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

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Computational study of transverse Peltier coolers for low temperature applications 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

Article history: Received 27 November 2012 Accepted 2 March 2013

Keywords: Thermoelectric Peltier cooler Transverse device Computational study Low temperature Bismuth telluride

a b s t r a c t Transverse thermoelectric effect can be produced artificially by stacking at an angle layers of a thermoelectric material with another material that may or may not be a thermoelectric material. In this exploratory computational study, a new meta-material, comprised of tilted alternating layers of an n-type thermoelectric alloy and a metal, is investigated to gain an understanding of how much cooling can be produced by transverse thermoelectric effect and the conditions under which maximum cooling is attainable. The governing conservation equations of energy and electric current, with the inclusion of thermoelectric effects, are solved on an unstructured mesh using the finite-volume method to simulate a transverse Peltier cooler under various operating conditions. First, the code is validated against experimental 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 aspect ratio are also established through parametric studies. Finally, it is shown that the DT can be amplified by constricting the phonon (heat) transport crosssection while keeping the electron (current) transport cross-section unchanged—a strategy that cannot be employed in conventional thermoelectric devices where electrons and phonons follow the same path. This makes transverse Peltier coolers particularly attractive for generating large DT without multi-stage cascading. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Solid-state Peltier coolers [1] are attractive because, unlike vapor-cycle refrigeration systems, they have no moving parts, and are lightweight. In the near future, cryogenic Peltier coolers are expected to supplant conventional vapor-cycle cooling systems used to cool far-infrared, X-ray, and c-ray sensors flown in space and weight-sensitive platforms such as UAVs and situational awareness satellites. For such applications, the relevant temperature range is 150–10 K, i.e., ultra low temperature. The theoretical performance of a conventional thermoelectric (TE) device is dictated by the non-dimensional figure of merit, ZT = R2rT/j, where T is the temperature, and R, r, and j are the Seebeck coefficient, the electrical conductivity, and the thermal conductivity, respectively, of the thermoelectric material being used [1,2]. A conventional Peltier cooler yields a maximum temperature depression of DTmax = (ZT)T/2 provided both n- and p-type materials of the TE alloy are available, and have similar ZT values. Otherwise, the DT is significantly reduced [1]. Recent research has ⇑ 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: [email protected] (S. Mazumder). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.03.018

doubled the ZT of thermoelectric materials [3] at temperatures higher than room temperature. When applied to cryogenic temperatures (less than 150 K), however, the same materials yield ZT values of much less than unity since ZT decreases following a T7/2 law [1,2]. To date, the most promising cryogenic thermoelectric material, n-type Bi95Sb5 doped with a resonant level, potassium, which has been developed by Heremans and co-workers [4], has been shown to yield ZT  0.7 at 100 K. However, such values are manifested only on n-type materials. In contrast, the best p-type material, Bi86Sb14, yields ZT  0.15 at 100 K [4,5]. For practical applications, more than 100 K cooling would be required. Even assuming that p-type materials with ZT  1 (at 100 K) will be developed in the future, simple estimates [6] show that with conventional designs that make use of the longitudinal (conventional) Peltier effect, an 11-stage cascade would be required to cool a device down from 300 K to 10 K, even disregarding transport losses. In practice, such a large number of stages of cascading produce diminishing returns due to parasitic losses such as poor contact and losses to the ambient. Thus, for ultra low temperature applications, even with the best-performing materials, there is a pressing need to explore designs that go beyond the conventional designs. Fundamentally, the performance must be enhanced by combining two approaches: (1) developing high-ZT materials and material combinations, and (2) developing new

374

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

Nomenclature d h j l q q_ J T

height of transverse device (m) convective heat transfer coefficient (W/m2/K) current density vector (A/m2) length of transverse device (m) heat flux vector (W/m2) joule heat per unit volume (W/m3) absolute temperature (K)

geometrical designs that cause phonons and electrons to travel along different and independent paths, thereby enabling us to redefine the conventional figure-of-merit. High-ZT material development, by itself, is not sufficient to meet the afore-stated goals. In this work, in an effort to re-define the conventional figureof-merit of a Peltier cooler, the so-called transverse Peltier cooler is explored. All thermoelectric materials exhibit small degrees of anisotropic Seebeck/Peltier 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 current) in the same direction as the heat flow. In most naturally occurring thermoelectric materials, anisotropic effects are insignificant, and the transverse current is not large enough to be of any practical interest, although it has been observed experimentally in crystalline systems [7,8]. The fact that the dominant longitudinal current is aligned with the heat current also suggests that the major heat carriers, namely phonons in a semiconductor, 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 performance of a conventional thermoelectric device. Any alteration of the phonon path via geometric alterations is tantamount to alteration of the charge carrier path, resulting in equal influences on the electrical and thermal conductance and unaltered ZT. On the other hand, if the transverse current could be amplified by some artificial means, it would suggest a device in which charge carriers and phonons flow through mutually independent paths. In such a scenario, the electrical or thermal conductance could be tuned independent of each other by altering the geometry, thereby creating an opportunity to enhance the effective ZT. Although anisotropic thermoelectric effect is weak in naturally occurring thermoelectric alloys, it is conceivable to force strong anisotropic effect artificially, as has been demonstrated by past research [9,10]. Babin et al. [11] first proposed the general concept of using a tilted multi-layered structure, as shown in Fig. 1, to generate anisotropic thermoelectric effect. This idea was later demonstrated by Gudkin et al. [12] for cooling applications. In recent years, Lengfellner and co-workers [9,10] have demonstrated conclusively through experimental observations that a meta-material comprised of alternating layers of a thermoelectric alloy and a metal can serve as a transverse Peltier cooler. A similar configuration was also employed by Mann and Huxtable [13] to construct a heat flux sensor. Although Lengfellner and co-workers [9,10] have performed some preliminary one-dimensional calculations using effective homogeneous medium theory to support their experimental findings, to date, no detailed computational analysis has been undertaken to shed light into the working mechanisms of such a meta-material. Specifically, although it is acknowledged that in such devices heat and current flow along independent paths [1,2,12], no research has been conducted to elucidate what role geometry can play in improving the performance of such devices. In this work, transverse Peltier coolers comprised of a metamaterial of the afore-stated type were modeled computationally.

Greek

a /

j r P R

tilt angle (degrees) electric potential (V) thermal conductivity (W/m/K) electrical conductivity (S/m) Peltier coefficient (V) Seebeck coefficient (V/K)

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

The bi-layer materials, layer thicknesses, and other operating conditions were chosen to match previously reported experimental studies. The governing conservation equations of energy and electric current, including thermoelectric effects, were solved numerically with high mesh resolution so that the individual layers of the meta-material are adequately resolved. After model validation, exploratory studies were undertaken to elucidate the effect of various geometric design parameters on the performance (DT) of such Peltier coolers. 2. Mathematical model The governing conservation equations are conservation equations of energy and current. Since the model is to be exercised for layers whose thicknesses are of the order of a millimeter (as will be discussed later), it is assumed that continuum is prevalent, and that the governing equations for equilibrium (or continuum) transport are valid. Under the continuum assumption, the governing equations at steady state are written as

Energy conservation : r  q ¼ q_ j

ð1aÞ

Current conservation : r  j ¼ 0

ð1bÞ

where q is the heat flux vector, j is the current density vector, and q_ J is the heat generated per unit volume due to Joule heating. As discussed earlier, most naturally occurring thermoelectric materials are only weakly anisotropic. Therefore, for modeling purposes, these materials are assumed to be isotropic. In a thermoelectric material, the heat flux is due to the combined effect of Fourier conduction and the Peltier effect. Under the isotropic assumption, the heat flux may be written as:

q ¼ jrT þ Pj

ð2Þ

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

375

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

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

j ¼ rr/  rRrT

ð3Þ

where / is the electric potential. The electrical conductivity is denoted by r, while the Seebeck coefficient is denoted by R. The Seebeck coefficient is related to the Peltier coefficient through the Thompson relation [1]: P = RT. Substitution of Eqs. (1b) and (2) into Eq. (1a) yields

Energy conservation : r  ðjrT  Prr/  PRrrTÞ ¼ q_ J ¼ 

jj

r

Current conservation : r  ðrr/  RrrTÞ ¼ 0

ð4aÞ ð4bÞ

Equations (4a) and (4b) represent a set of coupled elliptic partial differential equations with two dependent variables, namely temperature T and electric potential /. The presence of the Joule heating

Fig. 2. Case considered for the validation study: (a) schematic of experimental setup, (b) 2D model setup and boundary conditions and (c) computational mesh.

376

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

Table 1 Transport properties used for calculations in the present study. Transport property

Baseline properties Thermal conductivity, j Electrical conductivity, r Seebeck coefficient, R Modified properties Thermal conductivity, j Electrical conductivity, r Seebeck coefficient, R

Material

Source of data for Bi2Te3

Bismuth Telluride (Bi2Te3)

Lead (Pb)

2.3 W/m/K 105 S/m 200 lV/K

35 W/m/K 5  106 S/m 0

Goldsmid et al. [14] Goldsmid et al. [15] Reitmaier et al. [10]

1.2 W/m/K 1.1  105 S/m 287 lV/K

35 W/m/K 5  106 S/m 0

Takiishi et al. [16] Takiishi et al. [16] Tan et al. [17]

term on the right hand side of Eq. (4a) also makes it non-linear. Thus, these equations can only be solved numerically using an iterative procedure. In the present study, the governing equations were discretized using the finite-volume procedure on an unstructured mesh, and solved using the framework available in the commercial multi-physics code CFD-ACE+. 3. Results and discussion 3.1. Validation study Prior to conducting parametric studies, a validation study was undertaken to understand the limitations of the present modeling approach. Kyarad and Lengfellner [9] have performed experiments on a transverse Peltier cooler with a n-Bi2Te3–Pb meta-material, and recorded the temperature depression (DT) as a function of applied 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 of the 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 occurred 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 in this configuration, the final outcome is a colder (compared to 295 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 surrounding the device at the top and sides is electrically insulating, it is not an insulator thermally. However, the exact thermal conditions on these surfaces are unknown. Therefore, due to lack of better 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 immersed in a water bath at 295 K, which is stirred. Kyarad and Lengfellner [9] contend that although this scenario produces fairly efficient coupling of the bottom surface to the water bath, since the convective resistance is not zero, the temperature at the bottom surface will be somewhat larger than 295 K, and also be non-uniform. In an effort to mimic the actual setup, we treated the convective heat transfer coefficient at the bottom surface as a parameter. Indeed, it was found that the computed DT is dependent upon the value chosen for the convective heat transfer coefficient 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

assumptions are true in the experimental setup or not is also another unanswered question. Also, radiation loss is neglected. Finally, as shown in Fig. 2b, the model was assumed to be twodimensional (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 conducted with the 2D model to reduce computational time, thereby allowing a large number of parametric variations. Fig. 2c depicts the computational mesh used for the validation study. It is comprised of 50,270 triangular control volumes (or cells). This mesh was deemed appropriate after conducting a rigorous grid independence study 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-physical 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 Properties.’’ The values suggested by Reitmaier et al. [10] were obtained from different sources (see Table 1), and as discussed shortly, uncertainties remain with regard to the appropriateness 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 applied compressive 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. 3. Predicted and measured [9] DT in a transverse Peltier cooler with tilt angle of 25° and aspect ratio of 2 with baseline properties.

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

377

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.

values for the bottom surface. These computations 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

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

378

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

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.

not under isothermal bath conditions. Fig. 4 shows the actual temperature distribution in the transverse device under isothermal bottom surface and adiabatic bottom surface conditions for an applied current of 40 A. First, it is observed the temperature at the top surface (the target surface for cooling) is not uniform. Second, under adiabatic bottom surface conditions, although Fig. 3 might indicate best performance, Fig. 4b shows that the temperature at the 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 cooling is actually achieved. This is because under adiabatic bottom surface conditions, the Joule heat generated within the device has no place to go other than the left and right boundaries, and manifests itself as a temperature rise of the whole device. The DT obtained under isothermal conditions with baseline properties is significantly lower than the experimentally observed values (peak of 8 K versus 22 K). As mentioned earlier, there are significant uncertainties with regard to the transport properties used in the simulations. In an effort to investigate this issue, a second set of simulations were conducted with a different set of properties—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 properties under isothermal conditions. While the peak value of (DT) obtained with modified properties matches the experimentally observed values much better, the location of the peak (i.e., the current at which it occurs) is better predicted with the baseline properties. Therefore, the discrepancy between experimentally observed data and predicted results cannot be attributed to uncertainties in the transport properties alone. Most likely, it is a combination of uncertainties and assumptions in boundary conditions and material properties, such as (1) constant (temperature independent) transport properties, (2) negligible radiation loss, (3) isothermal bottom surface or bottom surface with constant convective heat transfer coefficient, (4) uniform current on the left boundary, and (5) negligible interface resistance. One final point to note is that the measured DT is at a location close to the central plane of the sample, and not an average. Nonetheless, since overall qualitative behavior of the predicted results is similar to the observed experimental results, the model was deemed suitable for subsequent parametric studies where the emphasis is on relative performance (i.e., trends) rather than the absolute predicted value of DT. 3.2. Effect of tilt angle

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

attributed to dominance of Joule heating at high current density over the Peltier heat flux since Joule heating is proportional to the 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 simulations under-predict the experimentally observed DT, while under adiabatic conditions, the simulations over-predict the observed DT. The best match to the experimental data is found to be for a convective heat transfer coefficient value 2000 W/m2/K—a value that is typically of forced convection by a liquid [18]. The findings clearly point to the fact that the heat transfer to the cold bath on the bottom surface affects the performance significantly, and also corroborates the contention by Kyarad and Lengfellner [9] that the bottom surface is probably

In the first set of parametric studies, the tilt angle, a, of the layers was varied. The baseline value used for the tilt angle was 25°, in keeping with the theoretical optimum reported by Reitmaier et al. [10]. In addition, two other angles, namely 15° and 35°, were explored in this study. Baseline aspect ratio of 2 was used. The predicted DT for various tilt angles is shown in Fig. 6 for both baseline and modified properties. First, in agreement with the findings reported by Reitmaier et al. [10], the best performance is manifested for tilt angle of 25°, irrespective of the transport properties used. Secondly, shallower tilt angles (i.e., 15°) appear to produce better performance than steeper tilt angles (i.e., 35°) at high currents (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 boundaries when the tilt angle is steep because there is no direct contact between the left and right surfaces through a single layer. This results in excessive heating of the overall device. At shallow angles, direct contact is established between the left and right boundaries by a single layer. This allows the Joule heat to escape laterally through the high thermal conductivity Pb strips.

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

379

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

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 conducted 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 ratios (1, 2, and 4). For both baseline and modified properties, larger aspect ratios result in higher peak DT. This is probably attributable to the fact that when the value of l is increased while keeping d constant, the thermal resistance between the hot and the cold surfaces (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 advantages of a transverse 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 constricting the area through which the heat flows without altering the current flow. A simple way to achieve this amplification 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 constriction effect is the development of a 3D model. As mentioned earlier, a 2D model, which is computationally more efficient, was exercised in the previous parametric studies since the geometry and boundary conditions permitted its use. For the trapezoidal geometry, this is not possible. A 3D model of the rectangular block shown in Fig. 2a was first created. 463,017 tetrahedral cells were used. Adiabatic boundary conditions were applied to the front and back surfaces to replicate a 2D scenario, and the results predicted by this model were compared with the 2D model. The comparison is shown in Fig. 8. Clearly, the 3D model replicates the results

380

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381

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.

produced by the 2D model barring minor differences due to differences in the exact mesh type and size used in the two cases. Having confirmed the validity of the 3D computations, the proposed trapezoidal Peltier cooler was simulated. The mesh used for this study comprised of 693,743 tetrahedral cells (Fig. 9a), and it required approximately 52 min of CPU time to obtain 4 orders of magnitude 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 rectangular device, while the bottom surface area was increased by a factor of 2, as shown in Fig. 9a. A typical temperature distribution is shown in Fig. 9b. The non-uniformity of temperature at the target surface is quite evident. The overall performance of the trapezoidal device in comparison to the rectangular device is shown in Fig. 10. As postulated, an improvement in DT is produced by constricting the area. However, both for baseline and modified properties, only a 25% increase in DT is produced at the optimum current, as opposed to the theoretically expected 100% increase (due to 100% area constriction). The less than expected gain is probably due to threedimensional effects within the tilted layers, which causes the heat to flow at an angle to the top and bottom surfaces rather than exactly perpendicular to these surfaces (as evidenced by Fig. 9b), and also due to end effects, i.e., losses through the left and right surfaces. Nonetheless, this detailed modeling study confirms that constriction of the area is a viable means to improving the performance of transverse Peltier coolers—a strategy that is ineffective in conventional Peltier coolers. 4. Summary and Conclusions A computational analysis based feasibility study was conducted to explore the working mechanism of a transverse Peltier cooler constructed out of a meta-material. This meta-material is comprised of alternating tilted layers of n-Bi2Te3 and Pb. The computational model was first validated against experimentally measured data. It was found that although the code reproduced the important trend of producing the best temperature depression (DT) at an intermediate current, it either over or under-predicted the

measured DT values depending on the kind of thermal boundary conditions used. Several sources of uncertainty and reasons for discrepancy between numerical predictions and experimental observations were discussed. Since the trends predicted by the model were in agreement with experimental observations, it was deemed suitable for further parametric 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 values were found to depend strongly on the applied current. Tilt angle of 15° performed better (produced larger DT) at high applied current, while tilt angle of 35° performed better at low applied current values. When the device aspect ratio was varied, it was found that larger aspect ratios (long with short height) produced the best performance. All of the afore-mentioned studies were conducted using 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 accomplish this goal, two different device geometries—one with a rectangular cross-section, and the other with a trapezoidal cross-section—were considered. It was found that constriction of the area (between the hot and cold surfaces) by a factor of 2 amplified the DT by 25% under optimum applied current conditions. 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 opportunity to develop large DT devices for cryogenic applications without multi-stage cascading.

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

References [1] H.J. Goldsmid, Introduction to Thermoelectricity, Springer Series in Material Science, 2010. [2] L.I. Anatychuk, Physics of Thermoelectricity, Institute of Thermoelectricity, Ukraine, 1998. [3] G.J. Snyder, E.S. Toberer, Complex thermoelectric materials, Nature Materials 7 (2008). 115-114. [4] J.P. Heremans, V. Jovovic, E.S. Toberer, A. Saramat, K. Kurosaki, A. Charoenphakdee, S. Yamanaka, J.G. Snyder, Enhancement of thermoelectric efficiency in PbTe by distortion of the electronic density of states, Science 321 (2008) 554. [5] B. Lenoir, A. Dauscher, M. Cassart, Y. Ravich, H. Scherrer, Effect of antimony content on the thermoelectric figure of merit of Bi1xSbx alloys, Journal of Physics and Chemistry of Solids 59 (1998) 129–134. [6] Snyder, G.J., presentation made at the AFRL/RV Workshop, Kirtland AFB, NM, 22–23 April 2009, organized by Charles Stein. [7] H. Lengfellner, G. Cremb, A. Schnellbogl, J. Betz, K.F. Renk, W. Prettl, Giant voltages upon surface heating in normal YBa2Cu3O7-5 films suggesting an atomic layer thermopile, Applied Physics Letters 60 (4) (1992) 501–503. [8] L.R. Testardi, Anomalous laser-induced voltages in YBa2Cu3Ox and off-diagonal thermoelectricity, Applied Physics Letters 64 (18) (1994) 2347–2349. [9] A. Kyarad, H. Lengfellner, Transverse Peltier effect in tilted Pb–Bi2Te3 multilayer structures, Applied Physics Letters 89 (2006). Article 192103. [10] C. Reitmaier, F. Walther, H. Lengfellner, Transverse thermoelectric devices, Applied Physics A 99 (2010) 717–722. [11] V.P. Babin, T.S. Gudkin, Z.M. Dashevskii, L.D. Dudkin, E.K. Iordanis, V.I. Kaidanov, N.V. Kolomoets, O.M. Narva, L.S. Stil’bans, Anisotropic synthetic thermoelements and their maximum capabilities, Soviet Physics Semiconductors—USSR 8 (4) (1974) 478–481. [12] T.S. Gudkin, E.K. Iordanishvili, V.A. Kudinov, E.E. Fiskind, Thermoelectric, galvanomagnetic, and thermomagnetic properties of heterogeneous layer media. 1. Calculation of transport-coefficients, Soviet Physics Semiconductors—USSR 16 (9) (1982) 1034–1036.

S.A. Ali, S. Mazumder / International Journal of Heat and Mass Transfer 62 (2013) 373–381 [13] B.S. Mann, S.T. Huxtable, Transverse Thermoelectric Effects for Cooling and Heat Flux Sensing, M.S. Thesis in Mechanical Engineering, Virginia Polytechnic Institute and State University. [14] H.J. Goldsmid, The thermal conductivity of Bismuth Telluride, Proceedings of the Physical Society B 69 (1956) 203–209. [15] H.J. Goldsmid, The electrical conductivity and thermoelectric power of Bismuth Telluride, Proceedings of the Physical Society B 71 (4) (1958) 633– 646. [16] M. Takiishi, S. Tanaka, K. Miyazaki, H. Tsukamoto, Thermal conductivity measurements of Bismuth Telluride thin films using the 3 Omega method, in:

381

Proceedings of the 27th Japan Symposium on Thermophysical Properties, Kyoto, Japan, Paper number J113, 2006. [17] J. Tan, K. Kalantar-zadeh, W. Wlodarski, S. Bhargava, D. Akolekar, A. Holland, G. Rosengarten, Thermoelectric properties of bismuth telluride thin films deposited by radio frequency magnetron sputtering, Proceedings of the SPIE, Smart Sensors, Actuators, and MEMS II 5836 (2005) 711–718. [18] T.L. Bergman, A.S. Lavine, F.P. Incropera, D.P. Dewitt, Introduction to Heat Transfer, Sixth Edition., John Wiley, New York, 2011.