Question

The boiling temperature of nitrogen at atmospheric pressure at sea level ( 1 atm pressure) is \(-196^{\circ} \mathrm{C}\). Therefore, nitrogen is commonly used in low-temperature scientific studies since the temperature of liquid nitrogen in a tank open to the atmosphere will remain constant at \(-196^{\circ} \mathrm{C}\) until it is depleted. Any heat transfer to the tank will result in the evaporation of some liquid nitrogen, which has a heat of vaporization of \(198 \mathrm{~kJ} / \mathrm{kg}\) and a density of \(810 \mathrm{~kg} / \mathrm{m}^{3}\) at 1 atm. Consider a 3-m-diameter spherical tank that is initially filled with liquid nitrogen at 1 atm and \(-196^{\circ} \mathrm{C}\). The tank is exposed to ambient air at \(15^{\circ} \mathrm{C}\), with a combined convection and radiation heat transfer coefficient of \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The temperature of the thin-shelled spherical tank is observed to be almost the same as the temperature of the nitrogen inside. Determine the rate of evaporation of the liquid nitrogen in the tank as a result of the heat transfer from the ambient air if the tank is \((a)\) not insulated, \((b)\) insulated with 5 -cm-thick fiberglass insulation \((k=0.035 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\), and (c) insulated with 2 -cm-thick superinsulation which has an effective thermal conductivity of \(0.00005 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).

Short answer

Answer: The rates of evaporation of liquid nitrogen in the spherical tank are (a) 1.051 kg/s (non-insulated), (b) 0.021 kg/s (fiberglass insulation), and (c) 0.00037 kg/s (superinsulation).

Step-by-step solution

(Step 1: Calculate the heat transfer rate for each case)

First, let's calculate the heat transfer rate for each case. For that, we need to know the temperature difference between the liquid nitrogen and the ambient air, and the surface area of the tank. The heat transfer rate formula for each case will be different depending on the insulation. For the non-insulated case (a), we will apply the combined convection and radiation heat transfer coefficient. For the insulated cases (b) and (c), we'll need to apply the formula for heat conduction through the insulation.

(Step 2: Find the temperature difference and tank's surface area)

Determine the temperature difference between the liquid nitrogen (\(T_N= -196°C\)) and the ambient air (\(T_A = 15°C\)), which is the driving force for heat transfer: \(\Delta T = T_A - T_N = 15 - (-196) = 211 K\) Now calculate the surface area of the 3-m diameter spherical tank: \(A = 4 \pi r^2\) Where \(r\) is the radius of the tank, \(r = \frac{D}{2} = \frac{3}{2} = 1.5 \thinspace m\). Calculate the surface area: \(A = 4 \pi (1.5)^2 = 28.27 m^2\)

(Step 3: Calculate the heat transfer rate for Case A)

For the non-insulated case (a), the heat transfer rate (\(\dot{Q}\)) will be given by: \(\dot{Q}_A = h \cdot A \cdot \Delta T\) Where \(h = 35 \frac{W}{m^{2} \cdot K}\). Calculate the heat transfer rate for case (a): \(\dot{Q}_A = 35 \cdot 28.27 \cdot 211 = 208012.95 \thinspace W\)

(Step 4: Calculate the heat transfer rate for Case B)

For the fiberglass insulated case (b), the heat transfer rate will be given by: \(\dot{Q}_B = \frac{k \cdot A \cdot \Delta T}{L}\) Where \(k = 0.035 \frac{W}{m \cdot K}\) and \(L = 0.05 m\). Calculate the heat transfer rate for case (b): \(\dot{Q}_B = \frac{0.035 \cdot 28.27 \cdot 211}{0.05} = 4205.17 \thinspace W\)

(Step 5: Calculate the heat transfer rate for Case C)

For the superinsulated case (c), the heat transfer rate will be given by: \(\dot{Q}_C = \frac{k \cdot A \cdot \Delta T}{L}\) Where \(k = 0.00005 \frac{W}{m \cdot K}\) and \(L = 0.02 m\). Calculate the heat transfer rate for case (c): \(\dot{Q}_C = \frac{0.00005 \cdot 28.27 \cdot 211}{0.02} = 73.85 \thinspace W\)

(Step 6: Determine the rate of evaporation for each case)

Finally, determine the rate of evaporation for each case by dividing the heat transfer rate by the heat of vaporization of liquid nitrogen: \(\dot{m} = \frac{\dot{Q}}{h_v}\) Where \(h_v = 198 \frac{kJ}{kg} = 198000 \frac{W \cdot s}{kg}\). Calculate the mass evaporation rate for case (a), (b), and (c): \(\dot{m}_A = \frac{208012.95}{198000} = 1.051 \thinspace kg/s\) \(\dot{m}_B = \frac{4205.17}{198000} = 0.021 \thinspace kg/s\) \(\dot{m}_C = \frac{73.85}{198000} = 0.00037 \thinspace kg/s\) The rates of evaporation of liquid nitrogen in the spherical tank for each case are \(\dot{m}_A = 1.051 \thinspace kg/s\) (non-insulated), \(\dot{m}_B = 0.021 \thinspace kg/s\) (fiberglass insulation), and \(\dot{m}_C = 0.00037 \thinspace kg/s\) (superinsulation).

Key concepts

Convection

Convection is a heat transfer process where thermal energy moves through fluids, which can be gases or liquids. When a fluid, like air, comes into contact with a surface that is at a different temperature, heat exchange occurs. This is because the molecules at the boundary layer of the fluid gain or lose energy, causing them to move and circulate.
When the air flows around the spherical nitrogen tank in our example, heat is transferred from the ambient air at 15°C to the liquid nitrogen inside. This happens due to convection, driven by the temperature difference of 211 K. A combined convection and radiation heat transfer coefficient of 35 W/m²·K is used to calculate how quickly this heat can be transferred.
This process is essential for understanding how non-insulated systems like case (a) lead to rapid heat gain and a higher rate of evaporation. Convection can be reduced significantly by using insulation, which decreases the contact between the ambient air and the tank's surface.

• Convection entails circulation of fluid molecules
• Driven by temperature differences
• Key factor in non-insulated environments

Radiation

Radiation is another mode of heat transfer that doesn't require a medium, like air or water, to carry heat from one body to another. Heat is transferred as electromagnetic waves, typically in the form of infrared light. Every object emits radiation based on its temperature, and hotter objects emit more radiation.
In our liquid nitrogen tank scenario, radiation works alongside convection to transfer heat from the warm ambient environment to the cooler nitrogen inside. Even in vacuum, radiation can occur, which makes it a critical factor for uninsulated tanks. The combined effect of convection and radiation means that insulating materials must block both for maximum effectiveness.
Radiation is substantially slowed by using superinsulation, which significantly reduces energy transfer. This efficient insulation minimizes the heat flux that might otherwise vaporize the liquid nitrogen at a higher rate. Insulation ensures that the surface of the tank does not easily exchange heat with its surroundings.

• Transfers heat via electromagnetic waves
• Occurs even in vacuum
• Slowed by effective insulation

Thermal Insulation

Thermal insulation reduces heat transfer between objects at different temperatures. Various materials achieve this by limiting conduction, convection, and radiation. In our exercise, two types of insulation are considered: fiberglass and superinsulation. Both serve to decrease the rate of heat enters the nitrogen tank, limiting nitrogen's evaporation rate.
Fiberglass insulation used in case (b) is a common material due to its affordability and effectiveness. It has a thermal conductivity of 0.035 W/m·K, which considerably slows heat transfer. Insulation with fiberglass decreases the mass evaporation rate from 1.051 kg/s (non-insulated) to 0.021 kg/s.
Superinsulation in case (c), with an even lower thermal conductivity of 0.00005 W/m·K, offers superior performance, primarily by forming a barrier against both conduction and radiation. It reduces the mass evaporation to a negligible 0.00037 kg/s. This highlights insulation's critical role in energy conservation and maintaining low temperatures in cryogenic applications.

• Reduces conduction, convection, and radiation
• Materials like fiberglass and superinsulation
• Essential for energy conservation

Heat of Vaporization

The heat of vaporization is the amount of energy required to turn a liquid into vapor without changing its temperature. It is an essential property for substances undergoing phase changes. For liquid nitrogen, the heat of vaporization is 198 kJ/kg, which is relatively high. This means that a significant amount of energy is required to convert it from liquid to gas.
Understanding this concept is vital in our exercise, as any heat entering the nitrogen tank through convection or radiation contributes to the evaporation process. As heat is absorbed, it goes into changing the phase of nitrogen from liquid to gas without increasing its temperature beyond -196°C.
In our scenario, the heat transfer rates directly determine how much nitrogen evaporates. A higher heat transfer rate results in a higher rate of evaporation. Thermal insulation helps control this by limiting the heat that enters the system, thereby preserving an efficient storage tank for the liquid nitrogen en route to conserving its quantity.

• Energy needed for liquid-to-vapor phase change
• Critical for phase change without temperature rise
• Directly affects evaporation rate with heat transfer