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applied
sciences
Article
Simulink Model of a Thermoelectric Generator for Vehicle
Waste Heat Recovery
Nicolae Vlad Burnete 1, Florin Mariasiu 1, * , Dan Moldovanu 1and Christopher Depcik 2
Citation: Burnete, N.V.; Mariasiu, F.;
Moldovanu, D.; Depcik, C. Simulink
Model of a Thermoelectric Generator
for Vehicle Waste Heat Recovery.
Appl. Sci. 2021,11, 1340. https://
doi.org/10.3390/app11031340
Academic Editor: Alberto Benato
Received: 18 January 2021
Accepted: 29 January 2021
Published: 2 February 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Automotive Engineering and Transports Department, Technical University of Cluj-Napoca,
400641 Cluj-Napoca, Romania; nicolae.vlad.burnete@auto.utcluj.ro (N.V.B.);
dan.moldovanu@auto.utcluj.ro (D.M.)
2Department of Mechanical Engineering, University of Kansas, Lawrence, KS 66045, USA; depcik@ku.edu
*Correspondence: florin.mariasiu@auto.utcluj.ro
Featured Application: The work presented in this study can be used in the design and develop-
ment process of thermoelectric generators for vehicle waste heat recovery.
Abstract:
More than 50% of the energy released through combustion in the internal combustion
engine (ICE) is rejected to the environment. Recovering only a part of this energy can significantly
improve the overall use of resources and the economic efficiency of road transport. One solution
to recoup a part of this otherwise wasted thermal energy is to use thermoelectric generator (TEG)
modules for the conversion of heat directly into electricity. To aid in development of this technology,
this effort covers the derivation of a respectively simple steady-state Simulink model that can be
utilized to estimate and optimize TEG system performance for ICEs. The model was validated against
experimental data found in literature utilizing water cooling for the cold side. Overall, relatively
good agreement was found with the maximum error in generated power around 10%. Following,
it was investigated whether air can be used as a cooling medium. It was established that, at the
same temperature as the water (18.4
◦
C), a flow velocity of 13.1 m/s (or 47.2 km/h) is required to
achieve a similar cold junction temperature and power output. Subsequently using the model with
air cooling, the performance of a TEG installed on a heavy-duty vehicle traveling at 50, 80, 90, and
120 km/h under different ambient temperatures was analyzed. It was determined that both a lower
temperature and a higher flow velocity can improve power output. A further increase of the power
output requires a larger temperature gradient across the module, which can be achieved by a higher
heat input on the hot side.
Keywords:
thermoelectric generator; thermoelectric module; waste heat recovery; internal combustion
engine; steady-state model
1. Introduction
With the advent of the internal combustion engine (ICE) in the late 19th century and
its continued development during the 20th century, humankind has benefited greatly
from its use for transportation. In fact, the 20th century can be defined in one word:
“Mobility”; local, regional, national, and international mobility of goods, people, and
ideas. However, owing to its construction and general operation, the ICE manages to only
partially transform fuel chemical energy into mechanical work. This is illustrated by the
thermal efficiency of modern ICEs that generally only reaches 40–45%. This respectively
low value of energy conversion directly influences the economic efficiency of the road
transport process. For example, a 40 t freight vehicle with a payload of 25 t operating with
an ICE has an overall transport process efficiency equal to 17–18% [
1
]. Furthermore, since
transportation has a significant influence in both national and global economies, increasing
transport process efficiency can immediately reduce operating costs while additionally
helping to decrease greenhouse gas emissions.
Appl. Sci. 2021,11, 1340. https://doi.org/10.3390/app11031340 https://www.mdpi.com/journal/applsci
Appl. Sci. 2021,11, 1340 2 of 33
Most of the thermal energy of the ICE (approx. 30%) is lost through the high tempera-
ture combustion exhaust gases that are discharged directly into the atmosphere. Therefore,
one solution to increase overall efficiency is to recover, at least a part of this significant heat
loss to the environment. For this purpose, research has targeted the design, development,
and implementation of thermoelectric generator modules for directly converting heat into
electricity, which can then be used in hybrid power systems to increase vehicle efficiency.
Thermoelectric devices are environmentally friendly and have numerous other advantages
including the lack of moving parts, the conversion of thermal energy directly into electrical
energy, high reliability, as well as low maintenance requirements. The main disadvantage
of such devices is their low efficiency owing to the small dimensionless figure of merit (
ZT
)
values currently found in commercial devices. However, there are promising solutions for
achieving higher ZT values even at low to mid temperatures (<150 ◦C) [2].
The research efforts focused on identifying the potential for converting thermal energy
into electricity, through thermoelectric generator (TEG) systems of different configurations
and sizes, installed on the engine’s exhaust path show that it is possible to obtain an
electrical power output of approximately 200–1000 W, depending on vehicle type and
operating conditions [
3
–
6
]. However, research has also shown that the vehicular use of
TEG systems depends on driving conditions with negative effects found when conditions
have a high changing frequency [7].
The development of numerical models investigating the efficient use of TEG systems
centers mainly on: Operating temperatures, optimizing geometric dimensions [
8
], different
materials in TEG construction [
9
], and design optimization of hot and cold side heat
exchangers [
10
]. Furthermore, TEG system design focuses on the energetic efficiency
of conductive and convective system types [
11
]. In this area, Fan et al. [
12
] proposed a
numerical model to determine the optimum values for the thermocouple leg lengths and
cross-sectional areas. Their analysis showed that, for given thermal boundary conditions,
the maximum output power can be improved by optimizing the leg length or cross-sectional
area. Other efforts by Yu et al. [
13
] developed a mathematical model to investigate the
effects of different vehicle engine operating regimes (e.g., cold-start and different loads)
on TEG performance. The major conclusion of their research was that TEG performance
is influenced more by vehicle speed than by ambient temperature. Wang et al. [
14
] used
a three-dimensional computational fluid dynamics simulation to analyze power output,
temperature distribution, and pressure drop of a TEG system with different internal
structures of the heat exchanger. Their model considered the coverage and placement of
the thermoelectric modules on the heat exchanger surface. The primary outcome was that
increasing the number of modules may eventually saturate the total power output. Du
et al. [
15
] investigated the design of a cooling channel for a thermoelectric module using a
CFD software program (Fluent). Their results showed that in general, the output power is
higher when using liquid as cooling fluid compared to air.
The analysis of this literature shows that there is a wide variety of numerical models
for TEG simulation with different types of sophistication. However, most models are
usually only briefly presented, with little detailing of the underlying equations and of the
implementation. Therefore, the aim of this study is to develop and validate a respectively
simple steady-state Simulink model that can be utilized to estimate and optimize TEG sys-
tem performance. The model is intended to aid in the design and development of thermal
energy recovery systems for ICEs, but it can be applied in other areas as well. Employing
the described model early in the design process can facilitate whether thermal energy
recovery is feasible for the TEG system under specific operating conditions. Furthermore,
the proposed Simulink model can be integrated into existing software solutions for the
analysis of different energy sources and powertrains.
2. Thermoelectric Model Equations
A thermoelectric couple (TC) consists of p-type and n-type semiconductor legs that are
connected thermally in parallel and electrically in series (Figure 1). When heat is applied
Appl. Sci. 2021,11, 1340 3 of 33
to produce electric current, the TC operates as a generator. If, however, current is applied
at the terminals, then the TC operates as a heat pump. For this effort, the focus is on
thermoelectric generation and the use of the model as a heat pump is left to future work.
Figure 1. Thermoelectric couple operating as a generator.
If the two junctions of the TC are subjected to different temperatures (
∆
T = T
h−
T
c
[K], where T
h
[K]—hot side temperature, T
c
—cold side temperature), a potential difference,
proportional to the temperature variation, will appear (V
OC,TC
[V]—open circuit voltage,
VLoad,TC [V]—potential difference across a load connected to the TC):
VOC,TC =αTC ·(Th−Tc)(1)
where αTC [V/K] is the relative Seebeck coefficient:
αTC =αp−αn. (2)
If the TC is connected to an external load with a resistance R
Load,TC
[
Ω
], a current flow
will appear (I [A]). Consequently, the potential difference VLoad,TC can be expressed as:
VLoad,TC =I·RLoad,TC. (3)
In addition, there is an internal electrical resistance of the TC, R
TC
[
Ω
], which is defined
based on the characteristics of the two legs: Electrical resistivity (
ρp
[
Ω
m]—electrical
resistivity of the p-type leg,
ρn
[
Ω
m]—electrical resistivity of the n-type leg), length (L
p
[m]—length of the p-type leg, L
n
[m]—length of the n-type leg), and cross-sectional area (A
p
[m2]—cross-section area of the p-type leg, An[m2]—cross-section area of the n-type leg).
RTC =ρp
Lp
Ap
+ρn
Ln
An(4)
Knowing that the relation between VLoad,TC and VOC,TC is
VLoad,TC =VOC,TC −IRTC (5)
which results in a derivation of the current:
Appl. Sci. 2021,11, 1340 4 of 33
I=αTC(Th−Tc)
RTC +RLoad,TC
. (6)
This flow of current leads to three additional effects: Peltier heating or cooling, Joule
heating, and the Thomson effect. The latter is relatively small and is usually neglected to
simplify the analysis, as is the case in this study.
If current flows through the junction, heat must be continuously added or rejected to
maintain a constant temperature. Furthermore, this heat flow is proportional to the current
flow and changes direction if the current is reversed. This is called the Peltier effect and is
defined by the Peltier coefficient πTC,junction [J/A]:
πTC,junction =
.
Qjunction
I(7)
where
.
Qjunction
[W] is the rate of heat absorption/rejection at the hot/cold junction. Us-
ing Kelvin relationships, the Peltier coefficient can be expressed based on the Seebeck
coefficient, which is respectively easier to measure:
πTC,junction =αTCTjunction (8)
where T
junction
[K] is the temperature of the hot side and cold side junction. Thus, the rate
of heat absorption/rejection at this junction can be expressed as:
.
Qjunction =αTCITjunction . (9)
In addition to Joule heating, there is another irreversible phenomenon that accom-
panies the reversible thermoelectric effect, namely the conduction of heat. In a TC, heat
conduction causes a transfer of heat from the hot side to the cold side as a means of
achieving thermal equilibrium. Considering the thermal conductivities of the two legs (k
p
[W/mK]—thermal conductivity of the p-type leg, k
n
[W/mK]—thermal conductivity of
the n-type leg), one can define the thermal conductance (KTC [W/K]) of the TC:
KTC =kp
Ap
Lp
+knAn
Ln. (10)
Joule heating,
.
QJoule,h/c
[W], is the process where electrical energy is converted into
thermal energy as current flows through a resistance. For a TC, Joule heating decreases the
rate of heat absorption at the hot junction and increases the rate of heat transfer to the cold
junction, in equal shares (i.e., half of the heat generated by the Joule effect passes to the hot
side and the other half to the cold side) [16]:
.
QJoule,h/c =1
2I2RTC. (11)
Neglecting the Thomson effect, by applying the first law of thermodynamics at the
hot junction of a TC, the rate of heat absorption, .
Qh,TC [W], is:
.
Qh,TC =αTCITh−1
2I2RTC +KTC(Th−Tc)(12)
whereas the rate of heat rejection, .
Qc,TC [W], at the cold side is:
.
Qc,TC =αTCITc+1
2I2RTC +KTC(Th−Tc). (13)
Overall, the power delivered to the load, Pel,TC [W], can be calculated in two ways:
Pel,TC =.
Qh,TC −
.
Qc,TC (14)
or as:
Appl. Sci. 2021,11, 1340 5 of 33
Pel,TC =I2RLoad,TC. (15)
The thermoelectric performance of the material is assessed using the figure of merit [1/K]:
Z=α2
ρk(16)
which, for a TC becomes
ZTC =α2
TC
hqρpkp+pρnkni2. (17)
This figure of merit, like the thermoelectric parameters, is temperature dependent. How-
ever, often, average values are assumed with errors found to be below 10% of the true
value [17].
The conversion (or thermal) efficiency of the TC is the ratio of the power output to
the heat input at the hot side. To simplify the final expression, the following notations are
employed for the load resistance ratio (m [-]) and Carnot efficiency (ηC[-]), respectively,
m=RLoad,TC
RTC
(18)
ηC=1−Tc
Th
. (19)
Thus, the conversion efficiency is:
ηth =ηCm
(1+m)−1
2ηC+1
2ZT (1+m)21+Tc
Th. (20)
Expressions of maximum parameters (current, voltage, power, and efficiency) are also
introduced here due to their importance in determining TEG parameters, which will be
shown later. The maximum voltage occurs at open circuit when I = 0:
VOC,TC,max =αTC(Th−Tc). (21)
The maximum current, however, occurs at short circuit, where R
Load,TC
= 0; thus,
leading to the following expression:
Imax =αTC(Th−Tc)
RTC
. (22)
To obtain the maximum power, it can be shown that the load resistance must equal
the internal resistance (thus: m = 1), which leads to the following expression:
Pel,max =α2
TC(Th−Tc)2
4RTC
(23)
For the maximum efficiency, the conversion efficiency is differentiated with respect to m
and set to 0. This leads to the following expression for the load resistance ratio:
m=p1+ZT (24)
where
ZT
[-] is the dimensionless figure of merit (
T=
1
/
2
·(Th+Tc)
[K] is the average
temperature across the TC). Consequently, the maximum conversion efficiency becomes:
ηth,max =1−Tc
Th·p1+ZT −1
p1+ZT +Tc
Th
. (25)
Of additional high importance is the efficiency at maximum power (
ηmp
), or, in other
words, at matching load resistance (m = 1). Based on Equation (20), ηmp is:
Appl. Sci. 2021,11, 1340 6 of 33
ηmp =ηC
2−1
2ηC+2
ZT 1+Tc
Th. (26)
The output of a single TC is generally small; e.g., well below 1 [W]. Therefore, many
TCs are connected to form a thermoelectric module (TEM) (Figure 2); thus, increasing
the power output. At this point it is worth recalling that a TEM can operate both as
a generator (often designated with TEG in literature) and/or as a heat pump (usually
TEC—thermoelectric cooler). In the present study, the TEM operates as a generator, but the
notation of TEM is used to avoid confusion with the acronym TEG that involves the overall
system including heat exchangers (shown later). The output of a TEM can be expressed
based on the number of thermocouples (n
TC
[-]) and the parameters of the TC, except for
electrical current that is independent of the number of thermocouples. Thus, the total rate
of heat absorption/rejection at the hot and cold sides are, respectively:
.
Qh,TEM =nTCαTC ITh−1
2I2RTC +KTC(Th−Tc)(27)
.
Qc,TEM =nTCαTC ITc+1
2I2RTC +KTC(Th−Tc). (28)
Appl. Sci. 2021, 11, 1340 7 of 36
Figure 2. Thermoelectric module operating as a generator (based on [18]).
3. TEM Properties Identification Model
Often TEM material properties are not available from the manufacturer and must be
determined, which requires specialized equipment [19]. Furthermore, it is not always
necessary to have high model accuracy throughout the entire operating range. For
example, when estimating the performance of a TEM system in the preliminary design
phase, it is helpful to have a respectively simple and physically based model that can be
used to run parametric studies. To solve this issue, Lee et al. [20] and Zhang [21] proposed
methods that use maximum parameter values, usually provided by manufacturers, to
estimate the electrical resistivity, Seebeck coefficient, thermal conductivity, and the figure
of merit. The authors call these effective parameters since, as compared to intrinsic
material properties, they are determined at the system (TEM) level; thus, they account for
the various losses caused by the manufacturing process (i.e., thermal and electrical contact
resistances), the temperature dependency (due to the Thomson effect), as well as imperfect
insulation (heat losses through radiation and convection). The basic idea is that, ideally,
the maximum current, voltage, power, and efficiency are functions of the material
properties, the thermocouple geometry (area, length), and the temperatures of the two
junctions (T
h
, T
c
).
3.1. Equations and Simulink Model
For the current study, the maximum parameters used to determine the outcomes are
current, efficiency, and power (only three of the four maximum parameters presented in
Equations (21)–(23) and (25) are required). By knowing the leg area (A = A
p
= A
n
) and
length (L = L
p
= L
n
), the effective Seebeck coefficient and the effective electrical resistivity
of the TC can be determined from Equations (22) and (23):
∗
ρ=
el,m ax
TC 2
TC max
A
4P
L
nI (30)
()
∗
α= −
el, max
TC
TC max h c
4P
nI T T . (31)
To determine the effective thermal conductivity, an additional parameter is required,
namely the effective figure of merit which can be obtained from Equation (25):
Figure 2. Thermoelectric module operating as a generator (based on [18]).
Using Equations (14), (27) and (28), the total power output of the module can
be calculated
:
Pel,TEM =nTChαTC I(Th−Tc)−I2RTCi. (29)
3. TEM Properties Identification Model
Often TEM material properties are not available from the manufacturer and must
be determined, which requires specialized equipment [
19
]. Furthermore, it is not always
necessary to have high model accuracy throughout the entire operating range. For example,
when estimating the performance of a TEM system in the preliminary design phase, it
is helpful to have a respectively simple and physically based model that can be used
to run parametric studies. To solve this issue, Lee et al. [
20
] and Zhang [
21
] proposed
methods that use maximum parameter values, usually provided by manufacturers, to
estimate the electrical resistivity, Seebeck coefficient, thermal conductivity, and the figure of
merit. The authors call these effective parameters since, as compared to intrinsic material
properties, they are determined at the system (TEM) level; thus, they account for the
various losses caused by the manufacturing process (i.e., thermal and electrical contact
resistances), the temperature dependency (due to the Thomson effect), as well as imperfect
insulation (heat losses through radiation and convection). The basic idea is that, ideally, the
Appl. Sci. 2021,11, 1340 7 of 33
maximum current, voltage, power, and efficiency are functions of the material properties,
the thermocouple geometry (area, length), and the temperatures of the two junctions
(Th, Tc).
3.1. Equations and Simulink Model
For the current study, the maximum parameters used to determine the outcomes are
current, efficiency, and power (only three of the four maximum parameters presented in
Equations (21)–(23) and (25) are required). By knowing the leg area (A = A
p
= A
n
) and
length (L = L
p
= L
n
), the effective Seebeck coefficient and the effective electrical resistivity
of the TC can be determined from Equations (22) and (23):
ρ∗
TC =4A
LPel,max
nTCI2
max
(30)
α∗
TC =4Pel,max
nTCImax (Th−Tc). (31)
To determine the effective thermal conductivity, an additional parameter is required,
namely the effective figure of merit which can be obtained from Equation (25):
Z∗
TC =1
T
1+ηmax
ηC
Tc
Th
1−ηmax
ηC!2
−1
. (32)
Based on Equation (16), the effective thermal conductivity can be expressed as:
k∗
TC =α∗
TC2
Z∗
TCρ∗
TC
. (33)
3.2. Model Validation
The model has been validated for two commercially available thermoelectric modules
which are presented in Table 1. This data has been used in the Simulink model presented
in Figure 3to determine the effective parameters.
1
3
Figure 3. Simulink model for determining effective properties (values for TGM-127-1.4-2.5).
Appl. Sci. 2021,11, 1340 8 of 33
Table 1. Thermoelectric parameters of commercially available thermoelectric modules.
TEM nTC [-] Pel,max [W] Imax [A] ηmax [%] Ap/n [mm2]Lp/n [mm] Th [◦C] Tc [◦C]
TGM-199-1.4-0.8 [22] 199 11.40 5.10 4.3 1.96 0.8 200 30
TGM-127-1.4-2.5 [23] 127 4.46 2.37 5.5 1.96 2.5 200 30
The effective parameters obtained with the model presented in Figure 3were used to
calculate the voltage, current, and power at matched load resistance conditions, as well as
the power, current, voltage, and efficiency at the maximum temperature gradient for various
values of the load resistance. The results show that there is a relatively good agreement be-
tween the measured and calculated values (Figures 4and 5). The major differences appearing
here can be attributed to the temperature dependency of the electrical resistivity, Seebeck
coefficient, and thermal conductivity, which is not captured in the proposed model. Despite
these differences, the predicted power output of the module is in relatively good agreement
with the measured values. A possible reason for this result is that the maximum power was
chosen as an input parameter in deference to the maximum voltage.
Figure 4.
Manufacturer data compared to calculated data at matched load resistance conditions for TGM-199-1.4-0.8:
(
a
) Module power; (
b
) module current; (
c
) module voltage; (
d
) module power and voltage with respect to current; and
(e) module efficiency.
Appl. Sci. 2021,11, 1340 9 of 33
Figure 5.
Manufacturer data compared to calculated data at matched load resistance conditions for TGM-127-1.4-2.5:
(
a
) Module power; (
b
) module current; (
c
) module voltage; (
d
) module power and voltage with respect to current; and
(e) module efficiency.
4. TEG Model for Exhaust Gas Waste Heat Recovery
For the current study, the system is analyzed under steady-state conditions. The do-
main is discretized along the flow direction in control volumes (CVs) having the same
length as a TEM (Figure 6). Initially, only one control volume is considered for the calcula-
tion; however, a greater domain can be built by adding more control volumes with each
taking input data from the previous CV. The changes in fluid properties with temperature
are considered separately for each control volume and are defined at the mean CV temper-
ature (average of inlet and outlet temperatures). For simplicity, thermoelectric parameters
are considered constant, but the model can be readily updated with equations for these
parameters, either from the literature or from the manufacturer.
Thermoelectric modules are often combined with heat exchangers (HX) to form ther-
moelectric generators (TEG), which are used to recover otherwise wasted thermal energy
from the exhaust gases of ICEs. The fins of the HX allow for an increased heat transfer
surface area, subsequently enhancing the amount of heat extracted through convection
from the hot gases. This heat is then transferred through conduction to the TEMs. An
Appl. Sci. 2021,11, 1340 10 of 33
example of such a TEG with all relevant dimensions of the finned heat exchanger (L
HX
[m]—heat exchanger length, W
HX
[m]—heat exchanger width, H
HX
[m]—heat exchanger
height, z
HX,fin-spacing
[m]—heat exchanger fin spacing, t
HX,fin
[m]—heat exchanger fin thick-
ness), as well as the positioning of the TEMs is presented in Figure 6. If the heat exchanger
is covered with TEMs on both sides, the CV must be modified accordingly (Figure 7).
Figure 6. Hot side heat exchanger.
Figure 7. Side view of module section.
4.1. Equations and Simulink Model
The layering of the TEG, which will be used for calculating the amount of heat trans-
ferred to the hot side and from the cold side, is presented in Figure 8. The TEG is composed
of two heat exchangers (hot and cold side), a thermal grease used to eliminate surface
Appl. Sci. 2021,11, 1340 11 of 33
imperfections in the interface area, and two ceramic plates acting as electric insulators. The
electrical conductor height is relatively small and, since it has a respectively high thermal
conductivity and low electrical resistivity, it was neglected in the present model.
Figure 8. Integration of thermoelectric module in a thermoelectric generator.
As a first step, all relevant areas for the heat transfer are calculated starting from the
area available for heat transfer in one TC (Aht,TC [m2]):
Aht,TC =Ap+An. (34)
For a TEM with n
TC
thermocouples, the area available for heat transfer (A
ht,TEM
[m
2
])
will be:
Aht,TEM =nTCAp+An. (35)
Thus, for a TEG with n
TEM
[-] TEMs, the total area of the HX base in a control volume
(Aht,CV,base-TEM [m2]) available for heat transfer will be:
Aht,CV,base−TEM =nTEMnTC Ap+An. (36)
The HX base area for one control volume (Aht,CV,base [m2]) is defined as:
Aht,CV,base =WHx ·LCV (37)
where L
CV
is the length of the CV. The part of the HX base area that is not covered by TEMs,
is considered to be insulated.
To calculate the amount of heat extracted from the hot gases, the total area (A
ht,CV,total
)
available for heat exchange with the hot gases in a control volume must be determined:
Aht,CV,total =nfinh2(tHx,fin +LCV )HHx +zHx,fin−spacingLCVi. (38)
Note that the equations presented are for a HX installed on the hot side, but the
equations can also be used for an HX on the cold side.
To show the fraction of the HX base surface that is used for heat transfer to the TCs, a
HX surface usage coefficient is proposed:
fHx,Area =Aht,CV,base−TEM
Aht,CV,base
. (39)
When only the flow velocity (V
gas
[m/s]) is available, the crossflow area of the HX is
required to compute the mass flow:
AHx,cross =nfinzHx,fin−spacing HHx (40)
consequently, the gas mass flow rate will be
Appl. Sci. 2021,11, 1340 12 of 33
.
mgas =ρgasVgas AHx,cross. (41)
If, however, the mass flow is available, then Equation (41) can be used to compute the
gas flow velocity.
The next step is to determine the forced convection heat transfer coefficient between
the gases and the HX surface. This is respectively difficult since the flow ranges from
developing to fully developed. On this topic, Teertstra et al. [
24
] proposed and validated a
model that uses a composite solution based on the limiting cases of flow between isothermal
parallel plates. In the transitional flow region, an average Nusselt number is calculated as
a function of the heat exchanger geometry and fluid flow velocity. To use this model, two
conditions must be met: Air must be contained within the channels (this is achieved since
a closed HX is considered) and the fin spacing must be considerably smaller than the fin
height (high aspect ratio). The reduced Reynolds number has an experimental range of
0.2 ≤Re∗≤200 [25] and is calculated as:
Re∗
Hx,channel =VgaszHx,fin−spacing
µgas
zHx,fin−spacing
LHx
. (42)
The proposed equation for the Nusselt number is:
NuHx,channel =
Re∗
Hx,channelPrgas
2!−3
+
0.664qRe∗
Hx,channelPr 1
3v
u
u
t1+3.65
qRe∗
Hx,channel
−3
−1
3
. (43)
If the flow is fully turbulent and Re * exceeds its range limits, then the Nusselt number
can be calculated using the Gnielinski relation [26]:
NuHx,channel =
fG
8(ReHx,channel −1000)Prgas
1+12.7qfG
83
qPr2
gas −1(44)
where Re
HX,channel
is the channel Reynolds number and f
G
is the friction factor in turbulent
flow in smooth tubes determined from the first Petukhov equation [27]:
ReHx,channel =VgaszHx,fin−spacing
µgas
(45)
fG=1
(0.79 ln ReHx,channel −1.64)2. (46)
Knowing the Nusselt number, the gas-fin heat transfer coefficient can then be calcu-
lated with the following expression:
hgas−fin =kgas
zHx,fin−spacing
NuHx,channel. (47)
To calculate the efficiency of the heat exchanger, as well as the thermal resistance, the
efficiency of a single fin must be determined [
28
], along with the thermal resistances of all
other intermediate layers. A single fin efficiency is calculated based on the m
fin
parameter
and the fin height, which, in this case is the same as the heat exchanger height:
mfin =shgas−fin ·2(LHx +tHx,fin)
kHxLHx tHx,fin
(48)
ηfin =tanh(mfinHHx )
mfinHHx
. (49)
The next step is to calculate the overall effectiveness of the fin (
ηo,fin
[-]), for which the
area of a single fin (Afin [m2]) and the number of fins (nfin [-]) must be known:
Appl. Sci. 2021,11, 1340 13 of 33
Afin =2HHx(LHx +tHx,fin )(50)
ηo,fin =1−nfinAfin
Aht,Hx,base
(1−ηfin). (51)
Figure 9b shows the thermal resistance network used to calculate the hot (T
h
[K]) and
cold side (Tc[K]) junction temperatures along with the heat exchanger efficiency. To keep
the model simple, and since it represents under 2% of the total heat transferred from the
hot side [
29
], radiation losses are neglected. The resistances in Figure 9are defined for
all layers presented in Figure 8based on their thickness (t
layer-name
), thermal conductivity
(k
layer-name
), and surface area (A
layer-name
). The equivalent gas-fin heat transfer resistance,
and the thermal resistances of the base, grease, and ceramic are calculated according to
Equations (52)–(55):
Rth,fin,eq =1
ηo,finhgas−fin Aht,Hx,total
(52)
Rth,Hx,base =tHx,base
kHx,baseAht,Hx,base
(53)
Rth,grease =tgrease
kgreaseAgrease (54)
Rth,ceramic =tceramic
kceramicAceramic
(55)
Figure 9.
(
a
) Thermoelectric generator (TEG) temperatures of interest; (
b
) thermal resistance network for a CV; (
c
) simplified
thermal resistance network for calculating Tc; and (d) simplified thermal resistance network for calculating Th.
The equivalent thermal resistance of the TEM is calculated based on the hot and cold
junction temperatures and the heat flow to the cold side:
Rth,TEM =Th−Tc
.
Qc
. (56)
Appl. Sci. 2021,11, 1340 14 of 33
Similarly, an equivalent load resistance is defined based on the hot junction temper-
ature and the temperature on the cold side (T
∞,c
= 1/2
∗
(T
∞,c,in
+ T
∞,c,out
) [K], where
T
∞,c,in
and T
∞,c,out
are the temperature of the gas at the cold side heat exchanger inlet and
outlet respectively):
Rth,Load,eq =Th−T∞,c
Pel,TEM
. (57)
Based on Equations (52)–(55), an equivalent thermal resistance of the hot side can
be determined:
Rth,h,eq =Rth,fin,eq +Rth,Hx,base +Rth,grease +Rth,ceramic. (58)
Similarly, the equivalent thermal resistance of the cold side is:
Rth,c,eq =Rth,ceramic +Rth,grease +Rth,Hx,base +Rth,fin,eq. (59)
Often, an aluminum block is used to act as a thermal spreader that is placed between
the HX and the TEM as in Figure 10. In this case, the thermal resistance of the base is
replaced by the thermal resistance of the aluminum block:
Rth,Al,block =tAl,block
kAl,blockAAl,block
. (60)
Figure 10. Integration of thermoelectric module in a thermoelectric generator.
It is important to note that, in such cases, the HX is usually also made of aluminum
which allows for a simplification. Namely, the addition of the HX base thickness to the
aluminum block thickness. They can, however, be introduced as separate blocks. In the
present study, the former solution has been adopted.
For the case presented in Figure 10, the heat exchanger efficiency is defined as:
ηHx =1
ηo,fin
+2hgas−finAht,Hx,total tgrease
kgreaseAgrease
+hgas−finAht,Hx,total tAl
kAlAAl . (61)
To compute the junction temperatures, one more thermal resistance is needed, namely
the equivalent thermal resistance of the out (output and heat rejection) side:
Appl. Sci. 2021,11, 1340 15 of 33
Rth,out,eq = 1
Rth,Load,eq
+1
Rth,TC +Rth,c,eq !−1
. (62)
Based on these resistances, the hot and cold side junction temperatures can be ex-
pressed with respect to T
∞,h
(T
∞,h
= 1/2
∗
(T
∞,h,in
+ T
∞,h,out
) [K], where T
∞,h,in
and T
∞,h,out
are the temperature of the gas at the hot side heat exchanger inlet and outlet respectively)
and T∞,c:
Th=Rth,out,eqT∞,h +Rth,h,eq T∞,c
Rth,out,eq +Rth,h,eq
(63)
Tc=Rth,c,eqTh+Rth,TC T∞,c
Rth,c,eq +Rth,TC
. (64)
The amount of heat extracted from the hot gases can be determined from the CV inlet
and outlet enthalpy represented by the heat capacity times the difference in temperature:
Qgas,Hx =.
mgascp,gas (T∞,h,in −T∞,h,out). (65)
However, in this effort, the outlet temperature is an output value. Neglecting the heat
losses through radiation Qgas,HX = Qh, therefore, the outlet temperature is:
T∞,h,out =T∞,h,in −Qh
.
mgascp,gas
. (66)
When installing a heat exchanger on the exhaust path of an ICE, there is also the issue
of the increased backpressure that negatively influences engine performance and emissions.
Consequently, the pressure drop must be estimated. Here, there are three effects that need
to be considered: Viscous drag effects, flow expansion, and flow contraction. For a CV, the
pressure loss due to this viscous drag effect is:
∆Ploss,Hx =∑
CV
fCVlength
Dhydr,channel
ρgas
V2
channel
2(67)
where f is the friction factor defined using the Churchill equation [30]:
f=8"8
Rechannel 12
+1
(A+B)1,5 #1
12
. (68)
The Churchill equation was chosen because it covers both developing and fully
developed internal flows. The Reynolds number is calculated using Equation (45), while A
and B are:
A=(−2.457 ln"7
Rechannel 0.9
+0.27 εsurface
Dhydr,channel #)16
(69)
B=37350
Rechannel 16
. (70)
To calculate A, two additional parameters are required, namely the surface rough-
ness (
εsurface
), which can be taken from literature and the channel hydraulic diameter
(Dhydr,channel), which is calculated from the channel cross-sectional area and perimeter:
Dhydr,channel =2zHx,fin−spacingHHx
zHx,fin−spacing +HHx
. (71)
For small gas density changes along the length of the heat exchanger, in case of n
CV
control volumes (CVlength = LHX/nCV), Equation (67) can be simplified to:
Appl. Sci. 2021,11, 1340 16 of 33
∆Ploss,Hx =fLHx
Dhydr,channel
ρgas
V2
channel
2. (72)
Changes in the pipe cross-section area along the flow direction also lead to pressure
losses by causing the flow to either expand or contract. To estimate these losses, the
Borda-Carnot correlations are used [
31
]. These correlations require knowledge about cross-
sectional areas, as well as the gas density and velocity. For a TEG installed on the exhaust
pipe of an ICE (Figure 11), these cross-sectional areas are: Exhaust pipe upstream of the
HX (A
inlet-pipe,cross
[m
2
]), HX inlet (A
HX,inlet,cross
[m
2
]) and outlet (A
HX,outlet,cross
[m
2
]), as
well as the exhaust pipe downstream of the HX (A
outlet-pipe,cross
[m
2
]). Usually, TEG heat
exchangers have a constant cross-section; thus, A
HX,inlet,cross
= A
HX,outlet,cross
= A
HX,cross
.
The inlet and outlet pipe cross-sectional areas are:
Ainlet−pipe,cross =π
d2
inlet−pipe
4(73)
Aoutlet−pipe,cross =π
d2
outlet−pipe
4. (74)
Figure 11. Areas used for calculating the pressure loss due to flow contraction and expansion.
The pressure loss due to flow expansion occurs before the HX, thus leading to
the expression:
∆Ploss,exp =−ρgas
Ainlet−pipe,cross
AHx,cross 1−Ainlet−pipe,cross
AHx,cross V2
gas (75)
whereas the pressure loss due to flow contraction occurs after exiting the HX:
Appl. Sci. 2021,11, 1340 17 of 33
∆Ploss,contr =1
2ρgas1
µcontraction
−12Ainlet−pipe,cross
AHx,cross 2
V2
gas (76)
with µcontraction as the contraction coefficient:
µcontraction =0.63 +0.37 AHx,cross
Ainlet−pipe,cross !3
. (77)
It must be noted that the pressure losses due to flow expansion and contraction are
calculated only for the HX inlet and outlet CVs, respectively.
4.2. Solution Method
The thermal circuit temperatures must be solved in an iterative manner due to the
dependence of the thermal resistances on the terminal temperatures and thermoelectric
properties. Therefore, before beginning the iterative calculation process, several temper-
atures must be estimated (i.e., T
h
, T
c
, T
∞,h,out
, and T
∞,c,out
). Using input data and these
estimated temperatures, the thermal resistances together with the other parameters are eval-
uated. Then, the values of T
h
, T
c
, T
∞,h,out
, and T
∞,c,out
are reevaluated and the convergence
is checked based on the relative error value:
Errorrel =∑
Ti−Ti+1
Ti
(78)
If Error
rel
is below a set value (10
−6
for this study), the calculation stops. If this is not
the case, the initially guessed values are updated with the current results and the process
starts again. The solution methodology flowchart for each CV is presented in Figure 12.
Figure 12. Solution methodology flow chart.
5. Results and Discussions
The model (Figure 13) was validated based on tests performed by Fagehi et al. [
32
]. In
their work, Fagehi et al. [
32
] used a liquid (water) heat exchanger on the cold side which
demanded a series of changes to the cold side heat exchanger model (not included in this
study, but available upon request).
The comparison between the simulated and measured values is presented in
Figure 14
.
Fagehi et al. [
32
] measured the junction temperatures, current, and voltage directly, while
they calculated the power using two methods. The power designated with P
el_Temp_I
is
calculated using the junction temperatures, electric current, and the ideal thermoelectric
equations; whereas, P
el_V_I
was determined basedon the measured voltage and current. Sim-
ilar to the results of Fagehi et al. [
32
] the simulation results were closer to the power values
Appl. Sci. 2021,11, 1340 18 of 33
determined using the junction temperatures, electric current, and the ideal thermoelectric
equations. The average and matched load resistance errors for the junction temperatures,
power output, current, voltage, and efficiency are listed in Table 2. The possible causes of
differences are: Measurement accuracy, the use of ideal equations, effective material prop-
erties, as well as constant values for both the thermoelectric properties and gas properties.
In addition, this effort neglects the Thomson coefficient, radiative heat transfer, and the
thermal spread resistance. Another possible source of error is the extraction of data from
the graphs presented in literature by using a digital tool. Despite all the uncertainties, there
is a reasonable agreement between the calculated and the measured values.
2
13
Figure 13.
Simulink model with two heat exchangers (red—hot side and blue—cold side), aluminum
blocks for thermal spread (light grey), the TEM between, and the thermal resistance network on
right side.
Table 2. Comparison of calculated and measured values at peak measured power.
Parameter Th[◦C] Tc[◦C] Pel [W] I [A] V [V] ηth [-]
Average error [%] 0.15 0.33 12.73/1.77 6.82 0.52 0.93
Error at matched load resistance [%] 0.11 0.45 10.46/3.50 2.99 0.38 1.8
As previously mentioned, Fagehi et al. [
32
] used a liquid (water) heat exchanger on
the cold side. The same cold side temperature can be achieved by using air as the cooling
fluid, but this poses an issue due to the very low thermal conductivity of air. This can be
resolved by increasing the flow velocity and, to test this theory, the water cold side heat
exchanger model (used only for the validation) was replaced with an air heat exchanger
(similar to the one used on the hot side). Then, the air flow velocity on the cold side was
adjusted to achieve a T
c
value as close as possible to that obtained with the validated model
(Figure 14). The required flow velocity was determined to be 13.1 m/s (or 47.2 km/h)
and appears feasible for highway trucks and buses running at a constant speed. Here, the
main advantages of utilizing air would be the lower weight and space requirements of the
system, as well as a reduced complexity. Furthermore, a proper design of the flow passage
to the cold side can allow for high air velocities; thus, leading to even greater temperature
gradients than in the case of the liquid coolant solution. The main disadvantages of this
solution are that the temperature gradient and, consequently, the power output depends
Appl. Sci. 2021,11, 1340 19 of 33
on vehicle speed and outside temperature (still less than the coolant temperature). Also,
this solution is not feasible for stationary engines.
Figure 14.
TEG model validation highlighting the difference between the simulated and measured values of (
a
) the electrical
power output, (
b
) hot and cold junction temperatures, (
c
) current, and (
d
) voltage for two different types of cold side heat
exchangers: Water (W_sim) and air (air_sim).
To test the possible output of such a TEG, 16 test cases (Table 3) were evaluated
with varying air temperatures (T
air
) and air flow velocities (V
air
). For the hot side, the
temperature of the exhaust gases was set at 250
◦
C with a flow velocity of 19.1 m/s (
12 L
engine running at 1500 min
−1
, with an exhaust pipe diameter of 100 mm). For brevity,
only the results for 50 and 90 km/h are presented here with the others are available upon
request. Regarding case notation, “V” stands for velocity and “T” for temperature. The
values following “V” and “T” are the velocity (in km/h) and temperature (in
◦
C) values for
that case (e.g., V50T-5 = velocity of 50 km/h and temperature of
−
5
◦
C). For this study it
was assumed that the hot and cold side fluid temperatures and flow velocities are uniform
and constant, but also that there is a uniform temperature distribution on the TEG and
TEM surface. Furthermore, the simulation was performed for only one TEM. Consequently,
when considering the desired number of TEMs this will result in an overprediction of the
power output.
Appl. Sci. 2021,11, 1340 20 of 33
Table 3. Test cases for cold side heat exchanger using air as cooling fluid.
Case Tair,-5 [◦C] Tair,10 [◦C] Tair,20 [◦C] Tair,30 [◦C]
Vair,50 = 13.9 [m/s]
−510 20 30
Vair,80 = 22.2 [m/s]
Vair,90 = 25.0 [m/s]
Vair,120 = 33.3 [m/s]
Table 4and Figures 15 and 16 show the results for the two different velocities 50 and
90 km/h. Analysis of the results finds that both the higher air flow velocity on the cold
side and a lower ambient temperature increase the output of the TEG. Decreasing the air
temperature raises the temperature gradient and the power output, but also reduces the
temperature of the hot side junction to that below the maximum allowable value of the
TEM. This is because the hot side heat flow rate is constant. To further augment the output
of the TEG, it would be necessary to increase the heat flow rate on the hot side (either by
increasing the exhaust temperature or the flow velocity). Consequently, a high TEG output
requires not only a good HX, but also a suitable heat extraction strategy.
Table 4. Results for cold side heat exchanger using air as cooling fluid.
Parameters
Case
V50T-5 V50T10 V50T20 V50T30 V90T-5 V90T10 V90T20 V90T30
Pmax [W] 4.023 3.562 3.260 2.972 4.568 4.043 3.702 3.376
Th[K] 449.2 453.5 456.4 459.3 444.6 449.0 452.1 455.2
Tc[K] 347.6 357.6 364.7 371.7 336.3 347.1 354.6 362.0
∆T [K] 101.7 95.9 91.8 87.6 108.3 101.9 97.5 93.1
Figure 15.
(
a
) Module power output, (
b
) junction temperature, (
c
) module current, and (
d
) module voltage with respect to
the load resistance ratio for a velocity of 50 km/h at various temperatures (−5, 10, 20, and 30 ◦C).
Appl. Sci. 2021,11, 1340 21 of 33
Figure 16.
(
a
) Module power output, (
b
) junction temperature, (
c
) module current, and (
d
) module voltage with respect to
the load resistance ratio for a velocity of 90 km/h at various temperatures (−5, 10, 20, and 30 ◦C).
6. Conclusions
Considering the vast amounts of wasted thermal energy from internal combustion
engines, the study of thermoelectric power generation is necessary as a possibility to
increase overall energy efficiency. Thermoelectric devices are environmentally friendly
and have numerous other advantages including the lack of moving parts, high reliability,
the direct conversion of thermal energy into electrical energy, as well as low maintenance
requirements. However, the main disadvantage of such devices is their low efficiency
owing to the small ZT values currently found in commercial devices.
In the present study, two Simulink models are presented (see Appendix Afor detailed
Simulink blocks). The first model was created based on the work of Lee et al. [
20
] to estimate
thermoelectric parameters of a thermoelectric module. The validation showed that, even
though the temperature dependency of the thermoelectric parameters was neglected, there
is a relatively good agreement with experimental data.
Furthermore, a TEG model was proposed and validated for power output estimations
in the case of exhaust gas thermoelectric generators. Model validation was accomplished
using the results of Fagehi et al. [
32
]. For the validation, the cold side heat exchanger model
was adapted to match the cooling method of the authors with reasonable agreement. The
next step was to study the possibility of using air as the cooling fluid and to achieve the
same cold side temperature, the flow velocity of the cold air was adjusted (an air velocity
of 13.1 m/s showed a relatively good agreement).
After determining that similar results can be obtained with air as the cooling fluid, a
parametric study was performed to show the potential of such a system for ICE exhaust
coupling. The test conditions (used as input data for the cold side heat exchanger) consid-
Appl. Sci. 2021,11, 1340 22 of 33
ered a vehicle (with a respectively large engine capacity) traveling at different speeds (50,
80, 90, and 120 km/h) under dissimilar ambient air temperatures (
−
5, 10, 20, and 30
◦
C).
The results showed that either a high air velocity or a low outside temperature leads to
greater TEG power output. For the analyzed cases, the peak power output was 4.568 W
for a temperature gradient of 108.3 K. It was also found that increasing the heat rejection
on the cold side requires a simultaneous growth in heat input at the hot side to facilitate
a favorable temperature gradient and, therefore, the power output of the TEG. Overall,
this demonstrates that it is feasible to install an air-cooled TEG on a heavy-duty vehicle
running at constant speed.
Author Contributions:
Conceptualization, N.V.B.; methodology, N.V.B.; software, N.V.B. and D.M.;
validation, F.M., C.D., and N.V.B.; formal analysis, N.V.B. and F.M.; investigation, N.V.B.; resources,
N.V.B.; data curation, N.V.B. and D.M.; writing—original draft preparation, N.V.B.; writing—review
and editing, N.V.B., F.M., and C.D.; visualization, N.V.B.; supervision, F.M.; project administration,
N.V.B. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not Applicable.
Informed Consent Statement: Not Applicable.
Data Availability Statement: Not Applicable.
Acknowledgments:
This paper was supported by the Project POCU/380/6/13/123927 “Entrepreneurial
competences and excellence research in doctoral and postdoctoral programs-ANTREDOC”, project
co-funded by the European Social Fund.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Detailed Simulink blocks (Figures A1–A14). Please note that the values that can be
seen in the figures were not necessarily used to validate the model. Also, the iterative
temperature calculation loop is not included here.
Heat exchanger–hot side (similar for cold side)
3
A1
Figure A1. Hot side heat exchanger block.
Appl. Sci. 2021,11, 1340 23 of 33
Figure A2. Hot side heat exchanger block—inputs.
Appl. Sci. 2021,11, 1340 24 of 33
Figure A3. Hot side heat exchanger block—HX heat transfer equations and outputs.
Appl. Sci. 2021,11, 1340 25 of 33
Figure A4. Hot side heat exchanger block—HX pressure drop equations and outputs.
Aluminum block—hot side (similar for cold side)
4
A5
Figure A5. Aluminum block.
Figure A6. Aluminum block—equations.
Appl. Sci. 2021,11, 1340 26 of 33
Thermoelectric module
5
A7
Figure A7. Thermoelectric module block.
Figure A8. Thermoelectric module block—inputs and outputs.
Appl. Sci. 2021,11, 1340 27 of 33
Figure A9. Thermoelectric module block—equations—part 1.
Appl. Sci. 2021,11, 1340 28 of 33
Figure A10. Thermoelectric module block—equations—part 2.
Appl. Sci. 2021,11, 1340 29 of 33
Thermal resistance network
6
A11
Figure A11. Thermal resistance network block.
Appl. Sci. 2021,11, 1340 30 of 33
Figure A12. Thermal resistance network block—equations.
Appl. Sci. 2021,11, 1340 32 of 33
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