Isothermal pumping analysis for high-altitude tethered balloons

High-altitude tethered balloons have potential applications in communications, surveillance, meteorological observations and climate engineering. To maintain balloon buoyancy, power fuel cells and perturb atmospheric conditions, fluids could be pumped from ground level to altitude using the tether as a hose. This paper examines the pumping requirements of such a delivery system. Cases considered include delivery of hydrogen, sulfur dioxide (SO₂) and powders as fluid-based slurries. Isothermal analysis is used to determine the variation of pressures and velocities along the pipe length. Results show that transport of small quantities of hydrogen to power fuel cells and maintain balloon buoyancy can be achieved at pressures and temperatures that are tolerable in terms of both the pipe strength and the current state of pumping technologies. To avoid solidification, transport of SO₂ would require elevated temperatures that cannot be tolerated by the strength fibres in the pipe. While the use of particle-based slurries rather than SO₂ for climate engineering can reduce the pipe size significantly, the pumping pressures are close to the maximum bursting pressure of the pipe.

1. Introduction

High-altitude tethered balloons are an emerging technology with potential applications in the fields of communications, meteorological measurements, surveillance, power harvesting and climate engineering. There are many engineering issues associated with the design of such a system [1], and this paper addresses one significant challenge: the pumping requirements for transporting materials along the length of the tether. Using the tether as a hose would provide a cost- and energy-efficient means of: (i) delivering lighter-than-air gases to the balloon, to maintain internal balloon pressures for long-term deployment; (ii) delivering materials that can be used to power fuel cells, thereby avoiding wires along the length of the tether; and (iii) delivering materials such as reflective particles that could be used to perturb atmospheric conditions and alter global temperatures. The latter forms the engineering concept of the SPICE (Stratospheric Particle Injection for Climate Engineering) Project, under which this work on pumping considerations was carried out. Alternative proposed climate engineering delivery systems include high-altitude aircraft, missiles, rigid towers and single-use balloons [2,3]. The use of a tether as a feed line for replenishment of lifting gases is not novel, having been described previously [4].

The basic engineering design of the system considered here is as follows: the balloon is flying in the stratosphere, at a height of about 20 km, probably in a region where relatively low wind speeds can be expected for a significant part of the year. The balloon tether serves both to fix the balloon and to contain a pipe through which the material is to be pumped. In this study, which constitutes an initial design assessment, only a single balloon at maximum altitude and a single, continuous tether will be considered. Further lift could be supplied by using multiple balloons or lifting devices attached along the tether, and further pumping capacity could be provided by using multiple tethers attached to a single balloon.

There are many engineering challenges associated with this system, but for the purposes of this paper it is assumed that the tether is formed from a pipe of circular cross-section that is wrapped with high strength fibres to resist the bursting pressures, which are in turn surrounded by axial fibres to carry the tension forces. This fibre arrangement is discussed by Causier [5]. The bursting pressures in the tether are largest at ground level, as this is where the greatest fluid pressure exists, and decrease with altitude, whereas the tension forces are smallest at ground level, and increase along the length of the tether to a maximum at the connection point to the balloon.

The tether fibres need to have high strength, low weight and to be electrically non-conducting to reduce the risk of lightning strikes. Causier [5] discusses candidate fibres and identifies Kevlar 49 and poly-p-phenylenebenzobisoxazole (PBO) as two options. As with many fibres, these materials will creep, and if the stresses are high enough they will creep to failure. As these processes are thermally activated, it will be taken as axiomatic that it is desirable to keep temperatures at or below ambient temperatures at ground level. Causier [5] analyses the stresses in a tether with separate hoop and axial fibres and shows that the maximum bursting pressure of the pipe depends on the ratio of the transverse and axial modulus of the fibres. The maximum bursting pressures are 139 MPa for Kevlar 49, and 130 MPa for PBO [5]. These values could be increased if the fibres are embedded in resin or if additional fibres are added between layers, and contain some uncertainty due to the limited data available on the transverse fibre modulus [5]. However, as a conservative estimate, these values will be used in this paper as the maximum allowable pressures in the pipe.

This paper presents an initial isothermal pumping analysis of such a tether system. Beginning with a number of representative materials for transportation, the implications for pumping pressures, temperatures and pipe design are discussed. The limitations of the isothermal assumption are discussed in §5.

2. Material selection

The representative materials considered here are hydrogen (H₂), sulfur dioxide (SO₂) and a mixture of titanium dioxide particles and nitrogen (TiO₂/N₂).

2.1 Hydrogen

Hydrogen is the lightest existing gas, making it an excellent candidate for use as a lifting gas in the balloon. While safety precautions will be required to manage the risk of explosion due to the flammability of hydrogen, it is nevertheless proposed as the preferred lifting gas over the inert and heavier helium due to its abundance and low manufacturing cost. The small size of the hydrogen molecule will cause diffusion through the balloon skin over time leading to a loss in pressure. By pumping hydrogen through the tether, the buoyancy of the balloon can be controlled for both long-term deployment and management of the balloon launch and recovery procedure. Hydrogen can also be used to run fuel cells, providing a clean, reliable and lightweight source of power at altitude.

The required mass flow rate of hydrogen is estimated based on a 10% loss of balloon gas volume per week, and the operation of a 3 kW fuel cell. The magnitude of the balloon gas loss is dependent on factors such as the resistance of the balloon skin material to gas diffusion and the integrity of the balloon skin. The assumed value of 10% loss per week can be considered a mid-range estimate, based on the experience of a high-altitude balloon specialist (Andy Elson 2011, personal communication). A balloon of 100 m radius, which will provide lift of 343 tonnes at 20 km, will require 0.69 m³ s⁻¹ of hydrogen to replace the lost gas, which equates to 4.9×10⁻³ kg s⁻¹ of hydrogen at ambient atmospheric conditions of −60°C and 6.3 kPa (absolute—equivalent to 800 Pa gauge pressure). A 3 kW fuel cell with 25% efficiency and calorific value of 39 kWh kg⁻¹ will require a flow rate of 8.5×10⁻⁵ kg s⁻¹ at an inlet pressure of 1 MPa. Combining these requirements, we will calculate the pumping requirements for a system that delivers 5.0 × 10⁻³ kg s⁻¹ of hydrogen at 1 MPa to 20 km altitude. This value is relatively insensitive to the power output of the fuel cell, as the flow rate required by the fuel cell is small compared with the balloon gas replenishment. To estimate the sensitivity of the results due to the chosen parameters, calculations are also performed for a 30% loss of balloon gas volume per week, and a 50 m radius balloon.

2.2 Sulfur dioxide

SO₂ is chosen as a candidate material due to its demonstrated cooling effect on global temperatures. The 1991 Mount Pinatubo event injected around 20 million tonnes of SO₂ and hydrogen sulfide (H₂S) into the stratosphere, resulting in an associated peak annual mean forcing of −2.97 W m⁻² [6], and a global mean temperature reduction of about 0.5°C for a period of about 2 years [7]. Climate engineering proposals seek to mimic this effect by spraying SO₂ aerosol particles into the stratosphere. SO₂ is readily available, non-flammable and not as toxic as H₂S, although significant safety measures would be required. There are a number of possible side effects of SO₂ injection [8,9].

The quantity of materials to be pumped over 1 year for climate engineering is taken as the amount of material required to achieve the same dimming effect as the Pinatubo eruption: 13.64 Mt of SO₂ [10]. This amount will increase the Earth's Bond albedo by 0.015. An estimated Bond albedo increase of about 0.018 is required to counteract the effects of a doubling of atmospheric carbon dioxide [11]. Simulation suggests one Pinatubo-size injection of SO₂ per year will produce a steady-state global mean temperature change of 5°C [6]. It will be assumed that this quantity of materials will be delivered by four tethered-balloon systems, each operating for 300 days per year. This gives a required mass flow rate of 132 kg s⁻¹ of SO₂ per pipe. To estimate the sensitivity of the results due to the chosen parameters, calculations are also performed for delivering the same total amount of material using 8 and 40 balloons, and maximum (exit) flow velocities of 10 and 50 m s⁻¹.

2.3 Titanium dioxide particles in nitrogen

Pope et al. [10] propose TiO₂ particles with an effective radius of 0.11 μm as an alternative cooling agent to SO₂. These particles give a magnified dimming effect compared to SO₂ aerosols due to increased light scattering and are already produced in large quantities at the submicrometre size. Other advantages of using TiO₂ over SO₂ include scope for relatively small-scale field experiments to study particle properties (equivalent studies for SO₂ would require significantly longer timescales and larger test areas), and potentially using surface coatings to minimize ozone depletion reactions, but much research needs to be carried out to confirm this [10]. For these reasons, TiO₂ will be considered as a candidate particle, although many of the arguments used can be transferred to other particles (including alumina, silica, silicon carbide, etc.) if they are preferred. The pumping of TiO₂ particles will require a carrier fluid. Using nitrogen as a carrier is considered here, as the introduction of nitrogen into the stratosphere will have no damaging environmental effect, poses no dangers to human health and is readily and cheaply available. H₂O is dismissed as a carrier fluid as, apart from the large pumping pressures that would be required due to the liquid density, the introduction of moisture into the dry environment of the stratosphere would have negative impacts of stratospheric chemistry [8].

The pumping of mixtures of gas and solids and liquids and solids is well established, and widely used in the agricultural, oil, mining and food industries. These industries often need to pump solids over long distances, both horizontally and vertically. For conveying particles in gas over long distances with high velocities, a dilute phase system is used. Typically, this involves conveying the material in suspension in the flowing gas, with a phase density ϕ(ratio of mass flow rate of solids to mass flow rate of gas) of less than 10, and usually in the range of 0.3–0.9 (Bill Layton 2012, personal communication). The primary effect of adding particles to the fluid is an increased head pressure, and the particles are so fine that they are considered to have no effect on the frictional flow losses. A phase density ratio of 0.5 is used in this paper. To keep the material in suspension, a minimum value of conveying line velocity of 13–15 m s⁻¹ is recommended [12]. To achieve an equivalent dimming effect as the Pinatubo eruption, 10.3 Mt of TiO₂ is required per year [10]. This gives a required mass flow rate of 298 kg s⁻¹ TiO₂/N₂ (phase density 0.5: 99.3 kg s⁻¹ TiO₂, 199 kg s⁻¹ N₂) for each of the four tethered-balloon systems operating for 300 days per year. To investigate the sensitivity of the results to the chosen parameters, calculations are also performed for delivering the same total amount of material using phase densities of 0.3 and 0.9, 8 and 40 balloons, and maximum (exit) flow velocities of 10 and 50 m s⁻¹.

3. Thermodynamic properties

The thermodynamic properties for the candidate materials are obtained from the NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP) [13].

3.1 Hydrogen

The phase diagram for H₂ is shown in figure 1a. The REFPROP database contains thermodynamic properties for the shaded region, which extends to 1000 K and 2000 MPa. These properties are calculated using an equation of state from Leachman et al. [14], and the solid–liquid line is constructed from data given in [15]. Figure 1b shows the variation of density with pressure for H₂ at 250 K. The uncertainty in the REFPROP data for density is approximately 0.04% [13]. To capture the variation of density with pressure and temperature, a suitable equation of state is required, and three commonly used equations of state are included in this figure: the ideal gas law; van der Waals equation; and the Peng–Robinson equation. The ideal gas law is based on the assumptions of negligible molecular size and intermolecular attractions, and as such is unsuitable for modelling gases at high densities and high pressures. For H₂ at 250 K and up to 50 MPa, the Peng–Robinson equation provides a good model of real gas behaviour.

3.2 Sulfur dioxide

The phase diagram for SO₂ is shown in figure 2a. The dark shaded region indicates the region of available data from the REFPROP, which is calculated based on equations of state from Lemmon & Span [16] and Ihmels et al. [17]. The data used to construct the solid–liquid line are from Hogenboom et al. [18]. At room temperature, the freezing of SO₂ occurs at 300 MPa [19,20]. However, apart from this value, there is little data available on the pressures and temperatures at which freezing of SO₂ occurs, which will be shown below to be critical. At 250 K, the density is largely invariant with pressure, as is expected for a liquid, and is approximately 1530 kg m⁻³. Figure 2b shows the variation of density with pressure for SO₂ at 350 K (gas phase). The uncertainty in the REFPROP data for density is approximately 0.5% [13]. For SO₂ at 350 K and pressures up to condensation at 1.67 MPa, the Peng–Robinson equation provides a good model of real gas behaviour.

3.3 Nitrogen

The properties of nitrogen at extremes of pressure and temperature are well documented, and extrapolation of data is not necessary for this case. The phase diagram for N₂ is shown in figure 3a, constructed using data from Span et al.[21]. At room temperature and pressure, N₂ exists as a gas and is considered a supercritical fluid when the pressure exceeds the critical pressure of 3.396 MPa. Figure 3b shows the variation of density with pressure for N₂ at 250 K, as given in REFPROP and calculated using various equations of state. The uncertainty in the REFPROP data for density is approximately 0.02% [13]. For nitrogen at pressures less than 20 MPa, the ideal gas equation of state provides a reasonably accurate model of the thermodynamic properties. At higher pressures, the Peng–Robinson equation provides the closest-fitting model.

4. Pumping analysis

This section presents a pumping analysis for the three fluids based on the assumption of isothermal flow through a constant-radius pipe. The assumption of constant-temperature conditions has been chosen as it provides a convenient means of simplifying the pumping analysis. Section 5 presents a discussion of the limitations of this assumption.

For isothermal flow through a vertical pipe of constant radius r, the change in fluid pressure P with height z is the sum of three contributions: the frictional pressure loss, the static head and the change in momentum, respectively, expressed as