chapter 4 optimal design
TRANSCRIPT
4-1
Chapter 4 Optimal Design
Contents
Chapter 4 Optimal Design ................................................................................................ 4-1
Contents ............................................................................................................................ 4-1
4.1 Introduction .............................................................................................................. 4-2
4.2 Optimal Design for Thermoelectric Generators ...................................................... 4-2
Example 4.1 Exhaust Thermoelectric Generators .......................................................... 4-15
4.3 Optimal Design of Thermoelectric Coolers ........................................................... 4-19
Example 4.2 Automotive Thermoelectric Air Conditioner ............................................ 4-34
Problems ......................................................................................................................... 4-40
References ....................................................................................................................... 4-43
The optimum design of thermoelectric devices (thermoelectric generator and cooler) in
conjunction with heat sinks is developed using dimensional analysis. New dimensionless groups
are defined to represent important parameters of the thermoelectric devices. In particular, use of
the convection conductance of a fluid in the denominators of the dimensionless parameters is
important, which leads to a new optimum design. This allows us to determine either the optimal
number of thermocouples or the optimal thermal conductance. 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 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.
4-2
4.1 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 [4-6], 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 constant junction
temperatures unless the thermal resistances of the heat sinks are zero, which is, of course,
impossible. This is the rationale that a new design is developed based on Lee (2013) [10].
4.2 Optimal Design for Thermoelectric Generators
Let us consider a simplified steady-state heat transfer on a thermoelectric generator module
(TEG) with two heat sinks as shown in Figure 4.1 (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, each of which consists of p-type and n-type thermoelements with the same
dimensions as shown in Figure 4.1 (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.
4-3
(a) (b)
Figure 4.1 (a) Thermoelectric generator module (TEG) with two heat sinks and (b)
thermocouple.
The basic equations for the TEG with two heat sinks are formulated as
111111 TTAhQ (4.1)
21
2
112
1TT
L
AkRIITnQ
(4.2)
21
2
222
1TT
L
AkRIITnQ
(4.3)
222222 TTAhQ (4.4)
4-4
RR
TTI
L
21
(4.5)
where np , np kkk , and np . Equations (4.1) – (4.5) 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. The dimensionless thermal conductance, the
ratio of thermal conductance to the convection conductance in fluid 2, is
222 Ah
LAknN k
(4.6)
The dimensionless convection, the ratio of convection conductance in fluid 1 to fluid 2, is
222
111
Ah
AhN h
(4.7)
The dimensionless electrical resistance, the ratio of the load resistance to the electrical resistance
of thermocouple, is
R
RR L
r (4.8)
The dimensionless temperatures are defined by
2
11
T
TT
(4.9)
2
22
T
TT
(4.10)
2
1
T
TT
(4.11)
4-5
The two dimensionless rates of heat transfer and the dimensionless power output are defined by
2222
11
TAh
(4.12)
2222
22
TAh
(4.13)
2222
TAh
WW n
n
(4.14)
It is noted that the above dimensionless parameters are based on the convection conductance in
fluid 2, which means that 2222 TAh must be initially provided. Using the dimensionless
parameters defined in Equations (4.6) – (4.11), Equations (4.1) – (4.5) reduce to two formulas as:
212
2
21212121
121TT
R
TTZT
R
TTTZT
N
TTN
rrk
h
(4.15)
212
2
21222122
121
1TT
R
TTZT
R
TTTZT
N
T
rrk
(4.16)
where Z is the figure of merit ( kZ 2 ). Equations (4.15) and (4.16) can be solved for
1T and
2T . The dimensionless temperatures are then a function of five independent dimensionless
parameters as
21 ,,,,
ZTTRNNfT rhk (4.17)
4-6
22 ,,,,
ZTTRNNfT rhk (4.18)
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:
11 TTNQ h (4.19)
122 TQ (4.20)
Then, we have the dimensionless power output as
21 QQWn (4.21)
Accordingly, the thermal efficiency is obtained by
1Q
Wnth
(4.22)
Defining AkILN I , the dimensionless current is obtained by
1
212
r
IR
TTZTN
(4.23)
Also, defining 2 TnVNV , the dimensionless voltage is obtained by
4-7
kI
nV
NN
WN
(4.24)
With the inputs (
T and 2ZT ), we begin developing the optimization with the dimensionless
parameters (Nk, Nh, and Rr) iteratively until they converge. It was learned from some initial
attempts that both Nk and Rr have optimal values for the dimensionless power output, while Nh
does not have an optimal since 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 a 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
and later the program for multiple computations. The dimensionless power output
nW and thermal
efficiency th are together plotted as a function of rR , and presented in Figure 4.2 (a). Both
nW
and th with respect to rR indeed have 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 1rR from Figure 4.2 (a) as usually
assumed for a TEG without heat sinks, but approximately at 7.1rR , because the dimensionless
temperatures
1T and
2T in Figure 4.2 (b) are no longer constant. This is often a confusing factor
in optimum design with a TEG of two heat sinks. We should not assume that rR is equal to one
for a TEG with heat sinks.
4-8
(a)
(b)
Figure 4.2 (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 .
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
0.15Wn*
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
4-9
With the dimensionless parameters obtained ( 1hN , 7.1rR , 6.2
T , and 0.12 ZT ) in
Figure 4.3 (a), we now plot the dimensionless power output
nW and the thermal efficiency th as
functions of the dimensionless thermal conductance kN defined in Equation (4.6). We find an
optimum
nW approximately at 3.0kN . Actually, the optimal values of kN and rR should be
iterated until the two converge. From 222 AhLAknN k as shown in Equation (4.6), the kN
actually determines the number of thermocouples n if the geometric ratio A/L and 222 Ah are
given or vice versa. The dimensionless power output
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 actually 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 dimensionless high and low junction temperatures are
presented in Figure 4.3 (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.
4-10
(a)
(b)
Figure 4.3 (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 .
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
4-11
Since there is an optimal combination of Nk and Rr for a given hN , we can plot the optimal
dimensionless power output
optW and optimal thermal efficiency opt as a function of
dimensionless convection hN , which is shown in Figure 4.4. It is very interesting to note that,
with increasing hN , opt barely changes, while
optW monotonically increases. According to
Equation (4.14), 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 . As a result of this, it is necessary to
examine the variety of the 222 Ah as a function of hN for the optimal power output. Figure 4.5
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. 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.
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
opt
W*opt opt
W*opt
Nh
4-12
Figure 4.4 Optimal dimensionless power output
optW and efficiency opt versus dimensionless
convection hN . This plot was generated using 6.2
T and 0.12 ZT .
Figure 4.5 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 .
For example, using Figure 4.4 and Figure 4.5, 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 30 , and 2
1 1000cmA and also an available maximum convection conductance in
fluid 2 (air) with 8.02 , KmWh 2
2 30 , 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. Each
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)
4-13
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 , 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 on hN . This
optimal design is indicated approximately at Point 1 in above. Note that there are ways to
improve the optimal power output, increasing either 222 Ah or hN or both, which obviously
depend on the feasible mechanical constraints.
The inputs and optimum results at Point 1 in above are summarized in Table 4.1. 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].
Table 4.1 Inputs and Results from the Dimensional Analysis for a TEG
Inputs Dimensionle
ss (
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 cm2 Rr = 1.7 RL = 1.7×n×R = 4.32
4-14
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 30 , 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 30 , and 2
1 1000cmA , but an available convection
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 4.5. The effect of hN on the high and cold
junction temperatures is also presented in Figure 4.6. 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 4.5, a significant temperature variation between 1hN and 1.0hN appears in Figure
4.6. 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
4-15
showing the variety of the mechanical constraints ( 111 Ah and 222 Ah ) even with the same
power outputs .
Figure 4.6 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 .
Example 4.1 Exhaust Thermoelectric Generators
A thermoelectric generator (TEG) module is designed using a newly developed material which is
so called TAGS-75 (AgSbTe/GeTe 75%) for exhaust waste heat recovery in a car. The whole
TEG device is shown in Figure E4.1-1a. The device has N modules, each of which consists of n
p- and n-type thermocouples. The material properties for thermoelements are estimated 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 thermoelements 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 consideration of 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
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
4-16
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 a 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 E4.1-1 (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 cm2 and 2 = 0.9, h1 = 1000 W/m2K, A1 = 50 cm2.
KWmKmWAh 8.4101000608.0 242
111
KWmKmWAh 5.4105010009.0 242
222
The cross-sectional area and pellet length of the thermoelement are
A = 2.4 × 10-6m2 and L = 1 × 10-3m
4-17
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 B.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 (4.6), the number of thermocouples is
95.219
104.26.2
1015.4305.0
26
3
222
mmKW
mKW
kA
LAhNn k
Using Equation (4.14), 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
KKTTT 58.415312332.1222
4-18
Using the dimensionless current in Equation (4.23), 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 (4.24), 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
4-19
Figure E4.1-2 Power output and efficiency versus current.
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.
4.3 Optimal Design of Thermoelectric Coolers
Let us consider a simplified steady-state heat transfer on a thermoelectric cooler module (TEC)
with two heat sinks as shown in Figure 4.7. 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
0 2 4 6 8 10 120
10
20
30
40
50
60
0
0.02
0.04
0.06
0.08
0.1
0.12Wn
th
Wn(W)th
Current
(A)
4-20
thermocouples, of which each thermocouple consists of p-type and n-type thermoelements with
the same dimensions.
Figure 4.7 Thermoelectric cooler module (TEC).
The basic equations for the TEC with two heat sinks are given by
111111 TTAhQ (4.25)
21
2
112
1TT
L
AkRIITnQ
(4.26)
21
2
222
1TT
L
AkRIITnQ
(4.27)
222222 TTAhQ (4.28)
4-21
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
222 Ah
LAknN k
(4.29)
The dimensionless convection, which is the ratio of convection conductance in fluid 1 to fluid 2,
is
222
111
Ah
AhN h
(4.30)
The dimensionless current is given by
LAk
IN I
(4.31)
The dimensionless temperatures are defined by
2
11
T
TT
(4.32)
2
22
T
TT
(4.33)
2
1
T
TT
(4.34)
The dimensionless cooling power, rate of heat liberated and electrical power input are defined by
4-22
2222
11
TAh
(4.35)
2222
22
TAh
(4.36)
2222
TAh
WW n
n
(4.37)
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 (4.29) – (4.34), Equations (4.25) – (4.28) reduce to two
formulas as:
21
2
2
11
2TT
ZT
NTN
N
TTN II
k
h (4.38)
21
2
2
22
2
1TT
ZT
NTN
N
T II
k
(4.39)
Equations (4.38) and (4.39) can be solved for
1T and
2T . The dimensionless temperatures are
then a function of five independent dimensionless parameters as
21 ,,,,
ZTTNNNfT Ihk (4.40)
22 ,,,,
ZTTNNNfT Ihk (4.41)
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
4-23
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 (4.42)
122 TQ (4.43)
1Q is called the dimensionless cooling power. Then, we have the dimensionless power input as
12 QQWn (4.44)
Accordingly, the coefficient of performance is obtained by
nW
QCOP 1
(4.45)
Defining 2 TnVNV , the dimensionless voltage is obtained by
kI
nV
NN
WN
(4.46)
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 have
optimal values for the dimensionless cooling power
1Q , while hN does not since 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. However, the dimensionless
convection hN actually has feasible mechanical constraints. Thus, we proceed with a typical
4-24
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 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 Figure 4.8 and Figure 4.9. 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 very small cooling power or sometimes even none, which is
impractical. Therefore, it is necessary to have a practical value 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.
4-25
Figure 4.8 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 .
Figure 4.9 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 an optimum kN in TECs has not been found in
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
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
4-26
the literature to the author’s knowledge. With Equation (4.29), 222 AhLAknN k , the
optimum of Nk = 0.3 implies that the module thermal conductance LAkn is at optimum since
the 222 Ah is given. This 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 the design of a
thermoelectric cooler module. It is good to know from Figure 4.10 that the dimensionless
temperature
1T becomes lowest at the optimal dimensionless cooling power
1Q , not at the
optimum COP.
Figure 4.10 Dimensionless temperatures versus dimensionless thermal conductance. This plot
was generated with 1hN , 5.0IN , 967.0
T and 0.12 ZT .
We also consider two heat sinks as a unit without a TEC to examine the limitation of the use of
the TEC (see Figure 4.11). The geometry of the unit is the same as the one shown in Figure 4.7
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.
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
4-27
Figure 4.11 Heat sinks without a TEC.
The basic equations for the unit can be expressed as
011110 TTAhQ (4.47)
202220 TTAhQ (4.48)
The dimensionless groups for the unit are
2
00
T
TT
(4.49)
2222
00
TAh
(4.50)
Using Equations (4.30), (4.34), (4.49) and (4.50), Equations (4.47) and (4.48) reduce to
4-28
1
10
h
h
N
TNT
(4.51)
The dimensionless cooling rate without heat sinks can be obtained by
100 TQ (4.52)
Figure 4.12 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 .
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
COP
Q1*
&
Q0*
COP
Q1*
Q0*
T
4-29
Figure 4.13 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 4.12. The crossing point in the
figure is
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 crossing point, there is no justification for use of the TEC
although the TEC still functions. This crossing point defines the maximum 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 the dimensionless convection hN ,
which is shown in Figure 4.13. It is seen that both
optQ ,1 and COPopt increase monotonically with
increasing hN . According to Equation (4.35), the actual optimal cooling power optQ ,1 is the
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 COPCOPopt
Nh
4-30
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 necessary to examine the variety of the 222 Ah as a function of hN for the optimal
cooling power. Figure 4.14 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 4.14 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 Figure 4.13and Figure 4.14, 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
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)
4-31
and also an available maximum convection conductance in fluid 2 (ambient 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 fin efficiencies, and h1 and h2 are 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 of 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 4.14. Note that there are
several ways to improve the optimal power output by increasing either 222 Ah or hN or both.
But this depends on the feasible mechanical constraints, whichever is available.
4-32
Figure 4.15 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 4.14 are summarized in the first two columns
of Table 4.2. 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
4.15. This causes from the fact that the hot and cold junction temperatures must decrease when
more heat is extracted from the limited input, which is a 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.
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
4-33
Table 4.2 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-optimised 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
Figure 4.2, 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.
4-34
The present work presents the optimal design of thermoelectric devices in conjunction with heat
sinks introducing new dimensionless parameters. The present optimum design includes the
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 the 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 feasible mechanical constraints.
Example 4.2 Automotive Thermoelectric Air Conditioner
Current automotive air conditioners have used R-134a for about two decades as a working fluid.
It 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
4-35
designed for the replacement as a green energy application (Figure E4.2a). 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 (Figure E4.2b). A newly developed nanostructured material, Bi2Te3, has the properties
p = −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 of 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 needed to meet a 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)
4-36
Figure E4.2. (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
1 = 0.8, h1 = 40 W/m2K, A1 = 1200 cm2 and 2 = 0.8, h1 = 800 W/m2K, A1 = 60 cm2.
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 × 10-6m2 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 (4.34) is
961.0307
295
2
1
K
K
T
TT
The dimensionless convection conductance from Equation (4.30) is
0.184.3
84.3
222
111 KW
KW
Ah
AhNh
(a) For the optimum cooling power:
Using Table B.2 for 12 ZT , we can approximately obtain the optimal parameters with both
96.0
T and 0.1hN as
4-37
NI = 0.510, Nk = 0.286, COP = 0.335, 036.01 Q , 924.01 T , 141.12 T , and NV = 0.727.
Using Equation (4.29), the number of thermocouples is
8.228105.24.2
1025.184.3286.0
26
3
222
mmKW
mKW
kA
LAhNn k
Using Equation (4.35), the cooling power is
WKKWTAhQQ 46.4230784.3036.0222211
The COP is already obtained from Table 2 as
COP = 0.335
The power input from Equations (4.35), (4.36), and (4.45) is
WW
COP
QWn 748.126
335.0
46.421
The cold and hot junction temperatures from Equations (4.32) and (4.33) are
KKTTT 80.283307924.0211
KKTTT 76.350307142.1222
Using Equation (4.31), 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 in Equation (4.46), 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:
4-38
Using Table B.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 (4.29), the number of thermocouples is
4.114105.24.2
1025.184.3143.0
26
3
222
mmKW
mKW
kA
LAhNn k
Using Equation (4.35), 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 (4.35), (4.36), and (4.45) is
WW
COP
QWn 48.14
385.1
05.201
The cold and hot junction temperatures from Equations (4.32) and (4.33) are
KKTTT 64.289307943.0211
KKTTT 364.316307030.1222
Using Equation (4.31), 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 in Equation (4.46), 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 E4.2 Summary of the Optimal cooling power and ½ optimal COP.
4-39
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 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
4-40
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 reality, 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.
Problems
4.1. 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 (Figure P4.1). A
4-41
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 consideration of 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 P4.1 (a) the arrangement of the whole TEG device, and (b) the TEG module.
4.2. An automotive thermoelectric air conditioner is designed, which consists of two air heat
sinks and a coolant heat exchanger as shown in Figure P4.2a. We design a unit module
of 4-cm × 4-cm base area, which represents a mean value of the whole system as a
4-42
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 (Figure P4-4b). 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)
4-43
Figure P4.2. (a) Thermoelectric air conditioner, and (b) TEC module.
References
1. Kraemer, D., et al., High-performance flat-panel solar thermoelectric generators with high
thermal concentration. Nat Mater, 2011. 10(7): p. 532-8.
2. Karri, M.A., E.F. Thacher, and B.T. Helenbrook, Exhaust energy conversion by
thermoelectric generator: Two case studies. Energy Conversion and Management, 2011.
52(3): p. 1596-1611.
3. Hsu, C.-T., et al., Experiments and simulations on low-temperature waste heat harvesting
system by thermoelectric power generators. Applied Energy, 2011. 88(4): p. 1291-1297.
4. Henderson, J. Analysis of a heat exchanger-thermoelectric generator system. in 14th
Intersociety Energy Conversion Engineering Conference. 1979. Boston, Massachusettes.
5. Stevens, J.W., Optimal design of small DT thermoelectric generation systems. Energy
Conversion and Management, 2001. 42: p. 709-720.
6. Crane, D.T. and G.S. Jackson, Optimization of cross flow heat exchangers for
thermoelectric waste heat recovery. Energy Conversion and Management, 2004. 45(9-10):
p. 1565-1582.
7. Rowe, D.M., Thermoelectrics handbook; macro to nano. 2006, Roca Raton: CRC Taylor
& Francis,.
8. Chein, R. and G. Huang, Thermoelectric cooler application in electronic cooling. Applied
Thermal Engineering, 2004. 24(14-15): p. 2207-2217.
9. Vining, C.B., An inconvenient truth about thermoelectrics. Nature Materials, 2009. 8: p.
83-5.
10. Lee, H., Optimal design of thermoelectric devices with dimensional analysis. Applied
Energy, 2013. 106: p. 79-88.