Stirling Engines and Irrigation Pumping
This document is an attempt to preserve the content of a document written by Colin West for the US Agency for International Development (USAID) in 1987, and made available by the Oak Ridge National Laboratory (ORNL) as a Portable Document Format (PDF) file.
Readers are advised that the PDF file from ORNL contains a number of digitization errors, and that this document contains changes to correct (only) digitization errors and to improve readability.
Readers are further cautioned that this document does not contain sufficient information to design practical fluidyne engines, that it contains serious errors of omission, and that (in my opinion) it makes false and misleading statements. As far as I've been able to determine, West’s paper has caused far too many worthwhile projects to be abandoned when developers found too large a gap between what West presented and their experimental results.
This report was prepared in support of the Renewable Energy Applications and Training Project that is sponsored by the U.S. Agency for International Development for which ORNL provides technical assistance. It briefly outlines the performance that might be achievable from various kinds of Stirling-engine-driven irrigation pumps. Some emphasis is placed on the very simple liquid-piston engines that have been the subject of research in recent years and are suitable for manufacture in less well-developed countries. In addition to the results quoted here (possible limits on M4 and pumping head for different-size engines and various operating conditions), the method of calculation is described in sufficient detail for engineers to apply the techniques to other Stirling engine designs for comparison.
1. Stirling Engine Power Output
If well-designed and constructed, conventional Stirling engines (see [R1] for a guide to Stirling technology) have a rather simple approximate relationship between the brake power, the piston stroke of the machine, the pressure of the working fluid, the frequency of operation, and the temperature of the heater and cooler:
[E1] Wo ≈ Wn * Pm * f * Vo * RT
A survey of 23 very different engines indicated the average value of Wn to be 0.25, and this is the number usually employed in making rough calculations or predictions of the performance of new Stirling engines. The Wn of 0.25 is only applicable if a consistent unit set (such as SI units) is used in [E1]. If the engine is used to drive a pump, then the power may be used to raise liquid (often water) against gravity:
[E2] Vp * H * ρ * g ≈ Wo ≈ 0.25 * Pm * f * Vo * RT
This relation does not take into account the loss of power in the pump itself. Actually, such an omission is quite legitimate because some of the engines surveyed to assign a value to Wn were pumping engines. Their measured power output referred to the actual pumped volume and head; that is, the pump efficiency (or inefficiency) has to some extent already been included in Wn.
2. Stirling Engine M4
For convenience, we can rearrange [E2] to calculate Vp * H, which is the quantity sometimes called M4. However, we also need to convert the pumping rate from m3/s (i.e., SI units) to m3/h (the units normally used, unfortunately, for M4). With these changes,
[E3] M4 = Vp * H = 3600 * 0.25 * Pm / (ρ * g) * f * Vo * RT.
Now, ρ = 103 kg/m3 (for water), g = 9.81 m/s2, and substituting these numbers into [E3] yields
[E4] M4 ≈ 0.092 * Pm * f * Vo * RT
Most Stirling engines operate at rather high pressure and high speed to maximize the specific power, but others, including the liquid-piston machines (fluidynes) to be described later, are inherently low-pressure, low-speed engines. Table 1 lists the relevant data for five different engines and the value of M4 calculated from [E4].
3. Fluidyne Liquid-Piston Stirling Engines
Fluidyne is the name given to a class of Stirling engines in which the pistons are actually columns of liquid (usually water) moving up and down in a set of U-tubes. The appendix, a reprint of a 1984 conference paper, describes the principles and practice of fluidynes, together with a brief history. Because the working parts of the fluidyne are water and the power output is available in the form of either pulsating pressure or movement of liquid in a tube, the most obvious application is as a pumping engine, for example, in irrigation systems. However, only experimental and demonstration machines have been built so far. The largest one had a throughput of more than 15 m3/h and a lift of almost 4 m. For more information on the technology of the fluidyne , see Ref. 2. In subsequent sections of this report, some estimates of the possible performance of fluidyne pumping systems are made.
3.1 Power Output of a U-Tube Fluidyne
Suppose the displacer is water in a U-tube of diameter D with a stroke of D, that is, a "square" engine. The uprights are separated by a thickness t of thermal insulation. To avoid excessive mixing and heat losses in the displacer liquid as it turns into the curved section of the U-tube, the liquid surface at bottom dead center of the piston movement is at a minimum height D above the curved portion of the U-tube (see Fig 1). We assume, reasonably, that the phase angle between the liquid motion in the two arms of the U-tube is 90° which is
the vector sum of the volume change column, is then the net volume change, at each end of the displacer column, is
[E5] Vo = sqrt(2) * π * D3 / 4
The operating frequency of a fluidyne is determined almost entirely by the length of the displacer column[R3]:
[E6] f = 1/(2 * π) * sqrt(2 * g / LD)
From the geometry of the displacer (Fig l), the length of the centerline along the column is calculated as
[E7] LD = 3 * D + π * (D/2 + t/2)
[E8] 1 = 1 / (2 * π) * sqrt(2 * g / (3 * D + π * (D / 2 + t / 2)))
We can now calculate the approximate power output of the fluidyne from [El], [E5], and [E8]:
[E9] Wo ≈ 0.25 * Pm * f * Vo * RT
It is not convenient to operate this kind of fluidyne at a mean pressure above atmospheric (0.1 MPa) because a higher pressure would expel the liquid from-the tuning line.
Substituting Pm = 105 Pa and g = -9.81 m/s2 into [E9] yields:
[E10] Wo ≈ 19600 * D3 / sqrt(3 * D + π * (D / 2 + t / 2)) * RT
Because of the large surface area of a high-power, atmospheric-pressure engine, good thermal insulation is essential to high efficiency. The insulation thickness should probably be at least equal to the U-tube diameter, that is, t = D. Then [E10] becomes
[E11] Wo ≈ 7900 * D5/2 * RT
Notice that [E8] is not very sensitive to the exact value of t, so that choice of an insulation thickness somewhat greater or less than the value assumed here would not significantly affect the conclusions. Equation [E11] could be applied approximately to a concentric cylinder fluidyne by defining D as the diameter of the innermost concentric cylinder.
3.2 Pumping Water with a U-Tube Fluidyne
A quantity of interest to many designers and sponsors of irrigation systems is the so-called M4 of a system:
[E12] M4 = the volume of water pumped X the lift = Vp * H
For irrigation purposes, a mixed unit set of metres and hours is usually adopted. Now,
Vp * H * ρ * g = energy added to the water/unit time.
In SI units, for a U-tube fluidyne, [E11] and [E12] indicate that
[E13] Vp * H * ρ * g = Wo = 7900 * D5/2 * RT
Changing the time units from seconds to hours gives
[E14] Vp * H * ρ * g = 3600 * 7900 * D5/2 * RT
M4 = Vp * H = 3600 * 7900 / (p * g) * D5/2 * RT
M4 = 3600 * 7900 / (103 * 9.81) * D5/2 * RT
[E15] M4 ≈ 2900 * D5/2 * RT
Assume Tc = 30°C (86°F). Then M4 can be calculated as a function of the displacer diameter and the hot-end temperature. A heater temperature of 350°C is practical with the simplest of materials and joining techniques (including adhesives). With more care paid to the choice of'materials and construction methods, but with no exotic technology, 550°C is easily reached with safety. Table 2 and Fig 2 illustrate the rapid increase of M4 as the displacer diameter is increased.
3.3 Pumping Head Available from a U-Tube Fluidyne
Staging the pumps by driving several pump arms from the same displacer[R2] can lift water to any desired height. However, each stage can lift the water no more than a certain maximum incretnental height because the maximum pressure available to drive the pump in an atmosphericpressure engine is quite limited.
An approximate calculation of the maximum head available is fairly straightforward. When pumping at close to the maximum available head, the volume pumped will be low with little change of volume in the engine during the pumping stroke. Therefore, the main sources of pressure variations in the working fluid of the engine will be the displacer action and the tuning-line action. According to [E2.3] of [R1], the peak-topeak pressure change due to displacer action is given by
[E16] ΔPd ≈ Pm * (Ve / Vm) * 2 * RT
The mean volume of working fluid Vm in the engine includes the heat exchangers and connecting ducts, as well as the volume in the displacer Ve. Typically in modern designs, the heat exchanger and duct volume (i.e., the unswept or dead volume) is such that Vm = 2.5 * Ve; see Table 2.1 of [R1].
The peak-to-peak pressure variation due to the volume change Vo is simply given by the ratio of the volume change to the mean volume:
[E17] ΔP ≈ P * Vo / Vm
As seen earlier, Vo ≈ sqrt(2) * Ve, and therefore
[E18] ΔP ≈ sqrt(2) * P * Ve / Vm
The two arms of the displacer were assumed to be moving with equal strokes and a 90° phase difference (which is why Vo = sqrt(2) * Ve), and the phase angle between the displacer action and the tuning-line or volume-changing action is therefore 45°.
The pressure change due to the displacer action is in phase with the displacer movement, but the pressure variations due to the volume change are 180° out-of-phase with the change (i.e., when the volume is minimum, the pressure is maximum and vice-versa). Therefore, the phase angle between ΔPd and ΔPt is 180° - 45° = 135°.
The total pressure variation ΔP is the sum, taking account of the phase angle between them, of ΔPd and ΔPt:
[E19] ΔP = sqrt(ΔPd2 + ΔPt2 - 2 * ΔPd * ΔPt * cos(135°))
Substitute for ΔPd and ΔPt from [E16] and [E18] recall that
Vm ≈ 2.5 * Ve, and simplify:
[E20] ΔP ≈ (2 * Pm / 2.5) * sqrt(RT2 * (1 / 2) + RT)
Only if the peak-to-peak pressure variation exceeds the hydrostatic pressure from the pumping head can the valves open and the water begin to flow. Therefore, the maximum possible lift (excluding any possible enhancement by dynamic effects) in a single stage is Hmax where
Hmax * ρ * g = ΔP
[E21] Hmax = (2 * Pm / (2.5 * ρ * g)) * sqrt(RT2 + RT + 1 / 2)
For an atmospheric engine and water pump, Pm = 105 Pa, ρ = 103 kg/m3, and g = 9.81 m/s2. Using [E21], the maximum possible lift for a single stage fluidyne with a hot-end temperature of 35O°C is 8.0 m. However, remember that the pumping rate falls toward zero as the maximum possible lift is approached, so that the practical limit is much lower than the theoretical one.
With a hot-end temperature of 550°C, the theoretical maximum head in a single stage is 8.8 m. As an example, Fig 3 combines the results of [E15] and [E21] to show the potential performance limits of a 300 mm (12-inch) displacer diameter fluidyne operating at 350°C and 550°C. The results are shown for a single-stage and a two-stage pumping system.
An existing correlation for the performance of Stirling engines can be used to make estimates of M4 for Stirling-powered irrigation pumps. Five engines, representing a wide range of different designs, have been evaluated on the basis of the correlation; the M4 values range from 20 m4/h for the smallest engine surveyed to 4000 m4/h for the largest. One particular type of Stirling machine, the fluidyne liquid-piston pump, was examined in greater detail. Depending on the size of the engine, M4 values in the range of 10 to 200 seem to be practical. The maximum theoretical pumping head is about 8 or 9 m (depending upon the operating temperature) for a single-stage design, or 16 m for a two-stage system. The flow rate falls rapidly as the maximum head is approached.
[R1] C. D. West, Principles and Applications of Stirling Engines, Van Nostrand Reinhold, New York, 1986.
[R2] C. D. West, Liquid Piston Stirling Engines, Van Nostrand Reinhold, New York, 1983.
[R3] C. D. West, "Dynamic Analysis of the Fluidyne", Paper 839126, Proc 18th IECEC, August 1983
Chinese Society of Engineering Thermophysics
Colin D. West
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37831 USA
Since the invention of the fluidyne engine in 1969, several research groups have explored and described the potential of liquid-piston Stirling machine designs for a wide variety of applications, including water pumping from solar heat, simple and long-lived fossil-fuel-fired irrigation pumps, and heat-powered heat pumps. A substantial amount of theoretical work has been been published, along with experimental results from a number of very different machines, and design data for the construction of experimental engines. This paper describes the projects that hae been done and the performance of existing systems, identifies outstanding research needs, and outlines some of the potential for further progress.
The fluidyne liquid-piston Stirling engine was invented at the Harwell Laboratory of the United kingdom Atomic Energy Authority in 1969, snd the first machines were operated there In 1970. Two internal reports were written by the Inventor [1,2] and a British patent was filed  covering the basic invention and some improvements. The Patent Specification was published In 1973, describing the basic theory and some early experimental results. In 1974, a paper describing the fluidyne concept was presented at a meeting of the International Solar Energy Society  and the attractive simplicity of the liquid-piston mschine with its potential for low-cost reliable water pumping was recognized; at that time, tha Hstal Ibx group of companies had already expressed an interest in collaborating with Harwell on the development of fluidyne irrigation prmps. In 1975, therefore, several small-scale research efforts were under way; over the next few years, research results confirmed the potential of the liquid-piston engine, while identifying some problems that lie in the way and offering possible solutions. Some of the major results of this rssearch are reviewed in this paper.
One of the simplest versions of the Fluidyne to construct and operate is the liquid-feedback machine shown diagramatically in Fig 1. Oscillation of liquid in the displacer U-tube unaccompanied by any movement of liquid in the tuning or output column U-tube must represent, in terms of a conventional Stirling engine, pure displacement; gas would be displaced between the hot and cold spaces but with no net change in gas volume. Movement of liquid in the tuning line does result in a net change of gas volume, as does the power piston of a conventional Stirling machine. For the machine to function as an engine, the phasing must be such that the liquid level in the open end of the tuning line is falling during the time that the liquid level in the hot side of the U-tube is higher than that in the cold side - this compresses the gas when most of it is in the cold space. Conversely, when the liquid level in the hot side is lower than in the cold, the level in the open end of the tuning line must be rising, thus expanding the gas. It is apparent that with such a phasing, the Fluidyne will operate as a Stirling-like engine in the alpha or Rider configuration. For a Fluidyne of the type shown in Fig 1, the positioning of the junction between the tuning line and the displacer U-tube is crucial. Usually the junction must be closer to the hot than to the cold end of the displacer for successful operation as an engine, although exceptions to this rule have been noted . With the arrangement shown in Fig 1, the two sections of the displacer-liquid column (between the junction and each free surface) are subjected to the same pressure difference, if the pressure drop in the gas flow across the regenerator/connecting tube is neglected. However, the liquid msss between the junction and the hot surface is less, because of its shorter length, than that between the junction and the cold surface.
As a consequence, the hot side of the displacer-liquid column responds to the pressure difference more readily than the cold side, and its movement will be more advanced in phase. This is exactly the relatlonship needed for a Stirling engine, in which the hot expansion-space vlolume verfation must lead the phase of the cold compression-epace volume. If the tuning line is driven externally, there will still, according to the above argument, be a positive phase difference between the resulting motions of the liquid surface In the short and long legs of the displacer U-tube. Consequently, heat will still be moved between the two cylinder volumes, and one will have a liquid-piston refrigerator or heat pump. The description given above of the liquid-feedback mechanism is greatly oversimplified. The liquid-feedback system was first proposed, on the basis of intuition, by Bake-Yarborough; but not until 1974 did theoretical explanation for Its operation become available when Elrod saw a description of the liquid-feedback Fluidyne and devised an elegantly simplified analysis of its principles . Elrod’s analysis vindicated Bake-Yarborough’s intuition. The theory has been subsequently extended to take account of loss and loading effects [7,8]. Several computer analyses have also been carried out and are included in the bibliography.
Other feedback systems have also been used or proposed, including systems in which the displacer is given, by one of several possible means, a rocking motion to mintain the amplitude of oscillation of the displacer liquid. Different configurations for the liquid columns have also been used, including a mUlticylinder arrangement of the Siemens type and a concentric machine in which one leg of the displacer U-tube forms an annulus around the other. Further details of these and other variants are given in Ref .
Although the Fluidyne is essentialby a Stirling engine (or at least has the same kind of volume variations), the use of liquid pistons gives an added freedom of design beyond that available with more conventional Stirling machines. The potential advantage is obvious in cost, simplicity, and maintenance requirements offered by pistons that must always fit their cylinders exactly, regardless of wear or manufacturing tolerances. It must be recognized, however, that the liquid pistons create or exaggerate effects that are absent or negligible in solid-piston engines. These include the effect (generally undesirable) of oscillating flow on viscous losses and on thermal leakage; the relative ease with which a desirable isothermalization of the cold cylinder can be introduced; the possibility (desirable or otherwise) of substantial evaporation in the hot cylinder; the undesirable limitation on stroke and frequency imposed by gravity-controlled oscillation and by the Rayleigh-Taylor instability of the surface; and the need to keep a more or less constant orientation of the engine so that gravity can hold the liquid in place. The major factors will be discussed in the next few sections.
Oscillating Flow Effects
The liquid flow rate in the displacer and tuning lines is not constant or even unidirectional; it varies approximately sinusoidally with two reversals in each cycle. For channels or tubes of the size normally used In a fluidyne, the behavior of the liquid, and especially the viscous drag, is greatly affected by this oscillation. In effect, the boundary layer never has time to develop fully hefore the flow reverses. Even in laminar flow conditions, the effect of the oscillitory nature of the flow is to anfine the velocity gradient in the liquid to a narrow region close to the wall . The thickness of this boundary layer is of the order sqrt(2 * ν / ω), where ν is the kinematic viscosity of the liquid and ω the angular frequency of the oscillation. For room temperature water oscillating at 1 Hz, this thickness is on the order of 1 mm.
There are two major consequences: First, the confinement of the velocity gradient to a narrow region means that within that region, the gradient will be higher than it would be if the shear stress were spread somewhat uniformly across the full width of the tube, as it is in normal Poiseuille flow. Therefore, the viscous flow losses are increased, on the other hand, the characteristic length for the Reynolds number of the oscillating flow is not the actual diameter of the tube, but the thickness of the boundary layer within which the shear effects are concentrated: this raises the flow velocity at which the onset of turbulence may be expected . Although the effect of turbulence on viscous flow losses in oscillating flow is not established, it is possible that once turbulence has set in, the distlnction between oscillating and unidirectional flow may be much reduced.
Although the influence of the oscillations on the kinetic or "minor pipe" losses is likewise unknown, it is reasonable to assume (in the absence of published experimeatal evidence to the contrary) that the correlations for unidirectional turbulent flow may be used. These correlations are established for conditions (turbulent flow) in which the time-averaged velocity profile in a straight tube section also shows a charactaristic narrow boundary layer near the wall and a fairly uniform velocity over the rest of the tube diameter.
Second, the oscillatory motion of the liquid also enhances, often by a very large factor , the conduction of heat along the liquid column. The magnitude of this undesirable effect depends, inter alia, on the ratio of thermal conductivity to the square root of kinematic viscosity. This ratio is rather large for water (almost 1000, in SI units, at WC, compared with only 10 for a typical oil, which Is unfortunate because water is often the liquid of choice for a fluidyne. The effect may be a major source of heat loss from the hot cylinder; although it can be greatly reduced by use of an insulating float on the water surface, such a float (if solid) makes it impractical to introduce extended surface area devices (such as fins or tubes) into the cylinder to isotheramlize the gas behavior.
Isothermalization and Transient Heat Transfer Losses
In an ideal Stirling cycle, all processes are isothermal. In practice, the gas in the cylinders, and sometimes in the connecting ducts as well, does not have time to exchange much heat with the walls during the course of a single cycle - that is, it behaves almost adiabatically. Consequently, the expansion and compression processes tend to lower and raise the gas temperature in these spaces (except in the regions very close to the wall, where there is a thermal boundary layer). Three major efficiency-loss effects can be attributed to this cause, although it should be realized that there are important interactions among the three.
First, heat flows across the temperature difference between the gas and the wall, which is an irreversible process. Although the net heat flow over a cycle will be zero, once the equilibrium condition has been reached, heat Is lost from the gas to the wall during the part of the cycle when the gas temperature is above the wall temperature and is returned to the gas during the part of the cycle when its temperature is lower. Giving up heat at a high temperature and regaining it at a lower temperature is obviously an inefficient process. This loss mechanism is often called the transient heat transfer loss, the hysteresis loss (because of its Importance in gas springs), or the cyclic heat-transfer loss. It was first recognized as a very important effect in the fluidyne by Ryden, whose report  was not however made public until 1983. For a truly adiabatic cylinder, the transient heat-transfer loss will be zero because, by definition, there is no heat exchange between the gas and the wall in an adiabatic space. On the other hand, neither will an ideally isothermal cylinder suffer from transient heat-transfer losses because there is, by definftion, no temperature difference between the gas and the wall in such a cylinder. It follows that there is a worst case somewhere in between the zero heat transfer coefficient and infinite heat transfer coefficient cases. In fact, Lee has shown  that the worst case will occur when the thermal boundary layer thickness is about equal to the hydraulic radius of the cylinder. The thermal boundary layer thickness is given by sqrt(2 * &alpha / ω), where α is the thermal diffusivity and ω the angular frequency of the oscillations. br 1 Hz oscillations in air at room temperature and pressure, the thickness calculated from the thermal diffusivity based on pure conduction is ~3 mm. Notice that the ratio of the flow boundary layer thickness to the thermal boundary layer thickness is u/a, the Prandtl number. For gases, the Prandtl number is of the order of 1, so that gas spaces large enough to show marked oscillating flow effects will tend to behave nearly adiabatically; nearly isothermal spaces, on the other hand, will not show very marked oscillating gas flow effects. In practice, the thermal diffusivity will be enhanced by convection and turbulence that will modify, and often increase, the transient heat-transfer loss.
By placing fins, tubes, or other area enhancements in the fluidyne cylinder, provided the spacing between them is less than a few millimeters, the gas behavior can be brought into the nearly isothermal regime. The transient heat-transfer loss may still be considerable; although the temperature fluctuations and therefore the heat transferred per unit area will be greatly reduced, this is offset by the increase in area. Nevertheless, as we shall see, the reduced temperature fluctuations favorably affect some other losses, and near-isothermalization is generally worthwhile if it can be achieved without an excessive increase in complexity or in flow losses. It is obviously easier with a liquid piston, where the liquid will invest the spaces between the fins regardless of tolerances, than with a solid piston, where close matching of moving and stationary fins would be needed.
The second major effect of the pressure-induced fluctuations in the cylinder gas temperature is that the mean gas temperature over the cycle is no longer equal to the temperature of the adjacent heat exchanger. During the compression phase when the gas temperature will be raised by adiabatic compression, most of the gas is In the cold compression cylinder. During the expansion phase when the gas temperature is falling, there is relatively little gas in the compression cylinder. Therefore, the mean temperature of the gas in the compression cylinder reflects the compressive heating more than the expansive cooling, and so the mean gas temperature in the compression cylinder is higher than that in the cooler. Similarly, the mean gas temperature In the expansion space is lower than the heater temperature. As a result, the effective temperature difference between the expansion and compression phases of the cycle Is less than the difference between the heater and cooler temperatures, so that efficiency calculations based on gas tempuatures l a the but exchangers will be overestimates. clearly, reducing the amplitude of the temperature fluctuation by partially isothermalizing the cylinders will usually reduce the efficiency loss due to this shift of mean temperature, although it may not decrease the transient heat transfer loss unless the isothermalization is very effective. The use of liquid pistons makes it easier to install attended surface area into the cylinder for this partial isothermalization.
The third affect, which can also be reduced by even partial isothermalization, arises from the instantaneous temperature difference between the heater (or cooler) and the gas in the adjacent cylinder. With ideal componants, gas leaving the heater and entering the expansion space will do so at a constant temperature - the heater wall temperature. The gas already within the cylinder will be at a different and usually lower temperature due to the pressure-induced temperature variations. There will therefore be a mixing of gas at two different temperatures - an ireversible process. Similarly, during the part of the cycle when gas is leaving the expansion space and entering the heater, it will generally do so at a temperature that is different from and, for most of the cycle, lower than the heater. Once again, heat is transmitted irreversibly across a finite temperature difference, leading to a loss of efffciency .
Effect of evaporatlon on Engine Performance
Unless the liquid In the expansion cylinder has a low vapor preosure at the operating temperature or lo osparated from the hot gas by some kind of insulating float, enough evaporation will take place from the liquid surface and wetted cylinder walls to modify substantially the composition and pressure of the working gas. Because the evaporation will naturally tend to take place mainly during the downstroke of the expansion piston when hot gas is entering the cylinder, it will raise the pressure during the expansion phase and thereby increaee the indicated power. Calculations have indicated that a several-fold increase may be attainable, and this is presumably the reason that small Fluidynes (which suffer from the large transient heat-transfer losses associated with the increased surface-to-volume ratio of small cylinders and the large flow losses associated with small-diameter tubes) will not work in the absence of evaporation . Much greater heat input is needed for the evaporative ("wet") cycle than for one in which evaporation is surpressed ("dry" Fluidyne). A conventional regenerator cannot recover much of this extra heat because the saturation temperature of the fluid is higher during the high pressure of the compression phase (when much of the heat stored in the regenerator should be returned to the workng fluid) than during the expansion phase, when the heat was stored in the regenerator. Without regeneration, an earlier study suggested that the brake efficiency of the evaporative cycle in a simple fluidyne may be limited to around 1x or 2x . However, with the careful control of the quantity and timing of the evaporation an indicated efficiency in the range of 3 to 9x at low temperatures (110°C to 130°C) may be achievable even without regeneration ; this may open the way to higher brake efficiency, moreover, a most ingenious proposal by Menfroe  that uses a hydrated-salt atorage medium In the regenerator has opened up the possibility of achieving at least some degree of effective regeneration and hence still higher efflciency in evaporative fluidynes.
At present, the general conclusions are that the theoretical efficiency advantage of the dry engine is likely ts be realized in practice only for larger mschines (with a cylinder diameter perhaps >SO mm) and that for small engines evaporation is essential for succeesful operation. Similarly, for operation across a small temperature difference - such as might be available from simple solar collectors - the greater indicated power output of the evaporative cycle may be needed. If the highest possible efficiency is the goal, then a large dry engine is needed.
As indicated earlier, transient heat transfer is a major source of loss in the fluidyne engine; if the expansion cylinder is filled with tubes or fins to isothermalize the space , then the liquid piston in that cylinder will be approximately the same temperature as the gas. Evaporation would then be an important effect unless a lower vapor pressure liquid were used in place of water or unless very low temperatures are used, but as already stated, it is difficult to achieve efficient low-temperature operation without evaporation.
Although the discussion here hes centered around machines in which the working fluid is either a permanent gas (usually air, although lower molecular weight gases have been used with good results) or a gas/vapor mixture, one could construct a fluidyne type of engine in which permanent gas is excluded from the working space so that the cycle depends entirely on steam or some other vapor . Apparently, no experimental results from such a machine have been published.
Frequency and stroke limitations
For the type of machine shown in Fig 1, the frequency of opeation is determined largely by the natural frequency of oscillation of the liquid in the displacer tube under the restraining force of gravity. This natural frequency is equal to sqrt(2 * g / L), where g is the acceleration due to gravity and L is the length of the liquid column. Some small demonstration models have been built with a displacer column as short as 100 mm, corresponding to a frequency of about 2 hz, but for larger engines the minimum length is limited by the diameter of the displacer tubing and the need to accomodate the stroke in each upright of the U-tube. For an engine with a 150 mm diameter displacer and a bore-to-stroke ratio of 1, it would be dlfficult to construct a displacer U-tube of much less than 1 m length (fig 2), corresponding to a frequency of only 0.7 Hz; in fact, both the engines of this size described in the literature [18,19] have a displacer somewhat larger than 1 m and operate at a frequency of ~0.55 hz.
There are ways to circumvent these limitations to a certain extent, such as using a displacer tube of nonuniform cross section or using a rocking-beam configuration in which the frequency is determined primarily by the motion of the displacer tube itself rather than the liquid column within it . A more fundamental approach is to increase the restoring force beyond that provided by gravity alone. In the multicylinder engine shown in Fig 3, each liquid column is subject to the restoring forces arising from the compression or expansion of gas in the adjacent cylinders as well as from gravity. Usually, these gas spring forces are much larger than the gravitationa1 ones, and so the resonant frequency Is much higher than it would be for the same column oscillatlng under gravity alone.
In a stationary liquid with a free surface, the direction of the acceleration due to gravity is from the dense fluid (liquid) toward the less dense
fluid (gas), and the interface is stable - as In a glass of water. In the last century, Lord Rayleigh showed that if the acceleration is in the opposite direction (e.g., if the glass Is rapidly accelerated downwards, or if it is turned upside down), the Interface is unstable and will break up - that is, small departures from a smooth flat surface till tend to grow. According to legend, the theory was reinvented by Taylor in the course of investigations into the problems encountered during WWII while trying to obtain a smooth and spherically symmetrlcal implosion of the core of an atomic bomb. In any case, it is now known as the Rayleigh-Taylor instability. For a sinusoidal movement of the liquid, the peak acceleration of the surface Is σ * ω / 2, where σ is the etroke and ω the angular frequency. If this exceeds the acceleration due to gravity, then the liquid surface will be subject to the Rayleigh-Taylor instability during the downstroke. For a 100 mm stroke, this will be the case for all frequencies higher than 2.2 Hz. This could pose a severe design constraint, espscially on a high-frequency multi-cylinder engine and on the output column of a liquid-feedback engine. (The output column is usually made from narrower tubing than the displacer to minimize its length, and it has a higher stroke so that the output column will reach the stability limit before the displacer column.) However, even beyond the limits of the Rayleigh-Taylor instability, the surface disturbances have a finite rate of growth, and the stability boundary can bs crossed without problems for at least a short time. Both the stability limit and the rate of growth are affected by surface tension and other effects that may be particularly important in narrow tubes; this may explain why Martini’s multi-cylinder heat pump fluldyne  has successfully operated with a frequency/stroke combination exceeding gravitational acceleration. This question has not been very deeply explored, especially not in a quantitative manner, although it is clear that some sort of limit must exist; it is, after all, well known that water will generally fall from a glass held upside down.
Having described some of the interesting theoretical aspects of the fluidyne engine, one may turn to the measured performance of some of the machines that have been built.
Table 1 lists the characteristics, in relation to the previous discussion, of several different engines along with the engine performance. This table includes only engines used for pumping. The wet mechines operate successfully even in small sizes, but no results have been published showing the performance of a large engine with evaporation. The three larger engines listed in the table all operate on a dry cycle, and although they show a higher efficiency, a higher operating temperature is needed. The effective cylinder temperature of the wet machines is not recorded, but it is presumably no greater than 100°C or so because there is usually no boiling during operation. All these machines had air as the permanent gas fraction of the working fluid.
Little has been published about the largest machine, designed and built by R. Pandey as part of a research program at the Metal Box Company of India; the program was ultimately aimed at developing coal-fired irrigation pumps. The engine is of concentric cylinder design with the tuning line constructed as a spiral on the base of the displacer (Fig 4 shows one possible layout of this type). The water pump is driven by the gas pressure variations in the
working space, a system known as gas-coupling. The Pandey engine operates at a mean gas pressure equal to atmspheric and at a fairly low heater temperature (375°C). Consequently, the specific output is low, even though the Beale -bar is 0.008, which ls quite respectable for such a low-temperature machine. A large, but simple, engine is therefore needed to generate the pumping power, which for a throughput of 4,400 US gal/h through a head of 12 ft is almost l/4 hp. No details of the later head design are available, and very little is known now about the basic dimensions of the engine. Nowever, it appears that the expansion cylinder, equipped with a float to minimize evaporation, may have a diameter of about 12 in. and the tuning line an effective diameter of 8 in. The operating frequency Is 0.625 Hz, Implying a displacer liquid column length of slightly more than 4 ft.
This author prepared an outline design, Including all major dimensions, for a pump intended to have a performance (80 gal/min through a head of 10 ft ) quite similar b Pandey’s machine. The design was prepared for the Briggs & Stratton Corporatlon, but it was never built; the company has kindly given their permission for its publication. Full details are available , but briefly the design proposes a U-tube displacer with an internal diameter of 13-l/2 in. and a tuning line internal diameter of 7-3/4 in. The heater is an annulus placed above the expansion cylinder and concentric with it. The regenerator matrix is 3 mm pitch aluminum honeycomb, and the cold cylinder ls filled with 5 mm pitch honeycomb for isothermalization. Evaporation and heat conduction from the expansion cylinder are minimlzed by means of a short lnsulation-filled float on the water surface.
The most obvious application for a liquid-piston engine is as a liquid pump, and most of the fluidyne research and development has been carried out with this in mind. Many people have thought of the use of solar heat or waste heat with this kind of engine, particularly for irrigation pumping. Small engines (15 mm diam or smaller displacer cylinders) have in fact been successfully operated from sunlight focused with an inexpensive plastic Fresnel lens of perhaps 300 cm2 area. In the quest for simplicity and reduced maintenance requirements, various fiuidyne pumps of 10 to 100 mm displacer diameter have been successfully operated with fluidlc valves, In a system having no solid moving parts (except, for a hot cylinder float in the 100 mm engine). For irrigation pumping, the simplest type of fiuldyne has a limitation; by choosing a mean working gas pressure equal to atmosphertc pressure, construction and operation are much simplified compared with a pressurized engine, but the pumping head is limited by the available pressure swing to no more than perhaps 15 ft. This is, of course, adequate for irrigation pumping in many areas of the world, but not in all. There, are ways to increase the pumping head by staging or by pressurizing the working fluid , but these methods have so far been little. explored.
Other pumping applications that have been proposed include water circulation in gas-fired hot water central heating systems (to reduce the dependence of gas-fired systems on electricity supplies), drainage pumping, domestic water supply in remote locations without benefit of utility electricity. combustion engine cooling with engine waste heat as the power source, and failsafe cooling of nuclear reactors after shutdown.
Ths mlticylinder configuration can, in principle, be oparated as a heat powered heat pump with some of the expansion gasses bsing heated externally and generating the power to set the system into oscillation. The remaining expansion cylinder(s) then act as the input side of a Stirling heat pump . For several years, there was a series of conflicting arguments about this scheme : some argued, on apparently impeccable grounds, that it could mrk; some argued, on apparently equally impeccable grounds, it could not. The immediate question was resolved by Martini, who built a small machine that worked . Martini’s small proof-of-principle mode1 does not give any guide to the ultimtely available performance , cost, or efficiency of these systems, but it does offer reassurance that the efficiency can be greater than zero.
As noted earlier, a fluldyne driven by external gas pressure pulsations would operate as a refrigerator, but it appears that no calculations, much less experiments, have been carried out on such a system.
Finally, it may be remarked that the fluidyne incorporates, in a working heat engine simple enough to be put together in the smallest laboratory or workshop, many of the important basic lessons of classical physics - including single- and two-phase thermdynamics, kinematics, the behavior of tuned oscillators, and fluid flow. It is therefore a good teaching and learning example for high school snd college students; many successful student projects on the fluldyne are known to the author, and pesumably there are others that have not been communicated,
The liquid-piston fluidyne is a form of Stirling engine sharing many of the characteristics of conventional kinematic and free-piston Stirling machines, The use of liquid pistons, however, gives it some unique advantages as well as certain problems that are not encountered or are not important in engines with solid pistons. Because the output is naturally available in the form of an osclllatlng liquid flow or a fluctuating pressure, the fluldyne is well-suited to liquid pumping, but other applications have also been considered.
ANALYTICAL STUDY OF LIQUID PISTON BEAT PUMP TECHNOLOGY, ANTI Interim Report to Argonne National Laboratory under Contract No. 31-109-38-6506 (1982)
DYNAMICS OF A PISTONLESS RNGINE CAPABLE OF UTILIZING SOLAR ENERGY, S c Bhargava et al, J Appl Phys 49(6), 3521-3526 (1978)
PASSIVE SOLAR WATER PUMP, G C Bell, Independent Project Arch 589, [University of New Mexico] 1979
SOLAR POWERED LIQUID PISTON STIRLING CYCLE IRRIGATION PUMP, G C Bell, [Sandia Laboratory] SAN-1894/l, 1979
THE FLUIDYNE ENGINE, A J. Bell and L F Goldberg, Final-year laboratory report, [University of the Witwatersrand] 1976
TEST REPORT OF TWO FLUID PISTON HEAT ENGINES, D Outler and C Hanke, [Chicago Bridge and Iron Co], CBI Res contract R-0268, 1979
AN INITIAL MODEL FOR THE FINITE DISPLACEMENT RESPONSE CHARACTERISTICS OF A FLUIDYNE PUMP, T M Drzewiecki, WL-TR-1868, 1979
A STUDY OF A FLUIDYNE HEAT ENGINE, R Nndlay and P Hook, Thesis, Department of Mechanical Engineering, [McGill University] Montreal, Canada, 1977
THE ONSET OF OSCILLATIONS IN A LOSSLESS FLUIDYNE, D J Ryden, AERE-M 2840, 1976
THE MATHEMATICAL MODELLING OF A JET-STREAM FLUIDYNE, P F Gill, RNEC-SERF-Fl-80, 1980
A PROTOTYPE LIQUID-PISTON FREE-DISPLACER STIRLING ENGINE, L F Goldberg and C J Rallis, Paper No. 799239, Proceedings of the 14th IECEC, 1979
SW EXPERIMENTAL RESULTS ON LABORATORY MODEL FLUIDYNE ENGINES, L F Goldberg et al, Paper No. 779255, Proceedings of the 12th IECEC, 1977
ANALOG SIMULATION OF A FLUIDYNE ENGINE, M Gosling and D Boast, BSc Project Report 382 [University of Bath] 1976
ASSESSMENT OF A HARD-COUPLED FLUIDYNE PUMP, T W Hensman, RNEC-SERF-F2-1980, 1980
THE DESIGN, MANUFACTURE, OPERATION AND PRELIMINARY TESTLNG OF A LIQUID PISTON; FREE DISPLACER, BACK-TO-BACK, GAMMA TYPE STIRLING ENGINE, S W Leigh, P C Lohrman, and R Swedlow, Final-year laboratory project [University of the Witwatersrand], 1979
OPERATION OF A JET-STREAM FEEDBACK FLUIDYNE, P D Lewis, RNEC-TR-78008, 1978
THE PUMPING CHARACTERISTICS OF A JET-STREAM FEEDBACK FLUIDYNE PUMP, S J Lloyd and T W Hensman, RNEC-SERG-4, 1979
EXPERIMENTAL AND COMPUTATIONAL EVALUATIONS OF ISOTHERTHERMALIZED STIRLING ENGINES, W R Martini, S G Hauser, and M W Martini, Paper 779250, Proceedings of the 12th IECEC, 1977
DESIGN STUDY OF ISOTHERMALIZED STIRLING J%GINES, W. R Martinl, Final draft report to Argonne National Laboratory under ANL contract 31-109-38-5304, 1980
REVIEW OF LIQUID PISTON PUMPS AND THEIR OPERATION WITH SOLAR ENERGY, C L Murphy, ASME-79-Sol-4, pp 2-8, 1979
THE FLUIDYNE - A NEW CLASS OF HEAT ENGINE, G T Reader et al, Paper Nr 19, Polytechnic Symp Thermodyn Heat Transfer, Leicester, England, 1979
MODELLING TM JET-STREAM FLUIDYNE, G T Reader et al, Paper No. 819792, Proceedings of the 16th IECEC, 1981
MODES OF OPERATION OF A JET-STREAM FLUIDYNE, G T Reader and P D Lewis, Paper No. 799238, Proceedings of the 14th IECEC, 1979
THE FLUIDYNE - A WATER IN GLASS HEAT ENGINE, G T Reader and P D Lewis, JNS, 5, (4) 240-245, 1979
STIRLING ENGINES, G T Reader and C Hooper, Spon, 1983
INVESTIGATIONS OF A PHYSICAL SYSTEM CAPABLE OF USING SOLAR ENERGY, R Rup, J Physics D - Applied Physics, 11, (18) L208-L209 1978
THE FLUID MECHANICS OF THE JET-STREAM FLUIDYNE, J R Singleton, RNEC-SERG-S-79, 1979
DEVELOPMENT OF THE INSTRUMENTATION TO ENABLE AN EXPERIMENTAL ANALYSIS TO BE MADE OF A JET-STREAM FEEDBACK FLUIDYNE, G C Thwaites, RNEC-SERG-5-79, 1979
C D West, E H Cooke-Yarborough and J C H Geisow, “Improvements in or Relating to Stirling Cycle Heat Engines”, British Patent 1 329 567, filed 1970 (describes basic fluidyne concept and numerous different configurations; also gives some experimental results)
C D West, “Improvements in or Relating to Stirling Cycle Heat Engine”, British Patant 1 487 332, filed 1974 (describes rocking beam fluidyne - drlven by bellows from working gas pressure variations)
C D West, “Improvements in or Relating to Stirling Cycle Heat Engines”, British Patent 1 507 678, filed 1974 (describes multicylinder fluidyne engines and refrigerators; applications also filed in India, Egypt and Kuwait)
C D West and J C H Geisow, “Improvements in or Relating to Stirling Cycle Engines”, British Patent 1 568 057, filed 1975 (describes isothermalizer techniques, cooling and heating through the liquid medium, use of a float to reduce evaporation in the hot cylinder, and design of an internal pump for cooling; applications also filed in India, Egypt and Kuwait)
C D West, J C H Geisow and R B. Pandey, “Improvements in or Relating to Stirling Cycle Heat Engines”, British Patent 1 581 748, filed 1976 (describes the concentric cylinder layout, folded tuning line, multicylinder concentric layout, and application of multicylinder liquid piston configuration to heat activated heat pumping; applications also filed in India and Kuwait)
C D West, J C H, Geisow and R B Pandey, “Improvements in or Relating to Stirling Cycle Heat Engines”, British Patent 1 581 749, filed 1977 (describes means for pressurizing the working fluid in a fluidyne pump; Applications also filed in India, Egypt, Kuwait, and Nigeria)
J Gerstmann and Y Friedman, “Liquid Piston Heat Actuated Heat Pump and Methods of Operating Same”, US Patent 4,148,195, filed 1977 (describes multicylinder, liquid-piston heat-actuated heat pump, and methods of switching between summer and winter operation)
E Franklin, “Improvements in or Relating to Pumps”, British Patent 21072278, filed 1979 (describes a fluidyne in which the working fluid is vapor only, with no permanent gas content)