International Journal of Heat and Mass Transfer 62 (2013) 373–381

Contents lists available at SciVerse ScienceDi rect

Internati onal Journ al of Heat and Mass Transfe r

journal homepage: www.elsevier .com/locate / i jhmt

Computational study of transverse Peltier coolers for low temperature applications

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.018

⇑ Corresponding author. Address: Department of Mechanical and Aerospace Engineering, The Ohio State University, Suite E410, Scott Laboratory, 201 West 19th Avenue, Columbus, OH 43210, USA. Tel.: +1 (614) 247 8099; fax: +1 (614) 292 3163.

E-mail address: mazumder.2@osu.edu (S. Mazumder).

Syed Ashraf Ali, Sandip Mazumder ⇑Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 November 2012 Accepted 2 March 2013 Available online 9 April 2013

Keywords:ThermoelectricPeltier cooler Transverse device Computational study Low temperature Bismuth telluride

Transverse thermoelectric effect can be produced artificially by stacking at an angle layers of a thermo- electric material with another material that may or may not be a thermoelectric material. In this explo r-atory computatio nal study, a new meta-material, comprised of tilted alternating layers of an n-typethermoelectric alloy and a metal, is investigated to gain an understanding of how much cooling can beproduced by trans verse thermoelectric effect and the condit ions under which maximum cooling is attain- able. The governing conservation equations of energy and electric current, with the inclusion of thermo- electric effects, are solved on an unstructured mesh using the finite-volume method to simulate atransverse Peltier cooler under various operating conditions. First, the code is validated against experi- mental data for a n-Bi2Te3–Pb meta-material, and subsequently explored. It is found that intermediate applied currents produce maximum temperature depression (DT). Optimum values of the geometric design parameters such as tilt angle and device aspec t ratio are also established through parametric stud- ies. Finally, it is shown that the DT can be amplified by constricting the phonon (heat) transport cross- section while keeping the electron (current) transport cross-section unchanged—a strategy that cannot be employed in conventional thermoe lectric devices where electrons and phonons follow the same path.This makes transverse Peltier coolers particularly attractive for generating large DT without multi-stage cascading.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Solid-state Peltier coolers [1] are attractive because, unlike va- por-cycle refrigeration systems, they have no moving parts, and are lightweight. In the near future, cryogenic Peltier coolers are ex- pected to supplant conventional vapor-cycle cooling systems used to cool far-infrared , X-ray, and c-ray sensors flown in space and weight-sens itive platforms such as UAVs and situational aware- ness satellites. For such applications, the relevant temperature range is 150–10 K, i.e., ultra low temperature .

The theoretical performanc e of a conventional thermoelectric (TE) device is dictated by the non-dimensional figure of merit,ZT = R2rT/j, where T is the temperat ure, and R, r, and j are the Seebeck coefficient, the electrical conductivity , and the thermal conductivity , respectivel y, of the thermoelectr ic material being used [1,2]. A conventional Peltier cooler yields a maximum tem- perature depression of DTmax = (ZT)T/2 provided both n- and p-typematerials of the TE alloy are available, and have similar ZT values.Otherwise, the DT is significantly reduced [1]. Recent research has

doubled the ZT of thermoelectric materials [3] at temperature shigher than room temperature. When applied to cryogeni c temper- atures (less than 150 K), however, the same materials yield ZT val-ues of much less than unity since ZT decreases following a T7/2 law[1,2]. To date, the most promising cryogenic thermoelectric mate- rial, n-type Bi95Sb5 doped with a resonant level, potassium, which has been developed by Heremans and co-worke rs [4], has been shown to yield ZT � 0.7 at 100 K. However, such values are mani- fested only on n-type materials. In contrast, the best p-type mate- rial, Bi86Sb14, yields ZT � 0.15 at 100 K [4,5].

For practical applications, more than 100 K cooling would berequired. Even assuming that p-type materials with ZT � 1 (at100 K) will be developed in the future, simple estimates [6] showthat with conventional designs that make use of the longitudina l(conventional) Peltier effect, an 11-stage cascade would berequired to cool a device down from 300 K to 10 K, even disregard- ing transport losses. In practice, such a large number of stages ofcascadin g produce diminishing returns due to parasitic losses such as poor contact and losses to the ambient. Thus, for ultra low tem- perature applicati ons, even with the best-perform ing materials,there is a pressing need to explore designs that go beyond the conventi onal designs. Fundamental ly, the performance must beenhanced by combining two approach es: (1) developing high- ZTmaterials and material combinati ons, and (2) developing new

Fig. 1. Schematic representation of a transverse Peltier cooler.

Nomenc lature

d height of transvers e device (m)h convective heat transfer coefficient (W/m2/K)j current density vector (A/m2)l length of transvers e device (m)q heat flux vector (W/m2)_qJ joule heat per unit volume (W/m3)T absolute temperat ure (K)

Greeka tilt angle (degrees)/ electric potential (V)j thermal conduc tivity (W/m/K)r electrical conductivity (S/m)P Peltier coefficient (V)R Seebeck coefficient (V/K)

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geometrical designs that cause phonons and electrons to travel along different and independent paths, thereby enabling us to re- define the conventi onal figure-of-merit. High- ZT material develop- ment, by itself, is not sufficient to meet the afore-sta ted goals.

In this work, in an effort to re-define the conventional figure-of-merit of a Peltier cooler, the so-called transverse Peltier cooler is explored . All thermoel ectric materials exhibit small degrees ofanisotropic Seebeck/Pel tier effect [1,2]. In the case of the Seebeck effect, this implies that heat flow in a certain direction produces a very small current (transverse current) in the direction normal to the heat flow, and a relatively large current (longitudinal cur- rent) in the same direction as the heat flow. In most naturally occurring thermoelectric materials , anisotropic effects are insignif- icant, and the transverse current is not large enough to be of any practical interest, although it has been observed experimental lyin crystalline systems [7,8]. The fact that the dominan t longitudinal current is aligned with the heat current also suggests that the ma- jor heat carriers, namely phonons in a semiconduc tor, follow the same path as the major charge carriers, namely electrons and holes. The implication of this is that the device geometry (i.e.,length, width, height) does not play any role in the performanc eof a conventional thermoelectric device. Any alteration of the pho- non path via geometric alterations is tantamount to alteration ofthe charge carrier path, resulting in equal influences on the electri- cal and thermal conductance and unaltered ZT. On the other hand,if the transverse current could be amplified by some artificialmeans, it would suggest a device in which charge carriers and pho- nons flow through mutually independent paths. In such a scenario,the electrical or thermal conductance could be tuned independen tof each other by altering the geometry, thereby creating an oppor- tunity to enhance the effective ZT.

Although anisotropic thermoel ectric effect is weak in naturally occurring thermoelectric alloys, it is conceiva ble to force strong anisotropic effect artificially, as has been demonst rated by past re- search [9,10]. Babin et al. [11] first proposed the general concept ofusing a tilted multi-layer ed structure , as shown in Fig. 1, to gener- ate anisotropic thermoelectr ic effect. This idea was later demon- strated by Gudkin et al. [12] for cooling applicati ons. In recent years, Lengfellner and co-worke rs [9,10] have demonstrat ed con- clusively through experimental observations that a meta-mater ial comprised of alternating layers of a thermoelectr ic alloy and a me- tal can serve as a transverse Peltier cooler. A similar configurationwas also employed by Mann and Huxtable [13] to construct a heat flux sensor. Although Lengfellner and co-workers [9,10] have per- formed some preliminary one-dimensional calculations using effective homogeneous medium theory to support their experi- mental findings, to date, no detailed computational analysis has been undertaken to shed light into the working mechanism s ofsuch a meta-mater ial. Specifically, although it is acknowledged that in such devices heat and current flow along independen t paths [1,2,12], no research has been conducte d to elucidate what role geometry can play in improving the performanc e of such devices.

In this work, transverse Peltier coolers comprised of a meta- material of the afore-stated type were modeled computational ly.

The bi-layer materials, layer thicknesses, and other operating condition s were chosen to match previously reported experimen- tal studies. The governing conservati on equations of energy and electric current, including thermoelectric effects, were solved numerica lly with high mesh resolution so that the individual lay- ers of the meta-material are adequately resolved. After model val- idation, explorator y studies were undertaken to elucidate the effect of various geometric design parameters on the performanc e(DT) of such Peltier coolers.

2. Mathematica l model

The governing conservation equations are conservation equa- tions of energy and current. Since the model is to be exercised for layers whose thicknesses are of the order of a millimeter (aswill be discussed later), it is assumed that continuu m is prevalent,and that the governing equations for equilibriu m (or continuum)transport are valid. Under the continuum assumpti on, the govern- ing equations at steady state are written as

Energy conservation : r � q ¼ � _qj ð1aÞ

Current conservation : r � j ¼ 0 ð1bÞ

where q is the heat flux vector, j is the current density vector, and _qJ

is the heat generated per unit volume due to Joule heating. As dis- cussed earlier, most natura lly occurring thermoel ectric mater ials are only weakl y anisotro pic. Therefore, for modeling purposes ,these material s are assum ed to be isotropic. In a thermoelect ric mater ial, the heat flux is due to the combined effect of Fourier con- duction and the Peltier effect. Under the isotrop ic assumption , the heat flux may be written as:

q ¼ �jrT þPj ð2Þ

where T is the absolute temperature. The thermal conduct ivity isdenoted by j, while the Peltier coefficient is denoted by P. The

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current density (or flux), on the other hand, is due to Ohmic conduc- tion, and the Seebeck effect, and is written as

j ¼ �rr/� rRrT ð3Þ

where / is the electric potenti al. The electrical conduct ivity is de- noted by r, while the Seebeck coefficient is denoted by R. The See- beck coefficient is related to the Peltier coefficient throug h the Thompson relation [1]: P = RT. Substitutio n of Eqs. (1b) and (2) intoEq. (1a) yields

Fig. 2. Case considered for the validation study: (a) schematic of experimental setu

Energy conservation : r � ð�jrT �Prr/�PRrrTÞ

¼ � _qJ ¼ �j � jr

ð4aÞ

Current conservation : r � ð�rr/� RrrTÞ ¼ 0 ð4bÞ

Equation s (4a) and (4b) represent a set of coupled elliptic partial differen tial equations with two dependent variables, namely tem- perature T and electric potential /. The presen ce of the Joule heating

p, (b) 2D model setup and boundary conditions and (c) computational mesh.

Table 1Transport propert ies used for calculat ions in the present study.

Transport property Material Source of data for Bi2Te3

Bismuth Telluride (Bi2Te3) Lead (Pb)

Baseline properties Thermal conductivity, j 2.3 W/m/K 35 W/m/K Goldsmid et al. [14]Electrical conductivity, r 105 S/m 5 � 106 S/m Goldsmid et al. [15]Seebeck coefficient, R 200 lV/K 0 Reitmaier et al. [10]Modified properties Thermal conductivity, j 1.2 W/m/K 35 W/m/K Takiishi et al. [16]Electrical conductivity, r 1.1 � 105 S/m 5 � 106 S/m Takiishi et al. [16]Seebeck coefficient, R 287 lV/K 0 Tan et al. [17]

Fig. 3. Predicted and measured [9] DT in a transverse Peltier cooler with tilt angle of 25� and aspect ratio of 2 with baseline properties.

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term on the right hand side of Eq. (4a) also makes it non-linea r.Thus, these equations can only be solved numerica lly using an iter- ative procedu re. In the present study, the governin g equations were discretize d using the finite-volume procedure on an unstructured mesh, and solved using the frame work available in the commerc ial multi-phy sics code CFD-ACE +.

3. Results and discussion

3.1. Validation study

Prior to conducting parametric studies, a validation study was undertaken to understand the limitatio ns of the present modeling approach. Kyarad and Lengfellner [9] have performed experiments on a transverse Peltier cooler with a n-Bi2Te3–Pb meta-ma terial,and recorded the temperature depression (DT) as a function of ap- plied current for a tilt angle of 25� and Bi2Te3 to Pb layer thickness ratio of unity. Typical layer thickness of 1 mm was used, and size ofthe sample was 2 cm in length, 1 cm in height, and 2 mm in width (Fig. 2a). Water-cooled leads were connected to the left and right surfaces of the sample, as a result of which current transfer oc- curred from the left to the right of the sample. The bottom surface of the sample was quenched into a bath of water at 295 K, while the top and side surfaces were exposed to ambient air. Since the Peltier heat moves from the top surface to the bottom surface inthis configuration, the final outcome is a colder (compared to295 K) top surface. The DT between the top and bottom surfaces were recorded for various current density values.

In terms of modeling the experimental setup described above,several unanswered questions arise. First, although the air sur- rounding the device at the top and sides is electricall y insulating,it is not an insulator thermally. However, the exact thermal condi- tions on these surfaces are unknown. Therefore, due to lack of bet- ter information , these surfaces were assumed to be adiabatic,especially since the convective loss from these surfaces is expected to be several orders of magnitude lower than the loss from the water-cooled bottom surface. Secondly , questions arise regarding the bottom surface. As mentioned earlier, the bottom surface is im- mersed in a water bath at 295 K, which is stirred. Kyarad and Leng- fellner [9] contend that although this scenario produces fairly efficient coupling of the bottom surface to the water bath, since the convectiv e resistance is not zero, the temperature at the bot- tom surface will be somewhat larger than 295 K, and also benon-uniform . In an effort to mimic the actual setup, we treated the convective heat transfer coefficient at the bottom surface as aparameter. Indeed, it was found that the computed DT is depen- dent upon the value chosen for the convective heat transfer coeffi-cient at the bottom surface, as will be discussed later. The current entering the system from the left is assumed to enter as a uniform (plug) distribution, while the right end is grounded, implying that the current density will not be uniform. Whether these

assumpti ons are true in the experimental setup or not is also an- other unanswered question. Also, radiation loss is neglected. Final- ly, as shown in Fig. 2b, the model was assumed to be two- dimensio nal (2D) since the front and back surfaces are expected to have similar conditions, both thermally and electrically.Although the same geometry was also modeled as a 3D block later in the study (see Section 3.4), initial studies were conducte d with the 2D model to reduce computational time, thereby allowing alarge number of parametri c variations. Fig. 2c depicts the compu- tational mesh used for the validation study. It is comprised of50,270 triangula r control volumes (or cells). This mesh was deemed appropriate after conducting a rigorous grid independen cestudy in which the mesh was refined successively in 4 stages until the results were found to be within 1% of each other.

For the thermo-phys ical properties of the materials involved,the values suggested by Reitmaier et al. [10] were used initially,and are summarized in Table 1 under the heading ‘‘Baseline Prop- erties.’’ The values suggested by Reitmaier et al. [10] were obtained from different sources (see Table 1), and as discussed shortly,uncertainti es remain with regard to the appropriate ness of these values, as well. Contact resistance between the layers was assumed to be zero due to lack of better information. Kyarad and Lengfellner [9] state that the layers were heat treated in a furnace under ap- plied compress ive force, and contend that the layers have ‘‘good’’thermal and electrical contact.

Fig. 3 depicts the temperature depression (DT) obtained using baseline properties and various convective heat transfer coefficient

Fig. 4. Predicted temperature distribution in a transverse thermoelectric device with tilt angle of 25�, aspect ratio of 2, and current of 40 A: (a) isothermal (295 K) bottom surface, and (b) adiabatic bottom surface.

Fig. 5. Predicted and measured [9] DT in a transverse Peltier cooler with tilt angle of 25� and aspect ratio of 2 using baseline and modified properties and isothermal (295 K) bottom surface.

Fig. 6. Effect of tilt angle on the predicted DT in a transverse Peltier cooler having aspect ratio of 2 using baseline and modified properties and isothermal (295 K)bottom surface.

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values for the bottom surface. These computati ons were performed for a tilt angle, a, equal to 25�, and an aspect ratio, l/d, equal to 2.For all simulations, the layer thickness ratio (of Pb to Bi2Te3) was

maintain ed at unity. The values reported in Fig. 3 are differenc esin average temperature between the top and bottom surface. The DT increases with increasing current and then decreases. This is

Fig. 7. Effect of device aspect on the predicted DT in a transverse Peltier cooler having a layer tilt angle of 25� using baseline and modified properties and isothermal (295 K) bottom surface.

Fig. 8. Comparison of the predicted DT in a transverse Peltier cooler using two- dimensional (2D) and three-dimensional (3D) models. Tilt angle is 25�, aspect ratio is 2, and the bottom surface is at 295 K.

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attributed to dominance of Joule heating at high current density over the Peltier heat flux since Joule heating is proportional tothe square of the current. Also, as the DT increases, the Fourier heat flux also increases, thereby negating any increase in the Peltier heat flux. It is also seen that under isothermal conditions for the bottom surface, the simulatio ns under-predict the experimental lyobserved DT, while under adiabatic condition s, the simulations over-predic t the observed DT. The best match to the experimental data is found to be for a convectiv e heat transfer coefficient value �2000 W/m 2/K—a value that is typically of forced convectio n by aliquid [18]. The findings clearly point to the fact that the heat transfer to the cold bath on the bottom surface affects the performanc e significantly, and also corroborates the contention by Kyarad and Lengfellner [9] that the bottom surface is probably

not under isothermal bath conditions. Fig. 4 shows the actual tem- perature distribution in the transverse device under isothermal bottom surface and adiabatic bottom surface conditions for an ap- plied current of 40 A. First, it is observed the temperature at the top surface (the target surface for cooling) is not uniform. Second, un- der adiabatic bottom surface conditions, although Fig. 3 mightindicate best performance, Fig. 4b shows that the temperat ure atthe top wall is actually very close to 295 K (the computed average is 292.3 K), with portions of the surface actually hotter than 295 K.In other words, although the DT is large, only 2.7 K of average cool- ing is actually achieved. This is because under adiabatic bottom surface condition s, the Joule heat generated within the device has no place to go other than the left and right boundaries, and manifest s itself as a temperat ure rise of the whole device.

The DT obtained under isothermal conditions with baseline propertie s is significantly lower than the experimental ly observed values (peak of 8 K versus 22 K). As mentioned earlier, there are significant uncertainties with regard to the transport propertie sused in the simulatio ns. In an effort to investigate this issue, a sec- ond set of simulations were conducted with a different set of prop- erties—henceforth referred to as ‘‘modified properties.’’ The modified properties and their sources are shown in Table 1.Fig. 5 shows the predicted results for both modified and baseline propertie s under isothermal conditions. While the peak value of(DT) obtained with modified properties matches the experimen- tally observed values much better, the location of the peak (i.e.,the current at which it occurs) is better predicted with the baseline propertie s. Therefore, the discrepancy between experimentally ob- served data and predicted results cannot be attributed to uncer- tainties in the transport propertie s alone. Most likely, it is acombinati on of uncertainties and assumptions in boundary condi- tions and material properties, such as (1) constant (temperature independen t) transport properties, (2) negligible radiation loss,(3) isothermal bottom surface or bottom surface with constant convectiv e heat transfer coefficient, (4) uniform current on the left boundary , and (5) negligible interface resistance. One final point tonote is that the measured DT is at a location close to the central plane of the sample, and not an average. Nonetheless , since overall qualitativ e behavior of the predicted results is similar to the ob- served experimental results, the model was deemed suitable for subsequent parametri c studies where the emphasis is on relative performanc e (i.e., trends) rather than the absolute predicted value of DT.

3.2. Effect of tilt angle

In the first set of parametric studies, the tilt angle, a, of the lay- ers was varied. The baseline value used for the tilt angle was 25�, inkeeping with the theoretical optimum reported by Reitmaier et al.[10]. In addition, two other angles, namely 15� and 35�, were ex- plored in this study. Baseline aspect ratio of 2 was used. The pre- dicted DT for various tilt angles is shown in Fig. 6 for both baseline and modified properties. First, in agreement with the find-ings reported by Reitmaier et al. [10], the best performanc e is man- ifested for tilt angle of 25�, irrespective of the transport propertie sused. Secondly, shallower tilt angles (i.e., 15�) appear to produce better performanc e than steeper tilt angles (i.e., 35�) at high cur- rents (higher than the optimum current), while the reverse is true at low currents. This is because at high currents, the excess Joule heat cannot be dissipated effectively to the left and right bound- aries when the tilt angle is steep because there is no direct contact between the left and right surfaces through a single layer. This re- sults in excessive heating of the overall device. At shallow angles,direct contact is established between the left and right boundari esby a single layer. This allows the Joule heat to escape laterally through the high thermal conductivity Pb strips.

Fig. 9. Transverse Peltier cooler of trapezoidal shape: (a) computational mesh, and (b) temperature distribution with baseline properties at 40 A.

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3.3. Effect of device aspect ratio

In the second set of parametric studies, the device aspect ratio,l/d, was varied. Since the afore-mentioned experiments were con- ducted for aspect ratio of 2, this value was considered the baseline value. In this parametric study, aspect ratio values of 1 and 4 were also explored. Fig. 7 shows the predicted DT for all three aspect ra- tios (1, 2, and 4). For both baseline and modified properties, larger aspect ratios result in higher peak DT. This is probably attributabl eto the fact that when the value of l is increased while keeping dconstant, the thermal resistance between the hot and the cold sur- faces (top and bottom) decreases since the thermal resistance roughly scales as d/(jl).

3.4. Effect of area constriction

As discussed earlier, one of the conceived advantag es of a trans- verse Peltier cooler of the kind investigated here over conventional (longitudinal) Peltier coolers is that heat (phonons) and current (charge carriers) travel along different paths—perpendicular, in

this particular case. This provides an opportunity for amplifying the DT by constrict ing the area through which the heat flows with- out altering the current flow. A simple way to achieve this ampli- fication is to have a trapezoid shaped device in which the target (top) surface is smaller in area than the reference (bottom) surface.Since the heat flux has to be conserved from the bottom to the top surface—disregarding left/right end effects—constriction of the area is expected to increase the DT. This idea was explored in this study.

The first step needed for exploration of the area constrict ion ef- fect is the developmen t of a 3D model. As mentioned earlier, a 2Dmodel, which is computationall y more efficient, was exercised inthe previous parametric studies since the geometry and boundary condition s permitted its use. For the trapezoidal geometry, this isnot possible. A 3D model of the rectangular block shown inFig. 2a was first created. 463,017 tetrahedr al cells were used. Adi- abatic boundary conditions were applied to the front and back sur- faces to replicate a 2D scenario, and the results predicted by this model were compared with the 2D model. The comparison isshown in Fig. 8. Clearly, the 3D model replicates the results

Fig. 10. Effect of area constriction by a factor of 2 on the performance of transverse Peltier coolers. Constant cross-section represents a rectangular geometry, while constricted cross-section represents a trapezoidal geometry.

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produced by the 2D model barring minor differences due to differ- ences in the exact mesh type and size used in the two cases.

Having confirmed the validity of the 3D computations , the pro- posed trapezoidal Peltier cooler was simulated. The mesh used for this study comprised of 693,743 tetrahedral cells (Fig. 9a), and itrequired approximately 52 min of CPU time to obtain 4 orders ofmagnitude convergence on a 3.4 GHz Intel Pentium 4 processor having 1 GB of RAM. The rectangular device that was investigated for comparison had a length l = 2 cm, height d = 1 cm, and width (front to back) of 1 cm. For the trapezoidal device, the top (target)surface area was retained to be the same as that of the rectangu lar device, while the bottom surface area was increased by a factor of2, as shown in Fig. 9a. A typical temperature distribution is shown in Fig. 9b. The non-uniform ity of temperat ure at the target surface is quite evident. The overall performanc e of the trapezoidal device in comparison to the rectangu lar device is shown in Fig. 10. As pos- tulated, an improvement in DT is produced by constrict ing the area. However, both for baseline and modified properties, only a25% increase in DT is produced at the optimum current, as opposed to the theoretical ly expected 100% increase (due to 100% area con- striction). The less than expected gain is probably due to three- dimensional effects within the tilted layers, which causes the heat to flow at an angle to the top and bottom surfaces rather than ex- actly perpendicular to these surfaces (as evidenced by Fig. 9b), and also due to end effects, i.e., losses through the left and right sur- faces. Nonetheless, this detailed modeling study confirms that con- striction of the area is a viable means to improvin g the performanc e of transverse Peltier coolers—a strategy that is inef- fective in conventional Peltier coolers.

4. Summary and Conclusion s

A computational analysis based feasibility study was conducted to explore the working mechanism of a transverse Peltier cooler constructed out of a meta-ma terial. This meta-ma terial is com- prised of alternatin g tilted layers of n-Bi2Te3 and Pb. The computa- tional model was first validated against experime ntally measure ddata. It was found that although the code reproduced the impor- tant trend of producing the best temperature depression (DT) atan intermedi ate current, it either over or under-predicte d the

measure d DT values depending on the kind of thermal boundary condition s used. Several sources of uncertainty and reasons for dis- crepancy between numerica l predictions and experimental obser- vations were discussed. Since the trends predicted by the model were in agreement with experimental observations , it was deemed suitable for further parametri c studies.

Two sets of parametric studies were conducted in which the layer tilt angle and the device aspect ratio were varied. It was found that an optimum tilt angle exists, and was found to be about 25�. For other tilt angles, namely 15� and 35�, the predicted DT val-ues were found to depend strongly on the applied current. Tilt an- gle of 15� performed better (produced larger DT) at high applied current, while tilt angle of 35� performed better at low applied cur- rent values. When the device aspect ratio was varied, it was found that larger aspect ratios (long with short height) produced the best performanc e. All of the afore-menti oned studies were conducte dusing a 2D model that allowed quick computational turnaround.

Finally, a 3D model of the transverse device was developed and validated. It was then exercised to explore the idea of amplifying DT by area constriction. In order to accomplis h this goal, two dif- ferent device geometries—one with a rectangular cross-section,and the other with a trapezoidal cross-sectio n—were considered.It was found that constrict ion of the area (between the hot and cold surfaces) by a factor of 2 amplified the DT by �25% under optimum applied current condition s. Such an effect is unique to transverse Peltier coolers in which the heat transport path can be altered without altering the current transport path. From an engineering standpoint, this finding is promising since it opens up the opportu- nity to develop large DT devices for cryogenic applications without multi-sta ge cascading.

Acknowled gments

The authors acknowledge Prof. Joseph P. Heremans for motivat- ing this computational study, and for helpful discussions . ESI Group is acknowledged for providing licenses of their commercial software CFD-ACE+ ™.

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