Liquid helium is stored at its boiling-point temperature of 4.2 K in a spherical container (r = 0.30 m). The container is a perfect blackbody radiator. The container is surrounded by a spherical shield whose temperature is 74 K. A vacuum exists in the space between the container and the shield. The latent heat of vaporization for helium is 2.1 × 104 J/kg. What mass of liquid helium boils away through a venting valve in one hour?
Use the stephan boltzman law to get power (energy/time). https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law
Then mass He/second=Hv*energy/second
mass in an hour: multiply by 3600sec/hr
To calculate the mass of liquid helium that boils away through a venting valve in one hour, we need to determine the rate of heat transfer from the liquid helium to the surroundings.
The rate of heat transfer is given by the equation:
Q = k * A * (Thot - Tcold) / d
Where:
Q is the rate of heat transfer (in watts),
k is the thermal conductivity of helium (which we will assume to be constant),
A is the surface area of the container,
Thot is the temperature of the hot body (in this case, the liquid helium),
Tcold is the temperature of the cold body (in this case, the shield surrounding the container), and
d is the thickness of the vacuum gap between the container and the shield.
We can use this equation to find the rate of heat transfer, and then calculate the mass of helium that boils away using the latent heat of vaporization.
1. Calculate the rate of heat transfer (Q):
We need to determine the values for k, A, Thot, Tcold, and d.
- Thermal conductivity, k:
The thermal conductivity of helium is approximately 145 J/(m*K) at atmospheric pressure. Since the pressure is not given, we will assume this value as an approximation.
- Surface area, A:
The surface area of the container can be calculated using the formula for the surface area of a sphere: A = 4πr^2, where r is the radius of the container.
Substituting the given radius, r = 0.30 m, we get:
A = 4π(0.30)^2
- Hot body temperature, Thot:
The hot body temperature is given as the boiling point of liquid helium, which is 4.2 K.
- Cold body temperature, Tcold:
The temperature of the shield surrounding the container is given as 74 K.
- Vacuum gap thickness, d:
The problem states that a vacuum exists in the space between the container and the shield, meaning there is no material between the two. Therefore, the thickness of the vacuum gap, d, is zero.
Now, substitute the values into the equation for Q:
Q = k * A * (Thot - Tcold) / d
(Note: Since d = 0, we need to modify the equation.)
Q = k * A * (Thot - Tcold) / ε
Where ε is the emissivity of the container. In this case, the container is a perfect blackbody radiator, so the emissivity, ε, is equal to 1.
Q = k * A * (Thot - Tcold) / 1
Q = k * A * (Thot - Tcold)
2. Calculate the mass of helium boiled away:
To calculate the mass of helium boiled away, we can use the equation:
Q = m * ΔH
Where:
m is the mass of the helium boiled away, and
ΔH is the latent heat of vaporization for helium, which is given as 2.1 × 10^4 J/kg.
Rearranging the equation, we get:
m = Q / ΔH
Substitute the value of Q obtained in step 1 and ΔH:
m = Q / 2.1 × 10^4
Finally, convert the time from one hour to seconds, as the rate of heat transfer is given in watts:
m = (Q * t) / 2.1 × 10^4
Where t is the time in seconds (t = 1 hour = 3600 seconds).
Substitute the known values and calculate the mass.