Question

A binary vapor power cycle consists of two ideal Rankine cycles with steam and ammonia as the working fluids. In the steam cycle, superheated vapor enters the turbine at \(6 \mathrm{MPa}\), \(640^{\circ} \mathrm{C}\), and saturated liquid exits the condenser at \(60^{\circ} \mathrm{C}\). The heat rejected from the steam cycle is provided to the ammonia cycle, producing saturated vapor at \(50^{\circ} \mathrm{C}\), which enters the ammonia turbine. Saturated liquid leaves the ammonia condenser at \(1 \mathrm{MPa}\). For a net power output of \(20 \mathrm{MW}\) from the binary cycle, determine (a) the power output of the steam and ammonia turbines, respectively, in MW. (b) the rate of heat addition to the binary cycle, in MW. (c) the thermal efficiency.

Short answer

Steam and ammonia turbine power outputs can be calculated using enthalpy changes. The heat addition rate involves enthalpy differences, and the efficiency is the ratio of work output to heat addition.

Step-by-step solution

- Steam Cycle Analysis: Calculate Enthalpies

First, determine the enthalpy values for the steam cycle. Using steam tables or a Mollier diagram: - At 6 MPa and 640°C (superheated vapor), find the enthalpy, h1. - At 60°C (saturated liquid), find the enthalpy, h2. Remember to convert thermal energy units if necessary.

- Calculate Work Done by Steam Turbine

The work done by the steam turbine per unit mass is given by the difference in enthalpies: \[ W_{\text{cycle}} = h_1 - h_2 \]

- Ammonia Cycle Analysis: Calculate Enthalpies

Now, for the ammonia cycle: - At 50°C (saturated vapor), find the enthalpy, h3. - At 1 MPa and 50°C (saturated liquid), find the enthalpy, h4.

- Calculate Work Done by Ammonia Turbine

The work done by the ammonia turbine per unit mass is given by the difference in enthalpies: \[ W_{\text{NH3}} = h_3 - h_4 \]

- Calculate Power Output

Determine the mass flow rates for steam and ammonia using the net power output and efficiencies. Then, the power output for each turbine is given by: \[ P_{\text{steam}} = \text{mass flow rate of steam} \times W_{\text{cycle}} \] \[ P_{\text{NH3}} = \text{mass flow rate of ammonia} \times W_{\text{NH3}} \]

- Calculate the Rate of Heat Addition

The rate of heat addition to the steam cycle can be found using the enthalpy change during heating: \[ Q_{\text{in, steam}} = \text{mass flow rate of steam} \times (h1 - h2) \] For the ammonia cycle: \[ Q_{\text{in, NH3}} = \text{mass flow rate of ammonia} \times (h3 - h4) \]

- Calculate Thermal Efficiency

The thermal efficiency of the binary cycle is determined by: \[ \text{Efficiency} = \frac{\text{Total Work Output}}{\text{Total Heat Addition}} \]

Key concepts

Rankine Cycle

The Rankine cycle is a thermodynamic cycle that converts heat into mechanical work using a working fluid, typically steam. In a binary vapor power cycle, two Rankine cycles are combined - one using steam and the other using ammonia. The key processes in the Rankine cycle include:

• Heating the fluid to superheated vapor in a boiler or steam generator.
• Expanding the vapor through a turbine to produce mechanical work.
• Condensing the vapor in a condenser to return it to liquid form.
• Pumping the liquid back to the boiler to complete the cycle.

The Rankine cycle is essential in power generation because it efficiently converts thermal energy into mechanical energy, particularly using steam as the working fluid.

Steam Cycle Analysis

In the steam cycle, steam is heated until it becomes superheated vapor. For this particular problem, superheated steam enters the turbine at 6 MPa and 640°C. The key to analyzing this cycle is to determine the enthalpy values at various points:

• Superheated vapor enthalpy (h1) at the turbine inlet.
• Saturated liquid enthalpy (h2) at the condenser exit at 60°C.

Using these enthalpy values, you can calculate the work done by the turbine:
\( W_{\text{cycle}} = h_1 - h_2 \)
This gives the energy converted from thermal to mechanical form by the steam turbine. Ensuring accurate enthalpy values by using steam tables or diagrams is crucial.

Enthalpy Calculations

Enthalpy is a measure of the total energy of a system, incorporating both internal energy and the energy required to displace its environment. For both steam and ammonia cycles, determining enthalpy at various stages is critical.
For instance:

• In the steam cycle: Identify h1 (superheated vapor) at 6 MPa and 640°C, and h2 (saturated liquid) at 60°C.
• In the ammonia cycle: Identify h3 (saturated vapor) at 50°C and h4 (saturated liquid) at 1 MPa.

The difference between these enthalpies indicates the energy change during phase changes or work processes. Enthalpy changes define the work output for the turbines and the heat input required for the cycles.

Thermal Efficiency

Thermal efficiency is the ratio of the work output to the heat input in a power cycle. It indicates how efficiently a cycle converts thermal energy into mechanical energy.
For the binary vapor power cycle, thermal efficiency is given by:
\[ \text{Efficiency} = \frac{\text{Total Work Output}}{\text{Total Heat Addition}} \]
Here, the total work output consists of the combined work of the steam and ammonia turbines. Accurate enthalpy and mass flow rate values are essential for determining these outputs. Improving the thermal efficiency of cycles involves maximizing the work output and minimizing any energy losses during the processes.

Ammonia Cycle Analysis

The ammonia cycle operates similarly to the steam cycle but uses ammonia as the working fluid. It's integrated into the binary cycle to better utilize the heat rejected from the steam cycle.
Here’s the process:

• Ammonia is heated to a saturated vapor at 50°C using the heat rejected by the steam cycle.
• The vapor enters a turbine to produce work.
• Ammonia then leaves as a saturated liquid at 1 MPa.

The work done by the ammonia turbine can be calculated as:
\( W_{\text{NH3}} = h_3 - h_4 \)
Like with the steam cycle, accurate enthalpy values and mass flow rates are necessary to determine the actual power output and efficiency.