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Performance improvement of variable-angle annular thermoelectric
generators considering different boundary conditions
Zebin Weng1, Furong Liu1, Wenchao Zhu1,2, Yang Li1,3, Changjun Xie1,2*, Jian
Deng1, Liang Huang1*
1 School of Automation, Wuhan University of Technology, Wuhan 430070, China.
2 Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of
Technology, Wuhan 430070, China.
3 Department of Electrical Engineering, Chalmers University of Technology, Gothenborg 41576,
Sweden.
* Corresponding author
E-mail address: jackxie@whut.edu.cn (C. Xie); huangliang@whut.edu.cn (L. Huang)
Abstract
Practical applications of thermoelectric generators are impeded by their low thermoelectric
conversion efficiency, and improving the efficiency is vital for the advancements of thermoelectric
technology. In this paper, a novel method is proposed for the performance analysis and improvement of
the annular thermoelectric generators with variable-angle PN legs (VATEGs). The influence of the PN
leg angle on the output performance of the VATEG is investigated by introducing an angle function.
Given the volume of the PN legs, the relationship of output performance between the VATEG and
traditional constant-angle ATEG (CATEG) is established under different boundary conditions based on
a proposed generic model of VATEG. The results are verified numerically using the finite element method.
Using the model, it is shown that the output performance of the VATEG is significantly affected by the
shape of the PN leg. Finally, the thermal stress on the PN leg is next investigated using a high-fidelity
3D model of the variable-angle PN legs implemented in COMSOL, and it is found that the shape
difference has a considerable influence on the thermal stability of VATEG. Under the condition of
constant heat flux on the hot side and constant temperature on the cold side of the thermoelectric modules,
it shows that when the radius factor is 2, the output performance can be improved by 35% with the
designed VATEG, at the expense of 30% higher maximum thermal stress on the PN legs.
Keywords: Variable-angle annular thermoelectric generator (VATEG); Finite element method; Shape
factor; Thermal stress; Energy efficiency; Heat recovery.
1. Introduction
Global energy crisis and environmental issues demand more sustainable and clean energy with
improved energy conversion efficiency [1]. Amongst various green energy harvest candidates, a
thermoelectric generator (TEG) couples thermal field and electric field to generate electrical power from
heat sources according to the Seebeck effect [2], and it is a promising technology due to its low-cost
production, environment-friendly operation, and free of moving parts [3]. TEGs can be potentially used
in various applications. For example, the thermal energy in the waste heat of automobile exhaust gas can
be recovered using TEGs [4]. The heat in solar photovoltaic panels can be utilized for TEGs to generate
additional electricity to supply remote villages [5]. In biomedicine, TEGs can be used for micro-heat
energy recovery by converting human waste heat to electricity for establishing a self-powered human
health detection system [6]. In artificial intelligence and aviation, waste heat can be recovered by
developing isotope radioactive TEGs [7] or micro thermoelectric components [8].
Unfortunately, practical applications of TEGs are impeded by the low thermoelectric conversion
efficiency, and improving the efficiency is vital for the advancements of thermoelectric technology. The
factors that influence the thermoelectric efficiency and the output power of TEGs can be generally
divided into internal and external factors. Internal factors are associated with the thermoelectric materials
used in TEG devices. To assess how the performance of TEG is affected by the internal factors, the figure
of merits ZT, a dimensionless parameter, is commonly used in the literature, and a larger ZT indicates
superior performance. Defined by ZT = α2σT/λ, the figure of merits is a function of the Seebeck
coefficient α, electrical conductivity σ, operating temperature T, and the thermal conductivity λ [9]. Since
these parameters are often intercorrelated, it is challenging to obtain optimal performance by adjusting
them independently. In the literature, the ZT of common thermoelectric materials ranges from 1 to 1.8
[10], and some studies show that the ZT can achieve over 2 [11]. The ZT of thermoelectric materials with
nanostructures can be increased to 1.7 [12]. With well-developed material processing technology, a ZT
of 1.8 can be achieved [13]. If the ZT can be increased further to 3, the thermoelectric conversion
efficiency can exceed 30%, which is promising for practical applications [14]. Thermoelectric materials
with ZT over 4 can receive substantial commercial interest in the future. Hence, the exploration of high
ZT is of great significance [15]. Some materials, such as ZnO [16] and graphene [17], have relatively
high electrical conductivity without processing. Other materials demand particular processing procedures
to increase the electrical conductivity or reduce the thermal conductivity. For example, Rogl et al. [18]
refined the skutterudites from commercial powders by cold pressing and high-pressure torsion at 850 K.
With this method, the thermal conductivity of skutterudites can be significantly reduced at the expense
of slightly reduced electrical conductivity, leading to a high figure of merits ZT = 2.1. Doping other
ingredients enables further improvement on ZT. For instance, Zhao et al. [19] concluded that ZT = 1.34
could be obtained by doping sodium in tin selenide samples. In addition, searching for thermoelectric
materials with high ZT values, such as Zintl phases [20] and half-Heusler alloys [21], is also an effective
way to enhance thermoelectric efficiency.
On the other hand, the operating temperature of PN legs, shape matching between the heat source
and the heat exchanger, and the geometric parameters belong to external factors. Some studies have
attempted to match the optimal operating temperatures with different materials via segmented [22] or
staged TEGs [23]. For instance, Chen et al. [24] designed a segmented TEG system. Compared to
conventional TEGs, the output power and efficiency of the TEG system can be improved by 21.94% and
14.05%, respectively. In a different study, Karana et al. [25] showed that the overall efficiency of a
segmented TEG can be increased by 5%. El-Genk et al. [26] proposed a segmented TEG with
skutterudites, and its thermoelectric efficiency can reach 14.7% when the cold and hot side temperatures
are 573 K and 300 K, respectively. In addition, the nonlinear physical properties of the materials are
taken into consideration to improve the performance of the TEGs further. In fact, the assumption of
constant physical properties is only applicable to micro TEGs or wearable thermoelectric energy harvest
devices with ignorable temperature gradients. Since a large temperature gradient exists in most industrial
applications, the dependence of the material parameters on local temperature should be considered during
the design of the TEG system. The impact of nonuniform temperature distribution on the output
performance of the PN legs can be investigated using the finite element method (FEM) [27]. For example,
Meng et al. [28] used a 1D numerical model to study the multi-irreversibility effects of TEGs.
Thermoelectric materials with nonlinear parameters can also be modeled with 3D commercial software
for numerical simulation, such as FLUENT, COMSOL, and ANSYS [29].
The shape of the heat exchanger is the second external factor that needs to be properly considered.
Well-matched heat source and heat exchanger can effectively reduce the heat loss arising from the contact
impedance. For cylindrical heat sources such as exhaust pipes in automobiles, the TEG is usually
designed into annular or ring-annular to match the cylindrical heat sources [30]. Bauknecht et al. [31]
investigated the performance of an ATEG under the condition of nonuniform temperature distribution,
and it was shown that the output performance can be significantly improved. Shen et al. [32] studied
both the annular and the flat TEGs by developing a steady-state model and revealed that the governing
equations of both designs are similar except for the mathematical expressions of the resistance and
thermal conductivity. Hence, an ATEG can be converted into a flat TEG to evaluate its performance.
Moreover, the research interest in flexible TEGs is growing in recent years, and these TEGs are mainly
used in wearable devices to match the complex structure of the heat source, i.e., the human body [33].
Cui et al. [34] examined several aspects in designing a flexible wearable TEG, such as the convective
heat transfer coefficient between the thermal layer and the environment, as well as the contact thermal
resistance between the thermal layer and the skin. Their results exhibited that the degree of flexible
bending is associated with the open-circuit voltage (OCV), and the maximum power density can be
achieved by properly selecting the length of the hot spot layer. The shape of TEG can also affect the
external thermal resistance and output power of the TEG. For example, Kim et al. [35] proposed a direct
contact TEG to reduce the contact thermal resistance, resulting in a 132% increase in the output power.
Apart from the steady-state performance, there are also many studies on the transient heat transfer process
of segmented ATEGs, and the research objective is usually to test the mechanical features of the materials.
For example, Asaadi et al. [36] investigated rectangular, triangular, and sinusoidal pulsed inputs to study
the performance of an ATEG, and compared the results with the steady-state performance under the same
operating conditions. The authors concluded that all these pulse heating strategies could enhance the
efficiency of ATEG, while the rectangular pulse heating strategy is the best with which the efficiency
can be improved by 249.36%. Similarly, Fan et al. [37] analyzed the thermoelectric and mechanical
properties of an ATEG under both transient and steady-state conditions. Their results showed that the
output power can be increased by 18.3% with a segmented structure, and the maximum von Mises stress
can be reduced by 12.5%. Shittu et al. [38] studied the electrical and mechanical characteristics of
segmented and asymmetric thermoelectric legs in the steady-state and transient, and the optimized
geometric parameters of the legs were obtained. The optimized TEG output power was increased by
117.11% with rectangular pulse heating inputs, and the thermal stress of an N-type asymmetric leg is
39.21% smaller than that of a P-type symmetric leg on the cold side. Samson et al. [39] investigated the
thermoelectric and mechanical properties of two ATEGs based on bismuth telluride and skutterudite.
The authors showed that when the temperature difference is above 200 K, the efficiencies of the
segmented ATEG are 21.7% and 82.9% higher than the designs without segmentation, and increasing
the leg length can effectively reduce the thermal stress and improve the electrical performance.
When the materials and segmentation method has been determined, optimization on various
geometric parameters of the PN legs can play an important role in achieving high TEG performance. A
large body of literature exists to investigate the geometric parameter optimization of TEGs from both
technical and economic aspects [40-42]. Furthermore, different boundary conditions applied to the model
can affect optimization results. Fan et al.[43] studied the influence of the variations of geometric
parameters on output power and energy conversion efficiency under different thermal boundary
conditions. It shows that the efficiency is unchanged at the boundary condition of constant temperature
and inversely proportional to the cross-sectional area of legs under the boundary condition of constant
heat transfer coefficient. He et al. [44] proposed a 1D model of the TEG based on the hill-climbing
algorithm, considering the size of the hot leg. It was verified by the 3D numerical simulation that for any
leg length, the maximum output power always increases with the increase of leg area. Chen et al. [42]
used the genetic algorithm to optimize the cross-sectional area and the length of legs, with conversion
efficiency being the objective function. Under a fixed temperature difference of 40 K, the output power
and efficiency corresponding to the optimized cross-sectional area and length are 51.9% and 5.4% higher
than the original TEG design.
Conventionally, PN legs are designed with a fixed leg angle. Recently, there is growing interest in
designing non-conventional PN legs. Fabián-Mijangos et al. [45] designed a thermoelectric module with
a PN leg in an asymmetric truncated square pyramid shape. Compared to the traditional design, the
thermoelectric quality factor can be doubled under the same operating condition. Sahin et al. [46] showed
that legs with linearly varying cross-sectional areas in the leg length direction could reduce the output
power. In fact, the cross-sectional area can be designed to be changed linearly, exponentially [47], or
irregularly [48] in the leg length direction. Bengisu et al. [49] conducted numerical simulations with
trapezoids-shaped and hourglass-shaped PN legs and considered different boundary conditions. The
results showed that the hourglass-shaped leg has twice as much potential and maximum power as the
traditional PN leg. At the same time, it showed that the influence of boundary conditions should be
considered when choosing the best shape. Zhang et al. [50] introduced the formula of thickness variation
along the leg length direction δ(r)=amrm to discuss the influence of variable cross-sectional areas on the
output characteristics of an ATEG, and the authors concluded that the power per unit mass could be
maximized when m = −1. Liu et al. [51] obtained the relationship between the thermal resistances of
variable cross-section leg and fixed cross-section leg for flat-plate TEGs. It was found that the conversion
efficiency can be improved with variable cross-sections, and the output power depends on both the
boundary conditions and the shape factor. However, the derived expression had not been verified with
different materials and at different working temperatures. Ali et al. [52] optimized the cross-sectional
area of the exponentially changed leg by introducing a dimensionless geometric parameter. It was found
that although the output efficiency can be improved, there is a tradeoff between the efficiency and the
output power. However, this paper does not quantitatively compare the output performance of the
proposed design with the traditional TEG. Liu et al. [53] developed a solar TEG model taking into
account the subsection and asymmetry of thermoelectric legs. It concluded that with the subsection, the
output power can be increased by 14.9%, and with the optimal cross-sectional area of the PN legs, the
output power can be increased by 16.6%. The performance of this design was evaluated using a 3D multi-
physics thermoelectric model, whereas the thermal stress and thermal stability of the thermoelectric
structure was not considered for different cross-sectional areas.
Electricity
Heat
Application field
Main challenge
Low thermoelectric
efficiency
Geometric
optimization Segment
or Stage
Thermoelectric
materials with
high ZT
Heat source
of
suitable shape
Thermoelectric
module
solution
this paper
Fig.1. Overview of application fields and research directions of thermoelectric generators for
thermal energy recovery.
The abovementioned research background and research directions are summarized in Fig. 1. Since
well-selected geometric parameters of the PN legs can effectively improve the thermoelectric conversion
efficiency, the primary research objective of this paper is to improve the thermoelectric conversion
performance from the perspective of geometric parameter optimization. Although extensive efforts have
been devoted to the research on geometric parameter optimization, several aspects of practical
consideration are rarely investigated in the existing studies. First, it is necessary to pay attention to the
temperature distribution of PN legs if the cross-sectional areas are variable [54]. For a practical system,
examining whether the designed shape difference can cause local over-temperature is essential to avoid
potential damage on thermoelectric material, although the procedure is usually ignored in many existing
works. Second, we notice that there is a lack of study on the influence of the change of the cross-section
area on the material and structural stability for practical use. In addition, theoretical derivation and
analysis based on a 3D model with the leg shape change in the ATEG are both rarely investigated. In
order to fill the research gaps, this paper proposes a new ATEG design with variable-angle PN leg
(VATEG) and establishes a generic geometric ATEG model to describe the change of PN leg angle in the
leg length direction. Specifically, given the leg volume, the relationship between the physical parameters
of VATEG and the corresponding constant-angle ATEG (CATEG) is first obtained from theoretical
analysis. The angle function of PN legs is next derived with different boundary conditions to compare
the output performance, and the results are verified using a high-fidelity 3D model implemented in
COMSOL. Finally, to confirm whether the stability of PN legs material is affected by the shape difference,
the developed 3D model is utilized to study the temperature distribution of the proposed VATEG and the
thermal stress on PN legs. Note that the proposed model is generic, and the analysis can be extended to
obtain the optimized design with other boundary conditions that are not studied in the present
investigation. The contributions of the proposed method are summarized as follows:
1) A generic ATEG model is first developed by introducing a two-parameter angle function to
analyze and simulate the heat recovery process for practical application.
2) Under the assumption of the same volume of PN legs, the relationship between the resistance and
thermal conductance of the PN leg is established for the first time, and the two parameters in the
angle function were characterized with a proposed shape factor.
3) In order to guarantee the suitability of the proposed research method for other application fields,
different scenarios are simulated and verified by considering different boundary conditions on the
cold and the hot sides. The relationship of the output performance between the proposed VATEG
and the traditional ATEG is derived under different boundary conditions.
4) Thermal stress and voltage distribution under different boundary conditions and leg lengths are
first investigated using a 3D model, and the results were compared with the traditional ATEG to
study how the shape of PN leg can affect the thermal stress.
2. Model description and theoretical analysis
Fig. 2(a) shows the 3D structure of the proposed VATEG, where the PN legs angle along the leg
length direction is not designed as a constant. Figs. 1(b) and (c) are the schematic diagrams of the
proposed and the conventional designs of the PN legs, respectively. For the ease of model development
and analysis, the following assumptions are made in this work.
1) All surfaces are insulated except for the hot and cold surfaces;
2) Only the heat transfer along the leg length direction is considered;
3) The Thomson effect is not considered;
4) The electrical resistance and the contact thermal resistance of the copper sheets are ignored;
5) The heat loss due to radiation in all directions is neglected;
6) The P-leg and the N-leg possess symmetrical geometric structures;
Nomenclature
Symbols
V
volume (m3)
Greek symbols
m
exponent coefficient of angle function
Rt
internal resistance (Ω)
α
Seebeck coefficient (V/K)
K
thermal conductance (W/K)
λ
thermal conductivity (W/(m∙K))
R
electrical resistance (Ω)
σ
electrical conductivity (S/m)
rh
inner radius of thermocouple (mm)
δ
thickness of thermocouple (mm)
rc
outer radius of thermocouple (mm)
θ
angle of PN legs (rad)
T
temperature (K)
ξ
heat flux (W∙m−2)
I
current(A)
η
conversion efficiency (%)
H
length of thermocouple (mm)
ρ
electrical resistivity (m/S)
f
Shape factor
Superscript
r
radius of thermocouple (m)
S
number of finite element discretization
k
index of elementary unit (k = 1, 2,⋯, S)
n
number of thermocouples
Subscript
P
power(W)
Q
heat flow(W)
χ
P-type or N-type leg
Qin
heat absorbed by thermocouple module
(W)
Qout
heat released from thermocouple module
(W)
e
ceramic layer
q
scaling coefficient of angle function
0
parameters of CATEG
h
convective heat transfer coefficient
(W∙m−1∙K−1)
P
P-type leg
A
cross-sectional area (m2)
N
N-type leg
Te
ambient temperature (K)
L
external load
U
Seebeck voltage (V)
h
hot side of PN leg
c
cold side of PN leg
7) The leg volume and the leg length of the proposed design are both the same as those in the
conventional design.
Fig. 2. Schematic diagrams of the proposed VATEG. (a) 3D view of VATEG and a PN leg with variable leg angles.
(b) 2D view of a pair of PN legs with variable leg angles. (c) 2D view of PN legs with a fixed leg angle of a
CATEG.
2.1. Shape factor of PN legs
As shown in Fig. 2(b), the inner radius and the outer radius of the PN legs are denoted by rh and rc,
respectively, and the leg length is H = rc − rh. In contrast to using a constant leg angle θ0 in the
conventional CATEG, the leg angle of a VATEG varies with respect to the radial coordinate r, denoted
by θ(r). The expression of θ(r) can be derived by equalizing the leg volumes of CATEG and VATEG, i.e.,
22
0
0( 1)
() 2
cc
hh
rr h
rr
r Sr
r r dr r dr
−
= =
(1)
where δ is the thickness of PN legs and Sr is the radius ratio defined as
/
ch
rrSr =
(2)
For the CATEG, the resistance and the thermal conductance of PN legs are calculated by [55]
,0 00
0
,0
ln( )
ln( )
c
h
r
leg r
leg
dr Sr
Rr
KSr
==
=
(3)
where σ and λ are the electrical conductivity and the thermal conductivity of PN legs, respectively. Note
(b)
(c)
(a)
that the subscript ‘0’ indicates the quantity for the corresponding CATEG design in this work.
Similarly, for the VATEG, with (2) and (3), the resistance and the thermal conductance are
0,0
,0
0
()
( ) ln( ) ( )
1
()
() ln( ) ( )
cc
hh
cc
hh
rr
leg leg
rr
leg
leg rr
rr
dr dr
RR
r r Sr r r
K
Kdr dr
rr Sr r r
= =
==
(4)
According to (2)−(4), the following relationships can be obtained
,0leg leg
R f R=
(5a)
,0leg
leg
K
Kf
=
(5b)
0
ln( ) ( )
c
h
r
r
dr
fSr r r
=
(5c)
where f is defined as the shape factor. Hence, an algebraic relationship exists between the resistances and
thermal conductances of PN legs of the VATEG and the CATEG, i.e.,
,0 ,0leg leg leg leg
K R R K=
. It can be
seen that the CATEG can be considered a particular VATEG with θ(r) = θ0 and f = 1.
In order to investigate the characteristics of VATEG based on the shape factor, it is expected that
the integral in (5c) can be expressed analytically. Hence, in the following investigation, we limit our
discussion to a special form of θ(r) so that the performance of the designed system can be analyzed both
quantitatively and qualitatively.
2.2. Analysis of PN leg
Here, a two-parameter angle function is introduced to represent θ(r), i.e.,
() m
r q r
=
(6)
where q > 0 and m are the scaling coefficient and the exponent coefficient, respectively. A reasonable
range of m for practical design is
22m−
. Using (1), (2), and (6), the scaling coefficient q can be
expressed as a function of m Sr, the angle θ0 of the corresponding CATEG, as well as the inner radius
rh, i.e.,
22
0
2
02
( 1) ,2
2 ln( )
( 2) ( 1) ,2
2 ( 1)
h
mm
h
r Sr m
Sr
qm Sr m
r Sr
+
− =−
=+ −
−
−
(7)
Using (1), (2), and (5c), the shape factor f can be expressed as,
2
2
1, { 2,0}
2 ( 1) ( 1) , ( 2,0) (0,2]
( 2) ln( ) ( 1)
mm
m
m
fSr Sr m
m m Sr Sr Sr
+
−
= − −
−
+ −
(8)
where f is a function of m
and Sr, whereas θ0 and rh do not influence f. According to (8), the relationship
between f, Sr, and m is plotted in Fig. 3, where it can be found that within the typical design range (1 ≤
Sr ≤ 3.5), one can increase f by increasing Sr or m. If f is less than 1, the resistance of the PN leg of
CATEG (m = 0) is higher than that of VATEG, and the thermal conductance of CATEG is lower than
that of VATEG. Fig. 4 presents the geometric relationships of PN leg in the polar coordinate with different
m.
Fig.3. Surface of the radius ratio Sr as a function of the coefficient m and the shape factor f for a VATEG.
Fig.4. Geometic relationships of PN legs of a VATEG with (a) m = −1, (b) m = 1, (c) m = 2, and (d) m = 0.
2.3 Boundary conditions
As observed in [51], the effect of geometric parameters on output performance also relies on the
boundary conditions (BCs) on both sides of PN legs. This work considers three BCs, namely the constant
temperature, constant heat flux, as well as constant convection heat transfer coefficient conditions,
respectively. The three BCs are used for the cold and hot ends of the PN leg to solve the distributed
temperature fields and to study the heat transfer process. Generally, we name the ceramic surface in direct
contact with the heat source as the hot side surface, and the corresponding end of the PN leg is the hot
end. On the other side, the surface in direct contact with the cold side radiator is called the cold side
surface, and the corresponding end of the PN leg is the cold end.
2.3.1 Condition 1: Constant temperature on both sides
In Condition 1, the temperatures of both ends of PN legs maintain constant. As shown in Fig. 1(b),
when T1 and T2 are kept constant, according to the 1D steady-state heat transfer equation, the heat
absorbed by the hot end and the heat released from the cold end of a pair of PN legs can be expressed by
[50]
2
2
( ) ( ) ( ) 0.5 ( )
( ) ( ) ( ) 0.5 ( )
in P N h P N h c N P
out P N c P N h c N P
Q I T K K T T I R R
Q I T K K T T I R R
= − + + − − +
= − + + − + +
(9)
where I is current, α is the Seebeck coefficient, K is thermal conductance, and the subscripts P and N
denote the P-type and the N-type legs of VATEG, respectively.
By ignoring the effects of the thermal and electrical contact resistances, Qin and Qout can be
expressed as
1 1 1
2 2 2
()
()
in e h
out e c
Q Q K T T
Q Q K T T
= = −
= = −
(10)
where K1e and K2e are the thermal conductances of the connection layers on the cold end and the hot end,
respectively. Q1 and Q2 are the heat absorbed and released on the cold end and hot end, respectively.
Hence, the temperatures at the two ends are
11
22
in
he
out
ce
Q
TTK
Q
TTK
=−
=+
(11)
Since the amount of heat absorbed by a single pair of PN legs is small, we assume K1e>>Qin and
K2e>>Qout, which gives
12
,
hc
T T T T
(12)
Consequently, the Seebeck voltage U and the current I of PN legs in the VATEG can be obtained as
0( ) ( )
P N h c
U U T T
− −
(13)
tL
U
IRR
=+
(14)
where Rt = RP + RN is the internal resistance, and RL is the external load. By combining (13) and (14)
with (5), the relationship between the output performance of CATEG and VATEG can be established, i.e.,
00
0
22
tt
UI
U
IR f R f
= = =
(15)
00
0IP
P U I U ff
= = =
(16)
2
0
02
00
( ) 0.5
/
( / ) ( / ) ( ) 0.5 ( / ) ( )
in h h c
h h c
PP
QI T K T T I R
Pf
I f T K f T T I f R f
==
+ − −
= + − −
(17)
0
=
(18)
where K = KN + KP and α = αN + αP.
For the ease of performance evaluation and comparison between different models and designs, we
define the power ratio P/P0 and the efficiency ratio η/η0. Clearly, under Condition 1, P/P0 = 1/f and η/η0
= 1.
2.3.2. Condition 2: constant heat flux on the hot side and constant temperature on the cold
side
In this condition, Q1 and T2 shown in Fig. 1 are constant. Considering that the Seebeck coefficient
of the PN leg is small (in the order of 10−4), the first and third terms on the RHS of (9) can be neglected,
which gives
0 0 0
( ) ( )
h c h c
K T T K T T − = −
(19)
The relationship between the temperature difference of VATEG and the temperature difference of
CATEG is
00
()
h c h c
T T f T T− = −
(20)
The output of the VATEG can be obtained as
2 2 2 2 2
00
0
( ) ( ) ( ) ( )
44
P N h c P N h c
tt
T T f T T
PR f R
− − − −
==
(21)
0
P f P=
(22)
0
f
=
(23)
Under Condition 2, the power ratio and the efficiency ratio are P/P0 = η/η0 = 1/f.
2.3.3. Condition 3: Constant convection coefficient on the hot side and constant temperature
on the cold side
In this condition, the heat on the hot side of VATEG and CATEG can be expressed as
1
11
1
10 0
1
() ()
() ()
eeh
e
eeh
e
hA K
Q T T
hA K
hA K
Q T T
hA K
= −
+
= −
+
(24)
where Q1 and Q10 are the heat absorbed by the PN leg of VATEG and CATEG, respectively. h and Te are
the convective heat transfer coefficient and ambient temperature, respectively. A is the contact area
between the hot side ceramic layer and the external heat fluid. Considering that (KP + KN)·(Th −Tc)>>(
P
−
N)·I
·
Th and
2
0.5 ( ) 0
NP
I R R +
, we have
1
10 0 0 0 0 0 0
( ) ( )
( ) ( )
in out P N h c
in out P N h c
Q Q Q K K T T
Q Q Q K K T T
= + −
= + −
(25)
With (10), (24), and (25), the temperature difference between the two ends of PN legs is governed
by
2( ) (1 )
e h c
T T T T K C− = − +
(26)
where the
12
1 1 1
()
ee
CK K hA
= + +
.
Similarly, for the CATEG,
2 0 0 0
( ) (1 )
e h c
T T T T K C− = − +
(27)
Hence, the relationship between the output powers, absorbed heat by the hot end, and efficiency of
CATEG and VATEG are as follows, i.e.,
2
0
00
1KC
Pf
P f K C
+
=
+
(28)
0
00
1
in
in
Q K C
Q f K C
+
=+
(29)
0
0 0 0 0
/1
/in
in
P Q K C
f
P Q f K C
+
= = +
(30)
2.4. Finite element analysis
Due to the highly nonlinear nature of the model and the coupled relationship between the properties
and the temperature distribution of PN legs, an analytical solution of the model does not exist. Hence, an
iterative numerical method based on FEM will be adopted to analyze the performance of VATEG. The
schematic diagram of the finite element analysis of a single PN leg is shown in Fig. 5, where the PN leg
is discretized into S elementary units. The subscript χ can be P or N, and k is the index of the elementary unit
after discretization.
Fig. 5. Finite element analysis for a PN leg of VATEG. The PN leg is discretized into S elementary units in
the radial direction.
The physical parameters of each elementary unit can be calculated by
()
( 1)
( ) ( 1) ( ) ( 1)
( ) ( ) ( )
( ) ( 1)
22
, ( ) , ( )
k
k
k k k k
T
T
k k k
kk T T T T
TT
dT TT
TT
+
++
+−−
==
= = =
−
(31)
Using (4), the thermal conductance and the resistance of each elementary unit can be obtained, i.e.,
( 1)
()
()
()
1
()
k
k
k
r
k
r
Kdr
rr
+
=
(32)
( 1)
()
()
() ()
k
k
k
r
k
r
dr
Rrr
+
=
(33)
The internal resistance and the Seebeck voltage of PN legs are
()
1
Sk
tk
RR
=
=
(34)
( ) ( ) ( 1)
1()
Sk k k
k
U T T
+
=
=−
(35)
Since a thermoelectric module consists of n pairs of PN legs connected in series, the heat absorbed
on the hot side and the heat released on the cold side can be expressed as
(1) (1) (1) (2) (1) (1)
( ) ( ) ( ) ( 1) ( ) ( )
1
( ( ) )
21
( ( ) )
2
in h
S S S S S S
out c
Q n IT K T T IR R
Q n IT K T T IR R
+
= + − −
= + − +
(36)
where αχ = αP − αN Th = Tχ(1), Tc = Tχ(S+1).
It should be noted that in order to obtain I, P, and η, the temperature distribution of PN legs must
be given. In order to obtain the temperature distribution, we consider the following continuity of heat
flow for a PN leg, i.e.,
( 1) ( ) ( 1) ( 1) ( ) ( 1) ( 1)
( ) ( ) ( ) ( ) ( 1) ( ) ( )
1
()
2
1
()
2
k k k k k k k
k k k k k k k
IT K T T IR R
IT K T T IR R
− − − − −
+
+ − +
= + − −
(37)
The temperature distribution of each elementary unit obtained in the iterative process can be
described by
( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( ) ( )
() ( ) ( 1) ( ) ( 1)
0.5 ( )
()
k k k k k k k k
kk k k k
K T K T I R R R R
TI K K
− − − + − −
−−
+ + +
=− + +
(38)
2.5 Model validation
The simulation and experimental results in Ref. [56] will be used to verify the accuracy of the
proposed VATEG model. The design in [56] can be considered a particular case of the proposed model
where the shape of PN legs is flat and the radius radio Sr is infinite. Fig. 6(a) compares the variation of
the predicted voltages due to temperature change using the proposed VATEG model and the simulated
and experimental data provided in [56]. The geometric parameters are the same as [56], and constant
temperatures are considered on both sides (Condition 1). It can be seen that the maximum error between
the experimental result and the result using the proposed method is less than 2%, and thus the model is
deemed to reproduce the experimental results with high accuracy.
In addition, the output performance of the proposed model under constant temperature boundary
conditions is verified using the data and parameters in Ref. [57] by controlling the temperature ratio Th/Tc
to 2. Fig.6(b) shows the relationship between the maximum efficiency and the leg length using different
models. It can be seen that that with the increase of leg length, the maximum efficiency varies slightly,
and compared with the results in [57], the maximum error is only 1.5%. Therefore, this model is
considered suitable for investigating the influence of leg shape on the output performance of the VATEG.
0 50 100 150 200 250 300 350
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Seebeck Voltage(V)
DT(K)
Present model
Simulation in ref.[56]
Experiments in ref.[56]
1.5 2.0 2.5 3.0
0.10
0.12
0.14
0.16
max
Sr(rc/rh)
Ref.[57]
Present model
error
0
1
2
3
4
5
error(%)
Fig. 6 Model validation. (a) Comparison between the simulated Seebeck voltage and experimental data
from [56]. (b) Comparison between simulated maximum efficiency and experimental data from [57].
3. Results and Discussion
In this section, the results from numerical analysis are presented based on the proposed VATEG
model. Bi2Te3-based PN legs were investigated in this work. Bi2Te3 is a type of low/medium-temperature
thermoelectric material suited for the operating temperature ranging from 300 K to 500 K, and under this
condition, the optimal performance can be obtained[58]. The model parameters are listed in Table 1. The
simulated results were obtained using the proposed VATEG model and solved by the iterative numerical
method in MATLAB.
Table 1
Parameters of PN legs of the VATEG [58].
P-type Semiconductor
N-type Semiconductor
(V/K)
16 4 13 3
9 2 6 4
4.9 10 9.17 10
3.2 10 2.3 10 2.6 10
pTT
TT
−−
− − −
= − + −
+ −
15 4 12 3
9 2 6 5
1.36 10 2.5 10
2.94 10 1.5 10 7.4 10
nTT
TT
−−
− − −
= − +
− +
11
(W m K )
−−
10 4 7 3
4 2 2
3.04 10 3.89 10
2 10 5.12 10 6.925
pTT
TT
−−
−−
= − +
− +
11 4 7 3
5 2 3
7.6 10 1.12 10
5.162 10 6.51 10 1.426
nTT
TT
−−
−−
= − + −
+ +
(Ω/m)
16 4 13 3
10 2 7 5
1.9 10 3.47 10
3.32 10 1.97 10 2.85 10
pTT
TT
−−
− − −
= − + −
+ −
16 4 13 3
10 2 7 5
2.98 10 4.92 10
3.82 10 1.79 10 1.96 10
nTT
TT
−−
− − −
= − + −
+ −
0
/ / /
h
nr
1/ 3 mm / / 1 mm
12
3.1 Power ratio and efficiency ratio under different boundary conditions
Table 2 provides the parameters used for different BCs. Furthermore, the thermal conductivity and
thickness of the ceramic layer are selected as ke = 30 W/(m2∙K) and re = 0.3 mm. The thermal
conductivity and thickness of the copper sheets are selected as kCu = 400 W/(m2∙K) and rCu = 0.1 mm.
They were used to calculate the thermal conductances Ke1 and Ke2 of the connection layers in (10).
Table 2
Boundary conditions and parameters [51,59].
Boundary
Condition
Hot End
Cold End
Condition 1
Th = 403.15 K
Tc = 303.15 K
Condition 2
= 30000 W∙m−2
Tc = 303.15 K
Condition 3
h = 3000 W∙m−1∙K−1
Te = 473.15 K
Tc = 303.15 K
1.0 1.5 2.0 2.5 3.0
1.00
1.02
1.04
1.06
1.08
1.10
1.12
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
1.00
1.03
1.05
1.07
1.10
Ratio of conversion efficiency(/0)
1.0 1.5 2.0 2.5 3.0
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
0.90
0.95
1.00
1.05
1.10
1.15
1.20
Ratio of conversion efficiency(/0)
1.0 1.5 2.0 2.5 3.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
0.90
0.95
1.00
1.05
1.10
1.15
1.20
Ratio of conversion efficiency(/0)
-2 -1 0 1 2
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Ratio of power(P/P0)
Exponent coefficient(m)
Sr=1.5
Sr=2
Sr=3
Pmax=1.013P0Pmax=1.04P0
Pmax=1.098P0
Fig. 7. Variations of power ratio P/P0 and efficiency ratio η/η0 with different exponent coefficient m under
Conditions 1: (a) m = −1; (b) m = 1; (c) m = 2; (d) Variation of power ratio with m under different Sr.
Under Condition 1, as illustrated in Fig. 7(a), the power ratio is over one at any leg length when m
= −1. It implies that when the angle function of PN legs is in the form of qr−1, higher output power can
be achieved by the VATEG design compared to the CATEG regardless of leg length. With m = −1, a
smaller shape factor f due to longer leg length would enhance the VATEG output power. Nevertheless,
Figs. 7(b) and (c) show that if the angle function of the PN leg is qr or qr2, the output power of CATEG
is always optimal regardless of the selection of leg length.
On the other hand, the conversion efficiency is hardly affected by the leg shape under Condition 1.
This can be explained using (16) and (18), from which it can be seen that the output power of VATEG is
inversely proportional to the shape factor f. When f < 1 and m = −1, increasing the leg length would
reduce f. Hence, the results on output performance obtained from theoretical analysis and finite element
(a)
(b)
(c)
(d)
simulation are highly consistent. In addition, Fig. 7(d) shows the variation of power ratio with m under
different Sr. It can be seen that when m < 0, the output power of VATEG is higher than that of CATEG,
and the benefits using variable-angle leg can be maximized when m = −1. Compared to the CATEG,
when Sr = 1.5, 2, and 3, the output power of VATEG can be increased by 1.3%, 4%, and 9.8%,
respectively.
1.0 1.5 2.0 2.5 3.0
0.90
0.93
0.95
0.98
1.00
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
0.90
0.93
0.95
0.98
1.00
Ratio of conversion efficiency(/0)
1.0 1.5 2.0 2.5 3.0
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical 1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
Ratio of conversion efficiency(/0)
1.0 1.5 2.0 2.5 3.0
1.0
1.2
1.4
1.6
1.8
2.0
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical 1.0
1.2
1.4
1.6
1.8
2.0
Ratio of conversion efficiency(/0)
-2 -1 0 1 2
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Ratio of power(P/P0)
Exponent coefficient(m)
Sr=1.5
Sr=2
Sr=3
Pmax=1.11P0
Pmax=1.35P0
Pmax=1.96P0
Fig.8. Variations of power ratio P/P0 and efficiency ratio η/η0 with different exponent coefficient m under
Conditions 2: (a) m = −1; (b) m = 1; (c) m = 2; (d) Variation of P/P0 with m under different Sr.
A similar analysis was carried out for Condition 2, and the results are shown in Fig. 8. It can be
seen from Fig. 8 and explained using (22) and (23) that, with a constant heat flux Qin, both the output
power and the conversion efficiency can be increased by increasing the shape factor f. Furthermore, f can
be increased with a higher Sr when m > 0, while it will be decreased when m < 0. It can be found that the
assumption made in Section 2 for theoretical analysis has become less valid in the simulation where the
leg length is long. Nevertheless, in practice, the assumption tends to be valid since a design with a long
leg length should be avoided to maintain structural stability. A quantitative analysis of the influence of
shape change on output performance was carried out, and the results are presented in Fig. 8(d). It is
observed that when m = 2, the output performance of VATEG is maximized at any leg length. In terms
of the output power, the performance of VATEG is 11% higher than that of CATEG when Sr = 1.5. The
(a)
(c)
(d)
(b)
improvement in output power can be further increased with a higher Sr, e.g., 35% if Sr = 2, and 96% if
Sr = 3.
1.0 1.5 2.0 2.5 3.0
1.00
1.03
1.05
1.07
1.10
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
0.90
0.95
1.00
1.05
1.10
Ratio of conversion efficiency(/0)
1.0 1.5 2.0 2.5 3.0
0.75
0.80
0.85
0.90
0.95
1.00
1.05
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
0.90
0.95
1.00
1.05
1.10
Ratio of conversion efficiency(/0)
1.0 1.5 2.0 2.5 3.0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Ratio of power(P/P0)
Sr
P/P0 /0
Simulation
Theoretical
0.70
0.80
0.90
1.00
1.10
1.20
1.30
Ratio of conversion efficiency(/0)
-2 -1 0 1 2 3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Ratio of power(P/P0)
Exponent coefficient(m)
P/P0 /0
Sr=1.5
Sr=2
Sr=3
Pmax=1.005P0Pmax=1.024P0
Pmax=1.072P0
max=1.0280
max=1.0460
max=1.0610
Pmin=0.56P0
Pmin=0.81P0
Pmin=0.95P0
min=0.9890
min=0.9960
min=0.9930
0.98
0.99
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
Ratio of conversion efficiency(/0)
Fig. 9. Variations of power ratio P/P0 and efficiency ratio η/η0 with different exponent coefficient m under
Conditions 3: (a) m = −1; (b) m = 1; (c) m = 2; (d) Variation of power ratio with m under different Sr.
Figs. 9(a)−(c) reveals the relationship between the power ratio, the efficiency ratio, and Sr with
different PN leg shape factors m under Condition 3. It can be seen that the curves obtained from the
theoretical analysis are again close to the simulation results based on the FEM. In Fig. 9(d), variations of
the power and efficiency ratios with the change of m show opposite trends, which indicates they cannot
be optimized simultaneously. Particularly, when m = −1, the output power of VATEG is maximized,
whereas m = −1 is the least efficient operating condition. Furthermore, when m = −1, the output power
can be improved by 7.2%, 2.4%, and 0.5%, respectively, when Sr = 3, 2, and 1.5, respectively, while the
corresponding efficiency decreases by 1.1%, 0.7%, and 0.4%, respectively. When m = 2, the efficiency
can be increased by 6.1%, 4.6%, and 2.8%, respectively, whereas the output power is decreased by 44%,
19%, and 5%, respectively.
3.2. Comparison of output performance under different boundary conditions
Under Condition 2, although increasing the leg length can improve the output performance, a
(a)
(c)
(b)
(d)
practical design shall also consider structural stability. For example, in the case of m = 2, with the increase
of Sr, the angle of PN legs increases exponentially. Given a constant volume as the design requirement,
the calculated hot side contact area of the PN leg will be very small if the leg length is long, leading to
structural instability. Hence, a high radius radio such as Sr = 3 is not suitable for practical applications.
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.00
10.00
20.00
30.00
40.00
U(mV)
I(A)
U P
m=0
m=-1
m=1
m=2
0
1
2
3
4
5
6
P(mW)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.00
0.01
0.02
0.03
0.04
0.05
I(A)
m=0
m=-1
m=1
m=2
Fig. 10. Comparison of output performance under different PN legs angle shapes when Sr = 2 under Condition 1:
(a) U−I and P−I relationships. (b) η−I relationship.
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
10.00
20.00
30.00
40.00
50.00
U(mV)
I(A)
U P
m=0
m=-1
m=1
m=2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P(mW)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.01
0.02
0.03
0.04
I(A)
m=0
m=-1
m=1
m=2
Fig. 11. Comparison of output performance under different PN legs angle shapes when Sr = 2 under Condition 2:
(a) U−I and P−I relationships. (b) η−I relationship.
0.0 0.2 0.4 0.6 0.8
0
10
20
30
40
50
60
U(mV)
I(A)
U P
m=0
m=-1
m=1
m=2
0
2
4
6
8
10
12
P(mW)
0.0 0.2 0.4 0.6 0.8
0.00
0.02
0.03
0.05
0.06
I(A)
m=0
m=-1
m=1
m=2
Fig. 12 Comparison of output performance under different PN legs angle shapes when Sr = 2 under Condition 3:
(a) U−I and P−I relationships. (b) η−I relationship.
(a)
(b)
(a)
(b)
(a)
(b)
Fig. 10 to Fig. 12 compare the U−I, P−I, and η−I relationships under different BCs when Sr = 2. In
Fig. 9, under Condition 1, the output power of a pair of PN legs in CATEG (m = 0) is 5.24 mW. In
contrast, the VATEG can achieve the maximum power of 5.45 mW when m = −1, which is 4% higher
than the CATEG. Also, the short-circuit current is the highest when m = −1, which is 0.59 A. Moreover,
the OCV, i.e., the Seebeck voltage U when I = 0, is unchanged.
Fig. 11 illustrates the output performance under Condition 2. It can be seen that when m = 2, both
the efficiency and the output power are significantly increased compared to CATEG: The maximum
output power is 3.1 mW, and the maximum efficiency is 3.8%, increased by 34% and 2.8%, respectively.
In contrast, the short-circuit current is hardly affected by the shape of PN legs. In addition, the shape
factor affects the OCV, and the maximum OCV is obtained when m = 2.
Fig. 12 compares output performance under Condition 3. The maximum output power of the
VATEG is 11 mW when m = −1, which is only 0.3 mW or 2.4% higher than the CATEG. In this case,
the improvement in the performance is limited through modifying the leg shape.
3.3. Influence of external parameters on the performance of PN legs under different boundary
conditions
In this subsection, we investigate the influence of external parameters on the output performance of
the VATEG under different BCs when Sr = 2. Fig. 13 presents the relationship between the power ratio
and the temperature difference with different m when the cold end temperature is fixed. It can be seen
from the results that the change of temperature difference has little impact on the VATEG output power
and efficiency regardless of the shape of PN legs. Therefore, the effect of temperature differences can be
ignored in the design for practical applications.
Fig. 14 illustrates the influence of the heat flux on the output performance under Condition 2. It
shows that the influence of heat flux on output performance depends heavily on the exponent coefficient
m of the variable-angle PN legs. It can be seen that compared to Fig. 13, the influence of heat flux on the
output performance is more significant. As the heat flux increases, there are different trends for different
m. Specifically, when m = −1, the output performance can be improved by increasing the heat flux. In
contrast, increasing the heat flux will cause reduced output power and efficiency when m = 1 or m =
2.The results show that although a design of PN leg shape with m = 2 can improve the output performance
compared to the conventional ATEG, for a practical system, the heat flux must be limited to avoid reduced
performance.
Fig. 15 shows the effect of the convective heat transfer coefficient h on the output performance of
PN legs with different shapes under Condition 3. It can be seen that when the convective heat transfer
coefficient varies from 1000 W/(m2·K) to 6000 W/(m2·K), the output power and the efficiency increase
as h increases when m = −1, and thus this shape of PN leg can be used for high convective heat transfer
coefficient. On the contrary, the output performance will deteriorate as h increases when m = 1 or m = 2,
and thus the corresponding PN leg is only suitable for the operating conditions with low convective heat
transfer coefficient.
0 50 100 150 200
1.03
1.04
1.05
1.06
1.07
100 150 200
1.0380
1.0385
1.0390
50 100 150
0.9994
0.9996
0.9998
Ratio of power(P/P0)
DT(K)
P/P0
/0
0.96
0.97
0.98
0.99
1.00
1.01
Ratio of conversion efficiency(/0)
0 50 100 150 200
0.89
0.90
0.91
0.92
80 120 160 1.0008
1.0012
1.0016
80 120160
0.8927
0.8928
Ratio of power(P/P0)
DT(K)
P/P0
/0
0.98
0.99
1.00
1.01
Ratio of conversion efficiency(/0)
0 50 100 150 200
0.74
0.75
0.76
0.77
100 150 200
1.000
1.002
1.004
Ratio of power(P/P0)
DT(K)
P/P0
/0
50 100150
0.7422
0.7424
0.7426
0.98
0.99
1.00
1.01
Ratio of efficency conversion(/0)
Fig.13. Impacts of temperature difference on output performance with various shapes of PN legs under Condition
1. (a) m = −1. (b) m = 1. (c) m = 2.
0 2 4 6
0.94
0.95
0.96
0.97
0.98
Ratio of power(P/P0)
q104(W/m2)
P/P0
/0
0.94
0.95
0.96
0.97
0.98
Ratio of conversion efficency(/0)
0 2 4 6
1.10
1.11
1.12
1.13
Ratio of power(P/P0)
q104(W/m2)
P/P0
/01.10
1.11
1.12
1.13
Ratio of conversion efficency(/0)
0 2 4 6
1.30
1.32
1.34
1.36
Ratio of power(P/P0)
q104(W/m2)
P/P0
/01.30
1.32
1.34
1.36
Ratio of conversion efficency(/0)
Fig.14. Impacts of heat flux on output performance with various shapes of PN legs under Condition 2. (a) m = −1.
(b) m = 1. (c) m = 2.
0 2 4 6
1.01
1.02
1.03
1.04
1.05
Ratio of power(P/P0)
h
103(W/(m2·K))
P/P0
/0
0.97
0.98
0.99
1.00
Ratio of conversion efficency(/0)
0 2 4 6
0.90
0.92
0.94
0.96
Ratio of power(P/P0)
h103(W/(m2·K))
P/P0
/01.01
1.02
1.03
Ratio of conversion efficency(/0)
0 2 4 6
0.75
0.80
0.85
0.90
Ratio of power(P/P0)
h103(W/(m2·K)
P/P0
/01.02
1.04
1.06
1.08
Ratio of conversion efficency(/0)
Fig.15. Impacts of convective heat transfer coefficient on output performance with various shapes of PN legs
under Condition 3. (a) m = −1. (b) m = 1. (c) m = 2.
3.4. Thermal stress analysis and temperature distribution of different PN leg shapes with
COMSOL
As the material characteristics are related to temperature, the shape of PN legs change would
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
significantly affect the variation of the temperature distribution on PN legs. It is necessary to investigate
the upper limit of the local temperature of the PN leg to avoid overheating problems. Fig. 16 compares
the simulated temperature distributions of a PN leg in the leg length direction obtained from COMSOL
and MATLAB when Sr = 2. The results show that the proposed algorithm based on finite element
analysis is accurate with different VATEG designs. It can be seen that when m = 2, the nonlinearity of
temperature distribution is significant under all boundary conditions, and only under Condition 2, the hot
side temperature of PN legs will increase obviously in this shape, which is nearly 30 K higher than the
hot side temperature of CATEG. This phenomenon can account for with the same leg volume, a smaller
hot side contact cross-sectional area of PN legs at m = 2 would obstruct the thermal conductivity, which
attributes to significant temperature differences along the PN legs. In conclusion, a suitable cross-section
of PN legs can avoid excessive temperature rise and damaging the materials.
Fig.16. Temperature distribution of PN legs with different shapes along the leg length direction. (a) Condition 1.
(b) Condition 2. (b) Condition 3.
Fig. 17 to Fig. 19 illustrate the Seebeck voltage of PN legs obtained from COMSOL with different
m and under different boundary conditions. Fig. 17 measures the Seebeck voltage under Condition 1. It
can be observed that the maximum Seebeck voltage, i.e., the OCVs are all about 37 mV with different
0.0 0.5 1.0 1.5 2.0 2.5 3.0
300
320
340
360
380
400
420
440
460
T(K)
H(mm)
COMSOL MATLAB
m=0
m=-1
m=2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
300
320
340
360
380
400
420
440
T(K)
H(mm)
COMSOL MATLAB
m=0
m=-1
m=2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
300
320
340
360
380
400
420
T(K)
H(mm)
COMSOL MATLAB
m=0
m=-1
m=2
(a)
(b)
(c)
shapes, which is consistent with the results shown in Fig. 10(a). Similar results can be observed under
Condition 3 as shown in Fig. 19: the OCVs are close, which complies with the results in Fig. 12. In
contrast, under Condition 2, the OCVs are quite different with different PN legs. The OCV reaches up to
49.08 mV when m = 2, and this is 15.68 mV or 50% higher than the CATEG.
Fig.17. Seebeck voltage of PN legs with different shapes under Condition 1.
(a) m = 0 (CATEG). (b) m = −1. (c) m = 2.
Fig.18. Seebeck voltage of PN legs with different shapes under Condition 2.
(a) m = 0 (CATEG). (b) m = −1. (c) m = 2.
Fig. 19. Seebeck voltage of PN legs with different shapes under Condition 3.
(a) m = 0 (CATEG). (b) m = −1. (c) m = 2.
To investigate how the shape of the PN legs can affect the stability of thermoelectric materials, Fig.
20 to Fig. 22 show the thermal stress distribution of various PN legs under different BCs when Sr = 2. It
is worth noting that the thermal stresses in all PN leg shapes are on the hot end of PN legs. Furthermore,
it can be seen that the coefficient m has little impact on the thermal stress and the stability of the materials
under the three applied BCs. Nevertheless, under Condition 2 as shown in Fig. 21, the thermal stress
increases dramatically when m = 2 (about 31% compared to m = 0). This is mainly because a higher
temperature can be achieved with this design at the hot end of the PN legs.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
Fig. 20. Thermal stress of PN legs with different shapes under Condition 1:
(a) m = 0 (CATEG). (b) m = −1. (c) m = 2.
Fig. 21. Thermal stress of PN legs with different shapes under Condition 2:
(a) m = 0 (CATEG). (b) m = −1. (c) m = 2.
Fig.22. Thermal stress of PN legs with different shapes under Condition 3.
(a) m = 0 (CATEG). (b) m = −1. (c) m = 2.
4. Conclusion
A variable-angle annular thermoelectric generator (VATEG) is proposed in this work. First, the
relationship between the output performance of VATEG and a conventional constant-angle annular
thermoelectric generator (CATEG) under three boundary conditions is investigated analytically and
numerically, including constant temperature on both sides of the PN leg (Condition 1), constant heat flux
on the hot side and constant temperature on the cold side (Condition 2), and constant convection
coefficient on the hot side and constant temperature on the cold side (Condition 3). Several parameters
are defined to indicate the geometrical characteristics of the VATEG, including shape factor f, radius
ratio Sr, and exponent coefficient m. Next, the optimal output performance of VATEG is obtained with
different radius ratios of PN legs. Finally, the open-circuit voltage and the thermal stress distribution of
the PN legs are investigated through 3D simulation in COMSOL. The main findings of this work are
summarized as follows.
1) Boundary conditions can affect the output performance of the VATEG significantly. For
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
example, under Condition 1, the maximum output power of the PN leg with variable leg angles
is inversely proportional to the shape factor f, while the maximum conversion efficiency is not
affected by the shape factor. Under Condition 2, the maximum output power and the maximum
conversion efficiency of the PN legs with variable angles are proportional to f. Furthermore,
the maximum output power is negatively correlated with f, while the corresponding conversion
efficiency is positively correlated with f under Condition 3.
2) The specific output performance of PN legs when Sr = 2 is analyzed. When m = −1, the output
power reaches its maximum value, 4% higher than that of CATEG under Condition 1. The
output power and conversion efficiency are 35% higher than CATEG under Condition 2 when
m = 2, while the thermal stress increases by 31%. Under Condition 3, although the maximum
output power of the VATEG is 2.4% higher than the CATEG, the conversion efficiency is
slightly reduced.
3) From theoretical analysis and numerical simulation, consistent temperature distributions are
obtained in the radial direction of the PN leg. The hot side temperature will increase only under
Condition 2 with m = 2, leading to an increase in the voltage.
4) The impacts of the external environment on the output performance are investigated with
various shapes of PN legs. The results show that the temperature difference has little influence
on the output performance. However, heat flux and convective heat transfer coefficient greatly
influence the output performance, and these two factors should be considered in practice.
5) The performance of the VATEG is superior to the conventional CATEG in terms of output
power and efficiency. Under Conditions 1 and 3, the VATEG efficiency is 4% and 2.4% higher
than CATEG, respectively, when m = −1. When m = 2, the efficiency and output power can be
increased by 34% under Condition 2.
Acknowledgments
This research was supported by the National Key Research and Development Program of China
(2020YFB1506802) and the National Natural Science Foundation of China (51977164).
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