The Stirling cycle consists of two isothermal stages and two...

The Stirling cycle consists of two isothermal stages and two isochoric (constant volume) stages; see Figure 11.4. (i) Show that if the heat capacity of the working fluid is a function of temperature alone, then the amount of heat entering the fluid in one of the isochoric processes is equal to the amount leaving in the other. (ii) Hence show that the thermal efficiency is $1-T_2 / T_1$, the same as that of a Carnot engine. It follows from (i) that the energy extracted when the fluid cools from $T_1$ to $T_2$ can be stored and used to provide the energy needed to heat the fluid from $T_2$ to $T_1$. Such an energy store is called a regenerator and is a common feature of Stirling cycle engines. Figure 11.4 Can't copy

The Stirling cycle consists of two isothermal stages and two isochoric (constant volume) stages; see Figure 11.4. (i) Show that if the heat capacity of the working fluid is a function of temperature alone, then the amount of heat entering the fluid in one of the isochoric processes is equal to the amount leaving in the other. (ii) Hence show that the thermal efficiency is $1-T_2 / T_1$, the same as that of a Carnot engine.
It follows from (i) that the energy extracted when the fluid cools from $T_1$ to $T_2$ can be stored and used to provide the energy needed to heat the fluid from $T_2$ to $T_1$. Such an energy store is called a regenerator and is a common feature of Stirling cycle engines.
Figure 11.4 Can't copy

Key Concepts

Regenerator

A regenerator is an energy storage system used in the Stirling cycle where the thermal energy released during one process (cooling) is stored and then reused to supply energy during a subsequent heating process. This technique improves the overall efficiency by recycling energy within the cycle.

Stirling Cycle

The Stirling cycle is a thermodynamic cycle that consists of alternate isothermal and isochoric processes. It is used to model heat engine behavior and, despite having reversible processes, it incorporates mechanisms like regeneration that help in achieving efficiencies comparable to the Carnot cycle.

Isothermal Process

An isothermal process is one in which the system’s temperature remains constant during the transformation. In such processes, any heat exchanged with the environment is converted entirely into work, or vice versa, keeping the internal energy constant.

Isochoric Process

An isochoric process is a thermodynamic process that occurs at constant volume. Because volume does not change, no work is done by or on the system, and any heat transfer changes the temperature of the working fluid based solely on its heat capacity.

Heat Capacity

Heat capacity defines the amount of heat required to change the temperature of a substance by a given amount. When treated as a function of temperature alone, it ensures that heat added or removed during isochoric processes can be mathematically related to specific temperature changes.

Thermal Efficiency

Thermal efficiency is a measure of how well a heat engine converts heat into work. It is defined as the net work output divided by the heat input, and it sets an upper limit on the performance of any engine operating between two temperatures.

Carnot Cycle

The Carnot cycle represents an idealized reversible engine that operates between two fixed temperatures and provides the maximum possible efficiency, given by 1 minus the ratio of the lower temperature to the higher temperature. It serves as a benchmark for comparing real engine cycles.

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