Question

Consider an ideal Ericsson cycle with air as the working fluid executed in a steady-flow system. Air is at \(27^{\circ} \mathrm{C}\) and \(120 \mathrm{kPa}\) at the beginning of the isothermal compression process, during which \(150 \mathrm{kJ} / \mathrm{kg}\) of heat is rejected. Heat transfer to air occurs at \(1200 \mathrm{K}\). Determine \((a)\) the maximum pressure in the cycle, \((b)\) the net work output per unit mass of air, and \((c)\) the thermal efficiency of the cycle.

Short answer

Question: Calculate the maximum pressure, net work output, and thermal efficiency for an ideal Ericsson cycle with air as the working fluid, given that the initial state of air is at 120 kPa and 27°C, the heat rejected during isothermal compression is 150 kJ/kg, and the temperature at the end of constant-pressure heat addition is 1200 K. Answer: The maximum pressure in the cycle is approximately 276 kPa, the net work output per unit mass of air is approximately \(\dot{m}(1005(1200- 300) - 150000)\,\text{J/kg}\), and the thermal efficiency of the cycle is approximately 16.7%.

Step-by-step solution

Determine the states of air at the end of each process

First, we need to identify the four processes involved in an ideal Ericsson cycle: 1. Isothermal compression 2. Constant-pressure heat addition 3. Isothermal expansion 4. Constant-pressure heat rejection We are given the initial state of the air at the beginning of the isothermal compression process. Let's denote the states at the end of each process as follows: - State 1: \(T_1 = 27^{\circ} \mathrm{C}\) (convert to Kelvin), \(P_1 = 120 \mathrm{kPa}\) - State 2: End of isothermal compression, \(T_2 = T_1\) (as it's isothermal), \(Q_{out} = 150 \mathrm{kJ/kg}\) - State 3: End of constant-pressure heat addition, \(T_3 = 1200 \mathrm{K}\) - State 4: End of isothermal expansion, \(T_4 = T_3\)

Calculate the maximum pressure in the cycle

As the maximum pressure would be reached at the end of the constant-pressure heat addition process, we know that \(P_3 = P_2\). Using the ideal gas law during the Isothermal compression process, we can determine the ratio of the volumes: \(\frac{V_2}{V_1} = \frac{n_1 R T_1}{P_1} \times \frac{P_2}{n_2 R T_2} = \frac{P_1}{P_2}\) Next, we'll use the heat rejected during the isothermal compression process to find the pressure ratio: \(Q_{out} = 150 \mathrm{kJ/kg} = n R T_1 \ln(\frac{V_2}{V_1})\) \(P_2=P_1+\frac{Q_{out}}{R \cdot T_1} =120+150\cdot 10^3/(8.314 \frac{\text{kJ}}{\text{kmol·K}}\cdot (27+273.15)\,\text{K})\approx 276\,\text{kPa}\) Thus, \(P_2=P_3\approx 276\,\text{kPa}\), which is the maximum pressure in the cycle.

Find the net work output per unit mass of air

Now, we'll find the net work output by subtracting the work input (isothermal compression) from the work output (isothermal expansion): \(W_{net} = W_{out} - W_{in}\) For an ideal Ericsson cycle, \(W_{in} = Q_{out}\), and \(W_{out} = n R T_2 \ln(\frac{V_3}{V_4}) = nR(T_3-T_1)\). Since it is a steady flow system, we can rewrite the equation in terms of specific heat capacities and mass flow rate (\(\dot{m}\)): \(W_{net} = \dot{m}(c_p(T_3 - T_1) - Q_{out}) \approx \dot{m}(1005(1200- 300) - 150000) \,\text{J/kg}\)

Compute the thermal efficiency of the cycle

We'll now compute the thermal efficiency of the cycle, which is given by the ratio of the net work output to the heat input: \(\eta = \frac{W_{net}}{Q_{in}}\) where \(Q_{in} = \dot{m}c_p(T_3 - T_2)\). \(\eta = \frac{c_p(T_3-T_1)-Q_{out}}{c_p(T_3-T_2)}\) Plugging in the values, we get: \(\eta \approx \frac{1005(1200-300)-150000}{1005(1200 - 300)}\approx 0.167\) So the thermal efficiency of the cycle is approximately 16.7%. In summary, the maximum pressure in the cycle is \(276 \mathrm{kPa}\), the net work output per unit mass of air is \(\dot{m}(1005(1200- 300) - 150000)\,\text{J/kg}\), and the thermal efficiency of the cycle is approximately 16.7%.

Key concepts

Isothermal Compression

Isothermal compression is a thermodynamic process where a gas is compressed at a constant temperature. In the context of the Ericsson cycle, it refers to the initial phase where the working fluid (air in our exercise) is compressed without changing its temperature. The fact that temperature remains constant is vital because it allows us to use specific equations derived from the ideal gas law to analyze the system.

During isothermal compression, the heat is typically rejected to maintain the temperature constant, which in our example is 150 kJ/kg. This process is performed reversibly, and according to the first law of thermodynamics, the work done on the gas is equal to the heat expelled. Isothermal processes are depicted as horizontal lines on a pressure-volume (P-V) diagram since the temperature, and thus the product of pressure and volume, does not change.

Constant-Pressure Heat Addition

Constant-pressure heat addition is another crucial step in thermodynamic cycles like the Ericsson cycle. It involves adding heat to the working fluid while maintaining pressure constant. The temperature of the gas increases as heat is added, which leads to a change in volume to keep the pressure steady.

This process is also known as isobaric and is represented by a horizontal line in a temperature-entropy (T-S) diagram and as an upward curve in a P-V diagram since the volume increases as heat is added at constant pressure. In our Ericsson cycle exercise, the constant-pressure heat addition continues until the air reaches a temperature of 1200 K, which denotes the completion of this phase and sets the stage for the subsequent isothermal expansion.

Thermodynamic Efficiency

Thermodynamic efficiency is a dimensionless measure of the effectiveness of a thermodynamic cycle, defined as the ratio of the net work output to the heat input. It essentially tells us how well the cycle converts heat energy into work. In our Ericsson cycle example, the net work is the difference between the work output during isothermal expansion and the work input during isothermal compression.

The ideal efficiency of any cycle can be improved by increasing the temperature at which heat is added or by reducing the temperature at which it is rejected. In real-world applications, several factors limit the efficiency, such as material constraints, heat losses, and non-reversible processes. The calculated efficiency of approximately 16.7% in our exercise provides insight into the potential and limitations of the Ericsson cycle for practical applications.

Ideal Gas Law

The ideal gas law is a fundamental equation in the study of thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. For a given amount of gas, the law is expressed as 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 absolute temperature.

In the context of the Ericsson cycle problem, we utilize the ideal gas law during the isothermal compression to relate the changes in pressure and volume when the temperature remains constant. By understanding and applying the ideal gas law, we can determine the maximum pressure in the cycle and calculate important variables such as work and heat transfer, which are essential for analyzing the cycle's efficiency and functionality.