Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
ISSN 1843-6188
ELECTRO-THERMAL ANALYSIS OF PELTIER COOLING USING FEM
D. ENESCU1, E.O. VÎRJOGHE2, M. IONEL1, M.F. STAN2
1
Electronic, Telecommunications and Energetics Department
Automatics, Informatics and Electrical Engineering Department
Valahia University of Targoviste, Electrical Engineering Faculty 18-20 Unirii Ave., 130082
E-mail: [email protected]
2
figure of merit (ZT), where T is the temperature of interest.
Therefore, equation (1) can be rewritten as:
Abstract: Peltier technology has been the subject of major
advances in recent years, due to the development of
semiconductors and the incorporation of the thermoelectric
devices into domestic appliances. According to the
environmental problems produced by chlorofluorocarbons, the
development of equipment based on the Peltier technology
increased in the last years. Thermoelectric systems are used for
measurement techniques and in other devices where a highprecision temperature control is essential (thermocouples and
thermopiles), for Peltier cooling (Peltier elements for CPU
cooling, refrigeration, temperature stabilization) and direct
energy conversion of heat (thermoelectric generators, driven by
waste heat, radioactive decay, combustion). In this paper an
implementation of thermoelectric effects in ANSYS
Multiphysics is described. The authors present a computational
model which simulates thermal and electric performance of a
thermoelectric cooling device based on thermoelectric
technology. The finite element method (FEM) is used here in
order to solve the system of thermoelectric equations providing
values for temperature distribution, thermal flux, temperature
gradient, and voltage distribution.
ZT 
1. INTRODUCTION
Generally, if a thermal gradient is applied to a solid, it
will always be accompanied by an electric field in the
opposite direction. This process is called as the
thermoelectric effect (TE). Thermoelectric material
applications include refrigeration or electric power
generation. The efficiency of a thermoelectric material is
given by the figure of merit, Z, which is defined as [1]:
 2   1 
k
, 
K 
k
(2)
An important point it is represented by achieving a high
value of ZT, this being carried out by increasing the
power factor (α2σ) and decreasing the thermal
conductivity (k).
One of the main applications of thermoelectrics is for
refrigeration purposes. An electrical current applied
across a material will cause a temperature differential
which can be used for cooling.
As it is known, metals are poor thermoelectric materials
because they have a low Seebeck coefficient and large
electron contribution to thermal conductivity k, so
electrical conductivity σ and thermal conductivity k will
cancel each other out. A low thermoelectric effect is
carried out by insulators which have a high Seebeck
coefficient and small electron contribution to thermal
conductivity, so their charge density and electrical
conductivity are low. The best thermoelectric materials are
between metals and insulators (i.e., semiconductors) [1].
The thermoelectric materials of choice for the steadystate simulations illustrated in this paper on a
thermoelectric element Peltier cooler are BismuthTellurium (Bi-Te) and Lead-Tellurium (Pb-Te). They
have a high Seebeck coefficient α, a good electric
conductivity σ, and a poor thermal conductivity k.
This paper presents the finite element formulation,
which, in addition to Joule heating, includes Seebeck,
Peltier and Thomson effects.
An implementation of thermoelectric effects in ANSYS
Multiphysics is described in the next sections. Numerical
results and their interpretation are provided and
compared with other literature results.
Keywords: thermoelectric cooler, Peltier elements, Seebeck
effect, finite element method FEM, thermal gradient, thermal
flux, voltage distribution.
Z
 2  T
(1)
where:
2. THERMOELECTRIC COOLER MODELING
α – material's Seebeck coefficient, V/K,
σ – electrical conductivity of material, S/m,
k – thermal conductivity of material, W/(m.K).
The numerator  2   in equation 1 is called the power
factor. Therefore, the most useful method in order to
describe and compare the quality and thermoelectric
efficiency of different material systems is the dimensionless
2.1 Thermoelectric Cooler
The basic unit of a thermoelectric (TE) cooler is
composed of two semiconductor elements connected at a
copper strap as shown in Figure 1. It consists of an n-type
and a p-type thermoelement connected electrically in
series by a conducting strap.
89
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
ISSN 1843-6188
E- electric field, V/m.
In the application shown in this paper, the n-type and ptype elements have a length 3.8 mm, and the element
width is 2.5 mm. The width of the copper strap is 1 mm.
Thermoelectric power generation results in order to
provide a temperature gradient across a material. Seebeck
coefficients are also central to Peltier cooling. Cooling
occurs by the absorption of heat as an electrical current
passes through a junction between materials with
different Seebeck coefficients.
The general equation of heat flow used in the
thermoelectric analysis is given by:
 T

 vT T     q  q

t


c 
(6)
where:
ρ – density, kg/m3,
c – specific heat capacity, J/(kg.K),
t – time, s,
 - represents the grad operator,
  - represents the divergence operator,
v x 
v  v y  - velocity vector for mass transport of heat
 
v z 
q - heat generation rate per unit volume, W/m3.
Fourier’s law of heat transfer by conduction is used in
order to relate the heat flux vector to the thermal
gradients:
q k T
Using eq. (6) and eq. (7), it results [4]:
Figure 1. The thermoelectric cooler (Source: [2]).
 T

 vT LT   LT k LT   q
 t

c 
2.2 Governing Equations of Thermoelectricity
The new set of ANSYS coupled-field elements developed
in [2] enables users to accurately and efficiently analyze
thermoelectric devices. The finite element method (FEM)
flexibly used here can model arbitrary shaped structures,
work with complex materials, and apply various types of
loading and boundary conditions. The method can easily
be adapted to different sets of equations, which makes it
particularly attractive for coupled-physics simulation.
The coupled thermoelectric equations are [3], [5]:
q ΠJ  k T
J   E  T
(8)
Developing eq. (8) it results:
 T
T
T
T 
  q 
 vx
 vy
 vz
x
y
z 
 t
c
(9)
  T    T    T 
   kz
  kx
  ky

x  x  y  y  z  z 
The velocity vector for mass transport of heat is zero, in
order to obtain the general equation of thermal
conduction.
(3)
(4)
Replacing Π  with T   in eq.3, it results:
q T  J  k T
(7)
c
T
  T    T    T 
   kz
 q   k x
   ky

t
x  x  y  y  z  z 
(5)
(10)
where:
Π  T   - Peltier coefficient matrix, V,
T - absolute temperature, K,
  - Seebeck coefficient matrix, V/K,
k  - thermal conductivity matrix, W/m.K
The continuity of electric charge equation is:

  D 
 J  
  0
 t 


q- heat flux vector, W/m ,
J - electric current density, A/m ,
( 11)
2
and the equation for a dielectric medium is given by
2
T  - thermal gradient, K/m,
 - electrical conductivity matrix, S/m,
D     E
90
( 12)
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
ISSN 1843-6188
k T  N      N dV

where:
D - electric flux density, C/m2,
  - dielectric permittivity matrix, F/m.
In the absence of time-varying magnetic fields, the
electric field E is irrotational   E  0 , and can be
derived from an electric scalar potential  :

- element specific heat matrix
C TT   CN N dV


E  
- element dielectric permittivity coefficient matrix
C  N   NdV

- vector Q of combined heat generation loads
- Peltier heat load vector:
(13)
Q P  N   J dV

Substituting eqs. (3)-(13) into eqs. (6)-(11), a system of
coupled equations of thermoelectricity is obtained [3]:

- electric power load vector:
Qe 

T
c
     J    k  T   q
(14)
t
 

     
      T          0 (15)
t 

- electric current load vector I.
Thermal loads (Q) can be in the form of imposed
temperature, point heat flow rate, surface heat flux,
convection, or radiation, as well as body heat generation
rate for causes other than electric power dissipation
(accounted for in Qe).
Electrical loads (I) can be in the form of imposed electric
potential and point electric current. Linear electric circuit
components (resistors, capacitors, and voltage or current
sources) can be connected to the finite element model to
simulate passive and active electrical loads [2], [5].
The ANSYS input of material matrices [k], [σ], [α], [ε] is
in the form of their diagonal terms, i.e., material
coefficients along the x, y, z axes. This input can be
combined with an arbitrarily oriented element coordinate
system to account for an alternative material orientation.
Electrical properties are input as resistivity and internally
converted into conductivity [σ], which is the
conductivity evaluated at zero temperature gradient. The
input [λ] is the thermal conductivity evaluated at zero
electric current J  0 . All material properties can be
temperature dependent. In particular, Thomson effect is
taken into account by specifying temperature dependent
Seebeck coefficients [α] [3].
The global matrix equation is assembled from the
individual finite element equations, and is nonsymmetric like eq. (18). Since the thermal load vector
depends on the electric solution, the analysis is nonlinear and requires at least two iterations to converge.
The solution yields temperatures (Te) and electric
potentials (φe) at unconstrained nodes, or reactions in the
form of heat flow rate and electric current at nodes with
imposed temperature and electric potential respectively.
The temperature gradient and electric field are calculated
as [3]:
where the heat generation term q in eq. (14) includes the
electric power J  E spend on Joule heating and on work
against the Seebeck field  T .
The system thermoelectric finite element equations can
be obtained by applying the Galerkin FEM procedure to
the coupled equations derived in the previous section.
This technique involves the following steps [4]:
a) approximate the temperature T and the electric scalar
potential  over a finite element as [3]:
T  N  T e
  N e
(16)
(17)

where:
N  - vector of element shapes functions,
T e - vector of nodal temperatures,
e - vector of nodal electric potentials;
b) write the system of eqs. (14) and (15) in a useful
form;
c) integrate the equations by parts;
d) take into account the Neumann boundary conditions.
The resulting system of thermoelectric finite element
equations is [2], [5]:
C TT 0  Te  k TT 0  Te  Q  Q P  Q e 


   

     T

k    e  
I
 0 C   e  k
(18)
where the element matrices and load vectors are obtained
by numerical integration over the element volume V.
The corresponding expressions are:
- element diffusion conductivity matrix:
k TT  N    N dV

k

T  N  Te ,
(19)
E  N e ,
(20)
and then substituted into Eqs. (3)-(12) to obtain the
values of J , q , D fields, and Joule heat generation
density for each element.
- element electrical conductivity coefficient matrix:

 NE J dV
 N   NdV

- element Seebeck coefficient coupling matrix
91
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
3.
ISSN 1843-6188
independent values are shown in Table 1. Here typical
values for Bismuth-Telluride, Lead-Telluride and copper
were taken from [6].
STEADY-STATE ANALYSIS OF A
THERMOELECTRIC ELEMENT
Numerical simulation is carried out by using a finite
element package ANSYS. This package operates with
three stages: preprocessor, solver and postprocessor. The
procedure for doing a static thermoelectricity analysis
consists of following main steps: create the physics
environment, build and mesh the model and assign
physics attributes to each region within the model, apply
boundary conditions and loads (excitation), obtain the
solution, review the results.
In order to define the physics environment for an
analysis, it is necessary to use the ANSYS preprocessor
(PREP7) and to establish a mathematical simulation
model of the physical problem [5].
In order to do this, the following steps are presented
below: set GUI Preferences, define the analysis title,
define element types and options, define element
coordinate systems, set real constants and define a
system of units, define material properties.
ANSYS includes three elements which can be used in
modeling the thermoelectricity phenomenon [5].
Element types establish the physics of the problem
domain. Depending on the nature of the problem, it is
necessary to define several element types to model the
different physics regions in the model.
In the present application, for modeling the electric and
thermal fields the SOLID227 element was chosen.
SOLID227 has the following capabilities: structuralthermal, piezoresistive, electroelastic, piezoelectric,
thermal-electric,
structural-thermoelectric,
thermalpiezoelectric. The element has ten nodes with up to five
degrees of freedom per node. Thermoelectric capabilities
include Seebeck, Peltier, and Thomson effects, as well as
Joule heating.
Table 1. Numerical material properties from [6], [10]
Material
properties
from
Seebeck
Coefficient
Electric
resistivity
Thermal
conductivity
Density
Heat
capacity
Units
measure
α
[V/K ]
ρ
[S/m ]
λ
[W/m/K ]
δ
[V/K ]
C
[V/K ]
BismuthTellurium
(Bi-Te)
p:200e-6
n:-200e-6
0.9 e-5
LeadTellurium
(Pb-Te)
p:175e-6
n:-175e-6
0.8 e-5
Cooper
6.5e-6
0.169 e-8
1.6
1.548
350
7740
8160
8920
154.4
156
385
Usually, those material properties depend on the
temperature and may be anisotropic. Here only isotropic
material properties are used at constant material
parameters. Thermoelectric (TE) materials based on
(Bi,Sb)2(Te,Se)3 are the best and, in fact, the only
materials used for cooling. These include bismuthtellurium (Bi-Te) and antimony-tellurium (Sb-Te)
compounds. More recently, nanostructured materials
have been investigated as candidates to increase the
performance of thermoelectric devices.
PbTe
nanocomposites have been prepared from PbTe
nanocrystals, synthesized via chemical route, by
compaction under high pressure and temperature. The
thermoelectric (TE) properties are found to vary with the
shape and size of the composites’ nanostructures.
Transport properties of PbTe nanocomposites have been
evaluated through temperature-dependent electrical
conductivity, Seebeck coefficient, room temperature, and
thermal conductivity measurements [7], [8].
1
1
NODAL SOLUTION
ELEMENTS
APR 6 2010
22:30:41
STEP=1
SUB =1
TIME=1
TEMP
(AVG)
RSYS=0
SMN =273.15
SMX =341.15
APR 6 2010
20:21:55
MN
Y
Z
Y
Z
X
X
MX
273.15
280.587
288.025
295.463
302.9
310.338
317.775
325.213
332.65
341.15
Figure 3. Distribution of temperature for Bi2Te3 material
Figure 2. Meshing the model with triangular elements
A calculation model that simulates the thermal and
electric performance of whole cooling based on
thermoelectric technology has been implemented. The
model inputs are: semiconductor materials and
geometry, Peltier pellet components type, electrical
voltage supplied to the Peltier pellet components and the
hot and cold side temperatures.
Next step in the preprocessor phase is mesh generation
and load application on the elements. It was used a mesh
with 6366 nodes and 3249 triangular elements. The finite
element mesh of the thermoelectric element is shown in
Figure 2.
The following examples show results of calculations for
typical thermoelectric applications. The material
properties for the calculations with temperature-
92
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
ISSN 1843-6188
The distribution of temperature for Bi2Te3 material is
shown in Figure 3 and the distribution of voltage for BiTe material is shown in Figure 4.
When an electric current is running from the cold end to
the hot end, the Joule heating is generated uniformly
inside the element. The dissipated heat must reach both
ends equally by conduction.
1
NODAL SOLUTION
APR 6 2010
22:36:05
STEP=1
SUB =1
TIME=1
TFSUM
(AVG)
RSYS=0
SMN =.858E-06
SMX =610888
MX
MN
Y
1
Z
NODAL SOLUTION
X
APR 6 2010
22:30:58
STEP=1
SUB =1
TIME=1
VOLT
(AVG)
RSYS=0
SMX =.0924
.858E-06
MX
66816
133632
200448
267264
400895
334080
534527
467711
610888
Figure 6. Distribution of thermal flux for Bi2Te3 material
Y
Z
The distribution of temperature for PbTe material is
shown in Figure 7.
MN
X
1
0
.010106
.020213
.030319
.040425
.050531
.060638
.070744
.08085
NODAL SOLUTION
APR 26 2010
00:18:07
STEP=1
SUB =1
TIME=1
TEMP
(AVG)
RSYS=0
SMN =273.15
SMX =336.15
.0924
Figure 4. Distribution of voltage for Bi2Te3 material
The cooler with Bi2Te3 material is designed to maintain
the cold junction at a temperature Tc=273.15 K and to
dissipate heat from the hot junction T h=341.15 K on the
passage of an electric current of magnitude I=0.535 A.
The positive direction of the current is from the n-type
material to the p-type material.
MN
Y
Z
MX
273.15
1
NODAL SOLUTION
APR 6 2010
22:52:20
STEP=1
SUB =1
TIME=1
TGSUM
(AVG)
RSYS=0
SMN =.153E-08
SMX =30456
X
280.041
286.931
293.822
300.713
307.603
314.494
321.384
328.275
336.15
Figure 7. Distribution of temperature for PbTe material
The distribution of voltage for PbTe material is shown in
Figure 8.
MN
1
MX
APR 26 2010
00:18:59
STEP=1
SUB =1
TIME=1
VOLT
(AVG)
RSYS=0
SMX =.086435
Y
Z
NODAL SOLUTION
X
MX
.153E-08
3331
6662
9993
13324
16655
19986
23318
26649
30456
Figure 5. Distribution of thermal gradient for Bi2Te3
material
Y
Z
After the simulation, the model-returned outputs are:
temperatures, heat flows, thermal gradient, thermal flux,
and voltage distribution. The distribution of thermal
gradient for Bi2Te3 material is shown in Figure 5 and the
distribution of thermal flux for Bi2Te3 material is shown
in Figure 6.
The cold junction is at a temperature of Tc=273.15 K
and dissipates heat from the hot junction Th=336.15 K
for PbTe material.
0
.009454
.018908
.028362
MN
X
.037815
.047269
.056723
.066177
.075631
.086435
Figure 8. Distribution of voltage for PbTe material
The temperature dependence of Seebeck coefficient,
electrical conductivity and power factor of the PbTe
material lie within the temperature range 400–600 K.
Thermal conductivity reduction has played a central role
in improving the thermoelectric figure-of merit, ZT, of
materials that already have a good power factor.
93
Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 1 (12)
ISSN 1843-6188
4. CONCLUSIONS
1
NODAL SOLUTION
APR 26 2010
00:20:06
STEP=1
SUB =1
TIME=1
TGSUM
(AVG)
RSYS=0
SMN =.153E-08
SMX =33024
When lead telluride PbTe is compared with bismuth
telluride Bi2Te3, a temperature difference of nearly 63 K
for cooler with PbTe and temperature difference of nearly
68 K for cooler with Bi2Te3 is noted. The voltage at the
upper electrode is 46.2 mV for cooler with Bi2Te3 and
43.2 mV for cooler with PbTe. This means that, although
the value of the figure of merit of PbTe is lower than for
Bi2Te3, the latter material is used. In fact, lead telluridebased materials have been used for a range of purposes in
the hot-junction temperature range 600 to 900 K [9].
Conversely, PbTe has been considered more as a material
for thermoelectric generation at moderately high
temperatures rather than for refrigeration at room
temperature and below [10]. PbTe thermoelectric
generators have been widely used by the US army, in
space crafts to provide onboard power, and in pacemaker
batteries. The application shown in this paper can be
useful to represent the characteristics of the Peltier
cooling through numerical models.
MN
MX
Y
Z
.153E-08
3612
7224
10836
X
14448
18060
21672
25284
28896
33024
Figure 9. Distribution of thermal gradient for PbTe
material
The distribution of the thermal gradient for PbTe
material is shown in Figure 9 and the distribution of
thermal flux for Bi-Te material is shown in Figure 10.
4. REFERENCES
1
NODAL SOLUTION
APR 26 2010
00:20:43
STEP=1
SUB =1
TIME=1
TFSUM
(AVG)
RSYS=0
SMN =.858E-06
SMX =601935
[1]
[2]
MX
[3]
MN
[4]
Y
Z
X
[5]
[6]
.858E-06
65837
131673
197510
263347
329183
395020
460856
526693
601935
Figure 10. Distribution of thermal flux for PbTe material
[7]
Materials and device characterization play a key role in
thermoelectric research. Materials composition and
parameters affect the achieved thermoelectric (TE)
performance (for example, functional properties and
figure-of-merit of materials, efficiency, coefficient of
performance, or sensitivity of devices).
Applications of the two materials Bi2Te3 and PbTe
demonstrate the Peltier effect for thermoelectric cooling.
It can be seen that a smaller thermal conductivity will
decrease the heat transfer between the two ends, a
smaller electrical resistivity will reduce the Joule
heating, and a larger Seebeck or Peltier coefficient will
enhance the heat removal. For most metals, the thermal
conductivity is too high and the Seebeck coefficient is
too small for refrigeration applications. Some insulators
can have a large Seebeck coefficient but their electrical
resistivity is too high for them to be used in
thermoelectric devices.
[8]
[9]
Kimmel, J. – Thermoelectric Materials, Physics 152,
Special Topics Paper, March 2, 1999
Angrist, S. W., Direct Energy Conversion, 3rd Edition,
Allyn and Bacon (Boston, 1976), pp. 140-166;
Antonova E.E, Looman D.C. – Finite Elements for
Thermoelectric Device Analysis in ANSYS, International
Conference on Thermoelectrics 2005, 0-7803-9552-2/05
IEEE;
Silvester P. P. and Ferrari, R. L., Finite Elements for
Electrical Engineers, 3rd Edition, University Press
(Cambridge, 1996);
ANSYS Release 11.0 Documentation;
Jaegle M. - Multiphysics Simulation of Thermoelectric
Systems - Modeling of Peltier - Cooling and
Thermoelectric Generation, Proceedings of the COMSOL
Conference 2008 Hannover;
Biplab Paul and Pallab Banerji - Grain Structure Induced
Thermoelectric Properties in PbTe Nanocomposites,
Nanoscience and Nanotechnology Letters, Vol. 1, 208–
212, 2009;
Zhuomin M. Zhang, Nano/Microscale Heat Transfer, Mc
Grow Hill, 2007;
D.M. Rowe, Ph.D., D.Sc., Thermoelectric Handbook
Macro to Nano, 2006 by Taylor & Francis Group, LLC;
[10] H. Julian Goldsmid, Introduction to Thermoelectricity,
Springer Series in Materials Science, 2009.
94