Applied Energy 253 (2019) 113557
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Applied Energy
journal homepage: www.elsevier.com/locate/apenergy
Stirling engine regenerators: How to attain over 95% regenerator effectiveness with sub-regenerators and thermal mass ratios T ⁎ 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
HIGHLIGHTS
• 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.
ARTICLE INFO ABSTRACT
Keywords: A combined theoretical and experimental approach is used to determine how to achieve a desired value for the Stirling engine Stirling engine regenerator effectiveness. A discrete one-dimensional heat transfer model is developed to de- Regenerator termine which parameters influence the effectiveness of Stirling engine regenerators and quantify how they ffi E ciency influence it. The regenerator thermal mass ratio and number of sub-regenerators were found to be the two Effectiveness parameters that influence the regenerator effectiveness, and the use of multiple sub-regenerators is shown to Heat transfer produce a linear temperature distribution within a regenerator, which enables the effectiveness to be increased Sub-regenerators above 50%. It is shown that increasing the regenerator thermal mass ratio and number of sub-regenerators results 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, provides sufficient thermal resistance to generate a temperature distribution throughout the regenerator. This is the first study to determine how Stirling engine designers can attain a desired value for the regenerator effectiveness and/or a desired value for the Stirling engine efficiency by selecting appropriate values of regenerator thermal mass ratio and number of sub-regenerators.
1. Introduction reject, respectively. Accordingly, regenerator design and development has received much attention in past research, and has been a focal point Stirling engines have a high potential to alleviate some of the in- in the process of enhancing Stirling engine performance and efficiency. creasing global demand for clean energy. This potential is due in part to Characteristics of an effective regenerator are a high thermal storage the fact that Stirling engines are external combustion engines that can and heat transfer capacity, a limited resultant pressure drop, and a therefore be powered by a wide range of heat sources, including sus- sufficient temperature distribution to ensure that the working fluid al- tainable options such as solar thermal, waste heat, and biofuels [1]. ternates between target temperatures, TH and TL. It was recently es- Additionally, Stirling engines have the ability to attain the highest tablished by Dai et al. [2] that in order for a regenerator to attain a theoretical engine efficiency, primarily due to the use of a regenerator, regenerator effectiveness higher than 50%, it is required that the re- which exchanges thermal energy with the working fluid after each generator be divided into discrete thermally isolated components, re- isothermal expansion and compression stroke, thus minimizing the ferred to as sub-regenerators. Interesting questions that arise in the amount of thermal energy the heater and cooler must either add or design of regenerators are: how does the regenerator thermal mass ratio
⁎ Corresponding author. E-mail address: [email protected] (B.D. MacDonald). https://doi.org/10.1016/j.apenergy.2019.113557 Received 8 March 2019; Received in revised form 3 July 2019; Accepted 15 July 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved. A.S. Nielsen, et al. Applied Energy 253 (2019) 113557
Nomenclature Tsys,L cold system settling temperature (K) Tsys system temperature (K) Symbols V regenerator volume (m3) 3 Vmax maximum volume of working fluid (m ) 3 Δt time step (s) Vmin minimum volume of working fluid (m ) ΔTreg regenerator temperature change (K) Nu Nusselt number (–) Greek Letters Pr Prandtl number (–) Re Reynolds number (–) ∊reg regenerator effectiveness (–) 2 ffi – A regenerator surface area (m ) ηCarnot Carnot e ciency ( ) – ffi – b Nusselt number correlation exponent ( ) ηStirling Stirling engine e ciency ( ) c specific heat capacity (J/kg K) γ regenerator thermal mass ratio (–) – C1 correction factor (Reynolds number) (–) λ compression ratio ( ) C2 correction factor (Sub-regenerators) (–) ω Stirling engine rotational speed (rpm) 3 cv constant volume specific heat capacity (J/kg K) ρ regenerator material density (kg/m ) D regenerator wire diameter (m) σ standard deviation of γ distribution (–) h heat transfer coefficient (W/m2 K) τ time constant for regeneration (s) k thermal conductivity (W/m K) ζ regeneration time ratio (–) m mass (kg) N number of sub-regenerators (–) Superscripts n number of regenerator passes (–) fi Qin heat addition (J) 1 rst sub-regenerator Qnon- reg heat not transferred by regenerator (J) N last sub-regenerator Qout heat rejection (J) p current time step R working fluid ideal gas constant (J/kg K) p + 1 successive time step r regenerator radius (m) T temperature (K) Subscripts t time (s) avg average properties TH heat source temperature (K) f working fluid TTHL/ target temperature ratio (–) final final state TL heat sink temperature (K) initial initial state treg engine speed regeneration time (s) r regenerator Tsys, H hot system settling temperature (K) and number of sub-regenerators influence the effectiveness of re- generation in wire mesh regenerators [25] and micro-channel re- generators? How can one design a regenerator to attain 95% effec- generators [26]. Past analytical studies have developed heat transfer tiveness or higher? Can parallel plates or other continuous regenerators models for counterflow regenerators in Stirling engine applications to attain higher than 50% effectiveness? Does stacking wire meshes re- examine the infl uence of the flush ratio [27,28] and the number of duce the regenerator to a single sub-regenerator and limit its effec- transfer units (NTU) [29] on regenerator effectiveness. Optimization tiveness to a maximum of 50%? In order to further improve the design analyses have varied the regenerator’s fill factor [30], wire diameter and our understanding of Stirling engine regenerators, a comprehensive [31,32], length [33], and mesh size [34] to maximize Stirling engine investigation to determine the influence of the regenerator thermal performance and efficiency. Investigations into the performance of mass ratio and number of sub-regenerators on the effectiveness of re- Stirling engines using an isothermal model [35,36] and a finite time generators is required to answer these questions. thermodynamic model [37] have varied the effectiveness of the re- In past literature, significant effort has been invested in quantifying generator without discussing what parameters would alter the re- the thermal and flow characteristics of various types of regenerators. generator effectiveness. Many studies have also assumed perfect re- Experiments have implemented steady flow conditions to quantify the generation when analyzing Stirling engines at maximum power pressure drop [3] and heat transfer characteristics [4] in wire mesh conditions [38,39], when comparing the net work output between regenerators, while other experiments have imposed oscillatory flow Stirling and Ericsson engines [40], and when assessing the efficiency conditions to measure the pressure drop and heat transfer rate in wire reduction in Stirling engines due to sinusoidal motion [41]. The pre- mesh regenerators [5,6], parallel-plate regenerators [7], and sponge vious studies have played a vital role in the advancement of Stirling metal regenerators [8]. Combined experimental and numerical studies engine technologies and have elucidated much about the effect of re- have quantified the pressure drop [9,10] and heat transfer rate [11] in generators on engine performance, but it is still not understood how the wire mesh regenerators subjected to oscillatory flow conditions. Other regenerator thermal mass ratio and number of sub-regenerators influ- experiments have investigated the influence of the geometry [12] and ence the effectiveness of regenerators, and how to design regenerators the type of material used in regenerators on the performance of Stirling to attain a desired effectiveness. engines, including copper and stainless steel [13], and aluminum and There have been limited studies that have examined the influence of Monel 400 [14,15]. Numerical studies have developed Nusselt number either the number of sub-regenerators or regenerator thermal mass [16,17] and friction factor [18,19] correlations for wire mesh re- ratio on regenerator effectiveness or Stirling engine performance and generators, while other numerical studies have developed similar cor- efficiency, and no study has investigated them in tandem. Dai et al. [2] relations for micro-channel regenerators [20], segmented involute-foil theoretically showed that a regenerator must have an unevenly dis- regenerators [21], and circular miniature-channel regenerators using tributed temperature to attain an effectiveness greater than 50%, and 3D CFD [22,23] and Reynolds-Averaged Navier-Stokes (RANS) model- that an unevenly distributed temperature can be accomplished using a ling [24]. Other numerical studies have examined the entropy number of smaller heat reservoirs to divide the regenerator into sub-
2 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557 regenerators. A study conducted by Jones [42] examined the effects of effectiveness. Experiments were conducted, and the experimental re- the regenerator thermal mass ratio on the ineffectiveness of the re- sults validated the heat transfer model. The experiments also demon- generator and power loss factor of the Stirling engine. The work found strated that stacking sub-regenerators, such as wire meshes, provides that regenerator thermal mass ratios between 20:1 and 40:1 can in- sufficient thermal resistance to generate a temperature distribution. As fluence the performance of Stirling engines, and was not concerned a result of this study, Stirling engine designers can select a value of with recommending an ideal value for the regenerator thermal mass regenerator thermal mass ratio and number of sub-regenerators to at- ratio. Zarinchang et al. [43] examined the pressure drop and enthalpy tain a specific regenerator effectiveness and/or a specific value of the flux through a wire mesh regenerator and indirectly found that, for Stirling engine efficiency. their system, the optimum value for the regenerator thermal mass ratio was 14.9:1. The GPU-3 and 4L23 Stirling engines [44,45] had re- 2. Mathematical modelling generator thermal mass ratios in the range of 205:1–235:1, and the GPU-3 engine used 308 stacked wire mesh sub-regenerators as its re- 2.1. Transient heat transfer model of a single regenerator generator. Such high values of regenerator thermal mass ratios and number of sub-regenerators require a considerable amount of re- In order to determine the thermal behaviour of a regenerator in a generator material, which inhibits the flow of the working fluid and Stirling engine we first consider a single regenerator (one sub-re- leads to higher pressure drops, ultimately reducing the efficiency and generator) along with the working fluid and model the two isochoric performance of Stirling engines. Consequently, it is important to de- (constant volume) regeneration processes of the Stirling cycle. During termine what regenerator thermal mass ratio and quantity of sub-re- these processes, energy is exchanged between the working fluid and the generators are required to complete the necessary heat exchange to Stirling engine’s regenerator, which is shown in Fig. 1, and their tem- attain a high effectiveness, while avoiding the use of unnecessarily high peratures change conversely to one another. This means that an in- values that cause increased pressure drops. These studies demonstrate cremental increase of energy in one of the mediums must be balanced that the regenerator thermal mass ratio and quantity of sub-re- by the decrease of energy in the other, which is expressed as: generators have a significant impact on the effectiveness of re- p+1 p p+1 p generators and Stirling engines, but to date, there has not been any − mcTTmcTTrr()(r −=r fvf, f −f ) (1) research that has determined the regenerator thermal mass ratio and where superscript p denotes the current time step, and p + 1 denotes number of sub-regenerators required to attain specific regenerator ef- the successive step. The rate of energy change must also be balanced, fectiveness values without employing unnecessarily high values of each such that the rate of heat transfer in the regenerator is equal to the and generating unnecessary pressure drops. The understanding of how convection at the surface of the regenerator. We assume local thermal to select these values will lead to enhanced regenerator designs capable equilibrium within the regenerator material and working fluid, thus of high effectiveness and low pressure drops. neglecting local conduction heat transfer within each medium, resulting In this paper, a discrete one-dimensional transient heat transfer in: model of a Stirling engine regenerator was developed to track the thermal response of the working fluid and regenerator, simultaneously. dTr ρVcr =−hA( Tfr T ) A parametric analysis of the transient model determined that only the dt (2) regenerator thermal mass ratio influences the settling temperature of For thermal equilibrium to be valid for the working fluid would the regenerator and working fluid, and that the remaining transient require sufficient mixing and small length scales, which we expect in parameters can be neglected. Accordingly, a heat transfer model was Stirling engine regenerators. To ensure that the thermal equilibrium developed to quantify the settling temperature of the regenerator and approximation is valid for the solid regenerator material, the following working fluid, and expressions for the regenerator effectiveness and condition must be satisfied: Stirling engine efficiency were derived to incorporate the influence of the regenerator thermal mass ratio and number of sub-regenerators. It hr < 0.1 was found that increasing the number of sub-regenerators and re- 2kr (3) generator thermal mass ratio increases regenerator effectiveness and That is, the Biot number must be below the threshold value of 0.1 Stirling engine efficiency, and it was determined that a uniform sub- [46]. The largest expected value of the Biot number for the work ana- regenerator mass distribution provides maximum regenerator 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.
3 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557 lowest thermal conductivity (stainless steel: kr = 16.2 W/m K), the lar- Eq. (1) is then rearranged as follows: gest radius (r = 0.04 mm), and the highest heat transfer coefficient TTγTTp+1 =−p ( p+1 −p) (h = 18, 000 W/m2 K), which results in a value of 0.022. Since this is f f r r (6) below the threshold value, the approximation will be valid. where the regenerator thermal mass ratio, γ, is: To model the transient thermal response of the working fluid and mc regenerator, Eq. (2) was discretized as follows: γ = rr mcfvf, (7) p+1 p ⎛TTr − r ⎞ p p ρVcr ⎜⎟=−hA( Tf Tr ) Eqs. (5) and (6) model the thermal response of the regenerator and ⎝ Δt ⎠ (4) working fluid, respectively, and they reveal that the parameters that influence the transient response of the system are the radius of the Assuming that the regenerator is made up of a cylindrical material, regenerator, r, material properties of the regenerator, c and ρ, the heat where the length is substantially larger than the radius, we can write r transfer coefficient, h, and the regenerator thermal mass ratio, γ. The V //2Ar= , and simplify Eq. (4) to: value of the time step, Δt, was set low enough to maintain stability of the simulation and to ensure it did not influence the temperatures. To p+1 2Δht p ⎛ 2Δht⎞ p Tr =+Tf ⎜⎟1 − Tr examine how each of the parameters influence the settling temperature ρcr r ⎝ ρcr r ⎠ (5) of 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 = 0.02 mm, constant regenerator material (stainless steel), and constant γ = 5:1, (b) varying r with constant h = 18,000 W/m2 K, constant regenerator material (stainless steel), and constant γ = 5:1, (c) varying regenerator material with constant r = 0.02 mm, constant h = 18,000 W/m2K, and constant γ = 5:1, and (d) varying γ, constant r = 0.02 mm, constant h = 18,000 W/m2 K, and constant regenerator material (stainless steel).
4 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557 undertaken. Eq. (9) corresponds to the time when the regenerator and working fluid temperatures are within 63.2% of the system settling temperature, 2.2. Transient parametric analysis of a single regenerator and we must therefore multiply the natural time scale by 5 to ensure that they are within 99% of the system settling temperature, hence: A parametric analysis of the transient heat transfer model was 5 ρc r 5τ = r conducted for a single regenerator (one sub-regenerator) to reveal how 2(1)hγ+ (11) the temperatures of the regenerator and working fluid are impacted by ffi the regenerator radius, heat transfer coefficient, material properties, To ascertain that there is su cient time for regeneration to occur, and regenerator thermal mass ratio. The regenerator radius, material, we evaluate: and heat transfer coefficient were varied in Fig. 2(a)–(c) to assess how 5τ < 1 they impact the settling temperature of the system, which is the settling treg (12) temperature of both the regenerator and working fluid. The influence of the regenerator thermal mass ratio, γ, was then examined in Fig. 2(d). which results the following expression, denoted as ζ : The values were chosen based on the specifications of a GPU-3 Stirling ωρc r ζ = r < 1 engine [44,45], which has a stainless steel regenerator with a wire ra- 6(1hγ+ ) (13) dius of r = 0.02 mm, and we calculated a heat transfer coefficient of Values of ζ less than one indicate that more than 99% of the approximately h = 18,000 W/m2 K for the entire regenerator mesh, available heat is being exchanged within the given window of time, using [47]: while values greater than one indicate that the regenerator’s specifi- 1 Pr 4 cations are inadequate to exchange 99% of the available heat with the hD b 0.36 ⎛ avg ⎞ NuR==CC21e Pravg ⎜⎟ working fluid. The ζ values are labelled in Fig. 2(a)–(d). Stirling engine kf Prr (8) ⎝ ⎠ designers can now evaluate whether or not the regenerator is trans- A regenerator thermal mass ratio of γ = 5:1 was selected for the ferring sufficient thermal energy within the given window of time, and plots in Fig. 2(a)–(c) to illustrate the influence of these parameters on can vary the regenerator’s specifications to ensure that they are max- the regenerator and working fluid settling temperatures, and the value imizing the heat transfer between the working fluid and regenerator of γ was then varied for Fig. 2(d). The results from a regeneration with the use of Eq. (13). In summary, Eqs. (3) and (13) are the two fi fi process of a single sub-regenerator with air moving from hot (TH = conditions that must be satis ed to con rm that only the regenerator fl 1000 K) to cold (TL = 300 K) are plotted in Fig. 2(a)–(c), and they re- thermal mass ratio in uences the settling temperature of the system for veal that the heat transfer coefficient, radius, and material have an a single regenerator, and correspondingly the effectiveness of the re- impact on the rapidity of the transient response but have no influence generator. Accordingly, a heat transfer model that considers only the on the settling temperature of an individual cycle. The GPU-3 Stirling regenerator thermal mass ratio is now developed and expanded to the engine operates at rotational speeds at or below a value of 3,500 rpm case of a regenerator composed of multiple sub-regenerators. [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 of 2.3. Heat transfer model for multiple sub-regenerators 0.004 s, which indicates that the regenerator temperature will reach the settling temperature during each regeneration process. Therefore, in Having shown that for a single regenerator the thermal mass ratio is these cases, the settling time is irrelevant and these parameters will not the only valid parameter, we now turn our attention to the case of have an influence on the effectiveness of the regenerator or perfor- multiple sub-regenerators. To model a regenerator with multiple sub- mance of the engine. The only way to influence the regenerator’s ef- regenerators we make the assumption that each sub-regenerator be- fectiveness is to change the value of the equilibrium temperature, which haves as a single regenerator and the whole regenerator is then com- can be accomplished by varying the remaining parameter, the re- posed of a series of thermally isolated sub-regenerators. This approx- generator thermal mass ratio. imation will be validated by experiments and shown to yield accurate The response of four different regenerator thermal mass ratios were temperature predictions for regenerators with multiple sub-re- plotted in Fig. 2(d) for a regeneration process with air moving from hot generators. Having determined that the regenerator thermal mass ratio fl (TH = 1000 K) to cold (TL = 300 K), and it is shown that the re- is the sole parameter that in uences the system settling temperature, generator thermal mass ratio not only impacts the rapidity of the only Eq. (6) is required to model the settling temperature of re- system response, but also the system settling temperature. The value of generators composed of a number of sub-regenerators and the working the system settling temperature is an important consequence of the fluid. Since the settling time has been shown to be irrelevant, Eq. (6) regenerator thermal mass ratio since it indicates how much additional can be rewritten without successive time steps, and the p superscript is thermal energy the heater and cooler of the Stirling engine must either replaced with a subscript denoting the initial temperature (indicating add or reject, respectively, to attain target temperatures, TH and TL. the temperature at the start of each regeneration process), while the Not all Stirling engines operate at 3,500 rpm, and it is desirable to (p + 1) superscript is replaced with a subscript denoting the final tem- ensure that the regenerator has the necessary specifications to exchange perature (indicating the temperature at the end of each regeneration thermal energy with the working fluid within the given time frame process). We then simplify the heat transfer model by rearranging Eq. determined by the engine’s rotational speed. To do this, we must (6) for a system settling temperature, which is the condition when the compare the natural time scale for regeneration, τ, and the time working fluid and each sub-regenerator attain the same temperature: available in the cycle for regeneration to occur, treg. The natural time TTsys== f,, final T r final (14) scale can be obtained from Eqs. (5) and (6) and is expressed as follows: ρc r and Eq. (6) becomes: τ = r 2(1hγ+ ) (9) TγTf,, initial+ r initial Tsys = γ + 1 (15) The time available for regeneration to occur is one quarter of a rotation in the Stirling cycle, which is calculated as: Four versions of Eq. (15) are required to model the system settling temperature for a regenerator with multiple sub-regenerators. To model 15 treg = one regenerative cooling process, where the regenerator is absorbing ω (10) thermal energy from the working fluid, we require the following two
5 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557 equations: first stage of step ① the working fluid and first sub-regenerator tem- T (1) fi (1) (1) peratures begin at H and Tsys, L respectively, and the rst sub-re- TγTH + sys, L fl T (1) = generator absorbs thermal energy from the working uid resulting in sys, H (1) (1) γ + 1 (16) both mediums settling at a temperature of Tsys,H , which is the re- generative cooling process for the first sub-regenerator. Subsequently, TγT(1)i− ()i ()i ()i sys, H + sys, L the working fluid travels through the remaining two sub-regenerators, Tsys, H = γ()i + 1 (17) as shown in Fig. 3, and both the regenerator and working fluid settle at a temperature of T (3) after the third and final sub-regenerator. The where Eq. (16) is the system settling temperature of the first sub-re- sys,H working fluid must attain a temperature of T before the subsequent generator for one regenerative cooling process, and Eq. (17) is the L isothermal compression process; therefore, the Stirling engine’s cooler system settling temperature of each of the subsequent sub-regenerators, must reject an additional amount of thermal energy that is proportional with their order denoted by the superscript i. To model one re- ff (3) to the remaining temperature di erence, TTsys, H − L (sinceN = 3), generative heating process, where the regenerator is returning thermal which is shown as step ② in Fig. 3. For the subsequent regenerative fl energy to the working uid, we require the following two equations: heating process, denoted by the circled ③, the working fluid travels ()N ()N from the cold side through the 3 sub-regenerators in reverse order, as TγTL + sys, H T ()N = sys, L ()N shown in Fig. 3. The working fluid and third sub-regenerator tem- γ + 1 (18) (3) peratures begin at TL and Tsys,H respectively, and the third sub-re- fl TγT(1)i+ + ()i ()i generator returns thermal energy to the working uid resulting in both ()i sys, L sys, H (3) Tsys, L = mediums settling at a temperature of T , which is the regenerative γ()i + 1 (19) sys, L heating process for the third sub-regenerator. Subsequently, the where Eq. (18) is the system settling temperature of the N th or last sub- working fluid travels through the remaining two sub-regenerators in regenerator, and Eq. (19) is the system settling temperature of each of reverse order, as shown in Fig. 3, and both the regenerator and working fl (1) fi the subsequent sub-regenerators in reverse order since the working uid settle at a temperature of Tsys, L after the nal sub-regenerator. The fl fluid is flowing back through the regenerator. The regenerator thermal working uid must attain a temperature of TH before the subsequent mass ratio for each sub-regenerator is a fraction of the total regenerator isothermal expansion process; therefore, the Stirling engine’s heater thermal mass ratio, and in the case of a uniformly distributed mass, must add an additional amount of thermal energy that is proportional ff (1) each value of γ(i) would be equal to γ/N . It should be noted that to the remaining temperature di erence, TTH − sys,L, which is shown as ④ modelling a system with one sub-regenerator requires only Eqs. (16) step . The amount of energy that the regenerator absorbs and returns ff and (18). The superscript i, is the sub-regenerator location (i.e. second is proportional to the temperature di erence from the regenerative ① ③ sub-regenerator, third sub-regenerator, etc.) and N is the total number cooling and heating processes, shown in steps and respectively, of sub-regenerators that the regenerator is composed of, where which we denote as ΔTreg: 1 <
∞ ∞ n n n (1) γ2(− 1) ⎛ γ(2− 1) γ ⎞ TT=+H TL + sys, H ∑∑(1+ γ )(2n− 1) ⎜ (1+ γ )2n (1+ γ )n ⎟ n=1 ⎝n=1 ⎠ (20) To simulate steady operating conditions, we set the number of re- generator passes to infinity (n = ∞), which reduces the hot system settling temperature to:
(1) (1)γ + γ TTsys H = HL+ T , (2γ + 1) (2γ + 1) (21) A similar methodology is used to model regenerators with multiple sub-regenerators using Eqs. (16)–(19). We can use Eqs. (16)–(19) to calculate a regenerator effectiveness based on the resultant system settling temperatures. Fig. 3 shows the system settling temperatures for a regenerator with 3 sub-regenerators (N = 3) across a complete Stirling engine cycle, where steady engine Fig. 3. Schematic diagram of the heat transfer model once steady engine op- operation has been reached. The first step is denoted by the circled ① in eration has been reached for one complete Stirling cycle, illustrating the re- Fig. 3, which shows the temperature of the working fluid as the working generative heating and cooling processes for a regenerator composed of 3 sub- fluid travels from the hot side through the 3 sub-regenerators. In the regenerators, N = 3, and regenerator thermal mass ratio, γ = 20:1.
6 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557
ff QQ− The regenerator e ectiveness is only a function of the target tem- η in out ()N (1) Stirling = peratures TH or TL and the system settling temperatures Tsys,H or Tsys, L,so Qin (27) we can substitute Eq. (21), which is the hot system settling temperature for a single regenerator after steady operating conditions have been where: reached, to yield the regenerator effectiveness for a single regenerator: Qin=+mRT f Hln( λ ) Qnon− reg (28) γ ∊=reg Q =+mRTln( λ ) Q 21γ + (24) out f L non− reg (29) and: Similarly, for a regenerator with multiple sub-regenerators, the re- generator effectiveness for steady operating conditions simplifies to: V λ = max V (30) Nγ min ∊=reg (1)NγN++ (25) An additional Qnon- reg term in Eqs. (28) and (29) is required since it represents the amount of supplemental thermal energy that the heater Since Stirling engines operate primarily in steady operating condi- and cooler of the Stirling engine must either add or reject, respectively, tions, with only a short warm-up period, the remainder of the analysis to attain target temperatures, TH and TL, due to non-ideal regeneration. will consider the thermal behaviour once steady operating conditions The regeneration process in the Stirling cycle is isochoric, hence the have been reached. additional heat added and rejected is:
Qnon− reg =−∊(1reg )mc f v, f ( T H − T L) (31) 2.4. Stirling engine efficiency model Eq. (31) is then substituted into Eqs. (28) and (29), which are then The Stirling engine efficiency is commonly listed as the Carnot ef- substituted into Eq. (27) to generate an expression for the thermal ef- ficiency: ficiency of the Stirling engine that incorporates the influence of the regenerator effectiveness (Eq. (25)), as follows: TL ηCarnot==−η Stirling, ideal 1 TH (26) RT()ln()HL− T λ ηStirling = RTHregvfHLln( λ )+−∊ (1 ) c, ( T − T ) (32) however, this efficiency considers ideal regeneration (∊=reg 1). The derivations in the previous section revealed that regenerator effec- It is shown in Eq. (32) that the efficiency of the Stirling engine is not tiveness is a function of the regenerator thermal mass ratio and number only dependent on the target temperatures, TH and TL, as seen in Eq. of sub-regenerators. For non-ideal regeneration, the Stirling engine ef- (26), but is also a function of the compression ratio, λ, the properties of
ficiency must also be a function of the regenerator thermal mass ratio the employed fluid, cvf, and R (specific gas constant), and the re- and number of sub-regenerators, and we derive an expression to cap- generator effectiveness, ∊reg, which is a function of the regenerator ture this dependence. The Stirling engine efficiency with non-ideal re- thermal mass ratio, γ, and number of sub-regenerators, N, according to generation is expressed as: Eq. (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) PID controller, (6) air-flow regulator, and (7) data acquisition module.
7 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557
3. Experimental apparatus distributed evenly across the sub-regenerators. It is shown that as the number of sub-regenerators increases, the temperature of the last sub- We designed and built a regenerator test apparatus to validate the regenerator and correspondingly the exiting working fluid becomes heat transfer model developed in Section 2.3. The regenerator test ap- closer to the cold target temperature, TL. Therefore the temperature paratus, which is shown in Fig. 4, consisted of two linear pneumatic difference between the exiting working fluid and cold target tempera- ()N ff actuators that were programmed to move the hot and cold pistons in an ture (TTsys, H − L) decreases and regenerator e ectiveness increases, oscillatory manner for a constant volume regeneration process at an according to Eq. (23). Fig. 6 also reveals that regenerators composed of equivalent rotational speed of 180 rpm. The apparatus also employed a multiple sub-regenerators maintain a linear temperature distribution, PID controlled electric heat source, an air-flow regulator to control the which is crucial for enhancing the regenerator effectiveness. This linear motion of the pistons, K-type thermocouples for temperature mea- temperature profile also indicates that average regenerator tempera- surements, and a data acquisition module to process the signal gener- tures should be evaluated using an arithmetic mean between the first ated by the thermocouples. The thermocouples had an accuracy of ± 1.5 and last sub-regenerators, rather than using a log-mean temperature K. Each sub-regenerator consisted of a 304 stainless steel wire mesh difference, as is often done in the literature. In summary, increasing the with a wire diameter of approximately 0.04 mm, which are the same number of sub-regenerators enables the last sub-regenerator and exiting specifications that were reported for the GPU-3 Stirling engine [44,45]. working fluid to achieve a temperature closer to the target temperature, The two regenerator configurations that were examined in the experi- which enhances the regenerator’seffectiveness. We will now quantify ments were spaced and unspaced sub-regenerators. The number of sub- how the number of sub-regenerators and regenerator thermal mass regenerators was set at both 12 and 16 sub-regenerators, which corre- ratio influence the effectiveness of Stirling engine regenerators. sponds to a regenerator thermal mass ratio of γ = 4.01:1 and γ = 5.35:1, respectively. An unspaced sub-regenerator configuration de- 4.2.2. Multiple sub-regenerator effectiveness notes that there is physical contact between neighbouring sub-re- The regenerator effectiveness from Eq. (25) is plotted against the generators, while a spaced configuration indicates that each sub-re- number of sub-regenerators with various thermal mass ratio values in generator was separated by a ceramic insulation ring to actively Fig. 7, which shows that increasing both the regenerator thermal mass minimize the local axial conduction between each sub-regenerator. ratio and number of sub-regenerators enhances the effectiveness of the Thermocouples were placed between every group of four sub-re- regenerator. To determine the maximum attainable effectiveness, an generators to record the temperature at the 2nd, 6th, 10th, and 14th infinite regenerator thermal mass ratio (γ =∞:1) is substituted into Eq. sub-regenerators. The heat source and heat sink temperatures were (25) to yield: measured using thermocouples that recorded the inside air tempera- N tures of both hot and cold cylinders, and their respective temperatures ∊=reg, max N + 1 (33) were 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 40 min which represents the maximum effectiveness that a regenerator can before recording measurements to ensure that the temperatures of the attain based on the number of sub-regenerators employed (N), analo- working fluid and regenerator reached steady operating conditions. The gous to the Carnot efficiency for heat engines. The maximum effec- results gathered from the experiments were then compared against tiveness from Eq. (33) is plotted in Fig. 7 to show the upper limitation those obtained from the heat transfer model to assess its validity. on the regenerator effectiveness value versus the number of sub-re- generators used. 4. Results and discussion We can also determine how to achieve a specific value for re- generator effectiveness based on the number of sub-regenerators (N) 4.1. Single regenerator analysis and regenerator thermal mass ratio (γ) used in the design of a Stirling engine regenerator. Fig. 8 is a plot of Eq. (25), which shows how to We first analyze a single regenerator to determine the limitations for select the values of both the number of sub-regenerators and the regenerator effectiveness of a single regenerator setup. The regenerator effectiveness described in Eq. (24) is only a function of the regenerator thermal mass ratio, and thus is plotted in Fig. 5 against a range of values for the regenerator thermal mass ratio. It can be seen in Fig. 5 that the maximum effectiveness a single regenerator can attain is 50%. As the regenerator thermal mass ratio increases, the regenerator effectiveness increases asymptotically towards a maximum value of 50%. This maximum 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 demonstrated that single regenerators, such as parallel plate regenerators, with spe- cifications that satisfy the thermal equilibrium approximation, cannot attain a regenerator effectiveness greater than 50%. The following sections in this study will reveal how to obtain a regenerator effec- tiveness greater than 50%, and how to design for an effectiveness greater than 95%.
4.2. Multiple sub-regenerator analysis
4.2.1. Influence of number of sub-regenerators Using Eqs. (16)–(19), the temperature distributions of regenerators composed of various numbers of sub-regenerators for a regenerative ff cooling process were plotted in Fig. 6. For this analysis the regenerator Fig. 5. Regenerator e ectiveness, ∊reg, of a single regenerator, N = 1,asa thermal mass ratio remained constant and the regenerator mass was function of the regenerator thermal mass ratio, γ.
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Fig. 6. Temperature distribution within regenerators with varying number of Fig. 8. Plot of the regenerator effectiveness curves (for 95%, 96%, 97%, and sub-regenerators, N, and constant regenerator thermal mass ratio, γ = 20:1, 98%) and the corresponding values of the regenerator thermal mass ratio, γ, during a regenerative cooling process for steady operating conditions. and number of sub-regenerators, N, required to produce a desired effectiveness.
4.2.3. Influence of mass distribution An 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, the regenerator thermal mass ratio values in Eqs. (16)–(19) were varied for each of the sub-regenerators over a wide range of distributions. The analysis revealed that the regenerator effectiveness is a function of the non-uniformity of the mass distribution, which is described by the standard deviation of the regenerator thermal mass ratio, σ, and the arrangement of the distribution did not matter. The regenerator effec- tiveness is plotted against the standard deviation in Fig. 9, which shows that as the standard deviation increases, which corresponds to an
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 the maximum regenerator effectiveness when γ =∞:1. regenerator thermal mass ratio to achieve regenerator effectiveness values of 95%, 96%, 97%, and 98%. Any values above and to the right of the effectiveness lines will result in a higher regenerator effective- ness. It is also shown in Fig. 8 that the effectiveness lines approach a vertical asymptote as the number of sub-regenerators decreases. This occurs since each regenerator effectiveness has a corresponding minimum required number of sub-regenerators to attain the specified effectiveness. For example, to attain 95% regenerator effectiveness, the minimum required number of sub-regenerators is 19, and it is im- possible for a regenerator with less than 19 sub-regenerators to attain 95% effectiveness. Stirling engine designers can use Eq. (25) to select a ff regenerator effectiveness they desire based on the selected regenerator Fig. 9. Regenerator e ectiveness, ∊reg, as a function of the standard deviation, thermal mass ratio and number of sub-regenerators employed. σ, of the distribution of the regenerator thermal mass ratio among the sub- regenerators for three different cases. The black points indicate the ∊reg values that coincide with a single regenerator.
9 A.S. Nielsen, et al. Applied Energy 253 (2019) 113557 increasingly uneven distribution of the sub-regenerator mass, the re- 4.3.1. Effect of number of sub-regenerators generator effectiveness decreases until it reaches the value for a single The number of sub-regenerators directly influences the effectiveness regenerator (with the corresponding regenerator thermal mass ratio). of the regenerator, which subsequently impacts the thermal efficiency The reason that the regenerator effectiveness approaches the value for a of the Stirling engine, as seen in Eq. (32). In this portion of the para- single regenerator as the standard deviation increases is because an metric analysis, the number of sub-regenerators was varied from 1 to 25 extremely uneven mass distribution approximates that of a single re- for various target temperature ratios, TTHL/ , while the compression generator. It is concluded from this analysis that the optimum sub-re- ratio was fixed at a value of 10:1, the employed working fluid was generator mass distribution for a Stirling engine regenerator is a uni- hydrogen, and the regenerator thermal mass ratio was fixed at a value form mass distribution. of 20:1. It is shown in Fig. 11(a) that as the number of sub-regenerators increases, Stirling engine efficiency also increases, as expected with an ff 4.2.4. Experimental validation increasing regenerator e ectiveness. It is seen, however, that the in- ffi Validation of the heat transfer model is necessary to ascertain that cremental increase in Stirling engine e ciency decreases as the number the derived expressions can generate accurate predictions of the of sub-regenerators increase, illustrated by the lines becoming pro- thermal behaviour in Stirling engine regenerators. To ensure that the gressively closer together, which indicates that there are diminishing experiments were eligible to be compared against the heat transfer returns as the number of sub-regenerators is increased. Therefore, model developed in Section 2.3, the conditions described in Eqs. (3) and Fig. 11(a) can be used by Stirling engine designers to select an (13) had to be satisfied. The highest resultant Biot number from the experiments was 6.5× 10−4, and the highest regeneration time ratio value, ζ , was 0.82, thus demonstrating that the experiments satisfied the conditions. The experimental and theoretical results of regenerative heating and cooling processes for spaced and unspaced sub-regenerator configurations 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 temperatures of TH = 353 K (80 °C) and TL = 303 K (30 °C), with percent differences ranging between 0.14% to 2.4% for 12 sub-regenerators (0.47 K to 7.73 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 that the temperature distribution within the regenerator is linear. Since there is good agreement between the heat transfer model and experi- mental results, it is confirmed that the heat transfer model has been validated and can be used in the design of Stirling engine regenerators to predict their effectiveness based on the selected values of regenerator thermal mass ratio and number of sub-regenerators.
4.2.5. How to thermally isolate sub-regenerators Another interesting question that arises from the results of this study is: how can a designer ensure that there is sufficient thermal isolation between sub-regenerators to yield a regenerator configuration that has multiple 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 meshes used 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 of 0.53% 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 attain the required thermal resistance to produce a sufficient temperature distribution, 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 important outcome when designing regenerators. We perform a parametric ana- lysis using Eq. (32), to examine the effects of the regenerator thermal Fig. 10. Comparison between the results predicted by the heat transfer model mass ratio and number of sub-regenerators on the efficiency of Stirling and experimental results for spaced and unspaced sub-regenerators for re- engines, and compare the resultant efficiencies with that of the Carnot generative heating and cooling processes with (a) N = 12 and γ = 4.01:1, and cycle (ideal regeneration). (b) N = 16 and γ = 5.35:1.
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fixed at a value of 25. It is shown in Fig. 11(b) that as the value of the regenerator thermal mass ratio increases, the Stirling engine’sefficiency also increases, as expected with an increasing regenerator effectiveness. There are also diminishing returns as the regenerator thermal mass ratio approaches ∞:1, as illustrated by the lines becoming progressively closer together. Similarly to the number of sub-regenerators, Fig. 11(b) can be used by Stirling engine designers to select an appropriate value of the regenerator thermal mass ratio required to achieve a desired value for the Stirling engine efficiency. The selection of an appropriate value of the regenerator thermal mass ratio will avoid the use of an unnecessary amount of regenerator mass.
5. Conclusions
In this work, a discrete one-dimensional heat transfer model was developed to determine which parameters influence the effectiveness of Stirling engine regenerators. The heat transfer model revealed that the regenerator thermal mass ratio and number of sub-regenerators were the two parameters that influence the regenerator effectiveness and quantify how they influence it. The use of multiple sub-regenerators was 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 ratio and number of sub-regenerators resulted in an increase in regenerator effectiveness and a corresponding increase in the Stirling engine effi- ciency. It was also found that the regenerator effectiveness has a maximum attainable value, which is limited by the number of sub-re- generators used. For example, to attain a regenerator effectiveness of 95%, a minimum of 19 sub-regenerators must be used. It was also de- termined that a uniform sub-regenerator mass distribution provides the highest regenerator effectiveness. Experiments using a regenerator test apparatus, with a temperature range of TH = 353 K (80 °C) and TL = 303 K (30 °C), validated the heat transfer model, and demonstrated that stacking sub-regenerators, such as wire meshes, provides sufficient thermal resistance to generate a temperature distribution throughout the regenerator, which enhances regenerator effectiveness. As a result of this study, Stirling engine designers can select a value of regenerator thermal mass ratio and number of sub-regenerators to attain a desired regenerator effectiveness and/or a desired value for the Stirling engine efficiency.
Acknowledgements
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