• Current global food demand growth is ~1.25% pa, while annual growth in supply has been falling and is now only ~1% pa.
• This means food prices are now rising (after decades of falling food prices). 2011 was a record profit year for US farmers. This is good news for renewed investment in the agricultural sector, but until supply can be increased, the poorest will suffer.
• Developing countries have by far the largest effect on food demand. Not only are they growing much faster than developed countries, but a much larger proportion of income increases are spent on food.
• Currently 85-90% of food is consumed in the country it is produced. However, most arable land in Asia is already used, so rising Asian demand will require large increases in productivity per hectare or large-scale food imports.
• The remaining unutilised arable land in the world is mostly in Sub-Saharan Africa and Latin America. Huge land grabs by foreign investors are occurring as a result. In many corrupt countries, the proceeds are going to the political classes, while the poor get dispossessed (even in those countries with property rights, many poor are not within the land title system). A 2008 Daewoo deal to lease 1.3 million hectares in Madagascar contributed to the overthrow of the government there.
• Nutrient deficiencies in the developing world are more severe than energy deficiencies (~15% of population in developing countries are deficient in energy, 31% in Vitamin A, 33% in iodine, 61% in iron). Effects of nutrient deficiencies on intellectual development constitute a poverty trap.
• Governments everywhere need to invest more in research on productivity-increasing sustainable farming methods, which may or may not include GMOs, to avoid excessive monopolisation of agricultural technology vital to food security. Patent reform may be required.

For any elastic balloon, the pressure inside is always a little greater than the pressure outside, since the rubber exerts a restoring force inwards. For maximum lift, we want to minimise the density of the gas inside the balloon, so this extra pressure is not desirable. My goal was to model how much of an effect this has on the lift produced.

I will work through an example using a Novalynx 400-8242 balloon. The reason I have chosen this balloon is that test data is readily available on the web page.

Let us write the pressure equation as:

p_out, the pressure outside the balloon, is a function of height and can be approximated from a standard atmospheric model. p_in can be calculated from the gas law, as a function of the number of moles of gas in the balloon (a constant throughout the ascent), the volume of the balloon, and temperature. In order to calculate Δp, we need a stress-strain model of the balloon rubber [1,2]. Here we use two well-known hyperelasticity models: the Mooney-Rivlin model and the Gent model. The Gent model has the advantage of modelling the stiffening that occurs as the rubber approaches breaking point, but the parameters are more difficult to determine accurately without performing destructive balloon measurements.

For the Mooney-Rivlin model we obtain:

We use the typical values given in [1] for the shear modulus μ and the parameter α (μ=300 kPa and α=10/11). In the Gent model, only the final term in parentheses is different; it includes a Jm parameter related to the maximum possible stretch.

r₀ (the unstretched radius) for the balloon is given by the manufacturer. Using r₀, the mass of the balloon, and the density of rubber, we can also approximate t₀ (the unstretched thickness), which works out as approximately 0.2mm. (In the same way we can estimate the thickness at bursting to be approximately 5μm. Other sources [3] suggest a value of approximately 3.4μm for the balloons used in their experiments, which may be due to differences in the unstretched dimensions or in the manufacturing process.)

By setting r to the initial inflation radius at h=0, we can obtain n (the number of moles of gas in the balloon). Then, we can solve the above equation for r at each altitude h.

Figure 1 shows the resulting graph of balloon radius vs height. As well as the Mooney-Rivlin model and the Gent model, the graph also includes a purely theoretical non-restoring model, in which the balloon exerts no restoring force at all (Δp=0).

The burst radius of the balloon is given by the manufacturer as 3.4m. This occurs at 27.5km for the non-restoring model, 28.2km for the Mooney-Rivlin model and 28.5km for the Gent model with the chosen parameters. All of these figures are close to the performance test data, which indicates a burst height of 28km. So it does not seem that elasticity has much of an effect on the ceiling in this particular example.

Figure 2 shows the pressure difference across the balloon membrane for each model:

For the type of balloon at hand (with given t₀/r₀), it can be seen that the membrane pressure is of the order of 100-200 Pa throughout the flight envelope. This means that the difference in pressure between inside and outside is negligible at low altitudes (sea level pressure is of the order of 100,000 Pa). However, at high altitudes it may start to become important. At 28km the pressure has fallen to 1600 Pa; at 40km it is 290 Pa.

The significance of this becomes more evident when we plot the lift curve:

In the non-restoring model with no pressure difference between inside and outside, constant lift is maintained. However, this is not true when elasticity is considered. In the case of the Gent model the free lift drops from 2.1kg to 1.5kg as the balloon approaches its burst point (the exact value has some uncertainty due to choice of Gent model parameters). In the present example this is not sufficient to arrest the ascent of the balloon, but it may be important to consider if the target altitude is higher and/or the balloon already has marginal lift performance.

Finally the ascent rate was calculated based on a 0.47 drag co-efficient for a sphere:

Notice that the ascent rate increases with height — the reason is that air density (and hence drag) drops more sharply with height than lift does. The initial ascent rate is 5m/s, or 300m/min, which is lower than the tabulated 400m/min. However, the average ascent rate is almost exactly 400m/min, so it is likely that this is what is shown in the performance table. (The whole ascent to 28.5km takes 71.4 minutes.)

The Octave/MATLAB functions used to plot these graphs are available here.

References:
[1] Ingo Müller and Peter Strehlow. Rubber and rubber balloons: paradigms of thermodynamics. Lecture Notes in Physics, Springer, 2004
[2] Landon M. Kanner. Inflation of strain-stiffening rubber-like thin spherical shells. Proceedings of the Virginia Space Grant Consortium Student Research Conference, April 2007
[3] Noboyuki Yayima et al. Scientific ballooning: technology and applications of exploration balloons floating in the stratosphere and the atmospheres of other planets. Lecture Notes in Mathematics, Springer, 2009

The argument that leads to the Ehrenfest paradox is as follows:

1. Assume that a rotating disc is a certain size in its own rotating reference frame.
2. Divide the circumference into small segments of angle dφ and length dσ = Rdφ.
3. Consider each segment in its own frame, which is instantaneously travelling tangentially at v = ωR relative to a stationary observer.
4. Assume that the length of the segment is still dσ in the segment’s frame.
5. Transform from the segment’s frame to the stationary frame, obtaining a contracted length dσ/γ (where γ is the Lorentz factor).
6. The segments do not cover the circumference of the disc in the stationary frame.

In my opinion the problem with this argument, surprisingly, is point 4. Whereas the segment was previously considered from the point of view of a rotating observer at the center of the disc (with angular velocity but no linear velocity), it is now being considered from the point of view of an observer moving with the segment (with linear velocity).

In fact it can be shown that an observer in the segment’s frame¹ measures longer lengths locally, according to the Langevin-Landau-Lifschitz metric:

Measuring the segment (of angle dφ) along the rim we obtain:

Thus, the corrected argument would be:

4. An observer in the segment’s frame would measure the length of the segment as γdσ.
5. Transform from the segment’s frame to the stationary frame, obtaining a contracted length dσ.
6. There is no paradox in the stationary frame.

Note that while each segment-riding observer locally measures an arc length greater than what might be expected, the same observer would see length contraction if observing the far side of the disc. Thus it is not meaningful to add up each observer’s local measurements into a whole greater than 2πR, and there is no paradox there either.

¹ The observer in the segment’s frame is usually known as a Langevin observer after Paul Langevin who originally considered the geometry of this reference frame. Nathan Rosen showed that the same metric also applies to the inertial frame that is instantaneously co-moving alongside the Langevin observer, which we can be more confident applying Lorentz transformations to.

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