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Applied Energy

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Stirling engine regenerators: How to attain over 95% regeneratoreffectiveness with sub-regenerators and thermal mass ratios

Anders S. Nielsen, Brayden T. York, Brendan D. MacDonald⁎

Faculty of Engineering and Applied Science, Ontario Tech University (UOIT), 2000 Simcoe Street North, Oshawa, ON L1G 0C5, Canada

H I G H L I G H T S

• Develop heat transfer model to predict temperature in regenerator and working fluid.

• Derive expressions for regenerator effectiveness and Stirling engine efficiency.

• Require minimum of 19 sub-regenerators to attain 95% regenerator effectiveness.

• Uniform sub-regenerator thermal mass ratio distribution maximizes effectiveness.

• Experimentally validate heat transfer model.

A R T I C L E I N F O

Keywords:Stirling engineRegeneratorEfficiencyEffectivenessHeat transferSub-regenerators

A B S T R A C T

A combined theoretical and experimental approach is used to determine how to achieve a desired value for theStirling engine regenerator effectiveness. A discrete one-dimensional heat transfer model is developed to de-termine which parameters influence the effectiveness of Stirling engine regenerators and quantify how theyinfluence it. The regenerator thermal mass ratio and number of sub-regenerators were found to be the twoparameters that influence the regenerator effectiveness, and the use of multiple sub-regenerators is shown toproduce a linear temperature distribution within a regenerator, which enables the effectiveness to be increasedabove 50%. It is shown that increasing the regenerator thermal mass ratio and number of sub-regeneratorsresults in an increase in regenerator effectiveness and a corresponding increase in the Stirling engine efficiency.A minimum of 19 sub-regenerators are required to attain a regenerator effectiveness of 95%. Experiments va-lidated the heat transfer model, and demonstrated that stacking sub-regenerators, such as wire meshes, providessufficient thermal resistance to generate a temperature distribution throughout the regenerator. This is the firststudy to determine how Stirling engine designers can attain a desired value for the regenerator effectivenessand/or a desired value for the Stirling engine efficiency by selecting appropriate values of regenerator thermalmass ratio and number of sub-regenerators.

1. Introduction

Stirling engines have a high potential to alleviate some of the in-creasing global demand for clean energy. This potential is due in part tothe fact that Stirling engines are external combustion engines that cantherefore be powered by a wide range of heat sources, including sus-tainable options such as solar thermal, waste heat, and biofuels [1].Additionally, Stirling engines have the ability to attain the highesttheoretical engine efficiency, primarily due to the use of a regenerator,which exchanges thermal energy with the working fluid after eachisothermal expansion and compression stroke, thus minimizing theamount of thermal energy the heater and cooler must either add or

reject, respectively. Accordingly, regenerator design and developmenthas received much attention in past research, and has been a focal pointin the process of enhancing Stirling engine performance and efficiency.Characteristics of an effective regenerator are a high thermal storageand heat transfer capacity, a limited resultant pressure drop, and asufficient temperature distribution to ensure that the working fluid al-ternates between target temperatures, TH and TL. It was recently es-tablished by Dai et al. [2] that in order for a regenerator to attain aregenerator effectiveness higher than 50%, it is required that the re-generator be divided into discrete thermally isolated components, re-ferred to as sub-regenerators. Interesting questions that arise in thedesign of regenerators are: how does the regenerator thermal mass ratio

https://doi.org/10.1016/j.apenergy.2019.113557Received 8 March 2019; Received in revised form 3 July 2019; Accepted 15 July 2019

⁎ Corresponding author.E-mail address: brendan.macdonald@uoit.ca (B.D. MacDonald).

Applied Energy 253 (2019) 113557

0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

T

and number of sub-regenerators influence the effectiveness of re-generators? How can one design a regenerator to attain 95% effec-tiveness or higher? Can parallel plates or other continuous regeneratorsattain higher than 50% effectiveness? Does stacking wire meshes re-duce the regenerator to a single sub-regenerator and limit its effec-tiveness to a maximum of 50%? In order to further improve the designand our understanding of Stirling engine regenerators, a comprehensiveinvestigation to determine the influence of the regenerator thermalmass ratio and number of sub-regenerators on the effectiveness of re-generators is required to answer these questions.

In past literature, significant effort has been invested in quantifyingthe thermal and flow characteristics of various types of regenerators.Experiments have implemented steady flow conditions to quantify thepressure drop [3] and heat transfer characteristics [4] in wire meshregenerators, while other experiments have imposed oscillatory flowconditions to measure the pressure drop and heat transfer rate in wiremesh regenerators [5,6], parallel-plate regenerators [7], and spongemetal regenerators [8]. Combined experimental and numerical studieshave quantified the pressure drop [9,10] and heat transfer rate [11] inwire mesh regenerators subjected to oscillatory flow conditions. Otherexperiments have investigated the influence of the geometry [12] andthe type of material used in regenerators on the performance of Stirlingengines, including copper and stainless steel [13], and aluminum andMonel 400 [14,15]. Numerical studies have developed Nusselt number[16,17] and friction factor [18,19] correlations for wire mesh re-generators, while other numerical studies have developed similar cor-relations for micro-channel regenerators [20], segmented involute-foilregenerators [21], and circular miniature-channel regenerators using3D CFD [22,23] and Reynolds-Averaged Navier-Stokes (RANS) model-ling [24]. Other numerical studies have examined the entropy

generation in wire mesh regenerators [25] and micro-channel re-generators [26]. Past analytical studies have developed heat transfermodels for counterflow regenerators in Stirling engine applications toexamine the influence of the flush ratio [27,28] and the number oftransfer units (NTU) [29] on regenerator effectiveness. Optimizationanalyses have varied the regenerator’s fill factor [30], wire diameter[31,32], length [33], and mesh size [34] to maximize Stirling engineperformance and efficiency. Investigations into the performance ofStirling engines using an isothermal model [35,36] and a finite timethermodynamic model [37] have varied the effectiveness of the re-generator without discussing what parameters would alter the re-generator effectiveness. Many studies have also assumed perfect re-generation when analyzing Stirling engines at maximum powerconditions [38,39], when comparing the net work output betweenStirling and Ericsson engines [40], and when assessing the efficiencyreduction in Stirling engines due to sinusoidal motion [41]. The pre-vious studies have played a vital role in the advancement of Stirlingengine technologies and have elucidated much about the effect of re-generators on engine performance, but it is still not understood how theregenerator thermal mass ratio and number of sub-regenerators influ-ence the effectiveness of regenerators, and how to design regeneratorsto attain a desired effectiveness.

There have been limited studies that have examined the influence ofeither the number of sub-regenerators or regenerator thermal massratio on regenerator effectiveness or Stirling engine performance andefficiency, and no study has investigated them in tandem. Dai et al. [2]theoretically showed that a regenerator must have an unevenly dis-tributed temperature to attain an effectiveness greater than 50%, andthat an unevenly distributed temperature can be accomplished using anumber of smaller heat reservoirs to divide the regenerator into sub-

Nomenclature

Symbols

tΔ time step (s)TΔ reg regenerator temperature change (K)

Nu Nusselt number (–)Pr Prandtl number (–)Re Reynolds number (–)A regenerator surface area (m2)b Nusselt number correlation exponent (–)c specific heat capacity (J/kg K)C1 correction factor (Reynolds number) (–)C2 correction factor (Sub-regenerators) (–)cv constant volume specific heat capacity (J/kg K)D regenerator wire diameter (m)h heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)m mass (kg)N number of sub-regenerators (–)n number of regenerator passes (–)Qin heat addition (J)Qnon reg- heat not transferred by regenerator (J)Qout heat rejection (J)R working fluid ideal gas constant (J/kg K)r regenerator radius (m)T temperature (K)t time (s)TH heat source temperature (K)T T/H L target temperature ratio (–)TL heat sink temperature (K)treg engine speed regeneration time (s)Tsys H, hot system settling temperature (K)

Tsys L, cold system settling temperature (K)Tsys system temperature (K)V regenerator volume (m3)Vmax maximum volume of working fluid (m3)Vmin minimum volume of working fluid (m3)

Greek Letters

∊reg regenerator effectiveness (–)ηCarnot Carnot efficiency (–)ηStirling Stirling engine efficiency (–)γ regenerator thermal mass ratio (–)λ compression ratio (–)ω Stirling engine rotational speed (rpm)ρ regenerator material density (kg/m3)σ standard deviation of γ distribution (–)τ time constant for regeneration (s)ζ regeneration time ratio (–)

Superscripts

1 first sub-regeneratorN last sub-regeneratorp current time step

+p 1 successive time step

Subscripts

avg average propertiesf working fluidfinal final stateinitial initial stater regenerator

A.S. Nielsen, et al. Applied Energy 253 (2019) 113557

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regenerators. A study conducted by Jones [42] examined the effects ofthe regenerator thermal mass ratio on the ineffectiveness of the re-generator and power loss factor of the Stirling engine. The work foundthat regenerator thermal mass ratios between 20:1 and 40:1 can in-fluence the performance of Stirling engines, and was not concernedwith recommending an ideal value for the regenerator thermal massratio. Zarinchang et al. [43] examined the pressure drop and enthalpyflux through a wire mesh regenerator and indirectly found that, fortheir system, the optimum value for the regenerator thermal mass ratiowas 14.9:1. The GPU-3 and 4L23 Stirling engines [44,45] had re-generator thermal mass ratios in the range of 205:1–235:1, and theGPU-3 engine used 308 stacked wire mesh sub-regenerators as its re-generator. Such high values of regenerator thermal mass ratios andnumber of sub-regenerators require a considerable amount of re-generator material, which inhibits the flow of the working fluid andleads to higher pressure drops, ultimately reducing the efficiency andperformance of Stirling engines. Consequently, it is important to de-termine what regenerator thermal mass ratio and quantity of sub-re-generators are required to complete the necessary heat exchange toattain a high effectiveness, while avoiding the use of unnecessarily highvalues that cause increased pressure drops. These studies demonstratethat the regenerator thermal mass ratio and quantity of sub-re-generators have a significant impact on the effectiveness of re-generators and Stirling engines, but to date, there has not been anyresearch that has determined the regenerator thermal mass ratio andnumber of sub-regenerators required to attain specific regenerator ef-fectiveness values without employing unnecessarily high values of eachand generating unnecessary pressure drops. The understanding of howto select these values will lead to enhanced regenerator designs capableof high effectiveness and low pressure drops.

In this paper, a discrete one-dimensional transient heat transfermodel of a Stirling engine regenerator was developed to track thethermal response of the working fluid and regenerator, simultaneously.A parametric analysis of the transient model determined that only theregenerator thermal mass ratio influences the settling temperature ofthe regenerator and working fluid, and that the remaining transientparameters can be neglected. Accordingly, a heat transfer model wasdeveloped to quantify the settling temperature of the regenerator andworking fluid, and expressions for the regenerator effectiveness andStirling engine efficiency were derived to incorporate the influence ofthe regenerator thermal mass ratio and number of sub-regenerators. Itwas found that increasing the number of sub-regenerators and re-generator thermal mass ratio increases regenerator effectiveness andStirling engine efficiency, and it was determined that a uniform sub-regenerator mass distribution provides maximum regenerator

effectiveness. Experiments were conducted, and the experimental re-sults validated the heat transfer model. The experiments also demon-strated that stacking sub-regenerators, such as wire meshes, providessufficient thermal resistance to generate a temperature distribution. Asa result of this study, Stirling engine designers can select a value ofregenerator thermal mass ratio and number of sub-regenerators to at-tain a specific regenerator effectiveness and/or a specific value of theStirling engine efficiency.

2. Mathematical modelling

2.1. Transient heat transfer model of a single regenerator

In order to determine the thermal behaviour of a regenerator in aStirling engine we first consider a single regenerator (one sub-re-generator) along with the working fluid and model the two isochoric(constant volume) regeneration processes of the Stirling cycle. Duringthese processes, energy is exchanged between the working fluid and theStirling engine’s regenerator, which is shown in Fig. 1, and their tem-peratures change conversely to one another. This means that an in-cremental increase of energy in one of the mediums must be balancedby the decrease of energy in the other, which is expressed as:

− − = −+ +m c T T m c T T( ) ( )r r rp

rp

f v f fp

fp1

,1

(1)

where superscript p denotes the current time step, and +p 1 denotesthe successive step. The rate of energy change must also be balanced,such that the rate of heat transfer in the regenerator is equal to theconvection at the surface of the regenerator. We assume local thermalequilibrium within the regenerator material and working fluid, thusneglecting local conduction heat transfer within each medium, resultingin:

= −ρVc dTdt

hA T T( )rr

f r (2)

For thermal equilibrium to be valid for the working fluid wouldrequire sufficient mixing and small length scales, which we expect inStirling engine regenerators. To ensure that the thermal equilibriumapproximation is valid for the solid regenerator material, the followingcondition must be satisfied:

<hrk2

0.1r (3)

That is, the Biot number must be below the threshold value of 0.1[46]. The largest expected value of the Biot number for the work ana-lyzed in this paper corresponds to the regenerator material with the

Fig. 1. Schematic of a Stirling engine with a regenerator comprised of five sub-regenerators, a hot piston, cold piston, heater, and cooler.

A.S. Nielsen, et al. Applied Energy 253 (2019) 113557

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lowest thermal conductivity (stainless steel: =k 16.2r W/mK), the lar-gest radius ( =r 0.04 mm), and the highest heat transfer coefficient( =h 18, 000 W/m2 K), which results in a value of 0.022. Since this isbelow the threshold value, the approximation will be valid.

To model the transient thermal response of the working fluid andregenerator, Eq. (2) was discretized as follows:

⎜ ⎟⎛⎝

− ⎞⎠

= −+

ρVcT T

thA T T

Δ( )r

rp

rp

fp

rp

1

(4)

Assuming that the regenerator is made up of a cylindrical material,where the length is substantially larger than the radius, we can write

=V A r/ /2, and simplify Eq. (4) to:

⎜ ⎟= + ⎛⎝

− ⎞⎠

+T h tρc r

T h tρc r

T2 Δ 1 2 Δrp

rfp

rrp1

(5)

Eq. (1) is then rearranged as follows:

= − −+ +T T γ T T( )fp

fp

rp

rp1 1

(6)

where the regenerator thermal mass ratio, γ , is:

=γ m cm c

r r

f v f, (7)

Eqs. (5) and (6) model the thermal response of the regenerator andworking fluid, respectively, and they reveal that the parameters thatinfluence the transient response of the system are the radius of theregenerator, r, material properties of the regenerator, cr and ρ, the heattransfer coefficient, h, and the regenerator thermal mass ratio, γ . Thevalue of the time step, tΔ , was set low enough to maintain stability ofthe simulation and to ensure it did not influence the temperatures. Toexamine how each of the parameters influence the settling temperatureof the regenerator and working fluid, a parametric analysis was

Fig. 2. Transient response of a single regenerator (one sub-regenerator) absorbing energy from the working fluid for: (a) varying h with constant r = 0.02mm,constant regenerator material (stainless steel), and constant γ = 5:1, (b) varying r with constant h = 18,000W/m2 K, constant regenerator material (stainless steel),and constant γ = 5:1, (c) varying regenerator material with constant r = 0.02mm, constant h = 18,000W/m2K, and constant γ = 5:1, and (d) varying γ , constant r= 0.02mm, constant h = 18,000W/m2 K, and constant regenerator material (stainless steel).

A.S. Nielsen, et al. Applied Energy 253 (2019) 113557

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undertaken.

2.2. Transient parametric analysis of a single regenerator

A parametric analysis of the transient heat transfer model wasconducted for a single regenerator (one sub-regenerator) to reveal howthe temperatures of the regenerator and working fluid are impacted bythe regenerator radius, heat transfer coefficient, material properties,and regenerator thermal mass ratio. The regenerator radius, material,and heat transfer coefficient were varied in Fig. 2(a)–(c) to assess howthey impact the settling temperature of the system, which is the settlingtemperature of both the regenerator and working fluid. The influence ofthe regenerator thermal mass ratio, γ , was then examined in Fig. 2(d).The values were chosen based on the specifications of a GPU-3 Stirlingengine [44,45], which has a stainless steel regenerator with a wire ra-dius of =r 0.02 mm, and we calculated a heat transfer coefficient ofapproximately h = 18,000W/m2 K for the entire regenerator mesh,using [47]:

⎜ ⎟= = ⎛⎝

⎞⎠

hDk

C C PrPrPr

Nu Ref

bavg

avg

r2 1

0.3614

(8)

A regenerator thermal mass ratio of γ = 5:1 was selected for theplots in Fig. 2(a)–(c) to illustrate the influence of these parameters onthe regenerator and working fluid settling temperatures, and the valueof γ was then varied for Fig. 2(d). The results from a regenerationprocess of a single sub-regenerator with air moving from hot (TH =1000 K) to cold (TL = 300 K) are plotted in Fig. 2(a)–(c), and they re-veal that the heat transfer coefficient, radius, and material have animpact on the rapidity of the transient response but have no influenceon the settling temperature of an individual cycle. The GPU-3 Stirlingengine operates at rotational speeds at or below a value of 3,500 rpm[44,45], which corresponds to a minimum regeneration time of ap-proximately 0.004 s. The settling time in Fig. 2(a)–(c) is a maximum of0.004 s, which indicates that the regenerator temperature will reach thesettling temperature during each regeneration process. Therefore, inthese cases, the settling time is irrelevant and these parameters will nothave an influence on the effectiveness of the regenerator or perfor-mance of the engine. The only way to influence the regenerator’s ef-fectiveness is to change the value of the equilibrium temperature, whichcan be accomplished by varying the remaining parameter, the re-generator thermal mass ratio.

The response of four different regenerator thermal mass ratios wereplotted in Fig. 2(d) for a regeneration process with air moving from hot(TH = 1000 K) to cold (TL = 300 K), and it is shown that the re-generator thermal mass ratio not only impacts the rapidity of thesystem response, but also the system settling temperature. The value ofthe system settling temperature is an important consequence of theregenerator thermal mass ratio since it indicates how much additionalthermal energy the heater and cooler of the Stirling engine must eitheradd or reject, respectively, to attain target temperatures, TH and TL.

Not all Stirling engines operate at 3,500 rpm, and it is desirable toensure that the regenerator has the necessary specifications to exchangethermal energy with the working fluid within the given time framedetermined by the engine’s rotational speed. To do this, we mustcompare the natural time scale for regeneration, τ , and the timeavailable in the cycle for regeneration to occur, treg. The natural timescale can be obtained from Eqs. (5) and (6) and is expressed as follows:

=+

τρc r

h γ2 (1 )r

(9)

The time available for regeneration to occur is one quarter of arotation in the Stirling cycle, which is calculated as:

=tω15

reg (10)

Eq. (9) corresponds to the time when the regenerator and workingfluid temperatures are within 63.2% of the system settling temperature,and we must therefore multiply the natural time scale by 5 to ensurethat they are within 99% of the system settling temperature, hence:

=+

τρc r

h γ5 5

2 (1 )r

(11)

To ascertain that there is sufficient time for regeneration to occur,we evaluate:

<τt5 1reg (12)

which results the following expression, denoted as ζ :

=+

<ζωρc r

h γ6 (1 )1r

(13)

Values of ζ less than one indicate that more than 99% of theavailable heat is being exchanged within the given window of time,while values greater than one indicate that the regenerator’s specifi-cations are inadequate to exchange 99% of the available heat with theworking fluid. The ζ values are labelled in Fig. 2(a)–(d). Stirling enginedesigners can now evaluate whether or not the regenerator is trans-ferring sufficient thermal energy within the given window of time, andcan vary the regenerator’s specifications to ensure that they are max-imizing the heat transfer between the working fluid and regeneratorwith the use of Eq. (13). In summary, Eqs. (3) and (13) are the twoconditions that must be satisfied to confirm that only the regeneratorthermal mass ratio influences the settling temperature of the system fora single regenerator, and correspondingly the effectiveness of the re-generator. Accordingly, a heat transfer model that considers only theregenerator thermal mass ratio is now developed and expanded to thecase of a regenerator composed of multiple sub-regenerators.

2.3. Heat transfer model for multiple sub-regenerators

Having shown that for a single regenerator the thermal mass ratio isthe only valid parameter, we now turn our attention to the case ofmultiple sub-regenerators. To model a regenerator with multiple sub-regenerators we make the assumption that each sub-regenerator be-haves as a single regenerator and the whole regenerator is then com-posed of a series of thermally isolated sub-regenerators. This approx-imation will be validated by experiments and shown to yield accuratetemperature predictions for regenerators with multiple sub-re-generators. Having determined that the regenerator thermal mass ratiois the sole parameter that influences the system settling temperature,only Eq. (6) is required to model the settling temperature of re-generators composed of a number of sub-regenerators and the workingfluid. Since the settling time has been shown to be irrelevant, Eq. (6)can be rewritten without successive time steps, and the p superscript isreplaced with a subscript denoting the initial temperature (indicatingthe temperature at the start of each regeneration process), while the

+p( 1) superscript is replaced with a subscript denoting the final tem-perature (indicating the temperature at the end of each regenerationprocess). We then simplify the heat transfer model by rearranging Eq.(6) for a system settling temperature, which is the condition when theworking fluid and each sub-regenerator attain the same temperature:

= =T T Tsys f final r final, , (14)

and Eq. (6) becomes:

=++

TT γT

γ 1sysf initial r initial, ,

(15)

Four versions of Eq. (15) are required to model the system settlingtemperature for a regenerator with multiple sub-regenerators. To modelone regenerative cooling process, where the regenerator is absorbingthermal energy from the working fluid, we require the following two

A.S. Nielsen, et al. Applied Energy 253 (2019) 113557

5

equations:

=+

+T

T γ Tγ 1sys H

H sys L,

(1)(1)

,(1)

(1) (16)

=+

+

−

TT γ T

γ 1sys Hi sys H

i isys L

i

i,( ) ,

( 1) ( ),

( )

( ) (17)

where Eq. (16) is the system settling temperature of the first sub-re-generator for one regenerative cooling process, and Eq. (17) is thesystem settling temperature of each of the subsequent sub-regenerators,with their order denoted by the superscript i. To model one re-generative heating process, where the regenerator is returning thermalenergy to the working fluid, we require the following two equations:

=+

+T

T γ Tγ 1sys L

N LN

sys HN

N,( )

( ),

( )

( ) (18)

=+

+

+

TT γ T

γ 1sys Li sys L

i isys H

i

i,( ) ,

( 1) ( ),

( )

( ) (19)

where Eq. (18) is the system settling temperature of the N th or last sub-regenerator, and Eq. (19) is the system settling temperature of each ofthe subsequent sub-regenerators in reverse order since the workingfluid is flowing back through the regenerator. The regenerator thermalmass ratio for each sub-regenerator is a fraction of the total regeneratorthermal mass ratio, and in the case of a uniformly distributed mass,each value of γ i( ) would be equal to γ N/ . It should be noted thatmodelling a system with one sub-regenerator requires only Eqs. (16)and (18). The superscript i, is the sub-regenerator location (i.e. secondsub-regenerator, third sub-regenerator, etc.) and N is the total numberof sub-regenerators that the regenerator is composed of, where

< <i N1 .The operation of a Stirling engine is cyclical, so the regenerative

processes described by Eqs. (16)–(19) must be repeated over a numberof cycles to simulate the thermal behaviour. The Stirling engine andregenerator will reach a steady operating condition after a number ofcycles where the system settling temperatures in Eqs. (16)–(19) nolonger change. To model the cyclical behaviour for a single regeneratorwe begin by assuming the working fluid is in the hot side at a tem-perature of TH and the regenerator is initially at a temperature of

=T Tsys L L,(1) . We then use Eq. (16) for the first regenerative cooling pro-

cess (we denote these with n, so for the first one =n 1), and substitutethe resulting hot system settling temperature (Tsys H,

(1) ) into Eq. (18) todescribe the subsequent regenerative heating process ( =n 2). The re-sulting cold system settling temperature (Tsys L,

(1) ) from Eq. (18) is thensubstituted into Eq. (16) to model the subsequent regenerative coolingprocess ( =n 3). This process is repeated and produces the followingsummation when rearranged for the hot system settling temperature:

∑ ∑= + ⎛

⎝⎜ + ⎞

⎠⎟

=

∞

+=

∞

+ +

−

−

−T T Tsys H H

n

γγ L

n

γγ

γγ,

(1)

1(1 )

1(1 ) (1 )

n

n

n

n

nn

2( 1)

(2 1)

(2 1)

2(20)

To simulate steady operating conditions, we set the number of re-generator passes to infinity ( = ∞n ), which reduces the hot systemsettling temperature to:

=++

++

T Tγγ

Tγ

γ( 1)

(2 1) (2 1)sys H H L,(1)

(21)

A similar methodology is used to model regenerators with multiplesub-regenerators using Eqs. (16)–(19).

We can use Eqs. (16)–(19) to calculate a regenerator effectivenessbased on the resultant system settling temperatures. Fig. 3 shows thesystem settling temperatures for a regenerator with 3 sub-regenerators

=N( 3) across a complete Stirling engine cycle, where steady engineoperation has been reached. The first step is denoted by the circled ① inFig. 3, which shows the temperature of the working fluid as the workingfluid travels from the hot side through the 3 sub-regenerators. In the

first stage of step ① the working fluid and first sub-regenerator tem-peratures begin at TH and Tsys L,

(1) respectively, and the first sub-re-generator absorbs thermal energy from the working fluid resulting inboth mediums settling at a temperature of Tsys H,

(1) , which is the re-generative cooling process for the first sub-regenerator. Subsequently,the working fluid travels through the remaining two sub-regenerators,as shown in Fig. 3, and both the regenerator and working fluid settle ata temperature of Tsys H,

(3) after the third and final sub-regenerator. Theworking fluid must attain a temperature of TL before the subsequentisothermal compression process; therefore, the Stirling engine’s coolermust reject an additional amount of thermal energy that is proportionalto the remaining temperature difference, −T Tsys H L,

(3) =N(since 3),which is shown as step ② in Fig. 3. For the subsequent regenerativeheating process, denoted by the circled ③, the working fluid travelsfrom the cold side through the 3 sub-regenerators in reverse order, asshown in Fig. 3. The working fluid and third sub-regenerator tem-peratures begin at TL and Tsys H,

(3) respectively, and the third sub-re-generator returns thermal energy to the working fluid resulting in bothmediums settling at a temperature of Tsys L,

(3) , which is the regenerativeheating process for the third sub-regenerator. Subsequently, theworking fluid travels through the remaining two sub-regenerators inreverse order, as shown in Fig. 3, and both the regenerator and workingfluid settle at a temperature of Tsys L,

(1) after the final sub-regenerator. Theworking fluid must attain a temperature of TH before the subsequentisothermal expansion process; therefore, the Stirling engine’s heatermust add an additional amount of thermal energy that is proportionalto the remaining temperature difference, −T TH sys L,

(1) , which is shown asstep ④. The amount of energy that the regenerator absorbs and returnsis proportional to the temperature difference from the regenerativecooling and heating processes, shown in steps ① and ③ respectively,which we denote as TΔ reg:

= − = −T T T T TΔ reg H sys HN

sys L L,( )

,(1)

(22)

The effectiveness of the regenerator is the fraction of energy ab-sorbed and returned by the regenerator in comparison to the energyrequired for the cycle (proportional to −T TH L), which is expressed as:

∊ =−T

T TΔ

regreg

H L (23)

Fig. 3. Schematic diagram of the heat transfer model once steady engine op-eration has been reached for one complete Stirling cycle, illustrating the re-generative heating and cooling processes for a regenerator composed of 3 sub-regenerators, =N 3, and regenerator thermal mass ratio, γ = 20:1.

A.S. Nielsen, et al. Applied Energy 253 (2019) 113557

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The regenerator effectiveness is only a function of the target tem-peraturesTH orTL and the system settling temperaturesTsys H

N,

( ) orTsys L,(1) , so

we can substitute Eq. (21), which is the hot system settling temperaturefor a single regenerator after steady operating conditions have beenreached, to yield the regenerator effectiveness for a single regenerator:

∊ =+γ

γ2 1reg(24)

Similarly, for a regenerator with multiple sub-regenerators, the re-generator effectiveness for steady operating conditions simplifies to:

∊ =+ +

NγN γ N( 1)reg

(25)

Since Stirling engines operate primarily in steady operating condi-tions, with only a short warm-up period, the remainder of the analysiswill consider the thermal behaviour once steady operating conditionshave been reached.

2.4. Stirling engine efficiency model

The Stirling engine efficiency is commonly listed as the Carnot ef-ficiency:

= = −η η TT

1Carnot Stirling idealL

H, (26)

however, this efficiency considers ideal regeneration (∊ = 1reg ). Thederivations in the previous section revealed that regenerator effec-tiveness is a function of the regenerator thermal mass ratio and numberof sub-regenerators. For non-ideal regeneration, the Stirling engine ef-ficiency must also be a function of the regenerator thermal mass ratioand number of sub-regenerators, and we derive an expression to cap-ture this dependence. The Stirling engine efficiency with non-ideal re-generation is expressed as:

= −η Q QQStirling

in out

in (27)

where:

= + −Q m RT λ Qln( )in f H non reg (28)

= + −Q m RT λ Qln( )out f L non reg (29)

and:

=λ VV

max

min (30)

An additional Qnon reg- term in Eqs. (28) and (29) is required since itrepresents the amount of supplemental thermal energy that the heaterand cooler of the Stirling engine must either add or reject, respectively,to attain target temperatures, TH and TL, due to non-ideal regeneration.The regeneration process in the Stirling cycle is isochoric, hence theadditional heat added and rejected is:

= − ∊ −−Q m c T T(1 ) ( )non reg reg f v f H L, (31)

Eq. (31) is then substituted into Eqs. (28) and (29), which are thensubstituted into Eq. (27) to generate an expression for the thermal ef-ficiency of the Stirling engine that incorporates the influence of theregenerator effectiveness (Eq. (25)), as follows:

= −+ − ∊ −

η R T T λRT λ c T T

( )ln( )ln( ) (1 ) ( )Stirling

H L

H reg v f H L, (32)

It is shown in Eq. (32) that the efficiency of the Stirling engine is notonly dependent on the target temperatures, TH and TL, as seen in Eq.(26), but is also a function of the compression ratio, λ, the properties ofthe employed fluid, cv f, and R (specific gas constant), and the re-generator effectiveness, ∊reg, which is a function of the regeneratorthermal mass ratio, γ , and number of sub-regenerators, N, according toEq. (25).

Fig. 4. Experimental setup of the regenerator test apparatus with (1) linear pneumatic actuators, (2) hot cylinder, (3) regenerator chamber, (4) cold cylinder, (5) PIDcontroller, (6) air-flow regulator, and (7) data acquisition module.

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3. Experimental apparatus

We designed and built a regenerator test apparatus to validate theheat transfer model developed in Section 2.3. The regenerator test ap-paratus, which is shown in Fig. 4, consisted of two linear pneumaticactuators that were programmed to move the hot and cold pistons in anoscillatory manner for a constant volume regeneration process at anequivalent rotational speed of 180 rpm. The apparatus also employed aPID controlled electric heat source, an air-flow regulator to control themotion of the pistons, K-type thermocouples for temperature mea-surements, and a data acquisition module to process the signal gener-ated by the thermocouples. The thermocouples had an accuracy of ± 1.5K. Each sub-regenerator consisted of a 304 stainless steel wire meshwith a wire diameter of approximately 0.04mm, which are the samespecifications that were reported for the GPU-3 Stirling engine [44,45].The two regenerator configurations that were examined in the experi-ments were spaced and unspaced sub-regenerators. The number of sub-regenerators was set at both 12 and 16 sub-regenerators, which corre-sponds to a regenerator thermal mass ratio of γ = 4.01:1 and γ =5.35:1, respectively. An unspaced sub-regenerator configuration de-notes that there is physical contact between neighbouring sub-re-generators, while a spaced configuration indicates that each sub-re-generator was separated by a ceramic insulation ring to activelyminimize the local axial conduction between each sub-regenerator.Thermocouples were placed between every group of four sub-re-generators to record the temperature at the 2nd, 6th, 10th, and 14thsub-regenerators. The heat source and heat sink temperatures weremeasured using thermocouples that recorded the inside air tempera-tures of both hot and cold cylinders, and their respective temperatureswere maintained at approximately TH = 353 K (80 °C) and TL = 303 K(30 °C) for every test case. Each experiment was run for at least 40minbefore recording measurements to ensure that the temperatures of theworking fluid and regenerator reached steady operating conditions. Theresults gathered from the experiments were then compared againstthose obtained from the heat transfer model to assess its validity.

4. Results and discussion

4.1. Single regenerator analysis

We first analyze a single regenerator to determine the limitations forregenerator effectiveness of a single regenerator setup. The regeneratoreffectiveness described in Eq. (24) is only a function of the regeneratorthermal mass ratio, and thus is plotted in Fig. 5 against a range of valuesfor the regenerator thermal mass ratio. It can be seen in Fig. 5 that themaximum effectiveness a single regenerator can attain is 50%. As theregenerator thermal mass ratio increases, the regenerator effectivenessincreases asymptotically towards a maximum value of 50%. Thismaximum effectiveness of a single regenerator applies to any re-generator composed of a continuous material with high thermal con-ductivities and small thicknesses, satisfying the conditions from Eqs. (3)and (13), such as parallel plates. From this analysis it is demonstratedthat single regenerators, such as parallel plate regenerators, with spe-cifications that satisfy the thermal equilibrium approximation, cannotattain a regenerator effectiveness greater than 50%. The followingsections in this study will reveal how to obtain a regenerator effec-tiveness greater than 50%, and how to design for an effectivenessgreater than 95%.

4.2. Multiple sub-regenerator analysis

4.2.1. Influence of number of sub-regeneratorsUsing Eqs. (16)–(19), the temperature distributions of regenerators

composed of various numbers of sub-regenerators for a regenerativecooling process were plotted in Fig. 6. For this analysis the regeneratorthermal mass ratio remained constant and the regenerator mass was

distributed evenly across the sub-regenerators. It is shown that as thenumber of sub-regenerators increases, the temperature of the last sub-regenerator and correspondingly the exiting working fluid becomescloser to the cold target temperature, TL. Therefore the temperaturedifference between the exiting working fluid and cold target tempera-ture ( −T Tsys H

NL,

( ) ) decreases and regenerator effectiveness increases,according to Eq. (23). Fig. 6 also reveals that regenerators composed ofmultiple sub-regenerators maintain a linear temperature distribution,which is crucial for enhancing the regenerator effectiveness. This lineartemperature profile also indicates that average regenerator tempera-tures should be evaluated using an arithmetic mean between the firstand last sub-regenerators, rather than using a log-mean temperaturedifference, as is often done in the literature. In summary, increasing thenumber of sub-regenerators enables the last sub-regenerator and exitingworking fluid to achieve a temperature closer to the target temperature,which enhances the regenerator’s effectiveness. We will now quantifyhow the number of sub-regenerators and regenerator thermal massratio influence the effectiveness of Stirling engine regenerators.

4.2.2. Multiple sub-regenerator effectivenessThe regenerator effectiveness from Eq. (25) is plotted against the

number of sub-regenerators with various thermal mass ratio values inFig. 7, which shows that increasing both the regenerator thermal massratio and number of sub-regenerators enhances the effectiveness of theregenerator. To determine the maximum attainable effectiveness, aninfinite regenerator thermal mass ratio ( = ∞γ :1) is substituted into Eq.(25) to yield:

∊ =+N

N 1reg max, (33)

which represents the maximum effectiveness that a regenerator canattain based on the number of sub-regenerators employed (N), analo-gous to the Carnot efficiency for heat engines. The maximum effec-tiveness from Eq. (33) is plotted in Fig. 7 to show the upper limitationon the regenerator effectiveness value versus the number of sub-re-generators used.

We can also determine how to achieve a specific value for re-generator effectiveness based on the number of sub-regenerators (N)and regenerator thermal mass ratio (γ) used in the design of a Stirlingengine regenerator. Fig. 8 is a plot of Eq. (25), which shows how toselect the values of both the number of sub-regenerators and the

Fig. 5. Regenerator effectiveness, ∊reg , of a single regenerator, =N 1, as afunction of the regenerator thermal mass ratio, γ .

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regenerator thermal mass ratio to achieve regenerator effectivenessvalues of 95%, 96%, 97%, and 98%. Any values above and to the rightof the effectiveness lines will result in a higher regenerator effective-ness. It is also shown in Fig. 8 that the effectiveness lines approach avertical asymptote as the number of sub-regenerators decreases. Thisoccurs since each regenerator effectiveness has a correspondingminimum required number of sub-regenerators to attain the specifiedeffectiveness. For example, to attain 95% regenerator effectiveness, theminimum required number of sub-regenerators is 19, and it is im-possible for a regenerator with less than 19 sub-regenerators to attain95% effectiveness. Stirling engine designers can use Eq. (25) to select aregenerator effectiveness they desire based on the selected regeneratorthermal mass ratio and number of sub-regenerators employed.

4.2.3. Influence of mass distributionAn interesting question that arises in the design of regenerators is:

does the distribution of the sub-regenerator mass influence the re-generator effectiveness? To simulate an uneven mass distribution, theregenerator thermal mass ratio values in Eqs. (16)–(19) were varied foreach of the sub-regenerators over a wide range of distributions. Theanalysis revealed that the regenerator effectiveness is a function of thenon-uniformity of the mass distribution, which is described by thestandard deviation of the regenerator thermal mass ratio, σ , and thearrangement of the distribution did not matter. The regenerator effec-tiveness is plotted against the standard deviation in Fig. 9, which showsthat as the standard deviation increases, which corresponds to an

Fig. 6. Temperature distribution within regenerators with varying number ofsub-regenerators, N, and constant regenerator thermal mass ratio, γ = 20:1,during a regenerative cooling process for steady operating conditions.

Fig. 7. Regenerator effectiveness, ∊reg , as a function of the number of sub-re-generators, N, and regenerator thermal mass ratio, γ , in comparison to themaximum regenerator effectiveness when = ∞γ :1.

Fig. 8. Plot of the regenerator effectiveness curves (for 95%, 96%, 97%, and98%) and the corresponding values of the regenerator thermal mass ratio, γ ,and number of sub-regenerators, N, required to produce a desired effectiveness.

Fig. 9. Regenerator effectiveness, ∊reg, as a function of the standard deviation,σ , of the distribution of the regenerator thermal mass ratio among the sub-regenerators for three different cases. The black points indicate the ∊reg valuesthat coincide with a single regenerator.

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increasingly uneven distribution of the sub-regenerator mass, the re-generator effectiveness decreases until it reaches the value for a singleregenerator (with the corresponding regenerator thermal mass ratio).The reason that the regenerator effectiveness approaches the value for asingle regenerator as the standard deviation increases is because anextremely uneven mass distribution approximates that of a single re-generator. It is concluded from this analysis that the optimum sub-re-generator mass distribution for a Stirling engine regenerator is a uni-form mass distribution.

4.2.4. Experimental validationValidation of the heat transfer model is necessary to ascertain that

the derived expressions can generate accurate predictions of thethermal behaviour in Stirling engine regenerators. To ensure that theexperiments were eligible to be compared against the heat transfermodel developed in Section 2.3, the conditions described in Eqs. (3) and(13) had to be satisfied. The highest resultant Biot number from theexperiments was × −6.5 10 4, and the highest regeneration time ratiovalue, ζ , was 0.82, thus demonstrating that the experiments satisfiedthe conditions. The experimental and theoretical results of regenerativeheating and cooling processes for spaced and unspaced sub-regeneratorconfigurations are shown in Fig. 10(a) and (b), for 12 and 16 sub-re-generators, respectively. It is shown that there is good agreement be-tween the experimental and theoretical results, for target temperaturesof TH = 353 K (80 °C) and TL = 303 K (30 °C), with percent differencesranging between 0.14% to 2.4% for 12 sub-regenerators (0.47 K to7.73 K), and 0.01% to 1.96% for 16 sub-regenerators (0.03 K to 6.68 K).The experimental results shown in Fig. 10(a) and (b) also confirm thatthe temperature distribution within the regenerator is linear. Sincethere is good agreement between the heat transfer model and experi-mental results, it is confirmed that the heat transfer model has beenvalidated and can be used in the design of Stirling engine regeneratorsto predict their effectiveness based on the selected values of regeneratorthermal mass ratio and number of sub-regenerators.

4.2.5. How to thermally isolate sub-regeneratorsAnother interesting question that arises from the results of this study

is: how can a designer ensure that there is sufficient thermal isolationbetween sub-regenerators to yield a regenerator configuration that hasmultiple sub-regenerators and is not behaving as a single regenerator?To examine this we performed the experiments with two sub-re-generator configurations: spaced and unspaced, where the wire meshesused to form the regenerator were either directly in contact or sepa-rated by a ceramic insulation ring, respectively. It is shown in Fig. 10(a)and (b) that the experimental values of spaced and unspaced config-urations are nearly identical, with an average percent difference of0.53% for 12 sub-regenerators (1.78 K), and 1.21% for 16 sub-re-generators (4.06 K). Fig. 10(a) and (b) reveal that unspaced sub-re-generators, such as wire meshes stacked directly against one another,provide sufficient thermal contact resistance to be considered as ther-mally isolated for the temperature range used in the experiments (TH =353 K (80 °C) and TL = 303 K (30 °C)). Therefore, Stirling engine de-signers can simply stack wire mesh sub-regenerators together to attainthe required thermal resistance to produce a sufficient temperaturedistribution, rather than needing to space each individual sub-re-generator or use other complex isolation strategies.

4.3. Stirling engine efficiency

The efficiency of Stirling engines is ultimately the most importantoutcome when designing regenerators. We perform a parametric ana-lysis using Eq. (32), to examine the effects of the regenerator thermalmass ratio and number of sub-regenerators on the efficiency of Stirlingengines, and compare the resultant efficiencies with that of the Carnotcycle (ideal regeneration).

4.3.1. Effect of number of sub-regeneratorsThe number of sub-regenerators directly influences the effectiveness

of the regenerator, which subsequently impacts the thermal efficiencyof the Stirling engine, as seen in Eq. (32). In this portion of the para-metric analysis, the number of sub-regenerators was varied from 1 to 25for various target temperature ratios, T T/H L, while the compressionratio was fixed at a value of 10:1, the employed working fluid washydrogen, and the regenerator thermal mass ratio was fixed at a valueof 20:1. It is shown in Fig. 11(a) that as the number of sub-regeneratorsincreases, Stirling engine efficiency also increases, as expected with anincreasing regenerator effectiveness. It is seen, however, that the in-cremental increase in Stirling engine efficiency decreases as the numberof sub-regenerators increase, illustrated by the lines becoming pro-gressively closer together, which indicates that there are diminishingreturns as the number of sub-regenerators is increased. Therefore,Fig. 11(a) can be used by Stirling engine designers to select an

Fig. 10. Comparison between the results predicted by the heat transfer modeland experimental results for spaced and unspaced sub-regenerators for re-generative heating and cooling processes with (a) =N 12 and =γ 4.01:1, and(b) =N 16 and =γ 5.35:1.

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appropriate value for the number of sub-regenerators required toachieve a desired value for the Stirling engine efficiency. The selectionof an appropriate number of sub-regenerators will avoid the use of anexcessive number of sub-regenerators, which could impede the flow ofthe working fluid through the regenerator and increase the pressuredrop.

4.3.2. Effect of regenerator thermal mass ratioThe regenerator thermal mass ratio also influences the effectiveness

of the regenerator, which subsequently impacts the thermal efficiencyof the Stirling engine, as seen in Eq. (32). In this portion of the para-metric analysis, the value of the regenerator thermal mass ratio wasvaried from 1:1 to ∞:1 for various target temperature ratios, T T/H L,while the compression ratio was fixed at a value of 10:1, the employedworking fluid was hydrogen, and the number of sub-regenerators was

fixed at a value of 25. It is shown in Fig. 11(b) that as the value of theregenerator thermal mass ratio increases, the Stirling engine’s efficiencyalso increases, as expected with an increasing regenerator effectiveness.There are also diminishing returns as the regenerator thermal massratio approaches∞:1, as illustrated by the lines becoming progressivelycloser together. Similarly to the number of sub-regenerators, Fig. 11(b)can be used by Stirling engine designers to select an appropriate valueof the regenerator thermal mass ratio required to achieve a desiredvalue for the Stirling engine efficiency. The selection of an appropriatevalue of the regenerator thermal mass ratio will avoid the use of anunnecessary amount of regenerator mass.

5. Conclusions

In this work, a discrete one-dimensional heat transfer model wasdeveloped to determine which parameters influence the effectiveness ofStirling engine regenerators. The heat transfer model revealed that theregenerator thermal mass ratio and number of sub-regenerators werethe two parameters that influence the regenerator effectiveness andquantify how they influence it. The use of multiple sub-regeneratorswas shown to produce a linear temperature distribution within a re-generator, which enables the effectiveness to be increased above 50%.It was determined that increasing the regenerator thermal mass ratioand number of sub-regenerators resulted in an increase in regeneratoreffectiveness and a corresponding increase in the Stirling engine effi-ciency. It was also found that the regenerator effectiveness has amaximum attainable value, which is limited by the number of sub-re-generators used. For example, to attain a regenerator effectiveness of95%, a minimum of 19 sub-regenerators must be used. It was also de-termined that a uniform sub-regenerator mass distribution provides thehighest regenerator effectiveness. Experiments using a regenerator testapparatus, with a temperature range of TH = 353 K (80 °C) and TL =303 K (30 °C), validated the heat transfer model, and demonstrated thatstacking sub-regenerators, such as wire meshes, provides sufficientthermal resistance to generate a temperature distribution throughoutthe regenerator, which enhances regenerator effectiveness. As a resultof this study, Stirling engine designers can select a value of regeneratorthermal mass ratio and number of sub-regenerators to attain a desiredregenerator effectiveness and/or a desired value for the Stirling engineefficiency.

Acknowledgements

The authors gratefully acknowledge the funding support from aNatural Sciences and Engineering Research Council of Canada (NSERC)Idea-to-Innovation (I2I) grant.

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