Performance Prediction of Commercial Thermoelectric Generator Modules using the Effective Material Properties

Transcription

1 Performance Prediction of Commercial Thermoelectric Generator Modules using the Effective Material Properties Abdulmunaem Elarusi Nithin Illendula Hassan Fagehi

2 ABSTARCT This work is employed to examine the validity of the thermoelectric modules performance predicted by formulating the effective thermoelectric material properties. The average temperature of the thermoelectric generator is used to formulate the three maximum parameters. The three maximum parameters which are taken from commercial thermoelectric generator module or measurements are used to define the effective material properties. These commercial modules were taken as to validate this work by substituting effective material properties in the simple ideal/ standard equation. The commercial performance curves provided by the manufacturer were compared with the present work result obtained by the simple standard equation with the effective material properties. The characteristics of the thermoelectric generators were represented using the normalised charts constructed by formulating the normalised parameters over the maximum parameters. The normalized charts would be universal for any given thermoelectric generator. 1

3 Table of Contents Introduction... 2 Thermoelectric Phenomenon... 2 Literature... 4 Thermoelectric Ideal Equations... 5 Performance Parameters of TEG module Maximum Parameters of TEGs Normalized Parameters.12 Effective material properties 14 Results and discussion TG12-4 by Marlow HZ-2 by Hi-Z TGM by Kryotherm TGM by Kryotherm Normalized curves Conclusion References Appendix MATHCAD Analytical

4 1. Introduction Thermo-Electrics is a science which is associated with thermal and electrical phenomena. Here, in the thermo-electrics we convert the electrical energy to the thermal energy and vice versa. In this we will use thermo-couple which consists of two dissimilar wires at two different temperatures and if we connect these two wires an electric potential is generated. Mainly, the whole thermo-electrics is depends on two devices only these are Thermo-Electric Generator and Thermo-Electric Cooler. There is a big advantage while using the Thermo- Electric devices is that there is no moving parts, no maintenance is required. The thermo-electric devices has a various applications the major application is that the producing the electricity these are used in the automobiles for the various application like cooling or heating the car seat which is now in the market as an extra fitting but installed in the high end models. We can also use the thermo-electric device (TEG) to produce the electricity by tapping the waste heat of exhaust gas which emitted after combustion of fuel. These thermoelectric devices are used in the space exploration robotic rovers for example the rover sent to the planet Pluto it uses the heat generated by the decay of radio isotope. We can also use the thermos electric devices (TEC) as a cooling device in the refrigerator by there are a lot advantages there is no pollution to environment due to danger of emitting the dangerous the greenhouse effect gases and also there is a quiet cooling operations as compared to conventional refrigerator. 1.1 Thermoelectric Phenomenon There are some governing effects on which the thermoelectric effect is depended: Seebeck Effect: The See back is the conversion of temperature difference into an electric current. Usually a temperature difference is maintained between a thermocouples so as to produce current. The relation between the voltage and temperature difference is: V=α*ΔT Here V is the voltage across the circuit, α is the Seebeck Coefficient, ΔT is the temperature difference of the hot and cold junctions 3

5 Figure 1: Seebeck effect in a circuit with two dissimilar materials with hot and cold junctions Peltier Effect: When a current flows across a junction between different wires i.e. thermo-couple, it is found that on the one side of junction heat is added and the other side heat is subtracted in order to keep its temperature constant. QPeltier = Π*I Here, QPeltier is the rate of heat transfer, I is the current generated in the circuit, and Π is the Peltier coefficient. Thomson Effect: Here, in this effect which is similar to the Peltier Effect in which the heat is liberated or added if an electric current flows across junction but in this liberation or addition of heat is also on depends on the flow direction of current. QThomson = τ*i*δt QThomson is the Thomson heat transfer, τ is the Thomson coefficient, I is the current flowing in the circuit, and ΔT is the temperature difference between hot and cold junctions. QThomson Wire A QPeltier Tc I Th QPeltier - + Wire B QThomson Figure 2: It shows both the Peltier without considering flow of current and the Thomson effect in it with considering flow of current. 4

6 Thermo-Couple: A Thermo-couple consists of a p-type and n-type semiconductor elements. In which these thermos-couples are connected thermally in parallel and electrically in series to form a thermo-electric module. Figure 3: The thermo-electric couple with two dissimilar material. 2. Literature As in the commercial modules the information provided by the different manufacturer is different like if one manufacturer shows a performance curve of COP v/s hot side temperature and other Manufacturer shows cooling power v/s Temperature difference this makes a lot for the person who buys the thermo-electric module for comparing. So, if they want to find the comparison then they have to take the modules and do experiments which cots a lot of time and money. So, an easy way to find the comparison between is by using the ideal equations even though in the ideal equations we cannot find the Seebeck coefficient, current I, Resistance R, Thermal Conductance K. Shiho Kim has find out the material properties analytically with function of internal plate temperature difference of p and n-type pellets. He formulated the heat absorbed and heat rejected in the hot and cold junctions respectively. And he formulated equations for thermal conductance for the overall module and for the pellets. He also find out equations for the intrinsic seebeck coefficient, electrical resistance for the internal plates. Using the heat equations first formulated he derived the ΔTe i.e. is the internal temperature difference. Then he taken the case when the load current is zero and he find out the ΔTe. Finally using the above 5

7 all values he find out he formulated equations for the material properties. He also drawn curves for current v/s voltage for various hot junction temperatures with temperature difference. Here, in this test performed by Shiho Kim he did not compared his values with the performance curves provided by the manufacturer. HE just find out the values analytically. But the D Angelo and Hogan has find out the values experimentally he used the vacuum enclosure and the constant heat source on the hot junction of module by the nickel chromium wire then with obtained results he drew the performance curves and those are in good agreement the curves provided by Tellurex. 3. Thermoelectric Ideal Equations: The general governing equations in this section are represented considering a nonuniformly heated thermoelectric material with isotropic material properties (Temperature independent). Therefore, the continuity equation for a constant current gives,. j = 0 (1) Where j is the current, and is the differential operator with respect to length. The electric field E (Or electric potential) is defined according to the ohm's low and Seebeck effect, because the electric field is influenced by the temperature gradient T and the current density j Thus, the electric field can be defined as, E = jρ + α T (2) Similarly, the heat flow (or heat flux) is effected by the electric field E and the temperature gradient T the heat flow density, consequently, is expressed considering the Thomson relationship and the Onsager's principle, as q = αtj K T (3) Where T is the temperature at the boundary through which the heat flux flows. The αtj is the Peilter heat contribution, and K T the heat transfer from the Fourier's low of conduction. The general heat diffusion equation as time dependent is given by, T. q + q = ρc p t Where q is the heat flux per unit volume, ρ is the density of the material, C p is the (4) 6

8 specific heat capacity, and T is the rate of change of temperature with respect to time. t Now, for steady state condition ( T goes to zero), we have, t q =. q (5) Since we can define the heat flux as a function in the electric power, q is expressed by, q = E. j = j 2 ρ + j. α T (6) Substituting equations (3) and (6) in equation (5) gives us,. (K T) + j 2 ρ T dα j. T = 0 (7) dt Where, from Thomson relation, τ = T dα is known as the Thomson coefficient. The second term in equation (7) is the Joule heating, and the third term is the Thomson heat. In special cases where the seebeck coefficient α is a temperature independent, the Thomson effect is negligible, which means that the Thomson coefficient τ is zero. As such, this work considers negligible Thomson effect and the seebeck coefficient is not a temperature independent. dt (a) 7

9 (b) Figure 4. (a) cutaway of a thermoelectric generator module, and (b) a p-type and n- type thermocouple. Figure.4 (a) illustrates a steady state one-dimensional thermoelectric module. This module contains on many p-type and n-type thermocouples as shown in figure.1b. Considering that the thermal and electric contact resistances are both negligible, no radiation or convection heat losses through the boundaries of the element, equation (7) can be reduced to, d dx (ka dt dx ) + I2 ρ A = 0 (8) Equation (8) is regarded as a differential equation that requires a set of boundary conditions to be solved. These boundary conditions can be defined simply ad functions of position x (T x=0 = T h and T x=l = T c ). Therefore, the solution for the temperature gradient can be found as, d dx (ka dt dx ) = I2 ρ A (9) By integrating equation (9), we get Which gives, ka d ( dt ) = dx I2 ρ dx (10) A dt dx = I2 ρ ka x + C 1 (11) 8

10 Where C 1 is the integrating constant. Equation (11) can be rearranged and integrated again as, T2 T1 dt = I 2 ρ L xdx ka 0 L + C 1 dx 0 (T 2 T 1 ) = I 2 ρ ka L2 + C 1 L (12) Equation (12) can be rewritten as, C 1 = (T 2 T 1 ) L + I 2 ρl 2kA 2 (13) Now, by substituting Equation (13) in Equation (11), (at x = 0) we get, dt = (T 2 T 1 ) dx x=0 L Similarly, at x = L Equation (11) yields, dt = (T 2 T 1 ) dx x=l L + I 2 ρl 2kA 2 (14) I 2 Equation (3) is expressed as following, ρl 2kA 2 (15) q x=0 = αt 1 I 1 2 ρ L A I2 + AK L (T 1 T 2 ) (16) More specifically, Equation (16) can be expressed for either p-type or n-type as, q px=0 = α p T 1 I 1 2 ρ p q n x=0 = α nt 1 I 1 2 ρ n L A I2 + Ak p L (T 1 T 2 ) (17) L A I2 + Ak n L (T 1 T 2 ) (18) By similar fashion the heat flux equation is carried out at x = L for the p-type and the n-type as following, q px=l = α p T 1 I ρ p q n x=l = α nt 1 I ρ n L A I2 + Ak p L (T 1 T 2 ) (19) L A I2 + Ak n L (T 1 T 2 ) (20) Then, the total heat flux at position (1) and (2) (see figure 4.a) can be found by taking the summation of Equations (17) and (18), (19) and (20) respectively as, 9

11 Q 1. Q 2. = n[(α p α n )T 1 I 1 2 I2 ( ρ pl p A P = n[(α p α n )T 2 I I2 ( ρ pl p A P + ρ nl n ) + ( k pa P + k na n ) (T A n L p L 1 T 2 )] (21) n + ρ nl n ) + ( k pa P + k na n ) (T A n L p L 1 T 2 )] (22) n Where n is the number of the thermocouples in the module. The material properties ae defined as, α = α p α n (23) R = ρ pl p A P k = k pa P L p + ρ nl n A n + k na n L n (24) (25) Where R and K are the total electric resistance and the total thermal conductance, respectively. This reduces Equations (21) and (22) to, Q 1. = n[αt 1 I 1 2 RI2 + K(T 1 T 2 )] (26) Q 2. = n[αt 2 I RI2 + K(T 1 T 2 )] (27) Equations (26) and (27) are known as the thermoelectric ideal equations. The first term in theses equations αt 1 I is known as Peltier (or Seebeck effect), and it is reversible. The second term 1 2 RI2 is the Joule heating term. The last term is known as the thermal conduction term. Both the Joule heating and the thermal conduction are irreversible. Figure.5 Thermoelectric generator attached to a load resistance. In the case of thermoelectric generator (TEG), the ideal equations can be modified with using respective hat and cold junction temperatures (Figure.5) as, 10

12 Q ḣ = n[αt h I 1 2 RI2 + K(T h T c )] (28) Q c. = n[αt c I RI2 + K(T h T c )] (29). Where Q ḣ And Q c Are the heat absorbed and dissipated at the hot and cold junctions of the TEG, respectively. Assuming that the p-type and n-type thermocouples are similar, we have R = ρ L and K = k A, where ρ = α A L p + α n and K = k p + k n. From the 1 st law of thermodynamics, the output power across the thermoelectric. generator module can be defined as W n = Q ḣ Q. c. Then, by substituting Equation (28) and (29) in the power equation, the output power can be expressed in terms of the internal properties as,. W n = n[αi(t h T c ) RI 2 ] (30) Also, the output power can be defined in terms of the load resistance R L as,. W n = ni 2 R L = IV (31) Now, equating Equations (30) and (31) yields to, ni 2 R L = IV = n[αi(t h T c ) RI 2 ] (32) Equation (32) can be reduced to, V n = nir L = n[α(t h T c ) RI] (33) Where V n is the voltage across the load resistance. 4. Performance Parameters of TEG module: The current can be obtained from equation (33) as, I = α(t h T c ) R L +R (34) It is clear form Equation (34) that the current is not a function of the number of thermocouples. Substituting Equations (34) in Equation (32) gives, V n = nα(t h T c ) R L R +1 ( RL R ) (35) 11

13 Inserting Equation (34) in Equation (19) gives, W n. = nα2 (T h T c ) 2 R R L R (1+ R L R )2 (36) Now, from the thermal efficiency definition η th = W n. Q ḣ, and by inserting Equation (34) in (31) then substituting the result with Equation (28) in the thermal efficiency equation we get, η th = (1 T c ) R L T h R (1+ R (1+ R 2 L T ) c L R ) 1 2 (1 T c R T )+ h T h ZTc (37) Where R L R is the resistance ratio, and ZT c is known as the dimensionless figure of merit, both of which are important in design of thermoelectric modules. 5. Maximum Parameters of TEGs: For any thermoelectric module, there are two types of maximum parameters, the first is the maximum parameter at the maximum power and the second is the maximum conversion parameters. These two modes can be obtained by modifying the resistance ratio R L R on the operating conditions. regarding The maximum current usually accrues at the short circuit, where the load resistance R L = 0. Thus, the maximum current is found from Equation (34) as, I max = α(t h T c ) R (38) Since the maximum voltage occurs at the open circuit, where I = 0, sitting the current to zero in Equation (33) yields, V max = nα(t h T c ) (39) For the maximum power, the output power Equation (36) is differentiated with respect to ( R L ) and set to zero, which gives, R 12

14 d(w n. ) d( R L R ) = 0 RL R = 1 (40) Now, by substituting R L R power equation as, = 1 in Equation (36), the result leads to the maximum W max = nα2 (T h T c ) 2 4R (41) In similar fashion, the maximum conversion efficiency can be obtained by differentiating Equation (37) with respect to ( R L ) and equating the result will give the R maximum conversion efficiency as, η max = (1 T c T h ) 1+ZT 1 1+ZT T c T h (42) Where T is the average temperature between the hot and the cold junction temperatures and is expressed by, T = (T h T c ) 2 (43) For the case of maximum parameters at the maximum power, the maximum power efficiency is obtained by setting R L R give, equals to one in Equation (37) and this will η mp = (1 T c T h ) T c 2 (1 T c T )+ h T h ZTc (44) Similarly, the voltage and current at the maximum power can be obtained by sitting R L R = 1 in Equations (35) and (34), respectively, and yields, V mp = nα(t h T c ) 2 I mp = α(t h T c ) 2R (45) (46) 13

15 Note that so far there are seven maximum parameters, which areη max,η mp,v max,v mp,i max, I mp, and W max. 6. Normalized Parameters: These normalized parameters can be obtained by dividing the active parameters by the maximum parameters, and by doing that we can represent the characteristics of the thermoelectric generator. The normalized output power can be defined by dividing Equation (36) by Equation (41), which gives, W = 4 W max R L R ( R L R +1)2 (47) For the normalized voltage, Equation (34) is divided by Equation (39), and the result is, V n V max = R L R R L R +1 (48) Equations (34) and (38) gives the normalized current as, I = 1 R I L max R +1 (49) Not so far that the three normalized parameters are only functions in the resistance ratio R L R. For the normalized efficiency, we divide equation (37) by Equation (42) and the result is, η th η max = R L R ( 1+ZT + T c T h ) [( R ( R L+1) L R +1) 1 2 (1 T c R )+ T h 2 (1+ T c) T h 2ZT ]( 1+ZT 1) (50) From Equation (50), it is clear that the normalized efficiency is not only a function in R L R, but also in the dimensionless figure of merit ZT and the temperature ratio T c T h. 14

16 Figure.6 Generalized TEG performance for T c Th = 0.5 and ZT = 1 Figure.6 above indicates the generalized performance characteristic for TEGs typical values of T c Th = 0.5 and ZT = 1. This chart was constructed using the normalized Equations from (47) to (50). It is seen from this chart that the maximum output power occurs at the resistance ration of R L R = 1, which has already been predicted early in this work. The maximum efficiency, however, occurs at a value of resistance ratio equals to 1 + ZT, or more specifically at R L R 7. Effective material properties: = 1.6 in this case The maximum parameters, represented in the previous section, are all functions in the material properties (seebeck, thermal conductivity, and electric resistivity) for known geometry factor (G = A L ) and junction temperatures T h, and T c. Therefore, the effective material properties can be defined in terms of these four maximum parameters (η max, W max, V max, and I max ), which are usually provided by the manufacturer or taken by measurements, and differ from one module to another. With this concept, the effective resistivity is formulated by using Equations (38) and (41), which give, 15

17 ρ = 4(A L )W max 4(I max ) (51) 2 The effective figure of merit can be formulated from Equation (42), which gives, Z = 2 T c (1+( T c T h ) 1 ) [( Where η c = 1 T c T h ηmax Tc 1+ ηc T h 1 η max ηc ) 2 1] (52) is Carnot efficiency. However, in cases of having the maximum power efficiency instead of the conversion efficiency provided by the manufacturer, the figure of merit can be defined as, Z = 4 Tc (T c T h ) η c ( 1 ηmp +1 2 ) 2 (53) Also, the effective seebeck coefficient can be obtained from Equations (38) and (50), which is, α = 4W max ni max (T h T c ) (54) From the definition of the thermal conductivity, we can define the effective thermal conductivity as a function of the other effective properties as, k = α 2 ρ Z (55) Note that the effective material properties are obtained using the ideal equations. Thus, they include different effects like Thomson effect, conduct resistances, and losses due to radiation and convection. These effective material properties are the total properties, so they should be divided by two to obtain the single p-type and n-type thermocouple properties. 8. Results and discussion In this project, the effective material properties (,, and k*) were calculated using the manufacture s maximum parameters (Wmax, Imax, and ηmax). Four different modules were chosen to check the status of ideal equation with the effective material properties as shown in Table 1. First, the effective material properties for each module were obtained using Equations 16

18 (51) (55). The cross-sectional (A) area and length (L) of thermoelement were either measured or provided by manufactures. As shown in Table 1, by Equations (38) (42) the maximum parameters (Wmax, Imax, ηmp, and Vmax) were recalculated using the effective material properties and compared with the manufacture s maximum parameters. They appear almost the same except the maximum voltage which is reasonable because it was not consider in our calculations (secondary parameter). Table.1 Comparison of the properties and dimensions for the commercial products of thermoelectric modules Description TEG Module (Bismuth Telluride) Symbols TG12-4 HZ-2 TGM TGM Tc =50 o C Tc =30 o C Tc =30 o C Tc =30 o C Th =170 o C Th =230 o C Th =200 o C Th =200 o C Number of thermocouples Manufacturers maximum parameters n W max (W) I max (A) η mp (%) V max (V) Rn ( ) Measured geometry of thermoelement A (mm 2 ) L (mm) G = A/L (cm) Dimension mm (W L H) 17

19 Effective material properties (calculated using commercial(wmax, Imax, and ηmax) The maximum parameters using effective material properties V/K cm k (W/cmK) ZT avr W max (W) I mam (A) η max (%) V max (V) TG12-4 by Marlow Figure.7 shows comparison between the calculations (solid lines) and the manufacturer s performance data (triangles) of Module TG12-4 with two different cold side temperature. The output power in Figure.7 (a) was calculated using Equation (41) and the effective material properties versus the hot side temperature. It is seen in Figure.7 (a) that the calculated output power is in good agreement with the manufacturer s performance curves. It is also seen that at high temperature the effective material properties could not accurately predict the power output. This occurs because of the temperature dependent of material properties. On the other hand, Figure.7 (b) shows that the errors on the voltage-vs-hot side temperature curves, decrease with increasing the hot side temperature. The performance chart for efficiency were not provided but the heat input values at corresponding hot side temperatures for the graph of power and voltage output were provided. So, the efficiency were predicted using equation (42) and has a good agreement as shown in Figure.7 (c). 18

20 Figure 7. (a) Output power versus hot side temperature Figure 7. (b) Voltage versus hot side temperature 19

21 Figure 7. (c) Efficiency versus hot side temperature 8.2.HZ-2 by Hi-Z Figures.8 (a), (b), (c), and (d) show comparison between the calculations and the performance data of Module HZ-2. In figure (a), (b), and (c), the output power, voltage, and efficiency versus temperature difference were compared with commercial data. In general, the calculations are in good agreement with the manufacturer s performance data. In figure.8 (d), the output power, efficiency, voltage versus current were accurate in predicting with commercial data. Figure 8. (a) Output power versus Temperature difference 20

22 Figure 8. (b) Voltage versus Temperature difference. Figure8. (c) Efficiency versus Temperature difference 21

23 Figure 8. (d) Output power, efficiency, and voltage versus current 8.3.TGM by Kryotherm Figures.9 (a), (b), (c), and (d) show comparison between the calculations and the performance data for different cold side temperature of Module TGM , with a good agreement. In figure.9 (b), the current showed discrepancies at regions of non-linearity. This was occurred because of the temperature independent material properties for ideal equation that only predicts a linear voltage output. The Open circuit voltage which represent the maximum voltage was compared with commercial data as shown in figure.9 (c). The voltage comparison at matched load conditions in Figure.9 (d) shows similar results to the commercial data. In figure.9 (e), the efficiency versus load resistance was in excellent agreement with commercial data. The output voltage and power versus current were exactly the same as the manufacture s data as shown in figure.9 (f). 22

24 Figure 9. (a) Output power versus hot side temperature Figure 9. (b) Current versus hot side temperature 23

25 Figure 9. (c) Open circuit voltage versus hot side temperature Figure 9. (d) Voltage versus hot side temperature 24

26 Figure 9. (e) Efficiency versus load resistance Figure 9. (f) Output voltage and power versus current 25

27 8.4. TGM by Kryotherm Figures.10 (a), (b), (c), (d), (e), and (f) predict comparison between the calculations and the performance data for different cold side temperature of Module TGM , with a good agreement. Figure 10. (a) Output power versus hot side temperature Figure 10. (b) Current versus hot side temperature 26

28 Figure 10. (c) Open circuit voltage versus hot side temperature Figure 10. (d) Voltage versus hot side temperature 27

29 Figure 10. (e) Efficiency versus load resistance Figure 10. (f) Output voltage and power versus current 9. Normalized curves: The normalized charts were created using the normalized parameters over the maximum parameters presented in this work. Using equation (50) the normalized efficiency η / ηmax was plotted versus the resistance ratio RL / R as shown in figures 11. (a) and (b). 28

30 The normalized efficiency is a function of the dimensionless figure of merit evaluated at the average temperature (ZTavr) and junction temperature ratio Tc / Th. These charts can be very helpful for designers to find the load resistance that gives the maximum possible efficiency. In figure 11 (a), the junction temperature ratio was fixed at 0.5 with various dimensionless figure of merit (evaluated at the average temperature) and plotted versus resistance ratio. In figure.11 (b), the dimensionless figure of merit (evaluated at the average temperature) was fixed at 1.5 with various junction temperature ratio and plotted versus resistance ratio. It is seen in figure.11 (b) that the normalized curves at different junction temperature ratio were close to each other. Figure 11. (a) Normalized Efficiency for TEGs for various ZTavr 29

31 Figure 11. (b) Normalized Efficiency for TEGs for various Tc/Th 10. Conclusion: To conclude, it is demonstrated that using the effective material properties technique along with the ideal equations is accurate as it is showing an acceptable agreement with the several manufacturer s performance data (which even can be provided by the manufacturer of obtained by measurements). Thus, this work has shown that once one has the maximum parameters of a module, the performance of this module can be evaluated analytically. Considering that the material properties are usually temperature dependents and the existing of the thermal and electrical contact resistances, usually make the thermoelectric analysis complicated. However, employing the analysis of the ideal equations with the effective material properties can be reliable and simple if the temperature difference was moderated. The normalized charts were obtained by using the maximum parameters defined in this study to present the characteristics of the four thermoelectric generators. These normalized charts can be standard for any thermoelectric generator at a given dimensionless figure of merit. The maximum parameters provided by the manufacturer (used to obtain the effective material properties) may not be accurate, yet the results may vary considering this factor. 30

32 References [1] H. Lee, Thermal Design: Heat Sinks, Thermoelectrics, Heat Pipes, Compact Heat Exchaners, and Solar Cells. Hoboken: John Wiley & Sons, Inc., 2010 [2] H. Lee, "The Thomson effect and the ideal equation on thermoelectric coolers," Energy, vol. 56, pp , [3] H. Lee, A. Attar, and S. Weera, "Performance evaluation of commercial thermoelectric modules using effective material properties," in 2014 International Conference on Thermoelectrics, Nashville, 2014, pp [4] S. Kim et al., "Thermoelectric power generation system for future hybrid vehicles using hot exhaust gas," Journal of Electronic Materials, vol. 40, no. 5, pp , [5] Weera, Sean Lwe Leslie, "Analytical Performance Evaluation of Thermoelectric Modules Using Effective Material Properties" (2014). Master's Theses. Paper

33 Appendix 32