Question

A system consists of \(2 \mathrm{~kg}\) of carbon dioxide gas initially at, state 1 , where \(p_{1}=1\) bar, \(T_{1}=300 \mathrm{~K}\). The system undergoes a power cycle consisting of the following processes: Process 1-2: constant volume to \(p_{2}, p_{2}>p_{1}\) Process 2-3: expansion with \(p v^{1.28}=\) constant Process 3-1: constant-pressure compression Assuming the ideal gas model and neglecting kinetic and potential energy effects, (a) sketch the cycle on a \(p-v\) diagram. (b) plot the thermal efficiency versus \(p_{2} / p_{1}\) ranging from \(1.05\) to 4 .

Short answer

(a) Sketch the cycle on a p-v diagram, including states and process lines. (b) Plot thermal efficiency vs \( \frac{p_2}{p_1} \).

Step-by-step solution

Analyze initial conditions and assumptions

Given: Mass of CO2, m = 2 kg, Initial pressure, \( p_1 = 1 \text{ bar} = 100 \text{ kPa} \), Initial temperature, \( T_1 = 300 \text{ K} \). Assume CO2 behaves as an ideal gas. The specific gas constant for CO2, \( R = 0.1889 \text{ kJ} / \text{kg} \cdot \text{K} \).

Process 1-2 (Constant volume)

The volume remains constant, so we use the ideal gas law to find \(V_1\) and then use it to find \( T_2 \). From the ideal gas law: \( p_1 V_1 = nRT_1 \), where \( n = \frac{m}{M} \). Rearranging, \( V_1 = \frac{mR T_1}{p_1} \). Use this in process 1-2.

Process 2-3 (polytropic process)

For process 2-3, \( p v^{1.28} = \text{constant} \). Using the relationship between state variables during polytropic processes and using values obtained from step 2, find \( V_3 \) in terms of the given parameters.

Process 3-1 (Constant pressure)

Since the pressure is constant, we use \( p V = nRT \). Find \( T_3 \) and \(T_1\) from the ideal gas law relationship and volumes determined from previous steps to close the cycle.

Sketch the cycle on a \(p-v\) diagram

On the \(p-v\) diagram, mark the states: 1 (at \( p_1, V_1 \)), 2 (at \( p_2, V_1 \)), 3 (at \( p_3, V_3 \)), and draw the process lines for 1-2, 2-3, and 3-1.

Calculate the work done and heat added

Compute the work done and heat added for each process using the first law of thermodynamics and the expressions for work done in polytropic, isochoric, and isobaric processes.

Determine the thermal efficiency

Efficiency, \( \eta = 1 - \frac{Q_{out}}{Q_{in}} \). Use the computed heat exchanges in the cycle to determine \( \eta \).

Plot thermal efficiency versus \(\frac{p_2}{p_1}\)

Vary \( \frac{p_2}{p_1}\) from 1.05 to 4 in steps and plot the corresponding thermal efficiency to get the desired graph.

Key concepts

constant volume process

In an ideal gas power cycle, a constant volume process is one where the volume of the gas remains unchanged. This is also known as an isochoric process. Since the volume doesn't change, any heat added to or removed from the system changes only the pressure and temperature of the gas.
For a constant volume process, we can use the ideal gas law: \( pV = nRT \), and since the volume is constant, an increase in temperature leads to an increase in pressure.
To find the final state temperature (\( T_2 \)) and pressure (\( p_2 \)) in our example, we use the relation:
\[ T_2 = T_1 \times \frac{p_2}{p_1} \]
This step helps us move forward to the next phases of our cycle analysis, involving different types of processes.

polytropic process

A polytropic process is a type of thermodynamic process that follows the relation \( pV^n = \text{constant} \), where \( n \) is the polytropic index.
This can represent a variety of processes, from adiabatic (\( n = \text{specific heat ratio} \)) to isothermal (\( n = 1 \)).
In our problem, process 2-3 follows this law with \( n = 1.28 \).
This effectively means that a specific combination of pressure and volume during this process remains constant.
Using the relationship between state variables during polytropic processes and values from the constant volume process, we can formulate:
\[ p_2 V_2^{1.28} = p_3 V_3^{1.28} \]
From here, with known values of \( p_2 \) and \( V_2 \) (constant from previous process), we can derive \( V_3 \) and further analyze our cycle.

thermal efficiency

Thermal efficiency, \( \text{η} \), is a measure of how well an engine converts heat from fuel into work. It highlights the efficiency of the power cycle and is defined by the ratio of work output to heat input.
In mathematical terms:
\[ \text{η} = 1 - \frac{Q_{out}}{Q_{in}} \]
For our power cycle:

• \(Q_{in} \) and \( Q_{out} \) represent the heat added and rejected during the cycle respectively.
• Using the first law of thermodynamics, we compute \( Q_{in} \) during the constant volume and polytropic expansion processes.
• \( Q_{out} \) is determined from the constant pressure compression process.

Once we have these values, we simply plug them into the efficiency formula. Finally, to analyze the impact, we plot thermal efficiency against the variable \( \frac{p_2}{p_1} \) to visualize how changes in compression ratio affect the efficiency of our cycle.
This understanding is crucial for optimizing power cycles in practical applications such as engines and power plants.