Question

A system consisting of \(1 \mathrm{~kg}\) of \(\mathrm{H}_{2} \mathrm{O}\) undergoes a power cycle composed of the following processes: Process 1-2: Constant-pressure heating at 10 bar from saturated vapor. Process 2-3: Constant-volume cooling to \(p_{3}=5\) bar, \(T_{3}=160^{\circ} \mathrm{C}\). Process 3-4: Isothermal compression with \(Q_{34}=-815.8 \mathrm{~kJ}\) Process \(4-1:\) Constant-volume heating. Sketch the cycle on \(T-v\) and \(p-v\) diagrams. Neglecting kinetic and potential energy effects, determine the thermal efficiency.

Short answer

Thermal efficiency can be calculated using the work done and heat input determined from the state changes in the cycle.

Step-by-step solution

Understand the System and Processes Involved

The system consists of 1 kg of \(\text{H}_2\text{O}\) undergoing a power cycle. The processes include constant pressure heating, constant volume cooling, isothermal compression, and constant volume heating.

Plot the Cycle on T-v and p-v Diagrams

To sketch the cycle on T-v and p-v diagrams, start by marking the state points based on the given conditions and then draw the respective processes: - Process 1-2: Starts at the saturated vapor line at 10 bar. - Process 2-3: Moves to 5 bar and 160°C with constant volume. - Process 3-4: Isothermal compression at 160°C. - Process 4-1: Returns with constant volume heating.

Determine State Properties

Identify the properties (pressure, temperature, specific volume) at each state using steam tables or a Mollier chart. - State 1: Saturated vapor at 10 bar. - State 2: Determine the temperature at 10 bar for saturated vapor. - State 3: 5 bar and 160°C, find specific volume and internal energy. - State 4: Using isothermal compression to return to initial state.

Apply First Law of Thermodynamics

Use the first law of thermodynamics to calculate the heat transfers and work done: - Process 1-2: Constant-pressure, use \(Q_{12} = m (h_2 - h_1)\). - Process 2-3: Constant-volume, use \(Q_{23} = m (u_3 - u_2)\). - Process 3-4: Isothermal compression with \(Q_{34} = -815.8 \text{ kJ}\). - Process 4-1: Constant-volume, use \(Q_{41} = m (u_1 - u_4)\).

Calculate Thermal Efficiency

Using the definition of thermal efficiency, \( \text{Efficiency} = \frac{(\text{Total Work Output})}{(\text{Total Heat Input})} \), where \(\text{Total Work Output} = W_{net} \) and \(\text{Total Heat Input} = Q_{in} \). Identify and sum the net work done and the total heat input in the cycle.

Key concepts

constant-pressure heating

Constant-pressure heating is a process in which the pressure of the system remains unchanged while the temperature and volume may vary. In the given exercise, the system undergoes constant-pressure heating from state 1 to state 2 at 10 bar, starting as a saturated vapor. During this phase, heat is added to the system, causing an increase in the internal energy and specific enthalpy of the water. The heat added can be found using the formula: \( Q_{12} = m \times (h_2 - h_1) \) where \( h_1 \) and \( h_2 \) are the specific enthalpies at states 1 and 2 respectively.

Using steam tables, you can find the enthalpy values for state 1 (saturated vapor at 10 bar) and state 2. This helps in graphically representing the process on T-v and p-v diagrams, where the line for this process will be horizontal due to the constant pressure.

constant-volume cooling

Constant-volume cooling involves reducing the temperature and pressure of the system while keeping the volume constant. In the exercise, this occurs from state 2 to state 3, where the system cools down to 5 bar and 160°C. Since the volume remains constant, we can track changes in internal energy rather than specific enthalpy. The heat removed during this process is calculated using the formula: \( Q_{23} = m \times (u_3 - u_2) \) where \( u_2 \) and \( u_3 \) are the specific internal energies at states 2 and 3 respectively.

Steam tables provide these internal energy values, enabling accurate calculations. Graphically, on a T-v diagram, the line remains vertical due to the constant volume. On a p-v diagram, the process may appear as a drop in pressure.

isothermal compression

Isothermal compression is a process during which the system undergoes a decrease in volume while maintaining a constant temperature. In the provided exercise, the system compresses isothermally from state 3 to state 4. During this process, heat is transferred out of the system and can be measured by the formula: \( Q_{34} = W_{compression} \) In this problem, the heat transferred out (\Q_{34}) is given as -815.8 kJ. Since the process is isothermal, the internal energy remains constant, which implies that the heat lost is equal to the work done by the system. This can be visualized on a T-v diagram as a horizontal line (indicating constant temperature) and on a p-v diagram as a curve.

thermal efficiency

Thermal efficiency is a measure of how effectively a thermodynamic cycle converts heat into work. It is defined as: \[ \text{Efficiency} = \frac{\text{Total Work Output}}{\text{Total Heat Input}} \] In this exercise, the total work output (\W_{net}) is the net work done by the system over one complete cycle. This can be computed by summing the work done in each process. The total heat input (Q_{in}) is the heat added during the heating phases.

Given the specific processes, you will use the steam tables to find the enthalpy and internal energy values required to calculate heat and work transfers. Finally, you determine the thermal efficiency by dividing the net work output by the total heat input.

steam tables

Steam tables are comprehensive charts that include the properties of water and steam at various temperatures and pressures. They are crucial for solving thermodynamic problems involving water and steam. In this exercise, steam tables are used to determine specific enthalpy, specific internal energy, and other properties at different states. For example, you would use the tables to find the specific enthalpy of saturated vapor at 10 bar (state 1) and specific internal energy values at states 2 and 3.

Steam tables typically consist of:

• Saturation tables, providing properties at the boiling point for different pressures.
• Superheated steam tables for properties at higher temperatures.

Accurate usage of these tables is essential for plotting T-v and p-v diagrams and calculating heat and work transfers in the cycle.