Performance Prediction of Commercial

Thermoelectric Generator Modules using

the Effective Material Properties

Abdulmunaem Elarusi

Nithin Illendula

Hassan Fagehi

1

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.

2

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

3

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 Thermo-

Electric 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 thermo-

electric 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

4

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 I Th 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.

5

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

6

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 non-

uniformly 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 E⃗⃗ (Or electric potential) is defined according to the ohm's low and Seebeck effect,

because the electric field is influenced by the temperature gradient ∇⃗⃗ 𝑇 and the current density

𝑗 Thus, the electric field can be defined as,

E⃗⃗ = 𝑗 𝜌 + 𝛼∇⃗⃗ 𝑇 (2)

Similarly, the heat flow (or heat flux) is effected by the electric field E⃗⃗ and the

temperature gradient ∇⃗⃗ 𝑇 the heat flow density, consequently, is expressed considering the

Thomson relationship and the Onsager's principle, as

𝑞 = 𝛼𝑇𝑗 − 𝐾∇⃗⃗ 𝑇 (3)

Where T is the temperature at the boundary through which the heat flux flows. The 𝛼𝑇𝑗

is the Peilter heat contribution, and 𝐾∇⃗⃗ 𝑇 the heat transfer from the Fourier's low of conduction.

The general heat diffusion equation as time dependent is given by,

−∇ ⃗⃗ ⃗. 𝑞 + �̇� = 𝜌𝐶𝑝𝜕𝑇

𝜕𝑡 (4)

Where �̇� is the heat flux per unit volume, 𝜌 is the density of the material, 𝐶𝑝 is the

7

specific heat capacity, and 𝜕𝑇

𝜕𝑡 is the rate of change of temperature with respect to time.

Now, for steady state condition (𝜕𝑇

𝜕𝑡 goes to zero), we have,

�̇� = ∇ ⃗⃗ ⃗. 𝑞 (5)

Since we can define the heat flux as a function in the electric power, �̇� is expressed by,

�̇� = E⃗⃗ . 𝑗 = 𝑗2 𝜌 + 𝑗 . 𝛼 ∇⃗⃗ 𝑇 (6)

Substituting equations (3) and (6) in equation (5) gives us,

∇ ⃗⃗ ⃗. (𝐾∇⃗⃗ 𝑇) + 𝑗2 𝜌 − 𝑇𝑑𝛼

𝑑𝑇𝑗 . ∇⃗⃗ 𝑇 = 0 (7)

Where, from Thomson relation, 𝜏 = 𝑇𝑑𝛼

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)

8

(b)

Figure 4. (a) cutaway of a thermoelectric generator module, and (b) a p-type and n-

type 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(𝑘𝐴

𝑑𝑇

𝑑𝑥) + 𝐼2

𝜌

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

𝑑𝑥(𝑘𝐴

𝑑𝑇

𝑑𝑥) = −𝐼2

𝜌

𝐴 (9)

By integrating equation (9), we get

𝑘𝐴∫𝑑 (𝑑𝑇

𝑑𝑥) = −𝐼2

𝜌

𝐴∫𝑑𝑥 (10)

Which gives,

𝑑𝑇

𝑑𝑥= −𝐼2

𝜌

𝑘𝐴𝑥 + 𝐶1 (11)

9

Where 𝐶1 is the integrating constant.

Equation (11) can be rearranged and integrated again as,

∫ 𝑑𝑇𝑇2

𝑇1= −𝐼2

𝜌

𝑘𝐴 ∫ 𝑥𝑑𝑥

𝐿

0+ 𝐶1 ∫ 𝑑𝑥

𝐿

0 → (𝑇2 − 𝑇1) = −𝐼2

𝜌

𝑘𝐴 𝐿2 + 𝐶1𝐿 (12)

Equation (12) can be rewritten as,

𝐶1 =(𝑇2−𝑇1)

𝐿+ 𝐼2

𝜌𝐿

2𝑘𝐴2 (13)

Now, by substituting Equation (13) in Equation (11), (at 𝑥 = 0) we get,

𝑑𝑇

𝑑𝑥|𝑥=0

=(𝑇2−𝑇1)

𝐿+ 𝐼2

𝜌𝐿

2𝑘𝐴2 (14)

Similarly, at 𝑥 = 𝐿 Equation (11) yields,

𝑑𝑇

𝑑𝑥|𝑥=𝐿

=(𝑇2−𝑇1)

𝐿− 𝐼2

𝜌𝐿

2𝑘𝐴2 (15)

Equation (3) is expressed as following,

𝑞𝑥=0 = 𝛼𝑇1𝐼 −1

2𝜌

𝐿

𝐴𝐼2 +

𝐴𝐾

𝐿(𝑇1 − 𝑇2) (16)

More specifically, Equation (16) can be expressed for either p-type or n-type as,

𝑞𝑝𝑥=0= 𝛼𝑝𝑇1𝐼 −

1

2𝜌𝑝

𝐿

𝐴𝐼2 +

𝐴𝑘𝑝

𝐿(𝑇1 − 𝑇2) (17)

𝑞𝑛𝑥=0= −𝛼𝑛𝑇1𝐼 −

1

2𝜌𝑛

𝐿

𝐴𝐼2 +

𝐴𝑘𝑛

𝐿(𝑇1 − 𝑇2) (18)

By similar fashion the heat flux equation is carried out at 𝑥 = 𝐿 for the p-type and the

n-type as following,

𝑞𝑝𝑥=𝐿= 𝛼𝑝𝑇1𝐼 +

1

2𝜌𝑝

𝐿

𝐴𝐼2 +

𝐴𝑘𝑝

𝐿(𝑇1 − 𝑇2) (19)

𝑞𝑛𝑥=𝐿= −𝛼𝑛𝑇1𝐼 +

1

2𝜌𝑛

𝐿

𝐴𝐼2 +

𝐴𝑘𝑛

𝐿(𝑇1 − 𝑇2) (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,

10

𝑄1. = 𝑛[(𝛼𝑝 − 𝛼𝑛)𝑇1𝐼 −

1

2𝐼2 (

𝜌𝑝𝐿𝑝

𝐴𝑃+

𝜌𝑛𝐿𝑛

𝐴𝑛) + (

𝑘𝑝𝐴𝑃

𝐿𝑝+

𝑘𝑛𝐴𝑛

𝐿𝑛) (𝑇1 − 𝑇2)] (21)

𝑄2. = 𝑛[(𝛼𝑝 − 𝛼𝑛)𝑇2𝐼 +

1

2𝐼2 (

𝜌𝑝𝐿𝑝

𝐴𝑃+

𝜌𝑛𝐿𝑛

𝐴𝑛) + (

𝑘𝑝𝐴𝑃

𝐿𝑝+

𝑘𝑛𝐴𝑛

𝐿𝑛) (𝑇1 − 𝑇2)] (22)

Where n is the number of the thermocouples in the module. The material properties ae defined

as,

𝛼 = 𝛼𝑝 − 𝛼𝑛 (23)

𝑅 =𝜌𝑝𝐿𝑝

𝐴𝑃+

𝜌𝑛𝐿𝑛

𝐴𝑛 (24)

𝑘 =𝑘𝑝𝐴𝑃

𝐿𝑝+

𝑘𝑛𝐴𝑛

𝐿𝑛 (25)

Where R and K are the total electric resistance and the total thermal conductance,

respectively. This reduces Equations (21) and (22) to,

𝑄1. = 𝑛[𝛼𝑇1𝐼 −

1

2𝑅𝐼2 + 𝐾(𝑇1 − 𝑇2)] (26)

𝑄2. = 𝑛[𝛼𝑇2𝐼 +

1

2𝑅𝐼2 + 𝐾(𝑇1 − 𝑇2)] (27)

Equations (26) and (27) are known as the thermoelectric ideal equations. The first term

in theses equations 𝛼𝑇1𝐼 is known as Peltier (or Seebeck effect), and it is reversible. The second

term 1

2𝑅𝐼2 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,

11

𝑄ℎ. = 𝑛[𝛼𝑇ℎ𝐼 −

1

2𝑅𝐼2 + 𝐾(𝑇ℎ − 𝑇𝑐)] (28)

𝑄𝑐. = 𝑛[𝛼𝑇𝑐𝐼 +

1

2𝑅𝐼2 + 𝐾(𝑇ℎ − 𝑇𝑐)] (29)

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

have 𝑅 = 𝜌𝐿

𝐴 and 𝐾 = 𝑘

𝐴

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) − 𝑅𝐼2] (30)

Also, the output power can be defined in terms of the load resistance 𝑅𝐿 as,

𝑊𝑛. = 𝑛𝐼2𝑅𝐿 = 𝐼𝑉 (31)

Now, equating Equations (30) and (31) yields to,

𝑛𝐼2𝑅𝐿 = 𝐼𝑉 = 𝑛[𝛼𝐼(𝑇ℎ − 𝑇𝑐) − 𝑅𝐼2] (32)

Equation (32) can be reduced to,

𝑉𝑛 = 𝑛𝐼𝑅𝐿 = 𝑛[𝛼(𝑇ℎ − 𝑇𝑐) − 𝑅𝐼] (33)

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)

𝑅𝐿+𝑅 (34)

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)

𝑅𝐿𝑅

+1(𝑅𝐿

𝑅) (35)

12

Inserting Equation (34) in Equation (19) gives,

𝑊𝑛. =

𝑛𝛼2(𝑇ℎ−𝑇𝑐)2

𝑅

𝑅𝐿𝑅

(1+𝑅𝐿𝑅

)2 (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,

η𝑡ℎ =(1−

𝑇𝑐𝑇ℎ

)𝑅𝐿𝑅

(1+𝑅𝐿𝑅

)−1

2(1−

𝑇𝑐𝑇ℎ

)+(1+

𝑅𝐿𝑅

)2𝑇𝑐𝑇ℎ

𝑍𝑇𝑐

(37)

Where 𝑅𝐿

𝑅 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)

𝑅 (38)

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

respect to (𝑅𝐿

𝑅) and set to zero, which gives,

13

𝑑(𝑊𝑛. )

𝑑(𝑅𝐿𝑅

)= 0 →

𝑅𝐿

𝑅= 1 (40)

Now, by substituting 𝑅𝐿

𝑅= 1 in Equation (36), the result leads to the maximum

power equation as,

𝑊𝑚𝑎𝑥 =𝑛𝛼2(𝑇ℎ−𝑇𝑐)

2

4𝑅 (41)

In similar fashion, the maximum conversion efficiency can be obtained by

differentiating Equation (37) with respect to (𝑅𝐿

𝑅) and equating the result will give the

maximum conversion efficiency as,

η𝑚𝑎𝑥 = (1 −𝑇𝑐

𝑇ℎ)

√1+𝑍�̅�−1

√1+𝑍�̅�−𝑇𝑐𝑇ℎ

(42)

Where �̅� is the average temperature between the hot and the cold junction

temperatures and is expressed by,

�̅� =(𝑇ℎ−𝑇𝑐)

2 (43)

For the case of maximum parameters at the maximum power, the maximum

power efficiency is obtained by setting 𝑅𝐿

𝑅 equals to one in Equation (37) and this will

give,

η𝑚𝑝 =(1−

𝑇𝑐𝑇ℎ

)

2−1

2(1−

𝑇𝑐𝑇ℎ

)+4𝑇𝑐𝑇ℎ

𝑍𝑇𝑐

(44)

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 =𝑛𝛼(𝑇ℎ−𝑇𝑐)

2 (45)

𝐼𝑚𝑝 =𝛼(𝑇ℎ−𝑇𝑐)

2𝑅 (46)

14

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=

4𝑅𝐿𝑅

(𝑅𝐿𝑅

+1)2 (47)

For the normalized voltage, Equation (34) is divided by Equation (39), and the result is,

𝑉𝑛

𝑉𝑚𝑎𝑥=

𝑅𝐿𝑅

𝑅𝐿𝑅

+1 (48)

Equations (34) and (38) gives the normalized current as,

𝐼

𝐼𝑚𝑎𝑥=

1𝑅𝐿𝑅

+1 (49)

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=

𝑅𝐿𝑅

(√1+𝑍�̅�+𝑇𝑐𝑇ℎ

)

[(𝑅𝐿𝑅

+1)−1

2(1−

𝑇𝑐𝑇ℎ

)+(𝑅𝐿𝑅

+1)2(1+

𝑇𝑐𝑇ℎ

)

2𝑍�̅�](√1+𝑍�̅�−1)

(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 𝑍�̅� and the temperature ratio

𝑇𝑐

𝑇ℎ.

15

Figure.6 Generalized TEG performance for 𝑇𝑐

𝑇ℎ⁄ = 0.5 and 𝑍�̅� = 1

Figure.6 above indicates the generalized performance characteristic for TEGs

typical values of 𝑇𝑐

𝑇ℎ⁄ = 0.5 and 𝑍�̅� = 1 . This chart was constructed using the

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 + 𝑍�̅� , 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

known geometry factor (𝐺 =𝐴

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,

16

𝜌∗ =4(𝐴 𝐿⁄ )𝑊𝑚𝑎𝑥

4(𝐼𝑚𝑎𝑥)2 (51)

The effective figure of merit can be formulated from Equation (42), which gives,

𝑍∗ =2

𝑇𝑐(1+(𝑇𝑐𝑇ℎ

)−1)[(

1+ η𝑚𝑎𝑥

η𝑐

𝑇𝑐𝑇ℎ

1− η𝑚𝑎𝑥

η𝑐

)

2

− 1] (52)

Where η𝑐 = 1 −𝑇𝑐

𝑇ℎ is Carnot efficiency. However, in cases of having the

maximum power efficiency instead of the conversion efficiency provided by the

manufacturer, the figure of merit can be defined as,

𝑍∗ =

4

𝑇𝑐(𝑇𝑐𝑇ℎ

)

η𝑐(1

η𝑚𝑝+

1

2)−2

(53)

Also, the effective seebeck coefficient can be obtained from Equations (38) and

(50), which is,

𝛼∗ =4𝑊𝑚𝑎𝑥

𝑛𝐼𝑚𝑎𝑥(𝑇ℎ−𝑇𝑐) (54)

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

𝜌∗𝑍∗ (55)

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

17

(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 TG12-4

Tc =50 oC

Th =170oC

HZ-2

Tc =30 oC

Th =230oC

TGM199-1.4-2.0

Tc =30 oC

Th =200oC

TGM127-1.4-2.5

Tc =30 oC

Th =200oC

Number of

thermocouples

n 127 97 199 127

Manufacturers’

maximum

parameters

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

Measured

geometry of

thermoelement

A (mm2) 1 2.1 1.96 1.96

L (mm) 1.17 2.87 2 2.5

G = A/L

(cm)

0.085 0.073 0.098 0.078

Dimension

(W×L×H)

mm 30 × 30 ×3.4 30 × 30 ×4.5 40 × 40 × 4.4 40 × 40 × 4.8

18

Effective

material

properties

(calculated using

commercial(Wmax

, Imax, and ηmax)

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

ZTavr 0.708 0.456 0.652 0.679

The maximum

parameters using

effective material

properties

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

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).

19

Figure 7. (a) Output power versus hot side temperature

Figure 7. (b) Voltage versus hot side temperature

20

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

21

Figure 8. (b) Voltage versus Temperature difference

. Figure8. (c) Efficiency versus Temperature difference

22

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).

23

Figure 9. (a) Output power versus hot side temperature

Figure 9. (b) Current versus hot side temperature

24

Figure 9. (c) Open circuit voltage versus hot side temperature

Figure 9. (d) Voltage versus hot side temperature

25

Figure 9. (e) Efficiency versus load resistance

Figure 9. (f) Output voltage and power versus current

26

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

27

Figure 10. (c) Open circuit voltage versus hot side temperature

Figure 10. (d) Voltage versus hot side temperature

28

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).

29

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

30

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.

31

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.

32

Appendix