Question

Refrigerant \(134 \mathrm{a}\) is the working fluid in a solar power plant operating on a Rankine cycle. Saturated vapor at \(60^{\circ} \mathrm{C}\) enters the turbine, and the condenser operates at a pressure of 6 bar. The rate of energy input to the collectors from solar radiation is \(0.4 \mathrm{~kW}\) per \(\mathrm{m}^{2}\) of collector surface area. Determine the \(\mathrm{min}\) imum possible solar collector surface area, in \(\mathrm{m}^{2}\), per \(\mathrm{kW}\) of power developed by the plant.

Short answer

The minimum possible solar collector surface area is calculated using \[ A = \frac{Q_{solar}}{0.4} \].

Step-by-step solution

Determine State Properties at the Turbine Inlet

Identify the properties at the turbine inlet. Given that the refrigerant enters the turbine as saturated vapor at 60°C, we can obtain the specific enthalpy (\(h_1\)) and specific entropy (\(s_1\)) from refrigerant tables.

Determine State Properties at the Condenser Exit

The condenser operates at 6 bar. The refrigerant leaves the condenser as saturated liquid. Using refrigerant tables, find the specific enthalpy (\(h_4\)) at this state.

Apply Energy Balance on the Turbine

For an isentropic process in the turbine, the specific entropy at the turbine exit (\(s_2\)) equals \(s_1\). Using the pressure at the condenser (6 bar), find \(h_2\) from the refrigerant tables.

Calculate the Work Done by the Turbine

Work done per unit mass by the turbine: \[ w_T = h_1 - h_2 \]

Calculate Heat Added in the Boiler

Heat added per unit mass in the boiler: \[ q_{in} = h_1 - h_4 \]

Determine Plant Efficiency

Efficiency of the Rankine cycle: \[ \text{Efficiency} = \frac{w_T}{q_{in}} \]

Calculate Power Developed

With a power output of 1 kW, determine the mass flow rate (\( \dot{m} \)) using the relation: \[ \dot{m} = \frac{1 \text{kW}}{w_T} \]

Determine the Energy Input to Collectors

The energy input to the collectors: \[ Q_{solar} = q_{in} \cdot \dot{m} \]

Calculate Minimum Possible Collector Area

Given the rate of energy input to the collectors from solar radiation is 0.4 kW per m², the collector area: \[ A = \frac{Q_{solar}}{0.4} \]

Key concepts

thermodynamic cycles

A thermodynamic cycle is a series of processes that involve the transfer and conversion of energy. These cycles are fundamental in mechanical and chemical engineering.
One key example is the Rankine cycle, commonly used in power plants.
The Rankine cycle consists of four main processes: compression, heating, expansion, and cooling.
Each process happens in different components of the system such as the pump, boiler, turbine, and condenser.

• Compression: Involves increasing the pressure of the working fluid.
• Heating: The fluid absorbs heat, leading to a phase change from liquid to vapor.
• Expansion: The high-pressure vapor expands in the turbine, producing work.
• Cooling: The fluid releases heat, condensing back into liquid.

A typical Rankine cycle with refrigerant 134a includes entering the turbine as saturated vapor, going through expansion, condensation, and then reheating in the boiler.
Understanding these steps is important for comprehending how energy conversion is optimized in various systems.

solar power plant

A solar power plant utilizes solar radiation to generate electricity.
It often involves the use of thermodynamic cycles, like the Rankine cycle, to convert thermal energy into electrical energy. The process begins with solar collectors that capture sunlight and convert it into heat energy.
This energy is used to heat a working fluid, which undergoes phase changes and work production in a turbine.
Key components of a solar power plant include:

• Solar Collectors: Capture and convert sunlight into thermal energy.
• Heat Exchanger: Transfers heat to the working fluid.
• Turbine and Generator: Convert thermal energy to mechanical and then to electrical energy.
• Condenser: Cools the working fluid, completing the cycle.

The efficiency of a solar power plant depends on factors like the quality of solar radiation, efficiency of heat exchangers, and the performance of the Rankine cycle.
The Rankine cycle's efficiency impacts how effectively solar energy is converted into electrical power, which directly relates to the plant's feasibility and cost-effectiveness.

energy balance

Energy balance is essential for analyzing and designing thermodynamic systems.
It involves calculating the energy input and output to ensure the system is properly optimized.
In the context of a Rankine cycle, energy balance helps determine the efficiency and performance of the system.
Key steps in performing energy balance for the Rankine cycle in a solar power plant:

• Calculate energy input from the solar collectors, typically given in kW per m² of collector area.
• Determine the heat added to the working fluid in the boiler.
• Calculate the work output from the turbine.
• Assess efficiency by comparing the work output to the heat input.
• Determine the mass flow rate of the working fluid, essential for scaling the system to desired power outputs.
• Calculate the minimum solar collector area required to produce a specific power output.

For example, the step-by-step solution provided shows how each part is computed:

• Using refrigerant tables to find enthalpy values.
• Applying equations like \( w_T = h_1 - h_2 \) for turbine work and \( q_{in} = h_1 - h_4 \) for heat input.
• Finding efficiency as \(\frac{w_T}{q_{in}}\right\).

Knowing energy balance helps in designing more efficient and effective solar power plants, ensuring optimal use of available resources.