3- Capter 3 ermoelectric Coolers Contents Capter 3 ermoelectric Coolers... 3- Contents... 3-3. deal Equations... 3-3. Maximum Parameters... 3-7 3.3 Normalized Parameters... 3-8 Example 3. ermoelectric Air Conditioner... 3-3.4 Effective Material Properties... 3-8 Problems... 3-3 References... 3-5 is capter formulates a set of simplified ideal equations for a termoelectric cooler, wit some assumptions, to exibit te general caracteristics of termoelectric coolers. e imum parameters are defined, wic are te imum current, imum temperature difference, imum cooling power, and imum voltage. en, te normalized parameters are plotted as general caracteristics for te coolers. e ideal equation is based on tree material properties: te Seebeck coefficient, electrical resistivity, and termal conductivity. ese material properties for commercial termoelectric cooler modules are not usually provided by te manufacturers since
3- tey consider tem proprietary information. erefore, te effective material properties are developed from te available imum parameters of te product wit fair agreement wit te measurements. ese are used for prediction of performance and design later. 3. deal Equations Since te discovery of termoelectric effects in te early nineteent century, a very essential equation for te rate of eat flow per unit area q was formulated as sown in Equation (.3), wic is q = αj k (3.) were is te Seebeck coefficient, j te current density, k te termal conductivity and te gradient. is equation relates te eat flow, te electric current and te termal conduction, leading to te steady-state eat diffusion equation (Equation (.7)), wic is rewritten ere as d k j j 0 d (3.) were is te electrical resistivity. e first term gives te termal conduction, te second term gives te Joule eating, and te tird term gives te omson effect wic results from te temperature-dependent Seebeck coefficient. e above two equations govern termoelectric penomena.
3-3 Heat Absorbed p n n-type Semiconductor p-type Semiconcuctor p n p p n p n Positive (+) Electrical nsulator (Ceramic) Heat Rejected Negative (-) Electrical Conductor (copper) Figure 3. Cutaway of a typical termoelectric module Figure 3. ermoelectric cooler wit p-type and n-type termoelements.
3-4 Consider a steady-state one-dimensional termoelectric cooler module as sown in Figure 3.. e module consists of many p-type and n-type termocouples, were one termocouple (unicouple) wit lengt L and cross-sectional area A is sown in Figure 3.. An electrical current is applied and induces a eat flow at te cold and ot junction temperatures as sown. Wit te assumptions tat te Seebeck coefficient is independent of temperature, no termal or electrical contact resistances, and no eat losses, Equation (3.) reduces to d dx ka d dx 0 A (3.3) e solution for te temperature gradient wit two boundary conditions ( x0 c and x L ) is d dx L A k x0 L c (3.4) Equation (3.) is expressed in terms of p-type and n-type termocouples. Q c = n [(α p α n ) c + ( ka d dx ) + ( ka d x=0 dx ) ] p x=0 n (3.5) were n is te number of termocouples and Substituting Equation (3.4) in (3.5) gives Q c is te rate of eat absorbed at te cold junction. Q c = n [(α p α n ) c ( ρ pl p A p + ρ nl n ) ( k pa p + k na n ) ( A n L p L c )] n (3.6) Finally, te cooling power at te junction of temperature c is expressed as
3-5 Q c = n [α c R K( c )] (3.7) were p n (3.8) pl R A p p nl A n n (3.9) K k p L A p p kn A L n n (3.0) f we assume tat te p-type and n-type termocouples are similar, we ave R = L/A and K = ka/l, were = p + n and k = kp + kn. Equation (3.7) is called te ideal equation wic as been widely used in science and industry. e rate of eat liberated at te ot junction is Q = n [α + R K( c )] (3.) Considering te st law of termodynamics across te termoelectric device, we ave W = Q Q c (3.) e amount of work per unit time across te module (rate of work) is obtained substituting Equations (3.7) and (3.) in (3.). W = n[α( c ) + R] (3.3) were te first term is te rate of work to overcome te termoelectric voltage, and te second term is te resistive loss. Since te power is W V, te voltage across te couple will be
3-6 V = n[α( c ) + R] (3.4) e COP is defined as te ratio of te cooling power to te input electrical power. COP = Q c = n [α c R K( c )] W n[α( c ) + R] (3.5) ere are two values of te current tat are of special interest: te current mp tat yields te imum cooling power and te current COP tat yields te imum COP. e imum cooling power can be obtained by differentiating Equation (3.7) wit respect to and setting it to zero. e current for te imum cooling power is found to be mp R c (3.6) e optimum COP can be obtained by differentiating Equation (3.5) and setting it to zero d(cop) = 0 d (3.7) We finally ave COP R Z (3.8) were = c, Z Z is expressed by k and is te average temperature of c and. n terms of,
3-7 Z Z (3.9) 3. Maximum Parameters Let us consider a termoelectric module sown in Figure 3. wit te teoretical imum parameters in te ideal equation. e module consists of a number of termocouples as sown. e ideal equation assumes tat tere are no te electrical or termal contact resistances, no omson effect, and no radiation or convection. t is noted tat te teoretical imum parameters migt differ from te manufacturers imum parameters tat are usually obtained troug measurements. e imum current is te current tat produces te imum possible temperature difference, wic always occurs wen te cooling power is at zero. is is obtained by setting Q c = 0 in Equation (3.7), replacing c wit ( ), taking te derivative of wit respect to and setting it to zero. e imum current is finally expressed by R Z Z (3.0) is te imum possible temperature difference wic always occurs wen te cooling power is zero and te current is imum. is is obtained by setting Q c = 0 in Equation (3.7), substituting bot and c by and, respectively, and solving for. e imum temperature difference is obtained as Z Z (3.) were te figure of merit Z (unit: K - ) is given by
3-8 Z or k Z RK (3.) Using Equation (3.), may be written as R (3.3) e imum cooling power Q c is te imum termal load wic occurs at = 0 and =. is can be obtained by substituting bot and c in Equation (3.7) by and, respectively, and solving for wit n termocouples is Q c. e imum cooling power for a termoelectric module Q c n R (3.4) e imum voltage is te DC voltage wic delivers te imum possible temperature difference at =. e imum voltage is obtained from Equation (3.4), wic is V n (3.5) 3.3 Normalized Parameters f we divide te actual values by te imum values, we can normalize te caracteristics of te termoelectric cooler. e normalized cooling power can be obtained by dividing Equation (3.7) by Equation (3.4), wic is
3-9 R n K R n Q Q c c (3.6) wic, in terms of te normalized current and normalized temperature difference, reduces to c c Z Q Q (3.7) were Z Z (3.8) e coefficient of performance in terms of te normalized values is Z COP (3.9) e normalized voltage is V V (3.30) e normalized current for te optimum COP is obtained from Equation (3.8).
3-0 COP Z (3.3) were Z is expressed using Equation (3.9) and Equation (3.8) by Z Z (3.3) Note tat te above normalized values in Equations (3.7), (3.9) and (3.30) are a function of tree parameters, wic are, and Z. Figure 3.3 and Figure 3.4 are based on te ideal equations using te normalized parameters. e tree imum parameters of,, and Q c are predictable inversely wit te effective material properties, we can ten use te normalized carts for estimation of te performance. e solid lines for te bot figures indicate te normalized prediction wit Z = 0.75 wic is a typical commercial value (see able 3.). Figure 3.3 sows te general caracteristics of ow cooling power and voltage depend on temperature difference. For example, te imum cooling power occurs at bot te zero of temperature difference and te imum current. e lower curve (red line) indicates te cooling power at te optimal COP, wic implies tat te optimal COP generally results in a low cooling power and te medium current exibits good design point in a practical view. Figure 3.4 sows tat te COP and cooling power versus te current along wit te temperature difference. e optimal COPs and imum cooling powers are clearly seen. e current may be properly arranged between te optimal COP and imum cooling power in a practical design.
3-0.9 / =.0 / =.0 0.9 0.8 0.7 0.6 0.8 0.6 0.8 0.6 0.8 0.7 0.6 0.5 0.5 Q c /Q c 0.4 V/V 0.4 0.4 0.4 0. 0.3 0.3 0. 0. 0. 0. 0. 0 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 / Figure 3.3 Normalized cart for termoelectric coolers: cooling power and voltage versus as a function of current. Z = 0.75 was used in calculations. e red line depicts te cooling power ratios at te optimum COP. []
3-3.7.4. 0. / = 0 0. / = 0 0. 0. 0.3 0.9 0.8 0.7.8 0.4 0.6 COP.5 0.3 0.5 0.5 Q c /Q c. 0.6 0.4 0.9 0.4 0.3 0.6 0.5 0.8 0. 0.6 0.3 0.8 0. 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 / Figure 3.4 Normalized cart for termoelectric coolers: cooling power and COP versus current as a function of. Z = 0.75 was used in calculations. [] Example 3. ermoelectric Air Conditioner A novel termoelectric air conditioner is designed as a green energy replacement for te conventional compressor-type air conditioner in a car. A termoelectric module wit eat sinks consists of n = 8 p- and n-type termocouples, one of wic is sown in Figure 3.5. e air conditioner as a number of te modules. Cabin cold air enters te upper eat sink, wile te outside ambient air enters te lower eat sink. An electric current is applied in a way tat a eat flow (cooling power) sould be absorbed at te cold junction temperature of 5 C and liberated at te ot junction temperature of 40 C. e EC material of bismut telluride (Bie3) is used aving te properties as p = n = 00 V/K, p = n =.0 0-3 cm, and kp = kn =.5 0 - W/cmK. e cross-sectional area and leg lengt of te termoelement are An = Ap = mm and
3-3 Ln = Lp = mm, respectively. Assuming tat te cold and ig junction temperatures are steadily maintained, answer te following questions (Use and calculations). (a) For te imum cooling power, compute te current, cooling power, and COP. (b) For te imum COP, compute te current, cooling power, and COP. (c) f te midpoint of te current between te imum cooling power and imum COP is used for te optimal design, compute te current, te cooling power and COP. (d) f te total cooling load of 630 W (per occupant) for te air conditioner is required, compute te number of modules to meet te requirement using te midpoint of current. (a ) (b) Figure 3.5 (a) A termoelectric module. (b) A p-type and n-type termocouple Solution: Material properties: =p n = 400 0-6 V/K, = p + n =.0 0-5 m, and k = kp + kn = 3.04 W/mK e number of termocouples is n = 8. e ot and cold junction temperatures are ( 40 73) K 33K and c ( 5 73) K 88K c 5K
3-4 e figure of merit is Z k 5.0 0 m3.04w mk 6 4000 V K 3.630 K and te dimensionless figure of merit is Z c 3.630 K 88K 0. 758 e internal resistance R and te termal conductance K are calculated as R L A 5 3.0 0 m 0 m 0.0 0 6 m K ka L 6.04W / mk 0 m 3 3 6.080 3 0 m W K (a) For te imum cooling power: Using Equation (3.6), te current for te imum cooling power is 6 4000 V K 88K 0.0 c mp. 56A R Using Equation (3.7), te imum cooling power is
3-5 Q cmp nc 65.567W mp mp R K W K 6 3 8 4000 V K 88K.56A.56A 0.0 6.080 5K Using Equation (3.3), te power input is W 84.8W nmp n mp c mp R 6 8 4000 V K.56A5K.56A 0.0 Using Equation (3.5), te COP at te imum cooling power is COP mp Q W cmp nmp 65.567W 84.8W 0.355 (b) For te imum COP: Z Z c 3 5K.630 K 0. 79 Using Equation (3.8), te current for te imum COP is 6 4000 V K5K COP. 956A R Z 0.0 0.79 Using Equation (3.7), te imum cooling power is
3-6 Q ncop nc 8.557W cop cop R K W K 6 3 8 4000 V K 88K.956A.956A 0.0 6.080 5K Using Equation (3.3), te imum power input is W 4.964W ncop n cop c cop R 6 8 4000 V K.956A5K.956A 0.0 Using Equation (3.5), te COP is COP Q W ncop ncop 8.557W 4.964W.4 (c) For te midpoint of te current between te imum cooling power and imum COP: e midpoint current is.56a.956a 7. mp COP mid 4 Using Equation (3.7), te imum cooling power is Q cmid nc 53.85W mid mid R K 6 3 8 4000 V K 88K 7.4A 7.4A 0.0 6.080 5K A W K
3-7 Using Equation (3.3), te imum power input is W 76.377W nmid n mid c mid R 6 8 4000 V K 7.4A5K 7.4A 0.0 Using Equation (3.5), te midpoint COP is COP mid Q W cmid nmid 53.85W 76.377W 0.705 e required cooling power is Q req 630W e number of EC modules required is N Q Q req cmid 630W 53.8W.7 able 3. Summary of te Results Max. Cool. Power Max. COP Midpoint Current mp =.56 A cop =.956 A mid = 7.4 A Cooling power Q cmp = 65.576 W Q cop = 8.557 W Q cmid = 53.85 W Power input W cnp = 84.8 W W ncop = 4.964 W W nmid = 76.377 W COP COP mp = 0.355 COP =.4 COP mid = 0.705 Number of modules N mp = 9.6 N cop = 33.9 N mid =.7 Design comments Uneconomical (oo ig power consumption) Uneconomical (oo many modules) Economical (reasonable design) Comments e results in able 3. are reflected in te COP and Qc versus current curves (Figure 3.6) plotted using Equations (3.7), (3.3), and (3.5) wit te material properties and inputs given in
3-8 te example description. t is grapically seen in Figure 3.6 tat te imum cooling power accompanies te very low COP, wile te imum COP accompanies very low cooling power. ese lead to te uneconomical results. e midpoint of current between te imum COP and imum cooling power gives reasonable values for bot. Automotive air conditioners intrinsically demand bot a ig COP and a ig cooling power. Figure 3.6 COP and Qc versus current for te given properties and inputs. 3.4 Effective Material Properties As mentioned before, teoretically, te four imum parameters (,, Q c and V) are exactly reciprocal wit te tree material properties (,, and k). n oter words, te tree material properties constitute te four imum parameters in a reciprocal manner. n order to predict te performance of termoelectric coolers, te material properties are, of course, required. However, we ave a dilemma in tat usually manufacturers do not provide te material properties as teir proprietary information but only te measured imum parameters as specifications of teir products. Using te reciprocal relationsip, we can easily formulate te tree material properties in terms of te four manufacturers imum parameters. wo
3-9 imum parameters ( and ) are essential and must be used, but tere is a coice of eiter Q c or V. eoretically tere is no difference weter eiter is selected but practically tere is a difference depending on te coice. According to te analysis (not sown ere), if we coose te imum cooling power, te errors between te ideal equation and real measurements tend to go to te voltages. On te oter and, if we coose te imum voltage, te errors tend to be distributed evenly to te cooling powers and voltages. t sould be noted tat tere is no longer te reciprocity between te four imum parameters and te tree material properties if we determine te material properties by extracting tem from te manufacturers imum parameters. e material properties extracted are called te effective material properties. e effective figure of merit is obtained from Equation (3.), wic is Z (3.33) e effective Seebeck coefficient is obtained using Equations (3.3) and (3.4), wic is n Q c e effective electrical resistivity can be obtained using Equation (3.3), wic is (3.34) A L (3.35) e effective termal conductivity is now obtained using Equation (3.), wic is k (3.36) Z e effective material properties include effects suc as te electrical and termal contact resistances, te temperature dependency of te material, and te radiative and convective eat losses. Hence, te effective figure of merit appears sligtly smaller tan te intrinsic figure of
3-0 merit as sown in able 3. Comparison of te Properties and Dimensions for te Commercial Products of ermoelectric Modules []. Since te material properties were obtained for a p-type and n-type termocouple, te material properties of a termoelement (eiter p-type or n-type) sould be attained by dividing by. Comparison of Calculations wit a Commercial Product e effective material properties can be calculated for any commercial termoelectric module modules as long as te four imum parameters are provided. Calculated effective material properties from te imum parameters for four commercial termoelectric modules are illustrated in able 3.. en, we can simulate te performance curves of te module wit tese effective material properties using te ideal equations. For example, we obtained te effective material properties for C-30-503 module in able 3. and compared te calculated performance curves wit te commercial performance curves, wic are sown in Figure 3.7(a) (c). t is found tat te calculated results are in good agreement wit te manufacturer s curves (wic are typically experimental values) able 3. Comparison of te Properties and Dimensions for te Commercial Products of ermoelectric Modules [] Description Symbols EC Module (Bismut elluride) Marlow RC-4 ( =98 K) ( =98 K) Laird CP0-7-05 ( =98 K) Kryoterm B-7-.0-.3 ellurex C-30-503 ( =300 K) # of termocouples n 7 7 7 7 Effective material V/K 89.. 04.5 08.5 properties cm 0.9 x 0-3.5 x 0-3.0 x 0-3.0 x 0-3 (calculated using k (W/cmK).6 x 0 -.7 x 0 -.6 x 0 -.7 x 0 - commercial,, and Qc) Z 0.744 0.673 0.776 0.758 Measured geometry of termoelement Dimension (W L H) A (mm ).0.0.0. L (mm).5.7.3.66 G=A/L (cm) 0.080 0.085 0.077 0.073 mm 30 30 3. 30 30 30 30 3.6 30 30 3.7 3.4 ( C) 67 66 (63) 69 68 (A) 3.9 3.7 3.6 3.5
3- Manufacturers imum parameters Q c (W) 34.3 36 34.5 34. V (V) 4.4 4.7 5.7 5.5 R ()-module 3.36 3. 3. 3.85 35 Cooling Power, Qc (W) 30 5 0 5 0 = 3.5 A 3 A.5 A A.5 A A Prediction Commercial product Optimal COP 5 0 0 0 0 30 40 50 60 70 (a) emperature Difference, ( C)
3- Voltage (V) 6 4 0 8 6 4 = 3.5 A 3 A.5 A A.5 A A Prediction Commercial product 0 0 0 0 30 40 50 60 70 (b) emperature Difference ( C) 3.5 = 0 C Prediction Commercial product COP.5 0 C 30 C 0.5 40 C 50 C 0 0 3 4 Current (A) (c) Figure 3.7 (a) Cooling power versus, (b) voltage versus, as a function of current, and (c) COP versus current as a function of. e original performance data (triangles) of te commercial module (ellurex C-30-503) are compared to te prediction (solid lines). e curve at te bottom in (a) indicates te cooling powers at te optimum COP.[]
3-3 Problems 3. A compact termoelectric air conditioner is developed as an ambitious green energy project. N = 0 termoelectric modules are installed between two eat sinks as sown in Figure P3-a. e module as n = 7 termocouples, eac of wic consists of p- and n- type termoelements as sown in Figure P3-b. Cabin air flows troug te top and bottom eat sinks, wile liquid coolant is routed troug a eat excanger at te centre of te device werein te coolant is cooled separately at te car radiator. Wit an effective design of bot te eat sinks and eat excanger, te cold and ot junction temperatures can be maintained at 4 C and 3 C, respectively. Nanostructured termoelectric properties of bismut telluride based are given as p = n = 38 V/K, p = n =.3 0-3 cm, and kp = kn = 0.945 0 - W/cmK. e cross-sectional area A and pellet lengt L are mm and. mm, respectively. Answer te following questions for te wole air conditioner (Use and calculations). (a) For te imum cooling power, compute te current, cooling power, and COP. (b) For te imum COP, compute te current, cooling power, and COP. (c) f te midpoint of te current between te imum cooling power and imum COP is used for te optimal design, compute te current, te cooling power and COP. (d) Draw te COP-and-cooling-power-versus-current curves wit te given properties and information (Use Matcad only for tis part). Briefly explain te design concept.
3-4 (a) Figure P3-. (a) A termoelectric air conditioner. (b) A p-type and n-type termocouple (b) 3. A compact termoelectric air conditioner is developed as an ambitious green energy project. N = 40 termoelectric modules are installed between two eat sinks as sown in Figure P3- (a). e module as n = 7 termocouples, eac of wic consists of p- and n-type termoelements as sown in Figure P3- (b). Cabin air flows troug te top and bottom eat sinks, wile liquid coolant is routed troug a eat excanger at te center of te device werein te coolant is cooled separately at te car radiator. Wit te effective design of bot te eat sinks and eat excanger, te cold and ot junction temperatures are maintained at 5 C and 30 C, respectively. t is found tat a commercial module (CP0-7-05) of bismut telluride is appropriate for tis purpose, wic as te imum parameters: cooling power of 34.3 W, temperature difference of 67 C, current of 3.9 C, and voltage of 4.4 V at a ot side temperature of 5 C. e cross-sectional area A and pellet lengt L are mm and.5 mm, respectively. Answer te following questions for te wole air conditioner (Use and calculations). (a) Obtain te effective material properties: te Seebeck coefficient, electrical resistance, and termal conductivity. (b) For te imum cooling power, compute te current, cooling power, and COP. (c) For te imum COP, compute te current, cooling power, and COP.
3-5 (d) f te midpoint of te current between te imum cooling power and imum COP is used for te optimal design, compute te current, te cooling power and COP. (e) Draw te COP-and-cooling-power-versus-current curves wit te given properties and information (Use Matcad only for tis part). Briefly explain te design concept. (a ) (b) Figure P3-. (a) A termoelectric air conditioner. (b) A p-type and n-type termocouple 3.3 Sow te derivation of Equation (3.4). 3.4 Derive Equation (3.7). 3.5 Sow te derivation of Equation (3.). 3.6 Develop te expressions and plots in Figure 3.3 and Figure 3.4 using Matcad. 3.7 Plot Figure 3.7 (a) (c) using Matcad. References. Lee, H., A.M. Attar, and S.L. Weera, Performance Prediction of Commercial ermoelectric Cooler Modules using te Effective Material Properties. Journal of Electronic Materials, 05. 44(6): p. 57-65.