Constant heat characterisation and geometrical optimisation of thermoelectric generators

Abstract

It is well known that for a thermoelectric generator (TEG) in thermal steady-state with constant temperature difference across it the maximum power point is found at half of the open-circuit voltage (or half of the short-circuit current). However, the effective thermal resistance of the TEG changes depending on the current drawn by the load in accordance with the parasitic Peltier effect. This article analyses the different case in which the input thermal power is constant and the temperature difference across the TEG varies depending on its effective thermal resistance. This situation occurs in most waste heat recovery applications because the available thermal power is at any time limited. The first part of this article presents the electrical characterisation of TEGs for constant-heat and it investigates the relationship between maximum power point and open-circuit voltage. The second part studies the maximum power that can be produced by TEGs with pellets (or legs) of different size and number, i.e. with different packing factors, and of different height. This work provides advice on the optimisation of the pellets geometrical parameters in order to increase the power generated, and consequently the thermodynamic efficiency, and to minimise the quantity of thermoelectric material used, for systems with limited input thermal power.

Full text

Constant heat characterisation and geometrical optimisation
of thermoelectric generators
Andrea Montecucco
⇑
, Jonathan Siviter, Andrew R. Knox
Thermoelectric Conversion Systems Ltd, UK
School of Engineering, College of Science and Engineering, University of Glasgow, UK
highlights

In most waste heat applications at any time the maximum available heat is limited.

The performance characterisation of TEGs with constant thermal power is presented.

The influence of the geometrical parameters is analysed.

The pellets number and geometry are optimised for limited heat systems.

The efficiency and output power are maximized and the material needed is minimised.
article info
Article history:
Received 12 August 2014
Received in revised form 20 February 2015
Accepted 23 March 2015
Keywords:
Thermoelectric
TEG
Heat transfer
Constant power
Characterisation
Optimization
abstract
It is well known that for a thermoelectric generator (TEG) in thermal steady-state with constant tempera-
ture difference across it the maximum power point is found at half of the open-circuit voltage (or half of
the short-circuit current). However, the effective thermal resistance of the TEG changes depending on the
current drawn by the load in accordance with the parasitic Peltier effect.
This article analyses the different case in which the input thermal power is constant and the tempera-
ture difference across the TEG varies depending on its effective thermal resistance. This situation occurs
in most waste heat recovery applications because the available thermal power is at any time limited.
The first part of this article presents the electrical characterisation of TEGs for constant-heat and it
investigates the relationship between maximum power point and open-circuit voltage. The second part
studies the maximum power that can be produced by TEGs with pellets (or legs) of different size and
number, i.e. with different packing factors, and of different height. This work provides advice on the opti-
misation of the pellets geometrical parameters in order to increase the power generated, and conse-
quently the thermodynamic efficiency, and to minimise the quantity of thermoelectric material used,
for systems with limited input thermal power.
Ó2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Thermoelectric generators (TEGs) are recently being utilised to
recover waste heat in a multitude of applications, ranging from low
power (sensors [1,2] and battery charging [3]) to medium power
(automotive [4,5], stoves [6,7], CHP systems [8], and combined to
TPV [9] or PV [10] systems) to high power (heavy-industry [11]
and geothermal [12]) because of their reliability, small size and
weight, and modular scalability [13].
For a given temperature difference the electrical power deliv-
ered by the TEG varies depending on the current drawn by the
electrical load connected to its terminals. The TEG can be electri-
cally modelled in thermal steady-state as a voltage source in series
with an internal resistance [14,15]. In available literature, to max-
imise the electrical power extracted from the TEG at any fixed tem-
perature difference the load’s impedance should equal the TEG’s
internal resistance, as stated by the ’maximum power transfer’
theorem [16]. Hence the maximum power point lies at half of
the open-circuit voltage V
OC
or equivalently at half of the short-
circuit current I
SC
.
A characterisation showing the relationship between electrical
power, voltage and current for a constant applied temperature dif-
ference is an established method to specify the performance of TEG
devices. When physically obtaining this characterisation it is
necessary to adjust the thermal power through the device because
http://dx.doi.org/10.1016/j.apenergy.2015.03.120
0306-2619/Ó2015 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
⇑
Corresponding author.
E-mail address: andrea.montecucco@glasgow.ac.uk (A. Montecucco).
Applied Energy 149 (2015) 248–258
Contents lists available at ScienceDirect
Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

a change in electrical load varies the effective thermal conductance
of the TEG due to the Peltier effect [17]. This method of character-
isation is referred to as constant temperature operation, and its use
effectively masks the complex and subtle device response to vari-
able load current. The internal resistance (R
int
) is the inverse slope
of the V-I line obtained from this electrical characterisation, and its
absolute value is dependent on the average temperature at which
the TEG is operating. When the TEG is operated to the left of the
maximum power point as shown in Fig. 1, reduced current flows
through the TEG and the effective thermal conductivity of the
TEG (which depends also on the current flow, due to the parasitic
Peltier effect) decreases. Under this condition the thermal energy
conducted via the TEG is less than that at the maximum power
point and hence a lower thermal load is imposed on the overall
system. This is advantageous in most circumstances since it leads
to increased thermal efficiency of the system. When the TEG is
operated to the right of the maximum power point the thermal
conductivity increases and the thermal energy conducted via the
TEG is greater than that which flows at the maximum power point.
Operation in the region to the right on Fig. 1 leads to a reduced
thermal efficiency of the system. For the module data (product
code: GM250-449-10-12 by European Thermodynamics Ltd.) shown
in Fig. 1, the maximum power is approximately 13.2 W with a
corresponding output voltage of 16.5 V (being half of the open-cir-
cuit voltage of 33 V).
In most practical applications, however, and especially in auto-
motive exhaust gas energy recovery systems, TEGs are subject to
limited thermal input energy rather than to a constant temperature
difference. This is referred to as ‘‘constant heat’’ operation. Kumar
et al. [18] observed strong variations of the electrical power genera-
tion with the exhaust gas flow rate and temperature, i.e. the input
thermal power. The available thermal energy may change with time,
but its rate of variation will be orders of magnitude slower than the
TEGs electrical response [19]. In considering the constant heat con-
dition, changing from open-circuit to at-load operation results in a
smaller temperature difference across the device, due to its greater
effective thermal conductivity. The change of temperature differ-
ence after a transient that could last for several seconds thus leads
the device to produce lower electrical power.
Mayer and Ram [20] firstly noticed that when the temperature
gradient across the TEG is not constant the optimum current is
lower than that required for constant temperature systems.
Moreover, they found that this optimum load also differs from
the load that maximises efficiency in constant temperature sys-
tems. They also provided guidance on the optimisation of the pellet
length per unit area. Similar results about optimum load condition
are reported by Gomez et al. [21]. They compared a model in which
the temperatures are constant with a model in which the tempera-
tures on the sides of the device vary depending on the load, while
the ambient temperature and the hot-source temperature (sepa-
rated from the TEG by thermal resistances) are constant. We
proposed similar results about the influence of the load on the
temperature profile -with a different analytical solution that can
also simulate time transients- in [22].
Yazawa and Shakouri [23] optimised the thermoelectric device
design together with its heat source and heat sink at constant tem-
peratures. They concluded that the optimum operating load is
when R
load
¼R
int
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZT
p, where R
int
is the internal resistance and
ZT is the figure of merit of the thermoelectric device. They also sug-
gested that using low fill (or packing) factors could increase the
electrical power output per unit mass. McCarty [24] confirmed
Yazawa’s result and provided equations to calculate the optimum
number of couples and pellets length-to-area ratio as functions
of the total thermal resistance of the system and for fixed electrical
load, hot-side temperature and material properties. However, the
value of ZT ¼
ra
2
T
AVG
=
j
depends on the electrical conductivity
(
r
), the Seebeck coefficient (
a
), the thermal conductivity (
j
) and
the average temperature of the semiconductor material, T
AVG
(in
degrees Kelvin), therefore it is difficult to calculate its correct value
before knowing the effective operating temperatures. Nemir and
Beck [25] explained that the maximum efficiency in constant tem-
perature is dependent only on the temperatures and ZT, but not on
the particular values of
r
;
j
and
a
. They also investigate the influ-
ence of thermal interfaces but they do not analyse the variations on
thermal power through the TEG. Apertet et al. [26] complemented
this analysis.
The previous literature reported here ([20,21,23–26])
effectively considered a system in which the thermal input power
varies depending on the load, because the hot-source temperature
and cold-source (often ambient) temperature are maintained con-
stant (ideal temperature sources). For such reason maximum
power and maximum efficiency are at different points. On the con-
trary, in constant heat systems the point that maximises the elec-
trical power output also guarantees maximum efficiency, as
introduced by Wang et al. [27] and further described in this article.
Despite high ZT values recently claimed by thermoelectric
material scientists, there is not much literature focused on improv-
ing the thermoelectric device design and architecture. Rezania
et al. [28] focused on the single thermoelectric element to optimise
the ratio of the cross-sectional areas of the p- and n-semiconductor
pellets; they concluded that maximum power generation occurs
when the area of the n-pellet is smaller than that of the p-pellet,
due to lower electrical resistance and higher thermal conductivity
of the n-type material considered. Lee [29] focused on the design of
thermoelectric devices in conjunction with the heat sinks perfor-
mance, asserting that there is an optimal ratio of system thermal
conductivities to provide maximum power output. Jang and Tsai
[30] and Favarel et al. [31] modify the spacing between TEG
modules (or the occupancy rate) placed on a heat exchanger of
fixed geometry to maximise the power output depending on the
available thermal power. Kajihara [32] presented an analysis of
the influence of legs’ width, height, gap (clearance) and number
Nomenclature
DTtemperature difference (K)
j
thermal conduction coefficient (W/mK)
llength (or height) of pellets (or legs) (mm)
Lthickness of the TEG (mm)
Kthermal conductance (W/K)
Relectrical resistance (
X
)
a
seebeck coefficient (
l
V/K)
a;b;cparameters for the calculation of VOC (V/K
2
, V/K and V,
respectively)
d;e;fparameters for the calculation of Rint (X=K2;X=K and
X
,
respectively)
Nnumber of pellets in a thermoelctric device
s
clearance space between pellets (mm)
x
side length of a pellet (mm)
Across-sectional area (mm
2
)
Vvolume (mm
3
)
/pellets packing (or fill) factor
ZT dimensionless thermoelectric figure of merit
A. Montecucco et al./ Applied Energy 149 (2015) 248–258 249

for a constant temperature difference. Wide pellets with small gap
between them lead to high power generation but also require great
amounts of heat through them and the author also considers
manufacturing constraints.
This work analyses the performance of a TEG device under
steady-state thermal conditions of constant heat through it (as
opposed to constant temperature difference), for varying number
and size (both width and height) of pellets, and clearance space
between them. This study is independent from the effect of the
heat exchangers in contact with the TEG, and the associated ther-
mal contact resistances. The design of the heat exchangers can be
undertaken independently.
This article firstly presents a steady-state theoretical analysis of
the power balance in constant heat TEG systems, which is used to
provide the electrical characterisation (V-vs-I and P-vs-I), which
has not been previously presented in literature. The optimum
operating point is located between the open-circuit and the maxi-
mum power point of ‘‘constant temperature’’ systems. In fact this
load point, which is characterised by a higher output voltage com-
pared to the maximum power point, has a reduced Peltier effect
which results in a greater temperature difference across the device
and consequently yields higher electrical power production and
greater system efficiency.
This article then provides a detailed analysis of the influence on
the thermoelectric performance of the fill (or packing) factor, which
is related to the number and cross-sectional area of the pellets (or
legs), and the clearance space between them. Firstly, a comparison
is provided for commercial TEG devices with same overall surface
area but different number of pellets of dissimilar size, which often
results in different fill factors. Secondly, a simulation tool is used to
optimise the number of pellets and clearance space between them
depending on the available heat power in constant heat TEG sys-
tems. Different combinations of pellets number and clearance
space result in optimum performance, with the same pellets pack-
ing factor in all instances.
The simulation tool is further improved to optimise the TEG
device geometry modifying both the pellets packing factor and
height (or thickness). The resulting simulation model provides an
optimisation tool to design the geometrical parameters of the
TEG device in order to increase power production and efficiency
while minimising the quantity of thermoelectric material used
only by acting on the physical architecture of the TEG pellets.
The results provided offer interesting insights on the break-
down of powers due to the thermoelectric effects and assist in
the design of the TEG device depending on its intended use and
available thermal power.
2. Constant-heat characterisation of TEGs
First consider a TEG sandwiched between a cold side at tem-
perature T
C
and a hot side at temperature T
H
, both of which can
vary their temperature. As explained by Min and Yatim [17], the
effective thermal conductance K
v
ar
of the TEG varies depending
on the amount of current drawn from it, and it can be calculated as
K
v
ar
¼Q
H
T
H
T
C
ð1Þ
where Q
H
is the thermal power flowing through the TEG, consid-
ered constant throughout this discussion. Moving from open-circuit
to at-load, K
v
ar
increases and the temperature difference decreases
as T
C
increases and T
H
decreases. If T
C
is maintained at a constant
temperature then T
H
must decrease by the deviation corrected for
in T
C
to keep
D
T¼T
H
T
C
as previously. Hence we can set T
C
as
a constant without significant loss of accuracy; the only small dif-
ference is represented by the fact that the thermal conductivity,
j
, and the electrical resistivity,
q
, slightly vary with the average
temperature of the device, but in this case the average temperature
is almost unchanged. Not only this assumption allows to maintain
the thermal power constant despite changes in T
H
, but this assump-
tion is also realistic: in most TEG systems the cold side temperature
remains almost constant with relatively small changes in thermal
power flowing into the cold side. Also, this analysis allows to
neglect the effect of the contact resistances, because the tempera-
ture difference can vary, depending on the effective TEG thermal
resistance and the thermal power flowing through it, with reference
to the constant T
C
.
Fig. 1. Electrical characterisation (V-I and P-I curves) of the thermoelectric device GM250-449-10-12 by European Thermodynamics Ltd.
250 A. Montecucco et al. / Applied Energy 149 (2015) 248–258

The steady-state thermal input power to the hot junction is
derived from the steady-state solution of the one-dimensional heat
conduction equation for solids with internal energy generation,
assuming constant temperatures at the hot and cold side as bound-
ary conditions [15]. In this analysis the Thomson effect is not con-
sidered due to its negligible effect [33,34]:
Q
H
¼
j
A
D
T
Lþ
a
T
H
I1
2R
int
I
2
ð2Þ
where
j
is the heat conduction coefficient, Ais the area and Lthe
thickness of the TEG,
D
Tis the temperature gradient,
a
is the
Seebeck coefficient and R
int
is the overall internal resistance of
the device. In Eq. (2) both R
int
and
a
vary with
D
T. It is possible to
express their variation with
D
Tusing a 2nd-order polynomial equa-
tion, so that the load voltage can be written as a function of I
load
and
D
T:
V
load
¼ða
D
T
2
þb
D
TþcÞðd
D
T
2
þe
D
TþfÞI
load
ð3Þ
where a;b;c;d;eand fare constant coefficients, different for each
TEG and obtained from experimental data [35]. They are listed in
Table 1. Using a¼a
D
Tþbþc=
D
Tand R
int
¼d
D
T
2
þe
D
Tþfin Eq.
(2) results in:
Q
H
¼KðT
H
T
C
ÞþaðT
H
T
C
Þ
2
þbðT
H
T
C
Þþc
T
H
T
C
T
H
I
dðT
H
T
C
Þ
2
þeðT
H
T
C
Þþf
2I
2
ð4Þ
where K¼jA=Lis the thermal conductance of the TEG in [W/K] at
open-circuit. The second term on the right side of Eq. (4) considers
the Seebeck coefficient for the whole device as V
OC
=
D
T; even if
a
should be divided by the number of pellets, it needs to be multiplied
again by the same number for the calculation of the Peltier power.
Eq. (4) in the variable T
H
has three real solutions, which can be
easily calculated by Matlab
1
and the correct solution thus identified.
Using Eq. (4) it is possible to obtain the variation of the steady-state
temperature difference across the TEG versus the load current, as it
will be shown in Fig. 2. A graph reporting the variation in tempera-
ture difference versus load change was presented in [17] for different
values of the figure of merit Z, considered constant over the tem-
perature range.
It is trivial to calculate the steady-state temperature difference
at open-circuit
D
T
0
, at which the TEG is producing the open-circuit
voltage V
OC
0
. If there were no changes in temperature difference
due to the Peltier effect and Joule heating then the maximum
power point would be found at V
load
¼V
OC
0
=2. In reality after draw-
ing current from the TEG the temperature difference decreases
exponentially to the steady-state value. Due to the long thermal
time constant of a typical thermoelectric system, it may take sev-
eral seconds to complete at least 90% of the transition.
A program was written in Matlab in order to find the value of I
that leads to the maximum power production: starting from open-
circuit and gradually increasing the load current Ithe correspond-
ing steady-state temperature difference is calculated solving Eq.
(4), and the output power is then obtained by multiplying Eq. (3)
by I. A single TEG with the thermal and electrical characteristic
of the device GM250-127-14-10
2
(characterised in [35]) was con-
sidered for this analysis. The value of thermal conductivity at
open-circuit is jOC ¼1:5W=mK;TC¼25 C and QH¼156 W. The
thermal conductivity is considered constant because it is difficult
to obtain confident measurements of it at different temperatures.
Its variation might slightly affect the results presented here.
Fig. 2 shows the resulting electrical characterisation computed
by Matlab for constant thermal power input, in steady-state; for
each load point (blue line), the green line plots the temperature
difference (on the secondary y-axis) and the red line the produced
power. It can be noted that the temperature difference decreases
significantly with increasing current loads due to an increase in
heat pumped from the hot to the cold side. When going from
open-circuit to maximum power the temperature difference drops
40 °C from
D
T
OC
¼195

Cto
D
T
MP
¼155

C, and the
D
Tat short-
circuit is 124 °C. The maximum power is obtained for
V
MP
¼4:32 V which is almost half of the initial open-circuit
steady-state voltage V
OC
init
¼8:95 V, which is the highest voltage
that the system can produce. This is due to the slight bend in the
voltage line which is in turn due to the not-exactly linear variation
of the Seebeck effect with the temperature difference (see Eq. (3)).
Hence it is found that the ‘‘real’’ MPP is around 0.483 times the
steady-state open-circuit voltage, V
OC
init
, and that this relationship
holds similarly for other TEG geometries.
In practical TEG systems for waste heat recovery maximum
power point tracking (MPPT) converters are employed to maximise
the power produced by the TEGs at any time. Due to the fast fre-
quency response of the electronic converters, both the fractional
open-circuit algorithm [36] as well as hill climbing algorithms
[37,38] set the operating load at half of the open-circuit voltage,
which is the optimum operating point of the constant temperature
difference electrical characterisation.
The next aim of the program is to compare the value of maxi-
mum power found before to the one that would be set if the load
voltage was to be continuously adjusted to half of the instanta-
neously resulting open-circuit voltage. This does not mean that
an open-circuit condition is applied to the load (in which case
the temperature difference would go back to
D
T
OC
¼195

C,
corresponding to V
OC
init
¼8:95 V, after a certain transient time).
The ‘‘instantaneous’’ V
OC
corresponds to the voltage that would
be established immediately after a sudden disconnection of the
load, and it is always smaller than V
OC
init
¼8:95 V. The program
uses a recursive loop to adjust the operating voltage V
load
at half
of the V
OC
calculated in the previous iteration. The recursive opera-
tion is needed because when a new voltage is applied the current
changes accordingly and so does the temperature difference. A
new V
OC
is established and V
load
must be updated again. The pro-
gram is considered to have achieved convergence when the differ-
ence between V
load
and V
OC
=2 is less than 1 mV. This V
OC
is marked
by a magenta circle on the primary y-axis of Fig. 2 (7.14 V), while
the open-circuit voltage related to the maximum power point is
marked by a black circle (7.39 V). As a result of this analysis the
load chosen (I
HV
¼1:86 A) is greater than I
MP
¼1:58 A, leading to
a smaller
D
T¼149:6

C and a power produced of 6.62 W, which
is 2.9% smaller than P
MP
¼6:82 W.
Summarising these results, the output voltage that leads to
maximum power production in case of fixed thermal input power
is greater than V
OC
=2 but correspondent to almost half of V
OC
init
. The
smaller current drawn reduces the Peltier effect, thus resulting in a
higher temperature difference across the device and a correspond-
ing higher power produced.
It is very important to re-iterate that these results are calculated
for the thermal steady-state under constant thermal input power,
Table 1
The a;b;c;d;e;fcoefficients used in the simulations.
V
OC
ðVÞR
int
ð
X
Þ
a(V=K
2
)b(V/K) c(V) d(
X
=K
2
)e(
X
=K) f(
X
)
710
5
0.0639 0:8536 910
6
0.0062 1.1972
1
www.mathworks.com.
2
by European Thermodynamics Ltd (www.europeanthermodynamics.com/index.
php/products/thermal/thermoelectrics/power-generation/gm250.html).
A. Montecucco et al. / Applied Energy 149 (2015) 248–258 251

which is usually reached after several minutes. In some
thermoelectric power generation applications the thermal input
power varies fairly rapidly with time, e.g., exhaust gas mass flow
and temperature. When varying the electrical operating point from
V
OC
=2toV
MP
, the change in hot-side temperature is quite slow –
dominated by the thermal time constants of the system – so that
the immediate change in power production could be negligible, if
not negative. However, for certain driving conditions, or in other
applications with slower variations of thermal input power, setting
the load at V
MP
instead of V
OC
=2 could prove beneficial.
3. Effect of the pellets packing factor
This section studies the electrical performance and thermal
behaviour exhibited by commercial TEG devices made of different
number of pellets (or legs) with dissimilar sizes, but with the same
overall size and height.
Firstly, assume that two TEGs can hypothetically be produced
without clearance space between pellets. If these two devices are
made of the same quantity of thermoelectric material but have pel-
lets of dissimilar size (and same height), they will produce the
same amount of electrical power and have the same thermal beha-
viour; what changes is the current and voltage rating. In practice it
is impossible to manufacture two thermoelectric devices like those
just described. The necessary clearance space between pellets usu-
ally leads to different packing (or fill) factors for devices of equal
overall dimensions but with different pellet sizes. The clearance
space, labelled hereafter
s
, usually ranges from 0.8 to 1.2 mm; as
a consequence, commercial devices with wider pellets contain a
greater quantity of thermoelectric material (considering constant
the pellets’ height). In a square thermoelectric device with total
surface area Acomprising of Nsquare pellets, the side length
x
of each pellet can be calculated as
x
¼ffiffiffi
A
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Nþ2
p
s
ffiffiffiffiffiffiffiffiffiffiffiffiffi
Nþ2
pð5Þ
where Nþ2 is used because two locations without pellets are occu-
pied by the two electrical wires. In Eq. (5) it was considered the
same number of pellets and clearance spaces in each side of the
device (half clearance space at each corner).
The packing factor, /, is defined as the ratio of surface area of
thermoelectric material (N
x
2
) over the total surface area (A)of
the module and obtained from
/¼
NAþðNþ2Þ
s
2
2ffiffiffi
A
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Nþ2
p
s
hi
AðNþ2Þð6Þ
In order to compare the thermoelectric performance of TEGs
with the same Abut different
x
, the parameters used in Section 2
are normalised to the mechanical parameters of device GM250-
127-14-10 (N
norm
¼254;A
norm
¼40 40 mm
2
;
x
norm
¼1:4 mm)
and then adapted to a TEG device with different number of pellets
Nand side length
x
. The same height of pellets is used. This proce-
dure is described next:
the pellet side length wis calculated from Eq. (5),
a;b;c(related to V
OC
) are scaled by N=254,
d;e;f(related to R
int
) are scaled by 1:4
2
N=254
x
2
,
the open-circuit thermal conductance of the pellets, K
pellets
, var-
ies linearly with the packing factor, hence it is scaled by
x
2
N
1:4
2
254
.
The parameters d;e;fare obtained experimentally therefore R
int
includes the effects of the electrical contact resistances (copper
tabs). The scaling of R
int
depends on the number of pellets to which
the electrical contact resistances are related, whence scaled appro-
priately. The thickness of the Alumina ceramic substrate is consid-
ered constant at 1 mm with thermal conduction coefficient
j
ceramic
¼36 W=mK. The thermal resistance of the thin copper con-
tacts is considered negligible, therefore the total thermal conduc-
tivity of the TEG device is calculated as 1=K
TEG
¼2=K
ceramic
þ
1=K
pellets
.
Table 2 presents the parameters of the three TEGs considered in
this section: the first is the TEG analysed in Section 2
(N¼254;
s
¼1:1 mm), the second is a TEG with N¼482 and
s
¼0:8 mm (GM250-241-10-12 by European Thermodynamics
Ltd), and the third is a TEG with N¼110 and
s
¼1 mm (equivalent
Fig. 2. Electrical characterisation of a TEG with fixed thermal input power Q
H
¼156 W. V-I line in blue, P-I curve in red,
D
T-I line in green. Point of maximum power
production: V
MP
¼4:32 V;I
MP
¼1:58 A;
D
T¼155

C with V
OC
¼7:39 V (black circle). Point set by continuously adjusting V¼V
OC
=2:V
HV
¼3:56 V;I
HV
¼1:86 A;
D
T¼150

C,
with V
OC
¼7:14 V (magenta circle). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
252 A. Montecucco et al. / Applied Energy 149 (2015) 248–258

to GM250-31-28-12 by European Thermodynamics Ltd scaled from
30 30 mm
2
to 40 40 mm
2
for comparison with the other two
devices). A comparison of the values of R
int
and V
OC
obtained with
this procedure to experimental results shows a maximum error of
6.94%. This can be due to different performance of the Bi
2
Te
3
used,
electrical and thermal contact resistances, manufacturing toler-
ances or test accuracy.
Table 3 presents the three sets of results related to the TEGs
used (Table 2) and obtained maintaining constant Q
H
¼156 W
and T
C
¼25

C. The first two devices have the same packing factor
/, which results in the same values for thermal conductivity Kand
temperature difference
D
Tacross the device. The maximum elec-
trical power produced for constant thermal power through the
device, Peltier power and Joule power are also equivalent. The third
device, characterised by a packing factor 1.7 times greater, demon-
strates worse performance because its higher Kleads to a con-
siderably lower
D
Tand consequently lower maximum electrical
power (1.5 times smaller). As a comparison with the results pre-
sented in Section 2(for the first TEG), in which the difference
D
P
out
between maximum power and power set by the half open-
circuit voltage method was 2.9%, for the third TEG this difference
rises to 4.6%. This difference is not related to the particular geome-
try of the TEG, but to the temperature difference and it increases
when the temperature difference decreases. In other words
D
P
out
decreases at greater
D
T, which is when the power output and the
thermal-to-electrical efficiency increase.
It was already introduced in Section 2that the maximum power
is extracted for a voltage greater than V
OC
=2. Consider the at-load
voltage as a fraction of the ’’instantaneous’’ open-circuit voltage:
V¼bV
OC
ð7Þ
In Table 3 b¼0:585 for the first two TEGs and b¼0:608 for the
third TEG. Running the same simulation for other values of Q
H
it
is noteworthy to discover that this relationship depends on
D
T:b0:62 at
D
T¼50

C and it varies linearly towards b0:55
at
D
T¼275

C. This variation in the optimum value for bmight
be related to the variation of the thermoelectric coefficients with
the temperature difference.
The power pumped from the hot side due to the Peltier effect
(P
Pelt
) remains almost constant, but this effect is further investi-
gated in Section 4. For all devices analysed P
Pelt
þK
D
TP
Joule
is
equal to Q
H
¼156 W (note that P
Joule
¼0:5RI
2
).
Table 4 compares the results obtained in this analysis with pre-
vious literature. For all three TEGs the optimum load is different
from the one proposed by Yazawa and McCarty
(R
lit
¼R
int
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZT
p)[23,24]. The difference is around 11% for the
first two TEGs and 17% for the third one. However, it must be
highlighted that these results cannot be used for a direct compar-
ison because they are obtained with constant heat flux but varying
temperature difference (as opposed to constant temperature dif-
ference and varying heat input as done in literature). ZT is calcu-
lated considering the variation of both the Seebeck coefficient,
a
,
and the electrical conductivity,
r
, with the temperature difference
(from Table 3 and Eq. (3)), while the thermal conduction coeffi-
cient,
j
, is maintained constant at 1.5 W/mK.
The ratio between the open-circuit and short-circuit tempera-
ture differences, mD
T
, agrees with the one proposed by Min [17],
mD
T
¼1þZT, in all cases.
The differences evinced from Table 3 highlight the great influ-
ence of the packing factor on the performance of commercial
thermoelectric devices. In particular its influence on the thermal
conductivity makes one device suited for a particular application,
depending on the available thermal energy, to establish a high
temperature difference, or to protect the TEG device from excessive
temperatures.
4. Optimisation of the pellets packing factor
This section presents an expansion of the analysis presented in
Section 3that can be used to design the geometry of the pellets and
the clearance space between them, for any preferred thermal
operating point and for constant pellets thickness (or height).
The same code used in the previous sections is inserted in a nested
for loop relative to the number of pellets Nand the clearance space
s
to calculate the geometry of TEG devices that leads to maximum
power generation in conditions of constant thermal power flowing
through the device. For each pair of values of Nand
s
the algorithm
calculates the pellet side length
x
and the packing factor /from
Eqs. (5) and (6), respectively. The algorithm rejects values of
x
smaller than a certain threshold (
x
min
¼0:75 mm), due to practical
manufacturing constrains. The impact of solder contacts on the
electrical resistance is included in this study because the values
of the parameters d, e, f (in Eq. (3) and Table 1) are obtained experi-
mentally, thus clearly including the electrical contact resistances.
The algorithm excludes data points that would establish hot side
temperatures greater than T
max
¼300

C for Bi
2
Te
3
. The algorithm
finally stores all the values of interest in arrays and plots the sur-
faces related to the xand yaxes of
s
and N, respectively.
A simulation was computed varying the number of pellets
between 50 and 600 and the clearance space between 0.6 and
2.4 mm, for Q
H
¼156 W and T
C
¼25

C. The resulting surfaces
for the maximum electrical power output, temperature difference,
Peltier power, pellets side length, fill factor and volume of
thermoelectric material needed are shown in Fig. 3 from the z-axis;
the granularity is due to the discrete values used for Nand
s
. It can
Table 2
Geometrical, thermal and electrical parameters of the three TEG devices considered.
N
s
(mm) w(mm) /K(W/K) R
int
(
X
)
TEG 1 254 1.1 1.4 0.31 0.8 1.94
TEG 2 482 0.8 1 0.31 0.8 6.97
TEG 3 110 1 2.8 0.53 1.37 0.184
Table 3
Simulation results to compare the effect of pellets’ dimensions and packing factor on the thermoelectric performance when considering a system with constant thermal input
power.
D
T(K)
a
(
l
V=K) I(A) V(V) V
OC
(V) P
elec
(W)
D
P
out
(%) P
Pelt
(W) P
Joule
(W)
TEG 1 155.5 187 1.58 4.32 7.39 6.82 2.90 34.1 2.4
TEG 2 155.0 187 0.83 8.2 14.0 6.81 2.92 33.9 2.4
TEG 3 92.8 190 4.08 1.18 1.94 4.80 4.59 33.3 1.6
Table 4
Comparison of results from this work with literature: Rlit ¼Rint ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZT
p[23],
mDT¼DTOC
DTSC [17].
ZT R
load
R
lit D
T
OC D
T
SC
m
D
T here
m
D
T lit
TEG 1 0.585 2.73 2.45 195.0 123.8 1.57 1.59
TEG 2 0.585 9.88 8.77 194.3 123.5 1.57 1.59
TEG 3 0.633 0.29 0.24 116.5 73.2 1.59 1.63
A. Montecucco et al. / Applied Energy 149 (2015) 248–258 253

be seen that there are several combinations of Nand
s
that lead to
almost identical performance. Fig. 3c shows that the pellets
side length progressively decreases for greater Nand
s
. In each plot
of Fig. 3 the points related to the TEGs of Table 2; the point
that provides optimum performance is also marked
(N¼180;
s
¼1:7mm;
x
¼1:27 mm).
Fig. 3b shows that the maximum allowed temperature differ-
ence of 275 °C can be reached with numerous combinations of N
and
s
, all correspondent to the same values of fill factor
(/¼0:18), as per Fig. 3d, and thermal conductivity
(K
TEG
¼0:47 W=K). The optimum point produces P
max
¼8:34 W at
V¼4:46 V;I¼1:87 A.
The Peltier power is similar for all TEGs and it seems to follow
the variation of the Seebeck coefficient with temperature, with
peak values at temperatures lower than T
max
for Bi
2
Te
3
. The power
due to the Joule effect increases when power generation is greater.
It can be seen from Fig. 3a that the commercial TEG with large
pellets produces almost half of what could potentially be extracted
in this constant Q
H
situation. This is due to its high packing factor
(/¼0:53), resulting from the use of large pellets (
x
¼2:8 mm)
and clearance space of similar value to that of the other two com-
mercial TEGs (
s
¼1 mm) that produce around 6.8 W. However, the
results presented in this section suggest that high power can still
be obtained using big pellets provided that the clearance space
between them is increases substantially, e.g.,
N¼80;
x
¼2mm;
s
¼2:4 mm, in order to keep the packing fac-
tor /(and consequently the thermal conductivity K) down.
However, this solution would lead to low voltage/high current rat-
ings for the device, which is usually more difficult to deal with in
the power electronics conditioning system. The volume of
necessary thermoelectric material remains constant for same val-
ues of packing factor.
Fig. 3a shows that an optimisation of the pellets geometry could
improve power production from the analysed constant heat condi-
tion of Q
H
¼156 W from 6.8 W to 8.3 W, which corresponds to an
increment of almost 20%.
The thermal to electrical efficiency of thermoelectric generators
is calculated as
g
¼P
elec
Q
H
ð8Þ
and Q
H
is constant, therefore an increase in electrical power output
directly improves
g
. In this case
g
would increase by around one
percentage from 4.36% to 5.34%.
Longer pellets reduce the thermal conductivity of the device,
thus improving the efficiency, but at the same time they increase
the electrical resistance, hence the current output decreases
because the open-circuit voltage remains the same. The effect of
pellet length (or thickness) is analysed in Section 5.
5. Optimisation of the pellets geometry
The aim of this section is to investigate the effect on perfor-
mance and material cost of a varying pellets thickness (or height).
Section 4highlighted that for a constant pellets thickness (or
height) a certain pellets packing factor leads to same performance,
irrespective of the values of Nand
s
. Therefore the simulation tool
is modified to analyse performance as a function of the pellets
packing factor, /, and height, h. For this analysis the results are
(a) Electrical Power Output (b) Temperature Difference
(c) Pellets Side Length (d) Pellets Fill Factor
Fig. 3. Surface plots (viewed from the z-axis) as functions of the number of pellets (y-axis) and the clearance space (x-axis). Note that the colours selected by Matlab for each
box are related to the values of the left corners. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
254 A. Montecucco et al. / Applied Energy 149 (2015) 248–258

obtained using N¼250;Q
H
¼156 W;
j
¼1:5W=mK;T
C
¼25

C
and his varied between 0.6 and 2.4 mm.
Fig. 4 shows the resulting surfaces, viewed from the z-axis, of
the electrical power output, temperature difference, internal resis-
tance and thermal conductance. It is easy to note that multiple
combinations of pellets height and packing factor lead to maxi-
mum performance, and the data cursors mark the calculated maxi-
mum power point for reference. However, a noteworthy result can
be obtained by calculating the volume of thermoelectric material
needed for each pellets design. This is plotted in Fig. 5, where it
can be seen that the volume is minimum for short pellets charac-
terised by a small fill factor, and maximum when using long pellets
with high packing factor. The difference between minimum and
maximum volumes leading to identical performance is almost
one order of magnitude, which constitutes a significant difference.
This simulation is run also for different values of input thermal
power: Q
H
¼50;100;150;200;250 W. Some important results are
listed in Table 5. In all cases, the simulation tool aims at finding
the TEG design that establishes a temperature difference of
275 °C across the TEG, therefore the open-circuit voltage, which
depends on Nand
a
, is always V
OC
¼11:2 V. Confirming the results
of Section 2, the load voltage for maximum power is always at
V
MP
¼6:2V.
The thermal-to-electrical efficiency and figure of merit are in all
cases
g
¼5:35%and ZT ¼0:455, respectively. The electrical power
output, P
out
, increases linearly with Q
H
because
g
is constant:
P
out
¼
g
Q
H
. The same trend is exhibited by the current produced
at the maximum power point: I
MP
¼P
out
=V
MP
.
It is very important to note from Table 5 that the optimum TEG
thermal conductance varies linearly with Q
H
. This does not mean
that the optimum Kcan simply be calculated by Q
H
=
D
T. In fact, this
occurs because the heat removed from the hot side by the Peltier
effect is always the same fixed percentage of Q
H
, equal to 20.2%
in all cases. The same occurs for the Joule power, whose percentage
of Q
H
is 2.15%. This result is noteworthy, because it allows to cal-
culate the optimum thermal conductance to achieve a desired tem-
perature difference
D
Tby
K
opt
¼Q
H
0:202 Q
H
þ0:0215 Q
H
D
T0:82 Q
H
D
Tð9Þ
The efficiency obtained (
g
¼5:35%) seems to agree with the
value proposed by Min [39]:
g
max
¼
D
T
T
H
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZT
p1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZT
pþT
C
=T
H
¼5:7%ð10Þ
Contrary to the comparison results obtained in Section 3(Table 4),
in Table 5 R
lit
¼R
int
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þZT
pR
load
, with a difference of only 2.5%.
This could be determined by the fact that the results in this section
are obtained for optimised values at maximum allowed tempera-
ture difference. The difference then widens for less optimised pel-
lets and lower temperatures.
Similar result trends were obtained also using N¼100 and
N¼400.
Summarising the results achieved in this section, the output
power and efficiency do not depend on the pellets geometry and
can be obtained by a multitude of combinations of the TEG geo-
metrical parameters (mainly pellets number, size and clearance
space between them). However, the quantity of thermoelectric
material can be minimised when using shorter pellets with small
fill factor.
(a) Electrical Power Output (b) Temperature Difference
(c) TEG Internal Resistance (d) TEG Thermal Conductance
Fig. 4. Surface plots (viewed from the z-axis) as functions of the pellets fill factor (y-axis) and height (x-axis), for Q
H
¼156 W.
A. Montecucco et al. / Applied Energy 149 (2015) 248–258 255

6. Discussion of results
The first part of this work presented the thermal and electrical
characterisation of TEGs under steady-state constant heat condi-
tions (Fig. 2). The relationships between the optimum operating
voltage and both the initial and the instantaneous open-circuit
voltages are provided in Sections 2 and 3.
The second part offered an innovative solution to calculate the
optimum thermal conductance (Eq. (9)) to obtain a desired tem-
perature difference across the TEG, for a constant heat input.
Also, the variation trend of thermoelectric material required for
any combination of pellets height and fill factor was plotted in
Fig. 5 to show that optimum values of the pellets geometrical
parameters can be selected to minimise the thermoelectric mate-
rial cost.
In this study the thermal contact resistances, as well as the
thermal resistance of the copper tabs and solder, were not consid-
ered. The variation of thermal conductivity with temperature was
not considered due to the difficulty in obtaining precise measure-
ments of it. Nonetheless, in the results presented here the tem-
perature difference between points around the optimum ones
does not vary significantly, hence the variation of thermal conduc-
tivity should not affect the results. For what concerns the constant
power characterisation, the temperature difference changes by
around 70 °C thus not leading to great variation in thermal
conductivity.
The electrical resistance of the copper tabs and solder was con-
sidered because the experimental values used in the simulations
implicitly include this resistance. This work included the variation
of both the Seebeck coefficient and the electrical resistivity with
the temperature as per Eq. (3), while the open-circuit thermal
conductivity has been considered constant. It must be noted that
the results presented in this article are related to the experimental
performance of the device tested. Nonetheless the results can be
extended to similar commercial devices. The results trend can be
considered of general validity, but particular values and coeffi-
cients depend on the materials used. This analysis is valid in
steady-state and transient periods between load changes are not
accounted for. The amount of time needed to reach steady-state
and the change in power output during the transient could be stud-
ied in future work to provide additional knowledge.
The results presented in this work are noteworthy because they
offer new insights on the behaviour of thermoelectric generators in
conditions of constant heat input. The relationships for maximum
power and efficiency, as much as the geometrical optimisation of
the TEG device differ from the case of constant temperature differ-
ence, which has already been thoroughly investigated in literature.
In this latter case the temperature sources across the TEG device
(or after the hot and cold heat exchangers) are treated as ideal,
i.e. capable of providing ’’unlimited’’ thermal power. However, as
explained in detail in Section 2, in most waste heat recovery appli-
cations the available heat energy is limited, therefore the resulting
temperatures depend significantly on the effective thermal resis-
tance of the TEG, which depends both on its geometrical design
and on the current it generates.
The results of the optimisation tool described in Section 5are
significant, because they suggest that same performance can be
obtained even if using different amounts of thermoelectric mate-
rial. The explanation behind this result is related to the variations
of the TEG internal resistance, R
int
(Fig. 4c), and the pellets thermal
conductance, K
legs
(Fig. 4d), with the height, h, and cross-sectional
area,
x
2
, of the pellets (or legs). Neglecting for this explanation
the electrical and thermal contact resistances, R
int
and K
legs
can
be written as
R
int
¼NR
int;pellets
¼N
q
h
x
2
K
legs
¼N
j
legs
x
2
hð11Þ
where R
int;pellets
and j
legs
are the internal resistance and thermal con-
ductivity coefficient of the pellets, respectively. In Eq. (11) it is pos-
sible to maintain R
int
and K
legs
constant if hand x
2
are varied by the
same amount. In particular, decreasing them the volume of
Fig. 5. Volume of thermoelectric material required to produce a TEG with N¼250 pellets of varying dimensions, as function of the pellets fill factor (y-axis) and height
(x-axis), for Q
H
¼156W.
Table 5
Optimisation of pellets packing factor and length of a TEG with N¼250 pellets, for
five values of constant thermal input power (QH¼50;100;150;200;250 W).
Q
H
R
int
KR
load
R
lit
50 11.63 0.15 14.30 14.03
100 5.80 0.30 7.18 7.00
150 3.90 0.45 4.82 4.70
200 2.90 0.60 3.59 3.49
250 2.32 0.74 2.86 2.79
256 A. Montecucco et al. / Applied Energy 149 (2015) 248–258

thermoelectric material, V
TE
¼Nhx
2
can be minimised, thus
achieving significant cost savings.
As a consequence, it is possible to design the pellets geometrical
parameters in order to minimise the thermoelectric material
required while still producing maximum power.
Sections 4 and 5 provided the optimum parameters for the TEG
pellets architecture that maximise power production for the
selected value of Q
H
. The obtained parameters might not maximise
the power output for other thermal powers. Using these parame-
ters with a greater Q
H
will result in an excessive temperature dif-
ference established across the TEG, therefore the system designer
should design for the worst case, or include means for removing
excess thermal energy, e.g., bypass valve in automotive exhaust
systems, and find the best overall compromise: the optimum
combination for N;
x
;hand
s
must be selected over the whole
expected thermal operating range and relative to the particular
system. Designing the TEG geometry as done in this work provides
meaningful results because the temperature difference is calcu-
lated considering the parasitic Peltier effect when the TEG is
operated.
7. Conclusions
This research study made use of a simulation tool based on
steady-state equations for thermoelectric generators and experi-
mental data to investigate behaviour and performance of TEGs
with constant heat across them, and to find the pellets optimum
geometrical design leading to best performance and lowest mate-
rial expenditure.
This work offered a new solution to set the optimum electrical
operating point in conditions of constant heat and explained how
the cost associated to the thermoelectric material can be min-
imised, while still producing maximum power at the selected heat
input condition.
This work firstly analysed the characterisation of TEG devices
for constant thermal input power. It provides useful information
on the performance that TEGs are likely to produce in practical
applications, because in most waste heat applications at any time
the maximum available heat is limited. Therefore changes in the
current extracted from the TEG influence the temperature differ-
ence established across it, due to the Peltier effect. Consequently,
the output power is a function of the heat power and the electrical
current. Its maximum value is obtained when the load voltage
approximates half of the steady-state open-circuit voltage and it
is higher than half of the instantaneous open-circuit voltage.
This work also offered an insight on how the number and size of
pellets, and the clearance space between them (hence the packing
factor) influence the power output when considering a TEG with a
fixed amount of thermal power through it. Furthermore, this work
described an algorithm that calculates the best architecture of TEG
pellets to maximise power production and minimise material cost
for a selected constant heat condition.
The results evinced from this work can improve the design and
the MPPT control of TEG devices for waste heat recovery applica-
tions, leading to enhancements in efficiency and power production.
Also, they stress the importance of the Peltier effect and the need
to consider not only the almost instantaneous change determined
by the TEG’s electrical response, but also the slower effect on the
temperature difference determined by changes in the electrical
load and thermal flux.
Future work will focus on experimental tests and transient sim-
ulations, applying the simulation tool described in [40], which con-
siders all thermoelectric phenomena from the system point of
view.
Acknowledgements
The authors would like to thank Dr. Gao Min of the University of
Cardiff and Dr. Lourdes Ferre Llin of the University of Glasgow for
the interesting discussions on this subject.
This work was partially supported by the Engineering and
Physical Sciences Research Council (EPSRC) under Grant EP/
K022156/1 (RCUK).
References
[1] Ramadass YK, Chandrakasan AP. A battery-less thermoelectric energy
harvesting interface circuit with 35 mV startup voltage. IEEE J Solid-State
Circuits 2011;46:333–41.
[2] Elefsiniotis A, Kokorakis N, Becker T, Schmid U. A thermoelectric-based energy
harvesting module with extended operational temperature range for powering
autonomous wireless sensor nodes in aircraft. Sens Actuat A: Phys
2014;206:159–64.
[3] Kinsella C, O’Shaughnessy S, Deasy M, Duffy M, Robinson aJ. Battery charging
considerations in small scale electricity generation from a thermoelectric
module. Appl Energy 2014;114:80–90.
[4] Crane D, LaGrandeur J, Jovovic V, Ranalli M, Adldinger M, Poliquin E, et al. TEG
on-vehicle performance and model validation and what it means for further
TEG development. J Electron Mater 2012.
[5] Risse S, Zellbeck H. Close-coupled exhaust gas energy recovery in a gasoline
engine. Res Therm Manage 2013;74:54–61.
[6] Champier D, Bedecarrats JP, Kousksou T, Rivaletto M, Strub F, Pignolet P. Study
of a TE (thermoelectric) generator incorporated in a multifunction wood stove.
Energy 2011;36:1518–26.
[7] O’Shaughnessy S, Deasy M, Kinsella C, Doyle J, Robinson A. Small scale
electricity generation from a portable biomass cookstove: prototype design
and preliminary results. Appl Energy 2013;102:374–85.
[8] Chen M, Lund H, Rosendahl La, Condra TJ. Energy efficiency analysis and
impact evaluation of the application of thermoelectric power cycle to todays
CHP systems. Appl Energy 2010;87:1231–8.
[9] Qiu K, Hayden ACS. Development of a novel cascading TPV and TE power
generation system. Appl Energy 2012;91:304–8.
[10] Sark WV. Feasibility of photovoltaic thermoelectric hybrid modules. Appl
Energy 2011;88:2785–90.
[11] Kaibe H, Makino K, Kajihara T, Fujimoto S, Hachiuma H. Thermoelectric
generating system attached to a carburizing furnace at Komatsu Ltd., Awazu
Plant. In: 9th European conference on thermoelectrics, 2011 (ECT’11), p. 524–
7.
[12] Suter C, Jovanovic Z, Steinfeld A. A 1;kWe thermoelectric stack for geothermal
power generation: modeling and geometrical optimization. Appl Energy
2012;99:379–85.
[13] Patyk A. Thermoelectric generators for efficiency improvement of power
generation by motor generators: environmental and economic perspectives.
Appl Energy 2013;102:1448–57.
[14] Rowe D, Min G. Evaluation of thermoelectric modules for power generation. J
Power Sources 1998;73:193–8.
[15] Lineykin S, Ben-Yaakov S. Modeling and analysis of thermoelectric modules.
IEEE Trans Ind Appl 2007;43:505–12.
[16] Laird I, Lovatt H, Savvides N, Lu D, Agelidis VG. Comparative study of
maximum power point tracking algorithms for thermoelectric generators. In:
Australasian universities power engineering conference, 2008 (AUPEC’08).
[17] Min G, Yatim NM. Variable thermal resistor based on self-powered Peltier
effect. J Phys D: Appl Phys 2008;41:222001.
[18] Kumar S, Heister SD, Xu X, Salvador JR, Meisner GP. Thermoelectric generators
for automotive waste heat recovery systems Part I: Numerical modeling and
baseline model analysis. J Electron Mater 2013;42:665–74.
[19] Chen L, Cao D, Yi H, Peng FZ. Modeling and power conditioning for
thermoelectric generation. In: Power electronics specialists conference. IEEE;
2008. p. 1098–103.
[20] Mayer P, Ram R. Optimization of heat sinklimited thermoelectric generators.
Nanoscale Microscale Thermophys Eng 2006;10:143–55.
[21] Gomez M, Reid R, Ohara B, Lee H. Influence of electrical current variance and
thermal resistances on optimum working conditions and geometry for
thermoelectric energy harvesting. J Appl Phys 2013:0–8.
[22] Montecucco A, Buckle JR, Knox AR. Solution to the 1-D unsteady heat
conduction equation with internal Joule heat generation for thermoelectric
devices. Appl Therm Eng 2012;35:177–84.
[23] Yazawa K, Shakouri A. Cost-efficiency trade-off and the design of
thermoelectric power generators. Environ Sci Technol 2011;45:7548–53.
[24] McCarty R. Thermoelectric power generator design for maximum power: its
all about ZT. J Electron Mater 2012;42:1504–8.
[25] Nemir D, Beck J. On the significance of the thermoelectric figure of merit Z. J
Electron Mater 2010;39:1897–901.
[26] Apertet Y, Ouerdane H, Glavatskaya O, Goupil C, Lecoeur P. Optimal working
conditions for thermoelectric generators with realistic thermal coupling. EPL
(Europhys Lett) 2012;97:28001.
[27] Wang Y, Dai C, Wang S. Theoretical analysis of a thermoelectric generator
using exhaust gas of vehicles as heat source. Appl Energy 2013;112:1171–80.
A. Montecucco et al. / Applied Energy 149 (2015) 248–258 257

[28] Rezania A, Rosendahl L, Yin H. Parametric optimization of thermoelectric
elements footprint for maximum power generation. J Power Sources
2014;255:151–6.
[29] Lee H. Optimal design of thermoelectric devices with dimensional analysis.
Appl Energy 2013;106:79–88.
[30] Jang J-Y, Tsai Y-C. Optimization of thermoelectric generator module spacing
and spreader thickness used in a waste heat recovery system. Appl Therm Eng
2013;51:677–89.
[31] Favarel C, Bédécarrats J-P, Kousksou T, Champier D. Numerical optimization of
the occupancy rate of thermoelectric generators to produce the highest
electrical power. Energy 2014;68:104–16.
[32] Kajihara T. Study of thermoelectric generation unit for radiant waste heat. In:
12th European conference on thermoelectrics, 2014 (ECT’14).
[33] Fraisse G, Ramousse J, Sgorlon D, Goupil C. Comparison of different modeling
approaches for thermoelectric elements. Energy Convers Manage
2013;65:351–6.
[34] Sandoz-Rosado EJ, Weinstein SJ, Stevens RJ. On the Thomson effect in
thermoelectric power devices. Int J Therm Sci 2013;66:1–7.
[35] Montecucco A, Siviter J, Knox AR. The effect of temperature mismatch on
thermoelectric generators electrically connected in series and parallel. Appl
Energy 2014;123:47–54.
[36] Montecucco A, Knox AR. Maximum power point tracking converter based on
the open-circuit voltage method for thermoelectric generators. IEEE Trans
Power Electron 2015;30:828–39.
[37] Kim R-Y, Lai J-S. A seamless mode transfer maximum power point tracking
controller for thermoelectric generator applications. IEEE Trans Power
Electron 2008;23:2310–8.
[38] Kim R-Y, Lai J-S, York B, Koran A. Analysis and design of maximum power point
tracking scheme for thermoelectric battery energy storage system. IEEE Trans
Ind Electron 2009;56:3709–16.
[39] Min G. Thermoelectric energy harvesting. In: Energy harvesting for
autonomous systems, vol. 413–414, Artech House; 2010, p. 135–57.
[40] Montecucco A, Knox AR. Accurate simulation of thermoelectric power
generating systems. Appl Energy 2014;118:166–72.
258 A. Montecucco et al. / Applied Energy 149 (2015) 248–258