Question

The Goodyear blimp Columbia (see Fig. \(15-20\) ) is cruising slowly at low altitude, filled as usual with helium gas. Its maximum useful payload, including crew and cargo, is \(1280 \mathrm{~kg} .\) How much more payload could the Columbia carry if you replaced the helium with hydrogen? Why not do it? The volume of the helium-filled interior space is \(5000 \mathrm{~m}^{3}\). The density of helium gas is \(0.160 \mathrm{~kg} / \mathrm{m}^{3}\) and the density of hydrogen is \(0.0810 \mathrm{~kg} / \mathrm{m}^{3}\).

Short answer

By replacing helium with hydrogen, the Columbia could carry approximately an extra 395 kilograms. However, this change is not put into practice due to the highly flammable nature of hydrogen, which would severely increase the risk of fires or explosions.

Step-by-step solution

Calculate the weight of helium

First, the weight of the helium occupying the internal volume needs to be calculated. This is done by multiplying the volume (V) of the blimp with the density (ρ) of helium, i.e., \(V_{He} * ρ_{He}\), which is equal to \(5000 m^3 * 0.160 kg/m^3\). This results in the total weight of the helium.

Find the weight of hydrogen

Now, calculate the weight of the hydrogen if the blimp were to be filled with it instead of helium. This follows the same formula as above but with the density of hydrogen, i.e., \(V_{H2} * ρ_{H2}\), which is equal to \(5000 m^3 * 0.0810 kg/m^3\) . This will provide us with the weight of the hydrogen.

Calculate the additional payload

Next, the extra payload the blimp can bear when filled with hydrogen can be determined by subtracting the weight of hydrogen from the weight of helium calculated in the above steps. The difference gives us the additional payload that can be carried when the blimp is filled with hydrogen instead of helium.

Answer the second part of the question

After doing the calculations, one might wonder why hydrogen isn't actually used instead of helium, despite maximizing the payload. This is primarily for safety reasons. Hydrogen is highly flammable and poses risks of explosion, as famously happened in 1937 with the Hindenburg disaster. Therefore, helium, which is a noble gas and non-combustible, is considered more suitable for use in airships.

Key concepts

Helium vs Hydrogen Buoyancy

Understanding the differences in buoyancy between helium and hydrogen is crucial in applications such as lighter-than-air travel, where lifting gases are selected for airships and balloons. Buoyancy itself is a force that arises from differences in density between a fluid (such as air) and a submerged object or gas. \(Archimedes' principle\) tells us that this force is equal to the weight of the fluid displaced by the object.

Helium and hydrogen are both lighter than air, which is why they are used to lift balloons and blimps like the Goodyear blimp. Helium, with a density of \(0.160 \mathrm{~kg}/\mathrm{m}^3\), is not as light as hydrogen, which has a density of \(0.0810 \mathrm{~kg}/\mathrm{m}^3\). This means under the same conditions, hydrogen can generate more lift because it displaces more air weight per unit volume. Despite hydrogen's superior lifting capacity, helium is preferred due to safety concerns, as hydrogen is highly flammable.

When we consider an airship cruising at low altitude, the lifting gas it carries must provide enough buoyant force to support the airship's own weight, as well as any payload. Replacing helium with hydrogen would, in theory, allow for a greater payload due to its lower density and thus higher buoyant force. However, safety outweighs the additional lifting potential.

Goodyear Blimp Physics

The physics behind the Goodyear blimp involve principles of buoyancy and gas density. The blimp achieves lift by using a gas that is less dense than the surrounding air. A careful calculation is made to ensure that the overall density of the blimp (including its structure, the gas, and the payload) is less than that of air. This is necessary for the blimp to float and manoeuvre through the air.

The blimp's envelope is filled with a gas like helium to provide the necessary buoyancy. For the Goodyear blimp, the maximum useful payload, including the crew and cargo, is dictated by the difference in weight between the volume of gas used for buoyancy and the equivalent volume of air. If helium is replaced with hydrogen, the blimp could carry more because the weight of the hydrogen is less, increasing the net buoyant force. However, the trade-off for this increased payload capacity is the augmented risk of using hydrogen, as its flammable nature can lead to catastrophic outcomes in the case of a leak or ignition source.

The understanding of such physics not only guides the design and operation of blimps but is also an example of applied fluid dynamics where the relationship between pressure, volume, temperature, and density determine the behavior of gases and their application in real-world technology.

Gas Density Calculations

Calculating gas density plays a pivotal role in various scientific and engineering fields. Density, which is mass per unit volume (\( \rho = \frac{m}{V} \)), is essential for understanding how gases will behave under different conditions, particularly in applications like aeronautics and meteorology.

In the case of the Goodyear blimp, the gas densities of helium and hydrogen determine the overall lift that the blimp can achieve. To calculate the weight of a gas within a volume, we use the formula:\[ W = V \cdot \rho \]where \(W\) is the weight, \(V\) is the volume the gas occupies, and \(\rho\) is the gas's density. The helium-filled blimp's interior space volume is \(5000 \mathrm{m}^3\), and by multiplying it with the density of helium, \(0.160 \mathrm{kg}/\mathrm{m}^3\), we find the weight of the helium. Likewise, we can determine the weight of hydrogen by using its density, \(0.0810 \mathrm{kg}/\mathrm{m}^3\), with the same volume.

The difference in these weights reflects the potential additional payload capacity when switching from helium to hydrogen. Such density calculations provide a basis for understanding buoyant forces in gases and enable engineers to design and operate airships efficiently and safely according to their intended use and the properties of available lifting gases.