The Stirling Cycle – Principles and Calculations

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The Stirling cycle is a thermodynamic cycle and a variation of the Carnot cycle. Rather than undergoing isentropic processes, the cycle undergoes isochoric processes. As a truly reversible cycle, it supplies mechanical power from heat, or heating and cooling from mechanical work. In this article, you will learn the principles of the Stirling cycle, calculations of efficiency, and applications.

Principles of the Stirling Cycle

The Stirling cycle operates in a reversible manner, undergoing isothermal and isochoric processes. Because of this, the cycle is incredibly heat efficient and flexible. It was one of the first thermodynamic cycles discovered. It came to prominence in 1816 as an alternative to the steam engine. The Stirling cycle does not apply in many everyday circumstances. It is heat efficient, however, the required temperature differential is too high for common use.

In most circumstances, it isn’t necessary to keep something well below zero, and for heating and power, there are better options. However, as humans are venture into space more this cycle could see more use in keeping things at a fixed temperature.

The cycle starts by undergoing an isothermal heat addition, putting heat into the system. Next, the cycle undergoes an isochoric heat removal, where the heat flows from one cylinder to another.  In the third step, the Stirling cycle undergoes isothermal compression as heat leaves the system. Finally, the cycle undergoes an isochoric heat addition. Overall the cycle produces symmetric functionality due to its ease of reversal.

Calculations of the Stirling Cycle

As the cycle operates under a set temperature differential, knowing the high(TH) and low(TL) temperatures allow for the determination of efficiency. The maximum and minimum volumes, and the maximum pressure also require determination.

Because of the way the Stirling cycle operates, determining efficiencies occurs relatively easily. Using the specific heats(Cp & Cv) of the working fluid and other constants of the fluid(R & k). Since the first stage operates isothermally and the temperature stays the same, the max pressure(P1) and the maximum(V2) and minimum(V1) volumes determine the pressure(P2) at stage 2.

    \[ P_2=\frac{V_1}{V_1} P_1 \]

The heat added(qin) into the Stirling cycle derives from the change in entropy(s). The Temperature(T1 & T2) equal each other, causing the left side term to drop from the equation.

    \[ s_2 - s_1 = \frac{q_{in}}{T_H} = c_p ln(\frac{T_2}{T_1} - R ln(\frac{P_2}{P_1}) \]

Because the first and second temperatures are the same the equation can be simplified. Just using the first(P1) and second(P2) pressures and the highest temperature(TH).

    \[ Q_{in} = T_H R ln(\frac{P_2}{P_1}) \]

Because of the reversibility of the cycle, the heat out(qout) of the system arrives by using a simple ratio that includes the lowest temperature(TL).

    \[ \frac{q_{in}}{T_H} = \frac{q_{out}}{T_L} \]

Determination of thermal efficiency(ηT) occures by using the heat in(qin) and heat out(qout).

    \[ \eta_T = 1- (\frac{q_{out}}{q_{in}}) \]

Determination of the mass of the working fluid occurs by using the initial pressure(P1), initial volume(V1), initial temperature(T1), and the gas constant(R).

    \[ m = \frac{P_1 V_1}{R T_1} \]

Finally, the heat transfer between the fluid and regenerator can be found.

    \[ Q_{reg} = M c_v (T_H - T_L) \]

Applications of the Stirling Cycle

Though the Stirling cycle is efficient, it requires a temperature differential so large that the materials of the engine itself become less reliable. This reduces its everyday use because these engines require more expensive materials. It can be used as an engine or even a heat pump because of its reversibility. This cycle has seen most of its use in industries where money isn’t an issue. For example in the space industry, it can sometimes be seen used.

There are several configurations and versions of the Stirling cycle which present the opportunity to be used for absolute zero conditions in space. The cycle is also mostly used in cryogenics where it is used as a high-efficiency heat pump. The cycle is able to cool things well below freezing which makes it better for extremely cold situations. Despite its longevity, modern uses are still being researched for this engine.

History of the Stirling Cycle

The Scottish engineer Robert Stirling discovered the principles of the Stirling cycle in 1816. The original use was to replace a steam engine to pump water at a quarry. However, the original engines couldn’t output much power and thus never obtained wide use. Later years brought more research around this engine, allowing for the development of more useful versions. Today, the cycles’ best application is to heat and cool things using a small amount of power.