A U-Shaped Oscillatory Liquid Piston Compression Air Conditioner Driven by Rotary Displacer Stirling Engine

: A rotary displacer, Stirling engine-driven, oscillatory liquid piston-compression air conditioning unit is analyzed, designed, and conceptually proved by experiments on its cooling capability and performance e ﬃ ciency. The engine itself aims to work compatibly with a conventional solar water heater as its power source, with a rotary displacer to manipulate its operation frequency. A U-shaped, oscillatory liquid piston, with the cylinder chamber vented to low pressure, aims to both reduce the sealing friction and serve as the refrigerant. During the experiments, the engine is fed with an electrically heated hot bath whose temperature proﬁle is determined according to a ﬁeld test of solar thermal collectors in summer conditions. Experimental results show that the coe ﬃ cient of performance (COP) of the oscillatory liquid piston air conditioning unit would be in the range of 1.3 to 1.5.


Introduction
Solar heating of buildings is the main reason for high electricity consumption for air conditioning in tropical and subtropical regions, thus, making solar cooling an especially attractive idea for application in these areas. Normally, solar air conditioning could be accomplished by three means: absorption cycle, desiccant cycle, and solar mechanical cycle [1]. The solar mechanical cycle might run on two main kinds of energy supply: electricity from a photovoltaic system or mechanical energy from a solar thermal engine. A solar thermal engine would quite often be an external combustion engine, due to its natural compatibility with all kinds of thermal energy as input. In particular, the Stirling engine has been receiving considerable attention in the solar mechanical application field.
A Stirling engine, operating on a closed regenerative thermodynamic cycle, with cyclic compression and expansion of the working fluid at different temperature levels, as described by Walker [2], relies on internal volume variation caused by repeated heating and cooling of the working fluid and its changing position to produce work output. Being able to approach the ideal cycle efficiency, i.e., the equivalent Carnot cycle efficiency, at designated temperatures with a suitably designed regenerator, along with other features such as low emissions and quietness, a Stirling engine would have great potential for application in residential spaces.
Generally, Stirling engines can be categorized into three types of configuration [3] as follows: 1. α-type: this type of configuration features two pistons, each in its own cylinder; 2.
β-type: this type of configuration has a piston and a displacer in the same cylinder; 3.
γ-type: this type of configuration has a piston and a displacer, each in its own cylinder. The displacer chamber is composed of upper and lower plates, and additional fin arrays are installed to increase the heat transfer efficiency. A cylindrical side wall with relatively lower heat conductivity ensures that the energy flow will pass through the working fluid rather than the wall itself. The bulk body of the displacer rotates within the chamber. To control the heat transfer sequence, the end plates should be covered with a heat insulation layer over 240° of their surface, leaving a 120° arc where the fin array is located for heat exchange. The two fin arrays should phase to each other by 60° intervals, as illustrated in Figure 2, and the displacer bulk should have a 60° notch to contain the working fluid, as shown in Figure 3.  The displacer chamber is composed of upper and lower plates, and additional fin arrays are installed to increase the heat transfer efficiency. A cylindrical side wall with relatively lower heat conductivity ensures that the energy flow will pass through the working fluid rather than the wall itself. The bulk body of the displacer rotates within the chamber. To control the heat transfer sequence, the end plates should be covered with a heat insulation layer over 240 • of their surface, leaving a 120 • arc where the fin array is located for heat exchange. The two fin arrays should phase to each other by 60 • intervals, as illustrated in Figure 2, and the displacer bulk should have a 60 • notch to contain the working fluid, as shown in Figure 3. The displacer chamber is composed of upper and lower plates, and additional fin arrays are installed to increase the heat transfer efficiency. A cylindrical side wall with relatively lower heat conductivity ensures that the energy flow will pass through the working fluid rather than the wall itself. The bulk body of the displacer rotates within the chamber. To control the heat transfer sequence, the end plates should be covered with a heat insulation layer over 240° of their surface, leaving a 120° arc where the fin array is located for heat exchange. The two fin arrays should phase to each other by 60° intervals, as illustrated in Figure 2, and the displacer bulk should have a 60° notch to contain the working fluid, as shown in Figure 3.    The liquid piston is housed in a U-shaped tube, which is connected to the displacer chamber at one end and sealed tight at the other. As shown in Figure 4, the sealed end may have two heat exchange surfaces, with the upper one as the condenser and the lower one as the evaporator. The liquid piston oscillates within the tube, while the liquid head should stay in the region between the condenser and the evaporator. As shown in Figure 5, when the liquid piston head goes down, the chamber above the piston head at the seal end is enlarged, causing a sudden pressure drop to encourage evaporation of the liquid at the contact surface of the liquid piston and the evaporator section. The liquid piston is housed in a U-shaped tube, which is connected to the displacer chamber at one end and sealed tight at the other. As shown in Figure 4, the sealed end may have two heat exchange surfaces, with the upper one as the condenser and the lower one as the evaporator. The liquid piston oscillates within the tube, while the liquid head should stay in the region between the condenser and the evaporator. As shown in Figure 5, when the liquid piston head goes down, the chamber above the piston head at the seal end is enlarged, causing a sudden pressure drop to encourage evaporation of the liquid at the contact surface of the liquid piston and the evaporator section.  The liquid piston is housed in a U-shaped tube, which is connected to the displacer chamber at one end and sealed tight at the other. As shown in Figure 4, the sealed end may have two heat exchange surfaces, with the upper one as the condenser and the lower one as the evaporator. The liquid piston oscillates within the tube, while the liquid head should stay in the region between the condenser and the evaporator. As shown in Figure 5, when the liquid piston head goes down, the chamber above the piston head at the seal end is enlarged, causing a sudden pressure drop to encourage evaporation of the liquid at the contact surface of the liquid piston and the evaporator section.   Though a small amount of heat absorption in the vapor housing in the upward chamber may reduce the heat absorption effect, this is considered relatively unimportant. When the liquid piston head goes up, the chamber above then decreases in volume, causing a pressure increase to encourage condensation, which thus, rejects the previously absorbed heat out of the conditioning unit. Then, with the cyclic oscillation behavior of the liquid piston, heat absorption and rejection take place alternately, thus, forming a typical piston compression cycle.

Rotary Displacer Analysis
According to Naotsugu Isshiki [4], the variation of the heat transfer area within a rotary displacer chamber, and even the temperature and pressure profiles, can be approximately simplified into sinusoidal functions. With the nature of the rotary displacer described above, a sinusoidal approximation of the real contact surface area , d real A , whose magnitude is denoted as d A , between the displacer cover plates and the gas within is shown in Figure 6. The effective contact area which renders as: This would suggest a heat flow profile as follows: where d Q is the heat transfer amount of the displacer gas; g T is the displacer chamber gas temperature; , d wall T is the contact area wall temperature; ω is the reference rotational velocity, Though a small amount of heat absorption in the vapor housing in the upward chamber may reduce the heat absorption effect, this is considered relatively unimportant. When the liquid piston head goes up, the chamber above then decreases in volume, causing a pressure increase to encourage condensation, which thus, rejects the previously absorbed heat out of the conditioning unit. Then, with the cyclic oscillation behavior of the liquid piston, heat absorption and rejection take place alternately, thus, forming a typical piston compression cycle.

Rotary Displacer Analysis
According to Naotsugu Isshiki [4], the variation of the heat transfer area within a rotary displacer chamber, and even the temperature and pressure profiles, can be approximately simplified into sinusoidal functions. With the nature of the rotary displacer described above, a sinusoidal approximation of the real contact surface area A d,real , whose magnitude is denoted as A d , between the displacer cover plates and the gas within is shown in Figure 6. The effective contact area A d,real is introduced as a sinusoidal function A d,e with a magnitude of A max , such that the time integral of the contact area would give the same value as the actual case. Thus, A d,e is defined as follows: which renders as: This would suggest a heat flow profile as follows: .
where Q d is the heat transfer amount of the displacer gas; T g is the displacer chamber gas temperature; T d,wall is the contact area wall temperature; ω is the reference rotational velocity, with t the designated time; h d is the convection coefficient between the displacer chamber gas and the contact area. Assuming the displacer chamber gas undergoes the ideal gas behavior, the energy conservation of gas within the displacer chamber would lead to: where U g is the internal energy of the displacer chamber gas, whose mass is m g ; the work output term is m g M R dT g dt , with R as the gas constant; M is the molecular mass of the chamber gas, and its specific heat at constant volume c v , if the volume variation is relatively small. This gives the relation of the temperature of the displacer chamber gas T g and T d,wall as follows: and where T H and T c are the temperatures of the hot end and cold end, respectively. Then, the pressure of the displacer chamber gas, P d , can be assumed as: where P d,0 and T g,0 are the initial pressure and temperature of the chamber gas, with k = c p /c v , the ratio of specific heat at constant pressure to that at constant volume.
where g U is the internal energy of the displacer chamber gas, whose mass is g m ; the work output term is and where H T and c T are the temperatures of the hot end and cold end, respectively.
Then, the pressure of the displacer chamber gas, d P , can be assumed as: where ,0 d P and ,0 g T are the initial pressure and temperature of the chamber gas, with p v k c c = , the ratio of specific heat at constant pressure to that at constant volume.

U-Tube Oscillatory Liquid Piston Air Conditioning Unit Analysis
According to Zhang [6], and defining x as the displacement of the liquid piston with the upward end at the right-hand side of the tube as positive, then, for the U-shaped tube oscillatory system shown in Figure 7, the momentum equation would be: the initial condition p is shear stress; ρ is the density of the liquid; d is the diameter of the tube; P p is the pressure in the chamber above the piston; P d is the pressure in the displacer; L is the liquid column length. system shown in Figure 7, the momentum equation would be: where p ϒ is shear stress; ρ is the density of the liquid; d is the diameter of the tube; p P is the pressure in the chamber above the piston; d P is the pressure in the displacer; L is the liquid column length. With the Hagen-Poiseuille law, it could be rearranged as: where ν is the effective kinetic viscosity of the liquid.
With sufficiently long operation time, the liquid piston should behave like a simple harmonic motion system, prompting a displacement of the liquid head as follows: where φ is the initial phase difference between the rotary displacer and the liquid piston phases; α is the vibration magnitude of the liquid piston head. This would render the pressure difference term as: With the Hagen-Poiseuille law, it could be rearranged as: ..
where ν is the effective kinetic viscosity of the liquid. With sufficiently long operation time, the liquid piston should behave like a simple harmonic motion system, prompting a displacement of the liquid head as follows: where φ is the initial phase difference between the rotary displacer and the liquid piston phases; α is the vibration magnitude of the liquid piston head. This would render the pressure difference term as: where As a fair approximation, with a delicate setting of the phase angle, the pressure difference between the piston chamber and the displacer is: The momentum equation is: ..
which renders the solution of the piston head as: This can be simplified as where Π and Θ are constants determined by the initial condition of vibration. The first term accounts for the effect of the initial vibration condition, by which energy put into the system is eventually dissipated through the damping force, and as the operating time increases, it may become more and more negligible. After a long operating period, then, the motion of the liquid piston is reduced to: which reconfirms the suggestion from Naotsugu Isshiki [4,5].
As to the vapor slug in the upper chamber on the top of the liquid piston, its ideal gas behavior gives: where L p0 is the original length of the vapor slug in the upper chamber above the piston; m p is the mass of the vapor slug; T p is the temperature of the vapor slug. By differentiation with respect to time, this becomes: Considering the first law of thermodynamics, the related energy equation of the vapor is: which can be further revised into: This suggests an interpretation of the vapor mass m p above the liquid piston as: where C is the integration constant. Its temperature would be readily obtained as: As the initial conditions may easily be set to T p0 and P p0 , the integration constant C 1 is simply: With P p defined in Equation (15), the temperature of the piston chamber gas is readily obtained from Equation (27).

System Performance Definition
During the experiment, the heat source for the displacer was simulated by an electrically heated hot water bath #1 to simulate the hot water from a solar thermal collector, with its temperature set as the maximum value of the temperature profile acquired from field tests of the performance of solar thermal collectors in summer conditions. The waste heat of the displacer was ejected into another water bath (#2) simulating room temperature water, which would be preheated and go into the hot water tank of the solar water heating system. The two heat exchange surfaces of the U-tube were subjected to water baths at room temperature, in which bath #3 represents the indoor space where the conditioning power would be of use and bath #4 stands for the outdoor space where the heat would be ejected. Each water bath was covered by a heat insulation layer to reduce heat loss. The arrangement of the water baths was as illustrated in Figure 8.
The performance of the system was evaluated in two parts: the efficiency of the Stirling engine and the coefficient of performance of the conditioning unit (COP).
The efficiency of the engine is defined as: where H use f ul is the available work to power the liquid piston; H in is the electrical work to keep the temperature in bath #1 around the initial condition; H waste is the energy gained in bath #2.
thermal collectors in summer conditions. The waste heat of the displacer was ejected into another water bath (#2) simulating room temperature water, which would be preheated and go into the hot water tank of the solar water heating system. The two heat exchange surfaces of the U-tube were subjected to water baths at room temperature, in which bath #3 represents the indoor space where the conditioning power would be of use and bath #4 stands for the outdoor space where the heat would be ejected. Each water bath was covered by a heat insulation layer to reduce heat loss. The arrangement of the water baths was as illustrated in Figure 8. The performance of the system was evaluated in two parts: the efficiency of the Stirling engine and the coefficient of performance of the conditioning unit (COP).
The efficiency of the engine is defined as: where useful H is the available work to power the liquid piston; in H is the electrical work to keep the temperature in bath #1 around the initial condition; waste H is the energy gained in bath #2.
The conditioning unit coefficient of performance could be defined as: The conditioning unit coefficient of performance could be defined as: where H removal is measured by the heat loss from bath #3, which is attached to the evaporator of the conditioning unit, and the work input W input is calculated as:

Experiment Settings
The whole experiment setup was as shown in Figure 8, along with the components of the Stirling engine and the oscillatory liquid conditioning unit. The whole inner system was initially vented to a low pressure around 2.3 kPa, which enabled the liquid water to boil at approximately 20 • C. Each section exposed to the atmosphere was covered with a heat insulation layer to reduce interference from outside heat sources, while only the heat exchanging sections would be in contact with the associated water baths. It should be mentioned that the topic of this paper is the prototype design of an oscillatory liquid piston conditioning unit powered by a Stirling engine containing rotary displacer with its cooling capability and compatibility with a conventional solar water heating system. Moreover, in the proposed design, the circumferential fin in rotary displacer as well as the heat absorption and rejection in bath #5 and #6, respectively, have the function of heat exchange through periodic variety of temperatures. The water loading of each water bath is listed in Table 1. The electrical heater for bath #1 was set at 200 W, which was able to keep the water temperature in bath #1 at around 85 • C throughout the experiment, which was set as the average maximum temperature according to the field study of solar thermal collectors. The water temperature in bath #2 was deliberately kept around 27 • C initially, as the same tests suggested that it was the average water temperature before it entered the solar collector if the water supply pipe was embedded within the concrete wall of the building, exposed to the outer atmosphere. The water temperatures in the other two baths were also kept around 27 • C initially to simulate the averaged shielded outdoor and indoor temperature for a room subject to indirect solar emission in a subtropical region, which was also suggested by the tests. The experiment was conducted with a rotary displacer revolving at 0.01, 0.03, 0.05, and 0.1 Hz.

Experiment Results and Discussions
The experimental results suggest that a rotary displacer may serve the purpose effectively, causing a fluctuating variation in the displacer chamber gas temperature profile T g , throughout the testing range of the operating frequencies. While the heat source, bath #1, was kept at around 85 • C and the heat sink, bath #2, rose from 27 • C to around 40 • C, the temperature of the displacer chamber gas T g tended to fluctuate by a 15 • C magnitude around the mean temperature between the two baths, which is quite close to the value expected from the prediction of Equation (5) with T d,wall , using the experimental readings of the contact surface temperature.
Though the illustration in Figure 9 only shows the result of 0.01 Hz operation, the same tendency is shared in operations at other frequencies. It is a fair guess that for a large enough concentric residential application, due to its constant dispensing of hot water and resupply of cold water, the preheated water, which is intended to serve as the displacer heat sink, may mostly stay at the same temperature. The average value of theoretical prediction and experimental readings of Tg are, respectively, 59.30 and 58.13 • C. The difference of theoretical prediction and experimental readings between the beginning and ending values of Tg are, respectively, 27.91 and 33.25 • C.
A smaller scale application may not always have fresh cold water undergoing the preheating process, so the water from bath #2 would not be returned to bath #1 like the regeneration process used in the present experiments.
Due to the pipe arrangement and the insulation, the motion of the water heat of the liquid piston may not be directly visible. However, judging from the reading of the fin array at the condenser section, as in the example of 0.01 Hz operation shown in Figure 10, along with the theoretical prediction from Equation (27), the temperature T p should indeed be a cyclic pressure and temperature change within the chamber above the liquid piston, causing heat ejection into water bath #4, as with a conventional piston compressor. The deviation between the theoretical estimate of T p and the experimental reading may come from all kinds of loss, especially the friction between the viscous liquid piston and the wall. The average value of theoretical prediction of Tp and condense fin array are, respectively, 29.70 and 29.51 • C. The difference of theoretical prediction of Tp and condense fin array between the beginning and ending values of Tg are, respectively, 3.27 and 3.07 • C. A smaller scale application may not always have fresh cold water undergoing the preheating process, so the water from bath #2 would not be returned to bath #1 like the regeneration process used in the present experiments.
Due to the pipe arrangement and the insulation, the motion of the water heat of the liquid piston may not be directly visible. However, judging from the reading of the fin array at the condenser section, as in the example of 0.01 Hz operation shown in Figure 10, along with the theoretical prediction from Equation (27), the temperature p T should indeed be a cyclic pressure and temperature change within the chamber above the liquid piston, causing heat ejection into water bath #4, as with a conventional piston compressor. The deviation between the theoretical estimate of   η . The engine efficiency tended to stay around 5.5%, while the ideal cycle efficiency at the designated temperature conditions might be around 14%, and the waste heat in bath #2 was not yet returned to heating bath #1 to fulfill the regeneration process due to experimental needs.   Table 2 summarizes the system performance under different operating frequencies. The performance of the Stirling engine is evaluated by the efficiency, η e f f . The engine efficiency tended to stay around 5.5%, while the ideal cycle efficiency at the designated temperature conditions might be around 14%, and the waste heat in bath #2 was not yet returned to heating bath #1 to fulfill the regeneration process due to experimental needs. The cooling capability of the oscillatory liquid piston conditioning unit is evaluated by the temperature drop in bath #3, which represents the indoor space cooled, and the COP of the oscillatory liquid piston air conditioning unit. The COP seemed to rise as the operation frequency increased, but not proportionally. Since the average temperature drops, normalized for 100 s, still tend to rise as the frequency goes higher, the increase in COP may not merely be due to the longer operation time, implying that, although the amount of heat transferred in a single cycle may be lower at higher operating frequency than at lower, it may be compensated by the increased number of cycles. It should be mentioned that, as the example shown in Figure 10, we take 0.01 Hz operation along with the theoretical prediction. This is because the target area will oscillate with the temperature of each cycle; we average the temperature of each cycle to observe the rising trend of the temperature.

Conclusions
An oscillatory liquid piston conditioning unit powered by a Stirling engine with a rotary displacer was designed, prototyped, and tested for its cooling capability and its compatibility with a conventional solar water heating system, using a simulated heat source. The rotary displacer mechanism with the arc fin array conceptually proved operational for the Stirling engine, capable of easy tuning to different operating frequencies to suit the demands of the location.
A further investigation of the potential to operate at higher temperature differences may help to boost the performance. The liquid piston conditioning unit with its COP of 1.3~1.5 could operate like a conventional piston compressor cooler whose COP normally ranges from 1 to 4. However, featuring fewer components and a flexible geometry arrangement, the oscillatory liquid piston conditioning unit may help to reduce maintenance demands.