Chapter 4 Optimal Design

Transcription

1 4- Capter 4 Optimal Design e optimum design of termoelectric devices (termoelectric generator and cooler) in conjunction wit eat sins was developed using dimensional analysis. ew dimensionless groups were properly defined to represent important parameters of te termoelectric devices. Particularly, use of te convection conductance of a fluid in te denominators of te dimensionless parameters was important, wic leads to a new optimum design. is allows us to determine eiter te optimal number of termocouples or te optimal termal conductance. t is stated from te present dimensional analysis tat, if two fluid temperatures on te eat sins are given, an optimum design always exists and can be found wit te feasible mecanical constraints. e optimum design includes te optimum parameters suc as efficiency, power, current, geometry or number of termocouples, and termal resistances of eat sins. 4. ntroduction ermoelectric devices (termoelectric generator and cooler) ave found compreensive applications in solar energy conversion [], exaust energy conversion [, 3], low grade waste eat recovery [4-6], power plants [7], electronic cooling [8], veicle air conditioners, and refrigerators [7]. e most common refrigerant used in ome and automobile air conditioners is R-34a, wic does not ave te ozone-depleting properties of Freon, but is neverteless a terrible greenouse gas and will be banned in te near future [9]. e pertinent candidate for te replacement would be termoelectric coolers. Many analyses, optimizations, even manufacturers performance curves on termoelectric devices ave been based on te constant

2 4- ig and cold junction temperatures of te devices. Practically, te termoelectric devices must wor wit eat sins (or eat excangers). t is ten very difficult to ave te constant junction temperatures unless te termal resistances of te eat sins are zero, wic is, of course, impossible. is is te rationale tat a new design is developed in tis section []. 4. Optimal Design for ermoelectric Generators Let us consider a simplified steady-state eat transfer on a termoelectric generator module (EG) wit two eat sins as sown in Figure 4. (a). Eac eat sin faces a fluid flow at temperature. Subscript and denote ot and cold quantities, respectively. We assume tat te electrical and termal contact resistances in te EG are negligible, te material properties are independent of temperature, and also te EG is perfectly insulated. e EG as a number of termocouples, of wic eac termocouple consists of p-type and n-type termoelements wit te same dimensions as sown in Figure 4. (b). t is noted tat te termal resistance of eat sin can be expressed by te reciprocal of te convection conductance A, were is te fin efficiency, is te convection coefficient, and A is te total surface area in te eat sin. We ereafter use te convection conductance rater tan te termal resistance.

3 4-3 (a) (b) Figure 4. (a) ermoelectric generator module (EG) wit two eat sins and (b) termocouple. e basic equations for te EG wit two eat sins are given by A A n R L A n R L A (4.) (4.) (4.3) (4.4) R L R (4.5)

4 4-4 were p n, p n, and p n. Equations (4.) (4.5) can be solved for and, providing te power output. However, in order to study te optimization of te EG, several dimensionless parameters are introduced. e dimensionless termal conductance, te ratio of termal conductance to te convection conductance in fluid, is n A L A (4.6) e dimensionless convection, te ratio of convection conductance in fluid to fluid, is A A (4.7) e dimensionless electrical resistance, te ratio of te load resistance to te electrical resistance of termocouple, is R r R (4.8) L R e dimensionless temperatures are defined by (4.9) (4.) (4.) e two dimensionless rates of eat transfer and te dimensionless power output are defined by

5 4-5 A (4.) A (4.3) A W W n n (4.4) t is noted tat te above dimensionless parameters are based on te convection conductance in fluid, wic means tat A sould be initially provided. Using te dimensionless parameters defined in Equations (4.6) (4.), Equations (4.) (4.5) reduce to two formulas as: R Z R Z r r (4.5) R Z R Z r r (4.6) were Z is called te figure of merit ( Z ). Equations (4.5) and (4.6) can be solved for and. e dimensionless temperatures are ten a function of five independent dimensionless parameters as,,,, Z R f r (4.7),,,, Z R f r (4.8) is te input and Z is te material property wit te input, and bot are initially provided. erefore, te optimization can be performed only wit te first tree parameters (,, and

6 4-6 Rr). Once te two dimensionless temperatures ( and ) are solved for, te dimensionless rates of eat transfer at bot ot and cold junctions of te EG can be obtained as: (4.9) (4.) en, we ave te dimensionless power output as W n (4.) Accordingly, te termal efficiency is obtained by t W n (4.) Defining L A, te dimensionless current is obtained by Z R r (4.3) Also, defining V V n, te dimensionless voltage is obtained by V Wn (4.4) Wit te inputs ( and Z ), we begin developing te optimization wit te dimensionless parameters (,, and Rr) iteratively until tey converge. t is learned from some initial attempts tat bot and Rr sow teir optimal values for te dimensionless power output, wile

7 4-7 does not sow te optimal value sowing tat te dimensionless power output monotonically increases wit increasing. is implies tat, if is given, te optimal combination of and Rr can be obtained. However, te dimensionless convection actually presents te feasible mecanical constraints. us, we first proceed wit a typical value of for illustration and later examine te variety of wit some practical design examples. Suppose tat we ave two initial inputs of. 6 (two fluid temperatures) and Z. (materials) along wit. We ten determine te optimal combination for and R r, wic may be obtained eiter grapically or using a computer program. We first use te grapical metod at tis moment and later te program for multiple computations. e dimensionless power output Wn and termal efficiency t are togeter plotted as a function of R r, wic are presented in Figure 4. (a). Bot Wn and t wit respect to Rr indeed sow teir optimal values tat appear close. We are interested primarily in te power output and secondly in te efficiency. However, since tey are close eac oter, we erein use te power output for te optimization. t sould be noted tat te dimensionless maximum power output does not occur at R from Figure 4. (a) as usually assumed for a EG witout eat sins, but approximately at R. 7, because te dimensionless temperatures and in Figure 4. (b) are no longer constant. is is often a confusing factor in optimum design wit a EG wit two eat sins. We sould not assume tat R r is equal to one for a EG wit eat sins. r r

8 4-8.5 W n *.5.4. W n *.3 t.9 t (a) R r * & * * W n *.4 W n * * (b) R r Figure 4. (a) Dimensionless power output Wn and efficiency t versus te ratio of load resistance to resistance of termocouple Rr and (b) dimensionless temperatures R r. ese plots were generated using. 3,,. 6 and Z.. and versus

9 4-9 Wit te dimensionless parameters obtained (, R r. 7,. 6, and Z. ), we now plot te dimensionless power output Wn as a function of te dimensionless termal conductance defined in Equation (4.6) along wit te termal efficiency t, wic is sown in Figure 4.3 (a). We find an optimum W approximately at. 3. Actually, te optimal values of and Rr sould be iterated until te two converge eac oter. From n A L A n as sown in Equation (4.6), te actually determines te number of termocouples n if te geometric ratio A/L and A are given or vice versa. e dimensionless power output Wn first increases and later decreases wit increasing. t is important to realize tat, if A is given, tere is an optimal number n of termocouples (or optimal termal conductance A L ) in te termoelectric module, wic is usually unnown. Pysically, te surplus number of termocouples virtually increases te termal conduction more tan te production of power, wic causes te net power output to decline. ere is anoter important aspect of te optimal dimensionless termal conductance of. 3, wic is tat te module termal conductance termal conductance n A L directly depends on te A. n oter words, te module n A L must be redesigned on te basis of te A to meet. 3. e dimensionless ig and low junction temperatures are presented in Figure 4.3 (b). As decreases towards zero, and approac and, respectively. is indicates tat te termal resistances of two eat sins approaces zero, wic never appens. t is noted tat te termal efficiency approaces te teoretical maximum efficiency of. for te given fluid temperatures as approaces zero.

10 W n *.5.4. W n *.3.5 t. t (a) 4. 3 t.5 * & * * *..5 t (b) Figure 4.3 (a) Dimensionless power output Wn and termal efficiency t versus dimensionless termal conductance, and (b) ig and low junction temperatures ( and ) versus dimensionless termal conductance. ese plots were generated using, R. 7, r.6 and Z..

11 4- Since tere is an optimal combination of and Rr for a given, we can plot te optimal dimensionless power output W opt and optimal termal efficiency opt as a function of dimensionless convection, wic is sown in Figure 4. t is very interesting to note in Figure 4 tat, wit increasing, opt barely canges, wile W opt monotonically increases. According to Equation (4.4), te actual optimal power output Wopt is te product of W and A, seemingly increasing linearly wit A. n practice, tere is a controversial tendency tat may decrease systematically wit increasing A. As a result of tis, it is needed to examine te variety of te A as a function of for te optimal power output. Figure 5 reveals te intricate relationsip between A and (or A ) along wit te optimum actual power output (not dimensionless), wic would lead system designers to a variety of possible allocations ( A and ) for teir optimal design. ote tat te analysis so far is entirely based on te dimensionless parameters. ow we loo into te actual optimal design wit te actual values. For example, using Figure 4.4 and Figure 4.5, we develop an optimal design for automobile exaust gas waste eat recovery. A termoelectric generator module wit a 5-cm 5-cm base area is subject to exaust gases at 5 C in fluid and air at 5 C in fluid. We estimate an available maximum convection conductance in fluid (exaust gas) wit. 8, 3W m K, and A cm and also an available maximum convection conductance in fluid (air) wit. 8, W m K 3, and A cm opt, wic gives A A 4. 8W K and. ote tat and are typical fin efficiencies, and and are te reasonable convection coefficients for exaust gas eating and air cooling. Eac area of A and A is based on fins ( sides) wit a fin eigt of 5 cm on a 5-cm 5-cm base area of te module. e typical termoelectric material properties are assumed to be p = n =

12 4- V/K, p = n =. -3 cm, and p = n =.4 - W/cmK. e above data approximately determines tree dimensionless parameters as,. 6 and Z opt.6. W* opt opt.4.. W* opt.9.8. Figure 4.4 Optimal dimensionless power output W opt and efficiency opt versus dimensionless convection. is plot was generated using. 6 and Z.. W opt (W) =. Point = =.. Point. η A (W/K)

13 4-3 Figure 4.5 Optimal power output W versus convection conductance A in fluid as a opt function of dimensionless convection. is plot was generated using = 5 C,. 6 and Z.. As mentioned before, te present dimensional analysis enables te tree dimensionless parameters (,. 6 and Z. ) to determine te rest two optimal parameters, wic are found to be. 3 and R. 7 as sown before. is leads to a statement tat, if r two individual fluid temperatures on eat sins connected to a termoelectric generator module are given, an optimum design always exists wit te feasible mecanical constraints tat present. is optimal design is indicated approximately at Point in above. ote tat tere are ways to improve te optimal power output, increasing eiter A or or bot, wic obviously depends on te feasible mecanical constraints, wicever is feasible. e inputs and optimum results at Point in above are summarized in able 4.. e inputs are te geometry of termocouple, te material properties, two fluid temperatures, and te available convection conductance in fluid. e dimensionless results are converted to te actual quantities as sown. e maximum power output is found to be 65. W for te 5cm 5cm base area of te module. e power density is calculated to be.6 W/cm, wic appears significantly ig compared to an available power density of ~ W/cm wit te similar operating conditions by EDO program (Japan) [7]. able 4. nputs and Results from te Dimensional Analysis for a EG nputs Dimensionle ss ( W, ) n opt Actual (Wn,opt) = 5 C, = 5 C, = 475 C =.3 n = 54 A = mm, L = mm = A = 4.8 W/K =.8, = 6 W/m K, A = cm R r =.7 R L =.7 n R = 4.3 A = 4.8 W/K. 6 = 5 C

14 4-4 Base area A b of module = 5 cm 5 cm Z. Z. p = - n = V/K. 7 = 374 C p = n =. -3 cm. 367 = 37 C p = n =.4 - W/cmK W. 45 W n = 65. W (Z = K - ) (R =. per termocouple) =.36 = 3.9 A ( 8 K 3, A cm ) V =.5 V = 6.7 V (Power Density P d = W n/a b) - P d =.6 W/cm n t t Air was used in fluid so far. However, we want to see te effect of A or by canging fluid from air to liquid coolant. Oterwise te same conditions were applied to as te previous example. We ten estimate an available convection conductance in fluid (exaust gas) wit te same one of. 8, W m K 3, and A cm, but an available convection conductance in fluid (liquid coolant) wit. 8, W m K 3, and A cm, wic gives A 4. 8W K and A 4W K, respectively, wic yields.. e area of A is based on fins ( sides) wit a fin eigt of 5 cm for te 5-cm 5-cm base area of te module and A is estimated to be one tent of A (liquid coolant does not require a large eat transfer area). ese inputs and optimum results give all te five dimensionless parameters as.7,., R r. 5,. 6 and Z., for wic te optimum at A 4W K is indicated at Point in Figure 4.5. e effect of on te ig and cold junction temperatures was also presented in Figure 4.6. t is interesting to see tat, altoug a small variation in te optimal power outputs between Point ( ) and Point (. ) appears in Figure 4.5, a significant temperature variation between and. appears in Figure 4.6. is may be an important factor particularly wen termoelectric materials are considered in te optimal design. e proximity of te power outputs between Points and is an example sowing te variety of te mecanical constraints ( A and A ) even wit te same power outputs.

15 opt ( C) opt... opt Figure 4.6 Hot and cold junction temperatures and optimal efficiency versus dimensionless convection. is plot was generated wit = 5 C,. 6 and Z.. Example 4. Exaust ermoelectric Generators A termoelectric generator (EG) module is designed using a newly developed material wic is so called AGS-75 (AgSbe/Gee 75%) for an exaust waste eat recovery in a car. e wole EG device is sown in Figure E4.-a. e device as number of modules, eac of wic consists of n number of p- and n-type termocouples. e material properties for termoelements are assumed to be similar as p = n = 8 V/K, p = n =. -3 cm, and p = n =.3 - W/cmK. e cross-sectional area and pellet lengt of te termoelement are An = Ap =.4 mm and Ln = Lp = mm, respectively. e exaust gases at 75 K enter eat sin wile coolant at 3 K flows in eat sin. e convection conductances for te flows, wit considering te effectiveness of te eat sins, are estimated on te 5-cm 5-cm base areas to be.8, W m K 6, and W m K, and A cm for eat sin (exaust gases) and. 9, A 5cm for eat sin (coolant).

16 4-6 (a) For te optimal design of one EG module, determine te number of termocouples n, power output, conversion efficiency, ot and cold junction temperatures, current and voltage. (b) f te total power output of W for te wole EG device is required, determine te number of modules (assuming tat te module represents te mean value of te modules in te device). (a) Figure E4.- (a) te wole EG device, and (b) te EG module. (b) Solution: Material properties of AGS-75: =p n = V/K, = p + n =. -5 m, and = p + n =.6 W/mK. e convection conductances are given as =.8, = 6 W/m K, A = cm and =.9, = W/m K, A = 5 cm W m K m 4. W K A W m K5 m 4. W K A 5 e cross-sectional area and pellet lengt of te termoelement are A =.4-6 m and L = -3 m e figure of merit is

17 4-7 Z and Z 5. m.6 W mk V K K 3K. 37 e dimensionless fluid temperature is 75K 3K e dimensionless convection conductance is A A 4.8W K 4.5W K.67 From able A4. for Z, we can approximately obtain te optimal parameters using bot.4 and. as Rr =.638, =.35, t =.96, W. 35,. 3,. 33, =.65, and V =.435. (a) Using Equation (4.6), te number of termocouples is n A L A n 3 4.5W K m 6.6W mk.4 m.35 Using Equation (4.4), te power output is W K3K 49. W Wn W n A 4 e conversion efficiency is t =.96 e ot and cold junction temperatures are.33k 633. K K 45. K 58 K 9.95 Using te dimensionless current in Equation (4.3), te current is

18 4-8 A 3. 79A L 6.6W mk.4 m V K m Using te definition of dimensionless voltage in Equation (4.4), te voltage is V K3K 3. V Vn Vn (b) Determine te number of modules if te total power output of W is required. W W n W 49.4W 4.4 Comments: We want to compare te above optimal power output of 49.4 W wit te maximum power output of about 54 W plotted in a conventional way (assuming te constant ot and cold junction temperatures) using te following equations as R L A m m.4 m 9.7 A L n R, n W n R, and t W n 6 5 W n.. 4 t.8 W n (W) t Current (A)

19 4-9 Figure E4.- Power output and efficiency versus current. e maximum power output of about 54 W at = 5 A from te figure appears iger tan te optimal power output of 49.4 W obtained wit = 3.79 A. However, it is noted tat te 54 W is attained under te assumption of te constant ot and cold junction temperatures wic are no longer true. is indicates tat te predictions wit te ideal equations witout eat sins (under assumption of te constant ot and cold junction temperatures) may lead to te appreciable errors. 4.3 Optimal Design of ermoelectric Coolers Let us consider a simplified steady-state eat transfer on a termoelectric cooler module (EC) wit two eat sins as sown in Figure 4.7. Eac eat sin faces a fluid flow at temperature. Subscript and denote te entities of fluid and, respectively. Consider tat an electric current is directed in a way tat te cooling power enters eat sin. We assume tat te electrical and termal contact resistances in te EC are negligible, te material properties are independent of temperature, and also te EC is perfectly insulated. e EC as a number of termocouples, of wic eac termocouple consists of p-type and n-type termoelements wit te same dimensions.

20 4- Figure 4.7 ermoelectric cooler module (EC). e basic equations for te EC wit two eat sins are given by A A n R L A n R L A (4.5) (4.6) (4.7) (4.8) were p n, p n, and p n. n order to study te optimization of te EC, several dimensionless parameters are introduced. e dimensionless termal conductance, wic is te ratio of termal conductance to te convection conductance in fluid, is

21 4- n A L A (4.9) e dimensionless convection, wic is te ratio of convection conductance in fluid to fluid, is A A (4.3) e dimensionless current is given by A L (4.3) e dimensionless temperatures are defined by (4.3) (4.33) (4.34) e dimensionless cooling power, rate of eat liberated and electrical power input are defined by A (4.35) A (4.36)

22 4- W n Wn A (4.37) t is noted tat te above dimensionless parameters are based on te convection conductance in fluid, wic means tat A sould be initially provided. Using te dimensionless parameters defined in Equations (4.9) (4.34), Equations (4.5) (4.8) reduce to two formulas as: Z (4.38) Z (4.39) Equations (4.38) and (4.39) can be solved for and ten a function of five independent dimensionless parameters as f,,,, Z. e dimensionless temperatures are (4.4) f,,,, Z (4.4) is te input and Z is te material property wit te input, and bot are initially provided. erefore, te optimization can be performed only wit te first tree parameters (,, and ). Once te two dimensionless temperatures ( and rates of eat transfer at bot junctions of te EC can be obtained as: ) are solved for, te dimensionless (4.4)

23 4-3 (4.43) is called te dimensionless cooling power. en, we ave te dimensionless power input as W n (4.44) Accordingly, te coefficient of performance is obtained by COP W n (4.45) Defining V V n, te dimensionless voltage is obtained by V Wn (4.46) Wit te inputs ( and Z ), we try to find te optimal combination for te dimensionless parameters (,, and ) iteratively until tey converge. t is found tat bot and te optimal values for te dimensionless cooling power value sowing tat te dimensionless cooling power, wile sow does not sow te optimal monotonically increases wit increasing. is implies tat, if any is given, te optimal combination of and However, te dimensionless convection can be obtained. actually presents te feasible mecanical constraints. us, we proceed wit a typical value of for illustration and later examine te variety of wit a practical design example. Suppose tat we ave. 967 (two arbitrary fluid temperatures) and Z. (materials) along wit as inputs. en, we can determine te optimal combination for and wic may be obtained eiter grapically or using a computer program. We first use te grapical,

24 4-4 metod at tis moment and later te program for multiple computations (a Matematical software Matcad was used). e optimal combination of and for eac maximum cooling power are found to be =.5 and =.3, respectively, wic are sown in Figure 4.8 and Figure 4.9. e maximum cooling power of. 37 in bot figures is actually te optimal dimensionless cooling power,opt. However, te COP also sows an optimal value at. 74, wic gives.6. e optimal COP usually gives a very small cooling power or sometimes even no exists, wic seems impractical, albeit te ig COP. erefore, it is needed to ave a practical point for te optimal COP, wic is determined in te present wor to be te midpoints of te optimum and. For example, te practical optimal COP in tis case occurs simultaneously at. 5 and. 5, wic leads to. 9 tat may be seen after re-plotting wit te two values.. W n *.5.8 * & W n *.6.4 COP *.5 COP Figure 4.8 Dimensionless cooling power, power input W n and COP versus dimensionless current. is plot was generated wit. 3,,. 967 and Z..

25 *.8 * COP. COP Figure 4.9 Dimensionless cooling power and COP versus dimensionless termal conductance. is plot was generated wit,. 5,. 967 and Z.. e existence of an optimum cooling power as a function of current is a well-nown caracteristic of ECs. However, te existence of te optimum in ECs as not been found n A L, A in te literature to te autor s nowledge. Wit Equation (4.9) tat is te optimum of =.3 implies tat te module termal conductance n A L is at optimum since te A is given, wic leads to te optimum n (te number of termocouples) if A L is given or vice versa. is is one of te most important optimum processes in design of a termoelectric cooler module. t is good to now in Figure 4. tat te dimensionless temperature optimum COP. becomes lowest at te optimal dimensionless cooling power, not at te

26 * *.4 * & *...3. * * Figure 4. Dimensionless temperatures versus dimensionless termal conductance. is plot was generated wit,. 5,. 967 and Z.. We also consider two eat sins as a unit witout a EC to examine te limitation of use of te EC, wic is sown in Figure 4.. e geometry of te unit is te same as te one sown in Figure 4.7 except tat tere is no EC between te eat sins. ere sould be a cooling rate wit given fluid temperatures, wic is. We want to compare te cooling power wit tis cooling rate to determine te limit of use of te EC.

27 4-7 Figure 4. Heat sins witout a EC. e basic equations for te unit can be expressed as A A (4.47) (4.48) e dimensionless groups for te unit are A (4.49) (4.5) Using Equations (4.3), (4.34), (4.49) and (4.5), Equations (4.47) and (4.48) reduce to a formula as (4.5) e dimensionless cooling rate witout eat sins can be obtained by (4.5)

28 * &.5 *..5 * COP * COP Figure 4. Dimensionless cooling power, cooling rate of te unit, and COP versus dimensionless fluid temperature. is plot was generated wit. 3,,. 5 and Z *,opt * COP COP opt Figure 4.3 Optimal (cooling power optimized) dimensionless cooling power and COP versus dimensionless convection. is plot was generated wit. 967 and Z..

29 4-9 e dimensionless cooling power and cooling rate of te unit versus te dimensionless fluid temperature along wit te COP are presented in Figure 4.. e cross point in te figure is found to be dimensionless fluid temperature =., wic is a design point as te limit of use of te EC. f te is iger tan te cross point, tere is no justification for use of te EC altoug te EC still functions. is cross point is defined as te maximum dimensionless temperature. ere is also a minimum point at. 83, were,, max wic is called te minimum dimensionless temperature, min. ote tat te EC can perform effective cooling witin a range from, min. 83 to, max =.. Since tere is an optimal combination of and for a given, we can plot te optimal dimensionless cooling power,opt and COPopt as a function of dimensionless convection, wic is sown in Figure 4.3. t is seen tat bot,opt and COPopt increase monotonically wit increasing product of. According to Equation (4.35), te actual optimal cooling power,opt, opt is te and A, seemingly increasing linearly wit A. n practice, tere is a controversial tendency tat may decrease systematically wit increasing A. As a result of tis, it is needed to examine te variety of te A as a function of for te optimal cooling power. Figure 4.4reveals te intricate relationsip between A and (or A ) along wit te optimum actual cooling power (not dimensionless), wic would lead system designers to a variety of possible allocations ( A and ) for teir optimal design. ote tat te analysis so far is entirely based on te dimensionless parameters. ow we loo into te actual optimal design wit te actual values.

30 ,opt (W) =. η A (W/K). = Point =. Figure 4.4 Optimal cooling power versus convection conductance in fluid as a function of dimensionless convection. is plot was generated wit = 3 C,. 967 and Z.. For example, using Figure 4.3and Figure 4.4, we develop an optimal design for an automobile air conditioner. A termoelectric cooler module wit a 5-cm 5-cm base area is subject to cabin air at C in fluid and ambient air at 3 C in fluid. We estimate an available maximum convection conductance in fluid (cabin air) wit. 8, W m K 6, and A cm and also an available maximum convection conductance in fluid (ambient air) wit. 8, 6W m K, and tat and A cm, wic gives A A 4. 8W K and are te fin efficiencies, and and are te reasonable convection coefficients for te cabin air cooling and te ambient air cooling, respectively. Eac area of A and A is based on fins ( sides) wit a fin eigt of 5 cm on a 5-cm 5-cm base area of te module.. ote e typical termoelectric material properties are assumed to be p = n = V/K, p = n =. -3 cm, and p = n =.4 - W/cmK. e above data approximately determines tree dimensionless parameters as,. 967 and Z..

31 4-3 As mentioned before, te present dimensional analysis enables te tree dimensionless parameters (,. 967 and Z. ) to determine te rest optimal parameters, wic are found to be. 3 and. 5 as sown before. is leads to a statement tat, if two individual fluid temperatures on eat sins connected to a termoelectric cooler module are given, an optimum design always exists wit te feasible mecanical constraints tat present is optimal design is indicated approximately at Point in above. ote tat tere are several ways to improve te optimal power output by increasing eiter A or or bot, wic apparently depends on te feasible mecanical constraints, wicever is available.. 8,opt 6 8 ( C) 4,opt 6 (W),opt 4. Figure 4.5 wo junction temperatures and cooling power versus convection conductance in fluid. is plot was generated wit, A 4. 8W K = 3 C,. 967 and Z.. e inputs and optimum results at Point in Figure 4.4 are summarized in te first two columns of able 4.. e inputs are te geometry of termocouple, te material properties, two fluid temperatures, and te available convection conductance in fluid. e dimensionless results are

32 4-3 converted to te actual quantities. e optimal cooling power is found to be 54.4 W for te 5cm 5cm base area of te module. e cooling power density is calculated to be.8 W/cm. Wen A is limited, simply decreasing A invoes decreasing, wic is sown in Figure 4.5. is attributes to te eat balance tat te ot and cold junction temperatures must decrease wen more eat is extracted from te limited input, wic is a caracteristic of te termoelectric cooler wit eat sins. e ot and cold junction temperatures are sometimes a design factor, noting tat te optimal cold junction temperature may cause icing and reducing te eat transfer., opt reaces zero Celsius at. 4, wic able 4. nputs and Results from te Dimensional Analysis for a ermoelectric Cooler Module nput,opt,opt COP / opt COP / opt (dimensionless) (Actual) (dimensionless) (Actual) = C, = 3 C =.3 n = 57 =.5 n = 8 A = mm, L = mm = A = 4.8 W/K = A = 4.8 W/K =.8, =6 W/m K, =.5 = 6.36 A =.5 = 3.8 A A = cm. 967 = C. 967 = C Base area A b = 5 cm 5 cm Z. Z. Z. Z. p = - n = V/K. 93 = 8.7 C. 949 = 4.4 C p = n =. -3 cm. 45 = 73.9 C. 3 = 39.4 C p = n =.4 - W/cmK. 37 = 54.4 W. 9 = 6.9 W (Z = K - ) COP =.35 COP =.49 (R =.per termocouple) V =.75 V n = 4.5 V V =.33 V n = 5.7 V, max.,max =, max.6,max = 49.7 C 9.3 C.83,min =.84,min = 9. C (Cooling Power Density P d = /A b), min.8 C - P d =.8 W/cm, min - P d =.8 W/cm As mentioned before, te optimal COP is sometimes in demand in addition to te optimal cooling power. However, te real optimal COP usually gives a very small value of te cooling

33 4-33 power, albeit te ig COP. erefore, in te present wor, te midpoints of te optimal and are used to provide approximately a alf of te optimal cooling power and at least four folds of te cooling-power-optimised COP. is modified optimal COP is called a alf optimal coefficient of performance COP/opt. ese results are also tabulated in te last two columns of Figure 4., so tat designers could determine wic optimum is better depending on te application. e optimum cooling power is usually selected wen te resources (electrical power or capacity of coolant) are abundant or inexpensive or te efficacy is not important as in microprocessor cooling, wile te alf optimum COP is selected wen te resources are limited or expensive or te efficacy is important as in automotive air conditioners. e present paper presents te optimal design of termoelectric devices in conjunction wit eat sins introducing new dimensionless parameters. e present optimum design includes te power output (or cooling power) and te efficiency (or COP) simultaneously wit respect to te external load resistance (or electrical current) and te geometry of termoelements. e optimal design provides optimal dimensionless parameters suc as te termal conduction ratio, te convection conduction ratio, and te load resistance ratio as well as te cooling power, efficiency and ig and low junction temperatures. e load resistance ratio (or te electrical current) is a well nown caracteristic of optimum design. However, it is found tat te load resistance ratio would be greater tan unity (.7 in te present case). is is a confusing factor in optimum design wit a EG. One sould not assume tat te load resistance ratio RL/R is equal to unity for a EG wit eat sin(s). e optimal termal conductance [(n)(a/l)()] consists of te number of termocouples, te geometric ratio, and te termal conductivity. t is important tat tere is an optimum number of termocouples n for a given te convection conductance A if te optimal termal conductance A/L is constant or vice versa. ese are te optimum geometry of termoelectric devices. e information of te optimal termal conductance is particularly important in design of microstructured or tin-film termoelectric devices. Furtermore, tere is a potential to improve te performance or to provide te variety of te geometry by reducing te termal conductivity.

34 4-34 Finally, it is stated from te present dimensional analysis tat, if two individual fluid temperatures on eat sins connected to a termoelectric generator or cooler are given, an optimum design always exists and can be found wit te feasible mecanical constraints. Example 4. Automotive ermoelectric Air Conditioner Current automotive air conditioners ave used R-34a for about two decades as a woring fluid, wic does not ave te ozone-depleting properties of Freon, but it is neverteless a terrible greenouse gas and will be banned in te near future. A termoelectric air conditioner is designed for te replacement as a green energy application, as sown in Figure E4.a. e analysis indicates tat te cooling load of 6 W per occupant is required. We consider a unit module of 7-cm 7-cm base area, wic represents a mean value of te modules performance of te air conditioner. Cabin air at C enters te upper and lower eat sins wile coolant at 34 C enters te central flat eat excanger, so tat te EC modules are installed between eac eat sin and te central excanger, wic is scematically sown in Figure E4.b. A newly developed nanostructured material of Bie3 as te properties asp = n = 8 V/K, p = n =. -3 cm, and p = n =. - W/cmK. e cross-sectional area and pellet lengt of te termoelement are An = Ap =.5 mm and Ln = Lp =.5 mm, respectively. e convection conductances on te 7-cm 7-cm base areas are estimated to be. 8, W m K 4, and A cm for eat sin (cabin air) and. 8, W m K 8, and eat sin (coolant). A 6cm for (a) For te optimal cooling power per one EC module, determine te number of termocouples n, cooling power, COP, power input, cold and ot junction temperatures, current and voltage. (b) For te ½ optimal COP per EC module, determine te number of termocouples n, cooling power, COP, power input, cold and ot junction temperatures, current and voltage.

35 4-35 (c) Estimate te number of EC modules to meet te cooling load of 6 W per occupant wit te ½ optimal COP (assuming tat te module obtained represents te mean value of te modules in te device). (a) Figure E4.. (a) ermoelectric air conditioner, and (b) EC module. (b) Solution: Material properties of Bie3: =p n = 46-6 V/K, = p + n =. -5 m, and = p + n =.4 W/mK. e convection conductances are given as =.8, = 4 W/m K, A = cm and =.8, = 8 W/m K, A = 6 cm W m K m 3. W K A W m K6 m 3. W K A 84 e cross-sectional area and pellet lengt of te termoelement are A =.5-6 m and L =.5-3 m e figure of merit is Z and 5. m.4 W mk 6 46 V K K

36 4-36 Z K 37K. 7 e dimensionless fluid temperature from Equation (4.34) is 95K 37K.96 e dimensionless convection conductance from Equation (4.3) is A A 3.84W K 3.84W K. (a) For te optimum cooling power: Using able A4. for Z, we can approximately obtain te optimal parameters wit bot.96 and. as =.5, =.86, COP =.335,. 36,. 94,. 4, and V =.77. Using Equation (4.9), te number of termocouples is n A L A W K.5 m W mk.5 m Using Equation (4.35), te cooling power is w K37K 4. W A 46 e COP is already obtained from able as COP = e power input from Equations (4.35), (4.36), and (4.45) is W W 4.46 n W COP.335 e cold and ot junction temperatures from Equations (4.3) and (4.33) are.9437k 83. K 8.437K 35. K 76 Using Equation (4.3), te current is

37 W mk.5 m 6 46 V K.5 m A A 3 L Using te definition of dimensionless voltage in Equation (4.46), te voltage is V K37K. V Vn V n 5 e number of modules for te cooling load of 6 W per occupant is 6W 6W 4.46W 4.3 (b) For te ½ optimal COP per one EC module: Using able 3 for Z, we can approximately obtain te optimal parameters wit bot.96 and. as =.55, =.43, COP =.385,. 7,. 943,. 3, and V =.34. Using Equation (4.9), te number of termocouples is n A L A W K.5 m W mk.5 m Using Equation (4.35), te cooling power is.73.84w K37K. W A 5 e COP is already obtained from able as COP = e power input from Equations (4.35), (4.36), and (4.45) is W W.5 n 4. 48W COP.385 e cold and ot junction temperatures from Equations (4.3) and (4.33) are.94337k 89. K K 36. K 364

38 4-38 Using Equation (4.3), te current is 6.4W mk.5 m 6 46 V K.5 m A A 3 L Using te definition of dimensionless voltage in Equation (4.46), te voltage is V K37K 5. V Vn Vn e number of modules for te cooling load of 6 W per occupant is 6W 6W.5W 9.9 able E4. Summary of te Optimal cooling power and ½ optimal COP. Per EC Module Optimal Cool. Power COP ½ opt n number of termocouples n = 8.8 n = 4.4 Current (A) = 5.88 A =.94 A Voltage (V) V n=.5 V V n= 5. V COP COP =.335 COP =.385 Cooling power (W) = 4.46 W =.5 W Power input (W) W n= 6.75 W W n = 4.48 W Cold junction temperature =.8 C = 6.6 C Hot junction temperature = 77.7 C = 43.4 C number of modules for 6 W (otal number of termocouples) (otal power consumption) (Design comments) = 4.3 ( n = 333) (6.75W 4.3 =,79 W) (oo ig power consumption for automobiles) = 9.9 ( n = 34) (4.48W 9. = 433 W) (Acceptable design) Comments: e optimal design of termoelectric coolers wit eat sins is callenging because te distinct optimal COP does not exist altoug te optimal cooling power clearly exists. erefore, a alf optimal COP is introduced as a reference for te optimal COP. However, tere is no a strict rule for te optimal COP wit eat sins. f tere is no eat sin, it becomes an ideal device wit te constant junction temperatures. e ideal maximum COP and te current are usually available in te literature, wic are

39 4-39 Z COP max and Z COP R ( ) Z Using te cold and ot junction temperatures (6.6 C and 43.4 C) obtained for te alf optimal COP information, we calculate te ideal optimal COP and current as R Z 5 3. m.5 m. COP.5 6 m K 36.36K 3.78 K. 993 max 89.64K 36.36K 89.64K V K43.4 C 6.6 C K 89.64K COP. 45A.436 e ideal optimal COP and current are.436 and.54 A, wic appear fortuitously close to te alf optimal COP and current (.385 and.94 A). However, if we calculate te optimal cooling power in te same way, we find tat tere are appreciable discrepancies between te optimal values wit eat sins and te ideal values witout eat sins. n real, te termoelectric device sould wor wit one or two eat sins and te ideal maximum COP and cooling power wit constant junction temperatures would cause significant errors.

40 4-4 Problems 4.. As a potential alternative for clean energy generation, a termoelectric generator (EG) is designed to recover exaust waste energy from a car. An array of = 36 modules (wic as te base area of 5-cm 5-cm) is installed on te exaust of te car (Figure P4.). One of te recently developed material to meet te ig temperature is nanostructured lead telluride (Pbe) aving te properties asp = n = 8 V/K, p = n =. -3 cm, and p = n =.8 - W/cmK. e cross-sectional area and pellet lengt of te termoelement are An = Ap =.6 mm and Ln = Lp =. mm, respectively. e exaust gases at 7 K enter eat sin wile coolant at 3 K enters eat sin. e convection conductances for te flows, wit considering te effectiveness of te eat sins, are estimated on te 5-cm 5-cm base areas to be. 8, W m K 45, and and. 8, W m K 9, and A cm for eat sin (exaust gases) A 5cm for eat sin (coolant). We want to now te optimal number of te termocouples per module for te given information. (a) For te optimal design of one EG module, determine te number of termocouples n, power output, conversion efficiency, ot and cold junction temperatures, current and voltage. (b) Determine te total power output (assuming tat te module obtained represents te mean value of te modules in te device).

41 4-4 (a) Figure P4. (a) te arrangement of te wole EG device, and (b) te EG module. (b) 4.. An automotive termoelectric air conditioner is designed, wic consists of two air eat sins and a coolant eat excanger as sown in Figure P4.a. We design a unit module of 4-cm 4-cm base area, wic represents a mean value of te wole system as a simplified concept. Cabin air at 96. K enters te upper and lower eat sins wile coolant at 3.8 K enters te central flat eat excanger, so tat te EC modules are installed between eac eat sin and te central excanger, wic is scematically sown in Figure P4-4b. A widely used material of Bie3 as te properties asp = n = V/K, p = n =. -3 cm, and p = n =.3 - W/cmK. e crosssectional area and pellet lengt of te termoelement are An = Ap =.8 mm and Ln = Lp =.9 mm, respectively. e convection conductances on te 4-cm 4-cm base areas are estimated to be. 8, W m K 4, and and. 8, W m K 96, and A 6cm for eat sin (cabin air) A 5cm for eat sin (coolant). (a) For te optimal cooling power per one EC module, determine te number of termocouples n, cooling power, COP, power input, cold and ot junction temperatures, current and voltage.

42 4-4 (b) For te ½ optimal COP per EC module, determine te number of termocouples n, cooling power, COP, power input, cold and ot junction temperatures, current and voltage. (c) Estimate te number of EC modules to meet te cooling load of 6 W per occupant wit te ½ optimal COP (assuming tat te module obtained represents te mean value of te modules in te device). (a) Figure P4.. (a) ermoelectric air conditioner, and (b) EC module. (b) References. Kraemer, D., et al., Hig-performance flat-panel solar termoelectric generators wit ig termal concentration. at Mater,. (7): p Karri, M.A., E.F. acer, and B.. Helenbroo, Exaust energy conversion by termoelectric generator: wo case studies. Energy Conversion and Management,. 5(3): p Hsu, C.-., et al., Experiments and simulations on low-temperature waste eat arvesting system by termoelectric power generators. Applied Energy,. 88(4): p Henderson, J. Analysis of a eat excanger-termoelectric generator system. in 4t ntersociety Energy Conversion Engineering Conference Boston, Massacusettes.

43 Stevens, J.W., Optimal design of small D termoelectric generation systems. Energy Conversion and Management,. 4: p Crane, D.. and G.S. Jacson, Optimization of cross flow eat excangers for termoelectric waste eat recovery. Energy Conversion and Management, 4. 45(9-): p Rowe, D.M., ermoelectrics andboo; macro to nano. 6, Roca Raton: CRC aylor & Francis,. 8. Cein, R. and G. Huang, ermoelectric cooler application in electronic cooling. Applied ermal Engineering, 4. 4(4-5): p Vining, C.B., An inconvenient trut about termoelectrics. ature Materials, 9. 8: p Lee, H., Optimal design of termoelectric devices wit dimensional analysis. Applied Energy, 3. 6: p