Pumping to the Stratosphere
To understand some of the important design parameters and engineering trade-offs for a Stratospheric Shield, we analyzed a system that could raise 100,000 tons of liquid a year from the ground to an elevation of 30 kilometers (100,000 feet). Delaying for a moment the question of how to support the hose, let’s concentrate first on the fluid flow, which for the purposes of this exercise we assume will be constant.
Although 100,000 tons a year sounds like a lot of liquid, when pumped continuously through a hose, that amounts to just 3.2 kilograms per second and, at a liquid SO2 density of 1.46 grams per cubic centimeter, a mere 34 gallons (150 liters) per minute. A garden hose with a ¾-inch inner diameter can
deliver liquid that fast.
It takes quite a bit of energy to lift material into the stratosphere: about 30 trillion Joules of potential energy, in fact, to lift 100,000 tons to a height of 30 kilometers. If the work is spread out over the course of a year, however, that energy translates to a required power of just 1,000 kilowatts. Inefficiencies and other practical considerations will increase this amount, possibly by several times; nonetheless, the power levels are not daunting by industrial standards.
To pump 34 gallons a minute up a 30-kilometer-long hose, the system must overcome both the gravitational head and flow resistance. The gravitational head, which is simply another way of talking about the potential energy considered previously, would amount to a pressure of 4,300 bar (62,000 p.s.i.) if the liquid has a constant density of 1.46 g/cm3—not taking into account the small attenuation in the strength of gravity with increasing altitude.
The density of the SO2 does not remain constant during its journey through the hose, however. That transit takes enough time that at any point in length of the hose, the temperature of the liquid inside the hose is not too far from the temperature of the air outside it, although friction from the flow will impart some heat to the fluid. Air temperature drops with altitude, and so will the temperature of the SO2; the density of the liquid thus increases with altitude. The magnitude of the density change will vary depending on the site of the Strato Shield as well as the season and time of day, but we can use the thermal profile of the Standard Atmosphere to estimate a typical value: between 1.40 g/cm3 and 1.57 g/cm3. This density range from bottom to top produces an overall gravitational head of 4,520 bar. There isn’t much we can do about gravity except fight it with pumping power.
We have more control over the second kind of impediment, flow resistance. This pressure arises from drag forces imposed on the fluid by the walls of the pipe. By selecting the diameter of the hose and other design characteristics, we can choose whether the flow resistance pressure is much greater than the gravitational head or much less than it. A lower flow resistance might seem always preferable, but it comes at a price: a larger diameter hose, which means more mass for the balloons to support.
The weight of both the hose itself and the fluid it contains increase quickly as hose diameter expands. Consider two designs, one using a hose with a diameter of 5⁄8 inch (1.6 cm), the other a hose 1½ inches (3.8 cm) in diameter. The 5⁄8-inch hose has a cross-sectional area of 1.98 cm2, which means that the flow velocity at the ground must be 11.4 m/s to achieve the required 34 gallons per minute delivery rate. (The flow velocity for this hose drops to 10.2 m/s at higher altitudes, due to cooling of the SO2.)
To calculate the resulting flow resistance, we need to factor in the flow’s Reynolds number and also the effect of pipe roughness. We’ll assume a wall roughness of ½ mil (13 micron). The Reynolds number, like density, is a function of temperature and thus altitude. It changes along the hose by more than a factor of two—from 320,000 to 810,000—due to the temperature-induced gradients in density, viscosity, and velocity.
Fortunately, this variation in the Reynolds number has very little effect. The flow resistance remains essentially constant along the hose, ranging from 1,000 to 1,100 bar/km. The total flow-induced pressure head for the 5⁄8-inch hose is thus 30,800 bar, much larger than the 4,500 bar gravitational head. For a 5⁄8-inch hose, drag forces thus largely determine our pumping power.
In contrast, a 1½-inch hose can deliver the payload at a flow rate under 2 m/s, which generates a markedly smaller flow resistance of just 360 bar. The price for this huge reduction in pumping requirements is, of course, the need to generate more lift to support a heavier hose. The SO2 alone in the 5⁄8-inch hose weighs 9.1 tons, whereas the liquid in the 1½-inch hose comes to a whopping 52.5 tons. The larger-bore hose will also weigh more than the thin hose, of course, but that difference is at least partially offset by the need to install more pumps (and electrical cable to run them) along the length of the thin hose. The choice of the optimum hose diameter thus requires a complexset of design trade-offs; one cannot simply peg the flow resistance to some percentage of the gravitational pressure head.
Option 1: A Big Pump on the Ground
Raising the fluid up the entire length of the hose with a single pump on the ground may seem impractical. A more feasible alternative, we thought, would be to distribute a series of small pumps at intervals along the hose. Each pump could then be of lower power, because it would only have to raise the liquid as far as the next pump. In effect, this does an endrun around gravity.
With further thought, however, we recognized that pumping from the ground, either using one large pump or a set of pumps in series, offers a number of advantages. Maintenance and replacement would be significantly easier, for example. Keeping the pumps on the ground would reduce the size of
the balloons required and could eliminate the need to run electrical wiring up the hose.
Another important, but less obvious, advantage of pumping from the ground is that in such a system the pump can support the mass of the SO2 liquid, through the pressure it delivers. Flow resistance will actually push up on the hose material and can be used to support part of its mass as well. These effects greatly reduce the lift necessary to raise the hose to altitude.
Unfortunately, however, a hose supported this way would be unstable to sideways wind forces, which can impose lateral momentum far greater than the upward momentum delivered by the flow of SO2. Supporting most of the system weight with ground pressure is also ill-advised because of the possibility that a disruption of pump operation could cause the StratoShield to fall precipitously. A system pumped from the ground would thus probably need enough external support to handle wind forces and pump failures safely.
Support issues aside, an obvious drawback to pumping only from the ground is that the resulting pressures inside the hose must be extremely large. The hose wall must be thickened to withstand the high pressure, and the density of the SO2 (which is a compressible fluid) will increase. The magnitude of this latter effect is not completely clear. Experimental data on the compressibility of liquid SO2 extends only up to about 350 bar, which is not even a tenth of the gravitational head in the StratoShield. What data there are show that SO2 has compressibility at 0 °C of 1.1 x 10-9/Pa (1.1 x 10-4/bar), a value about twice that of water. Using the existing data to fit an expression for the linear-secant modulus, we expect a 20% density increase at 4,500 bar and 0 °C.
At the lower temperatures encountered throughout most of the hose, the SO2 is stiffer. We estimate that the integrated pressure head, taking into account the pressure and temperature dependence of the compressibility, is about 5,000 bar. So, for a relatively fat hose, where the pressure is dominated by gravitational head, compressibility is not a major concern, even if we are pumping solely from a ground station. Compressibility becomes a much larger issue if the hose is narrow, due to the additive effect of flow resistance.
Hoses capable of containing pressures above 5,000 bar are already available commercially, so this does not seem to present a difficult technical challenge. High-pressure hoses are heavier, however. The question is whether the hose material and thickness required is compatible with a StratoShield system. Consider a hose made from a composite (possibly multilayered) material 10 mil (254 micron) thick with a mass of 400 g/m2. A layer of high-strength Zylon fibers woven into the hose wall contain the high fluid pressure and are designed to reduce the operating stress from 800,000+ psi to a longterm creep-resistant value of 340,000 psi.
The hose mass required to confine a pressure of 5,000 bar scales with the mass of the fluid and the ratio of pressure to fiber strength. For most of the hose’s length, the pressure-resistant mass dominates, requiring a hose mass of about 40% that of the SO2. This penalty is highest at the base and decreases with height as the pressure requirement falls. The hose, in other words, need not be as strong and heavy at the top as it is at the bottom, if all the pumping is done on the ground.
For large diameter hoses, the pressure is dominated by the gravitational head, and the hose weight is dominated by the large diameter of the hose rather than the thickness of the wall. For narrow hoses, flow resistance increases the pressure, the compressibility of the fluid, and hence the weight penalty
imposed by hose wall thickness. On the other hand, the overall mass can be lower for a narrow hose simply because it encloses a smaller volume.
Option 2: Smaller Pumps in the Air
Instead of relying solely on a big pump on the ground, we could place a series of pumps at intervals along the hose. Large pressures and fluid compressibility then cease to be concerns, and the hose can be lighter and have thinner walls. Each pump need deliver only modest pressure, and we could build extras into the chain so that the system can tolerate occasional pump failures. The total mass requiring support will be greater than what is shown in table 2, however, because it will include the
additional weight of the pumps themselves as well as the electrical cables that power them.
The total pumping power required for the distributed approach is, of course, very similar to that for a ground-based pump, but there are small differences. The absence of compressibility reduces the gravitational head, but for low diameter hoses this effect is more than offset by the fact that denser fluid requires lower flow velocities and hence incurs less flow resistance.
Up, up, and Away
Let us turn now to the question of how to raise the hose to the sky and hold it there. Others have suggested building enormous towers to support a hose, but this seems unnecessarily expensive and risky. A more practical way to support a hose to the sky is to harness atmospheric forces, either buoyancy or aerodynamic lift.
Balloons and blimps are well developed technologies, and are quite capable of lofting the hose weights presented in tables 1 and 2. As with pumping, we can choose among several strategies. One extreme is to lift only from the top of the hose, using a single long-duration balloon of 200 meters or more in diameter, flying at an altitude of about 30 kilometers. (A cluster of 100-meter-diameter balloons could work as well.) The hose material must then have sufficient tensile strength to support the entire system, or must be assisted by additional support cables. Because atmospheric density is low in the stratosphere, the balloon would have to be enormous to develop enough buoyancy.
At the opposite end of the range of strategies is an approach in which the hose itself is buoyant, so that every point along its length carries its own weight. (For a preliminary analysis of this option, see Blackstock et al. 2009.) In between these two extremes are intermediate strategies that use multiple balloons, each of which supports one segment of the hose. This approach allows the balloons to fly at lower altitudes and thus to be smaller (see illustration on next page). The hose itself need have minimal tensile strength, which translates to lighter weight. One benchmark that is useful is considering these options is NASA’s long-standing project to develop and demonstrate large, high-altitude balloons that are superpressurized with helium. A mission in December 2008 flew one such balloon that was 80 meters in diameter (200,000 m3 volume) to an altitude of 33 kilometers. NASA plans to fly even larger balloons, of over 600,000 m3 volume, in future missions.
The NASA balloons are not spherical, but rather are pumpkin- shaped for greater structural efficiency. The envelope has an isotensoid meridional profile and a multi-lobed, azimuthal shape. A thin-walled plastic material both contains the helium and transfers the internal gas pressure azimuthally to the meridional borders of each lobe. Global pressure loads are then handled by strong fibers running along each meridional cusp.
Table 3 shows the lifting capacity of such balloons as a function of their size and of altitude along the hose (see page 9). These figures show that a series of small balloons, each 20 to 30 meters in diameter and spaced roughly a kilometer from the next, should easily support typical hose weights of 1 to 2 ton/km along the lower half of the hose. Near the top of the hose, however, larger balloons of 60 to 70 meters in diameter would be needed (or alternatively more small balloons spaced closer together).
The distributed support strategy offers us considerable design freedom, because balloons need not be equal in size or set at equal intervals. Nor, given the tensile carrying capability of the hose, do lift and weight have to be balanced to close tolerances at each location. We could, for instance, elect to devote 0.2 ton/km of the hose mass to the strong Zylon fibers previously discussed; this strategy would yield a hose with a 30-ton carrying capacity, allowing very large offsets of lift to weight.
Blowing in the Wind
Given all these options, a support system would be straightforward to design—if only there were no wind. Unfortunately, winds at altitude are strong, often blow in different directions at different altitudes, and can change speed and direction rapidly. The need to deal with the static and dynamic forces imposed by wind will greatly influence the design of the hose’s aerial support.
The existence of winds prevent the question of top-hung vs. distributed support from being the open-and-shut case it would otherwise be. The most efficient way structurally to help a long, thin object such as the hose resist sideways deflection by the wind is to draw it taut—exactly what a giant balloon at the top would do. Moreover, the strongest and most variable winds do not occur in the stratosphere, but at intermediate altitudes of around 10 kilometers (33,000 feet)— altitudes where one might distribute smaller support balloons. Lofting balloons in the windiest part of the atmosphere will expose the system to more wind stress.
Wind speeds generally increase in altitude, reaching values around 60 m/s at heights of 10 to 15 kilometers. When convolved with the atmospheric density profile, the dynamic pressures generated by the wind peak at roughly 1,000 Pa in the vicinity of 10 km altitude.
The wind pushes both the balloons and the hose itself. These should be thus designed to minimize drag and to present the smallest cross-section to the wind achievable (particularly for segments near 10 km altitude, where the wind forces are highest).
The balloons pose the greater challenge because of their larger lateral area: a single spherical balloon 35 meters in diameter presents about 1,000 m2 of area to the wind, for example, which is about the same lateral area as the entire length of a hose 3 centimeters wide and 30 km long. Omitting balloons
from the hose in the region around 10 km altitude would reduce the dynamic pressure on the system. But if the hose is denuded of balloons in its middle, the balloons at higher altitudes must be correspondingly larger.
To illustrate the trade-off, let’s compare two designs for supporting a StratoShield that includes a hose 3 cm in diameter, pumped solely from the ground. For safety, let’s assume the balloons must support the full 50 tons of the lofted structure plus the SO2 payload, not just the weight of the empty hose. The first design balances lift and weight locally, as they vary along the hose, by placing balloons of appropriate size every half kilometer. The balloons range in diameter from 15 meters at the base to 56 meters at the top. Altogether, the balloons present an aggregate lateral area of 30,000 m2 to the wind—30 times the area of the hose itself. When convolved with the dynamic wind pressure, the aggregate side force (for a drag coefficient of 1) is 3.3 MNwt, which is more than six times the weight of the hose.
The second design balances lift and weight globally, by placing balloons only near the top of the hose, at a spacing of a half kilometer between the altitudes of 20 and 30 km. The balloons in this design are larger, ranging in diameter from 50 meters to 85 meters. Altogether, their aggregate lateral area is 45,000 m2, 50% larger than in the first case. When convolved with the dynamic wind pressure, however, the aggregate side force (again for a unit drag coefficient) is only 2.5 MNwt, about one quarter lower than in the first design. The side force is still much greater than the weight of the hose, however. Clearly we must find some way to drastically reduce the wind load.
One redeeming feature of wind forces is that they can provide aerodynamic lift as well as drag. We could take advantage of this by using kites or other lifting airfoils to help support the hose. Although they wouldn’t function all the time, they would provide lift at precisely the times it is most needed—when the wind is severe and pushing the hose sideways.
An even better solution may be to use buoyant lifting bodies, such as elongated balloons shaped like aerodynamic blimps rather than squat pumpkins. The balloons themselves can then combine the functions of static and dynamic lift. This approach offers three major advantages. First, an elongated
shape presents a much smaller frontal area to the wind for any given interior volume. Second, and even more important, is a reduction to the drag coefficient: for a typical blimp this is about 0.05, 1/20th that of a pumpkin-shaped balloon.
Finally, blimps can be designed to generate aerodynamic lift that greatly exceeds the drag force. JP Aerospace has designed large V-shaped blimps that reportedly can generate 20 times as much lift force as the drag imposed by incident wind. The company has even constructed prototypes. Although a high ratio of lift to drag doesn’t actually reduce the lateral force imposed by the wind, it would increase the hose tension, thereby reducing the deflection caused by the wind.
The one clear disadvantage of using blimp-like balloons is that they are less structurally efficient than pumpkin-shaped designs. That is, they have more wall mass per unit of buoyant lift, so they must be larger and made from more envelope material. These are affordable penalties, however, particularly since the gains in aerodynamic lift more than offset the losses in buoyancy.
We can similarly reduce the drag coefficient of the hose by giving it a streamlined shape or by surrounding it with a low-mass aerodynamic sheath. In either case, the wind will automatically twist the hose into the proper, drag-minimizing, orientation. It seems clear that sensible use of well understood strategies for producing aerodynamic lift and reducing aerodynamic drag can enable a StratoShield system to tolerate wind forces with only modest (albeit highly dynamic) deflection of the hose.
Intead of a Hose, an Elevator?
An “elevator” is another alternative for lifting mass to the stratosphere. Like the hose, it would use one or more lighterthan- air structures tethered to the ground and a dispersal system at the top of the tether, nominally at 30 km altitude. The elevator, however, would carry the payload liquid in discrete tanks carried by vehicles (“climbers”), which crawl up the tether cable.
The main advantage that an elevator offers over a hose is the elimination of flow resistance. In principal, an elevator could transport liquids much more quickly than a hose of equivalent static capacity. It is certainly reasonable to imagine designing a vehicle that climbs a cable at tens of meters per second, in contrast to the few meters per second envisioned above for a 1½-inch (3.8-centimeter) hose.
We could consider many design options for a stratospheric elevator system. Motive power could be delivered mechanically by a continuous loop of moving cable (similar to a ski lift) or by a winch; or via electric traction, using external power from the cable or beamed from the ground; or by self-powered motors on the vehicles themselves.
The system could use just one large-capacity climber or several smaller vehicles. A single-car system is simpler. Increasing the number of cars keeps the load on the cable closer to constant, however, as well as more evenly distributed. Multiple vehicles could travel on a single cable if “sidings” were placed to allow up- and down-traveling vehicles to pass one another.
Other options include:
vehicles that simply drop from the top of the cable and fall or glide back to Earth when empty;
separate cables going up and coming down. A challenge with this approach would keeping cables from tangling or vehicles from colliding, unless the cables were very widely space at the ground.
The simplest option is probably to send a single self-powered up and down a single stationary cable. The most efficient option is likely a “conveyor belt” with an endless loop of cable carrying many small tanks. The latter would require a large amount of engineering development, however.
The first choice of power plant for a self-powered climber would be a turboshaft engine—or perhaps a lightweight, turbocharged piston engine—driving the vehicle mechanically. Unfortunately, the upper portion of the cable is, at 25 to 30 km, too high for existing air-breathing engine designs; the Perseus-B used a triple turbocharger to run at 18 km (62,000 ft.) altitude, the current record. (See www.aurora.aero for details.)
We have considered other options that might work for short-duration climbs with minimal pollution into the stratosphere, including:
a monopropellant or bipropellant turbogenerator, e.g., using hydrogen peroxide plus a small amount of hydrocarbon fuel;
an air-breathing turbogenerator that operates from sea level to 15–18 km, at which point high-specific-energy lithium batteries provide main propulsive power;
high-efficiency electric motors driven exclusively by battery power.
A climber powered solely by batteries, if it is reasonably efficient, could climb to 30 km with about 50% payload fraction (~200 Wh/kg = 720 kJ/kg = 2.4 kg lifted to 30 km per 1 kg of battery). Outfitted with a lightweight motor to provide power for the first 15 km, it could have a payload fraction of about
70%. For long-term use of a battery-powered climber, however, batteries would have to endure many more than 1,000 chargedischarge cycles. If such options were not available, then a laseror microwave-beamed power system, or a moving cable, would offer the next most attractive and cost-effective approaches.
A Cable to the Sky
Finally, let’s consider what kind of cable would be required by a 30 km elevator. Zylon or similar cable of 1 cm2 thickness offers a usable tensile strength (with safety margins) of 2 GPa and a load rating of 20,000 kg at a cable mass of 156 kg/km (so 4,700 kg for 30 km). It may be necessary to use multiple thinner cables interconnected by webbing to provide both protection from single-point breaks and additional traction area. Indeed, this is the “ribbon” configuration beloved of those who advocate development of space elevators.
If we assume a top station (tanks, tank swap mechanism, sprayer) that weighs one metric ton, then the total mass to be lifted is 15,700 kg. That is less than one third of the weight of a hose system pumped solely from the ground.
A slightly more sophisticated elevator system capable of maintaining climb speeds of 50 m/s—or one that includes a relay station at around 15 km altitude so that two climbers can travel at once—could substantially reduce the cycle time and thus the system mass. A 6,000 kg vehicle and 10,000 kg total system weight would be a reasonable goal.
An elevator could offer other advantages over a hose besides lower weight. It would be easier to unload the system quickly in the event of high winds aloft or low-altitude storms. Unloading a 30 km hose might require more than an hour, compared to about 15 minutes for an elevator. A related advantage is the ease with which the system could be unloaded at night in order to reduce load on the balloons and maintain constant altitude. An elevator system is also probably easier to prototype at small scale (e.g. 10,000 tons per year delivery rates), whereas flow resistance makes this difficult to do with a long hose.
J. J. Blackstock, D. S. Battisti, K. Caldeira, D. M. Eardley, J. I. Katz, D. W. Keith, A. A. N. Patrinos, D. P. Schrag, R. H. Socolow and S. E. Koonin, Climate Engineering Responses to Climate Emergencies, Novim, 2009.