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Journal of Electronic Materials ISSN 0361-5235 Journal of Elec MateriDOI 10.1007/s11664-015-4201-y

Cost Scaling of a Real-World ExhaustWaste Heat Recovery ThermoelectricGenerator: A Deeper Dive

Terry J. Hendricks, Shannon Yee &Saniya LeBlanc

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Cost Scaling of a Real-World Exhaust Waste Heat RecoveryThermoelectric Generator: A Deeper Dive

TERRY J. HENDRICKS,1,4 SHANNON YEE,2 and SANIYA LEBLANC3

1.—Power and Sensors Section, Thermal Energy Conversion Group, NASA-Jet Propulsion Labo-ratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA.2.—G. W. W. School of Mechanical Engineering, Georgia Institute of Technology, Atlanta,GA 30332, USA. 3.—Department of Mechanical & Aerospace Engineering, George WashingtonUniversity, Washington, DC 20052, USA. 4.—e-mail: terry.j.hendricks@jpl.nasa.gov

Cost is equally important to power density or efficiency for the adoption of wasteheat recovery thermoelectric generators (TEG) in many transportation andindustrial energy recovery applications. In many cases, the system design thatminimizes cost (e.g., the $/W value) can be very different than the design thatmaximizes the system’s efficiency or power density, and it is important tounderstand the relationship between those designs to optimize TEG perfor-mance-cost compromises. Expanding on recent cost analysis work and usingmore detailed system modeling, an enhanced cost scaling analysis of a wasteheat recovery TEG with more detailed, coupled treatment of the heat exchangershas been performed. In this analysis, the effect of the heat lost to the environ-ment and updated relationships between the hot-side and cold-side conduc-tances that maximize power output are considered. This coupled thermal andthermoelectric (TE) treatment of the exhaust waste heat recovery TEG yieldsmodified cost scaling and design optimization equations, which are now stronglydependent on the heat leakage fraction, exhaust mass flow rate, and heat ex-changer effectiveness. This work shows that heat exchanger costs most oftendominate the overall TE system costs, that it is extremely difficult to escape thisregime, and in order to achieve TE system costs of $1/W it is necessary to achieveheat exchanger costs of $1/(W/K). Minimum TE system costs per watt generallycoincide with maximum power points, but preferred TE design regimes areidentified where there is little cost penalty for moving into regions of higherefficiency and slightly lower power outputs. These regimes are closely tied topreviously identified low cost design regimes. This work shows that the optimumfill factor Fopt minimizing system costs decreases as heat losses increase, andincreases as exhaust mass flow rate and heat exchanger effectiveness increase.These findings have profound implications on the design and operation of vari-ous TE waste heat recovery systems. This work highlights the importance ofheat exchanger costs on the overall TEG system costs, quantifies the possibleTEG performance-cost domain space based on heat exchanger effects, and pro-vides a focus for future system research and development efforts.

Key words: Thermoelectric systems, cost analysis, cost scaling, energyrecovery, waste heat recovery

List of symbols

VariablesA Total heat exchanger area (m2)

ATE One n-type + p-type TE element area, TEcouple area (m2)

C00 TE material area-dependent manufac-turing costs ($/m2)

C000 TE material volumetric-dependent costs ($/m3)

(Received July 8, 2015; accepted October 30, 2015)

Journal of ELECTRONIC MATERIALS

DOI: 10.1007/s11664-015-4201-y� 2015 The Minerals, Metals & Materials Society

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CHX,C Cold heat exchanger cost coefficient[$/(W/K)]

CHX,H Hot heat exchanger cost coefficient[$/(W/K)]

Cp Exhaust flow specific heat (J/kg K)F TEG fill factorFopt Optimum TEG fill factorG TE system total cost per output power ($/W)I Electrical current (A)k Thermal conductivity of TE material (W/

m K)KC TE system cold-side conductance (W/K)Kexh Exhaust conductance (= _mCpe) (W/K)KH Effective hot-side conductance (W/K)KHX Additional hot-side thermal conductance

(W/K)KTE TE module thermal conductance (W/K)Kk Parallel leakage thermal conductance (W/

K)LTE TE element length (m)_m Mass flow rate of exhaust (kg/s)N Number of TE couplesP TE power output (W)Q Heat input from exhaust (W)QC Heat rejected (W)QH TE hot-side thermal input (W)Qloss Parasitic thermal loss at heat exchanger/

TE interfaces (W)Qk Parallel leakage heat (W)R TE module electrical resistance (X)Spn Junction seebeck coefficient (= Sp � Sn) (V/

K)T1 TE hot-side junction temperature (K)T2 TE cold-side junction temperature (= Tcold)

(K)TC Cold sink temperature (K)Texh Hot source (exhaust) temperature (K)TH Hot heat exchanger temperature (K)Tm Mean junction temperature (T1 + T2)/2 (K)U Overall heat exchanger heat transfer

coefficient (W/m2 K)UAh Over a l l h o t - s i de h e at e xch an g e r

conductance (W/K)Z* Optimum thermoelectric module figure of

merit [= Spn2 /(R*KTE)] (1/K)

Greek letterse Heat exchanger effectivenessgTotal Total TE system efficiencygTE TE module efficiencyq Material electrical resistivity [= (qp + qn)/2]

(X m)r Heat loss fraction (= Qloss/Q)

INTRODUCTION

Various commercial and military vehicles andindustrial process systems create and dissipateenormous amounts of waste thermal energy globally

every year. Various research and developmentprojects have designed and developed advancedthermoelectric (TE) materials and systems torecover thermal energy in high-temperature trans-portation, industrial, and military energy systems.Thermoelectric generators (TEGs) for high-temper-ature, waste heat recovery applications have bene-fitted from significant TE materials advancementsin these projects. The advancements enable TEGswith power output capabilities on the order �100–1000 kW/m2 and system-level efficiency of >7%.The barrier to commercial TEG applications lies inoptimizing system cost, which should be approxi-mately $1/W1 to be competitive with currently usedpower technologies to foster market penetration.*Hence, system designs based on cost minimizationare particularly useful. Comprehensive and inte-grated TE system performance and cost modeling isan absolute requirement for optimizing energyrecovery systems, but little work has been donewith this coupled modeling. Consequently, there aremisconceptions and little quantifiable metrics toguide TE system designs in overcoming the systemcost barriers.

Previous work investigated the material, manu-facturing, and system component costs for TEGdesigns optimized for minimum system cost.1,2 Thework presented here utilizes the cost-performancemetrics and applies them to realistic applicationscenarios using detailed TEG system models basedon extensive prior work.3–5 Notably, the analysisemploys key, realistic system parameters by includ-ing heat lost to the environment from the exhaustsource, heat exchanger effectiveness, exhaust massflow rate effects, temperature-dependent TE prop-erties (for skutterudite materials), and multipleexhaust and cold-side temperatures. The impact ofcurrent, realistic heat exchanger costs and targetcosts are investigated and quantified for commonrepresentative waste heat recovery conditions.

MATHEMATICAL FORMULATION

The mathematical framework that describes thethermal and electrical operation of a TEG is wellestablished.6 Recently, this framework was revis-ited in an exhaust waste-heat recovery scenario,which allowed a fraction of the heat to leak from thehot-side and cold-side heat exchangers.3–5 Morerecently, cost-performance optimizations based onthe aforementioned original mathematical frame-work were developed.2 Herein, we demonstratemodifications to this cost-performance optimizationin the scenario of exhaust waste heat recoverywhere a fraction of the heat is allowed to leak fromthe hot-side and heat exchanger performance is

*U.S. Department of Energy, Office of Energy Efficiency andRenewable Energy, Vehicle Technologies Program, Multi-yearProgram Plan, 2011–2015, December 2010; US Department ofEnergy, Sunshot Vision Study, 2012.

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included. The thermal resistance network for thisscenario is shown in Fig. 1a.

Using the e-NTU methodology,7 the thermalconductance corresponding to the exhaust heatexchanger can be expressed as a product of theheat exchanger effectiveness, e, the exhaust flowspecific heat, Cp, and the mass flow rate of exhaust,_m. In this methodology, heat from the exhaust is

transferred to the hot-side heat exchanger, which isat intermediate temperature, TH, before the heat istransferred to the TE module. It is also at thistemperature that heat can be lost (i.e., leaked to thecold-sink). The fraction of heat that is lost, r, can beexpressed as the ratio of that heat lost, Qloss, to theheat delivered by the exhaust, Q. Performing anenergy balance at the heat exchanger temperaturenode yields an expression for the heat entering thehot-side of the TE module, QH, which is similar tothe expression given by Hendricks andLustbader.3,5,8

QH ¼ Texh � T1ð Þ1

_m�Cp�e� 1�rð Þ þ 1KHX

h i ð1Þ

From this expression, one can define an effectivehot-side conductance:

1

KH¼ 1

_m � Cp � e � 1 � rð Þ þ 1

KHX

¼ KHX þ _m � Cp � e � 1 � rð Þ_m � Cp � e � 1 � rð Þ �KHX

ð2Þ

that simplifies the thermal resistance network inFig. 1a into the more familiar (traditional) thermalresistance network in Fig. 1c. One can see that this

effective hot-side conductance behaves like a vari-able thermal resistor whose value depends on theheat loss fraction, r, mass flow rate of exhaust, _m,and heat exchanger effectiveness. This modificationfurther propagates, and energy balances around thehot and cold junctions yield a system of non-linearequations that can be numerically evaluated todetermine the exact junction temperatures.1,2

KH Texh � T1ð Þ � KTE þ Kjj� �

T1 � T2ð Þ

�S2

pn T1 � T2ð ÞT1

2RþS2

pn T1 � T2ð Þ2

8R¼ 0 ð3aÞ

S2pn T1 � T2ð Þ2

4R� KH Texh � T1ð Þ þ KC T2 � TCð Þ ¼ 0

ð3bÞ

These expressions are identical to those given byYee2 and LeBlanc1 with minor modification ofinterpretation of the subscripts; the hot-side sourcetemperature is now represented by Texh and KH is avariable conductance rather than a static conduc-tance. Finally, Hendricks4 showed that to maximizepower output in a TEG waste heat recovery system,the cold-side conductance should be 109–209greater than the hot-side thermal conductance.Thus, for a fixed (static) hot-side conductance, thereexists a maximum variable cold-side conductancethat should be targeted to maximize power output.This analysis demonstrates the design impactswhen hot-side heat exchanger thermal leakage,exhaust mass flow rate, and heat exchanger effec-tiveness are included.

Fig. 1. (a) Thermal resistance network for exhaust waste-heat recovery including leakage from the hot-side heat exchanger. (b) Heat andelectrical energy flows. (c) Equivalent (traditional) thermal circuit.

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To understand the cost-performance in this leaky,exhaust-waste-heat recover scenario, consider thetotal system cost normalized by the power output, P,given by Yee et al.2 and LeBlanc et al.1 The systemcost ($/W) is given by:

G $=Wð Þ ¼ Total TEG Costs

P¼ Total TEG Costs

gTE �QH

¼ Total TEG Costs

gTE � 1 � rð Þ �Q ; ð4Þ

and is directly related to the efficiency, gTE (= P/QH),and either the heat entering the hot-side of the TEmodule, QH after leakage, or the total heat deliveredincluding leakage, Q, given by Hendricks andLustbader.3–5 This thermal efficiency discountsthermal leakage from the hot-side heat exchangerand can be expressed as:

gTE ¼ P

QH¼ T1�T2

T1

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ ZTm

p� 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ ZTm

p� T2

T1

!ð5Þ

It is important to note that this expression for theefficiency is subject to the condition that the elec-trical load on the TEG is greater than the internalresistance of the TEG by a factor of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ ZTm

p,

which, to first-order, is the load that maximizes theefficiency of the TEG. It is also important to notethat T1 and T2 are the junction temperatures, notthe hot-source (exhaust) temperature, Texh, or thecold-sink temperature, Tsink. Finally, we re-empha-size that this is the efficiency discounting thermalleakage and is a factor of 1/(1 � r) larger than thetotal system efficiency (gTotal = P/Q).

The total TEG costs can be expressed as:

Total TEG Costs $ð Þ ¼ C000 � LTE þ C00ð Þ � A � Fþ CHX;H � KH þ CHX;C �KC; ð6Þ

which is very similar to the expression provided byLeBlanc et al.1 and Yee et al.,2 but instead the heatexchanger cost coefficient and heat exchanger con-ductance for the hot and cold side are allowed to bedifferent. However, since the heat exchanger costcoefficients are likely similar [i.e., �1–10 $/(W/K)],the cost will likely be dominated by the cold-sideheat exchanger costs based on work by Hendricks.4

CHX;H � CHX;C and KC � 20KH

Total TEG Costs $ð Þ � C000 � LTE þ C00ð Þ � A � Fþ CHX;C �KC ð7Þ

Furthermore, the total TEG costs are a function ofthe fill factor, F. It is important to remember thatthere is no fill factor that maximizes the efficiencyor minimizes the cost. However, as given by Yeeet al.,2 there exists an optimum fill factor (definedby a point of diminishing returns) where reducingthe fill factor below this value results in only a

marginal reduction in the $/W cost. The optimumfill factor given by Yee et al.2 is:

Fopt �1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCHX;HU2

H

C000k

sð8Þ

Here the optimum fill factor is determined by the hot-side heat exchanger values as the limiting condition,because they are smaller than the cold-side heatexchanger values. This shows that the optimum fillfactor is dependent on the thermal loss factor,r,exhaust mass flow rate, and heat exchanger effec-tiveness through the hot-side heat exchanger heattransfer coefficient UH. When heat is not lost,

r ! 0 ð9Þ

a larger fill factor allows that heat to be effectivelyconverted to power. Conversely, as more heat is lost,the device should be designed with smaller fillfactors to allow for better-cost savings of the system.

Next, returning to Eq. 4, the cost-performancevalue G can be explicitly expressed as:

G ¼ C000 � LTE þ C00ð Þ � A � F þ CHX;H � KH þ CHX;C � KC

1 � rð Þ � T1�T2

T1

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ Z�Tm

p�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ Z�Tm

pþT2

T1

� �� KH � Texh � T1ð Þ

ð10Þ

where Z* = Spn2 /(R*KTE). For the calculations that

follow, we set KC = 10 KH where we employ thisrelationship between KC and KH developed byHendricks4 to maximize power output. Equation 10gives one a first insight as to how real system heatlosses affect system-level costs per watt, withincreasing r generally increasing system costs perwatt by a factor on the order of 1/(1 � r)2, (i.e.,�¶G/¶r). Therefore, while it has been intuitivelyand mathematically well known for years thatminimizing system heat losses increases systemperformance, one can now also see that minimizingheat loss also minimizes system-level costs. In fact,cost per watt scaling directly with the non-dimen-sional factor 1/(1 � r) and cost per watt increasingas r increases by �¶G/¶r shown above. In this workwe generally set r = 0.1 in order to demonstrate thelowest attainable costs of a real system, but empha-size that in real systems there may be higher heatlosses that must be quantified. Furthermore, Eq. 10also provides one insight into how cost per wattdepends on heat exchanger mass flow rate andeffectiveness implicitly via the KH term throughEq. 2. However, cost per watt dependence on thesetwo heat exchanger parameters is complex becauseKH is in both the numerator (system cost) anddenominator (power itself). Finally, for simplifica-tion we let CHX,H = CHX,C.

To further illustrate the cost-scaling, Eq. 10 canbe reorganized into other non-dimensional groups tobetter illustrate the primary driving factors that

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influence cost as done by Yee et al.2 In thisapproach, it is necessary to define a characteristicthermal length:

LT ¼ k

UH; ð11Þ

by which all the system lengths scale. Given thisdefinition, by rearranging and unpacking Z * T inEq. 10, as done by Yee et al.,2 this work recovers amodified cost-scaling factor:

Go ¼ 16 � C000 � q � L2T

S2pn � T1 � T2ð Þ2

ð12Þ

This factor has units of $/W and is simply a result ofalgebraic manipulation of Eq. 10 into a non-dimen-sional group (e.g., G/G0 is a non-dimensionalgroup). This modified cost-scaling factor differsslightly from the one previously develop by Yeeet al.,2 in that it is a function of the junctiontemperatures and does not contain a simplificationrelating the junction temperatures to the reservoirtemperatures, and in that the electrical resistivity isused instead of the electrical conductivity for clean-liness of notation in this document.

KEY PARAMETRIC SENSITIVITIES

Equations 10 and 12 show the mathematicalrelationship of what design parameters enter intothe TEG system cost-per-watt analysis and howthey affect the total calculation. Further in-depthanalysis of the different design parameters andtheir relative magnitude of influence on overallcosts reveals that TEG system costs are controlledby the following six parameters:

� Texh

� T2/T1 (TE cold-side junction temperature to hot-side junction temperature ratio)

� TE module Z * T (primarily the module thermalconductivity k, and separately the module powerfactor)

� KH (and more importantly, KC/KH)� CHX

� r

KH and KC strongly influence G, as they are theprimary drivers for relating the reservoir tempera-tures to the junction temperatures. Furthermore,the exhaust temperature and the cold sink temper-atures are also key sensitivity parameters; higherexhaust temperatures or lower cold-sink tempera-tures will always result in lower $/W values scalingby Eq. 10. Therefore, waste heat recovery applica-tions with these characteristics will benefit fromthis fundamental cost-scaling dependency. Finally,it is necessary to recognize that LT is a function ofthe module’s effective thermal conductivity k. Thus,the module Z * T parameters strongly influence thecost scaling through LT and the power factor

recognized in Eq. 12. Because the controlling termsin Eq. 10 are the heat exchanger costs governed byCHX,C and CHX,H, their values are critical in Eq. 10.Real-world TEG system applications are often con-strained by potentially high heat exchanger costs,which must be controlled and minimized to makeTEG systems more economically attractive.

Heat Exchanger Costs

LeBlanc et al.1 and Yee et al.2 have demonstratedthat the cost of heat exchangers can be character-ized by CHX * (UA)HX, such that higher heatexchanger conductance (UA) results in larger heatexchanger costs within Eqs. 6 and 7. However, theUA term is itself a combination of two heatexchanger attributes: U (W/m2 K), the overall heattransfer coefficient that is described by Kays andLondon,7 and A (m2), the total exchanger heattransfer area. One generally pays for enhancementsin either one or both of these terms, as summarizedin Table I.

The costs of the various enhancements shown inTable I can impact the CHX values one actually usesin any given application, and can ultimately affector determine CHX sensitivity. Earlier work byLeBlanc et al.1 and Yee et al.2 has shown that CHX

is often in the range of $10/(W/K) to $20/(W/K) forreal-world heat exchangers. This work highlightsthis dependency in order to frame the ensuingdiscussions and guide future research and develop-ment work in bringing TEG systems to the forefrontof energy recovery solutions.

COST ANALYSIS RESULTSAND DISCUSSION

Several system cost analyses were conducted for773 K and 848 K hot-side exhaust temperatures,design parameters shown in Table II, and heatexchanger costs, generally focusing on a range ofheat exchanger cost from $1/(W/K) to $10/(W/K).These analyses focused on using the latest TEmaterial properties for skutterudite (SKD) materi-als, shown in Fig. 2,9,10 developed by NASA-JetPropulsion Laboratory (JPL) for advanced spacecraftpower system applications, which are now availablefor transportation and industrial waste energy recov-ery applications. Exhaust temperatures of 773 K and848 K are representative of engine exhaust and someindustrial process heat recovery applications, respec-tively. Current heat exchanger costs are on the orderof $10/(W/K), while target costs are on the order of $1/(W/K); an aggressive target requiring extensivefuture research and development investment toachieve. The key result is that TEG system costsare dominated by heat exchanger costs, even whenheat exchanger cost is as low as $1/(W/K). Figure 3shows where these two important realistic heatexchanger cost cases reside on the dimensionlesscost regime map developed by Yee et al.2 Thisillustrates that it is extremely difficult in real-world

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TEG systems, which require crucial heat exchangersfor high performance, to escape the heat exchangerdominated regime shown in Fig. 3.

The cost results demonstrate the controllingnature of the exhaust temperatures, where higherexhaust temperatures offer the opportunity to getmore heat out of the system and lead to lower cost.The minimum TEG system cost varies by approxi-mately 40% for the two exhaust temperatures.When the heat exchanger cost is $1/(W/K), theminimum TEG costs are $1.35/W and $1.00/W forexhaust temperatures of 773 K (e.g., engine exhaustapplications) and 848 K (e.g., industrial processapplications), respectively, for design conditions inTable II. These minimum cost values are consistentin the cases of either maximum efficiency or max-imum power. The minimum cost values are approx-imately one order of magnitude higher for heaterexchanger costs of $10/(W/K), since the TEG systemcost is dominated by the heat exchanger cost.

The results for the best cost scenario (exhausttemperature of 848 K and heat exchanger cost of$1/(W/K) are presented in Figs. 4, 5, 6, and 7 forthree representative cold-side temperatures (333 K,

363 K, 398 K), and thus three ratios of hot to cold-side temperatures. These cold-side temperatureswere generally selected to provide a reasonable,achievable range of cold-side temperatures toexpose and quantify critical design sensitivities inEq. 7. Figure 4 is simply the required TE materialarea (for a constant TE element length), and reflectshow the (LTE/ATE) changes for different TE hot-sidetemperatures shown in Figs. 5 and 6. It shows howATE is changing for different thermal and powerconditions, and its relationship to the maximum-power-point area requirement. Figure 5 shows theTE module efficiency versus power relationship andthe well-known tradeoff between hot-side heattransfer into the system and TE conversion effi-ciency in defining the maximum power point3–5 inthis map. The performance benefits of operating inthe ‘‘preferred TE design regime’’ are higher effi-ciency, higher power fluxes, and higher powerdensity.8 Figure 5 illustrates that when the tem-perature difference and temperature ratio are

Table I. Heat exchanger attributes governing costs

Heat exchanger attribute Enhancements that impact cost

U Complex fin and flow geometries, microchannel designs, two-phase flowA Footprint area, volume, material selections

Table II. Design parameters used in predicting representative system performance and cost

C000 ($/m3) C00 FTE elementlength (cm)

Exhaust massflow, _m (kg/s)

Heat lossfactor, r UAh (W/K)

8.66 9 10411 168.21,2 0.2 0.25 0.1 0.1 200, 400

Fig. 2. Recent TE material advances for high-temperature, high-performance spacecraft power systems. Materials currently availablefor transportation and industrial waste energy recovery.9,10 Fig. 3. Dimensionless cost versus dimensionless length showing

regions where each cost component dominates. The two casesmodeled here for realistic device and application parameters fallwithin the region where heat exchanger costs dominate.

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largest (i.e., cold-side temperature of 333 K), themaximum power generated increases dramaticallyto approximately 1020 W with a power flux of0.35 W/cm2 (3.5 kW/m2) under these system analy-sis conditions. The TE module efficiency at the peakpower point is approximately 5% with the p-typeand n-type skutterudite materials in Fig. 2 at thehot-side and cold-side temperature levels shown inFig. 5. In this chart, TE module efficiency can berelated to overall total TE system efficiency throughthe relation:

gTotal ¼P � 1 � rð Þ

QH: ð13Þ

Figures 5, 6, and 7 are used together to understandwhere power is maximizing and TEG system costsare minimizing in the possible design space for thecases analyzed, and how sensitive cost is to either thepower or efficiency. Figures 6 and 7 represent thenew information related to TE system costs from thiswork. Figure 6 shows how TE system costs vary withTE module efficiency and how those costs-per-wattminimize over a rather broad range of efficiencies,with system costs being relatively insensitive toefficiency over a broad range to either side of theminimum cost point. Because of this relationship,one can actually move into higher efficiency regimes

Fig. 4. Area of active thermoelectric material versus TEG power output for three cold-side temperatures [333 (red squares), 363 (blue triangles),398 (black circles) K].

Fig. 5. Maximum system efficiency versus TEG power output for three cold-side temperatures.

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(i.e., ‘‘preferred TE design regime’’) with very littlesystem cost penalty. However, one does not want togo too far, as power decreases will eventuallyincrease system costs-per-watt as illustrated inFig. 6. It is noteworthy that the ‘‘preferred TE designregime’’ in Figs. 5 and 6 coincide and show the lack ofsignificant cost penalty in this region, adding toalready noted benefits of higher TE efficiency andhigher TE system power density in this region.

The plots of TEG system cost versus power output(Fig. 7) and TEG system cost versus efficiency

(Fig. 6) demonstrate the impact of the heat exchan-ger cost. It is clear in Figs. 6 and 7 that maximumpower and minimum system cost points generallycoincide and occur at a lower system cost for lowercold-side temperatures. Since the heat exchangercost dominates the system cost, the system costvariation as a function of different cold-side tem-peratures is minimal around the minimum systemcost regions shown in Figs. 6 and 7, characteristic ofthe asymptotic behavior discussed by Yee et al.2

However, the system cost variation with cold-side

Fig. 6. Thermoelectric generator system cost in $/W versus maximum system efficiency for $1/(W/K) heat exchangers.

Fig. 7. Thermoelectric generator system cost in $/W versus power output for $1/(W/K) heat exchangers.

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temperature grows as one moves away from thisminimum system cost region. The dominance of theheat exchanger costs also collapses the cost relation-ship in Eq. 10 to one where essentiallyG � 1/P. If theheat exchanger cost did not dominate (e.g., if insteadthe TE material cost dominated), then the curves foreach cold-side temperature (or hot to cold-side tem-perature ratio) would separate out, and a strongersystem cost variation with cold-side temperaturewould be more apparent. Higher power results inlower system cost, as illustrated by the point at whicheach curve reverses (i.e., reaches its maximum powerand minimum cost point); after that point, more TEmaterial area and lower power results in highersystem cost-per-watt. It is clear that lower cold-sidetemperatures can drive TEG system costs down,emanating from the 1/P relation as lower cold sidesincrease power. Therefore TEG system designersshould strive to this goal for both cost and perfor-mance reasons in most waste heat recoveryapplications.

Additional system performance and cost analysiscases were performed for heat exchanger costs at$10/(W/K), with all other parameters in Table IIheld constant. The change in heat exchanger costsdoes not impact the results in Figs. 4 and 5.Figures 8 and 9 show the resulting TE system costsfrom Eq. 10 versus TE module efficiency and poweroutput, respectively, for $10/(W/K) heat exchangercosts. Although the basic relationships remainedsimilar to those in Figs. 6 and 7, it is clear thatsystem costs generally increase by roughly the orderof magnitude increase in heat exchanger costs. Inthis case, the TE system cost is even more domi-nated by the heat exchanger costs, as evidenced bythe even tighter collapsing of the cost relationshipsin Fig. 9. Unfortunately, at this point in time, heatexchanger costs of �$10/(W/K) are actually closer toreality and are the main driver as to why TE

systems generally run �$10/W in current applica-tions. In order to significantly decrease the TEsystem costs and bring them down to �$1/W, asdesired by many organizations worldwide, thisanalysis clearly shows that research and develop-ment to reduce heat exchanger costs by an order ofmagnitude are required.

Additional system cost analyses were also per-formed to investigate the effect of lower Texh andhigher UAh values. Higher UAh values lead tohigher KH values through higher heat exchangereffectiveness given by Ref. 7:

e ¼ 1 � exp�UAh

_m � Cp

� �ð14Þ

and Eq. 2. Table III gives the resulting TE systemcost-per-watt (i.e., G) and maximum power for fourdifferent system performance cases shown. At both

Fig. 8. Thermoelectric generator system cost in $/W versus maximum system efficiency for $10/(W/K) heat exchangers.

Fig. 9. Thermoelectric generator system cost in $/W versus poweroutput for $10/(W/K) heat exchangers.

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UAH values shown, increasing Texh from 773 K to848 K decreases the system cost-per-watt signifi-cantly at each cold-side condition. This is a directresult of Eq. 10, where increasing Texh in the finalterm of the denominator increases heat transfer intothe TE system, therefore increasing power outputand drivingG lower. One can therefore see that wasteenergy recovery applications with higher Texh condi-tions will necessarily have lower system costs.

In contrast, there is no effect on TE system costsper watt (i.e., G) from UAH increasing from 200 W/Kto 400 W/K at either Texh condition for any cold-sidecondition. However, the power output increasesdramatically as G stays constant when UAH, andtherefore KH, increases. This is also a direct resultof Eq. 10 for G. When the heat exchanger costsdominate in Eq. 10, KC/KH = 10–20, and KH andCHX,H dominate the costs, then one reaches a pointwhere the TEG costs in G (numerator of Eq. 10)become essentially proportional to KH throughCHX,H. At that point, both the TEG costs (numera-tor) and the TEG power (denominator) in Eq. 10 arebasically proportional to KH and by examination thevalue and effect of KH actually cancels out in Eq. 10;therefore, G mathematically becomes insensitive toKH, and, in fact, UAh as well through Eqs. 2 and 14.

The completely positive effect of minimizing TCold

(= T2) on both decreasing G and increasing poweroutput is also evident in the data of Table III, onceagain driving home the design requirement tominimize cold-side temperature conditions in indus-trial or transportation waste energy recovery appli-cations. This is a universal truth in waste energyrecovery.

Fill Factor Dependence on Thermal HeatLosses, Exhaust Mass Flow Rates,Heat Exchanger Effectiveness

The fill factor has direct influence over the cost. Inmany applications, especially for waste-heat recov-ery applications, the fill factor is fixed by othersystem considerations, such as mechanical robust-ness or pick-and-place machine tolerances. How-ever, if this parameter is allowed to be a free designvariable in addition to the TE leg length, then theperformance and cost of the TEG can be optimized

differently.2 To perform this optimization, one mustconsider the full free design domain and the result-ing $/W cost design space,2 which is a function ofboth the fill factor and leg length. In this designspace, there are two valleys; one that corresponds tothe optimum fill factor for a fixed leg length, andanother that corresponds to the optimum leg lengthfor a fixed fill factor. These two valleys converge,and this is the location of the preferred design TEdesign regime2,8 that minimizes the cost-per-wattvalue. This location is characterized by a fill factorgiven by Eq. 8 and by substituting Eq. 2 into Eq. 8,one can see the dependence of the fill factor to theheat loss and other thermal parameters.

Fopt � 12

ffiffiffiffiffiffiffiffiffiffiCHX;H

C000 �k

q� _m�Cp�e� 1�rð Þ�KHX

KHX�Aþ _m�Cp�e� 1�rð Þ�A

� �

) limKHX _m�Cp�eFopt ! 12

ffiffiffiffiffiffiffiffiffiffiCHX;H

C000 �k

q� _m�Cp�e

A

� �� 1 � rð Þ

ð15Þ

It is always desirable to maximize the interfaceconductance between the hot-side heat exchangerand the TE module; particularly into the regimewhere KHX _mCpe. Next, by Eq. 15, it is clear tosee that the optimum fill factor is linearly related toheat loss fraction, and has a maximum value whenthe heat loss is minimum. Thus, as discussedearlier, with more heat loss, the device should bedesigned with smaller fill factors using less activematerial in order to minimize the $/W system cost.It is also clear from Eq. 15 that the optimum fillfactor is dependent on the exhaust mass flow rate,_m, and heat exchanger effectiveness, e; the optimum

fill factor increases in most design cases as eitherthe exhaust flow rate or heat exchanger effective-ness increases. Hence, the optimum fill factor isinherently tied to the heat exchanger performanceand exhaust mass flow rate in any given wasteenergy recovery application.

CONCLUSIONS

This work has expanded on the TE system costanalyses by LeBlanc et al.1 and Yee et al.,2 byincorporating the results of Hendricks4 and thenincorporating these into TE system level analyses

Table III. Minimum TE system cost-per-watt point and maximum power as Texh and UAH vary—heatexchanger costs $1/(W/K)

Texh 773 K 848 K

T2 = Tcold 333 K 363 K 398 K 333 K 363 K 398 K

200 W/K $1.35/W740 W

$1.45/W690 W

$1.6/W630 W

$1/W1020 W

$1.07/W960 W

$1.2/W870 W

400 W/K $1.35/W850 W

$1.45/W795 W

$1.6/W730 W

$1/W1180 W

$1.07/W1110

$1.2/W1000 W

Multiply cost-per-watt values by 109 for heat exchanger costs of $10/(W/K).

Hendricks, Yee, and LeBlanc

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by Hendricks et al.3,5,8 The effects of heat exchangercosts and performance have been accounted for in anew, expanded analysis, and have shown andquantified the critical importance of heat exchangercosts in TE system cost analysis. TE system costsare generally governed by:

� Waste energy exhaust temperature, Texh

� TE cold-side junction temperature to hot-sidejunction temperature ratio, T2/T1

� TE module Z * T (primarily the module thermalconductivity k, and separately the module powerfactor)

� Hot-side thermal conductance, KH (and moreimportantly KC/KH)

� Heat exchanger cost factor, CHX [$/(W/K)]� Parasitic thermal losses, r

In general, TE system costs, G ($/W) are dominatedby heat exchanger costs everywhere in the range of$1/(W/K) to $10/(W/K) and above. It was discoveredthat to achieve a TE system cost of �$1/W, it isnecessary to achieve heat exchanger costs of �$1/(W/K); an aggressive cost goal that would requiresignificant research and development investment toachieve, since current costs are �$10/(W/K) orhigher. Minimum TE system costs generally occurat maximum power points, but not at maximum TEmodule or system efficiency or the highest powerdensity. Increasing Texh, decreasing T2/T1 andincreasing Z * T are all crucial to minimizing TEsystem costs-per-watt in all waste energy recoveryapplications. Therefore, energy recovery applica-tions with higher Texh benefit preferentially, andminimizing cold-side temperatures is required in allcases. Preferred TE design regimes have beenidentified where higher efficiency and higher powerdensity are possible, with little TE system cost-per-watt penalty and little power loss penalty. Thesepreferred TE design regimes coincide with theoptimum cost regions identified by Yee et al.2 Thisnew work has demonstrated that the optimum fillfactor, Fopt, minimizing TE system costs is depen-dent on parasitic thermal losses, exhaust mass flowrate, and heat exchanger effectiveness. Increasingexhaust mass flow rates and heat exchanger

effectiveness increase Fopt, while increasing para-sitic thermal losses decreases Fopt. This work high-lights the critical importance of the heat exchangercosts and performance in optimizing TE systemcosts in transportation and industrial waste heatrecovery applications worldwide.

ACKNOWLEDGEMENTS

This work was carried out under NASA Space ActAgreement No.43-17508, a contract between NASAand General Motors with funding from the U.S.Department of Energy, at the Jet Propulsion Labo-ratory, California Institute of Technology, under acontract to the National Aeronautics and SpaceAdministration.

REFERENCES

1. S. LeBlanc, S.K. Yee, M.L. Scullin, C. Dames, and K.E.Goodson, Renew. Sustain. Energy Rev. 32, 313 (2014).

2. S.K. Yee, S. LeBlanc, K.E. Goodson, and C. Dames, EnergyEnviron. Sci. 6, 2561 (2013).

3. T.J. Hendricks and J. Lustbader, in Proceedings of the 21stInternational Conference on Thermoelectrics (Long Beach,CA: IEEE Catalogue #02TH8657, 2002), p. 381.

4. T.J. Hendricks, in Proceedings, 1642, Materials ResearchSociety, mrsf13-1642-bb02-04 (2014). doi:10.1557/opl.2014.443.

5. T.J. Hendricks, J. Energy Resour. Technol. 129, 223 (2007).6. S.W. Angrist, Direct energy conversion, 4, Chapter 4 ed.

(Boston: Allyn and Bacon Inc., 1982), pp. 121–176.7. W.M. Kays and A.L. London, Compact heat exchangers, 3,

Chapter 2 ed. (New York: McGraw-Hill, 1984), pp. 11–78.8. T.J. Hendricks and D. Crane, Thermoelectric Energy

Recovery Systems: Thermal, Thermoelectric and StructuralConsiderations’’ CRC Press Handbook of Thermoelectricsand its Energy Harvesting: Modules, Systems, and Appli-cations in Energy Harvesting, Book 2, Section 3, Chapter 22(Boca Raton, FL: Taylor and Francis Group, 2012).

9. J.-P. Fleurial, S.K. Bux, B.C.-Y. Li, S. Firdosy, N.R. Keyawa,P.K. Gogna, D.J. King, J.M. Ma, K. Star, A. Zevalkink, andT. Caillat, in Symposium BB: thermoelectric materials–frombasic science to applications, Proceedings of 2013 MaterialsResearch Society Fall Meeting, Boston (2013).

10. S. Bux, J.-P. Fleurial, and T. Caillat, in Proceedings of 11thInternational Energy Conversion Engineering Conference(American Institute of Aeronautics and Astronautics, 2013).

11. J.-P. Fleurial and T. Caillat, Internal Jet Propulsion Labo-ratory Thermoelectric Material Studies Based on RawMaterial Suppliers (Washington, DC: Thermal EnergyConversion Technology Group, National Aeronautics andSpace Administration, 2010–2014).

Cost Scaling of a Real-World Exhaust Waste Heat Recovery Thermoelectric Generator: ADeeper Dive

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