3-1

Chapter 3 Thermoelectric

Coolers

Contents

Chapter 3 Thermoelectric Coolers .................................................................................... 3-1

Contents ............................................................................................................................ 3-1

3.1 Ideal Equations......................................................................................................... 3-2

3.2 Maximum Parameters .............................................................................................. 3-7

3.3 Normalized Parameters ............................................................................................ 3-8

Example 3.1 Thermoelectric Air Conditioner ................................................................ 3-12

3.4 Effective Material Properties ................................................................................. 3-18

Problems ......................................................................................................................... 3-23

References ....................................................................................................................... 3-25

This chapter formulates a set of simplified ideal equations for a thermoelectric cooler, with some

assumptions, to exhibit the general characteristics of thermoelectric coolers. The maximum

parameters are defined, which are the maximum current, maximum temperature difference,

maximum cooling power, and maximum voltage. Then, the normalized parameters are plotted as

general characteristics for the coolers. The ideal equation is based on three material properties: the

Seebeck coefficient, electrical resistivity, and thermal conductivity. These material properties for

commercial thermoelectric cooler modules are not usually provided by the manufacturers since

3-2

they consider them proprietary information. Therefore, the effective material properties are

developed from the available maximum parameters of the product with fair agreement with the

measurements. These are used for prediction of performance and design later.

3.1 Ideal Equations

Since the discovery of thermoelectric effects in the early nineteenth century, a very essential

equation for the rate of heat flow per unit area �⃗� was formulated as shown in Equation (2.3),

which is

q⃗⃗ = 𝛼𝑇𝑗 − 𝑘∇⃗⃗⃗T (3.1)

where is the Seebeck coefficient, 𝑗 the current density, k the thermal conductivity and ∇⃗⃗⃗ the

gradient. This equation relates the heat flow, the electric current and the thermal conduction,

leading to the steady-state heat diffusion equation (Equation (2.7)), which is rewritten here as

02 TjdT

dTjTk

(3.2)

where is the electrical resistivity. The first term gives the thermal conduction, the second term

gives the Joule heating, and the third term gives the Thomson effect which results from the

temperature-dependent Seebeck coefficient. The above two equations govern thermoelectric

phenomena.

3-3

Figure 3.1 Cutaway of a typical thermoelectric module

Figure 3.2 Thermoelectric cooler with p-type and n-type thermoelements.

p

n

p

n

np

p

pn

Positive (+)

Negative (-)

Heat Absorbed

Heat Rejected

Electrical Conductor (copper)Electrical Insulator (Ceramic)

p-type Semiconcuctor

n-type Semiconductor

3-4

Consider a steady-state one-dimensional thermoelectric cooler module as shown in Figure 3.1.

The module consists of many p-type and n-type thermocouples, where one thermocouple

(unicouple) with length L and cross-sectional area A is shown in Figure 3.2. An electrical current

is applied and induces a heat flow at the cold and hot junction temperatures as shown. With the

assumptions that the Seebeck coefficient is independent of temperature, no thermal or electrical

contact resistances, and no heat losses, Equation (3.2) reduces to

02

A

I

dx

dTkA

dx

d

(3.3)

The solution for the temperature gradient with two boundary conditions ( cx TT 0 and

hLx TT ) is

L

TT

kA

LI

dx

dT ch

x

2

2

0 2

(3.4)

Equation (3.1) is expressed in terms of p-type and n-type thermocouples.

𝑄𝑐 = 𝑛 [(𝛼𝑝 − 𝛼𝑛)𝑇𝑐𝐼 + (−𝑘𝐴𝑑𝑇

𝑑𝑥|𝑥=0

)𝑝

+ (−𝑘𝐴𝑑𝑇

𝑑𝑥|𝑥=0

)𝑛

] (3.5)

where n is the number of thermocouples and cQ is the rate of heat absorbed at the cold junction.

Substituting Equation (3.4) in (3.5) gives

𝑄𝑐 = 𝑛 [(𝛼𝑝 − 𝛼𝑛)𝑇𝑐𝐼 −1

2𝐼2 (

𝜌𝑝𝐿𝑝

𝐴𝑝+𝜌𝑛𝐿𝑛𝐴𝑛

) − (𝑘𝑝𝐴𝑝

𝐿𝑝+𝑘𝑛𝐴𝑛

𝐿𝑛) (𝑇ℎ − 𝑇𝑐)]

(3.6)

Finally, the cooling power at the junction of temperature Tc is expressed as

3-5

�̇�𝑐 = 𝑛 [𝛼𝑇𝑐𝐼 −1

2𝐼2𝑅 − 𝐾(𝑇ℎ − 𝑇𝑐)]

(3.7)

where

np (3.8)

n

nn

p

pp

A

L

A

LR

(3.9)

n

nn

p

pp

L

Ak

L

AkK

(3.10)

If we assume that the p-type and n-type thermocouples are similar, we have R = L/A and K =

kA/L, where = p + n and k = kp + kn. Equation (3.7) is called the ideal equation which has

been widely used in science and industry. The rate of heat liberated at the hot junction is

�̇�ℎ = 𝑛 [𝛼𝑇ℎ𝐼 +1

2𝐼2𝑅 − 𝐾(𝑇ℎ − 𝑇𝑐)]

(3.11)

Considering the 1st law of thermodynamics across the thermoelectric device, we have

�̇� = �̇�ℎ − �̇�𝑐 (3.12)

The amount of work per unit time across the module (rate of work) is obtained substituting

Equations (3.7) and (3.11) in (3.12).

�̇� = 𝑛[𝛼𝐼(𝑇ℎ − 𝑇𝑐) + 𝐼2𝑅] (3.13)

where the first term is the rate of work to overcome the thermoelectric voltage, and the second

term is the resistive loss. Since the power is IVW , the voltage across the couple will be

3-6

𝑉 = 𝑛[𝛼(𝑇ℎ − 𝑇𝑐) + 𝐼𝑅] (3.14)

The COP is defined as the ratio of the cooling power to the input electrical power.

𝐶𝑂𝑃 =�̇�𝑐

�̇�=𝑛 [𝛼𝑇𝑐𝐼 −

12 𝐼

2𝑅 − 𝐾(𝑇ℎ − 𝑇𝑐)]

𝑛[𝛼𝐼(𝑇ℎ − 𝑇𝑐) + 𝐼2𝑅]

(3.15)

There are two values of the current that are of special interest: the current Imp that yields the

maximum cooling power and the current ICOP that yields the maximum COP. The maximum

cooling power can be obtained by differentiating Equation (3.7) with respect to I and setting it to

zero. The current for the maximum cooling power is found to be

R

TI c

mp

(3.16)

The optimum COP can be obtained by differentiating Equation (3.15) and setting it to zero

𝑑(𝐶𝑂𝑃)

𝑑𝐼= 0

(3.17)

We finally have

11

TZR

TICOP

(3.18)

where ∆𝑇 = 𝑇ℎ − 𝑇𝑐, kZ 2 and T is the average temperature of cT and hT . In terms of Th,

TZ is expressed by

3-7

h

hT

TZTTZ

21

(3.19)

3.2 Maximum Parameters

Let us consider a thermoelectric module shown in Figure 3.1 with the theoretical maximum

parameters in the ideal equation. The module consists of a number of thermocouples as shown.

The ideal equation assumes that there are no the electrical or thermal contact resistances, no

Thomson effect, and no radiation or convection. It is noted that the theoretical maximum

parameters might differ from the manufacturers’ maximum parameters that are usually obtained

through measurements.

The maximum current Imax is the current that produces the maximum possible temperature

difference Tmax , which always occurs when the cooling power is at zero. This is obtained by

setting cQ = 0 in Equation (3.7), replacing Tc with (Th – T), taking the derivative of T with

respect to I and setting it to zero. The maximum current is finally expressed by

ZT

ZT

RI hh

11 2

2

max

(3.20)

Tmax is the maximum possible temperature difference Tmax which always occurs when the

cooling power is zero and the current is maximum. This is obtained by setting cQ = 0 in

Equation (3.7), substituting both I and Tc by Imax and Th – Tmax, respectively, and solving for

Tmax. The maximum temperature difference is obtained as

2

2

max

11hhh T

ZT

ZTT

(3.21)

where the figure of merit Z (unit: K-1) is given by

3-8

kZ

2

or RK

Z2

(3.22)

Using Equation (3.21), Imax may be written as

R

TTI h max

max

(3.23)

The maximum cooling power maxcQ is the maximum thermal load which occurs at T = 0 and I

= Imax. This can be obtained by substituting both I and Tc in Equation (3.7) by Imax and Th,

respectively, and solving for maxcQ . The maximum cooling power for a thermoelectric module

with n thermocouples is

R

TTnQ h

c2

2

max

22

max

(3.24)

The maximum voltage is the DC voltage which delivers the maximum possible temperature

difference Tmax at I = Imax. The maximum voltage is obtained from Equation (3.14), which is

hTnV max (3.25)

3.3 Normalized Parameters

If we divide the actual values by the maximum values, we can normalize the characteristics of

the thermoelectric cooler. The normalized cooling power can be obtained by dividing Equation

(3.7) by Equation (3.24), which is

3-9

RTTn

TKRIITTn

Q

Q

h

h

c

c

2

2

1

2

max

22

2

max

(3.26)

which, in terms of the normalized current and normalized temperature difference, reduces to

2

max

max

max

max

2

max

max

max

max

max

max

max

1

2

1

1

1

12

h

h

h

h

h

h

h

c

c

T

TZT

T

T

T

T

T

T

I

I

T

T

T

T

I

I

T

T

T

T

Q

Q

(3.27)

where

11

11

1

2

max

hhh ZTZTT

T

(3.28)

The coefficient of performance in terms of the normalized values is

2

max

max

max

max

max

max

max

max

2

max

max

max

max

max

1

1

12

11

I

I

T

T

I

I

T

T

T

T

T

TZT

T

T

T

T

I

I

T

T

I

I

T

T

T

T

COP

hh

h

h

h

hh

(3.29)

The normalized voltage is

max

maxmax

maxmax

1I

I

T

T

T

T

T

T

V

V

hh

(3.30)

The normalized current for the optimum COP is obtained from Equation (3.18).

3-10

111 max

max

max

max

TZT

T

T

T

T

T

I

I

h

hCOP

(3.31)

where TZ is expressed using Equation (3.19) and Equation (3.28) by

max

max

2

11

T

T

T

TZTTZ

h

h (3.32)

Note that the above normalized values in Equations (3.27), (3.29) and (3.30) are a function of three

parameters, which are maxTT , maxII and ZTh. Figure 3.3 and Figure 3.4 are based on the ideal

equations using the normalized parameters. The three maximum parameters of Tmax, Imax, and

maxcQ are predictable inversely with the effective material properties, we can then use the

normalized charts for estimation of the performance. The solid lines for the both figures indicate

the normalized prediction with ZTh = 0.75 which is a typical commercial value (see Table 3.2).

Figure 3.3 shows the general characteristics of how cooling power and voltage depend on

temperature difference. For example, the maximum cooling power occurs at both the zero of

temperature difference and the maximum current. The lower curve (red line) indicates the cooling

power at the optimal COP, which implies that the optimal COP generally results in a low cooling

power and the medium current exhibits good design point in a practical view.

Figure 3.4 shows that the COP and cooling power versus the current along with the temperature

difference. The optimal COPs and maximum cooling powers are clearly seen. The current may be

properly arranged between the optimal COP and maximum cooling power in a practical design.

3-11

Figure 3.3 Normalized chart I for thermoelectric coolers: cooling power and voltage versus T as

a function of current. ZTh = 0.75 was used in calculations. The red line depicts the cooling power

ratios at the optimum COP. [1]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I/Imax = 1.0I/Imax = 1.0

0.8

0.8

0.6

0.6

Qc/Qcmax 0.4 V/Vmax

0.4

0.2

0.2

T/Tmax

3-12

Figure 3.4 Normalized chart II for thermoelectric coolers: cooling power and COP versus current

as a function of T. ZTh = 0.75 was used in calculations. [1]

Example 3.1 Thermoelectric Air Conditioner

A novel thermoelectric air conditioner is designed as a green energy replacement for the

conventional compressor-type air conditioner in a car. A thermoelectric module with heat sinks

consists of n = 128 p- and n-type thermocouples, one of which is shown in Figure 3.5. The air

conditioner has a number of the modules. Cabin cold air enters the upper heat sink, while the

outside ambient air enters the lower heat sink. An electric current is applied in a way that a heat

flow (cooling power) should be absorbed at the cold junction temperature of 15 °C and liberated

at the hot junction temperature of 40 °C. The TEC material of bismuth telluride (Bi2Te3) is used

having the properties as p = −n = 200 V/K, p = n = 1.0 × 10-3 cm, and kp = kn = 1.52 × 10-

2 W/cmK. The cross-sectional area and leg length of the thermoelement are An = Ap = 2 mm2 and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1T/Tmax = 0

T/Tmax = 0

0.1

0.1

0.2

0.2

0.3

0.4

COP 0.3 0.5 Qc/Qcmax

0.6

0.4

0.5 0.8

0.6

0.8

I/Imax

3-13

Ln = Lp = 1 mm, respectively. Assuming that the cold and high junction temperatures are steadily

maintained, answer the following questions (Use hand calculations).

(a) For the maximum cooling power, compute the current, cooling power, and COP.

(b) For the maximum COP, compute the current, cooling power, and COP.

(c) If the midpoint of the current between the maximum cooling power and maximum COP

is used for the optimal design, compute the current, the cooling power and COP.

(d) If the total cooling load of 630 W (per occupant) for the air conditioner is required,

compute the number of modules to meet the requirement using the midpoint of current.

(a ) (b)

Figure 3.5 (a) A thermoelectric module. (b) A p-type and n-type thermocouple

Solution:

Material properties: =p − n = 400 × 10-6 V/K, = p + n = 2.0 × 10-5 m, and k = kp + kn

= 3.04 W/mK

The number of thermocouples is n = 128. The hot and cold junction temperatures are

KKTh 313)27340( and KKTc 288)27315(

KTTT ch 25

3-14

The figure of merit is

13

5

262

10632.204.3100.2

10400

K

mKWm

KV

kZ

and the dimensionless figure of merit is

758.028810632.2 13 KKZTc

The internal resistance R and the thermal conductance K are calculated as

01.0102

101100.226

35

m

mm

A

LR

K

W

m

mmKW

L

kAK 3

3

26

1008.6101

102/04.3

(a) For the maximum cooling power:

Using Equation (3.16), the current for the maximum cooling power is

AKKV

R

TI c

mp 526.1101.0

28810400 6

Using Equation (3.7), the maximum cooling power is

3-15

W

KK

WAAKKV

TKRIITnQ mpmpccmp

567.65

251008.601.0526.112

1526.1128810400128

2

1

326

2

Using Equation (3.13), the power input is

WAKAKV

RITTInW mpchmpnmp

8.18401.0526.1125526.111040012826

2

Using Equation (3.15), the COP at the maximum cooling power is

355.08.184

567.65

W

W

W

QCOP

nmp

cmp

mp

(b) For the maximum COP:

791.02

2510632.2

2

13

KK

TTZTZ ch

Using Equation (3.18), the current for the maximum COP is

AKKV

TZR

TICOP 956.2

1791.0101.0

2510400

11

6

Using Equation (3.7), the maximum cooling power is

3-16

W

KK

WAAKKV

TKRIITnQ copcopcncop

557.18

251008.601.0956.22

1956.228810400128

2

1

326

2

Using Equation (3.13), the maximum power input is

WAKAKV

RITTInW copchcopncop

964.1401.0956.225956.21040012826

2

Using Equation (3.15), the COP is

24.1964.14

557.18max

W

W

W

QCOP

ncop

ncop

(c) For the midpoint of the current between the maximum cooling power and maximum COP:

The midpoint current is

AAAII

ICOPmp

mid 241.72

956.2526.11

2

Using Equation (3.7), the maximum cooling power is

W

KK

WAAKKV

TKRIITnQ midmidccmid

815.53

251008.601.0241.72

1241.728810400128

2

1

326

2

3-17

Using Equation (3.13), the maximum power input is

WAKAKV

RITTInW midchmidnmid

377.7601.0241.725241.71040012826

2

Using Equation (3.15), the midpoint COP is

705.0377.76

815.53

W

W

W

QCOP

nmid

cmidmid

The required cooling power is

WQreq 630

The number of TEC modules required is

7.118.53

630

W

W

Q

QN

cmid

req

Table 3.1 Summary of the Results

Max. Cool. Power Max. COP Midpoint

Current Imp = 11.526 A Icop = 2.956 A Imid = 7.241 A

Cooling power Qcmp = 65.576 W Qcop = 18.557 W Qcmid = 53.815 W

Power input Wcnp = 184.8 W Wncop = 14.964 W Wnmid = 76.377 W

COP COPmp = 0.355 COPmax = 1.24 COPmid = 0.705

Number of modules Nmp = 9.6 Ncop = 33.9 Nmid = 11.7

Design comments Uneconomical

(Too high power

consumption)

Uneconomical

(Too many modules)

Economical

(reasonable design)

Comments

The results in Table 3.1 are reflected in the COP and Qc versus current curves (Figure 3.6)

plotted using Equations (3.7), (3.13), and (3.15) with the material properties and inputs given in

3-18

the example description. It is graphically seen in Figure 3.6 that the maximum cooling power

accompanies the very low COP, while the maximum COP accompanies very low cooling power.

These lead to the uneconomical results. The midpoint of current between the maximum COP and

maximum cooling power gives reasonable values for both. Automotive air conditioners

intrinsically demand both a high COP and a high cooling power.

Figure 3.6 COP and Qc versus current for the given properties and inputs.

3.4 Effective Material Properties

As mentioned before, theoretically, the four maximum parameters (Imax, Tmax, maxcQ and Vmax)

are exactly reciprocal with the three material properties (, , and k). In other words, the three

material properties constitute the four maximum parameters in a reciprocal manner. In order to

predict the performance of thermoelectric coolers, the material properties are, of course,

required. However, we have a dilemma in that usually manufacturers do not provide the material

properties as their proprietary information but only the measured maximum parameters as

specifications of their products. Using the reciprocal relationship, we can easily formulate the

three material properties in terms of the four manufacturers’ maximum parameters. Two

3-19

maximum parameters (Imax and Tmax) are essential and must be used, but there is a choice of

either maxcQ or Vmax. Theoretically there is no difference whether either is selected but practically

there is a difference depending on the choice. According to the analysis (not shown here), if we

choose the maximum cooling power, the errors between the ideal equation and real

measurements tend to go to the voltages. On the other hand, if we choose the maximum voltage,

the errors tend to be distributed evenly to the cooling powers and voltages. It should be noted

that there is no longer the reciprocity between the four maximum parameters and the three

material properties if we determine the material properties by extracting them from the

manufacturers’ maximum parameters. The material properties extracted are called the effective

material properties. The effective figure of merit is obtained from Equation (3.21), which is

2max

max2

TT

TZ

h

(3.33)

The effective Seebeck coefficient is obtained using Equations (3.23) and (3.24), which is

maxmax

max2

TTnI

Q

h

c

(3.34)

The effective electrical resistivity can be obtained using Equation (3.23), which is

max

max

I

LATTh

(3.35)

The effective thermal conductivity is now obtained using Equation (3.22), which is

Zk

2

(3.36)

The effective material properties include effects such as the electrical and thermal contact

resistances, the temperature dependency of the material, and the radiative and convective heat

losses. Hence, the effective figure of merit appears slightly smaller than the intrinsic figure of

3-20

merit as shown in Table 3.2 Comparison of the Properties and Dimensions for the Commercial

Products of Thermoelectric Modules [1]. Since the material properties were obtained for a p-type

and n-type thermocouple, the material properties of a thermoelement (either p-type or n-type)

should be attained by dividing by 2.

Comparison of Calculations with a Commercial Product

The effective material properties can be calculated for any commercial thermoelectric module

modules as long as the four maximum parameters are provided. Calculated effective material

properties from the maximum parameters for four commercial thermoelectric modules are

illustrated in Table 3.2. Then, we can simulate the performance curves of the module with these

effective material properties using the ideal equations. For example, we obtained the effective

material properties for C2-30-1503 module in Table 3.2 and compared the calculated

performance curves with the commercial performance curves, which are shown in Figure 3.7(a)

–(c). It is found that the calculated results are in good agreement with the manufacturer’s curves

(which are typically experimental values)

Table 3.2 Comparison of the Properties and Dimensions for the Commercial Products of

Thermoelectric Modules [1]

Description TEC Module (Bismuth Telluride)

Symbols Laird

CP10-127-05

(Th=298 K)

Marlow

RC12-4

(Th=298 K)

Kryotherm

TB-127-1.0-1.3

(Th=298 K)

Tellurex

C2-30-1503

(Th=300 K)

# of thermocouples n 127 127 127 127

Effective material

properties

(calculated using

commercial Tmax,

Imax, and Qcmax)

V/K 189.2 211.1 204.5 208.5

cm 0.9 x 10-3 1.15 x 10-3 1.0 x 10-3 1.0 x 10-3

k (W/cmK) 1.6 x 10-2 1.7 x 10-2 1.6 x 10-2 1.7 x 10-2

ZTh 0.744 0.673 0.776 0.758

Measured geometry

of thermoelement

A (mm2) 1.0 1.0 1.0 1.21

L (mm) 1.25 1.17 1.3 1.66

G=A/L (cm) 0.080 0.085 0.077 0.073

Dimension

(W×L×H)

mm 30 × 30 × 3.2 30 × 30 ×

3.4

30 × 30 × 3.6 30 × 30 × 3.7

Tmax (°C) 67 66 (63) 69 68

Imax (A) 3.9 3.7 3.6 3.5

3-21

Manufacturers’

maximum

parameters

Qcmax (W) 34.3 36 34.5 34.1

Vmax (V) 14.4 14.7 15.7 15.5

R ()-module 3.36 3.2 3.2 3.85

(a)

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

Temperature Difference, T (°C)

Cooli

ng P

ow

er, Q

c (W

)

I = 3.5 APrediction

Commercial product3 A

Optimal COP

2.5 A

2 A

1.5 A

1 A

3-22

(b)

(c)

Figure 3.7 (a) Cooling power versus T, (b) voltage versus T, as a function of current, and (c)

COP versus current as a function of T. The original performance data (triangles) of the

commercial module (Tellurex C2-30-1503) are compared to the prediction (solid lines). The

curve at the bottom in (a) indicates the cooling powers at the optimum COP.[1]

0 10 20 30 40 50 60 700

2

4

6

8

10

12

14

16

Temperature Difference (°C)

Vol

tage

(V)

I = 3.5 A

3 A

2.5 A

2 A

1.5 A

1 A

Prediction

Commercial product

0 1 2 3 40

0.5

1

1.5

2

2.5

3

Current (A)

CO

P

T = 10°C Prediction

Commercial product

20°C

30°C

40°C

50°C

3-23

Problems

3.1 A compact thermoelectric air conditioner is developed as an ambitious green energy

project. N = 20 thermoelectric modules are installed between two heat sinks as shown in

Figure P3-1a. The module has n = 127 thermocouples, each of which consists of p- and n-

type thermoelements as shown in Figure P3-1b. Cabin air flows through the top and

bottom heat sinks, while liquid coolant is routed through a heat exchanger at the centre of

the device wherein the coolant is cooled separately at the car radiator. With an effective

design of both the heat sinks and heat exchanger, the cold and hot junction temperatures

can be maintained at 14 °C and 32 °C, respectively. Nanostructured thermoelectric

properties of bismuth telluride based are given as p = −n = 238 V/K, p = n = 1.23 ×

10-3 cm, and kp = kn = 0.945 × 10-2 W/cmK. The cross-sectional area A and pellet length

L are 1 mm2 and 1.1 mm, respectively. Answer the following questions for the whole air

conditioner (Use hand calculations).

(a) For the maximum cooling power, compute the current, cooling power, and COP.

(b) For the maximum COP, compute the current, cooling power, and COP.

(c) If the midpoint of the current between the maximum cooling power and maximum

COP is used for the optimal design, compute the current, the cooling power and COP.

(d) Draw the COP-and-cooling-power-versus-current curves with the given properties

and information (Use Mathcad only for this part). Briefly explain the design concept.

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(a) (b)

Figure P3-1. (a) A thermoelectric air conditioner. (b) A p-type and n-type thermocouple

3.2 A compact thermoelectric air conditioner is developed as an ambitious green energy

project. N = 40 thermoelectric modules are installed between two heat sinks as shown in

Figure P3-2 (a). The module has n = 127 thermocouples, each of which consists of p- and

n-type thermoelements as shown in Figure P3-2 (b). Cabin air flows through the top and

bottom heat sinks, while liquid coolant is routed through a heat exchanger at the center of

the device wherein the coolant is cooled separately at the car radiator. With the effective

design of both the heat sinks and heat exchanger, the cold and hot junction temperatures

are maintained at 15 °C and 30 °C, respectively. It is found that a commercial module

(CP10-127-05) of bismuth telluride is appropriate for this purpose, which has the

maximum parameters: cooling power of 34.3 W, temperature difference of 67 °C, current

of 3.9 °C, and voltage of 14.4 V at a hot side temperature of 25 °C. The cross-sectional

area A and pellet length L are 1 mm2 and 1.25 mm, respectively. Answer the following

questions for the whole air conditioner (Use hand calculations).

(a) Obtain the effective material properties: the Seebeck coefficient, electrical resistance,

and thermal conductivity.

(b) For the maximum cooling power, compute the current, cooling power, and COP.

(c) For the maximum COP, compute the current, cooling power, and COP.

3-25

(d) If the midpoint of the current between the maximum cooling power and maximum

COP is used for the optimal design, compute the current, the cooling power and COP.

(e) Draw the COP-and-cooling-power-versus-current curves with the given properties

and information (Use Mathcad only for this part). Briefly explain the design concept.

(a ) (b)

Figure P3-2. (a) A thermoelectric air conditioner. (b) A p-type and n-type thermocouple

3.3 Show the derivation of Equation (3.4).

3.4 Derive Equation (3.7).

3.5 Show the derivation of Equation (3.22).

3.6 Develop the expressions and plots in Figure 3.3 and Figure 3.4 using Mathcad.

3.7 Plot Figure 3.7 (a) –(c) using Mathcad.

References

1. Lee, H., A.M. Attar, and S.L. Weera, Performance Prediction of Commercial

Thermoelectric Cooler Modules using the Effective Material Properties. Journal of

Electronic Materials, 2015. 44(6): p. 2157-2165.