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Efficiency and output power of thermoelectric module by taking into account corrected
Joule and Thomson heat
Hee Seok Kim, Weishu Liu, and Zhifeng Ren
Citation: Journal of Applied Physics 118, 115103 (2015);
View online: https://doi.org/10.1063/1.4930869
View Table of Contents: http://aip.scitation.org/toc/jap/118/11
Published by the American Institute of Physics
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JOURNAL OF APPLIED PHYSICS 118, 115103 (2015)
Efficiency and output power of thermoelectric module by taking into account
corrected Joule and Thomson heat
Hee Seok Kim, Weishu Liu,a) and Zhifeng Renb)
Department of Physics and Texas Center for Superconductivity, University of Houston, Houston, Texas 77204,
USA
(Received 5 July 2015; accepted 31 August 2015; published online 16 September 2015)
The maximum conversion efficiency of a thermoelectric module composed of p- and n-type materials has been widely calculated using a constant property model since the 1950s, but this conventional model is only valid in limited conditions and no Thomson heat is accounted for. Since
Thomson heat causes the efficiency under- or over-rated depending on the temperature dependence
of Seebeck coefficient, it cannot be ignored especially in large temperature difference between the
hot and cold sides. In addition, incorrect Joule heat is taken into consideration for heat flux evaluation of a thermoelectric module at thermal boundaries due to the assumption of constant properties
in the conventional model. For this reason, more practical predictions for efficiency and output
power and its corresponding optimum conditions of p- and n-type materials need to be revisited. In
this study, generic formulae are derived based on a cumulative temperature dependence model
including Thomson effect. The formulae reliably predict the maximum efficiency and output power
C 2015 AIP Publishing LLC.
of a thermoelectric module at a large temperature. V
[http://dx.doi.org/10.1063/1.4930869]
I. INTRODUCTION
Thermoelectric modules are environmentally benign,
quiet, and direct energy converters from thermal to electric
power.1–5 The energy conversion efficiency of a typical thermoelectric module consisting of p-type and n-type materials
connected thermally in parallel and electrically in series is a
function of the figure of merit [Z] of a p-n pair, defined as6
½Z ¼
ðSp Sn Þ2
;
ðRp þ Rn ÞðKp þ Kn Þ
(1)
where S, R, and K are the Seebeck coefficient, electrical
resistance, and thermal conductance, respectively, and the
subscript p, n, and square brackets denote p-type, n-type
materials, and p-n module property, respectively. The
efficiency of a thermoelectric module has been analytically
calculated based on an assumption that S, R, and K are temperature independent as6
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
DT 1 þ ½Z Tavg 1
g¼
(2)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ;
Th
c
1 þ ½Z Tavg þ
Th
where Th, Tc, DT, and Tavg are temperatures of hot end, cold
end, their difference, and average, respectively. This is reasonably accurate when its operating temperature difference
DT is small, or when [Z] is temperature independent over the
operating temperature range, but most thermoelectric materials have strongly temperature dependent Seebeck coefficient
S, electrical resistivity q, and thermal conductivity j. In
a)
Electronic mail: wliu11@uh.edu
Electronic mail: zren@uh.edu
b)
0021-8979/2015/118(11)/115103/9/$30.00
addition, thermoelectric modules as electric power generators are gaining attention in various applications such as
unmanned aerial vehicle (UAV),7,8 solar energy harvesting,9,10 and heat recovery of gasoline/diesel engine,11,12
which requires larger temperature gradient on thermoelectric
modules. Hence, the conventional model by Eq. (2) is not
reliable on prediction for practical conversion efficiency of a
module operating at large temperature difference between
hot and cold sides. In addition, Thomson effect is neglected
in the conventional model, which leads to shifting the efficiency and output power depending on the degree of
Thomson heat at a given temperature gradient. Sunderland
and Burak evaluated how Thomson heat affects output
power and conversion efficiency,13 and Min et al. developed
a modified ZT considering Thomson effect,14 but both
assumed that Thomson coefficient is constant. The conversion efficiency accounting for Thomson effect was reported
based on linear behavior of S, q, and j and temperature dependent Seebeck coefficient with constant q and j.15–17 Kim
et al. recently established new formulae for maximum efficiency of a homogeneous thermoelectric material to analytically calculate efficiency including Thomson effect at a large
temperature gradient.18 This model evaluated the maximum
efficiency of a single thermoelectric material, which is good
to first examine the characteristics of individual thermoelectric materials. However, usually thermoelectric modules operate with pairing of p- and n-type materials, which
requires a balance of thermal flow and electric current of dissimilar p-type and n-type materials. Here, we report formulae for maximum efficiency of thermoelectric modules based
on cumulative temperature dependence model leading to
more reliable predictions than the conventional model, and
define the maximum engineering figure of merit and
engineering power factor density as direct indicators of a
118, 115103-1
C 2015 AIP Publishing LLC
V
115103-2
Kim, Liu, and Ren
J. Appl. Phys. 118, 115103 (2015)
ð Th ð Th
thermoelectric module at practical temperature differences. The thermoelectric module performance is demonstrated in three different p-n pairings: (1) skutterudites
(SKU): Ce0.45Nd0.45Fe3.5Co0.5Sb12 (p-type)19 and Ba0.08
La0.05Yb0.04Co4Sb12 (n-type),20 (2) chalcogenides: K0.02
Pb0.98Te0.75Se0.25 (p-type)21 and Pb0.995SeCr0.005 (n-type),22
and (3) chalcogenides: SnSe (p-type)23 and PbSe (n-type).22
Tc
WJ;p or n ¼
Tc
ð Th
:
(9)
sp or n ðT ÞdT
Tc
where x is the distance from the hot side surface of the thermoelectric leg, and J is the current density. The temperature
dependent Thomson coefficient is defined as s(T) ¼ TdS(T)/
dT. Integrating Eq. (3) twice with respect to x with applying
the boundary conditions, Tjx¼0 ¼ Th and Tjx¼Lp ¼ Tc , yields
the conduction heat at Th of p-type leg as
ð
ð ð
dT Ap Th
I 2 Lp x
Ap jp ðT Þ ¼
jp ðT ÞdT q ðT Þdxdx
Lp Tc
Ap Lp 0 0 p
dx Th
ð ð
I Lp T
þ
sp ðT ÞdTdx;
(4)
Lp 0 Th
where Lp, Ap and I are the leg length, cross sectional area of
p-type leg and electric current, respectively. The conduction
heat of n-type leg is also obtained in the same way. From the
constitutive relation for heat Q and electric current I
(5)
the heat transferred into each thermoelectric leg at T ¼ Th
becomes
ð
Ap Th
jp ðT ÞdT WJ;p Ip2 Rp
Qh;p ¼ Ip Th Sp ðTh Þ þ
Lp Tc
ð Th
WT;p Ip
sp ðT ÞdT;
(6)
Tc
Qh;n
(8)
sp or n ðT ÞdTdT
T
DT
The one-dimensional governing equation for a steadystate heat flow in thermoelectric module is24
d
dT
dT
ð
Þ
jT
þ J 2 qðT Þ JsðT Þ
¼ 0;
(3)
dx
dx
dx
ð
An Th
¼ I n T h S n ðT h Þ þ
jn ðT ÞdT WJ;n In2 Rn
Ln Tc
ð Th
WT;n In
sn ðT ÞdT;
;
qp or n ðT ÞdT
Tc
ð Th ð Th
II. CUMULATIVE TEMPERATURE DEPENDENCE
MODEL OF P-N MODULE
dT
;
dx
ð Th
DT
WT;p or n ¼
Q ¼ ITS Aj
qp or n ðT ÞdTdT
T
By Eqs. (6) and (7), total input heat Qh (¼Qh,p þ Qh,n) at Th
becomes
ð
Ap Th
jp ðT ÞdT
Qh ¼ ITh Sp ðTh Þ Sn ðTh Þ þ
Lp Tc
ð
An Th
þ
jn ðT ÞdT ðWJ;p Rp þ WJ;n Rn ÞI2
Ln Tc
!
ð
ð
Th
Th
sp ðT ÞdT WT;n
WT;p
sn ðT ÞdT I;
Tc
where I ¼ Ip ¼ In. The output power accounting for
Thomson effect is (see Appendix A)
Pout ¼
2
Voc
m
;
Rint ð1 þ mÞ2
Tc
where S(Th) is the Seebeck coefficient at Th, and Rp and Rn
are DT-dependent electrical resistance at Ða given thermal
T
boundary, defined as R ¼ ðLA1 DT 1 Þ Tch qðTÞdT. The
terms on the right-hand side in Eqs. (6) and (7) represent
Peltier, conduction, Joule, and Thomson heat, respectively.
WJ and WT are DT-dependent weight factors for practical
fraction to the hot side of the total Joule and Thomson heat
associated with the temperature dependence of q(T) and
s(T), defined as18
(12)
Tc
m is the ratio of load (RL) to internal (Rint ¼ Rp þ Rn) electrical resistance, m ¼ RL/Rint. The conversion efficiency of the
module is expressed as the ratio of the output power to input
heat
g¼
Pout
:
Qh
(13)
Eqs. (10)–(13) yield
2
Th Sp ðTh Þ Sn ðTh Þ
m
1þm
g¼
6
þ ð Th
ð Th
1 þ m 4½ZT eng
Sp ðT ÞdT Sn ðT ÞdT
Tc
(11)
where Voc is the open circuit voltage expressed as
ð Th
ð Th
Sp ðTÞdT Sn ðTÞdT:
Voc ¼
Tc
(7)
(10)
Tc
WJ;p R0p
WJ;n R0n
þ
1þm
Tc
WT;p s0p
WT;n s0n
#1
;
(14)
where R0p ¼ Rp =Rint and R0n ¼ Rn =Rint . s0p and s0p are
expressed as
ð Th
sp or n ðT ÞdT
Tc
:
(15)
s0p or n ¼ ð Th
ð Th
Sp ðT ÞdT Sn ðT ÞdT
Tc
Tc
Here, the engineering dimensionless figure of merit of p-n
module [ZT]eng is defined as
115103-3
Kim, Liu, and Ren
J. Appl. Phys. 118, 115103 (2015)
ð Th
½ZT eng ¼
Sp ðT ÞdT Tc
Lp
Ap
ð Th
Tc
Ln
qp ðT ÞdT þ
An
ð Th
!
qn ðT ÞdT
Tc
The engineering power factor of p-n module [PF]eng in W
K1 is defined as
!2
ð
ð
Th
Th
Sp ðT ÞdT ½PFeng ¼
Lp
Ap
Sn ðT ÞdT
Tc
Tc
Tc
Ln
qp ðT ÞdT þ
An
ð Th
ð Th
:
ð Th
Tc
An
jp ðT ÞdT þ
Ln
ð Th
! DT:
Tc
Tc
jn ðT ÞdT
Substituting Eq. (21) into Eqs. (18) and (19) leads to the efficiency formula without Thomson effect on heat flux based
on the cumulative temperature dependence model. If S, q,
and j are assumed to be temperature-independent, Eq. (20)
is further reduced to
gc
i:
2
(22)
Due to [ZT]eng ¼ [Z]DT from Eq. (16) for temperature independent properties, Eq. (19) becomes
mopt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ ½ZTavg ;
Tc
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ ½ZT eng a1 g1
c 1
¼ gc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
þ
a
a0 1 þ ½ZT eng a1 g1
2
c
When S, q, and j are temperature independent,
(25) is reduced to
½Zeng
(26)
in Eq.
(23)
and its corresponding efficiency is identical to Eq. (2), which
indicates Eqs. (18)–(20) are generic formulae that can be
converted to the expression based on conventional as well as
the cumulative temperature dependence model.
The optimized efficiency by Eq. (18) can be maximized
by matching the leg size of p- and n-type materials for balancing thermal flow and electric current. Taking derivative
of the denominator of Eq. (16) and setting it equal to zero
yields be for the maximum [ZT]eng as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð Th
uð Th
u
u qn ðT ÞdT
jp ðT ÞdT
An Lp u
Tc
Tc
be ¼
¼u
;
ð
ð Th
Ap Ln u
t Th
qp ðT ÞdT
jn ðT ÞdT
Tc
(24)
Tc
and the corresponding maximum engineering figure of merit
of the module ½ZTeng and the maximum efficiency become
Sp ðT ÞdT ð Th
!2
Sn ðT ÞdT
Tc
¼ 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 DT;
ð Th
ð Th
ð Th
ð Th
@
jp ðT ÞdT
qp ðT ÞdT þ
jn ðT ÞdT
qn ðT ÞdT A
Tc
(16)
Tc
ai ¼ 1 ð Th
gmax
Ap
Lp
qn ðT ÞdT
where i ¼ 0, 1, and 2. When Thomson effect and weight
factors are not taken into consideration, i.e., sp or n ¼ 0 and
WJ,p or n ¼ WT,p or n ¼ 1/2, Eq. (20) is reduced to
Sp ðTh Þ Sn ðTh Þ DT
g
(21)
c i:
ai ¼ ð Th
ð Th
2
ð
Þ
ð
Þ
Sp T dT Sn T dT
½Z eng DT
Sn ðT ÞdT
Tc
Tc
where gc is the Carnot efficiency (DT/Th), and ai is defined as
Sp ðTh Þ Sn ðTh Þ DT
ai ¼ ð Th
ð Th
Sp ðT ÞdT Sn ðT ÞdT
Tc
Tc
WT;p s0p WT;n s0n gc i WJ;p R0p þ WJ;n R0n gc ; (20)
½ZT eng ¼
!2
(17)
By optimizing the ratio m satisfying dg/dm ¼ 0 regarding Eq.
(14), the optimized efficiency and its corresponding mopt are
obtained as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ ½ZT eng a1 g1
c 1
;
(18)
gopt ¼ gc qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a0 1 þ ½ZT eng a1 g1
þ
a
2
c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mopt ¼ 1 þ ½ZTeng a1 g1
(19)
c ;
Tc
ð Th
Tc
(25)
Tc
ðS S Þ2
½Z eng ¼ ½Z ¼ pffiffiffiffiffiffiffiffiffipffi pnffiffiffiffiffiffiffiffiffiffi2 ;
jp qp þ jn qn
(27)
where [Z]* is the optimal figure of merit for the maximum efficiency based on conventional model, and its maximum efficiency is6
115103-4
Kim, Liu, and Ren
gmax
J. Appl. Phys. 118, 115103 (2015)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
DT 1 þ ½ZT Tavg 1
¼
:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Tc
Th
1 þ ½ZT Tavg þ
Th
(28)
III. RESULTS AND DISCUSSION
The output power density Pd in W m2 by Eqs. (11) and
(17) is expressed as
Pd ¼
¼
Pout
Ap þ An
2
Voc
m
DT
ðAp þ An ÞRint ð1 þ mÞ2
!2
ð Th
ð Th
Sp ðT ÞdT Sn ðT ÞdT
m
DT
ð1 þ mÞ2
!
¼
ð
ð
Lp Th
Ln Th
q ðT ÞdT þ
q ðT ÞdT
ðAp þ An Þ
Ap Tc p
An Tc n
Tc
¼
Tc
½PFeng
m
DT
ðAp þ An Þ ð1 þ mÞ2
m
DT:
¼ ½PFeng;d
ð1 þ m Þ2
(29)
When the optimal value m ¼ 1 by taking derivative
m/(1 þ m)2 with respect to m and setting it to zero, the output
power density Pd is proportional to the engineering power
factor density [PF]eng,d at a given DT, so the maximum Pd
and its corresponding condition bp are obtained by taking derivative of the denominator of [PF]eng,d with respect to An/Ap
Pd;max ¼ ½PFeng;d
DT
4
ð Th
ð Th
!2
DT
4
Tc
Tc
¼ 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 ; (30)
ð Th
ð Th
@ L
q ðT ÞdT þ L
q ðT ÞdT A
Sp ðT ÞdT p
Sn ðT ÞdT
n
p
Tc
n
Tc
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u ð Th
u
uLn
qn ðT ÞdT
An u
Tc
u
bp ¼
¼
;
ð Th
Ap u
t
ð
Þ
qp T dT
Lp
Tc
where ½PFeng;d is the maximum engineering power factor
density of a module.
(31)
The DT-dependent maximum efficiency for p-SKU/nSKU is shown in Fig. 1(a), where Tc is fixed at 100 C and Th
is ramped up to 500 C. The efficiency predictions by the cumulative temperature dependence model and conventional
model have good agreement in whole temperature range and
show 2.6% and 2.5% of relative difference as compared
with results by the numerical simulation25,26 at DT ¼ 400 C
[Fig. 1(b)], respectively. The conventional model can show
the reliable efficiency prediction for p-SKU/n-SKU because
S, q, and j of p-type and n-type SKUs have weak temperature dependence,19,20 i.e., linear behavior of ZT. Thus, the
conventional model can be simply utilized in such limited
material systems. For p-PbTe/n-PbSe, however, the conventional model underestimates the maximum efficiency over
whole temperature range [Fig. 1(c)] and gives rise to 27% of
relative difference as compared with the simulation result at
DT ¼ 400 C [Fig. 1(d)], whereas the cumulative temperature
dependence model without considering Thomson effect on
the input heat flux shows more accurate prediction with the
relative difference by 10.3% mainly because of the cumulative effect of temperature dependent S, q, and j taken into
account. By accounting for Thomson effect, the efficiency
prediction becomes further accurate over the whole temperature range and leads to only 2.8% of underestimation as compared to the simulation result. For p-SnSe/n-PbSe paired
module shown in Figs. 1(e) and 1(f), the cumulative temperature dependence model without Thomson effect overrates
the efficiency by 4.6% at DT ¼ 400 C, and the prediction is
improved by accounting for Thomson effect leading to the
relative difference by 3% while the conventional model
overrates it by 17% as compared with the simulation. In
some cases, however, the conventional model shows more
accurate prediction. For example, in p-SnSe/n-PbSe if
DT ¼ 500 C, the conventional model only overrates by 2.5%
as shown in the inset of Fig. 1(e), while it predicts mostly
inaccurate efficiency in the rest of temperature range
[Fig. 1(e)]. This is because the averaged constant values of
S, q, and j at DT ¼ 500 C are by chance very close to each
FIG. 1. The maximum efficiency as a
function of DT, and its relative difference as compared to the numerical
simulation at DT ¼ 400 C for (a) and
(b) p-SKU/n-SKU, (c) and (d) p-PbTe/
n-PbSe, and (e) and (f) p-SnSe/n-PbSe.
The efficiency of p-SnSe/n-PbSe up to
DT ¼ 500 C is shown as the inset in
(e). The x-axis in (b), (d), and (e) represents A: numerical simulation, B:
CTD model with Thomson effect, C:
CTD model without Thomson effect,
and D: conventional model. CTD
stands for the cumulative temperature
dependence model.
115103-5
Kim, Liu, and Ren
J. Appl. Phys. 118, 115103 (2015)
equivalent value which includes the influence of temperature
dependent properties as well as Thomson effect taken into
account by the numerical simulation. Thus, the conventional
model is not reliable, since it sometimes has inadvertent
accuracy with lack of analogy.
To examine the effect of Thomson heat on the thermoelectric module based on the cumulative temperature
dependence model, Figs. 2(a) and 2(b) show the cumulative
effect of practical Joule and Thomson heat on the input heat
flux at hot side, respectively. The conventional model
assumes a half of the Joule heat returning to the hot and cold
side by the temperature-independent q. The concept of the
cumulative temperature dependence provides a practical
contribution of Joule heat on each thermal boundary of a
single thermoelectric leg as WJ, and its combination effect in
p-n pair module is defined by referring to Eq. (14)
WJ;pn ¼ WJ;p R0p þ WJ;n R0n :
(32)
In Fig. 2(a), WJ,pn of p-SKU/n-SKU and p-PbTe/n-PbSe is
over 1/2 indicating more Joule heat on the hot side than that
based on the conventional model, and it has an increasing
trend as DT increases due to dq/dT > 0 of each material. In
p-SnSe/n-PbSe, WJ,p–n fluctuates and becomes below 1/2 at
DT ¼ 300 C due to a large reduction of q of SnSe with
higher temperature.23 Figure 2(b) shows the combined effect
of Thomson heat for input heat flux, which is also defined as
WT;pn ¼ WT;p s0p WT;n s0n :
(33)
WT,p–n for p-SKU/n-SKU and p-PbTe/n-PbSe gradually
decreases with higher temperature due to the decreasing
FIG. 2. The cumulative effect of (a) Joule heat and (b) Thomson heat on the
input heat flux at the hot side.
tendency of djSj=dT indicating less Thomson effect is
associated. For p-SnSe/n-PbSe, WT,p–n shows a parabolic
behavior up to DT ¼ 350 C and decreases below zero. The
negative WT,p–n corresponds to the cumulative effect of heat
absorbed by Thomson effect and exerts an influence on
decreasing the conversion efficiency at the given thermal
boundary condition as compared with the model without
Thomson effect, as shown in Fig. 1(e). In contrast, the positive WT,pn for p-SKU/n-SKU and p-PbTe/n-PbSe has an
effect on increasing the efficiency compared with that in
absence of Thomson heat as shown in Figs. 1(a) and 1(c).
Figure 3 shows the maximum Pd with respect to DT for
three p-n module configurations in which the temperature
gradient is assumed to be 200 C mm1 for the leg length of
2 mm where Lp ¼ Ln in a typical p-shape module, and the
temperature profiles of each thermoelectric leg at
Th ¼ 500 C by the numerical computation. The analytical
prediction for p-SKU/n-SKU by the conventional model has
good agreement with the simulation results by 2.7% at
DT ¼ 400 C [Fig. 3(a)] due to the high linearity of ZT curves
of each SKU material, while the maximum Pd predictions
for p-PbTe/n-PbSe [Fig. 3(c)] and p-SnSe/n-PbSe [Fig. 3(e)]
lead to 23% and 42% of relative difference at DT ¼ 400 C
as compared to the simulation results, respectively. As the
conventional model is reliable only when thermoelectric
properties are not much varied with temperature such as
p-type and n-type SKUs, the cumulative temperature dependence model computes more accurate prediction through
the whole temperature range [Figs. 3(c) and 3(e)] and gives
rise to 0.5% and 4.9% of relative difference at DT ¼ 400 C
for p-PbTe/n-PbSe and p-SnSe/n-PbSe, respectively, which
shows Eq. (30) is more reliable by taking Thomson effect
into consideration. The differences between the cumulative
temperature dependence model and numerical simulations
are mainly caused by the assumption of the linear approximation for temperature gradient in evaluating Rint for
integrating q(T) with respect to x. In p-SKU/n-SKU, the temperature profiles by the simulations are over the linear
approximation in terms of x (solid line) in Fig. 3(b), for
which lower Rint based on the linear approximation is
obtained due to q / T in SKUs.19,20 This results in the overestimated output power density by this model compared by
the numerical simulation as shown in Fig. 3(a). In contrast,
since the temperature distributions of p-PbTe/n-PbSe by the
simulation are lower than the linear profile [Fig. 3(d)], the
overestimated Rint by this model leads to the lower output
power density [Fig. 3(c)] as q of p-PbTe and n-PbSe is
proportional to T.21,22 For p-SnSe/n-PbSe, the temperature
profiles of both p-type and n-type leg are below the linear
distribution with respect to x [Fig. 3(f)], but q of the p-type
SnSe is higher than that of n-type PbSe by two order of magnitude, and inversely proportional to T.23 This causes the
overestimated output power density by this model [Fig. 3(e)]
resulting from the underrated Rint that is dominant by Rp of
SnSe. Therefore, the cumulative temperature dependence
model including Thomson effect more correctly predicts the
output power generation than the conventional model, and
the main difference between the cumulative temperature
dependence model and the numerical analysis is caused by
115103-6
Kim, Liu, and Ren
J. Appl. Phys. 118, 115103 (2015)
FIG. 3. The maximum output power density as a function of DT up to 400 C and the temperature profile through each leg at Th ¼ 500 C for (a) and (b) pSKU/n-SKU, (c) and (d) p-PbTe/n-PbSe, and (e) and (f) p-SnSe/n-PbSe. The length of legs was determined based on the temperature gradient DTmax/
L ¼ 200 C mm1.
the linear approximation for the temperature distribution in
this model. This over- or under-estimate of output power
gives rise to the tendency of over- or under-rated maximum
efficiency prediction by this model shown in Fig. 1 as compared with the numerical results. Another possible reason for
the difference is that this model accounts for the cumulative
effect of the temperature dependence of each property at
thermal boundaries, which is path-independent according to
the temperature while the numerical model incorporates the
instantaneous temperature dependence that is path dependent
associated with the temperature.
Figures 4(a)–4(c) show the maximum efficiency with
respect to be for 1-pair module of p-SKU/n-SKU, p-PbTe/nPbSe, and p-SnSe/n-PbSe, respectively, at DT ¼ 400 C and
Tc ¼ 100 C with Lp ¼ Ln. The be by the model in this study
(open square and open diamond) has better agreement with
that by numerical simulation (solid line and open triangle in
inset) than the prediction by the conventional model (open
circle). The efficiency of the p-SKU/n-SKU is maximized
when An is about 20% smaller than Ap (be ¼ 0.802), while
p-PbTe/n-PbSe requires about 16% larger An (be ¼ 1.164) for
the maximum efficiency. For p-SnSe/n-PbSe, be ¼ 0.103
FIG. 4. The efficiency with respect to
the maximum efficiency condition be
of (a) p-SKU/n-SKU, (b) p-PbTe/nPbSe, and (c) p-SnSe/n-PbSe, where
numerical simulation (solid line and
open triangle in insets), conventional
model (open circle), the model in this
study without Thomson effect (open
square), and with Thomson effect
(open diamond). The area near peak
efficiencies is magnified as inset. (d)
Comparison of be (open symbols) and
bp (solid symbols) corresponding to
the maximum efficiency and maximum
output power density, respectively.
115103-7
Kim, Liu, and Ren
should be met for the maximum efficiency, which requires
the large difference between An and Ap. In Fig. 4(d), bp (solid
symbols) for the maximum output power density is obtained
by Eq. (31), and compared with be (open symbols). For small
variations of S, q, and j between p- and n-type materials like
p-SKU/n-SKU and p-PbTe/n-PbSe, the maximum efficiency
is not much different from those at Ap ¼ An (be ¼ 1), but for
p-SnSe/n-PbSe showing large property variations between pand n-type materials, the maximum conditions should be
examined instead of assuming be or bp ¼ 1. The relative
differences between bp and be for p-SKU/n-SKU, p-PbTe/
n-PbSe, and p-SnSe/n-PbSe are 1.5%, 14%, and 65%,
respectively, where the difference between them is caused
by the ratio of thermal conductivity of p-type and n-type
materials [Eqs. (24) and (31)]. However, the efficiencies
led by be and bp are close each other within 1% in p-SKU/
n-SKU and p-SnSe/n-PbSe, and within 4% in p-SnSe/nPbSe since they are on the plateau curve near the peak efficiency of each p-n configuration as shown in Figs.
4(a)–4(c). Thus, finding the range of be or bp is crucial for
a thermal and electric balance of p-n configuration, but it is
not a critical design parameter of a module to select either
be or bp unless a fine tuning of a module is necessary to see
what the module’s primary concern is, e.g., its efficiency
or output power.
J. Appl. Phys. 118, 115103 (2015)
Figure 5(a) shows ½ZTeng as a function of DT for pSKU/n-SKU, p-PbTe/n-PbSe, and p-SnSe/n-PbSe, and the
maximum efficiency as inset. Note that ½ZTeng has a close
correspondence to the maximum efficiency (inset), which
means that ½ZTeng is the practical indicator to judge which
p-n configuration is superior in terms of the efficiency. In
Fig. 5(b), ½PFeng;d is the intrinsic output power density per
DT as a material property, so it directly represents the power
generation at a given temperature boundaries and the ratio m
by showing similar tendency to Pd at matched load condition
(inset). Thus, ½PFeng;d is also the practical indicator for the
output power generation of each p-n module. Figures
5(c)–5(e) show ½ZTeng ; ðZTÞeng of p- and n-type materials,
and their averaged (ZT)eng for each p-n configuration, where
parentheses denotes a single material’s engineering figure of
merit. For p-SKU/n-SKU and p-PbTe/n-PbSe, the simply
averaged (ZT)eng of each p- and n-type materials has a good
agreement with ½ZTeng of their corresponding p-n module
within 1% as shown in Figs. 5(c) and 5(d), while the averaged (ZT)eng of p-SnSe/n-PbSe is overrated by 47% as compared to ½ZTeng of the module at DT ¼ 400 C shown in Fig.
5(e). The differences are caused by the correspondence of
the cumulative temperature-dependent S, q, and j for p- and
n-type materials in the p-n module (see Appendix B), so it
should be careful to predict a module efficiency by simply
FIG. 5. (a) The maximum engineering
figure of merit of p-n module ½ZTeng
and the efficiency (inset) at
Tc ¼ 100 C, and (b) the maximum engineering power factor density
½PFeng;d and output power density
(inset) as a function of DT of the p-n
configurations, where Tc ¼ 100 C and
Comparison
of
Lp ¼ Ln ¼ 2 mm.
½ZTeng (larger symbols), (ZT)eng of
individual p- (solid lines) and n-type
(dashed lines) materials, and their
averaged (ZT)eng (smaller symbols) in
(c) p-SKU/n-SKU, (d) p-PbTe/n-PbSe,
and (e) p-SnSe/n-PbSe.
115103-8
Kim, Liu, and Ren
J. Appl. Phys. 118, 115103 (2015)
averaging ZTs of p- and n-type materials when using
Eq. (28).
IV. CONCLUSION
We have established the maximum efficiency formulae of
thermoelectric module consisting of p-n pair based on the cumulative temperature dependence model including Thomson
heat showing how Thomson heat affects the output power and
conversion efficiency. The effect of the cumulative Joule and
Thomson heat enables the formulae to predict the efficiency
and output power more reliably than the conventional model.
The optimized engineering figure of merit ½ZTeng and engineering power factor density ½PFeng;d of modules are direct
indicators of practical efficiency and output power.
ACKNOWLEDGMENTS
This work was supported by the U.S. Department of
Energy under Contract No. DOE DE-FG02-13ER46917/DESC0010831.
The second term on the right-hand side for p-type becomes
ðL
ðL
dT
dSp ðT Þ dT
I sp ðT Þ dx ¼ I T
dx
dx
dT dx
0
0
ð Tc
dSp ðT Þ
dT
T
¼I
dT
Th
ð Th
¼ I Tc Sp ðTc Þ Th Sp ðTh Þ þ I
Sp ðT ÞdT:
Tc
(A5)
In the same manner for n-type
ð Th
ðL
dT
Sn ðT ÞdT:
I sn ðT Þ dx ¼ I ðTc Sn ðTc Þ Th Sn ðTh ÞÞ þ I
dx
Tc
0
(A6)
By substituting Eqs. (A5) and (A6) into Eq. (A4), and applyÐT
ÐT
oc
and Voc ¼ Tch Sp ðTÞdT Tch Sn ðTÞdT
ing I ¼ Rint Vð1þmÞ
!
ð
ð
Th
Tc
APPENDIX A: OUTPUT POWER PREDICTION
ACCOUNTING FOR THOMSON EFFECT
Assuming no parasitic electric and/or thermal losses
through thermoelectric configuration, the output power generation is expressed based on an energy balance as
Pout ¼ Qh Qc :
(A1)
Qh and Qc are the heat flux at hot and cold side, shown by
Eqs. (3)–(7)
ð
A Th
jp ðT ÞdT
Qh ¼ ITh Sp ðTh Þ Sn ðTh Þ þ
L Tc
ð
A Th
jn ðT ÞdT I2 ðWJ;p Rp þ WJ;n Rn Þ
þ
L Tc
!
ð
ð
Th
Th
sp ðT ÞdT WT;n
I WT;p
Tc
sn ðT ÞdT ;
(A2)
Tc
ð
A Th
Qc ¼ ITc Sp ðTc Þ Sn ðTc Þ þ
jp ðT ÞdT
L Tc
ð
A Th
þ
jn ðT ÞdT I2 ðWJ;p Rp þ WJ;n Rn Þ
L Tc
!
ð Th
ð Th
sp ðT ÞdT WT;n
sn ðT ÞdT
I WT;p
Tc
I
ðL
0
dT
sp ðT Þ dx dx
Tc
ðL
0
dT
sn ðT Þ dx
dx
Th
Sp ðT ÞdT Pout ¼ I
!
þ I2 Rint ; (A3)
where Rint is electric resistance of thermoelectric legs, and
the same geometry for p-type and n-type legs is considered.
Substituting Eqs. (A2) and (A3) into Eq. (A1) yields
Pout ¼ I Th Sp ðTh Þ Th Sn ðTh Þ Tc Sp ðTc Þ þ Tc Sn ðTc Þ
!
ðL
ðL
dT
dT
þI
sp ðT Þ dx sn ðT Þ dx I2 Rint : (A4)
dx
dx
0
0
Sn ðT ÞdT
I2 Rint
Tc
¼
2
2
Voc
Voc
Rint ð1 þ mÞ Rint ð1 þ mÞ2
¼
2
Voc
m
:
Rint ð1 þ mÞ2
(A7)
Eq. (A7) for the output power generation taking Thomson
effect into account is analytically exact expression based on
Rint with linear approximation of dx (L/DT)dT, and it is
proved that integrating the Seebeck coefficient with respect
to temperature gives rise to the open circuit voltage influenced by Thomson effect.
APPENDIX B: ½ZT eng VS. AVERAGED (ZT)eng
To examine the relative difference of averaged (ZT)eng
and ½ZTeng
i
1h
ðZT Þeng;p þ ðZT Þeng;n ½ZT eng
2
!
2
2
S^n
1 S^p
ðSp Sn Þ2
¼
DT þ
DT pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi2 DT;
^ pj
^p
^ nj
^n
q
2 q
qp jp þ qn jn
<¼
(B1)
where symbol with a hat indicates the integral with respect
to Th and Tc. Eq. (B1) is reduced to
2
3
sffiffiffiffiffiffiffiffiffiffi12
!2 0
2
2
^
^
^ pj
^p
S p DT 6
S q
^ nj
^ nA 7
S^n
@1 þ q
41 þ p2
5;
2 1
<¼
^
^p
^ pj
^p
q
2^
qpj
Sp
^ j
^
S^ q
n n n
(B2)
which is a function of the ratios of p-type to n-type properties.
< can be positive or negative value depending on the ratio of
^ n =^
^ p , e.g., < > 0 for p-SnSe/n-PbSe as
^ nj
qpj
S^n =S^p and q
shown in Fig. 5(e), and (ZT)eng and ½ZTeng become compara^ n =^
^ p 1.
^ nj
ble, i.e., < 0 when S^p =S^n 1 and q
qpj
115103-9
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