Optimal Design of Thermoelectric Devices with Dimensional...

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Optimal Design of Thermoelectric Devices with Dimensional Analysis

HoSung Lee

Mechanical and Aeronautical Engineering, Western Michigan University,

1903 W. Michigan Ave, Kalamazoo, Michigan 49008-5343, USA

Office (269) 276-3429

Fax (269) 276-3421

Email: [email protected]

Abstract

The optimum design of thermoelectric devices (thermoelectric generator and cooler) in

connection with heat sinks was developed using dimensional analysis. New dimensionless

groups were properly defined to represent important parameters of the thermoelectric devices.

Particularly, use of the convection conductance of a fluid in the denominators of the

dimensionless parameters was critically important, which leads to a new optimum design. This

allows us to determine either the optimal number of thermocouples or the optimal thermal

conductance (the geometric ratio of footprint of leg to leg length). It is stated from the present

dimensional analysis that, if two fluid temperatures on the heat sinks are given, an optimum

design always exists and can be found with the feasible mechanical constraints. The optimum

design includes the optimum parameters such as efficiency, power, current, geometry or number

of thermocouples, and thermal resistances of heat sinks.

Keywords: Optimal design; Dimensional analysis; Thermoelectric generator; thermoelectric

cooler; thermoelectric module

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Nomenclature

A cross-sectional area of thermoelement (cm2)

1A total fin surface area at fluid 1 (cm2)

2A total fin surface area at fluid 2 (cm2)

bA base area of heat sink (cm2)

COP the coefficient of performance

1h heat transfer coefficient of fluid 1 (W/m2K)

2h heat transfer coefficient of fluid 2 (W/m2K)

I electric current (A)

L length of thermoelement (mm)

k thermal conductivity (W/mK), np kkk

n the number of thermocouples

kN dimensionless thermal conductance, 222 AhLAknNk

hN dimensionless convection, 222111 AhAhNh

IN dimensionless current, LAkIN I

VN dimensionless voltage, 2 TnVN nV

1Q the rate of heat transfer entering into TEG (W)

2Q the rate of heat transfer leaving TEG (W)

dP power density (W/cm2)

R electrical resistance of a thermocouple ()

LR load resistance of a thermocouple ()

rR dimensionless resistance, RRR Lr

1T junction temperature at fluid 1 (°C)

2T junction temperature at fluid 2 (°C)

1T temperature of fluid 1 (°C)

2T temperature of fluid 2 (°C)

max,1T maximum temperature of fluid 1 (°C)

min,1T minimum temperature of fluid 1 (°C)

nV Voltage of a module (V)

nW power output (W)

nW power input (W)

Z the figure of merit

Greek symbols

Seebeck coefficient (V/K), np

electrical resistivity (cm), np

1 fin efficiencyof heat sink 1

2 fin efficiencyof heat sink 2

3

th thermal efficiencyof TEG

Subscripts

p p-type element

n n-type element

opt optimal quantity

1/2opt half optimal quantity

Superscript

dimensionless

1. Introduction

Thermoelectric devices (thermoelectric generator and cooler) have found comprehensive

applications in solar energy conversion [1], exhaust energy conversion [2,3], low grade waste

heat recovery [4,5,6], power plants [7], electronic cooling [8], vehicle air conditioners, and

refrigerators [7]. The most common refrigerant used in home and automobile air conditioners is

R-134a, which does not have the ozone-depleting properties of Freon, but is nevertheless a

terrible greenhouse gas and will be banned in the near future [9]. The pertinent candidate for the

replacement would be thermoelectric coolers. Many analyses, optimizations, even manufacturers’

performance curves on thermoelectric devices have been based on the constant high and cold

junction temperatures of the devices. Practically, the thermoelectric devices must work with heat

sinks (or heat exchangers). It is then very difficult to have the constant junction temperatures

unless the thermal resistances of the heat sinks are zero, which is, of course, impossible.

A significant amount of research related to the optimization of thermoelectric devices in

conjunction with heat sinks has been conducted as found in the literature [10-30]. It is well noted

from the literature that there is the existence of optimal conditions in power output or efficiency

with respect to the external load resistance for a thermoelectric generator (TEG) or the electrical

current for a thermoelectric cooler (TEC). Many researchers attempted to combine the theoretical

thermoelectric equations and the heat balance equations of heat sinks, and then to optimize

design parameters such as the geometry of heat sinks [10], allocation of the heat transfer areas of

heat sinks [12,13,18,19], thermoelement length [14], the number of thermocouples [15], the

geometric ratio of the cross-sectional area of thermoelement to the length [16], and slenderness

ratio (the geometric factor ratio of n-type to that of p-type elements) [17]. It can be seen from the

above literature that the geometric optimization of thermoelectric devices is important in design

and also formidable due to so many design parameters. The thermal conductance of

thermoelements that is the most important geometric parameter has been often addressed in

analysis, which is the product of three parameters: the number of thermocouples, the geometric

ratio, and the thermal conductivity. In order to reduce the optimum design parameters, obviously

dimensionless analyses were performed in the literature [21-26]. Yamanashi [21] developed

optimum design introducing dimensionless parameters for a thermoelectric cooler with two heat

sinks, wherein the thermal conductance appears twice in the nominators and fourth in the

denominators of the dimensionless parameters. Although his work led to a new approach in

dimensionless optimum design, the analysis encountered difficulties in optimizing the cooling

power with respect to the thermal conductance because the conductance is intricately related to

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the others. Later, researchers [22,24,25] reported optimum design using the similar

dimensionless parameters used by Yamanashi [21], presenting valuable optimum design features

as Xuan [22] optimized cooling power for a TEC as a function of thermoelement length, Pan et

al. [24] showed the optimum thermal conductance for a TEC with a given cooling power, and

Casano and Piva [25] presented the optimum external load resistance ratio with heat sinks for a

TEG which is greater than unity. There are also some experimental works [27-30] comparing

with the theoretical thermoelectric equations. Gou et al. [27] conducted experiments for low-

temperature waste heat recovery and demonstrated that the experimental results were in fair

agreement with the solution formulas originally derived by Chen et al. [15] from the general

theoretical thermoelectric equations. Chang et al. [28] and Huang et al. [29] conducted

experiments for a TEC from a heat source with air-cooling and water cooling heat sinks,

respectively. Casano and Piva [30] reported experimental work on a set of nine thermoelectric

generator modules with a heat source on one side and a heat sink on the other side. After

deliberately determined the heat leakage which turned out to be about 30% of the supplied heat

source, they demonstrated that the theoretical performance curves of the power output and

efficiency as a function of the external load resistance and temperature difference were in good

agreement with the measurements. It is realized from the above experimental works that the

theoretical thermoelectric equations with the heat balance equations of heat sinks can reasonably

predict the real performance. However, proper optimum design still remains questionable.

In spite of many efforts for optimum design, its applications seem greatly challenging to

system designers [1,2,3]. For example, Hsu et al. [3] in 2011 tested an exhaust heat recovery

system both experimentally using an automobile and mathematically using computer simulations.

They found a reasonable agreement between the measurements and the simulations. However,

they obtained the power output of 12.41 W over 24 thermoelectric generator modules with the

exhaust gas temperature of 573 K and the air temperature of 300 K. When the power output was

divided by the footprint of 24 thermoelectric generator modules, it gives the power density of

0.032 W/cm2, which seems unusually small. Karri et al. [2] in 2012 conducted a similar

experiment with an SUV automobile. This time they designed the exhaust heat recovery system

with an optimum coolant flow rate. They obtained the power output of 550 W over 16

thermoelectric generator modules with the exhaust gas temperature of 686 K and the coolant

temperature of 361 K, which provided the power density of 0.61 W/cm2. This shows a significant

improvement, indicating the importance of optimum design. A New Energy Development

Organization (NEDO) Program (Japan) [7] in 2003 also reported a similar experiment with a

passenger car, obtaining the power output of 240 W over 16 segmented-type modules with the

exhaust gas temperature of 773 K and the coolant temperature of 298 K, which provided the

power density of ~1 W/cm2. Notably, the power densities obtained are no way to evaluate how

good it is until the better comes because proper optimum design seems not available.

From the review of the above theoretical and experimental studies including optimum

design in the literature, it is summarized that the proper optimum design should be determined

basically not only by the power output for TEG (or cooling power for TEC) but also by the

efficiency (the coefficient of performance) simultaneously with respect to both the external load

resistance (or the electrical current) and the geometry of thermoelement which refers to the

number of thermocouples and the geometric ratio. The former (external load resistance) is well

attained in the literature but the latter (geometry) is vague. Therefore, the optimum design seems

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incomplete. This is the rationale why the present paper is to improve the optimum design

introducing new dimensionless parameters.

2. Thermoelectric Generator Let us consider a simplified steady-state heat transfer on a thermoelectric generator module

(TEG) with two heat sinks as shown in Figure 1 (a). Each heat sink faces a fluid flow at

temperature T∞. Subscript 1 and 2 denote hot and cold quantities, respectively. We assume that

the electrical and thermal contact resistances in the TEG are negligible, the material properties

are independent of temperature, and also the TEG is perfectly insulated. The TEG has a number

of thermocouples, of which each thermocouple consists of p-type and n-type thermoelements

with the same dimensions as shown in Figure 1 (b). It is noted that the thermal resistance of heat

sink 1 can be expressed by the reciprocal of the convection conductance 111 Ah , where 1 is the

fin efficiency, 1h is the convection coefficient, and 1A is the total surface area in the heat sink 1.

We hereafter use the convection conductance rather than the thermal resistance.

(a) (b)

Figure 1. (a) Thermoelectric generator module (TEG) with two heat sinks and (b) thermocouple.

The basic equations for the TEG with two heat sinks are given by

111111 TTAhQ (1)

21

2

112

1TT

L

AkRIITnQ (2)

21

2

222

1TT

L

AkRIITnQ (3)

6

222222 TTAhQ (4)

RR

TTI

L

21

(5)

where np , np kkk , and np . Equations (1) – (5) can be solved for T1 and

T2, providing the power output. However, in order to study the optimization of the TEG, several

dimensionless parameters are introduced. As mentioned in Section of Introduction, it is reminded

that optimum design should consider not only the power output but also the efficiency

simultaneously with respect to both the external load resistance and the geometry of

thermoelement which refers to the number of thermocouples and the geometric ratio. In order to

reveal the effect of thermal conductance LAkn , the thermal conductance is placed in the

nominator while the convection conductance 222 Ah in fluid 2 is placed in the denominator of

the parameter. The convection conductance 222 Ah and temperature 2T at fluid 2 are assumed

to be given. The dimensionless thermal conductance, the ratio of thermal conductance to the

convection conductance in fluid 2, is defined by

222 Ah

LAknN k

(6)

The dimensionless convection is an important geometry of heat sinks. Since the convection

conductance at fluid 2 is given as mentioned before, the dimensionless convection, the ratio of

convection conductance in fluid 1 to fluid 2, is defined by

222

111

Ah

AhNh

(7)

The dimensionless electrical resistance, the ratio of the load resistance to the electrical

resistance of thermocouple, is given by

R

RR L

r (8)

Since the fluid temperature 2T at fluid 2 is given as mentioned before, the dimensionless

temperatures are defined by

2

11

T

TT

(9)

2

22

T

TT

(10)

7

2

1

T

TT

(11)

The dimensionless power and heat transfer are defined by dividing by the product of the

convection conductance and the temperature of fluid 2 so that the quantities depend only on the

nominators not on the denominators since the denominator is assumed to be constant or given.

The two dimensionless rates of heat transfer and the dimensionless power output are defined by

2222

11

TAh

QQ

(12)

2222

22

TAh

QQ

(13)

2222

TAh

WW n

n

(14)

It is noted that the above dimensionless parameters are based on the convection conductance

in fluid 2, which means that 2222 TAh should be initially provided. Also note that the thermal

conductance LAkn appears only in Equation (6) among other parameters so that the thermal

conductance can be examined for optimization. Using the dimensionless parameters defined in

Equations (6) – (11), Equations (1) – (5) reduce to two formulas as:

212

2

21212121

121TT

R

TTZT

R

TTTZT

N

TTN

rrk

h

(15)

212

2

21222122

121

1TT

R

TTZT

R

TTTZT

N

T

rrk

(16)

where Z is called the figure of merit ( kZ 2 ). Equations (15) and (16) can be solved for

1T

and

2T . The dimensionless temperatures are then a function of five independent dimensionless

parameters as

21 ,,,,

ZTTRNNfT rhk

(17)

22 ,,,,

ZTTRNNfT rhk

(18)

T is the input and 2ZT is the material property with the input, and both are initially

provided. Therefore, the optimization can be performed only with the first three parameters (Nk,

Nh, and Rr). Once the two dimensionless temperatures (

1T and

2T ) are solved for, the

dimensionless rates of heat transfer at both hot and cold junctions of the TEG can be obtained as:

8

11 TTNQ h

(19)

122 TQ

(20)

Then, we have the dimensionless power output as

21 QQWn

(21)

Accordingly, the thermal efficiency is obtained by

1Q

Wnth

(22)

Defining AkILN I , the dimensionless current is obtained by

1

212

r

IR

TTZTN

(23)

Also, defining 2 TnVNV , the dimensionless voltage is obtained by

kI

nV

NN

WN

(24)

With the inputs (

T and 2ZT ), we begin developing the optimization with the dimensionless

parameters (Nk, Nh, and Rr) iteratively until they converge. It is found that both Nk and Rr show

their optimal values for the dimensionless power output, while Nh does not show the optimal

value showing that the dimensionless power output monotonically increases with increasing hN .

This implies that, if hN is given, the optimal combination of Nk and Rr can be obtained. However,

the dimensionless convection hN actually presents the feasible mechanical constraints. Thus, we

first proceed with a typical value of 1hN for illustration and later examine the variety of hN

with some practical design examples.

Suppose that we have two initial inputs of 6.2

T (two fluid temperatures) and 0.12 ZT (materials) along with 1hN . We then determine the optimal combination for kN

and rR ,

which may be obtained either graphically or using a computer program. We first use the

graphical method at this moment and later the program for multiple computations. The

dimensionless power output

nW and thermal efficiency th are together plotted as a function of

rR , which are presented in Figure 2 (a). Both

nW and th with respect to rR indeed show their

optimal values that appear close. We are interested primarily in the power output and secondly in

the efficiency. However, since they are close each other, we herein use the power output for the

optimization. It should be noted that the dimensionless maximum power output does not occur at

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1rR from Figure 2 (a) as usually assumed for a TEG without heat sinks, but approximately at

7.1rR , because the dimensionless temperatures

1T and

2T in Figure 2 (b) are no longer

constant. This is often a confusing factor in optimum design with a TEG with two heat sinks. We

should not assume that rR is equal to unity for a TEG with heat sinks.

(a)

(b)

Figure 2 (a) Dimensionless power output

nW and efficiency th versus the ratio of load

resistance to resistance of thermocouple rR and (b) dimensionless temperatures

1T and

2T versus

rR . These plots were generated using 3.0kN , 1hN , 6.2

T and 0.12 ZT .

With the dimensionless parameters obtained ( 1hN , 7.1rR , 6.2

T , and 0.12 ZT ),

we now plot the dimensionless power output

nW as a function of the dimensionless thermal

0 1 2 3 4 5 60

0.01

0.02

0.03

0.04

0.05

0

0.03

0.06

0.09

0.12

0.15

Wn*

th

Wn* th

Rr

0 1 2 3 4 5 60

1

2

3

4

0

0.02

0.04

0.06

0.08

T1*

& T2*

T1*

Wn*

Wn*

T2*

Rr

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conductance kN defined in Equation (6) along with the thermal efficiency th , which is shown in

Figure 3 (a). We find an optimum

nW approximately at 3.0kN . Actually, the optimal values of

kN and rR should be iterated until the two simultaneously converge.

(a)

(b)

Figure 3 (a) Dimensionless power output

nW and thermal efficiency th versus dimensionless

thermal conductance kN , and (b) high and low junction temperatures (

1T and

2T ) versus

dimensionless thermal conductance kN . These plots were generated using 1hN , 7.1rR ,

6.2

T and 0.12 ZT .

From 222 AhLAknNk as shown in Equation (6), the kN actually determines the

number of thermocouples n if the geometric ratio A/L and 222 Ah are given or vice versa. The

0 0.2 0.4 0.6 0.80

0.01

0.02

0.03

0.04

0.05

0.06

0

0.05

0.1

0.15

0.2

0.25

0.3

Wn*

Wn* th

th

Nk

0 0.2 0.4 0.6 0.80

1

2

3

4

0

0.05

0.1

0.15

0.2

th

T1* thT1*

& T2*

T2*

Nk

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nW first increases and later decreases with increasing kN . It is

important to realize that, if 222 Ah is given, there is an optimal number n of thermocouples (or

optimal thermal conductance LAk ) in the thermoelectric module, which is usually unknown.

Physically, the surplus number of thermocouples virtually increases the thermal conduction more

than the production of power, which causes the net power output to decline. There is another

important aspect of the optimal dimensionless thermal conductance of 3.0kN , which is that the

module thermal conductance LAkn directly depends on the 222 Ah . In other words, the

module thermal conductance LAkn must be redesigned on the basis of the 222 Ah to meet

3.0kN . The information of the optimal thermal conductance [(n)(A/L)(k)] is particularly

important in design of microstructured or thin-film thermoelectric devices. Furthermore, there is

a potential to improve the performance or to provide the variety of the geometry by reducing the

thermal conductivity k.

The dimensionless high and low junction temperatures are presented in Figure 3 (b). As kN

decreases towards zero,

1T and

2T approach

T and 1, respectively. This indicates that the

thermal resistances of two heat sinks approaches zero, which never happens. It is noted that the

thermal efficiency approaches the theoretical maximum efficiency of 0.2 for the given fluid

temperatures as kN approaches zero.

Since there is an optimal combination of Nk and Rr for a given hN , we can plot the optimal


optW and optimal thermal efficiency opt as a function of

dimensionless convection hN , which is shown in Figure 4. It is very interesting to note in Figure

4 that, with increasing hN , opt barely changes, while

optW monotonically increases. According

to Equation (14), the actual optimal power output optW is the product of

optW and 222 Ah ,

seemingly increasing linearly with 222 Ah . In practice, there is a controversial tendency that hN

may decrease systematically with increasing 222 Ah if 111 Ah is limited. As a result of this, it is

needed to examine the variety of the 222 Ah as a function of hN for the optimal power output.

Figure 5 reveals the intricate relationship between 222 Ah and hN (or 111 Ah ) along with the

optimum actual power output (not dimensionless), which would lead system designers to a

variety of possible allocations ( 222 Ah and hN ) for their optimal design. Now we look into the

actual optimal design with the actual values.

For example, using Figures 4 and 5, we develop an optimal design for automobile exhaust

gas waste heat recovery. A thermoelectric generator module with a 5-cm × 5-cm base area is

subject to exhaust gases at 500°C in fluid 1 and air at 25°C in fluid 2. We estimate an available

maximum convection conductance in fluid 1 (exhaust gas) with 8.01 , KmWh 2

1 60 , and 2

1 1000cmA and also an available maximum convection conductance in fluid 2 (air) with

8.02 , KmWh 2

2 60 , and 2

2 1000cmA , which gives KWAhAh 8.4222111 and

1hN . Note that 1 and 2 are typical fin efficiencies, and h1 and h2 are the reasonable

convection coefficients for exhaust gas heating and air cooling, of which the convection

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coefficients with exhaust gas or air typically have values ranging between 20 and 100 W/m2K

depending on the flow rate and the type of fin. The middle value of 60 W/m2K was used in the

present work. Each area of A1 and A2 is based on 20 fins (2 sides) with a fin height of 5 cm on a

5-cm × 5-cm base area of the module, which gives an total fin area (5cm × 5 cm × 2 sides × 20

fins =1000 cm2). The typical thermoelectric material properties are assumed to be p = −n =

220 V/K, p = n = 1.0 × 10-3

cm, and kp = kn = 1.4 × 10-2

W/cmK. The above data

approximately determines three dimensionless parameters as 1hN , 6.2

T and 0.12 ZT .

Figure 4. Optimal dimensionless power output

optW and efficiency opt versus dimensionless

convection hN . This plot was generated using 6.2

T and 0.12 ZT .

As mentioned before, the present dimensional analysis enables the three dimensionless

parameters ( 1hN , 6.2

T and 0.12 ZT ) to determine the rest two optimal parameters,

which are found to be 30.0kN and 7.1rR as shown before. This leads to a statement that, if

two individual fluid temperatures on heat sinks connected to a thermoelectric generator module

are given, an optimum design always exists with the feasible mechanical constraints that present

hN . This optimal design is indicated approximately at Point 1 in Figure 5. Note that there are

ways to improve the optimal power output, increasing either 222 Ah or hN or both, which

obviously depends on the feasible mechanical constraints, whichever is available.

The inputs and optimum results at Point 1 in Figure 5 are summarized in Table 1. The inputs

are the geometry of thermocouple, the material properties, two fluid temperatures, and the

available convection conductance in fluid 2. The dimensionless results are converted to the

actual quantities as shown. The maximum power output is found to be 65.0 W for the 5cm × 5cm

base area of the module. The power density is calculated to be 2.6 W/cm2, which appears

significantly high compared to an available power density of ~ 1 W/cm2 with the similar

operating conditions by NEDO program (Japan) [7].

0.1 1 100

0.02

0.04

0.06

0.08

0.1

0.08

0.09

0.1

0.11

0.12

0.13

optW*opt opt

W*opt

Nh

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Figure 5. Optimal power output optW versus convection conductance 222 Ah in fluid 2 as a

function of dimensionless convection hN . This plot was generated using T∞2 = 25°C, 6.2

T

and 0.12 ZT .

Table 1. Inputs and Results from the Dimensional Analysis for a TEG

Inputs Dimensionless

(

optnW , )

Actual

(Wn,opt)

T∞1 = 500°C, T∞2 = 25°C, T∞ = 475 °C Nk = 0.3 n = 254

A = 2 mm2, L = 1 mm Nh = 1 1 h1 A1 = 4.8 W/K

2 = 0.8, h2 = 60 W/m2K, A2 = 1000 cm

2 Rr = 1.7 RL = 1.7×n×R = 4.32

2 h2 A2= 4.8 W/K 6.2

T T∞1 = 500°C

Base area Ab of module = 5 cm × 5 cm 0.12 ZT 0.12 ZT

p = -n = 220 V/K 172.21 T T1 = 374°C

p = n = 1.0 × 10-3

cm 367.12 T T2 = 137°C

kp = kn =1.4 × 10-2

W/cmK 045.0

nW Wn = 65.0 W

(Z = 3.457 × 10-3

K-1

) 108.0th 108.0th

(R = 0.01 per thermocouple) NI = 0.306 I = 3.9 A

( 8.01 , KmWh 2

1 60 , 2

1 1000cmA ) NV = 0.50 V = 16.7 V

(Power Density Pd = Wn/Ab) - Pd = 2.6 W/cm2

Air was used in fluid 2 so far. However, we want to see the effect of 222 Ah or hN by

changing fluid 2 from air to liquid coolant. Otherwise the same conditions were applied to as the

previous example. We then estimate an available convection conductance in fluid 1 (exhaust gas)

with the same one of 8.01 , KmWh 2

1 60 , and 2

1 1000cmA , but an available convection

0.1 1 10 1000

20

40

60

80

100

120

140

160

180

200

Nh = 10

Nh = 1

Nh = 0.1

Wopt (W)

. . Point 1

Point 2

η2h2A2 (W/K)

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conductance in fluid 2 (liquid coolant) with 8.02 , KmWh 2

2 3000 , and 2

2 100cmA ,

which gives KWAh 8.4111 and KWAh 24222 , respectively, which yields 1.0hN . The

area of A1 is based on 20 fins (2 sides) with a fin height of 5 cm for the 5-cm × 5-cm base area of

the module and A2 is estimated to be one tenth of A1 (liquid coolant does not require a large heat

transfer area). These inputs and optimum results give all the five dimensionless parameters as

07.0kN , 1.0hN , 5.1rR , 6.2

T and 0.12 ZT , for which the optimum at

KWAh 24222 is indicated at Point 2 in Figure 5. The effect of hN on the high and cold

junction temperatures was also presented in Figure 6. It is interesting to see that, although a small

variation in the optimal power outputs between Point 1 ( 1hN ) and Point 2 ( 1.0hN ) appears

in Figure 5, a significant temperature variation between 1hN and 1.0hN appears in Figure 6.

This may be an important factor particularly when thermoelectric materials are considered in the

optimal design. The proximity of the power outputs between Points 1 and 2 is an example

showing the variety of the mechanical constraints ( 111 Ah and 222 Ah ) even with the same power

outputs . It is important to realize that, when 111 Ah is limited, simply increasing 222 Ah invokes

decreasing hN , which results in decreasing not only the high and cold junction temperatures but

also slightly the temperature difference as shown in Figure 6. The coexistence that the Seebeck

coefficient decreases with decreasing the temperature and reducing the temperature difference

diminishes the performance will cause the power output to decline. However, increasing 222 Ah

will directly increase the power output as mentioned earlier. The net power output of loss and

gain by increasing 222 Ah may be a role of system designer. Anyhow there will be a small

change in the efficiency.

Figure 6. Hot and cold junction temperatures and optimal efficiency versus dimensionless

convection. This plot was generated with T∞2 = 25°C, 6.2

T and 0.12 ZT .

0.1 1 100

100

200

300

400

500

0.08

0.09

0.1

0.11

0.12

0.13

T1

.

opt. optTopt (°C)

. T2

.

Nh

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3. Thermoelectric Cooler Let us consider a simplified steady-state heat transfer on a thermoelectric cooler module

(TEC) with two heat sinks as shown in Figure 7. Each heat sink faces a fluid flow at temperature

T∞. Subscript 1 and 2 denote the entities of fluid 1 and 2, respectively. Consider that an electric

current is directed in a way that the cooling power Q1 enters heat sink 1. We assume that the

electrical and thermal contact resistances in the TEC are negligible, the material properties are

independent of temperature, and also the TEC is perfectly insulated. The TEC has a number of

thermocouples, of which each thermocouple consists of p-type and n-type thermoelements with

the same dimensions.

Figure 7. Thermoelectric cooler module (TEC).

The basic equations for the TEC with two heat sinks are given by

111111 TTAhQ (25)

21

2

112

1TT

L

AkRIITnQ (26)

21

2

222

1TT

L

AkRIITnQ (27)

222222 TTAhQ (28)

where np , np kkk , and np . In order to study the optimization of the

TEC, several dimensionless parameters are introduced. The dimensionless thermal conductance,

which is the ratio of thermal conductance to the convection conductance in fluid 2, is

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222 Ah

LAknN k

(29)

The dimensionless convection, which is the ratio of convection conductance in fluid 1 to fluid 2,

is

222

111

Ah

AhNh

(30)

The dimensionless current is given by

LAk

IN I

(31)

The dimensionless temperatures are defined by

2

11

T

TT

(32)

2

22

T

TT

(33)

2

1

T

TT

(34)

The dimensionless cooling power, rate of heat liberated and electrical power input are defined by

2222

11

TAh

QQ

(35)

2222

22

TAh

QQ

(36)

2222

TAh

WW n

n

(37)

It is noted that the above dimensionless parameters are based on the convection conductance

in fluid 2, which means that 2222 TAh should be initially provided. Using the dimensionless

parameters defined in Equations (29) – (34), Equations (25) – (28) reduce to two formulas as:

21

2

2

11

2TT

ZT

NTN

N

TTN II

k

h

(38)

17

21

2

2

22

2

1TT

ZT

NTN

N

T II

k

(39)

Equations (38) and (39) can be solved for

1T and

2T . The dimensionless temperatures are then a

function of five independent dimensionless parameters as

21 ,,,,

ZTTNNNfT Ihk

(40)

22 ,,,,

ZTTNNNfT Ihk

(41)

T is the input and 2ZT is the material property with the input, and both are initially

provided. Therefore, the optimization can be performed only with the first three parameters (Nk,

Nh, and NI). Once the two dimensionless temperatures (

1T and

2T ) are solved for, the

dimensionless rates of heat transfer at both junctions of the TEC can be obtained as:

11 TTNQ h

(42)

122 TQ

(43)

1Q is called the dimensionless cooling power. Then, we have the dimensionless power input

as

12 QQWn

(44)

Accordingly, the coefficient of performance is obtained by

nW

QCOP 1

(45)

Defining 2 TnVNV , the dimensionless voltage is obtained by

kI

nV

NN

WN

(46)

With the inputs (

T and 2ZT ), we try to find the optimal combination for the dimensionless

parameters (Nk, Nh, and NI) iteratively until they converge. It is found that both kN and IN show

the optimal values for the dimensionless cooling power

1Q , while hN does not show the optimal

value showing that the dimensionless cooling power

1Q monotonically increases with increasing

hN . This implies that, if any hN is given, the optimal combination of kN and IN can be obtained.

18

However, the dimensionless convection hN actually presents the feasible mechanical constraints.

Thus, we proceed with a typical value of 1hN for illustration and later examine the variety of

hN with a practical design example.

Suppose that we have 967.0

T (two arbitrary fluid temperatures) and 0.12 ZT (materials) along with 1hN as inputs. Then, we can determine the optimal combination for IN

and kN , which may be obtained either graphically or using a computer program. We first use the

graphical method at this moment and later the program for multiple computations (a

Mathematical software Mathcad was used). The optimal combination of IN and kN for each

maximum cooling power are found to be IN = 0.5 and kN = 0.3, respectively, which are shown

in Figures 8 and 9. The maximum cooling power of 037.01 Q in both figures is actually the

optimal dimensionless cooling power

optQ ,1 . However, the COP also shows an optimal value at

074.0IN , which gives 006.01 Q . The optimal COP usually gives a very small cooling

power or sometimes even no exists, which seems impractical, albeit the high COP. Therefore, it

is needed to have a practical point for the optimal COP, which is determined in the present work

to be the midpoints of the optimum IN and kN . For example, the practical optimal COP in this

case occurs simultaneously at 25.0IN and 15.0kN , which leads to 019.01 Q that may be

seen after re-plotting with the two values.

Figure 8. Dimensionless cooling power

1Q , power input

nW and COP versus dimensionless

current IN . This plot was generated with 3.0kN , 1hN , 967.0

T and 0.12 ZT .

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0

0.5

1

1.5

2

2.5

Wn*

COP

Q1*

& Wn*

COP

Q1*

NI

19

Figure 9. Dimensionless cooling power

1Q and COP versus dimensionless thermal conductance

kN . This plot was generated with 1hN , 5.0IN , 967.0

T and 0.12 ZT .

The existence of an optimum cooling power as a function of current is a well known

characteristic of TECs. However, the existence of the optimum kN in TECs has not been found

in the literature to the author’s knowledge. With Equation (29) that is 222 AhLAknNk , the

optimum of Nk = 0.3 implies that the module thermal conductance LAkn is at optimum since

the 222 Ah is given, which leads to the optimum n (the number of thermocouples) if LAk is

given or vice versa. This is one of the most important optimum processes in design of a

thermoelectric cooler module. It is good to know in Figure 10 that the dimensionless temperature

1T becomes lowest at the optimal dimensionless cooling power

1Q , not at the optimum COP.

Figure 10. Dimensionless temperatures versus dimensionless thermal conductance. This plot was

generated with 1hN , 5.0IN , 967.0

T and 0.12 ZT .

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0

0.2

0.4

0.6

0.8

1

1.2

Q1*

Q1* COP

COP

Nk

0 0.2 0.4 0.6 0.8 10.9

1

1.1

1.2

1.3

1.4

0

0.01

0.02

0.03

0.04

0.05

Q1*

T2*

T1*

& T2*

Q1*

T1*

Nk

20

We also consider two heat sinks as a unit without a TEC to examine the limitation of use of

the TEC, which is shown in Figure 11. The geometry of the unit is the same as the one shown in

Figure 7 except that there is no TEC between the heat sinks. There should be a cooling rate with

given fluid temperatures, which is 0Q . We want to compare the cooling power 1Q with this

cooling rate 0Q to determine the limit of use of the TEC.

Figure 11. Heat sinks without a TEC.

The basic equations for the unit can be expressed as

011110 TTAhQ (47)

202220 TTAhQ (48)

The dimensionless groups for the unit are

2

00

T

TT (49)

2222

00

TAh

QQ

(50)

Using Equations (30), (34), (49) and (50), Equations (47) and (48) reduce to a formula as

1

10

h

h

N

TNT (51)

The dimensionless cooling rate without heat sinks can be obtained by

100 TQ (52)

21

Figure 12. Dimensionless cooling power, cooling rate of the unit, and COP versus dimensionless

fluid temperature. This plot was generated with 3.0kN , 1hN , 5.0IN and 0.12 ZT .

Figure 13. Optimal (cooling power optimized) dimensionless cooling power and COP versus

dimensionless convection. This plot was generated with 967.0

T and 0.12 ZT .

The dimensionless cooling power

1Q and cooling rate of the unit

0Q versus the

dimensionless fluid temperature

T along with the COP are presented in Figure 12. The cross

point in the figure is found to be

T = 1.2, which is a design point as the limit of use of the TEC.

If the dimensionless fluid temperature

T is higher than the cross point, there is no justification

for use of the TEC although the TEC still functions. This cross point is defined as the maximum

0.8 0.9 1 1.1 1.2 1.3 1.40

0.05

0.1

0.15

0.2

0.25

0.3

0

0.25

0.5

0.75

1

1.25

1.5

Q1*

&Q0*

COP

COP

Q1*

Q0*

T

0.1 1 100

0.01

0.02

0.03

0.04

0.05

0

0.2

0.4

0.6

0.8

1

Q*1,opt

Q*1 COP

COPopt

Nh

22

dimensionless temperature

max,T . There is also a minimum point at 83.0

T , where 01 Q ,

which is called the minimum dimensionless temperature

min,T . Note that the TEC can perform

effective cooling within a range from 83.0min,

T to

max,T = 1.2.

Since there is an optimal combination of NI and Nk for a given hN , we can plot the optimal

dimensionless cooling power

optQ ,1 and COPopt as a function of dimensionless convection hN ,

which is shown in Figure 13. It is seen that both

optQ ,1 and COPopt increase monotonically with

increasing hN . According to Equation (35), the actual optimal cooling power optQ ,1 is the product

of

optQ ,1 and 222 Ah , seemingly increasing linearly with 222 Ah . In practice, there is a

controversial tendency that hN may decrease systematically with increasing 222 Ah . As a result

of this, it is needed to examine the variety of the 222 Ah as a function of hN for the optimal

cooling power. Figure 14 reveals the intricate relationship between 222 Ah and hN (or 111 Ah )

along with the optimum actual cooling power (not dimensionless), which would lead system

designers to a variety of possible allocations ( 222 Ah and hN ) for their optimal design. Note that

the analysis so far is entirely based on the dimensionless parameters. Now we look into the

actual optimal design with the actual values.

Figure 14. Optimal cooling power versus convection conductance in fluid 2 as a function of

dimensionless convection. This plot was generated with T∞2 = 30°C,

967.0

T and 0.12 ZT .

For example, using Figures 13 and 14, we develop an optimal design for an automobile air

conditioner. A thermoelectric cooler module with a 5-cm × 5-cm base area is subject to cabin air

at 20°C in fluid 1 and ambient air at 30°C in fluid 2. We estimate an available maximum

convection conductance in fluid 1 (cabin air) with 8.01 , KmWh 2

1 60 , and 2

1 1000cmA

and also an available maximum convection conductance in fluid 2 (ambient air) with 8.02 ,

0.1 1 10 1000

10

20

30

40

50

60

70

80

90

100Nh = 10

Nh = 1

Nh = 0.1

. Q1,opt (W) Point 1

η2h2A2 (W/K)

23

KmWh 2

2 60 , and 2

2 1000cmA , which gives KWAhAh 8.4222111 and 1hN . Note

that 1 and 2 are the fin efficiencies, and h1 and h2 are the reasonable convection coefficients

for the cabin air cooling and the ambient air cooling, respectively. Each area of A1 and A2 is

based on 20 fins (2 sides) with a fin height of 5 cm on a 5-cm × 5-cm base area of the module.

The typical thermoelectric material properties are assumed to be p = −n = 220 V/K, p = n =

1.0 × 10-3

cm, and kp = kn = 1.4 × 10-2

W/cmK. The above data approximately determines three

dimensionless parameters as 1hN , 967.0

T and 0.12 ZT .

As mentioned before, the present dimensional analysis enables the three dimensionless

parameters ( 1hN , 967.0

T and 0.12 ZT ) to determine the rest optimal parameters, which

are found to be 3.0kN and 5.0IN as shown before. This leads to a statement that, if two

individual fluid temperatures on heat sinks connected to a thermoelectric cooler module are

given, an optimum design always exists with the feasible mechanical constraints that present hN .

This optimal design is indicated approximately at Point 1 in Figure 14. Note that there are

several ways to improve the optimal power output by increasing either 222 Ah or hN or both,

which apparently depends on the feasible mechanical constraints, whichever is available.

Figure 15. Two junction temperatures and cooling power versus convection conductance in fluid

2. This plot was generated with, KWAh 8.4222 T∞2 = 30°C, 967.0

T and 0.12 ZT .

The inputs and optimum results at Point 1 in Figure 14 are summarized in the first two

columns of Table 2. The inputs are the geometry of thermocouple, the material properties, two

fluid temperatures, and the available convection conductance in fluid 2. The dimensionless

results are converted to the actual quantities. The optimal cooling power is found to be 54.4 W

for the 5cm × 5cm base area of the module. The cooling power density is calculated to be 2.18

W/cm2. When 111 Ah is limited, simply decreasing 222 Ah invokes decreasing hN , which is

shown in Figure 15. This attributes to the heat balance that the hot and cold junction

temperatures must decrease when more heat is extracted from the limited input, which is a

0.1 1 1020

0

20

40

60

80

0

20

40

60

80

100

T2,opt

Q1,opt

T (°C) Q1(W)

T1,opt

Nh

24

characteristic of the thermoelectric cooler with heat sinks. The hot and cold junction

temperatures are sometimes a design factor, noting that the optimal cold junction temperature

optT ,1 reaches zero Celsius at 4.0hN , which may cause icing and reducing the heat transfer.

Table 2. Inputs and Results from the Dimensional Analysis for a Thermoelectric Cooler Module

Input

optQ ,1 (dimensionless)

optQ ,1

(Actual)

optCOP 2/1

(dimensionless)

optCOP 2/1

(Actual)

T∞1 = 20°C, T∞2 = 30°C Nk = 0.3 n = 257 Nk = 0.15 n = 128

A = 2 mm2, L = 1 mm Nh = 1 1 h1 A1 =

4.8 W/K

Nh = 1 1 h1 A1 = 4.8

W/K

2 = 0.8, h2 =60 W/m2K, NI = 0.5 I = 6.36 A NI = 0.25 I = 3.18 A

A2 =1000 cm2 967.0

T T∞1 = 20°C 967.0

T T∞1 = 20°C

Base area Ab = 5 cm × 5

cm 0.12 ZT 0.12 ZT 0.12 ZT 0.12 ZT

p = -n = 220 V/K 930.01 T T1 = 8.7°C 949.01 T T1 = 14.4°C

p = n = 1.0 × 10-3

cm 145.12 T T2 = 73.9°C 031.12 T T2 = 39.4°C

kp = kn =1.4 × 10-2

W/cmK 037.01 Q Q1 = 54.4 W 019.01

Q Q1 = 26.9 W

(Z = 3.457 × 10-3

K-1

) COP = 0.35

COP = 1.49

(R =0.01per

thermocouple)

NV = 0.715 Vn = 24.5 V NV = 0.332 Vn = 5.7 V

2.1max,

T T∞1,max =

91.3°C 06.1max,

T T∞1,max =

49.7°C

83.0min,

T T∞1,min =

−21.8°C 84.0min,

T T∞1,min =

−19.2°C

(Cooling Power Density

Pd = Q1/Ab)

- Pd = 2.18

W/cm2

- Pd = 1.08

W/cm2

As mentioned before, the optimal COP is sometimes in demand in addition to the optimal

cooling power. However, the real optimal COP usually gives a very small value of the cooling

power, albeit the high COP. Therefore, in the present work, the midpoints of the optimal IN and

kN are used to provide approximately a half of the optimal cooling power and at least four folds

of the cooling-power-optimized COP. This modified optimal COP is called a half optimal

coefficient of performance COP1/2opt. These results are also tabulated in the last two columns of

Table 2, so that designers could determine which optimum is better depending on the application.

The optimum cooling power is usually selected when the resources (electrical power or capacity

of coolant) are abundant or inexpensive or the efficacy is not important as in microprocessor

cooling, while the half optimum COP is selected when the resources are limited or expensive or

the efficacy is important as in automotive air conditioners.

4. Conclusions The present paper presents the optimal design of thermoelectric devices in conjunction with

heat sinks introducing new dimensionless parameters. The present optimum design includes the

25

power output (or cooling power) and the efficiency (or COP) simultaneously with respect to the

external load resistance (or electrical current) and the geometry of thermoelements. The optimal

design provides optimal dimensionless parameters such as the thermal conduction ratio, the

convection conduction ratio, and the load resistance ratio as well as the cooling power, efficiency

and high and low junction temperatures. The load resistance ratio (or the electrical current) is a

well known characteristic of optimum design. However, it is found that the load resistance ratio

would be greater than unity (1.7 in the present case). This is a confusing factor in optimum

design with a TEG. One should not assume that the load resistance ratio RL/R is equal to unity

for a TEG with heat sink(s). The optimal thermal conductance [(n)(A/L)(k)] consists of the

number of thermocouples, the geometric ratio, and the thermal conductivity. It is important that

there is an optimum number of thermocouples n for a given the convection conductance 222 Ah

if the optimal thermal conductance A/L is constant or vice versa. These are the optimum

geometry of thermoelectric devices. The information of the optimal thermal conductance is

particularly important in design of microstructured or thin-film thermoelectric devices.

Furthermore, there is a potential to improve the performance or to provide the variety of the

geometry by reducing the thermal conductivity.

Finally, it is stated from the present dimensional analysis that, if two individual fluid

temperatures on heat sinks connected to a thermoelectric generator or cooler are given, an

optimum design always exists and can be found with the feasible mechanical constraints.

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27

Example E-3

A thermoelectric generator (TEG) module is designed using a newly developed material which is

so called TAGS-75 (AgSbTe/GeTe 75%) for an exhaust waste heat recovery in a car. The whole

TEG device is shown in Figure E-3a. The device has N number of modules, each of which

consists of n number of p- and n-type thermocouples. The material properties for thermoelements

are assumed to be similar as p = −n = 218 V/K, p = n = 1.1 × 10-3

cm, and kp = kn = 1.3 ×

10-2

W/cmK. The cross-sectional area and pellet length of the thermoelement are An = Ap = 2.4

mm2 and Ln = Lp = 1 mm, respectively. The exhaust gases at 750 K enter heat sink 1 while

coolant at 312 K flows in heat sink 2. The convection conductances for the flows, with

considering the effectiveness of the heat sinks, are estimated on the 5-cm × 5-cm base areas to be

8.01 , KmWh 2

1 60 , and 2

1 1000cmA for heat sink 1 (exhaust gases) and 9.02 ,

KmWh 2

2 1000 , and 2

2 50cmA for heat sink 2 (coolant).

(a) For the optimal design of one TEG module, determine the number of thermocouples n,

power output, conversion efficiency, hot and cold junction temperatures, current and

voltage.

(b) If the total power output of 1200 W for the whole TEG device is required, determine the

number of modules N (assuming that the module represents the mean value of the

modules in the device).

(a) (b)

Figure E-3 (a) the whole TEG device, and (b) the TEG module.

Solution: Material properties of TAGS-75: =p − n = 436 × 10-6

V/K, = p + n = 2.2 ×

10-5

m, and k = kp + kn = 2.6 W/mK.

The convection conductances are given as

1 = 0.8, h1 = 60 W/m2K, A1 = 1000 cm

2 and 2 = 0.9, h1 = 1000 W/m

2K, A1 = 50 cm

2.

KWmKmWAh 8.4101000608.0 242

111

28

KWmKmWAh 5.4105010009.0 242

222

The cross-sectional area and pellet length of the thermoelement are

A = 2.4 × 10-6

m2 and L = 1 × 10

-3m

The figure of merit is

13

5

262

10323.36.2102.2

10436

K

mKWm

KV

kZ

and

037.131210323.3 13

2

KKZT

The dimensionless fluid temperature is

404.2312

750

2

1

K

K

T

TT

The dimensionless convection conductance is

067.15.4

8.4

222

111 KW

KW

Ah

AhNh

From Table 1 for 12 ZT , we can approximately obtain the optimal parameters using both

40.2

T and 0.1hN as

Rr = 1.638, Nk = 0.305, th = 0.096, 035.0

nW , 032.21 T , 332.12 T , NI = 0.265, and

NV = 0.435.

(a) Using Equation (6), the number of thermocouples is

95.219

104.26.2

1015.4305.0

26

3

222

mmKW

mKW

kA

LAhNn k

Using Equation (14), the power output is

WKKWTAhWW nn 14.493125.4035.02222

The conversion efficiency is

th = 0.096

The hot and cold junction temperatures are

KKTTT 98.633312032.2211

29

KKTTT 58.415312332.1222

Using the dimensionless current in Equation (23), the current is

AmKV

mmKW

L

kANI I 79.3

10110436

104.26.2265.0

36

26

Using the definition of dimensionless voltage in Equation (24), the voltage is

VKKVTnNV Vn 0.133121043695.219435.0 6

2

(b) Determine the number of modules N if the total power output of 1200W is required.

42.2414.49

12001200

W

W

W

WN

n

Comments: We want to compare the above optimal power output of 49.14 W with the maximum

power output of about 54 W plotted in a conventional way (assuming the constant hot and cold

junction temperatures) using the following equations as

32635 1017.9104.2101102.2 mmmALR

21

2

112

1TT

L

kARIITnQ , RITTInWn

2

21 , and 1Q

Wnth

The maximum power output of about 54 W at I = 5 A from the figure appears higher than the

optimal power output of 49.14 W obtained with I = 3.79 A. However, it is noted that the 54 W is

attained under the assumption of the constant hot and cold junction temperatures which are no

longer true. This indicates that the predictions with the ideal equations without heat sinks (under

assumption of the constant hot and cold junction temperatures) may lead to the appreciable errors.

0 2 4 6 8 10 120

10

20

30

40

50

60

0

0.02

0.04

0.06

0.08

0.1

0.12

Wn

th

Wn(W) th

Current (A)

30

Problem P-3

As a potential alternative for clean energy generation, a thermoelectric generator (TEG) is

designed to recover exhaust waste energy from a car. An array of N = 36 modules (which has the

base area of 5-cm × 5-cm) is installed on the exhaust of the car. One of the recently developed

material to meet the high temperature is nanostructured lead telluride (PbTe) having the

properties asp = −n = 218 V/K, p = n = 1.2 × 10-3

cm, and kp = kn = 1.18 × 10-2

W/cmK.

The cross-sectional area and pellet length of the thermoelement are An = Ap = 2.6 mm2 and Ln =

Lp = 1.2 mm, respectively. The exhaust gases at 720 K enter heat sink 1 while coolant at 300 K

enters heat sink 2. The convection conductances for the flows, with considering the effectiveness

of the heat sinks, are estimated on the 5-cm × 5-cm base areas to be 8.01 , KmWh 2

1 45 ,

and 2

1 1000cmA for heat sink 1 (exhaust gases) and 8.02 , KmWh 2

2 900 , and 2

2 500cmA for heat sink 2 (coolant). We want to know the optimal number of the

thermocouples per module for the given information.

(a) For the optimal design of one TEG module, determine the number of thermocouples n,

power output, conversion efficiency, hot and cold junction temperatures, current and

voltage.

(b) Determine the total power output (assuming that the module obtained represents the

mean value of the modules in the device).

(a) (b)

Figure P-3 (a) the arrangement of the whole TEG device, and (b) the TEG module.

31

Example E-4

Current automotive air conditioners have used R-134a for about two decades as a working fluid,

which does not have the ozone-depleting properties of Freon, but it is nevertheless a terrible

greenhouse gas and will be banned in the near future. A thermoelectric air conditioner is

designed for the replacement as a green energy application, as shown in Figure E-4a. The

analysis indicates that the cooling load of 600 W per occupant is required. We consider a unit

module of 7-cm × 7-cm base area, which represents a mean value of the modules’ performance

of the air conditioner. Cabin air at 21 °C enters the upper and lower heat sinks while coolant at

34 °C enters the central flat heat exchanger, so that the TEC modules are installed between each

heat sink and the central exchanger, which is schematically shown in Figure E-4b. A newly

developed nanostructured material of Bi2Te3 has the properties asp = −n = 208 V/K, p = n

= 1.1 × 10-3

cm, and kp = kn = 1.2 × 10-2

W/cmK. The cross-sectional area and pellet length of

the thermoelement are An = Ap = 2.5 mm2 and Ln = Lp = 1.25 mm, respectively. The convection

conductances on the 7-cm × 7-cm base areas are estimated to be 8.01 , KmWh 2

1 40 , and 2

1 1200cmA for heat sink 1 (cabin air) and 8.02 , KmWh 2

2 800 , and 2

2 60cmA for

heat sink 2 (coolant).

(a) For the optimal cooling power per one TEC module, determine the number of

thermocouples n, cooling power, COP, power input, cold and hot junction temperatures,

current and voltage.

(b) For the ½ optimal COP per TEC module, determine the number of thermocouples n,

cooling power, COP, power input, cold and hot junction temperatures, current and

voltage.

(c) Estimate the number of TEC modules to meet the cooling load of 600 W per occupant

with the ½ optimal COP (assuming that the module obtained represents the mean value

of the modules in the device).

(a) (b)

Figure E-4. (a) Thermoelectric air conditioner, and (b) TEC module.

Solution: Material properties of Bi2Te3: =p − n = 416 × 10-6

V/K, = p + n = 2.2 × 10-5

m, and k = kp + kn = 2.4 W/mK.

The convection conductances are given as

32

1 = 0.8, h1 = 40 W/m2K, A1 = 1200 cm

2 and 2 = 0.8, h1 = 800 W/m

2K, A1 = 60 cm

2.

KWmKmWAh 84.3101200408.0 242

111

KWmKmWAh 84.310608008.0 242

222

The cross-sectional area and pellet length of the thermoelement are

A = 2.5 × 10-6

m2 and L = 1.25 × 10

-3m

The figure of merit is

13

5

262

10278.34.2102.2

10416

K

mKWm

KV

kZ

and

007.130710278.3 13

2

KKZT

The dimensionless fluid temperature from Equation (34) is

961.0307

295

2

1

K

K

T

TT

The dimensionless convection conductance from Equation (30) is

0.184.3

84.3

222

111 KW

KW

Ah

AhNh

(a) For the optimum cooling power:

Using Table 2 for 12 ZT , we can approximately obtain the optimal parameters with both

96.0

T and 0.1hN as

NI = 0.510, Nk = 0.286, COP = 0.335, 036.01 Q , 924.01 T , 141.12 T , and NV = 0.727.

Using Equation (29), the number of thermocouples is

8.228105.24.2

1025.184.3286.0

26

3

222

mmKW

mKW

kA

LAhNn k

Using Equation (35), the cooling power is

WKKWTAhQQ 46.4230784.3036.0222211

The COP is already obtained from Table 2 as

COP = 0.335

33

The power input from Equations (35), (36), and (45) is

WW

COP

QWn 748.126

335.0

46.421

The cold and hot junction temperatures from Equations (32) and (33) are

KKTTT 80.283307924.0211

KKTTT 76.350307142.1222

Using Equation (31), the current is

A

mKV

mmKW

L

kANI I 88.5

1025.110416

105.24.2510.0

36

26

Using the definition of dimensionless voltage above Equation (46), the voltage is

VKKVTnNV Vn 25.21307104168.228727.0 6

2

The number of modules for the cooling load of 600 W per occupant is

13.1446.42

600600

1

W

W

Q

WN

(b) For the ½ optimal COP per one TEC module:

Using Table 3 for 12 ZT , we can approximately obtain the optimal parameters with both

96.0

T and 0.1hN as

NI = 0.255, Nk = 0.143, COP = 1.385, 017.01 Q , 943.01 T , 030.12 T , and NV = 0.342.

Using Equation (29), the number of thermocouples is

4.114105.24.2

1025.184.3143.0

26

3

222

mmKW

mKW

kA

LAhNn k

Using Equation (35), the cooling power is

WKKWTAhQQ 05.2030784.3017.0222211

The COP is already obtained from Table 2 as

COP = 1.385

The power input from Equations (35), (36), and (45) is

34

WW

COP

QWn 48.14

385.1

05.201

The cold and hot junction temperatures from Equations (32) and (33) are

KKTTT 64.289307943.0211

KKTTT 364.316307030.1222

Using Equation (31), the current is

A

mKV

mmKW

L

kANI I 94.2

1025.110416

105.24.2255.0

36

26

Using the definition of dimensionless voltage above Equation (46), the voltage is

VKKVTnNV Vn 0.5307104164.114342.0 6

2

The number of modules for the cooling load of 600 W per occupant is

92.2905.20

600600

1

W

W

Q

WN

Table E-4. Summary of the Optimal cooling power and ½ optimal COP.

Per TEC Module Optimal Cool. Power COP ½ opt

n number of thermocouples n = 228.8 n = 114.4

Current (A) I = 5.88 A I = 2.94 A

Voltage (V) V n= 21.25 V V n= 5.0 V

COP COP = 0.335 COP = 1.385

Cooling power (W) Q1 = 42.46 W Q1 = 20.05 W

Power input (W) Wn= 126.75 W Wn = 14.48 W

Cold junction temperatureT1 T1 = 10.8°C T1 = 16.6°C

Hot junction temperature T2 T2 = 77.7°C T2 = 43.4°C

N number of modules for 600 W

(Total number of thermocouples)

(Total power consumption)

(Design comments)

N = 14.13

(N × n = 3233)

(126.75W × 14.13 = 1,791 W)

(Too high power consumption

for automobiles)

N = 29.92

(N × n = 3422)

(14.48W × 29.02 = 433 W)

(Acceptable design)

Comments: The optimal design of thermoelectric coolers with heat sinks is challenging

because the distinct optimal COP does not exist although the optimal cooling power clearly

exists. Therefore, a half optimal COP is introduced as a reference for the optimal COP. However,

there is no a strict rule for the optimal COP with heat sinks. If there is no heat sink, it becomes

35

an ideal device with the constant junction temperatures. The ideal maximum COP and the

current are usually available in the literature, which are

11

1

2

1

1

22

1

12

1max

TZ

T

TTZ

TT

TCOP and 11

)( 12

TZR

TTICOP

Using the cold and hot junction temperatures (16.6 °C and 43.4 °C) obtained for the half

optimal COP information, we calculate the ideal optimal COP and current as

011.0105.2

1025.1102.226

35

m

mmR

993.0

2

36.31664.28910278.3 13

KK

KTZ

436.1

1993.01

64.289

36.316993.01

64.28936.316

64.289

2

1

2

1

max

K

K

KK

KCOP

ACCKV

ICOP 45.21993.01011.0

6.164.43104165.0

6

The ideal optimal COP and current are 1.436 and 2.54 A, which appear fortuitously close to

the half optimal COP and current (1.385 and 2.94 A). However, if we calculate the optimal

cooling power in the same way, we find that there are appreciable discrepancies between the

optimal values with heat sinks and the ideal values without heat sinks. In real, the thermoelectric

device should work with one or two heat sinks and the ideal maximum COP and cooling power

with constant junction temperatures would cause significant errors.

36

Problem P-4

An automotive thermoelectric air conditioner is designed, which consists of two air heat sinks

and a coolant heat exchanger as shown in Figure P-4a. We design a unit module of 4-cm × 4-cm

base area, which represents a mean value of the whole system as a simplified concept. Cabin air

at 296.2 K enters the upper and lower heat sinks while coolant at 311.8 K enters the central flat

heat exchanger, so that the TEC modules are installed between each heat sink and the central

exchanger, which is schematically shown in Figure P-4b. A widely used material of Bi2Te3 has

the properties asp = −n = 210 V/K, p = n = 1.1 × 10-3

cm, and kp = kn = 1.23 × 10-2

W/cmK. The cross-sectional area and pellet length of the thermoelement are An = Ap = 1.8 mm2

and Ln = Lp = 0.9 mm, respectively. The convection conductances on the 4-cm × 4-cm base areas

are estimated to be 8.01 , KmWh 2

1 40 , and 2

1 600cmA for heat sink 1 (cabin air) and

8.02 , KmWh 2

2 960 , and 2

2 25cmA for heat sink 2 (coolant).

(a) For the optimal cooling power per one TEC module, determine the number of

thermocouples n, cooling power, COP, power input, cold and hot junction temperatures,

current and voltage.

(b) For the ½ optimal COP per TEC module, determine the number of thermocouples n,

cooling power, COP, power input, cold and hot junction temperatures, current and

voltage.

(c) Estimate the number of TEC modules to meet the cooling load of 600 W per occupant

with the ½ optimal COP (assuming that the module obtained represents the mean value

of the modules in the device).

(a) (b)

Figure P-4. (a) Thermoelectric air conditioner, and (b) TEC module.

37

Table 1 for Optimal Dimensionless Parameters for TEGs with ZT∞2=1

Figure 1. Thermoelectric generator module (TEG) with two heat sinks.

Dimensionless thermal conductance:

222 Ah

LAknN k

Dimensionless convection: 222

111

Ah

AhNh

Dimensionless electrical resistance: R

RR L

r

Dimensionless temperatures:2

11

T

TT

, 2

22

T

TT , and

2

1

T

TT

Dimensionless rates of heat transfer: 2222

11

TAh

QQ

Dimensionless power output: 2222

TAh

WW n

n

Thermal efficiency:

1Q

Wnth

Dimensionless current: AkILN I

Dimensionless voltage: 2 TnVNV

38

Table 1 Optimal Power Output for ZT∞2=1

T∞* Nh Rr Nk ηth Wn* T1* T2* NI NV

1.0 0.1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.0 1.0 0.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000

1.0 10.0 0.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000

1.005 0.1 1.564 0.063 4.36E-04 9.72E-08 1.003 1 9.93E-04 1.55E-03

1.005 1 1.564 0.063 7.27E-04 2.71E-07 1.005 1 1.66E-03 2.60E-03

1.005 10 1.564 0.063 7.80E-04 3.12E-07 1.005 1 1.78E-03 2.78E-03

1.01 0.1 1.564 0.063 8.70E-04 3.88E-07 1.006 1 1.99E-03 3.11E-03

1.01 1 1.564 0.063 1.45E-03 1.08E-06 1.009 1.001 3.32E-03 5.19E-03

1.01 10 1.42 0.65 8.51E-04 3.89E-06 1.01 1.005 2.05E-03 2.92E-03

1.015 0.1 1.564 0.063 1.30E-03 8.73E-07 1.008 1.001 2.98E-03 4.66E-03

1.015 1 1.416 0.356 1.28E-03 4.82E-06 1.011 1.004 3.09E-03 4.38E-03

1.015 10 1.421 0.649 1.28E-03 8.75E-06 1.014 1.007 3.08E-03 4.38E-03

1.02 0.1 1.564 0.063 1.74E-03 1.55E-06 1.011 1.001 3.97E-03 6.21E-03

1.02 1 1.417 0.356 1.70E-03 8.55E-06 1.015 1.005 4.12E-03 5.84E-03

1.02 10 1.422 0.648 1.70E-03 1.55E-05 1.019 1.009 4.11E-03 5.84E-03

1.025 0.1 1.564 0.063 2.17E-03 2.42E-06 1.014 1.001 4.96E-03 7.76E-03

1.025 1 1.418 0.356 2.13E-03 1.34E-05 1.019 1.006 5.15E-03 7.30E-03

1.025 10 1.423 0.648 2.12E-03 2.43E-05 1.024 1.011 5.13E-03 7.30E-03

1.03 0.1 1.432 0.066 2.52E-03 3.50E-06 1.016 1.001 6.06E-03 8.68E-03

1.03 1 1.419 0.356 2.55E-03 1.92E-05 1.022 1.008 6.17E-03 8.76E-03

1.03 10 1.422 0.638 2.56E-03 3.49E-05 1.029 1.014 6.20E-03 8.82E-03

1.035 0.1 1.433 0.066 2.94E-03 4.76E-06 1.019 1.002 7.07E-03 1.00E-02

1.035 1 1.42 0.352 2.98E-03 2.61E-05 1.026 1.009 7.23E-03 0.01

1.035 10 1.423 0.638 2.98E-03 4.75E-05 1.033 1.016 7.24E-03 0.01

1.04 0.1 1.433 0.066 3.36E-03 6.21E-06 1.021 1.002 8.08E-03 0.012

1.04 1 1.421 0.352 3.41E-03 3.41E-05 1.03 1.01 8.26E-03 0.012

1.04 10 1.424 0.637 3.40E-03 6.20E-05 1.038 1.018 8.27E-03 0.012

1.045 0.1 1.434 0.066 3.77E-03 7.85E-06 1.024 1.002 9.08E-03 0.013

1.045 1 1.422 0.352 3.83E-03 4.32E-05 1.034 1.011 9.29E-03 0.013

1.045 10 1.425 0.636 3.83E-03 7.84E-05 1.043 1.02 9.30E-03 0.013

1.05 0.1 1.416 0.064 4.25E-03 9.69E-06 1.027 1.002 0.01 0.015

1.05 1 1.423 0.351 4.25E-03 5.32E-05 1.037 1.012 0.01 0.015

1.05 10 1.427 0.635 4.25E-03 9.67E-05 1.048 1.023 0.01 0.015

1.055 0.1 1.42 0.064 4.67E-03 1.17E-05 1.03 1.002 0.011 0.016

1.055 1 1.424 0.351 4.67E-03 6.44E-05 1.041 1.014 0.011 0.016

1.055 10 1.428 0.635 4.67E-03 1.17E-04 1.052 1.025 0.011 0.016

1.06 0.1 1.42 0.064 5.09E-03 1.39E-05 1.033 1.003 0.012 0.018

1.06 1 1.425 0.351 5.09E-03 7.65E-05 1.045 1.015 0.012 0.018

1.06 10 1.429 0.634 5.09E-03 1.39E-04 1.057 1.027 0.012 0.018

39


1.065 0.1 1.421 0.064 5.51E-03 1.64E-05 1.035 1.003 0.013 0.019

1.065 1 1.426 0.351 5.51E-03 8.98E-05 1.049 1.016 0.013 0.019

1.065 10 1.43 0.633 5.50E-03 1.63E-04 1.062 1.029 0.013 0.019

1.07 0.1 1.421 0.064 5.93E-03 1.90E-05 1.038 1.003 0.014 0.02

1.07 1 1.427 0.351 5.93E-03 1.04E-04 1.052 1.017 0.014 0.021

1.07 10 1.431 0.632 5.92E-03 1.89E-04 1.067 1.032 0.014 0.021

1.075 0.1 1.422 0.064 6.35E-03 2.17E-05 1.041 1.003 0.015 0.022

1.075 1 1.427 0.35 6.34E-03 1.19E-04 1.056 1.019 0.015 0.022

1.075 10 1.433 0.632 6.34E-03 2.17E-04 1.072 1.034 0.015 0.022

1.08 0.1 1.423 0.064 6.77E-03 2.47E-05 1.043 1.004 0.016 0.023

1.08 1 1.428 0.35 6.76E-03 1.36E-04 1.06 1.02 0.016 0.024

1.08 10 1.434 0.631 6.75E-03 2.46E-04 1.076 1.036 0.016 0.024

1.085 0.1 1.423 0.064 7.18E-03 2.79E-05 1.046 1.004 0.017 0.025

1.085 1 1.429 0.35 7.18E-03 1.53E-04 1.064 1.021 0.017 0.025

1.085 10 1.435 0.63 7.17E-03 2.78E-04 1.081 1.038 0.018 0.025

1.09 0.1 1.424 0.064 7.60E-03 3.12E-05 1.049 1.004 0.018 0.026

1.09 1 1.43 0.35 7.59E-03 1.72E-04 1.067 1.022 0.019 0.026

1.09 10 1.436 0.63 7.58E-03 3.11E-04 1.086 1.041 0.019 0.027

1.095 0.1 1.424 0.064 8.02E-03 3.48E-05 1.052 1.004 0.02 0.028

1.095 1 1.431 0.349 8.01E-03 1.91E-04 1.071 1.024 0.02 0.028

1.095 10 1.438 0.629 8.00E-03 3.46E-04 1.091 1.043 0.02 0.028

1.1 0.1 1.425 0.064 8.43E-03 3.85E-05 1.054 1.005 0.021 0.029

1.1 1 1.432 0.349 8.42E-03 2.11E-04 1.075 1.025 0.021 0.029

1.1 10 1.439 0.628 8.41E-03 3.83E-04 1.095 1.045 0.021 0.03

40

Table 1 Optimal Power Output for ZT∞2=1


1.00

0.1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.0 0.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000

10.0 0.000 0.000 0.000 0.000 1.000 1.000 0.000 0.000

1.20

0.1 1.435 0.064 0.017 0.000 1.108 1.009 0.041 0.058

1.0 1.449 0.345 0.017 8.34E-04 1.150 1.050 0.041 0.059

10.0 1.463 0.614 0.016 1.51E-03 1.191 1.090 0.041 0.060

1.40

0.1 1.455 0.064 0.032 5.95E-04 1.214 1.018 0.080 0.116

1.0 1.483 0.337 0.032 3.25E-03 1.298 1.098 0.081 0.119

10.0 1.510 0.589 0.032 5.85E-03 1.382 1.178 0.081 0.122

1.60

0.1 1.474 0.064 0.047 1.31E-03 1.319 1.027 0.118 0.174

1.0 1.515 0.330 0.046 7.11E-03 1.145 1.146 0.119 0.181

10.0 1.555 0.567 0.046 0.013 1.572 1.265 0.120 0.187

1.80

0.1 1.493 0.064 0.060 2.28E-03 1.422 1.036 0.155 0.231

1.0 1.547 0.323 0.060 0.012 1.594 1.194 0.157 0.243

10.0 1.599 0.546 0.059 0.022 1.763 1.350 0.159 0.254

2.00

0.1 1.511 0.063 0.073 3.49E-03 1.523 1.044 0.191 0.288

1.0 1.578 0.317 0.072 0.019 1.741 1.241 0.194 0.306

10.0 1.641 0.528 0.072 0.034 1.953 1.434 0.197 0.323

2.20

0.1 1.529 0.063 0.085 4.93E-03 1.623 1.053 0.226 0.345

1.0 1.608 0.311 0.084 0.026 1.887 1.287 0.230 0.370

10.0 1.682 0.511 0.084 0.047 2.144 1.516 0.234 0.394

2.40

0.1 1.546 0.063 0.097 6.57E-03 1.722 1.061 0.260 0.401

1.0 1.638 0.305 0.096 0.035 2.032 1.332 0.265 0.435

10.0 1.722 0.496 0.095 0.062 2.334 1.599 0.270 0.466

2.60

0.1 1.564 0.063 0.108 8.43E-03 1.820 1.070 0.293 0.457

1.0 1.667 0.300 0.107 0.045 2.177 1.377 0.300 0.500

10.0 1.761 0.482 0.105 0.080 2.524 1.678 0.306 0.540

2.80

0.1 1.580 0.063 0.118 0.010 1.916 1.078 0.325 0.513

1.0 1.695 0.295 0.117 0.056 2.322 1.422 0.334 0.566

10.0 1.799 0.469 0.115 0.099 2.714 1.758 0.342 0.615

3.00

0.1 1.596 0.063 0.128 0.013 2.011 1.086 0.356 0.569

1.0 1.722 0.290 0.126 0.067 2.466 1.466 0.367 0.633

10.0 1.836 0.457 0.125 0.119 2.904 1.836 0.377 0.691

3.20

0.1 1.612 0.063 0.138 0.015 2.105 1.094 0.387 0.624

1.0 1.749 0.286 0.136 0.080 2.610 1.510 0.400 0.700

10.0 1.872 0.446 0.134 0.141 3.094 1.914 0.411 0.769

3.40

0.1 1.628 0.062 0.147 0.018 2.199 1.102 0.417 0.679

1.0 1.776 0.282 0.144 0.093 2.745 1.553 0.432 0.768

10.0 1.907 0.435 0.142 0.164 3.284 1.991 0.445 0.848

41


3.60

0.1 1.643 0.062 0.156 0.020 2.291 1.111 0.447 0.734

1.0 1.802 0.278 0.153 0.108 2.896 1.596 0.464 0.836

10.0 1.942 0.425 0.150 0.189 3.474 2.067 0.478 0.929

3.80

0.1 1.658 0.062 0.164 0.023 2.382 1.119 0.475 0.788

1.0 1.827 0.274 0.161 0.123 3.039 1.639 0.495 0.905

10.0 1.976 0.416 0.158 0.215 3.664 2.143 0.511 1.010

4.00

0.1 1.673 0.062 0.172 0.026 2.473 1.126 0.504 0.843

1.0 1.852 0.270 0.169 0.138 3.181 1.681 0.526 0.974

10.0 2.009 0.407 0.166 0.242 3.854 2.218 0.544 1.092

42

Tables 2 and 3 for Optimal Dimensionless Parameters for TECs with ZT∞2=1

Figure 2. (a) Thermoelectric cooler module (TEC) and (b) Heat sinks without a TEC.


222 Ah

LAknN k

Dimensionless convection: 222

111

Ah

AhNh

Dimensionless electrical current: LAk

IN I


11

T

TT

, 2

22

T

TT , and

2

1

T

TT

Dimensionless cooling power: 2222

11

TAh

QQ

Dimensionless power input: 2222

TAh

WW n

n

Coefficient of Performance:

nW

QCOP 1


43

Table 2. Optimal Cooling Power for ZT∞2=1 T∞* Nh NI Nk COP Q1* T1* T2* NV T∞,min* T∞,max*

0.90

0.1 0.634 0.114 0.123 0.008 0.821 1.072 0.885 0.776 1.519

1.0 0.566 0.203 0.229 0.021 0.879 1.113 0.800 0.804 1.152

10.0 0.544 0.234 0.269 0.026 0.897 1.125 0.771 0.813 1.094

0.91

0.1 0.630 0.119 0.129 0.009 0.825 1.075 0.880 0.778 1.539

1.0 0.557 0.216 0.245 0.023 0.887 1.118 0.788 0.808 1.160

10.0 0.523 0.252 0.291 0.030 0.907 1.131 0.756 0.817 1.099

0.92

0.1 0.626 0.124 0.135 0.009 0.828 1.077 0.875 0.779 1.559

1.0 0.548 0.229 0.262 0.026 0.894 1.123 0.776 0.811 1.167

10.0 0.520 0.271 0.315 0.033 0.917 1.137 0.740 0.822 1.104

0.93

0.1 0.622 0.129 0.140 0.010 0.832 1.080 0.890 0.781 1.578

1.0 0.538 0.243 0.280 0.028 0.902 1.128 0.764 0.815 1.175

10.0 0.508 0.290 0.340 0.036 0.926 1.143 0.724 0.827 1.109

0.94

0.1 0.618 0.134 0.146 0.010 0.835 1.082 0.865 0.783 1.597

1.0 0.529 0.257 0.297 0.030 0.910 1.133 0.752 0.819 1.182

10.0 0.495 0.310 0.366 0.040 0.936 1.149 0.708 0.831 1.114

0.95

0.1 0.614 0.139 0.152 0.011 0.838 1.085 0.861 0.784 1.615

1.0 0.519 0.271 0.316 0.033 0.917 1.137 0.739 0.822 1.189

10.0 0.484 0.330 0.394 0.044 0.946 1.154 0.694 0.836 1.119

0.96

0.1 0.610 0.144 0.157 0.012 0.842 1.087 0.856 0.786 1.633

1.0 0.510 0.286 0.335 0.036 0.924 1.142 0.727 0.826 1.196

10.0 0.470 0.353 0.424 0.047 0.955 1.159 0.675 0.841 1.124

0.97

0.1 0.606 0.148 0.163 0.013 0.845 1.089 0.851 0.787 1.651

1.0 0.501 0.301 0.355 0.038 0.932 1.146 0.715 0.829 1.203

10.0 0.458 0.376 0.455 0.052 0.965 1.165 0.658 0.845 1.129

0.98

0.1 0.603 0.153 0.169 0.013 0.848 1.091 0.847 0.789 1.134

1.0 0.491 0.317 0.375 0.041 0.939 1.150 0.703 0.833 1.210

10.0 0.445 0.401 0.488 0.056 0.974 1.170 0.640 0.850 1.134

0.99

0.1 0.600 0.158 0.174 0.014 0.851 1.094 0.842 0.790 1.685

1.0 0.482 0.332 0.396 0.044 0.946 1.154 0.690 0.836 1.217

10.0 0.432 0.427 0.523 0.060 0.984 1.175 0.623 0.855 1.139

1.00

0.1 0.596 0.162 0.180 0.015 0.854 1.096 0.838 0.792 1.701

1.0 0.473 0.349 0.418 0.047 0.953 1.158 0.678 0.840 1.224

10.0 0.419 0.454 0.561 0.065 0.994 1.180 0.605 0.859 1.144

1.01

0.1 0.593 0.167 0.185 0.015 0.857 1.098 0.833 0.793 1.718

1.0 0.464 0.365 0.440 0.050 0.960 1.162 0.666 0.843 1.230

10.0 0.406 0.483 0.601 0.069 1.003 1.184 0.587 0.864 1.148

44

T∞* Nh NI Nk COP Q1* T1* T2* NV T∞,min* T∞,max*

1.02

0.1 0.589 0.172 0.191 0.016 0.860 1.100 0.829 0.795 1.734

1.0 0.454 0.383 0.463 0.053 0.967 1.166 0.653 0.847 1.237

10.0 0.393 0.514 0.645 0.074 1.013 1.189 0.569 0.869 1.153

1.03

0.1 0.586 0.176 0.196 0.017 0.863 1.102 0.824 0.796 1.749

1.0 0.445 0.401 0.487 0.056 0.974 1.170 0.641 0.850 1.243

10.0 0.379 0.548 0.691 0.079 1.022 1.193 0.551 0.873 1.158

1.04

0.1 0.582 0.181 0.202 0.017 0.866 1.104 0.820 0.797 1.765

1.0 0.436 0.419 0.512 0.059 0.981 1.174 0.628 0.853 1.250

10.0 0.366 0.583 0.741 0.084 1.032 1.198 0.532 0.878 1.163

1.05

0.1 0.579 0.185 0.208 0.018 0.868 1.106 0.816 0.799 1.780

1.0 0.427 0.438 0.538 0.062 0.988 1.177 0.616 0.857 1.256

10.0 0.353 0.622 0.796 0.089 1.041 1.202 0.513 0.883 1.168

1.06

0.1 0.576 0.190 0.213 0.019 0.871 1.108 0.813 0.800 1.795

1.0 0.417 0.458 0.566 0.065 0.995 1.118 0.603 0.860 1.236

10.0 0.388 0.664 0.856 0.095 1.051 1.206 0.494 0.888 1.172

1.07

0.1 0.573 0.194 0.218 0.020 0.870 1.109 0.809 0.801 1.810

1.0 0.408 0.478 0.594 0.068 1.002 1.184 0.590 0.863 1.268

10.0 0.324 0.709 0.921 0.100 1.060 1.210 0.474 0.892 1.177

1.08

0.1 0.569 0.199 0.224 0.020 0.877 1.111 0.804 0.803 1.824

1.0 0.399 0.500 0.624 0.072 1.008 1.187 0.578 0.866 1.275

10.0 0.310 0.759 0.993 0.106 1.069 1.214 0.454 0.897 1.182

1.09

0.1 0.566 0.203 0.229 0.021 0.879 1.113 0.800 0.804 1.838

1.0 0.390 0.522 0.655 0.075 1.015 1.190 0.565 0.870 1.281

10.0 0.296 0.814 1.072 0.112 1.079 1.217 0.434 0.902 1.187

1.10

0.1 0.563 0.207 0.235 0.022 0.882 1.115 0.796 0.805 1.853

1.0 0.380 0.545 0.687 0.079 1.021 1.193 0.552 0.873 1.287

10.0 0.282 0.874 1.159 0.118 1.088 1.220 0.414 0.907 1.191

1.11

0.1 0.560 0.212 0.240 0.023 0.884 1.116 0.792 0.807 1.866

1.0 0.371 0.569 0.721 0.082 1.028 1.196 0.539 0.876 1.293

10.0 0.267 0.941 1.258 0.124 1.098 1.223 0.393 0.912 1.196

1.12

0.1 0.557 0.216 0.246 0.023 0.887 1.118 0.788 0.808 1.880

1.0 0.362 0.594 0.757 0.086 1.034 1.199 0.526 0.879 1.299

10.0 0.253 1.017 1.368 0.131 1.011 1.226 0.372 0.917 1.201

1.13

0.1 0.554 0.221 0.251 0.024 0.890 1.120 0.784 0.809 1.894

1.0 0.353 0.621 0.794 0.089 1.041 1.202 0.514 0.883 1.305

10.0 0.238 1.102 1.493 0.137 1.116 1.229 0.351 0.921 1.205

1.14

0.1 0.550 0.225 0.257 0.025 0.892 1.121 0.780 0.810 1.907

1.0 0.343 0.648 0.834 0.093 1.047 1.204 0.501 0.886 1.310

10.0 0.223 1.199 1.637 0.144 1.126 1.232 0.329 0.926 1.210

45


1.15

0.1 0.547 0.229 0.262 0.026 0.895 1.123 0.776 0.811 1.920

1.0 0.334 0.677 0.875 0.097 1.053 1.207 0.488 0.889 1.316

10.0 0.208 1.312 1.803 0.151 1.135 1.234 0.307 0.931 1.215

1.16

0.1 0.544 0.234 0.268 0.026 0.897 1.125 0.772 0.813 1.933

1.0 0.325 0.708 0.919 0.100 1.060 1.209 0.475 0.892 1.322

10.0 0.192 1.443 1.997 0.158 1.144 1.237 0.285 0.937 1.220

46

Table 3. ½ OPtimal COP for ZT∞2=1 T∞* Nh NI Nk COP Q1* T1* T2* NV T∞,min* T∞,max*

0.90

0.1 0.317 0.057 0.470 0.004 0.860 1.012 0.470 0.805 1.167

1.0 0.283 0.102 0.717 0.008 0.892 1.020 0.412 0.821 1.049

10.0 0.272 0.117 0.782 0.010 0.899 1.022 0.395 0.827 1.030

0.91

0.1 0.315 0.060 0.515 0.004 0.865 1.013 0.463 0.806 1.173

1.0 0.279 0.108 0.812 0.010 0.900 1.022 0.400 0.824 1.051

10.0 0.262 0.126 0.897 0.011 0.909 1.024 0.381 0.830 1.032

0.92

0.1 0.313 0.062 0.559 0.005 0.870 1.014 0.457 0.807 1.180

1.0 0.274 0.115 0.912 0.011 0.909 1.023 0.388 0.826 1.054

10.0 0.260 0.136 1.022 0.013 0.919 1.026 0.367 0.833 1.033

0.93

0.1 0.311 0.065 0.605 0.005 0.875 1.015 0.450 0.807 1.186

1.0 0.269 0.122 1.019 0.013 0.917 1.025 0.376 0.828 1.056

10.0 0.254 0.145 1.158 0.015 0.928 1.028 0.353 0.836 1.035

0.94

0.1 0.309 0.067 0.650 0.006 0.880 1.015 0.444 0.808 1.192

1.0 0.265 0.129 1.134 0.014 0.926 1.026 0.365 0.831 1.059

10.0 0.248 0.155 1.308 0.017 0.938 1.030 0.339 0.839 1.037

0.95

0.1 0.307 0.070 0.696 0.007 0.885 1.016 0.438 0.809 1.198

1.0 0.260 0.136 1.255 0.016 0.934 1.028 0.353 0.833 1.061

10.0 0.242 0.165 1.472 0.019 0.948 1.032 0.325 0.842 1.038

0.96

0.1 0.305 0.072 0.743 0.007 0.890 1.017 0.432 0.810 1.204

1.0 0.255 0.143 1.385 0.017 0.943 1.030 0.342 0.835 1.063

10.0 0.235 0.177 1.653 0.021 0.958 1.034 0.312 0.846 1.040

0.97

0.1 0.303 0.074 0.790 0.008 0.894 1.017 0.427 0.811 1.209

1.0 0.251 0.151 1.524 0.019 0.951 1.031 0.331 0.838 1.065

10.0 0.229 0.188 1.653 0.021 0.958 1.034 0.312 0.849 1.040

0.98

0.1 0.302 0.077 0.837 0.008 0.899 1.018 0.421 0.812 1.215

1.0 0.246 0.159 1.673 0.021 0.959 1.033 0.320 0.840 1.068

10.0 0.223 0.201 2.075 0.026 0.977 1.039 0.284 0.852 1.043

0.99

0.1 0.300 0.079 0.885 0.009 0.903 1.019 0.415 0.813 1.220

1.0 0.241 0.166 1.833 0.023 0.967 1.035 0.309 0.843 1.070

10.0 0.216 0.214 2.323 0.029 0.987 1.041 0.270 0.856 1.045

1.00

0.1 0.298 0.081 0.934 0.009 0.907 1.019 0.410 0.814 1.225

1.0 0.237 0.175 2.004 0.025 0.975 1.037 0.298 0.845 1.072

10.0 0.210 0.227 2.600 0.032 0.997 1.044 0.257 0.859 1.046

1.01

0.1 0.297 0.084 0.983 0.010 0.912 1.020 0.405 0.815 1.231

1.0 0.232 0.183 2.188 0.027 0.983 1.039 0.287 0.847 1.074

10.0 0.203 0.242 2.911 0.035 1.007 1.047 0.243 0.863 1.048

47


1.02

0.1 0.295 0.086 1.033 0.010 0.916 1.021 0.399 0.816 1.236

1.0 0.227 0.192 2.387 0.029 0.991 1.041 0.277 0.850 1.076

10.0 0.197 0.257 3.263 0.038 1.016 1.049 0.230 0.867 1.050

1.03

0.1 0.293 0.088 1.083 0.011 0.920 1.021 0.394 0.816 1.241

1.0 0.223 0.201 2.601 0.031 0.999 1.043 0.266 0.852 1.079

10.0 0.190 0.274 3.664 0.041 1.026 1.052 0.216 0.870 1.051

1.04

0.1 0.291 0.091 1.135 0.012 0.924 1.022 0.389 0.817 1.246

1.0 0.218 0.210 2.833 0.033 1.007 1.045 0.256 0.855 1.081

10.0 0.183 0.292 4.122 0.045 1.036 1.055 0.203 0.874 1.053

1.05

0.1 0.290 0.093 1.186 0.012 0.928 1.023 0.384 0.818 1.251

1.0 0.214 0.219 3.084 0.035 1.015 1.047 0.245 0.857 1.083

10.0 0.177 0.311 4.649 0.048 1.045 1.059 0.189 0.878 1.054

1.06

0.1 0.288 0.095 1.239 0.013 0.932 1.023 0.379 0.819 1.256

1.0 0.209 0.229 3.356 0.038 1.022 1.049 0.235 0.860 1.085

10.0 0.194 0.332 5.259 0.052 1.055 1.062 0.176 0.882 1.056

1.07

0.1 0.287 0.097 1.292 0.013 0.936 1.024 0.375 0.820 1.260

1.0 0.204 0.239 3.653 0.040 1.030 1.051 0.225 0.862 1.087

10.0 0.162 0.355 5.974 0.056 1.064 1.066 0.163 0.886 1.058

1.08

0.1 0.285 0.100 1.346 0.014 0.939 1.025 0.370 0.820 1.265

1.0 0.200 0.250 3.976 0.030 1.037 1.053 0.216 0.865 1.089

10.0 0.155 0.380 6.818 0.060 1.074 1.069 0.151 0.890 1.059

1.09

0.1 0.283 0.102 1.401 0.015 0.943 1.025 0.365 0.821 1.270

1.0 0.195 0.261 4.329 0.045 1.045 1.056 0.206 0.868 1.091

10.0 0.148 0.407 7.824 0.065 1.084 1.073 0.138 0.895 1.061

1.10

0.1 0.282 0.104 1.456 0.015 0.947 1.026 0.361 0.822 1.274

1.0 0.190 0.273 4.715 0.048 1.052 1.058 0.196 0.870 1.093

10.0 0.141 0.437 9.037 0.070 1.093 1.077 0.125 0.899 1.062

1.11

0.1 0.280 0.106 1.513 0.016 0.950 1.027 0.356 0.823 1.279

1.0 0.186 0.285 5.140 0.051 1.059 1.061 0.187 0.873 1.095

10.0 0.134 0.471 10.520 0.075 1.103 1.082 0.113 0.904 1.064

1.12

0.1 0.279 0.108 1.570 0.017 0.954 1.027 0.352 0.824 1.283

1.0 0.181 0.297 5.608 0.053 1.067 1.063 0.177 0.875 1.097

10.0 0.127 0.509 12.363 0.080 1.112 1.087 0.101 0.908 1.066

1.13

0.1 0.277 0.111 1.628 0.017 0.958 1.028 0.347 0.824 1.288

1.0 0.177 0.311 6.125 0.056 1.074 1.066 0.168 0.878 1.099

10.0 0.119 0.551 14.694 0.086 1.121 1.092 0.089 0.913 1.067

1.14

0.1 0.275 0.113 1.687 0.018 0.961 1.029 0.343 0.825 1.292

1.0 0.172 0.324 6.699 0.059 1.081 1.068 0.159 0.881 1.101

10.0 0.112 0.600 17.706 0.092 1.131 1.097 0.078 0.918 1.069

48

T∞* Nh NI Nk COP Q1* T1* T2* NV T∞ min* T∞ max*

1.15

0.1 0.274 0.115 1.747 0.019 0.964 1.029 0.338 0.826 1.296

1.0 0.167 0.339 7.337 0.062 1.088 1.071 0.150 0.883 1.103

10.0 0.104 0.656 21.702 0.098 1.140 1.103 0.067 0.923 1.070

1.16

0.1 0.272 0.117 1.807 0.019 0.968 1.030 0.334 0.827 1.300

1.0 0.163 0.354 8.050 0.066 1.094 1.074 0.142 0.886 1.105

10.0 0.096 0.722 27.174 0.105 1.149 1.109 0.056 0.928 1.072

49

Table 4 for Optimal Dimensionless Parameters for TECs with ZT∞2=1

(Q1* is constant)

Figure 3. (a) Thermoelectric cooler module (TEC) and (b) Heat sinks without a TEC.


222 Ah

LAknN k

Dimensionless electrical current: LAk

IN I


11

T

TT

and 2

22

T

TT

Dimensionless cooling power: 2222

11

TAh

QQ

Dimensionless power input: 2222

TAh

WW n

n

Coefficient of Performance:

nW

QCOP 1


Two Governing Equations:

21

2

2

11

2TT

ZT

NTN

N

Q II

k

21

2

2

22

2

1TT

ZT

NTN

N

T II

k

50

Table 4 Optimal Lowest Junction Temperature T1* for ZT∞2=1 Q1* NI Nk T1* T2* Wn* COP NV T1,max*

0.02 0.571 0.197 0.875 1.11 0.09 0.221 0.806 1.074 0.022 0.562 0.209 0.883 1.115 0.093 0.236 0.795 1.078 0.024 0.554 0.22 0.889 1.12 0.096 0.251 0.784 1.081 0.026 0.546 0.232 0.896 1.124 0.098 0.266 0.774 1.084 0.028 0.538 0.243 0.902 1.128 0.1 0.28 0.763 1.087

0.03 0.53 0.255 0.908 1.132 0.102 0.295 0.754 1.09 0.032 0.523 0.266 0.914 1.136 0.104 0.309 0.744 1.093 0.034 0.516 0.277 0.92 1.139 0.105 0.324 0.734 1.096 0.036 0.508 0.289 0.926 1.142 0.106 0.338 0.725 1.099 0.038 0.501 0.3 0.931 1.146 0.108 0.353 0.716 1.101

0.04 0.495 0.311 0.936 1.149 0.109 0.368 0.707 1.104 0.042 0.488 0.322 0.942 1.152 0.11 0.382 0.698 1.106 0.044 0.481 0.334 0.947 1.155 0.111 0.397 0.689 1.109 0.046 0.475 0.345 0.952 1.158 0.112 0.412 0.681 1.111 0.048 0.469 0.356 0.957 1.16 0.112 0.428 0.672 1.113

0.05 0.462 0.368 0.961 1.163 0.113 0.443 0.664 1.116 0.052 0.456 0.379 0.966 1.165 0.113 0.459 0.656 1.118 0.054 0.45 0.391 0.971 1.168 0.114 0.474 0.647 1.12 0.056 0.444 0.402 0.975 1.17 0.114 0.49 0.639 1.122 0.058 0.438 0.414 0.979 1.173 0.115 0.506 0.631 1.124

0.06 0.432 0.426 0.984 1.175 0.115 0.522 0.623 1.126 0.062 0.426 0.438 0.988 1.177 0.115 0.539 0.615 1.128 0.064 0.421 0.45 0.992 1.179 0.115 0.556 0.608 1.13 0.066 0.415 0.463 0.996 1.181 0.115 0.573 0.6 1.132 0.068 0.409 0.475 1 1.183 0.115 0.59 0.592 1.134

0.07 0.404 0.488 1.005 1.185 0.115 0.608 0.585 1.136 0.072 0.398 0.501 1.008 1.187 0.115 0.626 0.577 1.137 0.074 0.393 0.514 1.012 1.189 0.115 0.644 0.569 1.139 0.076 0.387 0.527 1.016 1.191 0.115 0.662 0.562 1.141 0.078 0.382 0.54 1.02 1.193 0.115 0.681 0.555 1.143

0.08 0.377 0.554 1.024 1.194 0.114 0.7 0.547 1.145 0.082 0.372 0.568 1.028 1.196 0.114 0.72 0.54 1.146 0.084 0.366 0.582 1.031 1.198 0.114 0.74 0.533 1.148 0.086 0.361 0.596 1.035 1.199 0.113 0.76 0.525 1.15 0.088 0.356 0.611 1.038 1.201 0.113 0.781 0.518 1.151

0.09 0.351 0.626 1.042 1.202 0.112 0.802 0.511 1.153 0.092 0.346 0.641 1.046 1.204 0.112 0.824 0.504 1.154 0.094 0.341 0.657 1.049 1.205 0.111 0.846 0.497 1.156 0.096 0.336 0.673 1.052 1.207 0.111 0.869 0.49 1.158

51

Q1* NI Nk T1* T2* Wn* COP NV T1,max*

0.098 0.331 0.689 1.056 1.208 0.11 0.892 0.483 1.159 0.1 0.326 0.705 1.059 1.209 0.109 0.915 0.476 1.161

0.102 0.321 0.722 1.062 1.211 0.109 0.94 0.469 1.162 0.104 0.316 0.739 1.066 1.212 0.108 0.965 0.462 1.164 0.106 0.311 0.757 1.069 1.213 0.107 0.99 0.455 1.165 0.108 0.306 0.775 1.072 1.214 0.106 1.016 0.448 1.167

0.11 0.301 0.794 1.075 1.215 0.105 1.043 0.441 1.168 0.112 0.296 0.813 1.079 1.217 0.105 1.071 0.434 1.17 0.114 0.292 0.832 1.082 1.218 0.104 1.099 0.428 1.171 0.116 0.287 0.852 1.085 1.219 0.103 1.128 0.421 1.172 0.118 0.282 0.873 1.088 1.22 0.102 1.158 0.414 1.174

0.12 0.278 0.893 1.091 1.221 0.101 1.188 0.408 1.175 0.122 0.273 0.915 1.094 1.222 0.1 1.22 0.401 1.177

0.124 0.268 0.938 1.097 1.223 0.099 1.252 0.394 1.178 0.126 0.264 0.959 1.1 1.224 0.098 1.285 0.388 1.179 0.128 0.259 0.984 1.103 1.225 0.097 1.32 0.381 1.181

0.13 0.254 1.009 1.106 1.226 0.096 1.356 0.374 1.182 0.132 0.25 1.033 1.109 1.227 0.095 1.393 0.367 1.183 0.134 0.246 1.057 1.112 1.228 0.094 1.428 0.362 1.185 0.136 0.241 1.084 1.115 1.229 0.093 1.468 0.355 1.186 0.138 0.236 1.114 1.117 1.229 0.091 1.511 0.348 1.187

0.14 0.231 1.142 1.12 1.23 0.09 1.553 0.341 1.189 0.142 0.227 1.172 1.123 1.231 0.089 1.596 0.335 1.19 0.144 0.222 1.202 1.126 1.232 0.088 1.64 0.328 1.191 0.146 0.218 1.234 1.129 1.232 0.086 1.688 0.322 1.192 0.148 0.213 1.267 1.131 1.233 0.085 1.736 0.315 1.194

0.15 0.209 1.301 1.134 1.234 0.084 1.787 0.309 1.195 0.152 0.205 1.336 1.137 1.235 0.083 1.839 0.302 1.196 0.154 0.201 1.366 1.139 1.236 0.082 1.886 0.297 1.197 0.156 0.197 1.4 1.142 1.236 0.08 1.939 0.291 1.198 0.158 0.191 1.45 1.145 1.237 0.079 2.008 0.283 1.2

0.16 0.187 1.493 1.147 1.237 0.077 2.071 0.277 1.201 0.162 0.183 1.536 1.15 1.238 0.076 2.135 0.27 1.202 0.164 0.178 1.585 1.153 1.238 0.074 2.206 0.264 1.204 0.166 0.174 1.629 1.155 1.239 0.073 2.273 0.258 1.205 0.168 0.17 1.673 1.158 1.24 0.072 2.34 0.252 1.206

0.17 0.166 1.72 1.16 1.24 0.07 2.413 0.246 1.207

0.172 0.16 1.795 1.163 1.24 0.068 2.516 0.238 1.208 0.174 0.157 1.837 1.165 1.241 0.067 2.584 0.233 1.209 0.176 0.152 1.908 1.168 1.242 0.066 2.687 0.226 1.211 0.178 0.148 1.968 1.17 1.242 0.064 2.777 0.22 1.212

0.18 0.144 2.036 1.173 1.243 0.063 2.878 0.214 1.213

52

Q1* NI Nk T1* T2* Wn* COP NV T1,max*

0.182 0.14 2.105 1.175 1.243 0.061 2.982 0.208 1.214 0.184 0.135 2.181 1.178 1.243 0.059 3.095 0.201 1.215 0.186 0.132 2.252 1.18 1.244 0.058 3.204 0.196 1.216 0.188 0.128 2.331 1.182 1.245 0.057 3.32 0.19 1.217

0.19 0.124 2.409 1.185 1.245 0.055 3.437 0.185 1.218 0.192 0.121 2.488 1.187 1.246 0.054 3.557 0.18 1.219 0.194 0.117 2.584 1.189 1.246 0.052 3.698 0.174 1.22 0.196 0.113 2.68 1.192 1.247 0.051 3.84 0.168 1.221 0.198 0.106 2.866 1.194 1.246 0.048 4.119 0.158 1.223

0.2 0.104 2.928 1.196 1.247 0.047 4.231 0.155 1.223 0.202 0.1 3.055 1.199 1.248 0.046 4.426 0.149 1.224 0.204 0.094 3.266 1.201 1.247 0.043 4.722 0.14 1.226 0.206 0.092 3.352 1.203 1.248 0.042 4.874 0.137 1.227

0.208 0.083 3.743 1.205 1.246 0.038 5.438 0.123 1.229 0.21 0.08 3.912 1.208 1.247 0.037 5.668 0.119 1.23

0.212 0.077 4.068 1.21 1.248 0.036 5.913 0.115 1.23 0.214 0.073 4.3 1.212 1.248 0.034 6.265 0.109 1.231 0.216 0.069 4.559 1.214 1.248 0.032 6.653 0.103 1.232 0.218 0.065 4.846 1.217 1.249 0.031 7.088 0.097 1.233

Optimal Design of Thermoelectric Devices with Dimensional...

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