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20.50. A stirling-cycle Engine. the Otto cycle, except that the compression and expansion of the gas are done at constant temperature, not adiabatically as in the Otto cycle. The Stirling cycle is used in external combustion engines (in fact, burning fuel is not necessary; any way of producing a temperature difference will do -solar, geothermal, ocean temperature gradient, etc. \(.\) which means that the gas inside the cylinder is not used in the combustion process. Heat is supplied by burning fuel steadily outside the cylinder, instead of explosively inside the cylinder as in the Otto cycle. For this reason Stirling-cycle engines are quieter than Otto-cycle engines, since there are no intake and exhaust valves (a major source of engine noise). While small Stirling engines are used for a variety of purposes, Stiring engines for automobiles have not been successful because they are larger, heavier, and more expensive than conventional automobile engines. In the cycle, the working fluid goes through the following sequence of steps (Fig. 20.30\()\) : (i) Compressed isothermally at temperature \(T_{1}\) from the initial state \(a\) to state \(b\) , with a compression ratio \(r .\) (ii) Heated at constant volume to state \(c\) at temperature \(T_{2}\) . (iii) Expanded isothermally at \(T_{2}\) to state \(d\) . (iv) Cooled at constant volume back to the initial state \(a\) . Assume that the working fluid is \(n\) moles of an ideal gas (for which \(C_{V}\) is independent of temperature). (a) Calculate \(Q, W,\) and \(\Delta U\) for each of the processes \(a \rightarrow b, b \rightarrow c, c \rightarrow d,\) and \(d \rightarrow a\) . (b) In the Stirling cycle, the heat transfers in the processes \(b \rightarrow c\) and \(d \rightarrow a\) do not involve external heat sources but rather use regeneration: The same substance that transfers heat to the gas inside the cylinder in the process \(b \rightarrow c\) also absorbs heat back from the gas in the process \(d \rightarrow a\) . Hence the heat transfers \(Q_{b \rightarrow c}\) and \(Q_{d \rightarrow a}\) do not play a role in determining the efficiency of the engine. Explain this last statement by comparing the expressions for \(Q_{b \rightarrow c}\) and \(Q_{d \rightarrow a}\) calculated in part (a). (c) Calculate the efficiency of a Stirling-cycle engine in terms of the temperatures \(T_{1}\) and \(T_{2}\) . How does this compare to the efficiency of a Carnot-cycle engine operating between these same two temperatures? (Historically, the Stirling cycle was devised before the Carnot cycle.) Does this result violate the second law of thermodynamics? Explain. Unfortunately, actual Stirling-cycle engines cannot achieve this efficiency due to problems with the heat-transfer processes and pressure losses in the engine.

Step by step solution

01

Understanding Process a to b

For the isothermal compression from \(a\) to \(b\) at temperature \(T_1\), we use the formula for work done during isothermal processes: \(W_{a \rightarrow b} = -nRT_1 \ln(r)\), where \(r\) is the compression ratio. Since it's isothermal, \( \Delta U_{a \rightarrow b} = 0\), therefore \(Q_{a \rightarrow b} = -W_{a \rightarrow b}\).

02

Understanding Process b to c

During the heating at constant volume from \(b\) to \(c\), the change in internal energy \(\Delta U_{b \rightarrow c} = nC_V(T_2 - T_1)\). The heat added \(Q_{b \rightarrow c} = \Delta U_{b \rightarrow c}\) and work done \(W_{b \rightarrow c} = 0\) since volume is constant.

03

Understanding Process c to d

During the isothermal expansion from \(c\) to \(d\) at temperature \(T_2\), the work done \(W_{c \rightarrow d} = nRT_2 \ln(r)\). Since \(\Delta U_{c \rightarrow d} = 0\) for an isothermal process, the heat absorbed \(Q_{c \rightarrow d} = W_{c \rightarrow d}\).

04

Understanding Process d to a

During cooling at constant volume from \(d\) to \(a\), \(\Delta U_{d \rightarrow a} = nC_V(T_1 - T_2)\). The heat rejected \(Q_{d \rightarrow a} = \Delta U_{d \rightarrow a}\), and work done \(W_{d \rightarrow a} = 0\) since the volume is constant.

05

Regeneration in the Stirling Cycle

The heat transfer in processes \(b \rightarrow c\) and \(d \rightarrow a\) does not affect efficiency because the internal heat exchanged (regeneration) cancels out over a complete cycle, providing \(Q_{b \rightarrow c} = -Q_{d \rightarrow a}\). The effective external heat transfers are \(Q_{a \rightarrow b}\) and \(Q_{c \rightarrow d}\).

06

Efficiency of Stirling Cycle

The efficiency \( \eta \) for the Stirling cycle is \( \eta = 1 - \frac{T_1}{T_2} \), which is the same as the Carnot cycle's efficiency for the same temperatures, illustrating no violation of the second law of thermodynamics as the real-world system losses in Stirling engines prevent reaching this theoretical efficiency.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamic Cycles

The Stirling cycle, a type of thermodynamic cycle, plays a key role in understanding how thermal energy is converted into mechanical work. This cycle involves four distinct processes: two isothermal (constant temperature) and two isochoric (constant volume) processes. These cycles are essential for describing how engines operate using temperature differences to perform work.
The processes include:

• Isothermal compression where the engine reduces the gas volume at a steady temperature.
• Isochoric heating during which the temperature of the gas increases at constant volume.
• Isothermal expansion, where the gas expands at the higher temperature.
• Isochoric cooling that returns the gas to its initial temperature and volume.

External Combustion Engine

The Stirling cycle engine is an example of an external combustion engine. This means that fuel combustion, necessary for generating heat, occurs outside the engine itself. This differs from internal combustion engines like the Otto cycle where the heat is generated within the engine.
External combustion offers several distinct advantages:

• Quiet operation without the noise of explosive combustion inside the engine.
• The ability to utilize diverse heat sources, such as solar or geothermal energy.
• Reduced wear and tear since the explosive forces are not contained within the engine.

These attributes, though beneficial, also lead to increased size and complexity, which have hindered their widespread adoption in specific applications like automotive engines.

Regeneration Process

An essential feature of the Stirling cycle is the regeneration process. Here, the thermal energy from the gas during cooling is not wasted but stored temporarily and reused. This occurs between the steps where the gas is heated and cooled at constant volume.
The process operates by exchanging heat within the engine:

• During heating, the regenerator collects excess heat from the gas.
• During cooling, the stored heat is returned to the gas.

This method enhances the thermal efficiency of the cycle by conserving energy within the engine, reducing the need for continuous external heat supply.

Isothermal Process

In the Stirling cycle, isothermal processes are critical as they ensure that the temperature remains constant while the gas changes volume.

• Isothermal compression happens when gas compresses at temperature \( T_1 \), preventing a temperature drop.
• Isothermal expansion occurs at temperature \( T_2 \), allowing gas to expand without heat loss.

These processes involve precision in maintaining temperature by controlling heat exchange with a heat sink or source efficiently. The work done during these processes can be determined using formulas like \( W = nRT \ln(v_2/v_1) \), showing the importance of temperature control in thermodynamic performance.

Efficiency Calculation

Calculating the efficiency of the Stirling cycle involves comparing the work done by the engine to the heat input. The theoretical efficiency is expressed as:\[\eta = 1 - \frac{T_1}{T_2}\]where \( T_1 \) and \( T_2 \) are the absolute temperatures of the isothermal compression and expansion processes, respectively.
This formula resembles that of the Carnot cycle, indicating an ideal efficiency boundary.
While theoretically efficient, real-world efficiency is lower due to factors not accounted for, such as friction and imperfect heat transfer. Understanding these practical limitations helps in predicting and improving the performance of Stirling engines in various applications.