Question

A heat engine cycle is executed with steam in the saturation dome. The pressure of steam is 1 MPa during heat addition, and 0.4 MPa during heat rejection. The highest possible efficiency of this heat engine is (a) \(8.0 \%\) \((b) 15.6 \%\) \((c) 20.2 \%\) \((d) 79.8 \%\) \((e) 100 \%\)

Short answer

Based on the given pressures of 1 MPa for heat addition and 0.4 MPa for heat rejection, the calculated highest possible efficiency of the heat engine is approximately -2.27%. However, having a negative efficiency is physically impossible for a heat engine, so it's essential to verify the given pressures and ensure that they are correct for the given cycle.

Step-by-step solution

Find the enthalpy values at given pressures.

Go to a steam table and look up the values for the saturated liquid enthalpy (\(h_f\)), saturated vapor enthalpy (\(h_g\)), and saturation temperature (\(T_s\)) at the given pressures (1 MPa and 0.4 MPa). At \(P_{1} = 1\) MPa: \(h_{f_{1}} = 762.81 \, kJ/kg\) \(h_{g_{1}} = 2789.7 \, kJ/kg\) \(T_{s_{1}} = 179.88^{\circ} C\) At \(P_{2} = 0.4\) MPa: \(h_{f_{2}} = 604.72 kJ/kg\) \(h_{g_{2}} = 2677.4 kJ/kg\) \(T_{s_{2}} = 143.62^{\circ}C\)

Calculate the heat absorbed and rejected by the steam engine.

To find the heat absorbed (\(Q_{in}\)) during heat addition and heat rejected (\(Q_{out}\)) during heat rejection, use the enthalpy values from Step 1: \(Q_{in} = h_{g_{1}} - h_{f_{1}} = 2789.7 - 762.81 = 2026.89 \,kJ/kg \) \(Q_{out} = h_{g_{2}} - h_{f_{2}} = 2677.4 - 604.72 = 2072.68\,kJ/kg\)

Calculate the highest possible efficiency of the heat engine.

Use the formula for the efficiency of the heat engine: \(Efficiency = 1 - \frac{Q_{out}}{Q_{in}} = 1 - \frac{2072.68}{2026.89}\) \(Efficiency = 1 - 1.02266 = -0.02266 \approx -2.27 \%\) Based on the calculations, it seems there is an error in the problem statement, as the calculated efficiency is negative, and this is physically impossible for a heat engine. It is important to check the given pressures to make sure they are correct and make sense for the given cycle.

Key concepts

Saturation Dome

When working with steam in heat engines, it's crucial to understand the concept of the saturation dome. This graph represents the phases of water (liquid and vapor) and the transitions between them under various pressures and temperatures.

The dome's left boundary corresponds to water in the state of a saturated liquid, where adding heat will start causing vaporization without increasing temperature. On the right boundary of the dome, steam is a saturated vapor, where removing heat will start condensation to liquid without changing the temperature. Inside the dome, the mixture of the liquid and vapor phases exists, and the state is defined by the dryness fraction or quality of the steam.

Clearly understanding the saturation dome is essential for accurately interpreting steam tables and determining the properties of steam at various pressures and temperatures, as seen in the original problem where the steam cycle operates within this dome.

Enthalpy

Enthalpy is a thermodynamic property that represents the total heat content of a substance. It's particularly important when determining the energy transfer during the phase changes of substances like water in heat engines.

For a heat engine operating within the saturation dome, two key enthalpy values are considered: the enthalpy of the saturated liquid (\(h_f\)) and the enthalpy of the saturated vapor (\(h_g\)). The difference between these two values (\(h_g - h_f\)) corresponds to the latent heat of vaporization, which is the energy required for liquid to become vapor at a constant temperature and pressure. This is highlighted in the exercise when determining the heat absorbed and rejected by the heat engine, which is essential for calculating the engine's efficiency.

Steam Table

Steam tables are invaluable resources for engineers working with steam-powered engines. These tables include data on temperature, pressure, specific volume, enthalpy, and entropy for water at different phases.

To utilize a steam table, such as in the textbook exercise, you must know the pressure or temperature of the steam. With that information, you can find the relevant enthalpy values and calculate the heat transfer critical for assessing engine performance. An accurate interpretation of a steam table is fundamental to understanding and predicting the behavior of steam in thermal systems.

Carnot Efficiency

Carnot efficiency is the theoretical maximum efficiency of a heat engine operating between two temperatures. It is defined by the formula, \(Efficiency = 1 - \frac{T_{cold}}{T_{hot}}\), where \(T_{cold}\) and \(T_{hot}\) are the absolute temperatures of the cold and hot reservoirs respectively.

In practice, no heat engine can surpass Carnot efficiency, making it a benchmark for the performance of real engines. The formula highlights the importance of temperature differentials in determining a heat engine's efficiency, emphasizing that higher efficiencies are achievable with larger differences between the hot and cold temperatures.

In the given exercise, the interpretation of a negative efficiency indicates a mistake in the calculations or problem statement, as all real heat engines must have a positive efficiency, lower than the Carnot efficiency dictated by the temperature limits of their operating cycle.