research article exact optimum design of segmented...

Research ArticleExact Optimum Design of Segmented Thermoelectric Generators

M Zare1 H Ramin2 S Naemi1 and R Hosseini1

1Mechanical Engineering Department Amirkabir University of Technology 424 Hafez Avenue PO Box 15875-4413 Tehran Iran2Center of Excellence in Design and Optimization of Energy Systems School of Mechanical EngineeringCollege of Engineering University of Tehran Tehran Iran

Correspondence should be addressed to H Ramin hadiraminutacir

Received 11 December 2015 Revised 10 March 2016 Accepted 20 March 2016

Academic Editor Pouria Ahmadi

Copyright copy 2016 M Zare et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A considerable difference between experimental and theoretical results has been observed in the studies of segmentedthermoelectric generators (STEGs) Because of simplicity the approximate methods are widely used for design and optimizationof the STEGs This study is focused on employment of exact method for design and optimization of STEGs and comparison ofexact and approximate results Thus using new highly efficient thermoelectric materials four STEGs are proposed to operate inthe temperature range of 300 to 1300 kelvins The proposed STEGs are optimally designed to achieve maximum efficiency Designand performance characteristics of the optimized generators including maximum conversion efficiency and length of elementsare calculated through both exact and approximate methods The comparison indicates that the approximate method can cause adifference up to 20 in calculation of some design characteristics despite its appropriate results in efficiency calculationThe resultsalso show that the maximum theoretical efficiency of 2308 is achievable using the new proposed STEGs Compatibility factorof the selected materials for the proposed STEGs is also calculated using both exact and approximate methods The comparisonindicates a negligible difference in calculation of compatibility factor despite the considerable difference in calculation of reducedefficiency (temperature independence efficiency)

1 Introduction

Nowadays the use of thermoelectric cooler and generator hasbecome increasingly developed The thermoelectric appli-cations include electronic cooling portable refrigerator airconditioning high-precision temperature measurement andspace applicationsThe thermoelectric method has outstand-ing features compared to the other energy conversion meth-ods such as exclusion of moving parts high reliability andlong life span [1] In addition there are several studies on thecombined use of the thermoelectric technology with otherenergy conversion methods (eg steam and gas turbine) asan efficient way of harnessing waste heat in power plant andother industries [2] However the need for high temperatureheat source and low efficiency is a remarkable challenge inthe thermoelectric technology Today the first challenge hasbeen overcome thanks to the abundant waste heat sourceshigh temperature exhausted gas of diesel engine [3] and soforth However so many studies have been done to increasethe efficiency of thermoelectric applications and a lot is still

needed The effort to enhance thermoelectric efficiency canbe classified into three categories invention and developmentof highly efficient materials segmentation and cascadingThe first two of them are discussed in the current study

Invention andDevelopment ofHighly EfficientMaterialsTher-moelectric efficiency not only depends on temperature butalso depends on physical properties of the material such aselectrical and thermal conductivity and Seebeck coefficientThe effect of such properties on efficiency can be analysedusing figure of merit (FOM or 119885) or dimensionless figure ofmerit (119885119879) The higher the FOM the higher the conversionefficiency In this category the attempts are inclined toenhance the performance of thermoelectric applications bymodification of physical properties which results in theincrease of FOM that is increase in Peltier coefficientelectrical conductivity and thermal resistivity Many studieshave focused on this category In particular in recent yearswith the advent of nanotechnology many researchers havemade attempts to develop thermoelectric materials with

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2016 Article ID 6914735 11 pageshttpdxdoiorg10115520166914735

2 International Journal of Chemical Engineering

Heat source

Heat sink

x

I

x = 0

Lp1

Lp2

Lp3

Lp4

Lp5

Ln1

Ln2

Ln3

Ln4

RL

Th

Tc Tc

(a)

Heat source

Heat sink

L

I

Lp1

Lp2

Lp3

Lp4

Lp5

Ln1

Ln2

Ln3

Ln4

RL

Th

+

+

+

minus

minus

minus

Qc

Qh

W = RLI2

Tc Tc

(b)

Figure 1 Schematic of a segmented thermoelectric generator

higher FOM [4ndash7] The remarkable achievements of Bostonand MIT researchers can be mentioned [8 9]

Segmentation Considering the temperature dependence ofFOM so far no material has been found with a satisfactoryFOM within a sufficiently large temperature range (eg 300to 1300K) Different materials have a satisfactory FOM onlywithin a limited temperature range Hence the segmentedthermoelectric devices have emerged (see Figure 1(a) whichshows a segmented thermoelectric generator) In the seg-mented thermoelectric device each material is supposed tobe used in its own specific temperature range in which it hasa reasonable FOM Numerous researches have been done inthis category Chen et al [8] proposed a STEG for operationbetween 300 and 900K and achieved theoretical efficiency of15Kang et al [10] presented a119901-type thermoelectric leg andclaimed that 17 efficiency can be achieved experimentallySnyder and Caillat [11] presented a STEG considering com-patibility factor With the assumption of negligible contactresistance and adiabatic outside surface area they achieved181 efficiency theoretically in temperature range of 25 and1000∘C

Efficiency calculation is one of the most important partsof thermoelectric generators research as it is important inother energy conversion methods Due to importance of effi-ciency several researches have been conducted on the meth-ods of efficiency calculation of thermoelectric generatorsSherman et al [12] presented a comprehensive method forcalculation of performance of thermoelectric devices Withthe advent of segmented thermoelectric systems Swansonet al [13] offered an approximate method for calculation of

performance and design of segmented thermoelectric sys-tems Moore [14] calculated the performance characteristicsof a feasible STEG using the exact and Swanson et alrsquosmethods and compared the results Moore did not designor optimize a STEG he only focused on performance cal-culation of a predesigned generator El-Genk and Saber [15]improved Swanson et alrsquos method Their model uses volumeaverage to calculate thermoelectric properties rather thantemperature average of Swanson et alrsquos method Until nowdue to the widespread application of STEGs the researcherswork on a new methodology for calculation of the efficiencyof the STEGs [16 17]

Considering FOM as the only factor in design of STEGscan adversely affect the thermoelectric efficiency Thereforeanother parameter (called compatibility factor) must beconsidered in design of STEGs The compatibility factor alsoshows whether segmentation or cascading should be used indifferent conditions [18]

One of the most important steps in exergy analysis ofenergy systems is identification of waste heat sources inorder to minimize exergy destruction The thermoelectrictechnology is a promisingway to harness thewaste heatThusthe thermoelectric generators and specially STEGs are ofimportance in exergy optimization A considerable differencebetween experimental and theoretical results (approximatemethods) has been observed in the studies of STEGs [15]Although a major portion of the differences can result fromcontact resistance and experimental uncertainty it seemsnecessary to confirm the validity of theoretical results Up tonow all of the thermoelectric researchers use the approximatemethods for design and optimization of the STEGs because

International Journal of Chemical Engineering 3

200 400 600 800 1000 1200 1300

065

09

115

14

165

19

215

Temperature (K)

ZT

AgPb18SbTe20

La0377Yb044 Te0579

Bi03Ni005Co395Sb12

Si80Ge20P2

Ti05(Zr05Hf05)05NiSn0998Sb0002In02Ce02Co4Sb12

Bi2(Te094Se006)3

(a)

200 400 600 800 1000 1200 130006

08

1

12

14

16

18

Temperature (K)

ZT

AgSbTe 2

Yb14MnSb11

Bi04Sb16Te3

(Bi025Sb075)2Te3Pb013Ge087Te

CeFe3CoSb12

Na095Pb20SbTe22

(b)

Figure 2 Temperature dependence of dimensionless figure of merit for (a) 119899-type and (b) 119901-type materials

of their simplicity But this may be a cause of the differencebetween experimental and theoretical results There was lackof using exact method for optimization of STEGs in theliterature therefore application of exact method can bridgethe gap between theoretical and experimental results [15ndash17]Considering this fact the main objectives of this study can becategorized as follows

(i) In order to design and optimize STEGs at the firststep the chemically and physically modified thermo-electric materials are reviewed Thus the materialswhich have high efficiency in a temperature rangeare selected Then four STEGs are proposed foroptimization process

(ii) As the main objective of this study the proposedSTEGs are optimized using the exact method ratherthan the common simple approximate methodsThen the design and performance characteristics ofthe STEGs are calculated through exact method

(iii) Finally the design and performance characteristicsof the proposed STEGs are also calculated throughcommon approximate methods Then the results ofexact and approximate methods are compared

2 Selection of Materials

In this paper a major study has been done on the latestknown thermoelectric materials the best of which have beenapplied for the design of a thermoelectric generator Due tothe temperature dependence of FOM the materials with asatisfactory FOM in a specific temperature range have beenselected Figures 2(a) and 2(b) show the variation of FOMwith temperature for 119899-type and 119901-type legs respectivelyTwo arrangements have been optimized for each type ofleg Next by considering different combinations four STEGshave been proposed and analysed The materials used in thementioned arrangements as well as temperature ranges foreach arrangement are presented in Table 1

3 Optimization and Performance Calculations

31 Exact Method Performance calculation methods ofsingle-segmented generator were introduced by Sherman etal [12] Efficiency of a thermoelectric generator is defined bythe following relation

120578 =119882

119876ℎ

(1)

where 119882 and 119876ℎare output power and input heat respec-

tively (see Figure 1(b)) Assuming ideal condition

119882 = 1198771198711198682

= 119876ℎminus 119876119888 (2)

where119877119871 119868 and119876

119888are load resistance electrical current and

output heat respectively Using relations (1) and (2) for given119876ℎand 119876

119888 it is possible to calculate efficiency 119876

119888and 119876

ℎare

determined by energy balance in cold and hot junctions asfollows [12]

119876119888= 120587119888119868 + 119860

119901119896119901(119879119888)

119889119879119901

119889119909(119871) + 119860

119899119896119899(119879119888)119889119879119899

119889119909(119871)

119876ℎ= 120587ℎ119868 + 119860

119901119896119901(119879ℎ)

119889119879119901

119889119909(0) + 119860

119899119896119899(119879ℎ)119889119879119899

119889119909(0)

(3)

where119860 119896 and120587 are cross section area thermal conductivityand Peltier coefficient respectively The subscripts 119888 ℎ 119901and 119899 refer to cold junction hot junction 119901-type legand 119899-type leg respectively Equation (3) denotes that it isnecessary to determine temperature gradient as well as theelectrical current passing through the legs to calculate119876

119888and

119876ℎ Therefore it is essential to determine the temperature


Table 1 The materials of the proposed arrangements and their operation temperature range

1199011 1199012 1198991 1198992(temperature range) (temperature range) (temperature range) (temperature range)(Bi025Sb075)2Te3 [19] Bi04Sb16Te3 [20] Bi2(Te094Se006)3 [21] In02Ce02Co4Sb12 [22](300ndash440) (300ndash520) (300ndash440) (300ndash570)AgSbTe2 [23] Na095Pb20SbTe22 [24] AgPb18SbTe20 [25] AgPb18SbTe20 [25](440ndash670) (520ndash700) (440ndash800) (570ndash800)Pb013Ge087Te [26] Pb013Ge087Te [26] Si80Ge20P2 [27] Si80Ge20P2 [27](670ndash770) (700ndash770) (800ndash1200) (800ndash1200)CeFe3

CoSb12 [28] CeFe3CoSb12 [28] La0377Yb044Te0579 [29] La0377Yb044Te057 [29](770ndash920) (770ndash920) (1200ndash1300) (1200ndash1300)Yb14MnSb11 [30] Yb14MnSb11 [30] mdash mdash(920ndash1300) (920ndash1300)

distribution in legs The temperature distribution of a single-segmented leg is calculated through the following equation[31]

119889

119889119909[119896 (119879) 119879

1015840

(119909)] minus 120574 (119879) 1198691198791015840

(119909) + 120588 (119879) 1198692

= 0

120574 (119879) = 119879119889120572

119889119879 119869 =

119868

119860

(4)

In the above equation1198791015840 is derivative of temperaturewithrespect to 119909 120574 and 119869 are Thomson coefficient and currentdensity respectively It is evident that solving such equationrequires knowing exactly how the thermoelectric propertieschange with temperature Equation (4) is solved for each legusing the following change of variable

119910 (119879) = minus119896 (119879)

119869119889119909119889119879

or 119906 = minus 119869

119896 (119879) 119889119879119889119909

(5)

Hence (4) can be written as the following

119889119910

119889119879= minus120574 (119879) minus

120588 (119879)119870 (119879)

119910 (119879)

or 119889119906119889119879

= (120574 (119879) + 120588 (119879)119870 (119879) 119910 (119879)) 1199062

(6)

where 119906 represents relative current density In exact methodthe above equation can be solved using the variable substi-tution of (5) and considering an initial condition for 119906(119879)or 119910(119879) The solution results include determination of elec-trical current and temperature distribution two mentionednecessary factors for calculation of119876

119888and119876

ℎ Consequently

the efficiency is determined Next the efficiency can bemaximized by repeating the process with different initialconditions

In this study specific modifications must be applied tothis process due to the fact that more than one material isused in each leg and the Peltier heat is also presented ineach interface To solve the problem (4) is solved separatelyfor each segment and the following continuity conditions

are applied at the interfaces for 119899-type and 119901-type legsrespectively [14]

1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

+ 119879119868 (120572119895+1

minus 120572119895)

1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

minus 119879119868 (120572119895+1

minus 120572119895)

(7)

Electrical and thermal conductivity and Seebeck coefficientof the materials have been obtained from references given inTable 1 Then (4) has been solved numerically using Δ119879 = 1and assuming initial condition for 119906(119879

ℎ) or 119910(119879

ℎ)

Integrating of (4) gives (8) which is used for calculationof current density

int

119879ℎ

119879119888

119896 (119879) 119906 (119879) 119889119879 = minus119869119871 (8)

The length of each segment is calculated using (8) andconservation of electrical current

It is common that the cross section area of 119901-leg isassumed to be constant and the cross section of 119899-leg isoptimized Hence 119860

119899opt is obtained using current densityconservation of electrical current and the assumption of119860119901= 100mm2 Consequently the cross section area ratio

is calculatedThe internal resistance (119877int) is calculated using material

properties and relation 119877 = 120588119871119860 Finally 119877119871opt and the

resistance ratio are calculated through (2)

32 Swanson et alrsquos Method In approximate methods suchas Swanson et alrsquos method approximate equations are usedrather than (4) to determine the efficiency In fact Swansonet alrsquos method includes calculation of performance charac-teristics on the basis of temperature average properties Inthis method an approximate relation for efficiency is derivedin terms of temperature average properties The next stepis optimization of all characteristics including the ratios ofcross section area and resistance to maximize the efficiency(119860119899119860119901 119877119871119877int) The electrical current and temperature

distribution required in exact method are the current andelement lengths in Swanson et alrsquos methodThe calculation ofthe mentioned parameters through Swanson et alrsquos methodis based on solution of energy-balance equation on theinterfaces [13] which is different from solution of overallenergy-balance equation (4) in exact method


p1-legp2-legMaximum efficiency

0

005

01

015

02

025

Con

vers

ion

effici

ency

5 10 15 200Relative current density (Vminus1)

(a)

0

005

01

015

02

025

Con

vers

ion

effici

ency

n2-legn1-legMaximum efficiency


(b)

Figure 3 Variation of conversion efficiency versus 119906(119879ℎ

) (relative current density at 119879ℎ

) for (a) 1199011-leg and 1199012-leg and (b) 1198991-leg and 1198992-leg

33 Compatibility Factor (Exact and Approximate Methods)Compatibility factor is an important parameter which mustbe considered in addition to FOM in the design of STEGs[18] Compatibility factor indicates whether the use of twomaterials in an arrangement has a positive or negative effecton reduced efficiency The compatibility factor is the relativecurrent density in which reduced efficiency of a material orarrangement is maximized Reduced efficiency is an intrinsiccharacteristic of a material which eliminates the effect oftemperature range from the total efficiency It is defined inthe following form

120578119903=120578

120578119888

120578119888=Δ119879

119879ℎ

(9)

where 120578 and 120578119888are the absolute and Carnot efficiency respec-

tively In order to add a new material to an original materialor arrangement it is necessary to calculate the compatibilityfactor of the two elements If the compatibility factor of thenew material differs by a factor of two or less reduced effi-ciency of the resulting arrangement increases which meansthe two elements are compatible [32] Otherwise it is said thatthey are incompatible and segmentation leads to a decreasein reduced efficiency Compatibility factor of each element isthe relative current density of the element at its maximumreduced efficiency In order to calculate compatibility factor(4) is solved for each material by supposing a boundarycondition for 119910 and the reduced efficiency is obtained Nextby changing the boundary condition of 119910 reduced efficiencyis maximized The compatibility factor is the relative currentdensity corresponding to the maximized reduced efficiencyOwing to the fact that compatibility factor is temperaturedependent it is calculated at a specific temperature Alsothe compatibility factor should not change considerably withtemperature such materials are called self-compatible Inapproximate method the calculation of compatibility factor

is the same as exact method but the reduced efficiency iscalculated through the following relation [18]

120578119903=119906 (120588119896120572) (1 minus 119906 (120588119896120572))

119906 (120588119896120572) + 1119885119879

119885 =1205722

120588119896

(10)

At a specific temperature the curve of reduced efficiencyversus current density is plotted using (10) and then thecompatibility factor is obtained

In this study the proposed STEGs have been optimallydesigned to achievemaximumefficiency using both exact andapproximate methods (exact and Swanson et alrsquos methods)The design and performance characteristics of the optimizedSTEGs have been calculated using both methods and resultshave been compared Compatibility factors of the proposedarrangements have also been calculated through both exactand approximate methods and results have been compared

4 Results and Discussion

41 Analysis of the Proposed Generators In this paper twoarrangements have been suggested for each 119899-type and 119901-type leg using the best available thermoelectric materialsThen with the combination of them four STEGs have beenproposed Table 1 shows the arrangements of legs as wellas their temperature range The proposed STEGs have beenoptimized in order to achieve maximum efficiency Designand performance characteristics of the optimized STEGsincluding length of elements electrical current cross sectionratio and resistance ratio have been calculated using exactmethod The overall length of each leg and the cross sectionof 119901-type legs are assumed to be 10mm and 100mm2respectively

Figures 3(a) and 3(b) indicate the efficiency of 119901-typeand 119899-type legs versus relative current density respectively


Table 2 Design and performance characteristics of 1199011-1198991 and 1199011-1198992 generators at maximum efficiency calculated through both exact andSwanson et alrsquos [13] methods The differences between Swanson et alrsquos and exact results are shown in parenthesis 119860

119901

and 119871 are assumed asconstant inputs of the optimization process

Parameters Swanson method Exact method Swanson method Exact method(1199011-1198991) (1199011-1198991) (1199011-1198992) (1199011-1198992)

119871 (mm) (input) 10 10 10 10119860119901

(mm2) (input) 100 100 100 100Calculated 119876in (W) 3761 (41) 3924 3815 (12) 3863Calculated 119876rej (W) 3148 (29) 3058 3113 (48) 2971Calculated 119875

119890

(W) 857 (103) 865 912 (23) 892Residual power (119876in minus 119876rej minus 119875119890) (W) minus243 mdash minus210 mdashCalculated peak efficiency 120578 () 2277 (32) 2206 2391 (36) 2300119860119899opt (mm2) 8349 (77) 7751 9281 (145) 8108

1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2656 3263 15290808 17350 (505)

2636 3199 15140823 1827

2661 3262 15320807 1728 (54)

2636 3199 15140823 1827

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (912)

0447 5734 25431275

0533 6231 16671559 (809)

0493 6240 17111556

119868 (A) 2811 (71) 3027 2875 (501) 3027119877119871opt (mΩ) 108 (149) 94 110 (134) 97

which are calculated using exact method In this studyabsolute values of 119906 and 119910 have been used Sherman et al[12] mentioned that if the efficiency changes in the rangeof 03ndash003 the value of 119906 will change in the range of 09ndash357 (119910 028ndash112) As can be inferred from the figures theoptimum value of 119906 is 27 and 285 for 119899-type and 3125 and295 for 119901-type respectively which shows great agreementwith Sherman et alrsquos data The maximum efficiency for the1199011-leg and 1199012-leg equals 2256 and 2232 respectivelywhile it equals 2169 and 2348 for the 1198991-leg and 1198992-legrespectively

Four generators resulting from combination of the legshave been presented in Tables 2 and 3 The computed designcharacteristics of the generators at maximum efficiency usingthe exact method are shown in these tables as well Thehighest efficiency is 2308 which belongs to the generatorresulting from the combination of 1199011 and 1198992 which shows498 increase in comparison with Snyder arrangementwhich is 181 [11] The 1199011-1198992 generator has also beendesigned for operating between 300 and 973K in order formaking comparison with Fleurial and Caillatrsquos arrangement[15 33] The output power inlet heat and outlet heat ofthe mentioned generator are equal to 667W 3152W and24851W respectively in the same temperature range Thesevalues are by far less than those of Fleurial and Caillatrsquosgenerator at the same temperature range However theefficiency of the current generator is by far higher than that ofFleurial and Caillat (ie 2128 compared with 1329) Thementioned results are mainly caused by a decrease in thermalconductivity which has been a basis for most of the recentworks in the field of new thermoelectric materials especiallywhere nanotechnology is used

Since the amount of output power is of great impor-tance in most applications the 1199011-1198992 generator has alsobeen designed at maximum power (instead of at maximumefficiency) The results are shown in Table 4

Nevertheless it is explained in the following sectionthat such design cannot be an ideal design because thesystem is not so-called load-following [15] According toTable 2 the power generated by such generator is 892Wat maximum efficiency and 933W at maximum powerdesign Also the efficiency reduces from 2308 to 2272revealing an increase of 462 in power in exchange for158 decrease in efficiency It is remarkable that electricalcurrent has undergone 1733 increase and the optimumexternal resistance decreases by 2371 Figures 4(a) and 4(b)show the variation of generated power per unit cross sectionarea and the inlet heat flux versus relative current densityfor 119901-type and 119899-type legs respectively The solid circlesand open triangles show maximum efficiency and maximumpower respectively Generatorrsquos function in load-followingand non-load-following mode has been indicated by fulllines and dotted lines respectively Therefore the operationpoint of generator should be selected in fully lined portion ofthe curve According to the figures the design of generatorat maximum power results in loss of load-following whiledesign at maximum efficiency keeps the generator in a rangewith load-following [15]

42 Evaluation of Generators Using Compatibility FactorVariations of reduced efficiency with respect to relativecurrent density for 119899-type and 119901-type legs at 119879av have beencalculated by exact method and are presented with solidlines in Figures 5 and 6 respectively The solid circles showthe maximum of reduced efficiency The values of relativecurrent density corresponding to these maximum pointsrepresent the compatibility factor of each segment Thecurves of 119901-type and 119899-type legs reveal that in general thecompatibility factors of materials are better for 119901-type thanfor 119899-type However the curves cannot be used to investigatethe whole leg compatibility This is due to the fact thatwhen a new material is added to an original arrangement


Table 3 Design and performance characteristics of 1199012-1198991 and 1199012-1198992 generators at maximum efficiency calculated through both exact andSwanson et alrsquos [13] methods The differences of Swanson et alrsquos results compared to the exact ones are shown in parenthesis 119860

119901

and 119871 areassumed as constant inputs of the optimization process


119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg

Conversion efficiency ()

Maximum efficiencyMaximum power

16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)

Figure 4 Electrical output power per cross section area and input heat flux versus conversion efficiency for (a) 1199011-leg and 1199012-leg and (b)1198991-leg and 1198992-leg which have been calculated through exact method

the compatibility of the original arrangement as a whole andnew material must be assessed but the curves are plottedfor each material separately Therefore they are only usefulfor a general assessment of materialsrsquo compatibility In orderto clarify the subject the change of reduced efficiency oflegs with increasing number of elements has been shownin Table 5 The table reveals that the materials used in the1199011-leg are compatible and adding material increases thereduced efficiency of the resulting assembly Also in 1199012-leg the materials are compatible but the first two materialsare on the border of compatibility and incompatibility Forthe arrangements 1198991 and 1198992 only the first three materialsare compatible and the fourth material is incompatiblewith them therefore adding it to the arrangement leads to

a decrease in the reduced efficiency of the arrangement as awhole The materials used would be compatible if the 119899-legswere used in temperature range of 300 and 1200K Howeverowing to the fact that 119901-legs have been used in range of 300and 1300K 119899-legs should be used in the same temperaturerange to preserve consistency Our search for a material witha better compatibility factor did not lead to any satisfactoryresults in the mentioned temperature range Therefore thesearrangements have been proposed

43 Evaluation of Approximate Methods The optimizedgenerators have also been designed using Swanson et alrsquosmethod at maximum efficiency The calculated performanceand design characteristics are shown in Tables 2 and 3 and


2 4 6 8 10 12 14 160Relative current density (Vminus1)

0

005

01

015

02

025

Redu

ced

effici

ency

AgPb18SbTe20La0377Yb044 Te058Si08Ge20P2 Bi2(Te094Se006)3

(a)

0

005

01

015

02

025

Redu

ced

effici

ency

1 2 3 4 5 6 7 8 9 10 110Relative current density (Vminus1)

AgPb18SbTe20La0377Yb044 Te058Si08Ge20P2 In02C202Co4Sb12

(b)

Figure 5 Variation of reduced efficiency versus 119906(119879av) (relative current density at 119879av) for the materials used in (a) 1198991-leg and (b) 1198992-leg Thesolid and dashed lines indicate results of exact and approximate method respectively

Table 4 Design and performance characteristics of 1199011-1198992 generatorat maximum output power calculated through exact method

Parameters Exact (1199011-1198992)119871 (mm) (input) 10119860119901

(mm2) (input) 100Calculated 119876in (W) 4107Calculated 119876rej (W) 3173Calculated 119875

119890

(W) 933Calculated peak efficiency 120578 () 2272119860119899opt (mm2) 8403

1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74

Table 5 Reduced efficiency of arrangements with increasing num-ber of elements

Number of elements Legp1 p2 n1 n2

1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash

compared with those of exact method These tables showthat in Swanson et alrsquos method design and characteristicproperties of 119901-type and 119899-type legs depend on each otherwhile in the exact method 119901-type legs are designed separatelyand independently of 119899-type legsThe differences of Swansonet alrsquos results compared to exact ones are presented inparenthesis in these tables The tables indicate that Swanson

et alrsquos method could cause a difference of up to 41 ininlet heat 78 in outlet heat 58 in generated power 33in efficiency 203 in 119899-type leg cross section area 91in element lengths 71 in current and 140 in externalresistance In these tables the two highest differences belongto the external resistance and 119899-type leg cross sectionsarea (ie 14 and 203 resp) Swanson et al [13] hasreferred to such differences by optimizing the efficiency forthe two following cases constant 119877

119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890

variable 119860119899119860119901 and variable 119877

119871119877 constant 119860

119899119860119901

=

119900119901119905119894119898119906119898 V119886119897119906119890 In this paper the two preceding cases havebeen investigated in four generators and the results areshown in Figures 7(a)ndash7(d) through the curves of efficiencywith respect to electrical current In Swanson et alrsquos methodthe amount of calculated electrical power is not equal to thatof thermal power obtained through satisfaction of the energy-balance equation This is because of the approximations anduse of temperature average properties of materialsThereforethere exists residual power shown in Tables 2 and 3 Theexact method satisfies the overall energy-balance equationcompletely and thus there is no residual power

The compatibility factor of the materials used in 119899-type and 119901-type legs has also been calculated using theapproximate relation (10) and indicated with dashed linesin Figures 5 and 6 respectively A comparison of resultswith the ones of exact method reveals that relation (10)yields somewhat exact results and there is not a considerabledifference in compatibility factorwhereas there is a noticeabledifference in reduced efficiency

5 Conclusion

Four STEGs with the best known thermoelectric materi-als have been optimized for operating between 300 and1300K The maximum theoretically achievable efficiency hasbeen calculated assuming insulated outside surface area and


0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency

Relative current density (Vminus1)

AgSbTe 2Yb14MnSb11(Bi025Sb075)2Te3

Pb013Ge087TeCeFe3CoSb12

(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11

Pb013Ge087TeCeFe3CoSb12 Bi04Sb16Te3

Na095Pb20SbTe22

(b)


Variation of A ratio

0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency

26 28 30 32 3424Electrical current I (A)

Variation ratioRof(a)

02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency


Variation of A ratioVariation ratioRof

(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)

Figure 7 Variation of efficiency with respect to electrical current for the two following cases constant 119877119871

119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901

and variable 119877119871

119877 constant 119860119899

119860119901

= 119900119901119905119894119898119906119898 V119886119897119906119890 (a)ndash(d) represent 1199011-1198991 1199011-1198992 1199012-1198991 and 1199012-1198992 generators respectively


negligible contact resistance Design and performance char-acteristics of the generators have been calculated atmaximumefficiency using exact and Swanson et alrsquos methodMaximumefficiency of 2308 has been obtained for the best arrange-ment which shows 498 increase in comparison with thebest of the previous published results in a similar temperaturerange (298 and 1273K) Compatibility factor of the arrange-ments has been calculated through both exact and Swansonet alrsquos method Satisfactory compatibility has been observedfor the materials in each arrangement All the results of exactand Swanson et alrsquos method have been compared It hasbeen observed that Swanson et alrsquos method could cause adifference of up to 20 in some characteristics despite itssufficiently good results in calculation efficiency Enhancingthermoelectric efficiency opens the way to generate electricalpower especially by using waste heat as source Owing tothe great deal of attempt and investigation to improve ther-moelectric properties especially based on nanotechnologythis direct energy conversion method and consequently theexergy optimization are awaiting a bright future

Nomenclature

Variables

119860 Cross section area (m2)119868 Electrical current (A)119869 Electrical current density (Amminus2)119896 Thermal conductivity (Wmminus1Kminus1)119871 Length of legs (m)1198711199011 119871

1198991 Length of segments (m)

119875119890 Output electrical power (W)

119876 Heat power (W)119877 Electrical resistance (Ω)119879 Temperature (K)119906 Relative electrical current density (Vminus1)119882 Work output (W)119909 Coordinate originated from hot source

(m)119885 Figure of merit (FOM) (Kminus1)119885119879 Dimensionless figure of merit

Greek Symbols

120572 Seebeck coefficient (VKminus1)120578 Efficiency120578119888 Carnot efficiency

120578119903 Reduced efficiency

120574 Thomson coefficient (VKminus1)120587 Peltier coefficient (V)120588 Electrical resistivity (Ωsdotm)

Scripts1015840 Derivative with respect to 119909av Average119888 Cold sourceℎ Hot source

in Inputint Internal119895 Index for segment number119871 Electrical load119899 119899-legopt Optimum119901 119901-legrej Reject or output

Competing Interests

The authors declare that they have no competing interests

References

[1] S B Riffat and XMa ldquoThermoelectrics a review of present andpotential applicationsrdquoAppliedThermal Engineering vol 23 no8 pp 913ndash935 2003

[2] K Yazawa and A Shakouri ldquoThermoelectric topping cycleswith scalable design and temperature dependent material prop-ertiesrdquo Scripta Materialia vol 111 pp 58ndash63 2016

[3] H Tian X Sun Q Jia X Liang G Shu and X WangldquoComparison and parameter optimization of a segmentedthermoelectric generator by using the high temperature exhaustof a diesel enginerdquo Energy vol 84 pp 121ndash130 2015

[4] J P Heremans V Jovovic E S Toberer et al ldquoEnhancement ofthermoelectric efficiency in PbTe by distortion of the electronicdensity of statesrdquo Science vol 321 no 5888 pp 554ndash557 2008

[5] G J Snyder and E S Toberer ldquoComplex thermoelectricmaterialsrdquo Nature Materials vol 7 no 2 pp 105ndash114 2008

[6] L I Soliman M M Nassary H T Shaban and A S SalwaldquoInfluence of Se on the electron mobility in thermal evaporatedBi2

(Te1minus119909

Se119909

)3

thin filmsrdquo Vacuum vol 85 no 3 pp 358ndash3642010

[7] L T Hung N Van Nong G J Snyder et al ldquoHigh performancep-type segmented leg of misfit-layered cobaltite and half-Heusler alloyrdquo Energy Conversion and Management vol 99 pp20ndash27 2015

[8] G Chen M Dresselhaus and Z Ren ldquoNanocomposites withhigh thermoelectric figures of meritrdquo US Patent App 20090068 465 2008

[9] Z Ren B Poudel G Chen Y Lan D Wang and Q EA Hao ldquoMethods for high figure-of-merit in nanostructuredthermoelectric materialsrdquo US Patent App 11949 353 2007

[10] Y S Kang M Niino I A Nishida and J Yoshino ldquoDevel-opment and evaluation of 3-stage segmented thermoelectricelementsrdquo in Proceedings of the 17th International Conference onThermoelectrics (ICT rsquo98) pp 429ndash432 May 1998

[11] G J Snyder and T Caillat ldquoUsing the compatibility factor todesign high efficiency segmented thermoelectric generatorsrdquoMaterials Research Society Symposium Proceedings vol 793 pp37ndash42 2003

[12] B Sherman R R Heikes and R W Ure Jr ldquoCalculation ofefficiency of thermoelectric devicesrdquo Journal of Applied Physicsvol 31 no 1 pp 1ndash16 1960

[13] BW Swanson E V Somers and R R Heikes ldquoOptimization ofa sandwiched thermoelectric devicerdquo Journal of Heat Transfervol 83 no 1 pp 77ndash82 1961

[14] R G Moore ldquoExact computer solution of segmented thermo-electric devicesrdquo Advanced Energy Conversion vol 2 pp 183ndash195 1962


[15] M S El-Genk and H H Saber ldquoHigh efficiency segmentedthermoelectric unicouple for operation between 973 and 300KrdquoEnergy Conversion and Management vol 44 no 7 pp 1069ndash1088 2003

[16] C Hadjistassou E Kyriakides and J Georgiou ldquoDesigninghigh efficiency segmented thermoelectric generatorsrdquo EnergyConversion and Management vol 66 pp 165ndash172 2013

[17] T Ming Y Wu C Peng and Y Tao ldquoThermal analysis on asegmented thermoelectric generatorrdquo Energy vol 80 pp 388ndash399 2015

[18] T Ursell and G Snyder ldquoCompatibility of segmented ther-moelectric generatorsrdquo in Proceedings of the 21st InternationalConference on Thermoelectrics (ICT rsquo02) pp 412ndash417 LongBeach Calif USA 2002

[19] O Yamashita S Tomiyoshi and K Makita ldquoBismuth telluridecompounds with high thermoelectric figures of meritrdquo Journalof Applied Physics vol 93 no 1 pp 368ndash374 2003

[20] S Fan J Zhao J Guo Q Yan J Ma and H H Hng ldquoP-typeBi04 Sb16 Te3 nanocomposites with enhanced figure of meritrdquoApplied Physics Letters vol 96 no 18 Article ID 182104 2010

[21] O Yamashita and S Tomiyoshi ldquoHigh performance n-typebismuth telluride with highly stable thermoelectric figure ofmeritrdquo Journal of Applied Physics vol 95 no 11 pp 6277ndash62832004

[22] M Subramanian T He and J Krajewski ldquoHigh performancethermoelectric materials and their method of preparationrdquo USPatent Application 12785605 2010

[23] H Wang J-F Li M Zou and T Sui ldquoSynthesis and transportproperty of AgSbTe

2

as a promising thermoelectric compoundrdquoApplied Physics Letters vol 93 no 20 Article ID 202106 2008

[24] P F P Poudeu J DrsquoAngelo A D Downey J L ShortT P Hogan and M G Kanatzidis ldquoHigh thermoelectricfigure of merit and nanostructuring in bulk p-type Na1-xPbmSbyTem+2rdquo Angewandte Chemie-International Editionvol 45 no 23 pp 3835ndash3839 2006

[25] K F Hsu S Loo F Guo et al ldquoCubic AgPbmSbTe2+m bulkthermoelectric materials with high figure of meritrdquo Science vol303 no 5659 pp 818ndash821 2004

[26] Y Gelbstein B Dado O Ben-Yehuda Y Sadia Z Dashevskyand M P Dariel ldquoHighly efficient Ge-rich Ge

119909

Pb1minus119909

Tethermoelectric alloysrdquo Journal of Electronic Materials vol 39no 9 pp 2049ndash2052 2010

[27] X W Wang H Lee Y C Lan et al ldquoEnhanced thermoelectricfigure of merit in nanostructured n-type silicon germaniumbulk alloyrdquo Applied Physics Letters vol 93 no 19 Article ID193121 2008

[28] J-P Fleurial A Borshchevsky T Caillat D T Morelli and G PMeisner ldquoHigh figure ofmerit inCe-filled skutteruditesrdquo inPro-ceedings of the 15th International Conference on Thermoelectrics(ICT rsquo96) pp 91ndash95 March 1996

[29] A F May J-P Fleurial and G J Snyder ldquoOptimizing thermo-electric efficiency in La

3minus119909

Te4

via Yb substitutionrdquo Chemistry ofMaterials vol 22 no 9 pp 2995ndash2999 2010

[30] S R Brown S M Kauzlarich F Gascoin and G Jeffrey SnyderldquoYb14

MnSb11

new high efficiency thermoelectric material forpower generationrdquo Chemistry of Materials vol 18 no 7 pp1873ndash1877 2006

[31] C A Domenicali ldquoStationary temperature distribution in anelectrically heated conductorrdquo Journal of Applied Physics vol 25no 10 pp 1310ndash1311 1954

[32] G J Snyder and T S Ursell ldquoThermoelectric efficiency andcompatibilityrdquo Physical Review Letters vol 91 no 14 2003

[33] J-P Fleurial A Borshchevsky T Caillat and R Ewell ldquoNewmaterials and devices for thermoelectric applicationsrdquo in Pro-ceedings of the 32nd Intersociety Energy Conversion Engineer-ing Conference (IECEC rsquo97) vol 2 pp 1080ndash1085 HonoluluHawaii USA August 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Heat source

Heat sink

x

I

x = 0

Lp1

Lp2

Lp3

Lp4

Lp5

Ln1

Ln2

Ln3

Ln4

RL

Th

Tc Tc

(a)

Heat source

Heat sink

L

I

Lp1

Lp2

Lp3

Lp4

Lp5

Ln1

Ln2

Ln3

Ln4

RL

Th

+

+

+

minus

minus

minus

Qc

Qh

W = RLI2

Tc Tc

(b)

Figure 1 Schematic of a segmented thermoelectric generator

higher FOM [4ndash7] The remarkable achievements of Bostonand MIT researchers can be mentioned [8 9]

Segmentation Considering the temperature dependence ofFOM so far no material has been found with a satisfactoryFOM within a sufficiently large temperature range (eg 300to 1300K) Different materials have a satisfactory FOM onlywithin a limited temperature range Hence the segmentedthermoelectric devices have emerged (see Figure 1(a) whichshows a segmented thermoelectric generator) In the seg-mented thermoelectric device each material is supposed tobe used in its own specific temperature range in which it hasa reasonable FOM Numerous researches have been done inthis category Chen et al [8] proposed a STEG for operationbetween 300 and 900K and achieved theoretical efficiency of15Kang et al [10] presented a119901-type thermoelectric leg andclaimed that 17 efficiency can be achieved experimentallySnyder and Caillat [11] presented a STEG considering com-patibility factor With the assumption of negligible contactresistance and adiabatic outside surface area they achieved181 efficiency theoretically in temperature range of 25 and1000∘C

Efficiency calculation is one of the most important partsof thermoelectric generators research as it is important inother energy conversion methods Due to importance of effi-ciency several researches have been conducted on the meth-ods of efficiency calculation of thermoelectric generatorsSherman et al [12] presented a comprehensive method forcalculation of performance of thermoelectric devices Withthe advent of segmented thermoelectric systems Swansonet al [13] offered an approximate method for calculation of

performance and design of segmented thermoelectric sys-tems Moore [14] calculated the performance characteristicsof a feasible STEG using the exact and Swanson et alrsquosmethods and compared the results Moore did not designor optimize a STEG he only focused on performance cal-culation of a predesigned generator El-Genk and Saber [15]improved Swanson et alrsquos method Their model uses volumeaverage to calculate thermoelectric properties rather thantemperature average of Swanson et alrsquos method Until nowdue to the widespread application of STEGs the researcherswork on a new methodology for calculation of the efficiencyof the STEGs [16 17]

Considering FOM as the only factor in design of STEGscan adversely affect the thermoelectric efficiency Thereforeanother parameter (called compatibility factor) must beconsidered in design of STEGs The compatibility factor alsoshows whether segmentation or cascading should be used indifferent conditions [18]

One of the most important steps in exergy analysis ofenergy systems is identification of waste heat sources inorder to minimize exergy destruction The thermoelectrictechnology is a promisingway to harness thewaste heatThusthe thermoelectric generators and specially STEGs are ofimportance in exergy optimization A considerable differencebetween experimental and theoretical results (approximatemethods) has been observed in the studies of STEGs [15]Although a major portion of the differences can result fromcontact resistance and experimental uncertainty it seemsnecessary to confirm the validity of theoretical results Up tonow all of the thermoelectric researchers use the approximatemethods for design and optimization of the STEGs because


200 400 600 800 1000 1200 1300

065

09

115

14

165

19

215

Temperature (K)

ZT

AgPb18SbTe20

La0377Yb044 Te0579

Bi03Ni005Co395Sb12

Si80Ge20P2


Bi2(Te094Se006)3

(a)

200 400 600 800 1000 1200 130006

08

1

12

14

16

18

Temperature (K)

ZT

AgSbTe 2

Yb14MnSb11

Bi04Sb16Te3


CeFe3CoSb12

Na095Pb20SbTe22

(b)










120578 =119882

119876ℎ

(1)



119882 = 1198771198711198682

= 119876ℎminus 119876119888 (2)

where119877119871 119868 and119876




119888and 119876

ℎare


119876119888= 120587119888119868 + 119860

119901119896119901(119879119888)

119889119879119901

119889119909(119871) + 119860

119899119896119899(119879119888)119889119879119899

119889119909(119871)

119876ℎ= 120587ℎ119868 + 119860

119901119896119901(119879ℎ)

119889119879119901

119889119909(0) + 119860

119899119896119899(119879ℎ)119889119879119899

119889119909(0)

(3)


119888and







119889

119889119909[119896 (119879) 119879

1015840

(119909)] minus 120574 (119879) 1198691198791015840

(119909) + 120588 (119879) 1198692

= 0

120574 (119879) = 119879119889120572

119889119879 119869 =

119868

119860

(4)


119910 (119879) = minus119896 (119879)

119869119889119909119889119879

or 119906 = minus 119869

119896 (119879) 119889119879119889119909

(5)


119889119910

119889119879= minus120574 (119879) minus

120588 (119879)119870 (119879)

119910 (119879)

or 119889119906119889119879

= (120574 (119879) + 120588 (119879)119870 (119879) 119910 (119879)) 1199062

(6)


119888and119876

ℎ Consequently




1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

+ 119879119868 (120572119895+1

minus 120572119895)

1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

minus 119879119868 (120572119895+1

minus 120572119895)

(7)


ℎ) or 119910(119879

ℎ)


int

119879ℎ

119879119888

119896 (119879) 119906 (119879) 119889119879 = minus119869119871 (8)











0

005

01

015

02

025

Con

vers

ion

effici

ency


(a)

0

005

01

015

02

025

Con

vers

ion

effici

ency



(b)





120578119903=120578

120578119888

120578119888=Δ119879

119879ℎ

(9)




120578119903=119906 (120588119896120572) (1 minus 119906 (120588119896120572))

119906 (120588119896120572) + 1119885119879

119885 =1205722

120588119896

(10)








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2656 3263 15290808 17350 (505)

2636 3199 15140823 1827

2661 3262 15320807 1728 (54)

2636 3199 15140823 1827

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (912)

0447 5734 25431275

0533 6231 16671559 (809)

0493 6240 17111556

119868 (A) 2811 (71) 3027 2875 (501) 3027119877119871opt (mΩ) 108 (149) 94 110 (134) 97








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg



16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)







0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










200 400 600 800 1000 1200 1300

065

09

115

14

165

19

215

Temperature (K)

ZT

AgPb18SbTe20

La0377Yb044 Te0579

Bi03Ni005Co395Sb12

Si80Ge20P2


Bi2(Te094Se006)3

(a)

200 400 600 800 1000 1200 130006

08

1

12

14

16

18

Temperature (K)

ZT

AgSbTe 2

Yb14MnSb11

Bi04Sb16Te3


CeFe3CoSb12

Na095Pb20SbTe22

(b)










120578 =119882

119876ℎ

(1)



119882 = 1198771198711198682

= 119876ℎminus 119876119888 (2)

where119877119871 119868 and119876




119888and 119876

ℎare


119876119888= 120587119888119868 + 119860

119901119896119901(119879119888)

119889119879119901

119889119909(119871) + 119860

119899119896119899(119879119888)119889119879119899

119889119909(119871)

119876ℎ= 120587ℎ119868 + 119860

119901119896119901(119879ℎ)

119889119879119901

119889119909(0) + 119860

119899119896119899(119879ℎ)119889119879119899

119889119909(0)

(3)


119888and







119889

119889119909[119896 (119879) 119879

1015840

(119909)] minus 120574 (119879) 1198691198791015840

(119909) + 120588 (119879) 1198692

= 0

120574 (119879) = 119879119889120572

119889119879 119869 =

119868

119860

(4)


119910 (119879) = minus119896 (119879)

119869119889119909119889119879

or 119906 = minus 119869

119896 (119879) 119889119879119889119909

(5)


119889119910

119889119879= minus120574 (119879) minus

120588 (119879)119870 (119879)

119910 (119879)

or 119889119906119889119879

= (120574 (119879) + 120588 (119879)119870 (119879) 119910 (119879)) 1199062

(6)


119888and119876

ℎ Consequently




1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

+ 119879119868 (120572119895+1

minus 120572119895)

1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

minus 119879119868 (120572119895+1

minus 120572119895)

(7)


ℎ) or 119910(119879

ℎ)


int

119879ℎ

119879119888

119896 (119879) 119906 (119879) 119889119879 = minus119869119871 (8)











0

005

01

015

02

025

Con

vers

ion

effici

ency


(a)

0

005

01

015

02

025

Con

vers

ion

effici

ency



(b)





120578119903=120578

120578119888

120578119888=Δ119879

119879ℎ

(9)




120578119903=119906 (120588119896120572) (1 minus 119906 (120588119896120572))

119906 (120588119896120572) + 1119885119879

119885 =1205722

120588119896

(10)








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2656 3263 15290808 17350 (505)

2636 3199 15140823 1827

2661 3262 15320807 1728 (54)

2636 3199 15140823 1827

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (912)

0447 5734 25431275

0533 6231 16671559 (809)

0493 6240 17111556

119868 (A) 2811 (71) 3027 2875 (501) 3027119877119871opt (mΩ) 108 (149) 94 110 (134) 97








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg



16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)







0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119889

119889119909[119896 (119879) 119879

1015840

(119909)] minus 120574 (119879) 1198691198791015840

(119909) + 120588 (119879) 1198692

= 0

120574 (119879) = 119879119889120572

119889119879 119869 =

119868

119860

(4)


119910 (119879) = minus119896 (119879)

119869119889119909119889119879

or 119906 = minus 119869

119896 (119879) 119889119879119889119909

(5)


119889119910

119889119879= minus120574 (119879) minus

120588 (119879)119870 (119879)

119910 (119879)

or 119889119906119889119879

= (120574 (119879) + 120588 (119879)119870 (119879) 119910 (119879)) 1199062

(6)


119888and119876

ℎ Consequently




1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

+ 119879119868 (120572119895+1

minus 120572119895)

1198961198951198601198951198791015840

119895

= 119896119895+1119860119895+11198791015840

119895+1

minus 119879119868 (120572119895+1

minus 120572119895)

(7)


ℎ) or 119910(119879

ℎ)


int

119879ℎ

119879119888

119896 (119879) 119906 (119879) 119889119879 = minus119869119871 (8)











0

005

01

015

02

025

Con

vers

ion

effici

ency


(a)

0

005

01

015

02

025

Con

vers

ion

effici

ency



(b)





120578119903=120578

120578119888

120578119888=Δ119879

119879ℎ

(9)




120578119903=119906 (120588119896120572) (1 minus 119906 (120588119896120572))

119906 (120588119896120572) + 1119885119879

119885 =1205722

120588119896

(10)








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2656 3263 15290808 17350 (505)

2636 3199 15140823 1827

2661 3262 15320807 1728 (54)

2636 3199 15140823 1827

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (912)

0447 5734 25431275

0533 6231 16671559 (809)

0493 6240 17111556

119868 (A) 2811 (71) 3027 2875 (501) 3027119877119871opt (mΩ) 108 (149) 94 110 (134) 97








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg



16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)







0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

005

01

015

02

025

Con

vers

ion

effici

ency


(a)

0

005

01

015

02

025

Con

vers

ion

effici

ency



(b)





120578119903=120578

120578119888

120578119888=Δ119879

119879ℎ

(9)




120578119903=119906 (120588119896120572) (1 minus 119906 (120588119896120572))

119906 (120588119896120572) + 1119885119879

119885 =1205722

120588119896

(10)








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2656 3263 15290808 17350 (505)

2636 3199 15140823 1827

2661 3262 15320807 1728 (54)

2636 3199 15140823 1827

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (912)

0447 5734 25431275

0533 6231 16671559 (809)

0493 6240 17111556

119868 (A) 2811 (71) 3027 2875 (501) 3027119877119871opt (mΩ) 108 (149) 94 110 (134) 97








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg



16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)







0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2656 3263 15290808 17350 (505)

2636 3199 15140823 1827

2661 3262 15320807 1728 (54)

2636 3199 15140823 1827

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (912)

0447 5734 25431275

0533 6231 16671559 (809)

0493 6240 17111556

119868 (A) 2811 (71) 3027 2875 (501) 3027119877119871opt (mΩ) 108 (149) 94 110 (134) 97








119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg



16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)







0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901



119871 (mm) (input) 10 10 10 10119860119901


119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2570 3167 10501821 1382 (511)

2562 3132 10421808 1456

2573 3164 10521825 1376 (55)

2562 3132 10421808 1456

1198711198991

1198711198992

1198711198993

1198711198994

(mm) 0488 5716 25041283 (903)

0447 5734 25431275

0532 6232 16681560 (799)

0493 6240 17111556

119868 (A) 2801 (263) 2876 2865 (04) 2876119877119871opt (mΩ) 109 (79) 101 111 (67) 104

0 005 01 015 02 025

n2-legn1-leg



16

18

2

22

24

26

28

3

32

34

Inle

t hea

t (10

minus5

Wm

minus2 )

0

1

2

3

4

5

6

7

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(a)

0 005 01 015 02 025

p1-legp2-leg



1111213141516171819

2

Inle

t hea

t (10

minus5

Wm

minus2 )

0040812162242832364

Elec

tric

al p

ower

(10minus

4W

mminus

2 )

(b)







0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

005

01

015

02

025

Redu

ced

effici

ency


(a)

0

005

01

015

02

025

Redu

ced

effici

ency



(b)





119890


1198711199011

1198711199012

1198711199013

1198711199014

1198711199015

(mm) 2762 3274 1574 0841 18311198711198991

1198711198992

1198711198993

1198711198994

(mm) 0540 6521 1784 1589119868 (A) 3552119877119871opt (mΩ) 74



1 2061 2466 2032 24762 2587 2466 2535 28683 2741 2615 2842 30914 2846 2754 2819 30535 2932 2902 mdash mdash



119871119877 = 119900119901119905119894119898119906119898 V119886119897119906119890


119871119877 constant 119860

119899119860119901

=



5 Conclusion



0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 3 6 9 120

005

01

015

02

025

Redu

ced

effici

ency




(a)

0 3 6 9 12 15 160

005

01

015

02

025

Redu

ced

effici

ency


Yb14MnSb11


Na095Pb20SbTe22

(b)



0224

02248

02256

02264

02272

0228

Con

vers

ion

effici

ency



02355

02363

02371

02379

02387

02393

Con

vers

ion

effici

ency



(b)


0223

0224

0225

0226

Con

vers

ion

effici

ency


(c)


0234

0235

0236

0237

Con

vers

ion

effici

ency


(d)


119877 = 119900119901119905119894119898119906119898 V119886119897119906119890 variable119860119899

119860119901


119877 constant 119860119899

119860119901




Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Nomenclature

Variables






Greek Symbols






Competing Interests


References







(Te1minus119909

Se119909

)3




















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















2





119909

Pb1minus119909





3minus119909

Te4



MnSb11







RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article exact optimum design of segmented...

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