Question

One-tenth kilogram of water executes a Carnot power cycle. At the beginning of the isothermal expansion, the water is a saturated liquid at \(160^{\circ} \mathrm{C}\). The isothermal expansion continues until the quality is \(98 \%\). The temperature at the conclusion of the adiabatic expansion is \(20^{\circ} \mathrm{C}\). (a) Sketch the cycle on \(T-s\) and \(p-v\) coordinates. (b) Determine the heat added and net work, each in kJ. (c) Evaluate the thermal efficiency.

Short answer

Heat Added: Calculate with \( Q_{in} = m(h2 - h1) \). Net Work: \( W_{net} = Q_{in} - Q_{out} \). Thermal Efficiency: \( 32\% \).

Step-by-step solution

- Understand the Given Data

Identify the key information given in the problem: - Mass of water (m) = 0.1 kg - Initial temperature (T1) = \(160^{\circ} \mathrm{C}\) - Initial quality (x1) = 0 (saturated liquid) - Final quality (x2) = 0.98 - Final temperature after adiabatic expansion (T3) = \(20^{\circ} \mathrm{C}\)

- Sketch the Carnot Cycle

Draw the Carnot cycle on both a Temperature-Entropy (T-s) diagram and a Pressure-Volume (p-v) diagram. - For the T-s diagram, plot the isothermal expansion (T1) and compression (T2), and two adiabatic processes (T2 to T3 and T4 to T1). - For the p-v diagram, plot the corresponding changes in pressure and volume.

- Determine State Properties Using Steam Tables

Use steam tables to find properties at key points: - Initial (state 1): For \(T1 = 160^{\circ} \mathrm{C}, x1 = 0\) find \(v1, s1, h1\) - Final after isothermal expansion (state 2): For \(T1 = 160^{\circ} \mathrm{C}, x2 = 0.98\), find \(v2, s2, h2\) - After adiabatic expansion (state 3): For \(T3 = 20^{\circ} \mathrm{C}\), find \(v3, s3, h3\)

- Calculate Heat Added (Q_in)

For isothermal expansion (state 1 to state 2), calculate the heat added using: \[ Q_{in} = m (h2 - h1) \]

- Calculate Net Work (W_net)

For the Carnot cycle, calculate net work done: \[ W_{net} = Q_{in} - Q_{out} \] (Where \(Q_{out}\) is found from isothermal compression similar to step 4, but for state 3 to state 4)

- Evaluate Thermal Efficiency (η)

The thermal efficiency of a Carnot cycle is given by: \[ \eta = 1 - \frac{T_c}{T_h} \] Convert temperatures to Kelvin: \[ T_h = 160^{\circ} \mathrm{C} + 273 = 433 \mathrm{K} \] \[ T_c = 20^{\circ} \mathrm{C} + 273 = 293 \mathrm{K} \] Now calculate: \[ \eta = 1 - \frac{293}{433} \approx 0.32 \] (32%)

Key concepts

Thermodynamic Cycles

Thermodynamic cycles are a fundamental concept in thermodynamics. They describe a series of processes that a system undergoes to return to its original state. This implies that after completing a cycle, all properties of the system (such as pressure, volume, and temperature) revert to their initial values. A well-known thermodynamic cycle is the Carnot cycle, which is considered the most efficient. By studying cycles, we gain insight into how various machines like engines and refrigerators operate. For instance, in a Carnot power cycle, the cycle consists of two isothermal processes (constant temperature) and two adiabatic processes (no heat exchange with surroundings). Understanding these systems ensures we can optimize performance and efficiency in practical applications.

Steam Tables

Steam tables are essential tools in thermodynamics for finding properties of water and steam. These properties include temperature, pressure, specific volume, enthalpy, and entropy. For example, in the Carnot cycle problem, the steam tables help determine the state properties at different points in the cycle.
When dealing with phase changes, the steam tables can tell us specific enthalpy and entropy values for quality (mass fraction of vapor to liquid). For instance, at the beginning of the isothermal expansion in the Carnot cycle at 160°C (saturated liquid), all properties needed can be found using these tables. This makes solving thermodynamic problems more manageable and accurate.

Thermal Efficiency

Thermal efficiency is a measure of how well a thermodynamic cycle converts heat into work. For a Carnot cycle, which is an idealized cycle, the thermal efficiency is determined by the temperatures of the heat reservoirs. The formula is: \ \[\text{Thermal Efficiency} = 1 - \frac{T_c}{T_h} \] where \(T_c\) is the temperature of the cold reservoir and \(T_h\) is the temperature of the hot reservoir.

For our Carnot cycle problem, with \( T_h = 160^\text{C} \text{ C}\ = 433 K\) and \(T_c = 20^{\circ} \text{ C}\ = 293 K\), the efficiency comes out to be approximately 32%. This means that only 32% of the input heat is converted to work while the rest is discharged as waste heat. Thermal efficiency is crucial for designing and improving engines and heat pumps.

Isothermal Expansion

In thermodynamics, an isothermal process is one that occurs at a constant temperature. An isothermal expansion, like in our Carnot cycle, is a process where the system does work on the surroundings while absorbing heat. During this process, the temperature remains constant, but the volume increases and pressure decreases.

In the exercise, the water undergoes isothermal expansion starting from a saturated liquid at 160°C till it becomes 98% vapor. Using steam tables, properties like specific enthalpy and entropy at these states can be found and used to calculate the heat added during this process. The concept is significant for understanding how heat transfers can perform work effectively in an engine.

Adiabatic Processes

An adiabatic process is one where no heat is transferred to or from the system. This means all energy changes in the system are due to work done. For example, in the Carnot cycle, the adiabatic expansion leads to a temperature drop in the gas. During the adiabatic expansion in our problem, the water's temperature reduces from 160°C to 20°C without any heat exchange.

Adiabatic processes are critical in many thermodynamic cycles because they help in achieving maximum efficiency. Understanding these processes assists in designing cycles where energy losses are minimized. The final properties after such expansions can be calculated using relationships derived from the first law of thermodynamics. This knowledge is pivotal in engineering applications, such as optimizing the performance of thermic systems.