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.

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   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 P0). 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.

Fig. 1.
The spring engine. Labels 1 and 2 indicate the cold and hot zones at temperatures Th and Tc, respectively. The arrows indicate the movements of the piston and displacer, and z and x are their displacements from their equilibrium positions.
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The spring engine. Labels 1 and 2 indicate the cold and hot zones at temperatures Th and Tc, respectively. The arrows indicate the movements of the piston and displacer, and z and x are their displacements from their equilibrium positions.

Fig. 1.
The spring engine. Labels 1 and 2 indicate the cold and hot zones at temperatures Th and Tc, respectively. The arrows indicate the movements of the piston and displacer, and z and x are their displacements from their equilibrium positions.
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The spring engine. Labels 1 and 2 indicate the cold and hot zones at temperatures Th and Tc, respectively. The arrows indicate the movements of the piston and displacer, and z and x are their displacements from their equilibrium positions.

Close modal

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 Th, 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 Tc. The gas is never completely in either the hot or the cold zone, but its pressure 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 ϕ = π / 2⁠.

In this section, we offer a brief analysis of the theory behind the spring engine; a real device is not likely to behave so ideally. Because the gas pressure on both sides of the displacer is the same, the movement of the displacer will be due only to the spring. We model this as a harmonic oscillator with a frequency given by
(1)
with displacement
(2)
So far as the dynamics of the liquid are concerned, the pressures on its right side (P) and left side (P0) give rise to a net force proportional to P − 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,
(3)
where 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 V1 and V2 of the respective zones and on the total gas volume V = V 1 + V 2⁠; this is Eq. (18) of Ref. 1,
(4)
where α = T h / T c⁠. V10, V20, and V0 are the initial values of V1, V2, and V, respectively, and β is a parameter called polytropic index. Typical values of β are such that 1 ≤ β ≤ γ⁠, with γ = 1.4 for a diatomic gas. In Eq. (4), we have set the gas pressure inside of the engine as P0 when the volumes have their initial values.
Assuming that both the test tube and U-tube have the same cross-section area a, the volumes of the zones 1 and 2 can be expressed as (see Fig. 1)
(5)
where the displacements 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:
(6)
In this case, the left-hand sides of Eq. (6) can be treated as perturbations in Eq. (4). Keeping only the first terms in both z and x of the expansion and substituting in Eq. (3), a linear equation for z is obtained, namely,
(7)
where ξ = b / ( ρ l a ) and
(8)
Equation (7) describes a damped linear oscillator with an external forcing characterized by the frequency ωd, see Eq. (2). Its steady-state solution is
(9)
where
(10)
and
(11)
From Eq. (10), it is clear that the engine generates no work if α = 1 (that is, if there is no temperature difference between the reservoirs) or ϕ = 0⁠, and also that, from Eq. (11), the amplitude zs is maximized if ω p = ω d (⁠ ϕ = π / 2⁠). Moreover, when α → ∞⁠, zs tends to a finite value. The theoretical pressure–volume (P–V) diagram can be obtained using Eqs. (2), (4), (5), and (9).
The area enclosed by the steady-state curve of the P–V diagram represents the total work W generated by the cycle; this evaluates as
(12)
where the period is
(13)
Using Eqs. (2), (4), (5), (9), and (12) and the conditions of Eq. (6), an analytic expression for the first order approximation of work can be obtained; this is
(14)

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 τ ∼ 0.1   s⁠.

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 ϕ between the liquid and the displacer determines the power of the engine, with maximum work achieved at a resonance when ϕ = π / 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 → ∞⁠.

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 

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

1.
A.
Romanelli
, “
Alternative thermodynamic cycle for the Stirling machine
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R.
Darlington
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Strong
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Ludic Science, Marble Stirling engine <https://www.youtube.com/watch?v=5UzO16dCl6A>,
2015
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4.
See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0007153 for comments on experimental proposals.
© 2022 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2022
Author(s)

Supplementary Material

Supplementary materials- zip file