Journal of ELECTRONIC MATERIALS, Vol. 42, No. 7, 2013
DOI: 10.1007/s11664-013-2488-0
Ó 2013 TMS
Two-Dimensional Thermal Resistance Analysis of a Waste Heat
Recovery System with Thermoelectric Generators
GIA-YEH HUANG1 and DA-JENG YAO1,2,3
1.—Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu
30013, Taiwan, ROC. 2.—Institute of NanoEngineering and MicroSystems, National Tsing Hua
University, Hsinchu 30013, Taiwan, ROC. 3.—e-mail: djyao@mx.nthu.edu.tw
In this study, it is shown that two-dimensional (2D) thermal resistance
analysis is a rapid and simple method to predict the power generated from a
waste heat recovery system with thermoelectric generators (TEGs). Performance prediction is an important part of system design, generally being
simulated by numerical methods with high accuracy but long computational
duration. Use of the presented analysis saves much time relative to such
numerical methods. The simple 2D model of the waste heat recovery system
comprises three parts: a recovery chamber, the TEGs, and a cooling system. A
fin-structured duct serves as a heat recovery chamber, to which were attached
the hot sides of two TEGs; the cold sides were attached to a cooling system.
The TEG module and duct had the same width. In the 2D analysis, unknown
temperatures are located at the centroid of each cell into which the system is
divided. The relations among the unknown temperatures of the cells are based
on the principle of energy conservation and the definition of thermal resistance. The temperatures of the waste hot gas at the inlet and of the ambient
fluid are known. With these boundary conditions, the unknown temperatures
in the system become solvable, and the power generated by the TEGs can be
predicted. Meanwhile, a three-dimensional (3D) model of the system was
simulated in FloTHERM 9.2. The 3D numerical solution matched the solution
of the 2D analysis within 10%.
Key words: Thermal resistance, waste heat recovery system, thermoelectric
generator, modeling
INTRODUCTION
Thermoelectric generators (TEGs) have been
studied for more than a century. In many applications, the TEG converts heat into electricity.1–5 In
the reported work on models of waste heat recovery
systems, harvesting of energy from exhaust heat is
emphasized. Computational methods and models of
thermal resistance assist in analysis and improve
performance.6–9
In the present work, a model of a waste heat
recovery system was developed using two-dimensional thermal resistance analysis. This method
helps to solve the temperature gradient of the TEG
(Received July 19, 2012; accepted January 10, 2013;
published online February 20, 2013)
1982
modules. The solutions enable performance estimates of waste heat recovery systems. The same
systems were modeled using FloTHERM 9.2 software. The solutions of the thermal resistance model
are compared with the results of simulations.
TWO-DIMENSIONAL THERMAL
RESISTANCE
The concept of 2D thermal resistance is based on
one-dimensional (1D) thermal resistance. The
thermal resistance circuit is an electrical analogy
with heat transfer. The heat rate (Q) is analogous to
the current flow in an electric circuit. The analog of
the temperature difference (DT) is the voltage difference. In 1D thermal resistance analysis, the
thermal resistance is defined as
Two-Dimensional Thermal Resistance Analysis of a Waste Heat Recovery System with Thermoelectric
Generators
R¼
DT
:
Q
1983
(1)
For heat transfer by conduction, the thermal
resistance in one dimension is defined as
Rcond ¼
L
;
kA
(2)
where L denotes the length of material, A the contact area, and k the thermal conductivity. For convective heat transfer, the thermal resistance is
defined as
Rconv ¼
1
;
hA
(3)
where h denotes the convective heat transfer coefficient. Usually, thermoelectric effects are considered for thermoelectric devices. The thermal energy
flowing into a thermoelectric device is not equal to
the thermal energy flowing out of the other side, as
part of the thermal energy is converted into electric
power. Based on theoretical calculations for the
operating conditions and TEG module properties
considered in this study, the best efficiency of the
TEGs in the system with and without consideration
of thermoelectric effects is 5.21% and 5.64%. Due to
the small size of this difference, the thermoelectric
effects are neglected and the heat energy is assumed
to be conserved for the TEGs. Therefore, the principle of energy conservation is also applied in the 2D
thermal resistance analysis. In the 2D analysis, the
computational domain is divided into several cells;
the temperatures nodes are located at the centroid
of each cell, as shown in Fig. 1. For each cell, the
heat flowing in equals the heat flowing out. Moreover, the relation between two nodes is developed
based on the definition of thermal resistance as
represented in Eq. 4.
Ti Te Ti Tw Ti Tn Ti Ts
þ
þ
þ
¼ 0:
Re
Rw
Rn
Rs
(4)
The method for solving the estimated performance of a simple waste heat recovery system is as
follows. Figure 2 shows that the system comprises
three parts: a heat recovery chamber, the TEGs,
and a cooling system. To the fin-structured duct
that serves as a heat recovery chamber is attached
the hot sides of two TEGs. The properties of the
commercial TEG module (TMH400302055; Wise
Life Technology, Taiwan) considered in this work
are listed in Table I. The cold sides of the TEGs are
attached to a cooling system; in this case, heat sinks
serve as the cooling system. The TEG module and
the duct have the same width.
For the 2D analysis to solve the problem, nodes
with unknown temperatures are located at the
centroid of each cell into which the waste heat
Fig. 1. Relation developed among one node (Ti) and four other
nodes (Te, Tw, Tn, Ts) with a definition for thermal resistance.
recovery system has been divided. The relation
between one node and another is developed based on
the definition of thermal resistance. In Fig. 3, the
black nodes are the boundary conditions. The relations among wall nodes (square) are expressed as
Eq. 4, but the relations among the end nodes (triangle) must be corrected for their positions. When
the end nodes lack eastern nodes, Eq. 4 is rewritten
as Eq. 5. When the end nodes lack western nodes,
Eq. 4 is rewritten as Eq. 6.
Ti Tw Ti Tn Ti Ts
þ
þ
¼ 0;
Rw
Rn
Rs
(5)
Ti Te Ti Tn Ti Ts
þ
þ
¼ 0:
Re
Rn
Rs
(6)
For the interior chamber nodes (diamond), the
relation is expressed as Eq. 7.
mcp Ti ¼ mcp Ti1 q_ v ;
(7)
where q_ v is the heat flowing in the vertical direction.
The relations involve the matrix Eq. 8. Solving
this matrix, the unknown temperatures in the system are revealed. The estimated performance of the
waste heat recovery system is then predictable by
Eq. 9.
2
32
3 2
3
a11 a125
C1
T1
6 ..
.. 76 .. 7 6 .. 7
..
(8)
4 .
. 54 . 5 ¼ 4 . 5
.
T25
C25
a251 a2525
1984
Huang and Yao
Fig. 2. (a) Prototypical simple waste heat recovery system: (1) first TEG, (2) first heat sink, (3) second TEG, (4) second heat sink, (5) waste heat
recovery chamber; (b) front view of the system; (c) schematic diagram of the system.
Table I. Properties of a TE couple
Seebeck coefficient (V K1)
Resistivity (X m)
Thermal conductivity (W m1 K1)
Z (K1)
Thermal resistivity (K W1)
Contact area (m2)
Thickness (m)
P ¼ I 2 RL ¼
aDT
R
n-Type
p-Type
Copper
Ceramic
Solder
2.12 9 104
1.04 9 105
1.456
2.97 9 103
109.89
4 9 106
6.4 9 104
2.15 9 104
1.04 9 105
1.373
3.23 9 103
116.79
4 9 106
6.4 9 104
N/A
3.2 9 108
385
N/A
0.1443
9 9 106
5 9 104
N/A
1 9 1012
22
N/A
3.207
9 9 106
6.35 9 104
N/A
12.1 9 108
50
N/A
0.25
4 9 106
5 9 105
2
RL ;
(9)
where a is the Seebeck coefficient, DT is the temperature difference, R is the total electric resistance
in the circuit, and RL is the external load resistance.
ESTIMATION OF THE CONVECTIVE HEAT
TRANSFER COEFFICIENT
In the waste heat recovery system, convective
heat transfer plays an important role because heat
is transfered from the waste hot gas to the wall of
the recovery chamber by convection. Outside the
recovery chamber, the cooling mechanism is also
strongly related to convective heat transfer. With
these coefficients, the thermal resistances can be
defined and applied in the 2D analysis. In this
study, the convective heat transfer coefficient
applied in the 2D analysis is estimated as follows.
According to Benjan,10 the convective heat transfer
coefficient is estimated for external flow. When a
fluid flows over heat sinks, the fin efficiency is considered in the calculation of the thermal resistance.
For internal duct flow, there are two conditions; the
first is that fluid flows in a duct as shown in Fig. 4a,
for which the convective heat transfer coefficient is
estimated from Kurtbaş.11 The second is that the
fluid flows in a fin-structured duct as shown in
Fig. 4b. In this case, the convection heat transfer
coefficient is estimated from the method of Knight
et al.12
RESULTS
The performance of the simple waste heat recovery system of Fig. 2 was estimated using the
Two-Dimensional Thermal Resistance Analysis of a Waste Heat Recovery System with Thermoelectric
Generators
described 2D thermal resistance analysis. Also, the
same system was modeled using commercial software (FloTHERM 9.2). As we neglect thermoelectric
effects, FloTHERM is sufficient to model the system,
without thermoelectric effects. The dimensions and
material properties used in the simulation are listed
in Table II. Two factors are considered in this
model: the velocities of the internal hot gas (Vi) and
1985
of the external cold air (Ve). The operational and
boundary conditions are listed in Table III.
Figures 5 and 6 show the temperatures of the hot
side (Th) and the cold side (Tc) of the TEGs. On
average, the results from the 2D analysis and simulations differ by about 10%.
In this study, the performance of waste heat
recovery systems of three other types, as shown in
Fig. 4a–c, was also estimated by using the 2D
thermal resistance analysis and modeling of the
same systems using FloTHERM 9.2. The simple
waste heat recovery system is shown in Fig. 4d. For
the other three systems, the results from the 2D
analysis and the simulations differed by 10% on
average.
DISCUSSION
Influence of Flow Velocity
Fig. 3. The thermal resistance model in the simple waste heat
recovery system.
To develop the relationship between the performance and the flow velocity, we considered 12 conditions for the flow velocity. Applying the 2D
analysis to develop this relation requires only a few
minutes. The power generated by the system
increases with increasing velocity. The external flow
velocity of the cold gas affects the power generated
more than the internal velocity. When the external
flow velocity of cold gas increases from 6 m/s to
12 m/s, the power generated increases by 1.5 times,
whereas when the internal flow velocity of hot gas
increases from 12 m/s to 24 m/s, the power generated increases by 1.3 times, as shown in Fig. 7.
Fig. 4. Another prototypical waste heat recovery system: (a) type 1: a nonstructured duct works as a chamber for heat recovery, (b) type 2: a finstructured duct works as a chamber for heat recovery, (3) type 3: a nonstructured duct works as a chamber for heat recovery and TEGs are
attached to heat sinks, (d) type 4: a fin-structured duct works as a heat recovery chamber and TEGs are attached to heat sinks.
1986
Huang and Yao
Table II. Dimensions and material properties used in simulations (FloTHERM 9.2)
(a) Waste heat recovery chamber
Dimensions (mm3)
Number of internal fins
Fin thickness (mm)
Fin base thickness (mm)
Material
(b) Heat sink
Dimensions (mm3)
Number of internal fins
Fin thickness (mm)
Fin base thickness (mm)
Material
(c) TEG module
Dimensions (mm3)
Equivalent conductivity (W m1 K1)
270 (L) 9 54 (W) 9 56 (H)
6
2.5
5
Aluminum
52 (L) 9 54 (W) 9 40 (H)
12
2
6
Aluminum
52 (L) 9 54 (W) 9 2.8 (H)
3.2563
Table III. Operational and boundary conditions
Factor
Condition
1
Internal hot gas velocity (m s )
External cold air velocity (m s1)
Ambient temperature (°C)
Inlet temperature of hot gas (°C)
1
12
3 6
24
8 10 15
27
200
Influence of the Fin Structure
The performance of four system designs was
estimated quickly using the 2D analysis. Based on
the calculation results, both heat sinks and finstructured ducts enhance the heat transfer. Relative to nonstructured ducts (type 1), the power
generated with fin-structured ducts (type 2) is 1.1 to
2.3 times. When heat sinks are attached to the cold
sides of the TEGs (type 3), the power generated
increases about 25 times. With heat sinks and
fin-structured ducts (type 4), the power generated
increases over 100 times, as shown in Fig. 7; this
system design improves the performance most.
Influence of Thermal Resistance
Thermal resistance is an important factor affecting the power generated. The 2D analysis reveals
the relations among the thermal resistance and
other factors, including fin- or nonstructured ducts,
with or without attached heat sinks, as well as the
internal and external flow velocities. Figure 8
shows that the thermal resistance of fin-structured
ducts and heat sinks is smaller than for nonstructured ducts and no heat sink attached. Increasing
the flow velocities of the internal hot gas and the
external cold gas also might decrease the thermal
resistance, but the varied duct designs and heat
sink attachment are more effective. Attachment of
heat sinks effectively decreases the thermal resistance the most.
Fig. 5. Temperature of TEG modules with inlet velocity of 12 m/s:
(a) first TEG, (b) second TEG.
CONCLUSIONS
Two-dimensional thermal resistance analysis is a
rapid and simple method to predict the temperature
difference of TEGs, so that the performance of waste
heat recovery systems can be estimated. The results
of the 2D analysis are consistent with simulation
Two-Dimensional Thermal Resistance Analysis of a Waste Heat Recovery System with Thermoelectric
Generators
1987
Fig. 8. Relation among thermal resistance, velocity conditions, and
system designs.
results; the average difference between the results
from the 2D method and from simulation is about
10%. This method is efficient to estimate the power
generated from the system.
The numerical method provides a more accurate
prediction but requires much time; the 2D analysis
saves time. With varied parameters, the relations
between various factors and the performance are
readily revealed. The method helps to determine
quickly the most effective factor affecting the
performance.
Fig. 6. Temperature of TEG modules with inlet velocity of 24 m/s:
(a) first TEG, (b) second TEG.
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Fig. 7. Relation among power generated, velocity conditions, and
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