Inhomogeneous Thermal Conductivity Enhances Thermoelectric Cooling. Engineering, Tongji University, Shanghai , People's Republic of China

Transcription

1 Inhomogeneous Thermal Condutivity Enhanes Thermoeletri Cooling Tingyu Lu, Jun Zhou,, * Nianbei Li, Ronggui Yang, and Baowen Li,3,4 Center for Phononis and Thermal Energy Siene, Shool of Physis Siene and Engineering, Tongji University, Shanghai 9, People's Republi of China Department of Mehanial Engineering, University of Colorado, Boulder, Colorado 839, USA 3 Department of Physis, Center for Computational Siene and Engineering, and Graphene Researh Center, National University of Singapore, Singapore 7546, Republi of Singapore 4 NUS Graduate Shool for Integrative Sienes and Engineering, National University of Singapore, Singapore 7456, Republi of Singapore Abstrat We theoretially investigate the enhanement of thermoeletri ooling performane in thermoeletri devies made of materials with inhomogeneous thermal ondutivity, beyond the usual pratie of enhaning thermoeletri figure of merit ZT. The dissipation of Joule heat in suh thermoeletri devies is asymmetri whih an give rise to better thermoeletri ooling performane. Although the thermoeletri figure of merit and the oeffiient-of-performane are only slightly enhaned, both the imum ooling power and the imum ooling temperature differene an be enhaned signifiantly. This finding an be used to inrease the heat absorption at the * To whom orrespondene should be addressed. zhoujunzhou@tongji.edu.n

2 old end. The asymmetri dissipation of Joule heat also leads to thermal retifiation. KEYWORDS: inhomogeneous thermal ondutivity, thermoeletri ooling, ooling power, figure of merit, thermal retifiation PACS numbers: 84.6.Rb, 7..Pa, 85.8.Fi

3 There has been great interests in thermoeletri (TE devies that an diretly onvert eletriity into thermal energy for ooling or heating and an harvest solar and waste heat into eletri power [,]. The energy onversion effiieny of TE devies is determined by the figure of merit of TE materials [3,4] ZT = α T /( ρλ, where α is the Seebek oeffiient, T is the absolute temperature, ρ is the eletrial resistivity, and λ is the thermal ondutivity whih onsists of eletroni thermal ondutivity and lattie thermal ondutivity. High ZT materials are desirable for high effiieny TE devies. Even though TE devies have many advantages suh as reliability and salability, the ommerial available materials with ZT~ limits widespread appliations of thermoeletris. Great efforts in enhaning ZT have been made in past deades [5,6,7]. The performane of a TE ooler is evaluated with these three parameters: i. the imum ooling power ( q that desribes the imum rate at whih heat an be absorbed from the old end, ii the imum ooling temperature differene ( T whih an be reahed when the imum ooling power falls to zero, ( q =; and iii the imum oeffiient-of-performane (COP φ whih is the energy onversion effiieny. There have been many efforts in enhaning the performane of TE oolers through high ZT materials, system engineering [], and even transient ooling [8,9,]. In this work, we study the performane of TE devies made of materials with inhomogeneous thermal ondutivity. Assuming p- and n-type legs have same material properties, we only need to onsider a p-type branh with length L and ross setion area A as shown in Fig. (a 3

4 to evaluate devie performane []. The devie is operated with the temperature of T and T at the old and hot end, respetively. When an eletri urrent I flows aross the devie along x-diretion, heat an be absorbed at rate q ab = αit at the old end due to Peltier effet. As shown in Fig. (a, the absorbed heat an be partially anelled by the heat leakage due to the temperature differene between the hot and old ends q and the flow of a portion of Joule heat ( q Joule = I R generated inside DT the devie where R is the eletrial resistane. The net ooling power an then be expressed as q = q q γq, ( ab DT Joule where γ is defined as inhomogeneity fator of asymmetri Joule heat dissipation. It is rather straightforward that to enhane the devie performane ( q, ( T and φ [,], one needs to either enhane the Seebek oeffiient α or suppress q DT and q Joule. It is interesting to note that most past studies assume, by default, symmetri flow of the Joule heat to the old and hot ends, namely, γ = / in Eq. ( [,]. However, this assumption is valid only when all the transport oeffiients are not spatial-dependent. The fator γ an be very different from / in inhomogeneous materials, whih indeed gives rise to a great design freedom to improve the TE ooling performane. Indeed, the devies made of funtional graded TE materials (FGTM with inhomogeneous transport properties was first proposed by Ioffe [3] in 96 and then be widely studied by many researhers to enhane the devie performane [4,5,6,7,8,9]. For example, Bian et al. [] found that an 4

5 enhanement of ( T an be ahieved in FGTM with spatial-dependent Seebek oeffiient. In this Letter, we investigate the performane of TE devies made of inhomogeneous materials with varied transport oeffiients. By assuming spatial- and temperature-dependent eletrial resistivity ρ ( x, T, Seebek oeffiient α ( x, T, and thermal ondutivity λ ( x, T, the following equation will be solved to analyze the devie performane: d dt ( x I ρ( x, T I dα( x, T [ λ ( x, T ] = + T ( x dx dx A A dx, ( where x is the distane from the old end. The boundary onditions are hosen as T ( x = T = and T ( x L = T =. In Eq. (, the left term is the divergene of the Fourier heat urrent, while the first term on the right is the Joule heat generated by an eletri urrent I flowing through the devie, and the seond term is the Thomson heating or ooling due to the temperature- and spatial-dependent Seebek oeffiient. The temperature profile T (x an be solved with given λ ( x, T, ρ ( x, T, and α ( x, T. The ooling power an be then obtained as q = (, T IT λ(, T A[ dt ( x / dx] x= α with the temperature profile. 5

6 FIG. (olor online. (a Shemati diagram of a TE element with length L and ross-setional area A. The temperature at the old and hot end are kept at T and T, respetively. The ooling power q is the Peltier heat absorbed q ab = αit subtrated by onduted heat due to temperature different between the hot and old end q and a fration of Joule heat generated inside the TE element γ q Joule = γi R, DT where α is the Seebek oeffiient, R is the eletrial resistane, and γ is the inhomogeneity fator. (b Fration of the Joule heat flow to the old end (γ and that flow to the hot end ( γ as a funtion of parameter when the inhomogeneous thermal ondutivity is λ ( x = λ( + x / L. Without losing generality, we study here the enhanement on the ooling performane of TE devies by utilizing the inhomogeneous materials with spatial- and temperature-dependent lattie thermal ondutivities. For simpliity, the eletrial resistivity and the Seebek oeffiient are set as onstant values. Our model an be extended to the ase with varied Seebek oeffiients and eletrial resistivities. In order to unravel the underlying enhanement mehanism for the ooling performane, the spatial dependene of thermal ondutivities and temperature-dependene of thermal ondutivities are treated separately. The dependene of λ (T on 6

7 temperature is a well-known material property. The temperature-dependent λ (T an indue an intrinsi spatial-dependent λ ( T ( x sine the temperature T (x is spatial-dependent. Materials with expliit spatial-dependent λ (x, i.e. through mass gradient and other mehanisms [,], has reently been developed to realize thermal retifiation effet or thermal diode [3,4,5]. Table I lists several ommon analytial expressions of spatial- and temperature-dependent thermal ondutivities onsidered in this work. The first example is the inhomogeneous materials with linear spatial-dependent thermal ondutivity of λ ( x = λ( + x / L. Here λ is the referene thermal ondutivity at the old end and the slope denotes the strength of the spatial dependene or inhomogeneity. The ooling power for this kind of material an then be derived as: = αit βk T I R, (3 q γ where T = T T, K = A / L, and R = Lρ / A. Here we have introdued two λ new parameters of β and γ. The β = / ln( + is the normalized heat onduted by assuming a homogeneous material with λ, i.e. q DT / K T. When >, we have β whih means that more heat would be onduted from the hot end to the old end than q DT = K T due to a muh larger effetive thermal ondutivity. The inhomogeneity fator γ = / ln( + / whih denotes the distribution of Joule heat is no longer /. This inhomogeneity fator γ an now be tuned by the strength of spatial-dependent thermal ondutivity. In the limit of homogeneous ase with =, the familiar result of γ = / an be reovered as expeted. Figure (b shows the modulation of γ and γ as a funtion of 7

8 parameter. In general, the Joule heat will not flow to the old and hot ends symmetrially. The Joule heat flow to the old end γ dereases monotonially with inreasing. For example, when = whih means the thermal ondutivity varies from λ ( = λ to λ ( L = 4.45λ, γ is about.3 whih is muh less than γ =.7. The Joule heat prefers to flow along the diretion with inreasing thermal ondutivities. The disovery of this novel phenomenon enables us to manipulate the Joule heat flow to enhane the ooling performane of TE devies using inhomogeneous thermal ondutivities. The introdution of inhomogeneity modifies the expression of the imum ooling power ( q whih now beomes, ( q = B βk T, (4 γ when the imum eletri urrent = αt / γr is reahed, where B = ( αt /(R. In omparison with the homogeneous thermal ondutivity ase, when I m λ( x = aλ with a as an arbitrary oeffiient, there is one more fator / γ in I m and in the first term on the right side of Eq. (4, shown in Table I. The imum ooling temperature differene ( T is obtained by setting ( q = B ( T =. (5 γβ K It is obvious that both ( q and ( T an be enhaned when / γ >. In order to alulate the COP written as φ = q /( I T I α + R whih is the ratio between ooling power and total input power, we now redefine an effetive figure of 8

9 merit as ZTM = ZT M, (6 β where Z = α /( KR, and T M = ( γ T + γt is the mean temperature weighted by the inhomogeneity fator γ. Using suh an effetive figure of merit and weighted mean temperature and setting dφ / di =, the imum COP is obtained as φ T[( + ZT = ( T / β T [( + Z T M + T M / β M / T ] + ]. (7 In the limit of homogeneous ase when =, i. e. λ ( x = λ, the familiar results of β =, T M = ( T + T /, and the onventional expression of φ with homogeneous thermal ondutivity are reovered []. We perform the numerial alulations based on the above mentioned model for a TE element with L=5 mm and A=4mm. The typial material properties of p-type BiSbTe alloy [ 6 ] have been adopted as follows: the Seebek oeffiient α = μv/k, the eletrial resistivity ρ = 5 Ωm, and λ =.7W/(m K. The temperature at the hot end is fixed to be T = 3 K for all alulations. The temperature at the old end is hosen to be T = 9 K for the alulations of ( q and φ. In the alulation of ( T, T is obtained by solving Eq. (5 self-onsistently. 9

10 FIG. (olor online Enhanement of ( q, ( T, and φ by inhomogeneous materials with linear spatial-dependent thermal ondutivities λ ( x = λ( + x / L. (a q as a funtion of eletri urrent I for different γ. (b, ( and (d, ( q, ( T, and φ as a funtion of parameter (blue solid urve, respetively. For omparison, ( q, ( T, and φ with homogeneous thermal ondutivities, λ = βλ (red dashed urve and λ = λ ( + / (blak dotted urve, are also plotted. Relative ratio between ( T with inhomogeneous thermal ondutivity and ( T with homogeneous thermal ondutivity λ = βλ is plotted in (. Figure (a shows the ooling power q as a funtion of the eletri urrent I with linear spatial-dependent thermal ondutivities λ ( x = λ( + x / L when =,.85, and.4. The orresponding inhomogeneity fators are γ =.5 for =,

11 γ =.45 for =.85, and γ =.4 for =, respetively. The imum eletri urrent I m shifts from 5.A to 6.4A when γ dereases from.5 to.4 sine I / γ as m shown in Table I. In the mean time, the imum ooling power ( q inreases from.49w to.77w. When inreases, both the imum eletri urrent I m and normalized onduted heat β are inreased. When the inrease of α I m T / overomes the inrease of β K T, the overall effet is the enhanement of ( q, by realling Eq. (4. Figures (b-(d plot the imum ooling power ( q, the imum ooling temperature differene ( T, and the imum COP φ as a funtion of with inhomogeneous thermal ondutivity λ ( x = λ( + x / L, respetively. For omparison, ( q, ( T, and φ with homogeneous thermal ondutivities, λ = βλ and λ = λ ( + /, are also presented. These two ases of homogeneous thermal ondutivities are hosen for omparison beause: i both thermal ondutivities λ = βλ and λ ( x = λ( + x / L result in the same normalized onduted heat β ; ii λ = λ ( + / is the mean value of linear spatial-dependent thermal ondutivities over the length of devie, i. e. L λ ( + / = λ ( + x / L dx / L. Figure (b shows that the imum ooling power with inhomogeneous thermal ondutivity in Eq. (4 inreases as inreases. When >, we find that ( q is signifiantly larger than the imum ooling power with homogeneous thermal ondutivities whih are noted as ( q ' = B β K T for λ ( x = βλ and ( q " = B K T ( + / for λ ( x = λ( + /. These three imum

12 ooling powers satisfy the inequality q > q > q sine / γ > ( ( ' ( " and β < ( + /. For example, ( q is.89w whih is larger than ( q ' =.5W and ( q " =.5W when = 5. On the ontrary, negative results in a smaller ( q with inhomogeneous thermal ondutivities than both ( q ' and ( q " with homogeneous thermal ondutivities sine / γ < when <. Figure ( shows that the imum ooling temperature differene ( T with inhomogeneous thermal ondutivity in Eq. (5 dereases with inreasing. The reason is that the fator βγ inreases with inreasing. When >, we find that ( T is larger than that with homogeneous thermal ondutivities whih are noted as T ' = B /( βk for λ = βλ and T " = B /[ K ( + / ] for ( λ = λ( + / (. These three imum ooling temperature differenes satisfy the inequality example, ( T T sine / γ > and / β > /( +. For > ( T ' > ( " ( T is 46K whih is larger than ( T ' =36K and ( T " =3K when = 5. On the ontrary, negative results in a smaller ( T with inhomogeneous thermal ondutivity than both ( T ' and ( T " with homogeneous thermal ondutivity sine / γ < when <. Figure (d shows the imum COP with inhomogeneous thermal ondutivity dereases with inreasing. The reason is that φ is proportional to the figure of merit, φ ZT = Z T / β as shown in Eq. (6, whih dereases as inreases. ~ M M When >, we find that φ is larger than that with homogeneous thermal ondutivities whih are noted as φ ' and φ " for λ = βλ and

13 λ = λ( + /, respetively. The effetive figure of merit and mean temperature are Z / β and ( T + / in the alulation of φ ', Z /( + and ( T + / T T in the alulation φ ". The weighted mean temperature T M in the alulation of φ is slightly smaller than ( T + /. For instane, when = 5 whih leads to T γ =.36, T = 93.6K is.4k smaller than ( T + T / 95K. The relative M = differene between them is below.5%. Suh tiny differene makes φ slightly larger than φ '. Moreover, φ is larger than φ " beause that Z β > Z /( + results in a lager Z. On the ontrary, negative results in a / smaller φ with inhomogeneous thermal ondutivities than ' φ and φ " with homogeneous thermal ondutivities sine Z β < Z /( + and T M > ( T + T / when <. / Besides homogeneous thermal ondutivity λ = aλ and linear spatial-dependent thermal ondutivity λ ( x = λ( + x / L ases, we also investigate TE ooling performane of the expliit spatial-dependent thermal ondutivity with power law dependene d λ ( x = λ( x / L ( < d and exponential dependene λ ( x = λ exp[ gx / L] as shown in Table I. One an see that I m, ( T, and Z of power law and exponential spatial-dependent thermal ondutivities have the same expressions as that of linear spatial-dependent thermal ondutivities exept that the expressions of β and γ are hanged. More detailed numerial results are given in the Supplemental Material [7]. TABLE I. Expressions of the normalized onduted heat β, the inhomogeneity 3

14 fator γ, the imum eletri urrent I m, the imum ooling temperature differene ( T, and the effetive figure of merit ZT M for different expliit and intrinsi spatial-dependent thermal ondutivities λ ( x, T. Where β = q DT /( K T, B = ( αt /(R, Z α /( KR ( q = α T I m / qdt and =, T M ( γ T + γt φ inreases with inreasing Z. =. Remember that λ( x, T λ β γ I m ( T αt / R B / K Z Z a a / β β + x / L ln( + ln( + γ γβ β ( L d x / d d d γ γβ β exp ( gx / L g g ( e g e g γ γβ β ( T / T ( / T h T T (lnt lnt T h+ h+ T T h ( h + T T N/A N/A N/A N/A + bt / T T + T + b T β β From Fig. and Table I, the results with expliit spatial-dependent thermal ondutivity an be briefly desribed as follows: i the imum ooling power and the imum ooling temperature differene an be greatly enhaned while the imum COP is only slightly enhaned in TE devie; ii to enhane the ooling 4

15 performane, the thermal ondutivity lose to the old end should be smaller than the thermal ondutivity lose to hot end, whih results in a smaller fration of the Joule heat flow towards the old end, as noted by γ < / when >, d > g > as shown in Table I., and Table I also summarizes the results with temperature-dependent, or intrinsi spatial-dependent, thermal ondutivities with power law temperature-dependene ( T T h T / λ ( = λ ( h = and h are present separately in Table I and linear temperature-dependene ( T = λ ( + bt T λ where T is the room temperature. / The detailed numerial results an be found in Supplemental Material [7]. One important observation is that the intrinsi spatial-dependent thermal ondutivities due to its dependene on temperature do not lead to the asymmetri dissipation of Joule heat. In other words, γ is always equal to /. The Joule heat flowing towards the old end is exatly the same as the ase with homogeneous thermal ondutivity. Therefore the imum eletri urrent I m is the same as that with homogeneous thermal ondutivity. Only the normalized onduted heat β is modified. Furthermore, there is no simple expliit forms of ( T and Z for the ase with ( T T h T / λ ( = λ whih are noted as N/A in Table I. We believe that there is a fundamental differene between the expliit spatial-dependent thermal ondutivities ase and the temperature-dependent thermal ondutivities ase. The physial explanation is that spae inversion symmetry is broken for expliit spatial-dependent thermal ondutivities, but onserved for temperature-dependent thermal ondutivities. If we swap the boundary ondition, 5

16 T T, the heat transport proess and temperature profile after the reversion is exatly the same as that before the reversion. This might also be the reason why there is no thermal retifiation effet for homogeneous materials with temperature-dependent thermal ondutivities. Our earlier researh shows that it is ruial to utilize some kind of symmetry breaking mehanism to realize a thermal diode [4,5]. Sine the inversion symmetry is broken by the spatial-dependent thermal ondutivities, the resulted asymmetri Joule heat flow an also be used for novel design of thermal diodes. In partiular, without onsidering the Peltier effet, i.e. + α, the heat urrent flowing out of the devie hanges from q = β K T + I R γ to q = βk T + ( γ I R if the boundary ondition is swapped ( T T. Therefore, the thermal retifiation fator an be derived as [5]: + q q (γ R f = =, (8 q β / η + ( γ where η = I R /( K T denotes the normalized Joule heat. The retifiation fator R f varies from ~ for the ideal thermal diode [4,8]. 6

17 FIG. 3 (olor online Thermal retifiation fator R f versus parameter with inhomogeneous thermal ondutivity λ ( x = λ( + x / L for different normalized Joule heat η. It is obvious that any deviation from / for the inhomogeneity fator γ will indue a finite thermal retifiation effet for nonzero Joule heat. Figure 3 shows the thermal retifiation fator R f as a funtion of parameter of the linear spatial-dependent thermal ondutivities with λ ( x = λ( + x / L for different normalized Joule heat η. We find that the R is positive when < and negative f when >. Larger leads to an enhanement of R that means stronger retifiation. R f inreases with inreasing normalized Joule heat η sine the f ontribution of Joule heat to total heat urrent is enlarged. To summarize, we have disovered that thermoeletri ooling performane an be signifiantly enhaned through the manipulation of Joule heat flow with expliit spatial-dependent inhomogeneous thermal ondutivity. The flow of Joule heat towards the old end an be suppressed when the thermal ondutivity near the old end is smaller than that near the hot end. We found that the imum ooling power and the imum ooling temperature differene an be signifiantly enhaned while the oeffiient-of-performane is slightly enhaned. The intrinsi spatial-dependent thermal ondutivity due to its temperature dependene annot lead to suh enhanement. Our findings suggest that the materials with inhomogeneous thermal ondutivity used for thermal retifier/diode an be also used to improve the performane of thermoeletri ooling, whih in turn enrihes the appliations of 7

18 thermal retifier [5]. It should be pointed out that materials with inhomogeneous thermal ondutivity an be now be readily ahieved with nanotehnology. For example, the inhomogeneous nanotube [], thin diamond film in whih the inhomogeneity is due to spatially varying disorder assoiated with nuleation and grain oalesene [9], and thermal retifier with pyramid shaped LaCoO 3 /La.7 Sr.3 CoO 3 [3]. We expet that our investigation will inspire many follow-up works in realizing inhomogeneous thermal ondutivity and wide-spread appliations of thermal retifiers. Aknowledgments TL, JZ, NL, and BL are supported by the NSF China, with grant No NL is also supported by Shanghai Rising-Star Program with grant No. 3QA436. JZ is also supported by the program for New Century Exellent Talents in Universities grant no. NCET RY aknowledges the support from NSF and AFOSR from the United States. Referenes [] L. Bell, Siene 3, 457 (8. [] C. Vining, Nat. Mater. 8, 83 (9. [3] D. M. Rowe, in Thermoeletris Handbook: from Maro to Nano, edited by D. M. Rowe (CRC Press, Boa Raton, FL, 6 [4] D. M. Rowe and C. M. Bhandari, Modern Thermoeletris (Reston, Reston, VA,

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