Question

Water is the working fluid in a cogeneration cycle that generates electricity and provides heat for campus buildings. Steam at \(2 \mathrm{MPa}, 320^{\circ} \mathrm{C}\), enters a two-stage turbine with a mass flow rate of \(0.82 \mathrm{~kg} / \mathrm{s}\). A fraction of the total flow, 0.141, is extracted between the two stages at \(0.15 \mathrm{MPa}\) to provide for building heating, and the remainder expands through the second stage to the condenser pressure of \(0.06\) bar. Condensate returns from the campus buildings at \(0.1 \mathrm{MPa}, 60^{\circ} \mathrm{C}\) and passes through a trap into the condenser, where it is reunited with the main feedwater flow. Saturated liquid leaves the condenser at \(0.06\) bar. Each turbine stage has an isentropic efficiency of \(80 \%\), and the pumping process can be considered isentropic. Determine (a) the rate of heat transfer to the working fluid passing through the steam generator, in \(\mathrm{kJ} / \mathrm{h}\). (b) the net power developed, in \(\mathrm{kJ} / \mathrm{h}\). (c) the rate of heat transfer for building heating, in \(\mathrm{kJ} / \mathrm{h}\). (d) the rate of heat transfer to the cooling water passing through the condenser, in \(\mathrm{kJ} / \mathrm{h}\).

Short answer

Determine enthalpies from steam tables, then calculate heat transfers and net power using energy balance equations and mass flow rates.

Step-by-step solution

- Determine inlet and exit states of the steam for each process

Identify the pressures and temperatures at various points in the cycle. Use steam tables or Mollier charts to find the enthalpies and entropies at these states. For example, find the enthalpy and entropy at the initial turbine entry condition \( 2 \, \text{MPa}, \, 320^\circ \text{C} \).

- Calculate the enthalpy after the first turbine stage

Use the isentropic efficiency of the turbine to find the enthalpy after the first stage. This can be done using the formula: \( h_2 = h_1 - \text{Efficiency} \times (h_1 - h_{2s}) \), where \( h_{2s} \) is the enthalpy at the exit condition in an isentropic process.

- Calculate the enthalpy of the extracted flow

Since a fraction of the steam is extracted at \( 0.15 \, \text{MPa} \), you need to find the enthalpy at this state using steam tables. This is denoted \( h_3 \).

- Calculate the enthalpy after the second turbine stage

For the remaining steam that continues through the second turbine stage, use the isentropic efficiency again to determine the enthalpy at the exit state \( h_4 \).

- Calculate the rate of heat transfer to the working fluid

Use the relation for mass flow and enthalpy change to find the heat added in the steam generator: \[ \dot{Q}_{in} = \dot{m} (h_1 - h_f) \]. Convert the final result to \( \text{kJ} / \text{h} \).

- Calculate the net power developed

Find the net power developed by calculating the work done by both turbine stages, minus the work needed for the pump. The formula is: \[ W_{net} = \dot{m}_1 (h_1 - h_2) + \dot{m}_2 (h_3 - h_4) - \dot{m}_3 (h_f - h_p) \]

- Calculate the rate of heat transfer for building heating

For building heating, use the extracted steam and its enthalpy change: \[ \dot{Q}_{building} = \dot{m}_{extracted} (h_{extracted} - h_{return}) \].

- Calculate the rate of heat transfer to the condenser

Finally, for the condenser, the energy balance is: \[ \dot{Q}_{cond} = \dot{m} (h_4 - h_f) \].

Key concepts

Thermodynamic Cycles

Thermodynamic cycles are sequences of processes that involve heat and work interactions, bringing a working fluid back to its initial state at the end. Cogeneration cycles, like the one described, are particularly valuable because they generate electricity and provide heating simultaneously. In our example, water acts as the working fluid, undergoing various stages of pressurizing, heating, expanding, condensing, and reheating. Understanding each step in these cycles helps grasp how energy is conserved and transformed within power plants.
In a cogeneration cycle, important states include:

• The steam entering the turbine at high pressure and temperature.
• The partially expanded steam extracted for heating purposes.
• The final exhausted low-pressure steam directed to the condenser.

These states reflect different energy levels within the fluid and are critical for analyzing the cycle's performance.

Isentropic Efficiency

Isentropic efficiency measures how close a real device operates compared to an idealized, lossless version. For turbines, it is defined as the ratio of the actual work output to the work output if the process were isentropic (i.e., reversible and adiabatic). The formula is:
\[ \text{Efficiency} = \frac{h_1 - h_2}{h_1 - h_{2s}} \]
where \(h_1\) is the enthalpy at the turbine inlet, \(h_2\) is the actual exit enthalpy, and \(h_{2s}\) is the enthalpy at the exit under isentropic conditions. We use this to calculate how efficient the turbine stages are in converting thermal energy into work. Typical values range from 70% to 90%, with 80% used in this exercise.
Knowing the isentropic efficiency allows us to adjust our expectations and calculations to match real-world data, making it essential for accurate cycle analysis.

Steam Tables

Steam tables are indispensable tools in thermodynamic analyses, providing vital property data for water and steam at various temperatures and pressures. The key properties include enthalpy, entropy, specific volume, and internal energy. In our exercise, these tables help determine:

• The enthalpy of steam entering and exiting the turbine stages.
• The enthalpy of condensate and saturated liquid conditions.

For instance, by using the steam tables, we find the enthalpy at the turbine entry point (2 MPa, 320°C) and other critical states throughout the cycle.
This data is crucial as it allows us to compute the energy transfers, work, and efficiency of the system correctly. Without accurate steam tables, it would be practically impossible to analyze real thermodynamic cycles effectively.

Heat Transfer

Heat transfer in thermodynamic cycles refers to the exchange of thermal energy between the working fluid and its surroundings, which occurs in steam generators, condensers, and building heating systems. In the cogeneration cycle, key heat transfer processes include:

• Heat addition in the steam generator to convert liquid water into high-pressure steam.
• Heat rejection in the condenser to return steam to liquid state.
• Heat extracted for building heating purposes using steam at intermediate pressure.

The formula to calculate heat transfer in the steam generator is:
\[ \text{Q}_{in} = \text{m} \times (h_1 - h_f) \]
where \(\text{m}\) is the mass flow rate and \(h_1 - h_f\) represents the enthalpy change. Similarly, heat rejection in the condenser and heat provided to buildings are calculated considering the mass and enthalpy changes specific to those stages.
These calculations ensure efficient energy utilization in the cycle.

Energy Balance

Energy balance is the fundamental principle ensuring that energy entering and exiting a control volume accounts for all processes. This principle is embodied in the first law of thermodynamics, stating that energy can neither be created nor destroyed. In our cogeneration cycle, the energy balance is maintained by equating the energy supplied to the steam generator to the sum of the work done by turbines, heat extracted for buildings, and heat rejected in the condenser.
The main form of the energy balance equation for the cycle is:
\[ \text{Q}_{in} = \text{W}_{net} + \text{Q}_{building} + \text{Q}_{cond} \]
where:

• \(\text{Q}_{in}\) is the heat transfer to the working fluid.
• \(\text{W}_{net}\) is the net power developed.
• \(\text{Q}_{building}\) is the heat transfer for building heating.
• \(\text{Q}_{cond}\) is the heat rejected in the condenser.

Accurate energy balance is crucial for assessing the performance and efficiency of the cycle, ensuring all processes contribute correctly to the overall energy transformation.