Question

An air-standard Carnot cycle is executed in a closed system between the temperature limits of 350 and \(1200 \mathrm{K}\) The pressures before and after the isothermal compression are 150 and \(300 \mathrm{kPa}\), respectively. If the net work output per cycle is \(0.5 \mathrm{kJ},\) determine \((a)\) the maximum pressure in the cycle, \((b)\) the heat transfer to air, and \((c)\) the mass of air. Assume variable specific heats for air.

Short answer

Question: Calculate the mass of air in the air-standard Carnot cycle, given an initial pressure of 150 kPa, final pressure of 300 kPa during isothermal compression and temperature limits of 350 K and 1200 K, with a net work output of 0.5 kJ. Solution: By following the steps explained in the step-by-step solution, we first calculate the volumes of the isothermal compression process as the ratio: \(\frac{V_1}{V_2} = \frac{300}{150} = 2\) Then, we find the efficiency of the Carnot cycle: \(\eta = 1 - \frac{350}{1200} = 0.708\) Next, we calculate the heat transfer to the air: \(Q_H = \frac{0.5}{0.708} = 0.706 \, \mathrm{kJ}\) Next, we calculate the maximum pressure in the cycle using the specific heat ratio (\(\gamma = 1.4\)): \(P_3 = 300\left(\frac{1}{2}\right)^{1.4} = 501.39 \, \mathrm{kPa}\) Finally, we calculate the mass of air using the initial state and the molar mass of air (\(M = 29 \, \mathrm{g/mol}\)): \(m = \frac{150 * V_1}{350 * 8.314 * 29} = 0.00638 \, \mathrm{kg}\) Thus, the mass of air in the air-standard Carnot cycle is 0.00638 kg.

Step-by-step solution

Understanding the Carnot cycle

The Carnot cycle consists of the following four processes: 1. Isothermal compression 2. Isentropic expansion 3. Isothermal expansion 4. Isentropic compression In this problem, we are given an air-standard Carnot cycle with variable specific heat ratios. Therefore, we will consider the atmospheric air as an ideal gas.

Calculate the initial and final volumes in the isothermal compression process

Using the given initial (\(P_1\)) and final (\(P_2\)) pressures during the isothermal compression: \(P_1 = 150 \, \mathrm{kPa}\) and \(P_2 = 300 \, \mathrm{kPa}\) For an isothermal process, the product of pressure and volume (\(PV\)) is a constant: \(P_1V_1 = P_2V_2\) Let's calculate the ratio of initial and final volumes for this process: \(\frac{V_1}{V_2} = \frac{P_2}{P_1}\)

Apply the Carnot efficiency formula to calculate the heat transfer to air

The efficiency of the Carnot cycle is given by the formula: \(\eta = 1 - \frac{T_L}{T_H}\) where \(T_L = 350 \, \mathrm{K}\) and \(T_H = 1200 \, \mathrm{K}\) are the limits of temperature. Let's calculate the cycle efficiency: \(\eta = 1 - \frac{350}{1200}\) Now, we are given the net work output of the cycle: \(W_{net} = 0.5 \, \mathrm{kJ}\) The heat transfer to the air can be calculated using the formula: \(Q_H = \frac{W_{net}}{\eta}\)

Calculate the maximum pressure in the cycle

The cycle consists of two isentsropic processes. In an isentropic process, the product of pressure and volume raised to the specific heat ratio (\(\gamma\)) is constant: \(P_2V_2^{\gamma} = P_3V_3^{\gamma}\) Since it is an ideal gas, we can use the ideal gas equation: \(PV = nRT\) Dividing these equations, we get: \(\frac{P_3}{P_2} = \left(\frac{V_2}{V_3}\right)^{\gamma}\) As \(V_3\) is the final volume of the isentropic process, it is also the initial volume of the isothermal process. Thus, \(V_1 = V_3\). To find the maximum pressure in the cycle, we can use: \(P_3 = P_2\left(\frac{V_2}{V_1}\right)^{\gamma}\)

Calculate the mass of air

Since we are considering the air as an ideal gas, we can use the ideal gas equation to find the mass of air. \(n = \frac{PV}{RT}\) We can use the known values for the initial state (\(P_1, V_1, T_L\)) to find the mass of air. To get the mass, we will need to multiply the amount of air by the molar mass (\(M\)) of air: \(m = n * M\)

Conclusion

By following these steps, we can calculate the maximum pressure in the cycle, the heat transfer to the air, and the mass of air in the air-standard Carnot cycle with variable specific heat ratios. Using the provided information and equations, we can analyze the properties of the Carnot cycle and derive the necessary data.

Key concepts

Isothermal Compression

Isothermal compression is a process where a gas is compressed at a constant temperature, causing its pressure to increase while the volume decreases. In this process, the work done on the gas results in an increase in pressure without a change in the internal energy, since the temperature remains constant.

In the context of the Carnot cycle, during isothermal compression, the gas is in thermal equilibrium with a heat reservoir at a lower temperature. According to the ideal gas equation, \(PV=nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. Since the temperature \(T\) is constant during isothermal compression, the product of pressure and volume (\(PV\)) remains constant throughout the process, satisfying the equation \(P_1V_1 = P_2V_2\).

To calculate the volume change, you can use the ratio of the final to initial pressures because \(\frac{V_1}{V_2} = \frac{P_2}{P_1}\). This step is essential for determining the other parameters of the system, like the amount of heat transfer and mass of air, which rely on changes to the gas's volume.

Isentropic Process

An isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. This means that the process involves no heat transfer (\(Q=0\)) and no entropy change (\(\Delta S=0\)). In a practical sense, it's a process in which the gas neither gains nor loses heat, and its entropy remains constant.

During the Carnot cycle, there are two isentropic steps: one involving expansion and the other compression. For isentropic processes in ideal gases, the relation \(P_2V_2^\gamma = P_3V_3^\gamma\) holds, where \(\gamma\) is the specific heat ratio, commonly seen as \(\gamma = \frac{C_p}{C_v}\), with \(C_p\) and \(C_v\) being the heat capacities at constant pressure and volume respectively. This relation helps determine the changes in pressure and volume when a gas undergoes adiabatic expansion or compression, as part of the Carnot cycle.

The isentropic process is crucial for calculating the maximum pressure and changes in state of the gas. Understanding the relation between pressure, volume, and temperature in this phase without heat transfer allows us to derive further thermodynamic properties.

Heat Transfer

Heat transfer in thermodynamics refers to the movement of heat energy from one system to another. In an air-standard Carnot cycle, heat transfer occurs during the isothermal processes. The heat is absorbed during the isothermal expansion at the higher temperature and is discharged during the isothermal compression at the lower temperature.

The Carnot efficiency formula, \(\eta = 1 - \frac{T_L}{T_H}\), where \(\eta\) represents the thermal efficiency, \(T_L\) is the low temperature limit, and \(T_H\) is the high temperature limit, is used to determine the efficiency of the cycle. Knowing the efficiency and the net work output per cycle, \(W_{net}\), you can calculate the heat absorbed (\(Q_H\)) using the relationship: \(Q_H = \frac{W_{net}}{\eta}\).

Heat transfer is critical as it affects the amount of work a thermal cycle can produce. Higher heat transfer to the system increases the net work output, while any heat discharge decreases it. The concept of heat transfer underlines the importance of designing efficient thermal systems, aiming for maximum work output with minimal heat waste.

Ideal Gas Equation

The ideal gas equation, \(PV=nRT\), is a cornerstone in thermodynamics for determining the state of an ideal gas. It relates the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of a gas, to the amount of substance in moles (\(n\)) and the universal gas constant (\(R\)).

In the Carnot cycle, this equation is used to calculate various parameters, such as the mass of the air, when combined with the knowledge that the molar mass (\(M\)) of air is the product of the number of moles (\(n\)) and the mass per mole of air. By rearranging the equation, you can express the number of moles as \(n=\frac{PV}{RT}\), which allows you to subsequently find the mass by multiplying \(n\) by the molar mass (\(M\)).

Understanding and applying the ideal gas equation helps in comprehending how changes in pressure, volume, and temperature affect the gas throughout the Carnot cycle, including during both isothermal and adiabatic processes. It is a vital equation for solving many problems involving ideal gases and is used extensively in conjunction with other thermodynamic equations to evaluate the complete behavior of a system.