Performance Prediction of Commercial
Thermoelectric Generator Modules using
the Effective Material Properties
Abdulmunaem Elarusi
Nithin Illendula
Hassan Fagehi
ABSTARCT
This work is employed to examine the validity of the thermoelectric modules performance
predicted by formulating the effective thermoelectric material properties. The average
temperature of the thermoelectric generator is used to formulate the three maximum
parameters. The three maximum parameters which are taken from commercial thermoelectric
generator module or measurements are used to define the effective material properties. These
commercial modules were taken as to validate this work by substituting effective material
properties in the simple ideal/ standard equation. The commercial performance curves provided
by the manufacturer were compared with the present work result obtained by the simple
standard equation with the effective material properties. The characteristics of the
thermoelectric generators were represented using the normalised charts constructed by
formulating the normalised parameters over the maximum parameters. The normalized charts
would be universal for any given thermoelectric generator.
1
Table of Contents
Introduction .......................................................................................................................................... 2
Thermoelectric Phenomenon .......................................................................................................... 2
Literature .............................................................................................................................................. 4
Thermoelectric Ideal Equations.......................................................................................................... 5
Performance Parameters of TEG module ....................................................................................... 10
Maximum Parameters of TEGs ..... ………………………………………………………………………………………………11
Normalized Parameters ……………………………………………………………………………………………………………….12
Effective material properties ……………………………………………………………………………………………………14
Results and discussion ...................................................................................................................... 15
TG12-4 by Marlow ....................................................................................................................... 17
HZ-2 by Hi-Z ................................................................................................................................. 19
TGM199-1.4-2.0 by Kryotherm .................................................................................................. 21
TGM127-1.4-2.5 by Kryotherm ................................................................................................. 25
Normalized curves ............................................................................................................................ 27
Conclusion .......................................................................................................................................... 29
References .......................................................................................................................................... 30
Appendix ................................................................................................................................. 31
MATHCAD Analytical ..................................................................................................... 32
2
1. Introduction
Thermo-Electrics is a science which is associated with thermal and electrical
phenomena. Here, in the thermo-electrics we convert the electrical energy to the thermal energy
and vice versa. In this we will use thermo-couple which consists of two dissimilar wires at two
different temperatures and if we connect these two wires an electric potential is generated.
Mainly, the whole thermo-electrics is depends on two devices only these are Thermo-Electric
Generator and Thermo-Electric Cooler. There is a big advantage while using the ThermoElectric devices is that there is no moving parts, no maintenance is required.
The thermo-electric devices has a various applications the major application is that the
producing the electricity these are used in the automobiles for the various application like
cooling or heating the car seat which is now in the market as an extra fitting but installed in the
high end models. We can also use the thermo-electric device (TEG) to produce the electricity
by tapping the waste heat of exhaust gas which emitted after combustion of fuel. These thermoelectric devices are used in the space exploration robotic rovers for example the rover sent to
the planet Pluto it uses the heat generated by the decay of radio isotope.
We can also use the thermos electric devices (TEC) as a cooling device in the
refrigerator by there are a lot advantages there is no pollution to environment due to danger of
emitting the dangerous the greenhouse effect gases and also there is a quiet cooling operations
as compared to conventional refrigerator.
1.1 Thermoelectric Phenomenon
There are some governing effects on which the thermoelectric effect is depended:
Seebeck Effect: The See back is the conversion of temperature difference into an electric
current. Usually a temperature difference is maintained between a thermocouples so as to
produce current.
The relation between the voltage and temperature difference is:
V=α*ΔT
Here V is the voltage across the circuit, α is the Seebeck Coefficient, ΔT is the
temperature difference of the hot and cold junctions
3
Figure 1: Seebeck effect in a circuit with two dissimilar materials with hot and cold
junctions
Peltier Effect: When a current flows across a junction between different wires i.e.
thermo-couple, it is found that on the one side of junction heat is added and the other side heat
is subtracted in order to keep its temperature constant.
QPeltier = Π*I
Here, QPeltier is the rate of heat transfer, I is the current generated in the circuit, and Π is
the Peltier coefficient.
Thomson Effect: Here, in this effect which is similar to the Peltier Effect in which the
heat is liberated or added if an electric current flows across junction but in this liberation or
addition of heat is also on depends on the flow direction of current.
QThomson = τ*I*ΔT
QThomson is the Thomson heat transfer, τ is the Thomson coefficient, I is the current
flowing in the circuit, and ΔT is the temperature difference between hot and cold junctions.
QThomson
Wire A
QPeltier
Tc
Th
I
-
QPeltier
+
Wire B
QThomson
Figure 2: It shows both the Peltier without considering flow of current and the Thomson
effect in it with considering flow of current.
4
Thermo-Couple: A Thermo-couple consists of a p-type and n-type semiconductor
elements. In which these thermos-couples are connected thermally in parallel and electrically
in series to form a thermo-electric module.
Figure 3: The thermo-electric couple with two dissimilar material.
2. Literature
As in the commercial modules the information provided by the different manufacturer
is different like if one manufacturer shows a performance curve of COP v/s hot side
temperature and other Manufacturer shows cooling power v/s Temperature difference this
makes a lot for the person who buys the thermo-electric module for comparing. So, if they want
to find the comparison then they have to take the modules and do experiments which cots a lot
of time and money. So, an easy way to find the comparison between is by using the ideal
equations even though in the ideal equations we cannot find the Seebeck coefficient , current
I , Resistance R , Thermal Conductance K.
Shiho Kim has find out the material properties analytically with function of internal
plate temperature difference of p and n-type pellets. He formulated the heat absorbed and heat
rejected in the hot and cold junctions respectively. And he formulated equations for thermal
conductance for the overall module and for the pellets. He also find out equations for the
intrinsic seebeck coefficient, electrical resistance for the internal plates. Using the heat
equations first formulated he derived the ΔTe i.e. is the internal temperature difference. Then
he taken the case when the load current is zero and he find out the ΔTe. Finally using the above
5
all values he find out he formulated equations for the material properties. He also drawn curves
for current v/s voltage for various hot junction temperatures with temperature difference.
Here, in this test performed by Shiho Kim he did not compared his values with the
performance curves provided by the manufacturer. HE just find out the values analytically.
But the D’Angelo and Hogan has find out the values experimentally he used the vacuum
enclosure and the constant heat source on the hot junction of module by the nickel chromium
wire then with obtained results he drew the performance curves and those are in good
agreement the curves provided by Tellurex.
3. Thermoelectric Ideal Equations:
The general governing equations in this section are represented considering a nonuniformly heated thermoelectric material with isotropic material properties (Temperature
independent). Therefore, the continuity equation for a constant current gives,
⃗∇ . 𝑗 = 0
(1)
Where 𝑗 is the current, and ⃗⃗⃗
∇ is the differential operator with respect to length. The
electric field ⃗E (Or electric potential) is defined according to the ohm's low and Seebeck effect,
⃗ 𝑇 and the current density
because the electric field is influenced by the temperature gradient ∇
𝑗 Thus, the electric field can be defined as,
⃗ = 𝑗𝜌 + 𝛼∇
⃗𝑇
E
(2)
⃗ and the
Similarly, the heat flow (or heat flux) is effected by the electric field E
temperature gradient ⃗∇𝑇 the heat flow density, consequently, is expressed considering the
Thomson relationship and the Onsager's principle, as
⃗𝑇
𝑞 = 𝛼𝑇𝑗 − 𝐾∇
(3)
Where T is the temperature at the boundary through which the heat flux flows. The 𝛼𝑇𝑗
⃗ 𝑇 the heat transfer from the Fourier's low of conduction.
is the Peilter heat contribution, and 𝐾∇
The general heat diffusion equation as time dependent is given by,
⃗⃗⃗ . 𝑞 + 𝑞̇ = 𝜌𝐶𝑝 𝜕𝑇
−∇
𝜕𝑡
(4)
Where 𝑞̇ is the heat flux per unit volume, 𝜌 is the density of the material, 𝐶𝑝 is the
6
specific heat capacity, and
𝜕𝑇
𝜕𝑡
is the rate of change of temperature with respect to time.
𝜕𝑇
Now, for steady state condition ( 𝜕𝑡 goes to zero), we have,
𝑞̇ = ⃗⃗⃗
∇ .𝑞
(5)
Since we can define the heat flux as a function in the electric power, 𝑞̇ is expressed by,
𝑞̇ = ⃗E . 𝑗 = 𝑗 2 𝜌 + 𝑗 . 𝛼 ⃗∇𝑇
(6)
Substituting equations (3) and (6) in equation (5) gives us,
⃗⃗⃗ . (𝐾∇
⃗ 𝑇) + 𝑗 2 𝜌 − 𝑇 𝑑𝛼 𝑗 . ∇
⃗𝑇=0
∇
𝑑𝑇
(7)
𝑑𝛼
Where, from Thomson relation, 𝜏 = 𝑇 𝑑𝑇 is known as the Thomson coefficient. The
second term in equation (7) is the Joule heating, and the third term is the Thomson heat. In
special cases where the seebeck coefficient 𝛼 is a temperature independent, the Thomson effect
is negligible, which means that the Thomson coefficient 𝜏 is zero. As such, this work considers
negligible Thomson effect and the seebeck coefficient is not a temperature independent.
(a)
7
(b)
Figure 4. (a) cutaway of a thermoelectric generator module, and (b) a p-type and ntype thermocouple.
Figure.4 (a) illustrates a steady state one-dimensional thermoelectric module. This
module contains on many p-type and n-type thermocouples as shown in figure.1b.
Considering that the thermal and electric contact resistances are both negligible, no radiation
or convection heat losses through the boundaries of the element, equation (7) can be reduced
to,
𝑑
𝑑𝑥
𝑑𝑇
𝜌
(𝑘𝐴 𝑑𝑥 ) + 𝐼 2 𝐴 = 0
(8)
Equation (8) is regarded as a differential equation that requires a set of boundary
conditions to be solved. These boundary conditions can be defined simply ad functions of
position x (𝑇𝑥=0 = 𝑇ℎ 𝑎𝑛𝑑 𝑇𝑥=𝐿 = 𝑇𝑐 ). Therefore, the solution for the temperature gradient
can be found as,
𝑑
𝑑𝑇
𝜌
(𝑘𝐴 𝑑𝑥 ) = −𝐼 2 𝐴
𝑑𝑥
(9)
By integrating equation (9), we get
𝑑𝑇
𝜌
𝑘𝐴 ∫ 𝑑 ( ) = −𝐼 2 ∫ 𝑑𝑥
𝑑𝑥
𝐴
(10)
Which gives,
𝑑𝑇
𝑑𝑥
= −𝐼 2
𝜌
𝑘𝐴
𝑥 + 𝐶1
(11)
8
Where 𝐶1 is the integrating constant.
Equation (11) can be rearranged and integrated again as,
𝑇2
𝐿
𝜌
𝐿
𝜌
∫𝑇1 𝑑𝑇 = −𝐼 2 𝑘𝐴 ∫0 𝑥𝑑𝑥 + 𝐶1 ∫0 𝑑𝑥 → (𝑇2 − 𝑇1 ) = −𝐼 2 𝑘𝐴 𝐿2 + 𝐶1 𝐿
(12)
Equation (12) can be rewritten as,
𝐶1 =
2 𝜌𝐿
2𝑘𝐴2
(𝑇2 −𝑇1 )
+𝐼
𝐿
(13)
Now, by substituting Equation (13) in Equation (11), (at 𝑥 = 0) we get,
𝑑𝑇
|
𝑑𝑥 𝑥=0
=
(𝑇2 −𝑇1 )
𝐿
2 𝜌𝐿
2𝑘𝐴2
+𝐼
(14)
Similarly, at 𝑥 = 𝐿 Equation (11) yields,
𝑑𝑇
|
𝑑𝑥 𝑥=𝐿
=
(𝑇2 −𝑇1 )
𝐿
2 𝜌𝐿
2𝑘𝐴2
−𝐼
(15)
Equation (3) is expressed as following,
1
𝐿
𝐴𝐾
2
𝐴
𝐿
𝑞𝑥=0 = 𝛼𝑇1 𝐼 − 𝜌 𝐼 2 +
(𝑇1 − 𝑇2 )
(16)
More specifically, Equation (16) can be expressed for either p-type or n-type as,
1
𝐿
𝐴𝑘𝑝
2
𝐴
𝐿
𝑞𝑝 𝑥=0 = 𝛼𝑝 𝑇1 𝐼 − 𝜌𝑝 𝐼 2 +
(𝑇1 − 𝑇 2 )
1
𝐿
𝐴𝑘𝑛
2
𝐴
𝐿
𝑞𝑛 𝑥=0 = −𝛼𝑛 𝑇1 𝐼 − 𝜌𝑛 𝐼 2 +
(𝑇1 − 𝑇2 )
(17)
(18)
By similar fashion the heat flux equation is carried out at 𝑥 = 𝐿 for the p-type and the
n-type as following,
1
𝐿
𝐴𝑘𝑝
2
𝐴
𝐿
𝑞𝑝 𝑥=𝐿 = 𝛼𝑝 𝑇1 𝐼 + 𝜌𝑝 𝐼 2 +
(𝑇1 − 𝑇 2 )
1
𝐿
𝐴𝑘𝑛
2
𝐴
𝐿
𝑞𝑛 𝑥=𝐿 = −𝛼𝑛 𝑇1 𝐼 + 𝜌𝑛 𝐼 2 +
(𝑇1 − 𝑇2 )
(19)
(20)
Then, the total heat flux at position (1) and (2) (see figure 4.a) can be found by taking
the summation of Equations (17) and (18), (19) and (20) respectively as,
9
1
𝜌𝑝 𝐿𝑝
2
𝐴𝑃
1
𝜌𝑝 𝐿𝑝
𝑄1. = 𝑛[(𝛼𝑝 − 𝛼𝑛 )𝑇1 𝐼 − 𝐼 2 (
𝑄2. = 𝑛[(𝛼𝑝 − 𝛼𝑛 )𝑇2 𝐼 + 2 𝐼 2 (
𝐴𝑃
+
+
𝜌𝑛 𝐿𝑛
𝐴𝑛
𝜌𝑛 𝐿𝑛
𝐴𝑛
𝑘𝑝 𝐴𝑃
)+(
)+(
𝐿𝑝
𝑘𝑝 𝐴𝑃
𝐿𝑝
+
+
𝑘𝑛 𝐴𝑛
𝐿𝑛
) (𝑇1 − 𝑇2 )]
𝑘𝑛 𝐴𝑛
𝐿𝑛
) (𝑇1 − 𝑇2 )]
(21)
(22)
Where n is the number of the thermocouples in the module. The material properties ae defined
as,
(23)
𝛼 = 𝛼𝑝 − 𝛼𝑛
𝑅=
𝑘=
𝜌𝑝 𝐿𝑝
𝐴𝑃
𝑘𝑝 𝐴𝑃
𝐿𝑝
+
+
𝜌𝑛 𝐿𝑛
(24)
𝐴𝑛
𝑘𝑛 𝐴𝑛
(25)
𝐿𝑛
Where R and K are the total electric resistance and the total thermal conductance,
respectively. This reduces Equations (21) and (22) to,
1
2
(26)
1
2
(27)
𝑄1. = 𝑛[𝛼𝑇1 𝐼 − 2 𝑅𝐼 + 𝐾(𝑇1 − 𝑇2 )]
𝑄2. = 𝑛[𝛼𝑇2 𝐼 + 2 𝑅𝐼 + 𝐾(𝑇1 − 𝑇2 )]
Equations (26) and (27) are known as the thermoelectric ideal equations. The first term
in theses equations 𝛼𝑇1 𝐼 is known as Peltier (or Seebeck effect), and it is reversible. The second
1
term 2 𝑅𝐼 2 is the Joule heating term. The last term is known as the thermal conduction term.
Both the Joule heating and the thermal conduction are irreversible.
Figure.5 Thermoelectric generator attached to a load resistance.
In the case of thermoelectric generator (TEG), the ideal equations can be modified with
using respective hat and cold junction temperatures (Figure.5) as,
10
1
2
(28)
1
2
(29)
𝑄ℎ. = 𝑛[𝛼𝑇ℎ 𝐼 − 2 𝑅𝐼 + 𝐾(𝑇ℎ − 𝑇𝑐 )]
𝑄𝑐. = 𝑛[𝛼𝑇𝑐 𝐼 + 2 𝑅𝐼 + 𝐾(𝑇ℎ − 𝑇𝑐 )]
Where 𝑄ℎ. And 𝑄𝑐. Are the heat absorbed and dissipated at the hot and cold junctions
of the TEG, respectively. Assuming that the p-type and n-type thermocouples are similar, we
𝐿
𝐴
have 𝑅 = 𝜌 𝐴 and 𝐾 = 𝑘 𝐿 , where 𝜌 = 𝛼𝑝 + 𝛼𝑛 and 𝐾 = 𝑘𝑝 + 𝑘𝑛 .
From the 1st law of thermodynamics, the output power across the thermoelectric
generator module can be defined as 𝑊𝑛. = 𝑄ℎ. − 𝑄𝑐. . Then, by substituting Equation (28) and
(29) in the power equation, the output power can be expressed in terms of the internal properties
as,
𝑊𝑛. = 𝑛[𝛼𝐼(𝑇ℎ − 𝑇𝑐 ) − 𝑅𝐼 2 ]
(30)
Also, the output power can be defined in terms of the load resistance 𝑅𝐿 as,
𝑊𝑛. = 𝑛𝐼 2 𝑅𝐿 = 𝐼𝑉
(31)
Now, equating Equations (30) and (31) yields to,
𝑛𝐼 2 𝑅𝐿 = 𝐼𝑉 = 𝑛[𝛼𝐼(𝑇ℎ − 𝑇𝑐 ) − 𝑅𝐼 2 ]
(32)
Equation (32) can be reduced to,
(33)
𝑉𝑛 = 𝑛𝐼𝑅𝐿 = 𝑛[𝛼(𝑇ℎ − 𝑇𝑐 ) − 𝑅𝐼]
Where 𝑉𝑛 is the voltage across the load resistance.
4. Performance Parameters of TEG module:
The current can be obtained from equation (33) as,
𝐼=
𝛼(𝑇ℎ −𝑇𝑐 )
(34)
𝑅𝐿 +𝑅
It is clear form Equation (34) that the current is not a function of the number of
thermocouples.
Substituting Equations (34) in Equation (32) gives,
𝑉𝑛 =
𝑛𝛼(𝑇ℎ −𝑇𝑐 ) 𝑅𝐿
𝑅𝐿
+1
𝑅
( )
(35)
𝑅
11
Inserting Equation (34) in Equation (19) gives,
𝑊𝑛.
=
𝑅𝐿
𝑅
𝑅
(1+ 𝑅𝐿 )2
𝑛𝛼 2 (𝑇ℎ −𝑇𝑐 )2
𝑅
(36)
Now, from the thermal efficiency definition η𝑡ℎ =
𝑊𝑛.
𝑄ℎ.
, and by inserting Equation (34)
in (31) then substituting the result with Equation (28) in the thermal efficiency equation we
get,
𝑇
η𝑡ℎ =
Where
𝑅𝐿
𝑅
𝑅
(1−𝑇 𝑐 ) 𝑅𝐿
ℎ
(37)
𝑅 2𝑇
(1+ 𝐿 ) 𝑐
𝑅 𝑇ℎ
𝑅
1
𝑇
(1+ 𝑅𝐿 )−2(1−𝑇 𝑐 )+
𝑍𝑇𝑐
ℎ
is the resistance ratio, and 𝑍𝑇𝑐 is known as the dimensionless figure of
merit, both of which are important in design of thermoelectric modules.
5. Maximum Parameters of TEGs:
For any thermoelectric module, there are two types of maximum parameters, the first
is the maximum parameter at the maximum power and the second is the maximum conversion
parameters. These two modes can be obtained by modifying the resistance ratio
𝑅𝐿
𝑅
regarding
on the operating conditions.
The maximum current usually accrues at the short circuit, where the load resistance
𝑅𝐿 = 0. Thus, the maximum current is found from Equation (34) as,
𝐼𝑚𝑎𝑥 =
𝛼(𝑇ℎ −𝑇𝑐 )
(38)
𝑅
Since the maximum voltage occurs at the open circuit, where 𝐼 = 0, sitting the current
to zero in Equation (33) yields,
𝑉𝑚𝑎𝑥 = 𝑛𝛼(𝑇ℎ − 𝑇𝑐 )
(39)
For the maximum power, the output power Equation (36) is differentiated with
𝑅𝐿
respect to ( ) and set to zero, which gives,
𝑅
12
𝑑(𝑊𝑛. )
𝑅
𝑑( 𝑅𝐿 )
𝑅𝐿
=0→
𝑅
=1
Now, by substituting
𝑅𝐿
𝑅
(40)
= 1 in Equation (36), the result leads to the maximum
power equation as,
𝑊𝑚𝑎𝑥 =
𝑛𝛼 2 (𝑇ℎ −𝑇𝑐 )2
(41)
4𝑅
In similar fashion, the maximum conversion efficiency can be obtained by
𝑅𝐿
differentiating Equation (37) with respect to ( ) and equating the result will give the
𝑅
maximum conversion efficiency as,
η𝑚𝑎𝑥 = (1 −
𝑇𝑐
)
√1+𝑍𝑇̅ −1
(42)
𝑇ℎ √1+𝑍𝑇̅ − 𝑇𝑐
𝑇ℎ
Where
𝑇̅
is the average temperature between the hot and the cold junction
temperatures and is expressed by,
(𝑇 −𝑇 )
𝑇̅ = ℎ 𝑐
(43)
2
For the case of maximum parameters at the maximum power, the maximum
𝑅
power efficiency is obtained by setting 𝑅𝐿 equals to one in Equation (37) and this will
give,
𝑇
(1− 𝑐 )
η𝑚𝑝 =
𝑇ℎ
(44)
𝑇
4 𝑐
𝑇ℎ
1
𝑇𝑐
2− (1− )+
2
𝑇ℎ
𝑍𝑇𝑐
Similarly, the voltage and current at the maximum power can be obtained by sitting
𝑅𝐿
𝑅
= 1 in Equations (35) and (34), respectively, and yields,
𝑉𝑚𝑝 =
𝐼𝑚𝑝 =
𝑛𝛼(𝑇ℎ −𝑇𝑐 )
(45)
2
𝛼(𝑇ℎ −𝑇𝑐 )
(46)
2𝑅
13
Note that so far there are seven maximum parameters, which areη𝑚𝑎𝑥 ,η𝑚𝑝 ,𝑉𝑚𝑎𝑥 ,𝑉𝑚𝑝 ,𝐼𝑚𝑎𝑥 ,
𝐼𝑚𝑝 , and 𝑊𝑚𝑎𝑥 .
6. Normalized Parameters:
These normalized parameters can be obtained by dividing the active parameters by the
maximum parameters, and by doing that we can represent the characteristics of the
thermoelectric generator.
The normalized output power can be defined by dividing Equation (36) by Equation
(41), which gives,
𝑊
𝑊𝑚𝑎𝑥
=
𝑅
4 𝐿
𝑅
𝑅𝐿
( +1)2
𝑅
(47)
For the normalized voltage, Equation (34) is divided by Equation (39), and the result is,
𝑉𝑛
𝑉𝑚𝑎𝑥
=
𝑅𝐿
𝑅
𝑅𝐿
+1
𝑅
(48)
Equations (34) and (38) gives the normalized current as,
𝐼
𝐼𝑚𝑎𝑥
1
= 𝑅𝐿
𝑅
(49)
+1
Not so far that the three normalized parameters are only functions in the resistance ratio
𝑅𝐿
𝑅
.
For the normalized efficiency, we divide equation (37) by Equation (42) and the result is,
η𝑡ℎ
η𝑚𝑎𝑥
=
𝑅𝐿
𝑇
(√1+𝑍𝑇̅ +𝑇 𝑐 )
𝑅
ℎ
2
𝑅𝐿
𝑇
( +1) (1+ 𝑐 )
𝑅
𝑇ℎ
𝑅𝐿
1
𝑇𝑐
[( 𝑅 +1)−2(1−𝑇 )+
](√1+𝑍𝑇̅ −1)
̅
2𝑍𝑇
ℎ
(50)
From Equation (50), it is clear that the normalized efficiency is not only a function
in
𝑅𝐿
𝑅
, but also in the dimensionless figure of merit 𝑍𝑇̅ and the temperature ratio
14
𝑇𝑐
𝑇ℎ
.
𝑇
Figure.6 Generalized TEG performance for 𝑐⁄𝑇 = 0.5 and 𝑍𝑇̅ = 1
ℎ
Figure.6 above indicates the generalized performance characteristic for TEGs
typical values of
𝑇𝑐
⁄𝑇 = 0.5 and 𝑍𝑇̅ = 1 . This chart was constructed using the
ℎ
normalized Equations from (47) to (50). It is seen from this chart that the maximum
output power occurs at the resistance ration of
𝑅𝐿
𝑅
= 1, which has already been predicted
early in this work. The maximum efficiency, however, occurs at a value of resistance ratio
equals to √1 + 𝑍𝑇̅ , or more specifically at
𝑅𝐿
𝑅
= 1.6 in this case
7. Effective material properties:
The maximum parameters, represented in the previous section, are all functions
in the material properties (seebeck, thermal conductivity, and electric resistivity) for
𝐴
known geometry factor (𝐺 = 𝐿 ) and junction temperatures 𝑇ℎ , and 𝑇𝑐 . Therefore, the
effective material properties can be defined in terms of these four maximum parameters
( η𝑚𝑎𝑥 , 𝑊𝑚𝑎𝑥 , 𝑉𝑚𝑎𝑥 , and 𝐼𝑚𝑎𝑥 ), which are usually provided by the manufacturer or
taken by measurements, and differ from one module to another.
With this concept, the effective resistivity is formulated by using Equations (38)
and (41), which give,
15
𝜌∗ =
4(𝐴⁄𝐿 )𝑊𝑚𝑎𝑥
4(𝐼𝑚𝑎𝑥 )2
(51)
The effective figure of merit can be formulated from Equation (42), which gives,
𝑍∗ =
2
𝑇
𝑇𝑐 (1+( 𝑐 )−1 )
η𝑚𝑎𝑥 𝑇𝑐
η𝑐 𝑇ℎ
η𝑚𝑎𝑥
1−
η𝑐
1+
[(
𝑇ℎ
𝑇𝑐
Where η𝑐 = 1 −
𝑇
2
) − 1]
(52)
is Carnot efficiency. However, in cases of having the
ℎ
maximum power efficiency instead of the conversion efficiency provided by the
manufacturer, the figure of merit can be defined as,
4 𝑇𝑐
( )
𝑇𝑐 𝑇ℎ
1
1
η𝑐 (
+ )−2
η𝑚𝑝 2
𝑍∗ =
(53)
Also, the effective seebeck coefficient can be obtained from Equations (38) and
(50), which is,
𝛼∗ =
4𝑊𝑚𝑎𝑥
(54)
𝑛𝐼𝑚𝑎𝑥 (𝑇ℎ −𝑇𝑐 )
From the definition of the thermal conductivity, we can define the effective
thermal conductivity as a function of the other effective properties as,
𝑘∗ =
𝛼∗
2
(55)
𝜌∗ 𝑍 ∗
Note that the effective material properties are obtained using the ideal equations.
Thus, they include different effects like Thomson effect, conduct resistances, and losses
due to radiation and convection. These effective material properties are the total
properties, so they should be divided by two to obtain the single p-type and n-type
thermocouple properties.
8. Results and discussion
In this project, the effective material properties (,, and k*) were calculated using
the manufacture’s maximum parameters (Wmax , Imax, and ηmax). Four different modules were
chosen to check the status of ideal equation with the effective material properties as shown in
Table 1. First, the effective material properties for each module were obtained using Equations
16
(51) – (55). The cross-sectional (A) area and length (L) of thermoelement were either measured
or provided by manufactures.
As shown in Table 1, by Equations (38) – (42) the maximum parameters (Wmax , Imax,
ηmp, and Vmax) were recalculated using the effective material properties and compared with the
manufacture’s maximum parameters. They appear almost the same except the maximum
voltage which is reasonable because it was not consider in our calculations (secondary
parameter).
Table.1 Comparison of the properties and dimensions for the commercial products of
thermoelectric modules
Description
TEG Module (Bismuth Telluride)
Symbols
Number of
TG12-4
HZ-2
TGM199-1.4-2.0
TGM127-1.4-2.5
Tc =50 oC
Tc =30 oC
Tc =30 oC
Tc =30 oC
Th =170oC
Th =230oC
Th =200oC
Th =200oC
127
97
199
127
n
thermocouples
Manufacturers’
𝑊𝑚𝑎𝑥 (W)
2.12
2.6
7.3
4.4
𝐼𝑚𝑎𝑥 (A)
1.32
1.6
2.65
2.37
η𝑚𝑝 (%)
4.08
4.5
5.3
5.4
𝑉𝑚𝑎𝑥 (V)
6.5
6.53
11
7.7
Rn ()
6.32
4
3.7
3
maximum
parameters
Measured
A (mm2)
1
2.1
1.96
1.96
geometry of
L (mm)
1.17
2.87
2
2.5
G = A/L
0.085
0.073
0.098
0.078
30 × 30 ×3.4
30 × 30 ×4.5
40 × 40 × 4.4 40 × 40 × 4.8
thermoelement
(cm)
Dimension
mm
(W×L×H)
17
Effective
material
 V/K
210.769
167.526
162.856
171.981
cm
1.638 × 10-3
1.532 × 10-3
1.024 × 10-3
0.9672 × 10-3
k(W/cmK)
0.015
0.016
0.015
0.017
0.708
0.456
0.652
0.679
Wmax (W)
2.12
2.6
7.3
4.4
Imam (A)
1.32
1.6
2.65
2.37
ηmax (%)
4.1
4.5
5.3
5.4
Vmax (V)
6.424
6.5
11.019
7.426
properties
(calculated using
commercial(Wmax ZTavr
, Imax,
and ηmax)
The maximum
parameters using
effective material
properties
8.1. TG12-4 by Marlow
Figure.7 shows comparison between the calculations (solid lines) and the
manufacturer’s performance data (triangles) of Module TG12-4 with two different cold side
temperature. The output power in Figure.7 (a) was calculated using Equation (41) and the
effective material properties versus the hot side temperature. It is seen in Figure.7 (a) that the
calculated output power is in good agreement with the manufacturer’s performance curves. It
is also seen that at high temperature the effective material properties could not accurately
predict the power output. This occurs because of the temperature dependent of material
properties. On the other hand, Figure.7 (b) shows that the errors on the voltage-vs-hot side
temperature curves, decrease with increasing the hot side temperature. The performance chart
for efficiency were not provided but the heat input values at corresponding hot side
temperatures for the graph of power and voltage output were provided. So, the efficiency were
predicted using equation (42) and has a good agreement as shown in Figure.7 (c).
18
Figure 7. (a) Output power versus hot side temperature
Figure 7. (b) Voltage versus hot side temperature
19
Figure 7. (c) Efficiency versus hot side temperature
8.2.HZ-2 by Hi-Z
Figures.8 (a), (b), (c), and (d) show comparison between the calculations and the
performance data of Module HZ-2. In figure (a), (b), and (c), the output power, voltage, and
efficiency versus temperature difference were compared with commercial data. In general, the
calculations are in good agreement with the manufacturer’s performance data. In figure.8 (d),
the output power, efficiency, voltage versus current were accurate in predicting with
commercial data.
Figure 8. (a) Output power versus Temperature difference
20
Figure 8. (b) Voltage versus Temperature difference
. Figure8. (c) Efficiency versus Temperature difference
21
Figure 8. (d) Output power, efficiency, and voltage versus current
8.3.TGM199-1.4-2.0 by Kryotherm
Figures.9 (a), (b), (c), and (d) show comparison between the calculations and the
performance data for different cold side temperature of Module TGM199-1.4-2.0, with a good
agreement. In figure.9 (b), the current showed discrepancies at regions of non-linearity. This
was occurred because of the temperature independent material properties for ideal equation
that only predicts a linear voltage output. The Open circuit voltage which represent the
maximum voltage was compared with commercial data as shown in figure.9 (c). The voltage
comparison at matched load conditions in Figure.9 (d) shows similar results to the commercial
data. In figure.9 (e), the efficiency versus load resistance was in excellent agreement with
commercial data. The output voltage and power versus current were exactly the same as the
manufacture’s data as shown in figure.9 (f).
22
Figure 9. (a) Output power versus hot side temperature
Figure 9. (b) Current versus hot side temperature
23
Figure 9. (c) Open circuit voltage versus hot side temperature
Figure 9. (d) Voltage versus hot side temperature
24
Figure 9. (e) Efficiency versus load resistance
Figure 9. (f) Output voltage and power versus current
25
8.4. TGM127-1.4-2.5 by Kryotherm
Figures.10 (a), (b), (c), (d), (e), and (f) predict comparison between the calculations and
the performance data for different cold side temperature of Module TGM127-1.4-2.5, with a
good agreement.
Figure 10. (a) Output power versus hot side temperature
Figure 10. (b) Current versus hot side temperature
26
Figure 10. (c) Open circuit voltage versus hot side temperature
Figure 10. (d) Voltage versus hot side temperature
27
Figure 10. (e) Efficiency versus load resistance
Figure 10. (f) Output voltage and power versus current
9. Normalized curves:
The normalized charts were created using the normalized parameters over the
maximum parameters presented in this work. Using equation (50) the normalized efficiency
η / ηmax was plotted versus the resistance ratio RL / R as shown in figures 11. (a) and (b).
28
The normalized efficiency is a function of the dimensionless figure of merit evaluated
at the average temperature (ZTavr) and junction temperature ratio Tc / Th. These charts can be
very helpful for designers to find the load resistance that gives the maximum possible
efficiency. In figure 11 (a), the junction temperature ratio was fixed at 0.5 with various
dimensionless figure of merit (evaluated at the average temperature) and plotted versus
resistance ratio. In figure.11 (b), the dimensionless figure of merit (evaluated at the average
temperature) was fixed at 1.5 with various junction temperature ratio and plotted versus
resistance ratio. It is seen in figure.11 (b) that the normalized curves at different junction
temperature ratio were close to each other.
Figure 11. (a) Normalized Efficiency for TEGs for various ZTavr
29
Figure 11. (b) Normalized Efficiency for TEGs for various Tc/Th
10. Conclusion:
To conclude, it is demonstrated that using the effective material properties technique
along with the ideal equations is accurate as it is showing an acceptable agreement with the
several manufacturer’s performance data (which even can be provided by the manufacturer of
obtained by measurements). Thus, this work has shown that once one has the maximum
parameters of a module, the performance of this module can be evaluated analytically.
Considering that the material properties are usually temperature dependents and the
existing of the thermal and electrical contact resistances, usually make the thermoelectric
analysis complicated. However, employing the analysis of the ideal equations with the effective
material properties can be reliable and simple if the temperature difference was moderated.
The normalized charts were obtained by using the maximum parameters defined in this
study to present the characteristics of the four thermoelectric generators. These normalized
charts can be standard for any thermoelectric generator at a given dimensionless figure of merit.
The maximum parameters provided by the manufacturer (used to obtain the effective material
properties) may not be accurate, yet the results may vary considering this factor.
30
References
[1] H. Lee, Thermal Design: Heat Sinks, Thermoelectrics, Heat Pipes, Compact Heat
Exchaners, and Solar Cells. Hoboken: John Wiley & Sons, Inc., 2010
[2] H. Lee, "The Thomson effect and the ideal equation on thermoelectric coolers,"
Energy, vol. 56, pp. 61-69, 2013.
[3] H. Lee, A. Attar, and S. Weera, "Performance evaluation of commercial
thermoelectric modules using effective material properties," in 2014 International
Conference on Thermoelectrics, Nashville, 2014, pp. 1-5.
[4] S. Kim et al., "Thermoelectric power generation system for future hybrid vehicles
using hot exhaust gas," Journal of Electronic Materials, vol. 40, no. 5, pp. 778-784,
2011.
[5] Weera, Sean Lwe Leslie, "Analytical Performance Evaluation of Thermoelectric
Modules Using Effective Material Properties" (2014). Master's Theses. Paper 483.
31
Appendix
32