Thermal Matching of a Thermoelectric Energy Scavenger with the ...

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Gsph∆ Ttp= ∆T. (4)Gsph+ ktpa+ kair( A − a)The open circuit voltage is given by:α ∆T GsphV =, (5)( Gsph+ kairA)+ a ( ktp− kair)where α is Seebeck coefficient. Expressing the electricalresistance of a thermopile through its resistivity ρ, the poweron the matched electrical load can be written as:222spα ∆TG haPel=4 ⋅, (6)ρ2[( Gsph+ K air A)+ a(ktp− k air )]At its derivative dP el /da = 0, the maximal powercorresponds to a = a opt = (G sp h + k air A) / ( k tp – k air ), sothat:2α ∆TG hPmax=. (7)216ρ( ktp− kair)Substituting a opt into Eq. (4), the temperature difference∆T tp , corresponding to the maximal power, is:tp,opt22sp∆TG∆ T =, (8)sp⋅2 Gsp+ GTEG,0where G TEG,0 = k air A / h denotes the thermal conductance ofthe same TEG, but with no thermocouples in between theplates, i.e., of the empty TEG; this is the parasitic thermalconductance of a TEG. Eq. (8) shows that if the thermalconductance of the air in empty TEG G TEG,0 > Ramb,0(13) and RTEG 0R amb , opt,>> . (14)In the optimized device, Inequalities (13) and (14) shouldhold, at least in a weak form (with “much more” replacedwith “more”), furthermore, in typical situations of the energyscavengers, the ambient resistance does not vary greatlywith temperature. For these reasons, Eq. (12) instead of Eq.(11) can be usually used as a condition for optimizing thedevice.Considering that∆T TEG,0 = R TEG,0 W TEG,0 and ∆T TEG,opt = R TEG,opt W TEG,opt , (15)the condition of Eq. (12) can be rewritten as:R TEG,opt W TEG,opt = R TEG,0 W TEG,0 /2 . (16)In those cases, where the thermal resistance of the ambiencedominates, the heat flow does not depend on the thermalresistance of a TEG, and Eq. (16) simplifies to:R TEG,opt = R TEG,0 /2 , (17)thereby stating that the thermal resistance of thermocouplesand of the air are equal to each other. This condition iswidely used in designing the TEGs. As the goal of theoptimization is to make the thermal resistance of the TEGcomparable or larger than the one of the ambience, in mostcases, Eq. (17) cannot be used and must be replaced withEq. (12) or, even better, with Eq. (11).We proceed now with the optimization of the TEG. First,we replace ∆T TEG,opt and ∆T TEG,0 in Eq. (11) with Eqs. (15).Then we eliminate the heat flows using:∆TWTEG,0= , (18)R + RWTEG,optamb,0amb,optTEG,0∆T= . (19)R + RTEG,optAfter such replacements, we solve Eq. (11) for R TEG,opt andobtain:RTEG,optRamb,optRTEG,0= . (20)2Ramb,opt+ RTEG,0Eq. (20) can be solved by iterations. In the beginning, thevalue of R amb,0 can be used instead of R amb,opt . Uponobtaining the first approximation value of R TEG,opt , the valuesof W TEG,opt and R amb,opt can be recalculated more accurately;the latter then can be used in the next iteration. Only severaliterations are usually required for excellent accuracy.
As far as only two parallel thermal resistors compose theTEG, the required thermal resistance of the thermopile R tp,optcan be easily obtained from the value of ∆T TEG,opt . Theoptimal area of thermoelectric material a in the TEG is:a opt h / ktpRtp,opt= . (21)The minimal number of thermocouple legs, while satisfyingthe requirement for the output voltage on the matched loadV m , is given by:2Vmn = . (22)α ∆TTEG,optFinally, the required cross section s of thermopile legsshould not exceed s = a / n .Thermal matching: the application on manAs an example of application of the method described,let us consider a thermopile with the same minimal lateralleg dimensions as in [1] with their size of 80 µm × 80 µm ×600 µm. For calculations, we assume that the hot plate has athermal contact with the skin in the outer side of the wristover a circular area of 3.14 cm 2 , while the radiating area is 7cm 2 , so that the device body resembles a watch. The hot andcold plates are separated by 1.3 mm, which is the thickness(1), however, this is because in our device the Inequalities(13) and (14), even in their weak form, are not satisfied.Small-size thermopiles available on the market do not fitthe requirements for the thermopile legs coming out of themodeling of an optimal thermopile because their aspect ratiois much smaller than needed. An appropriate aspect ratiothen can be obtained by stacking thermopiles on top of eachother [2]. This increases the R TEG and hence the outputpower. For example, in case of a 10-stage thermopile, thepower increases in 5.7 times. Larger number of stages couldfurther increase the power, but the device would be too thickand therefore inconvenient for the users.The normalized power for a 10-stage TEG, Fig. 2,coincides with the similar curve for a one-stage device. Theheat flow is, however, different. Inequality (14), even in itsweak form, is not yet satisfied in the 10-stage TEG; theR TEG,0 is still 84% of R amb,opt , however, the change in heatflow is already 44% at the matching point (1). It furtherFigure 2. Thermal matching of a TEG to the ambience:normalized power (solid line) and normalized heat flow(dashed lines). The arrows show the heat flowscorresponding to three marked points.of the thermopile including the ceramic plates. The powerproduced depends on the number of thermocouple legs,therefore, at a certain number, corresponding to the optimalcross area of thermoelectric material, the matching conditionof Eq. (11) is satisfied. The normalized power computednumerically for an air temperature of 22 °C is shown in Fig.2 versus the ∆T TEG /∆T TEG,0 ratio. Each point of the curvecorresponds to a different value of a. The maximal power,point (1), is observed at ∆T TEG = ∆T TEG,0 /2.The analysis of numerical simulations shows that this isbecause the ambient resistance is only weakly dependent ontemperature. In addition, the thermal resistance of theambience is larger than the one of the generator. So, thedevice is in a condition, where the heat flow is nearlyindependent of R TEG , Fig. 2, so that Eq. (17) approximatelyholds. Using Eq. (17) for determining the number ofthermocouples corresponding to the optimal power wouldhave generated a small error giving the point (2), Fig. 2. Onemay see that it is not very far from the true matching pointFigure 3. The ratios of the thermal resistance of thethermopile and of the TEG to the one of the ambience versusthe number of stages at thermal matching, i.e., at a=a opt .increases after satisfying Inequality (14) in its weak form.This confirms that heat flow cannot be assumed as constantwhile modeling and that Eq. (17) cannot be used foroptimization: its use would have given the point indicated as(3) in Fig. 2, far away from real maximum.The ratios R TEG,opt /R amb,opt and R tp,opt /R amb,opt are shown inFig. 3 versus the number of stages. One can notice that thethermal resistance of a TEG does not halve at the matchingpoint (i.e., R TEG,opt ≠ R tp,opt /2), reflecting the difference withboth parallel and serial matching.Some aspects of thermal matching of TEGs on humansAs shown above, multi-stage thermopiles allow thethermal matching, but it occurs at a very high effectiveaspect ratio, which is the ratio of the length of all thethermocouple legs on top of each other to their width. Thereare, however, some other helpful practical ways of easilytuning the thermal resistance of the TEG components tofulfill the thermal matching requirements at smaller aspectratio or at smaller number of the stages.First, when applying thermal matching conditions to theTEG on human skin, Inequality (14) in its weak formdemands to provide a thermal plate-to-plate isolation of atleast 2 000 cm 2 K/W, which can be done only at theirdistance of about 1 cm from each other. Anyway, this spaceis required for a multi-stage thermopile. On the other hand,the free/forced convection layer near the body, e.g., around
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