Question

Submarines change their depth by adding or removing air from rigid ballast tanks, thereby displacing seawater in the tanks. Consider a submarine that has a \(700 \mathrm{m}^{3}\) air-ballast tank originally partially filled with \(100 \mathrm{m}^{3}\) of air at \(1500 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\). For the submarine to surface, air at \(1500 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C}\) is pumped into the ballast tank, until it is entirely filled with air. The tank is filled so quickly that the process is adiabatic and the seawater leaves the tank at \(15^{\circ} \mathrm{C}\). Determine the final temperature and mass of the air in the ballast tank.

Short answer

Answer: To find the initial mass of the air, we can rearrange the ideal gas law equation as follows: $$m = \frac{PV}{RT}$$ Given the initial conditions, we have \(P_1 =1.4\,\mathrm{bar}\), \(V_1 =100\,\mathrm{m}^3\), \(T_1 = 288\,\mathrm{K}\), and \(R = 0.287 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}}\). Note that we must convert the initial pressure to kPa by multiplying by 100: \(P_1 = 1.4\,\mathrm{bar} \cdot 100 = 140\,\mathrm{kPa}\). $$m = \frac{(140\,\mathrm{kPa})(100\,\mathrm{m}^3)}{(0.287 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}})(288\,\mathrm{K})}$$ Calculate the mass of the air, and provide your answer in kilograms (kg).

Step-by-step solution

Setup the ideal gas law equation for the initial conditions.

We'll use the initial conditions given to calculate the initial mass of the air, using the ideal gas law equation: $$PV = mRT$$ where \(P\) is the pressure, \(V\) is the volume, \(m\) is the mass, \(R\) is the specific gas constant for air, and \(T\) is the temperature in Kelvin. The specific gas constant for air is \(R = 0.287 \frac{\mathrm{kJ}}{\mathrm{kg} \cdot \mathrm{K}}\). Note that we must convert the initial temperature to Kelvin by adding 273: $$T_1 = 15^{\circ}\mathrm{C}+273=288\mathrm{K}.$$

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and amount of an ideal gas. It is usually expressed as
\(PV = nRT\)
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. For practical purposes, when dealing with specific amounts of a gas, we often use the mass \(m\) rather than the number of moles, and the specific gas constant \(R_{\text{specific}}\) instead of the universal gas constant. This modifies the equation to:
\(PV = mR_{\text{specific}}T\).
In this form, it is particularly useful for applications like the one in the exercise regarding the submarine's ballast tanks. It allows us to solve for any one of the variables if the others are known. Moreover, assuming the gas behaves ideally is reasonable when the conditions of temperature and pressure are not extreme, as with the air in the submarine's ballast tank.

Specific Gas Constant

The specific gas constant, symbolized as \(R_{\text{specific}}\), is an important property of a gas that differs for each gas type. It is related to the universal gas constant \(R\) but takes into consideration the individual gas's molecular weight. The specific gas constant is derived from the universal gas constant divided by the molecular mass (molar mass) of the gas:
\(R_{\text{specific}} = \frac{R}{M}\)
where \(M\) is the molar mass of the gas. For the air, which is essentially a mix of gases, an average value is used, typically \(R_{\text{specific}} = 0.287 \frac{kJ}{kg \cdot K}\). This property is critical when working with the ideal gas law, as seen in the submarine exercise. It allows you to calculate the mass of the gas when the pressure, volume, and temperature are known, or vice versa. To work any problem involving the ideal gas law, determining the appropriate specific gas constant for the gas in question is vital.

Submarine Ballast Tanks

Submarine ballast tanks are crucial for a submarine's ability to control its buoyancy and thus its depth in the water. These are large tanks that can be filled with water or air: filling with water increases the overall density of the submarine, making it sink, while filling with air reduces the density, allowing it to rise.
The exercise presented focuses on the process of surfacing, where air is added to the ballast tanks to displace water. This is a rapid procedure that can approximate an adiabatic process – a process where no heat is exchanged with the surroundings. In the context of the ideal gas law, the adiabatic nature informs us that the temperature change of the air in the tanks will be a result of the air compression, not an exchange of heat. Submarines are engineered to manage these tanks efficiently, enabling them to navigate through different water depths with precision. Understanding the physical principles behind the operation of ballast tanks, including the behavior of gases, thermodynamics, and fluid mechanics, is essential for the safe and effective operation of submarines.