## I. INTRODUCTION

In a previous paper published in this journal, we developed an alternative approach for the Stirling cycle that achieved theoretical results, which showed better agreement with experimental data than the usual textbook approach.^{1} Using expressions developed in that work and inspired by a toy Stirling engine known as the “marble” engine, we propose here a new theoretical Stirling engine.^{2} This engine, which we call the spring engine, possesses the same essential physics as the marble engine but can be analyzed with standard analytic techniques.

## II. FROM THE MARBLE TO THE SPRING ENGINE

The marble engine can be built very cheaply and easily, and its operation can be understood through the behavior of an ideal gas.^{3} In this device, the air displacer of a Stirling engine is replaced by glass marbles within a test tube; the marbles cyclically sweep the working gas (air) from one end of the tube to the other. Typical values of the period *τ* are experimentally found to be of the order of $ 1 \u2009 s$, and the engine can run for hours so long as there is a sufficient temperature difference between the tube ends. However, a detailed analysis of the performance of the marble engine is difficult as it involves chaotic dynamics.

In order to quantitatively address the essence of the marble engine, we make the following modifications (see Fig. 1): (i) The marbles are replaced by a single cylinder of mass *m*, length *d*, and a cross-section slightly smaller than the cross-section of the tube. This cylinder displaces the air in the tube from the cold side to the hot side and vice-versa. (ii) The tube is fixed and lies horizontally. Inside the tube is placed an ideal spring of constant *k* with one of its ends attached to the displacer and the other to one end of the tube. As described below, this arrangement produces a periodic movement of the displacer that simulates the recurrent movement of the marbles in the original device. By imagining that we can use some of the engine's work to secure the displacer's oscillation, we allow ourselves to disregard friction. (iii) The original power piston is replaced by a liquid piston consisting of a U-tube partially filled with liquid and connected to one end of the tube; the other end of the U-tube is open to the atmosphere (pressure *P*_{0}). The volume of the gas changes due to the motion of the liquid piston, and the work produced by the engine acts on the liquid.

Our theoretical analysis of the spring engine, which is summarized in Sec. III, assumes that there exists two zones that can be distinguished by their uniform temperatures in the volume occupied by the gas; we label these as 1 and 2. Zone 2 is the hot zone where the gas is in contact with the reservoir at the external temperature *T _{h}*, which is maintained by the candle, and zone 1 is the cold zone where the gas is in contact with the external environment at temperature

*T*. The gas is never completely in either the hot or the cold zone, but its pressure

_{c}*P*is the same everywhere. Consequently, the gas density must be different in each zone to keep the same pressure with different temperatures. Zone 1 includes the gas in the right side of the liquid.

To start the engine, we must set the liquid and/or the displacer in motion. One way to do this could be to start with the liquid at rest, move the displacer out of its equilibrium position, and then release it. With the displacer in motion, energy transfer between the hot and cold zones begins and can be described as follows; we focus on the movement of the displacer, supposing the liquid to be at rest. When the gas is swept by the displacer to the hot (cold) zone (through the loose fit that connects both chambers) it expands (contracts), the gas pressure increases (decreases), and the gas pushes the liquid down (up). The air pressure on the right side of the liquid depends on the displacer's motion, but that, at left side, remains at atmospheric pressure; this pressure difference sets the liquid in motion. If we now incorporate the motion of the liquid, the net motion of air between zones 1 and 2 depends on the phase of the relative motions of the displacer and the liquid, which results in a cyclical transfer of energy from the high temperature zone 2 to the liquid piston of zone 1. The phase of this motion is a crucial parameter for the efficiency and the power delivered by the engine. We show below that the optimum phase is around $ \varphi = \pi / 2$.

## III. DYNAMICS OF THE SPRING ENGINE

*P*) and left side (

*P*

_{0}) give rise to a net force proportional to $ P \u2212 P 0$. We also introduce an external force proportional to the velocity of the liquid which will be responsible for dissipating the useful power delivered by the gas to the liquid. Overall, the dynamics of the liquid are determined by Newton's second law applied to the displacement

*z*,

*ρ*is the liquid density,

*a*is the cross-sectional area,

*l*is the length of the liquid column,

*b*is a damping coefficient, and

*g*is the acceleration due to gravity. To solve this equation for

*z*(

*t*), an explicit expression for the dependence of

*P*on the parameters of the system is required. In a previous work, we obtained a generic analytic solution for the pressure inside a Stirling engine that depends on the volumes

*V*

_{1}and

*V*

_{2}of the respective zones and on the total gas volume $ V = V 1 + V 2$; this is Eq. (18) of Ref. 1,

*V*

_{10},

*V*

_{20}, and

*V*

_{0}are the initial values of

*V*

_{1},

*V*

_{2}, and

*V*, respectively, and

*β*is a parameter called polytropic index. Typical values of

*β*are such that $ 1 \u2264 \beta \u2264 \gamma $, with $ \gamma = 1.4$ for a diatomic gas. In Eq. (4), we have set the gas pressure inside of the engine as

*P*

_{0}when the volumes have their initial values.

*a*, the volumes of the zones 1 and 2 can be expressed as (see Fig. 1)

*x*and

*z*are taken from their initial positions when the engine is not yet running; we take these to be $ x = z = 0$. Substituting Eqs. (5) in Eq. (4) gives the explicit dependence of

*P*on

*x*and

*z*. This dependence is rather complicated and implies that Eq. (3) is a non-linear differential equation in

*z*. Therefore, Eq. (3) does not necessarily have periodic solutions in general, in which case, the system could not be used as a cyclical engine. However, the engine can be designed in such a way that the geometry of the hot and cold volumes satisfy the following reasonable relations:

*z*and

*x*of the expansion and substituting in Eq. (3), a linear equation for

*z*is obtained, namely,

*ω*, see Eq. (2). Its steady-state solution is

_{d}*α*= 1 (that is, if there is no temperature difference between the reservoirs) or $ \varphi = 0$, and also that, from Eq. (11), the amplitude

*z*is maximized if $ \omega p = \omega d$ ( $ \varphi = \pi / 2$). Moreover, when $ \alpha \u2192 \u221e$,

_{s}*z*tends to a finite value. The theoretical pressure–volume (

_{s}*P*–

*V*) diagram can be obtained using Eqs. (2), (4), (5), and (9).

*P*–

*V*diagram represents the total work

*W*generated by the cycle; this evaluates as

This result indicates that the work per cycle is constant, dictated by the initial volumes of the hot and cold zones of the tube and the initial position of the displacer.

We have run numerical simulations for the spring engine, setting dimensions and values for the various parameters to be similar to those of the marble engine. Results of these simulations, including a *P*–*V* diagram, can be found in the supplementary material.^{4} Our simulations indicate periods of the order of $ \tau \u223c 0.1 \u2009 s$.

## IV. FINAL REMARKS

The dynamics of the spring engine can be solved using standard techniques to linearize and solve a second-order differential equation. Our main theoretical conclusions are as follows: (i) The work per cycle is constant, dictated by the initial volumes of the hot and cold zones of the tube and the initial position of the displacer; (ii) the phase $\varphi $ between the liquid and the displacer determines the power of the engine, with maximum work achieved at a resonance when $ \varphi = \pi / 2$; and (iii) to extract work from the engine, the temperature ratio $ T h / T c$ must exceed unity, with the work becoming independent of this ratio as $ T h / T c \u2192 \u221e$.

The drawbacks of the spring engine are that it will not be as easy to construct as a marble engine, and since it will have a shorter period than the marble engine, any energy-loss compensation mechanism will have to be more sophisticated. Constructing such a device could be a good term project for advanced students. Possibly helpful ideas along this line are discussed in the supplementary material.^{4}

## ACKNOWLEDGMENTS

The author acknowledges the stimulating discussions with Anna Fló and Víctor Micenmacher. This work was partially supported by ANII and PEDECIBA (Uruguay).

## REFERENCES

*Stirling and Hot Air Engines*