- Email: [email protected]
Improvement of transient supercooling of thermoelectric coolers through variable semiconductor cross-section
Applied Energy 164 (2016) 501–508
Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Improvement of transient supercooling of thermoelectric coolers through variable semiconductor cross-section Hao Lv a,b, Xiao-Dong Wang a,b,⇑, Tian-Hu Wang c, Chin-Hsiang Cheng d a
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing 102206, China c School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China d Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan b
h i g h l i g h t s A new TEC design with variable semiconductor cross-sectional area is proposed. A multiphysics model is used to study the transient supercooling of the new design. Two additional effects are found and can be used to improve transient supercooling.
a r t i c l e
i n f o
Article history: Received 18 January 2015 Received in revised form 4 November 2015 Accepted 26 November 2015
Keywords: Thermoelectric cooler Transient supercooling Variable cross-section Simulation Minimum cold-end temperature
a b s t r a c t In this work, a new design of thermoelectric cooler (TEC) with variable semiconductor cross-sectional area is proposed to improve its transient supercooling characteristics. Four key evaluation indicators of transient supercooling for the conventional and new designs, including the minimum cold end temperature, maximum temperature overshoot, holding time of transient state, and recovery time ready for next steady-state, are examined and compared by a three-dimensional, transient, and multiphysics model. Two additional effects are observed in the TEC with variable semiconductor cross-sectional area. First, the variable cross-sectional area makes the thermal circuit asymmetric, so that Joule heat is preferentially conducted toward to the end with a larger cross-sectional area. Second, more Joule heat is produced close to the end with a smaller cross-sectional area. The present simulations find that these two effects can be utilized to achieve the desired evaluation indicators by changing the cross-sectional area ratio of hot end to cold end. When a lower cold end temperature, a smaller temperature overshoot, and/or a longer holding time are/is required, a larger cross-sectional area at the cold end is recommended. However, to achieve a shorter recovery time, a smaller cross-sectional area at the cold end is needed. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Thermoelectric devices can convert heat into electricity by Seebeck effect or electricity into heat by Peltier effect. With the development of a new generation of nanostructured thermoelectric materials, figure of merit of materials is improved significantly, which promotes rapid growth of studies on thermoelectric devices [1–11]. Thermoelectric coolers (TECs) have been widely employed in various cooling and refrigeration applications [1–11]. Compared with conventional cooling technologies, TECs have ⇑ Corresponding author at: State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China. Tel./fax: +86 10 62321277. E-mail address: [email protected] (X.-D. Wang). http://dx.doi.org/10.1016/j.apenergy.2015.11.068 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.
many advantages such as high reliability, compact volume, layout flexibility, large operating temperature range, and rapid temperature response, because the coolers do not use any moving parts and environmentally harmful fluids [12,13]. When a TEC operates at steady state with a constant hot end temperature, the lowest cold end temperature achievable is determined by the figure of merit of semiconductor materials, TEC structure, and input current [14,15]. However, when a current pulse with magnitude several times larger than the optimal steady-state one is applied to the TEC, an instantaneously lower cold end temperature than that reachable at steady-state can be achieved. This phenomenon is referred to as transient supercooling, which can be applied in many fields where extra cooling for a short time is needed [16,17].
502
H. Lv et al. / Applied Energy 164 (2016) 501–508
At least five indicators can be used to evaluate the transient supercooling characteristics [16]: maximum cold end temperature drop DTc,max1 = Tc,s Tc,min, maximum temperature overshoot DTc,max2 = Tc,max Tc,s, time to reach the minimum cold end temperature tmin, holding time of the supercooling state Dthold, and recovery time to the next new steady state Dtrec, where Tc,s is the cold end temperature reachable at steady-state, Tc,min and Tc,max are respectively the minimum and maximum cold end temperatures reachable when a pulse current is applied to the TEC. In recent years, many efforts have been devoted to investigating the transient supercooling [16,18–30]. These investigations found that for a specific pulse shape, pulse amplitude and width have significant effects on the transient supercooling. Various pulse shapes were also compared in Refs. [16,31–35]. The results showed that there exists an optimal pulse shape to achieve the maximum cold end temperature drop, however, the optimal shape obtained in Refs. [16,31–35] are different. Recently, our group has developed a multiphysics transient TEC model to investigate the effect of pulse shape [36]. The results showed that the optimal shape is only determined by the time to reach the minimum cold end temperature and the pulse width (s). For the pulses with tmin < s, a higher power pulse provides a lower cold end temperature, for the pulses with tmin = s, however, the trend is reversed. The results reasonably explained the divergence for the optimal pulse shape reported by the previous studies [16,31–35]. In the above studies [18–36], the p-type and n-type semiconductors were specified as regular cuboids or cylinders with constant cross-sectional areas. Hoyos et al. [37] proposed for the first time that it is possible to achieve a lower cold end temperature when variable semiconductor cross-sections are adopted. They fabricated a TEC with conical semiconductor legs and experimentally tested its transient supercooling characteristics. Their tests showed that with narrow pulse width and large amplitudes, additional cooling of the order of 45° below the steady-state maximum with recovery times in the range of 1–3 s was obtained. Following Hoyos et al.’s work, Yang et al. [16] developed a onedimensional heat conduction model to investigate the transient supercooling performance of a axisymmetric TEC element with variable semiconductor cross-sectional area. Their results showed that a lower minimum transient temperature but a shorter holding time are observed for the tapered axisymmetric semiconductor legs with smaller cross-sectional area at the cold end. Thus, they concluded that the increase of holding time for TEC legs with a larger cross-sectional area at the cold end can be potentially useful for the device to be operated for a longer time. It should be noted that in Yang et al.’s work, a freestanding TEC element was modeled with constant semiconductor properties, and only Joule heat was assumed as the internal heat source. Our previous study [36] has demonstrated that although the multiphysics model with constant and variable properties predict almost the same minimum cold end temperature, the model with constant properties underestimates the temperature overshoot by about 90 K. Accurate prediction of the temperature overshoot is very important for transient supercooling applications, because a larger temperature overshoot means that the TEC needs a longer time to return to the previous steady state. In additon, the larger temperature overshoot also could lead to burn-out of the electronic device that needs to be cooled. Thus, considering of variable properties is necessary for the accurate prediction of TEC transient supercooling performance. Furthermore, as expected, when the variable semiconductor cross-sectional areas are adopted, threedimensional current and temperature distributions may occur in p–n junction and hence the one-dimensional model may be improper. In addition, an actual TEC element is composed of a p–n junction, three metallic connectors, and two electrically insulating ceramic plates. The ceramic plates have large heat capacity,
hence, the transient response characteristics for the actual TEC element differs significantly from those for the freestanding TEC element. Based on the above analysis, a rigorous and comprehensive study on TEC shape effect on transient supercooling characteristics is quite lacking up to now. Therefore, the objective of this work is to investigate how variable semiconductor cross-sectional area influences the transient supercooling characteristics. To achieve this objective, a complete, three-dimensional, and multiphysics TEC model is firstly used to predict the steady-state TEC performance. The optimal steady-state currents are respectively obtained for the TEC with constant and variable cross-sectional semiconductor areas. Then, a pulse current with an amplitude several times larger than the optimal steady-state current is applied to the TECs to investigate and compare their transient supercooling characteristics. Finally, the effects of pulse amplitude and area ratio of hot end to cold end on the transient supercooling characteristics are investigated. 2. TEC with variable semiconductor cross-sectional area Generally, a TEC is composed of several tens or hundreds thermoelectric elements. These thermoelectric elements are connected thermally in parallel and electrically in series, and hence a thermoelectric element can be extracted as the computational domain (Fig. 1). The element consists of a p-type semiconductor leg, an n-type semiconductor leg, three metallic connectors, and two ceramic plates. Fig. 1(a) shows the schematic of a conventional TEC element, in which the thicknesses of ceramic plates, metallic connectors, and semiconductor legs are H0, H1, and H2, respectively, the p- and n-type semiconductor legs have the same square cross-section with the area of Asemi = L2 L2, and the two ceramic plates have the same rectangular cross-section with the area of Acer = (2L1 + 2L2) L2. Fig. 1(b) shows the schematic of a TEC element with variable semiconductor cross-sectional areas, in which the cross-sectional areas for both the p-type and n-type semiconductor legs change linearly with the leg thickness, while the other
(a)
Th H0
Ceramic plate
H1
Copper
n-type
H2
I, current outlet
L2 L1/2
z
p-type
L2 L1 Ceramic plate
y
L2
L1/2
Qc
x
Th
(b) H0 H1
I, current inlet
LH
n-type
Ceramic plate
Copper
L1
p-type
H2
I, current inlet
I, current outlet
LC L2
z
y x
Qc
Fig. 1. Schematics of the TEC element: (a) with constant cross-section; (b) with variable cross-section.
503
H. Lv et al. / Applied Energy 164 (2016) 501–508
geometric parameters are the same as those of the conventional TEC element. An area ratio of hot end to cold end is defined as c = Asemi,h/Asemi,c.
Asemi,h
Asemi,h
3. Simulation cases In this paper, the transient supercooling characteristics of the TEC element with variable semiconductor cross-sectional areas are firstly compared with that of the conventional TEC element. As shown in Fig. 2, design 1# is the convectional TEC element with H2 = 1.0 mm and Asemi = 0.5 0.5 mm2 with c = 1, design 2# has Asemi,c = 0.5 0.5 mm2 and Asemi,h = 0.5Asemi,c with c = 0.5, design 3# has Asemi,c = 0.5 0.5 mm2 and Asemi,h = 2Asemi,c with c = 2, design 4# has Asemi,c = 2Asemi,h and Asemi,h = 0.5 0.5 mm2 with c = 0.5, and design 5# has Asemi,c = 0.5Asemi,h and Asemi,h = 0.5 0.5 mm2 with c = 2. Subsequently, the transient supercooling characteristics of the TEC element with various c is studied. In these cases, to ensure a lower cost of semiconductor materials, the total volume of the semiconductor materials used in the TEC element should be smaller than the conventional design. Thus, when the hot end of the TEC element remains the same cross-sectional area as the conventional design, the cold end will be tapered and hence c is larger than 1 (Fig. 3). However, when the cold end remains the same crosssectional area as the conventional design, the hot end will be tapered with c < 1 (Fig. 4). 4. Numerical model
Asemi,c
Asemi,c
(a)
(b)
Fig. 3. Shape of p- or n-type semiconductor leg: (a) conventional design with c = 1; (b) design with c > 1.
Asemi,h
Asemi,h
Asemi,c
The three-dimensional, multiphysics, and transient TEC model includes energy equations and electric potential equations. These two sets of coupled equations need to be solved simultaneously to obtain the temperature and electric potential distributions within the TEC element. The model is described briefly in the following and more details can be found in our previous works [36,38]
@T J2 J rT ¼ r ðki rTÞ þ bi~ @t ri r ðri ðr/ ai rTÞÞ ¼ 0 ðqcp Þi
ð1Þ ð2Þ
!
E ¼ r/ þ ai rT
ð3Þ
!
~ J ¼ ri E
ð4Þ
where T is the temperature, t is the time, ~ J is the current density vector, q, cp, k, r, and b are the density, specific heat, thermal conductivity, electric conductivity, and Thomson coefficient, respectively, and / is the electric potential. The subscript i respectively denotes the
(a)
(b)
Asemi,c
Fig. 4. Shape of p- or n-type semiconductor leg: (a) conventional design with c = 1; (b) design with c < 1.
ceramic plate, metal connector, p-type semiconductor, or n-type semiconductor. The Thomson coefficient can be related to the Seebeck coefficient a by b = T(da/dT). For the ceramic plates, a and r are both assumed to be zero because of their ignorable thermoelectric effect and electrical insulation. It is worth noting that the present TEC model solves the electric potential equations and hence the current density vectors through the semiconductor legs and metallic connectors can be calculated by Eq. (4). Thus, the model can be used to investigate the effect of variable semiconductor cross-sectional area on the TEC transient supercooling characteristics. The TEC initially operates at steady state with an input current equal to the optimal steady-state value which can be determined by the steady state I–Tc curve of the TEC element. Subsequently, a current pulse with the amplitude two to five times higher than the optimal steady- state value is applied to the TEC at t = 0.05 s.
Asemi,h
Asemi,h
Asemi,c
Asemi,c
(a)
(b)
(c)
(d)
(e)
Fig. 2. Shape of p- or n-type semiconductor leg: (a) design 1#: conventional TEC with c = 1; (b)–(e) designs 2#–5# with variable semiconductor cross-sectional area.
H. Lv et al. / Applied Energy 164 (2016) 501–508
-30
P=2.5
(a)
τ =4.0 s
-40
Tc /°C
Hence, the initial conditions of temperatures and electric potentials for the TEC element can be determined by solving the steady-state model. An isothermal boundary condition (Th = 293 K) is applied to the hot end of the TEC element and a zero cooling load condition (Qc = 0) is specified to the cold end, while adiabatic boundary conditions are applied to the other outside surfaces of the TEC element. At the internal interfaces between adjacent materials, the temperature and heat flux are assumed to be continuous. The electric boundary conditions are as follows: a time-dependent current is specified to the current inlet of the TEC element, while a zero electric potential is applied to the current outlet. The n-type and p-type semiconductor legs are assumed to be Bi2(Te0.94Se0.06)3 and (Bi0.25Sb0.75)2Te3, respectively. The copper and Al2O3 ceramic are selected as the metallic connector and ceramic plate. The temperature-dependent properties for these materials can be found in our previous studies [14,36,38]. The coupled set of equations in association with the boundary conditions was solved iteratively using multiphysics package COMSOL 3.5a. Iteration criterion for convergence was set as 106. The independences of space and time grid sizes are examined for each simulation case to ensure the accuracy of simulated results.
-50 -60 -70 -80
Experimental data in Ref. [18] Numerical predictions
0
5
10
450
The experimental curve of Tc–t tested by Snyder et al. [18] is used to validate the present model. They used n-type Bi2Te2.85 Se0.15 and p-type Bi0.4Sb1.6Te3 to fabricate 5.8 mm tall thermoelectric elements with 1 mm2 cross-sectional areas. The cold end was soldered to a 35 lm thick copper foil to which was soldered a 25 lm diameter Chromel-Constantin thermocouple for measurement of the temperature. The hot end was soldered to an electrically isolated heat sink where the heat sink temperature could be adjusted from 55 to 35 °C. The optimal steady-state current was Iopt = 0.675 A. A step pulse with amplitude of P = I/Iopt = 2.5 and width of s = 4.0 s was applied to the thermoelectric elements. It should be noted that temperature-dependent properties for the semiconductor materials used in their experiments were not given in Ref. [18], and hence our simulation has to use constant properties. The properties of semiconductor legs reported in Ref. [18] were as follows: qcp = 1.20 106 J m3 K1, k = 1.20 W m1 K1, a = 2.35 104 V K1, and r = 5.11 104 X1 m1. For making a fair comparison, TEC dimension, material properties, and operating conditions used in our simulation are all the same as those in the experiment. As shown in Fig. 5(a), the simulation results agree well with the experimental data at the beginning stage but the deviation is observed for t > 5.5 s where the pulse has been terminated. The deviation can be attributed to the constant material properties used in the simulation. Generally, electric conductivity of Bismuth Telluride-based materials is negatively proportional to temperature. Ref. [14] has reported that, as compared with variable material properties, the TEC model with constant material properties predicts a lower temperature distribution across the semiconductor leg. Thus, the resultant larger electric conductivity leads to a weaker Joule heating. At the initial stage of supercooling, the temperature at the cold end is determined by Peltier effect rather than Joule heating due to finite thermal diffusion rate. Once the cold end reaches its lowest temperature, the Joule heat transferred back to the cold end becomes stronger than the Peltier cooling. Thus, the temperature begins to increase and develop into overshoot stage. Consequently, the weaker Joule heating predicted by the model with constant material properties leads to the underestimated temperature overshoot. Above analysis can be further justified in Fig. 5(b), which compares the dynamic responses of the cold end
20
25
τ =2.0 s P =5 qc=19685 W m-2
(b)
425 400
h=2000 W K-1m-2
375 350 325 300 275
5. Model validation
15 t /s
Tc /K
504
250
Model with variable properties Model with constant properties
0
1
2
3
t /s
4
5
6
Fig. 5. Model validation: (a) comparison of numerical predictions and experimental data in Ref. [18]; (b) comparison of dynamic responses of the cold end temperature predicted by models with constant and variable properties.
temperature predicted by the multiphysics models with constant and variable properties. The lowest cold end temperatures predicted by the both models are almost the same; however, the model with constant properties underestimates the temperature overshoot and recovery time, but overestimates the duration of the supercooling state. Therefore, it can be expected that if temperature-dependent material properties are used, our model can achieve an acceptable matching with the measured Tc–t curve. Thus, our model can be used to investigate the transient supercooling characteristics of TEC element.
6. Determination of the optimal steady-state current In order to study the transient supercooling characteristics, the optimal steady-state current, Iopt, needs to be determined first. Fig. 6 shows the I–Tc curves of the TEC elements with constant and variable semiconductor cross-sectional areas. The minimum cold end temperatures, Tc,min, for the five designs are 204.62, 204.79, 204.21, 204.84, and 204.17 K, respectively, indicating that Tc,min is almost independent of the TEC shape. This phenomenon is also observed for the TEC elements with various c (shown in Figs. 3 and 4), their Tc,min ranges from 203.10 to 204.93 K. However, the TEC elements with various semiconductor cross-sectional areas exhibit different Iopt from the conventional TEC element. Fig. 6 also shows that Iopt depends on the volume of semiconductor materials, designs 3# and 4# have larger semiconductor volume than the conventional design, their Iopt is higher, on the contrary, designs 2# and 5# have smaller semiconductor volume than the conventional design, so that their Iopt is lower. This result agrees well with the experiments conducted by Hoyos et al. [37] and is confirmed again in Fig. 7, where the TEC element with c – 1 has a smaller semiconductor volume so that its Iopt is lower than Iopt, conv of the
505
H. Lv et al. / Applied Energy 164 (2016) 501–508
Design 1#
285
Design 2#
Design 3#
Design 4#
Design 5#
T h=293K, Q c =0 W
270
Tc /K
255 240 225 210 195
design 1# (conventional design), design 3#, design 4#,
180
0.4
0.8
1.2
1.6
2.0
2.4
design 2#, design 5# 2.8
3.2
3.6
4.0
I /A Fig. 6. I–Tc curves for the TEC elements with different semiconductor shapes.
1.8 1.6
Iopt /A
1.4 1.2 1.0 0.8 0.6 0.4
0
1
2
3
4
5
6
7
8
γ =Asemi,h /Asemi,c Fig. 7. The optimal steady-state currents of TEC elements with various c.
conventional design. Furthermore, Figs. 6 and 7 also indicate when two TEC elements have the same semiconductor volume, their Iopt will be identical. 7. Transient supercooling characteristics 7.1. Comparison between various semiconductor shapes The transient supercooling characteristics of the TEC elements with constant and variable semiconductor cross-sectional areas are shown in Fig. 8. The step pulse with pulse amplitude of P = 5 and pulse width of s = 0.05 s is used here. As shown in Fig. 8, the changes of all Tc–t curves exhibit the similar trend: Tc fist keeps
its steady-state value of Tc,s at t < 0.005 s; it starts to decline after the step current is applied to the TEC element; when Tc reaches Tc,min it starts to rise; after maximum temperature overshoot occurs, Tc starts to decrease again and finally returns to Tc,s, ready for the next pulse. It should be noted that although designs 2# and 4# have different semiconductor shapes, they exhibit almost the same transient supercooling characteristics because the both have the same c = 0.5. Similarly, shapes 3# and 5# have the same c = 2, they also exhibit almost the same Tc–t curves. A better transient response should be one with higher amount of supercooling, weaker temperature overshoot, longer holding time of supercooling state, and shorter recovery time. Thus, these four key evaluation indicators for designs 1#–5# are listed in Table 1. It can be seen that a larger Asemi,c or c < 1 (designs 2# and 4#) ensures a larger DTc,max1, a smaller DTc,max2, and a longer Dthold, but it also leads to a longer Dtrec, as compared with the conventional design. Oppositely, a smaller Asemi,c or c > 1 (designs 3# and 5#) results in a shorter Dtrec, however, it also induces a worse DTc,max1, DTc,max2, and a shorter Dthold, as compared with the conventional design. Hoyos et al. [37] and Yang et al. [16] presented that the tapered (c < 1) or diverging (c > 1) semiconductor shape causes the thermal circuit asymmetric and Joule heat will preferentially be conducted toward to the end with the larger cross-sectional area (referred to as the heat conduction effect here). However, more Joule heat is also produced close to the end with the smaller cross-sectional area (referred to as the Joule heat effect here). This can be explained by the following equation:
dQ Joule ¼
ðI=AÞ2
r
!
Adh ¼
I2 dh rA
ð5Þ
H. Lv et al. / Applied Energy 164 (2016) 501–508
Design 1#
Design 2#
360
design 1# design 2# design 3# design 4# design 5#
340 320
Tc /K
300
Design 3#
Design 4#
Design 5#
208
204
Tc /K
506
200
196
280
192
260
0.01
0.02
0.03
0.04
0.05
0.06
t /s
240
T h =293 K
220
Q c =0 W
200
P=5 τ =0.05 s
180 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
t /s Fig. 8. Transient supercooling characteristics of TEC elements with five different shapes.
Table 1 The evaluation parameters of transient supercooling characteristics for TEC elements with designs 1#–5#. Parameter
Design 1#
Design 2#
Design 3#
Design 4#
Design 5#
DTc,max1 = Tc,s Tc,min/K DTc,max2 = Tc,max Tc,s/K Dthold/s Dtrec/s
8.86 98.95 0.049 4.133
9.40 80.56 0.058 5.448
7.46 147.87 0.033 3.466
9.79 81.96 0.057 5.225
6.96 146.01 0.033 3.513
where dQJoule denotes the Joule heat generated in a thin layer with a cross-sectional area of A and a thickness of dh. For a TEC element with constant applied current I, a smaller A corresponds to a larger dQJoule. Thus, Tc,min should be determined by the competition of these two effects. The present simulations show that a lower Tc,min can be achieved by the design with a larger Asemi,c, indicating that the Joule heat effect is dominant over the heat conduction effect. Based on the same reason, the larger Asemi,c weakens the temperature overshoot and lengthens the holding time. It is worth noting that the present simulations use a short pulse width of s = 0.05 s. After the step pulse is terminated, the applied current returns to Iopt, thus, the Joule heat effect becomes weaker and the heat conduction effect starts to dominate. As a result, the TEC with a larger Asemi,c requires a longer time to return to the steady state. Designs 3# and 4# use more semiconductor materials than the conventional design and designs 2# and 5#, however, their transient supercooling characteristics are found to be almost the same as that of design 5# and 2#, respectively, thus, these two designs are not recommended. Fig. 9 shows the four key evaluation indicators of transient supercooling characteristics as a function of pulse amplitude for designs 2# and 5#. As well known, for the conventional design with constant semiconductor cross-sectional area, the increase in
P means that more electric energy is supplied to the TEC element and Peltier cooling becomes stronger during the pulse period, so that a lower minimum cold end temperature, a stronger temperature overshoot, a shorter holding time of supercooling state, and a longer recovery time will be observed [16,36]. However, these results are not always followed by the TEC element with variable semiconductor cross-sectional area. As shown in Fig. 9, the temperature overshoot, holding time, and recovery time follow the same trend as the conventional design, but the minimum cold end temperature for design 5# with a smaller Asemi,c did not. This is because the variable semiconductor cross-sectional area causes two additional effects: the heat conduction effect and Joule heat effect as mentioned before. It is concluded that for design 5#, the heat conduction effect is dominant at smaller amplitude of P < 3, so that the increase in P decreases Tc,min, however, the Joule heat effect becomes dominant at larger amplitude of P > 3, which leads to Tc,min increased with P. Fig. 9 also shows that design 2# with a larger Asemi,c always provides a smaller minimum cold end temperature, a weaker temperature overshoot, a longer holding time, and a longer recovery time than those of design 5# with a smaller Asemi,c when P P 3. This indicates that for the both design, the Joule heat effect always dominates over the heat conduction effect during the pulse period,
507
H. Lv et al. / Applied Energy 164 (2016) 501–508
ΔTc,max1 /K
8
ΔTc,max2 /K
600 400 200 0
Δt hold /s
10
0.12 0.09 0.06 0.03 0.00 8
6
Th=293 K, Qc=0 W, τ =0.05 s
Δtrec /s
4
γ =1/2
γ =2
design 2#, γ =1/2 design 5#, γ =2
6 4 2
2
3
4 P
5
6
Fig. 9. Key evaluation parameters of transient supercooling characteristics as the function of pulse amplitude: (a) design 2# with c = 0.5; (b) design 5# with c = 2.
hence, more heat is transferred to the end that has the smaller cross-sectional area. However, this phenomenon is reversed after the pulse is terminated, and more heat is transferred to the end that has the larger cross-sectional area, as a result, design 2# requires a longer time to return to Tc,s ready for next steady-state. 7.2. Effect of cross-sectional area ratio of hot end to cold end
Δtrec /s
Δthold /s
ΔTc,max2 /K
ΔTc,max1 /K
In this section, the transient supercooling characteristics for the design with various c is discussed. The cross-sectional area ratio of hot end to cold end ranges from 1/7 to 7. Restated that c = 1 represents the conventional design with constant semiconductor crosssectional area. Four key evaluation indicators for various c are shown in Fig. 10. It is found that when the cross-sectional area at the cold end is smaller than that at the hot end (c > 1), use of a
10 8 6 4 2 900
smaller Asemi,c cannot achieve a higher amount of supercooling, meanwhile, it also leads to a stronger temperature overshoot and a shorter holding time of supercooling state. It should be noted that the present predictions agree well with Hoyos et al.’s experimental curves [37]. Thus, when a priority is given to these three evaluation parameters for a specific application, a smaller Asemi,c is not recommended. However, the advantage of smaller Asemi,c lies in its shorter recovery time, which is potentially useful for the device to be operated for a longer time. As shown in Fig. 10, when the cross-sectional area at the cold end is larger than that at the hot end (c < 1), there exists an optimal value of c = 1/3, at which the lowest Tc,min and the longest holding time can be achieved. The comparison between designs with c > 1 and c < 1 shows that when a lower minimum cold end temperature, a weaker temperature overshoot, and/or a longer holding
Th=293 K, Qc=0 W, τ =0.05 s, P=5
600 300 0 0.06 0.04 0.02 0.00 12 10 8 6 4 2
0
1
γ
10
Fig. 10. Key evaluation parameters of transient supercooling characteristics as the function of c.
508
H. Lv et al. / Applied Energy 164 (2016) 501–508
time are/is required, the design with a larger cross-sectional area at the cold end should be adopted; on the contrary, when a shorter recovery time needs to be preferentially considered, the design with a smaller cross-sectional area at the cold end is recommended. 8. Conclusions In this study, transient supercooling of the TEC with variable semiconductor cross-sectional area is investigated by a threedimensional, transient, and multiphysics model. The transient supercooling characteristics can be evaluated by the minimum cold end temperature, maximum temperature overshoot, and several time constants, such as the holding time of supercooling state and the recovery time ready for next steady-state. These evaluation indicators for the proposed designs with variable semiconductor crosssectional area are compared with the conventional TEC design. Compared with the conventional design, the design with variable semiconductor cross-sectional area produces two additional effects. Firstly, the variable cross-sectional area makes the thermal resistance asymmetric, and hence Joule heat is preferentially conducted toward to the end with a larger cross-sectional area, referred to as the heat conduction effect. Secondly, more Joule heat is produced close to the end with a smaller cross-sectional area, named as the Joule heat effect. The strength of these two effects depends on the cross-sectional area ratio of hot end to cold end (c = Asemi,h/Asemi,c), so that desired evaluation indicators can be achieved by changing the value of c. The present simulations show that the Joule heat effect always dominates over the heat conduction effect during the pulse period; hence, more heat is transferred to the end that has smaller crosssectional area. However, this phenomenon is reversed after the pulse is terminated, and more heat is transferred to the end with larger cross-sectional area. As a result, when a lower minimum cold end temperature, a weaker temperature overshoot, and/or a longer holding time are/is required, the design with a larger cross-sectional area at the cold end should be adopted; on the contrary, when a shorter recovery needs to be preferentially considered, the design with a smaller cross-sectional area at the cold end is recommended. Quantitatively, the maximum cold end temperature drop of DTc,max1 = 9.47 K, the weakest temperature overshoot of DTc,max2 = 78.13 K, and the longest holding time of Dthold = 63 ms are achieved at c = 1/3. Acknowledgments This study was partially supported by the National Natural Science Foundation of China (No. 51276060), the 111 Project (No. B12034), and the Fundamental Research Funds for the Central Universities (No. 13ZX13). References [1] Minnich AJ, Dresselhaus MS, Ren ZF, Chen G. Bulk nanostructured thermoelectric materials: current research and future prospects. Energy Environ Sci 2009;2:466–79. [2] Liu D, Zhao FY, Yang HX, Tang GF. Theoretical and experimental investigations of thermoelectric heating system with multiple ventilation channels. Appl Energy 2015;159:458–68. [3] Erturun U, Erermis K, Mossi K. Influence of leg sizing and spacing on power generation and thermal stresses of thermoelectric devices. Appl Energy 2015;159:19–27. [4] Chen WH, Huang SR, Lin YL. Performance analysis and optimum operation of a thermoelectric generator by Taguchi method. Appl Energy 2015;158:44–54. [5] Madan D, Wang ZQ, Wright PK, Evans JW. Printed flexible thermoelectric generators for use on low levels of waste heat. Appl Energy 2015;156:587–92. [6] Heghmanns A, Beitelschmidt M. Parameter optimization of thermoelectric modules using a genetic algorithm. Appl Energy 2015;155:447–54.
[7] Wang TH, Wang QH, Leng C, Wang XD. Parameter analysis and optimal design for two-stage thermoelectric cooler. Appl Energy 2015;154:1–12. [8] He W, Zhang G, Zhang XX, Ji J, Li GQ, Zhao XD. Recent development and application of thermoelectric generator and cooler. Appl Energy 2015;143: 1–25. [9] Massaguer E, Massaguer A, Montoro L, Gonzalez JR. Modeling analysis of longitudinal thermoelectric energy harvester in low temperature waste heat recovery applications. Appl Energy 2015;140:184–95. [10] Stevens RJ, Weinstein SJ, Koppula KS. Theoretical limits of thermoelectric power generation from exhaust gases. Appl Energy 2014;133:80–8. [11] Liang XY, Sun XX, Tian H, Shu GQ, Wang YS, Wang X. Comparison and parameter optimization of a two-stage thermoelectric generator using high temperature exhaust of internal combustion engine. Appl Energy 2014;130: 190–9. [12] Riffat SB, Ma X. Thermoelectrics: a review of present and potential applications. Appl Therm Eng 2003;23:913–35. [13] DiSalvo FJ. Thermoelectric cooling and power generation. Science 1999;285: 703–6. [14] Wang XD, Huang YX, Cheng CH, Lin DTW, Kang CH. A three-dimensional numerical modeling of thermoelectric device with consideration of coupling of temperature field and electric potential field. Energy 2012;47:488–97. [15] Chen WH, Liao CY, Huang CI. A numerical study on the performance of miniature thermoelectric cooler affected by Thomson effect. Appl Energy 2012;89:464–73. [16] Yang RG, Chen G, Kumar AR, Snyder GJ, Fleurial JP. Transient cooling of thermoelectric coolers and its applications for microdevices. Energy Convers Manage 2005;46:1407–21. [17] Zhou Q, Bian Z, Shakouri A. Pulsed cooling of inhomogeneous thermoelectric materials. J Phys D Appl Phys 2007;40:4376–81. [18] Snyder GJ, Fleurial JP, Caillat T, Yang RG, Chen G. Supercooling of Peltier cooler using a current pulse. J Appl Phys 2002;92:1564–9. [19] Buist RJ, Lau PG. Theoretical analysis of thermoelectric cooling performance enhancement via thermal and electrical pulsing. In: 15th International conference on thermoelectrics; 1996. p. 234–7. [20] Yang RG, Chen G, Snyder GJ, Fleuriel JP. Geometric effects on the transient cooling of thermoelectric coolers. Mater Res Soc Symp Proc 2002;691:281–6. [21] Chakraborty A, Ng KC. Thermodynamic formulation of temperature–entropy diagram for the transient operation of a pulsed thermoelectric cooler. Int J Heat Mass Transf 2006;49:1845–50. [22] Ju YS. Impact of interface resistance on pulsed thermoelectric cooling. J Heat Transfer 2008;130:014502. [23] Gupta MP, Sayer MS, Mukhopadhyay S, Kumar S. On-chip Peltier cooling using current pulse. In: 12th Intersociety conference on thermal and thermomechanical phenomena in electronic systems; 2010. [24] Gupta MP, Sayer MH, Mukhopadhyay S, Kumar S. Ultrathin thermoelectric devices for on-chip Peltier cooling. Trans Comp Packag Manufact Technol 2011;1:1395–405. [25] Shen LM, Xiao F, Chen HX, Wang SW. Numerical and experimental analysis of transient supercooling effect of voltage pulse on thermoelectric element. Int J Refrig 2012;35:1156–65. [26] Manno M, Wang P, Bar-Cohen A. Transient thermoelectric self-cooling of a germanium hotspot. In: 13th IEEE ITHERM conference; 2012. p. 413–20. [27] Shen LM, Chen HX, Xiao F, Yang YX, Wang SW. The step-change cooling performance of miniature thermoelectric module for pulse laser. Energy Convers Manage 2014;80:39–45. [28] Ma M, Yu JL. A numerical study on the temperature overshoot characteristic of a realistic thermoelectric module under a current pulse operation. Int J Heat Mass Transf 2014;72:234–41. [29] Sullivan O, Gupta MP, Mukhopadhyay S, Kumar S. Thermoelectric coolers for thermal gradient management on chip, vol. 4. In: Proceedings of the ASME 2010 international mechanical engineering congress and exposition; 2010. p. 199–207. [30] Redmond M, Manickaraj K, Sullivan O, Mukhopadhyay S, Kumar S. Hotspot cooling in stacked chips using thermoelectric coolers. Trans Comp Packag Manufact Technol 2013;3:759–67. [31] Thonhauser T, Mahan GD, Zikatanov L, Roe J. Improved supercooling in transient thermoelectrics. Appl Phys Lett 2004;85:3247–9. [32] Mitrani D, Salazar J, Turo A, García MJ, Chávez JA. Transient distributed parameter electrical analogous model of TE devices. Microelectron J 2009;40:1406–10. [33] Mao JN, Chen HX, Jia H, Qian XL. The transient behavior of Peltier junctions pulsed with supercooling. J Appl Phys 2012;112:014514. [34] Sullivan O, Gupta MP, Mukhopadhyay S, Kumar S. Array of thermoelectric coolers for on-chip thermal management. J Electron Packag 2012;134:021005. [35] Manno M, Wang P, Bar-Cohen A. Pulsed thermoelectric cooling for improved suppression of a germanium hotspot. Trans Comp Packag Manufact Technol 2013;4:602–11. [36] Lv H, Wang XD, Wang TH, Meng JH. Optimal pulse current shape for transient supercooling of thermoelectric cooler. Energy 2015;83:788–96. [37] Hoyos GE, Rao KR, Jerger D. Fast transient response of novel Peltier junctions. Energy Convers 1977;17:23–9. [38] Meng JH, Wang XD, Zhang XX. Transient modeling and dynamic characteristics of thermoelectric cooler. Appl Energy 2013;108:340–8.
Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Improvement of transient supercooling of thermoelectric coolers through variable semiconductor cross-section Hao Lv a,b, Xiao-Dong Wang a,b,⇑, Tian-Hu Wang c, Chin-Hsiang Cheng d a
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing 102206, China c School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China d Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan b
h i g h l i g h t s A new TEC design with variable semiconductor cross-sectional area is proposed. A multiphysics model is used to study the transient supercooling of the new design. Two additional effects are found and can be used to improve transient supercooling.
a r t i c l e
i n f o
Article history: Received 18 January 2015 Received in revised form 4 November 2015 Accepted 26 November 2015
Keywords: Thermoelectric cooler Transient supercooling Variable cross-section Simulation Minimum cold-end temperature
a b s t r a c t In this work, a new design of thermoelectric cooler (TEC) with variable semiconductor cross-sectional area is proposed to improve its transient supercooling characteristics. Four key evaluation indicators of transient supercooling for the conventional and new designs, including the minimum cold end temperature, maximum temperature overshoot, holding time of transient state, and recovery time ready for next steady-state, are examined and compared by a three-dimensional, transient, and multiphysics model. Two additional effects are observed in the TEC with variable semiconductor cross-sectional area. First, the variable cross-sectional area makes the thermal circuit asymmetric, so that Joule heat is preferentially conducted toward to the end with a larger cross-sectional area. Second, more Joule heat is produced close to the end with a smaller cross-sectional area. The present simulations find that these two effects can be utilized to achieve the desired evaluation indicators by changing the cross-sectional area ratio of hot end to cold end. When a lower cold end temperature, a smaller temperature overshoot, and/or a longer holding time are/is required, a larger cross-sectional area at the cold end is recommended. However, to achieve a shorter recovery time, a smaller cross-sectional area at the cold end is needed. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Thermoelectric devices can convert heat into electricity by Seebeck effect or electricity into heat by Peltier effect. With the development of a new generation of nanostructured thermoelectric materials, figure of merit of materials is improved significantly, which promotes rapid growth of studies on thermoelectric devices [1–11]. Thermoelectric coolers (TECs) have been widely employed in various cooling and refrigeration applications [1–11]. Compared with conventional cooling technologies, TECs have ⇑ Corresponding author at: State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China. Tel./fax: +86 10 62321277. E-mail address: [email protected] (X.-D. Wang). http://dx.doi.org/10.1016/j.apenergy.2015.11.068 0306-2619/Ó 2015 Elsevier Ltd. All rights reserved.
many advantages such as high reliability, compact volume, layout flexibility, large operating temperature range, and rapid temperature response, because the coolers do not use any moving parts and environmentally harmful fluids [12,13]. When a TEC operates at steady state with a constant hot end temperature, the lowest cold end temperature achievable is determined by the figure of merit of semiconductor materials, TEC structure, and input current [14,15]. However, when a current pulse with magnitude several times larger than the optimal steady-state one is applied to the TEC, an instantaneously lower cold end temperature than that reachable at steady-state can be achieved. This phenomenon is referred to as transient supercooling, which can be applied in many fields where extra cooling for a short time is needed [16,17].
502
H. Lv et al. / Applied Energy 164 (2016) 501–508
At least five indicators can be used to evaluate the transient supercooling characteristics [16]: maximum cold end temperature drop DTc,max1 = Tc,s Tc,min, maximum temperature overshoot DTc,max2 = Tc,max Tc,s, time to reach the minimum cold end temperature tmin, holding time of the supercooling state Dthold, and recovery time to the next new steady state Dtrec, where Tc,s is the cold end temperature reachable at steady-state, Tc,min and Tc,max are respectively the minimum and maximum cold end temperatures reachable when a pulse current is applied to the TEC. In recent years, many efforts have been devoted to investigating the transient supercooling [16,18–30]. These investigations found that for a specific pulse shape, pulse amplitude and width have significant effects on the transient supercooling. Various pulse shapes were also compared in Refs. [16,31–35]. The results showed that there exists an optimal pulse shape to achieve the maximum cold end temperature drop, however, the optimal shape obtained in Refs. [16,31–35] are different. Recently, our group has developed a multiphysics transient TEC model to investigate the effect of pulse shape [36]. The results showed that the optimal shape is only determined by the time to reach the minimum cold end temperature and the pulse width (s). For the pulses with tmin < s, a higher power pulse provides a lower cold end temperature, for the pulses with tmin = s, however, the trend is reversed. The results reasonably explained the divergence for the optimal pulse shape reported by the previous studies [16,31–35]. In the above studies [18–36], the p-type and n-type semiconductors were specified as regular cuboids or cylinders with constant cross-sectional areas. Hoyos et al. [37] proposed for the first time that it is possible to achieve a lower cold end temperature when variable semiconductor cross-sections are adopted. They fabricated a TEC with conical semiconductor legs and experimentally tested its transient supercooling characteristics. Their tests showed that with narrow pulse width and large amplitudes, additional cooling of the order of 45° below the steady-state maximum with recovery times in the range of 1–3 s was obtained. Following Hoyos et al.’s work, Yang et al. [16] developed a onedimensional heat conduction model to investigate the transient supercooling performance of a axisymmetric TEC element with variable semiconductor cross-sectional area. Their results showed that a lower minimum transient temperature but a shorter holding time are observed for the tapered axisymmetric semiconductor legs with smaller cross-sectional area at the cold end. Thus, they concluded that the increase of holding time for TEC legs with a larger cross-sectional area at the cold end can be potentially useful for the device to be operated for a longer time. It should be noted that in Yang et al.’s work, a freestanding TEC element was modeled with constant semiconductor properties, and only Joule heat was assumed as the internal heat source. Our previous study [36] has demonstrated that although the multiphysics model with constant and variable properties predict almost the same minimum cold end temperature, the model with constant properties underestimates the temperature overshoot by about 90 K. Accurate prediction of the temperature overshoot is very important for transient supercooling applications, because a larger temperature overshoot means that the TEC needs a longer time to return to the previous steady state. In additon, the larger temperature overshoot also could lead to burn-out of the electronic device that needs to be cooled. Thus, considering of variable properties is necessary for the accurate prediction of TEC transient supercooling performance. Furthermore, as expected, when the variable semiconductor cross-sectional areas are adopted, threedimensional current and temperature distributions may occur in p–n junction and hence the one-dimensional model may be improper. In addition, an actual TEC element is composed of a p–n junction, three metallic connectors, and two electrically insulating ceramic plates. The ceramic plates have large heat capacity,
hence, the transient response characteristics for the actual TEC element differs significantly from those for the freestanding TEC element. Based on the above analysis, a rigorous and comprehensive study on TEC shape effect on transient supercooling characteristics is quite lacking up to now. Therefore, the objective of this work is to investigate how variable semiconductor cross-sectional area influences the transient supercooling characteristics. To achieve this objective, a complete, three-dimensional, and multiphysics TEC model is firstly used to predict the steady-state TEC performance. The optimal steady-state currents are respectively obtained for the TEC with constant and variable cross-sectional semiconductor areas. Then, a pulse current with an amplitude several times larger than the optimal steady-state current is applied to the TECs to investigate and compare their transient supercooling characteristics. Finally, the effects of pulse amplitude and area ratio of hot end to cold end on the transient supercooling characteristics are investigated. 2. TEC with variable semiconductor cross-sectional area Generally, a TEC is composed of several tens or hundreds thermoelectric elements. These thermoelectric elements are connected thermally in parallel and electrically in series, and hence a thermoelectric element can be extracted as the computational domain (Fig. 1). The element consists of a p-type semiconductor leg, an n-type semiconductor leg, three metallic connectors, and two ceramic plates. Fig. 1(a) shows the schematic of a conventional TEC element, in which the thicknesses of ceramic plates, metallic connectors, and semiconductor legs are H0, H1, and H2, respectively, the p- and n-type semiconductor legs have the same square cross-section with the area of Asemi = L2 L2, and the two ceramic plates have the same rectangular cross-section with the area of Acer = (2L1 + 2L2) L2. Fig. 1(b) shows the schematic of a TEC element with variable semiconductor cross-sectional areas, in which the cross-sectional areas for both the p-type and n-type semiconductor legs change linearly with the leg thickness, while the other
(a)
Th H0
Ceramic plate
H1
Copper
n-type
H2
I, current outlet
L2 L1/2
z
p-type
L2 L1 Ceramic plate
y
L2
L1/2
Qc
x
Th
(b) H0 H1
I, current inlet
LH
n-type
Ceramic plate
Copper
L1
p-type
H2
I, current inlet
I, current outlet
LC L2
z
y x
Qc
Fig. 1. Schematics of the TEC element: (a) with constant cross-section; (b) with variable cross-section.
503
H. Lv et al. / Applied Energy 164 (2016) 501–508
geometric parameters are the same as those of the conventional TEC element. An area ratio of hot end to cold end is defined as c = Asemi,h/Asemi,c.
Asemi,h
Asemi,h
3. Simulation cases In this paper, the transient supercooling characteristics of the TEC element with variable semiconductor cross-sectional areas are firstly compared with that of the conventional TEC element. As shown in Fig. 2, design 1# is the convectional TEC element with H2 = 1.0 mm and Asemi = 0.5 0.5 mm2 with c = 1, design 2# has Asemi,c = 0.5 0.5 mm2 and Asemi,h = 0.5Asemi,c with c = 0.5, design 3# has Asemi,c = 0.5 0.5 mm2 and Asemi,h = 2Asemi,c with c = 2, design 4# has Asemi,c = 2Asemi,h and Asemi,h = 0.5 0.5 mm2 with c = 0.5, and design 5# has Asemi,c = 0.5Asemi,h and Asemi,h = 0.5 0.5 mm2 with c = 2. Subsequently, the transient supercooling characteristics of the TEC element with various c is studied. In these cases, to ensure a lower cost of semiconductor materials, the total volume of the semiconductor materials used in the TEC element should be smaller than the conventional design. Thus, when the hot end of the TEC element remains the same cross-sectional area as the conventional design, the cold end will be tapered and hence c is larger than 1 (Fig. 3). However, when the cold end remains the same crosssectional area as the conventional design, the hot end will be tapered with c < 1 (Fig. 4). 4. Numerical model
Asemi,c
Asemi,c
(a)
(b)
Fig. 3. Shape of p- or n-type semiconductor leg: (a) conventional design with c = 1; (b) design with c > 1.
Asemi,h
Asemi,h
Asemi,c
The three-dimensional, multiphysics, and transient TEC model includes energy equations and electric potential equations. These two sets of coupled equations need to be solved simultaneously to obtain the temperature and electric potential distributions within the TEC element. The model is described briefly in the following and more details can be found in our previous works [36,38]
@T J2 J rT ¼ r ðki rTÞ þ bi~ @t ri r ðri ðr/ ai rTÞÞ ¼ 0 ðqcp Þi
ð1Þ ð2Þ
!
E ¼ r/ þ ai rT
ð3Þ
!
~ J ¼ ri E
ð4Þ
where T is the temperature, t is the time, ~ J is the current density vector, q, cp, k, r, and b are the density, specific heat, thermal conductivity, electric conductivity, and Thomson coefficient, respectively, and / is the electric potential. The subscript i respectively denotes the
(a)
(b)
Asemi,c
Fig. 4. Shape of p- or n-type semiconductor leg: (a) conventional design with c = 1; (b) design with c < 1.
ceramic plate, metal connector, p-type semiconductor, or n-type semiconductor. The Thomson coefficient can be related to the Seebeck coefficient a by b = T(da/dT). For the ceramic plates, a and r are both assumed to be zero because of their ignorable thermoelectric effect and electrical insulation. It is worth noting that the present TEC model solves the electric potential equations and hence the current density vectors through the semiconductor legs and metallic connectors can be calculated by Eq. (4). Thus, the model can be used to investigate the effect of variable semiconductor cross-sectional area on the TEC transient supercooling characteristics. The TEC initially operates at steady state with an input current equal to the optimal steady-state value which can be determined by the steady state I–Tc curve of the TEC element. Subsequently, a current pulse with the amplitude two to five times higher than the optimal steady- state value is applied to the TEC at t = 0.05 s.
Asemi,h
Asemi,h
Asemi,c
Asemi,c
(a)
(b)
(c)
(d)
(e)
Fig. 2. Shape of p- or n-type semiconductor leg: (a) design 1#: conventional TEC with c = 1; (b)–(e) designs 2#–5# with variable semiconductor cross-sectional area.
H. Lv et al. / Applied Energy 164 (2016) 501–508
-30
P=2.5
(a)
τ =4.0 s
-40
Tc /°C
Hence, the initial conditions of temperatures and electric potentials for the TEC element can be determined by solving the steady-state model. An isothermal boundary condition (Th = 293 K) is applied to the hot end of the TEC element and a zero cooling load condition (Qc = 0) is specified to the cold end, while adiabatic boundary conditions are applied to the other outside surfaces of the TEC element. At the internal interfaces between adjacent materials, the temperature and heat flux are assumed to be continuous. The electric boundary conditions are as follows: a time-dependent current is specified to the current inlet of the TEC element, while a zero electric potential is applied to the current outlet. The n-type and p-type semiconductor legs are assumed to be Bi2(Te0.94Se0.06)3 and (Bi0.25Sb0.75)2Te3, respectively. The copper and Al2O3 ceramic are selected as the metallic connector and ceramic plate. The temperature-dependent properties for these materials can be found in our previous studies [14,36,38]. The coupled set of equations in association with the boundary conditions was solved iteratively using multiphysics package COMSOL 3.5a. Iteration criterion for convergence was set as 106. The independences of space and time grid sizes are examined for each simulation case to ensure the accuracy of simulated results.
-50 -60 -70 -80
Experimental data in Ref. [18] Numerical predictions
0
5
10
450
The experimental curve of Tc–t tested by Snyder et al. [18] is used to validate the present model. They used n-type Bi2Te2.85 Se0.15 and p-type Bi0.4Sb1.6Te3 to fabricate 5.8 mm tall thermoelectric elements with 1 mm2 cross-sectional areas. The cold end was soldered to a 35 lm thick copper foil to which was soldered a 25 lm diameter Chromel-Constantin thermocouple for measurement of the temperature. The hot end was soldered to an electrically isolated heat sink where the heat sink temperature could be adjusted from 55 to 35 °C. The optimal steady-state current was Iopt = 0.675 A. A step pulse with amplitude of P = I/Iopt = 2.5 and width of s = 4.0 s was applied to the thermoelectric elements. It should be noted that temperature-dependent properties for the semiconductor materials used in their experiments were not given in Ref. [18], and hence our simulation has to use constant properties. The properties of semiconductor legs reported in Ref. [18] were as follows: qcp = 1.20 106 J m3 K1, k = 1.20 W m1 K1, a = 2.35 104 V K1, and r = 5.11 104 X1 m1. For making a fair comparison, TEC dimension, material properties, and operating conditions used in our simulation are all the same as those in the experiment. As shown in Fig. 5(a), the simulation results agree well with the experimental data at the beginning stage but the deviation is observed for t > 5.5 s where the pulse has been terminated. The deviation can be attributed to the constant material properties used in the simulation. Generally, electric conductivity of Bismuth Telluride-based materials is negatively proportional to temperature. Ref. [14] has reported that, as compared with variable material properties, the TEC model with constant material properties predicts a lower temperature distribution across the semiconductor leg. Thus, the resultant larger electric conductivity leads to a weaker Joule heating. At the initial stage of supercooling, the temperature at the cold end is determined by Peltier effect rather than Joule heating due to finite thermal diffusion rate. Once the cold end reaches its lowest temperature, the Joule heat transferred back to the cold end becomes stronger than the Peltier cooling. Thus, the temperature begins to increase and develop into overshoot stage. Consequently, the weaker Joule heating predicted by the model with constant material properties leads to the underestimated temperature overshoot. Above analysis can be further justified in Fig. 5(b), which compares the dynamic responses of the cold end
20
25
τ =2.0 s P =5 qc=19685 W m-2
(b)
425 400
h=2000 W K-1m-2
375 350 325 300 275
5. Model validation
15 t /s
Tc /K
504
250
Model with variable properties Model with constant properties
0
1
2
3
t /s
4
5
6
Fig. 5. Model validation: (a) comparison of numerical predictions and experimental data in Ref. [18]; (b) comparison of dynamic responses of the cold end temperature predicted by models with constant and variable properties.
temperature predicted by the multiphysics models with constant and variable properties. The lowest cold end temperatures predicted by the both models are almost the same; however, the model with constant properties underestimates the temperature overshoot and recovery time, but overestimates the duration of the supercooling state. Therefore, it can be expected that if temperature-dependent material properties are used, our model can achieve an acceptable matching with the measured Tc–t curve. Thus, our model can be used to investigate the transient supercooling characteristics of TEC element.
6. Determination of the optimal steady-state current In order to study the transient supercooling characteristics, the optimal steady-state current, Iopt, needs to be determined first. Fig. 6 shows the I–Tc curves of the TEC elements with constant and variable semiconductor cross-sectional areas. The minimum cold end temperatures, Tc,min, for the five designs are 204.62, 204.79, 204.21, 204.84, and 204.17 K, respectively, indicating that Tc,min is almost independent of the TEC shape. This phenomenon is also observed for the TEC elements with various c (shown in Figs. 3 and 4), their Tc,min ranges from 203.10 to 204.93 K. However, the TEC elements with various semiconductor cross-sectional areas exhibit different Iopt from the conventional TEC element. Fig. 6 also shows that Iopt depends on the volume of semiconductor materials, designs 3# and 4# have larger semiconductor volume than the conventional design, their Iopt is higher, on the contrary, designs 2# and 5# have smaller semiconductor volume than the conventional design, so that their Iopt is lower. This result agrees well with the experiments conducted by Hoyos et al. [37] and is confirmed again in Fig. 7, where the TEC element with c – 1 has a smaller semiconductor volume so that its Iopt is lower than Iopt, conv of the
505
H. Lv et al. / Applied Energy 164 (2016) 501–508
Design 1#
285
Design 2#
Design 3#
Design 4#
Design 5#
T h=293K, Q c =0 W
270
Tc /K
255 240 225 210 195
design 1# (conventional design), design 3#, design 4#,
180
0.4
0.8
1.2
1.6
2.0
2.4
design 2#, design 5# 2.8
3.2
3.6
4.0
I /A Fig. 6. I–Tc curves for the TEC elements with different semiconductor shapes.
1.8 1.6
Iopt /A
1.4 1.2 1.0 0.8 0.6 0.4
0
1
2
3
4
5
6
7
8
γ =Asemi,h /Asemi,c Fig. 7. The optimal steady-state currents of TEC elements with various c.
conventional design. Furthermore, Figs. 6 and 7 also indicate when two TEC elements have the same semiconductor volume, their Iopt will be identical. 7. Transient supercooling characteristics 7.1. Comparison between various semiconductor shapes The transient supercooling characteristics of the TEC elements with constant and variable semiconductor cross-sectional areas are shown in Fig. 8. The step pulse with pulse amplitude of P = 5 and pulse width of s = 0.05 s is used here. As shown in Fig. 8, the changes of all Tc–t curves exhibit the similar trend: Tc fist keeps
its steady-state value of Tc,s at t < 0.005 s; it starts to decline after the step current is applied to the TEC element; when Tc reaches Tc,min it starts to rise; after maximum temperature overshoot occurs, Tc starts to decrease again and finally returns to Tc,s, ready for the next pulse. It should be noted that although designs 2# and 4# have different semiconductor shapes, they exhibit almost the same transient supercooling characteristics because the both have the same c = 0.5. Similarly, shapes 3# and 5# have the same c = 2, they also exhibit almost the same Tc–t curves. A better transient response should be one with higher amount of supercooling, weaker temperature overshoot, longer holding time of supercooling state, and shorter recovery time. Thus, these four key evaluation indicators for designs 1#–5# are listed in Table 1. It can be seen that a larger Asemi,c or c < 1 (designs 2# and 4#) ensures a larger DTc,max1, a smaller DTc,max2, and a longer Dthold, but it also leads to a longer Dtrec, as compared with the conventional design. Oppositely, a smaller Asemi,c or c > 1 (designs 3# and 5#) results in a shorter Dtrec, however, it also induces a worse DTc,max1, DTc,max2, and a shorter Dthold, as compared with the conventional design. Hoyos et al. [37] and Yang et al. [16] presented that the tapered (c < 1) or diverging (c > 1) semiconductor shape causes the thermal circuit asymmetric and Joule heat will preferentially be conducted toward to the end with the larger cross-sectional area (referred to as the heat conduction effect here). However, more Joule heat is also produced close to the end with the smaller cross-sectional area (referred to as the Joule heat effect here). This can be explained by the following equation:
dQ Joule ¼
ðI=AÞ2
r
!
Adh ¼
I2 dh rA
ð5Þ
H. Lv et al. / Applied Energy 164 (2016) 501–508
Design 1#
Design 2#
360
design 1# design 2# design 3# design 4# design 5#
340 320
Tc /K
300
Design 3#
Design 4#
Design 5#
208
204
Tc /K
506
200
196
280
192
260
0.01
0.02
0.03
0.04
0.05
0.06
t /s
240
T h =293 K
220
Q c =0 W
200
P=5 τ =0.05 s
180 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
t /s Fig. 8. Transient supercooling characteristics of TEC elements with five different shapes.
Table 1 The evaluation parameters of transient supercooling characteristics for TEC elements with designs 1#–5#. Parameter
Design 1#
Design 2#
Design 3#
Design 4#
Design 5#
DTc,max1 = Tc,s Tc,min/K DTc,max2 = Tc,max Tc,s/K Dthold/s Dtrec/s
8.86 98.95 0.049 4.133
9.40 80.56 0.058 5.448
7.46 147.87 0.033 3.466
9.79 81.96 0.057 5.225
6.96 146.01 0.033 3.513
where dQJoule denotes the Joule heat generated in a thin layer with a cross-sectional area of A and a thickness of dh. For a TEC element with constant applied current I, a smaller A corresponds to a larger dQJoule. Thus, Tc,min should be determined by the competition of these two effects. The present simulations show that a lower Tc,min can be achieved by the design with a larger Asemi,c, indicating that the Joule heat effect is dominant over the heat conduction effect. Based on the same reason, the larger Asemi,c weakens the temperature overshoot and lengthens the holding time. It is worth noting that the present simulations use a short pulse width of s = 0.05 s. After the step pulse is terminated, the applied current returns to Iopt, thus, the Joule heat effect becomes weaker and the heat conduction effect starts to dominate. As a result, the TEC with a larger Asemi,c requires a longer time to return to the steady state. Designs 3# and 4# use more semiconductor materials than the conventional design and designs 2# and 5#, however, their transient supercooling characteristics are found to be almost the same as that of design 5# and 2#, respectively, thus, these two designs are not recommended. Fig. 9 shows the four key evaluation indicators of transient supercooling characteristics as a function of pulse amplitude for designs 2# and 5#. As well known, for the conventional design with constant semiconductor cross-sectional area, the increase in
P means that more electric energy is supplied to the TEC element and Peltier cooling becomes stronger during the pulse period, so that a lower minimum cold end temperature, a stronger temperature overshoot, a shorter holding time of supercooling state, and a longer recovery time will be observed [16,36]. However, these results are not always followed by the TEC element with variable semiconductor cross-sectional area. As shown in Fig. 9, the temperature overshoot, holding time, and recovery time follow the same trend as the conventional design, but the minimum cold end temperature for design 5# with a smaller Asemi,c did not. This is because the variable semiconductor cross-sectional area causes two additional effects: the heat conduction effect and Joule heat effect as mentioned before. It is concluded that for design 5#, the heat conduction effect is dominant at smaller amplitude of P < 3, so that the increase in P decreases Tc,min, however, the Joule heat effect becomes dominant at larger amplitude of P > 3, which leads to Tc,min increased with P. Fig. 9 also shows that design 2# with a larger Asemi,c always provides a smaller minimum cold end temperature, a weaker temperature overshoot, a longer holding time, and a longer recovery time than those of design 5# with a smaller Asemi,c when P P 3. This indicates that for the both design, the Joule heat effect always dominates over the heat conduction effect during the pulse period,
507
H. Lv et al. / Applied Energy 164 (2016) 501–508
ΔTc,max1 /K
8
ΔTc,max2 /K
600 400 200 0
Δt hold /s
10
0.12 0.09 0.06 0.03 0.00 8
6
Th=293 K, Qc=0 W, τ =0.05 s
Δtrec /s
4
γ =1/2
γ =2
design 2#, γ =1/2 design 5#, γ =2
6 4 2
2
3
4 P
5
6
Fig. 9. Key evaluation parameters of transient supercooling characteristics as the function of pulse amplitude: (a) design 2# with c = 0.5; (b) design 5# with c = 2.
hence, more heat is transferred to the end that has the smaller cross-sectional area. However, this phenomenon is reversed after the pulse is terminated, and more heat is transferred to the end that has the larger cross-sectional area, as a result, design 2# requires a longer time to return to Tc,s ready for next steady-state. 7.2. Effect of cross-sectional area ratio of hot end to cold end
Δtrec /s
Δthold /s
ΔTc,max2 /K
ΔTc,max1 /K
In this section, the transient supercooling characteristics for the design with various c is discussed. The cross-sectional area ratio of hot end to cold end ranges from 1/7 to 7. Restated that c = 1 represents the conventional design with constant semiconductor crosssectional area. Four key evaluation indicators for various c are shown in Fig. 10. It is found that when the cross-sectional area at the cold end is smaller than that at the hot end (c > 1), use of a
10 8 6 4 2 900
smaller Asemi,c cannot achieve a higher amount of supercooling, meanwhile, it also leads to a stronger temperature overshoot and a shorter holding time of supercooling state. It should be noted that the present predictions agree well with Hoyos et al.’s experimental curves [37]. Thus, when a priority is given to these three evaluation parameters for a specific application, a smaller Asemi,c is not recommended. However, the advantage of smaller Asemi,c lies in its shorter recovery time, which is potentially useful for the device to be operated for a longer time. As shown in Fig. 10, when the cross-sectional area at the cold end is larger than that at the hot end (c < 1), there exists an optimal value of c = 1/3, at which the lowest Tc,min and the longest holding time can be achieved. The comparison between designs with c > 1 and c < 1 shows that when a lower minimum cold end temperature, a weaker temperature overshoot, and/or a longer holding
Th=293 K, Qc=0 W, τ =0.05 s, P=5
600 300 0 0.06 0.04 0.02 0.00 12 10 8 6 4 2
0
1
γ
10
Fig. 10. Key evaluation parameters of transient supercooling characteristics as the function of c.
508
H. Lv et al. / Applied Energy 164 (2016) 501–508
time are/is required, the design with a larger cross-sectional area at the cold end should be adopted; on the contrary, when a shorter recovery time needs to be preferentially considered, the design with a smaller cross-sectional area at the cold end is recommended. 8. Conclusions In this study, transient supercooling of the TEC with variable semiconductor cross-sectional area is investigated by a threedimensional, transient, and multiphysics model. The transient supercooling characteristics can be evaluated by the minimum cold end temperature, maximum temperature overshoot, and several time constants, such as the holding time of supercooling state and the recovery time ready for next steady-state. These evaluation indicators for the proposed designs with variable semiconductor crosssectional area are compared with the conventional TEC design. Compared with the conventional design, the design with variable semiconductor cross-sectional area produces two additional effects. Firstly, the variable cross-sectional area makes the thermal resistance asymmetric, and hence Joule heat is preferentially conducted toward to the end with a larger cross-sectional area, referred to as the heat conduction effect. Secondly, more Joule heat is produced close to the end with a smaller cross-sectional area, named as the Joule heat effect. The strength of these two effects depends on the cross-sectional area ratio of hot end to cold end (c = Asemi,h/Asemi,c), so that desired evaluation indicators can be achieved by changing the value of c. The present simulations show that the Joule heat effect always dominates over the heat conduction effect during the pulse period; hence, more heat is transferred to the end that has smaller crosssectional area. However, this phenomenon is reversed after the pulse is terminated, and more heat is transferred to the end with larger cross-sectional area. As a result, when a lower minimum cold end temperature, a weaker temperature overshoot, and/or a longer holding time are/is required, the design with a larger cross-sectional area at the cold end should be adopted; on the contrary, when a shorter recovery needs to be preferentially considered, the design with a smaller cross-sectional area at the cold end is recommended. Quantitatively, the maximum cold end temperature drop of DTc,max1 = 9.47 K, the weakest temperature overshoot of DTc,max2 = 78.13 K, and the longest holding time of Dthold = 63 ms are achieved at c = 1/3. Acknowledgments This study was partially supported by the National Natural Science Foundation of China (No. 51276060), the 111 Project (No. B12034), and the Fundamental Research Funds for the Central Universities (No. 13ZX13). References [1] Minnich AJ, Dresselhaus MS, Ren ZF, Chen G. Bulk nanostructured thermoelectric materials: current research and future prospects. Energy Environ Sci 2009;2:466–79. [2] Liu D, Zhao FY, Yang HX, Tang GF. Theoretical and experimental investigations of thermoelectric heating system with multiple ventilation channels. Appl Energy 2015;159:458–68. [3] Erturun U, Erermis K, Mossi K. Influence of leg sizing and spacing on power generation and thermal stresses of thermoelectric devices. Appl Energy 2015;159:19–27. [4] Chen WH, Huang SR, Lin YL. Performance analysis and optimum operation of a thermoelectric generator by Taguchi method. Appl Energy 2015;158:44–54. [5] Madan D, Wang ZQ, Wright PK, Evans JW. Printed flexible thermoelectric generators for use on low levels of waste heat. Appl Energy 2015;156:587–92. [6] Heghmanns A, Beitelschmidt M. Parameter optimization of thermoelectric modules using a genetic algorithm. Appl Energy 2015;155:447–54.
[7] Wang TH, Wang QH, Leng C, Wang XD. Parameter analysis and optimal design for two-stage thermoelectric cooler. Appl Energy 2015;154:1–12. [8] He W, Zhang G, Zhang XX, Ji J, Li GQ, Zhao XD. Recent development and application of thermoelectric generator and cooler. Appl Energy 2015;143: 1–25. [9] Massaguer E, Massaguer A, Montoro L, Gonzalez JR. Modeling analysis of longitudinal thermoelectric energy harvester in low temperature waste heat recovery applications. Appl Energy 2015;140:184–95. [10] Stevens RJ, Weinstein SJ, Koppula KS. Theoretical limits of thermoelectric power generation from exhaust gases. Appl Energy 2014;133:80–8. [11] Liang XY, Sun XX, Tian H, Shu GQ, Wang YS, Wang X. Comparison and parameter optimization of a two-stage thermoelectric generator using high temperature exhaust of internal combustion engine. Appl Energy 2014;130: 190–9. [12] Riffat SB, Ma X. Thermoelectrics: a review of present and potential applications. Appl Therm Eng 2003;23:913–35. [13] DiSalvo FJ. Thermoelectric cooling and power generation. Science 1999;285: 703–6. [14] Wang XD, Huang YX, Cheng CH, Lin DTW, Kang CH. A three-dimensional numerical modeling of thermoelectric device with consideration of coupling of temperature field and electric potential field. Energy 2012;47:488–97. [15] Chen WH, Liao CY, Huang CI. A numerical study on the performance of miniature thermoelectric cooler affected by Thomson effect. Appl Energy 2012;89:464–73. [16] Yang RG, Chen G, Kumar AR, Snyder GJ, Fleurial JP. Transient cooling of thermoelectric coolers and its applications for microdevices. Energy Convers Manage 2005;46:1407–21. [17] Zhou Q, Bian Z, Shakouri A. Pulsed cooling of inhomogeneous thermoelectric materials. J Phys D Appl Phys 2007;40:4376–81. [18] Snyder GJ, Fleurial JP, Caillat T, Yang RG, Chen G. Supercooling of Peltier cooler using a current pulse. J Appl Phys 2002;92:1564–9. [19] Buist RJ, Lau PG. Theoretical analysis of thermoelectric cooling performance enhancement via thermal and electrical pulsing. In: 15th International conference on thermoelectrics; 1996. p. 234–7. [20] Yang RG, Chen G, Snyder GJ, Fleuriel JP. Geometric effects on the transient cooling of thermoelectric coolers. Mater Res Soc Symp Proc 2002;691:281–6. [21] Chakraborty A, Ng KC. Thermodynamic formulation of temperature–entropy diagram for the transient operation of a pulsed thermoelectric cooler. Int J Heat Mass Transf 2006;49:1845–50. [22] Ju YS. Impact of interface resistance on pulsed thermoelectric cooling. J Heat Transfer 2008;130:014502. [23] Gupta MP, Sayer MS, Mukhopadhyay S, Kumar S. On-chip Peltier cooling using current pulse. In: 12th Intersociety conference on thermal and thermomechanical phenomena in electronic systems; 2010. [24] Gupta MP, Sayer MH, Mukhopadhyay S, Kumar S. Ultrathin thermoelectric devices for on-chip Peltier cooling. Trans Comp Packag Manufact Technol 2011;1:1395–405. [25] Shen LM, Xiao F, Chen HX, Wang SW. Numerical and experimental analysis of transient supercooling effect of voltage pulse on thermoelectric element. Int J Refrig 2012;35:1156–65. [26] Manno M, Wang P, Bar-Cohen A. Transient thermoelectric self-cooling of a germanium hotspot. In: 13th IEEE ITHERM conference; 2012. p. 413–20. [27] Shen LM, Chen HX, Xiao F, Yang YX, Wang SW. The step-change cooling performance of miniature thermoelectric module for pulse laser. Energy Convers Manage 2014;80:39–45. [28] Ma M, Yu JL. A numerical study on the temperature overshoot characteristic of a realistic thermoelectric module under a current pulse operation. Int J Heat Mass Transf 2014;72:234–41. [29] Sullivan O, Gupta MP, Mukhopadhyay S, Kumar S. Thermoelectric coolers for thermal gradient management on chip, vol. 4. In: Proceedings of the ASME 2010 international mechanical engineering congress and exposition; 2010. p. 199–207. [30] Redmond M, Manickaraj K, Sullivan O, Mukhopadhyay S, Kumar S. Hotspot cooling in stacked chips using thermoelectric coolers. Trans Comp Packag Manufact Technol 2013;3:759–67. [31] Thonhauser T, Mahan GD, Zikatanov L, Roe J. Improved supercooling in transient thermoelectrics. Appl Phys Lett 2004;85:3247–9. [32] Mitrani D, Salazar J, Turo A, García MJ, Chávez JA. Transient distributed parameter electrical analogous model of TE devices. Microelectron J 2009;40:1406–10. [33] Mao JN, Chen HX, Jia H, Qian XL. The transient behavior of Peltier junctions pulsed with supercooling. J Appl Phys 2012;112:014514. [34] Sullivan O, Gupta MP, Mukhopadhyay S, Kumar S. Array of thermoelectric coolers for on-chip thermal management. J Electron Packag 2012;134:021005. [35] Manno M, Wang P, Bar-Cohen A. Pulsed thermoelectric cooling for improved suppression of a germanium hotspot. Trans Comp Packag Manufact Technol 2013;4:602–11. [36] Lv H, Wang XD, Wang TH, Meng JH. Optimal pulse current shape for transient supercooling of thermoelectric cooler. Energy 2015;83:788–96. [37] Hoyos GE, Rao KR, Jerger D. Fast transient response of novel Peltier junctions. Energy Convers 1977;17:23–9. [38] Meng JH, Wang XD, Zhang XX. Transient modeling and dynamic characteristics of thermoelectric cooler. Appl Energy 2013;108:340–8.