Thermodynamic cycle analysis of heat driven elastocaloric cooling system

Journal Pre-proof Thermodynamic cycle analysis of heat driven elastocaloric cooling system Jianming Tan, Yao Wang, Shijie Xu, Huaican Liu, Suxin Qian PII:

S0360-5442(20)30368-6

DOI:

https://doi.org/10.1016/j.energy.2020.117261

Reference:

EGY 117261

To appear in:

Energy

Received Date: 28 October 2019 Revised Date:

21 January 2020

Accepted Date: 25 February 2020

Please cite this article as: Tan J, Wang Y, Xu S, Liu H, Qian S, Thermodynamic cycle analysis of heat driven elastocaloric cooling system, Energy (2020), doi: https://doi.org/10.1016/j.energy.2020.117261. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Thermodynamic cycle analysis of heat driven elastocaloric cooling system Jianming Tan1,2, Yao Wang3, Shijie Xu3, Huaican Liu1,2, Suxin Qian3* 1

State Key Laboratory of Air-Conditioning Equipment and System Energy Conservation Zhuhai, Guangdong, China, 519070 2 3

GREE Electric Appliances Inc., Zhuhai, Guangdong, China, 519070

Department of Refrigeration and Cryogenic Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, China, 710049 * Corresponding Author: [email protected]

Abstract The conventional elastocaloric cooling system is powered by mechanical drivers with more than 500 times mass over refrigerant mass, whereas the shape memory alloy actuator and heat driven cycle provide a new path for higher system compactness. Based on the thermodynamic and mechanical constraints between the actuator shape memory alloy and the refrigerant super-elastic alloy, the cycle model is implemented to investigate the characteristics of the cycle efficiency, mass ratio and driving temperature difference in terms of length ratio and cross-sectional area ratio. In addition, the impacts of Young’s modulus, transformation strain and Clausius-Clapeyron coefficient are studied. Based on the multi-objective optimization technique, regarding the three different combinations of actuator and refrigerant materials, the optimum normalized COP occurs when the MDTD ranges from 52 K to 59 K, which does not further increase with higher driving temperature, implying that low-grade thermal energy at a temperature less than 100°C is most economic to drive such a cycle. On the other hand, the heat driven cycle can be activated by MDTD down to 11 K, indicating a significant potential to harvest low-grade thermal energy. This study can promote future prototype development for solar-driven refrigerators and waste heat recovery for electronic devices. Keywords: not-in-kind cooling, solid-state cooling, thermoelastic cooling, caloric cooling, low-grade thermal energy, optimization

1. Introduction Solid-state caloric cooling technologies have experienced fast developments in the past decade [1]. Among them, elastocaloric cooling exploits the latent heat from shape memory alloy that is cyclically loaded and unloaded by stress variation [2], although the earliest prototype of this technology was built with natural rubber [3]. Usually, the refrigerant experiences four sequential steps, loading, heat rejection, unloading and cooling. The shape memory alloy could be operated by a single-stage cycle [4], active regeneration cycle [5], and cascaded cycle [6]. A single-stage water-based prototype has been developed by UMD with 4.7 K temperature lift using Ni-Ti tubes driven by compressive loading [7], whereas the non-fluid prototype developed by Saarland [8] and KIT implementing SMA sheets achieved more than 13 K temperature lift at zero cooling load condition [9,10]. These non-fluid based elastocaloric cooler has the advantage for chip-scale cooling application since low thermal inertia facilitates high cycling frequency and thus high cooling power density, while the efficiency of elastocaloric cooling is still intrinsically higher than that of the thermoelectric cooling technology [11]. On the other hand, the fluid-based active regeneration cycle has demonstrated 20 K temperature lift at zero load condition by a prototype developed at DTU [12,13] using stacked Ni-Ti plates, which reveals the potential for large scale applications. Cascaded cooler with three stages was developed with a no-load temperature lift of 15 K, however, that temperature lift is still less than the adiabatic temperature change of the SMA, and therefore significant efforts are still needed to address the heat transfer resistance between stages [6]. It should be mentioned that the reported temperature lift and cooling power from the above prototypes are significantly less than those estimated by numerical simulations [14– 16], and extensive studies are yet needed to bridge such gap. Although an elastocaloric cooling cycle requires four stages to complete, a system does not necessarily require a cyclic operation. Concept demonstrator developed by Maryland and Saarland show that continuous rolling of Ni-Ti wires or wire assemblies between two rotating disks with an inclined angle is applicable [2,17,18]. When applying air as the heat transfer fluid, such a concept fits well with the air-conditioner application since no additional heat transfer fluid is required. A continuous prototype was proposed by US Army Research Lab [19], in which the Ni-Ti wire is arranged in a loop configuration and bending is used to drive

the phase transformation. Other concept designs may be found in [20]. Nonetheless, the continuous prototype has a fundamental difference from the principle of the active regeneration cycle, and therefore paths for realistic temperature lift is needed for prototypes with continuous operation mechanism. Most of the developed elastocaloric cooling prototypes are driven by the linear motor, screw-jack or hydraulic driver uniaxially [21]. Recently, twisting was proposed as a new way to load the coil or spring made of rubber and SMA, which delivers similar adiabatic temperature change on the material level, empowering the application of conventional rotating motor while enabling application of active-regeneration cycle [22]. Other approaches, such as magnetostriction, has been proposed as a more compact linear driver option [23]. When considering the solid-state refrigerant, the SMA requires a large force (~102MPa) and a small displacement (~10-2 strain). This characteristic is not matched by the state-of-the-art driver in current prototypes, wherein the force and stroke are usually well-balanced, and consequently, an oversized driver is usually equipped. Quantitatively speaking, the driver to refrigerant mass ratio of current prototypes is still higher than 500:1 [24]. Heat driven elastocaloric cooling cycle was proposed to solve this challenge by providing the solid-state refrigerant a well-matched driver [24], which is a high temperature SMA actuator. Since both the driver and the refrigerant can be considered as the same category of materials, their force-displacement characteristics are similar. More precisely speaking, if the actuator SMA and the refrigerant SMA are both Ni-Ti for instance, their transformation stress will be equivalent, and ideally, they should have equal mass. This means that the driver to refrigerant mass ratio of the newly proposed cycle has a magnitude of 1:1, which is two magnitudes better than the state-of-the-art. In our previous study, numerical simulation demonstrated the applicability of the heat driven cycle [24]. However, numerical simulation is time-consuming, since additional resources are needed to compute the coupled phase transformation process between the driving SMA and refrigerant. The sufficiently large number of nodes and sufficiently small time steps are required to facilitate satisfactory accuracy, which leads to more than one hour to complete each case study. Such time consumption inhibits the optimum design of such a system, since more than 10 thousand model evaluations may be required by an optimization solver [25]. On the

other hand, thermodynamic analysis is a powerful tool to shed light on the fundamentals of the heat driven elastocaloric cooling cycle, and most importantly, is timewise efficient to evaluate. Therefore, the goal of this study is to better understand the performance potential of the heat driven elastocaloric cooling cycle using thermodynamic analysis and multi-objective optimization method.

2. Heat Driven Cycle In the heat driven cycle, the refrigerant SMA works between the ambient temperature Th and the cooling temperature Tc. In Figure 1 (a), beginning from state R1, the material is in full austenite phase at zero stress condition where martensite fraction xM equals zero. It can transform to martensite upon sufficient driving stress. On the contrary, beginning from state D1, the actuator SMA is in full martensite phase at ambient temperature Th. Once heated beyond the source temperature Tg, the actuator SMA transforms into austenite, following the path from D1 to D2. Due to the mechanical coupling between the actuator SMA and the refrigerant SMA, the transformation of actuator SMA provides the force and displacement to drive the transformation of the refrigerant SMA. Thus the actuator changes its state from D2 to D4 via D3 upon heating. This is corresponding to the loading process by a mechanical driver, accompanied by the R1 to R2 process for the refrigerant SE. When the actuator SMA is cooled down by ambient, it transforms back into martensite following the path from D4 to D1 via D5 and D6, while the force and displacement to the refrigerant SMA diminish. This is corresponding to the unloading process by a mechanical driver, i.e. state R4 to R5. The aforementioned cycle principle is presented in a temperature-stress diagram in Fig. 1 (a). This cycle can be operated in a system demonstrated in Fig. 1 (b), in which there are four assemblies of actuator SMA wires and one assembly of refrigerant SE plates. One side of the actuator SMA and refrigerant SE are mounted onto the static frame, while the other side is attached to a moving coupling to transfer force and displacement from the actuator. Upon heating, the actuator wires would shrink, thus allowing the moving coupling to move towards the actuator wires, which would stretch the refrigerant plates.

Stress

Refrigerant SE Driving force

R4

Actuator SMA xM = 1

xM = 1

xM = 0

R3 ∆Tad R2

xM = 0

D5 D3 D4

D6 R6 R1 D1

Ms AsR5 Tc

Th Ms As

D2

Temperature

Tg

(a) Cycle schematic

(b) Schematic of the mechanical design to operate the above cycle Figure 1: Cycle concept of a single-effect heat driven elastocaloric cooling cycle and system. (As is phase transformation temperature of SMA from martensite to austenite at zero stress state, vice versa for Ms, xM represents mass fraction of martensitic phase)

3. Thermodynamic Analysis Method 3.1 Mechanical Coupling In the heat driven cycle, the driving force from the actuator SMA is equal to the force required to induce transformation of the refrigerant SMA. The displacement of the actuator SMA and that of the refrigerant SMA are identical as well. These two constraints are stated in Eqs. (1-2). Force     

(1)

Displacement  |  |  |  |

(2)

The constitutive equation of SMAs states the relationship between the strain, the strain variation due to martensitic transformation, and the elastic deformation, as shown in Eqs. (3-4).

  , ∆, +

 

(3)

  , ∆, +

 

(4)

Whether or not driven by the SMA actuator, the refrigerant SMA always undergoes a full transformation to release the maximum latent heat, which is a common practice for the normal operation of an elastocaloric cooling system. In reality, fatigue has to be taken into account and a compromise is required [26]. For simplicity, the martensitic mass fraction is set to change from 0 to 1 for the refrigerant SMA, which yields Eq. (5). ∆,  1

(5)

Based on the above five equations, the variation of the martensitic mass fraction in the actuator SMA can be solved, as shown in Eq. (6). ∆, 

 ,     ∙ +! + ∙ "∙  ,    ,

(6)

Note that the actuator SMA cannot provide anymore driving force and displacement, when the martensitic phase fraction changes from 1 to 0, i.e. ∆, ≤ 1. When it is equal to unity, the actuator SMA provides the maximum force/stress for the refrigerant SMA. Consequently, the following maximum stress in Eq. (7) can be derived based on the aforementioned equilibrium analysis of the mechanical coupling between two SMAs. Full transformation is the only assumption involved here.

,$ %

  1 −   ∙  , ,  ∙      , + ∙  

(7)

3.2 Thermodynamic Coupling In addition to the mechanical coupling equations, the two SMAs follow the classic Clausius-Clapeyron equation for any first-order phase change process, as shown in Eq. (8). ',  ), '(

(8)

wherein Cr or Ca is the Clausius-Clapeyron coefficient of the refrigerant SMA or the actuator SMA. When applying the Clausius-Clapeyron equation to the actuator SMA, and given the phase transformation temperatures TAs and TAf, the following constraint equations in Eqs. (9-10) can be derived. ), *(+ − (,-, − ∆, .(,/, − (,-, 01   ,23  ),, *(4 − (-, − ∆, .(/, − (-, 01   ,523 

 

,23

 

,523

(9)

(10)

By subtracting Eq. (9) and Eq. (10), the driving temperature difference of the actuator SMA, i.e. Tg – Th, can be derived in Eq. (11). Note that Eqs. (9-10) imply two-phase conditions for the actuator SMA. If Tg increases beyond the threshold that the actuator SMA becomes single-phase austenite after heating, or if Th decreases below the threshold that the actuator SMA becomes single-phase martensite after cooling, the driving temperature difference will be larger than the temperature difference specified in Eq. (11). Therefore, based on the thermodynamic equilibrium analysis, Eq. (11) essentially states the minimum driving temperature difference (MDTD) of a heat-driven elastocaloric cooling cycle. (+ − (4 ≈

 ∆ ∙ + (,-, − (-, + ∆, .(,/, − (,-, + (-, − (/, 0

),,

(11)

3.3 Efficiency While the new cycle can be driven by limitless renewable heat sources, it is still important to understand the efficiency characteristics of such a cycle. More importantly, the heat driven cycle is a combination of the heat engine cycle and the refrigeration cycle, and therefore the theoretical efficiency limit of such a cycle can be given by the Carnot efficiency as described by Eq. (12), i.e. product of Carnot heat engine efficiency and Carnot refrigerator COP. )789 :;< 

(+ − (4 (= ∙ (+ (4 − (=

(12)

In each cycle, the heat absorbed by the actuator SMA is described by Eq. (13). >+  ?.(+ − (4 0@1 − ABC D + ∆ℎ ∙ ∆, + FG, ∙ HI

(13)

where ηHR is the heat recovery efficiency [27], and the MR is the mass ratio between the refrigerant SMA and the actuator SM. The MR is calculated by Eq. (14). HI 

  J ∙ ∙  J

(14)

The cooling provided by the refrigerant SE can be evaluated by Eq. (15). >=  ∆ℎ + FK, − ?L @(4 − (= D@1 − ABC D

(15)

The cycle efficiency is given by Eq. (16). )78 

>= ∙ HI >+

(16)

Note that the properties and important parameters are listed in Table 1 and Table 2. Table 1. Properties used in this study.

CA (MPa·K-1) CM (MPa·K-1) E (MPa) TAf (K) TAs (K) TMs (K) TMf (K) εT (-) c (J·g-1·K-1) ρ (kg·m-3) ∆h (J·g-1) w+ (J·g-1) w- (J·g-1)

Ni-Ti [28] Refrigerant (r) 7.6 7.3 30,000 288 268 258 238 0.03 0.5 6,500 10 5.6 4.1

Actuator (a) 7.6 7.3 30,000 343 323 313 293 0.03 0.5 6,500 10 NA NA

Cu-Zn-Al [5] Refrigerant (r) -1 CA (MPa·K ) 3.22 -1 CM (MPa·K ) 3.18 E (MPa) 80,000 TAf (K) 272 TAs (K) 268 TMs (K) 268 TMf (K) 264 εT (-) 0.056 c (J·g-1·K-1) 0.41 -3 ρ (kg·m ) 7,710 -1 ∆h (J·g ) 4.5 -1 w+ (J·g ) 1.58 -1 w- (J·g ) 0.78

Table 2. Nominal conditions for cycle analysis in this study. Variable

Value

Sr / Sa Lr / La

0.5 0.5 35

Th [°C] Tc [°C] ηHR

15 0.70

Actuator (a) 3.22 3.18 80,000 327 323 323 319 0.056 0.41 7,710 4.5 NA NA

4. Results 4.1 Verification Before discussing the cycle characteristics, the thermodynamic model is first validated by the numerical simulation results from the previous study [24]. The simulation model can predict the cooling performance degradation when the driving temperature Tg becomes insufficient for the actuator SMA to provide the minimum force inducing the complete phase transformation for the refrigerant SE. Consequently, when the cooling performance starts to decrease, the driving temperature is corresponding to the MDTD. In Fig. 2, the simulated cooling temperature lift (defined as Th - Tc at zero heat load condition) is plotted against different driving temperature Tg under different conditions. Comparing to the simulated, the predicted MDTD from the thermodynamic model in this study follows the trend with deviations of up to 2 K. Validated by the more fundamental simulation approach, the thermodynamic model and the results discussed in the following sections can be used to guide

(a)

30 20 10 0 45

55 65 75 Driving Temperature [° ° C]

Simulation by [24]

Temperature Lift [K]

Temperature Lift [K]

40

(c) 30 20 10 0 50 60 70 Driving Temperature [° ° C]

Simulation by [24]

(b) 30 20 10 0 50

Analytical MDTD

40

40

40

85

Analytical MDTD

60 70 80 Driving Temperature [° ° C]

Simulation by [24]

Temperature Lift [K]

Temperature Lift [K]

design of the heat driven elastocaloric cooling or heat pump systems.

90

Analytical MDTD

40

(d) 30 20 10 0 40

50 60 Driving Temperature [° ° C]

Simulation by [24]

Analytical MDTD

Figure 2. Simulated cooling temperature lift at zero heat load against different driving temperature to verify the predicted MDTD in Eq. (11). (a) Sr / Sa = 0.265, Lr / La = 0.333, MDTD = 35.2 K; (b) Sr / Sa = 0.397, Lr / La = 0.333, MDTD = 44.8 K; (c) Sr / Sa = 0.227, Lr / La = 0.333, MDTD = 31.8 K; (d) Sr / Sa = 0.177, Lr / La = 0.333, MDTD = 29.4 K. Red dashed lines refer to the MDTD predicted by the analytical thermodynamic model in this study. 4.2 Impact of Geometries According to the thermodynamic model from the previous section, cycle performance only depends on intensive parameters such as length ratio or cross-sectional area ratio, which are the two key geometric parameters to be studied. Regarding different combinations of actuator and refrigerant materials, the results vary significantly. Fig. 3 shows the results when Ni-Ti is implemented for both the actuator and the refrigerant. According to Fig. 3 (a), as the cross-sectional area of the actuator grows relatively to that of the refrigerant, the longer refrigerant is applicable to achieve a minimum of 600 MPa stress as driving stress, however, Lr / La does not exceed 0.6 under all investigated circumstances. From Fig. 3 (b), though the ideal mass ratio is one as pointed earlier, to satisfy the 600 MPa requirement for the refrigerant SE, the practical mass ratio ranges from 4 to 20. When considering the cycle efficiency, Fig. 3 (c) and (d) dictate a maximum COP of 0.2 when the driving temperature is approximately 80 K. The COP reduces with smaller Sr / Sa and Lr / La, resulting in an increment of mass ratio and consequently more heat input from the driving source. Meanwhile, the MDTD reduces to approximately 20 K. In other words, oversizing the actuator favors low driving temperature but suffers unfavorable COP. It should be noted that as the driving temperature difference reduces, so does the Carnot COP, and thus it is worthy to discuss the COP / COPCarnot in the next section. 20

600 MPa Contour Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1 Sr / Sa = 2

1600 1200

Actuator to refrigerant Mass ratio [-]

Max. stress [MPa]

2000

800 400 0

15

10

Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1 Sr / Sa = 2 600 MPa

5

0

0

0.2

0.4

0.6

Lr / La [-]

(a)

0.8

1

0

0.2

0.4

0.6

Lr / La [-]

(b)

0.8

1

0.3

Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1

Sr / Sa = 0.1 Sr / Sa = 0.2 Sr / Sa = 0.5 Sr / Sa = 0.8 Sr / Sa = 1 600 MPa contour

140 120

0.2

MDTD [K]

COP [-]

0.25

160

Sr / Sa = 0.2 Sr / Sa = 0.8 600 MPa contour

0.15 0.1

100 80 60 40

0.05

20

0

0 0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

Lr / La [-]

Lr / La [-]

(c)

(d)

0.5

Figure 3: Impact of length ratio and cross-sectional area ratio for Ni-Ti as actuator and refrigerant, plotted with 600 MPa contour to show the feasible domain. (a: maximum driving stress, b: mass ratio based on Eq. 14, c: COP based on Eq. 16, d: MDTD based on Eq. 11.)

When the refrigerant is switched to Cu-Zn-Al instead of Ni-Ti, the cycle characteristics become very different, as shown in Fig. 4. Since Cu-Zn-Al has a much larger transformation strain than that of Ni-Ti, a longer actuator is required, as Lr / La needs to be lower than 0.54 otherwise the actuator cannot provide any driving stress. While the COPs in Fig. 4 are similar to those values in Fig. 3, the corresponding MDTDs are much smaller, which are less than 45 K even for cases when Sr / Sa = 1. This is due to the fact that Cu-Zn-Al requires much less stress to induce phase transformation as a refrigerant, and therefore less temperature difference is needed to provide the driving force. However, the slope of MDTD versus Lr / La in Fig. 4 (d) is steeper than that of Fig. 3 (d), which is due to the larger transformation strain of the refrigerant that requires more displacement capacity from the actuator. 20 200 MPa Contour

Actuator to refrigerant mass ratio [-]

Maximum stress for refrigerant [MPa]

2000 Sr / Sa = 0.1

1600

Sr / Sa = 0.5 1200

Sr / Sa = 1 Sr / Sa = 2

800 400 0

15 Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1 Sr / Sa = 2 200 MPa contour

10

5

0 0

0.2

0.4

0.6

Lr / La [-]

(a)

0.8

1

0

0.2

0.4

0.6

Lr / La [-]

(b)

0.8

1

0.4

Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1

0.35

0.25 0.2 0.15

60

Sr / Sa = 0.2 Sr / Sa = 0.8 600 MPa contour

50 40 30

0.1

20

0.05

10

0

Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1

70

MDTD [K]

COP [-]

0.3

80

Sr / Sa = 0.2 Sr / Sa = 0.8 200 MPa contour

0 0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

Lr / La [-]

Lr / La [-]

(c)

(d)

0.4

0.5

Figure 4: Impact of length ratio for Ni-Ti as actuator and Cu-Zn-Al as refrigerant, plotted with 200 MPa contour to show the feasible domain. (a: maximum driving stress, b: mass ratio based on Eq. 14, c: COP based on Eq. 16, d: MDTD based on Eq. 11.)

When the Cn-Zn-Al alloy is implemented for actuator and Ni-Ti is used as the refrigerant, the cycle performance becomes different, as plotted in Fig. 5. Note that the actuator is capable to provide more displacement due to higher transformation strain, and consequently, the longer refrigerant becomes possible, i.e. Lr / La can be greater than 1 according to Fig. 5 (a). Subsequently, the mass ratio of this combination is larger than those of Fig. 3 and Fig. 4 at the same geometric ratios. Most importantly, the value of COP becomes much higher, in which the maximum COP reaches 0.89 when Sr / Sa = 2. This is because Cu-Zn-Al has smaller latent heat than that of the Ni-Ti. Comparing with the data from Fig. 3, the refrigerant side latent heat and cooling capacity do not change. When changing the actuator material from Ni-Ti to Cu-Zn-Al, smaller latent heat indicates that the input heat reduces. Consequently, with identical cooling but less input heat, the case in Fig. 5 becomes more efficient than the case in Fig. 3. Nonetheless, the tradeoff is a higher MDTD, as shown in Fig. 5 (d). While most cases the MDTD is larger than 50 K, oversizing the actuator towards Sr / Sa = 0.5 can lead to MDTD from 23 to 26 K, with a COP of less than 0.19. We will discuss the efficiency more systematically in the coming section.

20 600 MPa Contour Sr / Sa = 0.1 Sr / Sa = 0.5 Sr / Sa = 1 Sr / Sa = 2

1600 1200

Actuator to refrigerant mass ratio [-]

Maximum stress for refrigerant [MPa]

2000

800 400

15 Sr / Sa = 0.5 Sr / Sa = 1 Sr / Sa = 2 Sr / Sa = 0.1 600 MPa contour

10

5

0

0 0

0.5

1

1.5

2

0

0.2

0.4

Lr / La [-]

(a) 1.2

1

600

Sr / Sa = 0.5 Sr / Sa = 1.5 600 MPa contour

0.8 0.6 0.4 0.2

Sr / Sa = 0.1 Sr / Sa = 1 Sr / Sa = 2

500

MDTD [K]

COP [-]

0.8

(b)

Sr / Sa = 0.1 Sr / Sa = 1 Sr / Sa = 2

1

0.6

Lr / La [-]

Sr / Sa = 0.5 Sr / Sa = 1.5 600 MPa contour

400 300 200 100

0

0 0.1

0.3

0.5

0.7

0.9

1.1

0.1

0.3

0.5

0.7

Lr / La [-]

Lr / La [-]

(c)

(d)

0.9

1.1

Figure 5: Impact of length ratio for Cu-Zn-Al as actuator and Ni-Ti as refrigerant, plotted with 600 MPa contour to show the feasible domain. (a: maximum driving stress, b: mass ratio based on Eq. 14, c: COP based on Eq. 16, d: MDTD based on Eq. 11.)

4.3 Material Properties Geometric parameters matter when designing a new system. Meanwhile, improving material properties is of equal importance for material scientists. Transport properties such as thermal conductivity have been extensively studied in the literature [15,29,30]. In this section, we focus on the equilibrium point of view, investigating the impacts of Young’s modulus, transformation strain, and Clausius-Clapeyron coefficient. In this section, the baseline material is Ni-Ti alloy for both actuator and refrigerant. Fig. 6 shows the COP and MDTD variation trend as the ratio of Young’s modulus changes from 0.25 to 10. The monotonic trends indicate that stiffer alloy is favored for the actuator. This is because the elastic deformation of the actuator itself under the stretching condition during loading reduces the output displacement to the refrigerant, which should be minimized by choosing stiffer materials. With larger Young’s modulus for the actuator, less martensite needs to be transformed into austenite inside the actuator, resulting in less input heat, less

driving temperature difference and improved COP. On the contrary, the stiffer refrigerant is not favored. From Fig. 6, we may also tell that Young’s modulus ratio should be no less than 1 due to rapid performance degradation in terms of COP and MDTD. 0.12

45 40 35

0.08

MDTD [K]

COP [-]

0.1

0.06 0.04

25 20 15

Sr / Sa = 0.2, Lr / La = 0.2

Sr / Sa = 0.1, Lr / La = 0.5

10

Sr / Sa = 0.1, Lr / La = 0.5

Sr / Sa = 0.2, Lr / La = 0.5

5

Sr / Sa = 0.2, Lr / La = 0.5

Sr / Sa = 0.2, Lr / La = 0.2

0.02

30

0

0 0.1

1

10

0.1

Ea / Er [-]

1

10

Ea / Er [-]

Figure 6: Impact of Young’s modulus ratio on the COP and MDTD. (εta / εtr = 1, Ca / Cr = 1)

Fig. 7 shows that the COP is more sensitive to the transformation strain ratio than Young’s modulus ratio. The COP varies by more than 400% when the transformation strain ratio varies from 1 to 10 when the geometric parameters are fixed and assuming the latent heats are identical for refrigerant and actuator. As the transformation strain increases, the actuator experiences less mass fraction of martensitic transformation, when the output displacement and force remain constant, according to Eq. (3). Consequently, less heat input is needed since less latent heat is required, and less heat from heat source facilitates higher COP. Meanwhile, less transformation implies a lower driving temperature. By observing the convex characteristics in Fig. 7, applying a higher transformation strain ratio significantly benefits the COP, while reducing this ratio harms MDTD. 0.7

45 Sr / Sa = 0.2, Lr / La = 0.2

0.6

MDTD [K]

COP [-]

Sr / Sa = 0.1, Lr / La = 0.5

35

Sr / Sa = 0.2, Lr / La = 0.5

0.5

Sr / Sa = 0.2, Lr / La = 0.2

40

Sr / Sa = 0.1, Lr / La = 0.5

0.4 0.3 0.2

Sr / Sa = 0.2, Lr / La = 0.5

30 25 20 15 10

0.1

5

0

0 1

10

εta / εtr [-]

1

10

εta / εtr [-]

Figure 7: Impact of transformation strain ratio on the COP and MDTD. (Ea / Er = 1, Ca / Cr = 1)

Fig. 8 shows the variation of COP and MDTD in terms of the Clausius-Clapeyron coefficient. When Ca / Cr is greater than 1, the COP becomes saturated without much improvement, however, the MDTD increases significantly when Ca / Cr reduces, especially when Ca / Cr

drops below 1. When the coefficient of actuator reduces, higher driving temperature Tg is required to facilitate sufficient driving force for the refrigerant, which grows reciprocally with Ca. Consequently, higher driving temperature leads to more input heat, and thus less COP. With increasing Ca, though the MDTD keeps reducing, the input heat becomes dominated by the latent heat, and therefore the COP becomes saturated. 0.12

45 40 35

0.08

MDTD [K]

COP [-]

0.1

0.06 Sr / Sa = 0.2, Lr / La = 0.2

0.04 0.02

30 25 20 15

Sr / Sa = 0.2, Lr / La = 0.2

Sr / Sa = 0.1, Lr / La = 0.5

10

Sr / Sa = 0.1, Lr / La = 0.5

Sr / Sa = 0.2, Lr / La = 0.5

5

Sr / Sa = 0.2, Lr / La = 0.5

0

0 0.1

1

10

0.1

Ca / Cr [-]

1

10

Ca / Cr [-]

Figure 8: Impact of Clausius-Clapeyron coefficient ratio on the COP and MDTD. (εta / εtr = 1, Ea / Er = 1)

Considering the significantly different characteristics in section 4.2 for three combinations of actuator and refrigerant materials, the parametric results in this section indicate that there is no best actuator material among these candidates. While Cu-Zn-Al or similar copper-based SMA has superior COP due to larger transformation strain and smaller latent heat, their significantly smaller Clausius-Clapeyron coefficient leads to much higher MDTD. When applying them as refrigerant, the smaller Clausius-Clapeyron coefficient implies reduced transformation stress, yet smaller latent heat restricts cycle cooling performance. How to pair the actuator material and the refrigerant material properly varies case by case, where the parametric studies here can be used as basic guidelines to start with. 4.4 Constraint between MDTD and MR In section 4.2, we observe that both cycle COP and MDTD reduce when the mass ratio increases with smaller Lr / La or smaller Sr / Sa. Here, Fig. 9 visualizes such a compromise between MDTD and MR. From the data range in Fig, 8 (a) for Ni-Ti as both actuator and refrigerant, a minimum MDTD reaches 17 K when the mass ratio is 100, while a minimum MR spreads to 4.1 at a MDTD of 85 K. Recommended designs should balance the MR and MDTD. In the case of Fig. 9 (a), MR scatters from 10 to 30 and MDTD is in the range of 20 to 40 K, whereas Lr / La is between 0.2 to 0.4 and Sr / Sa is between 0.1 to 0.3. Further beyond

this range results in significant degradation of either MDTD or MR. Similar trend is observed in Fig. 9 (b) for the case where Ni-Ti is applied as the actuator material to drive Cu-Zn-Al, whereby the MDTD is lower due to reduced Clausius-Clapeyron coefficient of Cu-Zn-Al and therefore requires smaller stress and driving temperature. A significantly different distribution is shown in Fig. 9 (c) when Ni-Ti refrigerant is driven by the Cu-Zn-Al actuator. Each cluster of data plotted here represents a fixed Sr / Sa ratio between 0.1 and 0.5, wherein the Sr / Sa ratio becomes the determinant factor. In this case, Sr / Sa is recommended to be 0.1 while 1.1 is recommended for Lr / La, achieving a MDTD of 26 K and MR of 10.8. 100 90

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(c) Ni-Ti (r) + Cu-Zn-Al (a), , Lr / La from 0.1 to 2, Sr / Sa from 0.1 to 0.5.

Figure 9: Minimum driving temperature difference versus refrigerant mass ratio for a heat driven cycle.

The aforementioned figures are based on the uniform sampling of design variables, whereby the discrete and scatter distribution of performance metrics cannot demonstrate a clear path of Pareto front, i.e. a set of non-dominant solutions in terms of two objectives such as MDTD and MR. This gap can be improved by random sampling and applying an optimization algorithm, as shown in Fig. 10. The definition of optimization problems can be found in Table 3. NSGA-II is implemented as the algorithm [31]. The population is set to be 50. Each problem is solved three times to guarantee convergence. For example, for one solution to a specific problem in Fig. 10 (a), among the 5,650 evaluated design candidates, 606 final solutions constitute a clear track of Pareto front, representing the compromise between COP and MDTD. For most cases, the final solution converges after 100 to 130 iterations, however, for the two cases in Fig. 10 (c) and (d) more than 300 iterations are required. Table 3. Optimization problem definitions.

Objective functions

Problem #1 in Fig. 10 (a)

Problem #2 in Fig. 10 (b)

Min. MDTD

Min. MDTD

Max. COP

Max. COP / COPCarnot

Upper boundaries

Lr / La ≤ 2.0, Sr / Sa ≤ 2.0

Lower boundaries

Lr / La ≥ 0.1, Sr / Sa ≥ 0.1

Iteration

113, 127, 118 (solved 3 times)

112, 109, 106 (solved 3 times)

As mentioned earlier, both COP and its ratio to Carnot COP are important. Since COP is proportional to mass ratio, the MDTD and efficiency are intrinsically conflicting. Such a relation can be visualized in Fig. 10, where COP / COPCarnot is denoted as the 2nd law efficiency. For all the cases, higher MDTD is expected if higher COP is achieved. In Fig. 10 (a), as COP increases from 0.04 to 0.2, the MDTD increases from 17 K to 80 K. When considering the normalized COP, the distribution becomes slightly different in Fig. 10 (b), where the maximum COP / COPCarnot becomes 0.075 when the MDTD is at 59 K. Comparing with Fig. 10 (a), further improving COP is no longer beneficial since the cycle configuration in Fig. 1 cannot exploit the full potential when the driving temperature becomes higher and higher. If higher driving temperature is available, two-stage a.k.a. double-effect utilization of

heat source is a potential way to boost both COP and 2nd law efficiency [24]. Another interesting observation is the difference between design variables in terms of two optimization problems listed in Table 3. Regarding Fig. 10 (a), the maximum COP of 0.2 achieves when Sr / Sa = 0.75 and Lr / La = 0.32, while the maximum COP / COPCarnot of 0.075 occurs when both Sr / Sa and Lr / La equals 0.44 in Fig. 10 (b). Furthermore, because the properties of the actuator and refrigerant are identical, almost all of the Pareto front designs in

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Figure 10: Minimum driving temperature difference versus cycle efficiency. (Ranges of Sr / Sa and Lr / La are described in Table 3.)

When Ni-Ti drives Cu-Zn-Al refrigerant, a maximum COP of 0.27 is achieved when the driving temperature difference becomes 68 K, while the maximum COP / COPCarnot reaches 0.11 at MDTD of 55 K, corresponding to Sr / Sa of 1.38 and Lr / La of 0.36. The trend of Fig. 10 (c) and (d) are similar to Fig. 10 (a) and (b), respectively, yet the design variables are very dissimilar. In this case, we observe that higher efficiency favors larger Sr / Sa and smaller Lr / La, since Cu-Zn-Al refrigerant requires less stress and therefore facilitating larger Sr / Sa. Furthermore, we observe a significantly different distribution in Fig. 10 (e) and (f) when applying Cu-Zn-Al as the actuator. The COP in Fig. 10 (e) reaches notably high of 0.58 at a MDTD of 97 K compared with the previous cases due to the lower latent heat of Cu-Zn-Al. Once more, this is because of the reduction of input thermal energy to drive the cycle when the latent heat of the actuator’s material reduces. Meanwhile, the maximum COP / COPCarnot is 0.18 when the MDTD is 52 K, whereas the COP / COPCarnot varies by less than 0.001 when MDTD decreases from 52 K to 42 K. Unlike other two cases, when Cu-Zn-Al actuator drives Ni-Ti refrigerant, the unique outline of its Pareto front leads to two attractive designs. The first candidate is Sr / Sa = 0.107 and Lr / La = 0.83, corresponding to the COP / COPCarnot of 0.165 and MDTD of 26.5 K, if lower driving temperature is preferred. If higher efficiency is favored, one may choose the second design, i.e. Sr / Sa = 0.18 and Lr / La = 1.1, corresponding to the COP / COPCarnot of 0.18 and MDTD of 42 K. When comparing all three cases, one thing in common is that the COP / COPCarnot does not increase anymore when the MDTD is beyond 60 K, and more precisely speaking the optimum MDTD ranges from 52 K to 59 K under the cases simulated in this study. Considering the ambient temperature to be 20°C to 40°C, this MDTD indicates that low-grade thermal energy at a temperature between 70°C to 100°C is most promising in terms of the compromise between COP and proficient utilization of heat source. More efficient utilization of heat source at the higher temperature would require a more complicated configuration that

cascaded exploit the energy from heat source. In addition, the COP / COPCarnot for heat driven system is comparable to that of a conventional motor driven system, which ranges from 0.14 to 0.22 based on a previous study [32]. Implementing SMAs with less hysteresis, exploiting more efficient cycles and deploying more efficient work recovery, heat transfer, and regeneration designs could benefit the efficiency for both the conventional motor driven elastocaloric cooling system, as well as the heat driven elastocaloric cooling system. 4.5 Prospects As mentioned in the literature, the heat driven system may be constrained by low operating frequency because the heat driven actuator requires additional time to complete heat transfer and therefore actuation becomes slower than conventional mechanical drivers [33]. This drawback can be potentially solved by using miniature actuator wires with a diameter of up to 200µm, which could achieve a heat transfer time constant of fewer than 0.5 seconds using water as heat transfer fluid [34]. Another important aspect is the fatigue life. Fatigue is indeed the major challenge for SMA-based energy conversion devices including elastocaloric heat engine and cooling systems. Fortunately, there are material level solutions to improve the fatigue performance of SMAs. For example, precipitate phases in some SMAs may lead to better lattice compatibility [35], or induce preexisting sites to reduce friction during martensitic phase transformation [36]. Surface treatment such as mechanical polishing or electrochemical polishing could improve structural fatigue performance as well [37]. On the system level, pre-straining both actuator and refrigerant SMAs enabling operating around the middle of the transformation plateau, which could substantially improve the durability of the system [26]. When applying these techniques, the fatigue life would be improved for both the actuator and the refrigerant, which is important for heat driven elastocaloric cooling or heat pump systems.

5. Conclusions Heat driven elastocaloric cooling system offers an order of magnitude, or up to several orders of magnitude reduction in the actuator mass. However, considering the mechanical and thermodynamic couplings, the practical mass ratio does not always approach unity. The mechanical coupling constraint is a result of the elastic deformation of the actuator when sustaining the actuating force, which indicates a practical mass ratio above 4 for Ni-Ti alloy

implemented for both actuator and refrigerant. Results also reveal that the length ratio and cross-sectional area ratio are determined by the transformation strain and Young’s modulus of the actuator and refrigerant, that stiffer actuator with larger transformation strain is preferred. The larger transformation strain of the actuator qualifies the implementation of larger refrigerant to actuator’s length ratio. As a result, the mass ratio can reach 0.57 when Cu-Zn-Al actuator drives the Ni-Ti refrigerant. Furthermore, actuator material with a larger Clausius-Clapeyron coefficient leads to a smaller driving temperature difference and larger refrigerant to actuator’s cross-sectional area ratio. While neither copper-based SMA nor Ni-Ti dominates the other, optimization studies show that the COP at 40 K driving temperature difference is 0.11, 0.16, and 0.28, when Ni-Ti, Ni-Ti, and Cu-Zn-Al are actuators to drive Ni-Ti, Cu-Zn-Al and Ni-Ti, respectively, corresponding to a normalized COP ranging from 7.0% to 17.9%. In addition, the above three cases all have a mass ratio of less than 10, wherein the minimum mass ratio is 3.1 for the case that the Ni-Ti actuator drives the Cu-Zn-Al refrigerant. The optimum normalized COP occurs when the MDTD ranges from 52 K to 59 K, which does not further increase with higher driving temperature, implying that low-grade thermal energy at a temperature less than 100°C is most economic to drive such a cycle. On the other hand, the heat driven cycle can be activated by MDTD of 17 K, 11 K, and 23 K for the three material combinations, indicating a significant potential to harvest low-grade thermal energy. The optimization results can be served as design guidelines for future prototype development for solar-driven refrigerators and chip waste heat recovery devices.

Acknowledgments This research is funded by the State Key Laboratory of Air-Conditioning Equipment and System Energy Conservation (No. ACSKL2018KT1207), and the National Science Foundation of China under grant 51976149.

Nomenclature A

austenite

COP

coefficient of performance

C

Clausius-Clapeyron coefficient (MPa·K-1)

c

specific heat (J·g-1·K-1)

E

elastic modulus (MPa)

∆h

latent heat of transformation (J·g-1)

L

length (m)

MDTD

minimum driving temperature difference (K)

MR

mass ratio (-)

M

martensite

q

specific capacity (J·g-1)

T

temperature (K)

S

area (m2)

SMA

shape memory alloy

w

(un)loading work

x

mass fraction of martensite (-)

Greek symbols ε

strain (-)

η

efficiency (-)

ρ

density (g·cm-3)

σ

stress (MPa)

Subscripts a

actuator

g

source reservoir

c

cooling reservoir

f

finish

ld, uld

loading or unloading

r

refrigerant

h

ambient reservoir

s

start

T

transformation

HR

heat recovery

+,−

loading and unloading

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Highlights ·

Thermodynamic and mechanical constraints are derived for heat driven EC system.

·

11 K driving temperature difference for Ni-Ti actuator with Cu-Zn-Al refrigerant.

·

Maximum normalized COP is achieved when driving temperature difference is 55K.

Declaration of interests    ☑The authors declare that they have no known competing financial interests or personal relationships  that could have appeared to influence the work reported in this paper.    ☐The authors declare the following financial interests/personal relationships which may be considered  as potential competing interests: