight scenarios, we compared our numerical simulation results with experimental results of Experiment 1, as shown in Figure 13. By comparing numerical simulation results with experimental results, the variation in balloon flight altitude with flight time distributions was almost consistent. However, during the rapid climb phase, the measured actual flight altitude is greater than the simulated value. This is mainly because the upward drag coefficient is slightly smaller than the theoretical model established in this study, as the balloon that has not fully expanded will be blown into a streamlined shape by the incoming flow.

6. Results and Discussion

6.1. Tow Balloon Effect

Figure 14 describes the influence of the tow balloon on the climbing trajectory. The constrained-volume zero-pressure balloon designed in this paper takes 4.16 h to reach the predetermined flight altitude. The climbing time of dual-balloon system is reduced to 1.43 h. The results show that the zero-pressure balloon with a tow balloon reaches the predetermined altitude in a shorter time than the zero-pressure balloon without a tow balloon when the same predetermined altitude is set.

As shown in Figure 15, at the desired float altitude, the climbing velocity tends to zero, but the atmospheric environment may be attributed to the few effective uncertainties, and the floating velocity fluctuates around zero. It can be seen from Figure 15 that, for the zero-pressure balloon without a tow balloon, the climbing velocity begins to increase from the time of flight until reaching peak velocity, before it slowly decreases and fluctuates to zero. After 4.16 h, the balloon reached the predetermined altitude and began to float at this altitude. However, when we tested the zero-pressure balloon with a tow balloon, the climbing velocity began to increase from $v=1.8 \mathrm{m}/\mathrm{s}$ until the tow balloon was released in 85 min after the zero-pressure balloon was launched, the climbing velocity of balloon then plummets to a suspended velocity and the balloon floats at the predetermined altitude. While the buoyancy of the balloon increased, the climbing velocity increased so that the balloon reached the predetermined altitude earlier.

Figure 16 shows the influence of the wind on the balloon horizontal movement at different layers, different color lines represent the balloon trajectories in the climbing phase, respectively. The $m_{\mathrm{He}\_\mathrm{tow}}/m_{\mathrm{He}\_\mathrm{Balance}}$ set to 0.1 in Experiment 1. The focus will be on two degrees of freedom of the balloon, motion in the vertical plane, motion in the horizontal plane, motion in the vertical plane will be ignored in this study, as this degree of freedom is assumed to contribute little in terms of the dynamics balloon’s inertial position. The balloon’s motion in the horizontal plane was considered the most critical. As presented in Figure 16, the real flight data covered an altitude of 0 to 30 km, when the balloon climbs to a certain altitude, the horizontal motion of zero-pressure balloon without the tow balloon is farther away than that of zero-pressure balloon with the tow balloon. When the balloon drifts with the wind too far from the ground control station, the balloon payloads stop working; from this point of view, the zero-pressure balloon with tow balloon is more efficient.

Figure 17 shows the climbing trajectory with and without a tow balloon. It can be seen that the trajectory of the balloon differs greatly between the two cases. Due to the influence of wind direction and wind speed, the displacement of the balloon in the meridional direction is much larger than that in the zonal direction. For the zero-pressure balloon without the tow balloon, the climbing trajectory is more scattered and lengthy than that with the tow balloon, and the amplitude of fluctuation is larger. At the same time, in the case of the zero-pressure balloon with a tow balloon, the displacement in the meridional direction is much smaller than that without a tow balloon, and the zonal displacement of the zero-pressure balloon is relatively close in both cases.

6.2. Effects of Helium Mass in the Tow Balloon

In order to make a zero-pressure balloon climb to the predetermined altitude quickly, the key of flight experiments is to increase the vertical velocity of the balloon and shorten the time in the ascent phase. The effect of the helium mass in the tow balloon was studied when the payload was 2 kg.

The influences on the climbing velocity due to the change in the mass of helium filled in the tow balloon are shown in Figure 18, and $m_{\mathrm{He}\_\mathrm{Balance}}$ is the mass of the zero-pressure balloon. It can be seen that the mass of helium filled in the tow balloon increases, resulting in an observable increase in the climbing velocity, which is considered the main reason for the reduced climbing time. Figure 19 represents the vertical drag force of a zero-pressure balloon, showing that it decreases even more slowly with climbing time when the tow balloon is filled more helium. The drag force is directly proportional to the squared vertical velocity, and the mass of the helium in the tow balloon has a great effect on the climbing velocity (Figure 18).

Variation in the mass of the tow balloon helium in the ascent stage is the key point that causes the variation in the climbing parameters. By increasing the mass of the helium in the tow balloon, the balloon’s efficiency in the climbing phase is improved. Figure 20 indicates the volume of tow balloon variation in the climbing phase when the tow balloon is filled with different masses of helium. The tow balloon volume gradually increases with the increasing altitude, owing to the reduction in external atmospheric pressure, while the balloon volume slowly increases at low altitude compared to at a higher altitude.

Figure 21 and Figure 22 describe the impact on the climbing time and horizontal displacement due to the change in the mass of the helium in the tow balloon. The consumption of helium gas is considered as an important cost factor, as low-cost tow balloons have a relatively small impact on the cost of the dual-balloon system. When the mass of helium in the tow balloon is low, the balloon system takes a very long time to reach the float altitude, and the horizontal displacement is large. It is shown that the zero-pressure balloon with the tow balloon can work for longer at the predetermined altitude. When the mass of helium in tow balloon is relatively small, especially when $m_{\mathrm{He}\_\mathrm{tow}}/m_{\mathrm{He}\_\mathrm{Balance}}$ is less than 0.3, the climbing time and horizontal displacement decrease rapidly with the increase of mass of helium in tow balloon. However, it can also be seen from Figure 21 and Figure 22 that with the increase in mass of the helium in the tow balloon, the climbing time of the balloon first decreases rapidly and then tends to be flat. This result implies that as the mass of helium in tow balloon increases, the impact of the tow balloon on the decrease in climbing time and horizontal displacement gradually weakens in the climbing process. Therefore, considering the manufacturing complexity, operation difficulty, and cost, when designing the tow ball, it is necessary to select an appropriate volume, rather than a larger volume and helium mass.

6.3. Effects of the Payload Mass

Figure 23 shows the significant effect on the equilibrium height due to variations in the helium mass and the payload mass. It can be seen that the rise equilibrium height increases with the increase in the mass of the helium filling the zero-pressure balloon, and it also increases with the decreasing payload mass. In addition, with an increase in helium mass, the equilibrium height increases rapidly and then varies little.

As shown in Figure 24, the vertical drag force on the zero-pressure balloon decreases with increasing time. The curve drops more gently for the balloon with a heavier payload. The balloon system should throw out the tow balloon early and consume some time to reach the float altitude; the drag force drops and fluctuates to zero in this period. Consequently, the changes in the payload mass influence the climbing performance, and the balloon reaches the float altitude faster if carrying less payload.

7. Conclusions

Inspired by the space tethered satellite and tethered balloon, a dual-balloon scheme (mainly including tow balloon and balance balloon) is designed to reduce the climbing time and the horizontal movement. We conducted a numerical simulation and experimental investigation on the climbing performance of a high-altitude dual-balloon system. We considered horizontal movement and vertical climbing velocity, as well as the effect of the mass of the helium filling the tow balloon. The conclusions derived from our application of the dual-balloon system are summarized below:

(1)

Based on the verification results, it was found that the theoretical dynamic model for the dual-balloon system proposed in this paper can be utilized to study the climbing time and horizontal displacement of aerostat systems during the climbing phase.

(2)

By comparison with the given constrained-volume zero-pressure balloon, the dual-balloon system can reduce the climbing time by two-thirds. During takeoff, the horizontal flight distance can be reduced from 200 km to 50 km under a specific wind field environment.

(3)

When the mass of helium in the tow balloon is relatively small, especially when is less than 0.3, the climbing time and horizontal displacement decrease rapidly. However, as the mass of helium in the tow balloon increases, the impact of the tow balloon on the decrease in climbing time and horizontal displacement gradually weakens. Considering the manufacturing complexity, operation difficulty, and cost, when designing the tow ball, it is necessary to select an appropriate volume, rather than a larger volume and helium mass.

(4)

The equilibrium height of the dual-balloon system cannot be increased arbitrarily by increasing the mass of the helium in the balance balloon because the equilibrium height increases rapidly and then varies little with the increase in the helium mass when the payload mass is constant.