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International Summerschool on Advanced Materials and Thermoelectricity 1
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
One-dimensional modeling of TE devices
Daniel Mitrani and Juan A. ChávezElectrical Engineering Department, Universitat Politècnica de Catalunya
Barcelona, Spain. Email: mitrani@eel.upc.edu
International Summerschool on Advanced Materials and Thermoelectricity 2
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
• Overview
• TEM description and formulae
• Steady-state electrical models and simulations
• Dynamic electrical model and simulations
• Conclusions
Contents
Presentation contents
International Summerschool on Advanced Materials and Thermoelectricity 3
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
TEM Description
Negative (-)
Positive (+)MoistureProtection
Ceramic Plates
Thermocouples
T
p J
x
x=0 x=L
Th
TcI np
Qh
Qc
TE module characteristics
• Solid-state devices.
• Couples connected electrically in series and thermally in parallel.
• Peltier mode: heat pump.
• Seebeck mode: electrical power generation.
Single couple unitFree standing pellet
Th
Tc
International Summerschool on Advanced Materials and Thermoelectricity 4
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Interaction between thermaland electrical domains2T ds T
J JTx x dT x
1-D steady-state energy balance equation
Thomson Effect
Joule Effect
Fourier’s Law
TEM Formulae (I)
Tq sJT
x
Heat flow per unit area
Peltier Effect Fourier’s Law
TE J s
x
Electric field per unit length
Ohm’s Law Seebeck Effect
Electrical domain
Thermal domain
Peltier Effect
Thomson Effect
Seebeck Effect
ConductionConvection
Radiation
Joule Effect
Ohm’s Law
International Summerschool on Advanced Materials and Thermoelectricity 5
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
TEM Formulae (II)
( ) ( )h cL J L s T T
2 ( )
2h c
c c
J L T Tq s J T
L
2 2
2
( )T x J
x
Constant material properties
( 0) ( ) ( 0) 0Vc hT x T T x L T x
Dirichlet boundary conditions
2 22( )
2 2h c
c
J T T J LT x x x T
L
1-D temperature distribution
2 ( )
2h c
h h
J L T Tq s J T
L
222
2
( ) ( ) ( ) ( ) ( )( ) ( ) ( )
T x d T T x ds T T xT J T JT x
x dT x dT x
Heat flow per unit area at x=0 and x=L
Electrical potential at x=L
p-typeJ
xx=0 x=L
ThTc
qc qh
International Summerschool on Advanced Materials and Thermoelectricity 6
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
V h cI R S T T
212 ( )h h h cQ S I T I R K T T
2e h C sP Q Q I R I V
Heat absorbed at the cold side
212 ( )c c h cQ S I T I R K T T
Electrical power
Heat released at the hot side
Voltage across the terminalsp p n n
p n
A AK N
L L
p p n n
p n
L LR N
A A
p nS N s s
• Parallel thermal conductance of the N couples
• Serial electrical resistance of the N couples
• Seebeck coefficient of the N couples
TEM Formulae (III)
Qh
- +
np Inp np
Qc
International Summerschool on Advanced Materials and Thermoelectricity 7
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Lumped Electrical Model (I)
Electrical steady-state three-port model
+
-
V
ThPe
+-
R
Vs
Px
TEM model
Rth
+
-
Tc
Qc Qh
I
+
-
Thermal and electrical analogies
Thermal Domain Electrical Domain
Heal Flow (W) Electrical Current (A)
Temperature Difference (K) Voltage (V)
Thermal Resistance (K/W) Electrical Resistance ()
Thermal Mass, (J/K) Electrical Capacitance (F)
Vs h cS T T 2e h CP Q Q I R I V
Model expressions21
2x cP SIT I R
• Thermal processes described in electrical terms.
• Flexible boundary conditions.
• Simulation of control electronics and thermal elements.
Material parameters are set prior to simulation !!!
International Summerschool on Advanced Materials and Thermoelectricity 8
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Temperature profile for Qcmax case
25
30
35
40
45
50
55
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x (mm)
Tem
per
atu
re (
ºC)
Th=300K, Qc=Qcmax
Tm
Tc Th
Temperature profile for Tmax case
-40
-30
-20
-10
0
10
20
30
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x (mm)
Tem
per
atu
re (
ºC)
Th=300K, T=Tmax
Tm
Tavg
Tc
Th
2h c
avg
T TT
0
1( )
L
mT T x dxL
• Average temperature between hot and cold side • Mean module temperature
Lumped Electrical Model (II)
International Summerschool on Advanced Materials and Thermoelectricity 9
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
2
0
1 1( )
2 12
Lh c
m
T T I RT T x dx
L K
2h c
avg
T TT
• Mean module temperature:
• Hot side temperature
• Cold side temperature
• Average module temperature:
( ) ( )k
k T TS T S T ( ) ( )k
k T TR T R T ( ) ( )k
k T TK T K T
Material properties are calculated as:
+
-
V
Th
+
-
Tc
Qc Qh
I
+
-
Pe
+-Vs
Px
TEM model
Tcon+-
+ -
Vr
+-
S(Tk)+-
R(Tk)+-
K(Tk)+-
Tk
S R K Tk
Where Tk can be calculated as:
Tk
( )x c
conk
P QT
K T
( )r kV I R T
Additional VCVS’s are defined as:
Lumped Electrical Model (III)
International Summerschool on Advanced Materials and Thermoelectricity 10
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
+
-
V
Th
+
-Tc
Qc Qh
I
+
-
+
-T1Pe
Vs
Px
Tcon
+-
+ -
Vr
+ -
S1 R1 K1 Tm1
Pe
Vs
Px
Tcon
+-
+ -
Vr
+ -
S2 R2 K2 Tm2
+
-T2 Pe
Vs
Px
Tcon
+-
+ -Vr
+ -
S3 R3 K3 Tm3
+
-T3 Pe
Vs
Px
Tcon
+-
+ -
Vr
+ -
S4 R4 K4 Tm4
Steady-state equations are accurate as long as the thermoelectric properties do not vary over the region where they are applied.
• Divide the pellets of a TEM into many small segments• Each segment would be closer to meeting such criteria
Distributed Parameter Electrical Model
N
N-1
2
1
N
N-1
2
1
Tn
Tn-1
Qn-1
Qn
QN=Qh
Q0=Qc
TN=Th
T0=Tc
n nSn=S(Tn)Rn=R(Tn)Kn=K(Tn)
International Summerschool on Advanced Materials and Thermoelectricity 11
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Steady-State Simulation Setup and Results (I)
s(T) = -0.0024 T2 + 1.7062 T - 73.929R2 = 0.9992
0
50
100
150
200
250
0 100 200 300 400T (K)
s (1
0-6V
·K-1
)
Experimental data
s(T)
s(T) = 0.0024 T2 - 1.6293 T + 326.5
R2 = 0.9972
0
50
100
150
200
250
300
0 100 200 300 400T (K)
s 103
1m
1
Experimental datasT
(T) = 3E-05 T2 - 0.0193 T + 4.1157
R2 = 0.9966
1.00
1.50
2.00
2.50
3.00
3.50
0 100 200 300 400T (K)
·m1K
1
Experimental data(T)
(Bi0.5Sb0.5)2Te3
L
Th
Th=300 K
L=1 mm
A=1 mm2
International Summerschool on Advanced Materials and Thermoelectricity 12
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Temperature Difference vs. Electrical Current
ElecMod NumMod
NumMod
RelativeError 100T T
T
Model Tmax ITmaxRelative
Error
Lumped @ Th 59.83 31.21 4.06%
Lumped @ Tm 60.51 31.34 2.97%
Lumped @ Tavg 60.62 32.38 2.78%
Lumped @ Tc 59.65 35.68 4.34%
Distributed (10 FE) 62.34 32.97 0.03%
Numerical 62.36 32.99
0
10
20
30
40
50
60
70
0 10 20 30 40 50I (A)
T (
ºC)
0
2
4
6
8
10
12
14
Rel
ativ
e E
rro
r (%
)
Lumped @ThLumped @TmLumped @TavgLumped @TcDist. (10 FE)Numerical
Th=300K, Qc=0
58
59
60
61
62
63
30 31 32 33 34 35 36I (A)
T (
ºC)
Lumped @Th Lumped @Tc
Lumped @Tavg Lumped @Tm
Distributed (10 FE) Numerical
Th=300K, Qc=0
Steady-State Simulation Setup and Results (II)
International Summerschool on Advanced Materials and Thermoelectricity 13
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 10 20 30 40 50
I (A)
Qc
(W)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Rel
ativ
e E
rro
r (%
)
Lumped @Th=Tc=TavgLumped @TmDistributed (10 FE)Numerical
Th=300K, T=0
1.229
1.230
1.231
1.232
1.233
1.234
1.235
1.236
1.237
36.5 36.7 36.9 37.1 37.3 37.5 37.7 37.9
I (A)
Qc
(W)
Lumped @ Th=Tc=Tavg
Lumped @ Tm
Distributed (10 FE)
Numerical
Th=300K, T=0
Cooling Power vs. Electrical Current
ElecMod NumMod
NumMod
RelativeError 100Q Q
Q
Model Qcmax IQcmaxRelative
Error
Lump. @ Th =Tc=Tavg 1.237 W 37.74 A 0.37%
Lumped @ Tm 1.230 W 36.95 A 0.18%
Distributed (10 FE) 1.232 W 37.14 A 0.01%
Numerical 1.232 W 37.15 A
Steady-State Simulation Setup and Results (III)
International Summerschool on Advanced Materials and Thermoelectricity 14
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
218.50
212.56
208.88
195.61
190
195
200
205
210
215
220
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x (mm)
s (1
0-6V
·K-1
)
Lumped @ ThLumped @ TmLumped @ TavgLumped @ TcDistributed (10 FE)Numerical
17.37
16.16
15.44
13.20
12
13
14
15
16
17
18
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x (mm)
10
6 ·m
Lumped @ ThLumped @ TmLumped @ TavgLumped @ TcDistributed (10 FE)Numerical
1.325
1.336
1.402
1.31
1.33
1.35
1.37
1.39
1.41
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x (mm)
·m1K
1
Lumped @ Th
Lumped @ Tm
Lumped @ Tavg
Lumped @ Tc
Distributed (10 FE)
Numerical
2.074
2.110
2.116
2.068
2.05
2.06
2.07
2.08
2.09
2.10
2.11
2.12
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x (mm)
z (1
0-3K
-1)
Lumped @ ThLumped @ TmLumped @ TavgLumped @ TcDistributed (10 FE)Numerical
Spatial profiles for material parameters s(x), (x), (x), and z(x)
2sz
Steady-State Simulation Setup and Results (IV)
International Summerschool on Advanced Materials and Thermoelectricity 15
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Proposed distributed parameter transient electrical model
Qpc Qph
+
-
Th
+
-
Tc
Qc Qh
Finite Elem. 1 Finite Elem. N-1 Finite Elem. N
+
-V
I
C
21 Rth 2
1 Rth
R Vs
+ -
QJ
C C
21 Rth 2
1 Rth
R Vs
+ -
QJ
C C
21 Rth 2
1 Rth
R Vs
+ -
QJ
CC
21 Rth 2
1 Rth
R Vs
+ -
QJ
C
Finite Elem. 2
2 2
2 2
T I T
x A t
No analytical
solution !!!
I → Electrical current → Electrical resistivity → Thermal conductivity → Thermal diffusivity
• Start-up and shut-down periods.• Operating conditions are varied with time.• Fast-response heat sources.• Similar TEC and Heat load thermal time constants• Pulse cooling analysis.
Dynamic Distributed Parameter Electrical Model
One-dimensional heat flow equationJustification
16
210
220
230
240
250
260
270
-5 0 5 10 15 20 25 30 35 40
Time (s)
Tc
(K)
SPICE Model
Parabolic Solution
Linear SolutionT pulse @ t min
T postpulse
T ss
Return to steady-state
Th=300 KP=2.5
Pulse Cooling
t ret
205
210
215
220
225
230
235
240
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
time (s)
Tc
(K)
t0
t1/2
t1
t2
Th=300 K
P=5
Pulse cooling simulation analysis examples
0
5
10
15
20
25
30
35
2 3 4 5 6 7 8 9 10
P = Ipulse/Imax
Tpu
lse
(K)
0.0
0.5
1.0
1.5
2.0
2.5
t min (
s)
SPICE ModelParabolic SolutionLinear Solution
Tpulse curves
tmin curves
Th =300 K
205
210
215
220
225
230
235
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
time (s)
Tc
(K)
L=1 mm
L=5 mm
L=10 mm
P=5
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8 9 10P ellet L ength (mm)
t min
(s)
P =2.5P =5
P =10
Th=300 K
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-5 0 5 10 15 20 25 30 35 40
Time (s)
Cur
rent
(A
)
I max (curent for maximum stedy-state T )
I pulse = P·I max Current Pulse
210
225
240
255
270
285
300
0 50 100 150 200 250 300Time (s)
Tc (
K)
Initial condition (Tc=Th)
Steady-state (Tc=Tcss)
Transient cooling (Tc=Tcmin)
Post pulse heating
Return to steady-state
International Summerschool on Advanced Materials and Thermoelectricity 17
One-dimensional modeling of TE devices using SPICEOne-dimensional modeling of TE devices using SPICE
Conclusions
Conclusions
• Based on 1-D steady-state analysis we propose
– Lumped parameter model
– Distributed parameter model
• Simulation of electrical and thermal domains with a single tool
– Control electronics
– Thermal elements
• Material parameters chosen according to different module temperatures
• Dynamic Distributed Parameter Electrical Model
– Start-up and shut down periods
– Similar TEM and heat load time constants
– Transient cooling operation
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