Question

\(3-\mathrm{kg}\) of helium gas at \(100 \mathrm{kPa}\) and \(27^{\circ} \mathrm{C}\) are adiabati cally compressed to 900 kPa. If the isentropic compression efficiency is 80 percent, determine the required work input and the final temperature of helium.

Short answer

Answer: The required work input is 25,495,652 J, and the final temperature of the helium is 983.22 K.

Step-by-step solution

Convert Temperature to Kelvin

We need to convert the given initial temperature from Celsius to Kelvin. \(T_1 = 27^\circ\mathrm{C} + 273.15 = 300.15\, \mathrm{K}\)

Find the Specific Heat Ratio (k) for Helium

For helium gas, the specific heat ratio is \(k = 1.67\). We'll need this value in subsequent calculations.

Calculate the Initial Volume (V1) Using the Ideal Gas Law

Use the ideal gas law, \(PV = mRT\), to find the initial volume (V1). Rearrange the formula to solve for V1: \(V_1 = \frac{mRT_1}{P_1}\) Where: \(P_1 = 100\, \mathrm{kPa} \times 10^3\, \mathrm{Pa/kPa} = 1 \times 10^5\,\mathrm{Pa}\) \(R = 2077\, \mathrm{J/(kg \cdot K)}\) (specific gas constant for helium) \(m = 3\,\mathrm{kg}\) \(T_1 = 300.15\, \mathrm{K}\) Plug in the given values: \(V_1 = \frac{3\,\mathrm{kg} \times 2077\,\mathrm{J/(kg \cdot K)} \times 300.15\,\mathrm{K}}{1 \times 10^5\,\mathrm{Pa}} = 18.63\,\mathrm{m^3}\)

Calculate Final Volume and Temperature Assuming Isentropic Compression

Use the relationship for the adiabatic process: \(\frac{P_1}{P_2} = \left(\frac{V_2}{V_1}\right)^k\) Rearrange the formula to solve for V2: \(V_2 = V_1 \left(\frac{P_1}{P_2}\right)^{\frac{1}{k}}\) Where \(P_2 = 900\, \mathrm{kPa} \times 10^3\, \mathrm{Pa/kPa} = 9 \times 10^5\, \mathrm{Pa}\) Plug in the given values: \(V_2 = 18.63\,\mathrm{m^3} \left(\frac{1 \times 10^5\,\mathrm{Pa}}{9 \times 10^5\,\mathrm{Pa}}\right)^{\frac{1}{1.67}} = 5.23\,\mathrm{m^3}\) Now we can find the isentropic final temperature \(T_{2s}\) using the ideal gas law: \(T_{2s} = \frac{P_2 V_2}{mR} = \frac{9 \times 10^5\,\mathrm{Pa} \times 5.23\,\mathrm{m^3}}{3\,\mathrm{kg} \times 2077\,\mathrm{J/(kg \cdot K)}} = 847.55\,\mathrm{K}\)

Find the Actual Final Temperature (T2) Using Isentropic Efficiency

Use the isentropic efficiency, \(\frac{T_{2s} - T_1}{T_2 - T_1} = \eta_c\), to find the actual final temperature (T2). Rearrange the formula to solve for T2: \(T_2 = T_1 + \frac{T_{2s} - T_1}{\eta_c}\) Where \(\eta_c = 0.8\) (isentropic efficiency) Plug in the given values: \(T_2 = 300.15\,\mathrm{K} + \frac{847.55\,\mathrm{K} - 300.15\,\mathrm{K}}{0.8} = 983.22\,\mathrm{K}\)

Calculate the Required Work Input

Use the equation for work input, \(W = mC_v(T_2 - T_1)\), and the specific heat at constant volume for helium, \(C_v = \frac{R}{k-1}\), to find the required work input. \(C_v = \frac{2077\,\mathrm{J/(kg \cdot K)}}{1.67 -1} = 12460.72\,\mathrm{J/(kg \cdot K)}\) \(W = 3\,\mathrm{kg} \times 12460.72\,\mathrm{J/(kg \cdot K)} \times (983.22\,\mathrm{K} - 300.15\,\mathrm{K}) = 25495652\,\mathrm{J}\) The required work input is \(25495652\,\mathrm{J}\), and the final temperature of the helium is \(983.22\,\mathrm{K}\).

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics that connects the pressure, volume, temperature, and amount of moles of a gas. It's often expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, and \(R\) is the ideal gas constant. However, in many practical applications, we use the form \(PV = mRT\), where \(m\) represents the mass of the gas, and \(R\) is the specific gas constant, unique to each gas.

For example, in our exercise, we used the ideal gas law to calculate the initial volume of helium gas by rearranging the law to solve for volume: \(V_1 = \frac{mRT_1}{P_1}\). Knowing the starting conditions of the gas and its specific gas constant allowed us to definitively determine the volume it occupies before the compression process.

Specific Heat Ratio

The specific heat ratio, also known as the adiabatic index or gamma \(\gamma\), is the ratio of the specific heat at constant pressure \(C_p\) to the specific heat at constant volume \(C_v\), typically denoted as \(k\). It's symbolized by \(k = \frac{C_p}{C_v}\) and is crucial in thermodynamics when dealing with processes like isentropic compression, which is a type of adiabatic process.

For gases like helium, the specific heat ratio affects how the temperature and pressure change during compression. Helium, being a monatomic gas, has a higher specific heat ratio (in this case, \(k = 1.67\)) compared to diatomic gases. This means it heats up substantially during compression.

Thermodynamics

Thermodynamics is the branch of physics dealing with heat and its relation to other forms of energy and work. It describes how thermal energy is converted to and from other forms of energy and how it affects matter.

The exercise we're examining is based on the first and second laws of thermodynamics, which tell us energy cannot be created or destroyed and that systems tend to move towards increased entropy, respectively. Isentropic processes, a key part of this exercise, assume a special case where the entropy remains constant, typically in idealized, perfectly efficient systems or when considering rapid processes like the one in our example.

Isentropic Efficiency

Isentropic efficiency is a measure of the deviation of an actual compressive or expansive process from the ideal isentropic process. Since real-life processes aren't perfectly efficient due to factors like friction and heat loss, isentropic efficiency gives us a way to quantify this by comparing the work output (or input) of an isentropic process to that of the actual process.

In our problem, we use the isentropic efficiency of 80 percent to find the actual final temperature \(T_2\) of helium after compression. This efficiency value means the actual process requires more work input than the isentropic process to achieve the same pressure increase.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. When a gas is compressed adiabatically, the work done on the gas increases its internal energy, which in turn raises its temperature, assuming no heat is lost to the environment.

The adiabatic process is described by the formula \(PV^k = \text{constant}\), which we used in the exercise. This equation, along with the ideal gas law, allows us to calculate changes in volume and temperature throughout a compression or expansion that happens without heat transfer, reflecting the unique relationship between pressure, volume, and temperature for a specific heat ratio \(k\) and gas.