Question

A cryogenic storage container holds liquid helium, which boils at \(4.2 \mathrm{~K}\). Suppose a student painted the outer shell of the container black, turning it into a pseudoblackbody, and that the shell has an effective area of \(0.50 \mathrm{~m}^{2}\) and is at \(3.0 \cdot 10^{2} \mathrm{~K}\). a) Determine the rate of heat loss due to radiation. b) What is the rate at which the volume of the liquid helium in the container decreases as a result of boiling off? The latent heat of vaporization of liquid helium is \(20.9 \mathrm{~kJ} / \mathrm{kg} .\) The density of liquid helium is \(0.125 \mathrm{~kg} / \mathrm{L}\).

Short answer

Answer: The rate of heat loss due to radiation is 81 watts, and the rate at which the volume of liquid helium in the container decreases is 31.0 x 10^{-3} L/s.

Step-by-step solution

Find the rate of heat loss due to radiation

To find the rate of heat loss due to radiation, we will use the Stefan-Boltzmann law, which states that the power \(P\) radiated by a blackbody is given by: \(P = \sigma A T^4\) , where \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W} / \mathrm{m^2 K^4}\)), \(A\) is the surface area and \(T\) is the temperature in Kelvin. In this case, the effective area \(A = 0.50 \mathrm{m}^2\) and the temperature \(T = 3.0 \times 10^2 \mathrm{K}\).

Calculate the power radiated

Using the values provided and the Stefan-Boltzmann law, we can calculate the power radiated: \(P = \sigma A T^4\) \(P = (5.67 \times 10^{-8} \mathrm{W} / \mathrm{m^2 K^4}) \cdot (0.50 \mathrm{m}^2) \cdot (3.0 \times 10^2 \mathrm{K})^4\) \(P = 81 \mathrm{W}\) The rate of heat loss due to radiation is 81 watts.

Calculate the rate of mass loss

Now that we have found the rate of heat loss, we can determine the mass loss of helium due to boiling. To do this, we will use the latent heat of vaporization, which is given as \(20.9 \mathrm{kJ/kg}\). We will convert the power from watts to kilojoules per second: \(P = 81 \mathrm{W} \cdot \frac{1 \mathrm{kJ}}{1000 \mathrm{W} \cdot 1 \mathrm{s}} = 0.081 \mathrm{kJ/s}\) Now we can find the mass loss rate (in kg/s) by dividing the power loss by the latent heat of vaporization: \(mass \ loss \ rate = \frac{P}{latent \ heat} = \frac{0.081 \mathrm{kJ/s}}{20.9 \mathrm{kJ/kg}} = 3.88 \times 10^{-3}\ \mathrm{kg/s}\)

Calculate the rate of volume decrease

Finally, we can determine the rate at which the volume of the liquid helium in the container decreases. To do this, we will use the given density of liquid helium, which is \(0.125 \mathrm{kg/L}\). Since we found the mass loss rate to be \(3.88 \times 10^{-3} \mathrm{kg/s}\), we can now find the volume loss rate: \(volume \ loss \ rate = \frac{mass \ loss \ rate}{density} = \frac{3.88 \times 10^{-3} \mathrm{kg/s}}{0.125 \mathrm{kg/L}} = 31.0 \times 10^{-3} \mathrm{L/s}\) The rate at which the volume of the liquid helium in the container decreases is \(31.0 \times 10^{-3} \mathrm{L/s}\).

Key concepts

Heat Loss Due to Radiation

Understanding how objects lose heat through radiation is crucial in various scientific and engineering applications. The Stefan-Boltzmann law provides a cornerstone in calculating this thermal radiation. In the context of our exercise, a painted cryogenic storage container will emit energy in the form of heat radiation due to its increased temperature.

The Stefan-Boltzmann law equation we use is: \(P = \sigma A T^4\), where:\

\
• \(P\) is the thermal power radiated in watts (W),\
• \(\sigma\) is the Stefan-Boltzmann constant, roughly \(5.67 \times 10^{-8} \mathrm{W} / \mathrm{m^2 K^4}\),\
• \(A\) is the emitting surface area in square meters (m²), and\
• \(T\) is the temperature of the emitting surface in Kelvin (K).\

For the exercise, multiplying the constant by the effective area of the container and the fourth power of its temperature yielded the rate at which the container loses heat: 81 watts. This process is particularly important for understanding the thermal management of systems and designing containers like the one holding liquid helium where maintaining low temperatures is essential.

Latent Heat of Vaporization

As we examine the transformation of liquid helium into gas, the concept of latent heat of vaporization becomes vital. This is the amount of energy needed to change a substance from the liquid phase to the gaseous phase at constant temperature and pressure.

In the given scenario, the latent heat of vaporization for helium is \(20.9 \mathrm{kJ/kg}\). To determine how quickly the helium is evaporating, we need to consider the rate of heat loss due to radiation, which influences the rate of mass loss. By dividing the power radiated in kilojoules per second by the latent heat in kilojoules per kilogram, we obtain the mass loss rate.

This calculation helps in determining not just the rate of evaporation but also in planning for refill schedules and understanding the thermal performance of the storage container. Additionally, it is essential for the safe handling and storage of cryogenic substances, where rapid phase changes can lead to pressure build-up and potential hazards.

Cryogenic Storage Container

Cryogenic storage containers are designed to store extremely cold materials, like liquid helium, at temperatures well below those of standard refrigeration. The design of these containers must minimize heat transfer to keep the cryogenic material in its liquid state for as long as possible.

The effectiveness of a cryogenic storage container is evident from our exercise, where we calculate the heat loss due to radiation and the rate at which the helium boils off. This is why painting the container black—which alters its emissivity, thus increasing the rate of thermal radiation—is a significant change. A black surface approximates what is known as a 'blackbody', an ideal surface that absorbs and emits the maximum amount of radiation possible.

It is imperative that cryogenic containers are engineered with materials that provide high thermal insulation, prevent heat conduction, and manage the thermal radiation effectively. These containers are pivotal in industries working with low-temperature physics and chemistry, as well as in the medical field, where cryogenic storage is critical for preserving biological samples.