One-dimensional modeling of TE devices
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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: [email protected]
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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
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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
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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
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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
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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
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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 !!!
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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)
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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)
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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)
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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
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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)
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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)
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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)
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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
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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
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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