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NSF GRANT # 0838874 NSF PROGRAM NAME: Control Systems

Dynamic Modeling of a Regenerator for the Control-Based Design of Free-Piston Stirling Engines Eric J. Barth P.I. Department of Mechanical Engineering Vanderbilt University, Nashville, TN37212 Mark Hofacker Graduate Student Research Assistant Department of Mechanical Engineering Vanderbilt University, Nashville, TN37212 Abstract: This paper describes a new approach to modeling and designing free-piston Stirling engines with the goal of building a working prototype. The Stirling cycle is recast as a dynamic system where control design tools and techniques can be applied to determine optimal manufacturable parameters for the engine. A simplified state-space analysis exhibits the merits of this perspective by revealing a link between pole locations and power production. The results of this analysis suggest the potential of a new, compact free-piston Stirling engine configuration using elastomeric pistons. An engine of this type is described analytically with a physics-based, high-order, lumpedparameter, dynamic model. To support one component of the model, a regenerative heat exchanger is described using a lumped parameter, nth-order model generalizable to an arbitrary number of sections within the regenerator dependent on its physical aspect ratio. Finally, the heat exchanger model is experimentally verified. 1. Introduction: Stirling engines provide clean, reliable, mechanical power when provided only with a temperature gradient. Unlike combustion engines, Stirling engines do not require a distilate fuel like gasoline and can therefore run on heat from any source such as geothermal, solar, biomass or nuclear energy. The Stirling cycle operates by shuttling a compressed gas between two chambers separated by a lightweight piston. The engine cycle is driven by the transport of heat across a static temperature difference on either side of the chamber. One of these chambers is connected to a larger “power piston” which transmits power by moving out of phase from the smaller piston. A “free piston” Stirling engine is a Stirling engine in which the two pistons are connected through dynamics as opposed to a kinematic linkage seen in more traditional

kinematic Striling engines. Free piston Stirling engines offer low noise, low maintenance operation at scales and power outputs that rival more traditional internal combustion engines. [1] Stirling engines are theoretically capable of Carnot efficiency, but are not used in mainstream today because they have typically been heavy and inefficient compared to their alternatives. To understand why this has been the case in the past, and why a new approach holds the potential to improve their performance, a brief description of the Stirling cycle is in order. As seen in Fig. 1, the displacer piston in a Stirling engine serves to transport heat carried in a compressible fluid from a heat source on one side of the engine to a sink on the other. When the displacer piston is nearer to the bottom, more of the gas is on the hot side of the engine, and the pressure inside of the engine increases. When the displacer is nearer to the top, the majority of the gas is cooled and the pressure decreases. Adding to these effects is the movement of the power piston which interacts directly with the displacer via some amount of damping or spring force, or indirectly by via pressure. By delicately balancing the area and masses of the pistons, the dynamic relationship between the pistons, the mass flow restriction from one side to the other, the heat transfer, and the load dynamics, a self sustaining cycle can be obtained to transform heat into useful work that is extracted from the power piston.

Proceedings of 2009 NSF Engineering Research and Innovation Conference, Honolulu, Hawaii

Grant #0838874

Figure 1: Example schematic diagram of a freepiston Stirling cycle engine with a linear alternator for energy extraction and a damping mechanism of interaction between the displacer and power piston. Most Stirling engines use a regenerative heat exchanger or simply “regenerator” to increase efficiency. The regenerator works as a thermal capacitor where heat is absorbed and released from the gas as it passes from one chamber to the other. This heat transfer occurs cyclically at the operating frequency of the engine. Traditionally, Stirling engines were designed using a purely kinematic and thermodynamic approach, but these approaches are insufficient and inappropriate because the relationship between analytical design and physical performance currently remains inadequately characterized to the point of reducing most design efforts to a process of trial and error. Often times in the past, engines with designs that appeared feasible on paper would not run when built. To show the value of dynamic modeling and analysis for free-piston Stirling engines, it is useful to perform a simplified examination of a Sunpower B-10B free-piston Stirling demonstrator shown in Fig. 2. The engine weighs just under a kilogram and produces approximately 1 watt of power. The nomenclature used in this diagram and used in the following equations is described as follows: A Ar b bd bp g kd kp ld lp

displacer area displacer rod area damping coefficient between displacer rod and piston damping coefficient between displacer and wall damping coefficient between piston and wall acceleration of gravity spring constant for displacer spring constant for piston equilibrium length of displacer spring equilibrium length of piston spring

Figure 2: Picture and schematic of Sunpower B-10B free-piston Stirling demonstrator md mp P Patm P Te Tc Ve Ve0 Vc Vc0 Vr x x x y y y

mass of displacer mass of piston pressure in working gas atmospheric pressure initial pressure in working gas temperature in hot side temperature in cold side volume in hot side initial volume in hot side volume in cold side initial volume in cold side volume in regenerator displacer position displacer velocity displacer acceleration piston position piston velocity piston acceleration

2. Motivation for control-based approach to freepiston Stirling engines: A simplified dynamic model of the Sunpower engine suggests that there is a correlation between the locations of the closed loop poles in the right half plane and the operating characteristics of an engine. Performing a balance of forces on the displacer piston and the power piston shown in Fig 2. yields the following equations of motion:

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md x   PA  P A  Ar   Patm Ar  kd x  ld   bd x  bx  y   md g

m p y   P A  Ar   Patm Ar  k p  y  l p 



 bd y  b y  x   m p g

x  Ax  Bu

Accounting for initial conditions and linking the piston positions to the corresponding volumes yields the following:

Ve  Ve 0  Ax


Vc  Vc 0   A  Ar x   A  Ar  y


The commonly accepted method for describing the temperature dependent pressure in the engine, as described by Schmidt [2], permits the following equation:

 V V V lnTh / Tk  Vh Ve    Pˆ  mR  c  k  r Th  Tk  Vh Th   Tk Tk



where Ve , Vh , Vr , Vk , and Vc is the volume of the hot, heater, regenerator, cooler and cold spaces respectively; Te  Th , Th , Tr , Tk , and Tc  Tk are the temperatures in each of those spaces, m is the mass of gas, and R is the gas constant. For the device considered, volumes Vh , Vr , and Vk are small compared to Ve and Vc and can be neglected. Linearizing Equation (5) about an equilibrium pressure P0 and equilibrium volumes Vc 0 , and combining with Equations (3) and (4) yields the following expression,

P  P   C A  C A  C A x  C  A  A y 0



design problem, thereby offering a constructive controlbased design methodology for the closed-loop dynamics of the system. The linearized system dynamics of equations (1), (2), and (6) can be represented in state-space as:






where C1 and C 2 represent constants dependent on operating temperatures and volumes: V V  C1  mR e 0  c 0   Th Tk 


V V  C2  mR e 0  c 0   Th Tk 


1   T   h


1  T  k


   

It is important to note that Equation (6) can be interpreted as an input term to the coupled system dynamics of Equations (1) and (2). It indicates that the pressure developed above or below the equilibrium pressure is a function of the positions of the piston and displacer. Further, this “input term” is a function of two of the states of the system, x and y, and are effectively feedback gains which can be altered by manipulating the parameters of the system. This interpretation serves to cast the problem as a standard feedback control

 x   0  x  kd     md  y   0   0  y 

1 ( bd  b ) md

0 0



b mp

kd mp

0  x  0      Ar  b md   x    md u 1  y  0   A Ar  ( b p  b )    mp    y   m p 


with an input term given by the state-dependent feedback equation: u  Kx


u  ( P  P0 )  x  x   C1 A  C 2 ( A  Ar ) 0 C 2 ( A  Ar ) 0   y    y 

This state space representation was used in combination with the following estimated system parameters: A = 10 cm2,

Ar = 1.58,

mp=465 g,

bp = 10

kd = 450

N m,

md=75 g,

N s m ,

kp = 577.5

N m,

bd = 10

N s m ,

Tk = 300 K,

Th = 600 K The location of the closed loop poles under these conditions reveals valuable insight into the operation of a Stirling engine viewed as a dynamic system subject to a state-dependent physical feedback law. As shown below in Fig. 3., two of the closed loop poles are in the right-half plane, and their location corresponds to the operating frequency of the engine. This interpretation reveals that a free-piston Stirling engine produces power and maintains self-sustained oscillation if the system has unstable closed-loop poles. This linearized interpretation is valid about a small operating region of the equilibrium values, whereas the real nonlinear dynamics will limit the amplitude of oscillation as the states depart far enough from the equilibrium.

Proceedings of 2009 NSF Engineering Research and Innovation Conference, Honolulu, Hawaii

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as a design tool. Essentially control tools can be utilized to design the feedback gains of the system which, in turn, relate directly to physical parameters serving as design variables. As an illustration of this, the following engine configuration was found to be unstable with reasonable physical parameters and will be referenced throughout the paper:

Figure 3: Closed-loop poles of the Sunpower B-10B are located at 7.6±80j and -85±9.3j.

Operating Frequency

Imaginary Axis

Using logical tests such as changing the operating temperatures or piston damping reveals that there is also a connection between the depth of the poles into the right half plane and the power output of the engine. Indeed, if the temperature gradient is too small, or the damping is too large, the poles enter the left half plane and results in an engine that does not run. Conversely, increasing the temperature makes the poles more unstable and results in an engine that produces more power. It is therefore possible to predict both operating frequency and power production using the closed loop poles of a linearized dynamic model of a free-piston Stirling Engine as shown below in Figure 4.

Power Output

Real Axis Figure 4: Qualitative diagram showing the relationship between closed loop poles and engine operation

Although the precise locations and number of poles vary, this correlation between the locations of the unstable poles and engine performance holds true even when higher order models of the engine are used. Moreover, the closed-loop pole locations can be used to

Figure 5: Schematic of liquid piston free-piston Stirling engine

The displacer piston and power piston shown are liquid pistons formed by trapping water between elastomeric membranes. The damping and spring elements of the system are compactly contained within the elasticity of the membrane material. Heat is added to the working fluid through the walls surrounding the expansion space and it is rejected from the walls surrounding the compression space. Power is extracted by way of PV work to the right of the power piston. This engine configuration was chosen because it would be easy to manufacture, highly practical, and is novel. These advantages are primarily due to the elastomeric liquid pistons (acting as free pistons) which do not require tight tolerances, have long fatigue life, and have low damping. There are no recorded attempts using liquid pistons in this configuration known to the authors. This is most likely because free-piston Stirling engines typically have a connecting spring or damper between the pistons which would be difficult to reproduce with liquid pistons. Preliminary dynamic analysis shows that the proper balancing of the piston size and regenerator characteristics would eliminate this connection. More generally, the system dynamic and control related perspective taken here provides the appropriate insight regarding the operation of such non-traditional configurations and allows new configurations to be attempted with a higher degree of knowledge of their operation than before. 3. Advances in Dynamic Modeling: Given the goal of maximizing power output, an engineer would strive to

Proceedings of 2009 NSF Engineering Research and Innovation Conference, Honolulu, Hawaii

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drive the closed loop poles as far into the right half plane as possible. To effectively use control tools to increase this “instability robustness” therefore requires an accurate dynamic model of the engine’s behavior. The previous analysis, and work by Riofrio [3] made several simplifying assumptions that are useful in capturing the operating frequency of an engine, but detract from its utility as a design tool and its physical approximation of engine behavior. Both build upon the work of Schmidt [2] which assumes that the compression and expansion chambers remain at equal pressure throughout engine operation. Making this assumption requires one to disregard significant factors such as flow restrictions in the regenerator. More useful design tools must account for these factors, and therefore a higher fidelity model is needed. Conveniently, the free-piston Stirling engine is well suited to dynamic modeling because its operation is determined solely by dynamics as opposed to cams or kinetic linkages seen in traditional Stirling engines, and it is not subject to discontinuities such as combustion pulses seen in IC engines. A simple free-piston Stirling engine such as the one shown in Fig. 5 consists of five different components: the displacer piston, the power piston, the compression space including the cooler pipe, the expansion space including the heating pipe, and the regenerator. The displacer and power pistons are wellsuited to system dynamic analysis because together they form fourth-order mass spring damper system as shown in Fig 6.

Figure 6: Schematic of Stirling engines pistons as spring mass damper system.

The expansion and compression spaces that exert pressure on the pistons can be well described by performing a power balance on their respective control volumes as shown in prior work [4]. The energy rate in a control volume of ideal gas can be expressed as: U j  H j  Q j  W j (11) where U describes the rate of change of internal energy inside of the control volume, H is the net

enthalpy flow into the CV, Q is the heat flow into the control volume, and j is the subscript index used to identify a particular control volume . These four terms can be expanded in the following way:

U j 

1 P jV j  PjV j  j 1

H j   m j c p

in / out

 T 


in / out

Q j  h j Aj T wall  j  T j





W j  PjVj


where P , T , and V are the pressure, temperature, and volume of the control volume. The variables c pin / out and Tin / out are the specific heat at constant pressure and the temperature of an individual mass flow m entering or leaving the CV, and γ is the ratio of specific heats. The temperature, surface area, and heat transfer coefficient of the control volume wall are T wall , A , and h respectively. By rearranging Eqs. (12) through (15) the following equation is obtained: Pj 

 m c j

pin / out

 T  j

in / out


 PjVj  h j Aj T wall  j  T j Vj


Although the piston and control volume behavior is well understood, there is no precedent for lumped parameter dynamic modeling a regenerator. The regenerator is not well suited for lumped parameter modeling due to the pressure and temperature gradient contained within it. 4. Regenerators in free piston Stirling engines: The regenerator, also known as a regenerative heat exchanger, is a thermal reservoir that alternately absorbs and releases heat to the working fluid. It is a sealed chamber that is connected on one side to the expansion chamber and on the other to the compression chamber. It generally contains some kind of matrix material such as steel or glass wool to store thermal energy. Although the expansion and compression chambers can be directly connected, entirely omitting the regenerator, many Stirling engines will not operate without one, and even those engines that can function without one perform more efficiently with a regenerator [5].

In ideal operation, hot gas enters the regenerator from the compression side at a uniform high temperature Thigh, deposits its heat to the matrix material, and exits to the expansion space at a uniform Tlow. The cycle is then reversed where the gas enters at Tlow from the

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expansion side, the energy from the matrix is restored to the gas, and leaves into the compression side at a uniform Thigh. A schematic showing the operation of this idealized regenerator is shown below in Fig. 7.

Figure 7: Schematic of an ideal Stirling engine regenerator and boundary conditions

This ideal regenerator operation is difficult or impossible to achieve in a real machine because the requisite device would have to satisfy multiple conflicting requirements. The volume inside the regenerator does not contribute to piston displacement and therefore is considered dead volume. An effective regenerator must quickly absorb large amounts of heat from the working fluid and quickly dissipate that heat. Therefore, a regenerator must have high thermal capacity and as much surface area as possible for heat transfer. While a small, dense regenerator is desirable for minimizing dead volume, reducing the size of the regenerator also reduces flow. In moving gas from one chamber to another, the associated pressure drop results in a parasitic fluid friction loss. The ideal regenerator would cause no pressure drop, have zero volume, and transfer heat instantaneously. In order to actually build an engine, it must be accepted that the regenerator will always be less than ideal and must be modeled accordingly [1]. There is very little available theoretical assistance in modeling these departures from ideal operation [6]. The traditional thermodynamics approach to heating and cooling processes operate under the assumption that boundary conditions are steady state or slowly fluctuating. These models are well established and can precisely predict the temperatures in steady state operating conditions, but in practice the regenerator operates between two control volumes that are fluctuating at the operating frequency of the engine. A system dynamics approach is therefore necessary to describe the boundary conditions of the regenerator. In this instance, Eq. (16) can be used to succinctly describe the dynamic nature of the pressure in the

expansion and compression spaces that surround the regenerator. This kind of state space representation lends itself well to lumped parameter estimation, but there is no established method for describing the significant temperature gradient that exists across the regenerator [5]. This gradient is not necessarily undesirable but regardless, it cannot be avoided because the regenerator must physically connect the heat source and heat sink. If the end point temperatures of the regenerator matrix during steady state operation were known, it would not be difficult to reproduce its behavior using simple models, but such models would offer little help in designing an engine given that end point temperatures are results of dynamic engine conditions rather than parameters of a given design. Indeed, work by Willmott [7] has shown that the internal states of a heat exchanger can be accurately predicted in rapidly changing conditions if the frequency, temperature, and pressure are known, but this technique is not useful for our purposes if there are mutual dependencies between the regenerator and its environment. As is true with most aspects of the Stirling engine, the development and range of this temperature gradient depends on the behavior of the rest of the engine. A useful model must therefore have a gradient that is developed and influenced by engine operation. 4.2 Dynamic Model of the Regenerator: Assuming that the regenerator is well insulated, a black box approach would only require that the model provide the mass flow rate and temperature of the inlet and exit gas for given pressure boundary conditions. Although there is some interdependency between the mass flow and temperatures of the internal gas and matrix materials, it is assumed that the mass is described by lumping the distributed mass flow restriction into one location.

The mass flow rate in and out of the regenerator depends on the upstream and downstream pressure and upstream temperature. According to Eq. (11), the sign convention is such that mass flow rate is declared positive for mass entering any given control volume (CV). This mass flow rate relationship is given as: m j  a j j Pu , Pd , Tu  P  Cd C f 1 u  Tu    1 u  P P P  C d C f 2 u  d  1   d  Tu  Pu   Pu

  


Pd  Pcr Pu


Pd  Pcr Pu

 u 1  u


where Cd is a nondimensional discharge coefficient of the valve, a is the effective area of the orifice, Pu and

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Pd are the upstream and downstream pressures, Tu is the upstream temperature,  u is the ratio of specific

heats of the upstream substance, and C f 1 , C f 2 and Pcr are substance-specific constants given by

Cf1 

u 

2    Ru   u  1 

Cf2 

 u 1  u 1


Q j  k j ASA j Tr  j  Tv  j

k j ASA j Tv  Tr

m  c 




Ru  u  1  u  u 1

 2   (20) Pcr      u 1 where Ru is the gas constant of the upstream substance.

The regenerator is modeled as having two distinct components: a control volume space representing the regenerator void space and the matrix material. This representation is shown below in Fig. 8.


where Tr and ASA are the temperature and surface area of the matrix material. The matrix itself is modeled as a thermal mass that is driven by the temperature difference between it and the gas inside the void: Tr 

2 u







where mr and c pr are the mass and heat capacity of the matrix material. Combining the enthalpy (Eq. 1) and heat transfer (Eq. 14 and Eq. 22) into Eq. 21 shows a more complete picture of the energy storage rate, and consequently the gas temperature, inside of the regenerator void Tj 

 m c  T   m c  T  m c  j

pin / out


in / out









 k j Tv  Tr j


 (24)


It may also be important to model the temperature gradient throughout both the gas and matrix material within the regenerator with some degree of spatial resolution, depending on the regenerator’s aspect ratio. This can be accomplished by establishing lumpedparameter discrete temperatures within the control volume and matrix material. Gas leaves each chamber at the temperature of the given control volume and enters at the temperature of the upstream volume. Gas inside of the chambers exchanges heat with its adjacent matrix material. The matrix materials in turn exchange heat with each other. These interactions are shown below in Fig. 9. Figure 8: Schematic of Stirling engine regenerator and boundary conditions

Equation 12 can be alternatively expressed as Eq. 21 and is useful in describing the regenerator void space as a control volume,

 

U j  m j c v  j TV  j  m j c v  j TV



where TV , is the temperature of the control volume cv is the constant volume specific heat of the gas inside the control volume. The only power flow to the gas in the regenerator, under the assumption that the control volume is of constant volume and adiabatic, is enthalpy associated with mass flow in or out of the space and heat transfer between the gas and the matrix material. The heat transfer inside of the regenerator has significant convective and conductive components that cannot be easily separated [8]. Their combined effects will be designated as a combined heat transfer coefficient “k”, as shown in Eq. 22:

Figure 9: Multiple discrete spatial section model of regenerator

The heat transfer between the sections of the matrix occurs through conduction. Accounting for the multiple sections of matrix material, Eq. 23 becomes:

k j ASA j Tv j  Tr j  Tr j 

k r j AX j Tr j 1  Tr j 1  2Tr j

mr  j c p


2 * Lr j



where k r j is the conductive heat transfer coefficient of the matrix material and Lr is the length of each section.

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Temperatures Tr and j 1

Trj 1 are the upstream and

downstream matrix temperatures surrounding Tr . j

4.3 Motivation for a Graphite Regenerator: Steel and glass wool are the most common materials used in regenerators, but these are not ideal candidates for characterization because of their unorganized nature; an organized matrix can provide more surface area with less flow restriction. Graphite was chosen as a matrix material because it is available as uniform cylinders which could be arranged to provide a relatively uninhibited flow path. Additionally, given that graphite has a lower coefficient of thermal expansion than steel or other metals, its geometry changes less and hence the flow path changes less throughout the range of possible engine temperatures. In our experiments, the amount of flow restriction imposed by the regenerator changed by a negligible amount while testing across a range of temperatures.

Graphite has no history as a regenerator matrix, but has found widespread application as a heat exchanger. Companies such as Apex and Fareast sell graphite heat exchangers for industrial applications and show excellent heat transfer characteristics. A regenerator is very similar in function to a heat exchanger except the gas from which the heat is taken is also one to which the heat is reabsorbed. Graphite heat exchangers have demonstrated heat transfer coefficients over two orders of magnitude greater than current metallic designs [9]. They are capable of temperatures in excess of 300Ëš C, have high thermal efficiency and are currently in use in some thermal power plants [10]. Like regenerators, the study of heat exchangers has been largely confined to the study of thermodynamics and has not received very much attention from a system dynamics perspective. Regenerators and heat exchangers share almost identical goals such as low weight and pressure drop and high rates of heat transfer. This would appear to indicate that the because graphite is an excellent medium for heat transfer in a heat exchanger it would also work well as a matrix regenerator provided that the heat capacity is high enough. As shown below in Fig. 10, graphite has a higher heat capacity than other matrix material candidates such as steel, aluminum, or glass.

Figure 10: Thermal conductivity and heat capacity for candidate regenerator materials

There are currently no Stirling engines that utilize a graphite matrix regenerator, providing an opportunity to characterize a novel device that could improve current Stirling Engines. 4.4 Experimental validation and characterization: To characterize a regenerator and validate the model, the experimental setup shown in Figs. 11 and 12 was constructed.

Figure 11: schematic of experimental apparatus

Proceedings of 2009 NSF Engineering Research and Innovation Conference, Honolulu, Hawaii

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In future applications the regenerator housing would also be built of graphite with an insulting outer coating to aid in heat storage and transfer, but for testing it was desirable that that the heat was stored primarily in a matrix of uniform graphite rods. Utilizing ABS, it was hoped that a negligible amount of heat would be lost through the ABS walls due to their differences in thermal conductivity (24.0 vs. 1.96 W/m-K). This however was not the case and the heat loss to the ABS was measureable and therefore modeled.

Figure 12: Picture of experimental apparatus

A mass flow controller was connected to a high pressure source to provide air to the arrangement. Two solenoid valves were arranged so that air could be directed either into a heater or directly into the regenenerator. This was necessary so that the air flowing into the regenerator could change temperature without the need for high temperature valves. Flow restrictions were placed on either side of the regenerator so that the fluctuations in mass flow could be calculated instantaneously given the known geometries and the measured pressure drops and temperatures using the Eqs. 17-20. The mass flow controller was used to help find the effective discharge coefficient of the lumped parameter flow restriction. The temperature was measured using a series of small diameter (40 AWG), exposed junction thermocouples capable of a 50ms temperature measurement response time. Graphite cylinders were oriented longitudinally within the regenerator in order to minimize drag and maximize convective heat transfer surface area. The regenerator housing was built of ABS plastic due to its low thermal conductivity. A diagram of this regenerator is shown in Fig. 13

Figure 13: Computer generated diagram of regenerator

It is difficult to artificially reproduce the conditions that a regenerator would see as part of a larger Stirling engine because the rapid fluctuations in bi-directional mass flow that would occur in a running engine are difficult to create and measure. For testing, mass flow was held constant at 10 standard liters per minute and the incoming temperature was manipulated by means of the electronic valves shown in Fig. 12. During the test shown in Figs. 14 and 15, hot air was directed into the control volume for 400 seconds starting at 50 seconds. Sharp transitions in temperature were difficult to achieve because heat was stored in and transferred from the pipe connecting the source gas to the regenerator. Known physical and thermal properties of the graphite, air, and ABS plastic were used in the simulation. The combined heat transfer coefficients “k” were found by trial and error through different tests. These values can be seen below in Table 1. Table 1: Physical and thermal properties used in simulation Material


ABS Plastic

Combined Heat Transfer Coefficient “k” (W/m-K)



Thermal Conductivity (W/m/K)



Mass (kg)



Surface Area (m^2)



Specific Heat Capacity (J/g/K)



Figure 14: Inlet Temperature compared to Exit Temperature and simulated Exit Temperatrue

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The simulated exit temperature shown in Fig 14., was genenerated using Eqs. 24 and 25 three times in series to produce a regenerator model of three discrete control volumes and thermal matrix masses. Reducing the order of the model produced a less accurate models of exit gas temperature as shown in Fig. 15.

Figure 15: Actual regenerator exit temperature vs. Model output

Testing occurred over long periods of time to reduce the effect of unknown initial conditions. Although long amounts of time elapse between changes in incoming and exiting temperature, this is of a similar time scale as the observed regenerator exit gas temperature behavior. In essence, the regenerator is a low-pass filter to inlet temperature changes. This agrees with the observation that Stirling engines must be warmed up for several minutes to allow for the regenerator to heat up before they can operate. The heat transfer coefficients found experimentally would be difficult to formulate analytically because they combine the effects of conduction and convection which are themselves highly dependent on geometry and flow conditions. Testing a regenerator with this method and modeled as described in this paper would give a designer a clear picture of how the regenerator would behave as part of a Stirling engine. With this knowledge one could intelligently design the pistons and control volumes to maximize engine performance. The ability to use established values of thermal properties to roughly describe engines behavior would be highly useful in the design of an engine because it would allow for a minimum uncertainty when designing a regenerator. This appears possible because combined heat transfer coefficient “k” of the graphite was found to be 50 W/m-K which is close to its value of thermal conductivity. This is encouraging because it suggests that convective heat transfer rates on the matrix material can be mostly ignored. Since testing occurred at the maximum flow rate expected for this size of a regenerator integrated into a free-piston

Stirling engine, the effect of convective heat transfer would be even less significant for lower flow rates. 5. Conclusions: A control-based approach holds promise for helping design new and potentially useful free piston Stirling engine configurations through the placement of unstable closed loop poles. Critical to the success of this approach is an accurate dynamic model with parameters that can be reproduced physically. As one of the critical components of this overall dynamic engine model, a regenerator was modeled and experimentally verified. 6. References: [1] Walker, G., Stirling Engines, 1st ed., Oxford University Press, New York, Chap. 7, pp. 140151, 1980. [2] Urieli, I., and Berchowitz, D., “Linear Dynamics of Free Piston Stirling Engines,” IMECE, pp. 203213, 1985. [3] Riofrio, J., Khalid, A., Hofacker, M., and Barth, E., “Control-Based Design of Free-Piston Stirling Engines” Proceedings of the 2008 American Controls Conference (ACC), WeC09.4, June, pp. 1533-1538, 2008. [4] Riofrio, J., and Barth, E., “Design and Analysis of a Resonating Free Liquid-Piston Engine Compressor” Proceedings of the 2007 International Mechanical Engineering Congress and Exposition (IMECE), WeC09.4, November, pp. 1533-1538, 2007. [5] Senft, R., Ringbom Stirling Engines, 1st ed, Oxford University Press, New York, pp. 143, 1993. [6] Walker, G., Reader, G., Fauvel, O., and Bingham, E., The Stirling Alternative, 1st ed., Gordon and Breach, 1400 Yverdon, Switzerland, Chap. 1, pp. 51-52, 1994. [7] Willmot, J., Dynamics of Regenerative Heat Transfer, Taylor & Francis, New York, 2002. [8] Oldson, J.C., Knowles, T.R., Rauch, J., “Pulsed Single Blow Regenerator Testing” IECEC, August, 1992. [9] Klett, J., and Conway, B., “Thermal Management Solutions Utilizing High Thermal Conductivity Graphite Foams”. Bridging the Centuries with SAMPE’s Materials and Processes Technology, Vol. 45, 2000. [10] Shah, R., Thonon, B., Benforado, D., “Opportunities for heat exchanger applications in environmental systems”. Applied Thermal Engineering, May, pp. 641, 1999.

Proceedings of 2009 NSF Engineering Research and Innovation Conference, Honolulu, Hawaii

Grant #0838874

Dynamic Modeling of a Regenerator for the Control-Based Design of Free-Piston Stirling Engines  

Control modeling of regenerator