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Energy 114 (2016) 52e63

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Energy

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Thermal and stress analyses in thermoelectric generator with taperedand rectangular pin configurations

Bekir Sami Yilbas a, b, *, S.S. Akhtar a, A.Z. Sahin a, b

a Department of Mechanical Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabiab Center of Research Excellence in Renewable Energy, Research Institute, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

a r t i c l e i n f o

Article history:Received 24 May 2016Received in revised form29 July 2016Accepted 31 July 2016Available online 9 August 2016

Keywords:Thermoelectric generatorThermal stressPin geometryTapered pins

* Corresponding author. Department of MechanUniversity of Petroleum & Minerals, Dhahran 31261,

E-mail address: bsyilbas@kfupm.edu.sa (B.S. Yilba

http://dx.doi.org/10.1016/j.energy.2016.07.1680360-5442/© 2016 Elsevier Ltd. All rights reserved.

a b s t r a c t

Thermal stress developed in thermoelectric generators is critical for long service applications. Hightemperature gradients, due to a large temperature difference across the junctions, causes excessive stresslevels developed in the device pins and electrodes at the interfaces. In the present study, a thermoelectricgenerator with horizontal pin configuration is considered and thermal stress analysis in the device ispresented. Ceramic wafer is considered to resemble the high temperature plate and copper electrodesare introduced at the pin junctions to reduce the electrical resistance between the pins and the high andlow temperature junction plates during the operation. Finite element code is used to simulate temper-ature and stress fields in the thermoelectric generator. In the simulations, convection and radiation lossesfrom the thermoelectric pins are considered and bismuth telluride pin material with and withouttapering is incorporated. It is found that von Mises stress attains high values at the interface between thehot and cold junctions and the copper electrodes. Thermal stress developed in tapered pin configurationattains lower values than that of rectangular pin cross-section.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

A thermoelectric generator is one of the green energy devices,which constitutes a simple mechanical system without move-ments. Although the efficiency of the thermoelectric generator islow, research into improvement of the device efficiency and theoutput power through modifying pin geometric configurations [1]and pin materials [2] are in progress. Geometrically tapering of thethermoelectric pins improves the thermal efficiency of the device[3] and increasing junction temperature of the thermoelectricgenerator enhances the device output power [4]. In general, thesize of the thermoelectric generators is small, and temperaturedifference across the device junctions is high because of the questfor achieving the high Carnot efficiency of the system. The thermalefficiency and the device output power improve significantly whentemperature difference increases across the thermoelectric junc-tions within the range of operating conditions of the thermoelectricgenerator pin material [4]. High temperature difference results inhigh strain in the thermoelectric pins, particularly in the junction

ical Engineering, King FahdSaudi Arabia.s).

regions. Thermal strains give rise to development of thermalstresses in the pins and across the pin connections in these regions.These causes pin material failure under the high thermal stressconditions during the operation. Therefore, operational life ishighly influenced by the stress states in the pins, which in turn limitthe practical applications of the device. Although engineered pinconfigurations, such as tapering, improve the device efficiency, itmay have an adverse effect on the stress levels in the pins.Consequently, investigation of the thermal stress development inthe thermoelectric device with and without engineered pin con-figurations becomes essential.

Considerable research studies were carried out to examinethermal management of thermoelectric generators. The perfor-mance of a thermoelectric generators coupled with a solar pondwas studied by Ding et al. [5]. They indicated that thermoelectricgenerators could be used to extract heat stored from the solarponds; however, the thermal-electrical conversion efficiency of thethermal system was low, i.e. in the range of 1%e1.5%. A waste heatrecovery from the exhausted cryogenic nitrogen by using thermo-electric power generator was investigated by Weng et al. [6]. Theyshowed that the thermoelectric generation system worked suc-cessfully despite the system efficiency remained low. The effect ofradiation view factors on thermoelectric performance was

Nomenclature

A Cross-section area of the thermoelectric generator(m2)

E Energy gain (J/m3)I Electrical current (A)k Thermal conductivity (W/mK)ke,n Thermal conductivity of n-type semi-conductor (W/

mK)ke,p Thermal conductivity of p-type semi-conductor (W/

mK)K Total thermal conductance in of thermoelectric

generator (U)Kn Thermal conductance of n-type semi-conductor (W/K)Kp Thermal conductance of p-type semi-conductor (W/K)L Length of leg of thermoelectric generator (m)R Total electrical resistance in of thermoelectric

generator (U)RL External load resistance (U)Rn Electrical resistance of n-type leg of semi-conductor

(U)Rp Electrical resistance of p-type leg of semi-conductor

(U)R0 Reference electrical resistance (U)

T1 Hot junction temperature of the thermoelectricgenerator (K)

T2n Cold junction temperature of the thermoelectricgenerator for n-type pin (K)

T2p Cold junction temperature of the thermoelectricgenerator for p-type pin (K)

Tav Average temperature (K)W Power output of the thermoelectric generator (W)Z Figure of merit (1/K)ZT Dimensionless Figure of merit (�)a Net Seebeck coefficient of thermoelectric generator (V/

K)an Seebeck coefficient of n-type semi-conductor (V/K)ap Seebeck coefficient of p-type semi-conductor (V/K)aT Thermal expansion coefficient (1/K)DT Temperature difference at cold junction due to p-type

and n-type pins (K){Dε} Overall strain vector{Dεel} Elastic strain vector{Dεth} Thermal strain increment vector{Dεpl} Plastic strain increment vectorh Efficiencyl Temperature parametert1 ¼ T1/Tmax Hot junction temperature ratiot2 ¼ T2/Tmax Cold junction temperature ratio

B.S. Yilbas et al. / Energy 114 (2016) 52e63 53

examined by Barry et al. [7]. The findings revealed that radiationview factor behaved non-linearly with increasing temperature ra-tio. In addition, increasing the leg height to width ratio of thethermoelectric pins decreased radiation view factor monolithically.The influence of Thomson effect on the performance of a two stagethermoelectric generator was studied by Manikandan and Kaushik[8]. They demonstrated that the exergy efficiency of two stagethermoelectric generator was greater than the energy efficiency ofthe device. Energetic and exergetic performance analyses of a solarenergy-based integrated system incorporating the thermoelectricgenerators were investigated by Islam et al. [9]. They indicated thatthe maximum work done by the thermoelectric generator andcooler was increased considerably through increasing the temper-ature of the heat transferring fluid. The analysis of thermoelectricgenerator performance incorporating the pin tapering was carriedout by Yilbas and Ali [10]. They demonstrated that the first andsecond law efficiencies were significantly influenced by the pingeometry; in which case, increasing tapering of the thermoelectricpins within the range of 2 � a � 4 (a being the tapering slope)resulted in improved first and second law efficiencies. A multi-physics simulation of a thermoelectric generator was carried outby Li et al. [11]. They showed that the thermal radiation effect onthe thermal and electric performance was negligible and the tem-perature at the junction of the thermoelectric module remainednon-uniform. Optimal design of a novel thermoelectric generatorwith linear-shaped structure under different operating tempera-ture conditions was presented by Jia and Gao [12]. They indicatedthat decreasing total length and/or increasing pin height couldimprove the device output power. Thermal characteristics ofcombined thermoelectric generator and refrigeration cycle werestudied by Yilbas and Sahin [13]. They demonstrated that thelocation of the thermoelectric generator in between the condenserand the evaporator decreased coefficient of performance of thecombined system. Alternatively, the location of thermoelectricdevice in between the condenser and its ambient enhanced

coefficient of performance of the combined system. Thermoelectricdevice with coaxial rotated-leg configuration was examined byErturun and Mossi [14]. They showed that the effect of rotated-legconfiguration on thermal stresses and conversion efficiencies wereless than 1.2% and 0.3%, respectively; therefore, considerations ofrotating-leg configurations for the thermoelectric device designwas not promising.

Thermal stress development in thermoelectric generator underthe various operating conditions is important for the operationalsustainability of the device. Therefore, investigation of the thermalstress development in thermoelectric generators becomes essen-tial. Considerable research studies were carried out to examinethermal stress development in thermoelectric generators. A nu-merical simulation of the temperature gradient and thermal stressfields in a thermoelectric generator was carried out by Wu et al.[15]. They showed that under high heat flux imposing upon the hotend, the thermal stress was considerably high and it had a decisiveeffect on the life expectation of the device. Cracking and thermalshock resistance of the bismuth telluride based thermoelectricmaterial was examined by Wang et al. [16]. They indicated that themaximum stress intensity factor increased with increasing of thethickness of the plate and the thermal shock resistance wasimproved when the plate became thinner. Thermal stress analysisof thermoelectric power generator and influence of pin geometryon device performance was studied by Al-Marbati et al. [17]. Thefindings revealed that the thermal efficiency improved for certaingeometric configuration of the device; inwhich case; themaximumthermal stress developed in the pin reduced slightly indicatingimproved life expectation of the device. Thermally induced inter-facial shearing stress in a thermoelectric module with low frac-tional area coverage was investigated by Ziabari et al. [18]. Theydemonstrated that the shearing stress could be effectively reducedby using thinner (smaller fractional area coverage) and longer (inthe through thickness direction of the module) legs. Thermalstresses development in a multilayered thin film thermoelectric

B.S. Yilbas et al. / Energy 114 (2016) 52e6354

device was examined by Jin [19]. He indicated that the thermalstress calculated using the laminate model had a significantlyhigher magnitude than that predicted by the strength of materialsmodel.

Although thermal stress development in a thermoelectricgenerator was studied earlier [17,20], the parallel pin configura-tions were considered and the thermal stress development forhorizontally extended pins incorporating the electrodes was left forthe future study. In the present study, thermal analysis of a ther-moelectric generator with horizontally extended pin configurationsis carried out. This pin arrangement provides to alter the coldjunction temperature independent of the pin type. The thermalefficiency and power output of the thermoelectric generator isformulated after considering two different cold junction tempera-tures. In this case, temperature parameter is introduced to accountfor the cold junction temperature differences between p-type andn-type device pins, while keeping a single high temperature junc-tion. The thermal stress analysis is also carried out numericallyincorporating the finite element code ABAQUS. The study isextended to include the tapering effect of the pin geometry on thethermal stress field.

2. Heating and thermal stress analysis

The thermal efficiency is carried out using the analyticalformulation and thermal stress analysis are simulated numerically.The mathematical analysis related to the efficiency analysis andnumerical simulations is provided under the appropriate sub-headings.

2.1. Thermal efficiency and device output analysis

Consider the thermoelectric generator as shown in Fig. 1 withhigh temperature junction at temperature T1 and low temperatureends at T2p and T2n, respectively. Let T2p ¼ T2 andT2n ¼ T2p� DT¼ T2 � DT. The objective is to investigate the effect ofchanging one of the low end temperatures (i.e. reducing the endtemperature of the n-type pin, for example) on the device efficiencyand output power.

The efficiency of the thermoelectric power generator withgeometrically identical pins, as shown in Fig. 1, is given as

h ¼ I2RLaIT1 þ Kp

�T1 � T2p

�þ KnðT1 � T2nÞ � 12I2R

(1)

where

Kp ¼ ALkp; and Kn ¼ A

Lkn (2)

are the thermal conductances of p- and n-pins, respectively, and

Fig. 1. Schematic view of thermoelectric power generator.

R ¼ LA

�1ke;p

þ 1ke;n

�(3)

is the overall electrical resistivity of the thermoelectric generator.a ¼ ap � an is the Seebeck coefficient.

The current I is a function of Seebeck coefficienta ¼ ap � an, theupper and lower junction temperatures (T1 and T2), the electricalresistance R and the external load resistance RL and can be writtenas

I ¼ ap�T1 � T2p

�� anðT1 � T2nÞRL þ R

(4)

Noting that T2p ¼ T2 and T2n ¼ T2p � DT ¼ T2 � DT, Equation (4)can be re-written as

I ¼ laðT1 � T2ÞRL þ R

; (5)

where

l ¼ 1� ana

DTT1 � T2

; (6)

in which a ¼ ap � an.On the other hand the term Kp(T1 � T2p) þ Kn(T1 � T2n) in

Equation (4) can be written as

Kp�T1 � T2p

�þ KnðT1 � T2nÞ ¼ gKðT1 � T2Þ (7)

where

g ¼ 1þ Kn

KDT

T1 � T2; (8)

in which K ¼ Kp þ Kn.Substituting Equations (5) and (7) in Equation (1) the efficiency

becomes

h ¼ l2a2ðT1 � T2ÞRLla2T1ðRL þ RÞ þ gKðRL þ RÞ2 � 1

2l2a2ðT1 � T2ÞR

(9)

Assuming that g z l, Equation (9) becomes

h ¼ la2ðT1 � T2ÞRLa2T1ðRL þ RÞ þ KðRL þ RÞ2 � 1

2 la2ðT1 � T2ÞR

(10)

Equation (10) can be written in dimensionless form as

h ¼lZTaveðt1 � t2Þ

�RLR

�

ZTavet1

�RLR þ 1

�þ�RLR þ 1

�2

� 12 lZTaveðt1 � t2Þ

(11)

where

ZTave ¼ a2

KR

�T1 þ T2

2

�(12)

and

t ¼ TTave

and Tave ¼ T1 þ T22

: (13)

Since, t1þt22 ¼ 1 therefore t ¼ 22 � t1 and the efficiency of the

thermoelectric device, finally becomes

B.S. Yilbas et al. / Energy 114 (2016) 52e63 55

h ¼2lZTaveðt1 � 1Þ

�RLR

�

ZTavet1

�RLR þ 1

�þ�RLR þ 1

�2� lZTaveðt1 � 1Þ

(14)

The efficiency becomes a function of four parameters, l, ZTave, t1and RL/R as can be seen from Equation (14).

On the other hand, the power generation from the thermo-electric power generator is given as

_W ¼ I2RL ¼l2a2ðT1 � T2Þ2

ðRL þ RÞ2RL (15)

or

_W ¼l2ðKTaveÞZTaveðt1 � t2Þ2

�RLR

��RLR þ 1

�2 (16)

or, in dimensionless form

_WKTave

¼ 4l2ZTaveðt1 � 1Þ2

�RLR

��RLR þ 1

�2 (17)

which is also a function of the four parameters mentioned above.

2.2. Temperature and thermal stress analysis

Heating analysis pertinent to thermoelectric generator isinvolved with the 3-dimensional transient solid body heat con-duction and convection and radiation boundary conditions at theouter surfaces of the device. Since copper electrodes are introducedat the pin interfaces in the hot and cold junction plates to reducethe electrical resistance, temperature and heat flux continuity areassumed at the interfaces. In the analysis, rectangular and taperedpin configurations are incorporated. Fig. 2a and b show the ther-moelectric geometric configurations. The Fourier heat transferequation for the heating process can be written as:

DEDt

¼ ðVðkVTÞÞ (18)

where E is the energy gain of the substrate material and k is thethermal conductivity. The boundary conditions for the thermalanalysis include constant temperature boundary at the heat (523 K)and sink (300 K) plates, which are made from ceramic. The thermalproperties of the hot and cold junction plates and copper electrodesare given in Tables 1 and 2. Natural convection and radiation fromthe outer surfaces of pins, electrodes, and junction plates areconsidered, where they are assumed to be subjected to the atmo-spheric ambient. The heat transfer coefficient for natural convec-tion and ambient temperature are considered to be h ¼ 20 W/m2

and 300 K, respectively. Initially, it is assumed that all the ther-moelectric parts including junction plate, copper electrodes, andpins are in thermal equilibrium with same temperature at 300 K.The pin material, which is bismuth telluride and copper electrodeproperties are given in Table 3. Equation (18) is solved numericallywith the appropriate boundary conditions to predict the temper-ature field in the thermoelectric generator.

Temperature variation in the thermoelectric generator results inthermal stress filed development in the device materials. The

formulation of the total strain vector, {Dε}, may be expressed asfollows:

{Dε} ¼ {Dεel} þ {Dεth} þ {Dεpl} (19)

where {Dεel} is the elastic strain increment vector, {Dεth} is the

thermal strain increment vector, {Dεpl} is the plastic strain incre-

ment vector. The incremental thermal strain vector, {Dεth} arises

from the volume changes that accompany the temperature incre-ment,DT, which is calculated by the thermal analysis. It is normallyaccounted for in stress analyses through a temperature-dependentdifferential thermal expansion coefficient, aT. In the ABAQUS/Standard analysis, a spatially varying thermal expansion can bedefined for homogeneous solid continuum elements by using adistribution which includes the tabulated values for the thermalexpansion [21]. ABAQUS uses an implicit backward-differencescheme for time integration of both temperature and displace-ments at every material integration point [21].

A sequentially coupled heat transfer analysis is used in whichthe stress/deformation field depends on the temperature field.Nodal temperatures are stored as a function of time in the heattransfer results and are read into the stress analysis as a predefinedfield; the temperature varies with position and time. In thesequential thermal-elastic-plastic stress analysis, 91572 elementsare used to create the model using two element types; for the heattransfer analysis, mesh used elements of type DC3D8 (8-node linearheat transfer tetrahedron) and stress analysis used correspondingto the continuum C3D8R (8-node Linear 3D stress tetrahedron),which is compatible with the heat transfer element type. C3D8element type can handle complex nonlinear analyses involvingplasticity, and large deformations.

3. Validation

To validate the predictions of thermal stresses in the thermo-electric pin, the simulations are extended to include the casepresented in the previous study [22] and the simulations condi-tions are changed to be in line with the previous study [22]. In thiscase, the heating situation is introduced for a paralleled pin lengthbismuth telluride thermoelectric device with pin length ofdifferent sizes, i.e. 0.5 mme5 mm. The thermal stress predictedfrom the present study and the results of the previous study isshown in Fig. 3. It can be observed that both results are in goodagreement. Moreover, the validation study is extended to includethe comparison of the simulation results with the data presentedin the previous study [22]. In this case, the simulation conditionsare modified to the conditions presented in the previous study for1 mm pin length and 100 �C temperature differences between thecold and hot junctions. The maximum von Mises stress predictedfor the static thermal loading condition is 57 MPa and the datapresented in the previous study is 54 MPa [22]. The findingsrevealed that both results are in good agreement. The small dif-ference in the results can be attributed to the mesh used in thepresent study and assumption of isotropy in the previous study[22]. Nevertheless, this difference is considerably small. Thethermal stress developed in the device pin is further simulated tocompare the predictions of the maximum equivalent stress to thatof the previous data obtained for the rectangular pin shape [23].The simulations conditions are altered to accommodate the rect-angular pin configurations as similar to those reported in theprevious study [23]. In this case, the followings are incorporatedto describe the pin geometric configurations: the pin spacing (s) is1.25 mm, pin height (h) is 3.6 mm and pin width (w) is 1.2 mm[23]. The current predictions revealed that the maximum equiv-alent stress in the pin is in the order of 56.1 MPa while it is

Fig. 2. a. Thermoelectric generator assembly for a rectangular pin configuration. b. Thermoelectric generator assembly for a taper pin configuration.

B.S. Yilbas et al. / Energy 114 (2016) 52e6356

Table 1aThermal properties of ceramic.

Temperature, K Thermal conductivity, (W/m K) Specific heat, (J/kg K)

300 37.06 786.19400 28.19 939.53500 21.81 1019.86600 17.23 1071.45

Table 3Properties of pin material (bismuth telluride) used in the simulations. TheSeebeck coefficient e 170 mV/K for the n-type material and 160 mV/K for thep-type pin material.

Thermal expansion coefficient (1/K) 1.3E-05Yield Stress (MPa) 100Modulus of Elasticity (GPa) 40Poisson Ratio 0.33Density 7730Thermal conductivity (W/mK) 2Specific Heat (J/kgK) 16

Fig. 3. Validation of present simulations with the previous study [22].

Fig. 4. Efficiency of thermoelectric generator as function of the cold junction temper-ature parameter l for different hot junction temperature t1. (ZTave ¼ 1 and RL/R ¼ 10).

B.S. Yilbas et al. / Energy 114 (2016) 52e63 57

reported as 54.3 MPa in the previous study [23]. Consequently, themodel incorporated in the present study has the sound base topredict the thermal stress fields in the thermoelectric generator.

4. Results and discussion

The thermoelectric generator and the thermal analysis pertinentto the device efficiency and output power is presented. The thermalstress development during the device operation is also simulated.The simulations of the stress field are extended to include the pintapering and thermal stress development.

The modified design of the thermoelectric power generatorenables to operate the device between one hot and two differentcold junction temperatures. However, the arrangement of two cold-junction temperature for n and p pins of the device alters the ef-ficiency and the power generation. Consequently, in the presentstudy, the efficiency of thermoelectric power generator is examinedfor different cold junction temperatures ratios and external to in-ternal load parameters ratios. The cold junction temperatures of n-type and p-type pins of the thermoelectric generator are consid-ered to be different and temperature parameter is defined throughl ¼ 1� an

aDT

T1�T2where DT is temperature difference between cold

junctions of p and n pins, T1 is the hot junction temperature, whichis the same for n-type and p-type pins, and T2 is the cold junctiontemperature of n-pin. The external load parameter ratio is definedas external resistivity over the electrical resistivity of the thermo-electric generator RL/R.

Fig. 4 shows the thermal efficiency of the thermoelectricgenerator with temperature parameter (l) for different tempera-ture ratios (t1 ¼ T1/Tave where T1 is the hot junction temperature)and load parameter (RL/R) ratio of 10. The efficiency increases withincreasing temperature parameter. This is more pronounced withincreasing temperature ratio. Since the Seebeck coefficients of n-type and p-type pins remain the same, the term an/a becomesconstant for all temperature parameters. Therefore, increasing theefficiency with temperature parameter is associated with the termDT

T1�T2, since T1 represents hot junction temperature and T2 is the

Table 1bMechanical properties of ceramic.

Temperature, K Modulus of elasticity, E, Pa Poisson's ratio Yield stress, (Pa)

300 3.80E þ 11 0.27 4.70E þ 09400 3.75E þ 11 0.272 4.5E þ 09500 3.71E þ 11 0.274 4.20E þ 09600 3.66E þ 11 0.276 3.60E þ 09

Table 2Properties of copper used as an electrode in between pin material and hot and cold junctions. Density ¼ 8860 kg/m3, Thermal Conductivity ¼ 69.21 W/mK, and Specificheat ¼ 377 J/kg K.

Temperature, K Modulus of elasticity, E, Pa Poisson's ratio Yield stress, (Pa) Expansion coefficient (1/K)

477 9.10E þ 10 0.34 2.10E þ 08 1.54E-05533 8.89E þ 10 0.34 1.57E þ 08 1.73E-05589 8.61E þ 10 0.34 1.50E þ 08 1.75E-05

Fig. 6. Temperature variation along the thermoelectric pin. x ¼ 0 represents the mid-location of the ceramic hot junction and x ¼ 0.0085 m corresponds to the mid-locationof the ceramic cold junction as shown in the small frame.

B.S. Yilbas et al. / Energy 114 (2016) 52e6358

cold junction temperature of p-type pin, which are set constant.Therefore, difference in cold junction temperatures of p-type and n-type pins is the influencing parameter on the efficiency. Howeverincreasing the difference between the cold junction temperaturesmeans lowering cold junction temperature of n-type pin. Conse-quently, reducing cold junction temperature of the n-type pin en-hances the efficiency considerably. The variation of the efficiencywith temperature parameter is almost linear, which is true for alltemperature ratios. Therefore, the effect of cold junction tempera-ture difference has almost a linear effect on the device efficiency.This effect magnifies with increasing temperature ratio, whichrepresents the ratio of hot junction temperature to average tem-perature. In other words, increasing temperature ratio enhancesthe difference between hot and cold junction temperatures. Sincethe slope of the efficiency variation with temperature parameterincreases with increasing temperature ratio the influence of tem-perature parameter on the device efficiency signifies at high tem-perature ratios.

Fig. 5 shows non-dimensional thermoelectric power withtemperature parameter (l) for different values of temperatureratios (t1). It should be noted that external load parameter is set RL/R ¼ 0. The dimensionless power generated increases withincreasing temperature parameter (l) for all the values of thetemperature ratio. Increase in the power generation is more pro-nounced at high temperature ratios Moreover, variation ofdimensionless power generation with temperature parameterappears to be almost linear for temperature ratio t1 � 1.2. In thiscase, the coupling effect of temperature parameter (l) and tem-perature ratio t1 is not significant as compared to effect of tem-perature parameter alone. However, as temperature ratioincreases, the non-linear behavior in dimensionless power gen-eration with temperature parameter takes place. In this case, thecoupling effect of temperature parameter and temperature ratioon the power generated becomes important. In accordance withEquation (17), the non-linear behavior is due to the presence oftemperature parameter square (l2) and (t1 �1) square. Therefore,increasing temperature ratio (t1) increases the term (t1 � 1) andthe influence of (l2) becomes dominant in Equation (17). In anycase, increasing temperature difference between the cold junc-tions of p-type and n-type pins enhances the power generation ofthe thermoelectric device. However, this increase becomes sub-stantial for high values of ratio of the hot junction temperature tocold junction temperature.

Fig. 6 shows temperature variation along the length for rect-angular and tapered pin geometric configurations and geometric

Fig. 5. Dimensionless power of thermoelectric generator as function of the coldjunction temperature parameter l for different hot junction temperature t1. (ZTave ¼ 1and RL/R ¼ 10).

location of the x-axis while Fig. 7 shows temperature contours inthe thermoelectric generator with rectangular (Fig. 7a) andtapered (Fig. 7b) pin configurations. It should be noted that inorder to make the assessment of radiation losses from the hightemperature junction, the thermal analysis simulations arerepeated for the cases incorporating with and without thermalradiation losses from the thermoelectric generator. Temperaturevariation across the hot junction is shown in Fig. 8 while Fig. 9shows the temperature contours with and without radiationloses cases. Temperature along the y-axis (Fig. 8a) and z-axis(Fig. 8b) varies slightly due to the natural convection incorporatedin the simulations. In this case, the peak temperature remains atthe center of the hot junction and temperature decays graduallytowards the jot junction edges. However, heat transfer from thehigh temperature ceramic junction is not significant to alter thetemperature distributions along the x and z-axes. The temperaturedifference due to with and without radiation cases is small alongboth axes. In the case of temperature variation along the centerlineof the pin for rectangular pin geometric configuration (Fig. 6),temperature decay appears to remain almost steady along the pinlength. This behavior is attributed to the thermal conductivity ofthe pin material, which changes slightly in a linear form withincreasing temperature. Consequently, this suppresses the sharpchanges of temperature along the pin length. In the case of taperedpin geometric configuration, temperature variation along the pinlength differs slightly as compared to that of the rectangular pingeometric configurations. Although the pin length for rectangularand tapered cross-sections is kept same in the simulations, tem-perature variation along the pin length is attributed to the con-vection losses from the pin surfaces. Therefore, convection lossesfrom the rectangular cross-section are larger than those corre-sponding to the tapered pin geometry. This gives rise to slightlyhigher temperatures for tapered configuration than that of rect-angular configuration. In the case of Fig. 9a, in which temperaturecontours are shown in three-dimensional pin geometries, thegradual decay of temperature along the pin length is notable.Moreover, temperature distribution across the cross-section of thepin is almost uniform despite the fact that the natural convectioncurrent lowers temperature in the close region of the pin surface.However, the temperature scale given in the Figure does notdemonstrate this situation clearly. It should be noted that tem-perature variation in the pin varies in between 300 Ke523 K,therefore, temperature scale is large and does not demonstrate thesmall changes in temperature across the pin cross-section. The

Fig. 7. Temperature contours in the thermoelectric generator: a) rectangular pin configuration, and b) tapered pin configuration.

B.S. Yilbas et al. / Energy 114 (2016) 52e63 59

similar arguments are also true for the tapered pin cross-section asshown in Fig. 9b.

Fig. 10 shows von Mises stress distribution along the length forrectangular and tapered cross-sectioned thermoelectric pins. Itshould be noted that x ¼ 0 represents the mid-location of the hightemperature source. von Mises stress remains almost same in theheat source, which is considered to be a ceramic material, becauseof the uniform temperature inside the ceramic material. A coppersheet is considered to be a wafer in between ceramic and the pinmaterial (bismuth telluride), which minimizes the electrical andthermal resistances between the heat source and the pin material[24]. Consequently, thermal expansion across the ceramic and thepin material causes excessive von Mises stress increase in thecopper electrode. In addition, sharp change in temperature acrossceramic and copper film (Fig. 6) gives rise to high thermal strainand stresses in this region. Consequently, the peak value of vonMises stress occurs at the interface between the copper film andthe ceramic heat source. Moreover, the mismatches in thermal

expansion coefficients between copper and pin material givecontributes to attainment of high von Mises stress in this region.However, von Mises stress reduces as the distance increase to-wards the pinmaterial. In addition, vonMises stress reduces belowthe steady value in the close region of the pin edge. This isattributed to the temperature gradient in this region, which is lessthan that of at the copper-pin material interface. It should be notedthat the mismatch of the thermal properties between copper andpin material causes sharp change in the temperature gradientbecause of large difference in thermal conductivities. Since theheating reaches quasi-steady during the operation of the ther-moelectric generator, heat flux across the copper-pin materialinterface remains the same for both surfaces of copper and pinmaterial. The temperature gradients in copper and pin material atthe interface are defined trough the thermal conductivity ratiosdue to the flux continuity at the interface. Consequently, sharpvariation in temperature gradient results in attainment of highthermal stress at the interface. Since the temperature gradient

Fig. 8. Temperature variation across the hot junction ceramics with and without ra-diation losses: a) along y-axis, and b) along z-axis.

Fig. 9. Temperature contours across the high temperature junction: a) with radiation,and b) without consideration of thermal radiation from the junction surfaces.

B.S. Yilbas et al. / Energy 114 (2016) 52e6360

reduces slightly as the distance increases towards the pin materialfrom the interface, this lowers von Mises stress in this region. Asthe distance further increases along the pin length, some smallvariations of von Mises stress are observed. However, towards theedge of the pin, where the low temperature junction is located, vonMises stress shows small variation in this region. Since the copperfilm is located in between the pin material and the ceramic sink(Fig. 2), the thermal properties mismatch and heat flux continuityacross the interface result in temperature gradient variation in thisregion. This gives rise to variation in vonMises stress in this region.Consequently, the mismatch in between the material propertiesgives rise to attainment of high values of the von Mises stress inthe interface region of the thermoelectric pin. This is true for theinterfaces in between the hot and cold junctions with the pinmaterial.

Fig. 11 shows von Mises stress contours in the thermoelectricgenerator for rectangular (Fig. 11a) and tapered (Fig. 11b) cross-sections while Fig. 12 shows von Mises stresses contours acrossthe pin cross-section at the interface locations of copper and pinmaterial for hot (Fig. 12a) and cold (Fig. 12b) junctions. von Misesstress attains high values at the pin joints and at the interface be-tween the copper electrode and the ceramic heat source as well asat the interface of pin and the copper electrode. However, vonMises stress attains the highest vales in the region close to theedges of the ceramic and copper electrode at the interface of heat

source and the copper electrode, which is also true for across theinterface between the copper electrode and the pin material. Thisbehavior is associated with the mismatch of thermal expansioncoefficients of the materials in the thermoelectric junctions andhigh temperature gradients across the interfaces. von Mises stressalso attains high values in the close region of the interface betweenthe pin material and copper electrode at the cold junction. Mis-matching of thermal properties of the pin material and the copperelectrode is responsible for the attainment of von Mises stresses inthis region. The close examination of stress contours at the inter-face of the cod junction demonstrates that von Mises remains highin the corners of the pin material. Consequently, convection coolingplays an important role in this region, which gives rise to attain-ment of large von Mises in this region. In the case of tapered pingeometry (Fig. 11b), similar arguments are also applied providedthat von Mises stress attains slightly low values in the thermo-electric pin and at the interfaces. Moreover, stress distribution

Fig. 10. von Mises stress along the x-axis for rectangular and tapered pinconfigurations.

B.S. Yilbas et al. / Energy 114 (2016) 52e63 61

across the interface and pin cross-section differs from that corre-sponding to the rectangular pin cross-section. In this case, vonMises at the side edges of the interface remains higher than that ofthe corners as observed for the rectangular pin configuration.Consequently, changing pin shape from rectangular to tapered re-duces von Mises stress levels and stress distribution particularly at

Fig. 11. von Mises contours in thermoelectric generator: a) recta

the interfaces.

5. Conclusion

Thermal analysis of the thermoelectric generator withextended pin configuration is carried out. The thermal efficiencyand device output power are formulated and the influence oftemperature difference at the cold junction of n- and p-type pinson the thermoelectric performance in terms of the efficiency andoutput power is examined. The stress distribution in the ther-moelectric generator is simulated using the finite element nu-merical code ABAQUS. The effect of pin tapering on the thermalstress field is also investigated. In the simulations, horizontallyextending pin geometry is adopted and copper electrodes arelocated at the junctions between the high and low temperaturejunction plates and the pin material. It is found that the effect ofcold junction temperature difference has almost a linear effect onthe device efficiency; in which case, the thermal efficiency in-creases with increasing temperature difference, which is morepronounced with at high temperature ratios. von Mises stressremains high in the region close to the hot junction interface. Theproperties mismatch between the ceramic plate, which acts as aheat source, and the copper electrode, which reduces the elec-trical resistance at the interface, is responsible for the attainmentof high temperature gradients and stress levels in this region.Since temperature and flux continuity are assumed at the hot andcold junctions and copper electrode interface, temperature

ngular pin configuration, and b) tapered pin configuration.

Fig. 12. von Mises contours in the close region of copper electrodes: a) rectangular pin configuration, and b) tapered pin configuration.

B.S. Yilbas et al. / Energy 114 (2016) 52e6362

gradient in the copper and ceramic differ because of differences inthermal conductivity of both materials. Consequently, high tem-perature gradients give rise to high thermal strain and stresses inthis region. von Mises stress remains lower in the cold junctionregion than that of the hot junction region. This is attributed to thelow values of temperature and its gradient in the cold junctionregion. von Mises stress attains high values in the edges of the pincross section, particularly in the corner regions, where convectionand radiation heat transfer from the pin towards its surroundingsis high. This occurs in the region close to the hot junction of thethermoelectric generator where temperature is high. von Misesdistribution changes slightly in the cold temperature region; inwhich case, von Mises stress distribution becomes almost uniformaround the edges of the pin. In the case of tapered pin configu-ration, von Mises stress remains lower along the pin as comparedthat of the rectangular cross-section. In addition, von Mises stressdistribution in the interface region and cross-section of thetapered pin differs from that of the rectangular pin cross-section.The present study provides in depth analysis of thermal stresses in

thermoelectric generators, which adds useful information to thethermoelectric generator research.

Acknowledgments

The authors acknowledge the funded project RG 1501 via sup-port of Thermoelectric Group formed by the Deanship of ScientificResearch at King Fahd University of Petroleum and Minerals,Dhahran, Saudi Arabia for this work.

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