2010 NMThermoPhysEng Yao TE Cooler

8/6/2019 2010 NMThermoPhysEng Yao TE Cooler

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DESIGN AND ANALYSIS OF AN IN-PLANE

THERMOELECTRIC MICROCOOLER

Da-Jeng Yao1, Gang Chen2, and Chang-Jin Kim3

1Institute of NanoEngineering and MicroSystems, National Tsing Hua

University, Hsinchu, Taiwan, R.O.C.2Mechanical Engineering Department, Massachusetts Institute of Technology,

Cambridge, Massachusetts3Mechanical and Aerospace Engineering Department, University of California at

Los Angeles, Los Angeles, California

Thin-film thermoelectric devices have potentially greater efficiency than bulk devices because

of quantum and classical size effects involving electrons and phonons. We discuss criteria for

the design of thin-film thermoelectric microcoolers to achieve high performance. The devices

consideredare membrane structures based on electron transport along the film plane. A model

is developed to include the effects of heat loss and leg shape. The design is optimized based on

the modeling results and used to guide microcooler fabrication.

KEY WORDS: thermoelectric, microcooler, in-plane, contact resistance

INTRODUCTION

Many electronic and optoelectronic components dissipate much heat in a small

area, creating a temperature rise that affects the device performance and reliability [1].

A thermoelectric microcooler (-TEC) is a candidate device to decrease the operatingtemperature locally and to absorb heat generated by these devices. The efficiency of a

thermoelectric (TE) device is determined by the materials used in its manufacture. An

important TE material property, the figure of merit (ZT), which is proportional to the

square of the Seebeck coefficient and the electric conductivity of material but inversely

proportional to its thermal conductivity, is used to assess the performance of both a

thermoelectric cooler and a power generator. Several theories to increase the figure ofmerit of TE materials have been developed, and the effectiveness of some approaches

are being proved by experiment results [27]. The geometries fall into two categories.

In one case the current and heat flow perpendicular to the film plane; i.e., the cross-

plane device, which is similar to the traditional TE device. In the other case, both heat

and current flow parallel to the film plane; i.e., the in-plane device. For the latter

devices, an increased ZT is calculated to arise from a several factors, including an

increase in the electronic density of states per unit volume for small well widths (several

Nanoscale and Microscale Thermophysical Engineering, 14: 95109, 2010

Copyright Taylor & Francis Group, LLC

ISSN: 1556-7265 print / 1556-7273 online

DOI: 10.1080/15567265.2010.484008

Manuscript received May 5, 2009

Address correspondence to Da-Jeng Yao, Institute of NanoEngineering and Microsystems, National

Tsing Hua University, No. 101, Sec. 2, Kuang-Fu Rd., Hscinchu, Taiwan 30013., R.O.C. E-mail: djyao@

mx.nthu.edu.tw

95


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nanometers) and an increased carrier mobility if modulation doping is exploited [8]. A

prospective difficulty in obtaining high performance of such devices is that the inert

spacer or supporting material does not contribute to the Peltier heat flow but does

contribute to the heat flow that lowers the effective ZTof the structure [9]. In the cross-

plane device, decreased thermionic emission and phonon thermal conductivity has been

conjectured as possible ZTenhancement mechanisms. Significantly decreased thermal

conductivity in both in-plane and cross-plane directions in quantum structures have

been theoretically predicted and experimentally observed. In the device area, thin-filmdevices based on transport in both directions are being investigated, with each having its

own advantage and disadvantages. The cross-plane devices face the challenge of estab-

lishing a large temperature difference across films of thickness a few micrometers. The

in-plane devices must solve the issue of heat leakage through the supporting substrate.

For some cooling applications, as like Laser cooling and local spot cooling, the

in-plane type of thermoelectric cooling microdevice would a one of potential candi-

dates. Thus, we consider the design issues for an in-plane-TEC illustrated in Figure 1.The TE elements are not only the thermoelectric components but also the supporting

components for the cooling spot membrane. By applying current to the system, the

cooled spot membrane becomes cool, and the frame functions as a heat sink. Such a

configuration was studied previously by Gao [10]. Our work differs from the previouswork in that we consider the effects of the TE leg shape, heat leakage through the

buffer on which superlattices are deposited, and the distribution of the legs. Based on

the modeling results, we determine an optimal design for the fabrication of a -TEC.

ANALYTIC MODEL OF A -TEC

Figure 2 shows a top view and a cross-sectional view of the -TEC from Figure 1.The microdevice consists of 4n pairs of TE legs (width b and length l) connected in

series, and extending from the cold spot membrane (temperature Tc) around the cold

membrane (area F) to the frame rim (temperature T1). The membrane is supported bythe TE legs, made of the n-type (thickness tn) and p-type (thickness tp) Si/Ge

NOMENCLATURE

A area, m2

a length of cooled membrane, mm

b width of the TE leg, mC design parameterE Youngs modulus, N m-2

F area of cooled membrane, m2

h gap between cooled spot andbottom substrate, m

I current input, Ak thermal conductivity, W m-1 K-1

l length of the TE leg, mmN heat load, mWn No. of pairs of TE legs around the

cooled membraneP perimeter of cooled membrane, mQ heat flow, mW

R resistance of the TE leg, rec Electric contact resistance , m

2

T temperature, Kt thickness of films, mZT non-dimensional figure of merit

Greek Symbols Seebeck coefficient, V K-1

natural convection coefficient,W m-2 K-1

heat transfer coefficient, W m-2 K-1

emissivity resistivity of the TE leg , m-1

Stefan-Boltzmann constant,W m-2 K-4

96 D.-J. YAO ET AL.


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superlattice films with a buffer layer (thickness tb) [11]. To grow an effective super-lattice film, n- and p-type superlattice films should be grown separately by molecular-

beam epitaxy (MBE) [12, 13]. A buffer layer (Si0.5Ge0.5) must provide stress release,

and the layer has a continuously graded concentration of silicon from 100% Si at the

substrate interface to Si0.5Ge0.5 at the region at which the superlattice growth com-

mences [14, 15]. The buffer layer might be eliminated during fabrication if it is

unnecessary for the microdevice, but removing the 3-m buffer layer while retainingthe 0.4-m superlattice film is difficult; we hence include it in the model. The electricconnection layer (Au), which serves also as a bonding layer, is in either the cold or hot

region; because it does not run from hot to cold, it might be neglected in this model. An

additional Joule heating generated at junctions due to the electric contact resistance

(rec) must be retained in the model, which can degrade the performance of themicrodevice.

The thickness of the TE leg is limited by the thin-film growth technique to order

1m. For this reason the temperature distribution in the thickness direction can be ignoredbecause of its low thermal resistance compared with the one along the length direction.

Furthermore, because of the symmetric design of the-TEC, analysis of one pair of the TElegs, shown in Figure 3, is sufficient to predict the performance of the -TEC.

A uniform temperature Tc over the cooled area (F/4n) is assumed because a

material (Si) of great thermal conductivity is adopted. As current Ipasses through TE

legs, Peltier heat Qc is generated at the cold junction (at which x l), which is given by

Qc n pITc 1

n-type

TE element

p-type

TE element

electric

power input

frame

(heat sink)

cold spot

Figure 1 Schematic view of in-plane thermoelectric microcooler.

IN-PLANE THERMOELECTRIC MICROCOOLER 97


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Cooled area

F/4n

l=x0=x

T cT

Hot frame

(heat sink)

T

)(xT

dV

TE bridge

(p-type and n-type

with buffer layer) Joule heat Peltier heat

Qc

Heat load

N

Figure 3 Schematic diagram of the analytic -TEC model.

l

l

la 2 a

la 2

F

+

-

b

Current

input

P-type TEelements

N-type TE

elements

Cool spot

membrane

Hot frame

Electric

connection

(a)

P-type TE

elements

N-type TE

elements

Electric

connection

cooling

targetpt

ntBuffer layer bt

(b)

Figure 2 (a) Top view and (b) cross-sectional view of the -TEC.

98 D.-J. YAO ET AL.


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in which n and p are Seebeck coefficients of the n-type and p-type TE legs. Joule heatis also generated; the resistance of the TE leg is given by

R ntn

p

tp l

beq 2

in which idenotes resistivity, and beq denotes the equivalent width of the TE leg that isdefined later.

The governing equation for a simple fin model as shown in Figure 3 is

kftfbeqd2Tx T1

dx2 I

2R

l 2 8"T31beqTx T1 0 3

in which kf

tf

kn

tn

kp

tp

2 kb

tb. k

idenotes thermal conductivity, t

idenotes

thickness, and subscripts and n, p, and b represent n-type TE leg, p-type TE leg, and a

buffer layer. The third term of Eq. (3) describes the heat absorbed by convection and

radiation with the heat transfer coefficient 2 8"T31, in which is the naturalconvection coefficient of the surrounding air, is the emissivity of the TE legs, and is theStefan-Boltzmann constant. When a condition Tx T1=T1


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gs 1 1coshs 8

These two functions depend on the length of the TE legs (l) and the heat transfer

coefficient ratio (p). The optimal current for a maximum temperature difference and amaximum pumping capacity are found from Eq. (6) to be

Iopt n pTc2R

sfsgs 9

in which R R gssfsh i

Rec. On substituting Eq. (9) into Eq. (6), one achieves themaximum temperature difference Tmax when there is no heat load (N), and the

maximum pumping capacity Nmax is obtained when there is no temperature difference

on the entire -TEC.

Tc T1max n p2T2c

4R

sfsgs

1kftfbeq

ls

fs F F4n

24

35 10

Nmax nn p2

T21R

sfsgs

11

Eqs. (6)(11) serve to define an optimal design for a -TEC to achieve a maximum

temperature difference or a maximum heat pumping capacity. Detailed analysis anddiscussion follows.

If the buffer layer, electric contact resistance and heat leakages are all neglected,

and Eqs. (10)(11) become

Tmax ZT2c

212

Nmax 2nT

2c

n

tnP

l; 13

which are results well known for an idealized TEC system.

DESIGN AND ANALYSIS

To facilitate our discussion, we define several design parameters, shown in

Figure 4, as follows.

A square shape of the cooled area might not be best; C1 serves to define the ratio

of cooled area over a square shape with fixed length (a), which gives F C1a2. Forexample, if the cooled area has a circular shape, C1 /4.

The spacing between the TE legs is an important design parameter. C2 serves todefine the occupancy of the TE legs on each side of the cooled area, which makes

100 D.-J. YAO ET AL.


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a C2 nb. For example, if the spacing between two TE legs is the same as the width ofthe TE leg, C2 2. The TE legs might have any shape, not merely a rectangular shapeas shown in Figure 2(a). To simplify the analysis, if fixed width b and length lof the TE

legs are assumed, and the width of the TE legs near the hot frame is b2, then C3 (b b2)/2b defines an equivalent width beq C3 b to convert the top area (Atop C3 bl) andthe cross - sectional area (Across C3 b tf) of the TE legs. The proof of this assumptionis derived in an Appendix.

The electric contact resistance (rec) plays an important role in the design of a

-TEC [9, 16]. The total electric resistance ( "R) across the cold and hot regions servesfor an assessment of the extent to which the performance of -TEC is degraded. Ingeneral, rec is of order 10

6 to 108 cm2; a contact area Ac (50 m)2 is assumed. Theresistance ratio is listed in Table 1. In most cases, Rec,,"R; the electric contact

resistance is thus negligible, if the resistance of the TE leg is large in a thin-film type

of-TEC.

Table 1 Influence of electric contact resistance

l(mm) Resistance ratio (Rgssfs) Rec in (%)

0.3 48.2 0.00080.08

0.5 50.9 0.000790.079

1.0 51.2 0.000780.078

Perimeter (P)

Area

(F)

(a) C1=1, C

2=2, C

3=1 (b) C

1=1, C

2=2, C

3=2

(c) C1=/4, C

2=2, C

3=2 (d) C

1=5/9, C

2=2, C

3=2

Figure 4 Geometry design of the TE legs and cooled area.



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The importance of thermal contact resistance on microdevices can be estimated

by comparing the thermal resistance of substrate or legs. Taking the thermal conduc-

tivity of the leg as ,1 W/mK, the thermal resistance of the leg varies from 3 to 10 cm 2

K/W for various length of leg from 0.3 to 1 mm. The thermal boundary resistance is

typically on the order of 103

cm2

K/W, which is much smaller compared with thethermal resistance of legs. This suggests that thermal contact resistance is very impor-

tant when high thermal conductivity substrates or legs are used.

To simplify the analytic model, we assume the same material properties and sizes

of the n- and p-type TE legs: n p, kn kp, n -p, and tn tp. Eqs. (2) and (9)(11)become

R 2nlC3btn

14

Iopt

nT

cR

sf

sgs

n

Tc

2n

C3

btn

l

sf

sgs 15

Tmax ZT2c

4

1

1 kbtbkntn

gsf2s Fkntn

C1C2a2

2C3

lP

gssfs

24

35 16

Nmax 2nT

21

2n

C3tn

C2

P

l

sfsgs

17

in which Z 2

nnkn is the figure of merit.Tmax is proportional to Z T[17], as shown

in Eq. (16).

With the size of-TEC and material properties listed in Table 2, we examine theheat leakage effect, through both the TE legs and the cooled area. The heat transfer

coefficient Fis calculated from heat conduction, natural heat convection, and thermal

Table 2 Material properties and size parameters used in

the calculation

a (mm) 1

tn tp (m) 0.4l(mm) 0.3, 0.5, 1

h (mm) 0.5

4n 20

T1

(K) 300

n -p (V K-1) 200n p (10-5 m) 0.6kn kp, kb, kair (W m-1 K-1) 2, 2, 0.042ZT1

1

, F (worse case) 1(W m2 K1) 5rec ( m

2) 1010

(W m2 K4) 5.67x108



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radiation. Even in a vacuum, thermal radiation should be included in the model. The

range ofF can vary from 10 to 100 W m1 K1, and is sometimes even larger, which

might degrade the system. In addition, another air gap size (h) can make F differ, asshown in Table 3. Figures 5 and 6 compare the performance of a -TEC in a vacuum

(only thermal radiation) and in an air environment (F 110). According to theseplots and based on a fixed length of the TE leg on each curve, the maximum tempera-

ture difference (Tmax) and the maximum heat pumping capacity (Nmax) vary as a

function of design parameters: C C1 C2/C3 for Tmax, and C C3/C2 for Nmax..Nmax is achieved when there is no temperature difference and depends only on the

resistance of the TE leg. The buffer layer does not affect Nmax because there is no

thermoelectric effect, but thermal conduction in the buffer layer is assumed.

Comparing curves in two sets, with and without a buffer layer, we see that the buffer

layer effect makesTmax of-TEC decrease, shown obviously as the design parameteris less than unity. Removal of the buffer layer during fabrication is thus preferable if a

large

Tmax is required.

Tmax of -TEC is also degraded through heat loss whenoperating in an air environment.

Table 3 Heat transfer coefficient

h (mm) kair(g) 2 8 F T31 F (W m

2 K1)

0.5 84 10 12.5 108.5

1 42 10 12.5 64.5

0.0 0.5 1.0 1.5 2.0 2.5

0

5

10

15

20

25

30

35

40

45

50

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5C1: Cold island area ratio

C2: TE leg occupancy C3: Equivalent leg width

T

max

(K)

C=C1*C

2/C

3forT

maxand C=C

3/C

2for N

max

ZT= 1

l=0.3

l=0.5

l=1.0

With and w/o

buffer

With

buffer

Without

buffer

Nmax(mW)

Figure 5 Performance of the -TEC in vacuum.



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Figure 7 is plotted to predict the performance of a -TEC in vacuum with varieddesign, shown in Figure 4 for l 0.5 mm; fixed lengths of cooled area (a2) and of widthof TE legs (b) are assumed. Design parameters C1, C2, and C3 define the optimal

geometry of the TE legs and the cooled area:

1. The smaller the cooled area (F), the less heat leakage there is from the cooled area.

C1 should be designed as small as possible, shown in curves ofTmax in Figure 7,

but C1 is independent ofNmax because it is achieved when there is no temperature

difference, which means no heat loss from the cooled area.

2. C2 is designed for the occupancy of the TE legs on the perimeter of the cooled area.

The smaller C2, is, the larger is the occupancy of the TE legs, which correspondswith the performed TE effects on the cooled area, shown in Figure 7.

3. C3 serves calculate the equivalent width of the TE legs. A larger C3 also increases the

occupancy of the TE legs, so it also increases the performance of the -TEC, shownin Figure 7.

In general, to decrease the cooled area (C1) or to increase the optimal usage of the

perimeter of the cooled area (C3/C2) increases the performance of the designed

-TEC. Table 4 lists the calculated design parameters for both Tmax and Nmax,which demonstrate that either design (c) or design (d) can be the most advanta-

geous choice.

Because the cooled spot is suspended only by microscale TE legs, thestructural robustness of a -TEC device needs to be examined. We assume 50%

0.0 0.5 1.0 1.5 2.0 2.5

0

5

10

15

20

25

30

35

40

0.0

0.5

1.0

1.5

2.0

2.5

3.0C1: Cold island area ratio


T

max

(K)

C=C1*C

2/C

3forT

maxand C=C

3/C

2for N

max

ZT= 1

l=0.3

l=0.5

l=1.0

With and w/o

buffer

With

buffer

Without

buffer

Nmax(mW)

Figure 6 Performance of the -TEC in an air environment.



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Si and 50% Ge for the TE material [18], equivalent width beq 50 m for the TElegs, and that the mass of the TE legs is much smaller than that of the cooled spot.

See the device design in the previous section for other dimensions. We furtherassume that Youngs modulus of the TE legs is a combination of Esi 160 GPaand EGe 130 GPa [19]. The natural frequency of the suspended cooled spot isestimated as:

wn 12

ffiffiffiffiffiffiffiffiffiffikleg

mcold

s% 32kHz 19

The failure strength is also estimated to be about 30 mN, assuming the yield stress of

the TE legs to be 1 GPa, 1% of Youngs modulus [20]. The maximum shock before

failure is estimated to be well beyond the kG range, because of the small mass. Weexpect that the -TEC under current design will be sufficiently robust.

Table 4 Design parameters for cases ad in Figure 4

Case a Case b Case c Case d

for Tmax:C C1 C2/C3 2 1 /4 5/9for Nmax:C C3/C2 1/2 1 1 1

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

0

5

10

15

20

25

30

35

40

45

50

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

A

B,C,D

D

C

A

B

C1: Cold island area ratio


T

max

(K)

C=C1*C

2/C

3forT

maxand C=C

3/C

2for N

max

Tmax

w/o buffer

Tmax

with buffer

Nmax

ZT= 1

Nmax(mW)

Figure 7 Performance variation with shape design from Figure 4 in an air environment for l 0.5 mm.



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CONCLUSION

We have developed a complete analytic model including the effects of heat loss

and leg shape. Both a buffer layer and the heat leakage effect can degrade the

performance of a -TEC. The electric contact resistance becomes neglected becausethe resistance of the TE legs is large in a -TEC of thin-film type. According to ourresults for this analytic model, a thin-film thermoelectric microcooler (-TEC) can befabricated to achieve high performance.

REFERENCES

1. R.C. Chu, and R.E. Simons, Application of Thermoelectrics to Cooling Electronics: Review

and Prospects, 18th International Conference on Thermoelectrics, August 29September 2,

Baltimore, MD, 1999, pp. 270279.

2. I. Yamasaki, and R. Yamanaka, Thermoelectric Properties of Bi2Te3/Sb2Te3 SuperlatticeStructure, 17th International Conference on Thermoelectrics, May 2428, Nagoya, Japan,

1999, pp. 210213.

3. A. Sipatov, and V. Volobuev, Thermoelectric Characteristics of PhS/EuS Superlattices, 18th

International Conference on Thermoelectrics, August 29September 2, Baltimore, MD, 1999,

pp. 198200.

4. G.S., Nolas, G.A. Slack, J.L. Cohn, and S.B. Schujman, The Next Generation of

Thermoelectric Materials, 17th International Conference on Thermoelectrics, May 2428,

Nagoyo, Japan, 1998, pp. 294296.

5. G.D. Mahan, and J.O. Sofo, The Best Thermoelectric, Proceedings of the National Academy

of Sciences of the United States, Vol. 93, No. 15, pp. 74367439, 1996.

6. L.D. Hicks, and M.S. Dresselhaus, Effect of Quantum-Well Structures on the Thermoelectric

Figure of Merit, Physical Review B - Condensed Matter, Vol. 47, No. 19, pp. 1272712731,

1993.

7. D.J. Bergman, and L.G. Fel, Enhancement of Thermoelectric Power Factor in Composite

Thermoelectrics, Journal of Applied Physics, Vol. 85, No. 12, pp. 82058216, 1999.

8. M.S. Dresselhaus, et al., Thermoelectricity in Bi Nanowires, American Physical Society,

1998.

9. D.-J. Yao, G. Chen, and C.-J. Kim, Spot Cooling Using Thermoelectric Microcoolers, 17th

International Conference on Thermoelectrics, August 29September, Baltimore, MD, 1999,

pp. 256259.

10. G. Min, D.M. Rowe, and F. Volklein, Integrated Thin Film Thermoelectric Cooler.

Electronics Letters, Vol. 34, No. 2, pp. 222223, 1998.

11. A. Balandin, and K.L. Wang, Effect of Phonon Confinement on the Thermoelectric Figureof Merit of Quantum Wells, Journal of Applied Physics, Vol. 84, No. 11, pp. 61496153,

1998.

12. A. Boyer, E. Cisse, and Y. Azzouz, Medium-Power Thermophiles Using Thin-Film

Technology, Sensors and Actuators A - Physical, Vol. 24, No.3, pp. 217220, 1990.

13. A. Boyer, and E. Cisse, Properties of Thin Film Thermoelectric Materials-Application to

Sensors Using the Seebeck Effect, Materials Science and Engineering B -Solid State Materials

for Advanced Technology, Vol. 13 No. 2, pp. 103111, 1992.

14. D.K. Nayak, J.S. Park, J.C.S. Woo, K.L. Wang, and I.C. Ivanov, Interface Properties of Thin

Oxides Grown on Strained Ge, Journal of Applied Physics, Vol. 76, No. 2, pp. 982986, 1994.

15. D.K. Nayak, J.C.S. Woo, J.S. Park, K.L. Wang, and I.C.K.P. MacWilliams, Hole

Confinement in a Si/GeSi/Si Quantum Well on SIMOX, IEEE Transactions on Electronic

Devices, Vol. 43, No. 1, pp. 180186, 1996.



13/15

16. J.-P. Fleurial, A. Borshchevsky, M.A. Ryan, W. Philips, E. Kolawa, T. Kacisch, and

R. Ewell, Thermoelectric Microcoolers for Thermal Management Applications, 16th

International Conference on Thermoelectrics, August 2629, Dresden, Germany, 1997, pp.

641645.

17. H.J. Goldsmid, Section V: Thermoelectric system and application, Chapter 59:Thermoelectric Refrigeration for Mass-Market Applications, pp. 591,598,

Thermoelectric Refrigeration, in (eds.) D.K. Mendelssohn and D.K.D. Timmerhaus,

Plenum Press, 1964.

18. R. Hull, and J.C. Bean, Germanium Silicon: Physics and Materials, in R.K. Willardson and E.

R. Weber (eds.), Semiconductors and Semimetals, Vol. 56, pp. 825856, Academic Press, 1999.

19. E. Kasper, Properties of Strained and Relaxed Silicon Germanium, Short Run Press, Exeter,

UK, 1995.

20. S.P. Timoshenko, Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, 1959.

21. F.P. Incropera, and D.P.D. Witt, Fundamentals of Heat and Mass Transfer, John Wiley,

New York, 1990.

APPENDIX

In this article, to simplify the derivation of analytic model, we use control

parameter C3 to define beq, which serves to convert arbitrarily shaped TE legs into

rectangularly shaped TE legs. This approximation is validated on solving the fin-

governing equation with various cross-sectional areas as follows.

Instead of using control parameter C3, the side slope of the TE legs S, S (b2-b)/l, can serve to define a cross-sectional area Ac(x) (S x b) tfand surface area As(x) x (S x 2 b) tf of the TE legs. The resistance of the TE legs is obtained on integratingalong the length direction with a varying cross-sectional area, according to

R Zl

0

dx

dAcx

tSlnSl b

b A1

The resistance of the TE legs should thus be modified because of the variously shaped

TE legs, and the resistance is assumed to be constant as the shape of the TE legs is fixed.

Because of the varying cross section of the TE legs, the fin-governing equation,

Eq. (3) [21], becomes modified to

d2fkfAcxTx T1gdx

2

I2R

l

4"T3

1dAsx

dx Tx T1 0 A2

On substituting Ac(x) and As(x) into Eq. (A2), it is transformed into

kftfbx d2Tx T1

dx2 kftfSdTx T1

dx I

2R

l

2 8"T31bxTx T1 0A3

in which b(x) S x b is the width of the TE legs at position x. Using two boundaryconditions,

T

x

T1

xlj

0 and kftfbeqdTxT1

dx x0j

Qc

N

I2Rec

F F4n Tl T1, we obtain this solution:



14/15

Tc T1Db; l; F; S FF

4n n pITc N

I2RCb; l; F; S RecA4

in which Db; l:F; S and Cb; l:F; S are complicated functions in terms of the size ofthe TE legs and heat transfer coefficient:

Db; l; F; S ffiffiffiffiffiffiffiffiffiffiffiffibFp bY1AJ0B J1AY0BY0BJ0A J0BY0AS

ffiffiffiffiA

p A5

Cb; l; F; S bffiffiffiffi

Ap

Y0AJ1AJ1AY0B Y0AJ1AY1AJ0B J0AY1AJ1AY0B J0AY1AY1AJ0B

b ffiffiffiffiBp Y0BJ1BY1AJ0A J0BY1BY1AJ0A J1AY0BY0AJ1B J1AJ0BY1BY0A

=slffiffiffiffi

Ap

Y0BJ0A J0BY0A

A6

in which A b / kftfS2 and B (l S b)/kftfS2. Jnx and Ynx are Bessel functions of thefirst kind and of order n, defined as

Jnx Jn2ffiffiffiffiffiffiffiffiffiffiffiffiFb

p ffiffiffix

p A7

Length of TE leg l (mm)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

2

4

6

8

10

12

14

16

18

20

22

24

0

2

4

6

8

10

12

b=50m

b2=100m

Simplified case

Real case

1=

ZT

110=F

b=50m

b2=100m

max(mW

)T

max

(K)

Figure A1 Performance of the -TEC with a buffer layer as a function of lin an air environment for widthb 50 m of the TE leg.



15/15

Ynx Yn2ffiffiffiffiffiffiffiffiffiffiffiffiFb

p ffiffiffix

p A8

Eq. (A3) has a form similar to Eq. (6) above, but with more complicated

functions involved. The maximum temperature difference and maximum pumping

capacity are obtained following the same derivation in the article. Figure A1 serves to

compare results from both a real case and a simplified case. Control parameter C3 is

thus suitable for use in our model.


Documents

2010 NMThermoPhysEng Yao TE Cooler