Question

In a heat engine, heat \(\left(q_{\mathrm{h}}\right)\) is absorbed by a working substance (such as water) at a high temperature \(\left(T_{\mathrm{h}}\right)\) Part of this heat is converted to work \((w),\) and the rest \(\left(q_{1}\right)\) is released to the surroundings at the lower temperature ( \(T_{1}\) ). The efficiency of a heat engine is the ratio \(w / q_{\mathrm{h}}\). The second law of thermodynamics establishes the following equation for the maximum efficiency of a heat engine, expressed on a percentage basis. $$\text { efficiency }=\frac{w}{q_{\mathrm{h}}} \times 100 \%=\frac{T_{\mathrm{h}}-T_{1}}{T_{\mathrm{h}}} \times 100 \%$$ In a particular electric power plant, the steam leaving a steam turbine is condensed to liquid water at \(41^{\circ} \mathrm{C}\left(T_{1}\right)\) and the water is returned to the boiler to be regenerated as steam. If the system operates at \(36 \%\) efficiency, (a) What is the minimum temperature of the steam \(\left[\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\right]\) used in the plant? (b) Why is the actual steam temperature probably higher than that calculated in part (a)? (c) Assume that at \(T_{\mathrm{h}}\) the \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) is in equilibrium with \(\mathrm{H}_{2} \mathrm{O}(1) .\) Estimate the steam pressure at the temperature calculated in part (a). (d) Is it possible to devise a heat engine with greater than 100 percent efficiency? With 100 percent efficiency? Explain.

Short answer

The minimum temperature of the steam used in the plant is calculated in the first step. It's practically higher than this because the actual efficiency is less due to losses in the system. An estimation of the steam pressure associated with this temperature can be found using a steam table or Antoine's equation. According to the second law of thermodynamics, achieving 100% efficiency in a heat engine is impossible.

Step-by-step solution

Calculate the minimum necessary temperature of the steam

The efficiency of the heat machine can be expressed using the formula given in the problem statement. For part (a) this formula can be rearranged: \(T_{\mathrm{h}}=\frac{T_{1}}{1-\frac{Efficiency}{100}}\). Substitute the known values, \(T_{1} = 41+273.15\) (Converted the temperature to Kelvin scale) K and Efficiency \(=36\%\). The result is minimum temperature of the steam before condensation, \(T_{h}\).

Explain why the actual temperature is higher

For part (b), the ideal calculation done in part (a) is based on the assumption that the engine is working with maximum efficiency which, in a real case scenario, never happens due to losses happened in the system (like friction, heat losses etc.). Therefore, the actual temperature has to be greater than the calculated one in part (a).

Estimate the steam pressure

For part (c), the vapor pressure correlates directly with the temperature. The steam table or Antoine's equation can be used to find the steam pressure at the temperature calculated in part (a)

Determine the possibility of achieving greater efficiency

For part (d), firstly, the efficiency of a heat engine can never be 100% or greater according to the second law of thermodynamics. Some heat energy will always be lost in the form of exhaust heat and some other losses, hence some part of the heat energy will not be converted into work.

Key concepts

Understanding the Second Law of Thermodynamics

The second law of thermodynamics is a fundamental principle that provides deep insight into the direction of spontaneous processes and the feasibility of converting heat into work. It states that in a closed system, the total entropy can only increase over time. This increase in entropy translates to the spread of energy and brings about the concept of heat flow from hot to cold without external work applied.

When it comes to heat engines, the second law restricts how much useful work we can extract from thermal energy. The conversion of heat into work within a heat engine cannot be done with perfect efficiency because some of the heat will always be transferred to a cooler reservoir. This residual heat transfer is a manifestation of increased entropy. Essentially, the second law tells us that there is a theoretical limit to the efficiency of heat engines, which is determined by the temperatures of the heat source and sink.

The equation given in the original exercise, \(\text{efficiency} = \frac{w}{q_{\mathrm{h}}} \times 100\% = \frac{T_{\mathrm{h}} - T_{1}}{T_{\mathrm{h}}} \times 100\%\), is derived from this law. It reveals that to calculate the maximum possible efficiency of a heat engine, we must know the temperatures of its hot source (\(T_{\mathrm{h}}\)) and cold sink (\(T_{1}\)).

Deciphering Maximum Efficiency of a Heat Engine

The maximum efficiency that a heat engine can achieve is a key concept in thermodynamics, closely tied to the second law. This efficiency is the highest possible ratio of work extracted from the engine to the heat energy supplied.

The formula highlighted in the original exercise describes the maximum theoretical efficiency of an ideal heat engine without any energy loss, which is the case for a Carnot engine. In practice, actual engines can never achieve this ideal efficiency due to real-world inefficiencies like friction and heat loss. Thus, the calculated efficiency from the equation \(\frac{T_{\mathrm{h}} - T_{1}}{T_{\mathrm{h}}}\), gives us the upper limit of what is thermodynamically feasible.

For example, if an engine is said to operate at 36% efficiency, it means that 36% of the heat energy is turned into work, while the rest is wasted, typically as heat released into the environment. In reality, the efficiency of the engine will likely be lower than calculated due to the aforementioned inefficiencies. Improvements in engine design and operations are constantly sought to approach this maximum efficiency within the bounds set by the second law of thermodynamics.

Steam Turbine Operations and Efficiency

Steam turbines are a critical component in power plants and various industrial processes. Their operation is a sophisticated example of heat engines in action, where high-pressure steam is expanded through a turbine to generate work, typically in the form of electricity.

The efficiency of steam turbines is contingent upon the temperatures and pressures at which they operate. The higher the steam's initial temperature and pressure (before expansion in the turbine), the more potential work can be extracted from it. However, as pointed out in the exercise solution, the actual temperature will usually be higher than the ideal necessary temperature due to real-world constraints such as irreversible processes and non-ideal material behaviors.

Estimating the steam pressure at a given temperature, as mentioned in part (c) of the problem, involves consulting steam tables or employing Antoine's equation. This allows engineers to closely monitor and adjust the conditions at which the steam turbine operates to maintain optimal efficiency within the limits set by material strength and safety considerations.

Ultimately, the overarching principle governing steam turbine operations is the same as with any heat engine: the second law of thermodynamics. The actual operational efficiency will always be lower than the maximum theoretical efficiency due to the inevitable entropy increase and energy dispersion in any real process. Steam turbine technology continues to evolve with advances aimed at reducing energy losses and improving overall efficiency.